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2022","referrer_id_s":"_pLAIARb7flo","language_s":"ja","size_i":111933,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":21,"references_internal_i":21,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735073964839927808,"text_t":"歴史：[慶應義塾]. このページはJavaScriptを使用しています。JavaScriptを有効にしてご覧ください。. 本文へジャンプ ナビゲーションへジャンプ サイトマップへジャンプ ヘッダーの始まり. ホーム 慶應義塾について 歴史. 歴史. 本文の始まり. 写真で見る慶應義塾の歴史. 慶應義塾豆百科. 慶應義塾年表. ↓. 開塾～創立49年. ↓. 創立50年～74年. ↓. 創立75年～89年. ↓. 創立90年～99年. ↓. 創立100年～124年. ↓. 創立125年～. 開塾～創立49年. 西暦（年号）. 月（日）. 事歴. 1858年. （安政5年）. 10月. 開塾. 福澤諭吉、藩命により江戸築地鉄砲洲の中津藩中屋敷内（現在の東京都中央区明石町、聖路加国際病院のあたり）に蘭学塾を開く。これが慶應義塾の起源で、時に諭吉数え年25歳。いまはこの地に、創立100年を記念して「慶應義塾発祥の地記念碑」が建てられている。. 1863年. （文久3年）. 蘭学塾より英学塾に転向. 福澤は開塾の翌安政6年、横浜を見物してオランダ語が役立たぬことを知り、独学で英語を修めたばかりか、やがて教えるようになる。. 1867年. （慶應3年）. 12月. 芝新銭座に新校地購入. 塾はこれより先、文久元年に鉄砲洲から芝新銭座にうつり、同3年また鉄砲洲へともどっていたのであるが、さらにこの年、芝新銭座の有馬家控屋敷（現在の東京都港区浜松町1丁目、神明小学校のあたり）を手に入れ、翌4年はじめに再び移転する。現在のその地に塾跡を示す記念碑が建っている。. 1868年. （慶應4年＝明治元）. 4月. 塾を「慶應義塾」と命名、近代私学として新発足. 新銭座の新校舎が完成すると、時の年号をとって塾名を「慶應義塾」と定め、同時に組織も西洋の共立学校の制度にならって、志を同じくするものが共同で社を結び、家塾から脱して近代私学として発足、その主義精神を宣言する『芝新銭座慶應義塾之記』を公にした。. 1871年. （明治4年）. 3月. 三田に移転. 現在本拠の三田の地はもと島原藩の中屋敷があったところで、前年に邸地約1万2千坪を貸し下げられ、翌5年には払い下げを受けたものである。. 1873年. （明治6年）. 4月. 学課を整備し、正則・変則の両科新設. アメリカ風の教育課程を採用して修業年限7年の正則科を設け、別に変則科は満17歳以上を入学の条件として修業年数を定めなかった。そののち、両科は本科・予備または正科・別科などと変ったが、基本的にはこの学制が大学の確立するまでつづいた。また、卒業の制度もでき、そのとき創立から同年までの在籍者数約2千名中、およそ4年間在学して当時後進の生徒を導いた158名をえらんで卒業生とし、翌7年4月からはじめて実施されるようになった。. 1874年. （明治7年）. 幼稚舎創立. 1875年. （明治8年）. 5月1日. 三田演説館竣工、開館. 演説は福澤およびその門下の人々によって創始されたもので、この建物はわが国最初の演説会堂として大正4年に東京府史跡、昭和35年には都の重宝、同42年6月さらに国の重要文化財に指定されている。当初はいまの塾監局の北辺にあたり、現在の位置へは関東大震災後の大正13年に移されたのである。. 1880年. （明治13年）. 11月. 「慶應義塾維持法案」を発表する. 明治10年の西南戦争前後から生徒数が激減して、義塾の経営は危機に直面した。このため、福澤は一時は義塾の廃止の止むなきを決意されたほどであったが、小幡篤次郎、浜野定四郎などの社中の人々の力で「慶應義塾維持法案」ができ、これにより社中協力してよく艱難を乗り越えることができた。. 1881年. （明治14年）. 1月23日. 「慶應義塾仮憲法」制定. 慶應4年に組織を改めて同志が共同で社を結んだときから義塾は一種の法人のかたちをなし、福澤が社頭の地位にあったが、明治4年の三田移転に際し、「慶應義塾社中之約束」を定めて運営の機構をいっそう明らかにし、福澤はこれにより引きつづき社頭に推された。. 1885年. （明治18年）. 「ペンの記章」発足. このころ、塾生たちの間でペンの記章をつけはじめる. 1889年. （明治22年）. 8月. 「慶應義塾規約」制定. 明治14年の仮憲法に代え、社中に頒布して、これにもとづき評議員がえらばれることになり、10月に第1期第1回評議会が催された。その後、規定の内容には若干の改定はあったが、このとき以来それが今日に及んでいる。. 1890年. （明治23年）. 1月. 大学部発足、文学・理財・法律の三科を置く. 修業年限3年。私立として最初の総合大学である。なお、これに伴いこれまでの課程を普通部と呼ぶようになった。. 1892年. （明治25年）. 5月. 体育会創設. 当初は剣術・柔術・野球・端艇・弓術・操練(兵式体操)・徒歩の7部で組織されていた。. 1898年. （明治31年）. 5月. 学制を改革し、一貫教育制度を樹立。政治科増設. 大学部を中心として、大学科（5年）、普通学科（5年）、幼稚舎（6年）の16年の一貫教育体系を発足。同時に大学科に政治学部(政治科)を増設する。幼稚舎は明治7年1月から設けられていたものである。. 1899年. （明治32年）. 8月. 第1回海外留学生派遣. 私学として最初のことである。. 1901年. （明治34年）. 2月3日. 創立者福澤諭吉死去. 数え年68歳。法名を大観院独立自尊居士という。. 1903年. （明治36年）. 11月. 第1回早慶野球試合が行われた. 綱町運動場約4千坪を購入、11月21日ここで第1回早慶野球試合が行われ、11-9で慶應が勝利した。. 1904年. （明治37年）. 3月. 塾歌制定. 作詞は角田勤一郎（浩々）、作曲は金須嘉之進。. 1906年. （明治39年）. 4月. 大学院設置. 学問研究の深さをいっそう究めようとする気運がしきりに義塾内に動いてきて、その設置となったもの。. 創立50年～74年. 西暦（年号）. 月（日）. 事歴. 1907年. （明治40年）. 4月21日. 創立50年記念式典挙行. 記念事業として図書館の建設が企てられ、また『慶應義塾五十年史』が刊行された。. 1909年. （明治42年）. 4月23日. 開校記念日制定. 芝新銭座から、三田へ移転した日を太陽暦に換算して、4月23日として実施することになる。. 1912年. （明治45=大正元）. 5月18日. 創立50年記念図書館開館式挙行. 三田山上の図書館（旧館）がそれで、曽禰達蔵・中條精一郎の設計によるゴシック式れんが造りの壮麗な構築で、昭和44年3月には重要文化財として指定された。書庫は昭和2年8月と昭和36年11月との2回にわたり増築。. 1913年. （大正2年）. 1月10日. 福澤先生記念日改める. 命日2月3日から誕生日1月10日に改められ、翌3年から実施されるにいたった。記念日当日は休業とし、福澤先生誕生記念会が開催される。1月10日というのは、諭吉の誕生した天保5年12月12日を西暦(太陽暦)になおして、1835年1月10日としたことによる。. 1917年. （大正6年）. 4月. 医学科開設. さきに明治6年から同13年まで塾内には医学所が開かれていたことがあり、その後もしばしば理科系学科の設置が要望されていたが、創立60年にあたって実現、まず予科の授業を三田に開始した。. 1918年. （大正7年）. 4月. 医学科附属看護婦養成所の開設. 義塾における看護教育の開始。この後1922（大正11）年には産婆養成所も併設し、その後看護婦産婆養成所となる。. 1920年. （大正9年）. 4月. 大学令による大学として新発足、文学・経済学・法学・医学の四学部から成る総合大学となり、予科・大学院を付設. 大学令の公布によって大学は官私にかかわらず平等に取り扱われることになったので、全国の諸学校中、最初にその認可を得た。修業年限は予科3年、学部3年(医学部は4年)、大学院は年限を定めず。. 創立75年～89年. 西暦（年号）. 月（日）. 事歴. 1932年. （昭和7年）. 5月9日. 創立75年記念式典挙行. 天皇陛下の名代として秩父宮殿下が臨席、義塾出身の総理大臣犬養毅ほか参列者3千名に及び、皇室からは教育奨励のため1万円を下賜された。また、記念事業として日吉建設計画を発表、神奈川県橘樹郡日吉村(現在は横浜市港北区日吉)のおよそ13万坪の新校地に理想的な学園建設がすすめられることとなり、また『慶應義塾七十五年史』も刊行された。. 1934年. （昭和9年）. 11月2日. 福澤諭吉誕生100年ならびに日吉開校記念祝賀会開催. 日吉の新校舎には予科の新入生が5月から授業を開始していたが、福澤諭吉誕生100年と兼ねて、このときに改めてそれを披露し祝ったのである。. 1939年. （昭和14年）. 6月17日. 日吉に藤原工業大学開校式. 義塾出身の藤原銀次郎が私財を投じたもので、工学部の前身である。7月8日に入学式挙行。. 1941年. （昭和16年）. 1月10日. 新塾歌制定、発表. さきに明治37年3月につくられた旧塾歌は、すでにほとんど歌われていなかった。新塾歌の作詞は富田正文、作曲は信時潔。. 1944年. （昭和19年）. 8月. 工学部開設. さきに昭和14年6月に日吉構内に開校した藤原工業大学が寄付されて、そのまま工学部として引きつがれたものである。. 1945年. （昭和20年）. 4、5月. 日吉キャンパス、四谷医学部、三田キャンパス罹災. 三田地区では、施設の半分以上を焼失した。特に5月26日の東京大空襲で慶應義塾図書館も半壊し、屋根が焼失するなど甚大な被害があったが、全焼は免れ、一部の書庫は無事だった。. 9月8日. 日吉キャンパス施設、アメリカ軍に接収. 1946年. （昭和21年）. 5月. 医学部の一部、武蔵野町に一時移転. 戦災による施設不足のため、東京都北多摩郡武蔵野町に一時校舎を借用して一部移り、同23年にはこれを武蔵野分校と称した。. 創立90年～99年. 西暦（年号）. 月（日）. 事歴. 1947年. （昭和22年）. 5月24日. 創立90年記念式典挙行. 戦禍のあとの三田山上に天皇陛下のご臨席を仰いで挙行、速やかな再建を期した。10日間にわたり催しを開催。. 11月. 通信教育部を設け、翌23年1月から開講. 文学・経済学・法学の3学部に通信教育課程を実施することとなった。. 1949年. （昭和24年）. 2月. 学事振興資金規程制定. 昭和17年にすでに資金が設定されて実施されていたが、戦争のため一時廃絶のかたちとなっていたのを改めたものである。. 3月. 工学部小金井に移転. 日吉が接収されてのち、工学部は目黒区三田の目黒仮校舎や川崎市本鴨居町の溝ノ口仮校舎を転々としていたが、新たに東京都北多摩郡小金井町(現在は小金井市)に校地を求めた。. 4月. 新制大学発足、文学・経済学・法学・工学の4学部開設. 10月1日. 日吉返還式挙行. 日吉の接収が解かれ、新制の高等学校が日吉移転、翌25年4月からは大学の一部も日吉で授業をすることになった。. 1950年. （昭和25年）. 4月. 医学部附属厚生女子学院の発足. 看護婦産婆養成所を継承し、修業年限3年の専門学校として新制高校卒業生を対象とする看護教育を開始。. 1951年. （昭和26年）. 2月. 義塾が学校法人となる. 4月. 新制大学院(修士課程)開設. 文学・経済学・法学・社会学・工学の5研究科設置。また文学部に図書館学科が付設された。. 1952年. （昭和27年）. 4月. 新制大学医学部設置. 1953年. （昭和28年）. 3月31日. 秋草文壺、新国宝指定. 日吉台付近の白山古墳近くから昭和17年に出土した秋草文壼は昭和24年2月に国宝の指定があったが、その後、文化財保護法により重要文化財に指定がえになっていたものである。. 1956. （昭和31年）. 5月15日. 「福澤先生ウェーランド経済書講述記念日」制定. 明治維新の際、慶應4年5月15日の彰義隊の戦争当日、福澤先生がその騒乱をよそに平常どおり土曜日の日課であるウェーランド経済書の講義をつづけていた故事をしのび、毎年この日に記念講演または行事を催すことにしたものである。. 1957. （昭和32年）. 4月. 商学部増設. 創立100年～124年. 西暦（年号）. 月（日）. 事歴. 1958年. （昭和33年）. 4月23日. 慶應義塾発祥の地記念碑除幕. 創立100年建設事業の一つとして創設の地築地(東京都中央区明石町、聖路加国際病院のあたり)に建設した。. 11月8日. 創立100年記念式典挙行. 天皇陛下のご臨席を再び仰いで、新設の日吉記念館に盛大な記念式典を挙行。国の内外から招かれて参列するものおよそ5千名を数えた。「慶應義塾創立100年記念祝典曲」（堀口大学作詞、団伊玖磨作曲）演奏。また、この1年を通じて多くの記念行事が催され、『福澤諭吉全集』（全21巻、昭和39年2月完成）『慶應義塾百年史』（付録共全6巻、昭和44年3月完成)等の記念出版も刊行の運びとなった。. 1959年. （昭和34年）. 5月. 三田南校舎・西校舎の落成。また、同時に南側に新設の門が正門となった. 1961年. （昭和36年）. 7月. 福澤諭吉記念基金（現在は福澤諭吉記念慶應義塾学事振興基金）を設置. 1962年. （昭和37年）. 4月. 大学ビジネス・スクールの設立. 7月. 創立100年記念建設事業完成披露会挙行. 1964年. （昭和39年）. 2月. 塾旗の規準制定. 塾旗がいまの三色旗に定められたのは明治31年ごろと伝えられているが、その色合い・寸法・ペン章の形など従来はまちまちであったのを統一することになった。. 1965年. （昭和40年）. 1月. 財政建て直しのため塾債の発行を決定. 1967年. （昭和42年）. 3月. 小泉信三記念慶應義塾学事振興基金規程制定（4月施行）. 6月. 三田演説館、重要文化財に指定される. 1968年. （昭和43年）. 5月15日. 慶應義塾命名100年記念式典挙行. 1969年. （昭和44年）. 3月. 『慶應義塾百年史』完成. 18年の歳月をかけて編纂された。. 慶應義塾図書館、重要文化財に指定される. 11月. 三田、研究室の落成. これで人文社会科学系の研究室が全部1ヵ所に集約されたわけである。. 1970年. （昭和45年）. 4月. 研究・教育情報センター発足. 研究・教育用の図書・資料及びサービス業務の一元化を図る。. 1972年. （昭和47年）. 3月. 工学部新校舎が日吉矢上台に完成. 昭和45年9月に工学部の日吉矢上台復帰再建計画のため地鎮祭を執行したが、見事にその校舎新築工事が完成したのである。. 1973年. （昭和48年）. 7月. 立科山荘開荘. 長野県立科町よりの借地約6万坪に設けられた教育施設である。. 1974年. （昭和49年）. 4月. 伊勢慶應病院開設. 12月10日. 図書館のステンドグラスが復元され、除幕式を行った. 製作を担当したのは大竹竜蔵である。. 1975年. （昭和50年）. 5月1日. 三田演説館開館100年記念式挙行. 1976年. （昭和51年）. 5月. 「小泉信三賞」を設ける. 義塾の生んだ偉大な学者・教育者・文筆家小泉信三博士の没後10年を記念して設けられたもので、全国の高校生を対象とした小論文コンテストの最優秀者に毎年贈られる。第1回授賞式は翌52年1月10日第142回福澤先生誕生記念日に挙行された。. 1977年. （昭和52年）. 4月. 月が瀬リハビリテーションセンター開設. 1978年. （昭和53年）. 4月. 大学の学年暦の変更. 従来の学年暦を改めて、前期授業 4月8日～7月14日、試験 7月15日～7月21日、後期授業 9月22日～1月22日、試験 1月23日～2月5日とする。（医学部専門課程を除く）. 1981年. （昭和56年）. 4月1日. 大学理工学部の新設. 工学部を発展させ、この1月16日文部省から義塾の申請を許可された。. 1982年. （昭和57年）. 4月. 三田に図書館（新館）開館. 創立125年～. 西暦（年号）. 月（日）. 事歴. 1983年. （昭和58年）. 5月15日. 創立125年記念式典挙行. 1984年. （昭和59年）. 5月12日. 日吉開設50年記念式典挙行. 1985年. （昭和60年）. 1月10日. 福澤諭吉生誕150年記念式典挙行. 1月. 大学（日吉）新図書館・事務棟竣工. 2月. 三田大学院校舎竣工. 1988年. （昭和63年）. 4月. 看護短期大学の開設. 医学部附属厚生女子学院を改組し、3年制短期大学として新たな看護教育を始める。. 1990年. （平成2年）. 4月. 湘南藤沢キャンパスに総合政策学部・環境情報学部を開設. 9月29日. 「大学部」開設100年記念式典挙行. 10月23日. ニューヨーク学院（高等部）開校式典挙行. 1992年. （平成4年）. 4月. 湘南藤沢中等部・高等部開校. 1994年. （平成6年）. 4月. 大学院政策・メディア研究科開設. 湘南藤沢キャンパスに、「自律・分散・協調」を掲げ、総合政策学部・環境情報学部を基礎学部とする大学院政策・メディア研究科が開設された。. 1999年. （平成11年）. 12月. 矢上に創想館竣工. 2000年. （平成12年）. 4月. 新川崎タウンキャンパス開設. 12月. 第2回世紀送迎会. 1900年に開催された世紀送迎会に続き、20世紀から21世紀に変わる歴史的節目を祝う第2回世紀送迎会が、三田キャンパスで開催された。. 2001年. （平成13年）. 1月. 「世紀をつらぬく福澤諭吉」展開催. 福澤諭吉没後100年を記念する「世紀をつらぬく福澤諭吉」展が、東京と大阪において開催された。. 4月. 看護医療学部の開設. 看護短期大学を改組し、看護医療学部を開設した。信濃町における看護教育の歴史を継承しながら、湘南藤沢キャンパスを主たるキャンパスとして教育研究の新たな展開をはかる。. 5月. 鶴岡タウンキャンパス開設. 2002年. （平成14年）. 1月. 日吉に来往舎竣工. 2003年. （平成15年）. 9月. 伊勢慶應病院閉院. 2004年. （平成16年）. 4月. 法科大学院（大学院法務研究科）の開設. 新たな法曹養成の開始. 2005年. （平成17年）. 3月. 三田に南館竣工. 4月. 大学院健康マネジメント研究科の開設. 湘南藤沢キャンパスに開設。看護学、医療マネジメント、スポーツマネジメントの3専修から成る新たな分野の研究教育。. 2008年. （平成20年）. 4月. 学校法人共立薬科大学との合併により薬学部・大学院薬学研究科開設. 芝共立キャンパス、浦和共立キャンパスが新たに加わった。. 大学院システムデザイン・マネジメント研究科、大学院メディアデザイン研究科の開設. 5月. 慶應大阪リバーサイドキャンパス開設. 8月. 日吉に協生館竣工. 学生と世代を超えたさまざまな人々が集い交流し、連携を実現していく場としての役割を果たす、慶應義塾における社会・地域連携の拠点のひとつ。. 11月. 『慶應義塾史事典』刊行. 11月8日. 創立150年記念式典挙行. 2009年. （平成21年）. 1月～9月. 「未来をひらく福澤諭吉展」開催. 慶應義塾創立150年を記念して「未来をひらく福澤諭吉展」が、東京・福岡・大阪において開催された。. 3月. 日吉に第4校舎独立館竣工. 2010年. （平成22年）. 4月4日. 湘南藤沢キャンパス（SFC）20周年記念式典挙行. 7月16日. 福澤諭吉生誕175年記念式典挙行. 12月. 『福澤諭吉事典』刊行. 2011年. （平成23年）. 3月. 三田に南校舎建て替え竣工. 9月. 月が瀬リハビリテーションセンター閉院. 2012年. （平成24年）. 5月. 慶應義塾図書館開館100年記念式典開催. 2013年. （平成25年）. 4月. 横浜初等部開校. 5月. 慶應大阪シティキャンパス開設. （慶應大阪リバーサイドキャンパスを移転）. 2015年. （平成27年）. 10月. 大学部開設125年記念式典・講演会開催. 2016年. （平成28年）. 4月. 殿町タウンキャンパス開設. 2020年. （令和2年）. 3月. 新・日吉記念館竣工. 2021年. （令和3年）. 4月. 慶應義塾ミュージアム・コモンズ（KeMCo）開館. 2021年. （令和3年）. 7月. 福澤諭吉記念慶應義塾史展示館開館. 関連コンテンツ. →. 塾長メッセージ. →. 理念. →. 卒業生ネットワーク. →. キャンパス. →. 歴史ある取り組み. →. もっと知る. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. 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さまざまなデータベース等で教員・研究者のプロフィールや研究業績情報を公開しています。. →. 教員プロフィール. 所属別に教員・研究者のプロフィールを紹介しています。. →. 研究者情報データベース（K-RIS）. 外部サイトへリンク. 慶應義塾研究者情報データベース(K-RIS)では、慶應義塾の研究者の研究業績やプロフィールを公開しています。. →. 研究者情報データベース（Pure）. 外部サイトへリンク. 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このページはJavaScriptを使用しています。JavaScriptを有効にしてご覧ください。. 本文へジャンプ ナビゲーションへジャンプ サイトマップへジャンプ ヘッダーの始まり. ホーム 教育 留学・国際プログラム. 留学・国際プログラム. 本文の始まり. From Keio. 慶應から世界へ. →. 海外留学プログラム. 交換留学、短期海外研修などの慶應からの海外留学をご案内します。. →. 海外協定校. →. 国際センター短期プログラム. →. 協定校等による短期プログラム. →. 私費による長期留学. →. ダブルディグリープログラム. 慶應義塾大学と海外の協定校の両方で学び、修了時に両方の大学から学位が取得できるダブルディグリープログラムをご案内します。. To Keio. 世界から慶應へ. →. 英語のみで学位が取得できるプログラム. 慶應義塾大学には英語による授業（語学を除く）が955あり、また英語による学位課程も開設されています（日本人学生も履修可能）。. →. 短期プログラム[English]. 外部サイトへリンク. 慶應義塾での短期的な学習や研究を希望する留学生へのご案内です。. Keio Summer Program. 外部サイトへリンク. →. 交換留学[English]. 外部サイトへリンク. 海外の協定校から慶應義塾で交換留学生として学びたい学生へのご案内です。. →. 別科・日本語研修課程. 外部サイトへリンク. 留学生を対象に日本語ならびに日本研究のための講座を開講しています。留学目的に合わせて多様な学習段階・科目が用意されています。. 留学生向けプロモーションビデオ. 関連コンテンツ. →. 海外留学のための奨学金. 外部サイトへリンク. →. 留学のてびき. 外部サイトへリンク. 関連コンテンツ. →. 外国人留学生のサポート. 外部サイトへリンク. →. 海外留学生のための留学奨学金. 外部サイトへリンク. →. 国際センター. 外部サイトへリンク. 世界各地と協定を結び、お互いに研究者や学生を派遣・交流している慶應義塾の国際交流の拠点です。義塾で学ぶ留学生や海外への留学を希望する学生をサポートしています。国際センターのサイトでは、留学、在学研修プログラム、センター設置の研究講座などについてご案内しています。. →. 国際交流データ. 外部サイトへリンク. →. 海外協定校一覧. 外部サイトへリンク. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. 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アート・センター 外部サイトへリンク","→ 会計研究室 外部サイトへリンク","→ 外国語教育研究センター 外部サイトへリンク","→ 教職課程センター 外部サイトへリンク","→ 教養研究センター 外部サイトへリンク","塾内公募型研究センター","研究プロジェクト","→ グローバルリサーチインスティテュート（KGRI） 外部サイトへリンク","→ 経済研究所 外部サイトへリンク","→ 言語文化研究所 外部サイトへリンク","→ 国際センター 外部サイトへリンク","→ 産業研究所 外部サイトへリンク","→ システムデザイン・マネジメント研究所（SDM研究所） 外部サイトへリンク","→ 自然科学研究教育センター 外部サイトへリンク","→ 斯道文庫 外部サイトへリンク","→ 新川崎先端研究教育連携スクエア 外部サイトへリンク","→ スポーツ医学研究センター 外部サイトへリンク","→ 先端科学技術研究センター(KLL) 外部サイトへリンク","→ 先端生命科学研究所 外部サイトへリンク","→ 鶴岡タウンキャンパス 外部サイトへリンク","→ 総合医科学研究センター 外部サイトへリンク","→ 体育研究所 外部サイトへリンク","→ デジタルメディア・コンテンツ統合研究センター 外部サイトへリンク","→ 殿町先端研究教育連携スクエア 外部サイトへリンク","→ 日本語・日本文化教育センター 外部サイトへリンク","→ 東アジア研究所 外部サイトへリンク","→ 福澤研究センター 外部サイトへリンク","→ 法学研究所 外部サイトへリンク","→ 保健管理センター 外部サイトへリンク","→ メディア・コミュニケーション研究所 外部サイトへリンク","→ メディアデザイン研究所（KMD研究所） 外部サイトへリンク","→ GICセンター 外部サイトへリンク","→ SFC研究所 外部サイトへリンク","慶應義塾公式グッズ 外部サイトへリンク","","","","","","寄付 外部サイトへリンク"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 13:37:59 UTC 2022","fresh_date_dt":"Thu Jun 23 06:19:35 UTC 2022","referrer_id_s":"_pLAIARb7flo","language_s":"ja","size_i":103647,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":25,"references_internal_i":25,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735073967171960832,"text_t":"研究所・センター：[慶應義塾]. このページはJavaScriptを使用しています。JavaScriptを有効にしてご覧ください。. 本文へジャンプ ナビゲーションへジャンプ サイトマップへジャンプ ヘッダーの始まり. ホーム 研究 研究所・センター. 研究所・センター. 本文の始まり. 慶應義塾は、学部・研究科以外の幅広い活動の場として研究所・附属施設を多数設置し、従来の学問領域にとどまらない柔軟な研究環境を構築しています。ユニークで独創的な活動は内外から注目されています。. （50音順で掲載しています）. アート・センター. 慶應義塾唯一の芸術を専門研究領域とする研究所。博物館相当施設として、アート・スペース（三田キャンパス南別館1階）にて年間を通じて展覧会を開催しています。また、研究会・公演などの各種催事の開催、アーカイヴの構築、国内外における芸術資料の研究的活用の促進などの活動を通して、現代における芸術の問題に取り組んでいます。. アーカイヴでは、主として日本の戦後芸術に関する資料を閲覧に供しており、学内外の研究者が利用しています（予約制）。. [三田キャンパス]. →. アート・センター. 外部サイトへリンク. 会計研究室. 商学部の附属機関である会計研究室は、会計のプロフェッショナル（公認会計士・税理士）を目指す塾生ないし目指す可能性がある塾生を対象として、ガイダンス、講演会、監査法人見学会など、さまざまな啓蒙的イベントを企画・実施しています。. [三田キャンパス]. →. 会計研究室. 外部サイトへリンク. 外国語教育研究センター. 慶應義塾の外国語教育を全体的に捉え、活動する研究機関。外国語教育に関する研究を推進するほか、特色ある外国語科目の開講や自律学習環境の提供を行い、国際舞台で通用する言語運用能力の習得をめざす塾生を支援しています。塾生対象の英語論文コンテストや、広く一般の学生や社会人を対象とした語学学習のための公開講座「慶應外語」等のプログラムも実施しています。. [三田キャンパス・日吉キャンパス]. →. 外国語教育研究センター. 外部サイトへリンク. 教職課程センター. 教職課程の運営のほか、教員養成に関する調査・研究を推進。学生は教職課程センターに登録（例年300名から400名が新規登録）し、所定の科目を履修することで中学校および高等学校の教員普通免許状に必要な単位を修得できます（一部学部除く）。. [三田キャンパス]. →. 教職課程センター. 外部サイトへリンク. 教養研究センター. 多分野にわたる教養教育に関する研究活動を推進することを目的に設置。大学教育カリキュラムの研究、身体と言語のあり方をめぐって新しい「知」を探り、言語力育成を目指す研究や実験授業、学問的・知的活動のためのスキル・考え方を身につけるアカデミック・スキルズを学部横断的に提供しています。. [日吉キャンパス]. →. 教養研究センター. 外部サイトへリンク. グローバルリサーチインスティテュート（KGRI）. 大学のグローバル化をより一層推進し、世界に貢献する国際研究大学となるための基盤として設立され、学内の関連する教育研究分野と密接に連携しながら、「長寿」「安全」「創造」の3つのクラスターにおいて文理融合研究や領域横断研究を推進し、その成果を広く国際的に発信していきます。. また、KGRI内には、学部・研究科横断的な研究拠点（センター）である. 塾内公募型研究センター や、その他の. 研究プロジェクト が、多数設置されています。. [三田キャンパス]. →. グローバルリサーチインスティテュート（KGRI）. 外部サイトへリンク. 経済研究所. 大学部理財科発足125年を記念して設立された経済研究所は、経済学の学術研究の発展と、その成果を広く社会と共有することを目的としています。その活動は、研究成果の発信、学内外の研究者ネットワークの構築、研究支援のための資源・情報の共有と活用、内外の研究機関との連携協力の窓口、外部研究資金獲得等、多岐にわたります。. [三田キャンパス]. →. 経済研究所. 外部サイトへリンク. 言語文化研究所. 世界の諸言語、諸地域や各時代の文化・思想、言語学・言語理論などの研究を推進。学部では開設していない個別言語の講座を含むサンスクリット、アラビア語、古代エジプト語、ヘブル語、タイ語などの12言語におよぶ講座を開設しています。. [三田キャンパス]. →. 言語文化研究所. 外部サイトへリンク. 国際センター. 国際交流活動の拠点として世界各地の協定大学と研究者や学生の交流を推進。義塾で学ぶ留学生や留学を希望する学生をサポートし、短期海外研修プログラムも運営。また、「国際センター講座」として英語による科目を年間約80科目設置しています。. [三田キャンパス]. →. 国際センター. 外部サイトへリンク. 産業研究所. 主に経済・法律・行動科学の3分野の学際的研究機関。世界・日本の計量経済モデル、地域産業連関分野、環境の産業連関分析、資金循環分析、人的資源管理、社会心理分析テストの開発など、海外のさまざまな機関とグローバルな活動を行っています。. [三田キャンパス]. →. 産業研究所. 外部サイトへリンク. システムデザイン・マネジメント研究所（SDM研究所）. SDM研究所は、企業をはじめ社会のさまざまな組織におけるシステムデザイン・マネジメントの理解とマネジメント能力の向上のために、先端的・実践的研究や教育成果を広く社会に還元することを目的としています。研究所スタッフは、安心・安全・環境共生をキーワードに、産官学の連携を図りながら、グローバルな環境共生と異文化・異民族の協生を基本とする社会・技術システム再構築を先導するための研究・教育活動を実践しています。. [日吉キャンパス]. →. システムデザイン・マネジメント研究所（SDM研究所）. 外部サイトへリンク. 自然科学研究教育センター. 自然科学の研究と教育を促進し、さらなる質の向上を意識しながら、世界文明に貢献することを目的としています。学部、キャンパス、研究領域を超えるだけでなく、一貫教育校とも連携しながら、自然科学における研究と教育活動を行っています。. [日吉キャンパス]. →. 自然科学研究教育センター. 外部サイトへリンク. 斯道文庫（しどうぶんこ）. 麻生太賀吉氏寄贈の図書を基に、和書・漢籍と基本研究書など約17万冊を所蔵し、国内外の和漢書の現地調査、マイクロフィルムやデジタル撮影による副本作成、本の比較校勘などを行っている東洋古典籍学（書誌学）の研究所です。. [三田キャンパス]. →. 斯道文庫. 外部サイトへリンク. 新川崎先端研究教育連携スクエア. 川崎市との協働のもと、先端的な産学官共同研究を担う組織として新川崎タウンキャンパスが開設され、運営組織として新川崎先端研究教育スクエアが置かれています。従来型の研究・教育機能を持つキャンパスに対し、連携的かつ学部・研究科横断の研究重点型キャンパスです。. [新川崎キャンパス]. →. 新川崎先端研究教育連携スクエア. 外部サイトへリンク. スポーツ医学研究センター. 体育会学生を中心に、慶應義塾内外の運動選手の競技力向上、健康・体調管理のための医学的なサポートを行っています。また、生活習慣病の予防を目的とした、運動、食事に関する先導的な活動、研究を行っています。. [日吉キャンパス]. →. スポーツ医学研究センター. 外部サイトへリンク. 先端科学技術研究センター(KLL). KLLは、様々な研究活動を通して世界の科学技術分野の健全な発展に寄与し、慶應義塾から生まれた科学技術に関わる研究の成果を広く社会に還元し、科学技術分野における優れた人材を育成することによって、社会の先導的役割を担うことを目的としています。. [矢上キャンパス]. →. 先端科学技術研究センター(KLL). 外部サイトへリンク. 先端生命科学研究所(IAB). 鶴岡タウンキャンパス（山形県鶴岡市）に設置された、「統合システムバイオロジー」という新しい生命科学を中心とする研究所です。最先端のバイオテクノロジーを用いて細胞活動を網羅的に計測・分析し、コンピュータで解析・シミュレーションして医療・環境・食品などの分野に応用しています。. [鶴岡タウンキャンパス]. →. 先端生命科学研究所. 外部サイトへリンク. →. 鶴岡タウンキャンパス. 外部サイトへリンク. 総合医科学研究センター. 総合医科学研究センターは、医学・医療を中心とする分野において、慶應義塾大学の関連する研究分野と密接に協力しながら、生命医科学の研究成果を広く社会に還元し、その健全な発展に寄与することを目的としています。. [信濃町キャンパス]. →. 総合医科学研究センター. 外部サイトへリンク. 体育研究所. 健康・運動・スポーツに関する研究と教育を目的とした組織。塾生・教職員を対象とした各種大会や講演会など、スポーツの振興を目的とする活動も行っています。体育科目の授業のほか、地域住民への感動教育・体験教育の一助も担っています。. [日吉キャンパス]. →. 体育研究所. 外部サイトへリンク. デジタルメディア・コンテンツ統合研究センター. デジタルメディア・コンテンツに関連する研究教育活動等を通して、義塾の総合力を活用した人文科学、社会科学、理工学、医学などの融合による新しい知の創造と流通の国際的な先導を実現し、新たな産業、教育、文化、芸術分野等の創出に貢献しています。. [日吉キャンパス]. →. デジタルメディア・コンテンツ統合研究センター. 外部サイトへリンク. 殿町先端研究教育連携スクエア. 殿町タウンキャンパスは、2016年4月、殿町国際戦略拠点キングスカイフロント内に開設されました。キングスカイフロントに集積する企業および研究機関等との連携・協力によるイノベーションの創出をめざし、慶應義塾スーパーグローバル事業における「長寿」クラスターの研究や、科学技術振興機構リサーチコンプレックス推進プログラムを活用したウェルビーイング研究などを推進します。. [殿町タウンキャンパス]. →. 殿町先端研究教育連携スクエア. 外部サイトへリンク. 日本語・日本文化教育センター. 留学生に対する日本語・日本文化に関する教育の発展を目的として設置。別科・日本語研修課程、学部の第二外国語としての日本語の授業を運営。文学研究科・国文学専攻日本語教育学分野にも協力しています。別科・日本語研修課程では約180名の留学生が学んでいます。. [三田キャンパス]. →. 日本語・日本文化教育センター. 外部サイトへリンク. 東アジア研究所. 慶應義塾の東アジア研究を国内的・国際的ネットワークを通して発展させるため、学際的共同研究を推進しています。そのために、内外の研究機関や研究者との学術交流を図り、歴史研究と現状分析、および基礎研究と応用研究を重層的に展開しています。研究所内に、現代中国研究センター、現代韓国研究センターが設置されています。. [三田キャンパス]. →. 東アジア研究所. 外部サイトへリンク. 福澤研究センター. 福澤諭吉と慶應義塾を視野に置き、近代日本研究に資することを目的に設立。福澤および義塾の歴史に関する資料の収集・整理・保存や、福澤門下生などの義塾出身者についての調査研究のほか、大学設置講座、講演会なども行っています。. [三田キャンパス]. →. 福澤研究センター. 外部サイトへリンク. 法学研究所. 法律学・政治学の研究・教育機関です。主に法曹および国家公務員を目指す学生に対して、実務家ゼミ、法曹体験ゼミ、集中講座、学習到達度評価、合格者座談会等の各種プログラムの実施に加えて、自習室・専用ロッカー・専門図書の使用などに関する支援を行っています。なお、これらのプログラムへの参加および施設利用については研修生登録が必要です。. [三田キャンパス]. →. 法学研究所. 外部サイトへリンク. 保健管理センター. 健康診断や、健康の保持増進のための研究・教育に関する業務や健康診断・健康相談などを行っています。また、予防と医療を連携させるためにセンターの医師が慶應義塾診療所の医師を兼務し診療にあたっています。. [各キャンパス、本部：日吉キャンパス]. →. 保健管理センター. 外部サイトへリンク. メディア・コミュニケーション研究所. ジャーナリズムとメディア・コミュニケーションの研究と教育を実施しています。研究生には、基礎科目や、メディア心理学、政治社会学、法学、歴史学などに関するゼミ、業界の最前線で活躍している実務家による新聞・放送・広告・PRなどについての少人数形式の講義や演習を開講しています。研究生は「研究会」が必修です。. [三田キャンパス]. →. メディア・コミュニケーション研究所. 外部サイトへリンク. メディアデザイン研究所（KMD研究所）. 大学院メディアデザイン研究科附属メディアデザイン研究所は、学術的貢献に加え、社会的インパクトと経済的インパクトを重視して、セキュリティ、IoT、サービスデザイン、 ポップカルチャー、クリエイティブ教育、スポーツなど幅広い分野の研究を国内外の多くのパートナーと連携して推進しています。また、VRやARの研究拠点である国際バーチャルリアリティ研究センターおよびインタラクティブメディアの研究拠点として、シンガポールと日吉にKeio-NUS CUTE Centerを設置して活動しています。. [日吉キャンパス]. →. メディアデザイン研究所（KMD研究所）. 外部サイトへリンク. GICセンター. GICセンター（Center for Global Interdisciplinary Courses）は国際的かつ学際的な人材の育成を主たる目的として2014年11月に設立されました。GIC科目では、講義はもとより、原則として質疑応答、ディスカッション、レポート、試験等の一切が英語（またはその他の外国語）で行われます。約60科目のGICセンター設置科目と約400科目のGIC認定科目は全ての学部生が履修することができます。GIC科目を40単位以上取得した学生にはその努力を称えGIC Program修了証が授与されます。. [日吉キャンパス]. →. GICセンター. 外部サイトへリンク. SFC研究所. 湘南藤沢キャンパスの3学部2研究科の附属研究所。教育・研究活動と外部機関との協調関係を育みながら産官学による研究を推進するほか、外部人材の活用、SFC Open Research Forumを通じて対外的に研究成果の公開を行っています。. [湘南藤沢キャンパス]. →. SFC研究所. 外部サイトへリンク. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. Keio Times. お問い合わせ. 受験生. 在学生. 卒業生. 寄付. 外部サイトへリンク. 塾内サイト."}
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－ FAQ・お問い合わせ. 本文の始まり. 一般選抜 FIT入試 帰国生入試 IB入試 留学生入試 その他のお問い合わせ 一般選抜 - 出願に関する質問（昨年度参考）. ↓. Q. 複数の学部へ出願しようと思っています。調査書等は学部ごとに必要ですか？. A. 複数の学部へ出願する場合でも、調査書等は1通でかまいません。. ↓. Q. 卒業してから5年が経過しているため、高等学校で調査書を発行してもらえません。どのような書類を提出すればよいですか？. A. 「出願資格を証明する書類」として、以下のものを提出してください。なお、②も発行できない場合は、①のみを提出してください。. ①高等学校の卒業（修了）証明書. ②高等学校在学中の成績証明書. ↓. Q. 出願状況を知りたいのですが、どうすればよいですか？. A. 例年、1月中旬頃から、志願者状況を慶應義塾大学ウェブサイトの入学案内ページでお知らせします。. ↓. Q. 大学入学共通テストや民間の英語4技能試験の結果を利用する方式で出願することはできますか？. A. 大学入学共通テストや民間の英語4技能試験の結果を利用して出願することはできません。. 一般選抜 - 入学試験に関する質問（昨年度参考）. ↓. Q. 試験会場はどこですか？. A. 試験会場は、原則として日吉キャンパスか三田キャンパスのどちらかです。出願の学部・学科・方式・選択科目によって異なります。ご自身の試験会場は、必ず受験票で確認してください。. ↓. Q. 現役生と既卒者によって入学試験の有利、不利はありますか？. A. ありません。選考は全く同一の基準で行われます。. ↓. Q. 選択受験科目によって有利、不利はありますか？. A. ありません。選択受験科目の難易度の違いから生じる不公平については、通常は統計的処理により得点の補正を行うことによって対処しています。. ↓. Q. マークシートを使用している学部はありますか？. A. 2021 年度一般選抜では経済学部、法学部、商学部、理工学部、総合政策学部、環境情報学部、薬学部にて一部使用しました。詳しくは一般選抜要項で確認してください。. ↓. Q. 面接を行っている学部はありますか？. A. 2022 年度一般選抜で面接を行う学部は、医学部、看護医療学部です。いずれも第 2 次試験で実施します。. ↓. Q. 大学で受験時の宿泊先の斡旋は行っていますか？. A. 行っていません。ただし、. 慶應義塾大学生協のウェブサイト に宿泊案内が掲載されます。宿泊が必要な方は各自で手配してください。. 日吉キャンパスと三田キャンパスのどちらの試験会場になるか確定できない段階では、最終的にどちらのキャンパスになっても利用可能な宿泊先を手配することが多いようです。. ↓. Q. 飛行機の予約の関係があるのですが、最終時限終了後、すぐに退室することはできますか？. A. 試験時間が終了しても、監督者の指示があるまでは退室できません。また、安全上の理由により、退場に制限をかけるため、試験終了から退場までに30分から60分程度かかる場合があります。さらに、試験会場周辺の公共交通機関に大幅な乱れ、遅れが生じた場合には、全体の試験開始時刻を繰り下げることもありますので、それらも留意の上、十分に余裕をもった計画を立てることをお薦めします。. ↓. Q. 入試当日試験会場内に荷物を預ける場所はありますか？. A. ありません。荷物は自分の机の下または空いている横の座席下に置くことになります。大型のものは置けませんので注意してください。. ↓. Q. 試験会場に自動販売機やコンビニはありますか？. A. あります（コンビニは日吉キャンパスのみ）。以下の地図をご参照ください。. 三田キャンパス 日吉キャンパス 一般選抜 - 合格発表・入学手続に関する質問（昨年度参考）. ↓. Q. 合格発表はどのような形で行いますか？. A. インターネット出願システム上で行います。キャンパス内での掲示は行いません。. ↓. Q. 補欠について教えてください。. A. 合格発表の際に、正規合格者と同時に補欠者も発表します。ただし、医学部・看護医療学部の第 1 次合格発表には補欠はありません。. 補欠者は入学が許可されるとは限りません。合格者の入学手続状況により欠員が生じた場合に限り、順次入学が許可されます。. ↓. Q. 入学に必要な費用の振替（学部の振替）について教えてください。. A. ある学部（以下学部Aとします）への入学手続をした後、さらに他の学部（以下学部Bとします）から入学を許可され、かつ学部Bに入学を希望する場合、学部Aに納入した費用を、学部Bの費用として振り替えることができます。ただし、合格発表日から入学金等振込期限までの期間が両学部間で3日以上重複している場合には、入学に必要な費用の振替手続をすることができません。詳しくは一般選抜要項や入学手続要項で確認してください。. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. Keio Times. お問い合わせ. 受験生. 在学生. 卒業生. 寄付. 外部サイトへリンク. 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新型コロナウイルス感染症拡大に伴う支援制度、家計急変・災害被災時の奨学金に関するご案内 外部サイトへリンク","→ 奨学金リーフレット2022 PDF","→ 学問のすゝめ奨学金 外部サイトへリンク","→ 指定寄付奨学金（経済状況重視型） 外部サイトへリンク","→ 慶應義塾大学給費奨学金 外部サイトへリンク","→ 慶應義塾維持会奨学金 外部サイトへリンク","→ その他の奨学金（国際センター） 外部サイトへリンク","→ 石井・石橋基金 慶應義塾大学グローバルフェローシップ（私費留学助成） 外部サイトへリンク","→ 研究のすゝめ奨学金 外部サイトへリンク","研究助成 \n            学術研究支援部では、塾内の研究者の様々な研究活動を支援しています。 \n            上下","→ 民間団体・地方公共団体の奨学金 外部サイトへリンク","→ 国による「高等教育の修学支援新制度」（文部科学省） 外部サイトへリンク","→ 【受験生・在学生対象】国による「高等教育の修学支援新制度」について 外部サイトへリンク","→ 高校予約採用（学部入学予定者のみ） 外部サイトへリンク","→ 第一種奨学金　緊急採用／第二種奨学金 応急採用 外部サイトへリンク","→ 第二種奨学金　海外 外部サイトへリンク","→ 第一種奨学金・第二種奨学金定期採用 外部サイトへリンク","→ 第二種奨学金　短期留学 外部サイトへリンク","→ トビタテ！留学JAPAN（返済不要） 外部サイトへリンク","→ 奨学金以外の経済支援制度 外部サイトへリンク","→ 奨学金制度総合案内（塾生サイト） 外部サイトへリンク","→ Scholarships for International Students (English) 外部サイトへリンク","→ 外国人留学生向け奨学金（日本語） 外部サイトへリンク","→ 慶應義塾大学の奨学金（学内奨学金）へのご寄付をお考えの皆様 外部サイトへリンク","慶應義塾公式グッズ 外部サイトへリンク","","","","","","寄付 外部サイトへリンク"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 13:38:01 UTC 2022","fresh_date_dt":"Thu Jun 23 06:19:38 UTC 2022","referrer_id_s":"_pLAIARb7flo","language_s":"ja","size_i":112348,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":27,"references_internal_i":27,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735073969143283712,"text_t":"奨学金：[慶應義塾]. このページはJavaScriptを使用しています。JavaScriptを有効にしてご覧ください。. 本文へジャンプ ナビゲーションへジャンプ サイトマップへジャンプ ヘッダーの始まり. ホーム 学生生活 奨学金. 奨学金. 本文の始まり. 慶應義塾大学独自の奨学金. 慶應義塾大学は、学業に専念できる環境を整え、良識ある知識人として活躍できる人材育成の支援を目的に、成績優秀者を対象とした奨学金、家計状況が急変した際の支援、地方出身者や海外でのグローバルな学習活動への支援など、学生のニーズに対応するさまざまな奨学金を用意しています。. →. 新型コロナウイルス感染症拡大に伴う支援制度、家計急変・災害被災時の奨学金に関するご案内. 外部サイトへリンク. →. 奨学金リーフレット2022. PDF. リスト. 豊富な学内奨学金. 奨学金の種類　約. 110. 種類. 奨学金支援　約. 2,450. 名. 奨学金総額　. 11.8. 億円. 入学前予約. 学問のすゝめ奨学金. 制度を大幅に改定. 首都圏外から入学をさらに強力にサポート. 研究のすゝめ奨学金. 年間支給総額　. 1. 億. 500. 万円. 国内最大規模の. 奨学基金240億円. 返済不要の給付型奨学金を. 一人あたり平均. 47. 万円(学部生)支給. 給付型奨学金をさらに拡充. 卒業生や篤志家等の. 寄付による奨学金. 指定寄付奨学金. 慶應義塾維持会奨学金. 留学. 海外留学支援(派遣). 外国人留学生の支援(受入). 経済支援. 大規模自然災害被災学生支援. 慶應義塾大学修学支援奨学金. 慶應義塾大学東日本大震災被災学生復興支援奨学金. 奨学金の種類. 学内奨学金 （返済不要）. 慶應義塾大学は成績優秀者を対象とした奨学金、家計状況が急変した際の支援、地方出身者や海外でのグローバルな学習活動への支援など、学生のニーズに対応するさまざまな奨学金を用意しています。すべて返済の必要のない給付型の奨学金です。ほとんどの奨学金の給付期間は1年間ですが、原則再申請することができます。1年単位で申請ができるため、それだけ多くの学生にチャンスがあります。. 奨学金の種類. 学部生. 大学院生. 首都圏外から経済的な不安を抱えることなく入学したい（入学前申請）. 入学前に予約ができる地方出身者対象の奨学金です。. 上下. →. 学問のすゝめ奨学金. 外部サイトへリンク. 対象者は、東京・神奈川・千葉・埼玉以外の高等学校出身で両親の合計収入が1000万円未満の大学一般選抜受験予定者です。書類選考により一般選抜出願前に候補者認定を受け、一般選抜合格、入学手続き、入学後の所定の手続きを経て奨学金が給付されます。. 申請時期（予定）：10月下旬〜11月. 採用候補者数（予定）：550名以上. 奨学金額：年額90万円（医学部） \/ 年額80万円（薬学部） \/ 年額60万円（その他の学部）. 受給期間：入学年度を含め4年（医学部・薬学部薬学科は6年）※毎年の継続審査による. 入学時加算：入学初年度には、奨学金額に入学金相当額（20万円）を加算して支給. 成績優秀者増額：入学2年目以降、成績優秀者には奨学金額を増額して支給. 増額後　年額150万円（医学部）\/年額120万円（薬学部）\/. 年額90万円（理工学部、総合政策学部、環境情報学部、看護医療学部）\/. 年額80万円（文学部、経済学部、法学部、商学部）. 経済的に困ったとき. 家計の急変や継続的な経済困窮、または大規模自然災害の被災によって学業の継続が著しく困難になった学部生を支援する奨学金です。. 上下. →. 慶應義塾大学修学支援奨学金. 外部サイトへリンク. 勉学の意欲を持ちながらも、急激な家計状況の変化（大規模自然災害による被災を含む）、あるいは継続的な困窮のため経済的に修学が困難な者を支援します。. 申請時期（予定）：6月、11月. 学部生採用者数（予定）：120名程度. 奨学金額：学費の範囲内（年額平均約30万円）. 受給期間：1年（再申請可）. →. 指定寄付奨学金（経済状況重視型）. 外部サイトへリンク. 卒業生の団体である三田会等、多くの塾関係者からの寄付による奨学金です。. 申請時期（予定）：4月～5月. 学部生採用者数（予定）：130名程度. 奨学金額：年額10～60万円（奨学金により異なる）. 受給期間：1年（再申請可）. →. 慶應義塾大学東日本大震災被災学生復興支援奨学金. 外部サイトへリンク. 東日本大震災により被災し、経済的に就学が困難である学生を支援する奨学金です。. 申請時期（予定）：6月、11月. 奨学金額：学費の範囲内. 受給期間：1年（再申請可）. さらに勉学に励みたい. 勉学の意欲を持ち、人物・学業成績ともに優秀で、かつ経済的理由により修学が困難な学部生を対象とした奨学金です。. 上下. →. 慶應義塾大学給費奨学金. 外部サイトへリンク. 経済的に困難であると認められ、勉学の意欲を持ち、成績・人物ともに優秀な2年生以上の学部生が対象の奨学金です。. 申請時期（予定）：9月下旬～10月. 採用者数（予定）：310名程度. 奨学金額：年額50万円もしくは25万円. 受給期間：1年（再申請可）. →. 慶應義塾維持会奨学金. 外部サイトへリンク. 慶應義塾維持会会員のご厚意により設立された維持会基金の運用益によって運用されており、愛塾心に富み、人物・学業成績ともにきわめて優秀で、経済的に学業に専念することが困難な学部生を支援しています。申請に際しては、学力要件があり、地方出身者を優先します（首都圏の塾生も応募可能）。. 申請時期（予定）：4月. 採用者数（予定）：135名. 奨学金額：80万円（医学部、理工学部、看護医療学部、薬学部）、50万円（その他の学部）. 受給期間：1年（再申請可）. 全国の奨学生のメッセージを出身地別に掲載しています。. →. 慶應義塾維持会 奨学生のメッセージ. 外部サイトへリンク. →. 指定寄付奨学金（育英型）. 外部サイトへリンク. 卒業生の団体である三田会等、多くの義塾関係者からの寄付による奨学金です。. 申請時期（予定）：4月～5月. 学部生採用者数（予定）：30名程度. 奨学金額：年額10～50万円（奨学金により異なる）. 受給期間：1年（再申請可）. 留学したいとき. 留学することに意欲を持ちながらも、経済的理由により留学することが困難な学部生を対象とした奨学金です。. 上下. 慶應義塾大学は海外での修学経験者を増やすための取り組みに力を入れており、留学のための奨学金も充実しています。. 学位取得を目的としない留学のための奨学金. →. 慶應義塾大学創立150年記念奨学金海外学習支援. 外部サイトへリンク. 経済的に困難であると認められ、勉学の意欲を持ち、成績・人物ともに優秀な学部生が対象の奨学金です。. 申請時期（予定）：年3回. 採用者数（予定）：75名程度. 奨学金額：年額10～30万円（参加プログラムや渡航先の国・地域に応じる）. →. 指定寄付奨学金（留学支援）. 外部サイトへリンク. 卒業生の団体である三田会等、多くの義塾関係者からの寄付による奨学金です。. 申請時期（予定）：4月～5月. 採用者数（予定）：若干名. 奨学金額：年額10～30万円（奨学金により異なる）. →. その他の奨学金（国際センター）. 外部サイトへリンク. 学位取得を目的とする留学のための奨学金. →. 石井・石橋基金 慶應義塾大学グローバルフェローシップ（私費留学助成）. 外部サイトへリンク. 海外の大学院で学位を取得するための私費留学を予定する学部生・大学院生が対象の奨学金です。. ※次年度については未定. 　参考：2022年度募集時の条件等. 申請時期（予定）：10月～11月. 採用者数（予定）：若干名. 奨学金額：年額 最大500万円（留学先の学費・生活費等）. →. その他の奨学金（国際センター）. 外部サイトへリンク. 入学前に予約したい（入学前申請）. 一部の研究科では入学前に奨学金を予約できます。. 上下. →. 研究のすゝめ奨学金. 外部サイトへリンク. 将来、研究者としての活躍が期待される優秀な大学院生（修士課程、博士課程、後期博士課程の大学院生で、留学生も含む）を対象とした奨学金です。大学院入学前あるいは入学後に審査が行われます。他の学内外の奨学金との併用も可能です。. 奨学金額：年額30万円・50万円・70万円. 受給期間：1年. 詳細は研究科により異なります。. 経済的に困ったとき. 家計の急変や継続的な経済困窮、または大規模自然災害の被災によって学業の継続が著しく困難になった大学院生を支援する奨学金です。. 上下. →. 慶應義塾大学修学支援奨学金. 外部サイトへリンク. 勉学の意欲を持ちながらも、急激な家計状況の変化（大規模自然災害による被災を含む）、あるいは継続的な困窮のため経済的に修学が困難な者を支援します。. 申請時期（予定）：6月、11月. 大学院生採用者数（予定）：30名程度. 奨学金額：学費の範囲内（平均年額約30万円）. 受給期間：1年（再申請可）. →. 指定寄付奨学金（経済状況重視型）. 外部サイトへリンク. 卒業生の団体である三田会等、多くの塾関係者からの寄付による奨学金です。. 申請時期（予定）：4月～5月. 大学院生採用者数（予定）：10名程度. 奨学金額：年額10～50万円（奨学金により異なる）. 受給期間：1年（再申請可）. →. 慶應義塾大学東日本大震災被災学生復興支援奨学金. 外部サイトへリンク. 東日本大震災により被災し、経済的に就学が困難である学生を支援する奨学金です。. 申請時期（予定）：6月、11月. 奨学金額：学費の範囲内. 受給期間：1年（再申請可）. さらに勉学に励みたい. 勉学の意欲を持ち、人物・学業成績ともに優秀で、かつ経済的理由により修学が困難な大学院生を対象とした奨学金です。. 上下. →. 慶應義塾大学大学院奨学金. 外部サイトへリンク. 研究の意欲を持ち、経済的に修学困難で学業成績・人物ともに優秀な大学院生を支援する奨学金です。. 申請時期（予定）：4月入学者　4～5月、9月入学者　9月下旬～10月. 採用者数（予定）：200名程度. 奨学金額：年額50万円、60万円（研究科により異なる）. 受給期間：1年（再申請可）. →. 指定寄付奨学金（育英型）. 外部サイトへリンク. 卒業生の団体である三田会等、多くの義塾関係者からの寄付による奨学金です。. 申請時期（予定）：4月～5月. 大学院生採用者数（予定）：10名程度. 奨学金額：年額10万円～学費相当分（奨学金により異なる）. 受給期間：1年（再申請可）. 研究科独自の奨学金. 慶應義塾大学大学院奨学金、指定寄付奨学金の他に、研究科独自の奨学金があります。. 研究助成. 学術研究支援部では、塾内の研究者の様々な研究活動を支援しています。. 上下. 留学したいとき. 留学することに意欲を持ちながらも、経済的理由により留学することが困難な大学院生を対象とした奨学金です。. 上下. 慶應義塾大学は海外での修学経験者を増やすための取り組みに力を入れており、留学のための奨学金も充実しています。. →. 学位取得を目的としない留学のための奨学金（国際センター）. 外部サイトへリンク. →. 学位取得を目的とする留学のための奨学金（国際センター）. 外部サイトへリンク. →. 石井・石橋基金 慶應義塾大学グローバルフェローシップ（私費留学助成）. 外部サイトへリンク. 海外の大学院で学位を取得するための私費留学を予定する学部生・大学院生が対象の奨学金です。. ※次年度については未定. 　参考：2022年度募集時の条件等. 申請時期（予定）：10月～11月. 採用者数（予定）：若干名. 奨学金額：年額 最大500万円（留学先の学費・生活費等）. 民間団体・地方公共団体奨学金. （学部生・大学院生共通）. 財団法人・公益財団法人・民間企業・都道府県や市区町村の自治体など、多くの奨学団体が奨学生を募集しています。慶應義塾大学では、例年約110におよぶ民間団体・地方公共団体から学生の推薦依頼をいただいています。多くの奨学金は、一度採用されると卒業時まで継続的に給付または貸与されます。. →. 民間団体・地方公共団体の奨学金. 外部サイトへリンク. 国による「高等教育の修学支援新制度」. （学部生のみ）. 国による学びを支援する制度で、日本学生支援機構による「給付型奨学金」と文部科学省による「授業料等減免」の2つの制度で実施されています。家計状況や通学形態に応じた区分での支援が受けられますが、支援要件や支援内容等詳細は文部科学省のページをご確認ください。. →. 国による「高等教育の修学支援新制度」（文部科学省）. 外部サイトへリンク. 慶應義塾大学の受験希望者ならびに在学生向けのページもあわせてご確認ください。. →. 【受験生・在学生対象】国による「高等教育の修学支援新制度」について. 外部サイトへリンク. 日本学生支援機構奨学金. （学部生・大学院生共通）. 国の奨学制度で、主として政府からの借入金と卒業生の返還金によって運用されている貸与奨学金です。多くの学生が支援を受けており、慶應義塾大学でも9人に1人が利用しています。採用されると原則標準修業年限まで貸与を受けることができます。. →. 高校予約採用（学部入学予定者のみ）. 外部サイトへリンク. →. 第一種奨学金　緊急採用／第二種奨学金. 応急採用. 外部サイトへリンク. →. 第二種奨学金　海外. 外部サイトへリンク. →. 第一種奨学金・第二種奨学金定期採用. 外部サイトへリンク. →. 第二種奨学金　短期留学. 外部サイトへリンク. →. トビタテ！留学JAPAN（返済不要）. 外部サイトへリンク. 奨学金以外の経済支援制度. （学部生・大学院生共通）. 慶應義塾大学教育ローン制度. 金融機関と提携の制度です。提携金融機関から入学金・授業料等の学費を借り入れることができるもので、入学手続き時の納付金に利用することができます。※ただし、金融機関等の審査により融資を受けられない場合があります。. 国の教育ローン（日本政策金融公庫）. 学費やアパートの家賃等を保護者に融資する公的な制度です。日本学生支援機構奨学金と重複して利用できます。. 生活福祉資金　教育支援資金. 低所得世帯（市町村民税非課税程度）に対して、大学に就学するにあたり必要な経費の貸付を行っています。市区町村の社会福祉協議会が窓口となって実施しています。. →. 奨学金以外の経済支援制度. 外部サイトへリンク. 奨学金の申請・選考. どの奨学金も「学力」「家計状況」「人物」により選考します。申請条件は奨学金の種類によって異なりますので、詳細は以下をご覧ください。. →. 奨学金制度総合案内（塾生サイト）. 外部サイトへリンク. →. Scholarships for International Students (English). 外部サイトへリンク. →. 外国人留学生向け奨学金（日本語）. 外部サイトへリンク. 慶應義塾への支援をお考えの皆様. 慶應義塾への支援をお考えの方は以下をご覧ください。. →. 慶應義塾大学の奨学金（学内奨学金）へのご寄付をお考えの皆様. 外部サイトへリンク. 関連リンク. →. 学費. 豊富な学内奨学金. 学内には多種多様な奨学金があり、学部・研究科独自のものも含めると、その数は110以上（奨学金総額11.8億円、受給者数のべ2,450名　2021年度実績）。全て返済不要の給付型奨学金です。より多くの学生に奨学金を給付できるよう、さらに拡充を続けていきます。. →. 主な学内奨学金. 外部サイトへリンク. 入学前予約. 首都圏以外に住む受験生に安心して慶應義塾大学を進学先に選んでもらえるよう、一般選抜受験前に採否の結果がわかる予約型の奨学金を用意しています。学部によっては、国立大学と同程度もしくはそれ以下の学費負担での進学が可能になります。. →. 学問のすゝめ奨学金. 外部サイトへリンク. 受給候補者. 毎年550名以上採用. 奨学金額. 年間60万円～90万円（原則卒業まで受給可）. ※入学初年度は入学金相当額（20万円）を加算して支給. 成績優秀者. 入学2年目以降成績優秀者の奨学金額を増額. （増額後）年間80万円～150万円. 大学院入学に際しても、入学前に奨学生候補者となることができる給付型奨学金を用意しています（要件は研究科ごとに異なります）。. →. 研究のすゝめ奨学金. 外部サイトへリンク. 国内最大規模の. 奨学基金240億円. 奨学金を目的とする基金をさらに拡大しています。その総額は国内最大規模の240億円。運用益は返済不要の給付型奨学金の財源として使われます。既に国内の大学では標準的になりつつある給付型奨学金ですが、慶應義塾大学は早期から財源の安定化についても積極的に取り組んでいます。. 卒業生や篤志家等の寄付による奨学金. 慶應義塾大学の魅力は「社中協力」の精神が根付いていること。「塾員」（卒業生）で結成される様々な「三田会」（同窓会）や篤志家の皆様からのご支援による奨学金が数多くあります。奨学生は「義塾社中」による支援を実感しながら学業に励むことができます。. →. 指定寄付奨学金. 外部サイトへリンク. 35奨学金、支給総額5,000万円、給付予定人数150人以上（2022年度予定）. →. 慶應義塾維持会奨学金. 外部サイトへリンク. 支給額8,400万円、給付予定人数135人（2022年度予定）. 留学. グローバル社会の先導役を担う大学として、国際化を強力に推進するため、「留学」に係る奨学金を数多く用意しています。. →. 学位取得を目的としない留学（交換留学等）奨学金. 外部サイトへリンク. 慶應義塾大学交換留学生（派遣）奨学金. 慶應義塾創立150年記念奨学金 海外学習支援. →. 学位取得を目的とする留学奨学金. 外部サイトへリンク. 石井・石橋基金 慶應義塾大学グローバルフェローシップ（私費留学助成）. →. 外国人留学生向け奨学金. 外部サイトへリンク. 未来先導国際奨学金. 経済支援. 家計の急変や継続的な収入不足、大規模自然災害被災など、経済的な困窮を解消できない理由は学生によって様々です。最近では、新型コロナウイルス感染症の拡大による家計への影響も問題となっています。これらのような経済的な理由で、学生が勉学の機会を失うことがないよう、困窮の理由を限定せず幅広くサポートする奨学金を設けています。. →. 家計急変・災害被災時の奨学金. 外部サイトへリンク. 慶應義塾大学修学支援奨学金. 2022年度受給者数（予定）100名程度（年額平均約30万円）. 慶應義塾大学東日本大震災被災学生復興支援奨学金. CLOSE. →. ←. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. 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ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP EN JP KR CN BOOK NOW ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP JP. EN. KR. CN. 2022年5月26日. 《ひょうご旅しようキャンペーン＋》　兵庫県民割6月まで延長になります。. ゲストハウス萬家は「ひょうご旅しようキャンペーン」対象施設です。. 「ふるさと応援！ひょうごを旅しようキャンペーン＋（プラス）」を. 6月30日（木）まで期間延長（7月1日（金）チェックアウト分まで）します。. ・対象を兵庫県民に加え、滋賀県・京都府・大阪府・和歌山県・鳥取県・岡山県・徳島県・香川県民に拡大します。. ・【6月1日（水）以降】ワクチン3回接種済またはPCR検査、抗原定量検査（出発日3日以内）、抗原定性検査（出発日1日以内）が陰性であることが必要です。. 県民割クーポンを申請することで、いつもよりお気軽にお泊りいただけます。. ご予約が完了した時点ではキャンペーンの割引は適用されません。. 予約後に割引クーポン発行サービス（STAY NAVI）に移動し、. 割引クーポン発行の手続きを行ってください。. クーポン発行後、チェックイン時にフロントにで発行された割引クーポン番号を提示することにより割引が適用となります。. 県民割キャンペーンの詳細に関しては下記をご参考いただきますように宜しくお願い致します。. ↓　↓　↓. 県民割キャンペーン利用方法 Category. ニュース イベント Archive. 2022年5月 2022年4月 2022年3月 2021年8月 2021年5月 2020年6月 2020年4月 2020年3月 2020年2月 2019年8月 2019年7月 2019年5月 2019年4月 2019年2月 2019年1月 2018年11月 2018年10月 2018年9月 2018年7月 2018年5月 2018年4月 2018年3月 2018年1月 2017年12月 2017年9月 2017年7月 2017年6月 ゲストハウス萬家(MAYA). 〒657-0836. 兵庫県神戸市灘区城内通4丁目4-10. 予約・問い合わせ：090-9856-8280（受付時間 8:00-22:00）. TEL: 078-940-2809（受付時間 8:00-22:00）. e-mail: info@kobemaya.com. 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La Parole Errante, 9 rue François Debergue, MONTREUIL. Entrée Libre et Gratuite*. * des contrôles pourront être effectués à l’entrée. Programme En direct Le 5ème salon “Des livres et l’alerte” se déroulera les 22\/23\/24 novembre 2019. à la “Parole Errante” à Montreuil. Nous accueillerons cette année encore de nombreux acteurs du monde de l’alerte dans leur diversité, lanceuses et lanceurs d’alertes, auteurs, journalistes, chercheurs, universitaires…. toutes et tous engagés pour l’intérêt général. . Ce salon indépendant est l’unique lieu où se rencontrent largement celles et ceux qui enquêtent, s’informent et réfléchissent sur les alertes. et les enseignements qu’elles recèlent. Connus ou inconnus,. toutes et tous ont fait le choix d’obéir à leur conscience de l’intérêt général. , souvent au prix de représailles injustifiables, mais toujours. avec la fierté de pouvoir « se regarder dans la glace ». et d’avoir transmis les preuves si importantes pour notre société. Dossier de Presse Vidéo réalisée par. Julia Blagny Julian Assange. ,. Edward Snowden. ,. Reality Winner. , et toutes celles et ceux qui sont victimes de représailles, d’emprisonnement et, de menaces pour avoir alerté,. doivent pouvoir trouver refuge dans les pays démocratiques. pour ne pas être victimes de meurtres. Daphne Caruana Galizia. ,. Jàn Kuciak et Martina Kusnirova. ,. Berta Caceres. et tant d’autres assassiné.e.s pour avoir informé, resteront au coeur de notre salon. La qualité des intervenants et la diversité des thèmes abordés font de cette rencontre un véritable évènement ancré dans l’actualité. Daniel Ibanez. co-fondateur du salon. Intervenants. Ils seront présents pour cette édition 2019. Patrick ACKERMANN ,. Gaétan ALIBERT ,. Emmanuelle AMAR ,. Matthieu ARON ,. Raymond AVRILLIER ,. Isabelle BADOUREAUX ,. Gérard BAPT ,. Frédérique BASSET ,. Delphine BATHO ,. Karim BEN ALI ,. Jean-François BERNARDINI ,. Marie BLANDIN ,. David BOILLEY ,. Juliette BOUDRE ,. François BOULO ,. William BOURDON ,. Dominique BOURG ,. Céline BOUSSIE ,. Thomas BOUT ,. Denis BRETEAU ,. Gilles BRUEY ,. Guylain CABANTOUS ,. Patrick CAHEZ ,. Alexandre CALVEZ ,. Jean-Pierre CANET ,. Paul CASSIA ,. Caroline CHAUMET ,. Pascal CLOUAIRE ,. Xavier CŒUR-JOLLY ,. Collectif d’artistes “LANCER L’ALERTE” ,. Hélène CONSTANTY ,. Valérie CUSSAGUET ,. Marc DANTAN ,. Guillaume de MORANT ,. Jean-Marie DELARUE Intervention annulée ,. Jean René DELÉPINE ,. Gérard DELTEIL ,. Antoine DELTOUR ,. Mathias DESTAL ,. Thomas DIETRICH ,. Ivan DU ROY ,. David DUFRESNE ,. Anny DUPÉREY ,. Jacques DUPLESSY ,. Juliette DUQUESNE ,. Marc ENDEWELD ,. Jean Philippe FOEGLE ,. Jacques FOLLOROU ,. Nicolas FORISSIER ,. Stéphane FOUCART ,. Paul FRANÇOIS ,. Mauricio GARCIA PERREIRA ,. Quentin GUILLEMAIN ,. Irène GUNEPIN ,. HK (Kaddour HADDADI) ,. Daniel IBANEZ ,. Raphaël KEMPF ,. Hella KHERIEF ,. Nadezda KUTEPOVA ,. Vanessa LANGARD ,. Alexandre LANGLOIS ,. Léna LAZARE ,. Claire LE GAL ,. Frédéric LEMAIRE ,. Catherine LEMOINE BRUEY ,. Roger LENGLET ,. Inès LERAUD ,. Didier LESTRADE ,. Ingrid LEVAVASSEUR ,. Priscillia LUDOSKY ,. Marine MARTIN ,. Florence MÉRÉO ,. Caroline MICHEL-AGUIRRE ,. Emmanuelle MIGNATON ,. Aymeric MONVILLE ,. Françoise NICOLAS ,. Béatrice PANNIER ,. Philippe PASCOT ,. Denis PERREAU ,. Olivier PETITJEAN ,. Marie PEZÉ ,. Maurice PICHON ,. Frédéric PIERUCCI ,. Michel PINÇON ,. Monique PINÇON-CHARLOT ,. Alexis POULIN ,. Mathilde RÉGIS ,. Maxime RENAHY ,. Fabrice RIZZOLI ,. Denis ROBERT ,. Florence RODHAIN ,. Olivier ROUZET ,. Haruko SAKAGUCHI ,. Elisabeth SCHNEITER ,. Lionel SERRE ,. Akli TADJER ,. Annie THÉBAUD MONY ,. Marine TURCHI ,. Elise VAN BENEDEN ,. Jean-Michel VERNE ,. Nicolas VESCOVACCI ,. Patricia VIOUX ,. Denis ZMIROU-NAVIER Les intervenants en détail Le programme en détails Comment s’y rendre. La Parole Errante. 9 rue François Debergue, Montreuil. Métro: Croix de Chavaux – Ligne 9. Y aller Afficher une carte plus grande Nos Soutiens. Seul on va plus vite, ensemble on va plus loin. Anticor Cafés citoyens santé et travail Citoyens Résistants d'Hier et d'Aujourd'hui Crim'HALT Julia Blagny Le Presse Papier Libre et Solidaire Maison des Lanceurs d'Alerte La Montreuilloise Not a Number Satoriz Transparency International France Vivre et Agir en Maurienne Extra Muros Réseau \"Souffrance & Travail\" Association Henri Pézerat Ligue des Droits de l'Homme Nous voulons des coquelicots Libre et Solidaire Amis de la Terre Syndicat des Avocats de France SAF La Conquête du Pain SEPTIEME FACTORY ARROBE HAUTEVILLE PRODUCTIONS TVNET CITOYENNE Les Mutins de Pangée Gis Démocratie & Participation MSH Paris Nord Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Revue Dessinée @LaRevueDessinee 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec  @RevueDessineeIT,  @ElianePatriarca, Grégory Mardon,  @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans.  https:\/\/t.co\/teOnjQTYZC","","","","","3","6","Facebook","Twitter"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:44:31 UTC 2022","fresh_date_dt":"Wed Jun 08 14:44:32 UTC 2022","language_s":"fr","size_i":242857,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078153537191936,"text_t":"Programme Thématique 2021 | Rencontres Annuelles des Lanceurs d'Alerte. Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. 6. ème. Rencontre Annuelle des Lanceurs d'Alerte. 12, 13 & 14 Novembre 2021. à la Maison des Sciences de l'Homme Paris Nord. (MSH Paris Nord) La Plaine St-Denis. Métro Ligne 12 \"Front populaire\". Entrée Libre et Gratuite*. Les rencontres se dérouleront dans le respect des gestes barrières et des règles limitant la propagation des virus. Programme Dossier Presse Le programme Thématique 2021. Vous pouvez consulter le. programme complet ici. Santé Protection Affaires & Finances Démocratie Environnement Publier l'alerte Projections 13 Nov 2021 15:00-16:30. panoramique. Les soins, les soignants et l’accès aux soins. Caroline FIAT. ,. Députée. Pierre MERIAUX. ,. Maire-adjoint Grenoble. Olivia MOKIEJEWSKI. ,. Journaliste. Cyrille VENET. ,. Anesthésiste. Chacun d’entre nous a été alerté sur les conditions de protection des soignants et les difficultés dans les établissements de soins dès la première vague et lors du premier confinement. Depuis de longs mois, les professions de santé avaient alerté les pouvoirs publics sur les conséquences que pouvaient avoir les réductions de postes, de lits, […]. Lire la suite 13 Nov 2021 17:30-18:30. Salle Panoramique. Santé publique : qu’a révélé la pandémie ? Marie BLANDIN. ,. Ancienne sénatrice. Raphaël RUFFIER-FOSSOUL. ,. Journaliste. Bruno TOUSSAINT. ,. Membre de la rédaction de Prescrire. La controverse scientifique mise en scène sur les plateaux audiovisuels, sur laquelle se sont exprimés journalistes, essayistes, commentateurs, conduit une partie de la population à douter de la démarche scientifique et du monde médical. Les conflits d’intérêts ressurgissent et renforcent le doute. Dès lors toutes les théories se développent sur les imprécisions, les omissions et […]. Lire la suite 14 Nov 2021 11:00-12:30. auditorium 300. Il y a l’alerte et puis après Film “Droit au cœur” (procès Mediator). Irène FRACHON. ,. Pneumologue, lanceuse d'alerte de l'affaire du Médiator,. Anne RICHARD. ,. Documentariste, réalisatrice du film « Droit au cœur ». Nous projetterons le film d’Anne RICHARD qui a suivi Irène FRACHON pendant ces longues années et son documentaire retrace ces années jusqu’au procès. Nous pouvons projeter ce film grâce au soutien de Paris-Brest Productions. Et après l’alerte ? Nombreux sont les lanceurs d’alerte qui déclarent si j’avais su … Il est certain qu’Irène Frachon ne […]. Lire la suite 14 Nov 2021 11:00-12:30. Salle Panoramique. Écrans et apprentissage numérique Alerte pour les enfants ? Sabine DUFLO. ,. Psychologue clinicienne. Pablo IBAÑEZ CRUCEYRA. ,. Chercheur. Karine MAUVILLY GRATON. ,. Philippe MEIRIEU. ,. Professeur émérite. Les écrans à l’école et le numérique comme moyen d’apprentissage s’est imposé avec des slogans comme “lutter contre la fracture numérique” ou encore sur le présupposé que les enfants doivent manipuler l’informatique le plus tôt possible pour être adapté à la société de demain. Combien un enfant passe-t-il d’heure devant des écrans chaque jour, de […]. Lire la suite 12 Nov 2021 16:30-18:30. Amphithéâtre. Définition Transposition de l’alerte Propositions des lanceurs d’alerte. Ugo BERNALICIS. ,. député du Nord. Francis CHATEAURAYNAUD. ,. Sociologue Chercheur. Jean-Michel FOURNIAU. ,. Sociologue, Président GIS démocratie&Participation. Guillaume GONTARD. ,. Sénateur. Virginie ROZIERE. ,. Ex-députée européenne, porteuse de la directive européenne de protection. Nathalie TEHIO. ,. Avocate. Sylvain WASERMAN. ,. Député du Bas-Rhin (sous réserve). Les États membres de l’Union Européenne doivent transposer en droits national la Directive sur “la protection des personnes qui signalent des violations du droit de l’Union” avant la fin de cette année 2021. Un projet de loi a été déposé et doit être voté avant la fin de l’année. Selon la Déclaration des Droits de […]. Lire la suite 13 Nov 2021 11:00-12:30. auditorium 300. Carte Blanche à Erri DE LUCA. Erri DE LUCA. ,. Auteur. Philippe MEIRIEU. ,. Professeur émérite. Erri DE LUCA est l’auteur italien le plus vendu en France, il est l’observateur engagé de notre monde, ses mots sont précis et lorsqu’il utilise le mot sabotage, qui lui est reproché par les promoteurs du projet Lyon-Turin, il écrit “La parole contraire” revendiquant l’usage du mot et l’usage des mots. Cette procédure bâillon à […]. Lire la suite 14 Nov 2021 13:30-15:00. auditorium 300. 2021 Paroles de lanceurs d’alertes. Amar BENMOHAMED. ,. Policier. Marie BLANDIN. ,. Ancienne sénatrice. Ariane DENOYEL. ,. Journaliste Enseignante école de journalisme de Grenoble. Ludovic FAYOLLE. ,. Policier. Maureen KEARNEY. ,. Emmanuelle MIGNATON. ,. Lanceuse d'alerte, affaire de l'Androcur. Valérie MURAT. ,. Fondatrice de \"Alerte aux Toxiques\". Philippe PIARD. ,. Paysan. Denise SCHNEIDER. ,. présidente de l'association. Comme chaque année nous donnons la parole aux lanceuses et lanceurs d’alertes, celles et ceux qui ont informé la population en transmettant des informations pour prévenir ou corriger les situations de risque ou les dysfonctionnements. Venez écouter le courage, la citoyenneté et l’obéissance à l’intérêt général. Lire la suite 14 Nov 2021 16:00-17:30. Salle Panoramique. Alerte : Bilan de l’année. Juliette ALIBERT. ,. Avocate. Francis CHATEAURAYNAUD. ,. Sociologue Chercheur. Nicolas FORISSIER. ,. Lanceur d'alerte Affaire UBS, Metamorphosis. Elise VAN BENEDEN. ,. Avocate, administratrice d'ANTICOR. Deux années particulièrement exceptionnelles se sont écoulées depuis la dernière rencontre annuelle des lanceurs d’alertes, des nouvelles alertes sont apparues, mais aussi un fort enjeu avec la transposition de la directive européenne dite de protection des lanceurs d’alertes. Faire le bilan des actions menées est déterminant pour agir et protéger l’alerte et celles et ceux […]. Lire la suite 13 Nov 2021 13:00-14:30. Amphithéâtre. Les trous noirs des financements publics. Raymond AVRILLIER. ,. Citoyen engagé. Daniel IBANEZ. ,. Co-fondateur des rencontres. Patrice SALINI. ,. Économiste des transports, Ancien directeur de cabinet ministériel. Françoise VERCHERE. ,. Maire et conseillère générale honoraire. A quoi servent les investissements publics ? La question peut paraître incongrue, mais peut-on y répondre sans évaluer ces investissements a posteriori ? On parle beaucoup de la responsabilité des décideurs, voire de l’impunité dans l’utilisation de l’argent public mais comment sont évaluées les réalisations, y-a-t-il des obligations de les évaluer ? La Déclaration des […]. Lire la suite 14 Nov 2021 13:30-15:00. Salle Panoramique. Affaires d’État : ALSTOM, SNCF, Pôle emploi …. Elisabeth BORREL. ,. Ex magistrate. Denis BRETEAU. ,. Lanceur d'alerte des marchés SNCF \/ IBM. Yann GAUDIN. ,. Lanceur d'alertes sur Pôle emploi. Frédéric PIERUCCI. ,. auteur. Nicolas VESCOVACCI. ,. Journaliste, co-auteur de « Vincent tout puissant ». La liste des participants au « dîner du siècle » a été publiée et les citoyens ne peuvent que s’interroger sur ce que l’on appelle l’entre soi, la porosité entre les sphères politiques, judiciaires, économique et médiatiques. Le lobby est organisé au niveau Européen et national. Identifier ces réseaux, leur fonctionnement nous amène une fois […]. Lire la suite 13 Nov 2021 11:00-12:30. auditorium 300. Carte Blanche à Erri DE LUCA. Erri DE LUCA. ,. Auteur. Philippe MEIRIEU. ,. Professeur émérite. Erri DE LUCA est l’auteur italien le plus vendu en France, il est l’observateur engagé de notre monde, ses mots sont précis et lorsqu’il utilise le mot sabotage, qui lui est reproché par les promoteurs du projet Lyon-Turin, il écrit “La parole contraire” revendiquant l’usage du mot et l’usage des mots. Cette procédure bâillon à […]. Lire la suite 13 Nov 2021 13:00-14:30. panoramique. La surveillance pour quelle sûreté ? Antonio CASILLI. ,. Professeur Chercheur. Francis CHATEAURAYNAUD. ,. Sociologue Chercheur. Julie. ,. Thomas LEBONNIEC. ,. Olivier TESQUET. ,. Journaliste. Tous les problèmes de notre société semblent devoir recevoir comme réponse une surveillance accrue. Du contrôle biométrique dans les établissements scolaires ou la proposition de bracelets de présence électroniques pour les élèves dans certains établissements, aux surveillances des contestataires des grands projets comme à Bure par exemple, en passant par la surveillance des « actions […]. Lire la suite 13 Nov 2021 13:30-16:30. auditorium 300. Julian ASSANGE : l’acharnement Film “Hacking justice”. Olivier AZAM. ,. Les mutins de pangée. Clara LÓPEZ RUBIO. ,. Réalisatrice. Aymeric MONVILLE. ,. Juan PANCORBO. ,. joournaliste. Le journaliste Julian ASSANGE est enfermé depuis bientôt dix ans pour avoir publié les informations sur les pratiques de l’armée de coalition et les meurtres de civils notamment avec la vidéo “colateral murder” https:\/\/collateralmurder.wikileaks.org\/. L’ONU a qualifié le traitement qui lui est infligé de torture. Après la projection du documentaire de Clara LOPEZ RUBIO “Hacking […]. Lire la suite 13 Nov 2021 15:30-17:00. Amphithéâtre. La banlieue délaissée de la République. Amal BENTOUNSI. ,. Fondatrice \"Urgence notre police assassine\". Gilles BRUEY. ,. Libraire. Alexandre CALVEZ. ,. Organisateur. Eve SZEFTEL. ,. Journaliste. Parler de la banlieue c’est entendre « racaille » « radicalisé » « zone de non droit » « racisme » « discrimination » « violence » « abus des droits sociaux » “violence policière” “délit de sale gueule”… La banlieue est-elle soumise aux lois fondamentales de la République ou à des lois locales d’exception. […]. Lire la suite 13 Nov 2021 17:30-19:00. auditorium 300. États d’urgence \/ État de droit \/ État d’Alerte. Arié ALIMI. ,. Avocat. Daniel IBANEZ. ,. Co-fondateur des rencontres. Asma MHALLA. ,. Maître de conférence. Serge PORTELLI. ,. Ancien magistrat. Après l’état d’urgence prolongé dans les affaires terroristes et les attentats de 2015, la France connaît maintenant l’état d’urgence sanitaire avec des limitations exceptionnelles dans la liberté de circuler, celle de se rassembler. Durant cette période on a pu avoir le sentiment que les fonctionnaires de police faisaient la loi. Limitations de circulation, contrôle des […]. Lire la suite 13 Nov 2021 17:30-19:00. auditorium 140. Une police républicaine ? Christian CHOUVIAT. ,. Père de Cédric Chouviat. Vanessa LANGARD. ,. Gilet Jaune mutilée. Mathieu MOLARD. ,. Journaliste. Nathalie TEHIO. ,. Avocate. Laurent THINES. ,. Chirurgien. Une chose est sûre, nous aurions pu inviter bien plus d’intervenant e s pour cette rencontre témoignant des dérives d’une partie de celles et ceux qui portent l’uniforme. Le bilan des victimes est un indicateur des dysfonctionnements en matière de maintien de l’ordre, la nature des blessures et des mutilations infligées lors des manifestations par […]. Lire la suite 14 Nov 2021 13:00-14:00. Amphithéâtre. Covid-19 les pauvres paupérisés double peine. Philippe BREUVART. ,. citoyen engagé. Laurence CHAMPIER. ,. Banques Alimentaires. Manuel DOMERGUE. ,. Fondation Abbé Pierre. La crise sanitaire du printemps, nous a alertés sur les difficultés pour le monde économique et les salariés. Le rôle primordial pour la société des « invisibles », infirmières, aides-soignantes, caissières, livreurs, femmes de ménage, … a été mis en lumière. La situation des plus pauvres, travailleurs précaires, travailleurs saisonniers, travailleurs au noir, petits boulots […]. Lire la suite 14 Nov 2021 13:30-15:00. Salle Panoramique. Affaires d’État : ALSTOM, SNCF, Pôle emploi …. Elisabeth BORREL. ,. Ex magistrate. Denis BRETEAU. ,. Lanceur d'alerte des marchés SNCF \/ IBM. Yann GAUDIN. ,. Lanceur d'alertes sur Pôle emploi. Frédéric PIERUCCI. ,. auteur. Nicolas VESCOVACCI. ,. Journaliste, co-auteur de « Vincent tout puissant ». La liste des participants au « dîner du siècle » a été publiée et les citoyens ne peuvent que s’interroger sur ce que l’on appelle l’entre soi, la porosité entre les sphères politiques, judiciaires, économique et médiatiques. Le lobby est organisé au niveau Européen et national. Identifier ces réseaux, leur fonctionnement nous amène une fois […]. Lire la suite 12 Nov 2021 17:00-20:00. auditorium 300. Climat : Le mur, on s’en approche ou on est dedans ? Camille ÉTIENNE. ,. Etudiante. Gilbert GOURRAUD. ,. Référent Terre de Lien Savoie. Marie Monique ROBIN. ,. Journaliste. Un film sur les arbres et le risque de leur disparition. Le documentaire “Le génie des arbres” d’Emmanuelle NOBECOURT que nous projetterons en première partie grâce au soutien de Hauteville Productions démontre avec simplicité et clarté ce génie naturel des arbres et leur rôle essentiel mais il montre également les atteintes que porte l’espèce humaine […]. Lire la suite 13 Nov 2021 10:30-12:00. Amphithéâtre. L’emploi, limite de l’alerte ? Annie COLLOMBET. ,. Co-présidente de Vivre et Agir en Maurienne. Florent COMPAIN. ,. Ancien président Les Amis de la Terre. Pedro DA FONSECA. ,. Journaliste. Jean-Michel FOURNIAU. ,. Sociologue, Président GIS démocratie&Participation. Quentin GUILLEMAIN. ,. Asso Parents victimes de Lactalis. Karim Ben Ali accusé de mettre en péril l’emploi, la promotion de grands projets au nom de l’emploi, l’emploi et la santé publique, la pollution mais l’emploi… Peut-on conclure que l’emploi est un frein réel à l’alerte ? L’alerte doit-elle disparaître au nom de la préservation de l’emploi ? Les organisations professionnelles sont-elles dans une […]. Lire la suite 13 Nov 2021 11:00-12:30. panoramique. Alerte : déficience des services de l’Etat. Guillaume GONTARD. ,. Sénateur. Gérald LE CORRE. ,. Affaire LUBRIZOL. Pierre MERIAUX. ,. Maire-adjoint Grenoble. Gabriel ULLMANN. ,. expert environnement. Les affaires Lubrizol ou Lactalis font apparaître des dysfonctionnements majeurs et démontrent que la politique d’autocontrôle prônée atteint des limites et présentent de graves dangers pour la population. On constate dans ces dossiers que les risques viennent finalement de de la déficience des services de l’Etat à laquelle s’ajoute l’absence de lanceurs d’alertes. La réduction […]. Lire la suite 14 Nov 2021 15:00-17:30. Amphithéâtre. Pesticides : quelles limites ? Daniel CUEFF. ,. Ancien Maire de Langouët. Florian FERJOUX. ,. Avocat. Fabien GRACIA. ,. Maire de La Montagne. Françoise VERCHERE. ,. Maire et conseillère générale honoraire. L’année 2019 a été marquée par un large débat sur l’utilisation des « pesticides » pour les uns, « produits phytopharmaceutiques » pour les autres ; des maires ont interdit par arrêté leur utilisation à moins de 150 mètres des habitations, la FNSEA considère que la limite est de zéro mètre puisque les produits sont […]. Lire la suite 14 Nov 2021 16:00-17:30. auditorium 300. En même temps, Bâillonner l’alerte. Brigitte GOTHIERE. ,. Daniel IBANEZ. ,. Co-fondateur des rencontres. Morgan LARGE. ,. Journaliste. Inès LERAUD. ,. Journaliste, autrice de « Algues vertes l’histoire interdite ». Fin 2019, Le ministre de l’Intérieur annonce la création d’une cellule de gendarmerie dont l’objectif serait la protection du monde agricole avec 3 objectifs déclarés : « améliorer notre coopération avec le monde agricole et de recueillir des renseignements ; mieux connaître les groupes extrémistes à l’origine des atteintes et de pouvoir anticiper et prévenir […]. Lire la suite 14 Nov 2021 11:00-12:00. Amphithéâtre. L’alerte en Bande Dessinée. Louis-Antoine DUJARDIN. ,. Editeur de BD. Sylvain INSERGUEIX. ,. Libraire. Flore TALAMON. ,. Autrice. La publication de l’alerte connaît une évolution avec l’apparition de quelques bandes dessinées, le succès de l’enquête d’Inès Léraud sur les algues vertes et les pressions qu’elle suscitent montrent que cette forme d’édition de l’alerte est efficace. Quels sont les canaux d’édition et comment s’organise une publication de Bande Dessinée, comment les lanceurs d’alertes peuvent-ils […]. Lire la suite 14 Nov 2021 17:30-19:00. Amphithéâtre. Journalisme et réseaux sociaux : Alerte aux complots ? Jean-Pierre CANET. ,. journaliste. Daniel IBANEZ. ,. Co-fondateur des rencontres. Étienne MERLE. ,. Journaliste. Françoise NICOLAS. ,. ancienne fonctionnaire des affaires étrangères. La défiance envers la presse et les journalistes est un sujet d’inquiétude, les réseaux sociaux sont l’objet de critiques du monde politique et médiatique. Comment l’alerte trouve-t-elle, peut-elle trouver sa place ? Comment les lanceurs d’alertes peuvent-ils publier, faire publier les informations qu’ils considèrent comme graves pour notre société ? Comment un journaliste, avec les […]. Lire la suite Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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{"id":"YXq71si7ikli","host_id_s":"i7ikli","sku":"https:\/\/lanceurs-alerte.fr\/intervention\/nucleaire-et-apres","host_s":"lanceurs-alerte.fr","host_organization_s":"lanceurs-alerte","url_chars_i":58,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["intervention"],"url_file_name_s":"nucleaire-et-apres","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Nucléaire et après ? | Rencontres Annuelles des Lanceurs d'Alerte"],"last_modified":"Wed Jun 08 14:44:33 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Sun Nov 24 00:00:00 UTC 2019","Wed Dec 11 00:00:00 UTC 2013"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-4132102767645286631,"exact_signature_unique_b":true,"fuzzy_signature_l":-3991057679741499030,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["lanceurs-alerte.fr\/wp-content\/uploads\/2018\/09\/logo-deslivresetlalerte-300x300.jpg","lanceurs-alerte.fr\/wp-content\/uploads\/2018\/09\/logo-deslivresetlalerte-150x150.jpg"],"icons_rel_sxt":["icon apple-touch-icon","icon"],"icons_sizes_sxt":["192x192","32x32"],"h1_txt":["Nucléaire et après ?"],"h2_txt":["Salle Librairie, 24 Nov 2019 16:00-17:30","Intervenants"],"imagescount_i":3,"images_protocol_sxt":["000-https","001-https","002-https"],"images_urlstub_sxt":["lanceurs-alerte.fr\/wp-content\/uploads\/2020\/08\/logo-lanceursdalerte.png","pbs.twimg.com\/profile_images\/789132615601254400\/mw17e7Qn_normal.jpg","pbs.twimg.com\/media\/FUGAJ7YXwAIxq1u.jpg"],"images_alt_sxt":["Rencontres Annuelles des Lanceurs d'Alerte","Lanceurs d'alertes","Lanceursalertes photo"],"images_height_val":[41,-1,-1],"images_width_val":[41,-1,-1],"images_text_t":"lanceurs alerte fr wp content uploads 2020 08 logo lanceursdalerte png Rencontres Annuelles des Lanceurs d'Alerte pbs twimg com profile images 789132615601254400 mw17e7Qn normal jpg mw 17e 7 Qn Lanceurs d'alertes media FUGAJ7YXwAIxq1u FUGAJ YXw AIxq 1u Lanceursalertes photo","responsetime_i":3335,"text_t":"Nucléaire et après ? | Rencontres Annuelles des Lanceurs d'Alerte. Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Nucléaire et après ? Salle Librairie, 24 Nov 2019 16:00-17:30. Malgré les discours rassurants, le nuage de Tchernobyl ne s’est pas arrêté aux frontières et des millions de mètres cube d’eau contaminée sont déversés chaque jour dans l’océan à Fukushima. La France avec ses installations nucléaires vieillissantes et son passé d’expériences plus ou moins contrôlées présente-t-elle des risques et une menace pour ses voisins ? La population doit-elle craindre un accident et les conséquences irréversibles ? En dehors des accidents, le nucléaire laisse-t-l des traces irréversibles à l’échelle humaine ? Notre civilisation peut-elle enfouir ses déchets et laisser au futur la gestion des conséquences de ses choix ou de ses erreurs ? Alors que les coûts de l’EPR à Flamanville explosent et que l’on renationalise le risque nucléaire, débattre en transparence du choix nucléaire n’est pas superflu. Relater, analyser et informer sur ce qui s’est passé dans les installations nucléaires, de Maïak à Fukushima, et sur effets toujours visibles, permet de mieux approcher ce qui peut, au-delà des aspects purement techniques, se jouer dans la société toute entière. Intervenants. David BOILLEY Irène GUNEPIN Nadezda KUTEPOVA Haruko SAKAGUCHI Patricia VIOUX Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. 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En coédition avec  @RevueDessineeIT,  @ElianePatriarca, Grégory Mardon,  @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans.  https:\/\/t.co\/teOnjQTYZC","","","","","3","6","Facebook","Twitter"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:44:36 UTC 2022","fresh_date_dt":"Wed Jun 08 14:44:36 UTC 2022","referrer_id_s":"Wd6AgNi7ikli","language_s":"fr","size_i":228179,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":1,"references_i":5,"references_internal_i":5,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078157817479168,"text_t":"Le programme 2019 | Rencontres Annuelles des Lanceurs d'Alerte. Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Des Livres et l'Alerte. 5ème salon du livre de lanceuses et lanceurs d'alerte. 22,23 & 24 Nov 2019. La Parole Errante, 9 rue François Debergue, MONTREUIL. Programme de l’édition 2019. 22 Nov 2019 19:00-20:30. Salle Gradin. ActUp, 30 ans d’alertes. Xavier CŒUR-JOLLY. ,. Ancien vice-président d’ActUp Paris. Didier LESTRADE. ,. Journaliste co-fondateur d’ActUp Paris. Alexis POULIN. ,. Journaliste co-fondateur LeMondeModerne.media. Il était de bon ton dans les années 80 d’ironiser à propos du SIDA, maladie réservée aux « autres ». Sans ActUp et son rôle essentiel dans la prise de conscience que cette maladie mortelle touchait tout le monde, les pouvoirs publics seraient sans doute restés plus longtemps dans une position attentiste. Souvent critiquée pour […]. Lire la suite 23 Nov 2019 10:30-12:00. Salle Gradin. Projection XY CHELSEA (entrée gratuite). En ouverture de la journée du samedi nous proposons au public d’assister à la projection du fil XY CHELSEA en sortie nationale depuis le 30 octobre. L’entrée est gratuite. En 2010, le soldat Manning divulgue à Wikileaks des documents sur les bavures de l’armée américaine en Irak, en Afghanistan et l’espionnage de la diplomatie mondiale, […]. Lire la suite 23 Nov 2019 12:00-13:00. Salle Librairie. Lanceurs d’alertes : leurs droits de divulguer. William BOURDON. ,. Avocat, fondateur de l'ONG SHERPA. William BOURDON, acteur incontournable de l’alerte, comme avocat mais également au sein des associations dont il est membre. Son cabinet défend notamment Rui PINTO qui a été emprisonné pour avoir transmis les informations sur l’évasion fiscale pratiquée par le monde du football. Nous aborderons avec lui la situation des lanceurs d’alertes, Edward SNOWDEN qui est […]. Lire la suite 23 Nov 2019 12:00-13:00. Salle Étage. Évasion fiscale, qui consent à l’impôt et qui n’y consent pas ? Maxime RENAHY. ,. Fondateur de « Lanceuralerte.org », ancien espion de la DGSE. Les chiffres les plus fous courent sur le montant annuel de l’évasion fiscale. Dès le début de la contestation des taxes par les Gilets jaunes, le discours officiel a souvent repris la thèse du refus du consentement à l’impôt. De l’optimisation fiscale à la fraude en passant par des opérations de défiscalisation, qui consent et […]. Lire la suite 23 Nov 2019 13:00-14:30. Salle Gradin. Gilets Jaunes : Alerte pour la République ? François BOULO. ,. Avocat. Paul CASSIA. ,. Universitaire en Droit public. Gérard DELTEIL. ,. Écrivain. Priscillia LUDOSKY. ,. Gilet Jaune. Nicolas VESCOVACCI. ,. Journaliste, co-auteur de « Vincent tout puissant ». En qualifiant les Gilets Jaunes d’« émeutiers » (Laurent Nuñez), de « complices du pire » (Emmanuel Macron) ou de « séditieux » (Christophe Castaner), de complotistes ou de factieux, le Gouvernement, les personnalités politiques ou médiatiques exprimaient-ils leur peur POUR la République ou une peur DE la République ? A la lumière, notamment des […]. Lire la suite 23 Nov 2019 13:30-15:00. Salle Librairie. Journalisme en péril, journalisme en question. Hélène CONSTANTY. ,. Journaliste. Mathias DESTAL. ,. Journaliste. Frédéric LEMAIRE. ,. Membre d'ACRIMED. Les convocations de journalistes par la Direction générale de la surveillance intérieure (DGSI), leurs gardes à vues, les agressions verbales ou physiques dont ils sont victimes, venant de personnalités politiques, de la force publique ou de la population posent la question de savoir si les journalistes peuvent aujourd’hui exercer librement leur mission d’information. Les poursuites […]. Lire la suite 23 Nov 2019 14:30-16:00. Salle Étage. La démocratie locale, premier niveau d’alerte ? Pascal CLOUAIRE. ,. Membre de la CNDP (Commission nationale du débat public), adjoint au maire de Grenoble. Fabrice RIZZOLI. ,. Élu local, administrateur ANTICOR, président Crim’HALT. Gabriel ULLMANN. ,. expert environnement. A la veille des élections municipales 2020, il est important de s’interroger sur le rôle des citoyens dans la gestion des affaires municipales et locales. Le Président de la République déclarait le 10 décembre 2018 que les maires étaient les premiers à pouvoir porter les revendications des citoyens. Les élus représentent-ils leurs concitoyens devant les […]. Lire la suite 23 Nov 2019 15:30-17:00. Salle Gradin. Climat, pollution, qualité de l’air, Subir ou Agir ? Delphine BATHO. ,. Députée, ex-Ministre de l’Environnement. Dominique BOURG. ,. Universitaire, ancien président du conseil scientifique de la fondation Nicolas-Hulot. Léna LAZARE. ,. Le climato-scepticisme n’a fait que retarder la prise de conscience des désastres environnementaux dont les humains sont responsables par leurs activités. Pourtant, en 2006 déjà, le film « Une vérité qui dérange » laissait peu de place au doute et présentait la courbe exponentielle du réchauffement climatique ou la fonte accélérée de la banquise comme […]. Lire la suite 23 Nov 2019 16:00-17:00. Salle Librairie. Lanceuses d’alertes, « Les résistantes ». Juliette BOUDRE. ,. Anny DUPÉREY. ,. Marine MARTIN. ,. Lanceuse d'alerte, affaire de la Dépakine. Florence MÉRÉO. ,. Journaliste, auteure de « Les Résistantes ». “Marine, Chantal, Marion, Aurélie et les autres n’avaient rien à voir avec le milieu médical… jusqu’à ce qu’un médicament ne fasse totalement basculer leur vie et\/ou celle de leurs enfants. Refusant d’être des victimes silencieuses, elles ont alors attaqué, alerté, mobilisé, créé des mouvements inédits. Sans réseaux, sans argent, elles ont usé d’armes encore plus […]. Lire la suite 23 Nov 2019 17:00-18:00. Galerie. Vernissage de l’exposition « Lanceurs d’alertes ». Collectif d’artistes “LANCER L’ALERTE”. ,. Un collectif d’artistes propose de découvrir à travers leurs œuvres ces personnes indispensables à notre société que sont les lanceurs d’alerte. Les artistes du collectif sont: Alexandre Edeline — www.alexandre-edeline.wixsite.com\/noth Alex Rimbault — https:\/\/alex-rimbault.tumblr.com\/ Alexis Liddell — http:\/\/alexisliddell.blogspot.com Alexis Wemaëre — http:\/\/wemaerealexis.blogspot.com Anaïs Marmonier — www.artstation.com\/sia-the-kat Camille Monnier — youkin-error.tumblr.com Clémence Liberge — www.artstation.com\/clemenceliberge Cynthia […]. Lire la suite 23 Nov 2019 17:30-19:00. Salle Étage. Lancement d’alerte. Bilan de l’année et perspectives. Antoine DELTOUR. ,. Lanceur d’alerte LuxLeaks. Jean Philippe FOEGLE. ,. Porte-parole de la Maison des lanceurs d'alerte. Nicolas FORISSIER. ,. Lanceur d'alerte Affaire UBS, Metamorphosis. Elise VAN BENEDEN. ,. Avocate, administratrice d'ANTICOR. Denis ZMIROU-NAVIER. ,. Médecin, président de la Commission nationale de la déontologie et des alertes en matière de santé publique et d'environnement. Nombre d’évènements sont survenus en quelques années sur le front de l’alerte : protection européenne des lanceurs d’alertes, jugements correctionnels, emprisonnement de Julian Assange. Nombre de collectifs et d’associations se sont également constitués. Pour cette 5e édition du salon, nous avons voulu que collectifs et d’associations constitués depuis plusieurs années puissent faire le bilan des […]. Lire la suite 23 Nov 2019 18:00-19:30. Salle Gradin. Paroles de Lanceurs d’alertes. Emmanuelle AMAR. ,. Epidémiologiste et directrice du Registre des malformations congénitales. Guylain CABANTOUS. ,. Lanceur d'alerte de l'affaire Haemonetics. Mauricio GARCIA PERREIRA. ,. Lanceur d'alerte des abattoirs de Limoges. Hella KHERIEF. ,. Lanceuse d'alerte, maltraitances en EHPAD. Françoise NICOLAS. ,. ancienne fonctionnaire des affaires étrangères. Philippe PASCOT. ,. Auteur, ex-conseiller régional et maire-adjoint à Évry. Le salon « des Livres et l’Alerte » promeut le passage de l’alerte à la lettre. Les lanceuses et lanceurs d’alertes recueillent les preuves de situations dont la gravité et les atteintes à l’intérêt général exigent qu’elles soient portées à la connaissance du public pour que soient apportées les réponses urgentes. Leur laisser la parole […]. Lire la suite 23 Nov 2019 18:00-19:30. Salle Librairie. L’alerte face aux pollutions et risques industriels. Karim BEN ALI. ,. Lanceur d’alerte de l'affaire ARCELOR MITTAL. Pedro DA FONSECA. ,. Journaliste. Roger LENGLET. ,. Journaliste, philosophe. Maurice PICHON. ,. Maraîcher bio. Mathilde RÉGIS. ,. Journaliste. Annie THÉBAUD MONY. ,. D’Arcelor à Lactalis en passant par Servier ou Lafarge, des informations d’intérêt général ne sont pas publiées au risque de porter atteinte à la santé des personnes ou à l’environnement. On comprend, sans l’accepter, les raisons des dirigeants de cacher de telles informations, mais pourquoi ces informations n’ont-elles pas « fuité » dans la presse […]. Lire la suite 24 Nov 2019 11:00-12:30. Salle Gradin. Après l’alerte, la correctionnelle. Patrick ACKERMANN. ,. Délégué syndical Sud PTT. Caroline CHAUMET. ,. Journaliste, fondatrice « Passeurs d’Alerte ». Alors que peu d’alertes font l’objet d’un procès au pénal, pourquoi et comment certaines affaires comme les suicides chez France Télécom ou le scandale du Médiator y aboutissent ? La procédure pénale est-elle l’objectif de l’alerte ? Deux lanceurs d’alertes impliqués dans des dossiers d’actualité montreront que la persévérance, la rigueur, la preuve sont au […]. Lire la suite 24 Nov 2019 11:00-12:00. Salle Étage. Atelier Lanceur d’alerte : l’alerte anonyme ou publique. Maxime RENAHY. ,. Fondateur de « Lanceuralerte.org », ancien espion de la DGSE. Rester anonyme pour être protégé et protéger ses proches ou bien apparaître publiquement ? De nombreuses plateformes proposent de transmettre des alertes de façon anonyme. Avec les moyens actuels de surveillance, peut-on garantir l’anonymat des lanceurs d’alertes, faut-il rester anonyme et quelles peuvent être les conséquences d’une apparition publique ? Lire la suite 24 Nov 2019 11:30-12:30. Salle Librairie. Littérature jeunesse, sensibiliser à l’alerte pour agir plus vite. Thomas BOUT. ,. Éditeur, co-dirigeant des Editions Rue de l'échiquier. Valérie CUSSAGUET. ,. Directrice des Editions Les fourmis rouges. Catherine LEMOINE BRUEY. ,. Libraire, Le Presse Papier Argenteuil. Les initiatives de la jeunesse et les mouvements des lycéens pour la sauvegarde du climat, les interpellations aux « adultes » et aux politiques montrent que la jeunesse est parfaitement réceptive aux alertes. La littérature jeunesse développe des collections pour sensibiliser les plus jeunes et faire la part des choses, les intéresser aux questions que […]. Lire la suite 24 Nov 2019 13:00-14:00. Salle Étage. Atelier lanceur d’alerte : Écrire l’alerte, la preuve au cœur des livres. Publier un livre oblige à déposer le manuscrit à la bibliothèque nationale de France (BNF). A partir du dépôt le contenu du livre peut faire l’objet de poursuites judiciaires au titre de la loi sur la presse datant de 1881. Toute affirmation sans preuve génère un risque pour l’auteur et l’éditeur. Dès lors, qu’est-ce qu’une […]. Lire la suite 24 Nov 2019 13:30-15:00. Salle Gradin. Forces de l’ordre ou Gardiens de la Paix ? Gaétan ALIBERT. ,. Gardien de la paix, syndicat Sud Intérieur. Raphaël KEMPF. ,. Avocat. Alexandre LANGLOIS. ,. Secrétaire général du syndicat Vigi-Ministère de l'intérieur. La Déclaration des Droits de l’Homme et du Citoyen définit le rôle de la « force publique » : « La garantie des droits de l’Homme et du Citoyen nécessite une force publique : cette force est donc instituée pour l’avantage de tous, et non pour l’utilité particulière de ceux auxquels elle est confiée. ». […]. Lire la suite 24 Nov 2019 13:30-15:00. Salle Librairie. Travail et Suicide, du Mal être à l’Alerte ? Patrick CAHEZ. ,. Policier en retraite, syndicaliste Sud Intérieur. Jean René DELÉPINE. ,. Syndicaliste SUD Rail. Ivan DU ROY. ,. Journaliste, auteur de \"Orange stressée\". Denis PERREAU. ,. Secrétaire national de la Confédération Paysanne. Marie PEZÉ. ,. Docteur en psychologie, réseau de consultations Souffrance & Travail. Patricia VIOUX. ,. Consultante secteur médico-social et social. Procès France Télécom, vague de suicides dans la police, à la SNCF, à l’inspection du travail, chez les agriculteurs … la liste n’est pas exhaustive. Autant de suicides qui ont pris valeur d’alertes. Lorsque des personnes décident de mettre fin à leurs jours sur leur lieu de travail ou à cause de leur travail, est-ce […]. Lire la suite 24 Nov 2019 15:00-16:30. Salle Étage. Médicaments et principe de précaution. Le rôle des citoyens. Gérard BAPT. ,. Ancien député. Thomas DIETRICH. ,. Ancien secrétaire général de la conférence nationale de santé. Marine MARTIN. ,. Lanceuse d'alerte, affaire de la Dépakine. Victimes de médicament, elles ont alerté, elles ont fait face au discrédit, à la dérision, leurs symptômes ont été minimisés. L’alerte sur les médicaments et les risques encourus est-elle du seul ressort de spécialistes vêtus d’une blouse blanche? Marine MARTIN et Emmanuelle MIGNATON, comme Marion LARAT, qui avait été accueillie lors de la première édition […]. Lire la suite 24 Nov 2019 16:00-17:30. Salle Gradin. Chimie agricole, Traitement du vivant ou extermination ? Stéphane FOUCART. ,. Journaliste, auteur de « Et le monde devint silencieux - Comment l'agrochimie a détruit les insectes ». Paul FRANÇOIS. ,. Agriculteur, président de l’association « Phyto-victime ». HK (Kaddour HADDADI). ,. Chanteur, auteur de « Dounia, L’or bleu ». Les risques en matière de santé publique des produits utilisés depuis de longues années dans l’agriculture font l’objet de reconnaissances judiciaires. Les avancées et les reculs sur les interdictions de ces produits, qui attaquent également la biodiversité, interrogent jusqu’aux places boursières. Le principe de précaution est soulevé dans tous les débats sur le sujet. L’une […]. Lire la suite 24 Nov 2019 16:00-17:30. Salle Librairie. Nucléaire et après ? David BOILLEY. ,. Président de l'ACRO. Irène GUNEPIN. ,. Retraitée, militante BureStop. Nadezda KUTEPOVA. ,. Réfugiée russe de Maïak. Haruko SAKAGUCHI. ,. Réfugiée de Fukushima. Patricia VIOUX. ,. Consultante secteur médico-social et social. Malgré les discours rassurants, le nuage de Tchernobyl ne s’est pas arrêté aux frontières et des millions de mètres cube d’eau contaminée sont déversés chaque jour dans l’océan à Fukushima. La France avec ses installations nucléaires vieillissantes et son passé d’expériences plus ou moins contrôlées présente-t-elle des risques et une menace pour ses voisins ? […]. Lire la suite Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Maxime RENAHY Fondateur de « Lanceuralerte.org », ancien espion de la DGSE. anemptytextlline. En dédicace samedi 23 novembre de 13h à 14h30 et dimanche 24 novembre de... Florence RODHAIN anemptytextlline. En dédicace dimanche 24 novembre de 13H30 à 15h puis de 17h30 à 18h30... Catherine LEMOINE BRUEY Libraire, Le Presse Papier Argenteuil. anemptytextlline. Libraire indépendante elle dirige avec Gilles Bruey la librairie Le Presse Papier à Argenteuil.... Paul FRANÇOIS Agriculteur, président de l’association « Phyto-victime ». anemptytextlline. Agriculteur charentais, Paul François est intoxiqué en 2004 par un herbicide de Monsanto. Après... Jean René DELÉPINE Syndicaliste SUD Rail. anemptytextlline. Cheminot, syndicaliste à SUD-Rail, et juriste de formation et de métier, Jean-René DELEPINE s'investit... Mathilde RÉGIS Journaliste. anemptytextlline. Journaliste indépendante pour la presse écrite, Mathilde Régis a rencontré de nombreux lanceurs d'alerte... 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Frédéric PIERUCCI auteur. anemptytextlline. Frédéric Pierucci, ancien patron d’une des filiales d’Alstom, après avoir été longtemps contraint au... Gilles BRUEY Libraire. anemptytextlline. Libraire indépendant à Argenteuil, Gilles Bruey a ouvert Le Presse Papier en 1981 avec... Michel PINÇON anemptytextlline. En dédicace avec Monique PINÇON-CHARLOT, dimanche 24 novembre de 14h à 15h pour plusieurs... François BOULO Avocat. anemptytextlline. En dédicace samedi 23 novembre de 14h30 à 16h pour « La ligne jaune... Léna LAZARE anemptytextlline. Âgée de 21 ans, Léna Lazare est étudiante en mathématiques et en physique à... Guillaume de MORANT anemptytextlline. En dédicace sur le stand EXTRA-MUROS sur toute la durée du salon pour «... Delphine BATHO Députée, ex-Ministre de l’Environnement. anemptytextlline. En dédicace samedi 23 novembre de 17h à 18h pour « Ecologie intégrale, le... Aymeric MONVILLE anemptytextlline. Aymeric Monville est philosophe, directeur de collection aux éditions Delga et rédacteur en chef... Antoine DELTOUR Lanceur d’alerte LuxLeaks. anemptytextlline. Ex-auditeur chez PricewaterhouseCoopers (PwC) au Luxembourg, il est l’auteur de la fuite de plusieurs... Mauricio GARCIA PERREIRA Lanceur d'alerte des abattoirs de Limoges. anemptytextlline. En dédicace samedi 23 novembre de 17h à 18h pour « Ma vie toute... Raymond AVRILLIER Citoyen engagé. anemptytextlline. « Administré engagé », il est à l’origine des révélations de plusieurs affaires politico-financières... Alexandre CALVEZ Organisateur. anemptytextlline. Alexandre Calvez est fonctionnaire et administrateur de l'association Anticor. A ce titre, il intervient... Pascal CLOUAIRE Membre de la CNDP (Commission nationale du débat public),... anemptytextlline. Pascal Clouaire est adjoint au maire de Grenoble. Après un parcours dans les mouvements de... Ivan DU ROY Journaliste, auteur de \"Orange stressée\". anemptytextlline. Journaliste, co-fondateur de Basta ! et du site multinationales.org. Il s’est formé au management... Jean-François BERNARDINI anemptytextlline. En dédicace samedi 23 novembre de 16h à 17h pour « L’autre enquête corse... Isabelle BADOUREAUX anemptytextlline. En dédicace samedi 23 novembre de 17h à 18h, avec Roger LENGLET pour «... Thomas DIETRICH Ancien secrétaire général de la conférence nationale de santé. anemptytextlline. En dédicace dimanche 24 novembre de 16h30 à 18h30 pour « Les enfants de... Collectif d'artistes \"LANCER L'ALERTE\" anemptytextlline. Réuni-e-s au sein d'un collectif d'artistes illustratrices-teurs, ils ont décidé de créer des illustrations... Didier LESTRADE Journaliste co-fondateur d’ActUp Paris. anemptytextlline. En dédicace vendredi 22 novembre de 20h à 21h Didier Lestrade, fondateur d’Act Up-Paris et... Quentin GUILLEMAIN Asso Parents victimes de Lactalis. anemptytextlline. Quentin Guillemain est le père d'une petite fille intoxiquée par le lait de Lactalis.... Nicolas VESCOVACCI Journaliste, co-auteur de « Vincent tout puissant ». anemptytextlline. En dédicace samedi 23 novembre de 14h30 à 16h, avec Jean-Pierre CANET pour «... Jean Philippe FOEGLE Porte-parole de la Maison des lanceurs d'alerte. anemptytextlline. Jean-Philippe Foegle est doctorant en droit public spécialiste du statut juridique du lanceur d’alerte,... Maurice PICHON Maraîcher bio. anemptytextlline. Maraîcher en agriculture biologique depuis 35 ans, Maurice Pichon est co-président de l’association Santé... Raphaël KEMPF Avocat. anemptytextlline. En dédicace dimanche 24 novembre de 15h à 16h30 pour « Ennemis d’Etat » Avocat... Jean-Pierre CANET journaliste. anemptytextlline. Jean-Pierre Canet est journaliste d'enquête, avec Nicolas VESCOVACCI ils sont les auteurs « Vincent... Denis ROBERT Journaliste, rédacteur en chef de Le Media. anemptytextlline. Denis Robert est l’auteur de nombreux romans, de films documentaires et d’essais. Il est... Florence MÉRÉO Journaliste, auteure de « Les Résistantes ». anemptytextlline. En dédicace samedi 23 novembre de 17h à 18h pour « Les résistantes » Florence... Gérard DELTEIL Écrivain. anemptytextlline. En dédicace samedi 23 novembre de 14h30 à 16h pour plusieurs livres dont le... Denis PERREAU Secrétaire national de la Confédération Paysanne. anemptytextlline. Denis Perreau est paysan en Côte d’Or, en grades cultures et ovins. Il est... Frédéric LEMAIRE Membre d'ACRIMED. anemptytextlline. Frédéric Lemaire est membre de la rédaction de Action Critique Médias (Acrimed), et doctorant... Patrick CAHEZ Policier en retraite, syndicaliste Sud Intérieur. anemptytextlline. Inspecteur de police à la DST et la PJ mis à la retraite pour... Mathias DESTAL Journaliste. anemptytextlline. En dédicace samedi 23 novembre de 15h à 16h30, avec Marine TURCHI pour «Marine... Filter - All. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Rencontre Annuelle des Lanceurs d'Alerte. 12, 13 & 14 Novembre 2021. à la Maison des Sciences de l'Homme Paris Nord. (MSH Paris Nord) La Plaine St-Denis. Métro Ligne 12 \"Front populaire\". Entrée Libre et Gratuite*. Les rencontres se dérouleront dans le respect des gestes barrières et des règles limitant la propagation des virus. Programme Dossier Presse  La 6. ème. Rencontre annuelle des Lanceurs d’Alertes se déroulera les 12\/13\/14 novembre 2021 à la MSH Paris Nord à La Plaine Saint-Denis (Maison des Sciences de l’Homme). Les années 2020 et 2021 resteront marquées dans les esprits comme celles de la pandémie et du confinement, elles sont également celles où la fonte de la calotte glaciaire du Groënland a dépassé le point de non-retour, celle d’un nouvel état d’urgence, mais sanitaire cette fois avec ses contrôles de « pass » partout…. En 2020, nous avons dû comme tant d’autres initiatives, annuler notre rendez-vous annuel. Organiser en 2021 la 6. ème. rencontre annuelle n’a pas été plus simple au regard des différentes interprétations de la loi et de la réglementation. Ces conditions particulières nous ont demandé de l’imagination mais avec l’aide du GIS démocratie & participation et de chaque intervenant, nous sommes une fois encore en mesure de proposer un programme de rencontres riche et diversifié qui reste incomplet au regard de la multitude de sujets à traiter, malgré les 23 rencontres et plus de 80 intervenants. Nous accueillerons, cette année encore, de nombreux acteurs du monde de l’alerte dans leur diversité, auteurs, journalistes, chercheurs, universitaires et bien sûr les lanceuses et lanceurs d’alertes…. toutes et tous engagés pour l’intérêt général. . Notre volonté depuis l’origine est que ces rencontres uniques par leur forme et leur envergure permettent de partager les expériences, les analyses et les réflexions, de mutualiser les savoirs et de débattre ou exprimer des divergences. De la santé à la publication des alertes, de la protection des lanceurs d’alertes à l’environnement sans oublier les droits de l’Homme, nous souhaitons illustrer les trois expressions essentielles qui sont le fil conducteur des rencontres depuis 2015 :. « Délivrer l’alerte – Délivrez l’Alerte ! – Des Livres et l’Alerte ». .  . Transposer l’alerte dans la loi française. . En espagnol « denunciantes » (dénonciateurs), en italien « informatori » (informateurs),. en roumain. « . avertizori ». (dénonciateurs), en tchèque « . oznamovatelé » (notifiants)…. montrent que les approches sont bien différentes selon les pays et que le vocabulaire illustre la perception de l’acte citoyen d’alerter. Cela met en valeur. l’intelligence du travail de Francis Chateauraynaud qui a défini le terme de « lanceurs d’alerte ». Il n’est pas anodin non plus que le texte de la directive n’ait pas retenu « lanceurs d’alerte » mais la formule « . personnes qui signalent des violations du droit de. l’Union.  ». De cette formulation surgissent effectivement des interrogations, le droit de l’Union serait-il infaillible, prétend-il à l’exhaustivité ? On le voit le vocabulaire n’est pas neutre et les définitions non plus. Dans le cas présent, la prépondérance du droit de l’Union est mise en avant mais ses limites aussi… La jurisprudence de la Cour Européenne des Droits de l’Homme (CEDH) a, elle aussi, un rôle prépondérant dans la protection des lanceuses et lanceurs d’alerte en s’appuyant sur. l’article 10 de la convention ou. l’article 11 de la charte des droits fondamentaux Ces simples constats montrent que les enjeux de la transposition sont importants et que la précision des analyses est essentielle pour la rédaction d’une loi adaptée à l’alerte. L’alerte relève-t-elle d’un droit à protéger ou d’un devoir citoyen à encourager ? Face au risque de poursuites, le lanceur d’alerte doit-t-il se référer à un droit d’alerte ou à un devoir de citoyen ? On le voit, la définition du mot alerte est primordiale et nous vous proposons de nous y attacher lors de ces rencontres. Dossier de Presse Vidéo réalisée par Olivier BERARDI \/. TVNET Citoyenne Images Olivier Berardi \/ Danishout. Lors de cette 6. ème. rencontre, nous accueillerons lanceuses et lanceurs d’alertes, auteurs, journalistes, chercheurs, universitaires…. toutes et tous engagés pour l’intérêt général. Ils échangeront avec le public que nous espérons nombreux. . Nous restons attachés à notre indépendance et remercions tous les bénévoles qui organisent cette initiative unique où le public peut rencontrer une centaine d’intervenants qui enquêtent, informent et réfléchissent sur les alertes. et les enseignements qu’elles recèlent. Connus ou inconnus,. toutes et tous ont fait le choix d’obéir à leur conscience de l’intérêt général. , souvent au prix de représailles injustifiables, mais toujours. avec la fierté de pouvoir « se regarder dans la glace ». pour avoir alerté notre société sur des dysfonctionnements graves. Dès maintenant, nous vous invitons à inviter vos amis et connaissances à découvrir ces rencontres et à y participer. Elles  constituent. un évènement unique ancré dans l’actualité. . Les alertes ne portent que si le public est au rendez-vous et nous vous y attendons nombreu-ses-x. Compte tenu des restrictions qui peuvent nous être imposées dans le cadre de la protection des personnes contre la transmission du virus COVID-19, nos rencontres se dérouleront dans le respect des gestes barrières et des règles limitant la propagation des virus. Daniel Ibanez. co-fondateur du salon. Intervenants. Ils seront présents pour cette édition 2021. Juliette ALIBERT ,. Arié ALIMI ,. Raymond AVRILLIER ,. Olivier AZAM ,. Amar BENMOHAMED ,. Amal BENTOUNSI ,. Ugo BERNALICIS ,. Marie BLANDIN ,. Elisabeth BORREL ,. Denis BRETEAU ,. Philippe BREUVART ,. Gilles BRUEY ,. Alexandre CALVEZ ,. Jean-Pierre CANET ,. Antonio CASILLI ,. Laurence CHAMPIER ,. Francis CHATEAURAYNAUD ,. Christian CHOUVIAT ,. Annie COLLOMBET ,. Florent COMPAIN ,. Daniel CUEFF ,. Pedro DA FONSECA ,. Erri DE LUCA ,. Ariane DENOYEL ,. Manuel DOMERGUE ,. Sabine DUFLO ,. Louis-Antoine DUJARDIN ,. Camille ÉTIENNE ,. Ludovic FAYOLLE ,. Florian FERJOUX ,. Caroline FIAT ,. Nicolas FORISSIER ,. Jean-Michel FOURNIAU ,. Irène FRACHON ,. Yann GAUDIN ,. Guillaume GONTARD ,. Brigitte GOTHIERE ,. Gilbert GOURRAUD ,. Fabien GRACIA ,. Quentin GUILLEMAIN ,. Daniel IBANEZ ,. Pablo IBAÑEZ CRUCEYRA ,. Sylvain INSERGUEIX ,. Julie ,. Maureen KEARNEY ,. Vanessa LANGARD ,. Gérald LE CORRE ,. Thomas LEBONNIEC ,. Inès LERAUD ,. Clara LÓPEZ RUBIO ,. Karine MAUVILLY GRATON ,. Philippe MEIRIEU ,. Pierre MERIAUX ,. Étienne MERLE ,. Asma MHALLA ,. Emmanuelle MIGNATON ,. Olivia MOKIEJEWSKI ,. Mathieu MOLARD ,. Aymeric MONVILLE ,. Valérie MURAT ,. Françoise NICOLAS ,. Juan PANCORBO ,. Philippe PIARD ,. Frédéric PIERUCCI ,. Serge PORTELLI ,. Anne RICHARD ,. Marie Monique ROBIN ,. Virginie ROZIERE ,. Raphaël RUFFIER-FOSSOUL ,. Patrice SALINI ,. Denise SCHNEIDER ,. Eve SZEFTEL ,. Flore TALAMON ,. Nathalie TEHIO ,. Olivier TESQUET ,. Laurent THINES ,. Bruno TOUSSAINT ,. Gabriel ULLMANN ,. Elise VAN BENEDEN ,. Cyrille VENET ,. Françoise VERCHERE ,. Nicolas VESCOVACCI ,. Sylvain WASERMAN Les intervenants en détail Le programme en détails Comment s’y rendre. Maison des Sciences de l’Homme Paris Nord. 20, avenue George Sand. 93210 La Plaine St-Denis. Métro:. 12 Front Populaire. sortie n°3 MSH Paris Nord. *des contrôles de sécurité peuvent avoir lieu. Y aller Afficher une carte plus grande Nos Soutiens. Seul on va plus vite, ensemble on va plus loin. Anticor Cafés citoyens santé et travail Citoyens Résistants d'Hier et d'Aujourd'hui Crim'HALT Julia Blagny Le Presse Papier Libre et Solidaire Maison des Lanceurs d'Alerte La Montreuilloise Not a Number Satoriz Vivre et Agir en Maurienne Extra Muros Association Henri Pézerat Ligue des Droits de l'Homme Libre et Solidaire Amis de la Terre Syndicat des Avocats de France SAF ARROBE HAUTEVILLE PRODUCTIONS TVNET CITOYENNE Les Mutins de Pangée Gis Démocratie & Participation MSH Paris Nord Secrets Toxiques Primevère BBD éditions L'Arrière-cour GREENPEACE REMERA CRIIRAD Paris Brest Productions Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés. Share This. Facebook. Twitter. Gmail. Yahoo Mail. LinkedIn."}
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Des livres et l’alerte. 4ème salon du livre des lanceurs et lanceuses d’alerte. 16, 17 et 18 Novembre 2018, Paris. Suivre. Suivre Suivre. Suivre Entrée Libre et Gratuite. La Maison des Métallos. 94 rue Jean-Pierre Timbaud, 75011 Paris. Métro : Couronnes (ligne 2) ou Parmentier (ligne 3). Télécharger le programme Dédicaces des auteurs Dossier de Presse Le 4ème salon “Des livres et l’alerte”. se déroulera les 16\/17\/18 novembre à la Maison des métallos à Paris. Grâce à l’engagement des co-organisateurs : la librairie indépendante « Le Presse Papier », « la Maison des métallos » et l’équipe de bénévoles. Cette année encore,. le salon accueillera des personnalités du monde de l’alerte. , des auteurs et des journalistes, toutes et tous engagés pour l’intérêt général. Ce salon indépendant est l’unique lieu où se rencontrent largement celles et ceux qui enquêtent, s’informent et réfléchissent sur les alertes. et les enseignements qu’elles recèlent. Connus ou inconnus,. toutes et tous ont fait le choix d’obéir à leur conscience de l’intérêt général. , souvent au prix de représailles injustifiables, mais toujours. avec la fierté de pouvoir « se regarder dans la glace ». et d’avoir transmis les preuves si importantes pour notre société. Délivrer l’alerte. Les principes fondamentaux de la République seront au cœur du 4ème salon. « Des Livres et l’Alerte » l’année où le Conseil Constitutionnel a dû rappeler que le principe de fraternité et de solidarité s’impose au nom de l’idéal exprimé dans la devise de la République « Liberté, Égalité, Fraternité ». Nous l’évoquerons avec. Annette Beaumanoir, Christiane Hessel et Christiane Taubira. qui nous font l’honneur de participer au salon, car les régimes qui se sont imposés, notamment sous l’occupation, ont bafoué les principes fondamentaux de la République par la discrimination et la négation de la liberté. Protéger les « lanceurs d’alerte » qui osent publier les informations. essentielles pour la santé, pour l’environnement ou plus largement pour l’ensemble du fonctionnement de la société, est au centre des discussions. Cette protection ne passe-telle pas plutôt par. le droit fondamental et constitutionnel d’informer qu’il faut sanctuariser. , pour que celles et ceux qui l’exercent ne soient pas réprimés ? Analyser l’actualité, le fonctionnement de notre société ou encore. la loi « Secret des affaires » à la lumière de ces principes fondamentaux. , sera aussi à l’ordre du jour du salon. Des livres. et l’alerte. Un salon résolument tourné vers l’alerte et des lanceurs d’alerte, avec une spécificité toutefois : le livre pour « passer de l’alerte à la lettre ». Le livre d’alerte, instrument incomparable pour l’information ou la formation, la révélation ou l’accusation, est toujours précis, probant et rigoureux, en cela il se distingue d’autres médias et permets de rassembler le monde de l’alerte. Pour sa 4. ème. édition, le salon « Des Livres et l’Alerte » tiendra le rôle que nous lui avons assigné de « Délivrer l’Alerte », avec tous les invités et le public nous dirons haut et fort :. Délivrez l’Alerte ! Julian Assange. ,. Edward Snowden. ,. Reality Winner. , et toutes celles et ceux qui sont victimes de représailles, d’emprisonnement et, de menaces pour avoir alerté,. doivent pouvoir trouver refuge dans les pays démocratiques. pour ne pas être victimes de meurtres. Daphne Caruana Galizia. ,. Jàn Kuciak et Martina Kusnirova. ,. Berta Caceres. et tant d’autres assassiné.e.s pour avoir informé, resteront au coeur de notre salon. Tous les thèmes de l’alerte ne seront pas couverts cette année encore mais la richesse des sujets nous imposent des choix. Daniel Ibanez. co-fondateur du salon. Les. Intervenants. de la. 4ème édition. (Liste par ordre alphabétique pouvant encore évoluer). Isabelle. ATTARD. (députée 2012 – 2017, plaignante agression sexuelle). , Raymond. AVRILLIER. (administré engagé). , Christophe. BASSONS. (ancien cycliste professionnel). , Edmond. BAUDOIN. (écrivain, dessinateur). , Annette. BEAUMANOIR. (résistante, juste parmi les Nations). , Karim. BEN ALI. (chauffeur, lanceur d’alerte). , Laurent. BENEZECH. (ancien pilier de l’équipe de France de Rugby). , Andréa. BESCOND. (danseuse, comédienne, metteure en scène),. Pierrick. BOURGAULT. (journaliste). , Juan. BRANCO. (avocat). , Gilles. BRUEY. (libraire indépendant, co-fondateur du salon). , Marie-George. BUFFET. (ministre jeunesse et sports 1997 – 2002),. Alexandre. CALVEZ. (co-organisateur du salon). , Bruno. CHAREYRON. (ingénieur CRIIRAD). , Francis. CHATEAURAYNAUD. (directeur d’études EHESS). , Caroline. CHAUMET. (journaliste). , Marc. CREPON. (philosophe, directeur de recherche). , Pedro. DA. FONSECA. (journaliste). , Arnaud. DE BLAUWEN. (rédacteur en chef Que Choisir ?). , Antoine. DELTOUR. (lanceur d’alerte). , Bénédicte. DESFORGES. (lieutenant de police en disponibilité). , James. DUNNE. (informaticien lanceur d’alerte). , Lionel. DUROY. (écrivain). , Alexandre. DUSSOT HEZEZ. (co-fondateur « La parole libérée »). , Mahaut. FANCHINI. (maîtresse de conférence Paris Dauphine). , Marc. FIEVET. (ex-aviseur des douanes). , Jacques. FOLLOROU. (journaliste). , Nicolas. FORISSIER. (banquier, lanceur d’alerte UBS). , Irène. FRACHON. (pneumologue, lanceuse d’alerte). , Daphné. GASTALDI. (journaliste). , Alain. GAUTIER. (syndicaliste, lanceur d’alerte). , Jean-Charles. GÉRARD. (éditeur). , Christophe. GREBERT. (journaliste, élu local Puteaux). , Philippe. HALIMI. (chef du service radiologie Hôpital G Pompidou). , Serge. HERCBERG. (docteur en médecine, spécialiste en nutrition). , Christiane. HESSEL. (Présidente d’honneur EJE). , Claire. HOCQUET. (avocate). , Dominique. HUEZ. (médecin du travail retraité). , Daniel. IBANEZ. (co-fondateur et co-organisateur du salon). , Les voix d’. I MUVRINI. (groupe vocal corse). , Alexandre. LANGLOIS. (secrétaire général VIGI Ministère Intérieur). , Serge. LATOUCHE. (professeur émérite Droit, Économie, Gestion). ,. Geoffrey. LIVOLSI. (journaliste). ,. Pierre-Alain. MANNONI. (ingénieur Université de Nice). , Yasmine. MOTARJEMI. (lanceuse d’alerte, ex-Nestlé). , Amélie. MOUGET. (éditrice). , Claire. NOUVIAN. (présidente BLOOM, Goldman Prize). , Fabrice. OLIVET. (directeur ASUD). , Patrick. PELLOUX. (urgentiste, Président AMUHF). , Raoul. PERICHON. (professeur, collectif Combalinky). , Jean-Christophe. PICARD. (président ANTICOR). , Frédéric. PIERRU. (sociologue CERAPS, CNRS). , Frédéric. PLOQUIN. (journaliste). , Camille. POLLONI. (journaliste). , Serge. PORTELLI. (magistrat retraité). , Alexis. POULIN. (journaliste co-fondateur LeMondeModerne.media). , Mathilde. RÉGIS. (journaliste). , Fabrice. RIZZOLI. (docteur en sciences politiques, spécialiste des mafias). , Denis. ROBERT. (journaliste). , Carole. ROUAUD. (Co-organisatrice du salon). , Raphaël. RUFFIER. (journaliste). , Isabelle. SAPORTA. (journaliste). , Samuel. SAUVAGE. (président HOP). , Omar. SLAOUTI. (professeur Physique Chimie). , Christiane. TAUBIRA. (ministre de la Justice, mai 2012 – janvier 2016),. Annie. THEBAUD-MONY. (sociologue de la santé, présidente association Henri Pézerat). , Pierre-Marie. THEVENIAUD. (enseignant Biologie humaine). , Jean-Luc. TOULY. (syndicaliste, Lanceur d’alerte). , Bruno. TOUSSAINT. (directeur éditorial revue Prescrire). , Elise. VAN BENEDEN. (avocate). , Tatiana. VENTOSE. (LeFild’Actu). , Françoise. VERCHERE. (élue territoriale 1983 – 2015), Nicolas. VESCOVACCI. (journaliste). LE FIL D’ACTU, CRIIRAD, LES AMIS DE LA TERRE, LIBRE & SOLIDAIRE, ANTICOR, HOP, CRIM’HALT, VIVRE ET AGIR EN MAURIENNE, CAFÉ CITOYEN SANTÉ TRAVAIL, CRHA,.  . PASSEUR D’ALERTES, LILO, MAX MILO éditeur, REVUE DESSINÉE, SCIENCES CITOYENNES, LE LANCEUR, DISCLOSE. *. Nos. Soutiens. Register Now! Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque placerat sapien condimentum imperdiet cursus. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque placerat sapien condimentum imperdiet cursus. Ticket Info Contactez Nous. Nom. Email. Message. Submit. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Des livres et l’alerte. 4ème salon du livre des lanceurs et lanceuses d’alerte. 16, 17 et 18 Novembre 2018, Paris. Entrée Libre et Gratuite. La Maison des Métallos. 94 rue Jean-Pierre Timbaud, 75011 Paris. Métro : Couronnes (ligne 2) ou Parmentier (ligne 3). Télécharger le programme Dédicaces des auteurs Dossier de Presse Revoir les . rencontres . de la. 4ème édition. Vendredi 16 novembre. 18h00-21h00 salle Claire |. Ouverture – Projection débat “V. igilance, engagement, résistance”. 18h : Projection du film “Meeting Snowden” de Flore Vasseur (49 mn, 2016). 19h : Rencontre avec Anne Beaumanoir (médecin, résistante, Juste parmi les nations), Lionel Duroy (écrivain) et Mme la ministre Christiane Taubira.  .  . Animée par Daniel . Ibanez.  . (co-fondateur du salon, Affaire Lyon-Turin).  . et Denis Robert.  (journaliste, écrivain). . Samedi 17 novembre. 14h00-14h45 salle Noire |. Spectacle I Muvrini. Extraits de textes et polyphonies. I . Muvrini.  (. Les petits mouflons.  en . corse ) est un . groupe  de . musique corse  créé par les frères Alain et Jean-François Bernardini à la fin des années 1970. Lors de leur tournée 2017 en France et en Belgique ils ont fait le choix engagé de faire monter sur scène les héros anonymes de la société civile : Céline Boussié et Valérie Pouilly à Agen , Ahmad Darkanzali  à Tours, Pierre-Alain Mannoni et Hubert Jourdan à Aix, Florent Compain à Nancy.  . 14h00-15h00 salle Claire |. Des consommateurs suffisamment alertés ? Pour sa . 4e.  édition, le salon des livres de lanceurs d’alerte rend un hommage particulier aux revues . Que choisir.  et . Prescrire . dont le rôle en matière d’alerte n’est plus à démontrer. Au même titre que le livre, la revue, en posant l’alerte par écrit, la qualifie. . Arnaud d. e . Blauwe. (. rédacteur en chef de . Que choisir. ) et Bruno Toussaint (. chef éditorial de . Prescrire. ) . nous expliqueront leur conception du rôle de protection des consommateurs.  . Avec Arnaud de Blauwe (. Que Choisir. ) et Bruno Toussaint (revue . Prescrire. ). Animée par Irène Frachon (pneumologue, affaire Mediator). 14h-15h Salle Pavillon |. Tous sous surveillance. Et vous, avez-vous quelque chose à cacher ? La collecte de vos données personnelles ne vous dérange pas ? IMSI-Catchers, notes blanches, données GPS : la technologie ouvre des opportunités, inédites dans l’histoire, de contrôle social par nos dirigeants. Ces nouveaux enjeux appellent de nouveaux contre-pouvoirs. Existent-ils ? Comment s’assurer que la révolution technologique que nous traversons actuellement ne se transforme pas en cauchemar demain ? Avec James Dunne (lanceur d’alerte, affaire Qosmos) et Camille Polloni (journaliste, Les Jours.fr). Animée par Jacques Follorou (journaliste, . Le Monde. ). 14h00-15h00 salle 3 |. Décryptage : désobéir pour lancer l’alerte ? Les lanceurs d’alerte sont parfois décrits comme faisant acte de désobéissance civile. Faut-il nécessairement désobéir pour lancer l’alerte ? A quelles conditions l’acte transgressif permet-il de mieux lancer l’alerte ? Et quelles conséquences pour le lanceur d’alerte ? . Marc Crépon dont les travaux interrogent l’idée et les f. ormes de violence en démocratie . propo. se de réfléchir à ces questions.  . Masterclasse de Marc Crépon (philosophe, ENS). Animée Mahaut Fanchini (maîtresse de conférences en sciences de gestion, université Paris-Dauphine). 15h00-16h30 salle Noire |. Migrations : quand la solidarité devient un délit. Tant de femmes et d’hommes poursuivis par la justice pour avoir aidé des personnes en détresse, pour avoir fait preuve de solidarité. Par quels mécanismes notre société a-t-elle pu renoncer à ses principes humanistes fondamentaux ? Le principe de fraternité, reconnu récemment par le Conseil constitutionnel, permettrait-il d’entrevoir le crépuscule du « délit de solidarité » ? Avec Edmond Baudoin (artiste, auteur de BD), Christiane Hessel (militante des droits de l’Homme). et Pierre-Alain Mannoni (enseignant-chercheur au CNRS, désobéissant civil). Animée par Alexis Poulin (journaliste). 15h30-16h30 salle Claire |. Décryptage : gagner après l’alerte. Les lanceurs d’alerte ne sont pas condamnés d’avance ! Ces combats en valent vraiment la peine et lorsqu’ils aboutissent, c’est toute la société qui en bénéficie. Qu’il s’agisse du retrait du Mediator ou du vote par le Parlement européen de l’interdiction de la pêche électrique, les combats d’Irène Frachon et de Claire Nouvian nous éclairent sur les ressorts d’une démarche victorieuse. Avec Irène Frachon (affaire Mediator) et Claire Nouvian (association Bloom). Animée par Daniel Ibanez (co-fondateur du salon, Affaire Lyon-Turin). 15h30-16h30 Salle 3 |. Objets connectés : alerte sur les ondes. Avec l’explosion de l’usage des téléphones mobiles, antennes-relais, wifi, wimax, baby phones, bluetooth etc., les consommateurs sont-ils correctement informés face aux dépassements des seuils d’exposition des organismes des champs électromagnétiques liés à ces technologies sans fil ? Déjà plusieurs téléphones ont été retirés discrètement du marché pour dangerosité. Etat des lieux d’un scandale sanitaire en puissance….  . Raoul Périchon (électrosensible, collectif COMBALINKY) et Pierre-Marie Théveniaud (Association les Robins des Toits). Animée par Pierrick Bourgault (journaliste). 15h30-16h30 Salle Pavillon  |. Éditer l’alerte . Écrire l’alerte, c’est s’inscrire dans la chaîne rigoureuse de l’édition. C’est aussi prendre un risque, s’engager, entrer dans le temps long. De l’auteur jusqu’au lecteur, quelles sont les étapes pour éditer l’alerte ? Avec Romain Perrusset (éditions Robert Laffont), Jean-Charles Gérard (éditions Max Milo) et Amélie Mougey (. La Revue Dessinée. ). et le témoignage vidéo d’un coopérant de justice ayant appartenu à la mafia corse. Animée par Gilles Bruey (co-fondateur du salon, librairie Le Presse Papier). 17h-18h Salle Pavillon  |. Faire face au lobby agroalimentaire. Les lobbies sont omniprésents dans la prise de décision publique. La démission de Nicolas Hulot du gouvernement l’a rappelé brutalement. L’industrie agroalimentaire n’est pas épargnée par le phénomène, bien au contraire. Quelles sont les méthodes utilisées par ces multinationales de l’alimentation ? Qui sont ceux qui s’y opposent ? Comment leur faire face pour qu’ « assiette » ne rime plus avec « poison » ? Avec Serge Hercberg (programme Nutriscore), Yasmine Motarjemi (lanceuse d’alerte, affaire Nestlé) et Isabelle Saporta (journaliste, RTL). Animée par Alexandre Calvez (co-organisateur du salon). 17h-18h Salle 3 |. Atelier : délivrer l’alerte. Présentation du guide du lanceur d’alerte du Défenseur des Droits. Des référents répondent à 5 témoignages de la salle avec orientations et conseils en direct. Un lanceur d’alerte est une personne qui veut signaler un danger ou un risque, mettre fin à une action illégale ou irrégulière en interpelant les pouvoirs en place ou en suscitant une prise de conscience. Les simples salariés ou citoyens engagés, se sentent souvent démunis et seuls. Cet atelier permet au public de rencontrer des experts pour être aiguillés dans leurs parcours. Avec Elise Van Beneden (association Anticor), Nicolas Forissier (Collectif Metamorphosis) Raphaël Ruffier (Le Lanceur.fr). et Carole Rouaud (co-organisatrice du salon et membre de l’association Anticor). 17h30-19h Salle Claire |. Violences sexuelles : briser la loi du silence. Harcèlement, viol, intimidation, pédophilie, les agressions sexuelles et leurs auteurs se retrouvent dans toutes les activités. Politiques, sportives, religieuses, professionnelles toutes les sphères sont touchées. L’affaire Weinstein et le mouvement Time’s Up ont marqué à l’orée de 2018 un tournant dans la prise de conscience des violences sexuelles. La flambée médiatique, soutenue par les réseaux sociaux (le #metoo) a permis l’expression des victimes. Toutes les victimes. Ces paroles libérées changeront-elles la situation à long terme ? Avec Isabelle Attard (députée EELV du Calvados, affaire Denis Baupin), André Bescond (actrice et réalisatrice du film “Les chatouilles”) et Alexandre Dussot-Hezez (association La Parole libérée). Animée par Daphné Gastaldi (journaliste). 18h30-19h30 Salle Pavillon |. Des associations pour construire l’alerte.  . Henri Pézerat, scientifique militant, dénonça les ravages de l’amiante. Son fils Pierre continue son œuvre. L’association Henri Pézerat se bat pour la santé des personnes en lien avec le travail et l’environnement. Comment des associations lancent l’alerte ? Et comment viennent-elles en aide aux victimes des grands scandales sanitaires ?  . Avec Bruno Chareyron (CRIIRAD), Dominique Huez (médecin du travail), Geoffrey Livolsi (journaliste, cofondateur Disclose.ngo) et Annie Thébaud-Mony (chercheuse, association Henri Pézerat). Animée par Daniel Ibanez (co-fondateur du salon, Affaire Lyon-Turin). Dimanche 18 novembre. 14h00-15h30 salle Noire |. Alerte rouge sur l’hôpital. La situation de l’hôpital public en France se dégrade. Qu’il s’agisse des services des urgences saturés ou des dysfonctionnements récurrents du management, les patients et les personnels des hôpitaux sont en souffrance. Quelles sont les conséquences sur la santé des usagers ? Comment sortir de la course à la rentabilité ? Quelle voix donner au personnel soignant ? Les invités proposeront leurs solutions pour maintenir ce service public que tant de pays nous envient. Avec Philippe Halimi (professeur de médecine), Patrick Pelloux (médecin, association des Médecins urgentistes de France) et Frédéric Pierru (sociologue, CNRS). Animée par Caroline Chaumet (journaliste, réalisatrice, Passeursdalerte.org). 14h00-15h30 salle Claire |. Journaliste d’investigation, un métier à risques. En 2017, 65 journalistes (incluant les journalistes professionnels, les non-professionnels ainsi que les collaborateurs de médias) ont été tués dans le monde parce qu’exerçant leur mission. Les assassinats successifs de Daphnée Caruana Galizia à Malte et de Ján Kuciak, en Slovaquie nous rappellent que le continent européen n’est pas épargné par le phénomène. Les journalistes d’investigation se considèrent-ils menacés? Comment se protège-t-on pour faire éclater une vérité qui dérange ? Témoignages de journalistes d’investigation. Avec Antoine Deltour (lanceur d’alerte Luxleaks, soutien de The Signals), et Nicolas Vescovacci (journaliste, affaire Bolloré). Animée par Tatiana Ventôse (journaliste youtubeuse, le Fil d’Actu). 14h00-15h30 salle Pavillon |. Prohibition des drogues : alerte sur une criminalité légale ? D’après l’ONU, la prohibition est un échec. La prohibition des drogues, ce dispositif légal, a-t-elle amélioré la santé publique ? Respecte-t-elle les droits fondamentaux ? Quid de la sécurité ? Si la réponse est non : faut-il lancer l’alerte ? Bénédicte Desforges (ex-policière), Marc Fiévet (ex-agent infiltré de la Douane française). Fabrice Olivet (association Auto-support des Usagers de Drogues – ASUD). Animée par Fabrice Rizzoli (professeur de géopolitique critique des criminalités, association Crim’HALT). 15h30-16h30 salle 3 |. Atelier : délivrer l’alerte. Présentation du guide du lanceur d’alerte du Défenseur des Droits. Des référents répondent à 5 témoignages de la salle avec orientations et conseils en direct. Un lanceur d’alerte est une personne qui veut signaler un danger ou un risque, mettre fin à une action illégale ou irrégulière en interpelant les pouvoirs en place ou en suscitant une prise de conscience. Les simples salariés ou citoyens engagés, se sentent souvent démunis et seuls. Cet atelier permet au public de rencontrer des experts pour être aiguillés dans leurs parcours. Avec Raymond Avrillier (lanceur d’alerte), Fabrice Rizzoli (association Crim’HALT). Mathilde Régis (Le Lanceur.fr) et Françoise Verchère (association Anticor).  . 16h-17h30 salle Noire |. Projection-débat : “Moi, salarié lanceur d’alerte”. “. Moi, salarié lanceur d’alerte. ” (documentaire de Pedro da Fonseca – Premières lignes, 35 mn). Chacun d’eux, un jour, a pris la décision de rendre publiques les activités illégales de son employeur. Sans imaginer certainement à quel point cette décision allait durablement transformer leur existence. De salariés, ils sont devenus lanceurs d’alerte, avec toutes les difficultés professionnelles (procédure de licenciement, poursuites judiciaires) et personnelles que cela implique. Comment ont-ils vécu leur expérience ? Se sont-ils sentis piégés ? Se sont-ils sentis aidés ? Et si c’était à refaire ? Avec Karim Ben Ali (ex-chauffeur poids lourds, affaire Arcelor-Mittal), Pedro da Fonseca (journaliste, Agence Premières lignes), Alain Gautier (syndicaliste, Affaire Vortex), Claire Hocquet (avocate)Jean Christophe Picard (association Anticor). et Jean-Luc Touly (syndicaliste, affaire Veolia). Animée par Carole Rouaud (co-organisatrice du salon). 16h-17h salle Pavillon |. Dopage : vaincre l’omerta. Très régulièrement, la problématique du dopage revient sur le devant de l’actualité. Que ce soit dans le sport amateur ou professionnel, quelle que soit la discipline, l’ampleur du phénomène reste difficile à cerner en raison des difficultés à briser l’omerta. Christophe Bassons, cycliste professionnel dans les années 1990 et Laurent Benezech, ex-joueur du XV de France, ont eu le courage pour dénoncer le recours au dopage. Ils échangeront avec Marie-George Buffet, ministre de la Jeunesse et des sports lorsqu’a éclaté l’affaire Festina. Avec Christophe Bassons (lanceur d’alerte, ex-Coureur cycliste), Laurent Benezech (ancien rugbyman). et Mme la Ministre Marie-George Buffet. Animée par Alexandre Calvez (co-organisateur du salon). 16h-18h salle Claire |. Police, où est le malaise. Les récentes statistiques et l’actualité nous interrogent sur le rôle de la police. Pression du chiffre, pression des médias, pression des citoyens, pression des dirigeants politiques, violence institutionnelle, quel est le rôle de la police dans une démocratie ; faut-il le redéfinir ? Au regard de l’affaire Benalla, quelle est la portée du devoir d’obéissance des fonctionnaires de police ? Quel équilibre entre protection des citoyens et protection des institutions ? Avec Alexandre Langlois (syndicat VIGI – ministère de l’Intérieur \/ Police Nationale), Frédéric Ploquin (journaliste, . Marianne. ). et Omar Slaouti (comité Vérité et Justice pour Ali Ziri). Animée par Christophe Grébert (journaliste). 17h30-18h30 salle Pavillon |. Halte à l’obsolescence programmée ? La surproduction et la surconsommation de produits conçus pour ne pas durer ont des conséquences écologiques insoutenables à long terme. L’obsolescence programmée retire par ailleurs aux citoyens la liberté de consommer de manière durable et responsable. Dans un monde aux ressources en cours d’épuisement, comment lutter contre les entreprises coupables d’obsolescence programmée ?  . Dialogue entre Serge Latouche (professeur émérite d’économie) et Samuel Sauvage (association HOP). 18h-19h30 salle Noire |. Protéger le lancement de l’alerte. Les affaires récentes (Dieselgate, Luxleaks, Panama Papers etc.) démontrent le rôle essentiel que jouent les lanceurs d’alerte dans la mise au jour d’activités portant préjudice à l’intérêt général et au bien-être des citoyens et de la société. En écho, la Loi Sapin II, la directive européenne pour la protection des lanceurs d’alerte, mais aussi la loi sur le secret des affaires, se télescopent. Faut-il protéger les lanceurs d’alertes ou protéger le droit de lancer l’alerte, d’informer ? Avec Juan Branco (avocat, affaire Julian Assange), Francis Chateauraynaud (EHESS), et Serge Portelli (magistrat). Animée par Daniel Ibanez (co-fondateur du salon, Affaire Lyon-Turin). En marge du. salon. Un moment de. poésie. Un moment de poésie volé au salon Des Livres et l’Alerte 2018, des membres de la chorale gospel Voice2Gether et Madame Christiane Taubira. Lien vidéo Au salon du livre des lanceurs d’alerte |. par Radio-Parleur. Lien page Interview Antoine Deltour “Contre la résignation” |. par MetaMorphosis. “[…] le vrai risque, est que ce débat tourne à la résignation. A force de la multiplication de ces scandales, il y a un risque de lassitude des opinions, et ça c’est le vrai danger.” Antoine Deltour, lanceur d’alerte Luxleaks. Le 15 mai 2018 il fut définitivement acquitté pour l’ensemble des chefs d’inculpation concernant la copie et l’utilisation des documents des LuxLeaks. La juridiction lui a reconnu le statut de lanceur d’alerte au sens de la Cour européenne des droits de l’homme. (. https:\/\/reporterre.net\/Le-lanceur-d-a… ). Lien vidéo Interview Geoffrey Livolsi “Lanceur d’enquêtes”. “Il y a un risque que des lanceurs d’alerte qui auraient, il y a quelques années dénoncé des scandales, ne le fassent plus quand ils voient qu’ils ne sont pas protégés derrière…Et ça, ça va poser un problème pour les journalistes mais d’abord pour les citoyens parce que c’est de l’information et l’information est un bien public et cette information là, doit être défendue”. Geoffrey Livolsi, journaliste, co-fondateur de Disclose.ngo (. https:\/\/disclose.ngo\/ ). Lien vidéo Interview de Karim Ben Ali “lanceur d’alerte malgré lui”. “Et voilà comment on recycle les déchets à Florange (…) on balance de l’acide”, dénonçait Karim BEN ALI lanceur d’alerte ArcelorMittal dans sa vidéo (. https:\/\/www.youtube.com\/watch?v=Ear17… ) postée sur les réseaux sociaux. C’était en 2016. Lien vidéo Interview Pedro Brito Da Fonseca “témoin du fracas”. “[…] Il s’est fracassé avec son alerte” nous répond Pedro Brito Da Fonseca, journaliste. Il s’etait penché sur le cas ArcelorMittal et Karim Ben Ali- et avait enquêté Sous le radar – Pollution, Alertes citoyennes sur le net? que vous retrouverez ici :. https:\/\/www.pltv.fr\/sous-le-radarpoll… Lien vidéo Irène Frachon |. par L’Humanité. Irène Frachon, qui a révélé le scandale du Mediator, dédicace son livre à la Maison des Métallos à Paris (11e), où se tient, durant tout le week-end le deuxième Salon du livre des lanceurs et lanceuses d’alerte. Le film retraçant le combat de la pneumologue contre le laboratoire pharmaceutique Servier et pour l’indemnisation des victimes, intitulé “La Fille de Brest », est sorti au cinéma mercredi. Lien vidéo “Ils vous pourrissent votre vie, c’est un combat permanent.” |. par France 3 Paris Ile-de-France. “Le pot de terre contre le pot de fer. Le 4e salon du livre dédié aux lanceurs d’alerte à Paris met en avant ces personnes qui choisissent de révéler les turpitudes de leur entreprise. Intervenants : Nicolas Forissier, ancien cadre de la banque UBS; James Dunn, ancien salarié de Qosmos; Daniel Ibanez, organisateur du salon Des livres et l’alerte”. Lien vidéo Register Now! Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque placerat sapien condimentum imperdiet cursus. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque placerat sapien condimentum imperdiet cursus. Ticket Info Contactez Nous. Nom. Email. Message. Submit. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Rencontre Annuelle des Lanceurs d'Alerte. 12, 13 & 14 Novembre 2021. à la Maison des Sciences de l'Homme Paris Nord. (MSH Paris Nord) La Plaine St-Denis. Métro Ligne 12 \"Front populaire\". Entrée Libre et Gratuite*. Les rencontres se dérouleront dans le respect des gestes barrières et des règles limitant la propagation des virus. Programme Dossier Presse Les Intervenants de l’édition 2021. Erri DE LUCA Auteur. anemptytextlline. Erri De Luca est un écrivain, poète et traducteur italien, en 1968, il embrasse... Camille ÉTIENNE Etudiante. anemptytextlline. Camille Etienne s’engage depuis longtemps mais la cause écologique n’a pas toujours été son... Ariane DENOYEL Journaliste Enseignante école de journalisme de Grenoble. anemptytextlline. Ariane Denoyel, journaliste indépendante, diplômée de l’IEP de Paris, professeure associée de journalisme à... Frédéric PIERUCCI auteur. anemptytextlline. Frédéric Pierucci, ancien patron d’une des filiales d’Alstom, après avoir été longtemps contraint au... Olivia MOKIEJEWSKI Journaliste. anemptytextlline. Journaliste et présidente de l’association Collectif 9471. Olivia Mokiejewski réalise des documentaires engagés... Virginie ROZIERE Ex-députée européenne, porteuse de la directive européenne de protection. anemptytextlline. En dédicace samedi 23 novembre de 16h à 17h, puis de 19h à 19h30... Antonio CASILLI Professeur Chercheur. anemptytextlline. Antonio A. Casilli est professeur de sociologie à Telecom Paris (Institut Polytechnique de Paris)... Jean-Michel FOURNIAU Sociologue, Président GIS démocratie&Participation. anemptytextlline. Jean-Michel Fourniau est sociologue, directeur de recherche à l'Université Gustave Eiffel et enseignant à... Sylvain INSERGUEIX Libraire. anemptytextlline. Passionné de bande dessinée depuis toujours, Sylvain INSERGUEIX a fondé en 1971 le fanzine... Marie Monique ROBIN Journaliste. anemptytextlline. Marie-Monique Robin, journaliste et réalisatrice, est lauréate du Prix Albert-Londres (1995). Depuis trente ans,... Annie COLLOMBET Co-présidente de Vivre et Agir en Maurienne. anemptytextlline. Co-présidente de l'association \"Vivre et Agir en Maurienne\" Annie COLLOMBET, est une inlassable actrice... Étienne MERLE Journaliste. anemptytextlline. Étienne Merle, journaliste, 25 ans. Après un premier contrat chez Premières Lignes Télévision (Cash... Christian CHOUVIAT Père de Cédric Chouviat. anemptytextlline. Par sept fois Cédric Chouviat, plaqué au sol par trois policiers, a crié \"j'étouffe\"... Ludovic FAYOLLE Policier. anemptytextlline. Ludovic FAYOLLE, 35 ans, originaire du Mâconnais (71), fonctionnaire d'État, affecté au commissariat de... Yann GAUDIN Lanceur d'alertes sur Pôle emploi. anemptytextlline. Yann Gaudin ne pourra malheureusement être avec nous. Yann Gaudin, 44 ans, ex-conseiller Pôle emploi... Sylvain WASERMAN Député du Bas-Rhin (sous réserve). anemptytextlline. Sylvain WASERMAN député du Bas-Rhin, vice-président de l'Assemblée Nationale, il est porteur d'un projet... Denise SCHNEIDER présidente de l'association. anemptytextlline. Denise Schneider été présidente durant une vingtaine année de l'Association de protection et de... Valérie MURAT Fondatrice de \"Alerte aux Toxiques\". anemptytextlline. Valérie Murat est porte-parole d'Alerte aux Toxiques, une association d’information sur les dangers des... Eve SZEFTEL Journaliste. anemptytextlline. Journaliste à l’AFP, ancienne correspondante à Bobigny, Eve Szeftel est l'autrice de “Le maire... Quentin GUILLEMAIN Asso Parents victimes de Lactalis. anemptytextlline. Quentin Guillemain est le père d'une petite fille intoxiquée par le lait de Lactalis.... Arié ALIMI Avocat. anemptytextlline. Arié Alimi, avocat au barreau de Paris et membre de la ligue des Droits... Philippe MEIRIEU Professeur émérite. anemptytextlline. Philippe Meirieu est né le 29 novembre 1949 à Alès dans le Gard, au... Pedro DA FONSECA Journaliste. anemptytextlline. Journaliste réalisateur, Pedro da Fonseca travaille régulièrement avec l'agence Premières Lignes. Il a suivi... Jean-Pierre CANET journaliste. anemptytextlline. Jean-Pierre Canet est journaliste d'enquête, avec Nicolas VESCOVACCI ils sont les auteurs « Vincent... Denis BRETEAU Lanceur d'alerte des marchés SNCF \/ IBM. anemptytextlline. Denis Breteau, 53 ans, ex-cadre à la direction des achats de la SNCF ayant... Julie anemptytextlline. Ancienne micro-travailleuse ayant travaillé sur l'assistant vocal de Microsoft, les tâches qui m'étaient assignées... Elise VAN BENEDEN Avocate, administratrice d'ANTICOR. anemptytextlline. Militante au sein de l'association Anticor depuis 2008, Elise Van Beneden est présidente de... Fabien GRACIA Maire de La Montagne. anemptytextlline. Fabien GRACIA a été élu maire de LA MONTAGNE le 15 mars 2020? Il... Pierre MERIAUX Maire-adjoint Grenoble. anemptytextlline. Pierre Meriaux est inspecteur du travail depuis 1985 et militant syndical dans la fonction... Olivier TESQUET Journaliste. anemptytextlline. Né en 1987, Olivier Tesquet est journaliste à Télérama. Depuis dix ans, il s’intéresse... Francis CHATEAURAYNAUD Sociologue Chercheur. anemptytextlline. Sociologue, Francis Chateauraynaud est directeur d’études à l’EHESS. Il est notamment l’auteur, avec Josquin... Daniel CUEFF Ancien Maire de Langouët. anemptytextlline. Un maire d’une petite commune peut-il agir quand l’État renonce à exercer ses responsabilités... Gilbert GOURRAUD Référent Terre de Lien Savoie. anemptytextlline. Gilbert Gourraud a exercé diverses professions dans l'univers sportif et montagnard. Il s'intéresse à... Nathalie TEHIO Avocate. anemptytextlline. Nathalie Tehio, membre du bureau national de la LDH et de l’Observatoire parisien des... Amal BENTOUNSI Fondatrice \"Urgence notre police assassine\". anemptytextlline. Amal Bentounsi a fondé le collectif des familles des victimes tuées par la police.... Gilles BRUEY Libraire. anemptytextlline. Libraire indépendant à Argenteuil, Gilles Bruey a ouvert Le Presse Papier en 1981 avec... Florian FERJOUX Avocat. anemptytextlline. Mathieu MOLARD Journaliste. anemptytextlline. Mathieu Molard est rédacteur en chef du site d'information indépendant StreetPress.com. Depuis plusieurs années... Marie BLANDIN Ancienne sénatrice. anemptytextlline. Après avoir présidé la région Nord Pas de Calais (et entamé dès 1993 le... Raphaël RUFFIER-FOSSOUL Journaliste. anemptytextlline. Journaliste spécialisé dans la politique et l’investigation locale, Raphaël Ruffier-Fossoul a été rédacteur en... Philippe BREUVART citoyen engagé. anemptytextlline. Philippe Breuvart « citoyen engagé ». Ingénieur, cadre dans le domaine de la... Nicolas VESCOVACCI Journaliste, co-auteur de « Vincent tout puissant ». anemptytextlline. En dédicace samedi 23 novembre de 14h30 à 16h, avec Jean-Pierre CANET pour «... Brigitte GOTHIERE anemptytextlline. Juan PANCORBO joournaliste. anemptytextlline. Juan Pancorbo est journaliste et vit à Berlin. Il a réalisé avec Clara López... Vanessa LANGARD Gilet Jaune mutilée. anemptytextlline. Auxiliaire de vie, Vanessa Langard soutient le mouvement des Gilets Jaunes dès le début.... Louis-Antoine DUJARDIN Editeur de BD. anemptytextlline. Louis-Antoine Dujardin est éditeur de bandes dessinées pour les éditions Delcourt. Intéressé par les... Ugo BERNALICIS député du Nord. anemptytextlline. Ugo BERNALICIS, député du Nord, est le porteur d'un projet de loi de transposition... Gérald LE CORRE Affaire LUBRIZOL. anemptytextlline. Gérald LE CORRE est inspecteur du travail à Rouen et responsable des questions de... Aymeric MONVILLE anemptytextlline. Aymeric Monville est philosophe, directeur de collection aux éditions Delga et rédacteur en chef... Olivier AZAM Les mutins de pangée. anemptytextlline. En 2005, Olivier AZAM est cofondateur et membre de la coopérative Les Mutins de... Florent COMPAIN Ancien président Les Amis de la Terre. anemptytextlline. À 44 ans, Florent COMPAIN est un militant écologiste de terrain se battant pour... Alexandre CALVEZ Organisateur. anemptytextlline. Alexandre Calvez est fonctionnaire et administrateur de l'association Anticor. A ce titre, il intervient... Serge PORTELLI Ancien magistrat. anemptytextlline. Après sa formation à l’École Nationale de la Magistrature, Serge PORTELLI a débuté sa... Amar BENMOHAMED Policier. anemptytextlline. Le 27 juillet 2020, le site d’informations StreetPress a dévoilé le témoignage accablant du... Anne RICHARD Documentariste, réalisatrice du film « Droit au cœur ». anemptytextlline. \"Journaliste et réalisatrice, Anne Richard a rencontré Irène Frachon dès le début de l’affaire... Laurent THINES Chirurgien. anemptytextlline. Le Professeur Laurent Thines est l'un des auteurs de la tribune demandant l'interdiction des... Emmanuelle MIGNATON Lanceuse d'alerte, affaire de l'Androcur. anemptytextlline. Emmanuelle HUET-MIGNATON, 55 ans, est une victime de l'Androcur (laboratoire Bayer), ayant développé 5... Bruno TOUSSAINT Membre de la rédaction de Prescrire. anemptytextlline. Bruno Toussaint, ex-médecin généraliste, est depuis 1995 membre de la rédaction de la revue... Asma MHALLA Maître de conférence. anemptytextlline. Asma MHALLA est experte en économie numérique et maitre de Conférences à Sciences Po... Karine MAUVILLY GRATON anemptytextlline. Karine Mauvilly est essayiste. Formée en droit et en sciences politiques, elle a été... Daniel IBANEZ Co-fondateur des rencontres. anemptytextlline. Économiste des Procédures collectives, dirigeant d’un cabinet de conseil, il s’est intéressé au projet... Nicolas FORISSIER Lanceur d'alerte Affaire UBS, Metamorphosis. anemptytextlline. Pour Nicolas Forissier \"La vérité se meurt dans le silence\". C'est d'ailleurs le moto... Caroline FIAT Députée. anemptytextlline. Députée de Meurthe et Moselle, Caroline Fiat est membre de La France insoumise. Première... Françoise VERCHERE Maire et conseillère générale honoraire. anemptytextlline. Au sein du Collectif d’élus et d'élues doutant de la pertinence de l’aéroport de... Philippe PIARD Paysan. anemptytextlline. Paysan et porte parole de Secrets Toxiques, il se bat pour une agriculture saine... Patrice SALINI Économiste des transports, Ancien directeur de cabinet ministériel. anemptytextlline. Consultant indépendant spécialisé sur le secteur des transports Dynamicien des systèmes Ancien membre du Conseil... Gabriel ULLMANN expert environnement. anemptytextlline. Gabriel ULLMANN, docteur-ingénieur, docteur en droit, MBA de HEC, est expert judiciaire pour les... Laurence CHAMPIER Banques Alimentaires. anemptytextlline. Laurence Champier ne pourra malheureusement être avec nous. Directrice fédérale des Banques Alimentaires. Cyrille VENET Anesthésiste. anemptytextlline. Dr Cyrille Venet anesthésistes réanimateurs chef de service au CHU de Grenoble-Voiron et syndicaliste. Parti prenante dans... Pablo IBAÑEZ CRUCEYRA Chercheur. anemptytextlline. Pablo IBÁÑEZ CRUCEYRA, docteur en génétique et chercheur à l’École Supérieure de Physique et... Juliette ALIBERT Avocate. anemptytextlline. Juliette ALIBERT, avocate au Barreau de Paris, exerce dans la défense des libertés fondamentales... Françoise NICOLAS ancienne fonctionnaire des affaires étrangères. anemptytextlline. Dans un texte publié en juillet 2017 par le site Mondafrique sous le titre... Maureen KEARNEY anemptytextlline. Maureen Kearney, Irlandaise, était professeur d'anglais chez Areva depuis 1987. Elle était représentante cadre... Manuel DOMERGUE Fondation Abbé Pierre. anemptytextlline. Manuel Domergue est directeur des études à la Fondation Abbé Pierre, dont il coordonne... Sabine DUFLO Psychologue clinicienne. anemptytextlline. Je mène depuis plusieurs années un combat contre les graves dangers de la surexposition... Irène FRACHON Pneumologue, lanceuse d'alerte de l'affaire du Médiator,. anemptytextlline. Docteur en médecine spécialisée en pneumologie, elle est celle par qui l'affaire du Mediator... Guillaume GONTARD Sénateur. anemptytextlline. Elu Sénateur en 2017, et Vice-Président de la Commission Aménagement du Territoire et Développement... Thomas LEBONNIEC anemptytextlline. Thomas Le Bonniec a travaillé entre mai et juillet 2019 pour un sous-traitant d'Apple... Clara LÓPEZ RUBIO Réalisatrice. anemptytextlline. Clara LÓPEZ RUBIO est réalisatrice du film \"Hacking Justice\" que nous projetons grâce au... Inès LERAUD Journaliste, autrice de « Algues vertes l’histoire interdite ». anemptytextlline. En dédicace dimanche 24 novembre de 12h30 à 13h30 puis de 17h30 à 18h30... Flore TALAMON Autrice. anemptytextlline. Flore TALAMON est la scénariste de la bande dessinée qui vient d'être publiée chez... Raymond AVRILLIER Citoyen engagé. anemptytextlline. « Administré engagé », il est à l’origine des révélations de plusieurs affaires politico-financières... Elisabeth BORREL Ex magistrate. anemptytextlline. Élisabeth Borrel est magistrate à la retraite. Son mari, magistrat également et alors coopérant... Filter - All. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Mathieu MOLARD. Mathieu MOLARD. Journaliste. Mathieu Molard est rédacteur en chef du site d’information indépendant StreetPress.com. Depuis plusieurs années il consacre une partie de son travail à mettre en lumière les dysfonctionnements de la police. Il est notamment co-auteur du documentaire Gilets jaunes une répression d’État et de plusieurs enquêtes sur des violences policières. Ainsi, en juillet 202O, il révèle grâce au témoignage d’un lanceur d’alerte, un système de maltraitance et de racisme dans les cellules du tribunal de paris. Il est également l’auteur de nombreuses enquêtes sur l’extrême droite et (avec Robin D’Angelo) du livre Le système Soral, enquête sur un facho business. Rencontres. \"Une police républicaine ?\" auditorium 140, 13 Nov 2021 17:30 - 19:00. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. 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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Guillaume GONTARD. Guillaume GONTARD. Sénateur. Elu Sénateur en 2017, et Vice-Président de la Commission Aménagement du Territoire et Développement Durable, il est devenu président du groupe écologiste au Sénat. Architecte de profession, il a également été maire du Percy, une commune du Trièves (Isère) de 170 habitants pendant plus de dix ans. Son engagement au Sénat est l’accompagnement des territoires dans la transition écologique et sociale avec une attention particulière pour les communes rurales et les territoires de montagne. Persuadé qu’il est urgent de mettre en place de nouvelles pratiques plus en lien avec la réalité et avec les besoins des habitants. 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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Julie. Julie. Ancienne micro-travailleuse ayant travaillé sur l’assistant vocal de Microsoft, les tâches qui m’étaient assignées m’ont mise en première ligne pour constater le haut degré d’intrusion dans la vie privée que les assistants vocaux rendent possible, ainsi que l’omniprésence du micro-travail derrière l’IA, les conditions de travail précaires qu’il implique, et le manque de sécurité pour les utilisateurs. En 2018 j’ai alerté avec La Quadrature du Net en partageant mon expérience dans les coulisses de l’Intelligence Artificielle. Rencontres. \"La surveillance pour quelle sûreté ?\" panoramique, 13 Nov 2021 13:00 - 14:30. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. 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En coédition avec  @RevueDessineeIT,  @ElianePatriarca, Grégory Mardon,  @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans.  https:\/\/t.co\/teOnjQTYZC","","","","","3","6","Facebook","Twitter"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:44:43 UTC 2022","fresh_date_dt":"Wed Jun 08 14:44:44 UTC 2022","referrer_id_s":"Wd6AgNi7ikli","language_s":"fr","size_i":233583,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":9,"references_internal_i":9,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078165450063872,"text_t":"Programme 2021 | Rencontres Annuelles des Lanceurs d'Alerte. Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. 6. ème. Rencontre Annuelle des Lanceurs d'Alerte. 12, 13 & 14 Novembre 2021. à la Maison des Sciences de l'Homme Paris Nord. (MSH Paris Nord) La Plaine St-Denis. Métro Ligne 12 \"Front populaire\". Entrée Libre et Gratuite*. Les rencontres se dérouleront dans le respect des gestes barrières et des règles limitant la propagation des virus. Programme Dossier Presse Le programme Complet 2021. Vous pouvez aussi consulter le. programme par thématiques 12 Nov 2021 16:30-18:30. Amphithéâtre. Définition Transposition de l’alerte Propositions des lanceurs d’alerte. Ugo BERNALICIS. ,. député du Nord. Francis CHATEAURAYNAUD. ,. Sociologue Chercheur. Jean-Michel FOURNIAU. ,. Sociologue, Président GIS démocratie&Participation. Guillaume GONTARD. ,. Sénateur. Virginie ROZIERE. ,. Ex-députée européenne, porteuse de la directive européenne de protection. Nathalie TEHIO. ,. Avocate. Sylvain WASERMAN. ,. Député du Bas-Rhin (sous réserve). Les États membres de l’Union Européenne doivent transposer en droits national la Directive sur “la protection des personnes qui signalent des violations du droit de l’Union” avant la fin de cette année 2021. Un projet de loi a été déposé et doit être voté avant la fin de l’année. Selon la Déclaration des Droits de […]. Lire la suite 12 Nov 2021 17:00-20:00. auditorium 300. Climat : Le mur, on s’en approche ou on est dedans ? Camille ÉTIENNE. ,. Etudiante. Gilbert GOURRAUD. ,. Référent Terre de Lien Savoie. Marie Monique ROBIN. ,. Journaliste. Un film sur les arbres et le risque de leur disparition. Le documentaire “Le génie des arbres” d’Emmanuelle NOBECOURT que nous projetterons en première partie grâce au soutien de Hauteville Productions démontre avec simplicité et clarté ce génie naturel des arbres et leur rôle essentiel mais il montre également les atteintes que porte l’espèce humaine […]. Lire la suite 13 Nov 2021 10:30-12:00. Amphithéâtre. L’emploi, limite de l’alerte ? Annie COLLOMBET. ,. Co-présidente de Vivre et Agir en Maurienne. Florent COMPAIN. ,. Ancien président Les Amis de la Terre. Pedro DA FONSECA. ,. Journaliste. Jean-Michel FOURNIAU. ,. Sociologue, Président GIS démocratie&Participation. Quentin GUILLEMAIN. ,. Asso Parents victimes de Lactalis. Karim Ben Ali accusé de mettre en péril l’emploi, la promotion de grands projets au nom de l’emploi, l’emploi et la santé publique, la pollution mais l’emploi… Peut-on conclure que l’emploi est un frein réel à l’alerte ? L’alerte doit-elle disparaître au nom de la préservation de l’emploi ? Les organisations professionnelles sont-elles dans une […]. Lire la suite 13 Nov 2021 11:00-12:30. panoramique. Alerte : déficience des services de l’Etat. Guillaume GONTARD. ,. Sénateur. Gérald LE CORRE. ,. Affaire LUBRIZOL. Pierre MERIAUX. ,. Maire-adjoint Grenoble. Gabriel ULLMANN. ,. expert environnement. Les affaires Lubrizol ou Lactalis font apparaître des dysfonctionnements majeurs et démontrent que la politique d’autocontrôle prônée atteint des limites et présentent de graves dangers pour la population. On constate dans ces dossiers que les risques viennent finalement de de la déficience des services de l’Etat à laquelle s’ajoute l’absence de lanceurs d’alertes. La réduction […]. Lire la suite 13 Nov 2021 11:00-12:30. auditorium 300. Carte Blanche à Erri DE LUCA. Erri DE LUCA. ,. Auteur. Philippe MEIRIEU. ,. Professeur émérite. Erri DE LUCA est l’auteur italien le plus vendu en France, il est l’observateur engagé de notre monde, ses mots sont précis et lorsqu’il utilise le mot sabotage, qui lui est reproché par les promoteurs du projet Lyon-Turin, il écrit “La parole contraire” revendiquant l’usage du mot et l’usage des mots. Cette procédure bâillon à […]. Lire la suite 13 Nov 2021 13:00-14:30. panoramique. La surveillance pour quelle sûreté ? Antonio CASILLI. ,. Professeur Chercheur. Francis CHATEAURAYNAUD. ,. Sociologue Chercheur. Julie. ,. Thomas LEBONNIEC. ,. Olivier TESQUET. ,. Journaliste. Tous les problèmes de notre société semblent devoir recevoir comme réponse une surveillance accrue. Du contrôle biométrique dans les établissements scolaires ou la proposition de bracelets de présence électroniques pour les élèves dans certains établissements, aux surveillances des contestataires des grands projets comme à Bure par exemple, en passant par la surveillance des « actions […]. Lire la suite 13 Nov 2021 13:00-14:30. Amphithéâtre. Les trous noirs des financements publics. Raymond AVRILLIER. ,. Citoyen engagé. Daniel IBANEZ. ,. Co-fondateur des rencontres. Patrice SALINI. ,. Économiste des transports, Ancien directeur de cabinet ministériel. Françoise VERCHERE. ,. Maire et conseillère générale honoraire. A quoi servent les investissements publics ? La question peut paraître incongrue, mais peut-on y répondre sans évaluer ces investissements a posteriori ? On parle beaucoup de la responsabilité des décideurs, voire de l’impunité dans l’utilisation de l’argent public mais comment sont évaluées les réalisations, y-a-t-il des obligations de les évaluer ? La Déclaration des […]. Lire la suite 13 Nov 2021 13:30-16:30. auditorium 300. Julian ASSANGE : l’acharnement Film “Hacking justice”. Olivier AZAM. ,. Les mutins de pangée. Clara LÓPEZ RUBIO. ,. Réalisatrice. Aymeric MONVILLE. ,. Juan PANCORBO. ,. joournaliste. Le journaliste Julian ASSANGE est enfermé depuis bientôt dix ans pour avoir publié les informations sur les pratiques de l’armée de coalition et les meurtres de civils notamment avec la vidéo “colateral murder” https:\/\/collateralmurder.wikileaks.org\/. L’ONU a qualifié le traitement qui lui est infligé de torture. Après la projection du documentaire de Clara LOPEZ RUBIO “Hacking […]. Lire la suite 13 Nov 2021 15:00-16:30. panoramique. Les soins, les soignants et l’accès aux soins. Caroline FIAT. ,. Députée. Pierre MERIAUX. ,. Maire-adjoint Grenoble. Olivia MOKIEJEWSKI. ,. Journaliste. Cyrille VENET. ,. Anesthésiste. Chacun d’entre nous a été alerté sur les conditions de protection des soignants et les difficultés dans les établissements de soins dès la première vague et lors du premier confinement. Depuis de longs mois, les professions de santé avaient alerté les pouvoirs publics sur les conséquences que pouvaient avoir les réductions de postes, de lits, […]. Lire la suite 13 Nov 2021 15:30-17:00. Amphithéâtre. La banlieue délaissée de la République. Amal BENTOUNSI. ,. Fondatrice \"Urgence notre police assassine\". Gilles BRUEY. ,. Libraire. Alexandre CALVEZ. ,. Organisateur. Eve SZEFTEL. ,. Journaliste. Parler de la banlieue c’est entendre « racaille » « radicalisé » « zone de non droit » « racisme » « discrimination » « violence » « abus des droits sociaux » “violence policière” “délit de sale gueule”… La banlieue est-elle soumise aux lois fondamentales de la République ou à des lois locales d’exception. […]. Lire la suite 13 Nov 2021 17:30-19:00. auditorium 300. États d’urgence \/ État de droit \/ État d’Alerte. Arié ALIMI. ,. Avocat. Daniel IBANEZ. ,. Co-fondateur des rencontres. Asma MHALLA. ,. Maître de conférence. Serge PORTELLI. ,. Ancien magistrat. Après l’état d’urgence prolongé dans les affaires terroristes et les attentats de 2015, la France connaît maintenant l’état d’urgence sanitaire avec des limitations exceptionnelles dans la liberté de circuler, celle de se rassembler. Durant cette période on a pu avoir le sentiment que les fonctionnaires de police faisaient la loi. Limitations de circulation, contrôle des […]. Lire la suite 13 Nov 2021 17:30-19:00. auditorium 140. Une police républicaine ? Christian CHOUVIAT. ,. Père de Cédric Chouviat. Vanessa LANGARD. ,. Gilet Jaune mutilée. Mathieu MOLARD. ,. Journaliste. Nathalie TEHIO. ,. Avocate. Laurent THINES. ,. Chirurgien. Une chose est sûre, nous aurions pu inviter bien plus d’intervenant e s pour cette rencontre témoignant des dérives d’une partie de celles et ceux qui portent l’uniforme. Le bilan des victimes est un indicateur des dysfonctionnements en matière de maintien de l’ordre, la nature des blessures et des mutilations infligées lors des manifestations par […]. Lire la suite 13 Nov 2021 17:30-18:30. Salle Panoramique. Santé publique : qu’a révélé la pandémie ? Marie BLANDIN. ,. Ancienne sénatrice. Raphaël RUFFIER-FOSSOUL. ,. Journaliste. Bruno TOUSSAINT. ,. Membre de la rédaction de Prescrire. La controverse scientifique mise en scène sur les plateaux audiovisuels, sur laquelle se sont exprimés journalistes, essayistes, commentateurs, conduit une partie de la population à douter de la démarche scientifique et du monde médical. Les conflits d’intérêts ressurgissent et renforcent le doute. Dès lors toutes les théories se développent sur les imprécisions, les omissions et […]. Lire la suite 14 Nov 2021 11:00-12:00. Amphithéâtre. L’alerte en Bande Dessinée. Louis-Antoine DUJARDIN. ,. Editeur de BD. Sylvain INSERGUEIX. ,. Libraire. Flore TALAMON. ,. Autrice. La publication de l’alerte connaît une évolution avec l’apparition de quelques bandes dessinées, le succès de l’enquête d’Inès Léraud sur les algues vertes et les pressions qu’elle suscitent montrent que cette forme d’édition de l’alerte est efficace. Quels sont les canaux d’édition et comment s’organise une publication de Bande Dessinée, comment les lanceurs d’alertes peuvent-ils […]. Lire la suite 14 Nov 2021 11:00-12:30. auditorium 300. Il y a l’alerte et puis après Film “Droit au cœur” (procès Mediator). Irène FRACHON. ,. Pneumologue, lanceuse d'alerte de l'affaire du Médiator,. Anne RICHARD. ,. Documentariste, réalisatrice du film « Droit au cœur ». Nous projetterons le film d’Anne RICHARD qui a suivi Irène FRACHON pendant ces longues années et son documentaire retrace ces années jusqu’au procès. Nous pouvons projeter ce film grâce au soutien de Paris-Brest Productions. Et après l’alerte ? Nombreux sont les lanceurs d’alerte qui déclarent si j’avais su … Il est certain qu’Irène Frachon ne […]. Lire la suite 14 Nov 2021 11:00-12:30. Salle Panoramique. Écrans et apprentissage numérique Alerte pour les enfants ? Sabine DUFLO. ,. Psychologue clinicienne. Pablo IBAÑEZ CRUCEYRA. ,. Chercheur. Karine MAUVILLY GRATON. ,. Philippe MEIRIEU. ,. Professeur émérite. Les écrans à l’école et le numérique comme moyen d’apprentissage s’est imposé avec des slogans comme “lutter contre la fracture numérique” ou encore sur le présupposé que les enfants doivent manipuler l’informatique le plus tôt possible pour être adapté à la société de demain. Combien un enfant passe-t-il d’heure devant des écrans chaque jour, de […]. Lire la suite 14 Nov 2021 13:00-14:00. Amphithéâtre. Covid-19 les pauvres paupérisés double peine. Philippe BREUVART. ,. citoyen engagé. Laurence CHAMPIER. ,. Banques Alimentaires. Manuel DOMERGUE. ,. Fondation Abbé Pierre. La crise sanitaire du printemps, nous a alertés sur les difficultés pour le monde économique et les salariés. Le rôle primordial pour la société des « invisibles », infirmières, aides-soignantes, caissières, livreurs, femmes de ménage, … a été mis en lumière. La situation des plus pauvres, travailleurs précaires, travailleurs saisonniers, travailleurs au noir, petits boulots […]. Lire la suite 14 Nov 2021 13:30-15:00. auditorium 300. 2021 Paroles de lanceurs d’alertes. Amar BENMOHAMED. ,. Policier. Marie BLANDIN. ,. Ancienne sénatrice. Ariane DENOYEL. ,. Journaliste Enseignante école de journalisme de Grenoble. Ludovic FAYOLLE. ,. Policier. Maureen KEARNEY. ,. Emmanuelle MIGNATON. ,. Lanceuse d'alerte, affaire de l'Androcur. Valérie MURAT. ,. Fondatrice de \"Alerte aux Toxiques\". Philippe PIARD. ,. Paysan. Denise SCHNEIDER. ,. présidente de l'association. Comme chaque année nous donnons la parole aux lanceuses et lanceurs d’alertes, celles et ceux qui ont informé la population en transmettant des informations pour prévenir ou corriger les situations de risque ou les dysfonctionnements. Venez écouter le courage, la citoyenneté et l’obéissance à l’intérêt général. Lire la suite 14 Nov 2021 13:30-15:00. Salle Panoramique. Affaires d’État : ALSTOM, SNCF, Pôle emploi …. Elisabeth BORREL. ,. Ex magistrate. Denis BRETEAU. ,. Lanceur d'alerte des marchés SNCF \/ IBM. Yann GAUDIN. ,. Lanceur d'alertes sur Pôle emploi. Frédéric PIERUCCI. ,. auteur. Nicolas VESCOVACCI. ,. Journaliste, co-auteur de « Vincent tout puissant ». La liste des participants au « dîner du siècle » a été publiée et les citoyens ne peuvent que s’interroger sur ce que l’on appelle l’entre soi, la porosité entre les sphères politiques, judiciaires, économique et médiatiques. Le lobby est organisé au niveau Européen et national. Identifier ces réseaux, leur fonctionnement nous amène une fois […]. Lire la suite 14 Nov 2021 15:00-17:30. Amphithéâtre. Pesticides : quelles limites ? Daniel CUEFF. ,. Ancien Maire de Langouët. Florian FERJOUX. ,. Avocat. Fabien GRACIA. ,. Maire de La Montagne. Françoise VERCHERE. ,. Maire et conseillère générale honoraire. L’année 2019 a été marquée par un large débat sur l’utilisation des « pesticides » pour les uns, « produits phytopharmaceutiques » pour les autres ; des maires ont interdit par arrêté leur utilisation à moins de 150 mètres des habitations, la FNSEA considère que la limite est de zéro mètre puisque les produits sont […]. Lire la suite 14 Nov 2021 16:00-17:30. auditorium 300. En même temps, Bâillonner l’alerte. Brigitte GOTHIERE. ,. Daniel IBANEZ. ,. Co-fondateur des rencontres. Morgan LARGE. ,. Journaliste. Inès LERAUD. ,. Journaliste, autrice de « Algues vertes l’histoire interdite ». Fin 2019, Le ministre de l’Intérieur annonce la création d’une cellule de gendarmerie dont l’objectif serait la protection du monde agricole avec 3 objectifs déclarés : « améliorer notre coopération avec le monde agricole et de recueillir des renseignements ; mieux connaître les groupes extrémistes à l’origine des atteintes et de pouvoir anticiper et prévenir […]. Lire la suite 14 Nov 2021 16:00-17:30. Salle Panoramique. Alerte : Bilan de l’année. Juliette ALIBERT. ,. Avocate. Francis CHATEAURAYNAUD. ,. Sociologue Chercheur. Nicolas FORISSIER. ,. Lanceur d'alerte Affaire UBS, Metamorphosis. Elise VAN BENEDEN. ,. Avocate, administratrice d'ANTICOR. Deux années particulièrement exceptionnelles se sont écoulées depuis la dernière rencontre annuelle des lanceurs d’alertes, des nouvelles alertes sont apparues, mais aussi un fort enjeu avec la transposition de la directive européenne dite de protection des lanceurs d’alertes. Faire le bilan des actions menées est déterminant pour agir et protéger l’alerte et celles et ceux […]. Lire la suite 14 Nov 2021 17:30-19:00. Amphithéâtre. Journalisme et réseaux sociaux : Alerte aux complots ? Jean-Pierre CANET. ,. journaliste. Daniel IBANEZ. ,. Co-fondateur des rencontres. Étienne MERLE. ,. Journaliste. Françoise NICOLAS. ,. ancienne fonctionnaire des affaires étrangères. La défiance envers la presse et les journalistes est un sujet d’inquiétude, les réseaux sociaux sont l’objet de critiques du monde politique et médiatique. Comment l’alerte trouve-t-elle, peut-elle trouver sa place ? Comment les lanceurs d’alertes peuvent-ils publier, faire publier les informations qu’ils considèrent comme graves pour notre société ? Comment un journaliste, avec les […]. Lire la suite Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Mentions légales. Ce site est publié et crée par le collectif de bénévoles qui organise le salon «Des livres et l’alerte». Son contenu ne peut être reproduit sans autorisation. «Des livres et l’alerte» ®. Domiciliation :. Librairie Le Presse Papier 28 avenue Gabriel Péri 95100 Argenteuil. Contact :. contact@deslivresetlalerte.fr. Hébergeur du site :. OVH. Avertissement général. La consultation du site Des livres et l’alerte (. https:\/\/lanceurs-alerte.fr ) est proposée aux internautes à titre gratuit et sans aucune garantie de la part de son éditeur. Les informations disponibles sur le site Des livres et l’alerte (. https:\/\/lanceurs-alerte.fr ) qui proviennent de sources extérieures ne saurait garantir qu’elles sont exemptes d’erreurs, ni garantir leur complétude, leur actualité, leur exhaustivité ou autre. En conséquence, il appartient à l’internaute d’utiliser les informations figurant sur le site Internet à ses risques et de procéder à toute vérification sous sa responsabilité exclusive. Les informations fournies sur le site sont susceptibles d’être modifiées sans préavis dans le cadre de l’organisation du salon. Des livres et l’alerte (. https:\/\/lanceurs-alerte.fr ) ne saurait être tenu responsable de l’interprétation faite par les internautes, quelle qu’elle soit, de tous les renseignements, formations et conseils proposés sur son blog, et des conséquences pouvant en découler. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. 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Revue Dessinée @LaRevueDessinee 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec  @RevueDessineeIT,  @ElianePatriarca, Grégory Mardon,  @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans.  https:\/\/t.co\/teOnjQTYZC","","","","","3","6","Facebook","Twitter"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:44:39 UTC 2022","fresh_date_dt":"Wed Jun 08 14:44:42 UTC 2022","referrer_id_s":"Wd6AgNi7ikli","language_s":"fr","size_i":173212,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":8,"references_internal_i":8,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078161835622400,"text_t":"Qui sommes-nous ? | Rencontres Annuelles des Lanceurs d'Alerte. Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Une. équipe bénévole réduite. associée à une. librairie indépendante.  . C’est au printemps 2015 qu’émerge l’idée du premier salon du livre des lanceuses et lanceurs d’alertes entre Catherine et Gilles Bruey, libraires indépendants à Argenteuil, et Daniel Ibanez. Le nom du salon s’est formé en associant les différents enjeux de l’alerte, publication, transmission et protection. C’est la déclinaison du nom du salon « Des Livres et l’Alerte » en « Délivrer l’alerte » et « Délivrez l’Alerte ! » qui a finalement imposé ce choix. Après cinq éditions, nous avons fait évolué le nom de ces rencontres et notre initiative s’appelle dorénavant. “Rencontre Annuelle des Lanceurs d’Alerte”. . Nous espérons que ce nom toujours associé à l’idée de Délivrer l’Alerte sera plus identifiable pour le public et la presse. La Rencontre annuelle des Lanceurs d’alerte est co-organisé par une équipe réduite de personnes bénévoles engagées au quotidien pour le respect des valeurs de la République et la librairie indépendante « Le Presse Papier » à Argenteuil. A ces quelques bénévoles viennent s’ajouter de nombreuses personnes ou associations qui nous aident tout au long de l’année en contribuant à la programmation par leurs propositions, idées et contacts. Parmi ces associations, Crim’Halt, Anticor, Citoyens Résistants d’Hier et Aujourd’hui (CRHA) et Vivre et Agir en Maurienne (VAM) qui est l’association partenaire de notre initiative. Pendant les rencontres, ce sont encore soixante bénévoles de plus qui assurent le bon déroulement de l’accueil, de la technique, de la restauration… et permettent au public d’assister gratuitement aux rencontres, aux dédicaces, aux projections et aux ateliers proposés. Dès janvier nous réfléchissons aux thèmes et aux personnalités à qui nous souhaitons proposer une participation. C’est donc très en amont des rencontres que nous programmons, contactons et organisons. Le choix de la programmation est toujours difficile et risqué : “nous aurions peut-être dû ou encore peut-être n’aurions-nous pas dû”… sont les éternelles questions que nous nous posons. Notre initiative est indépendante de toute association ou organisation et son financement est assuré à la fois grâce à la participation des intervenants qui assument leurs frais de déplacement, des organisations qui présentent leurs activités durant le salon, aux dons des organisateurs et du public. Nous recevons des dons de personnes et de l’association VAM qui est liée à la « Rencontre Annuelle des Lanceurs d’Alerte » depuis son origine. Nous sommes très attachés à l’indépendance de notre initiative et notre liberté de programmation qui reposent sur une équipe attentive à l’actualité et aux réflexions menées dans le domaine de l’alerte. Promouvoir le passage « de l’alerte à la lettre » est l’un des objectifs de cette rencontre annuelle, et nous faisons notre possible pour qu’au cours du week-end, les lanceuses et lanceurs d’alerte puissent non seulement expliquer, transmettre les informations d’intérêt général qu’ils ont pu découvrir, mais aussi envisager de les publier dans des livres. Nous souhaitons aussi que la réflexion sur ce qu’est l’alerte et ses ressorts soit présente. Nous voulons donc à la fois des rencontres d’information et de formation. Nous souhaitons y rappeler et y faire vivre les valeurs et principes fondamentaux de la République en analysant les alertes, les actualités et les évolutions de notre société à leur lumière. L’histoire de la première édition en novembre 2015. Dès que l’idée d’un salon regroupant les lanceuses et lanceurs d’alerte est évoquée, le libraire du « Presse Papier » à Argenteuil recherche les publications sur le sujet et comprend que le livre d’alerte constitue un secteur de l’édition à part entière. À partir de là, les choses s’enchaînent avec une grande fluidité. Daniel Ibanez approche les premières personnalités qui acceptent immédiatement de participer à ces rencontres, la liste s’allongera et chaque fois, c’est avec enthousiasme que le projet de salon est accueilli. Restait à trouver un lieu qui puisse accueillir une telle initiative rassemblant près d’une centaine de personnalités impliquées dans le lancement d’alertes. Nous rencontrons Jean-Jacques Hocquard, le responsable de « La Parole Errante » à Montreuil, lors d’un rendez-vous à l’improviste. Après avoir écouté notre présentation et les noms des invités ayant déjà donné leur accord, la réponse ne tarde pas avec le même enthousiasme : « Je ne vois pas ce qui pourrait faire que ce salon ne se déroule pas ici ». Le premier salon « Des Livres et l’Alerte » se déroulera donc « La Parole errante » à Montreuil. Tout au long de l’organisation du salon et à chaque nouvelle confirmation d’un participant, nous nous disions « tout se passe trop bien, quel est le problème que nous n’avons pas anticipé, quelle tuile va nous tomber sur la tête ! ». Le 13 novembre 2015 en fin d’après-midi, le salon est installé pour les 14 et 15 novembre, les bénévoles ont travaillé pour installer, le buffet est prêt à accueillir le public que nous espérons nombreux. Nous découvrons, le soir, comme toute la France effarée, les carnages et la tragédie des attentats de Paris et Saint-Denis. Le premier salon n’aura pas lieu, même si après de longues discussions tout au long de la nuit, nous décidons d’accueillir les personnalités qui arrivaient de province. Après le recueillement, les lanceuses et lanceurs d’alerte se sont tenus chaud dans une atmosphère pesante. A l’issue de cette journée de rencontres et de réflexions, tous les présents se sont engagés à participer à un nouveau salon, en espérant qu’après le désastre humain du 13 novembre qui suivait celui du 7 janvier, les organisateurs gardent assez d’énergie et d’élan. Nous avons tenu cet engagement et chaque nouvelle édition nous incite à poursuivre. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. RSS. © Des livres et l'alerte - tous droits réservés."}
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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. David BOILLEY. David BOILLEY. Président de l'ACRO. Agrégé de physique et docteur en physique nucléaire, David Boilley est actuellement Président de l’Association pour le Contrôle de la Radioactivité dans l’Ouest (ACRO). A ce titre et auprès d’autres associations, il participe aux travaux du Comité d’orientation des recherches (COR) en sûreté nucléaire de l’IRSN et au Haut Comité pour la Transparence et l’Information sur la Sécurité Nucléaire (HCTISN). En juillet 2019, s’appuyant sur des données du Ministère de la Santé, l’ACRO publiait une carte interactive de la pollution de l’eau du robinet au tritium. David Boilley effectue un suivi de la catastrophe de Fukushima sur un blog dédié: fukushima.eu.org. Rencontres. \"Nucléaire et après ?\" Salle Librairie, 24 Nov 2019 16:00 - 17:30. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. 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{"id":"_RB0ksQ1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/who.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":30,"url_protocol_s":"https","url_paths_count_i":0,"url_file_name_s":"who","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Libreboot – Who develops Libreboot?"],"last_modified":"Sat Apr 02 12:48:53 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-2311006821265510,"exact_signature_unique_b":true,"fuzzy_signature_l":5278436095000739934,"fuzzy_signature_unique_b":true,"h1_txt":["Who develops Libreboot?","Navigate this page:","Leah Rowe (founder, lead developer)","Developers wanted!"],"bold_txt":["Learn how to contribute patches on the git page","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":399,"text_t":"Libreboot – Who develops Libreboot? Who develops Libreboot? Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Leah Rowe (founder, lead developer) Developers wanted! Return to index The purpose of this page is to clearly define who works on Libreboot, who runs the project, how decisions are made, and in general how the project functions. You can find information about major contributions made to Libreboot, on this page which lists such people:. List of contributors Leah Rowe (founder, lead developer). [link] Leah Rowe is the founder of the Libreboot project, and currently the only full-time developer. Leah oversees all development of Libreboot, reviewing outside contributions, and has the final say over all decisions. Leah owns and operates the libreboot.org servers from her lab in the UK. The servers are running Libreboot of course! You can learn more about Leah’s involvement with Libreboot, by reading her entry on the. page listing all contributors, past and present Developers wanted! [link] Leah is the only developer right now. This section will be updated when more developers join the project. Learn how to contribute patches on the. git page All are welcome to join in on development. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/who.md Site map This HTML page was generated by the. untitled static site generator .","wordcount_i":233,"linkscount_i":20,"linksnofollowcount_i":0,"inboundlinkscount_i":16,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/who.html","libreboot.org\/contrib.html","libreboot.org\/news\/policy.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/logo-license.html","libreboot.org\/who.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Leah Rowe (founder, lead developer)","List of contributors","Binary blob policy","License","Template","Logo","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:40 UTC 2022","fresh_date_dt":"Tue Jul 12 03:51:34 UTC 2022","language_s":"en","size_i":4923,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078539809521664}
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{"id":"_xD0Y2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/gnulinux\/","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":36,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","gnulinux"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Libreboot – GNU+Linux guides"],"last_modified":"Sat Apr 02 12:48:35 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":620724126406324251,"exact_signature_unique_b":true,"fuzzy_signature_l":713036041879170235,"fuzzy_signature_unique_b":true,"h1_txt":["GNU+Linux guides","Navigate this page:","Useful links","Guix, Parabola, Trisquel","Rebooting system in case of freeze","Fedora won’t boot?"],"h2_txt":["linux16 issue","BLS issue"],"bold_txt":["all","copylefted","are","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":400,"text_t":"Libreboot – GNU+Linux guides. GNU+Linux guides. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Useful links Guix, Parabola, Trisquel Rebooting system in case of freeze Fedora won’t boot? linux16 issue BLS issue Return to previous index This page is useful for those who wish to use the GRUB GRUB payload directly. If you’re using SeaBIOS, the boot process will work similarly to traditional BIOS systems; refer to the SeaBIOS documentation on. https:\/\/seabios.org\/SeaBIOS GNU+Linux is the operating system of choice, for Libreboot development. It is highly recommended over any other operating system, precisely because it consists of. Free Software (free as in freedom). There. are. other free operating systems, such as BSD, but most of the software in a typical GNU+Linux system is also. copylefted. . To learn more about the importance of copyleft, read this page on the GNU website:. https:\/\/www.gnu.org\/licenses\/copyleft.html Useful links. [link] Refer to the following pages:. How to Prepare and Boot a USB Installer in Libreboot Systems Modifying the GRUB Configuration in Libreboot Systems Installing Hyperbola GNU+Linux, with Full-Disk Encryption (including \/boot) Installing Debian or Devuan GNU+Linux-Libre, with Full-Disk Encryption (including \/boot) Installing Guix System, with Full-Disk Encryption (including \/boot) How to Harden Your GRUB Configuration, for Security Guix, Parabola, Trisquel. [link] These guides were outdated, so they were deleted. You can find links to them here:. https:\/\/notabug.org\/libreboot\/lbwww\/issues\/4 The above issue page is the same as this entry on the TODO page:. ..\/..\/tasks\/#move-all-distro-fdeboot-guides-to-distro-wikimanuals The Debian guide has been retained, because it’s currently up to date. The Hyperbola guide is already on the Hyperbola website, and the above is just a link. In general, it is recommended that you use SeaBIOS but if you want extra security, GRUB payload is recommended where you can then have a fully encrypted \/boot directory. TODO: Nuke. all. distro-specific guides on libreboot.org. Instead, move these instructions to the wiki pages of these projects, on their websites. The reasons are explained in the above issue page. Rebooting system in case of freeze. [link] Linux kernel has a feature to do actions to the system any time, even with it freezes, this is called a. Magic SysRq keys . You can do these actions with Alt + Sysrq + Command. These are the actions:. Alt + SysRq + B: Reboot the system. Alt + SysRq + I: Send SIGKILL to every process except PID 1. Alt + SysRq + O: Shut off the system. If some of them don’t work, you have to enable it in the kernel command line paramter. So append. sysrq_always_enabled=1. to your. GRUB_CMDLINE_LINUX_DEFAULT. in. \/etc\/default\/grub. You can also run. # sysctl kernel.sysrq=1. to enable them. Fedora won’t boot? [link] This may also apply to CentOS or Redhat. Chroot guide can be found on. fedora website linux16 issue. [link] When you use Libreboot’s default GRUB config, and libreboot’s grub uses fedora’s default. grub.cfg. (in. \/boot\/grub2\/grub.cfg. ), fedora by default makes use of the. linux16. command, whereas it should be saying. linux. Do this in fedora:. Open. \/etc\/grub.d\/10_linux. Set the. sixteenbit. variable to an empty string, then run:. grub2-mkconfig -o \/boot\/grub2\/grub.cfg. BLS issue. [link] With. newer versions of fedora , scripts from grub package default to generating. BLS instead of. grub.cfg. . To change that behaviour add following line to. \/etc\/default\/grub. (or modify existing one if it already exists):. GRUB_ENABLE_BLSCFG=false. Then generate. grub.cfg. with:. grub2-mkconfig -o \/boot\/grub2\/grub.cfg. Binary blob policy Edit this page Who develops Libreboot? 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{"id":"-5nP3sQ1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/license.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":34,"url_protocol_s":"https","url_paths_count_i":0,"url_file_name_s":"license","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Libreboot – License"],"last_modified":"Sat Apr 02 12:48:44 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Mon Nov 03 00:00:00 UTC 2008"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-642361184935145505,"exact_signature_unique_b":true,"fuzzy_signature_l":6045584287193454037,"fuzzy_signature_unique_b":true,"h1_txt":["License","Navigate this page:","GNU Free Documentation License","ADDENDUM: How to use this License for your documents"],"h2_txt":["0. PREAMBLE","1. APPLICABILITY AND DEFINITIONS","2. VERBATIM COPYING","3. COPYING IN QUANTITY","4. MODIFICATIONS","5. COMBINING DOCUMENTS","6. 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RELICENSING"],"bold_txt":["unmodified","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":381,"wordcount_i":3977,"linkscount_i":25,"linksnofollowcount_i":0,"inboundlinkscount_i":16,"outboundlinkscount_i":9,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/license.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/template-license.html","libreboot.org\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/license.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","GNU Free Documentation License","Binary blob policy","Who develops Libreboot?","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","www.fsf.org\/","www.gnu.org\/licenses\/fdl-1.3.en.html","www.gnu.org\/licenses\/fdl-1.3.md","fsf.org\/","www.gnu.org\/licenses\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","Free Software Foundation","","","","","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:41 UTC 2022","fresh_date_dt":"Tue Jul 12 03:51:40 UTC 2022","referrer_id_s":"lfDPysQ1CANx","language_s":"en","size_i":30845,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":6,"references_internal_i":6,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078540947226624,"text_t":"Libreboot – License. License. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. GNU Free Documentation License 0. PREAMBLE 1. APPLICABILITY AND DEFINITIONS 2. VERBATIM COPYING 3. COPYING IN QUANTITY 4. MODIFICATIONS 5. COMBINING DOCUMENTS 6. COLLECTIONS OF DOCUMENTS 7. AGGREGATION WITH INDEPENDENT WORKS 8. TRANSLATION 9. TERMINATION 10. FUTURE REVISIONS OF THIS LICENSE 11. RELICENSING ADDENDUM: How to use this License for your documents Return to index Unless otherwise stated, every page and image (e.g. JPG\/PNG files) on libreboot.org or in the repository that it is built on, is released under the terms of the GNU Free Documentation License, either version 1.3 or (at your option) any newer version as published by the. Free Software Foundation , with no Invariant Sections, no Front Cover Texts and no Back Cover Texts. These pages are static HTML, generated from Markdown files, which you can find a link to at the bottom of each HTML page, for the corresponding Markdown file. The terms of this license are written below, unmodified, except to change the formatting so that the text would integrate nicely on this website. You can also find the license here:. https:\/\/www.gnu.org\/licenses\/fdl-1.3.en.html The markdown version, hosted by the GNU project, can be found here:. https:\/\/www.gnu.org\/licenses\/fdl-1.3.md The. unmodified. license text is as follows:. GNU Free Documentation License. [link] Version 1.3, 3 November 2008. Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. . https:\/\/fsf.org\/ Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE. [link] The purpose of this License is to make a manual, textbook, or other functional and useful document “free” in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of “copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. 1. APPLICABILITY AND DEFINITIONS. [link] This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. A “Modified Version” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and\/or translated into another language. A “Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document’s overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none. The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words. 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These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. 2. VERBATIM COPYING. [link] You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3. You may also lend copies, under the same conditions stated above, and you may publicly display copies. 3. 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It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document. 4. MODIFICATIONS. [link] You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:. A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission. B. 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To do this, add their titles to the list of Invariant Sections in the Modified Version’s license notice. These titles must be distinct from any other section titles. You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. 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[link] You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document. 7. AGGREGATION WITH INDEPENDENT WORKS. [link] A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation’s users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate. 8. TRANSLATION. [link] Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title. 9. TERMINATION. [link] You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License. However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it. 10. FUTURE REVISIONS OF THIS LICENSE. [link] The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See. https:\/\/www.gnu.org\/licenses\/ . Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Document. 11. RELICENSING. [link] “Massive Multiauthor Collaboration Site” (or “MMC Site”) means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A “Massive Multiauthor Collaboration” (or “MMC”) contained in the site means any set of copyrightable works thus published on the MMC site. “CC-BY-SA” means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization. “Incorporate” means to publish or republish a Document, in whole or in part, as part of another Document. 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Permission is granted to copy, distribute and\/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled \"GNU Free Documentation License\". If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with … Texts.” line with this:. with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. The license text ended in the previous paragraph. Now you see the generic footer generated for every page on this website:. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/license.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"21fM12Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":36,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Libreboot – Hardware compatibility list"],"last_modified":"Sat Apr 02 12:48:39 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-3268992645201256679,"exact_signature_unique_b":true,"fuzzy_signature_l":-5544051077256980729,"fuzzy_signature_unique_b":true,"h1_txt":["Hardware compatibility list","Navigate this page:","Supported hardware","EC update on i945 (X60, T60) and GM45 (X200, X301, T400, T500, R400, W500, R500)"],"h2_txt":["How to find what EC version you have (i945\/GM45)"],"h3_txt":["Desktops (AMD, Intel, x86)","Servers\/workstations (AMD, x86)","Laptops (ARM)","Laptops (Intel, x86)"],"bold_txt":["Libreboot 20210522 and later","Libreboot 20160907 only","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":407,"text_t":"Libreboot – Hardware compatibility list. Hardware compatibility list. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Supported hardware Desktops (AMD, Intel, x86) Servers\/workstations (AMD, x86) Laptops (ARM) Laptops (Intel, x86) EC update on i945 (X60, T60) and GM45 (X200, X301, T400, T500, R400, W500, R500) How to find what EC version you have (i945\/GM45) Return to previous index This sections relates to known hardware compatibility in libreboot. For installation instructions, refer to. ..\/install\/ . NOTE: For T60\/R60 thinkpads, make sure that it has an Intel GPU, not an ATI GPU because coreboot lacks native video initialization for the ATI GPUs on these machines. (for later machines like T500, T400, ATI GPU doesn’t matter, because it also has an Intel GPU, and Libreboot uses the Intel one). Supported hardware. [link] Libreboot supports the following systems in this release:. Desktops (AMD, Intel, x86). [link] Gigabyte GA-G41M-ES2L motherboard Acer G43T-AM3 (. Libreboot 20210522 and later. ). Intel D510MO and D410PT motherboards Intel D945GCLF (D945GCLF2D also reported working by a user). Apple iMac 5,2 Servers\/workstations (AMD, x86). [link] ASUS KCMA-D8 motherboard ASUS KGPE-D16 motherboard ASUS KFSN4-DRE motherboard Laptops (ARM). [link] ASUS Chromebook C201 (. Libreboot 20160907 only. ). Laptops (Intel, x86). [link] ThinkPad X60 \/ X60S \/ X60 Tablet. ThinkPad T60 (with Intel GPU). Lenovo ThinkPad X200 \/ X200S \/ X200 Tablet Lenovo ThinkPad X301 (. Libreboot 20210522 and later. ). Lenovo ThinkPad R400 Lenovo ThinkPad T400 \/ T400S Lenovo ThinkPad T500 Lenovo ThinkPad W500 Lenovo ThinkPad R500 Apple MacBook1,1 and MacBook2,1 ‘Supported’ means that the build scripts know how to build ROM images for these systems, and that the systems have been tested (confirmed working). There may be exceptions; in other words, this is a list of ‘officially’ supported systems. EC update on i945 (X60, T60) and GM45 (X200, X301, T400, T500, R400, W500, R500). [link] It is recommended that you update to the latest EC firmware version. The. EC firmware is separate from libreboot, so we don’t actually provide that, but if you still have Lenovo BIOS then you can just run the Lenovo BIOS update utility, which will update both the BIOS and EC version. See:. ..\/install\/#flashrom http:\/\/www.thinkwiki.org\/wiki\/BIOS_update_without_optical_disk NOTE: this can only be done when you are using Lenovo BIOS. How to update the EC firmware while running libreboot is unknown. Libreboot only replaces the BIOS firmware, not EC. Updated EC firmware has several advantages e.g. better battery handling. How to find what EC version you have (i945\/GM45). [link] In GNU+Linux, you can try this:. grep 'at EC' \/proc\/asound\/cards. Sample output:. ThinkPad Console Audio Control at EC reg 0x30, fw 7WHT19WW-3.6. 7WHT19WW is the version in different notation, use search engine to find out regular version - in this case it’s a 1.06 for x200 tablet. Binary blob policy Edit this page Who develops Libreboot? 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If flashrom is complaining about \/dev\/mem , it means that you have CONFIG_STRICT_DEVMEM enabled in your kernel. Reboot with the following kernel parameter added in your bootloader: iomem=relaxed and try again with the above instructions. DO NOT continue until the above works, and you see the expected flashrom output as indicated above.","can","other","2","TS","Top Swap","BUC","Backup Control","should","I forgot to actually add the flashrom patches in the Libreboot 20211122 release. When you see the notes below about _sst and _mx , for now just use the util archive from Libreboot 20160907. That release has a utils archive with pre-compiled flashrom binaries, including patches binaries for Macronix and SST flash chips on these machines. Bucts is also included, pre compiled. They are statically linked binaries, so they should work on any distro. 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Installation instructions. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. About ROM image file names Init types and display mode libgfxinit corebootfb (libgfxinit) txtmode (libgfxinit) vgarom vesafb (vgarom) txtmode (vgarom) normal Payload names grub seabios seabios_withgrub seabios_grubfirst Which systems are supported? MAC address on GM45+ICH9M hardware (ThinkPad X200\/R400\/T400\/T500\/W500) Install via host CPU (internal flashing) Run flashrom on host CPU Flash chip size Howto: read\/write\/erase the boot flash (PLEASE CHECK LIST OF EXCEPTIONS BELOW BEFORE YOU ATTEMPT THIS!!!!) Exceptions If your boot flash is currently write-protected ASUS Chromebook C201 (Libreboot 20160907 only) Lenovo ThinkPad X200\/X200S\/X200T\/T400\/T400S\/T500\/W500\/R400\/R500 running non-free Lenovo BIOS Intel D510MO and D410PT running non-free Intel BIOS Gigabyte GA-G41M-ES2l (any firmware) Macbook1,1 running non-free Apple EFI firmware ASUS KFSN4-DRE? ASUS KGPE-D16 running non-free ASUS BIOS ASUS KCMA-D8 running non-free ASUS BIOS ASUS D945GCLF running non-free Intel BIOS ThinkPad X60\/X60S\/X60T\/T60 with Lenovo BIOS Install using external flashing equipment How to use external flashing equipment ASUS KFSN4-DRE TARGET: Apple Macbook2,1, Macbook1,1 and iMac5,2 (i945 platform) TARGET: Gigabyte GA-G41M-ES2L mainboard TARGET: Intel D510MO and D410PT mainboards TARGET: Intel D945GCLF mainboard TARGET: ASUS KGPE-D16 mainboard TARGET: ASUS KCMA-D8 mainboard TARGET: ASUS Chromebook C201 laptop TARGET: Lenovo ThinkPad X60 laptop TARGET: Lenovo ThinkPad X60 Tablet laptop TARGET: Lenovo ThinkPad T60 laptop TARGET: Lenovo ThinkPad X200 laptop TARGET: Lenovo ThinkPad X200S or X200 Tablet laptop TARGET: Lenovo ThinkPad T400 laptop TARGET: Lenovo ThinkPad T400S laptop TARGET: Lenovo ThinkPad R400 laptop TARGET: Lenovo ThinkPad T500 or W500 laptop TARGET: Lenovo ThinkPad R500 laptop Return to previous index This section relates to installing Libreboot on supported targets. NOTE: if running. flashrom -p internal. for software based flashing, and you get an error related to. \/dev\/mem. access, you should reboot with. iomem=relaxed. kernel parameter before running flashrom, or use a kernel that has. CONFIG_STRICT_DEVMEM. not enabled. Libreboot flashing can be risky business. Please ensure that you have external flashing equipment, in case anything goes wrong. The general rule of thumb with firmware is this: if it’s non-free, replace it, but if you’re already running free firmware and it works nicely for you, you do not need to update it. However, you might want to tweak it or try out newer releases of Libreboot if they have bug fixes for your board, and\/or new security fixes. If you’re already running libre firmware on your board, you should decide for sure whether you wish to risk it. See changelogs on the. release announcements via the news page and decide for yourself. About ROM image file names. [link] Init types and display mode. [link] NOTE: On Libreboot 20211122,. libgfxinit. in the only init type provided on the pre-compiled ROM images, but the build system does support other types defined below. NOTE: regardless of init type, on desktops, an external\/add-on GPU can always be used. On laptop hardware in Libreboot, libgfxinit will always be used. On desktop\/server hardware, if available, libgfxinit will also always be used by default (but in that setup, SeaBIOS can be used if you want to use an add-on graphics card, e.g. on KCMA-D8, KGPE-D16, GA-G41M-ES2L). This means that on desktop hardware such as KCMA-D8, KGPE-D16, G43T-AM3, GA-G41M-ES2L and others, you can use either the internal GPU or an add-on PCI-E graphics card. Simply use a ROM image that starts with SeaBIOS, and you can use both. On desktop\/server hardware, libgfxinit simply means that you CAN use the internal graphics chip, but you don’t have to; external add-on GPUs will also still work! However, if libgfxinit is enabled, that disables coreboot from loading\/executing PCI option ROMs which means you MUST use SeaBIOS if you wish to use the add-on cards! libgfxinit. [link] In this setup, on supported systems, coreboot’s own native video initialization code is used. This is referred to generically as libgfxinit, which is coreboot’s library in. 3rdparty\/libgfxinit. but not all boards with native video initialization use libgfxinit; some of them are using coreboot’s older style of video initialization method, written purely in C. corebootfb (libgfxinit). high resolution coreboot framebuffer used on startup. txtmode (libgfxinit). int10h text mode is used on startup. vgarom. [link] NOTE: no configs in libreboot are currently available that use this method. With this method, coreboot is finding, loading and executing a VGA option ROM for your graphics hardware. This would not be done on laptops, because that implies supplying non-free binary blobs in Libreboot, so this setup would only ever be provided on desktop hardware where no GPU exists or where it is desirable for you to use an external\/add-on graphics card. vesafb (vgarom). high resolution VESA framebuffer used on startup. txtmode (vgarom). int10h text mode is used on startup. normal. [link] int10h text mode startup is implied here. In this setup, coreboot is neither implementing libgfxinit \/ native graphics initialization nor is it finding\/loading\/executing VGA option ROMs. In this setup, SeaBIOS would most likely be used for that. The. normal. setup is supported in the Libreboot 20211122 build system, but not currently used. It is there for desktop hardware that will be added in the future, where those desktop boards do not have an onboard GPU and therefore an add-on GPU is always used.. Payload names. [link] grub. [link] ROM images with just. grub. in the file name will start first with the GNU GRUB payload. They may or may not also provide other payloads in the menu, such as memtest86+, SeaBIOS, Tianacore and so on. seabios. [link] ROM images with just. seabios. in the file name will start first with the SeaBIOS payload. They will only contain SeaBIOS, but may also contain memtest as an option in the boot menu. seabios_withgrub. [link] ROM images that have. seabios_withgrub. in the file name start with SeaBIOS first, but also have GNU GRUB available in the boot menu when you press ESC. seabios_grubfirst. [link] ROM images that have. seabios_grubfirst. in the file name start with SeaBIOS, but SeaBIOS is configured via special. bootorder. file in CBFS so as to ONLY load GNU GRUB. This setup would be most useful on desktops, where you wish to only have GNU GRUB available, but want to use an add-on GPU while also having the option to use libgfxinit, if a supported GPU\/framebuffer chip is present on your board. Which systems are supported? [link] Refer to the hardware compatibility page MAC address on GM45+ICH9M hardware (ThinkPad X200\/R400\/T400\/T500\/W500). [link] The MAC address is stored in a region of the boot flashed called. GbE NVM. which is short for. gigabit ethernet non-volatile memory. . Refer to the following article:. ich9utils documentation Libreboot puts a default MAC address in the available ROM images, but this is a generic MAC address and it’s identical on every ROM image. Technically, you can use it but if you encounter other Libreboot users on the same ethernet switch, using the same physical network as you, you will encounter a MAC address conflict. NOTE: R500 thinkpads do not have an Intel gigabit ethernet NIC, so on that laptop you can just flash the default ROM and you do not have to worry. There are also some Intel X4X platforms that use an ICH10 southbridge, supported in Libreboot, but these are flashed in a. descriptorless. setup, which means that the MAC address is irrelevant (either there will be an Intel PHY module that is now unusable, and you use an add-on card, or it doesn’t use an Intel PHY module and the onboard NIC is usable). Install via host CPU (internal flashing). [link] On all mainboards is a built-in programmer, which can read, erase and rewrite the boot flash. However, it is not always usable by default. For example, it may be configured to restrict write privileges by the host CPU. In some situations, the host CPU can rewrite\/erase\/dump the boot flash. This is called. internal flashing. . This means that you will run software, namely. flashrom. , to read\/erase\/write the contents of the boot flash from a running operating system on the target device. NOTE: please also read the sections further down this page. On some systems, external flashing is required. This means that you power the system down and use a special tool that connects to and reprograms the boot flash. NOTE: in some cases, external flashing is possible but special steps are required. This depends on your mainboard. Again, please read this page carefully. Run flashrom on host CPU. [link] You can simply take any ROM image from the Libreboot project, and flash it. Boot a GNU+Linux distribution on the target device, and install flashrom. In some cases, this is not possible or there are other considerations. Please read this section. carefully. . Flash chip size. [link] Use this to find out:. flashrom -p internal. In the output will be information pertaining to your boot flash. Howto: read\/write\/erase the boot flash (PLEASE CHECK LIST OF EXCEPTIONS BELOW BEFORE YOU ATTEMPT THIS!!!!). [link] How to read the current chip contents:. sudo flashrom -p internal:laptop=force_I_want_a_brick,boardmismatch=force -r dump.bin. You should still make several dumps, even if you’re flashing internally, to ensure that you get the same checksums. Check each dump using. sha1sum. How to erase and rewrite the chip contents:. sudo flashrom -p internal:laptop=force_I_want_a_brick,boardmismatch=force -w libreboot.rom. If you are re-flashing a GM45+ICH9M laptop (e.g. ThinkPad X200\/X200S\/X200T, T400, T500, R400, W500 etc - but not R500), you should run the ich9gen utility to preserve your mac address. Please read the ich9utils documentation:. \/docs\/install\/ich9utils.html NOTE:. force_I_want_a_brick. is not scary. Do not be scared! This merely disables the safety checks in flashrom. Flashrom and coreboot change a lot, over the years, and sometimes it’s necessary to use this option. If you’re scared, then just follow the above instructions, but remove that option. So, just use. -p internal. . If that doesn’t work, next try. -p internal:boardmismatch=force. . If that doesn’t work, try. -p internal:boardmismatch=force,laptop=force_I_want_a_brick. . So long as you. ensure. you’re using the correct ROM for your machine, it will be safe to run flashrom. These extra options just disable the safetyl checks in flashrom. There is nothing to worry about. If successful, it will either say. VERIFIED. or it will say that the chip contents are identical to the requested image. NOTE: there are exceptions where the above is not possible. Read about them in the sections below:. Exceptions. [link] If your boot flash is currently write-protected. You must flash it externally ASUS Chromebook C201 (Libreboot 20160907 only). Ignore this section. Instead, refer to the following guide:. ASUS Chromebook C201 installation guide Lenovo ThinkPad X200\/X200S\/X200T\/T400\/T400S\/T500\/W500\/R400\/R500 running non-free Lenovo BIOS. If you’re running one of these, it cannot be flashed internally if you’re still running the non-free Lenovo BIOS firmware. You must flash it externally See notes further down on this page. We have guides for specific thinkpads, related to disassembly and reassembly so that you can access the flash. Please also see notes about the built-in MAC address inside the boot flash, for the onboard NIC (ethernet one); not relevant on R500, which doesn’t use an Intel NIC. Intel D510MO and D410PT running non-free Intel BIOS. You must flash it externally D410PT is more or less the same board as D510MO, but we would like more info about this board. If you have a D410PT mainboard, please contact the Libreboot project via IRC and ping. leah. before you flash it. When you do so, please reference this paragraph on this web page. Gigabyte GA-G41M-ES2l (any firmware). Ignore this section. Internal flashing. is. possible, but there are two chips and you must flash both chips. Refer to the guide:. Gigabyte GA-G41M-ES2L installation guide Macbook1,1 running non-free Apple EFI firmware. This laptop requires external flashing. Remove the mainboard and refer to the. external flashing guide ; if Libreboot is already running, you can flash internally. MacBook2,1 can be flashed internally. ASUS KFSN4-DRE? Simply boot GNU+Linux with the default vendor firmware, and flash it internally, but before you do: take a push pin, remove the metal pin, and superglue the plastic part to the chip. Then remove the chip after you booting your GNU+Linux system. Install a new chip, and flash. that. . This board uses LPC flash in a PLCC32 socket. This coreboot page shows an example of the push pin as a proof of concept:. http:\/\/www.coreboot.org\/Developer_Manual\/Tools#Chip_removal_tools ASUS KGPE-D16 running non-free ASUS BIOS. You must flash it externally ASUS KCMA-D8 running non-free ASUS BIOS. You must flash it externally ASUS D945GCLF running non-free Intel BIOS. You must flash it externally ThinkPad X60\/X60S\/X60T\/T60 with Lenovo BIOS. I forgot to actually add the flashrom patches in the Libreboot 20211122 release. When you see the notes below about. _sst. and. _mx. , for now just use the. util. archive from Libreboot 20160907. That release has a utils archive with pre-compiled flashrom binaries, including patches binaries for Macronix and SST flash chips on these machines. Bucts is also included, pre compiled. They are statically linked binaries, so they should work on any distro. Use those binaries, but with the ROM images from the Libreboot 20211122 release! Here are a list of targets:. ThinkPad X60\/X60S\/X60T: flash the X60 ROM. ThinkPad T60 with Intel GPU: flash the T60 ROM. ThinkPad T60 with ATI GPU: flash the Headless T60 ROM (no video init, but you can get a serial console on the RS232 port if you use the Advanced Dock or Advanced Mini Dock. Connect to it from another machine, using null modem cable and USB serial adapter; GNU Screen can connect to the serial console and you will run it at 115200 baud rate. agetty\/fgetty in GNU+Linux can give you a serial console in your OS). Download and build flashrom, using the instructions on. the Git page , and download the. bucts. software using the notes on that very same page. You can replace Lenovo BIOS with Libreboot, using flashrom running on the host CPU. However, there are some considerations. Firstly, make sure that the yellow CMOS battery is installed, and functioning correctly. You could check the voltage. The battery is a CR2032 coin cell and it. should. be providing a 3V signal. You should check this while it is connected to the board, because this will give a more accurate reading (if the battery is weak, it will have severe voltage drop when there is any load on it, which there will be. This coincell powers the real-time clock and CMOS memory). Lenovo BIOS restricts write access, but there is a weakness in it. With a specially patched flashrom binary, you can easily flash it but the top 64KiB region of the boot flash, containing your bootblock, cannot be flashed just yet. However, there is a register called the. Backup Control. or. BUC. register and in that register is a status bit called. Top Swap. or. TS. . There are. 2. bootblocks possible. The. other. bootblock is below the upper 64KiB one, which can’t be flashed, but the lower one can. By using bucts, you can set the machine to boot using that lower 64KiB bootblock, which is read-write. You do this by setting the BUC.TS register to 1, using the. bucts. program referenced below. Libreboot ROM images already have the upper 64KiB bootblock copied to the lower one, so you don’t have to worry about copying it yourself. If you build flashrom using the libreboot build system, there will be three binaries:. flashrom. flashrom_i945_sst. flashrom_i945_mx. It’s these last two binaries that you should use. Now compile bucts (just run. make. in the bucts source directory). Run the bucts tool:. sudo .\/bucts 1. Ensure that your CMOS battery is connected too. Now you must determine whether you have Macronix or SST. An X60\/T60 thinkpad will have either an SST or a Macronix chip. The Macronix chip will have “MX” written on the chip. You will use. flashrom_i945_sst. for the SST chip, and. flashrom_i945_mx. for the Macronix chip. Now run flashrom (for SST):. sudo .\/flashrom_i945_sst -p internal -w coreboot.rom. Or Macronix:. sudo .\/flashrom_i945_mx -p internal -w coreboot.rom. NOTE: you. can. just run both. One of them will succeed. It is perfectly harmless to run both versions of flashrom. In fact, you should do so! You’ll see a lot of errors. This is normal. You should see something like:. Reading old flash chip contents... done. Erasing and writing flash chip... spi_block_erase_20 failed during command execution at address 0x0 Reading current flash chip contents... done. Looking for another erase function. spi_block_erase_52 failed during command execution at address 0x0 Reading current flash chip contents... done. Looking for another erase function. Transaction error! spi_block_erase_d8 failed during command execution at address 0x1f0000 Reading current flash chip contents... done. Looking for another erase function. spi_chip_erase_60 failed during command execution Reading current flash chip contents... done. Looking for another erase function. spi_chip_erase_c7 failed during command execution Looking for another erase function. No usable erase functions left. FAILED! Uh oh. Erase\/write failed. Checking if anything has changed. Reading current flash chip contents... done. Apparently at least some data has changed. Your flash chip is in an unknown state. If you see this, rejoice! It means that the flash was successful. Please do not panic. Shut down now, and wait a few seconds, then turn back on again. WARNING: if flashrom complains about. \/dev\/mem. access, please run. sudo .\/bucts 0. . If flashrom is complaining about. \/dev\/mem. , it means that you have. CONFIG_STRICT_DEVMEM. enabled in your kernel. Reboot with the following kernel parameter added in your bootloader:. iomem=relaxed. and try again with the above instructions. DO NOT continue until the above works, and you see the expected flashrom output as indicated above. If you. did. run flashrom and it failed to flash, but you set bucts to 1 and shut down, don’t worry. Just remove the yellow coin-cell battery (it’s underneath the keyboard, connected to the mainboard), wait a minute or two, reconnect the coin-cell and try again from scratch. In this instance, if flashrom didn’t do anything, and didn’t flash anything, it means you still have Lenovo BIOS but if bucts is set to 1, you can flush it and set it back to 0. BUC.TS is stored in volatile memory, powered by that CR2032 coin-cell battery. Assuming that everything went well:. Flash the ROM for a second time. For this second flashing attempt, the upper 64KiB bootblock is now read-write. Use the. unpatched. flashrom binary:. sudo .\/flashrom -p internal -w libreboot.rom. To reset bucts, do this:. sudo .\/bucts 0. ONLY set bucts back to 0 if you’re sure that the upper 64KiB bootblock is flashed. It is flashed if flashrom said VERIFIED when running the above command. If it said VERIFIED, shut down. If it didn’t say VERIFIED, make sure bucts is still set to 1, and consult the libreboot project on IRC for advice, and avoid shutting down your system until you get help. If all went well, Libreboot should now be booting and you should be able to boot into your operating system. If you messed up, there are external flashing instructions. See main navigation menu on this page. These “external” instructions teach you how to flash externally, using special equipment (requires disassembling your laptop and removing the mainboard). Install using external flashing equipment. [link] In many situations, the host CPU is restricted from rewriting\/erasing\/dumping the boot flash. In this situations, you must re-flash the chip (containing the boot firmware) externally. This is called. external flashing. . DO NOT buy CH341A! Read the above link, which explains why you shouldn’t use it. CH341A will damage your flash chip, and other components on your mainboard. How to use external flashing equipment. [link] Refer to the following article:. Externally rewrite 25xx NOR flash via SPI protocol ASUS KFSN4-DRE. [link] The KFSN4-DRE has an LPC chip. Most people have been flashing these internally, hot-swapping the chip out after boot, preserving the original chip, and using flashrom on a new chip as described above. TODO: Document PLCC32 (LPC) flashing. The. FlexyICE has been used to flash these chips, but it is hard to find now. A custom flasher may be made such as. flashrom serprog stm32 or. teensy flasher TARGET: Apple Macbook2,1, Macbook1,1 and iMac5,2 (i945 platform). [link] iMac5,2 is essentially the same board as Macbook2,1, and it is compatible with Libreboot. Refer to the following article:. Macbook2,1 and MacBook1,1 installation guide iMac5,2 isn’t documented but you can find the flash chip on that board quite easily. See the generic flashing guide:. Externally rewrite 25xx NOR flash via SPI protocol TARGET: Gigabyte GA-G41M-ES2L mainboard. [link] Refer to the following article:. Gigabyte GA-G41M-ES2L TARGET: Intel D510MO and D410PT mainboards. [link] Refer to the following article:. Intel D510MO and D410PT boards TARGET: Intel D945GCLF mainboard. [link] Refer to the following article:. Intel D945GCLF TARGET: ASUS KGPE-D16 mainboard. [link] Refer to the following article:. ASUS KGPE-D16 TARGET: ASUS KCMA-D8 mainboard. [link] Refer to the following article:. ASUS KCMA-D8 TARGET: ASUS Chromebook C201 laptop. [link] Refer to the following article:. ASUS Chromebook C201 TARGET: Lenovo ThinkPad X60 laptop. [link] Refer to the following article:. ThinkPad X60 TARGET: Lenovo ThinkPad X60 Tablet laptop. [link] Refer to the following article:. ThinkPad X60 Tablet TARGET: Lenovo ThinkPad T60 laptop. [link] Refer to the following article:. ThinkPad T60 TARGET: Lenovo ThinkPad X200 laptop. [link] Refer to the following article:. ThinkPad X200 TARGET: Lenovo ThinkPad X200S or X200 Tablet laptop. [link] Software-wise, identical to regular X200 but SMD rework skills are required. You must de-solder the default flash chip, and replace it with another one. Refer to the following article:. 25xx NOR flashing guide That guide, linked above, has instructions for how to deal with these machines. TARGET: Lenovo ThinkPad T400 laptop. [link] Refer to the following article:. ThinkPad T400 TARGET: Lenovo ThinkPad T400S laptop. [link] Software-wise, identical to regular T400 but SMD rework skills are required. You must de-solder the default flash chip, and replace it with another one. Refer to the following article:. 25xx NOR flashing guide TARGET: Lenovo ThinkPad R400 laptop. [link] Refer to the following article:. ThinkPad R400 TARGET: Lenovo ThinkPad T500 or W500 laptop. [link] These two laptops have identical mainboard, except for a few minor changes. Refer to the following article:. ThinkPad T500\/W500 TARGET: Lenovo ThinkPad R500 laptop. [link] Refer to the following laptop:. ThinkPad R500 Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/index.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"xDOjF2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/bsd\/","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":31,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","bsd"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Libreboot – BSD operating systems"],"last_modified":"Sat Apr 02 12:48:33 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-6606567722245336424,"exact_signature_unique_b":true,"fuzzy_signature_l":4449757075727079318,"fuzzy_signature_unique_b":true,"h1_txt":["BSD operating systems","Navigate this page:","Video modes","Booting BSD"],"h2_txt":["SeaBIOS payload","GNU GRUB payload"],"bold_txt":["can","especially","bare metal","also","either","just work","need","kernel mode setting","text mode","Free Software","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":575,"text_t":"Libreboot – BSD operating systems. BSD operating systems. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Video modes Booting BSD SeaBIOS payload GNU GRUB payload Return to previous index This section is largely x86-centric, pertaining to use of BSD operating systems. Although not as popular, BSD systems are also (in most cases). Free Software. , but they are non-copyleft. Libreboot is capable of booting many BSD systems. This section mostly documents the peculiarities of Libreboot as it pertains to BSD; you can otherwise refer to the official documentation for whatever BSD system you would like to use. Video modes. [link] For BSD systems, it is desirable that you boot in. text mode. . ROM images with. txtmode. in the file name, on x86 systems, boot up with int10h text mode in use. This is the most “compatible” option, and BSD operating systems have excellent support for text-mode startup. Many of them also support. kernel mode setting. (KMS) nowadays, which you. need. if you want a graphical desktop on the X window system. The reason is that Libreboot does not currently implement int10h VGA modes on x86 systems. However, basic video initialization is provided on all platforms (int10h text mode, or coreboot framebuffer). Combined with the use of SeaBIOS payload, BSD systems (and any other OS that can boot in text mode) will. just work. . If your BSD system supports kernel mode setting, it can set up a framebuffer without making use of int10h VGA modes. In this case, the driver (e.g. Intel video driver) will set modes directly, and implement its own framebuffer. Booting with a coreboot framebuffer will also work well on most BSD systems. These ROM images have. corebootfb. in the filename, on recent Libreboot releases. In this setup, you should make sure that your BSD system has a. corebootfb. driver (to make use of the coreboot framebuffer), but when switching to X, your video driver (e.g. Intel video driver) may already support kernel mode setting which means that the coreboot framebuffer will no longer be used at that point. Booting BSD. [link] On x86 platforms, Libreboot currently provides the choice of GNU GRUB and\/or SeaBIOS payload. You can use. either. payload, to boot BSD operating systems. SeaBIOS payload. [link] It is highly recommended that you use the SeaBIOS payload. ROM images are available in the latest Libreboot release, which start with the SeaBIOS payload. The ROM images with GNU GRUB. also. have SeaBIOS available in the boot menu. GNU GRUB, when compiled as a coreboot payload, runs on. bare metal. and it can boot any other coreboot payload if you use the. chainloader. command. The way to use SeaBIOS is fairly self-explanatory. SeaBIOS functions the way you would expect on a typical computer. Libreboot currently lacks any sort of documentation for SeaBIOS, but you can refer to their website:. https:\/\/seabios.org\/SeaBIOS SeaBIOS is. especially. recommended if you’re doing an encrypted installation. The benefit to using SeaBIOS is that it’s basically more reliable. For example, ZFS support is less reliable in GRUB, but a FreeBSD system booted in SeaBIOS would work just fine because you’d be using FreeBSD’s own bootloader in that instance. GNU GRUB payload. [link] GRUB can directly boot many BSD kernels, but support for this is quite unreliable compared to its support for booting Linux kernels. However, you. can. use GRUB. When you use GNU GRUB directly, in this way, the various BSD bootloaders are bypassed entirely. We have separate pages for each BSD system:. How to install NetBSD on x86 GNU GRUB payload How to install OpenBSD on x86 GNU GRUB payload How to install FreeBSD on a x86 GNU GRUB payload Binary blob policy Edit this page Who develops Libreboot? 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{"id":"vBX1T2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/build\/","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":33,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","build"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Libreboot – Build from source"],"last_modified":"Sat Apr 02 12:48:34 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-6272609407285049704,"exact_signature_unique_b":true,"fuzzy_signature_l":-9159277165465662523,"fuzzy_signature_unique_b":true,"h1_txt":["Build from source","Navigate this page:","GNU Make","Build without using GNU Make"],"h2_txt":["First, install build dependencies","Second, download all of the required software components","Third, build all of the modules:","Fourth, build all of the payloads:","Fifth, build the ROMs!"],"bold_txt":["clean","can","still","compatibility","every","snapshots","that","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":635,"wordcount_i":1117,"linkscount_i":22,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/build\/","libreboot.org\/docs\/maintain\/","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/build\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/build\/index.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","GNU Make","Libreboot maintenance manual","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:42 UTC 2022","fresh_date_dt":"Tue Jul 12 03:51:46 UTC 2022","referrer_id_s":"lfDPysQ1CANx","language_s":"en","size_i":12892,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078541994754048,"text_t":"Libreboot – Build from source. Build from source. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. GNU Make Build without using GNU Make First, install build dependencies Second, download all of the required software components Third, build all of the modules: Fourth, build all of the payloads: Fifth, build the ROMs! Return to previous index Libreboot’s build system is named. lbmk. , short for. Libreboot Make. , and this document describes how to use it. With this guide, you can know how to compile Libreboot from the available source code. This version, if hosted live on libreboot.org, assumes that you are using the. lbmk. git repository, which you can download using the instructions on. the code review page . If you’re using a release archive of Libreboot, please refer to the documentation included with. that. release. Libreboot releases are only intended as. snapshots. , not for development. For proper development, you should always be working directly in the Libreboot git repository. The following document describes how. lbmk. works, and how you can make changes to it:. Libreboot maintenance manual GNU Make. [link] Libreboot Make includes a file called. Makefile. . You can still use the. lbmk. build system directly, or you can use GNU Make. The. Makefile. simply runs. lbmk. commands. However, using. lbmk. directly will offer you much more flexibility; for example, the Makefile currently cannot build single ROM images (it just builds all of them, for all boards). You must ensure that all build dependencies are installed. If you’re running Ubuntu or similar distribution (Debian, Trisquel, etc) you can do this:. sudo make install-dependencies-ubuntu. One exists specifically for Debian:. sudo make install-dependencies-debian. Another exists for Arch:. sudo make install-dependencies-arch. Now, simply build the coreboot images like so:. make. This single command will build ROM images for. every. board integrated in Libreboot. If you only wish to build a limited set, you can use. lbmk. directly:. .\/build boot roms x200_8mb. You can specify more than one argument:. .\/build boot roms x200_8mb x60. ROM images appear under the newly created. bin\/. directory in the build system. For other commands, simply read the. Makefile. in your favourite text editor. The. Makefile. is simple, because it merely runs. lbmk. commands, so it’s very easy to know what commands are available by simply reading it. Standard. clean. command available (cleans all modules except. crossgcc. ):. make clean. To clean your. crossgcc. builds:. make crossgcc-clean. To build release archives:. make release. Build without using GNU Make. [link] The. Makefile. is included just for. compatibility. , so that someone who instictively types. make. will get a result. Actual development\/testing is always done using. lbmk. directly, and this includes when building from source. Here are some instructions to get you started:. First, install build dependencies. [link] Libreboot includes a script that automatically installs apt-get dependencies in Ubuntu 20.04. It works well in other apt-get distros (such as Trisquel and Debian):. sudo .\/build dependencies ubuntu2004. Separate scripts also exist:. sudo .\/build dependencies debian sudo .\/build dependencies arch sudo .\/build dependencies void. Technically, any GNU+Linux distribution can be used to build Libreboot. However, you will have to write your own script for installing build dependencies. Libreboot Make (lbmk) automatically runs all necessary commands; for example. .\/build payload grub. will automatically run. .\/build module grub. if the required utilities for GRUB are not built, to produce payloads. As a result, you can now (after installing the correct build dependencies) run just a single command, from a fresh Git clone, to build the ROM images:. .\/build boot roms. or even just build specific ROM images, e.g.:. .\/build boot roms x60. If you wish to build payloads, you can also do that. For example:. .\/build payload grub .\/build payload seabios. Previous steps will be performed automatically. However, you can. still. run individual parts of the build system manually, if you choose. This may be beneficial when you’re making changes, and you wish to test a specific part of lbmk. Therefore, if you only want to build ROM images, just do the above. Otherwise, please continue reading! Second, download all of the required software components. [link] If you didn’t simply run. .\/build boot roms. (with or without extra arguments), you can still perform the rest of the build process manually. Read on! You can read about all available scripts in. lbmk. by reading the. Libreboot maintenance manual ; lbmk is designed to be modular which means that each script. can. be used on its own (if that’s not true, for any script, it’s a bug that should be fixed). It’s as simple as that:. .\/download all. The above command downloads all modules defined in the Libreboot build system. However, you can download modules individually. This command shows you the list of available modules:. .\/download list. Example of downloading an individual module:. .\/download coreboot .\/download seabios .\/download grub .\/download flashrom. Third, build all of the modules:. [link] Building a module means that it needs to have already been downloaded. Currently, the build system does not automatically do pre-requisite steps such as this, so you must verify this yourself. Again, very simple:. .\/build module all. This builds every module defined in the Libreboot build system, but you can build modules individually. The following command lists available modules:. .\/build module list. Example of building specific modules:. .\/build module grub .\/build module seabios .\/build module flashrom. Commands are available to. clean. a module, which basically runs make-clean. You can list these commands:. .\/build clean list. Clean all modules like so:. .\/build clean all. Example of cleaning specific modules:. .\/build clean grub .\/build clean cbutils. Fourth, build all of the payloads:. [link] Very straight forward:. .\/build payload all. You can list available payloads like so:. .\/build payload list. Example of building specific payloads:. .\/build payload grub .\/build payload seabios. The build-payload command is is a prerequsite for building ROM images. Fifth, build the ROMs! [link] Run this command:. .\/build boot roms. Each board has its own configuration in. lbmk. under. resources\/coreboot\/. which specifies which payloads are supported. By default, all ROM images are built, for all boards. If you wish to build just a specific board, you can specify the board name based on the directory name for it under. resources\/coreboot\/. . For example:. .\/build boot roms x60. Board names, like above, are the same as the directory names for each board, under. resources\/coreboot\/. in the build system. That’s it! If all went well, ROM images should be available to you under bin\/. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/build\/index.md Site map This HTML page was generated by the. untitled static site generator ."}
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Code review. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Libreboot repositories lbmk (libreboot-make) lbwww and lbwww-img Name not required Licenses Send patches Return to index Libreboot repositories. [link] Information about who works on Libreboot and who runs the project can be found on. who.html Libreboot has 5 Git repositories:. Build system:. https:\/\/notabug.org\/libreboot\/lbmk Website (+docs):. https:\/\/notabug.org\/libreboot\/lbwww Images (for website):. https:\/\/notabug.org\/libreboot\/lbwww-img Bucts (utility):. https:\/\/notabug.org\/libreboot\/bucts ich9utils (utility):. https:\/\/notabug.org\/libreboot\/ich9utils You can download any of these repositories, make whatever changes you like, and then submit your changes using the instructions below. It is recommended that you build Libreboot (all parts of it) in a GNU+Linux distribution. For example, the build system (lbmk) is untested on BSD systems. Install. git. in your GNU+Linux system, and download one of the repositories. Libreboot development is done using the Git version control system. Refer to the. official Git documentation if you don’t know how to use Git. The. bucts. repository is hosted by the Libreboot project, because the original repository on. stuge.se. is no longer available, last time we checked. The. bucts. program was written by Peter Stuge. You need. bucts. if you’re flashing internally a Libreboot ROM onto a ThinkPad X60 or T60 that is currently running the non-free Lenovo BIOS. Instructions for that are available here:. Libreboot installation guides The. ich9utils. repository is used heavily, by the. lbmk. build system. However, you can also download. ich9utils. on its own and use it. It generates ICH9M descriptor+GbE images for GM45 ThinkPads that use the ICH9M southbridge. It may also work for other systems using the same platform\/chipset. Documentation for. ich9utils. is available here:. ich9utils documentation lbmk (libreboot-make). [link] This is the core build system in Libreboot. You could say that. lbmk. is. Libreboot! Download the Git repository:. git clone https:\/\/notabug.org\/libreboot\/lbmk. The. git. command, seen above, will download the Libreboot build system. lbmk. . You can then go into it like so:. cd lbmk. Make whatever changes you like, or simply build it. For instructions on how to build. lbmk. , refer to the. build instructions . Information about the build system itself, and how it works, is available in the. lbmk maintenance guide . lbwww and lbwww-img. [link] The. entire. Libreboot website and documentation is hosted in a Git repository. Download it like so:. git clone https:\/\/notabug.org\/libreboot\/lbwww. Images are hosted on. https:\/\/av.libreboot.org\/ and available in a separate repository:. git clone https:\/\/notabug.org\/libreboot\/lbwww-img. Make whatever changes you like. See notes below about how to send patches. The entire website is written in Markdown, specifically the Pandoc version of it. The static HTML pages are generated with. Untitled . Leah Rowe, the founder of Libreboot, is also the founder of the Untitled static site generator project. If you like, you can set up a local HTTP server and build your own local version of the website. Please note that images will still link to the ones hosted on. https:\/\/av.libreboot.org\/ , so any images that you add to. lbwww-img. will not show up on your local. lbwww. site if you make the image links (for images that you add) link to av.libreboot.org. However, it is required that such images be hosted on av.libreboot.org. Therefore, if you wish to add images to the website, please also submit to the. lbwww-img. repository, with the links to them being. https:\/\/av.libreboot.org\/path\/to\/your\/new\/image\/in\/lbwww-img for each one. When it is merged on the libreboot website, your images will appear live. For development purposes, you might make your images local links first, and then adjust the URLs when you submit your documentation\/website patches. Instructions are on the Untitled website, for how to set up your local version of the website. Download untitled, and inside your. untitled. directory, create a directory named. www\/. then go inside the www directory, and clone the. lbwww. repository there. Configure your local HTTP server accordingly. Again, instructions are available on the Untitled website for this purpose. Name not required. [link] Contributions that you make are publicly recorded, in a Git repository which everyone can access. This includes the name and email address of the contributor. In Git, for author name and email address, you do not have to use identifying data. You can use Libreboot Contributor and your email address could be specified as contributor@libreboot.org. You are permitted to do this, if you wish to maintain privacy. We believe in privacy. If you choose to remain anonymous, we will honour this. Of course, you can use whichever name and\/or email address you like. Legally speaking, all copyright is automatic under the Berne Convention of international copyright law. It does not matter which name, or indeed whether you even declare a copyright (but we do require that certain copyright licenses are used - read more about that on this same page). If you use a different name and email address on your commits\/patches, then you should be fairly anonymous. Use. git log and. git show to confirm that before you push changes to a public Git repository. Licenses. [link] We require all patches to be submitted under a free license:. https:\/\/www.gnu.org\/licenses\/license-list.html . GNU General Public License v3 is highly recommended. For documentation, we require GNU Free Documentation License v1.3 or higher. Always. declare a license on your work! Not declaring a license means that the default, restrictive copyright laws apply, which would make your work non-free. GNU+Linux is generally recommended as the OS of choice, for Libreboot development. However, BSD operating systems also boot on Libreboot machines. Send patches. [link] Make an account on. https:\/\/notabug.org\/ and navigate (while logged in) to the repository that you wish to work on. Click. Fork. and in your account, you will have your own repository of Libreboot. Clone your repository, make whatever changes you like to it and then push to your repository, in your account on NotABug. You can also do this on a new branch, if you wish. In your Notabug account, you can then navigate to the official Libreboot repository and submit a Pull Request. The way it works is similar to other popular web-based Git platforms that people use these days. You can submit your patches there. Alternative, you can log onto the Libreboot IRC channel and notify the channel of which patches you want reviewed, if you have your own Git repository with the patches. Once you have issued a Pull Request, the Libreboot maintainers will be notified via email. If you do not receive a fast enough response from the project, then you could also notify the project via the #libreboot channel on Libera Chat. Another way to submit patches is to email Leah Rowe directly:. leah@libreboot.org is Leah’s project email address. However, for transparency of the code review process, it’s recommended that you use Notabug, for the time being. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/git.md Subscribe to RSS for this site Site map This HTML page was generated by the. untitled static site generator ."}
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Tasks. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Help the Libreboot project Current tasks (more will be added soon) Move all distro FDE+\/boot\/ guides to distro wiki\/manuals Document other RPi GNU+Linux distros for SPI flashing bug: crossgcc not included in src archive if not already build ThinkPad R60 support Investigate u-boot GRUB: add BLS support Create a board-status repo, like coreboot TODO: i945: test framebuffer(non-i915) init during S3 resume 61a3c8a005 payloads\/tianocore: Add Kconfig to set boot timeout Document the following boards Acer G43T-AM3 Generate ICH10 descriptor\/nvm Re-do desktop boards Add the following boards lenovo\/g505s Intel x4x platform Dell Optiplex 760 asrock\/g41c-gs asus\/p5qc asus\/p5ql-em asus\/p5qpl-am foxconn\/g41s-k intel\/dg41wv intel\/dg43gt lenovo\/thinkcentre_a58 Intel Pineview platform GM45 \/ ICH9M lenovo\/x301 (thinkpad x200 variant) Intel i945 asus\/p5gc-mx getac\/p470 gigabyte\/ga-945gcm-s2l ibase\/mb899 kontron\/986lcd-m roda\/rk886ex AMD Fam10h \/ Fam15h advansus\/a785e-i amd\/bimini_fam10 amd\/mahogany_fam10 amd\/serengeti_cheetah_fam10 amd\/tilapia_fam10 asus\/m4a785-m asus\/m4a785t-m asus\/m4a78-em asus\/m5a88-v avalue\/eax-785e gigabyte\/ma785gm gigabyte\/ma785gmt gigabyte\/ma78gm hp\/dl165_g6_fam10 iei\/kino-780am2-fam10 jetway\/pa78vm5 msi\/ms9652_fam10 supermicro\/h8dmr_fam10 supermicro\/h8qme_fam10 supermicro\/h8scm_fam10 tyan\/s2912_fam10 gm45 thinkpad w700 Linuxboot payload Fork coreboot 4.11 and maintain, for fam10h\/15h boards Test SeaBIOS option: etc\/usb-time-sigatt SST+macronix patches for flashrom on X60\/T60 Warnings about option ROMs Use coreboot’s memtest86+ fork Gemini site for libreboot Tor site for libreboot 2nd HTTP site i2p site Fix GRUB bugs Security patch: spectre MSR fixes for Fam15h boards Document teensy SPI flasher TODO: document use of schottky diode for VCC on SPI flash (ISP) Document alternative external flashing method for X200S\/X200T Handle SATA power in ultrabay on gm45 thinkpads document serial\/lpt\/pcie bus enable\/disable on GA-G41M-ES2L Document quad-core mods on GM45 thinkpads Return to previous index Help the Libreboot project. [link] This page is very new. It’s intended to serve those who ask: what can I do to help Libreboot? You could try implementing some of the tasks listed on this page or you could submit new tasks to this page! Current tasks (more will be added soon). [link] Move all distro FDE+\/boot\/ guides to distro wiki\/manuals. [link] The Guix, Fedora, Parabola and Trisquel guides were outdated and therefore deleted. The Debian guide should also be deleted, even though it’s up to date. The hyperbola one is actually a link to a guide on the Hyperbola site. These are guides for fully encrypted GNU+Linux systems, including \/boot, but it’s desirable for these to be documented instead by each distro, because then they will more likely be properly maintained. We constantly have to update them, on libreboot.org. It is unsustainable. Move them to other projects and let them deal with it. Libreboot’s only job is to boot you into a payload. The rest is up to you! This concerns GRUB payload on x86 targets. For SeaBIOS, it’s fairly easy to just push a button and the distro installer boots, or the installed distro just boots up as normal. Here are links to the guides that were deleted:. https:\/\/notabug.org\/libreboot\/lbwww\/src\/8844c201ef0d1ab856fed2aa5148b89100fffe0d\/site\/docs\/gnulinux\/guix_system.md https:\/\/notabug.org\/libreboot\/lbwww\/src\/8844c201ef0d1ab856fed2aa5148b89100fffe0d\/site\/docs\/gnulinux\/encrypted_trisquel.md https:\/\/notabug.org\/libreboot\/lbwww\/src\/8844c201ef0d1ab856fed2aa5148b89100fffe0d\/site\/docs\/gnulinux\/encrypted_parabola.md https:\/\/notabug.org\/libreboot\/lbwww\/src\/8844c201ef0d1ab856fed2aa5148b89100fffe0d\/site\/docs\/gnulinux\/configuring_parabola.md The Trisquel one will be almost identical to the Debian one, with perhaps a few extra considerations taken. It’s recommended to focus on Debian first, and then adapt that to Trisquel. However, Trisquel is based on Ubuntu, so the guide can also be adapted for the Ubuntu site. This will cover most Ubuntu and Debian based distros. The remaining Debian guide is here:. https:\/\/notabug.org\/libreboot\/lbwww\/src\/8844c201ef0d1ab856fed2aa5148b89100fffe0d\/site\/docs\/gnulinux\/encrypted_debian.md Document other RPi GNU+Linux distros for SPI flashing. [link] See:. ..\/docs\/install\/spi.html#caution-about-rpi RPi’s default distro, Raspbian, no longer can be trusted to be secure. TODO: document how to use other distros, to configure the RPi for SPI flashing. bug: crossgcc not included in src archive if not already build. [link] fix this. in practise, i always build the roms and then run the release scripts which means crossgcc will have been built, but this bug should still be fixed. this is so that you can simply run the release build scripts right after downloading the git repository. ThinkPad R60 support. [link] macc24 on IRC ported it. add it! Investigate u-boot. [link] e.g. Pine64 ROCKPro64, which was added in coreboot 4.14 but it’s also supported by uboot. Lots of ARM hardware supported in coreboot, and lots of non-coreboot hardware out there with free firmware, but using uboot (not coreboot). Pinebook computers look interesting:. Some of their computers look like they will be suitable for Libreboot, but they are ARM and most of them don’t have coreboot support (instead, they use uboot exclusively). GRUB: add BLS support. [link] Resources:. The systemd’s Boot Loader Specification The freedesktop.org’s Boot Loader Specification systemd-boot - uefi app. Create a board-status repo, like coreboot. [link] See:. https:\/\/review.coreboot.org\/plugins\/gitiles\/board-status\/ For testing boards in Libreboot (and osboot-libre), it would be nice to have reports like on coreboot board-status entries. This is especially important. now. , because lots of boards are being added to both Libreboot and osboot-libre. It will. especially. be important for osboot-libre, after the Libreboot release, because osboot-libre will start to focus on being a rolling release, bleeding edge coreboot distro, while Libreboot focuses on stable release. board-status. entries like these will be invaluable to both projects. TODO: i945: test framebuffer(non-i915) init during S3 resume. [link] See notes here:. https:\/\/doc.coreboot.org\/releases\/coreboot-4.8.1-relnotes.html video init is skipped on i945 now, during S3 resume, to save time, and the i915 linux kernel driver can handle that, but other drivers should be tested. e.g. generic corebootfb driver, drivers in various BSD systems, etc. 61a3c8a005 payloads\/tianocore: Add Kconfig to set boot timeout. [link] this is from the coreboot git log. looks interesting. investigate. Document the following boards. [link] These boards are added, but not documented yet. Acer G43T-AM3. [link] See:. flashrom_read_me_disable.log.txt This is from Michael Büchler, whom I emailed, asking for info about this board. This is the person who ported the board to coreboot. Michael states the following:. I'm also attaching a flashrom read log. The filename suggests that I had the ME disable pin set.. so this was with the \"-p internal\" flasher, but the SPI_ROM1 header also works. The pinout is 1:1 the same as the EEPROM. This reminds me that I wanted to create a page for this board on the coreboot documentation. There you would have found this info. I should do it soon. ME disable pin? Probably setting GPIO33 or something. I’ve replied to Michael, encouraging that person to document this board on the coreboot website. indeed, next to the southbridge is a jumper and the silk screen says “ME disable” so I’m guessing this is actually just GPIO33 being grounded. so it’s not simply disabling the ME, but the intel flash descriptor (which also disables NVM, not just the ME). Here are some photos: TODO: add photos that michael sent me. i’m waiting for michael to confirm what license. for now see these photos that i pulled from a search engine:. https:\/\/i5.walmartimages.com\/asr\/7ded9e88-73e6-4bc4-9b2a-ff22313c7172_2.9abea30734ddf03fc15b7188cb3e92cd.jpeg For flashing instructions:. Refer to. https:\/\/av.libreboot.org\/g43t-am3\/soic8.jpg - a proper photo is not available under a free license, or could not be found, so this diagram was made. NOTE: It might not be possible to do ISP flashing. Several other X4X desktop mainboards are problematic. FOR EXAMPLE:. https:\/\/doc.coreboot.org\/mainboard\/intel\/dg43gt.html That’s another X4X board, and it recommends to de-solder the flash. It might be that this board, linked above, can be flashed ISP-style, but the person who wrote that page was using a 3.3V rail from a flasher like RPi or whatever, and maybe the flash chip shares a common rail with the southbridge or something else that draws a lot of current. On GA-G41M-ES2L, it’s possible to power on the board, then turn it off but leave it plugged in, and a 3.3V rail from the ATX PSU will be active, powering the chip and providing more than enough current. In that situation, you connect your SPI flasher without using your SPI flasher’s 3.3V rail. That may also be the case on this board, and the one linked dabove. I couldn’t find exact schematics\/boardviews, but I did find this:. http:\/\/download.ecs.com.cn\/dlfileecs\/manual\/mb\/eng\/p4\/G43T_MV10\/G43T-M_V20.pdf 2MiB flash chip according to:. https:\/\/review.coreboot.org\/plugins\/gitiles\/board-status\/+\/refs\/heads\/master\/acer\/g43t-am3\/4.12-4089-gb7e591e2da\/2020-11-17T18_20_46Z\/config.txt and. https:\/\/review.coreboot.org\/plugins\/gitiles\/board-status\/+\/refs\/heads\/master\/acer\/g43t-am3\/4.12-3211-gfb623a02c5\/2020-10-11T11_24_19Z\/config.txt NOTE: i think this is ICH10. Kconfig mentions IFD. flash it descriptorless based on intel\/dg43gt port using “motherboard porting guide”. coreboot ba49d859eeaeced032403b2da6a5f34ea2a93a94 added it. The following is from that coreboot revision, in the commit log:. Same board as Aspire M5800 (same vendor BIOS image). Similar mainboards by Acer: G41T-AM, G43T-AM, G43T-AM4, Q45T-AM, to name a few. ECS has some models that are obiously based on the same design, e.g. G43T-WM and G43T-M. Working (ignore the note about Windows. Libreboot project doesn’t care about that. This is just copied from the coreboot git log):. CPUs from Pentium Dual-Core E2160 to Core 2 Quad Q9550 at FSB1333. Native raminit. All four DIMM slots at 1066 MHz (tested 2x2GB + 2x4GB). PS\/2 mouse. PS\/2 keyboard (needs CONFIG_SEABIOS_PS2_TIMEOUT, tested: 500). USB ports (8 internal, 4 external). All six SATA ports. Intel GbE. Both PCI ports with various cards (Ethernet, audio, USB, VGA). Integrated graphics (libgfxinit). HDMI and VGA ports. boot with PCIe graphics and SeaBIOS. boot with PCI VGA and SeaBIOS. Both PCIe ports. Flashing with flashrom. Rear audio output. SeaBIOS 1.14.0 to boot slackware64. SeaBIOS 1.14.0 to boot Windows 10 (needs VGA BIOS). Temperature readings (including PECI). Super I\/O EC automatic fan control. S3 suspend\/resume. Poweroff. Not working:. Resource issues with the VGA BIOS of a PCI rv100-based card. Super I\/O voltage reading conversions. Untested:. The other audio jacks or the front panel header. On-board Firewire. EHCI debug. VBT (was extracted and added, but don’t know how to test). Super I\/O GPIOs. Generate ICH10 descriptor\/nvm. [link] Coreboot has a few Intel X4X boards with ICH10 southbridge. These can be booted descriptorless, but in some cases those boards will use an Intel gigabit NIC, which means that the NIC will be useless in a descriptorless setup. Basically. ich9utils. but for ich10. However, it’s preferable to generate it using bincfg. Intel provides some limited information about ICH10 descriptors in public datasheets. The rest can be guessed at like it was for ICH9M in libreboot. Re-do desktop boards. [link] Right now, the configs make no sense. VGA ROM setup (for external GPU) also runs libgfxinit, and vice versa, on KCMA-D8 \/ KGPE-D16, and in many configs, both coreboot and seabios run pci roms. There needs to be consistency here. I think there should be separation:. libgfxinit. coreboot doesn’t load pci roms. seabios loads them. vgarom-only setup, where coreboot runs pci roms. seabios doesn’t load them. Add the following boards. [link] NOTE: some of these might not be suitable for Libreboot. Each one will be checked, before adding it to Libreboot. TODO: also check under “variants” for each board, and add more to this list if any are found. These lists are generated by greping Kconfig files but sometimes multiple boards are specified in a single Kconfig file. For example, macbook21 Kconfig also specifies imac52 and macbook11 without any code changes. lenovo\/g505s. [link] Last I checked, video init was a problem on this laptop. (binary blob, but there was some work to implement free video initialization). It might still be worth looking into. Intel x4x platform. [link] NOTE: some use ICH7 southbridge. NOTE: others use ICH10, and some of. those. have Intel ME + descriptor. others have descriptorless setup (no Intel ME). all. of them can boot descriptorless, so it’s possible to nuke the Intel ME on all of them (ICH7 ones never have ME to begin with). NOTE: this is the same platform used by Gigabyte GA-G41M-ES2L which Libreboot already supports. that one uses an ICH7 southbridge. Dell Optiplex 760. vitali64 on IRC is porting this to coreboot, and has it almost fully working. asrock\/g41c-gs. Variants:. g41c-gs-r2. g41m-gs. g41m-s3. g41m-vs3-r2. asus\/p5qc. Variants:. p5q_pro. p5ql_pro. p5q. asus\/p5ql-em. No variants specified in Kconfig. asus\/p5qpl-am. Variants:. p5g41t-m_lx. foxconn\/g41s-k. Variants:. g41m. intel\/dg41wv. No variants specified in Kconfig. intel\/dg43gt. No variants specified in Kconfig. lenovo\/thinkcentre_a58. No variants specified in Kconfig. Intel Pineview platform. [link] NOTE: same platform as Intel D510MO \/ D410PT. foxconn\/d41s. GM45 \/ ICH9M. [link] lenovo\/x301 (thinkpad x200 variant). ThinkPad X200 variant. Use standand ICH9M descriptor+nvm image. Intel i945. [link] same platform as X60\/T60 thinkpads. some of these are desktops, so there will be some differences. it’s unlikely that Intel ME will be an issue on any of them. asus\/p5gc-mx. No variants specified by Kconfig. getac\/p470. No variants specified by Kconfig. gigabyte\/ga-945gcm-s2l. Variants:. ga-945gcm-s2c. ibase\/mb899. No variants specified by Kconfig. kontron\/986lcd-m. No variants specified by Kconfig. roda\/rk886ex. No variants specified by Kconfig. AMD Fam10h \/ Fam15h. [link] These boards are not a priority at the moment, but they will be added at some point (. after. the post-2016 release). These were all deleted from coreboot in version 4.11 (they are fam10h\/15h boards). On this TODO page is an entry asking whether to fork coreboot 4.11 and maintain it, backporting newer features from coreboot, making it work with newer GCC toolchains, and so on. NOTE: some of these are a. big. if, but many of them will work nicely without binary blobs when booting. NOTE: use of a VGA option ROM is implied, and Libreboot won’t provide these, but the user could install an add-on graphics card and coreboot\/seabios would just run whatever is on the card. There is no problem with Libreboot running those, because they could be free or non-free, we just don’t know. In practise, most of these probably don’t have native video initialization in coreboot for the onboard GPU (if present), because it’s probably an AMD\/ATI one and libgfxinit doesn’t have good support for those (it mostly has excellent support for Intel video chipsets). This doesn’t mean Libreboot can’t support them. It just means that we will have to provide ROM images that don’t use libgfxinit. Instead, the ROMs provided will always run VGA option ROMs present on the GPU. Here we mean add-on video cards, which means there’s no way for the Libreboot project to predict what hardware will be used. It means that any GPU could be used. It probably implies use of SeaBIOS, but coreboot itself is able to run those VGA ROMs which enables other payloads (such as GNU GRUB) to be used reliably (with text mode startup). Where external VGA ROMs are concerned, Libreboot prefers for coreboot to run them, and for SeaBIOS to run run them, OR, for SeaBIOS to run it but be the main payload. In cases where coreboot runs the VGA ROM, it can also run other PCI ROMs, and SeaBIOS doesn’t need to do anything (and in fact shouldn’t do anything). On boards that. do. have libgfxinit support, coreboot isn’t running any PCI ROMs, which means no PCI ROMs for GRUB, which means you should use the SeaBIOS payload, either as the main payload or chainloaded from GRUB. Also: it’s still possible to use a serial console. You could use any of these boards in a headless server setup, with no graphics card. Also: there are USB VGA adapters available. Driver support in the Linux kernel is flaky for a lot of them, but you might be able to get some sort of desktop usage out of these boards, if you used one of them (there would be no display during early boot, but you would see something when booting your kernel). With llvmpipe driver you could actually get good use out of these. They are usually a simple framebuffer chip inside. advansus\/a785e-i. No variants specified by Kconfig. amd\/bimini_fam10. No variants specified by Kconfig. amd\/mahogany_fam10. No variants specified by Kconfig. amd\/serengeti_cheetah_fam10. No variants specified by Kconfig. amd\/tilapia_fam10. No variants specified by Kconfig. asus\/m4a785-m. No variants specified by Kconfig. asus\/m4a785t-m. No variants specified by Kconfig. asus\/m4a78-em. No variants specified by Kconfig. asus\/m5a88-v. No variants specified by Kconfig. avalue\/eax-785e. No variants specified by Kconfig. gigabyte\/ma785gm. No variants specified by Kconfig. gigabyte\/ma785gmt. No variants specified by Kconfig. gigabyte\/ma78gm. No variants specified by Kconfig. hp\/dl165_g6_fam10. No variants specified by Kconfig. iei\/kino-780am2-fam10. No variants specified by Kconfig. jetway\/pa78vm5. No variants specified by Kconfig. msi\/ms9652_fam10. No variants specified by Kconfig. supermicro\/h8dmr_fam10. No variants specified by Kconfig. supermicro\/h8qme_fam10. No variants specified by Kconfig. supermicro\/h8scm_fam10. No variants specified by Kconfig. tyan\/s2912_fam10. No variants specified by Kconfig. gm45. [link] thinkpad w700. http:\/\/www.thinkwiki.org\/wiki\/Category:W700 might be fun to work on. probably doesn’t require much modification in coreboot, if any. buy one and port it to coreboot. There are photos on this page:. http:\/\/web.archive.org\/web\/20210510205738\/https:\/\/notabug.org\/libreboot\/libreboot\/issues\/573 Linuxboot payload. [link] Linuxboot is a busybox+linux system available here:. https:\/\/www.linuxboot.org\/ It goes in bootflash. It provides a bootloader program called u-root, which uses kexec to boot other kernels. It also provides some UEFI features, and it can parse GNU GRUB configuration files. It requires a large amount of flashing space (at least 12MiB, but it might be possible to squeeze it into 8MiB). The problem is: it is using the upstream Linux kernel. TODO: fork Linuxboot and make the fork use linux-libre. Check other packages too. With this, a fully libre (by FSF standards) busybox+linux distro can be made, based on Linuxboot. Linuxboot-libre is the working name for this new project. It will absolutely knock the wind out of GRUB and anything else, on setups where it’s possible to use this payload. Other payloads will still be retained, of course. Fork coreboot 4.11 and maintain, for fam10h\/15h boards. [link] Nowadays, coreboot removes boards. For example, KCMA-D8 and KGPE-D16 (and others were removed) from coreboot because they don’t support relocatable ramstage,. C_ENVIRONMENTAL_BOOTBLOCK. , postcar and a few other features are required now in coreboot ports. For libreboot purposes, it’s mostly AMD Fam10\/15h boards that were removed. These were maintained based on AMD’s AGESA codebase, which was never properly integrated. It was just bolted on to coreboot, without honouring coreboot’s native coding style and maintaining it was very difficult. The person maintaining fam10h\/15h boards (in particular KCMA-D8 and KGPE-D16) had stopped doing work on those boards at that point. Libreboot currently does not fork coreboot, and it never has. Rather, it has been a downstream distribution of coreboot, de-blobbing it and patching it when necessary. This was sustainable before, because more or less just one revision could be used. There are mainly 2 choices:. Re-add deleted boards to coreboot. Fork older coreboot revisions with those other boards, and keep backporting newer features from later coreboot revisions (for instance, coreboot now has the ability to clear all DRAM on every bootup but this configuration option is unavailable on KCMA-D8 and KGPE-D16 mainboards). In practise, since this mostly affects fam10h\/fam15h boards, it’s probably. easier. to do the latter; fork older coreboot revision (version 4.11 in this case) and start backporting newer features; the current code works well, and only minor fixes will be needed here and there over time (e.g. patch it up to work on newer GCC versions when building). Forking the. entire. coreboot project and maintaining it for more than just a few boards isn’t really practical. It is best to cooperate with upstream, but in this case we are talking only about boards that were deleted. It’s always possible to bring the code on those deleted boards up to date again in the future, for re-entry into the coreboot master repository. Test SeaBIOS option: etc\/usb-time-sigatt. [link] default is 500ms. setting it higher like 1000s might make USB drives work in SeaBIOS on KFSN4-DRE. see notes on. https:\/\/www.seabios.org\/Runtime_config#Option_ROMs SST+macronix patches for flashrom on X60\/T60. [link] These binaries are referenced in the documentation currently not actually available and the build system (lbmk) does not produce them. Warnings about option ROMs. [link] They’re bad because they’re non-free. They violate the four freedoms. Libreboot enables automatic loading of PCI option ROMs in some setups, simply for the purpose of technical correctness, because there’s no rule that says an option ROM must be non-free. It’s possible that an option ROM might actually be free software. Banning option ROMs in Libreboot desktops would be like banning all software from executing in an operating system, just because those programs might be non-free. Instead, the. correct. solution ethically is to just tell people not to use non-free software, and for the. libreboot project. to never directly recommend, distribute or document non-free software. Use coreboot’s memtest86+ fork. [link] The current version used does build, but it doesn’t run, or it glitches out. That version of memtest is designed to be run on a normal BIOS system, so it might actually work with the SeaBIOS payload, but we want to use a memtest version that is guaranteed to work on bare metal, which is more common on Libreboot systems. Gemini site for libreboot. [link] Gemini is a popular alternative to the web. See:. https:\/\/gemini.circumlunar.space\/ I’ve noticed a lot of projects starting to offer this, in addition to a regular website. pandox2gem i’m told is a good tool that could integrate with the current static site generator, which uses pandoc (the pages are written in markdown). Tor site for libreboot. [link] hidden onion service. host it separately from the main site, on a different server. that way, there is another website just in case. 2nd HTTP site. [link] Have different DNS records for ns2. specifically, different IPv4+6 for the site. When the main ns1 is down, the new website will kick in. (ns1 and ns2 are both currently hosted on the same network as the website). i2p site. [link] I probably won’t, but someone is welcome to do this and libreboot.org will link to it. Fix GRUB bugs. [link] Many of these bugs only happen in bare metal, and only on devices supported by libreboot. See:. http:\/\/web.archive.org\/web\/20210510213902\/https:\/\/notabug.org\/libreboot\/libreboot\/issues\/561 Security patch: spectre MSR fixes for Fam15h boards. [link] See:. http:\/\/web.archive.org\/web\/20210510214458\/https:\/\/notabug.org\/libreboot\/libreboot\/issues\/440 Document teensy SPI flasher. [link] The following page has information, which can be assimilated:. https:\/\/trmm.net\/SPI_flash\/ Also see:. https:\/\/www.flashrom.org\/Teensy_3.1_SPI_%2B_LPC\/FWH_Flasher Also see this interesting firmware here:. https:\/\/github.com\/urjaman\/frser-teensyflash NOTE: i’ve made a local git clone of this. TODO: document use of schottky diode for VCC on SPI flash (ISP). [link] this type of diode has minimal voltage drop. most flashes run close to their specified 3.3v, sometimes a bit higher, but the tolerated range is between 2.7 and 3.6v. notes about use of a diode is already specified in the external flashing guide but those notes should be improved. x200 (after cutting solder bridge - R405 - between flash chip and ICH9M). x200s\/x200t\/w700 - 25xx flash Vcc is hardwired :( (to be confirmed on production board). t400\/t400s\/t500\/x301 - 25xx flash Vcc is hardwired, as everything else on UCI\/lenovo boards. Document alternative external flashing method for X200S\/X200T. [link] GNUtoo wrote this interested guide:. https:\/\/framagit.org\/GNUtoo\/coreboot-scripts\/-\/tree\/master\/flash-wson8 It still requires external flashing, but no soldering. TODO: what about bucts? the bootblock is protected by PR4, but is it possible to use BUCTS and init from another bootblock? NOTE TO SELF: a local git clone has been made of the above. Handle SATA power in ultrabay on gm45 thinkpads. [link] See:. http:\/\/web.archive.org\/web\/20201022210929\/https:\/\/notabug.org\/libreboot\/libreboot\/issues\/484 document serial\/lpt\/pcie bus enable\/disable on GA-G41M-ES2L. [link] See:. http:\/\/web.archive.org\/web\/20210510214317\/https:\/\/notabug.org\/libreboot\/libreboot\/issues\/469 This might be why graphics cards and add-on network cards didn’t work on mine, last time i tested it. it’s a config option that must be enabled in coreboot? Document quad-core mods on GM45 thinkpads. [link] NOTE: MAX_CPUS=4 is now the default, in coreboot, for these machines. There’s a mod for T500 thinkpads, but it will work on any gm45 thinkpad supported in libreboot. Just have to study the schematics and boardview, then adapt the info available online for T500. NOTE: max CPUs has to be set in coreboot. Document a. clean. way to do it. The current guides online have you drilling holes in the CPU socket! That’s why they won’t be linked here. Some notes are already written here. expand upon them:. http:\/\/web.archive.org\/web\/20210307234010\/https:\/\/notabug.org\/libreboot\/libreboot\/issues\/340 Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/tasks\/index.md Site map This HTML page was generated by the. untitled static site generator ."}
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Project contributors. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Leah Rowe Alyssa Rosenzweig Andrew Robbins Arthur Heymans Damien Zammit Denis Carikli Jeroen Quint Joshua Gay Klemens Nanni Lisa Marie Maginnis Marcus Moeller Patrick “P. J.” McDermott Paul Kocialkowski Paul Menzel Peter Stuge Steve Shenton Swift Geek Timothy Pearson vitali64 Vladimir Serbinenko Return to index This list does not necessarily reflect who is currently working on the project, but it lists some people who have contributed to the project in meaningful ways. If we forgot to mention you here, let us know and we’ll add you. (or if you don’t want to be mentioned, let us know and we’ll remove your entry). Information about who works on Libreboot, and how the project is run, can be found on this page:. who.html You can know the history of the Libreboot project, simply by reading this page. It goes into detail about all of the major contributions to the project, and in general how the project was created (and who helped create it). Leah Rowe. [link] Founder of the Libreboot project, and currently the lead developer. Leah works on all aspects of Libreboot, such as:. General management. Leah handles all outside contributions to Libreboot, reviews pull requests, deals with bug reports, delegates tasks when necessary or desirable. Leah controls the libreboot.org server infrastructure, hosted in her lab (of course it runs Libreboot!). Leah has the final say on all decisions, taking input via discussion with members of the public, mostly on IRC. Leah oversees releases of Libreboot, and generally keeps the project going. Without Leah, there would be no Libreboot! The build system (lbmk, short for Libreboot Make). This is the automated build system that sits at the heart of Libreboot; it downloads, patches, configures and compiles the relevant components like coreboot, GNU GRUB and generates the Libreboot ROM images that you can find in release archives. Upstream work on coreboot, when necessary (and other projects that Libreboot uses). This means also working with people from outside of the Libreboot project, to get patches merged (among other things) on the upstream projects that Libreboot uses. Providing user support on IRC. Commercial. user support via her company listed on. the suppliers page Leah is also responsible for. osboot.org which is heavily based on Libreboot, but with different project goals. Other people are listed below, in alphabetical order:. Alyssa Rosenzweig. [link] Switched the website to use markdown in lieu of handwritten HTML and custom PHP. Former libreboot project maintainer (sysadmin for libreboot.org). Alyssa wrote the original static site generator (bash scripts converting markdown to html, via pandoc) for libreboot.org. This static site generator has now been heavily modified and forked into a formal project, by Leah Rowe:. https:\/\/untitled.vimuser.org\/ (untitled is Leah’s work, not Alyssa’s, but it’s based on Alyssa’s original work on the static site generator that Libreboot used to use; the Libreboot website is now built with Untitled). Andrew Robbins. [link] Worked on large parts of Libreboot’s old build system and related documentation. Andrew joined the Libreboot project as a full time developer during June 2017, until his departure in March 2021. I, Leah Rowe, am very grateful to Andrew Robbins for his numerous contributions over the years. Arthur Heymans. [link] Merged a patch from coreboot into libreboot, enabling C3 and C4 power states to work correctly on GM45 laptops. This was a long-standing issue before Athur’s contribution. Arthur also fixed VRAM size on i945 on GM45 systems, allowing maximum VRAM allocation for the onboard GPUs on these systems, another longstanding issue in libreboot. Arthur also did work on the Libreboot build system, when he was a member of the project. He still works on coreboot, to this day, and Libreboot greatly benefits from his work. His contributions to the coreboot project, and Libreboot, are invaluable. Damien Zammit. [link] Maintains the Gigabyte GA-G41M-ES2L coreboot port, which is integrated in libreboot. Also works on other hardware for the benefit of the libreboot project. Damien didn’t work directly on Libreboot itself, but he worked heavily with Leah Rowe, integrating patches and new board ports into Libreboot, based on Damien’s upstream work on coreboot. Denis Carikli. [link] Based on the work done by Peter Stuge, Vladimir Serbineko and others in the coreboot project, got native graphics initialization to work on the ThinkPad X60, allowing it to be supported in libreboot. Denis gave a lot of advice and helped found the libreboot project. Denis was a mentor to Leah Rowe in the early days, when she founded the Libreboot project. A lot of the decision decisions taken, especially with the Libreboot build system (lbmk), were inspired from talks with Denis. Denis taught Leah about registers used by Intel GPUs for backlight control. In the early days, the ThinkPad X60 and T60 laptops in Libreboot did not have backlight control working, so the brightness was always 100%. With Denis’s help, Leah was able to get backlight controls working by reverse engineering the correct values to write in those registers. Based on this, a simple fix was written in coreboot; however, the fix just wrote directly to the register and didn’t work with ACPI based brightness controls. Others in coreboot later improved it, making ACPI-based backlight controls work properly, based on this earlier work. Jeroen Quint. [link] Contributed several fixes to the libreboot documentation, relating to installing Parabola with full disk encryption on libreboot systems. Joshua Gay. [link] Joshua is former FSF staff. Joshua helped with the early founding of the Libreboot project, in his capacity (at that time) as the FSF’s licensing and compliance manager. It was his job to review products sent into to the FSF for review; the FSF has a certification program called. Respects Your Freedom. (RYF) where the FSF will promote your company’s products if it comes with all Free Software. I, Leah Rowe, was initially just selling ThinkPad X60 laptops with regular coreboot on them, and this included CPU microcode updates. At the time, I didn’t think much of that. Joshua contacted me, in his capacity at the FSF, and asked if I would be interested in the FSF’s RYF program; I was very surprised that the FSF would take me seriously, and I said yes. This is what started the early work on Libreboot. Joshua showed me all the problems my products had, and from that, the solution was clear:. A project needed to exist, providing a fully free version of coreboot, without any binary blobs. At the time (and this is still true today), coreboot was not entirely free software and shipped with binary blobs by default. In particular, CPU microcode updates were included by default, on all x86 machines. Working with Joshua who reviewed my work, I created a fully free version of coreboot. At first, it wasn’t called Libreboot, and the work was purely intended for my company (at that time called Gluglug) to be promoted by the FSF. Joshua used his media connections at the FSF to heavily promote my work, and on December 13th, 2013, the Libreboot project was born (but not called that). Joshua made sure that everyone knew what I was doing! A few months later, the name. Libreboot. was coined, and the domain name. libreboot.org. was registered. At that point, the Libreboot project (in early 2014) was officially born. Once again, Joshua provided every bit of help he could, heavily promoting the project and he even wrote this article on the FSF website, announcing it:. https:\/\/www.fsf.org\/blogs\/licensing\/replace-your-proprietary-bios-with-libreboot Klemens Nanni. [link] Made many fixes and improvements to the GRUB configuration used in libreboot, and several tweaks to the build system. Leah Rowe initially helped Klemens get his project, autoboot, off the ground. Autoboot (website autoboot.org) is no longer online, but was a fork of Libreboot with different project goals; in late 2020, Leah Rowe decided to create her own new fork of Libreboot called. osboot. , heavily inspired by Klemens’s earlier work. See:. https:\/\/osboot.org\/ The following is an archive of autoboot.org, from when it was online back in 2016:. http:\/\/web.archive.org\/web\/20160414205513\/http:\/\/autoboot.org\/ (the autoboot website went offline a few months later, after Klemens abandoned the project). Lisa Marie Maginnis. [link] Lisa is a former sysadmin at the Free Software Foundation. In the early days of the project, she provided Leah with a lot of technical advice. She initially created Libreboot IRC channel, when Leah did not know how to use IRC, and also handed +F founder status to Leah for the channel. As an FSF sysadmin, it was Lisa’s job to maintain a lot of the infrastructure used by Libreboot; at the time, mailing lists on the GNU Savannah website were used by the Libreboot project. Lisa was also the one who originally encouraged Leah to have Libreboot join the GNU project (a decision that was later, rather regrettably, reversed). When Paul Kocialkowski was a member of the project in 2016, she helped him get help from the FSF; he was the leader of the Replicant project at the time, which had funding from the FSF, and the FSF authorized him to use some of that funding for his work on Libreboot, thanks to Lisa’s encouragement while she worked at the FSF. Lisa also stepped in when Leah Rowe missed her LibrePlanet 2016 talk. Leah was scheduled to do a talk about Libreboot, but didn’t show up in time. Lisa, along with Patrick McDermott (former Libreboot developer, who was present at that conference) did the talk in Leah’s place. The talk was never recorded, but the Free Software Foundation has these photos of that talk on their LibrePlanet website (the woman with the blue hair is Lisa, and the long-haired dude with the moustache is Patrick):. https:\/\/media.libreplanet.org\/u\/libreplanet\/m\/session-02-c-mws-png-libreplanet-2016-sessions\/ (archive link:. http:\/\/web.archive.org\/web\/20170319043913\/https:\/\/media.libreplanet.org\/u\/libreplanet\/m\/session-02-c-mws-png-libreplanet-2016-sessions\/ ). https:\/\/media.libreplanet.org\/u\/libreplanet\/m\/session-02-c-wide-png-libreplanet-2016-sessions\/ (archive link:. http:\/\/web.archive.org\/web\/20170319043915\/https:\/\/media.libreplanet.org\/u\/libreplanet\/m\/session-02-c-wide-png-libreplanet-2016-sessions\/ ). Fun fact: Patrick is also the lead developer of ProteanOS, an FSF-endorsed embedded OS project:. http:\/\/proteanos.com\/ (uses BusyBox and Linux-libre). Leah Rowe ran. 2. LibrePlanet workshops; one in 2015 and another in 2016, while visiting Boston, MA, USA on both occasions to attend these conferences. These workshops were for Libreboot installations. People came to both workshops, to have Libreboot installed onto their computers. As FSF sysadmin, at that time, Lisa provided all of the infrastructure and equipment used at those workshops. Without her help, those workshops would have not been possible. When the ASUS KGPE-D16 mainboard (high-end server board) was ported to Libreboot, Leah, working with Timothy Pearson (the one who ported it), shared patches back and forth with Lisa around mid 2016, mostly raminit patches, to get the board running at the FSF offices. This work ultimately lead to a most wonderful achievement:. The. https:\/\/www.gnu.org\/ and. https:\/\/www.fsf.org\/ websites now run on Librebooted ASUS KGPE-D16 based servers, on a fully free GNU+Linux distro. This means that the FSF now has full software freedom for their hosting infrastructure. The FSF also provides access to this infrastructure for many other projects (besides GNU projects); for example, Trisquel uses a D16 provided by the FSF for their development server used for building Trisquel releases and testing changes to the Trisquel GNU+Linux distribution. Trisquel is a fully free GNU+Linux distribution, heavily promoted by the FSF. Lisa was a strong supporter of Libreboot in the very early days of the project, and her contributions were invaluable. I, Leah Rowe, owe her a debt of gratitude. Marcus Moeller. [link] Made the libreboot logo. Patrick “P. J.” McDermott. [link] Patrick also did a lot of research and wrote the libreboot FAQ section relating to the. Intel Management Engine , in addition to making several improvements to the build system in libreboot. Former libreboot project maintainer. In 2016, Leah Rowe ran a Libreboot installation workshop at the FSF’s LibrePlanet conference. Working alongside Leah, Patrick helped run the workshop and assisted with installing Libreboot onto people’s machines. Paul Kocialkowski. [link] Ported the ARM (Rockchip RK3288 SoC) based. Chromebook. laptops to libreboot. Also one of the main. Replicant developers. Paul Menzel. [link] Investigated and fixed a bug in coreboot on the ThinkPad X60\/T60 exposed by Linux kernel 3.12 and up, which caused 3D acceleration to stop working and video generally to become unstable. The issue was that coreboot, when initializing the Intel video chipset, was mapping. GTT Stolen Memory. in the wrong place, because the code was based on kernel code and the Linux kernel had the same bug. When Linux fixed it, it exposed the same bug in coreboot. Paul worked with Libreboot on this, sending patches to test periodically until the bug was fixed in coreboot, and then helped her integrate the fix in libreboot. Peter Stuge. [link] Helped write the. FAQ section about DMA , and provided general advice in the early days of the project. Peter was a coreboot developer in those days, and a major developer in the. libusb. project (which flashrom makes heavy use of). Peter also wrote the. bucts. utility used to set Backup Control (BUC) Top Swap (TS) bit on i945 laptops such as ThinkPad X60\/T60, which is useful for a workaround to flash Libreboot without using external hardware; on this machine, with Lenovo BIOS present, it’s possible to flash everything except the main bootblock, but Intel platforms have 2 bootblocks, and you specify which one is to be used by setting the TS bit. You then boot with only one bootblock flashed (by the coreboot project’s bootblock on that machine), and afterwards you reset bucts before flashing the ROM again, to flash the main bootblock. Libreboot hosts a copy of his work, because his website hosting bucts is no longer responsive. Steve Shenton. [link] Steve did the early reverse engineering work on the Intel Flash Descriptor used by ICH9M machines such as ThinkPad X200. He created a C struct defining (using bitfields in C) this descriptor region. With some clever tricks, he was able to discover the existence of a bit in the descriptor for. disabling. the Intel ME (management engine) on those platforms. His initial proof of concept only defined the descriptor, and would do this:. Read the default descriptor and GbE regions from a Lenovo X200 ROM (default firmware, not coreboot). Disable the ME, by setting 2 bits in the descriptor. Disable the ME region. Move descriptor+GbE (12KiB in total) next to each other. Allocate the remaining flash space to the BIOS region. Generated the 12KiB descriptor+GbE region, based on this, to insert into a coreboot ROM image. In the early days, before Libreboot supported GM45+ICH9M platforms such as ThinkPad X200\/T400, you could use those machines but to avoid the Intel ME you had to flash it without a descriptor region. This worked fine in those days, because the ME only handled TPM and AMT on those machines, and the system would work normally, but that Intel Flash Descriptor also handles the Intel GbE NVM region in flash, which is used for the Intel Gigabit Ethernet interface. So you either had Intel ME, or no ethernet support. Steve figured out how to disable the Intel ME via 2 toggle bits in the descriptor, and also how to remove the Intel ME region from flash. Based on his research, I, Leah Rowe, working alongside Steve, also reverse engineered the layout of the Intel GbE NVM (non-volatile memory) region in the boot flash. This region defines configuration options for the onboard Intel GbE NIC, if present. Based on this, I was able to take Steve’s initial proof of concept and write the. ich9gen. utility, which generates an Intel Flash Descriptor and GbE NVM region, from scratch, without an Intel ME region defined. It is this tool, the. ich9gen. tool, that Libreboot uses to provide ROM images for GM45+ICH9M platforms (such as ThinkPad X200\/T400\/T500\/W500), with a fully functional descriptor and functional Gigabit Ethernet, but. without. needing Intel Management Engine (ME) firmware, thus making those machines. libre. (the ME is fully disabled, when you use a descriptor+gbe image generated by. ich9gen. ). With. my. ich9gen. tool (Steve’s tool was called. ich9deblob. ), you didn’t need a dump of the original Lenovo BIOS firmware anymore! I could not have written this tool, without Steve’s initial proof of concept. I worked with him, extensively, for many months. All GM45+ICH9M support (X200, T400, etc) in Libreboot is made possible because of the work he did, back in 2014. Swift Geek. [link] Contributed a patch for ich9gen to generate 16MiB descriptors. After that, Swift Geek slowly became more involved until he became a full time developer. Swift Geeks contributions were never really in the form of. code. , but what he lacked in code, he made up for in providing excellent support, both to users and other developers, helping others learn more about technology at a low level. When Swift Geek was a member of the project, his role was largely providing user support (in the IRC channel), and conducting research. Swift Geek knows a lot about hardware. Swift Geek also did some upstream development on GNU GRUB. Swift Geek has provided technical advice on numerous occasions, to Leah Rowe, and helped her to improve her soldering skills in addition to teaching her some repair skills, to the point where she can now repair most faults on ThinkPad mainboards (while looking at the schematics and boardview). Swiftgeek left the project in March 2021. I, Leah Rowe, wish him all the best in his endeavours, and I’m very grateful to his numerous contributions over the years. Timothy Pearson. [link] Ported the ASUS KGPE-D16 board to coreboot for the company Raptor Engineering of which Timothy is the CEO. Timothy maintains this code in coreboot, helping the project with the libreboot integration for it. This person’s contact details are on the raptor site, or you can ping. tpearson. on the Libera IRC network. vitali64. [link] Added cstate 3 support on macbook21, enabling higher battery life and cooler CPU temperatures on idle usage. vitali64 on irc. Vladimir Serbinenko. [link] Ported many of the thinkpads supported in libreboot, to coreboot, and made many fixes in coreboot which benefited the libreboot project. Vladimir wrote a lot of the original video initialization code used by various Intel platforms in Libreboot, when flashing it (now rewritten by others in Ada, for libgfxinit in coreboot, but originally it was written in C and included directly in coreboot; libgfxinit is a 3rdparty submodule of coreboot). Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/contrib.md Site map This HTML page was generated by the. untitled static site generator ."}
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Libreboot news. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to previous index News about libreboot, both technical and organisational. Releases are also announced here. Subscribe to RSS:. \/feed.xml Binary blob extermination policy 2 January 2022 (updated 23 January 2022). New Hampshire (USA) may soon enshrine Software Freedom into law. YOUR HELP IS NEEDED! 8 January 2022. Translations wanted 11 December 2021. Libreboot 20211122 released! 22 November 2021. Libreboot 20210522 released! 22 May 2021. Libreboot 20160907 release 7 September 2016. Libreboot 20160902 release 2 September 2016. Libreboot 20160818 release 18 August 2016. Libreboot 20150518 release 18 May 2015. Libreboot 20150208 release 8 February 2015. Libreboot 20150126 release 26 January 2015. Libreboot 20150124 release 24 January 2015. Libreboot 20141015 release 15 October 2014. Libreboot 20140911 release 11 September 2014. Libreboot 20140903 release 3 September 2014. Libreboot 20140811 release 11 August 2014. Libreboot 20140729 release 29 July 2014. Libreboot 20140720 release 20 July 2014. Libreboot 20140716 release 16 July 2014. Libreboot 20140711 release 11 July 2014. Libreboot 20140622 release 22 June 2014. Libreboot 20140611 release 11 June 2014. Libreboot 20140605 release 5 June 2014. Libreboot 20140309 release 9 March 2014. Libreboot 20140221 release 21 February 2014. Libreboot 20131214 release 14 December 2013. Libreboot 20131213 13 December 2013. Libreboot 20131212 12 December 2013. Binary blob policy Edit this page Who develops Libreboot? 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Template license. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to index This website is written in Markdown, and compiled into static HTML using Pandoc. An HTML template is used, Copyright (c) 2014–2017, John MacFarlane. Modifications to it are copyright 2021 Leah Rowe, released under the terms of Creative Commons Zero license version 1.0 universal, which you can find here:. https:\/\/creativecommons.org\/publicdomain\/zero\/1.0\/legalcode You can find the template file here:. \/template.include The file. template.include. is the modified version (modified by Leah Rowe). The original version can be found here:. \/template.original Other modified templates may be used, on specific pages. Check for this on the Git repository for the Libreboot website. The original template file named. template.original. by John MacFarlane was released under these conditions:. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and\/or other materials provided with the distribution. Neither the name of John MacFarlane nor the names of other contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. Binary blob policy Edit this page Who develops Libreboot? 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Downloads. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. GPG signing key NEW KEY OLD KEY: Git repository HTTPS mirrors RSYNC mirrors HTTP mirrors FTP mirrors Statically linked Return to index New releases are announced in the. main news section . If you’re more interested in libreboot development, go to the. libreboot development page , which also includes links to the Git repositories. The page on. \/docs\/maintain\/ describes how Libreboot is put together, and how to maintain it. If you wish to build Libreboot from source,. read this page . GPG signing key. [link] The latest release is Libreboot 20211122, under the. testing. directory. NEW KEY. [link] Full key fingerprint:. 98CC DDF8 E560 47F4 75C0 44BD D0C6 2464 FA8B 4856. The above key is for Libreboot 20211122, and subsequent releases. Download the key here:. lbkey.asc Libreboot releases are signed using GPG. OLD KEY:. [link] This key is for Libreboot 20160907 and all older releases:. Full key fingerprint: CDC9 CAE3 2CB4 B7FC 84FD C804 969A 9795 05E8 C5B2. The GPG key can also be downloaded with this exported dump of the pubkey:. lbkeyold.asc . sha512sum -c sha512sum.txt gpg --verify sha512sum.txt.sig. Git repository. [link] Links to regular release archives are listed on this page. However, for the absolute most bleeding edge up-to-date version of Libreboot, there is a Git repository that you can download from. Go here:. How to download Libreboot from Git HTTPS mirrors. [link] The latest release is Libreboot 20211122, under the. testing. directory. These mirrors are recommended, since they use TLS (https:\/\/) encryption. You can download Libreboot from these mirrors:. https:\/\/rsync.libreboot.org\/ (Libreboot project official mirror, UK). https:\/\/www.mirrorservice.org\/sites\/libreboot.org\/release\/ (University of Kent, UK). https:\/\/mirrors.mit.edu\/libreboot\/ (MIT university, USA). https:\/\/mirror.math.princeton.edu\/pub\/libreboot\/ (Princeton university, USA). https:\/\/mirror.libremind.org\/libreboot\/ (libremind.org, Iceland). https:\/\/mirror.splentity.com\/libreboot\/ (Splentity Software, USA). https:\/\/mirror.sugol.org\/libreboot\/ (sugol.org) (formerly nephelai.zanity.net\/mirror\/libreboot). https:\/\/mirrors.qorg11.net\/libreboot\/ (qorg11.net, Spain). https:\/\/elgrande74.net\/libreboot\/ (elgrande74.net, France). https:\/\/mirror.koddos.net\/libreboot\/ (koddos.net, Netherlands). https:\/\/mirror.swordarmor.fr\/libreboot\/ (swordarmor.fr, France). https:\/\/mirror-hk.koddos.net\/libreboot\/ (koddos.net, Hong Kong). https:\/\/mirror.cyberbits.eu\/libreboot\/ (cyberbits.eu, France). RSYNC mirrors. [link] Useful for mirroring Libreboot’s entire set of release archives. You can put an rsync command into crontab and pull the files into a directory on your web server. If you are going to mirror the entire set, it is recommended that you allocate at least 25GiB. Libreboot’s rsync is currently about 12GiB, so allocating 25GiB will afford you plenty of space for the future. At minimum, you should ensure that at least 15-20GiB of space is available, for your Libreboot mirror. It is highly recommended that you use the libreboot.org mirror. , if you wish to host an official mirror. Otherwise, if you simply want to create your own local mirror, you should use one of the other mirrors, which sync from libreboot.org. Before you create the mirror, make a directory on your web server. For example:. mkdir \/var\/www\/html\/libreboot\/. Now you can run rsync, for instance:. rsync -avz --delete-after rsync:\/\/rsync.libreboot.org\/mirrormirror\/ \/var\/www\/html\/libreboot\/. It’s extremely important to have the final forward slash (\/) at the end of each path, in the above rsync command. Otherwise, rsync will behave very strangely. If you wish to regularly keep your rsync mirror updated, you can add it to a crontab. This page tells you how to use crontab:. https:\/\/man7.org\/linux\/man-pages\/man5\/crontab.5.html The following rsync mirrors are available:. rsync:\/\/rsync.libreboot.org\/mirrormirror\/ (Libreboot project official mirror). rsync:\/\/rsync.mirrorservice.org\/libreboot.org\/release\/ (University of Kent, UK). rsync:\/\/mirror.math.princeton.edu\/pub\/libreboot\/ (Princeton university, USA). rsync:\/\/rsync.libremind.org\/libreboot\/ (libremind.org, Iceland). rsync:\/\/qorg11.net\/mirrors\/libreboot\/ (qorg11.net, Spain). rsync:\/\/ftp.linux.ro\/libreboot\/ (linux.ro, Romania). rsync:\/\/mirror.koddos.net\/libreboot\/ (koddos.net, Netherlands). rsync:\/\/mirror-hk.koddos.net\/libreboot\/ (koddos.net, Hong Kong). Are you running a mirror? Contact the libreboot project, and the link will be added to this page! You can make your rsync mirror available via your web server, and also configure your. own. mirror to be accessible via rsync. There are many resources online that show you how to set up an rsync server. HTTP mirrors. [link] The latest release is Libreboot 20211122, under the. testing. directory. WARNING: these mirrors are non-HTTPS which means that they are unencrypted. Your traffic could be subject to interference by adversaries. Make especially sure to check the GPG signatures, assuming that you have the right key. Of course, you should do this anyway, even if using HTTPS. http:\/\/mirror.linux.ro\/libreboot\/ (linux.ro, Romania). http:\/\/mirror.helium.in-berlin.de\/libreboot\/ (in-berlin.de, Germany). FTP mirrors. [link] The latest release is Libreboot 20211122, under the. testing. directory. WARNING: FTP is also unencrypted, like HTTP. The same risks are present. ftp:\/\/ftp.mirrorservice.org\/sites\/libreboot.org\/release\/ (University of Kent, UK). ftp:\/\/ftp.linux.ro\/libreboot\/ (linux.ro, Romania). Statically linked. [link] Libreboot includes statically linked executables in some releases, built from the available source code. Those executables have certain libraries built into them, so that the executables will work on many GNU+Linux distros. Libreboot 20160907 was built in Trisquel GNU+Linux, version 7.0 64-bit. Some older Libreboot releases will have been built in Trisquel 6.0.1. To comply with GNU GPL v2, Trisquel 6 and 7 source ISOs are supplied by the Libreboot project. You can find these source ISOs in the. ccsource. directory on the. rsync. mirrors. Libreboot releases past version 20160907 do not distribute statically linked binaries. Instead, these releases are source-only, besides pre-compiled ROM images for which the regular Libreboot source code archives suffice. These newer releases instead automate the installation of build dependencies, with instructions in the documentation for building various utilities from source. These executables are utilities such as. flashrom. . Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/download.md Site map This HTML page was generated by the. untitled static site generator ."}
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This is still under discussion, and none of this is proven.","malicious software running as root could update this firmware, or the firmware could already be malicious","“the receiver of the FIS”","“The specific implementation of the buffer identifier and buffer\/address validation is not specified”","“Upon receiving a DMA Setup, the receiver of the FIS shall validate the received DMA Setup request.”","A drive can also send DMA Setup","“transfer of data from the host to the device, and goes on to say: Upon receiving a DMA Activate, if the host adapter’s DMA controller has been programmed and armed, the host adapter shall initiate the transmission of a Data FIS and shall transmit in this FIS the data corresponding to the host memory regions indicated by the DMA controller’s context.”","“DMA Activate FIS - Device to Host”","flashed","payload","beginning","-w","-a","coreboot\/util\/nvramtool\/","nvramtool","BIOS setup","CMOS","Rockchip RK3288","switchable graphics","the libreboot project recommends avoiding all modern AMD hardware. 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Frequently Asked Questions. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Important issues How to compile Libreboot from source How does the build system work? Do not use CH341A! Flashrom complains about DEVMEM access The backlight is darker on the left side of the screen when lowering the brightness on my ThinkPad X200\/X200S\/X200T, T400, T500, R400, W500, R500 and other Intel laptops The ethernet doesn’t work on my X200\/T400\/X60\/T60 when I plug in it My KCMA-D8 or KGPE-D16 doesn’t boot with the PIKE2008 module installed How to save kernel panic logs on thinkpad laptops? Machine check exceptions on some Montevina (Penryn CPU) laptops Hardware compatibility What systems are compatible with libreboot? Why is the latest Intel hardware unsupported in libreboot? Intel Management Engine (ME) Firmware Support Package (FSP) CPU microcode updates Intel is uncooperative Why is the latest AMD hardware unsupported in libreboot? AMD Platform Security Processor (PSP) AMD IMC firmware AMD SMU firmware AMD AGESA firmware AMD CPU microcode updates AMD is incompetent (and uncooperative) What can. I use, then? Will libreboot work on a ThinkPad T400 or T500 with an ATI GPU? Will desktop\/server hardware be supported? Hi, I have , is it supported? What about ARM? General questions How do I install libreboot? How do I program an SPI flash chip? How do I write-protect the flash chip? How do I change the BIOS settings? How do I pad a ROM before flashing? Do I need to install a bootloader when installing a distribution? Do I need to re-flash when I re-install a distribution? What does a flash chip look like? Who did the logo? What other firmware exists outside of libreboot? External GPUs EC (embedded controller) firmware HDD\/SSD firmware NIC (ethernet controller) CPU microcode Sound card Webcam USB host controller WWAN firmware Operating Systems Can I use GNU+Linux? Fedora won’t boot? (may also be applicable to Redhat\/CentOS) Can I use BSD? Are other operating systems compatible? Does libreboot make my machine 100% free? Where can I learn more about electronics Return to index AKA Frequently Questioned Answers. Important issues. [link] How to compile Libreboot from source. [link] Refer to the. lbmk build instructions . How does the build system work? [link] Refer to the. lbmk maintenance instructions . Do not use CH341A! [link] This SPI flasher will damage your chip, and the mainboard that it is connected to. Read the notes about CH341A on. docs\/install\/spi.html to learn more. Flashrom complains about DEVMEM access. [link] If running. flashrom -p internal. for software based flashing, and you get an error related to \/dev\/mem access, you should reboot with. iomem=relaxed. kernel parameter before running flashrom, or use a kernel that has. CONFIG_STRICT_DEVMEM. and. CONFIG_IO_STRICT_DEVMEM. not enabled. Example flashrom output with both. CONFIG_STRICT_DEVMEM. and. CONFIG_IO_STRICT_DEVMEM. enabled:. flashrom v0.9.9-r1955 on Linux 4.11.9-1-ARCH (x86_64) flashrom is free software, get the source code at https:\/\/flashrom.org Calibrating delay loop... OK. Error accessing high tables, 0x100000 bytes at 0x000000007fb5d000 \/dev\/mem mmap failed: Operation not permitted Failed getting access to coreboot high tables. Error accessing DMI Table, 0x1000 bytes at 0x000000007fb27000 \/dev\/mem mmap failed: Operation not permitted. The backlight is darker on the left side of the screen when lowering the brightness on my ThinkPad X200\/X200S\/X200T, T400, T500, R400, W500, R500 and other Intel laptops. [link] We don’t know how to detect the correct PWM value to use in coreboot-libre, so we just use the default one in coreboot which has this issue on some CCFL panels, but not LED panels. You can work around this in your distribution, by following the notes at. docs: backlight control . The ethernet doesn’t work on my X200\/T400\/X60\/T60 when I plug in it. [link] This was observed on some systems using network-manager. This happens both on the original BIOS and in libreboot. It’s a quirk in the hardware. On debian systems, a workaround is to restart the networking service when you connect the ethernet cable:. sudo service network-manager restart. On Parabola, you can try:. sudo systemctl restart network-manager. (the service name might be different for you, depending on your configuration). My KCMA-D8 or KGPE-D16 doesn’t boot with the PIKE2008 module installed. [link] Libreboot 20160818, 20160902 and 20160907 all have this behaviour: in SeaBIOS, PCI options ROMs are loaded when available, by default. This is not technically a problem, because an option ROM can be free or non-free. Loading the option ROM from the PIKE2008 module on either ASUS KCMA-D8 or KGPE-D16 causes the system to hang at boot. It’s possible to use this in the payload (if you use a linux kernel payload, like linuxboot), or to boot (with SeaGRUB and\/or SeaBIOS) from regular SATA and then use it in GNU+Linux. The Linux kernel is capable of using the PIKE2008 module without loading the option ROM. Refer to the. linuxboot website . This is a special busybox+linux system, which goes in the boot flash as a coreboot payload. You can insert it into your Libreboot ROM image using cbfstool, if it’s big enough. On KCMA-D8\/KGPE-D16 it’s trivial to upgrade the boot flash to 16MiB, which is more than enough to fit Linuxboot. See:. External flashing guide LinuxBoot has many advanced features. It provides a bootloader called uroot, which can boot other Linux kernels using kexec. It can parse GNU GRUB configs, and it can also provide some basic UEFI features. Because it’s using the Linux kernel, this means that LinuxBoot can make use of the PIKE2008 module. TODO: Integrate this in Libreboot, as a payload option, but also:. TODO: Fork LinuxBoot, and make a version of it that uses linux-libre. Ensure that it is a fully free distribution, so that the FSF can endorse it. LinuxBoot is. perfect. , especially if you’re setting up one of these systems to be used as a server. LinuxBoot is. much. more flexible and configurable than GNU GRUB, which right now is the payload that Libreboot prefers. How to save kernel panic logs on thinkpad laptops? [link] The easiest method of doing so is by using the kernel’s netconsole and reproducing the panic. Netconsole requires two machines, the one that is panicky (source) and the one that will receive crash logs (target). The source has to be connected with an ethernet cable and the target has to be reachable at the time of the panic. To set this system up, execute the following commands as root on the source (. source#. ) and normal user on the target (. target$. ):. Start a listener server on the target machine (netcat works well):. target$ nc -u -l -p 6666. Mount configfs (only once per boot, you can check if it is already mounted with. mount | grep \/sys\/kernel\/config. . This will return no output if it is not). source# modprobe configfs. source# mkdir -p \/sys\/kernel\/config. source# mount none -t configfs \/sys\/kernel\/config. find source’s ethernet interface name, it should be of the form. enp*. or. eth*. , see. ip address. or. ifconfig. output. source# iface=\"enp0s29f8u1\". change this. Fill the target machine’s IPv4 address here:. source# tgtip=\"192.168.1.2\". change this. Create netconsole logging target on the source machine:. source# modprobe netconsole. source# cd \/sys\/kernel\/config\/netconsole. source# mkdir target1; cd target1. source# srcip=$(ip -4 addr show dev \"$iface\" | grep -Eo '[0-9]+\\.[0-9]+\\.[0-9]+\\.[0-9]+'). source# echo \"$srcip\" local_ip. source# echo \"$tgtip\" remote_ip. source# echo \"$iface\" dev_name. source# arping -I \"$iface\" \"$tgtip\" -f | grep -o '..:..:..:..:..:..' remote_mac. source# echo 1 enabled. Change console loglevel to debugging:. source# dmesg -n debug. Test if the logging works by e.g. inserting or removing an USB device on the source. There should be a few lines appearing in the terminal, in which you started netcat (nc), on the target host. Try to reproduce the kernel panic. Machine check exceptions on some Montevina (Penryn CPU) laptops. [link] Some GM45 laptops have been freezing or experiencing a kernel panic (blinking caps lock LED and totaly unresponsive machine, sometimes followed by an automatic reboot within 30 seconds). We do not know what the problem(s) is(are), but a CPU microcode update in some cases prevents this from happening again. See the following bug reports for more info:. T400 Machine check: Processor context corrupt X200 Machine check: Processor context corrupt Unrelated, RAM incompatibility and suspend-to-ram issues on X200 Hardware compatibility. [link] What systems are compatible with libreboot? [link] See the. hardware compatibility list . Why is the latest Intel hardware unsupported in libreboot? [link] It is unlikely that any post-2008 Intel hardware will ever be supported in libreboot, due to severe security and freedom issues; so severe, that. the libreboot project recommends avoiding all modern Intel hardware. If you have an Intel based system affected by the problems described below, then you should get rid of it as soon as possible. . The main issues are as follows:. Intel Management Engine (ME). [link] NOTE: The information below is slightly out of date. Nowadays, Intel ME does not run on an ARC coprocessor, but instead runs on a modified Intel 486 based architecture, with the ME firware written for x86 based on the Minix operating system. However, the overall design philosophy and operation is mostly the same. On. most. current Intel platforms that have Intel ME, it is now possible to disable Intel ME after BringUp. See:. https:\/\/github.com\/corna\/me_cleaner On all Libreboot systems that have an Intel ME in it (GM45+ICH9M laptops and X4X+ICH10 desktops), the ME firmware is not needed in the boot flash. Either a modified descriptor is used, which disables the ME and removes the region for it in the boot flash, or a descriptorless setup is used. Now onto the main topic:. Introduced in June 2006 in Intel’s 965 Express Chipset Family of (Graphics and) Memory Controller Hubs, or (G)MCHs, and the ICH8 I\/O Controller Family, the Intel Management Engine (ME) is a separate computing environment physically located in the (G)MCH chip. In Q3 2009, the first generation of Intel Core i3\/i5\/i7 (Nehalem) CPUs and the 5 Series Chipset family of Platform Controller Hubs, or PCHs, brought a more tightly integrated ME (now at version 6.0) inside the PCH chip, which itself replaced the ICH. Thus, the ME is. present on all Intel desktop, mobile (laptop), and server systems since mid 2006. . The ME consists of an ARC processor core (replaced with other processor cores in later generations of the ME), code and data caches, a timer, and a secure internal bus to which additional devices are connected, including a cryptography engine, internal ROM and RAM, memory controllers, and a. direct memory access (DMA) engine. to access the host operating system’s memory as well as to reserve a region of protected external memory to supplement the ME’s limited internal RAM. The ME also has. network access. with its own MAC address through an Intel Gigabit Ethernet Controller. Its boot program, stored on the internal ROM, loads a firmware “manifest” from the PC’s SPI flash chip. This manifest is. signed with a strong cryptographic key. , which differs between versions of the ME firmware. If the manifest isn’t signed by a specific Intel key, the boot ROM won’t load and execute the firmware and the ME processor core will be halted. The ME firmware is compressed and consists of modules that are listed in the manifest along with secure cryptographic hashes of their contents. One module is the operating system kernel, which is based on a. proprietary real-time operating system (RTOS) kernel. called “ThreadX”. The developer, Express Logic, sells licenses and source code for ThreadX. Customers such as Intel are forbidden from disclosing or sublicensing the ThreadX source code. Another module is the Dynamic Application Loader (DAL), which consists of a. Java virtual machine. and set of preinstalled Java classes for cryptography, secure storage, etc. The DAL module can load and execute additional ME modules from the PC’s HDD or SSD. The ME firmware also includes a number of native application modules within its flash memory space, including Intel Active Management Technology (AMT), an implementation of a Trusted Platform Module (TPM), Intel Boot Guard, and audio and video DRM systems. The Active Management Technology (AMT) application, part of the Intel “vPro” brand, is a Web server and application code that enables remote users to power on, power off, view information about, and otherwise manage the PC. It can be. used remotely even while the PC is powered off. (via Wake-on-Lan). Traffic is encrypted using SSL\/TLS libraries, but recall that all of the major SSL\/TLS implementations have had highly publicized vulnerabilities. The AMT application itself has. known vulnerabilities , which have been exploited to develop rootkits and keyloggers and covertly gain encrypted access to the management features of a PC. Remember that the ME has full access to the PC’s RAM. This means that an attacker exploiting any of these vulnerabilities may gain access to everything on the PC as it runs: all open files, all running applications, all keys pressed, and more. Intel Boot Guard is an ME application introduced in Q2 2013 with ME firmware version 9.0 on 4th Generation Intel Core i3\/i5\/i7 (Haswell) CPUs. It allows a PC OEM to generate an asymmetric cryptographic keypair, install the public key in the CPU, and prevent the CPU from executing boot firmware that isn’t signed with their private key. This means that. coreboot and libreboot are impossible to port. to such PCs, without the OEM’s private signing key. Note that systems assembled from separately purchased mainboard and CPU parts are unaffected, since the vendor of the mainboard (on which the boot firmware is stored) can’t possibly affect the public key stored on the CPU. ME firmware versions 4.0 and later (Intel 4 Series and later chipsets) include an ME application for. audio and video. DRM called “Protected Audio Video Path” (PAVP). The ME receives from the host operating system an encrypted media stream and encrypted key, decrypts the key, and sends the encrypted media decrypted key to the GPU, which then decrypts the media. PAVP is also used by another ME application to draw an authentication PIN pad directly onto the screen. In this usage, the PAVP application directly controls the graphics that appear on the PC’s screen in a way that the host OS cannot detect. ME firmware version 7.0 on PCHs with 2nd Generation Intel Core i3\/i5\/i7 (Sandy Bridge) CPUs replaces PAVP with a similar DRM application called “Intel Insider”. Like the AMT application, these DRM applications, which in themselves are defective by design, demonstrate the omnipotent capabilities of the ME: this hardware and its proprietary firmware can access and control everything that is in RAM and even. everything that is shown on the screen. . The Intel Management Engine with its proprietary firmware has complete access to and control over the PC: it can power on or shut down the PC, read all open files, examine all running applications, track all keys pressed and mouse movements, and even capture or display images on the screen. And it has a network interface that is demonstrably insecure, which can allow an attacker on the network to inject rootkits that completely compromise the PC and can report to the attacker all activities performed on the PC. It is a threat to freedom, security, and privacy that can’t be ignored. Before version 6.0 (that is, on systems from 2008\/2009 and earlier), the ME can be disabled by setting a couple of values in the SPI flash memory. The ME firmware can then be removed entirely from the flash memory space. libreboot. does this on the Intel 4 Series systems that it supports, such as the. Libreboot X200 and. Libreboot T400 . ME firmware versions 6.0 and later, which are found on all systems with an Intel Core i3\/i5\/i7 CPU and a PCH, include “ME Ignition” firmware that performs some hardware initialization and power management. If the ME’s boot ROM does not find in the SPI flash memory an ME firmware manifest with a valid Intel signature, the whole PC will shut down after 30 minutes. Due to the signature verification, developing free replacement firmware for the ME is basically impossible. The only entity capable of replacing the ME firmware is Intel. As previously stated, the ME firmware includes proprietary code licensed from third parties, so Intel couldn’t release the source code even if they wanted to. And even if they developed completely new ME firmware without third-party proprietary code and released its source code, the ME’s boot ROM would reject any modified firmware that isn’t signed by Intel. Thus, the ME firmware is both hopelessly proprietary and “tivoized”. In summary, the Intel Management Engine and its applications are a backdoor with total access to and control over the rest of the PC. The ME is a threat to freedom, security, and privacy, and the libreboot project strongly recommends avoiding it entirely. Since recent versions of it can’t be removed, this means avoiding all recent generations of Intel hardware. More information about the Management Engine can be found on various Web sites, including. me.bios.io ,. unhuffme ,. coreboot wiki , and. Wikipedia . The book. Platform Embedded Security Technology Revealed describes in great detail the ME’s hardware architecture and firmware application modules. If you’re stuck with the ME (non-libreboot system), you might find this interesting:. http:\/\/hardenedlinux.org\/firmware\/2016\/11\/17\/neutralize_ME_firmware_on_sandybridge_and_ivybridge.html Also see (effort to disable the ME):. https:\/\/www.coreboot.org\/pipermail\/coreboot\/2016-November\/082331.html - look at the whole thread. Firmware Support Package (FSP). [link] On all recent Intel systems, coreboot support has revolved around integrating a blob (for each system) called the. FSP. (firmware support package), which handles all of the hardware initialization, including memory and CPU initialization. Reverse engineering and replacing this blob is almost impossible, due to how complex it is. Even for the most skilled developer, it would take years to replace. Intel distributes this blob to firmware developers, without source. Since the FSP is responsible for the early hardware initialization, that means it also handles SMM (System Management Mode). This is a special mode that operates below the operating system level. It’s possible that rootkits could be implemented there, which could perform a number of attacks on the user (the list is endless). Any Intel system that has the proprietary FSP blob cannot be trusted at all. In fact, several SMM rootkits have been demonstrated in the wild (use a search engine to find them). CPU microcode updates. [link] All modern x86 CPUs (from Intel and AMD) use what is called. microcode. . CPUs are extremely complex, and difficult to get right, so the circuitry is designed in a very generic way, where only basic instructions are handled in hardware. Most of the instruction set is implemented using microcode, which is low-level software running inside the CPU that can specify how the circuitry is to be used, for each instruction. The built-in microcode is part of the hardware, and read-only. Both the circuitry and the microcode can have bugs, which could cause reliability issues. Microcode. updates. are proprietary blobs, uploaded to the CPU at boot time, which patches the built-in microcode and disables buggy parts of the CPU to improve reliability. In the past, these updates were handled by the operating system kernel, but on all recent systems it is the boot firmware that must perform this task. Coreboot does distribute microcode updates for Intel and AMD CPUs, but libreboot cannot, because the whole point of libreboot is to be 100%. free software . On some older Intel CPUs, it is possible to exclude the microcode updates and not have any reliability issues in practise. All current libreboot systems work without microcode updates (otherwise, they wouldn’t be supported in libreboot). However, all modern Intel CPUs require the microcode updates, otherwise the system will not boot at all, or it will be extremely unstable (memory corruption, for example). Intel CPU microcode updates are. signed. , which means that you could not even run a modified version, even if you had the source code. If you try to upload your own modified updates, the CPU will reject them. The microcode updates alter the way instructions behave on the CPU. That means they affect the way the CPU works, in a very fundamental way. That makes it software. The updates are proprietary, and are software, so we exclude them from libreboot. The microcode built into the CPU already is not so much of an issue, since we can’t change it anyway (it’s read-only). Intel is uncooperative. [link] For years, coreboot has been struggling against Intel. Intel has been shown to be extremely uncooperative in general. Many coreboot developers, and companies, have tried to get Intel to cooperate; namely, releasing source code for the firmware components. Even Google, which sells millions of. chromebooks. (coreboot preinstalled) have been unable to persuade them. Even when Intel does cooperate, they still don’t provide source code. They might provide limited information (datasheets) under strict corporate NDA (non-disclosure agreement), but even that is not guaranteed. Even ODMs and IBVs can’t get source code from Intel, in most cases (they will just integrate the blobs that Intel provides). Recent Intel graphics chipsets also. require firmware blobs . Intel is. only going to get worse when it comes to user freedom. Libreboot has no support recent Intel platforms, precisely because of the problems described above. The only way to solve this is to get Intel to change their policies and to be more friendly to the. free software community. Reverse engineering won’t solve anything long-term, unfortunately, but we need to keep doing it anyway. Moving forward, Intel hardware is a non-option unless a radical change happens within Intel. Basically, all Intel hardware from year 2010 and beyond will never be supported by libreboot. The libreboot project is actively ignoring all modern Intel hardware at this point, and focusing on alternative platforms. Why is the latest AMD hardware unsupported in libreboot? [link] It is extremely unlikely that any modern AMD hardware will ever be supported in libreboot, due to severe security and freedom issues; so severe, that. the libreboot project recommends avoiding all modern AMD hardware. If you have an AMD based system affected by the problems described below, then you should get rid of it as soon as possible. . The main issues are as follows:. We call on AMD to release source code and specs for the new AMD Ryzen platforms! We call on the community to put pressure on AMD. Click here to read more AMD Platform Security Processor (PSP). [link] This is basically AMD’s own version of the. Intel Management Engine . It has all of the same basic security and freedom issues, although the implementation is wildly different. The Platform Security Processor (PSP) is built in on the AMD CPUs whose. architecture is Late Family 16h (Puma), Zen 17h or later (and also on the AMD GPUs which are GCN 5th gen (Vega) or later). On the CPUs, a PSP controls the main x86 core startup. PSP firmware is cryptographically signed with a strong key similar to the Intel ME. If the PSP firmware is not present, or if the AMD signing key is not present, the x86 cores will not be released from reset, rendering the system inoperable. The PSP is an ARM core with TrustZone technology, built onto the main CPU die. As such, it has the ability to hide its own program code, scratch RAM, and any data it may have taken and stored from the lesser-privileged x86 system RAM (kernel encryption keys, login data, browsing history, keystrokes, who knows!). To make matters worse, the PSP theoretically has access to the entire system memory space (AMD either will not or cannot deny this, and it would seem to be required to allow the DRM “features” to work as intended), which means that it has at minimum MMIO-based access to the network controllers and any other PCI\/PCIe peripherals installed on the system. In theory any malicious entity with access to the AMD signing key would be able to install persistent malware that could not be eradicated without an external flasher and a known good PSP image. Furthermore, multiple security vulnerabilities have been demonstrated in AMD firmware in the past, and there is every reason to assume one or more zero day vulnerabilities are lurking in the PSP firmware. Given the extreme privilege level (ring -2 or ring -3) of the PSP, said vulnerabilities would have the ability to remotely monitor and control any PSP enabled machine completely outside of the user’s knowledge. Much like with the Intel Boot Guard (an application of the Intel Management Engine), AMD’s PSP can also act as a tyrant by checking signatures on any boot firmware that you flash, making replacement boot firmware (e.g. libreboot, coreboot) impossible on some boards. Early anecdotal reports indicate that AMD’s boot guard counterpart will be used on most OEM hardware, disabled only on so-called “enthusiast” CPUs. AMD IMC firmware. [link] Read. https:\/\/www.coreboot.org\/AMD_IMC . AMD SMU firmware. [link] Handles some power management for PCIe devices (without this, your laptop will not work properly) and several other power management related features. The firmware is signed, although on older AMD hardware it is a symmetric key, which means that with access to the key (if leaked) you could sign your own modified version and run it. Rudolf Marek (coreboot hacker) found out how to extract this key. in this video demonstration , and based on this work, Damien Zammit (another coreboot hacker). partially replaced it with free firmware, but on the relevant system (ASUS F2A85-M) there were still other blobs present (Video BIOS, and others) preventing the hardware from being supported in libreboot. AMD AGESA firmware. [link] This is responsible for virtually all core hardware initialization on modern AMD systems. In 2011, AMD started cooperating with the coreboot project, releasing this as source code under a free license. In 2014, they stopped releasing source code and started releasing AGESA as binary blobs instead. This makes AGESA now equivalent to. Intel FSP . AMD CPU microcode updates. [link] Read the Intel section practically the same, though it was found with much later hardware in AMD that you could run without microcode updates. It’s unknown whether the updates are needed on all AMD boards (depends on CPU). AMD is incompetent (and uncooperative). [link] AMD seemed like it was on the right track in 2011 when it started cooperating with and releasing source code for several critical components to the coreboot project. It was not to be. For so-called economic reasons, they decided that it was not worth the time to invest in the coreboot project anymore. Unfortunately they haven’t even shared the source code of AGESA library for a Family 15h Steamroller\/Excavator architectures (which, like the earlier fam15h Bulldozer\/Piledriver, do not have a PSP) and released it to a coreboot project only as a binary. For a company to go from being so good, to so bad, in just 3 years, shows that something is seriously wrong with AMD. Like Intel, they do not deserve your money. Given the current state of Intel hardware with the Management Engine, it is our opinion that all performant x86 hardware newer than the AMD Family 15h CPUs (on AMD’s side) or anything post-2009 on Intel’s side is defective by design and cannot safely be used to store, transmit, or process sensitive data. Sensitive data is any data in which a data breach would cause significant economic harm to the entity which created or was responsible for storing said data, so this would include banks, credit card companies, or retailers (customer account records), in addition to the “usual” engineering and software development firms. This also affects whistleblowers, or anyone who needs actual privacy and security. What. can. I use, then? [link] Libreboot has support for AMD hardware of Family 15h (Bulldozer or Piledriver, ~2012 gen) and some older Intel platforms like Napa, Montevina, Eagle Lake, Lakeport (2004-2006). We also have support for some ARM chipsets (rk3288). On the Intel side, we’re also interested in some of the chipsets that use Atom CPUs (rebranded from older chipsets, mostly using ich7-based southbridges). Will libreboot work on a ThinkPad T400 or T500 with an ATI GPU? [link] Short answer: yes. These laptops also have an Intel GPU inside, which libreboot uses. The ATI GPU is ignored by libreboot. These laptops use what is called. switchable graphics. , where it will have both an Intel and ATI GPU. Coreboot will allow you to set (using nvramtool) a parameter, specifying whether you would like to use Intel or ATI. The ATI GPU lacks free native graphics initialization in coreboot, unlike the Intel GPU. Libreboot modifies coreboot, in such a way where this nvramtool setting is ignored. Libreboot will just assume that you want to use the Intel GPU. Therefore, the ATI GPU is completely disabled on these laptops. Intel is used instead, with the free native graphics initialization (VBIOS replacement) that exists in coreboot. Will desktop\/server hardware be supported? [link] Libreboot now supports desktop hardware:. (see list) (with full native video initialization). A common issue with desktop hardware is the Video BIOS, when no onboard video is present, since every video card has a different Video BIOS. Onboard GPUs also require one, so those still have to be replaced with free software (non-trivial task). Libreboot has to initialize the graphics chipset, but most graphics cards lack a free Video BIOS for this purpose. Some desktop motherboards supported in coreboot do have onboard graphics chipsets, but these also require a proprietary Video BIOS, in most cases. Hi, I have , is it supported? [link] Most likely not. First, you must consult coreboot’s own hardware compatibility list at. http:\/\/www.coreboot.org\/Supported_Motherboards and, if it is supported, check whether it can run without any proprietary blobs in the ROM image. If it can: wonderful! Libreboot can support it, and you can add support for it. If not, then you will need to figure out how to reverse engineer and replace (or remove) those blobs that do still exist, in such a way where the system is still usable in some defined way. For those systems where no coreboot support exists, you must first port it to coreboot and, if it can then run without any blobs in the ROM image, it can be added to libreboot. See:. Motherboard Porting Guide (this is just the tip of the iceberg!). Please note that board development should be done upstream (in coreboot) and merged downstream (into libreboot). This is the correct way to do it, and it is how the libreboot project is coordinated so as to avoid too much forking of the coreboot source code. What about ARM? [link] Libreboot has support for some ARM based laptops, using the. Rockchip RK3288. SoC. Check the libreboot. hardware compatibility list , for more information. General questions. [link] How do I install libreboot? [link] See. installation guide How do I program an SPI flash chip? [link] Refer to:. Externally rewrite 25xx NOR flash via SPI protocol It’s possible to use a 16-pin SOIC test clip on an 8-pin SOIC chip, if you align the pins properly. The connection is generally more sturdy. How do I write-protect the flash chip? [link] By default, there is no write-protection on a libreboot system. This is for usability reasons, because most people do not have easy access to an external programmer for re-flashing their firmware, or they find it inconvenient to use an external programmer. On some systems, it is possible to write-protect the firmware, such that it is rendered read-only at the OS level (external flashing is still possible, using dedicated hardware). For example, on current GM45 laptops (e.g. ThinkPad X200, T400), you can write-protect (see. ICH9 gen utility ). It’s possible to write-protect on all libreboot systems, but the instructions need to be written. The documentation is in the main git repository, so you are welcome to submit patches adding these instructions. TODO: Document PRx based flash protection on Intel platforms, and investigate other methods on AMD systems. How do I change the BIOS settings? [link] Libreboot actually uses the. GRUB payload . More information about payloads can be found at. coreboot.org\/Payloads . SeaBIOS is also available. The. CMOS. config is hardcoded in Libreboot. Libreboot inherits the modular payload concept from coreboot, which means that pre-OS bare-metal. BIOS setup. programs are not very practical. Coreboot (and libreboot) does include a utility called. nvramtool. , which can be used to change some settings. You can find nvramtool under. coreboot\/util\/nvramtool\/. , in the libreboot source archives. The. -a. option in nvramtool will list the available options, and. -w. can be used to change them. Consult the nvramtool documentation on the coreboot wiki for more information. In practise, you don’t need to change any of those settings, in most cases. Libreboot locks the CMOS table, to ensure consistent functionality for all users. You can use:. nvramtool -C yourrom.rom -w somesetting=somevalue. To get a full list of available options, do this:. nvramtool -C yourrom.rom -a. This will change the default inside that ROM image, and then you can re-flash it. How do I pad a ROM before flashing? [link] It is advisable to simply use a larger ROM image. This section was written mostly for ASUS KCMA-D8 and KGPE-D16 mainboards, where previously we only provided 2MiB ROM images in Libreboot, but we now provide 16MiB ROM images. Other sizes are not provided because in practise, someone upgrading one of these chips will just use a 16MiB one. Larger sizes are available, but 16MiB is the maximum that you can use on all currently supported Libreboot systems that use SPI flash. Required for ROMs where the ROM image is smaller than the flash chip (e.g. writing a 2MiB ROM to a 16MiB flash chip). Create an empty (00 bytes) file with a size the difference between the ROM and flash chip. The case above, for example:. truncate -s +14MiB pad.bin. For x86 descriptorless images you need to pad from the. beginning. of the ROM:. cat pad.bin yourrom.rom yourrom.rom.new. For ARM and x86 with intel flash descriptor, you need to pad after the image:. cat yourrom.rom pad.bin yourrom.rom.new. Flash the resulting file. Note that cbfstool will not be able to operate on images padded this way so make sure to make all changes to the image, including runtime config, before padding. To remove padding, for example after reading it off the flash chip, simply use dd(1) to extract only the non-padded portion. Continuing with the examples above, in order to extract a 2MiB x86 descriptorless ROM from a padded 16MiB image do the following:. dd if=flashromread.rom of=yourrom.rom ibs=14MiB skip=1. With padding removed cbfstool will be able to operate on the image as usual. Do I need to install a bootloader when installing a distribution? [link] Libreboot integrates the GRUB bootloader already, as a. payload . This means that the GRUB bootloader is actually. flashed. , as part of the boot firmware (libreboot). This means that you do not have to install a boot loader on the HDD or SSD, when installing a new distribution. You’ll be able to boot just fine, using the bootloader (GRUB) that is in the flash chip. This also means that even if you remove the HDD or SSD, you’ll still have a functioning bootloader installed which could be used to boot a live distribution installer from a USB flash drive. See. How to install GNU+Linux on a libreboot system Nowadays, other payloads are also provided. If you’re using the SeaBIOS payload, then the normal MBR bootsector is used on your HDD or SSD, like you would expect. So the above paragraphs only apply to the GNU GRUB payload. Do I need to re-flash when I re-install a distribution? [link] Not anymore. Recent versions of libreboot (using the GRUB payload) will automatically switch to a GRUB configuration on the HDD or SSD, if it exists. You can also load a different GRUB configuration, from any kind of device that is supported in GRUB (such as a USB flash drive). For more information, see. Modifying the GRUB Configuration in Libreboot Systems If you’re using the SeaBIOS payload, it’s even easier. It works just like you would expect. SeaBIOS implements a normal x86 BIOS interface. What does a flash chip look like? [link] You can find photos of various chip types on the following page:. External 25xx NOR flashing guide Who did the logo? [link] See the. license information . The Libreboot logo is available as a. bitmap , a. vector , or a. greyscale vector . Libreboot Inside stickers are available as a. PDF or a. vector You can find all of the available logos by browsing this directory:. https:\/\/av.libreboot.org\/logo\/ What other firmware exists outside of libreboot? [link] The main freedom issue on any system, is the boot firmware (usually referred to as a BIOS or UEFI). Libreboot replaces the boot firmware with fully free code, but even with libreboot, there may still be other hardware components in the system (e.g. laptop) that run their own dedicated firmware, sometimes proprietary. These are on secondary processors, where the firmware is usually read-only, written for very specific tasks. While these are unrelated to libreboot, technically speaking, it makes sense to document some of the issues here. Note that these issues are not unique to libreboot systems. They apply universally, to most systems. The issues described below are the most common (or otherwise critical). Dealing with these problems will most likely be handled by a separate project. External GPUs. [link] The Video BIOS is present on most video cards. For integrated graphics, the VBIOS (special kind of OptionROM) is usually embedded in the main boot firmware. For external graphics, the VBIOS is usually on the graphics card itself. This is usually proprietary; the only difference is that SeaBIOS can execute it (alternatively, you embed it in a coreboot ROM image and have coreboot executes it, if you use a different payload, such as GRUB). On current libreboot systems, instead of VBIOS, coreboot native GPU init is used, which is currently only implemented for Intel GPUs. Other cards with proper KMS drivers can be initialized once Linux boots, but copy of VBIOS may be still needed to fetch proper VRAM frequency and other similar parameters (without executing VBIOS code). In configurations where SeaBIOS and native GPU init are used together, a special shim VBIOS is added that uses coreboot linear framebuffer. NOTE: on desktop\/server mainboards, coreboot is configured to load the option ROM from an add-on GPU if present. This is the default behaviour on such systems in Libreboot. EC (embedded controller) firmware. [link] Most (all?) laptops have this. The EC (embedded controller) is a small, separate processor that basically processes inputs\/outputs that are specific to laptops. For example:. When you flick the radio on\/off switch, the EC will enable\/disable the wireless devices (wifi, bluetooth, etc) and enable\/disable an LED that indicates whether it’s turned on or not. Listen to another chip that produces temperature readings, adjusting fan speeds accordingly (or turning the fan(s) on\/off). Takes certain inputs from the keyboard, e.g. brightness up\/down, volume up\/down. Detect when the lid is closed or opened, and send a signal indicating this. Etc. EC is present on nearly all laptops. Other devices use, depending on complexity, either EC or variant with firmware in Mask ROM - SuperIO. HDD\/SSD firmware. [link] HDDs and SSDs have firmware in them, intended to handle the internal workings of the device while exposing a simple, standard interface (such as AHCI\/SATA) that the OS software can use, generically. This firmware is transparent to the user of the drive. HDDs and SSDs are quite complex, and these days contain quite complex hardware which is even capable of running an entire operating system (by this, we mean that the drive itself is capable of running its own embedded OS), even GNU+Linux or BusyBox\/Linux. SSDs and HDDs are a special case, since they are persistent storage devices as well as computers. Example attack that malicious firmware could do: substitute your SSH keys, allowing unauthorized remote access by an unknown adversary. Or maybe substitute your GPG keys. SATA drives can also have DMA (through the controller), which means that they could read from system memory; the drive can have its own hidden storage, theoretically, where it could read your LUKS keys and store them unencrypted for future retrieval by an adversary. With proper IOMMU and use of USB instead of SATA, it might be possible to mitigate any DMA-related issues that could arise. Some proof of concepts have been demonstrated. For HDDs:. https:\/\/spritesmods.com\/?art=hddhack&page=1 For SSDs:. http:\/\/www.bunniestudios.com\/blog\/?p=3554 Viable free replacement firmware is currently unknown to exist. For SSDs, the. OpenSSD project may be interesting. Apparently, SATA drives themselves don’t have DMA but can make use of it through the controller. This. http:\/\/www.lttconn.com\/res\/lttconn\/pdres\/201005\/20100521170123066.pdf (pages 388-414, 420-421, 427, 446-465, 492-522, 631-638) and this. http:\/\/www.intel.co.uk\/content\/dam\/www\/public\/us\/en\/documents\/technical-specifications\/serial-ata-ahci-spec-rev1_3.pdf (pages 59, 67, 94, 99). The following is based on discussion with Peter Stuge (CareBear\\) in the coreboot IRC channel on Friday, 18 September 2015, when investigating whether the SATA drive itself can make use of DMA. The following is based on the datasheets linked above:. According to those linked documents, FIS type 39h is. “DMA Activate FIS - Device to Host”. . It mentions. “transfer of data from the host to the device, and goes on to say: Upon receiving a DMA Activate, if the host adapter’s DMA controller has been programmed and armed, the host adapter shall initiate the transmission of a Data FIS and shall transmit in this FIS the data corresponding to the host memory regions indicated by the DMA controller’s context.”. FIS is a protocol unit (Frame Information Structure). Based on this, it seems that a drive can tell the host controller that it would like for DMA to happen, but unless the host software has already or will in the future set up this DMA transfer then nothing happens. A drive can also send DMA Setup. . If a DMA Setup FIS is sent first, with the Auto-Activate bit set, then it is already set up, and the drive can initiate DMA. The document goes on to say. “Upon receiving a DMA Setup, the receiver of the FIS shall validate the received DMA Setup request.”. - in other words, the host is supposed to validate; but maybe there’s a bug there. The document goes on to say. “The specific implementation of the buffer identifier and buffer\/address validation is not specified”. - so noone will actually bother. “the receiver of the FIS”. - in the case we’re considering, that’s the host controller hardware in the chipset and\/or the kernel driver (most likely the kernel driver). All SATA devices have flash-upgradeable firmware, which can usually be updated by running software in your operating system;. malicious software running as root could update this firmware, or the firmware could already be malicious. . Your HDD or SSD is the perfect place for a malicious adversary to install malware, because it’s a persistent storage device as well as a computer. Based on this, it’s safe to say that use of USB instead of SATA is advisable if security is a concern. USB 2.0 has plenty of bandwidth for many HDDs (a few high-end ones can use more bandwidth than USB 2.0 is capable of), but for SSDs it might be problematic (unless you’re using USB 3.0, which is not yet usable in freedom. See. Use of USB is also not an absolute guarantee of safety, so do beware. The attack surface becomes much smaller, but a malicious drive could still attempt a “fuzzing” attack (e.g. sending malformed USB descriptors, which is how the tyrant DRM on the Playstation 3 was broken, so that users could run their own operating system and run unsigned code). (you’re probably safe, unless there’s a security flaw in the USB library\/driver that your OS uses. USB is generally considered one of the safest protocols, precisely because USB devices have no DMA). Other links:. https:\/\/www.vice.com\/en_us\/article\/ypwkwk\/the-nsas-undetectable-hard-drive-hack-was-first-demonstrated-a-year-ago It is recommended that you use full disk encryption, on HDDs connected via USB. There are several adapters available online, that allow you to connect SATA HDDs via USB. Libreboot documents how to install several distributions with full disk encryption. You can adapt these for use with USB drives:. Full disk encryption with Debian Full disk encryption with Parabola The current theory (unproven) is that this will at least prevent malicious drives from wrongly manipulating data being read from or written to the drive, since it can’t access your LUKS key if it’s only ever in RAM, provided that the HDD doesn’t have DMA (USB devices don’t have DMA). The worst that it could do in this case is destroy your data. Of course, you should make sure never to put any keyfiles in the LUKS header. Take what this paragraph says with a pinch of salt. This is still under discussion, and none of this is proven. NIC (ethernet controller). [link] Ethernet NICs will typically run firmware inside, which is responsible for initializing the device internally. Theoretically, it could be configured to drop packets, or even modify them. With proper IOMMU, it might be possible to mitigate the DMA-related issues. A USB NIC can also be used, which does not have DMA. CPU microcode. [link] Implements an instruction set. See description. Here we mean microcode built in to the CPU. We are not talking about the updates supplied by the boot firmware (libreboot does not include microcode updates, and only supports systems that will work without it) Microcode can be very powerful. No proof that it’s malicious, but it could theoretically. There isn’t really a way to solve this, unless you use a CPU which does not have microcode. (ARM CPUs don’t, but most ARM systems require blobs for the graphics hardware at present, and typically have other things like soldered wifi which might require blobs). CPUs often on modern systems have a processor inside it for things like power management. ARM for example, has lots of these. Sound card. [link] Sound hardware (integrated or discrete) typically has firmware on it (DSP) for processing input\/output. Again, a USB DAC is a good workaround. Webcam. [link] Webcams have firmware integrated into them that process the image input into the camera; adjusting focus, white balancing and so on. Can use USB webcam hardware, to work around potential DMA issues; integrated webcams (on laptops, for instance) are discouraged by the libreboot project. USB host controller. [link] Doesn’t really apply to current libreboot systems (none of them have USB 3.0 at the moment), but USB 3.0 host controllers typically rely on firmware to implement the XHCI specification. Some newer coreboot ports also require this blob, if you want to use USB 3.0. This doesn’t affect libreboot at the moment, because all current systems that are supported only have older versions of USB available. USB devices also don’t have DMA (but the USB host controller itself does). With proper IOMMU, it might be possible to mitigate the DMA-related issues (with the host controller). WWAN firmware. [link] Some laptops might have a simcard reader in them, with a card for handling WWAN, connecting to a 3g\/4g (e.g. GSM) network. This is the same technology used in mobile phones, for remote network access (e.g. internet). NOTE: not to be confused with wifi. Wifi is a different technology, and entirely unrelated. The baseband processor inside the WWAN chip will have its own embedded operating system, most likely proprietary. Use of this technology also implies the same privacy issues as with mobile phones (remote tracking by the GSM network, by triangulating the signal). On some laptops, these cards use USB (internally), so won’t have DMA, but it’s still a massive freedom and privacy issue. If you have an internal WWAN chip\/card, the libreboot project recommends that you disable and (ideally, if possible) physically remove the hardware. If you absolutely must use this technology, an external USB dongle is much better because it can be easily removed when you don’t need it, thereby disabling any external entities from tracking your location. Use of ethernet or wifi is recommended, as opposed to mobile networks, as these are generally much safer. On all current libreboot laptops, it is possible to remove the WWAN card and sim card if it exists. The WWAN card is next to the wifi card, and the sim card (if installed) will be in a slot underneath the battery, or next to the RAM. Operating Systems. [link] Can I use GNU+Linux? [link] Absolutely! It is well-tested in libreboot, and highly recommended. See. installing GNU+Linux and. booting GNU+Linux . Any recent distribution should work, as long as it uses KMS (kernel mode setting) for the graphics. Fedora won’t boot? (may also be applicable to Redhat\/CentOS). [link] On Fedora, by default the grub.cfg tries to boot linux in 16-bit mode. You just have to modify Fedora’s GRUB configuration. Refer to. the GNU+Linux page . Can I use BSD? [link] Absolutely! Libreboot has native support for NetBSD, OpenBSD and LibertyBSD. Other distros are untested. See:. docs\/bsd\/ Are other operating systems compatible? [link] Unknown. Probably not. Does libreboot make my machine 100% free? [link] Libreboot on all devices only provides host hardware init firmware images, that can be written 25XX SPI NOR Flash. But on many systems there are a lot more computers running blob firmware. Some of them are not practicable to replace due to being located on Mask ROM. Some devices have EC firmware being build as well. Additionally, besides software components, there are hardware ones (from ICs to boards) that are not released on OSHW licenses. We do not have a single device that would be “100% free”, and such absolutes are nearly impossible to reach. Notable proprietary blobs (not a complete list):. All devices. SATA\/PATA Hard Drive\/Optical Disc Drive Firmware (. often contain powerful ARM based computer ). Pendrives and any USB peripherals - they contain a computer with code running to at the very least handle the USB protocol. ThinkPads:. EC Firmware (H8S until including Sandy Bridge, later ARC based MEC16xx). TrackPoint Firmware (8051). Penabled devices contain µPD78F0514 MCU on wacom subboard, and Atmega (AVR) on led indicator\/button board. Battery BMS, bq8030 (CoolRISC C816). Chomebooks C201PA\/C100PA:. Battery BMS, bq30z55. Elan Touchpad. eMMC. flash memory controller firmware. Where can I learn more about electronics. [link] Basics of soldering and rework by PACE. Both series of videos are mandatory regardless of your soldering skill. Basic Soldering Rework and Repair edX course on basics of electronics In most countries contents of this course is covered during middle and high school. It will also serve well to refresh your memory if you haven’t used that knowledge ever since. Impedance intro. Similiarities of Wave Behavior Reflections in tranmission line Stubs:. Wikipedia article on stubs Polar Instruments article on stubs With external SPI flashing we only care about unintended PCB stubs. How to accurately measure header\/connector pitch Other YouTube channels with useful content about electronics. EEVblog Louis Rossmann mikeselectricstuff bigclive ElectroBOOM Jeri Ellsworth Boardview files can be open with. OpenBoardview , which is free software under MIT license. Use of youtube-dl with mpv would be recommended for youtube links. Lastly the most important message to everybody gaining this wonderful new hobby -. Secret to Learning Electronics Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/faq.md Subscribe to RSS for this site Site map This HTML page was generated by the. untitled static site generator ."}
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Звільни свій BIOS сьогодні! Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled English |. українська Libreboot це. поважаюча свободу прошивка. , яка виконує ініціалізацію апаратного забезпечення (такого як - контролер памяті, центральний процесор, переферія) на. деяких комп’ютерах Intel\/AMD x86 та розпочинає завантажувач для вашої операційної системи. GNU+Linux та. BSD добре підтримуються. Замінює пропрієтарну прошивку BIOS\/UEFI. Допомога доступна через. #libreboot на. Libera IRC. Остання версія. Libreboot 20211122 , була випущена 22 листопада 2021 року. Приєднуйтесь до нас зараз і прошивайте прошивку! [link] У вас є права. Право на конфіденційність, свобода думки, свобода мови а також право читати. Вільне програмне забезпечення дає вам ці права. Ваша свобода має значення. Право на ремонт має значення. Багато людей використовує. пропрієтарну прошивку, навіть якщо вони використовують. GNU+Linux . Невільна прошивка часто. містить лазівки , та може бути забагованою. Libreboot був заснований в грудні 2013 року, з метою зробити вільне програмне забезпечення на рівні прошивки доступним для нетехнічних користувачів. Можна сказати, що Libreboot з відкритим вихідним кодом,. але вам варто називати його вільне програмне забезпечення . Libreboot використовує. coreboot для. апаратної ініціалізації . Coreboot неординарно складно встановити для більшості нетехнічних користувачів; він виконує лише базову ініціалізацію та перестрибує до окремої програми. корисного навантаження (таких як -. GRUB ,. Tianocore ), які також потрібно налаштовувати. Libreboot вирішує цю проблему. ; це. дистрибутив coreboot. з. автоматизованою системою побудови , який створює. ROM образи. , для більш міцної установки. Документація надається. Libreboot виключає бінарні блоби, доставляючи тільки вільне програмне забезпечення і, власним станом, підтримує тільки невелику кількість пристроїв з coreboot. Ви можете ознайомитись з політикою відсутності двійкових блобів Libreboot. Сторінка політики блобів в Libreboot . Чим Libreboot відрізняється від звичайного coreboot? [link] Всупереч поширеній думці, первинна мета Libreboot не полягає в тому, щоб забезпечити налаштування вичищеного від двійкових блобів coreboot; це просто одна з політик Libreboot, і вона важлива, але це не зовсім значний аспект Libreboot. Таким самим чином, як Trisquel є дистрибутивом GNU+Linux, Libreboot є. дистрибутивом coreboot. . Якщо ви хочете створити образ ROM від нуля, вам в інакшому випадку треба виконати експертну конфігурацію coreboot, GRUB та будь-якого іншого програмного забезпечення, яке вам потрібно, щоб підготувати образ ROM. З. Libreboot. ви можете буквально завантажити з Git або архіву вихідного коду, та запустити. make. , і таким чином будуть побудовані всі образи ROM. Автоматична система побудови Libreboot, названа. lbmk. (Libreboot MaKe), будує ці образи ROM автоматично, без будь-якого введення або втручання користувача. Конфігурація вже була виконана заздалегідь. Якщо складати звичайний coreboot, без використання автоматизованої системи побудови Libreboot, це потребувало би набагато більше інтервенцій та пристойних технічних знань для створення робочої конфігурації. Регулярні бінарні релізи Libreboot надають ці ROM образи попередньо зібраними, і ви можете просто встановити їх, не маючи спеціальних знань або навичок, крім можливості дотримуватися. спрощених інструкцій, написаних для нетехнічних користувачів . Як допомогти. [link] Перевірити сторінку. завдань і підібрати завдання на роботу. Ви також можете перевірити помилки, перераховані на. трекері помилок . Якщо ви помітите помилку і маєте виправлення,. ось інструкції щодо того, як відправити патчі , також ви можете повідомити про це. Також, цей сайт написаний в Markdown і розміщений в. окремому репозиторію , де можна надсилати патчі. Обговорення розробки Libreboot та підтримка користувачів відбувається в IRC каналі. Більше інформації на. сторінці зворотнього зв’язку . Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown:. https:\/\/libreboot.org\/index.uk.md Індекс сайта Ця сторінка була створена з. untitled static site generator ."}
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Contact. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. User support Development discussion IRC chatroom Social media Twitter and Mastodon Reddit Return to index TODO: mailing lists, mastodon server and peertube account. User support. [link] IRC or Reddit are recommended, if you wish to ask for support (IRC recommended). See below for information about IRC and Reddit. Development discussion. [link] Mailing lists are planned for the future. For now, see notes on. the Git page for information about how to assist with development. Instructions are also on that page for sending patches (via pull requests). IRC chatroom. [link] IRC is the main way to contact the Libreboot project. #libreboot. on Libera IRC. Webchat:. https:\/\/web.libera.chat\/#libreboot Libera is one of the largest IRC networks, used for Free Software projects. Find more about them here:. https:\/\/libera.chat\/ If you wish to connect using your preferred client (such as weechat or irssi), the connection info is as follows:. Server:. irc.libera.chat. Channel:. #libreboot. Port (TLS):. 6697. Port (non-TLS):. 6667. We recommend that you use port. 6697. with TLS encryption enabled. It is recommend that you use SASL for authentication. These pages on the Libera website tells you how:. WeeChat SASL guide:. https:\/\/libera.chat\/guides\/weechat Irssi SASL guide:. https:\/\/libera.chat\/guides\/irssi HexChat SASL guide:. https:\/\/libera.chat\/guides\/hexchat In general, you should check the documentation provided by your IRC software. Social media. [link] Libreboot exists officially on many places. Twitter and Mastodon. [link] News announcements:. https:\/\/twitter.com\/libreboot\/ The founder and lead developer, Leah Rowe, is also on Twitter and Mastodon:. https:\/\/twitter.com\/n4of7 (use nitter to avoid non-free JavaScript). https:\/\/mas.to\/@libreleah Leah can also be contacted by her email address:. leah@libreboot.org Reddit. [link] Mostly used as a support channel, and also for news announcements:. https:\/\/www.reddit.com\/r\/libreboot\/ Binary blob policy Edit this page Who develops Libreboot? 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{"id":"lfDPysQ1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":22,"url_protocol_s":"https","url_paths_count_i":0,"content_type":["text\/html"],"crawldepth_i":0,"collection_sxt":["user"],"title":["Libreboot – Free your BIOS today!"],"last_modified":"Sat Apr 02 12:48:53 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Mon Nov 22 00:00:00 UTC 2021"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-3997163402613940010,"exact_signature_unique_b":true,"fuzzy_signature_l":-6152495941177222570,"fuzzy_signature_unique_b":true,"h1_txt":["Free your BIOS today!"],"h2_txt":["Join us now and flash the firmware!","How does Libreboot differ from regular coreboot?","How to help"],"bold_txt":["coreboot distribution","Libreboot","Libreboot excludes binary blobs, shipping only Free Software and, as such, only supports a handful of machines from coreboot. 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Libreboot 20211122 , released on 22 November 2021. Join us now and flash the firmware! [link] You have rights. The right to privacy, freedom of thought, freedom of speech and the right to read. Free software gives you these rights. Your freedom matters. Right to repair matters. Many people use. proprietary boot firmware, even if they use. GNU+Linux . Non-free firmware often. contains backdoors , and can be buggy. Libreboot was founded in in December 2013, with the express purpose of making Free Software accessible for non-technical users at the firmware level. Libreboot can be called Open Source,. but you should call it Free Software . Libreboot uses. coreboot for. hardware initialization . Coreboot is notoriously difficult to install for most non-technical users; it handles only basic initialization and jumps to a separate. payload program (e.g. GRUB ,. Tianocore ), which must also be configured. Libreboot solves this problem. ; it is a. coreboot distribution. with an. automated build system that builds complete. ROM images. , for more robust installation. Documentation is provided. Libreboot excludes binary blobs, shipping only Free Software and, as such, only supports a handful of machines from coreboot. You can read Libreboot’s zero-blobs policy on the. Libreboot blob policy page . How does Libreboot differ from regular coreboot? [link] Contrary to popular opinion, Libreboot’s primary purpose is not to provide a de-blobbed coreboot setup; it is merely one of Libreboot’s policies, and an important one, but it is nonetheless a minor aspect of Libreboot. In the same way that Trisquel is a GNU+Linux distribution, Libreboot is a. coreboot distribution. . If you want to build a ROM image from scratch, you otherwise have to perform expert-level configuration of coreboot, GRUB and whatever other software you need, to prepare the ROM image. With. Libreboot. , you can literally download from Git or a source archive, and run. make. , and it will build entire ROM images. Libreboot’s automated build system, named. lbmk. (Libreboot MaKe), builds these ROM images automatically, without any user input or intervention required. Configuration has already been performed in advance. If you were to build regular coreboot, without using Libreboot’s automated build system, it would require a lot more intervention and decent technical knowledge to produce a working configuration. Reguar binary releases of Libreboot provide these ROM images pre-compiled, and you can simply install them, with no special knowledge or skill except the ability to follow. simplified instructions, written for non-technical users . How to help. [link] Check the. tasks page and pick a task to work on. You can also check bugs listed on the. bug tracker . If you spot a bug and have a fix,. here are instructions for how to send patches , and you can also report it. Also, this entire website is written in Markdown and hosted in a. separate repository where you can send patches. Libreboot development discussion and user support are all done on the IRC channel. More information is on the. contact page . Binary blob policy Edit this page Who develops Libreboot? 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YOUR HELP IS NEEDED!","Binary blob extermination policy","Tasks","Template license","Libreboot logo license","Project contributors","Who develops Libreboot?","","Subscribe to RSS for this site","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:39 UTC 2022","fresh_date_dt":"Tue Jun 14 11:42:29 UTC 2022","language_s":"en","size_i":12401,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078539361779712,"text_t":"Libreboot – Site map. Site map. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to index Explore the vast mountains of libreland that is libreboot.org! Bots are very much welcome, in this land. \/docs\/bsd\/freebsd.html:. How to install FreeBSD on x86 GNU GRUB payload \/docs\/bsd\/:. BSD operating systems \/docs\/bsd\/netbsd.html:. How to install NetBSD on x86 GNU GRUB payload \/docs\/bsd\/openbsd.html:. How to install OpenBSD on x86 GNU GRUB payload \/docs\/build\/:. Build from source \/docs\/depthcharge\/:. Depthcharge payload \/docs\/gnulinux\/grub_boot_installer.html:. Installing GNU+Linux \/docs\/gnulinux\/grub_cbfs.html:. Modifying grub.cfg in CBFS \/docs\/gnulinux\/grub_hardening.html:. Hardening GNU GRUB \/docs\/gnulinux\/encrypted_debian.html:. Encrypted Debian GNU+Linux \/docs\/gnulinux\/guix.html:. Encrypted Guix GNU+Linux \/docs\/gnulinux\/:. GNU+Linux guides \/docs\/grub\/:. GRUB payload \/docs\/hardware\/c201.html:. ASUS Chromebook C201 \/docs\/hardware\/d510mo.html:. Intel D510MO and D410PT desktop boards \/docs\/hardware\/imac52.html:. Apple iMac 5,2 \/docs\/hardware\/kcma-d8.html:. ASUS KCMA-D8 desktop\/workstation board \/docs\/hardware\/kfsn4-dre.html:. ASUS KFSN4-DRE server\/workstation board \/docs\/hardware\/mac_address.html:. Changing the MAC address \/docs\/hardware\/r400.html:. ThinkPad R400 \/docs\/hardware\/r500.html:. ThinkPad R500 \/docs\/hardware\/t400.html:. ThinkPad T400 \/docs\/hardware\/t500.html:. ThinkPad T500 \/docs\/hardware\/ga-g41m-es2l.html:. Gigabyte GA-G41M-ES2L desktop board \/docs\/hardware\/kgpe-d16.html:. ASUS KGPE-D16 server\/workstation board \/docs\/hardware\/macbook21.html:. MacBook2,1 and MacBook1,1 \/docs\/hardware\/d945gclf.html:. Intel D945GCLF desktop board \/docs\/hardware\/x200.html:. ThinkPad X200 \/docs\/hardware\/:. Hardware compatibility list \/docs\/install\/ga-g41m-es2l.html:. GA-G41M-ES2L flashing tutorial \/docs\/install\/ich9utils.html:. ich9utils \/docs\/install\/kgpe-d16.html:. KGPE-D16 external flashing instructions \/docs\/install\/r400_external.html:. Flashing the ThinkPad R400 \/docs\/install\/t400_external.html:. Flashing the ThinkPad T400 externally \/docs\/install\/t60_unbrick.html:. ThinkPad T60 Recovery guide \/docs\/install\/x200_external.html:. First-time ThinkPad X200 flashing \/docs\/install\/x60_unbrick.html:. ThinkPad X60 Recovery guide \/docs\/install\/x60tablet_unbrick.html:. ThinkPad X60 Tablet Recovery guide \/docs\/install\/t500_external.html:. Flashing the T500 with a BeagleBone Black \/docs\/install\/c201.html:. ASUS Chromebook C201 installation guide \/docs\/install\/:. Installation instructions \/docs\/install\/d945gclf.html:. Intel D945GCLF flashing tutorial \/docs\/install\/d510mo.html:. D510MO flashing tutorial \/docs\/install\/spi.html:. Read\/write 25XX NOR flash via SPI protocol \/docs\/maintain\/:. lbmk maintenance manual \/docs\/misc\/codenames.html:. Product Codenames \/docs\/misc\/:. Miscellaneous \/docs\/release\/:. Libreboot releases \/docs\/release.html:. Libreboot releases \/docs\/:. Documentation \/license.html:. License \/news\/libreboot20131212.html:. Libreboot 20131212 \/news\/libreboot20131213.html:. Libreboot 20131213 \/news\/libreboot20131214.html:. Libreboot 20131214 release \/news\/libreboot20140221.html:. Libreboot 20140221 release \/news\/libreboot20140309.html:. Libreboot 20140309 release \/news\/libreboot20140605.html:. Libreboot 20140605 release \/news\/libreboot20140611.html:. Libreboot 20140611 release \/news\/libreboot20140622.html:. Libreboot 20140622 release \/news\/libreboot20140711.html:. Libreboot 20140711 release \/news\/libreboot20140716.html:. Libreboot 20140716 release \/news\/libreboot20140720.html:. Libreboot 20140720 release \/news\/libreboot20140729.html:. Libreboot 20140729 release \/news\/libreboot20140811.html:. Libreboot 20140811 release \/news\/libreboot20140903.html:. Libreboot 20140903 release \/news\/libreboot20140911.html:. Libreboot 20140911 release \/news\/libreboot20141015.html:. Libreboot 20141015 release \/news\/libreboot20150124.html:. Libreboot 20150124 release \/news\/libreboot20150126.html:. Libreboot 20150126 release \/news\/libreboot20150208.html:. Libreboot 20150208 release \/news\/libreboot20150518.html:. Libreboot 20150518 release \/news\/libreboot20160818.html:. Libreboot 20160818 release \/news\/libreboot20160902.html:. Libreboot 20160902 release \/news\/libreboot20160907.html:. Libreboot 20160907 release \/news\/libreboot20210522.html:. Libreboot 20210522 released! \/news\/libreboot20211122.html:. Libreboot 20211122 released! \/news\/translations.html:. Translations wanted \/news\/usa-libre.html:. New Hampshire (USA) may soon enshrine Software Freedom into law. YOUR HELP IS NEEDED! \/news\/policy.html:. Binary blob extermination policy \/news\/:. Libreboot news \/tasks\/:. Tasks \/template-license.html:. Template license \/contact.html:. Contact \/logo-license.html:. Libreboot logo license \/contrib.html:. Project contributors \/git.html:. Code review \/who.html:. Who develops Libreboot? \/:. Звільни свій BIOS сьогодні! \/download.html:. Downloads \/:. Free your BIOS today! \/faq.html:. Frequently Asked Questions Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/sitemap.md Subscribe to RSS for this site Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"6jDYPsQ1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":27,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["docs"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Libreboot – Documentation"],"last_modified":"Sat Apr 02 12:48:44 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-2374067243150857752,"exact_signature_unique_b":true,"fuzzy_signature_l":-6352585930564631222,"fuzzy_signature_unique_b":true,"h1_txt":["Documentation","Installing libreboot","Documentation related to operating systems","Information for developers","Other information","How do I know what version I’m running?"],"bold_txt":["commitid","upstream","build\/roms\/helper: add version information to CBFS","Libreboot 20160907 only","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":395,"text_t":"Libreboot – Documentation. Documentation. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to previous index Always check. libreboot.org for the latest updates to Libreboot. News, including release announcements, can be found in the. main news section . Answers to Frequently Asked Questions about Libreboot . Installing libreboot. [link] What systems can I use libreboot on? How to install libreboot Documentation related to operating systems. [link] GNU+Linux Guides How to install BSD on a libreboot system Information for developers. [link] How to compile the libreboot source code Build system developer documentation Depthcharge payload (. Libreboot 20160907 only. ). GRUB payload Other information. [link] Miscellaneous List of codenames How do I know what version I’m running? [link] If you are at least 127 commits after release 20150518 (commit message. build\/roms\/helper: add version information to CBFS. ) (or you have any. upstream. stable release of libreboot after 20150518), then you can press C at the GRUB console, and use this command to find out what version of libreboot you have:. cat (cbfsdisk)\/lbversion. Alternatively, you may run this command in GRUB:. lscoreboot. If you’re using SeaBIOS, information is provided there aswell. This will also work on non-release images (the version string is automatically generated, using. git describe --tags HEAD. ), built from the git repository. A file named. version. will also be included in the archives that you downloaded (if you are using release archives). If it exists, you can also extract this. lbversion. file by using the. cbfstool. utility which libreboot includes, from a ROM image that you either dumped or haven’t flashed yet. In your distribution, run cbfstool on your ROM image (. libreboot.rom. , in this example):. .\/cbfstool libreboot.rom extract -n lbversion -f lbversion. You will now have a file, named. lbversion. , which you can read in whatever program it is that you use for reading\/writing text files. For git, it’s easy. Just check the git log. For releases on or below 20150518, or snapshots generated from the git repository below 127 commits after 20150518, you can find a file named. commitid. inside the archives. If you are using pre-built ROM images from the libreboot project, you can press C in GRUB for access to the terminal, and then run this command:. lscoreboot. You may find a date in here, detailing when that ROM image was built. For pre-built images distributed by the libreboot project, this is a rough approximation of what version you have, because the version numbers are dated, and the release archives are typically built on the same day as the release; you can correlate that with the release information in. release announcements on the news page . For 20160818, note that the lbversion file was missing from CBFS on GRUB images. You can still find out what libreboot version you have by comparing checksums of image dumps (with the descriptor blanked out with 00s, and the same done to the ROMs from the release archive, if you are on a GM45 laptop). There may also be a ChangeLog file included in your release archive, so that you can look in there to figure out what version you have. Binary blob policy Edit this page Who develops Libreboot? 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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Serge PORTELLI. Serge PORTELLI. Ancien magistrat. Après sa formation à l’École Nationale de la Magistrature, Serge PORTELLI a débuté sa carrière, en 1974, en tant que juge d’instruction au tribunal de grande instance de Melun. À partir de 1994, il a exercé ces mêmes fonctions à Créteil, s’intéressant plus particulièrement aux affaires économiques et financières, mais également aux affaires concernant les mineurs. De plus, il présidait la commission d’indemnisation des victimes. En 2000, il est nommé Vice-Président au tribunal de grande instance de Paris au sein duquel il préside la 12ème chambre correctionnelle. De 2012 jusqu’en 2018, Serge Portelli a été le Président de chambre près la cour d’appel de Versailles. 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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Camille ÉTIENNE. Camille ÉTIENNE. Etudiante. Camille Etienne s’engage depuis longtemps mais la cause écologique n’a pas toujours été son combat. Après s’être intéressée adolescente à la délinquance juvénile, elle porte ensuite son attention sur la crise migratoire. Au lycée, elle crée notamment dans son village l’antenne régionale pour la jeunesse d’Amnesty International et se rend avec le Secours Catholique à Calais. Suite à des soucis de santé, elle renonce finalement à une carrière sportive dans le ski de fond et le biathlon qu’elle avait pratiqués en compétition. Une vidéo virale illustre son propos et ses actions :. https:\/\/www.youtube.com\/watch?v=1Mw5ADaHyFo&feature=emb_title « J’ai beaucoup d’humilité face à la nature. Et encore plus avec un père guide de haute montagne, dont certains confrères y ont perdu la vie… Cela rappelle qu’elle est plus forte que nous, et nous rend insignifiants. Pour moi il était inenvisageable que nous puissions menacer l’environnement. Mais en arrivant à Sciences Po Paris j’ai réalisé que mes notions, qui me semblaient basiques comme savoir faire pousser une carotte, ne l’étaient pas pour mes camarades, plus ‘citadins’. » source https:\/\/dailygeekshow.com\/camille-etienne\/. Rencontres. \"Climat : Le mur, on s’en approche ou on est dedans ?\" auditorium 300, 12 Nov 2021 17:00 - 20:00. Lanceurs d'alertes ·. @Lanceursalertes Nous organisons une rencontre sur ce média. @LaRevueDessinee avec. @ElianePatriarca @ileraud et. @AmelieMougey RESERVEZ LES 11\/12\/13 novembre 2022 à la. @MSHParisNord pour une vingtaine de rencontres et de projections, des échanges avec les lanceuses et lanceurs d'alerte,. https:\/\/t.co\/aIURc4dILp View on Twitter La Revue Dessinée. @LaRevueDessinee. 🚞 Coûteux, titanesque et surtout contesté, le nouveau tunnel ferroviaire Lyon-Turin risque à tout moment de dérailler. En coédition avec @RevueDessineeIT, @ElianePatriarca, Grégory Mardon, @albertopi et Lorena Canottiere enquêtent sur un projet qui dure depuis 30 ans. https:\/\/t.co\/teOnjQTYZC. 3. 6. Follow @Lanceursalertes Mentions légales Facebook. Twitter. 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Programme Programme par Thématiques EN DIRECT Intervenants Soutiens Qui sommes-nous ? Archives Edition 2019 Programme 2019 Intervenants 2019 Edition 2018 Intervenants Revoir les rencontres Le live Sélectionner une page. Juliette BOUDRE. Juliette BOUDRE. En dédicace samedi 23 novembre de 17h à 18h pour « Maman, ne me laisse pas m’endormir ». Juliette Boudre, 50 ans, mère de trois garçons, vit à Paris. Elle est l’auteure du livre, Maman, ne me laisse pas m’endormir, dans lequel elle nous livre le récit tragique et bouleversant de la descente aux enfers de son fils Joseph, décédé en 2016 d’une overdose d’opiacés. Un ouvrage qui alerte sur l‘addiction aux médicaments chez les jeunes. Elle consacre désormais sa vie à l’écriture et à la prévention contre l’addiction ravageuse des benzodiazépines (anxiolytiques) ou opiacés (codéine, morphine, fentanyl) – qui foisonnent dans nos armoires à pharmacie ; Elle lance un cri d’alarme sur les dysfonctionnements des services de santé publics face à ces drogues de plus en plus répandues. Maman, ne me laisse pas m’endormir est son premier ouvrage. Rencontres. \"Lanceuses d’alertes, « Les résistantes »\" Salle Librairie, 23 Nov 2019 16:00 - 17:00. Livres liés à cet intervenant. \"Maman ne me laisse pas m’endormir\". 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website","SHOP","","","Faculty of medicine","Editorial","Management","Regulatory acts","Code of ethics","Divisions","Senior management","Health Courses Division","Human Resources Department","Procurement and Finance Division","Research and Development Division","Rental of conference facilites","Courses","Medical studies","Degree and inter-university degree catalogue","PASS and L.AS","Second cycle [years 2-6 of medical studies]","Third cycle [year 7 onwards of medical studies]","Midwifery studies","Paramedical studies","Speech and language therapy","Orthoptics","Psychomotricity","Research","Research units","MU 75 - Paris Brain Institute","MU 80 - Institut de la Vision","MRSU 933 - Laboratory of childhood genetic diseases","MRSU 938 – Saint-Antoine Research Center","MRSU 959- Immunology-Immunopathology-Immunotherapy","MRSU 974 – Center of Research in Myology","MRSU 1135 - Center for Immunology and Microbial Infections","MRSU 1135 - Centre for Immunology and Infectious Diseases","MRSU 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Sorbonne University, an attractive environment for doctoral candidates Doctoral College Application and admission Doctoral Course Career plans and professional development Milestones Administrative procedures Culture and outreach Culture and outreach. Cultural Events Noteworthy Places, Collections and Bequests Noteworthy Places, Collections and Bequests. Architecture and works of art Outstanding collections and bequests An Interdisciplinary Approach to Benefit Society An Interdisciplinary Approach to Benefit Society. Multidisciplinary thematic institutes University for All Ages University for All Ages. Summer University The Inter-Age University Community Outreach Community Outreach. The Science House The Patients’ University The Participatory Science Portal My Life as a Researcher The Teddy Bear Hospital Bookshop Bookshop. The Sorbonne University Press Other publications Foundation Foundation. \"Welcome to the Future\" \"Welcome to the Future\". Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Parismus. Parismus. Publié le 1\/07\/2020 - Mis à jour le 1\/07\/2020. Parismus is the international student association of Sorbonne University. The organization was created in 2007 by two students who, on coming back from their own Erasmus experience, wanted to welcome international students the same way they had been welcomed during their Erasmus year. The association has for missions to help international students sort out their move and to integrate them into the Parisian life. Thus, for 12 years now, we have been offering international and French students from the three faculties the opportunities to meet, exchange and have fun together! Parismus website . PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. Let's build the network together Join a club or association Alumni portraits Contact us Education Education. Study at Sorbonne University Study at Sorbonne University. Our Academic Programs Degree-seeking Students Exchange Students Continuing Education Ensure successful studies Ensure successful studies. Office of the Registrar Tutoring Student Services Student internships Tools and Resources Reorientation and caesura Student life Student life. Grants and financial assistance Health and Handicap Services Sports and Cultural Activities Student associations Work Spaces Student Life and Cultural Spaces Cafeteria and Food Service Housing Prepare Your Professional Goals Prepare Your Professional Goals. Employability Facts and Figures Student Entrepreneruial Training Sorbonne University Librairies Sorbonne University Librairies. Consult online resources Use Sorbonne University services About the Sorbonne University Libraries Commitment to Open Science Research and innovation Research and innovation. Research strategy Research strategy. Research themes and programs Sorbonne University in Europe and in the world Commitment to open science Multidisciplinary institutes and initiatives The HRS4R Label Research partners Research partners. Sorbonne University Alliance Sorbonne University in Europe Sorbonne University Worldwide Partner companies Research structures and technical platforms Research structures and technical platforms. Research units Observatories Technical platforms Innovation and exploitation Innovation and exploitation. Knowledge transfer Collaboration with a research team Setting up a business Carnot Institutes The Doctorate The Doctorate. Sorbonne University, an attractive environment for doctoral candidates Doctoral College Application and admission Doctoral Course Career plans and professional development Milestones Administrative procedures Culture and outreach Culture and outreach. Cultural Events Noteworthy Places, Collections and Bequests Noteworthy Places, Collections and Bequests. Architecture and works of art Outstanding collections and bequests An Interdisciplinary Approach to Benefit Society An Interdisciplinary Approach to Benefit Society. Multidisciplinary thematic institutes University for All Ages University for All Ages. Summer University The Inter-Age University Community Outreach Community Outreach. The Science House The Patients’ University The Participatory Science Portal My Life as a Researcher The Teddy Bear Hospital Bookshop Bookshop. The Sorbonne University Press Other publications Foundation Foundation. \"Welcome to the Future\" \"Welcome to the Future\". Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Foundation Our Organization Missions. Missions. Officially established on August 27, 2009, our Foundation is based on the humanist tradition and the scientific excellence of Sorbonne University. With a firm determination to create greater interdisciplinarity, it works to develop concrete solutions for major social issues and ensure the academic influence of Sorbonne University. Publié le 11\/12\/2019 - Mis à jour le 18\/11\/2021. Missions. Three questions for Barthélémy Jobert, President of the Sorbonne University Foundation. Can you tell us about the Foundation’s history? It was created in the Fall of 2017, at the time of the finalization of the merger between Paris-Sorbonne and Pierre & Marie Curie Universities. Based on the old UPMC Foundation which had been in existence for 9 years, the new Foundation has continued to pursue the same principal goals: creating a community of private, corporate and institutional donors around the university who participate in meaningful and clearly structured projects. Beyond the financial assistance they provide, this diverse group of individuals contribute to the university’s mission as well as its influence, while simultaneously respecting the autonomy of its teaching and research teams. The current SU Foundation is particularly active in medicine, science, technology, arts and humanities, and the human and social sciences. Singular in its ability to cover so many different fields, it is the leading French university foundation and one of the largest in Europe in terms of funds raised and activities undertaken. The SU Foundation also stands out for its interdisciplinary vocation, precisely calibrated to meet the challenges of the contemporary world. What actions have already been taken? The Foundation is based on a group of initiatives that it hopes to expand through the Welcome to the Future campaign. They concern academics, particularly with the unique Passport to the Master’s program. Selected based on an application process and interviews and in accordance with academic and social criteria, students are able to access a fully-funded 5-year higher education program, Master’s degree included, all the while receiving the support of a personalized outside mentor. Research chairs have been created in particularly important areas such as new materials, quantum computing, and oncology. The university’s patrimony has also benefitted from crowdfunding campaigns: private donors have financed, among other things, the restauration and preservation of the zoological collection in the Faculty of Sciences; patient-care manikins, destined for use in pediatric oncology in the Faculty of Medicine, have been purchased to help train students. What are the Foundation’s goals for the Welcome to the Future campaign?​​​​​​​. First of all we hope in the next five years to double the €50 million already raised in order to reach €100 million by 2022. This would make us the most successful of all similar campaigns launched by continental European universities. In doing so, we hope to develop and reinforce a veritable community around Sorbonne University. We believe that a universal institution of knowledge, naturally open to young people and thus to the future and the entire world, will bring our community of donors together. This community will also be encouraged to participate in an increasingly personal way in the life of the university, in order to better participate in the pursuit of knowledge that we undertake on a daily basis. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. 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Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. Let's build the network together Join a club or association Alumni portraits Contact us Education Education. Study at Sorbonne University Study at Sorbonne University. Our Academic Programs Degree-seeking Students Exchange Students Continuing Education Ensure successful studies Ensure successful studies. Office of the Registrar Tutoring Student Services Student internships Tools and Resources Reorientation and caesura Student life Student life. Grants and financial assistance Health and Handicap Services Sports and Cultural Activities Student associations Work Spaces Student Life and Cultural Spaces Cafeteria and Food Service Housing Prepare Your Professional Goals Prepare Your Professional Goals. Employability Facts and Figures Student Entrepreneruial Training Sorbonne University Librairies Sorbonne University Librairies. Consult online resources Use Sorbonne University services About the Sorbonne University Libraries Commitment to Open Science Research and innovation Research and innovation. Research strategy Research strategy. Research themes and programs Sorbonne University in Europe and in the world Commitment to open science Multidisciplinary institutes and initiatives The HRS4R Label Research partners Research partners. Sorbonne University Alliance Sorbonne University in Europe Sorbonne University Worldwide Partner companies Research structures and technical platforms Research structures and technical platforms. Research units Observatories Technical platforms Innovation and exploitation Innovation and exploitation. Knowledge transfer Collaboration with a research team Setting up a business Carnot Institutes The Doctorate The Doctorate. Sorbonne University, an attractive environment for doctoral candidates Doctoral College Application and admission Doctoral Course Career plans and professional development Milestones Administrative procedures Culture and outreach Culture and outreach. Cultural Events Noteworthy Places, Collections and Bequests Noteworthy Places, Collections and Bequests. Architecture and works of art Outstanding collections and bequests An Interdisciplinary Approach to Benefit Society An Interdisciplinary Approach to Benefit Society. Multidisciplinary thematic institutes University for All Ages University for All Ages. Summer University The Inter-Age University Community Outreach Community Outreach. The Science House The Patients’ University The Participatory Science Portal My Life as a Researcher The Teddy Bear Hospital Bookshop Bookshop. The Sorbonne University Press Other publications Foundation Foundation. \"Welcome to the Future\" \"Welcome to the Future\". Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home University About us The great adventure of knowledge. The great adventure of knowledge. Sorbonne University is a public university, created on 1 January 2018, by the merger of the Paris-Sorbonne University and Pierre and Marie Curie University. It is structured by three faculties, in the fields of arts and humanities, science and engineering, and medicine. Publié le 25\/11\/2019 - Mis à jour le 4\/05\/2022. The great adventure of knowledge. As a multidisciplinary research university in the heart of Paris, Sorbonne University is resolutely global. In cooperation with its partners, it creatively and innovatively fulfils its missions of teaching, research, innovation, mediation and technology transfer for the benefit of the common good. . Sorbonne University is a pioneer in the pathways of knowledge and tackles major contemporary challenges such as climate change and sustainability, digital transformation and the data revolution, personalised medicine, open science, and more. On the strength of its shared values—quality and integrity, freedom, transparency and collegiality, diversity, the exchange of ideas—it ambitiously pursues its public service mission and dedicates itself to the creation and development of knowledge, at the heart and at the intersection of disciplines. Sorbonne University promotes equal access to knowledge, from bachelor's degrees to doctorates and in life-long learning. At the end of high school, when returning to school or throughout a career, it provides demanding teaching based on the spirit and methods of research, and allows students to develop their own multidisciplinary career path. On its campuses, in Paris and in its marine stations in the regions, it boasts state-of-the-art equipment conducive to successful work. . Because there can be no success without the chance to flourish, Sorbonne University helps to improve the living conditions of students who are in need, offers a wide range of sports and cultural activities and encourages student initiatives. Sharing knowledge. Sorbonne University contributes to current public debates through a host of events open to everyone. On its campuses, all kinds of people have the chance to discover and share knowledge: courses, conferences, events, the discovery of unique collections, shows or concerts, etc. Innovation and entrepreneurship. The transfer of technology and expertise is a priority for Sorbonne University. It promotes entrepreneurship and supports promising startups from its community through multiple value-adding and investment instruments. Sorbonne University's economic partnerships, from startups to large companies, further strengthen the economic and societal impact of its research. . An open and global university. To perform its missions and succeed in its projects, Sorbonne University is developing a dynamic partnership policy. Its laboratories are co-managed by national research organisations and leading academic partners. In 2018, it created the Sorbonne University Alliance, which among other things, defines the orientations of the Excellence Initiative for the implementation of the Sorbonne University Strategic Plan. . In the French regions of Île-de-France, Brittany, Provence-Alpes-Côte d'Azur and Occitania, it works closely with local authorities and businesses, contributing to the economic development of the French regions. Particularly active within the European university and scientific area, and thanks to its strategic international partnerships, Sorbonne University contributes to the influence of French science and humanities. Committed to research and to the success of its students, with a spirit of diversity and openness, Sorbonne University participates in the great adventure of knowledge at the service of society. Statuts de Sorbonne Université - pdf - 1.55 MB. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Education Ensure successful studies Tutoring. Tutoring. Since students often work better in pairs or groups, tutoring offers free personalised support to reinforce student knowledge and facilitate their success. Publié le 2\/12\/2019 - Mis à jour le 4\/05\/2022. Tutoring. The tutors are students in the higher years, who discuss with the student any aspects of courses that may be posing problems, and help them to organise their work. In addition, throughout their studies, students can benefit from the pedagogical support provided by referring teachers, to receive pedagogical advice and help in the development of their educational plans.  To find out about the tutoring and support systems available to you, contact your faculty or your teaching department. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Université. Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Organised by the Faculty of Arts and Humanities, Sorbonne Summer University opens its doors to a broad public from all over the world, students or academic researchers wishing to discuss current research in the humanities and sciences, tourists wishing to make the most of their stay in Paris, and anyone with a desire to learn more.  . Useful links. Summer University. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. 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MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Our portraits Olivier Chaline. Olivier Chaline. Ocean Historian. Publié le 9\/02\/2022 - Mis à jour le 17\/02\/2022. Reality is always more complicated than the idea we have of it, but it is more exciting to study. What is the link between Baroque music, the history of Central Europe and the ocean? A historian from Sorbonne University, Olivier Chaline, is the bridge between these seemingly faraway shores. Portrait of one of the captains of the Ocean Institute. Olivier Chaline's biography is linked to the history of the ocean. Born in Neuilly-sur-Seine in 1964, the director of the Maritime History and Archaeology Research Federation. 1. was raised in Rouen, a city steeped in history, thanks in particular to its imposing medieval cathedral. And when your parents are historians, freshly recruited by the university of the Norman capital, it is difficult to escape history. “In this city with its considerable architectural heritage, history was immediately present, and this influenced me from a very young age,\". recalls Olivier Chaline. The idea of researching and teaching history was an obvious one. A fellow countryman also stimulated this interest. The young Olivier Chaline studied at the Lycée Pierre Corneille, in the same 17th and 18th century buildings where the illustrious playwright was educated and he was also a history buff. \"I thought to myself that at my age, he was there, like me. It's strange to have a kind of complicity with one of the great French literary authors, but I didn't feel intimidated by this character who seemed very close to me. We frequented the same city, the same places, I saw things that he had seen. As if he was my direct ancestor.”. In 1982, Olivier Chaline obtained his literary baccalaureate and, after two years of preparatory classes, entered the École normale supérieure (ENS) on rue d'Ulm, in Paris, where he obtained the agrégation. 2. in history in 1987. But vacancies in higher education were scarce and he was advised to go to the École nationale d'administration. He was stubborn, and the future rewarded him for it. \"History made me dream, and continues to make me dream.”. In 1989, his schooling and his military service completed, the ENS called him back to become a \"caïman\", an old hand who prepares his young comrades for the aggregation exam. At the same time, he prepared and obtained, in 1995, his doctorate on the history of the parliament of Normandy. In the wake of this, he was elected associate professor in modern history at the ENS. In 1998, he then obtained his habilitation to direct research with a work on the Thirty Years' War, a conflict that began in 1618 with the uprising of the Czech Protestant community against the House of Habsburg. He has a real love for the Czech Republic, whose language he learned in evening classes at the ENS during his years as a caïman. \"As a child, I discovered Smetana's Vltava at school and found this composer at home. Smetana and Dvorak undoubtedly oriented me towards their country and the waters of the Vltava have never ceased to irrigate my work.\". In 1999, after ten years at the ENS, it was time to see other horizons. \"I wanted a change of scenery.”. A position as a professor of history became available at the University of Rennes 2, where he stayed for two years, before Sorbonne University offered him the chair of Central European history. The Great Plunge. He was soon asked to take over the maritime history seminar at Sorbonne University, a rarity in France and the only seminar of this type in Paris. He returned to the halls of the university where, in 1986, his first-year Master’s thesis led him to study the fitting out of ships in Le Havre in the 18th century for the first time. It seemed to always come back to the sea. With a team in place, he worked on what would soon become, under his aegis, a research federation. After studying parliaments and magistrates in France under the Ancien Régime and then in Central Europe, he turned to the history of the maritime. His work on maritime wars enabled him to collaborate with the French Navy and in particular with the Naval School. At the National Archives, he immersed himself in the campaign diaries of the army commanded by Admiral de Grasse, the naval force sent by Louis XVI to secure the independence of the United States. \"I knew the history of the Navy, but when you don't have the practice at sea, you miss a lot. It was important to surround myself with sailors to explain the elements of navigation. This is a good example of how historians and sailors can work together. Just as I had to learn Czech, with the ocean, I learned a language, a vocabulary.\". The Navy even invited him aboard one of the schooners, the Belle Poule, which sailed from Brest to the Grande Armada de Rouen in 2013. \"I had never sailed before, except on ferries, and I learned to read a nautical chart, what a night watch is, an aber, a tidal current... I knew this route from the archives. But when you have the chance to go to the very places you have studied, you really appreciate and collect images, not only in the memory of your camera, but real sensations, too. The real thing is always more complicated than the idea we have of it, but it's more exciting to study.\". Due to his experience, he was nominated by the Faculty of Arts and Humanities to discuss this topic with the Faculty of Science and Engineering and the National Museum of Natural History. This led him to be quickly associated with the start of the Ocean Institute established by the Sorbonne University Alliance. So it’s only natural that he is also involved in the organization of the Oceans Forum 2022. Oceans Forum 2022. On March 2, the Ocean Forum, a challenge for European scientific expertise, begins in Paris, in line with the Decade of Ocean Sciences for Sustainable Development launched in 2021 by UNESCO. Prepared by the Ocean Institute and the Mare Institute in Lisbon, it will alternate conferences and round tables on global changes, new knowledge from research and the relationship between science and society to the Big Blue. Then the forum will move to Lisbon, from September 2, to address Franco-Portuguese themes, both geopolitical and gastronomic. At 57, Olivier Chaline sees his future in blue. \"It is likely that it will be maritime. Since 1989, while studying history, I have the impression that I have been in different jobs. I wouldn't be surprised if in the next ten years there are new metamorphoses.\". 1 The Maritime History and Archaeology Research Federation brings together Sorbonne University, the Musée national de la Marine, the Naval School and the Naval Archaeology Research Group. 2 Agrégation: certification to teach in higher education. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. 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The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. 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We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. Let's build the network together Join a club or association Alumni portraits Contact us Education Education. Study at Sorbonne University Study at Sorbonne University. Our Academic Programs Degree-seeking Students Exchange Students Continuing Education Ensure successful studies Ensure successful studies. Office of the Registrar Tutoring Student Services Student internships Tools and Resources Reorientation and caesura Student life Student life. Grants and financial assistance Health and Handicap Services Sports and Cultural Activities Student associations Work Spaces Student Life and Cultural Spaces Cafeteria and Food Service Housing Prepare Your Professional Goals Prepare Your Professional Goals. Employability Facts and Figures Student Entrepreneruial Training Sorbonne University Librairies Sorbonne University Librairies. 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Sorbonne University, an attractive environment for doctoral candidates Doctoral College Application and admission Doctoral Course Career plans and professional development Milestones Administrative procedures Culture and outreach Culture and outreach. Cultural Events Noteworthy Places, Collections and Bequests Noteworthy Places, Collections and Bequests. Architecture and works of art Outstanding collections and bequests An Interdisciplinary Approach to Benefit Society An Interdisciplinary Approach to Benefit Society. Multidisciplinary thematic institutes University for All Ages University for All Ages. Summer University The Inter-Age University Community Outreach Community Outreach. The Science House The Patients’ University The Participatory Science Portal My Life as a Researcher The Teddy Bear Hospital Bookshop Bookshop. The Sorbonne University Press Other publications Foundation Foundation. \"Welcome to the Future\" \"Welcome to the Future\". Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Foundation \"Welcome to the Future\" Alumni Chapters. Alumni Chapters. Welcome to the Future can count on a vast network of alumni around the world, organized both geographically and thematically. Our development campaign is the chance to give an active role to this large community and reinforce our graduates’ sense of belonging to Sorbonne University. They are called on to mobilize their personal and professional networks, to involve their business or become donors themselves. Publié le 12\/12\/2019 - Mis à jour le 24\/11\/2020. Alumni Chapters. Medical Alumni. The view of Professor Serge Uzan, Special Advisor to the President of Sorbonne University, Director of the University Institute of Oncology, National Advisor to the Order of Doctors, ex-Dean of the Faculty of Medicine. “The clinical and theoretical training provided in the Faculty of Medicine at Sorbonne University is of the highest quality, and internationally recognized.” For Professor Serge Uzan, this demanding status must be supported if the Faculty is to continue to play a pioneering role in the evolution of the medical disciplines. It is thus necessary to expand and accelerate its activities to create the medicine of the future. In this regard, as Professor Uzan emphasizes, the fundraising campaign of Sorbonne University constitutes a crucial tool: “I call on Sorbonne University’s Medical Alumni Chapter because I know how attached they are to their campus, with a very strong sense of belonging. By helping it to prosper and grow, you will give new meaning to your years of study while helping to position your Faculty of Medicine among the best in the world.” The Medical Alumni chapter was among the first to be mobilized, to make the Faculty more attractive, visible, and renowned, through the funding of projects such as a chair dedicated to Alzheimer’s research, the acceleration of immunotherapy work, the use of artificial intelligence as a diagnostic aid, and even the expansion of its digital and technological tools. USA Alumni – East Coast. Dr. Aurélie Jean, PhD - scientifique numéricienne et entrepreneuse - Promotion 2005. UK Alumni. Following the creation of the London Alumni chapter, Sorbonne University has now hit the ground running in the United Kingdom. The campaign is in full swing with a particular commitment on the part of our former students in favor of scholarships, and in particular the “Passeport Puri’elles” project, focused on supporting young women to study science. Annie Barthélémy, ambassadrice du club des Alumni UK. \"An Alumni Chapter is always a strategic group that brings great benefit not only to the higher education institution but also to the students who attended it. And of course the Sorbonne University Alumni Chapter in London is part of this logic. This Chapter has existed for a few years now, but it formally started to meet in April 2017. London is a very dynamic environment, but it is also very demanding and expensive. So while maintaining an active Chapter can be a real challenge, it represents hopes for future fundraising for our University.\". Read also. Join a club or association PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Read the 2018 Activity Report. Rapport d'activité 2018 Fondation - pdf - 2.54 MB. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. Let's build the network together Join a club or association Alumni portraits Contact us Education Education. Study at Sorbonne University Study at Sorbonne University. Our Academic Programs Degree-seeking Students Exchange Students Continuing Education Ensure successful studies Ensure successful studies. Office of the Registrar Tutoring Student Services Student internships Tools and Resources Reorientation and caesura Student life Student life. Grants and financial assistance Health and Handicap Services Sports and Cultural Activities Student associations Work Spaces Student Life and Cultural Spaces Cafeteria and Food Service Housing Prepare Your Professional Goals Prepare Your Professional Goals. Employability Facts and Figures Student Entrepreneruial Training Sorbonne University Librairies Sorbonne University Librairies. 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Sorbonne University, an attractive environment for doctoral candidates Doctoral College Application and admission Doctoral Course Career plans and professional development Milestones Administrative procedures Culture and outreach Culture and outreach. Cultural Events Noteworthy Places, Collections and Bequests Noteworthy Places, Collections and Bequests. Architecture and works of art Outstanding collections and bequests An Interdisciplinary Approach to Benefit Society An Interdisciplinary Approach to Benefit Society. Multidisciplinary thematic institutes University for All Ages University for All Ages. Summer University The Inter-Age University Community Outreach Community Outreach. The Science House The Patients’ University The Participatory Science Portal My Life as a Researcher The Teddy Bear Hospital Bookshop Bookshop. The Sorbonne University Press Other publications Foundation Foundation. \"Welcome to the Future\" \"Welcome to the Future\". Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Research and innovation The doctorate Sorbonne University, an attractive environment for PhDs. Sorbonne University, an attractive environment for PhDs. Pursuing a doctorate at Sorbonne University means being educated in and through research in one of the best universities in the world, and being mentored by renowned researchers to become a scientific expert but also rich in generalist skills. It also means forging a strategic vision of the world that will enable you to position yourself as a key player in society. . A world-class multidisciplinary intensive research university in the heart of Paris. 3 faculties, 115 research units, numerous technical platforms. Languages, history, arts, humanities and social sciences, science and engineering and medicine, Sorbonne University is known and recognised for the excellence of its academic research in all these disciplinary fields. Sorbonne University's priority is to respond to today's challenges and prepare for the future. . An inter-disciplinary approach for understanding, learning and engaging. Our. many interdisciplinary projects contribute to meeting the greatest challenges facing society, such as global warming, the aging population, the digital revolution and the place of democracy, health and education. Sorbonne University offers. doctoral programs that mobilise science and technology, human and social sciences and medicine at the intersection and very heart of disciplines. A university open to the world. Sorbonne University is equipped with the resources worth of a world-class university, notably through. strategic partnerships with other universities around the world (notably with the. 4EU+ Alliance ). Demanding and high-quality doctoral supervision. The quality of doctoral research and the smooth running of research are essential aspects of the doctoral policy set up by Sorbonne University. Thus, Sorbonne University is very attentive to the. quality of supervision and monitoring of doctoral candidate. s . It ensures that each thesis director devotes the necessary time to guiding and training the doctoral candidates they supervise. Sorbonne University encourages co-supervision, with a clear distribution of duties. To support thesis directors in this task,. specific ongoing education schemes. are offered by the Sorbonne University doctoral college to candidates for the HDR (accreditation to direct research) but also to all researchers who are keen to improve their supervision practices. A doctoral program full of activities. From the Induction Day, the first official meeting that symbolises the start of the doctorate, to the doctoral graduation ceremony marking its end, the doctoral program allows Sorbonne University doctoral candidates to assume the status of researcher within a scientific community. The doctoral program is punctuated by a series of. milestones during which doctoral candidates can meet and exchange ideas, whether during disciplinary seminars, festive events or other activities offered by the doctoral schools and the doctoral college. Doctoral candidates involved in community life. Sorbonne University encourages its young researchers to get involved in university life at the institutional, cultural or associative level. Whether as representatives and elected officials in the various bodies, councils and commissions, as. members of an association , particularly those attached to Sorbonne University, such as Doc'Up or Le collectif Doctoral, or as active members of the community of young researchers, Sorbonne University invites doctoral candidates to. actively participate in doctoral life and to be a driving force in the strategic choices of their doctoral schools and the doctoral college, but also to contribute to the organisation and implementation of events or training. Personalised support to help PhDs consider their career path. Developing a career plan in line with one's personality, qualities, desires, motivations and skills takes time. This is why the Sorbonne University's doctoral teaching teams encourage doctoral candidates from the first year of their doctorate to question themselves while offering support for the development of their career plans. Whatever their plans, they will have the opportunity to establish an individual training plan adapted to and coherent with their future needs. More information Opportunities to meet and collaborate with the world of business. To help doctoral candidates identify potential business sectors and the different professions they could focus on, the doctoral college schedules numerous meetings between doctoral candidates and PhDs from various backgrounds. It also organises visits to companies to enable them to broaden their career prospects. . In addition to providing food for thought for young researchers on their career development, all these events contribute to the creation and development of a professional network. The \"Mission d'expertise\" scheme gives doctoral candidates the opportunity to perform, within the framework of their doctoral contract,. a complementary paid activity. within a company, a local authority, an administration, an association or a foundation. Sometimes little-known, these expertise missions are an opportunity to discover a working environment outside the academic sector and also to create a link between a potential employer and a future doctor. A university at the service of society. Within the framework of its mission of service to society, Sorbonne University strongly encourages its doctoral candidates to disseminate their research, and promotes the values of openness illustrated in 3 areas: open science, innovation & dissemination of research and scientific knowledge. Open Science. Sorbonne University is committed to the. open science movement . To ensure the visibility, accessibility and sustainability of scientific publications from its laboratories, Sorbonne University makes its researchers aware of best practices for making their work, data and research results freely available. . Innovation & Dissemination of Research. To enhance the value of its research and help an. innovative ecosystem prosper, Sorbonne University maintains special links with a number of industrial partners. Thus, Sorbonne University promotes the entrepreneurial spirit in its community by encouraging and supporting researchers who are committed to entrepreneurship. Doctoral candidates who wish to carry out a business creation project can apply to the. PEPITE scheme to benefit from support. Dissemination of scientific knowledge. Sorbonne University promotes scientific culture, and works towards introducing the world of research and presenting its various professions to candidates and teachers but also to citizens. Doctoral candidates actively participate in major national operations such as the Fête de la science or international competitions such as MT180. We also encourage them to make their voices heard in public debate by writing \"general public\" articles in the online publication. The Conversation. . Doctoral Careers & Alumni Network. Sorbonne University regularly conducts. surveys to find out the professional future and career development of its PhDs . Follow-up surveys carried out by doctoral schools and the doctoral college show that Sorbonne University's doctoral candidates occupy high-level jobs (researchers, senior managers, entrepreneurs, scientific advisors, R&D) in all socio-economic sectors. To maintain a closer link with its graduates, Sorbonne University has created an. Alumni Department , a team whose goal is to build and coordinate a powerful network with a worldwide influence. . Every year, newly graduated PhDs join the Doctors' Club, a club that allows them to develop extended professional interactions (formal and informal exchanges, new collaborations, career boosts, ...) or simply share some pleasant moments. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. 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The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Programs in French Programs in English PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. 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CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Bureau des stages (SCUIOIP). Campus Sorbonne - Escalier F, 2e étage, salle F362. Contact by email. In Medicine. Scolarité du 2e cycle.  . Campus Saint-Antoine. Contact by email. In Science and Engineering. Bureau des études, pôle Stage et suivi de l’étudiant.  . Campus Pierre et Marie Curie, Tour Zamansky, 20e  étage, bureau 2008. Contact by email. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. 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CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Education Prepare your career plan Employability Facts and Figures. Employability Facts and Figures. The establishment regularly carries out integration studies to meet the needs of business sectors and adjust its courses accordingly. Publié le 7\/12\/2019 - Mis à jour le 2\/09\/2020. Employability Facts and Figures. The Faculties of Arts, Medicine and Science and Engineering conduct public surveys every year to find out what happens to their graduates. The professional integration of graduates of the Faculty of Arts . Survey of all master's graduates in 2013, situation on 01\/12\/2015, 30 months after graduation (source: intelligence unit of professional integration and career paths). Integration rate (graduates from master's degrees in employment \/ all graduates of master's degrees in employment or job search): 90%. Immediate (in 2013-2014) PhD continuation rate: 7%. Master's degrees in employment 30 months after graduation:. 73% are executives (including Civil Service \"A\" Executives). 40% work in the private or voluntary sector, 51% work in the civil service. 73% are employed on an indeterminate basis (permanent contracts and civil servants). Graduate opinion:. 86% are satisfied or very satisfied with their job. 62% use the skills acquired in their M2. 84% are satisfied or very satisfied with the level of responsibilities entrusted to them. Find out more. The professional integration of graduates of the Faculty of Science and Engineering. BVA survey conducted among general bachelor's graduates in 2015, 12 months after graduation and among master's graduates in 2014, 30 months after graduation. Rate of professional integration: 93%, of which 35% are doctoral students. Rate of continuation of studies after the general bachelor's degree: 94%. . Master's degrees in employment 30 months after graduation:. 86% are executives (including Civil Service \"A\" Executives). 70% work in the private sector. 81% have an indeterminate job (permanent contracts and civil servants). 85% have a job in line with their level of training. 73% have a job in line with the content of their training.  . PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. 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University Press | Sorbonne Université. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム サイトマップ. サイトマップ. サイトマップ. 京大について 総長からのメッセージ 湊 長博総長からのメッセージ 任期中の基本方針 ―世界に輝く研究大学を目指して― 湊長博 総長 略歴 総長の動き 式辞・挨拶等 理念・沿革・運営・将来構想 理念・憲章 沿革 将来像・長期目標 京都大学の改革と将来構想（WINDOW構想） 中期目標・中期計画・中期目標期間評価 年度計画・年度評価 指定国立大学法人としての取組 大学改革 総長選考・監察会議 シンボル 大学評価 自己点検・評価 認証評価 国立大学法人評価 データ集（学生数、入学状況等） 職員数 学生数 入学状況 卒業者・修了者数 進路・就職状況 患者数等（医学部附属病院） 入館者数 財務状況 国際交流 外部資金等 発明・特許・知的財産 土地面積 建物面積 蔵書数等 役員等・組織・諸規程 役員等 部局長 組織・管理運営機構 京都大学規程集 規程集未掲載の規程等 就業規則 栄誉等 京都大学名誉博士 京都大学名誉フェロー 栄誉 歴代総長 キャンパス・施設案内 キャンパスの概要 国内の研究所・附属研究施設等 海外の拠点 On-site Laboratory 大学の施設 福利厚生施設・課外活動施設 写真・動画で見る京都大学 広報活動 刊行物・資料請求 メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 物品・役務等の調達情報について 建設・工事等 資産の有効活用（不動産の売却・貸付等） 契約に係る情報等の公表について 調達に関する本学の取り組みについて 【取引先の皆様へ】本学への取引先登録、誓約書の提出および消費税の適格請求書発行事業者の登録申請について 本学における発注・検収・検査について 入札への参加資格について 電子メールによる振込通知サービスについて 入札情報（文部科学省共済組合京都大学支部） 取引金融機関の選定（メインバンク契約）について 法人としての取組 新型コロナウイルス感染症への対応 熊本地震への対応 環境への取組 東日本大震災への対応 人権 障害に関すること 障害者雇用促進への取組 安全への取組 情報環境の整備 公益通報 監査 その他 男女共同参画について アクションプラン 女性活躍推進法について 意識・実態調査 改革 次世代育成支援対策推進 育児・介護支援制度 公表事項 国立大学法人ガバナンス・コード 自己点検・評価（教員評価を含む） 認証評価 業務方法書 会議報告 国立大学法人法等による公表事項 教育情報の公表 贈与報告書 資金運用について 労働施策総合推進法に基づく中途採用比率の公表 情報公開 開示請求 個人情報ファイル簿 匿名加工情報に関する提案の募集 特定個人情報保護評価 入試・高大連携 一般選抜 一般選抜等学部入試における選抜方法等の変更 一般選抜入学者選抜要項 一般選抜学生募集要項 一般選抜インターネット出願 入学手続 一般選抜に係る情報開示について 一般選抜の試験問題等 入学者選抜実施状況 平成29年度京都大学一般入試　理科（物理）における入試ミスについて 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 出願状況と選考結果 よくある質問 その他の学部入学試験 外国学校出身者のための選考 編入学試験・学士入学試験 留学生のための入試 学部で学位をとる 学部入学案内 大学院で学位をとる 大学院入学案内 交換留学・短期留学・研究生等 大学院入試 大学院募集要項（外国人留学生特別入試を含む） 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る オープンキャンパス 大学案内冊子「知と自由への誘い 」 高大連携 高大連携・入試広報事業 学びコーディネーター事業 ELCAS（エルキャス） 高大接続・入試センター 関連情報 資料請求 受験生サポート 新入生の方へ（国際高等教育院Webサイト内）（外部リンク） 入学前後のスケジュール（国際高等教育院Webサイト内）（外部リンク） 教育・学生支援 教育の体制と内容 京都大学の基本理念と方針 学部新入生の方へ（国際高等教育院のサイトへ） アカデミックカレンダー 教養・共通教育と学部専門教育 大学院の教育 成績評価とGPA制度 補助金による教育プログラム 中学・高校の教員免許 リカレント教育（生涯教育等） 経済支援 高等教育の修学支援新制度について 授業料の免除／入学料の免除と徴収猶予 さまざまな奨学金 急に困った時に 海外留学を希望する京大生へ 留学の目的・種類 京大の留学プログラム 留学に関する予備知識 海外へ留学する京大生向け奨学金 海外留学のリスク管理 京都で学べる英語の授業 海外留学のための相談窓口 大学行事 入学式・卒業式・学位授与式 学園祭など 音楽会・能楽鑑賞会 課外活動支援 課外活動学生団体の紹介 課外活動のための物品貸出など 学生が利用できる本学の遠隔地施設 近隣の文化施設の優待制度 「学生ボランティア」学校サポート事業 学生表彰など 総長賞（すべての学生個人・団体） たちばな賞（若手女性研究者・学生） 久能賞（理系の女子学生） SPEC（京大生チャレンジコンテスト） 学生相談・障害学生支援・就職 学生相談窓口 障害学生のための支援 キャリア形成・就職支援 キャンパスライフ支援 学生生活で注意してほしいこと 健康管理 在学中の保険加入 住まい・食と購買 アルバイト 学生支援の窓口・証明書発行など 学生支援窓口のご案内 入学料・授業料などの納付 各種証明書や学生証などの発行 学生関係諸規程一覧 事例別相談窓口一覧 学生支援の情報発信 Campus Life Information（学生便覧） 学生生活支援情報公式twitter（外部リンク） Campus Life News 学生意見箱 情報公開連絡会 大学から学生への連絡について 災害等に伴う休講等の措置等に関する取扱い 京都大学に在籍する留学生の方へ 日本での学生生活 留学生のための奨学金 在留資格変更・在留期限更新・再入国・休学・復学・退学等 手続担当窓口早見表 日本語教育 研究・産官学連携 研究活動を知る 最新の研究成果を知る 研究力向上に資する取り組み 各賞の受賞 京都大学 教育研究活動データベース 学術情報リポジトリKURENAI 京大新刊情報ポータル K.U.RESEARCH 研究成果の公開ポリシー オープンアクセス方針 研究データ管理・公開ポリシー 産官学連携（共同研究・知的財産・ベンチャー支援） 京都大学に寄附する 京都大学基金 寄附講座・寄附研究部門 その他のご寄附 京都大学で研究したい 採用情報 白眉プロジェクト（グローバル型） 京阪神次世代グローバル研究リーダー育成コンソーシアム（K-CONNEX） 卓越研究員事業「白眉プロジェクト（部局連携型）」 研究員制度 研究支援体制について 研究戦略タスクフォース 学術研究支援室（KURA） 研究活動と研究者生活を支える組織 研究者が受けられる研究費・研究支援 研究資金・賞に関する公募等 研究活動に役立つ支援情報「研究者の歩きかた」 学外の研究者も利用できる設備 研究上の倫理・安全 公的研究費の適正管理 公正な研究活動の推進 京都大学における軍事研究に関する基本方針 安全保障輸出管理 利益相反マネジメント ライフサイエンス研究等に係る倫理の保持と安全の確保 研究の情報公開（オプトアウト等） 安全衛生管理 教職員に兼業を依頼したい 国際交流 世界とつながる京都大学 京都大学の国際化推進基本コンセプト 戦略的パートナーシップ 海外の機関との交流協定 大学間国際ネットワーク 国際シンポジウム 表敬訪問 部局の国際交流活動 海外の同窓会 海外拠点 On-site Laboratory 教育の国際化推進事業 京都大学への留学を希望する方へ 京都大学で学ぶ 留学生へのキャリア支援 留学生ラウンジ「きずな」 在留資格・学費・奨学金・住まい 学部・大学院研究科等事務室一覧 FAQ 便利な情報 留学に関する情報 国際化を支える学内組織 学術研究支援室 国際交流サービスオフィス 学部・大学院研究科等事務室一覧 外国人研究者へのサポート 保険について 海外留学を希望する京大生へ 社会連携 公開講座・講演会等 京都大学春秋講義 京都大学未来フォーラム 京大の知 京都大学地域講演会 京都大学 － 稲盛財団合同京都賞シンポジウム 公開講座お知らせメール受付 京大の研究・教育を知る 京都大学白浜水族館　展示 京都大学総合博物館　展示 京都大学大学文書館　歴史展示室 京都大学アカデミックデイ 京大ウィークス オープンキャンパス 京都大学オープンコースウェア（OCW） KoALA（Kyoto University Online for Augmented Learning Activities） edX 京都大学（KyotoUx） 社会人向け教育 リカレント教育（生涯教育等） 社会人のための大学院コース 社会人・一般向け講座 京都大学エグゼクティブ・リーダーシップ・プログラム（京都大学ELP） 教育・研究に関する情報・資料公開 京都大学研究資源アーカイブ（総合博物館） 京都大学貴重資料デジタルアーカイブ（図書館機構） 京都大学学術情報リポジトリKURENAI 京都大学大学文書館 京都大学教育研究活動データベース K.U. 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KyotoU GLOBAL ENGAGEMENT. 若手人材海外派遣事業 ジョン万プログラム. 国際交流サービスオフィス. Graduate Admission Guide for International Applicants. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（繁体中国語）. 京都大學概況. 校長的話. 大學介紹. 校內安全措施. 教育與學生支援服務. 學生生活. 留學・國際教育. 研究・產官學合作. 支援研究體制. Security Export Control. 國際交流. 學部與院係. ヘッダー セカンダリーリンク（日本語以外）. 交通資訊 聯絡我們 捐贈 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 画像. Resources for prospective international students. 説明. A two-minute video about Kyoto University's graduate education, featuring footage of research activities, facilities, and international students discussing the institution. 画像. 舉行了2022學年度研究所入學典禮。（2022年4月7日）. 説明. 2022學年度研究所入學典禮於2022年4月7日（週四），在京都市勸業館「Miyako Messe」舉行，各理事與副校長、各部門領導出席了典禮。. 画像. 舉行了2022學年度大學入學典禮。（2022年4月7日）. 説明. 2022學年度大學入學典禮於2022年4月7日（週四），在京都市勸業館「Miyako Messe」舉行，有井松本紘前校長、山極壽一前一任校長、各理事與副校長、各部門領導出席。. 画像. 舉行了2021學年度研究所學位授予儀式。（2021年3月24日）. 説明. 2022年3月24日，2021學年度研究所學位授予儀式於京都市勸業館Miyako Messe舉行， 各理事與副校長、各部門領導出席了儀式。. 画像. 舉行了2021學年度畢業典禮。（2022年3月24日）. 説明. 2022年3月24日，2021學年度畢業典禮於京都市勸業館Miyako Messe舉行，井村裕夫前校長、各理事與副校長、各部門領導出席了儀式。. 画像. 已發布2021學年度研究所秋季入學典禮的賀詞. 説明. 2021學年度研究所秋季入學典禮，受到新冠病毒疫情影響決定不舉辦，改由發布湊長博校長的賀詞。. 画像. 舉行了2021學年度研究所秋季學位授予儀式。. 説明. 2021年9月24日（星期五），研究所秋季學位授予儀式在百周年時計台紀念館舉行，湊長博校長、理事兼副校長、各研究科長、學館長、學舍長、教育部長、卓越研究所學程主持人等出席了儀式。. 画像. 舉行了2021學年度大學入學典禮。（2021年4月7日）. 説明. 2021學年度大學入學典禮於2021年4月7日（週三），在京都市勸業館「Miyako Messe」舉行，有井村裕夫前校長、松本紘前校長、山極壽一前一任校長、各理事與副校長、各部門領導出席。典禮上，湊長博校長發表了致辭。. 画像. 舉行了2021學年度研究所入學典禮。（2021年4月7日）. 説明. 2021學年度研究所入學典禮於2021年4月7日（週三），在京都市勸業館「Miyako Messe」舉行，松本紘前校長、山極壽一前一任校長、各理事與副校長、各部門領導出席了典禮。. chevron_left. play_arrow. pause. chevron_right. Announcements. Statement. Statement regarding the Situation in Ukraine. COVID-19. Regarding the novel coronavirus: Kyoto University’s Response to the Novel Coronavirus (COVID-19). Latest research news (available in English). 發表日期. 2022\/06\/03. COVID-19 Research Using Bronchial Organoids and Drug Discovery Applications. 發表日期. 2022\/05\/31. Invention of technology to create mature skeletal muscle cells from iPS cells without genetic manipulation. 發表日期. 2022\/05\/30. Breathtaking solution for a breathless problem. 發表日期. 2022\/05\/27. Some monkeys might pull a poker face. More research news KyotoU News. アイキャッチ画像. 發表日期. 2022\/02\/18. タグ. Global Engagement. Education. 舉辦了留學生休息室KI-ZU-NA「線上活動『摺紙製作聖誕飾品囉！』」. アイキャッチ画像. 發表日期. 2022\/02\/18. タグ. Global Engagement. Education. 開辦了全球環境人才培育課程. 發表日期. 2022\/04\/11. タグ. University News. 舉行了2022學年度研究所入學典禮。（2022年4月7日）. 發表日期. 2022\/04\/08. タグ. University News. 舉行了2022學年度大學入學典禮。（2022年4月7日）. 發表日期. 2022\/04\/05. タグ. University News. 舉行了2021學年度研究所學位授予儀式。（2021年3月24日）. 發表日期. 2022\/03\/29. タグ. University News. 舉行了2021學年度畢業典禮。（2022年3月24日）. 最新新聞 Upcoming Events. 最新活動 Features. Kyoto University's125th Anniversary. 網上公布課程. 京都大學開放課程（OCW) 講座影片内容. 大學介紹. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通資訊 フッター セカンダリーリンク. 站點地圖 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 京大の『学び』を知る. 京大の『学び』を知る. サイドメニュー. 一般選抜 特色入試 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る オープンキャンパス 大学案内冊子「知と自由への誘い 」 高大連携 関連情報 3階層目以降ページ用メニュー. オープンキャンパス. 大学案内冊子「知と自由への誘い 」. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム EVENTS. EVENTS. 条件で絞り込む. 日付で絞り込む. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. 要申し込み. - すべて -. 必要. 不要. 開催地. - すべて -. 吉田キャンパス. 宇治キャンパス. 桂キャンパス. 東京オフィス・京都アカデミアフォーラム. その他の地域. オンライン. 開催日. 前月. 1月. 2月. 3月. 4月. 5月. 6月. 7月. 8月. 9月. 10月. 11月. 12月. 翌月. 日. 月. 火. 水. 木. 金. 土. 絞り込み条件をリセット. アイキャッチ画像. 京都大学創立125周年記念 附属図書館所蔵貴重資料展示. 開催日. 2022年06月01日〜12月22日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 産官学連携. 京大MBA×日経ビジネススクール共同講座「サービス経営のフロンティアとセオリー」. 開催日. 2022年06月09日、他. 開催地. オンライン. Events. タグ. 入試. 令和5（2023）年度入学者京都大学大学院総合生存学館（思修館）博士課程学生募集入試説明会. 開催日. 2022年06月10日、他. 開催地. 吉田キャンパス. 桂キャンパス. その他の地域. Events. タグ. 125周年. 公開講座. 京都大学創立125周年記念 市民講座「物理と宇宙」. 開催日. 2022年06月11日、他. 開催地. オンライン. Events. タグ. 125周年. 研究. 京都大学創立125周年記念 研究の魅力を見出す”鏡”プロジェクト. 開催日. 2022年06月11日. 開催地. オンライン. Events. タグ. 公開講座. 人文研アカデミー2022「ヴィリエ・ド・リラダンとフランス象徴主義 －『残酷物語』と『未来のイヴ』が現代に語りかけてくるもの」. 開催日. 2022年06月11日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 研究. 2022年度 総合生存学ミニワークショップ［春］. 開催日. 2022年06月13日、他. 開催地. 吉田キャンパス. オンライン. Events. タグ. 公開講座. 多様性を多様な視点で捉え、行動する ～京都大学と考えるD&Iセミナー～. 開催日. 2022年06月14日. 開催地. オンライン. Events. タグ. 教育. 公開講座. オンラインコース「教育評価の基礎講座」2022. 開催日. 2022年06月15日、他. 開催地. オンライン. Events. タグ. 研究. 第256回アフリカ地域研究会「南アフリカの社会保障制度改革とケアの政治」. 開催日. 2022年06月16日. 開催地. その他の地域. オンライン. Events. タグ. 入試. オンライン個別進学相談（2022年6月開催）. 開催日. 2022年06月17日、他. 開催地. オンライン. Events. タグ. 教育. グローバル教育展開オフィス主催：2022年度レクチャーシリーズ「ダイバシティー時代の「日本型教育」の行方 」第1回. 開催日. 2022年06月17日. 開催地. オンライン. Events. 創立125周年記念 アカデミックマルシェ. 開催日. 2022年06月18日. 開催地. その他の地域. Events. 体育会学生主催イベント 「京大体育会、まるごと魅せます」. 開催日. 2022年06月19日. 開催地. その他の地域. オンライン. Events. タグ. 教育. 公開講座. 教育学研究科 E.FORUM 2022 全国スクールリーダー育成研修「スクールリーダー育成のためのオンライン・リレー講座」. 開催日. 2022年06月19日、他. 開催地. オンライン. Events. タグ. 公開講座. 京都大学金融セミナー「『金融リテラシーが未来を拓く』2022 ～シン・金融教育元年を迎えて～」. 開催日. 2022年06月20日. 開催地. オンライン. Events. タグ. 研究. 学術情報メディアセンターセミナー「教育DXと音声言語処理」. 開催日. 2022年06月21日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 入試. 2023年度 教育学研究科・教育学部学士入学 入試説明会. 開催日. 2022年06月25日. 開催地. オンライン. Events. タグ. 入試. 高校生のための進学ガイダンス2022（岐阜）. 開催日. 2022年06月26日. 開催地. その他の地域. Events. タグ. 研究. ヒト生物学高等研究拠点 研究室紹介「ヒトは、いかにしてヒトとなりえたのか？」. 開催日. 2022年06月27日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 公開講座. プラットフォーム学連続セミナーVol. 12「モビリティとプラットフォーム学」～シェアリング・MaaS・数理最適化から考える「これからの移動のかたち」～. 開催日. 2022年06月30日. 開催地. オンライン. ページ送り. カレントページ. 1 ページ. 2 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"z3uZu9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/about\/foundation\/coronavirus","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":57,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["ja","about","foundation"],"url_file_name_s":"coronavirus","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["新型コロナウイルス感染症への対応 | 京都大学"],"description_txt":["本学では、新型コロナウイルス感染症について下記のように対応しています。\n\n情報は随時、追加・更新をいたします。（2022年6月8日更新）"],"last_modified":"Wed Jun 08 16:18:23 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":3409407914053493993,"exact_signature_unique_b":true,"fuzzy_signature_l":-7498491407690353107,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["新型コロナウイルス感染症への対応"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","新型コロナウイルス感染症への対応について","問い合わせ先","関連リンク","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h4_txt":["新着情報"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":522,"text_t":"新型コロナウイルス感染症への対応 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 法人としての取組 新型コロナウイルス感染症への対応. 新型コロナウイルス感染症への対応. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 新型コロナウイルス感染症への対応 学生の皆さんへ 教職員の皆さんへ 総長メッセージ 本学における新型コロナウイルス感染症罹患者の発生について 武田／モデルナ社製 新型コロナワクチンへの異物混入の報道について（お知らせ） 熊本地震への対応 環境への取組 東日本大震災への対応 人権 障害に関すること 障害者雇用促進への取組 安全への取組 情報環境の整備 公益通報 監査 その他 男女共同参画について 公表事項 情報公開 In other languages. English 本学では、新型コロナウイルス感染症について下記のように対応しています。. 情報は随時、追加・更新をいたします。（2022年6月8日更新）. 新着情報. （2022年6月8日）. 本学における新型コロナウイルス感染症罹患者の発生について （2022年6月1日）. 新型コロナウイルスに対する本学の方針について（第15版）ー海外渡航等を中心とした対応についてー （2022年6月1日）. 新型コロナウイルス感染症の感染拡大防止のためのマスクの着用について （2022年6月1日）. 本学における新型コロナウイルス感染症罹患者の発生について （2022年5月25日）. 本学における新型コロナウイルス感染症罹患者の発生について 新型コロナウイルス感染症への対応について. e-Learning. （2020年11月13日）. 大学生活における新型コロナウイルス感染症対策 ※ 教材動画は、本学構成員以外の方も「. 京都大学YouTubeチャンネル 」にてご覧いただくことができます。. 本学の活動制限ガイドライン、方針等. （2021年9月30日）. 新型コロナウイルス感染拡大防止に伴う活動制限のガイドラインにおける対応レベルの変更について ※. 活動制限のガイドラインにおける対応レベルは、「レベル1」です。 （2022年6月1日）. 新型コロナウイルスに対する本学の方針について（第15版）ー海外渡航等を中心とした対応についてー ※. 海外渡航に関する安全対策手引き（国際的に懸念される感染症への対策を含む） ※. 学生の海外でのフィールドワーク実施に関する安全手引き （2022年2月2日）. 学生・教職員の新型コロナウイルス感染確認時等における対応について（第8版） （2022年4月1日）. 感染拡大予防マニュアル－令和4年度前期授業等の実施における配慮について－（第8版） （2022年6月1日）. 新型コロナウイルス感染症の感染拡大防止のためのマスクの着用について 総長メッセージ 学生の皆さんへ ※外国人留学生向け通知含む. 教職員の皆さんへ ※外国人研究者等向け通知含む. イベントについてのお知らせ 利用制限中の主な施設. 総合博物館 図書館機構 百周年時計台記念館 東京オフィス 京大生協 環境安全保健機構 健康管理部門からのお知らせ. （2020年7月2日）. 夏季における熱中症予防と新型コロナウイルス感染症対策について （2020年6月12日）. 新型コロナウイルス感染症に関する情報（第5報） 帰国・接触者相談センター等の連絡先 本学における新型コロナウイルス感染症罹患者の発生状況 武田\/モデルナ社製 新型コロナワクチンへの異物混入の報道について（お知らせ） 問い合わせ先. 総務部企画管理主幹付リスク管理掛. E-mail：830riskkanrikakari＊mail2.adm.kyoto-u.ac.jp（＊を@に変えてください）. 関連リンク. 文部科学省「新型コロナウイルスに関連した感染症対策に関する対応について」 首相官邸「新型コロナウイルス感染症に備えて～一人ひとりができる対策を知っておこう～」 外務省 海外安全ホームページ 厚生労働省「新型コロナウイルス感染症について」 世界保健機関（WHO）「Coronavirus」（英語ページ） 京都府「新型コロナウイルス感染症に関連する情報について」 京都市情報館「新型コロナウイルス感染症について」 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Publié le 20\/05\/2022 - Mis à jour le 31\/05\/2022. On Thursday, May 19, the Smart ArM team participated in the first edition of the Cybathlon Challenge 2022, a sports competition for people with disabilities using assistive technology. With a new, shorter format, this international competition gave teams and their pilots the opportunity to test their devices, demonstrate their achievements and evaluate their performances. Five teams, five pilots, five prosthetic arms. Five teams competed to win the Cybathlon Challenge's upper limb prosthesis race: Smart ArM from France, e-OPRA and x-OPRA from Sweden, Karla Bionics from Indonesia and ARM2u from Spain. The goal for each pilot: to perform a single task correctly and as quickly as possible. The competitors had to position eight everyday objects into their respective slots in a dedicated box. Each team was given two attempts. After three thrilling minutes, Christophe Huchet, Smart ArM's pilot, won the competition, finishing only two seconds ahead of the e-OPRA team led by Magnus Niska. Watch their incredible performance! CYBATHLON Challenges ARM Introduction. Read more. Recherche Retour sur le Cybathlon 2020 : les jeux paralympiques de demain seront-ils bioniques ? Research Isir's Smart ArM Team Takes on Cybathlon 2020 PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Consult online resources Use Sorbonne University services About the Sorbonne University Libraries Commitment to Open Science Research and innovation Research and innovation. Research strategy Research strategy. Research themes and programs Sorbonne University in Europe and in the world Commitment to open science Multidisciplinary institutes and initiatives The HRS4R Label Research partners Research partners. Sorbonne University Alliance Sorbonne University in Europe Sorbonne University Worldwide Partner companies Research structures and technical platforms Research structures and technical platforms. Research units Observatories Technical platforms Innovation and exploitation Innovation and exploitation. Knowledge transfer Collaboration with a research team Setting up a business Carnot Institutes The Doctorate The Doctorate. Sorbonne University, an attractive environment for doctoral candidates Doctoral College Application and admission Doctoral Course Career plans and professional development Milestones Administrative procedures Culture and outreach Culture and outreach. Cultural Events Noteworthy Places, Collections and Bequests Noteworthy Places, Collections and Bequests. Architecture and works of art Outstanding collections and bequests An Interdisciplinary Approach to Benefit Society An Interdisciplinary Approach to Benefit Society. Multidisciplinary thematic institutes University for All Ages University for All Ages. Summer University The Inter-Age University Community Outreach Community Outreach. The Science House The Patients’ University The Participatory Science Portal My Life as a Researcher The Teddy Bear Hospital Bookshop Bookshop. The Sorbonne University Press Other publications Foundation Foundation. \"Welcome to the Future\" \"Welcome to the Future\". Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Culture and outreach Community Outreach The Science House. The Science House. The Paris Île-de-France Maison pour la science (“The Science House”) offers the primary and secondary school teachers in Paris, a series of professional development courses in science, bringing them closer to the science and technologies being developed in research laboratories. . Publié le 7\/12\/2019 - Mis à jour le 18\/03\/2021. The Science House. The fruits of a partnership between Sorbonne University and the Paris Rectorate, the Paris-Île-de-France The Science House joins the existing network of 10 Regional Houses, supported by the Academy of Sciences and coordinated by the Fondation La main à la pâte. Professional development for Parisian teachers. Starting in the 2019-2020 academic year, The Science House will be implementing educational courses based on an investigative approach, questioning our daily lives dealing with scientific and interdisciplinary themes such as \"Climate and me\", \"Teaching biodiversity in the city\" or \"Food and health\", which are among the seven themes offered to Parisian teachers. Through these actions, The Science House helps bring together the educational, scientific and industrial communities, made up of engineers, researchers and academic researchers with teachers from the French National Education system to co-construct coherent actions that reflect current research and are in line with school curricula. The Paris Île-de-France Maison pour la science was inaugurated on 16 October 2019 on the Pierre and Marie Curie campus of Sorbonne University. Read also. Formation Une Maison pour la science à Sorbonne Université PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home University Rental of Conference Facilities The International Conference Centre. The International Conference Centre. On the Pierre and Marie Curie campus, the Sorbonne University International Conference Center (CICSU) hosts professional events for internal and\/or external audiences (conferences, symposia, training courses, meetings, etc.), plus events for the general public (Science Festival, concerts, etc.). Publié le 24\/12\/2019 - Mis à jour le 4\/05\/2022. The International Conference Centre. Capacity. The CICSU offers:. 1 auditorium: 485 seats + 12 reduced mobility seats. 5 conference rooms: 80 to 120 seats. 3 meeting rooms: 16 to 30 seats. 2 reception areas:. - the foyer of the auditorium: 300 people for cocktails. - the reception hall 102: 150 people for cocktails. Covered Patio for 44-55 people. Availability. Monday to Friday, from 7:30 am to 6 pm. In the evening, until 10 pm - On request. Weekends - On request. Annual closures: from the last week of July to the 3rd week of August inclusive and Christmas holidays. Address and contact. 4 place Jussieu. 75005 Paris. Contact by email. Presentation of the CICSU (pdf) (in French). Booking the CICSU (pdf) (in French). Auditorium. Rooms. Sorbonne Université - Faculté des Sciences et Ingénierie. Campus Pierre et Marie Curie. 4 place Jussieu 75005 Paris. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. Let's build the network together Join a club or association Alumni portraits Contact us Education Education. Study at Sorbonne University Study at Sorbonne University. Our Academic Programs Degree-seeking Students Exchange Students Continuing Education Ensure successful studies Ensure successful studies. Office of the Registrar Tutoring Student Services Student internships Tools and Resources Reorientation and caesura Student life Student life. Grants and financial assistance Health and Handicap Services Sports and Cultural Activities Student associations Work Spaces Student Life and Cultural Spaces Cafeteria and Food Service Housing Prepare Your Professional Goals Prepare Your Professional Goals. Employability Facts and Figures Student Entrepreneruial Training Sorbonne University Librairies Sorbonne University Librairies. Consult online resources Use Sorbonne University services About the Sorbonne University Libraries Commitment to Open Science Research and innovation Research and innovation. Research strategy Research strategy. Research themes and programs Sorbonne University in Europe and in the world Commitment to open science Multidisciplinary institutes and initiatives The HRS4R Label Research partners Research partners. Sorbonne University Alliance Sorbonne University in Europe Sorbonne University Worldwide Partner companies Research structures and technical platforms Research structures and technical platforms. Research units Observatories Technical platforms Innovation and exploitation Innovation and exploitation. Knowledge transfer Collaboration with a research team Setting up a business Carnot Institutes The Doctorate The Doctorate. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Research and innovation Research strategy Commitment to open science. Commitment to open science. Publié le 14\/01\/2020 - Mis à jour le 5\/05\/2022. Commitment to open science. Commitment to open science and open access to scientific publications. Sorbonne University is committed to open science and free access to scientific publications. Its researchers are encouraged to file their publications in the Sorbonne University HAL open archive as soon as they are published. To support research staff, the Publications & Open Access Department of the Sorbonne University Library answers questions relating to the publication of their work, its open access dissemination, and offers training and awareness-raising activities on the issues, tools and best open science practices. Deposit a publication with the HAL Sorbonne Université Charter for open access to publications. By mutual agreement, Sorbonne University, its research staff and doctoral students agree to respect a charter concerning open access to publications. By depositing the full text of any article in the open archive HAL at Sorbonne University as soon as it’s published and then referencing the author’s other types of publications, research staff and PhD students contribute to the open access dissemination of their publications. Sorbonne University supports them by facilitating the depositing of publications, by ensuring their long-term archiving and by training the University community in the best practices of open access. The Sorbonne University Charter on open access to publications provides the following:. An awareness of the academic community of the societal and scientific issues of open access and open science . Local transposition of national policies for the implementation of open access and the facilitation of the depositing and reporting of publications in the Sorbonne University HAL. Supporting research staff and doctoral students in the publication process. The promotion of \"bibliodiversity\" in the spirit of the Jussieu Call. . The research staff and doctoral students of Sorbonne University commit to:. Act in favour of open science and open access to publications in the exercise of their various responsibilities . Check the adequacy of publishing contracts with the Sorbonne University's policy on open access. Privilege open access over a hybrid publication when an article is subject to publication charges (Article Processing Charges). Create and use IdHAL and ORCID identifiers that allow them to unambiguously identify themselves as authors of their publications and promote the visibility of their scientific production. . Download the charter , voted at the Research Commission of the Academic Council on 28 March 2019 . Sorbonne Declaration on Research Data Rights. The International Research Data Rights Summit, was held at Sorbonne University on Monday, January 27, 2020. This initiative brought together intensive research universities from nine major networks in major regions of the world. It was an opportunity to sign the \"Declaration of the Sorbonne\" on the research data rights. This text energetically affirms the universities' willingness to share their data while firmly calling on governments to adopt a clear legal framework to regulate this sharing and to provide the means to put it in place. Download . the Sorbonne Declaration on Research Data Law .  . Co-organized by Sorbonne University, the Universiteit van Amsterdam (UvA) and the University College of London (UCL), this data summit is unprecedented in its scale: nine networks representing more than 160 of the main research-intensive universities in the world. The signing of the “Sorbonne Declaration” is an ambitious message to the international scientific community and to research funding bodies and governments. This framework document, intended to promote the sharing and proper use of data, calls on the public and private research communities to join this commitment. It also confronts governments and research funders with their responsibilities of providing universities with the financial means and the appropriate legal framework to open data. This is an essential issue for the quality and transparency of research. It is also a crucial economic issue: funded largely by public money, research data represents tens of billions of euros worldwide. The objective is therefore to make these data accessible in order to accelerate scientific discoveries and economic development. For example, in Europe, according to a PwC1 report produced by the European Commission, sharing and better managing research data would save 10.2 billion euros per year in Europe, with an additional potential of 16 billion euros of added value by the innovation generated. The Participants are driving this change by proposing concrete solutions to restore confidence in public research and accelerate the development of knowledge and the economy. [1]. Cost of not having FAIR research data , PwC EU Services, March 2018. The signatory networks:. Association of American Universities (AAU) : Founded in 1900, the Association of American Universities is composed of the 65 largest American research universities. African Research Universities Alliance (ARUA) : The ARUA was inaugurated in Dakar in March 2015, bringing together sixteen of Africa's leading research universities. Coordination of French Intensive Research Universities (CURIF)  : The CURIF brings together the 18 most important French research-intensive universities. German U15 : German U15 represents 15 top-level intensive research universities, among the most distinguished and internationally visible institutions of the German university system. League of European Research Universities (LERU) : The LERU is a network of 23 leading European universities pushing the frontiers of research. Canadian Association of Research Universities (U15)  : The Canadian Association of Research Universities is a group of several of Canada's major research universities. RU11 Japan : RU11 is a consortium created in November 2009, composed of 11 of Japan's top research universities, very active in the international academic community. Russell Group : The Russell Group represents 24 world-renowned UK research-intensive universities. The Group of Eight (Go8) : The Group of Eight is a coalition of world-class research-intensive universities based in Australia. Training researchers in open science. The Sorbonne University Library and the Doctoral Education Institute are committed to implementing the systematic training of scientific personnel, especially young researchers, in Open Science. Training in Open Science issues. The objective of these modules is to understand how to use the HAL Sorbonne University portal to deposit works as soon as they are published. In practice, participants are trained in the legal conditions of the open access repository and in the use of the portal to implement this repository. The different aspects of the HAL are presented: authors' affiliations, interactions with PubMed, PubMed Central, arXiV and OpenAIRE, complementary tools (IdHAL, CV HAL, AureHAL), among others. Catalogue and registrations on the intranet Training in the depositing of publications in HAL. The objective of these modules is to understand how to use the HAL Sorbonne Université portal to deposit works as soon as they are published. In practice, participants are trained in the legal conditions of the open access repository and in the use of the portal to implement this repository. The different aspects of the HAL are presented: authors' affiliations, interactions with PubMed, PubMed Central, arXiV and OpenAIRE, complementary tools (IdHAL, CV HAL, AureHAL), among others. Find out more about the intranet Support in writing a Data Management Plan (DMP). The library offers personalized support to write a data management plan. Sessions can be organized on request for a laboratory. Individual meetings, face-to-face or remotely, can also be scheduled to help draft the three versions of a DMP. The library administers the. OPIDoR DMP tool, which can be used to write team DMPs, and can make presentations. Policy for promoting open science actions in the internal evaluation of research. Since its creation, the Sorbonne University Alliance has chosen to be fully involved in the open science movement, allowing access to knowledge for all. With this in mind, it is committed to opening up the results of the research produced within its institutions and promoting their dissemination, in order to share with citizens the fruits of their contribution to public research and to develop a trustworthy science. Download the text presented during the Research Commission on June 24, 2021:. Policy for promoting open science actions in the internal evaluation of research Training for researchers and academic researchers. The Sorbonne University Library, in partnership with the Staff Training Department, offers support on the principles, tools and practices of open science. If you are wondering what open science is, how to meet the requirements of funders, how to make a data management plan, how to share and archive your data and source codes, or how to manage your academic digital identifiers, the library team is at your disposal to accompany you. Download the. 2021-2022 training catalog (in French .). T. o register for training courses, go to the. Sorbonne University intranet. Contacts. Departement of Publications & Open Access at the Sorbonne University Library:. For any questions on the referencing and uploading of publications in Sorbonne University HAL:. hal@sorbonne-universite.fr For all other questions related to publications:. publications@sorbonne-universite.fr Research Data Missions:. data-bsu@sorbonne-universite.fr PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? 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Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. 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We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Research and innovation Innovation and exploitation Knowledge transfer. Knowledge transfer. The results of research conducted in Sorbonne Université laboratories lead to numerous innovations with applications in various fields. Whether for direct transfer to companies or the creation of start-ups, Sorbonne Université is open to society by responding to both societal issues and the needs of industrial stakeholders. . Publié le 8\/12\/2019 - Mis à jour le 5\/05\/2022. Knowledge transfer. A network of actors is mobilised to enable technologies and projects resulting from research structures to be exploited or marketed. Sorbonne Université personnel are supported in all the necessary steps for negotiating and setting up public and private research projects at regional, national and international levels, and in overseeing their implementation. . The Société d’Accélération et de Transfert de Technologie (SATT) Lutech. The University has entrusted Lutech with the management of its intellectual property (patents, software, know-how, etc.) as well as the negotiation of contracts for the transfer of its innovations. Lutech offers a portfolio of opportunities to companies and contributes the excellence of Sorbonne Université laboratories to lift certain technological barriers and facilitate their growth. Lutech supports project leaders through scientific and business maturation programmes. The shareholders of SATT Lutech are Sorbonne Université, the CNRS, the Compiègne University of Technology (UTC), the Muséum national d’Histoire naturelle (MNHN), the Université Panthéon-Assas, the École nationale supérieure de création industrielle (ENSCI) and the Caisse des Dépôts group. The SATT Lutec website Quadrivium Ventures. Sorbonne Université is behind Quadrivium Ventures, a company with a private equity fund of approximately €60 million. Quadrivium Ventures enables the start-up of young companies with innovative projects that are primarily derived from, or linked to, the laboratories of the Sorbonne Université ecosystem. They work together to detect, evaluate and accompany high-potential projects. 4 partner incubators. Sorbonne Université benefits from the services of incubators located in Île-de-France, Brittany, Occitania and Provence-Alpes-Côte d'Azur. It works closely with four incubators in the Paris region:. Agoranov which it hosts at its premises on boulevard Raspail -. Silver Innov' , dedicated to the Silver economy in Ivry-sur-Seine and two incubators within laboratories, at the. Institut de la Vision (Sorbonne Université\/CNRS\/Inserm) and the. Institut du Cerveau   (Sorbonne Université\/CNRS\/Inserm). Consortium French Tech Seed Paris Parc. Sorbonne Université, Lutech, and Quadrivium Ventures, are part of the French Tech Seed Paris Parc consortium, which seeks to promote synergies between all stakeholders in the ecosystem in terms of innovation. To achieve this, it relies on close partners: incubators, nurseries, networks of investors, and support for business creation. It is one of the first French Tech Seed consortia to be accredited by Bpifrance. Sorbonne Université, Lutech, et Quadrivium Ventures, sont regroupés au sein du consortium French Tech Seed Paris Parc qui a pour ambition de favoriser les synergies de tous les acteurs de l’écosystème en matière d’innovation. Pour cela, il s’appuie sur des partenaires proches : incubateurs, pépinières, réseaux d’investisseurs, accompagnateurs de la création d’entreprise. Il fait partie des premiers consortium French Tech Seed labellisés par Bpifrance. . Key figures (2017). 2, 000 research contracts with companies. 25 operating licenses on protected assets. 200 invention disclosures. €2 M in licensing revenues. 16 start-ups created (2016 and 2017). PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム アクセス. アクセス. 2階層目ページ用メニュー. 採用情報 アクセス 見学にいらっしゃる方 各キャンパスについて（吉田・宇治・桂） 各学部・研究科、センター等の場所について 食堂、レストラン、ショップなど キャンパスマップ（ダウンロード用） キャンパス間の移動（学内者向け） 連絡バスの運行 In other languages. English 京都大学の位置. 広域アクセス図　. 拡大図はこちら 空港からのアクセス図　. 拡大図はこちら 空港から京都へのアクセス. 空港からJR京都駅へは、電車、バス、タクシー等の交通手段を利用できます。所要時間や料金については各空港のサイトをご覧ください。. 関西国際空港から. 詳しくは. 関西国際空港公式Webサイトの「アクセス」ページ をご覧ください。. 大阪国際空港（伊丹空港）から. 詳しくは. 大阪国際空港（伊丹空港）公式Webサイトの「アクセス」ページ をご覧ください。. 成田国際空港から. 詳しくは. 成田国際空港公式Webサイトの「交通アクセス」ページ をご覧ください。. 京都大学の3つのキャンパス. 拡大地図はこちら 京都大学には、大きく分けて吉田キャンパス、宇治キャンパス、桂キャンパスの三つのキャンパスがあります。. 吉田キャンパス 宇治キャンパス 桂キャンパス  . 研究所・附置研究施設等. 本学の研究所や附置研究施設等は、日本全国に点在しています。研究所や附置研究施設等についての詳細は、. 「京都大学概要」の「研究所・附属研究施設等位置図」 をご覧ください。. 京都府内の施設. A. 舞鶴水産実験所（フィールド科学教育研究センター） B. 牧場（農学研究科附属） C. 芦生研究林（フィールド科学教育研究センター） D. 上賀茂試験地（フィールド科学教育研究センター） E. 花山天文台（理学研究科附属） F. 流域災害研究センター 宇治川オープンラボラトリー（防災研究所） G. 栽培植物起原学研究室（農学研究科） H. 宇治構内. I. 農場（農学研究科附属 ）. J. フィールド研北白川試験地（フィールド科学教育研究センター） K. 植物園（理学研究科） 全11施設、2022年4月現在. 全国の施設. 1. 北海道研究林［標茶区］（フィールド科学教育研究センター） 2. 北海道研究林［白糠区］（フィールド科学教育研究センター） 3. 大潟波浪観測所（防災研究所 附属流域災害研究センター） 4. 上宝観測所（防災研究所 附属地震予知研究センター） 5. 穂高砂防観測所（防災研究所 附属流域災害研究センター） 6. 飛騨天文台（理学研究科附属） 7. 北陸観測所（防災研究所 附属地震予知研究センター） 8. 犬山キャンパス. 9. 生態学研究センター 10. 流域圏総合環境質研究センター（工学研究科附属） 11. 信楽MU観測所（生存圏研究所 ） 12. 逢坂山観測所（防災研究所 附属地震予知研究センター） 13. 阿武山観測所（防災研究所 附属地震予知研究センター） 14. 複合原子力科学研究所 15. 屯鶴峯観測所（防災研究所 附属地震予知研究センター） 16. 紀伊大島実験所（フィールド科学教育研究センター） 17. 潮岬風力実験所（防災研究所 附属流域災害研究センター） 18. 瀬戸臨海実験所（フィールド科学教育研究センター） 19. 白浜海象観測所（防災研究所 附属流域災害研究センター） 20. 和歌山研究林（フィールド科学教育研究センター） 21. 鳥取観測所（防災研究所 附属地震予知研究センター） 22. 岡山天文台（理学研究科附属） 23. 徳山試験地（フィールド科学教育研究センター） 24. 徳島観測所（防災研究所 附属地震予知研究センター） 25. 徳島地すべり観測所（防災研究所 附属斜面災害研究センター） 26. 地球熱学研究施設（理学研究科附属） 27. 火山研究センター（理学研究科 附属地球熱学研究施設） 28. 熊本サンクチュアリ（野生動物研究センター） 29. 宮崎観測所（防災研究所 附属地震予知研究センター） 30. 幸島観察所（野生動物研究センター） 31. 桜島火山観測所（防災研究所 附属火山活動研究センター） 32. 生活・森林圏シミュレーションフィールド（生存圏研究所） 33. 屋久島観察所（野生動物研究センター） 全33施設、2022年4月現在. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 国際交流 外国人研究者へのサポート. 外国人研究者へのサポート. サイドメニュー. 世界とつながる京都大学 京都大学への留学を希望する方へ 国際化を支える学内組織 外国人研究者へのサポート 保険について 海外留学を希望する京大生へ 京都大学が提供する生活・研究サポートの情報や、日本での生活に必要となる手続きの情報へのリンク集です。. 生活サポート. 国際交流サービスオフィス（KUISO：Kyoto University International Service Office）. 京都大学に在籍するまたは在籍予定の留学生・外国人研究者およびその家族の方が必要とする査証（ビザ）に関する手続き、京都大学国際交流会館の入居手続き、日本での生活をサポートするための情報提供等を行っています。. 国際交流サービスオフィス ビザについて. （日本入国前）ビザ取得のための在留資格認定証明書交付代理申請について （日本滞在中）在留期間の更新と在留資格の変更手続きに関するご案内 住まい. 大学宿泊施設（国際交流会館）の情報 賃貸契約時に必要となる「保証人」を委託できる保証会社の紹介 外国語対応可能な不動産業者の情報 ハウジングガイドブック 日本での生活. 住民登録、社会保障・税番号制度、子供の保育・学校に関する情報 病院・医療 日本語を学ぶ 防犯・防災について その他の情報. 保険について 京都大学に在籍する研究者が加入できる保険についての案内です。 常勤雇用研究者、「招へい外国人学者」、「外国人共同研究者」等が加入できる保険についての情報を掲載しています。. 外国人研究者ハンドブック 外国人研究者ハンドブック（PDF版）をこちらからダウンロードできます。渡日前、滞在中、帰国前に必要な手続き等、役立つ情報が掲載されています。. 研究サポート. 学術研究支援室（KURA：Kyoto University Research Administration Office）. 学術研究支援室は、高度な専門知識・ 経験を有する「リサーチ・アドミニストレーター（URA）」で構成される組織で、多様な研究支援ニーズに取り組んでいます。学術研究支援室のWebサイトでは外部資金獲得支援、学内ファンド提供等の研究支援情報を掲載しています。. 学術研究支援室 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"5952V9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/site-policy\/policy","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":47,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","site-policy"],"url_file_name_s":"policy","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["サイトポリシー | 京都大学"],"description_txt":["京都大学ホームページ　運用指針 \n\n目的\n\n京都大学ホームページ（以下「ホームページ」という。）の円滑な運用を図るため、この指針を定める。\n\n定義\n\n本指針におけるホームページとは、「kyoto-u.ac.jp」のドメイン名を持つ以下のサイト、コンテンツおよびページを指す。\n\n\n\t京都大学公式ホームページ（https:\/\/www.kyoto-u.ac.jp\/）（日・英・中（繁、簡）・韓ページ）とその配下の各種コンテンツ、ページ、およびリンクする京都大学（部局、その他、京都大学に設置されている組織を含む）が管理運営する特設サイト\n\t京都大学に設置される部局等が運営するホームページ（以下、「部局等ホームページ」に同じ。）\n\n\n管理責任者\n\n京都大学公式ホームページについては、京都大学広報委員会ホームページ企画専門部会（以下｢専門部会」という。）が管理を行う。部局等ホームページについては、当該部局等が管理を行う。ただし、全学的な観点から必要があるときは、専門部会が部局等ホームページに関して意見を述べることができる。"],"last_modified":"Wed Jun 08 16:08:07 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-2949417382219155326,"exact_signature_unique_b":true,"fuzzy_signature_l":5762623642668969809,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["サイトポリシー"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","京都大学ホームページ　運用指針","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["目的","定義","管理責任者","運用責任者","掲載依頼者","公序良俗に反する情報","改善勧告","雑則"],"bold_txt":["京都大学ホームページ　運用指針"],"italic_txt":["menu","search","close"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1412,"text_t":"サイトポリシー | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 広報にかかる基本方針（広報戦略等） サイトポリシー. サイトポリシー. 京都大学ホームページ　運用指針. 目的. 京都大学ホームページ（以下「ホームページ」という。）の円滑な運用を図るため、この指針を定める。. 定義. 本指針におけるホームページとは、「kyoto-u.ac.jp」のドメイン名を持つ以下のサイト、コンテンツおよびページを指す。. 京都大学公式ホームページ（https:\/\/www.kyoto-u.ac.jp\/）（日・英・中（繁、簡）・韓ページ）とその配下の各種コンテンツ、ページ、およびリンクする京都大学（部局、その他、京都大学に設置されている組織を含む）が管理運営する特設サイト. 京都大学に設置される部局等が運営するホームページ（以下、「部局等ホームページ」に同じ。）. 管理責任者. 京都大学公式ホームページについては、京都大学広報委員会ホームページ企画専門部会（以下｢専門部会」という。）が管理を行う。部局等ホームページについては、当該部局等が管理を行う。ただし、全学的な観点から必要があるときは、専門部会が部局等ホームページに関して意見を述べることができる。. 運用責任者. 京都大学公式ホームページについては、総務部広報課が運用を行う。部局等ホームページについては、当該部局ごとに運用責任者を1名置き、運用を行うとともに、専門部会との連絡調整にあたるものとする。. なお、ここでいう総務部広報課が行う運用とは、運用責任者が自ら掲載する情報、または後に定める掲載依頼者から依頼があった情報を精査し、京都大学公式ホームページに公開することを指す。. 掲載依頼者. ホームページへの掲載依頼者（以下「依頼者」という。）は、不特定多数に向けた速報性を伴った情報を発信する、および情報を蓄積する等のホームページの特性を踏まえて、本学の公表すべき事項を正確に伝える、または本学の魅力を迅速かつ効果的に伝える手段としてホームページを理解し、掲載すべき情報の内容の精査を予め十分に行う。また、情報掲載後もその内容を常に最新かつ正確な状態に保つよう努める。. 公序良俗に反する情報. ホームページに掲載する情報は、京都大学が行う教育、研究、社会貢献に関するものとし、例えば、公序良俗に反するもの、商業的行為や政治･宗教活動等を目的とするものは除外する。. 改善勧告. ホームページに掲載された記事の内容が前項に反するものと専門部会が判断したときは、専門部会はその記事の掲載者およびその個所の運用責任者に対し、改善の勧告を行う。また、公序良俗に反するホームページへリンクすることも同様な措置の対象とする。. 雑則. その他、ホームページの運用について必要な事項は専門部会が定める。. （平成9年10月27日 ホームページ専門委員会制定）. （平成27年2月24日 広報委員会ホームページ企画専門部会改正）. （平成19年6月15日 京都大学広報委員会ホームページ企画専門部会改正）. （平成27年4月 1日 広報委員会ホームページ企画専門部会改正）. （平成29年4月 1日 広報委員会ホームページ企画専門部会改正）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 国際交流 教育 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. ターゲット. 在学生の方 公開日. 2022年06月01日. 　留学生ラウンジ「きずな」では、5月の月例イベントとして、「お気に入りマンガ・アニメ プレゼン大会」を2022年5月24日に開催しました。. 　今回のイベントは「きずな」の学生チューターによる進行のもと、参加者同士が、自分の好きなマンガやアニメの魅力を紹介しながら、文化交流を深めることを目的に企画しました。. 　参加者はそれぞれのお気に入りシーンや台詞、キャラクターグッズを解説しながら、情熱たっぷりにプレゼンテーションを展開しました。. 　他にも、きずなチューターによる京都を舞台にしたマンガ・アニメ作品の紹介等もあり、穏やかな雰囲気の中、皆で国際交流を楽しめる温かいイベントとなりました。. きずなチューターとイベント参加者. 関連リンク. 留学生ラウンジ「きずな」 きずな月例イベント（過去のイベント情報等） Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携. 入試・高大連携. 学生募集案内や各種入試に関する情報、高大連携についての情報を掲載しています。. オープンキャンパス情報もこちら。. メインナビゲーション. 一般選抜 一般選抜等学部入試における選抜方法等の変更 一般選抜入学者選抜要項 一般選抜学生募集要項 一般選抜インターネット出願 入学手続 一般選抜に係る情報開示について 一般選抜の試験問題等 入学者選抜実施状況 平成29年度京都大学一般入試　理科（物理）における入試ミスについて 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 出願状況と選考結果 よくある質問 その他の学部入学試験 外国学校出身者のための選考 編入学試験・学士入学試験 留学生のための入試 学部で学位をとる 学部入学案内 大学院で学位をとる 大学院入学案内 交換留学・短期留学・研究生等 大学院入試 大学院募集要項（外国人留学生特別入試を含む） 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る オープンキャンパス 大学案内冊子「知と自由への誘い 」 高大連携 高大連携・入試広報事業 学びコーディネーター事業 ELCAS（エルキャス） 高大接続・入試センター 関連情報 資料請求 受験生サポート 新入生の方へ（国際高等教育院Webサイト内）（外部リンク） 入学前後のスケジュール（国際高等教育院Webサイト内）（外部リンク） NEWS. 公開日. 2022年06月03日. タグ. 入試. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年05月31日. タグ. 入試. 令和5年度 特色入試入学者選抜方法における変更について（農学部森林科学科、食品生物科学科）（予告）. 公開日. 2022年05月31日. タグ. 入試. アジア・アフリカ地域研究研究科 第1回入学試験のお知らせ. 公開日. 2022年05月26日. タグ. 入試. 2023年度 薬学研究科修士・一貫制博士・博士課程（4年制）学生募集について. 最新ニュース一覧を見る EVENTS. タグ. 入試. 令和5（2023）年度入学者京都大学大学院総合生存学館（思修館）博士課程学生募集入試説明会. 開催日. 2022年06月10日、他. 開催地. 吉田キャンパス. 桂キャンパス. その他の地域. タグ. 入試. オンライン個別進学相談（2022年6月開催）. 開催日. 2022年06月17日、他. 開催地. オンライン. タグ. 入試. 2023年度 教育学研究科・教育学部学士入学 入試説明会. 開催日. 2022年06月25日. 開催地. オンライン. タグ. 入試. 高校生のための進学ガイダンス2022（岐阜）. 開催日. 2022年06月26日. 開催地. その他の地域. 最新イベント一覧を見る 関連タグ. 入試 高大連携 Features. 受験生ナビゲーション. 特色入試. 学びコーディネーター. 国際高等教育院. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 特色入試. 特色入試. サイドメニュー. 一般選抜 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 出願状況と選考結果 よくある質問 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 お知らせ. 現在公開中のものは、令和4年度特色入試の情報です。. 本学は、平成28年度入学者選抜から高等学校における幅広い学習に裏付けられた総合力と学ぶ力および高い志を評価し、個々の学部が定めたカリキュラムと教育コースを受けるにふさわしい学力と意欲を備えた者を選抜する「京都大学特色入試」を導入しました。. 「高大接続型」特色入試を受験される皆さんへ. 京都大学は、来る令和4年に創立125周年を迎える長い歴史と伝統を誇る総合大学です。京都大学の基本理念は自由の学風と創造の精神であり、自然科学分野ではアジア最多のノーベル賞やフィールズ賞受賞者を輩出し、人文社会科学分野では京都学派といわれる独自の思想・文化の潮流を作り出すなど、広い学術研究領域で輝かしい成果をあげてきました。私たちが尊重するのは独創性と多様性であり、若い皆さんの持つ大きな潜在的可能性を引き出し育んでいくことが責務であると考えています。. 大きな可能性を秘めた受験生が少しでも多く京都大学に集まっていただけるよう、京都大学では一般選抜に加え特色入試制度を開始し、全ての学部・学科で実施しています。これは、京都大学の基本精神を体現し、将来様々な分野でリーダーシップをとって社会に貢献できる優れた人材を発掘し育成することを目的として、高校での学修における特徴的な活動や成果を重視し、各学部・学科が独自に定める資質や意欲・志などを総合的に評価する本学独自の選抜方式です。. 京都大学は、特色入試と一般選抜で入学された多様な可能性を秘めた学生達が同じキャンパスで学業や課外活動に励み、対話と切磋琢磨のなかで成長していただくことを強く期待しています。様々なタイプの学生達が、お互いに親しく交流し友情を育んでいくことによって、きっと新しい考えの芽生えや自己発見があることでしょう。. 受験生の皆さんには、この特色入試という本学への新しい扉の趣旨を理解し利用してこの伝統ある京都大学に集い、新しい自分の可能性を発見していく旅にチャレンジしていただくことを期待しています。. 京都大学総長　湊 長博.  . 特色入試の基本方針. 本特色入試では、高大接続と個々の学部の教育を受ける基礎学力を重視し、. 高等学校での学修における行動と成果の判定. 個々の学部におけるカリキュラムや教育コースへの適合力の判定. を行い、1と2の判定を併せて、志願者につき高等学校段階までに育成されている学ぶ力及び個々の学部の教育を受けるにふさわしい能力並びに志を総合的に評価して選抜します。. 1. については、高大接続を重んじるという観点から、高等学校での学修における行動や成果を丁寧に評価するため、「調査書」に加え高等学校長等の作成する「学業活動報告書」や「推薦書」を提出していただきます。そこには、出願者の高等学校在学中の顕著な活動歴（例えば、数学オリンピックや国際科学オリンピック出場、各種大会における入賞、教育委員会賞、国際バ力ロレアディプロマコース・SAT・TOEFL・TOEIC・英検の成績など）を記していただき、志願者が受験科目以外にどういったことを学んできたか、どういった活動を実践してきたかを見ます。さらに、志願書が作成する「学びの設計書」等をもとに、高等学校での活動内容から本学において何を学びたいのか、卒業後どういった仕事に就きたいのかといった、志願者自らの学ぶ意欲や志について書類審査を通じて評価します。. 2. については、学部が定めた力リキユラムの内容を修得するのに必要とされる基礎学力や個々の学部における教育コースにとって望ましい能力を重んじるという観点から、書類審査に加えて、大学入学共通テストの成績、学部ごとの能力測定考査、論文試験、面接試験、口頭試問等を組み合わせて実施します。. 京都大学の基本理念 京都大学入学者受け入れの方針（アドミッション・ポリシー） 京都大学各学部が特色入試を通じて求める人物像 選抜の概要. チラシ（PDF） 詳細については、7月下旬に本サイトに掲載予定の特色入試学生募集要項で必ず確認してください。. 問い合わせ. 特色入試に関する問い合わせ. 教育推進・学生支援部 入試企画課. Tel： 075-753-2522. 各学部・学科に関する問い合わせ. 各学部・学科の教務担当窓口. 学部・学科. 担当掛. 電話番号. 総合人間学部. 教務掛. 075-753-6506. 文学部. 第一教務掛. 075-753-2709. 教育学部. 教務掛. 075-753-3010. 法学部. 教務掛. 075-753-3107. 経済学部. 教務掛. 075-753-3406. 理学部. 学部教務掛. 075-753-3616. 医学部 医学科. 学部教務掛. 075-753-4325. 医学部 人間健康科学科. 教務掛. 075-753-9313. 薬学部. 教務掛. 075-753-4514. 工学部. 教務掛. 075-753-5039. 農学部. 学部教務掛. 075-753-6012. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"t8vXB9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/about\/gender-equality","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":50,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","about"],"url_file_name_s":"gender-equality","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["男女共同参画について | 京都大学"],"description_txt":["基本理念\n\n新たな千年紀の開始を前にした1999年、国会で、男女共同参画社会基本法（The Basic Law for a Gender-Equal Society）が、衆参両院とも全会一致で可決成立しました。男女共同参画＝ジェンダー平等（gender equality）の原理に基づいた社会形成をめざすこの基本法は、その前文において、男女共同参画社会の実現は、「21世紀我が国社会を決定する最重要課題」であると明記しています。男女が、性別にかかわらず、意思決定も含めてあらゆる分野に対等に参加・参画する機会が確保され、その結果生み出される成果を対等に分かち合い、責任を共有し合う社会の形成が、今、本格的に求められようとしているのです。\n\n基本法では、その基本理念として\n\n\n\t男女の人権の尊重\n\t社会における制度または慣行の見直し\n\t政策等の立案および決定への共同参画\n\t家庭生活における活動と他の活動の両立\n\t国際的協調\n\n\n\nをうたっています。"],"last_modified":"Wed Jun 08 15:27:04 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":805533711496436704,"exact_signature_unique_b":true,"fuzzy_signature_l":8311198084530590870,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["男女共同参画について"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","基本理念","基本方針","京都大学男女共同参画推進センター","問い合わせ","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":574,"text_t":"男女共同参画について | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 男女共同参画について. 男女共同参画について. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について アクションプラン 女性活躍推進法について 意識・実態調査 改革 次世代育成支援対策推進 育児・介護支援制度 公表事項 情報公開 基本理念. 新たな千年紀の開始を前にした1999年、国会で、男女共同参画社会基本法（The Basic Law for a Gender-Equal Society）が、衆参両院とも全会一致で可決成立しました。男女共同参画＝ジェンダー平等（gender equality）の原理に基づいた社会形成をめざすこの基本法は、その前文において、男女共同参画社会の実現は、「21世紀我が国社会を決定する最重要課題」であると明記しています。男女が、性別にかかわらず、意思決定も含めてあらゆる分野に対等に参加・参画する機会が確保され、その結果生み出される成果を対等に分かち合い、責任を共有し合う社会の形成が、今、本格的に求められようとしているのです。. 基本法では、その基本理念として. 男女の人権の尊重. 社会における制度または慣行の見直し. 政策等の立案および決定への共同参画. 家庭生活における活動と他の活動の両立. 国際的協調. をうたっています。. この理念は、具体的にはそれぞれ、性差別的取り扱いの禁止、固定的役割分担意識など男女の性別に中立的でない制度や慣行の見直し、あらゆる分野での意思決定における男女の対等な参画の推進、男女両性の職業生活・家庭生活・地域活動のバランスの確保と男女の協力に基づく家庭運営、さらに国連を中心としたジェンダー平等の国際的取組みとの密接な連携などを意味しています。. 男女共同参画＝ジェンダー平等とは、社会的に構築された性別（ジェンダー）に基づくさまざまな偏見・差別を克服し、ひとりひとりの個性と多様性を認め合える社会を目指すものです。そのためにも、生物学的な性差に対する十分な配慮とともに、差別や排除の持つ問題性に常に敏感に対応し、ジェンダーによる固定的な決めつけを理由にした不利益や差別が生じないようにするための措置が何よりも求められます。. こうした男女共同参画＝ジェンダー平等の重要性という観点に立って、日本の大学のおかれた現状をみるとき、ここにも多くの問題が潜んでいることが理解できると思います。セクシュアルハラスメントやアカデミックハラスメントのような人権問題、研究や就業と家庭生活との両立の困難性、女性教員や重要なポストについている女性の割合の低さ、そして何よりも、こうした構造をささえる古い制度や慣行の存在など、解決すべき課題は山積みです。国際的にみても、男女共同参画＝ジェンダー平等という点で、日本の大学のおかれた状況はきわめて問題を含んだものになっています。. 本学は、その創設以来、自由の学風と批判的精神に基づく創造性あふれる学知の探求によって国際的にも広く知られてきました。そして今、時代の大きな転換点にあたって、自由な討論と多様性に開かれた学風をより発展深化させ、創造的かつ人類の幸福に資する学問の発展を図るためにも、男女共同参画＝ジェンダー平等に基づいた大学を構築していく必要があります。以上のような観点に立って、私たちは、以下のような基本方針を確認し、京都大学における男女共同参画の一層の推進を目指したいと思います。. 基本方針. 男女共同参画の視点に立った教育・研究および就業の確立. 教育・研究および就業と家庭生活との両立支援. 男女共同参画に資する教育・学習・研究の充実. 性差別への敏感な対応と迅速な解決. 教職員・学生への啓発活動の推進. 国・地方自治体、企業や市民セクターとの連携. 国際的な連携の促進. 京都大学男女共同参画推進センター. 男女共同参画に関する各種支援制度等については、以下のWebサイトをご覧ください。. 京都大学男女共同参画推進センター 問い合わせ. 人事部職員育成課ダイバーシティ推進掛. E-mail： g-e＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 国際交流 国際化を支える学内組織. 国際化を支える学内組織. サイドメニュー. 世界とつながる京都大学 京都大学への留学を希望する方へ 国際化を支える学内組織 学術研究支援室 国際交流サービスオフィス 学部・大学院研究科等事務室一覧 外国人研究者へのサポート 海外留学を希望する京大生へ 国際戦略本部. 国際戦略に係る施策の企画立案および調整を行うとともに、国際交流に係る支援、 海外拠点の設置・運営等その他国際化の推進に関する業務を行います。. International Strategy Office Kyoto University (iSO-KU) （in English）. Annual report academic year 2018（PDF） （1.6 MB）. 京都大学全学海外拠点. ASEAN拠点 （タイ・バンコク）. 欧州拠点 （ドイツ・ハイデルベルク）. 北米拠点 （アメリカ合衆国・ワシントンD.C.）. 学術研究支援室. 大学の改革、研究力強化、国際化等を戦略的に支援・推進事業を行います。. 京都大学 学術研究支援室  . 事務組織. 企画部 国際交流課. 国際交流に関する事務の統括および連絡調整を行います。. 問い合わせ先： kokusa850＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください。). 国際学術協定、国際交流事業・助成、表敬訪問対応に関する業務を行います。. 問い合わせ先： ipromo＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください。). 全学海外拠点の運営支援、国際危機管理に関する業務を行います。. 問い合わせ先：i_strategy850＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください。). 国際交流サービスオフィス. 外国人研究者・留学生向け宿舎・ビザに関する業務を行います。. 京都大学国際交流サービスオフィス 教育推進・学生支援部 国際教育交流課. 国際教育交流に関する事務の総括および連絡調整を行うとともに、外国人留学生の受入れおよび学生の海外留学に関する業務を行います。. 問い合わせ先： 850inter-c＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください。）. 学部・大学院研究科等の事務室一覧 国際教育アドミニストレーター. 国際教育アドミニストレーターによる教育の国際推進事業を行います。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 キャンパスライフ支援. キャンパスライフ支援. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生生活で注意してほしいこと 健康管理 在学中の保険加入 住まい・食と購買 アルバイト 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 3階層目以降ページ用メニュー. 学生生活で注意してほしいこと. キャンパスライフに潜んでいる様々な危険・あやうさをよく知り、自身がその危険に陥らないように注意することが大切です。その主なものを示します。 健康管理. 本学には学生の健康のために「健康科学センター」が設置され、「保健診療所」で診療を行っています。また、すべての学生に対して健康診断を実施しています。 在学中の保険加入. 大学生の生活には、正課・課外活動・日常生活の様々な場面で危険が潜んでいます。それに大人として備えるために、本学ではすべての学生が保険に加入することを原則としています。その概要を説明します。 住まい・食と購買. キャンパスライフの基礎となる住まい・食・書籍や物品の購入については、本学は主として京都大学生活協同組合に業務委託しています。 アルバイト. アルバイトをする場合は、日常の勉学に支障のないよう心掛けるとともに、危険を伴う作業や健康を害する恐れのある職種は避けてください。 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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Google Analyticsサービスの利用について","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1086,"text_t":"プライバシーポリシー | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム プライバシーポリシー. プライバシーポリシー. 京都大学は、高度情報通信社会における個人情報保護の重要性を認識し、京都大学のホームページ（. https:\/\/www.kyoto-u.ac.jp\/ ）において、その扱いに当たっては、法令を遵守し、. 京都大学における個人情報の保護に関する規程 （以下「規程」という。）とともに以下の方針に基づき個人情報の保護に努めます。. 1. 個人情報の取得・保有について. 京都大学は、適法かつ公正な手段によって、個人情報を取得致します。. 京都大学は、本学の業務の遂行上必要な場合で、かつ、その目的をできる限り特定し、その特定された利用目的の達成に必要な範囲を超えて個人情報を保有しません。. 個人情報を取得する場合には、取得する情報の利用目的、取得した情報の管理者を表示します。. 2. 個人情報の利用について. 京都大学は、法令に基づく場合を除き、個人情報を、取得の際に示した利用目的以外の目的のために利用し、又は第三者に提供しません。. 京都大学は、法令に定める場合を除き、個人情報を、事前に本人の同意を得ることなく、第三者に提供しません。. 京都大学は、法令に基づいて、個人情報を第三者に提供し、又は個人情報の取扱いに係る業務を第三者に委託する場合には、当該第三者につき、当該個人情報の安全確保、管理の能力等について厳正な調査を行い、個人情報を取り扱わせるに当たっては適正な監督を行います。. 3. 個人情報の管理について. 京都大学は、保有する個人情報について正確性を保つとともに、その紛失、破壊、改ざん及び漏えいなどを防止するため、不正アクセス、コンピュータウイルス等に対する適正な情報セキュリティ対策を講じます。. 4. 個人情報の開示・訂正・利用停止・消去について. 京都大学は、本人が自己の個人情報について、開示・訂正・利用停止・消去等を求める権利を有していることを確認し、これらの請求については、「規程」の定めるところにより、速やかに対応します。. 京都大学の個人情報の取扱いにつきましてご意見、ご質問がございましたら、京都大学の個人情報に関する開示請求の窓口（総務部法務室） 830jyohokoukai. ＊. mail2.adm.kyoto-u.ac.jp. （＊を@に変えてください）. までご連絡願います。. 開示請求方法について 5. 個人情報保護規程の策定・実施・維持・改善. この方針に定めるもののほか、京都大学における個人情報の保護については、「規程」の定めるところにより適切に実施します。. 6. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について データ集（学生数、入学状況等）. データ集（学生数、入学状況等）. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 職員数 学生数 入学状況 卒業者・修了者数 進路・就職状況 患者数等（医学部附属病院） 入館者数 財務状況 国際交流 外部資金等 発明・特許・知的財産 土地面積 建物面積 蔵書数等 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 データから見る京都大学. カテゴリ. データ. 歴代総長  . 創立以来の比較  . 職員数 1. 職員数（特定有期雇用教職員を除く）. 2. 学系別教員数. 3. 特定有期雇用教職員数. 4. 外国人職員数. 学生数 1. 大学院学生数. 2. 学部学生数. 3. 各種研修員等受入数. 入学状況 1. 大学院入学状況. 2. 専門職学位課程入学状況. 3. 学部入学状況. 卒業者・修了者数 1. 博士学位授与者数. 2. 修士課程修了者数. 3. 専門職学位課程修了者数. 4. 学部卒業者数. 進路・就職状況 1. 進路状況. 2. 産業別就職状況. 患者数等（医学部附属病院）  . 入館者数 1. 総合博物館入館者数. 2. 瀬戸臨海実験所水族館（白浜水族館）入館者数. 財務状況. 決算 財務状況 国際交流. 1. 外国人研究者・受入留学生 2. 学術交流協定校等 3. 派遣実績等 4. 海外交流拠点等位置図 外部資金等 1. 科学研究費助成事業. 2. 外部資金受入状況. 3. 寄附講座等. 発明・特許・知的財産 1. 発明件数. 2. 特許出願件数・取得件数. 3. 知的財産（件数・収入）. 土地面積  . 建物面積等 1. 部局別建物面積. 2. 構内別建物面積. 3. 工事概要. 蔵書数等  . 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 研究・産官学連携 本学教職員への兼業依頼について. 本学教職員への兼業依頼について. サイドメニュー. 研究活動を知る 研究成果の公開ポリシー 産官学連携（共同研究・知的財産・ベンチャー支援） 京都大学に寄附する 京都大学で研究したい 研究支援体制について 研究者が受けられる研究費・研究支援 研究上の倫理・安全 教職員に兼業を依頼したい 本学の教職員が兼業を行う場合は、学内規程で定める許可基準に基づき、各部局の審査を経て、大学から事前に許可を受ける必要があります。本学の教職員に兼業を依頼される際は、以下の要領により手続きください。. 【学内規程】. 国立大学法人京都大学に勤務する教職員の兼業に関する指針 依頼方法について. 本学の教職員に兼業を依頼される場合は、依頼状を作成のうえ、当該教職員の所属する部局の担当部署宛に送付ください。メール添付による送付も受け付けます。. 部局別の担当部署連絡先 依頼状の様式例. 様式の具体な取扱いや、その他必要な添付資料については、部局毎に定めています。詳細については. 各部局の担当部署 にご確認ください。. 営利企業の場合. 役員の職. 研究成果活用企業の役員・発起人 技術移転関連事業者（TLO等）の役員 株式会社の監査役 株式会社の社外取締役 非役員の職. アドバイザーや委員会委員等、非役員の職 営利企業以外の場合. 非常勤講師、行政機関等の委員会委員等 ※. 兼業を許可できる期間は、原則として3年以内です。（法令や定款等に任期の定めのある職につく場合は6年以内）. 許可期間を超えて引き続き兼業を行う場合は、許可期間の終了時に改めて依頼してください。. 依頼時期について. 兼業許可にあたっては会議附議等、学内手続きに時間を要しますので、予め余裕をもって依頼してください。. 特に、研究科長等の部局の長が行うものや営利企業の役員の職にかかるものについては、通常よりも審査に時間を要しますので、ご留意ください。. 依頼時期の目安. 部局の長が行うもの、営利企業の役員の職にかかるもの…兼業開始日の2～3か月前. 上記以外のもの…兼業開始日の1～2か月前. ※. 依頼日から兼業開始日までの期間が短い場合、開始日までに許可できないことがあります。. 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EP2\/4阻害薬による抗腫瘍作用の機序解明―新しいがん免疫治療薬の開発に向けて―. 研究成果. 公開日. 2022年06月08日. 廃棄物から生分解性プラスチック素材へ―大気窒素活用型発酵生産への展開―. 研究成果. 公開日. 2022年06月08日. 自ら流体中を泳ぐ「奇弾性体」の発見―生き物らしい自律的なマイクロマシンの仕組み―. 研究成果. 公開日. 2022年06月06日. 再発性多発軟骨炎の再発リスク因子の同定. 研究成果. 公開日. 2022年05月31日. 気管支オルガノイドを用いた新型コロナウイルス研究とその創薬応用. 研究成果. 公開日. 2022年05月31日. 遺伝子操作を加えずヒトiPS細胞から成熟骨格筋細胞を作製する技術を発明. 研究成果. 公開日. 2022年05月27日. タンパク質の立体構造解析に新たなモデルを提唱―より正確な立体構造の観測や予測を実現！生命科学研究の進展に寄与―. 研究成果. 公開日. 2022年05月27日. ゼロカーボンバイオ産業創出の基盤となるタンパク質繊維試作品の完成. 研究成果. 公開日. 2022年05月27日. ゼロカーボン水産養殖用飼料の作出に成功. 研究成果. 公開日. 2022年05月27日. ゼロカーボン窒素肥料の作出に成功. 研究成果. 公開日. 2022年05月27日. 新型コロナウイルスの超高感度・全自動迅速検出装置の開発－汎用的な感染症診断装置としての社会実装に期待－. 研究成果. 公開日. 2022年05月26日. 光誘導物質による新しい体内時計の同調制御機構―光により誘導される神経細胞の活動抑制物質の発見―. 研究成果. 公開日. 2022年05月24日. 表情豊かなコモンマーモセット―新しい表情解析ツール―. 研究成果. 公開日. 2022年05月24日. 酸素が反応に寄与するニッケル系電池材料 高エネルギー密度リチウム電池実現に期待―新規不規則岩塩型ニッケル系材料開発と次世代蓄電池への応用―. 研究成果. 公開日. 2022年05月23日. 「てこ」と「ドミノ倒し」で巧妙に開くイオンの経路―新規不整脈治療薬へ向けた手がかりを提示―. 研究成果. 公開日. 2022年05月20日. マイクロ流体デバイスの製造に革新をもたらす新手法. ページ送り. カレントページ. 1 ページ. 2 ページ. 3 ページ. 4 ページ. 5 ページ. 6 ページ. 7 ページ. 8 ページ. 9 &hellip;. 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. 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Education. \"AY2021 IAfP Short-term online exchange programs – Addis Ababa University – \/ AY2021 Online lecture series on lifelong sciences\". Published on. 2022\/04\/13. タグ. Global Engagement. Education. \"Short-term Intern Program in Primatology and Wildlife Science\" held online (1-9 March 2022). More news Upcoming Events. タグ. Research. An Ex-Cell Press Editor presents: Telling the Story in a Scientific Paper. Date. 2022\/06\/24. Venue. Online. More events Features. Graduate Admission Guide for International Applicants. KyotoUx Free online courses. Kyoto University OpenCourseWare. University profile. Kyoto iUP (Kyoto University International Undergraduate Program). Public relations. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. 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on Facebook","Share on Twitter","","","","","安否確認システム"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:43:33 UTC 2022","fresh_date_dt":"Wed Jun 08 17:07:41 UTC 2022","referrer_id_s":"PrXlG9U8j7Mo","language_s":"ja","size_i":86651,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085641827352576,"text_t":"令和4年度 京都大学CFプロジェクト奨学金（在学採用）の申請について | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 令和4年度 京都大学CFプロジェクト奨学金（在学採用）の申請について. 学生支援 令和4年度 京都大学CFプロジェクト奨学金（在学採用）の申請について. ターゲット. 在学生の方 公開日. 2022年04月20日. ※ 「令和4年度京都大学CFプロジェクト奨学金（在学採用）募集要項」のファイルを変更、「対象学生、支給額、支援期間、募集人数等」の記載を一部削除しました。（2022年4月22日）. 　京都大学CFプロジェクトは、建築家の安藤忠雄氏、株式会社ニトリホールディングス会長の似鳥昭雄氏をはじめとする方々の御賛同とご支援をいただき、立ち上がったプロジェクトです。. 　CFプロジェクトでは、民間からの寄附金による人材育成基金により、京都大学での多様な分野の優秀な次世代研究者の育成を支援していきます。向学心に富み強い創造心や研究意欲をもつ学生が、経済的理由などでその志を途中であきらめなくてすむようにすることが、CFプロジェクトの目的です。. 　本奨学金の申請に際しては. 「令和4年度京都大学CFプロジェクト奨学金（在学採用）募集要項」 をご確認ください｡. 　. （「3、奨学金給付額、支給期間」専門職学位課程学生に係る出願対象学年について追記）. 対象学生、支給額、支援期間、募集人数等. 対象学生（区分）. 出願対象学年. 支給額、支援期間. 募集枠、採用予定者数. 学部学生. （3、4年次）. 学部3年次在籍者に限る. 月額：5万円. 年額：60万円. 期間：2年間. ニトリchair（35名程度）. 6年制の学部学生. （4、5、6年次）. 学部4年次在籍者に限る. 月額：5万円. 年額：60万円. 期間：3年間. 修士. *1. 、. 専門職学位課程学生. 大学院（修士・専門職学位課程）1年次在籍者に限る. 月額：10万円. 年額：120万円. 期間：最短修業年限まで. ニトリchair（11名程度）、. 安藤忠雄＋オフィシャルサポータ－chair（12名程度）. 博士後期課程学生、. 博士課程学生（4年）、. 一貫制博士課程学生（3年次以上）. 博士後期課程、博士課程1年次在籍者に限る、一貫制博士課程においては、博士後期課程の第1年次に相当する年次の在籍者に限る. 月額：10万円. 年額：120万円. 期間：最短修業年限まで. 安藤忠雄＋オフィシャルサポータ－chair（6名程度）. 修士課程には、一貫制博士課程の1年次・2年次に在籍の者を含みます。なお、この場合の支給期間は一貫制博士課程の1年次・2年次となります。. なお、一貫制博士課程の学生が、3年次以降も本奨学金を希望する場合は、改めて一貫制博士課程学生（3年次以上）を対象とする区分に申請する必要があります。. 出願資格. 次の（1）から（3）を満たす者. 本学の正規の教育課程に在籍の者。. 学業優秀である者。学業基準は別表のとおり。. 日本学生支援機構第一種奨学金の家計基準を満たす者。または、日本学生支援機構第一種奨学金の家計基準を満たす者に準じると本学が認めた者。. ただし、次に該当する者は申請できません。. 日本学術振興会の特別研究員（DC1､DC2）として研究奨励金の支給を受けている者。なお、特別研究員に申請中の者は、本奨学金に申請可能ですが、特別研究員に採用された場合は、本奨学金の受給資格を失います。. 科学技術振興機構（JST）科学技術イノベーション創出に向けた大学フェローシップ創設事業に採用されている者。なお、同プログラムに申請中の者は、本奨学金に申請可能ですが、採用された場合は、本奨学金の受給資格を失います。. 科学技術振興機構（JST）次世代研究者挑戦的研究プログラム（京都大学大学院教育支援機構プログラム）に採用されている者。なお、同プログラムに申請中の者は、本奨学金に申請可能ですが、採用された場合は、本奨学金の受給資格を失います. 申請時に休学している者（ただし、休学理由が留学の者を除く）。. 出願時または出願にかかる学期の開始前6ヶ月以内に京都大学通則第32条第1項（第53条及び第53条の15において準用する場合を含む）の規定による懲戒処分を受けている者及び処分中の者。. なお、出願後、本奨学金の支給決定までの間に懲戒処分を受けた場合は、当該出願資格を無効とします。. 在留資格に関する要件（日本国籍でない場合）. 外国籍の学生は、在留資格等によって申し込みができない場合があります。. 出願の際は、「在留資格」及び「在留期限（在留期間の満了日）」を申告し、支給対象となる在留資格であることの証明書を提出する必要があります。. ※. 申し込みが可能な在留資格. 法定特別永住者、永住者、日本人の配偶者等、永住者の配偶者等、定住者. 法定特別永住者及び永住者の者は、在留期間が記載されている書類の提出は必要ありません。. 申請方法. 申請には､以下の 1. および 2. の2つの手続きが必要です（1. または 2. のみの手続きでは申請は完了しません）｡. 「民間奨学金等申請システム」を利用したWeb申請で受け付けを行います。. KULASIS（京都大学教務情報システム）のトップ画面－（画面右下）リンク集「民間奨学金等申請システム」. 「民間奨学金等申請システム」は、2022年4月28日（木曜日）にオープン予定です。. 「民間奨学金等申請システム」で作成した「地方公共団体・民間団体奨学金申込書」および必要書類によって申し込み手続きを行ってください。. 必要書類は、. 「令和4年度京都大学CFプロジェクト奨学金（在学採用）募集要項」 を確認してください。. （「3、奨学金給付額、支給期間」専門職学位課程学生に係る出願対象学年について追記）. 申請期間. 「民間奨学金等申請システム」によるWeb申請. 申請期間. 2022年4月28日（木曜日）～5月19日（木曜日）. 提出書類. 提出期間. 2022年年4月28日（木曜日）～5月20日（金曜日）17時00分まで（時間厳守・郵送必着）. 土曜日、日曜日、祝日は閉室していますのでご注意ください。. 持参・郵送のいずれも可（学内便は不可）。. 提出先. 〒606-8501 京都市左京区吉田本町（総合研究10号館）. 京都大学 教育推進・学生支援部 学生課 奨学掛. 本部・西部構内マップ ［60］. 郵送の場合. 簡易書留やレターパックライトなど配達の記録が残るものでの送付をお願いします。また、宛先住所には（総合研究10号館）まで記載してください。. 封筒の表面（レターパックライトでは｢品名｣欄）には赤字で｢CFプロジェクト奨学金申請書類在中｣と記載してください｡. 持参の場合. 上記提出先の窓口まで持参ください。. 問い合わせ. 教育推進・学生支援部学生課奨学掛. 総合研究10号館（窓口9時00分～17時00分、土日・休日閉室）. E-mail： 840scholarship＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Privacy policy. Privacy policy. In other languages. 日本語 In recognition of the growing importance of privacy protection in today's highly advanced information and communication environment, Kyoto University strives to comply with all applicable laws and to protect personal information on its website (. https:\/\/www.kyoto-u.ac.jp\/ ) in accordance with its. Regulations Concerning the Handling of Personal Information (hereinafter, \"the Regulations\") and the Privacy Policy, the latter of which is described below. 1. Collection and retention of personal information. Kyoto University shall use legitimate and fair means when collecting personal information. Kyoto University shall retain personal information for administrative purposes only and within the limits of clearly defined objectives. When collecting personal information, Kyoto University shall disclose the intended use for such information and the person(s) to be in charge of managing the information obtained. 2. Use of personal information. Unless required by law, Kyoto University shall not use personal information for purposes other than those for which it was collected, or pass it on to a third party. Unless required by law, Kyoto University shall not submit personal information to a third party without prior consent from the subject of that information. When submitting personal information to a third party as required by law, or outsourcing a task that involves the use of such information, Kyoto University shall assess, in advance, the privacy protection and information management capabilities of the party in question, and ensure that its handling of the entrusted information is subject to adequate supervision. 3. Management of personal information. In order to ensure the accuracy of the personal information it has obtained, and to safeguard such information against loss, destruction, falsification, or improper dissemination, Kyoto University shall have security measures in place aimed at blocking unauthorized access and avoiding computer viruses. 4. Disclosure, correction, suspension of use, and deletion of personal information. Kyoto University shall strive to ensure that those whose personal information is in its possession are aware of their rights to request the disclosure, correction, suspension of use, and deletion of that information, and to promptly respond to such requests in accordance with the Regulations. Comments and questions about Kyoto University's handling of personal information may be directed to the Legal Affairs Division, General Affairs Department (email , replacing [at] with @). Requesting the disclosure of retained personal information 5. Formulation, implementation, maintenance, and updating of the Regulations and Privacy Policy. In addition to this Policy, the handling of personal information at Kyoto University is subject to the terms of the Regulations. 6. Google Analytics. Google Analytics services are employed to collect and analyze usage data for the purpose of website improvement. Google Analytics uses first-party cookies. To learn about how Google Analytics handles personal information, please see. this page .  . Approved 19 March 2005. Kyoto University Public Relations Committee. Website Subcommittee. Amended 7 November 2016. Website Task Force. Amended 1 April 2017. Website Task Force. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 男女共同参画について 女性活躍推進法について. 女性活躍推進法について. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について アクションプラン 女性活躍推進法について 意識・実態調査 改革 次世代育成支援対策推進 育児・介護支援制度 公表事項 情報公開 女性の個性と能力が十分に発揮できる社会を実現するために、平成28年4月1日より女性活躍推進法が施行されました。同法に基づき、本学では同法で定められた基礎項目を毎年度公表し、本学における男女共同参画推進の状況を周知することとしています。. 京都大学行動計画（令和2年4月1日～令和6年3月31日） 基礎項目（令和4年1月1日時点） 過去の行動計画. 行動計画（平成28年4月1日～平成32年3月31日） 基礎項目（平成28年1月1日時点） 基礎項目（平成29年1月1日時点） 基礎項目（平成30年1月1日時点） 基礎項目（平成31年1月1日時点） 基礎項目（令和2年1月1日時点） 基礎項目（令和3年1月1日時点） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 栄誉等 京都大学名誉フェロー. 京都大学名誉フェロー. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 京都大学名誉博士 京都大学名誉フェロー 栄誉 歴代総長 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 京都大学名誉フェローの称号授与について. 平成25年3月19日に制定された称号で、本学の国際交流に寄与した功績が特に顕著である者、本学に教職員、学生等で所属または在学した者であって、特に優れた業績により国内外で高い評価を受けた者、本学の運営および経営に特に顕著な貢献があった者、またはその他総長が特に顕彰することが必要と認める者を対象とし、推薦に基づいて総長および理事で構成する会議の議を経て、総長が授与を決定します。. 第5号. 氏名. 柳井 正. 国籍. 日本. 授与時の年齢. 71歳. 授与決定年月日. 令和2年6月29日. 授与時の所属等. 株式会社ファーストリテイリング代表取締役会長兼社長. 授与理由. 2020年に柳井正 氏より、本学が誇るノーベル賞受賞者の本庶佑医学研究科附属がん免疫総合研究センター長および山中伸弥iPS細胞研究所長に、寄附いただいた。. 本庶センター長への寄附については、本庶センター長が進める「PD-1 阻害がん免疫療法」を推進するとともに、次世代を担うがん免疫療法の研究者を育成することを目的として、京大基金：柳井基金を設置するものである。これにより、同センターにおけるがん免疫分野における研究の推進および人材育成拠点としての発展が期待される。. 山中所長への寄附については、山中所長が進める「新型コロナウイルス研究プロジェクト」への支援をいただくもので、新型コロナウイルスの感染拡大防止を目的に、検査システムの構築や感染把握の調査、治療薬およびワクチンの開発を進めるため、iPS細胞技術を用いた研究を推進していくために活用されるものである。. このような多大なる支援は、本学の教育研究活動・社会貢献に大きく寄与するものであり、本学の運営および経営に特に顕著な貢献があったと認め、名誉フェローの称号を授与することとなった。. 第4号. 氏名. 船井 哲良. 国籍. 日本. 授与時の年齢. 87歳. 授与決定年月日. 平成26年9月8日. 授与時の所属等. 船井電機株式会社 取締役会長. 授与理由. 2005年に船井哲良氏より、教育研究環境の充実に寄与することで我が国の学術研究の発展や今後期待される日本産業の競争力強化に貢献したいと、「船井哲良記念講堂」および「船井交流センター」を寄附いただいた。両施設は2007年度に完成し、学会・講演会・シンポジウムの開催のほか、共同研究施設や実験施設における共同研究の実施により、本学に蓄積された知の資産と伝統を社会に開くことで研究を活性化させ、高度で先端的な研究の推進を先導すべく、知の融合・交流の場としての役割を担っている。. また、2012年に「京都大学大学院思修館」の目的等に共鳴され、合宿型研修施設の建設費を寄附いただいた。. この寄附を通して、思修館が目的とする国際社会でグローバルリーダーとして活躍できる人材の育成の場として学生が24時間起臥を共にし、異文化・異分野出身の学生同士や教員との議論を通じて多様な思考と実践力を身につけさせる学習環境整備に寄与することが期待される。. このように、国際的な学術研究の交流の場と研究成果の発信の場を新たに本学に配することができ、結果、本学の学術研究が発展・推進したことから、本学の運営および経営に特に顕著な貢献があったと認め、名誉フェローの称号を授与することとなった。. 第2号. 氏名. 稲盛 和夫. 国籍. 日本. 授与時の年齢. 81歳. 授与決定年月日. 平成25年7月23日. 授与時の所属等. 京セラ株式会社 名誉会長、公益財団法人稲盛財団 理事長. 授与理由. 2007年に公益財団法人稲盛財団より、国際社会において21世紀のさらなる学術・文化の発展に貢献していくため、「稲盛財団記念館」を建設・寄附いただいた（2008年竣工）。. 同館にはアジア・アフリカ地域研究研究科、東南アジア研究所等が入居し、アジア・アフリカ地域をはじめとする世界的地域研究の拠点が形成されることとなった。. このように、国際的な学術研究の交流の場と研究成果の発信の場を新たに本学に配することができ、結果、本学の学術研究が発展・推進したことから、本学の運営および経営に特に顕著な貢献があったと認め、名誉フェローの称号を授与することとなった。. 第2号. 氏名. 山内 溥. 国籍. 日本. 授与時の年齢. 85歳. 授与決定年月日. 平成25年7月23日. 授与時の所属等. 任天堂株式会社 相談役. 授与理由. 2006年に医学部附属病院に対して、「大学病院の使命にふさわしい病棟を」という趣旨で病棟建設費を寄附いただいた。これにより、高度先進医療・最先端医療を実践する病棟「積貞棟（せきていとう）」を建設した（2010年竣工）。. この病棟の建設により、医学部附属病院の医療の質的向上の推進が図られ、結果、病院経営の安定に繋がったことから、本学の運営および経営に特に顕著な貢献があったと認め、名誉フェローの称号を授与することとなった。. ※ 第2号については五十音順. 第1号. 氏名. アウン・サン・スー・チー Aung San Suu Kyi. 国籍. ミャンマー連邦共和国. 授与時の年齢. 67歳. 授与決定年月日. 平成25年4月9日. 授与時の所属等. ミャンマー連邦共和国 連邦議会 下院議員、国民民主連盟（NLD）議長. 授与理由. アウン・サン・スー・チー氏は、1985～86年に東南アジア研究センター（現 東南アジア研究所）に外国人研究員として在籍され、石井米雄教授の指導のもと、父アウンサン将軍についての研究に従事された。. 本学に研究員として在籍した同氏が、「民主主義と人権回復のための非暴力闘争」という業績によりノーベル平和賞を受賞される等、国内外で高い評価を受けたため、名誉フェローの称号を授与することとなった。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 受賞. タグ:. 受賞. すべて. Events. News. News. 公開日. 2022年05月26日. タグ. 受賞. 山下真由子 数理解析研究所助教が第1回羽ばたく女性研究者賞（マリア・スクウォドフスカ＝キュリー賞）を受賞しました. News. 公開日. 2022年05月16日. タグ. 受賞. 藤野修 理学研究科教授が2022年度日本数学会代数学賞を受賞しました. News. 公開日. 2022年05月02日. タグ. 受賞. 2022年春の叙勲受章者が発表されました. News. 公開日. 2022年04月22日. タグ. 受賞. 飯間麻美 医学部附属病院助教が第27回日本女性科学者の会 奨励賞を受賞しました. News. 公開日. 2022年04月11日. タグ. 受賞. 文部科学省「令和4年度科学技術分野の文部科学大臣表彰」に本学から11名が選ばれました. News. 公開日. 2022年03月23日. タグ. 受賞. 「歴史的化学論文大賞（Citation for Chemical Breakthrough Awards）」を受賞しました. News. 公開日. 2022年03月23日. タグ. 受賞. 令和3年度「総長賞表彰式」を挙行しました. News. 公開日. 2022年03月17日. タグ. 受賞. 村上 章 理事・副学長、野田 進 工学研究科教授が日本学士院賞を受賞（2022年3月14日）. News. 公開日. 2022年03月17日. タグ. 受賞. 研究. 黒田浩一 農学研究科准教授らが日本農芸化学会の2021年B.B.B.論文賞を受賞しました. News. 公開日. 2022年03月11日. タグ. 受賞. 北島薫 農学研究科教授が「みどりの学術賞」を受賞しました. News. 公開日. 2022年03月08日. タグ. 受賞. 第14回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催しました. News. 公開日. 2022年03月02日. タグ. 受賞. 乾 晴行 工学研究科教授がフンボルト賞（Humboldt Research Award）を受賞しました. News. 公開日. 2022年02月25日. タグ. 受賞. 時任宣博 理事・副学長が、2021年度「第74回日本化学会賞」を受賞しました. News. 公開日. 2022年02月15日. タグ. 受賞. 第14回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式の開催について. News. 公開日. 2022年02月08日. タグ. 受賞. 第15回湯川記念財団・木村利栄理論物理学賞授賞式を開催しました. News. 公開日. 2022年02月08日. タグ. 受賞. 2021年度 京都大学久能賞受賞者が決定しました. ページ送り. カレントページ. 1 ページ. 2 ページ. 3 ページ. 4 ページ. 5 ページ. 6 ページ. 7 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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高等教育の修学支援新制度について 授業料の免除／入学料の免除と徴収猶予 さまざまな奨学金 日本学生支援機構（JASSO）奨学金 大学を通じて申請する他の奨学金（民間団体と地方公共団体） 本人が直接申請する奨学金 急に困った時に 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 以下に日本学生支援機構（JASSO）奨学金の諸手続きについて説明しています。また、手続きに必要な様式をダウンロードすることができます。2020年度より高等教育の修学支援新制度が始まり、奨学金制度と授業料免除制度が連動することになりました。あわせてご確認ください。. 高等教育の修学支援新制度について ※ 募集、各手続きの最新情報については、. さまざまな奨学金 ページのお知らせ欄を参照してください。また. 日本学生支援機構奨学金 のWebサイトも併せてご覧ください。. 申し込み・採用後の手続きについて 「適格認定」について 異動・月額変更手続きについて 貸与終了（後）にかかる手続きについて （大学院生）第一種奨学金の特に優れた業績による返還免除について 書類提出時の注意. ： 押印箇所はスタンプ印不可です。原則、印鑑持参のうえ教育推進・学生支援部学生課奨学掛窓口に提出してください。ただし卒業・退学後や休学・留学中の場合は郵送での提出を受け付けます。郵送で提出する場合は、不備のないよう特に注意してください。不備があった場合は処理が遅れることがあります。. 申し込み・採用後の手続きについて. 奨学金の種類と給付・貸与月額について. 日本学生支援機構（JASSO）奨学金は、給付型奨学金（学部生のみ対象）および無利子貸与の第一種奨学金と有利子貸与の第二種奨学金があります。. 学部生・給付月額. 通学形態. 給付月額. 自宅通学. 29,200円、19,500円、9,800円、20,000円. （※）. 自宅外通学. 66,700円、44,500円、22,300円、30,000円. （※）. （※）は、2018、2019年度入学時に採用された旧給付奨学生のみ。（授業料免除を受けた場合は減額となります）.  . 学部生・貸与月額. 種別. 通学形態. 貸与月額. 第一種. 自宅通学. 20,000円. （※）. 、30,000円、45,000円. 自宅外通学. 20,000円. （※）. 、30,000円、40,000円. （※）. 、51,000円. 第二種.  . 20,000円～120,000円（10,000円ごと）. 第一種奨学金と給付奨学金（新制度）を併用する場合は、. こちら に記載のとおり貸与額が調整されます。. （※）は、2018年度以降入学者のみ. 大学院生. 課程. 種別. 貸与月額. 修士課程. （専門職学位課程、一貫制博士課程修士相当含む）. 第一種. 50,000円、 88,000円. 第二種. 50,000円、 80,000円、100,000円、130,000円、150,000円. ※ 法科大学院は、ほかに190,000円、220,000円もあります。. 博士（後期）課程. 第一種. 80,000円、122,000円. 第二種. 50,000円、80,000円、100,000円、130,000円、150,000円. ※ 入学時特別増額貸与奨学金（有利子）：. 第一種・第二種奨学金と併せ、（編）入学時に申し込むことができます（一時金、10万円、20万円、30万円、40万円、50万円から選択）。ただし申し込み要件は、一定の収入金額以下である場合、または国の教育貸付（日本政策金融公庫の教育ローン）を申し込んで貸付を受けることができなかった場合です。. 奨学生の採用について. 採用の種類として、次のとおり「在学採用」「予約採用」「緊急・応急採用（貸与）」「家計急変採用（給付・学部生のみ）」等があります。. 在学採用. 給付奨学金： 原則毎年、春および秋に募集します。. 貸与奨学金： 原則毎年春に募集を行いますが、秋にも募集することがあります。. 予約採用. 学部生については、高等学校等で申し込みを行い採用候補者として決定され、入学後に一定の手続を経て採用されるものです。. 大学院生については、入学試験合格後、入学予定の大学で申し込みを行い採用候補者として決定され、入学後に一定の手続を経て採用されるものです（研究科によっては実施していないところもあります）。. ※ 「採用候補者決定通知」を紛失した場合は「再交付願」を教育推進・学生支援部学生課奨学掛に提出してください。. 「再交付願」 （学部生用）／. 「再交付願」 （大学院生用）. 緊急・応急採用. 家計急変採用. 家計の急変、風水害等の災害等により、緊急に奨学金の貸与が必要となった学生を対象とする制度です。随時募集を行っていますが、家計が急変してから12ヶ月以内（給付奨学金は3ヶ月以内）に申し込む必要があります。詳しくは教育推進・学生支援部学生課奨学掛窓口に相談してください。. 災害救助法適用地域一覧 （日本学生支援機構Webサイト）. 編入学・転入学等により入学した場合、以前在籍した学校で給付・貸与された奨学金を継続できる場合があります。詳しくは入学試験合格後、入学前に教育推進・学生支援部学生課奨学掛に連絡してください。. 海外の大学等に留学、進学する学生に対し、第二種奨学金を貸与する制度があります。. 【学部生・貸与奨学金】申し込みにかかる本学所定様式（該当者のみ）. 年収見込み申告書 （2021年1月2日以降に転職・退職した場合等）. 無収入申立書 （公的書類により無収入であることを証明できない場合等）. ※ 2021年1月2日以降に無職となり、離職票や退職証明書等で証明できない場合に限り使用可能。. 長期療養費支出状況申告書 ／. 【長期療養】領収書貼付台紙 （長期療養控除を希望する場合）. 主たる家計支持者の別居支出状況申告書_ ／. 【主たる家計支持者別居】領収書等貼付台紙 （家計支持者別居控除を希望する場合）. 【貸与奨学金のみ】採用手続きについて. 採用された場合は、奨学生証、返還誓約書等を受け取り、所定の期日までに返還誓約書を作成し、添付書類とともに提出してください。（書類交付は採用月下旬）. 返還誓約書添付書類一覧. 人的保証選択者. 機関保証選択者. 奨学生本人の住民票※（コピー不可）. 申込時に奨学生本人のマイナンバーを提出している場合は、提出不要. 連帯保証人の印鑑登録証明書※（コピー不可）. 連帯保証人の収入証明書（コピー可）. （所得証明書、源泉徴収票等）. 保証人の印鑑登録証明書※（コピー不可）. 奨学金申し込み日から3ヶ月前以降に発行されたものを提出すること. 奨学生本人の住民票※（コピー不可）. 申込時に奨学生本人のマイナンバーを提出している場合は、提出不要. 保証依頼書（兼保証委託契約書）. 奨学金申し込み日から3ヶ月前以降に発行されたものを提出すること. 返還誓約書にかかる様式. 印字事項を訂正（一部追記含む）する場合：. 返還誓約書記載事項訂正届 （成年者用）、. 返還誓約書記載事項訂正届 （未成年者用）. 65歳以上の保証人を選任する場合：. 返還保証書 ※. 保証人（または連帯保証人）に4親等以内の親族以外の者を選任する場合：. 返還保証書 ※. ※「返還保証書」には資産等に関する証明書（コピー可）の添付が必要です。また、次のとおり「貸与予定総額の返還を確実に保証できる資力」を有することが証される必要があります。. 「貸与予定総額の返還を確実に保証できる資力」とは、次のいずれかに該当する人です。. 所得証明書等：. （給与所得者）年間収入320万円以上. （給与所得者以外）年間所得220万円以上. 預貯金残高証明書、固定資産評価証明書等：（預金残高＋評価額）が貸与予定総額以上. ※ 1、2複合の場合は、2の合計額を「16」（平均返還予定年数）で除した額を1の金額へ加算し、判定. 「適格認定」について. 適格認定（学業成績等）. 年1回（例年12月中頃）奨学生が「奨学金継続願」を提出した上で実施され、継続が認められると翌年度（4月から）も貸与が受けられます。手続きを怠った場合や、奨学生として適格でないと判断された場合は「停止」「廃止」等の処置がなされることがあります。また、奨学生として不適切であると認定される事由が生じた場合は、その都度、「適格認定」が行われます。. 適格認定（家計）【給付奨学生のみ】. 年1回（例年9月頃）、奨学生本人及び生計維持者（父母等）の経済状況に応じた支援区分の見直しが行われ、10月以降の1年間の支援区分が決まります。. 在籍報告【給付奨学生のみ】. 給付奨学生は、適格認定に加えて、4月と7月と10月の年3回「在籍報告」を提出する必要があります。（採用年度の4月は在籍報告の必要なし）. 手続きを怠った場合や、奨学生として適格でないと判断された場合は「停止」「廃止」等の処置がなされることがあります。. 異動・月額変更手続きについて. 異動手続きに該当する事由が発生した場合、速やかに手続を行ってください。所定期日までに提出がない場合は、延滞金・超過振込金が発生したり、貸与開始の遅延等、不利益を被る場合がありますので注意してください。. 2022年度異動等手続きスケジュール 異動手続きについて. 次に該当する場合は、該当様式を作成の上、所定の期日までに教育推進・学生支援部学生課奨学掛に提出してください（. 書類提出時の注意 参照）。. 異動（学籍異動）に関すること. 異動始期：原則、日付が月の初日の場合はその月、それ以外の場合は翌月. 異動の日付は各自で記入すること（休学日、退学日、復活日、卒業期、留学期間）。なお、学籍内容と異なる場合は修正が入ることがあります。. 給付奨学金と貸与奨学金を併用している場合は、両方の手続きが必要です。. 異動内容. 貸与型. 奨学金. 給付型. 奨学金（新制度）. 休学する場合. ※ 休学の事由が留学の方は「留学する場合」を参照. 異動届（願） 様式1-1（休止、退学、辞退等）. 様式1-2（復活）. 【1－(3)】休止の異動願 退学する場合. 【1－(1)】給付終了の異動願及び認定報告 奨学金を辞退する場合. （給付奨学金では「辞退」の申請はできません。下段の「停止の手続きについて」を参考ください）. 短縮修了する場合. 【1－(1)】給付終了の異動願及び認定報告 復学または留学を終了する場合. 【1－(5)】休止からの復活の異動願 留学する場合. 【3ヶ月未満の留学について】. 学籍上の身分（休学・留学・在学）を問わず、奨学金の継続貸与が認められます。. 【3ヶ月以上の留学について】. 留学中の学籍上の身分が「留学」または「在学」の場合は、奨学金の継続貸与が認められます。. 3ヶ月以上の留学かつ休学する場合は、 《貸与継続を希望する》. →. 留学奨学金継続願（様式8-1）   を提出。原則、承認希望月の前月3日までに提出すること. 《貸与継続を希望しない》. →. 異動届（願） を提出して休止（留学）すること. 学籍上の身分が「留学」または「在学」であり、かつ下記の2～4のいずれにも当てはまらない場合. →届け出の提出は不要です。引き続き支給されます。. 学籍上の身分が「休学」の場合. →. 【1－(3)】休止の異動願  を提出. 留学中に海外留学支援制度（協定派遣）を利用する場合. →学籍上の身分が「休学」の場合は2.の手続きと同じです。. →休学でない場合は. 【1－(2)】停止の異動願 を提出. （※ 停止の期間中は、奨学金の支給はありませんが、給付を受けた期間として通算されます。）. 留学中に官民協働海外留学支援制度（トビタテ！留学JAPAN日本代表プログラム）の支給を受ける場合. →学籍上の身分が「休学」の場合は2.の手続きと同じです。. →休学でない場合は1.の手続きと同じです。. 停止の手続きについて. （貸与奨学金では「停止」の申請はできません）. 自己の都合や、併給禁止の他奨学金を受ける等により「停止」が必要な場合は. 【1－(2)】停止の異動願  を提出。. 停止から復活する場合は、. 【1－(4)】停止からの復活の異動願 を提出。. 停止の期間中は、奨学金の支給はありませんが、給付を受けた期間として通算されます。. 旧制度の給付奨学生は、下記問い合わせ先まで確認ください。. 所属・登録内容の変更に関すること. 異動内容. 貸与型. 奨学金. 給付型. 奨学金（新制度）. 転学部（研究科）した場合. 転学部（科）届【貸与用】 転学部（科）届【給付用】 改氏名した場合. 改氏名届 同左. 口座変更をしたい場合. 奨学金振込口座変更届 同左. 連帯保証人・保証人を変更する場合. 連帯保証人・保証人変更届 （該当者のみ. 返還保証書 も必要）. ―. 住所が変わる場合. 本人. ※. の. 住民票. 住所、連帯保証人等の住所を変更した場合. →. 住所変更届 ※本人分は、2019年度以降採用者でマイナンバー提出済の場合は提出不要です。. ただし、通学形態に変更がある場合は、あわせて下段の手続きが必要です。. 本人の. 現住所. 、生計維持者の現住所が変更した場合. 4・7・10月の在籍報告の際にWeb上で変更してください。. ただし、通学形態に変更がある場合は、あわせて下段の手続きが必要です。. 通学形態が変わる場合. （自宅通学⇔自宅外通学の変更）. 大学院生は、必要な手続きはありません。. 学部生は、下記を確認してください。. 【第一種奨学金（学部生）】. 《給付奨学金を併用している者》. 給付奨学金と同じ様式で手続きできます。. 《給付奨学を併用していない者》. 現在（変更前）の貸与額が、通学形態変更後も選択できる額であるか確認ください。同じ額を貸与できない場合は新たに貸与額を選び、下記の「月額変更手続きについて」に記載の書類を提出してください。. 【第二種奨学金（学部生）】.  必要な手続きはありません。. 事由発生後すみやかに手続きが必要です。在籍報告時ではありませんのでご注意ください。. 自宅外通学⇒自宅通学への変更. →. 通学形態変更届（自宅外→自宅） を提出すること。. 自宅⇒自宅外通学への変更. →. 通学形態変更届（兼自宅外証明書送付状） 、証明書類（賃貸借契約書等）を提出すること。. 第二種奨学金の「利率の算定方法」を変更したい場合. 第二種奨学金「利率の算定方法」変更届 ―. 第一種奨学金の変換方式を変更したい場合. 選択している保証制度が機関保証の者. 人的保証制度選択者については本様式と別の様式での申請となるため下記問い合わせ先へご連絡ください。. 第一種奨学金返還方式変更届 ―. 緊急採用（第一種奨学金）の貸与終了予定者で4月以降、貸与継続を希望する場合. 緊急採用（第一種）奨学金継続願 奨学金継続に係る申告書（学部用） 奨学金継続に係る申告書（大学院用） ―. 旧制度の給付奨学生は、下記問い合わせ先まで確認ください。. 上記以外の手続については、教育推進・学生支援部学生課奨学掛に問い合わせてください。. 【貸与奨学金のみ】月額変更手続きについて. 月額変更にかかる注意事項 を必ず読み、対応する様式を作成の上、所定の期日までに教育推進・学生支援部学生課奨学掛に提出してください（提出は表面のみで可）。. 書類提出時の注意 参照）. 種別. 増額. 減額. 添付書類について. 第一種. 第一種奨学金貸与月額変更願（届） 「. 月額変更にかかる注意事項 」裏面の「変更後の借用金額（総額）の計算方法」もしくは「年度内精算が可能かどうかの計算方法」に記入の上、「月額変更願」とともに提出してください。. 第二種. 第二種奨学金貸与月額変更願（届） 給付奨学生については、通学形態の変更に伴って支給額が変わります。詳細は、上記の「住所が変わる場合」を参照ください。.  貸与終了（後）にかかる手続きについて. 貸与終了手続きについて. 貸与終了（満期、辞退、退学等）に伴い、手続きが必要です。返還確認票、「返還のてびき」等の交付を受け、金融機関にて振替口座（リレー口座）加入の手続きを行ったうえで、加入申込書（預貯金者控）のコピーを教育推進・学生支援部学生課奨学掛窓口に提出してください。. 貸与終了後も引き続き在学する場合の手続き（在学猶予）について. 貸与終了後も引き続き在学する場合、願出により返還期限が猶予されます。希望者は「在学猶予願」をスカラネット・パーソナルにて提出してください（詳細は、. さまざまな奨学金 ページのお知らせ欄を参照してください）。. 留年や休学等により、最短修業年限を超えて在学する場合は、あらためて1年ごとに「在学猶予願」を提出する必要があります。. 短縮修了や退学等により在学期間が短くなった場合は、「在学期間短縮願」をスカラネット・パーソナルにて提出してください。. その他の貸与終了後の諸手続きについて. 上記以外の貸与終了後の諸手続きについては、直接、日本学生支援機構とやりとりしてください。様式は、貸与終了手続時に交付された「返還のてびき」または、. 日本学生支援機構 のWebサイトより取得してください。. （大学院生）第一種奨学金の特に優れた業績による返還免除について. 大学院において第一種奨学金の貸与を受けた学生で、貸与期間中に特に優れた業績を挙げたとして認定された場合は、貸与期間終了時に奨学金の全部または一部の返還が免除されます。対象は当該年度に貸与を終了した学生ですが、貸与終了手続を行わないと願出ができませんので注意してください。提出期間、申込方法などは所属研究科によって異なります。. また、当該年度の早い時期に貸与が終了し、本免除の認定結果が出る前に返還期日が到来する場合、「奨学金返還期限猶予願」を提出することにより、返還期限の猶予が受けられます。希望者は教育推進・学生支援部学生課奨学掛に提出してください（ただし引き続き在学する場合は「在学猶予願」の提出により猶予されるため、提出の必要はありません）。. 「奨学金返還期限猶予願」 問い合わせ先. 教育推進・学生支援部学生課奨学掛. 〒606-8501 京都市左京区吉田本町. （吉田キャンパス 本部構内 総合研究10号館1階）. Tel： 075-753-2535. Fax： 075-753-2563. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート. 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 ページ別. 各事業年度の年次活動報告（PDF）（2.71MB） 尾池総長の言葉～式辞より～（PDF）（9.38MB） 総長としての5年間を終えて～学生との対談～（PDF）（7.14MB） 各界からのメッセージ：尾池総長を支えてきた方々[前半]（PDF）（7.82MB） 各界からのメッセージ：尾池総長を支えてきた方々[後半]（PDF）（6.80MB） 各界からのメッセージ：協定締結大学の総長･学長（PDF）（6.61MB） 学外での活動（PDF）（375KB） 主な特徴. 平成20年9月30日をもって退任しました尾池和夫前総長の「アクションレポート」として作成したものです。. 総長在任時（平成15年12月16日～20年9月30日）の主な出来事や、式辞、学生との対談、各界から寄せられたメッセージなどにより構成されています。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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On-site Laboratory. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 キャンパスの概要 国内の研究所・附属研究施設等 海外の拠点 On-site Laboratory 大学の施設 福利厚生施設・課外活動施設 写真・動画で見る京都大学 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 On-site Laboratoryの取り組みについては以下のページをご覧ください。. On-site Laboratory 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 総長からのメッセージ 湊 長博総長からのメッセージ. 湊 長博総長からのメッセージ. サイドメニュー. 総長からのメッセージ 湊 長博総長からのメッセージ 任期中の基本方針 ―世界に輝く研究大学を目指して― 湊長博 総長 略歴 総長の動き 式辞・挨拶等 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 京都大学は1897年、科学・技術の社会における役割の急速な拡大に伴い、教育だけでなく研究も大学が担うべきであるという機運の中で、我が国で２番目の帝国大学として創立されました。爾来、京都大学は、自由で創造的研究を尊び、新たな知的価値の創出とそれを担う人材の養成によって「地球社会の調和ある共存に貢献すること」を基本理念として、長い道のりを歩んできました。この基本理念の下で本学は、アジア諸国の中で最多のノーベル賞やフィールズ賞の受賞者を輩出するなど、我が国を代表する研究型大学として大きな成果をあげてきました。2017年には指定国立大学法人の指定を受け、新たな道のりを歩き始めています。. 本学は、約5,500名の教職員に加え、23,000名近くの学生（学部、大学院）を擁しており、教育においても大きな責任を担っています。本学の教育の伝統は、創造的な研究活動の中での実践的教育であり、自らに課題を課し自学自習によりその解決に向け努力することを促すことによって、学生の創造的精神の涵養をはかることにあります。その前提となるのは、確実な科学的知識と幅広い豊かな人間的素養であり、国際高等教育院では、入学した全ての学生にその将来の専門領域を問わず全学共通教育を課し、健全な知的市民としての成長を求めています。創造性の根源は知的好奇心と探求心であり、「研究を通して教育する」という伝統的なモットーの下、高度の専門能力を有した人材や次世代を担う創造的研究者が育っていくことを、強く期待しています。. 今日私達は、予想を超えるテンポで進行する地球の気候変動と大規模な自然災害や地球環境悪化、様々な国際的対立抗争の激化や格差の拡大、さらには昨今の新型コロナウイルスに代表される感染症の拡大など、地球上の人々の生命と健康を脅かす多くの困難な課題に直面しています。今私達は、改めて本学の基本理念に深く思いを致し、これらの地球社会における多元的で困難な諸課題の解決に向けて真摯かつ果敢に挑戦し、着実にその成果を社会に発信していく必要があると思います。. 京都大学は、来る2022年に創立125周年を迎えます。これまで本学は、独創性と多様性を尊ぶ開拓精神の学風の下、多くの豊穣な果実を結んできましたが、未来に向けて美しい実を結び続けていくためには、同時に新しい苗木を植え正しく育てていく必要があります。120年余にわたる京都大学の力強い歩みを確実に未来に繋ぎ、新たな知的価値の創出と豊かな人材の養成によって、社会に貢献する大学であり続けるために、全力を尽くしていく所存です。.  . 2020年10月. 京都大学総長. 湊 長博. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション（繁体中国語）. 京都大學概況. 校長的話. 大學介紹. 校內安全措施. 教育與學生支援服務. 學生生活. 留學・國際教育. 研究・產官學合作. 支援研究體制. Security Export Control. 國際交流. 學部與院係. ヘッダー セカンダリーリンク（日本語以外）. 交通資訊 聯絡我們 捐贈 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 導航連結. Home 網站政策. 網站政策. In other languages. 日本語 京都大學網站  營運準則. 目的. 為順利運用京都大學網站（以下稱「網站」），特制定本準則。. 定義. 本準則中的網站指的是網域名稱為「kyoto-u.ac.jp」的以下網站、內容及頁面。. 京都大學官網（. http:\/\/www.kyoto-u.ac.jp\/ ）（日文\/英文\/中文繁體\/中文簡體\/韓文頁面）與其下的各種內容、頁面及連結到的由京都大學（包括學系院所及其他設置於京都大學的組織）管理營運的專門網站。. 由設置於京都大學的學系院所等所營運的網站（以下同「學系院所等網站」）。. 管理主管. 京都大學官網由京都大學公共關係委員會網站企劃專門部會（以下稱「專門部會」）進行管理，而學系院所等網站由相關學系院所等進行管理。但是，從全校的角度看認為有必要時，專門部會可針對學系院所等網站發表意見。. 營運主管. 京都大學官網由總務部公共關係課進行營運，而學系院所等網站，由各學系院所設置1名營運主管進行營運，並負責與專門部會聯繫協調。. 此處所謂由總務部公共關係課進行營運，指的是要詳細審查營運主管自行刊載的資訊，或者受後述規定之刊載委託者委託刊載的資訊內容，再發布於京都大學官網。. 刊載委託者. 網站刊載委託者（以下稱「委託者」）了解網站有向不特定多數人發布速報資訊，以及累積資訊等特性，理解網站是正確傳播本校應公告事項或迅速有效傳達本校魅力的手段，事前應充分詳細審查應刊載資訊的內容，而資訊刊載後，要致力於維護相關內容隨時在最新且正確的狀態。. 妨害公共秩序或善良風俗的資訊. 網站所載資訊應是與京都大學所推動的教育、研究、社會貢獻有關，不得刊載妨害公共秩序或善良風俗的內容，以及以商業行為、政治、宗教活動等為目的的內容。. 改善建議. 當專門部會判斷網站所載報導內容違反前項規定時，專門部會將對該報導的刊載者及該網站的營運主管提出改善建議，若有連結到違反公共秩序或善良風俗之網站的行為，也將採取相同措施。. 其他規定. 其他與網站營運有關之必要事項由專門部會規定。. （1997年10月27日網站專門委員會制定）. （2015年2月24日公共關係委員會網站企劃專門部會修訂）. （2007年6月15日京都大學公共關係委員會網站企劃專門部會修訂）. （2015年4月1日公共關係委員會網站企劃專門部會修訂）. （2017年4月1日公共關係委員會網站企劃專門部會修訂）. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通資訊 フッター セカンダリーリンク. 站點地圖 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"-Gew19U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/about\/publication\/disclosure\/accounting","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":68,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["ja","about","publication","disclosure"],"url_file_name_s":"accounting","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["決算 | 京都大学"],"description_txt":["毎事業年度の財務諸表等を公表しています。"],"last_modified":"Wed Jun 08 15:46:10 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-3786842113945124305,"exact_signature_unique_b":true,"fuzzy_signature_l":507716014225306351,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["決算"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","2020年度 決算／FY2020 Financial Statements","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["バックナンバー／Archives"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":678,"text_t":"決算 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 公表事項 国立大学法人法等による公表事項 決算. 決算. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 国立大学法人ガバナンス・コード 自己点検・評価（教員評価を含む） 認証評価 業務方法書 会議報告 国立大学法人法等による公表事項 理事 業務方法書 役員報酬・退職金の支給基準 職員給与・退職金の支給基準 国立大学法人等の役員に就いている退職公務員等の状況等の公表について 役員の報酬等及び職員給与の水準の公表について 決算 2020年度 決算／FY2020 Financial Statements 会計検査院決算検査報告 教育情報の公表 贈与報告書 資金運用について 労働施策総合推進法に基づく中途採用比率の公表 情報公開 2020年度 決算／FY2020 Financial Statements. 2020年度 決算／FY2020 Financial Statements アニュアルレポートはこちら バックナンバー／Archives. 2019年度 決算／FY2019 Financial Statements 2018年度 決算／FY2018 Financial Statements 2017年度 決算／FY2017 Financial Statements 2016年度 決算 2015年度 決算 2014年度 決算 2013年度 決算 2012年度 決算 2011年度 決算 2010年度 決算 2009年度 決算 2008年度 決算 2007年度 決算 2006年度 決算 2005年度 決算 2004年度 決算 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求 入手方法. 入手方法. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 広報誌の入手には以下の方法があります。. 学外の方・学生の皆さんの場合. 1．ダウンロード（PDF）. 各ページよりPDFファイルをダウンロードください。. 2．直接、取りに行く. 広報センター（正門横）、百周年時計台記念館1階、本部棟1階にあります。. 3．郵送. メール（または、Fax）でご連絡いただきましたら、送付します。. 総務部広報課で送付希望を受け付けているものは、京都大学概要、紅萠です。. そのほかの刊行物については、それぞれの刊行物の発行元にお問い合わせください。. 送料は着払いでご負担いただいております。. 必要事項. 郵便番号・住所・電話番号・氏名・ご希望の広報誌（広報誌名・号数）・部数・着払いでの送付了承（の旨明記）. メール送付先（Fax送付先）. 総務部広報課. E-mail： kohho52＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. Fax： 075-753-2094. 教職員が業務や来客用に必要な場合. 所属・ご希望の広報誌とその部数・必要理由をご記入の上、前日までにメールください。. 総務部広報課. E-mail： kohho52＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. 教職員の方へのお願い. 大量に必要な場合はなるべく早くお申し出ください。. 今後の発行計画・必要性を検討する資料とするため、用途を記録しています。. 業務で必要な場合は広報センター等から無断で持ち出さないでください。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"aeNUy9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/event?audience=45&target%5B45%5D=45","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":64,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["ja"],"url_file_name_s":"event","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["EVENTS | 京都大学"],"last_modified":"Wed Jun 08 16:01:56 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":6192878362717452990,"exact_signature_unique_b":true,"fuzzy_signature_l":-2358167653387146879,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["EVENTS"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","令和5（2023）年度入学者京都大学大学院総合生存学館（思修館）博士課程学生募集入試説明会","オンライン個別進学相談（2022年6月開催）","2023年度 教育学研究科・教育学部学士入学 入試説明会","高校生のための進学ガイダンス2022（岐阜）","主要大学説明会2022（札幌、横浜、名古屋、大阪、広島、福岡）","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":12,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/styles\/width_scale_80\/public\/2022-05\/20220526-0616.pdf_-7c2410a5376b38e8d64b0f7d56bd42e3.png?itok=SpzaQAry","www.kyoto-u.ac.jp\/sites\/default\/files\/styles\/width_scale_80\/public\/default_images\/default-featured-image.png?itok=6EkSO4G-","www.kyoto-u.ac.jp\/sites\/default\/files\/styles\/width_scale_80\/public\/2022-05\/r4-kyoiku-setsumeikai-78dc7fb39d4ba20289c8b5b4d845aae8.jpg?itok=AbB5e3OG","www.kyoto-u.ac.jp\/sites\/default\/files\/styles\/width_scale_80\/public\/2022-06\/220626-gifu2022-f4c73380fefa1f178d4b23c0919e91b2.jpg?itok=f6S6_Zav","www.kyoto-u.ac.jp\/sites\/default\/files\/styles\/width_scale_80\/public\/2022-05\/2022-shuyodaigaku-setumeikai-977c8a6a4781cbe0b70376b15e31826f.jpg?itok=2MRyB8vH","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","","","","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,462,167,463,453,453,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,320,320,320,320,320,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal sites default files styles width scale 80 public 2022 05 20220526 0616 pdf 7c2410a5376b38e8d64b0f7d56bd42e3 png itok SpzaQAry 7c 2410a 5376b 38e 8d 64b 0f 7d 56bd 42e 3 Spza QAry featured image 6EkSO4G 6 Ek SO 4 G r4 kyoiku setsumeikai 78dc7fb39d4ba20289c8b5b4d845aae8 jpg AbB5e3OG r 78dc 7fb 39d 4ba 20289c 8b 5b 4d 845aae 8 Ab B 5e OG 06 220626 gifu2022 f4c73380fefa1f178d4b23c0919e91b2 f6S6 Zav gifu f 4c 73380fefa 1f 178d 4b 23c 0919e 91b 2 S shuyodaigaku setumeikai 977c8a6a4781cbe0b70376b15e31826f 2MRyB8vH 977c 8a 6a 4781cbe 0b 70376b 15e 31826f MRy 8v H icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1356,"text_t":"EVENTS | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム EVENTS. EVENTS. 条件で絞り込む. 日付で絞り込む. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. 要申し込み. - すべて -. 必要. 不要. 開催地. - すべて -. 吉田キャンパス. 宇治キャンパス. 桂キャンパス. 東京オフィス・京都アカデミアフォーラム. その他の地域. オンライン. 開催日. 前月. 1月. 2月. 3月. 4月. 5月. 6月. 7月. 8月. 9月. 10月. 11月. 12月. 翌月. 日. 月. 火. 水. 木. 金. 土. 絞り込み条件をリセット. Events. タグ. 入試. 令和5（2023）年度入学者京都大学大学院総合生存学館（思修館）博士課程学生募集入試説明会. 開催日. 2022年06月10日、他. 開催地. 吉田キャンパス. 桂キャンパス. その他の地域. Events. タグ. 入試. オンライン個別進学相談（2022年6月開催）. 開催日. 2022年06月17日、他. 開催地. オンライン. Events. タグ. 入試. 2023年度 教育学研究科・教育学部学士入学 入試説明会. 開催日. 2022年06月25日. 開催地. オンライン. Events. タグ. 入試. 高校生のための進学ガイダンス2022（岐阜）. 開催日. 2022年06月26日. 開催地. その他の地域. Events. タグ. 入試. 主要大学説明会2022（札幌、横浜、名古屋、大阪、広島、福岡）. 開催日. 2022年07月30日、他. 開催地. その他の地域. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 海外留学を希望する京大生へ 京大の留学プログラム 交換留学. 交換留学. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 留学の目的・種類 京大の留学プログラム 交換留学 交換留学の流れ 大学間学生交流協定一覧 学内応募について 最新応募情報 ケーススタディ 交換留学Q&A 交換留学体験談 短期留学プログラム 留学に関する予備知識 海外へ留学する京大生向け奨学金 海外留学のリスク管理 京都で学べる英語の授業 海外留学のための相談窓口 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 交換留学（派遣）とは. 交換留学（派遣）とは、京都大学が海外の大学と締結している大学間・部局間の学生交流協定に基づく、1学期以上～1年以内の留学です。. 留学先での在学期間も京都大学における修業年限に通算されるので、必要単位を取得すれば、留年せずに所定修業年限内で卒業することも可能です。. 留学先. 大学間学生交流協定校 、所属学部・研究科の部局間学生交流協定校. 期間. 1学期～1年間. 時期. 留学先大学の学期による. 対象者. 学部生、大学院生（受入不可の大学もあり）. 授業料. 派遣先での授業料は免除され、京都大学にのみ納める. 単位互換・認定制度. あり. ※. ※. 留学前に所属学部・研究科の教務担当掛や指導教員と十分に相談してください。. この交換留学は、本学と交流協定校との契約に基づいて、過去の学生たちが築き上げてきた信頼関係の上に成り立っています。それを後輩たちに伝えていくためにも、派遣学生は京都大学の代表であるという自覚を持って留学期間を過ごしてください。. 交換留学の流れ 学内応募について 最新応募情報 ケーススタディ 交換留学Q&A 交換留学体験談 参考. 留学に関する予備知識 冊子「海外留学の手引き」 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム Kyoto U Research News. Kyoto U Research News. In other languages. 日本語 Web edition. Kyoto U Research News interactive web edition. 2021 Winter. vol. 9（2021 Winter ：5.29MB） Contents. 03. Backstage at the lab. Crossroads of the world and the mind. 08. KyotoU moving forward. 10. In Covid’s wake. 11. Cutting edge. A special elemental magic. How do birds understand ‘foreign’ calls? 12. From bacteria to you: the biological reactions that sustain our rhythms. 13. Two for the price of one. Flaring, massively. 14. Two for the price of one. Order from chaos. 15. Changing ties that naturally bind. 16. Reducing the high social cost of death. Detect with PKAch. 17. Terahertz zaps alter gene activity in stem cells. 18. How does the spider spin its self-assembled silk? 19. KyotoU today. Science with industry: The forest and the trees: science supporting ASEAN. 20. New strategic partnerships established. 21. News from overseas centers. 22. Student voices. 23. Eternal aesthetic. 2020 Spring. Vol. 8（2020 Spring : 4.90 MB) Contents. 03. Backstage at the lab. Into the human unknown. 10. Cutting edge. Bridging the scales of the brain. An overactive cerebellum causes issues across the brain. 11. Why fruit flies eat practically anything. 12. Lend me a flipper. Using a chip to find better cancer fighting drugs. 13. The little auks that lived in the Pacific. 14. Reconstructing the clock of human development. 15. KyotoU today. Science with industry: KyotoU venture bringing light to patients’ eyes. 16. News from overseas centers. 18. Student voices. 19. Eternal aesthetic. 2019 Autumn. Vol. 7（2019 Autumn : 4.34 MB) Contents. 03. Backstage at the lab. An Asian Humanities, for the future. 10. Cutting edge. A petrifying virus key to evolution. Let there be light-activated genes. 11. Infertility’s roots in DNA packaging. 12. Prince Charming’s kiss unlocking brain’s regenerative potential? Chimps caught crabbing. 13. A study in scarlet Japanese macaques. 14. Testing corneal cell quality? Apply physics. The worms that roared. 15. KyotoU today. So tell us... 16. News from overseas centers. 17. On-Site Laboratory network established. 18. Student voices. 19. Eternal aesthetic. 2019 Spring. Vol. 6（2019 Spring : 5.38 MB) Contents. 03. Nobel Prize 2018. Congratulations, Honjo-sensei! 04. Defining ‘Kyotoness’ in the research environment. 08. Driven by curiosity. 10. Cutting edge. New gears in your sleep clock. What does the koala genome tell us about the taste of eucalyptus? 11. Meet ERICA. She wants to listen to you. 12. Giant, recently extinct seabird also inhabited Japan. Japanese in Canada, with mixed feelings. 13. Finding success through failure. Using microcredit to increase rice yield in Bangladesh. 14. Making an eye for you. 15. KyotoU today. Science with industry: Microbes are us. But how can we see them? 16. News from overseas centers. 18. Student voices. 19. Eternal aesthetic. 2018 Autumn. Vol. 5（2018 Autumn : 4.76 MB) Contents. 03. Backstage at the lab. The high-tech astronomy of Seimei. 08. Personnel profiles. Understanding the machines, Kyoto style. 12. Cutting edge. High achievement not always based on high student initiative. How does HIV escape cellular booby traps? 13. Visualizing danger from songbird warning calls. 14. What a handsome schnoz! Electro-mechano-optical NMR detection. 15. Twisting graphene into spirals. 16. Studying the human impact on sea urchin abundance. Capturing the rare Yanbaru whiskered bat. 17. Using geometry and qubits to build real universes. 18. KyotoU today. KURNe: research news in motion. 19. News from overseas centers. 20. Science with industry: Mining answers from the data mountain. 21. So tell us... what do you do and how did you get to Japan? 22. Student voices. 23. Eternal aesthetic. 2018 Spring. Vol. 4（2018 Spring : 2.18 MB) Contents. 03. Backstage at the lab. Humanities 101: A Kyoto approach to history, society, and the study of humanity. 12. Cutting edge. Special delivery : macromolecules via spider’s ‘bite’. Cells pumping iron to prevent anemia. 13. Black hole pair born inside a dying star? How am I feeling? Ask my house. 14. Lightning, with a chance of antimatter. 15. Winds blowing off a dying star. The secret lives of ancient land plants. 16. What grosses out a chimpanzee? 17. Cells rockin’ in their DNA. 18. News from overseas centers. 20. Student voices. 22. Global outreach. 23. Eternal aesthetic. 2017 Autumn. Vol. 3（2017 Autumn : 2.93 MB) Contents. 03. Backstage at the lab. Earth, sky, space, and everything in between. 08. Personnel Profiles. ARN:student perspectives. 12. Cutting edge. Energizing immune cells to combat cancer. Visualizing nuclear radiation. Choosing a simpler path to drug discovery. Charting the skies of history. Born to love superheroes. High stakes, high risk, and a bad bet. You don’t see what I see? Disentangling chloroplast genetics. Untangling the knots in cell stress. Take a look, and you’ll see, into your imagination. Nicking in new nucleotides. 18. International activities. 19. News from overseas centers. 20. Student voices. 22. Global outreach. 23. Eternal aesthetic. 2017 Spring. Vol. 2（2017 Spring : 2.43 MB) Contents. 03. Backstage at the lab. Kyoto into Africa. 08. Personnel Profiles. The genetics of wildlife breeding and conservation. 12. Cutting edge. Smiling baby monkeys and the roots of laughter. Homosexual termite regicide. Is it your second cousin? Cotton swabs may tell you. Drinking green tea to prevent artery explosion. Great apes can “read minds”. Mt Aso could erupt much sooner, scientists warn. Aging bonobos in the wild could use reading glasses too. A big nano boost for solar cells. Proteins at the movies. The sea roils and life returns. Safety in darkness. 18. International activities. 20. Student voices. 22. Global outreach. 23. Eternal aesthetic. 2016 Autumn. Vol. 1（2016 Autumn : 2.57 MB) Contents. 03 from the President. 04. On the cover: Backstage at the lab. Reading beyond time and across disciplines. 08. Personnel Profiles. Working in CiRA labs. 11 News from overseas centers. 12 Cutting edge. The search for happiness: using MRI to find where happiness happens. Fish oil helps burn fat by transforming fat-storage cells into fat-burning cells. \"Seeing\" black holes with home-use telescopes. To hear a pit ter patter from afar: catching hear tbeats with milimeter-wave radar. Genetically modified E.coli pump out morphine precursor. \"Popular girls\" have less lice . in the monkey world. Australopithecus fossils found east of the Great Rif t Valley. 'Slow' NZ seabed quake sheds light on tsunami-earthquake mechanism. Cancer cell immunity in the crosshairs: wor th the expense? Flipping a protein switch to illuminate brain functions. \"Big mama\" bonobos help younger females stand up for themselves. Did the LIGO gravitational waves originate from primordial black holes? 18 International activities. 20 Student voices. 22 Global outreach. 23 Eternal aesthetic. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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5月. 工業教員養成所設置. 37年4月. 経済研究所附置. 38年4月. 教養部設置. 4月. 数理解析研究所附置. 4月. 原子炉実験所附置. 40年4月. 東南アジア研究センター設置. 41年4月. 保健管理センター設置. 42年6月. 霊長類研究所附置. 6月. 結核研究所を結核胸部疾患研究所と改称. 44年4月. 大型計算機センター設置. 45年3月. 工業教員養成所廃止. 46年4月. 放射性同位元素総合センター設置. 4月. 工学研究所を原子エネルギー研究所と改称. 47年5月. 体育指導センター設置. 50年4月. 医療技術短期大学部設置. 51年5月. ヘリオトロン核融合研究センター設置. 5月. 放射線生物研究センター設置. 52年4月. 環境保全センター設置. 7月. 埋蔵文化財研究センター設置. 53年4月. 情報処理教育センター設置. 55年4月. 医用高分子研究センター設置. 56年4月. 超高層電波研究センター設置. 61年4月. アフリカ地域研究センター設置. 63年4月. 遺伝子実験施設設置. 4月. 結核胸部疾患研究所を胸部疾患研究所と改称. 12月. 国際交流センター設置. 平成. 2年3月. 医用高分子研究センター廃止（10年時限）. 6月. 生体医療工学研究センター設置. 6月. 留学生センター設置（国際交流センター廃止）. 3年4月. 大学院人間・環境学研究科設置. 4月. 生態学研究センター設置. 4月. 木材研究所を木質科学研究所と改称. 4年10月. 総合人間学部設置. 5年3月. 教養部廃止. 6年6月. 高等教育教授システム開発センター設置. 8年3月. アフリカ地域研究センター廃止（10年時限）. 4月. 大学院エネルギー科学研究科設置. 4月. アフリカ地域研究資料センター設置. 4月. 学生懇話室設置. 5月. 原子エネルギー研究所とヘリオトロン核融合研究センターを統合しエネルギー理工学研究所に改組・転換. 9年4月. 総合博物館設置. 4月. 総合情報メディアセンター設置（情報処理教育センター廃止）. 10年4月. 大学院アジア・アフリカ地域研究研究科設置. 4月. 大学院情報学研究科設置. 4月. 胸部疾患研究所と生体医療工学研究センターを統合し再生医科学研究所に改組・転換. 11年4月. 大学院生命科学研究科設置. 6月. 学生懇話室をカウンセリングセンターに改組. 12年4月. 超高層電波研究センターを宙空電波科学研究センターに改組. 11月. 大学文書館設置. 13年4月. 食糧科学研究所廃止（大学院農学研究科と統合）. 4月. 国際融合創造センター設置. 14年3月. 大学情報収集・分析センター設置. 4月. 大学院地球環境学舎・学堂設置. 4月. 大型計算機センターと総合情報メディアセンターを統合し学術情報メディアセンターに改組・転換. 4月. 低温物質科学研究センター設置. 4月. 福井謙一記念研究センター設置. 15年4月. 高等教育研究開発推進機構設置. 4月. 高等教育研究開発推進センター設置. 4月. 高等教育教授システム開発センター廃止. 4月. フィールド科学教育研究センター設置. 4月. 農学研究科附属演習林廃止. 4月. 体育指導センター廃止. 10月. 医学部保健学科設置. 16年4月. 国立大学法人京都大学設立.  . 法学研究科基礎法学専攻、公法専攻、民刑事専攻、政治学専攻を法政理論専攻、国際公共政策専攻（修士課程）、法曹養成専攻（修士課程、法科大学院）に改組. 4月. 木質科学研究所と宙空電波科学研究センターを統合し生存圏研究所に改組・転換. 4月. 東南アジア研究センターを廃止･転換し、東南アジア研究所附置. 4月. 遺伝子実験施設廃止. 12月. 大学情報収集・分析センター廃止. 17年4月. 環境安全保健機構設置. 4月. 国際イノベーション機構設置. 4月. 国際交流推進機構設置. 4月. 情報環境機構設置. 4月. 図書館機構設置. 4月. 留学生センターを国際交流センターに改組. 18年4月. 大学院公共政策連携研究部・公共政策教育部設置. 4月. 大学院経営管理研究部・経営管理教育部設置. 4月. 地域研究統合情報センター設置. 4月. ナノメディシン融合教育ユニット設置. 4月. 生存基盤科学研究ユニット設置. 7月. 次世代開拓研究ユニット設置. 9月. 女性研究者支援センター設置. 19年4月. こころの未来研究センター設置. 4月. 先端医工学研究ユニット設置. 4月. 生命科学系キャリアパス形成ユニット設置. 4月. 医療技術短期大学部廃止. 7月. 国際イノベーション機構廃止. 7月. 産官学連携本部設置. 7月. 国際融合創造センターを産官学連携センターに改組. 10月. 物質－細胞統合システム拠点設置. 20年4月. 野生動物研究センター設置. 4月. 埋蔵文化財研究センター廃止. 4月. 文化財総合研究センター設置. 4月. 宇宙総合学研究ユニット設置. 10月. 先端技術グローバルリーダー養成ユニット設置. 21年9月. 次世代研究者育成センター設置. 22年3月. 学際融合教育研究推進センター設置. 4月. iPS細胞研究所設置. 4月. 産官学連携センター廃止（産官学連携本部に統合）. 4月. ナノメディシン融合教育ユニット廃止. 4月. 先端医工学研究ユニット廃止. 4月. 生命科学系キャリアパス形成ユニット廃止. 23年4月. 放射性同位元素総合センター廃止（環境安全保健機構に統合）. 4月. 環境保全センター廃止（環境安全保健機構に統合）. 4月. 国際交流センター廃止（国際交流推進機構に統合）. 4月. 保健管理センター廃止（環境安全保健機構に統合）. 4月. 生存基盤科学研究ユニット廃止. 4月. 次世代開拓研究ユニット廃止. 24年4月. 次世代研究者育成センターを白眉センターと改称. 25年4月. 大学院総合生存学館設置. 4月. 高等教育研究開発推進機構廃止. 4月. 国際高等教育院設置. 4月. 先端技術グローバルリーダー養成ユニット廃止. 4月. 宇宙総合学研究ユニット廃止. 8月. カウンセリングセンター廃止. 8月. 学生総合支援センター設置. 26年4月. 女性研究者支援センター廃止. 4月. 男女共同参画推進本部設置. 27年4月. 研究連携基盤設置. 28年4月. 低温物質科学研究センター廃止（環境安全保健機構に統合）. 4月. 国際交流推進機構廃止. 4月. 国際戦略本部設置. 4月. 高等研究院設置. 10月. 再生医科学研究所とウイルス研究所を統合しウイルス・再生医科学研究所に改組. 29年1月. 東南アジア研究所と地域研究統合情報センターを統合し東南アジア地域研究研究所に改組. 4月. 物質−細胞統合システム拠点を高等研究院物質−細胞統合システム拠点に改組. 30年4月. 放射線生物研究センター廃止（大学院生命科学研究科に統合）. 4月. 原子炉実験所を複合原子力科学研究所と改称. 4月. 大学院横断教育プログラム推進センター設置. 10月. 高等研究院ヒト生物学高等研究拠点設置. 31年4月. 文化財総合研究センター廃止（文学研究科に統合）. 令和. 1年7月. オープンイノベーション機構設置.  . 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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2022","referrer_id_s":"N7P-X9U8j7Mo","language_s":"ja","size_i":74627,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085635213983744,"text_t":"短期留学プログラム | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 海外留学を希望する京大生へ 京大の留学プログラム 短期留学プログラム. 短期留学プログラム. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 留学の目的・種類 京大の留学プログラム 交換留学 短期留学プログラム 京都大学体験型海外渡航支援制度 ～鼎会プログラム「おもろチャレンジ」～ 学生海外研究活動助成金 米国短期留学プログラム「Kingfisher Global Leadership Program with Kyoto University」（旧「Kyoto-DC Global Leadership Program」） 過去のプログラム 留学に関する予備知識 海外へ留学する京大生向け奨学金 海外留学のリスク管理 京都で学べる英語の授業 海外留学のための相談窓口 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 本学では海外留学促進の一環として、3ヵ月未満の各種短期留学プログラムを企画・実施しています。自分に合ったプログラムを見つけて、まずは短期で留学してみませんか？. 英語研修 異文化交流・実地研修（現地語学習） リーダー育成・社会奉仕等 プログラム説明会・イベント等の情報は、以下のページをご覧ください。. 留学フェア「留学のススメ」 お知らせ. 2022年度のプログラムについては、プログラムごとに渡航を伴う実施の可否判断を行い、一部のプログラムをオンラインで実施します。. 1.英語研修. 協定校や海外の大学等と教育連携し、現地で3～4週間の大学生活を体験しながら世界に通用する英語能力の向上およびコミュニケーション能力の養成を目指します。参加者の語学習得状況に合わせて複数のプログラムを用意しています。英語能力に不安のある場合でも参加しやすいプログラムも用意しています。. 2022年度実施予定のプログラム. プログラム／留学先. 実施予定期間. （変更の可能性有）. 対象者. 語学レベル. 募集時期. （オンライン）マギル大学 短期英語研修（カナダ）. ※渡航を伴うプログラムは中止。オンラインにて代替プログラムを実施。. 2022年8月8日～8月26日. （3週間）. B・M・D. 語学力は問いません。プレースメントテストによるクラス分けあり。. 募集終了. （オンライン）オークランド大学 短期英語研修（ニュージーランド）. ※渡航を伴うプログラムは中止。オンラインにて代替プログラムを実施。. 2022年8月15日～9月9日. （4週間）. ※日程を変更しました。. B・M・D. 語学力は問いません。プレースメントテストによるクラス分けあり。. 募集中（応募締切：6月30日17時）. プログラムの研修報告書はこちら（英語研修 報告書） 2.異文化交流・実地研修（現地語学習）. 本学学生が各国の大学生と、現地で交流を深め、国際性を養い、外国語運用能力を向上させる機会として、提供されています。語学学習、教員による特別講義、現地学生との共学、フィールドトリップ等、カリキュラムは先方大学との協力の下で開発され、充実した内容になっています。. 2022年度実施予定のプログラム（変更の可能性あり）. プログラム・内容. 実施予定期間. （変更の可能性有）. 対象者. 語学レベル. 募集時期. 多文化共学短期［派遣］留学プログラム（タイ・チュラーロンコーン大学サマースクール）. 2023年2月中旬～3月上旬（2週間）. B・M・D. 応募時の語学レベルは問わない。事前会話教室（必須）あり。. 未定. 多文化共学短期［派遣］留学プログラム（ベトナム国家大学ハノイ校サマースクール）. 2023年2月中旬～3月上旬（2週間）. B・M・D. 応募時の語学レベルは問わない。出発時にベトナム語日常会話レベル以上。. 未定. 多文化共学短期［派遣］留学プログラム（インドネシア大学スプリングスクール）. 2023年2月中旬～3月上旬（2週間）. B・M・D. 応募時の語学レベルは問わない。事前会話教室（必須）あり。. 未定. 多文化共学短期［派遣］留学プログラム（国立台湾大学スプリングスクール）.  . 2023年3月（3週間）. B・M・D. 英語：講義が理解できるレベル以上. 中国語：応募時の語学レベルは問わない。事前会話教室（必須）あり。. 未定. 多文化共学短期［派遣］留学プログラム（中国・浙江大学スプリングスクール）.  . 2023年3月上旬～中旬（2週間）. B・M・D. 応募時の語学レベルは問わない。事前会話教室（必須）あり。. 未定. 多文化共学短期［派遣］留学プログラム（韓国・延世大学校スプリングスクール）.  . 2023年3月（3週間）. B・M・D. 応募時の語学レベルは問わない。事前会話教室（必須）あり。. 未定. ウィーン大学 1857～1914年の学術成果と文化を知る研修プログラム（オーストリア）. 2023年3月（2週間）. B・M・D. 講義が理解できるレベル. 11月頃. 報告書等. 各種報告書（アジア研究教育ユニットWebサイト） 3.リーダー育成・社会奉仕等. 大学連携組織等が主催の留学プログラムや業者主催のインターンシッププログラム等も提供しています。海外の学生と英語での討論を行う機会や、現地の企業・機関等を訪れる機会を設けています。. 2022年度実施予定のプログラム（変更の可能性あり）. 留学先. プログラム・内容. 実施予定期間. （変更の可能性有）. 対象者. 語学レベル. 募集時期. ワシントンD.C.諸機関. （アメリカ合衆国）. Kingfisher Global Leadership Program 2022年9月3日～9月18日（2週間）. B・M・D. IELTS6.0以上、TOEFL iBT80点以上. 募集終了. VIA（Volunteer in Asia）スタンフォード大学（アメリカ合衆国）. スタンフォードVIA. Exploring Social Innovation(ESI). Exploring Health Care（EHC）. 1．2023年2月上旬～中旬（2週間）. 2．2023年3月中旬～下旬（2週間）. 1. B・M・D. 2. 医学部・薬学部. IELTS6.0以上、TOEFL iBT80点程度（Webによる英語面接あり）. 11月頃. 加盟大学等. （オンライン）USRNサマープログラム. Generation-C SERVE. 未定. B. 英語でのコミュニケーションが円滑に取れる. 未定. 加盟大学. （英国・日本）. RENKEI. （日英産学連携プログラム）. 不定期. B・M・D. （プログラムによる）. 英語でのコミュニケーションが円滑に取れる. 未定. ASEAN各国の大学（ASEAN）. ASEAN+3UNet. （アセアンプラススリーユーネット）. 教育フォーラム・スピーチコンテスト. 未定. B. 英語でのコミュニケーションが円滑に取れる. 未定. 加盟大学（ドイツ・日本）. HeKKSaGOn. （日独6大学コンソーシアム）. 不定期. B・M・D. （プログラムによる）. 英語でのコミュニケーションが円滑に取れる. 未定. ※ プログラムによって、大学から費用助成を受けられる可能性があります。. ※ 各プログラムの公募情報はKULASIS留学情報ページにてお知らせします。. 　 . KULASIS 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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Graduate School of Advanced Leadership Studies Inter-Graduate School Program for Sustainable Development and Survaivable Societies Training Program of Leaders for Integrated Medical System Collaborative Graduate Program in Design Leading Graduate Program in Primatology and Wildlife Science 研究所. 化学研究所 人文科学研究所 再生医科学研究所 能源理工学研究所 生存圈研究所 防灾研究所 基础物理学研究所 病毒研究所／微生物研究所 经济研究所 数理分析研究所 核反应堆实验所 灵长类研究所 东南亚研究所 iPS细胞研究所 Kyoto University Institute for Advanced Study (KUIAS). 物质细胞统合系统中心 Institute for the Advanced Study of Human Biology (ASHBi) 教育研究机构. 学术信息媒体中心 射线生物研究中心 生态学研究中心 区域研究综合情报中心 野生动物研究中心 高等教育研究开发推进中心 综合博物馆 低温物质科学研究中心 (Japanese) 野外调查科学教育研究中心 福井谦一纪念研究中心 心的未来研究中心 文物综合研究中心 学生综合支援中心 大学档案馆 非洲区域研究资料中心 白眉中心 跨学科的教育研究推进中心 (Japanese) 机构. 国际高等教育院 (Japanese) 环境安全保健机构 (Japanese) 国际交流推进机构 (Japanese) 信息环境机构 图书馆机构 产官学合作本部 Features. Kyoto University's 125th Anniversary. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. 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News. 公開日. 2022年03月11日. タグ. 受賞. 北島薫 農学研究科教授が「みどりの学術賞」を受賞しました. 研究成果. 公開日. 2022年03月10日. オープンソースで野外植物フェノタイピング用ローバーを開発 －狭い場所でもスイスイ計測－. 研究成果. 公開日. 2022年03月04日. 意図せず侵入する外来植物の侵入個体数と定着成功との関係を解明 －穀物輸入港と非穀物輸入港間の植物種組成の比較－. News. 公開日. 2022年03月03日. タグ. 教育. 国際交流. オンライン短期交流プログラム「変容する東アジアの食料・生命・環境と先端研究」を実施しました. ページ送り. カレントページ. 1 ページ. 2 ページ. 3 ページ. 4 ページ. 5 ページ. 6 ページ. 7 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home 教職員の方へ. 教職員の方へ. 教育 研究 国際交流 会議･委員会 採用・公募 イベント情報 総長選考関係（学内限定） その他通知・手続き 学内の問い合わせ先等 教職員が利用できる施設 京都大学教職員グループウェア 教育. 大学教育支援プログラム FD（ファカルティ・ディベロップメント）（FD専門委員会） 京都大学のFD 研究. 外部資金 競争的資金等の適正管理 研究倫理・生命倫理 研究成果発表（記者発表の手続き） 国際交流. 国際交流事務手続き（学内限定） 会議・委員会. 会議報告・委員会報告 採用・公募. 教職員公募一覧 イベント情報. EVENTS （イベントカレンダー）. 総長選考関係. 総長選考関係 その他通知・手続き. 監査. 内部監査 公益通報 総務. 諸規程 法務（学内限定）. 「コンプライアンスガイドブック（日本語）」および「Compliance Guidebook（英語）」について. 教職員ポータル内の以下の場所に掲載しています。. 京都大学教職員ポータル 日本語：ファイル管理＞文書共有＞総務部＞05法務室＞03コンプライアンスガイドブック＞コンプライアンスガイドブック. 英語：ファイル管理＞文書共有＞総務部＞05法務室＞03コンプライアンスガイドブック＞Compliance Guidebook. （2020年10月）担当部署名の変更、法令や学内規程等の改正に伴う部分的な改訂を行いました。（日本語／英語）. 広報. 刊行物等の請求方法 記者発表 広報用データ（学内限定）. ビジュアル・アイデンティティ マップ等 ホームページ・京大広報への情報掲載. 掲載に関する手続きは、「京都大学教職員グループウェア」の「ファイル管理」内に掲載しています。. 京都大学教職員ポータル ファイル管理＞文書共有＞総務部＞02広報課＞京大広報・ホームページ等への情報掲載手続き. 職員・人事・共済. 就業規則一覧 人事に関する情報（学内限定） その他 （人事制度・異動関係のお知らせ、人事異動、職員研修・自己啓発支援、兼業、障害者雇用等） ハラスメントの防止と対応 叙位・叙勲 育児支援制度について 財務. 京都大学における発注・検収について 環境安全衛生（学内限定）. 環境管理 安全衛生管理 情報環境. 情報セキュリティ 学内の問い合わせ先等. 公益通報 ハラスメント 学生総合支援センター 研究活動上の不正行為 競争的資金等適正管理 人事（学内限定） 教職員が利用できる施設. 学内施設 学内で使用できる実験室・会議室等 京都大学教職員グループウェア. 役職者内線番号一覧. 本部座席表. 資産の供用公募. などについては、. 京都大学教職員ポータル からご覧いただけます。. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home NEWS 举行了2022学年本科入学典礼。（2022年4月7日）. University News 举行了2022学年本科入学典礼。（2022年4月7日）. In other languages. 日本語 ターゲット. Current Students & Staff Visitors 发表日期. 2022\/04\/08. 　2022学年本科入学典礼于2022年4月7日（星期四）在京都市劝业馆“Miyako Messe”举行，松本纮原校长、山极寿一前校长、各理事和副校长、以及各部门领导出席了典礼。典礼上，凑长博校长进行了. 致辞 。本学年入学的本科生为2,938名。. 本科入学人数 凑 长博校长的致辞. 2022学年本科入学典礼 致辞（2022年4月7日） Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 发表日期. 2022\/04\/11. 举行了2022学年研究生院入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/08. 举行了2022学年本科入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/05. 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. 发表日期. 2022\/03\/29. 举行了2021学年毕业典礼。（2022年3月24日）. 最新新闻 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home NEWS 举行了2022学年研究生院入学典礼。（2022年4月7日）. University News 举行了2022学年研究生院入学典礼。（2022年4月7日）. In other languages. 日本語 ターゲット. Current Students & Staff Visitors 发表日期. 2022\/04\/11. 　2022学年研究生院入学典礼于2022年4月7日（星期四）在京都市劝业馆“Miyako Messe”举行，各理事和副校长、以及各部门领导出席了典礼。. 　本学年考入研究生院的学生为硕士课程2,291名、专业学位课程325名、博士课程895名。. 硕士课程与专业学位课程入学人数 博士课程入学人数 凑 长博校长的致辞. 2022学年研究生院入学典礼 致辞（2022年4月7日） Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 发表日期. 2022\/04\/11. 举行了2022学年研究生院入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/08. 举行了2022学年本科入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/05. 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. 发表日期. 2022\/03\/29. 举行了2021学年毕业典礼。（2022年3月24日）. 最新新闻 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 2022年度京都大学短期派遣プログラムKingfisher Global Leadership Programの参加者募集について. 学生支援 2022年度京都大学短期派遣プログラムKingfisher Global Leadership Programの参加者募集について. ターゲット. 在学生の方 公開日. 2022年05月11日. ポスター（PDF） 　米国短期留学プログラム「Kingfisher Global Leadership Program」の2022年度の参加者を募集します。. 　このプログラムは、本学の久能祐子理事（国際渉外、海外同窓会担当／京都大学大学院工学研究科博士後期課程修了、工学博士）とワシントンDC京都大学同窓会のご支援により企画されました。. 　詳細については、全学生向け共通掲示板KULASISの募集要項をご確認ください。. プログラム概要. 米国のワシントンD.C.において、各種機関および現地企業・財団（直近の例：国務省、NASA、NIH、世界銀行等）を訪問し、そこで働く研究者やプロフェッショナルによる本学学生のための講義とディスカッションに参加します。. 実施期間：2022年9月3日（土曜日）～9月18日（日曜日）. 派遣先国・地域：米国・ワシントンD.C. 募集人数：10名. 参加費用：参加者の負担は原則なし. 応募資格・要件. 応募からプログラム参加に至るまで、以下（1）～（5）の資格・要件をすべて満たすこと. 本学の学部または大学院における正規学生であること（休学中の応募・参加不可）. 優秀な学業成績を有すること. 高い英語力を有すること. 京都大学の代表としての自覚を持ち、専門分野以外にも関心を持ち見聞を広める意欲のあること. 米国および日本への入国に際し有効と認められるワクチンの2回接種を完了し、3回目接種を出発日の15日前までに完了していること. 応募の手順. 応募方法や応募書類、提出期限等の詳細は、全学生向け共通掲示板KULASISの募集要項をご確認ください。. 問い合わせ先. 国際・共通教育推進部 国際教育交流課 海外留学掛. E-mail： koryusien＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. 関連リンク. 京大コラボ 米国短期留学プログラム「Kingfisher Global Leadership Program with Kyoto University」（旧「Kyoto-DC Global Leadership Program」） Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home 研究・行政・民间合作 Security Export Control. Security Export Control. サイドメニュー（簡体中国語）. 支援研究体制 Security Export Control 国际交流 In other languages. 日本語 What is \"Security Export Control\"? In order to maintain both national and international peace and security, the Government of Japan controls military sensitive goods and technologies, including relevant dual-use goods and technologies. Based on the Foreign Exchange and Foreign Trade Act (1949) and its relevant legislations, exporters, including universities in Japan, are requested by the Ministry of Economy, Trade and Industry (METI) to apply for an export license in case:. - Items or technologies are listed in the control list; or. - The export is under the provision of end-use control such as 'catch-all control.'. The export of goods and technology designated in international export control agreements without permission from METI is subject to criminal and legal penalties. Such penalties may be applied not only to individual(s) but also to institutions involved. In the course of the various research and educational activities undertaken by universities, certain activities, such as the exportation or carrying of research equipment or materials abroad, collaborative research with foreign researchers or institutions, the hosting and education of foreign researchers or students and the exchange of sensitive information with foreign researchers, could be subject to security export control. Furthermore, the export control legislation of the US, such as the Export Administration Regulations (EAR), must also be observed when research activities involve the receipt of materials or technologies from US researchers, institutions, industries or government. \"Security Export Control Regulations of Kyoto University\" (internal) (PDF) \"Complying The Security Export Control Laws\" (internal) (PDF) \"Security Trade Control Guidelines for Academic Exchange Agreements at Kyoto University\" (internal) (PDF) Further information. Security Export Control Policy Division, Trade Control Department,. The Ministry of Economy, Trade and Industry (English Homepage) (external link). http:\/\/www.meti.go.jp\/policy\/anpo\/englishpage.html Guidance for the Control of Sensitive Technologies for Academic Institutions February 2010 (PDF) Export Administration Regulations (US) (external link). https:\/\/www.bis.doc.gov\/index.php\/regulations\/export-administration-regulations-ear Contact. Research Promotion Division, Research and International Affairs Department. (Security Export Control Section). Phone: ext. 2297, 2298. Email: info-yusyutu*mail2.adm.kyoto-u.ac.jp (replace * with @). 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home 研究・行政・民间合作 国际交流. 国际交流. サイドメニュー（簡体中国語）. 支援研究体制 Security Export Control 国际交流 刊物 3階層目以降ページ用メニュー（簡体中国語）. 刊物. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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大学間学生交流協定に基づく派遣留学については、「新型コロナウイルス感染症の影響下における大学\n間学生交流協定に基づく交換留学（派遣）の渡航可否判断について（通知）」（2022年3月14日付副\n学長通知、 閲覧は教職員限定）に基づき、2022年4月1日以降に渡航する学生を対象に、外務省感染症\n危険情報がレベル２または3の国・地域であっても、渡航の必要性と渡航可としうる相当な理由・状況\nがある場合に限り、渡航期間の制限を設けず、例外的に渡航を認める緩和措置が実施されています。\nなお、国際教育交流課が所掌する短期派遣プログラムについては、プログラムごとに渡航を伴う実施の\n可否判断を行うこととなっています。\n詳細については、国際・共通教育推進部国際教育交流課国際教育企画掛\n（840kikaku-kanri*mail2.adm.kyoto-u.ac.jp *を＠に変えてください）にお問い合わせください。\n その他の部局間交流協定に基づく派遣等における渡航を認める緩和措置については、渡航許可にかかる\n権限者が同通知に準ずる等して判断し、派遣を実施する場合は同通知で定める確認項目等を参考に十分\nな安全対策を講じて万全を期してください。\n学生 【コロナ禍における例外的な緩和措置について】\n教職員\n① 業務上やむを得ない渡航の必要性が認められるか。\n② 渡航先への交通手段（国内移動含む）があるか。\n③ 渡航先における行動制限により、渡航の目的が達せられない恐れはないか。\n④ 渡航先の感染状況や治安が悪化していないか。\n⑤ 渡航国・渡航先機関で十分な防疫措置がとられているか。\n⑥ 現地関係機関との連携体制は十分か。 特に緊急時の連絡体制を整備しているか。\n⑦ 渡航先で必要な生活物資が確保できるか。\n⑧ 渡航先で万一病気等に罹患した場合に十分な医療を受けられる状況か。\n⑨ （外国籍の方）渡航後、日本への再入国にかかる制限はないか。\n⑩ 渡航国および日本帰国後の自宅・宿泊施設等における隔離、待機による業務上の支障がないか。\n【コロナ禍における渡航可否判断時の要検討事項】\n外務省 危険情報・\n感染症危険情報\n学生 教職員\nレベル1\n十分注意してください。\n原則可\nただし、渡航を中止すべき相当な理由・状\n況があれば不可\n原則可\nただし、渡航を中止すべき相当な理由・状況\nがあれば不可\nレベル2\n不要不急の渡航は止め\nてください。\n原則不可\nただし、渡航の必要性と渡航可としうる相\n当な理由・状況があれば可\n要検討\n渡航可としうる相当な理由・状況があれば可\n※コロナ禍における要検討事項は以下を参照\nレベル3\n渡航は止めてください。\n(渡航中止勧告)\n不可\n※コロナ禍における感染症危険情報レベル３にか\nかる例外的な緩和措置は以下を参照\n原則不可\nただし、渡航の必要性と渡航可としうる相当\nな理由・状況があれば可\n※コロナ禍における要検討事項は以下を参照\n更新箇所：青字部分\n2018年3月23日 国際戦略本部運営協議会決定\n教職員については、感染症危険情報レベル２で「要検討」、レベル３で「原則不可」ですが、コロナ禍にお\nいて渡航を検討する場合は、渡航許可にかかる権限者が以下の事項等を慎重に検討の上、可否判断してください。\n【海外渡航にかかる可否判断基準（一部抜粋）】\n2. コロナ禍における渡航時の安全対策の強化について\nコロナ禍においては、海外渡航時の通常の安全対策（安全情報の収集、現地機関との連携体制の構築、緊\n急連絡網の整備、事件等発生時の対応フローやマニュアルの整備、外務省たびレジの登録等）に加え、渡\n航前に必ず以下の事項を確認の上、危険状況に応じて安全対策を強化してください。\n① 外国人による入国が停止されていたり、査証発給が停止されていたりしませんか。\n② 入国時に必要な査証や書類（新型コロナワクチン接種証明書、陰性証明書等）は準備しましたか。\n③ 飛行機搭乗時のルール（陰性証明書の提示義務、マスク着用義務、消毒剤等の持込制限等）を確認しましたか。\n④ 入国時の検査受検、隔離の必要性について確認しましたか。\n⑤ 移動制限はないですか。航空便を含む公共交通機関の運行が停止していませんか。\n⑥ 行動制限により渡航の目的が達成されない恐れはありませんか。\n⑦ 休業命令、集会禁止、マスク着用義務、病院などの施設や飲食店におけるワクチン接種証明書の提示義務等の現地の規制、規則、罰則を確認しましたか。\n⑧ （ワクチン未接種の場合）航空会社の搭乗規制や、渡航国への入国規制、渡航先機関における入構規制、イベント参加規制、施設・店舗（病院、飲食店、公共施設）の利用制限を確認しましたか。\n【入国制限・行動制限】\n【事前の安全対策】\n⑨ 外務省や現地関係機関等から渡航先の最新の感染状況、安全情報、現地関係機関の衛生環境や感染対策情報を収集の上、感染予防や安全対策を確認しましたか。\n⑩ 万一感染が疑われる場合の現地における公的な相談・報告先や、救急連絡先、受診可能な医療機関、大使館\/総領事館及び家族等の緊急連絡先を確認し、関係者間で共有しましたか。\n⑪ 万一感染した場合の報告義務や消毒等の現地で必要な対応を確認の上、関係者間で共有しましたか。\n⑫ 現地関係機関が対応可能な渡航者への支援を確認の上、万一感染した場合の対応フローやマニュアル等を整備しましたか。\n⑬ 「海外渡航に関する安全対策手引き」 で「国際的に懸念される感染症への対策」を含めて安全対策を確認しましたか。\n⑭\n治療・救援費用補償が無制限で、かつ新型コロナウイルス感染症の治療等にも対応している海外旅行\n保険に加入しましたか。国によっては入国時にコロナに対応していることが明示されている保険加入\n者証または付保証明書の提示を求められることがありますので、必要に応じて保険会社から取得して\nください。\n⑮ 感染症対策として、マスク、体温計、消毒剤等を、行動制限対策として、食料品、生活用品等の備蓄を準備しましたか。\ncheck\n【渡航中の安全対策】\n⑯ 渡航中は、渡航者、現地関係機関、および日本側の大学関係者が定期的に連絡をとり、渡航者の健康状態や安全状況を確認してください。\n⑰ 海外で発熱や咳等の症状が出た場合は、事前に調べておいた現地報告・相談先に連絡の上、現地医療機関で受診してください。受診結果については、速やかに所属部局に報告をしてください。\n2\n【日本帰国時の準備】\n⑱ 国際便の減便のおそれがある場合は、早めに帰国便を確保してください。また、空港までの交通手段も早\nめに確保してください。\n⑲ 最新の日本政府の水際対策と大学の方針を確認しましたか。\n(参考）厚生労働省：水際対策に係る新たな措置について\n⑳ PCR等の検査を予約・受検し、飛行機搭乗時、日本入国時に必要な陰性証明書を準備しましたか。\n㉑ 空港で長時間待機する場合に備え、準備をしましたか（防寒具、飲料、簡易食品等）。\n3. 海外からの帰国、入国後について\n3\n入国・帰国時には日本政府による最新の水際対策を確認の上、指示に従ってください。\n【入国・帰国時の手続き】※ファストトラックおよびVisit Japan Webサービスの利用を推奨\n 入国時のWEB質問票への登録（QRコードの提示）、陰性証明書の提示\n 検査受検（適用がある場合）、誓約書の提出（検疫所が確保する宿泊施設または自宅等待機対象の場合）、\n有効なワクチン接種証明書の提示（検疫措置（検査・待機期間）の緩和を受ける場合）\n 政府指定期間の待機要請および健康報告（適用がある場合）※入国日の翌日を1日目としてカウントします。\n 政府指定待機期間の公共機関（航空機、新幹線、電車、タクシー、乗り合いバス等）の不使用\n※ただし、自宅等待機場所への移動については空港検査から24時間以内で最短距離あれば公共交通機関の\n使用が認められます。\n スマートフォンへの位置情報確認アプリ等のインスト―ル 等\n（参考）厚生労働省 水際対策\n【注】新規渡日外国人留学生、研究者については、別途受入手続きが定められています。渡日手続き開始時\nに必ず大学からの指示を確認してください。\n（参考）文部科学省 日本への入国申請（受入機関の皆様へ）\n② 入国時に、滞在した国、日本への帰国・入国日、帰国・入国時点の健康状態（発熱・咳症状・咽頭痛・息\n切れ・全身倦怠感・下痢などの有無、解熱剤、咳止めの服用の有無）について、速やかに、所属部局に報\n告をしてください。発熱や咳等の症状がある場合には、必ず空港等の検疫官に申告してください。\n③ 【待機ありの場合】\n 海外から帰国・入国後、政府指定の待機期間は、検疫所指定施設または自宅や登録待機場所となって\nいる宿泊施設に待機し、大学には立ち入らないでください。当該期間は自宅学習、在宅勤務等とします。\n 待機期間は発熱・咳・咽頭痛・息切れ・全身倦怠感・下痢などの症状がないか必ず経過観察（体調、体\n温など健康状態の記録）をしてください。\n（参考）海外からの帰国者・入国者の健康状態の継続管理、報告システムの一元化、並びに情報共有について（教職員限定）\n 待機終了後、大学への入構前に以下の書類をメールで所属部局に提出して、待機が解除となった旨を\n申し出てください。\n①様式1 （待機期間の健康状態の記録）\n②入国者健康確認センターから受信した「待機解除」通知画面の写し\n【待機なしの場合】\n大学への入構前に以下の書類をメールで所属部局に提出して検疫所で待機なしとなった旨を申し出てください。\n●様式1 （健康状態の記録（入国日のみ））\n※待機あり、なしにかかわらず、入国後10日間を経過するまでは、入国者自身で検温など健康状態の確\n認を行ってください。（厚生労働省 検疫措置の詳細 Q＆A参照）\n④ 体調等について、所属部局と緊密に連絡を取り合うとともに、体調に変化がある場合は、医療機関には\n直接行かず、身近な医療機関又は相談窓口に電話で相談のうえ、その指示に従うとともに、必ず所属部\n局及び本学保健診療所(075-753-2405)へ報告をしてください。\n⑤ 新型コロナウイルスに感染（類似症状等を含む）した場合、「学生・教職員の新型コロナウイルス感染\n確認時等における対応について」に従い対応してください。\n渡航許可にかかる権限者は、以下のような場合には渡航者の安全を最優先して、渡航の延期、中止や緊急帰\n国の指示を検討してください。\n 移動制限、国境閉鎖や航空便の休止により渡航者が出国困難となる可能性がある場合\n 移動制限、行動制限により通常の生活を送ることが困難となる恐れがある場合や、研究・教育の目的が達\n成できない恐れがある場合\n 現地の医療体制が脆弱で、新型コロナウイルス感染症やその他の傷病について十分な医療が受けられない\n恐れがある場合\n渡航の延期、中止、緊急帰国\n4\n 厚生労働省 0120-565653   （フリーダイヤル・日本語のみ・9時00分から21時00分）\n 観光庁 050-3816-2787（外国人旅行者向けコールセンター・日本語・英語・中国語・韓国語・ 24時間可）\n 京都府 075-414-5487  （きょうと新型コロナ医療相談センター・日本語・英語・中国語・韓国語・ポルトガ\nル語・スペイン語(24時間可)・ベトナム語(10-20時)・タイ語(9-18時））\n※発熱症状などがある場合は、まずは身近な医療機関に電話で相談してください。\n※夜間・休日など受診可能な医療機関がない場合は「きょうと新型コロナ医療相談センター」に相談してください。\n※京都府以外にお住まいの場合は居住地域の相談窓口に相談してください。\n相談窓口\n 京都大学環境安全保健機構 健康管理部門／健康科学センター\n 文部科学省 新型コロナウイルスに関連した感染症対策に関する対応について\n 首相官邸 新型コロナウイルス感染症対策について 日本語 英語 中国語\n 外務省 海外安全ホームページ 各国・地域の入国制限措置及び入国後の行動制限\n 厚生労働省 新型コロナウイルス感染症について 日本語 英語 中国語 多言語情報\n 検疫所 新型コロナウイルス感染症における検疫の強化体制について\n 世界保健機関 Coronavirus（英語ページ）\n 京都府 新型コロナウイルス感染症に関連する情報について 日本語 英語 中国語 ベトナム語\n 京都市 情報館\n問合せ先：Email: intl_kiki_kanri＊mail2.adm.kyoto-u.ac.jp ＊を＠に変えてください。\n（本件通知について） 京都大学 総務部企画管理主幹付リスク管理掛 Tel.075-753-2226\n（渡航可否判断基準、入国時の手続き等）企画部国際交流課海外拠点掛 Tel.075-753-2604\n（学生の国際危機管理について） 国際・共通教育推進部国際教育交流課国際教育企画掛 Tel.075-753-2482\n"}
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CLOSE. search. menu. グローバルナビゲーション（繁体中国語）. 京都大學概況. 校長的話. 大學介紹. 校內安全措施. 教育與學生支援服務. 學生生活. 留學・國際教育. 研究・產官學合作. 支援研究體制. Security Export Control. 國際交流. 學部與院係. ヘッダー セカンダリーリンク（日本語以外）. 交通資訊 聯絡我們 捐贈 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 導航連結. Home NEWS 舉行了2021學年度研究所學位授予儀式。（2021年3月24日）. University News 舉行了2021學年度研究所學位授予儀式。（2021年3月24日）. In other languages. 日本語 ターゲット. Current Students & Staff Visitors 發表日期. 2022\/04\/05.     2022年3月24日，2021學年度研究所學位授予儀式於京都市勸業館Miyako Messe舉行， 各理事與副校長、各部門領導出席了儀式。.     本次學位授予儀式的對象為，取得碩士學位的2,211名同學、取得（專業）碩士學位的176名同學、取得（專業）法律博士學位的129名同學，以及分別於2021年11月24日、2022年1月24日和3月23日取得課程博士學位的510名同學、論文博士的44名同學。.     12名同學修畢了始於2012年的博士班教育領導學程。.     12名同學修畢了始於2018年的卓越研究所學程。.     湊長博校長將學位證書頒給學生代表後發表. 致辭 。. 博士學位授予人數一覽（2020年11月24日、2022年1月24日、2022年3月23日） 碩士學位、（專業）碩士學位、（專業）法律博士學位授予人數一覽 湊長博校長的致辭. 2021學年度學位授予儀式 致辭 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 發表日期. 2022\/04\/11. 舉行了2022學年度研究所入學典禮。（2022年4月7日）. 發表日期. 2022\/04\/08. 舉行了2022學年度大學入學典禮。（2022年4月7日）. 發表日期. 2022\/04\/05. 舉行了2021學年度研究所學位授予儀式。（2021年3月24日）. 發表日期. 2022\/03\/29. 舉行了2021學年度畢業典禮。（2022年3月24日）. 最新新聞 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通資訊 フッター セカンダリーリンク. 站點地圖 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. 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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home About Health, safety, and environment Kyoto University’s Response to the Novel Coronavirus (COVID-19). Kyoto University’s Response to the Novel Coronavirus (COVID-19). サイドメニュー（英語）. International strategy Global profile Global partners International risk & crisis management Safety for students, faculty, and staff traveling abroad Framework to ensure safety in international activities Safety information and tips for travelers abroad Kyoto University's Response to the Novel Coronavirus (COVID-19) Global University Social Responsibility KyotoU International Concierge and information resources In other languages. 日本語 Recent Updates (Last Updated: 1 June 2022). (1 June 2022). Kyoto University’s Policy on the Novel Coronavirus (COVID-19) (Ver. 15)－Regarding Overseas Travel, Etc.－ (1 June 2022). Regarding the Wearing of Masks to Prevent the Spread of COVID-19 (1 April 2022). Manual of Measures to Prevent the Spread of the Novel Coronavirus (COVID-19):Precautions to be Taken Regarding Classes in the First Semester of 2022 (Ver.8) (10 February 2022). Application for Special Provisions during the First Semester of 2022  . form Kyoto University's Response to the Novel Coronavirus (COVID-19). e-Learning. (November 13, 2020). “COVID-19 Prevention Measures for Students” e-Learning course Please note that the e-learning course is also accessible by the general public through the Kyoto University YouTube Channel below. 　. Kyoto University YouTube channel Activity restriction measures to prevent the spread of the novel coronavirus \/ Kyoto University’s Policy on the Novel Coronavirus (COVID-19). (30 September 2021). Reduction of the level of activity restriction to prevent the spread of the novel coronavirus (COVID-19) *The Current level of activity restriction is. \"level 1\". (1 June 2022). Kyoto University’s Policy on the Novel Coronavirus (COVID-19) (Ver. 15)－Regarding Overseas Travel, Etc.－ *. Safe Travel Abroad (leaflet for students, faculty, and staff traveling abroad) 　The latest edition includes the measures to prevent infectious diseases that spread internationally. *. Safety Guide for Students Engaged in Overseas Fieldwork (2 February 2022) . Procedures to be Followed if a Student, Faculty Member, or Staff Member is Confirmed as Infected with the Novel Coronavirus (Ver. 8) (1 April 2022). Manual of Measures to Prevent the Spread of the Novel Coronavirus (COVID-19):Precautions to be Taken Regarding Classes in the First Semester of 2022 (Ver.8) (1 June 2022). Regarding the Wearing of Masks to Prevent the Spread of COVID-19 Messages from the President For students (including announcements for international students). For faculty and staff members (including announcements for international researchers, faculty, and staff). Event Calendar (in Japanese). Facilities currently subject to usage restrictions. Kyoto University Museum (in Japanese). Kyoto University Library Network Clock Tower (in Japanese). Tokyo Office (in Japanese). Kyoto University CO-OP Notices Issued by the Kyoto University Health Service. Division of Health Management\/Kyoto University Health Service, Agency for Health, Safety and Environment Consultation Services Health Ministry report: foreign materials found in some doses of the Moderna COVID-19 vaccine Contact. Risk Management Section, General Affairs Department. Email: 830riskkanrikakari*mail2.adm.kyoto-u.ac.jp（Please replace * with @.）. Links. Ministry of Education, Culture, Sports, Science and Technology (MEXT): Responses to the novel coronavirus infection (in Japanese). Website of the Office of the Prime Minister of Japan: [COVID-19] Government Responses on the Coronavirus Disease 2019. (in English). http:\/\/japan.kantei.go.jp\/ongoingtopics\/_00013.html (in Chinese). http:\/\/www.kantei.go.jp\/cn\/japan_intro\/2020\/_00044.html Overseas Travel Safety Information provided by Japan’s Ministry of Foreign Affairs (in Japanese). Website of Japan’s Ministry of Health, Labour and Welfare (MHLW): About Coronavirus Disease 2019 (COVID-19). (in English). https:\/\/www.mhlw.go.jp\/stf\/seisakunitsuite\/bunya\/newpage_00032.html (in Chinese). https:\/\/www.mhlw.go.jp\/stf\/seisakunitsuite\/bunya\/newpage_09534.html World Health Organization (WHO): Coronavirus  (in English). Kyoto Prefecture Website: Information about 2019 Novel Coronavirus. (COVID-19) I. nfection. (in English). https:\/\/www.pref.kyoto.jp\/kokusai\/coronavirus_update.html (in Chinese). https:\/\/www.pref.kyoto.jp\/kokusai\/coronavirus_cn.html Kyoto City Official Website: Information on the novel coronavirus infection (in Japanese). 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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2021","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-9083757294326255655,"exact_signature_unique_b":true,"fuzzy_signature_l":-3154098365029448151,"fuzzy_signature_unique_b":true,"wordcount_i":24,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:44:27 UTC 2022","fresh_date_dt":"Wed Jul 13 12:09:48 UTC 2022","referrer_id_s":"w5KXo9U8j7Mo","publisher_t":"Adobe PDF Library 15.0","language_s":"ja","size_i":694699,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085699069116416,"text_t":"各学部が特色入試を通じて 求める人物像\n■ 総合人間学部\n総合人間学部は、異質な学問分野の接触と相互作用を基盤とした新たな「人間の学」の創出\nを目ざしています。積み上げられてきた既成の学問の成果を生かしながら問題を発見し追究\nすること、問題の解決に向けて学問間の連携を図り新たな体系を創出することが、本学部の\n目的です。こうした意味での「総合」、「文理融合」の意義を理解し、これを積極的に進めよう\nとする学生諸君が門をたたくことを望んでいます。\n■ 文学部\n文学部は、哲学・歴史学・文学・行動科学に関わる諸問題を学び考え、自由の学風を重んじ\nる本学の基本理念を踏まえながら、新たな知的価値を創出することをめざす学生を求めてい\nます。特色入試では、基礎学力を十分に備え、これからの文学部での勉学についてプランと\n展望を持ち、意欲を持って広い意味での勉学に励む人を求めています。\n■ 教育学部\n教育学部は、教育と心・人間・社会にかかわる多様な事象を対象とした諸科学を学ぶこと\nで、教育や心理についての専門的知見、さらに、広い視野と異質なものへの理解、多面的・総\n合的な思考力と批判的判断力を育成する教育を行っています。\n　そこで、次のような人物を求めています。\n・\b教科の学習及び総合的な学習の時間などにおいて学習を深め、テーマを設定して探究活動\nを行い、卓越した学力を身につけ、成果をあげた者、あるいは、学校内外の活動で豊かな経\n験を積み、創造的な熟達を通して、深い洞察を得ている者\n・\b人間と社会、教育や心理について関心を持ち、論理的・批判的に思考し、問題を解決する能\n力とコミュニケーション能力を持つ者\n・将来、教育や心理にかかわる専門的識見を発揮して、社会に貢献する志を持つ者\n■ 法学部\n法学部の特色入試（後期日程）では、特に次に掲げる能力及び資質等を備える人材を求めて\nいます。\n・\b様々な分野で、多様な人々と協働し、指導的な役割を果たす基盤とするため、多様な考え\n方を理解し、論拠を示して自らの意見を述べることができる基本的なコミュニケーション\n能力、とりわけ論理的な文章を書く力の基本を身につけていること\n・\bグローバルな視野から国家・社会に関する事象に強い関心を持ち、このような事象を本質\nから理解しようとする知的探究心を有すること\n・\b人々が協働し共に生きる社会の実現を志す倫理性と責任感を持ち、未だ答えのない課題等\nを自ら見いだし、文献や資料等を調査して、徹底して考え抜こうとする自学自習の姿勢を\n有していること\n■ 経済学部\n経済学部は、総合的な学力とともに、長文読解力、問題発見力、論理的思考、柔軟な思考と\n創造性、そして高い自学自習の能力を持つ人材を求めています。\n■ 理学部\n理学部は、自由な雰囲気の下での学問的創造および自律的学修を大切にしており、１学科\n制の下、緩やかな専門化を経て、各学生の興味や適性に合致した専門分野に向う教育を行っ\nています。\n　理学部では、望む学生像として次の２項目を掲げています。（1）自由を尊重し、既成の概念\nを無批判に受け入れることなく、自ら考え、新しい知を吸収し創造する姿勢を持つ人。（2）高\n等学校の教育課程により培われる十分な科学的素養、論理的合理的思考力と語学能力を有\nし、粘り強く問題解決を試みる人。そのような人の中で、数理科学入試では特に数理科学の\n分野において極めて優れた才能を持つ人、また生物科学入試では生物科学の分野に志向性を\n持ち科目に偏らない幅広い学力を持つ人の受験を望んでいます。\n■ 医学部医学科\n京都大学は創立以来、自由の学風、闊達な対話を重視し、世界トップレベルの先端的学術\n研究を推進してきました。医学部医学科は、社会が大きく変革している現代においても、京\n都大学の学問環境で学び、研究することで、世界をリードできる人材、具体的には以下の要\n件を満たす人材を全国から広く求めます。\n１．\b医学・生命科学に深い関心を持ち、かつ真摯な姿勢、熱意を持って真理を探究できる将来\nの世界の医学をリードするような医学研究者としての資質・適性を持つ人材。\n２．\b１．以外でも自然科学の少なくとも１領域において傑出した能力を有し、かつ医学研究\n者としての資質・適性を持つ人材も考慮します。\n３．本学が提供するMD-PhDコースへの進学を希望する人材。\n■ 医学部人間健康科学科\n医学部人間健康科学科は、国家資格をベースに、国際的視野を兼ね備えた医療専門職の\nリーダーや研究者の養成を推進します。特色入試ではこのうち看護師、保健師、理学療法士、\n作業療法士へのモチベーションを持った学生を募集します。多面的かつバランスの良い思考\n力で課題や問題を抽出し解決していく医療のエキスパートとなり得る人材を求めています。\n■ 薬学部薬科学科\n京都大学薬学部薬科学科では、創薬に関係する幅広い分野において、リーダーとして国際\n的に活躍することのできる人材を育成しています。そのため、特色入試では、特に次のよう\nな要件を満たす人材を全国から求めます。\n１．\b創薬科学研究に関心と興味を持ち、大学院薬学研究科で博士の学位を取得した後に、世\n界の創薬科学の革新と発展をリードする研究者となる志と資質を有する人\n２．\b知的好奇心に基づいた科学的探求力と英語力に優れ、自発的な自然科学の学習や探求活\n動を実施してきた人\n３．\b十分な基礎学力とコミュニケーション能力を有し、自ら目標を設定して挑戦できる行動\n力と最後までやり遂げる力がある人\n■ 薬学部薬学科\n京都大学薬学部薬学科では、医療薬学や先端医療に関係する幅広い分野において、国際的\nなリーダーとして薬学研究を牽引できる人材を育成しています。そのため、特色入試では、\n特に次のような要件を満たす人材を全国から求めます。\n１．\b医療薬学研究に関心と興味を持ち、大学院薬学研究科で博士の学位を取得した後に、薬\n剤師資格を背景にして世界の薬学の発展をリードする研究者となる志と資質を有する人\n２．\b知的好奇心に基づいた科学的探求力に優れ、医療人であるとともに教育者・研究者とな\nるにふさわしい人間性を備えており、倫理観を持って自発的な科学の学習や探求活動を\n実施してきた人\n３．\b十分な基礎学力とコミュニケーション能力を有し、自ら目標を設定して挑戦できる行動\n力と最後までやり遂げる力がある人\n■ 工学部地球工学科\n持続的な発展をめざす人類社会のより良い未来に向けて、社会インフラの構築や都市・地\n域の開発と防災、資源・エネルギーの高度利用と新規開発、地球・地域環境の保全と改善に関\nする革新的技術を自ら創り出すことに強い意欲をもち、それに必要な高い基礎学力を有し、\n将来、国際的視野をもったリーダーとなり得る人材を求めています。\n■ 工学部建築学科\n建築学科は、自然科学だけでなく、人文科学、社会科学、さらには芸術にも深い関心をも\nち、それらの特筆すべき能力、高い基礎学力とリーダーシップを持つ人材を求めています。\n■ 工学部物理工学科\n次世代の画期的な機械システム、新材料、エネルギーシステムを開発すること、宇宙空間\nへの活動の場を拡げていくことに強い関心を持ち、これらの課題の実現に向け、物理学を基\n礎とした工学を学ぶ強い意欲、特筆すべき能力、リーダーシップと高い基礎学力を持つ人を\n求めます。\n■ 工学部電気電子工学科\n自然現象や科学技術、特に電気電子工学関連の学術分野に対する強い興味と旺盛な探究心\nをもつとともに、専門教育を受けるのに十分な高い基礎学力と論理的思考力を有しており、\nかつ科学技術に関する特筆すべき能力やリーダーシップを備えた創造性豊かな人材を求め\nます。\n■ 工学部情報学科\n情報学科では、高度情報社会における幅広い問題を情報学の知識と数理的思考により解決\nすることに強い関心を持つ人を求めています。特色入試ではさらに、授業科目の一環として\n実施した課題研究や、科学に関する課外活動で顕著な実績を上げた経験を有する人を求めます。\n■ 工学部工業化学科\n化学および化学に関連する工学の基礎学理を理解し、広い視野から物事を論理的に解決す\nる能力、未知なる課題に取り組む意欲と創造性、高い倫理観、優れたコミュニケーション能\n力とリーダーシップを有し、将来、国際的に活躍する研究者、技術者になり得る人材を求め\nています。\n■ 農学部資源生物科学科\n資源生物科学科では、知的好奇心を出発点として最先端の基礎生物学に取り組み、幅広い\n応用分野に展開をしていく研究と、現在の農水畜産業などが抱える諸問題の解決を目指して\n資源生物と関連分野に関する知を究めていく研究を推進しています。これら二種類の出発点\nの異なる研究から得られる基礎と応用の知を統合することで、当学科の究極目標であるより\n豊かで持続的な社会の実現を目指した教育と研究を行っています。\b\n　そこで、特色入試では、対象とする様々な生物と自ら向き合い、自ら問い、自ら考えて解決\nの道を切り拓く強い意欲を持ち、学問の基礎を大切にする人材を求めます。また、農水畜産\n業などの応用的課題について取り組む視野の広い人材も求めています。特に、将来、博士の\n学位を取得して、豊かな知的好奇心と科学的探求心をもち世界をリードする研究者となるこ\nとを目指す人材を求めています。\n■ 農学部応用生命科学科\n生命現象や生物の機能を化学的なアプローチによって解明・活用することに興味があり、\n歴史を踏まえて自身と社会の未来を設計することができ、多くの分野にまたがる領域にも飛\nび込んでいける人材を求めています。\n■ 農学部地域環境工学科\n食料・環境・農業などの分野において、高度な専門知識と工学的問題解決能力を持って社\n会のリーダーとなるべき人材を求めています。\n■ 農学部食料・環境経済学科\n食料・環境・農業などの分野において、大学卒業後、大学院へ進学する意欲を持ち、高度な\n専門知識を持って国際社会のリーダーとなるべき人材を求めています。\n■ 農学部森林科学科\n森林科学（森林や木質資源に関する基礎及び応用科学）の分野について学びつつ、研究を深\nめることに強い関心があり、高度な専門知識と問題解決能力を習得する強い意欲と能力のあ\nる人材を求めています。また卒業後は、森林管理や生物多様性保全といった面から、あるい\nは木材由来のさまざまな材料を駆使して持続可能な社会の確立に貢献する人材となること\nを期待しています。森林科学のキーワードは多岐にわたります。森林管理、森林生態系、生物\n多様性、ランドスケープ、緑化、土砂災害、森林由来の生物資源や機能性材料、木造建築、木質\n文化財、木材と感性、バイオマス、セルロースナノファイバー、きのこ、などです。\n■ 農学部食品生物科学科\n大学卒業後、大学院修士・博士後期課程に進学して博士の学位を取得し、大学院修了後、研\n究において世界のトップレベルで活躍する学究的な人材を求めています。\n"}
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{"id":"xAYMH9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/about\/public\/issue\/ryugaku-annai","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":61,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["ja","about","public","issue"],"url_file_name_s":"ryugaku-annai","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["京都大学への留学案内 | 京都大学"],"description_txt":["京都大学への留学案内です。\n京都大学に留学するためのさまざまな情報のほか、日本での学生生活などについて説明しています。"],"last_modified":"Wed Jun 08 14:43:26 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-1459757664062634370,"exact_signature_unique_b":true,"fuzzy_signature_l":7824086938756593063,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["京都大学への留学案内"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","バックナンバー","English version","CHINESE（簡体）、KOREAN version バックナンバー","問い合わせ","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["全ページ","章別"],"italic_txt":["menu","search","close"],"imagescount_i":9,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/admissons-guide-ia-2022-7a0dd389e12093d0371b9e39d8503110.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,212,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,150,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button sites default files inline admissons guide ia 2022 7a0dd389e12093d0371b9e39d8503110 jpg 7a 0dd 389e 12093d 0371b 9e 39d 8503110 facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":770,"text_t":"京都大学への留学案内 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求 京都大学への留学案内. 京都大学への留学案内. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 京都大学への留学案内 バックナンバー 京都大学への留学案内（CHINESE（簡体）、KOREAN） バックナンバー 「京都大学への留学案内」の請求 紅萠（くれなゐもゆる） Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 In other languages. English 京都大学への留学案内です。. 京都大学に留学するためのさまざまな情報のほか、日本での学生生活などについて説明しています。. 全ページ. 2022\/2023 （3.54MB）. 章別. 1. 大学紹介 （1.0MB）. 2. 私費留学生として入学するためには. 2-1. 学士課程 （624KB）. 2-2. 修士課程・博士課程・専門職学位課程 （649KB）. 2-3. English-Taught Degree Programs （232KB）. 2-4. ダブルディグリー／ジョイントディグリープログラム・研究生・交換留学生・短期プログラム （6171KB）. 3. 日本政府（文部科学省）奨学金留学生として入学するためには （622KB）. 4. 学納金・奨学金制度 （512KB）. 5. 留学生支援・生活費・住まい （539KB）. 6. 査証・在留資格・諸手続き （455KB）. 7. 資料. 7-1. 私費留学生対象奨学金⼀覧 （523KB）. 7-2. 事務室連絡先 （829KB）. 7-3. キャンパス・アクセス・京都の魅力と生活 （872KB）. バックナンバー. 京都大学への留学案内 バックナンバー English version. Admissions Guide for International Applicants CHINESE（簡体）、KOREAN version バックナンバー. ※ 「京都大学への留学案内」のCHINESE（簡体）およびKOREANは現在作成していません。. 同冊子の最新版は日本語版または英語版を参照してください。. 京都大学への留学案内（CHINESE（簡体）、KOREAN）バックナンバー 問い合わせ. 「京都大学への留学案内」の請求 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求 京都大学概要. 京都大学概要. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京都大学概要 2020（10月1日～） 京都大学概要 2020（～9月30日） 京都大学概要 2019 京都大学概要 2018 京都大学概要 2017 京都大学概要 2016 京都大学概要 2015 京都大学概要 2014（10月1日～） 京都大学概要 2014（～9月30日） 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 京都大学概要 2021. 京都大学概要2021 全ページPDF：8.50MB. ページ別PDF. 基本理念・目次 （166KB）. ご挨拶 （229KB）. 沿革 （369KB）. 歴代総長・創立以来の比較 （128KB）. 組織 （685KB）. 役員・役職者等 （1.10MB）. 大学院・学部 （664KB）. 附置研究所 （129KB）. 教育研究施設等 （171KB）. 栄誉 （152KB）. 職員数 （116KB）. 学生数等 （283KB）. 入学状況 （280KB）. 卒業者・修了者数等 （351KB）. 進路・就職状況 （118KB）. 国際交流 （3.35MB）. 財務状況 （175KB）. 外部資金 （216KB）. 発明・特許・知的財産 （83KB）. 土地面積 （138KB）. 建物面積等 （213KB）. 蔵書数 （365KB）. 患者数 （537KB）. 総合博物館・水族館入館者数 （296KB）. 所在地一覧 （723KB）. 研究所・附属研究施設等位置図 （430KB）. 教職員像・環境憲章 （336KB）. アクセスマップ・アカデミックカレンダー （328KB）. バックナンバー. 京都大学概要 2020（10月1日～） 京都大学概要 2020（～9月30日） 京都大学概要 2019 京都大学概要 2018 京都大学概要 2017 京都大学概要 2016 京都大学概要 2015 京都大学概要 2014（10月1日～） 京都大学概要 2014（～9月30日） 京都大学概要 2013 京都大学概要 2012 京都大学概要 2011 京都大学概要 2010 京都大学概要 2009 京都大学概要 2008 京都大学概要 2007 京都大学概要 2006 京都大学概要 2004-2005 京都大学概要 2002 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム アクセス キャンパス間の移動（学内者向け）連絡バスの運行. キャンパス間の移動（学内者向け）連絡バスの運行. 2階層目ページ用メニュー. 採用情報 アクセス 見学にいらっしゃる方 各キャンパスについて（吉田・宇治・桂） 各学部・研究科、センター等の場所について 食堂、レストラン、ショップなど キャンパスマップ（ダウンロード用） キャンパス間の移動（学内者向け） 連絡バスの運行 連絡バスの運行. 連絡バス利用時の留意点. 感染症拡大防止のため、連絡バスご利用の際は以下の点について、ご理解・ご協力のほどお願いいたします。. 連絡バスに乗車の際はマスクを着用ください。. 車内での会話は控えてください。. バス停では、前の人と間隔を空けて並んでください。. 時刻表. 本部～宇治構内間 本部～桂構内間 宇治～桂構内 ダウンロード用時刻表. 連絡バス時刻表（PDF） 運行状況. 平日. 運行. 土日祝. 運休. 創立記念日 （6月18日）. 夏季一斉休業日. （8月第3週の月曜日、火曜日および水曜日の3日間）. 年末年始 （12月29日から1月3日）. 乗車できる者の範囲. 教職員、学部学生、大学院生等（通勤・通学に使用することは認めません。）. ※ なお、乗車定員に達した時点で乗車をお断りします。また、交通事情等により時刻表どおり運行しない場合もありますので、ご了承願います。. 連絡バス運行時間に関する問い合わせ先. 施設部 プロパティ運用課. Tel： 075-753-2151. 本部～宇治構内間. 発着場所. 本部構内（本部構内バス停）. ↑　↓. 宇治構内（宇治地区研究所本館東側）. ※ 学外の停車場所から乗車される方は、バスが近づきましたら、手を挙げて運転手にお知らせください。. 本部～宇治構内間発着場所 （本部→）宇治構内行. 停車場所. 宇治構内行. 第1便. 第2便. 第3便. 第4便. 第5便. 第6便. 第7便. 第8便. 本部構内. 9:30. 10:40. 12:05. 13:10. 15:15. 16:40. 17:40. 18:45. 近衛. 9:31. 10:41. 12:06. 13:11. 15:16. 16:41. 17:41. 18:46. 宇治構内. 10:20. 11:30. 12:55. 14:00. 16:05. 17:30. 18:30. 19:35. 所要時間. 約50分. （宇治→）本部構内行. 停車場所. 本部構内行. 第1便. 第2便. 第3便. 第4便. 第5便. 第6便. 第7便. 第8便. 宇治構内. 8:30. 9:30. 11:00. 12:00. 13:50. 15:00. 16:25. 17:40. 近衛. 9:19. 10:19. 11:49. 12:49. 14:39. 15:49. 17:14. 18:29. 本部構内. 9:20. 10:20. 11:50. 12:50. 14:40. 15:50. 17:15. 18:30. 所要時間. 約50分. 本部～桂構内間. 運行ルート. 本部構内～東大路通～丸太町通～川端通～御池通～堀川通～五条通～国道9号線～桂構内. ※ 学外の停車場所から乗車される場合は、バスが近づきましたら、手を挙げて運転手にお知らせください。. なお、京都リサーチパーク（KRP）付近（五条七本松）での発着時間については、バスが順調に運行した際での時間設定となっていますので、交通事情によっては遅れが生じることがありますことを予めご了承願います。. 発着場所. 本部構内（本部構内バス停）. ↑　↓. KRP付近（五条七本松）. ↑　↓. 桂構内（Aクラスター正門前）. ↑　↓. （Cクラスター前駐車場）. 本部～桂構内間発着場所 （本部→）桂構内行. 停車場所. 桂構内行. 第1便. 第2便. 第3便. 第4便. 第5便. 第6便. 第7便. 第8便. 本部構内. 8:45. 9:15. 9:55. 10:40. 11:05. 11:50. 12:25. 13:35. 五条七本松. （降車、乗車可）. 9:00. 9:30. 10:10. 10:55. 11:20. 12:05. 12:40. 13:50. 桂構内. 9:40. 10:10. 10:50. 11:35. 12:00. 12:45. 13:20. 14:30. 停車場所. 桂構内行. 第9便. 第10便. 第11便. 第12便. 第13便. 第14便. 第15便. 本部構内. 14:25. 15:15. 15:35. 16:20. 17:00. 18:10. 18:45. 五条七本松. （降車、乗車可）. 14:40. 15:30. 15:50. 16:35. 17:15. 18:25. 19:00. 桂構内. 15:20. 16:10. 16:30. 17:15. 17:55. 19:05. 19:40. （桂→）本部構内. 停車場所. 本部構内行. 第1便. 第2便. 第3便. 第4便. 第5便. 第6便. 第7便. 第8便. 桂構内. 9:15. 9:55. 10:40. 11:05. 11:50. 12:25. 13:35. 14:25. 五条七本松. （降車、乗車可）. 9:25. 10:05. 10:50. 11:15. 12:00. 12:35. 13:45. 14:35. 本部構内. 10:10. 10:50. 11:35. 12:00. 12:45. 13:20. 14:30. 15:20. 停車場所  . 本部構内行. 第9便. 第10便. 第11便. 第12便. 第13便. 第14便. 桂構内. 15:10. 15:35. 16:25. 17:00. 18:10. 18:45. 五条七本松. （降車、乗車可）. 15:20. 15:45. 16:35. 17:10. 18:20. 18:55. 本部構内. 16:05. 16:30. 17:20. 17:55. 19:05. 19:40. 宇治～桂構内間. 運行ルート. 宇治構内～宇治バイパス～京都縦貫道自動車道～桂構内. ※ 宇治・桂構内の直行便とします。（途中の停車はしません。）. 発着場所. 宇治構内（宇治地区研究所本館東側）. ↑　↓. 桂構内（Aクラスター正門前）. ↑　↓. （Cクラスター前駐車場）. 宇治～桂構内間発着場所 （宇治→）桂構内行. 停車場所. 桂構内行. 第1便. 第2便. 第3便. 宇治構内. 9:00. 11:40. 15:30. 桂構内. 9:45. 12:25. 16:15. （桂→）宇治構内行. 停車場所. 宇治構内行. 第1便. 第2便. 第3便. 桂構内. 10:35. 12:50. 16:55. 宇治構内. 11:20. 13:35. 17:40. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 栄誉等 歴代総長. 歴代総長. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 京都大学名誉博士 京都大学名誉フェロー 栄誉 歴代総長 第26代 総長 山極壽一 第25代 総長 松本紘 第24代総長 尾池和夫 第23代 総長 長尾真 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 （令和2年10月1日現在）. 第27代. 湊 長博 （令和2年10月1日就任）. 第26代. 山極 壽一 （平成26年10月1日就任）. 第25代. 松本 紘 （平成20年10月1日就任）. 第24代. 尾池 和夫 （平成15年12月16日就任）. 第23代. 長尾 真 （平成9年12月16日就任）. 第22代. 井村 裕夫. （平成3年12月16日就任）. 第21代. 西島 安則. （昭和60年12月16日就任）. 第20代. 沢田 敏男. （昭和54年12月16日就任）. 第19代. 岡本 道雄. （昭和48年12月16日就任）. 第18代. 前田 敏男. （昭和44年12月16日就任）. 第17代. 奥田 東. （昭和38年12月16日就任）. 第16代. 平澤 興. （昭和32年12月16日就任）. 第15代. 瀧川 幸辰. （昭和28年12月11日就任）. 第14代. 服部 峻治郎. （昭和26年11月1日就任）. 第13代. 鳥養 利三郎. （昭和20年11月1日就任）. 第12代. 羽田 亨. （昭和13年11月25日就任）. 第11代. 濱田 耕作. （昭和12年6月30日就任）.  .  . 平野 正雄. ［事務取扱］. （昭和13年7月25日就任）. 第10代. 松井 元興. （昭和8年7月7日就任）. 第9代. 小西 重直. （昭和8年3月22日就任）.  .  . 山本 美越乃. ［事務取扱］. （昭和8年6月30日就任）. 第8代. 新城 新蔵. （昭和4年3月22日就任）. 第7代. 荒木 寅三郎. （大正4年6月15日就任）. 第6代. 山川 健次郎. ［兼任］※※. （大正3年8月19日就任）. 第5代. 澤柳 政太郎. （大正2年5月9日就任）.  .  . 荒木 寅三郎. ［事務取扱］. （大正3年4月28日就任）. 第4代. 久原 躬弦. （明治45年5月13日就任）. 第3代. 菊池 大麓. （明治41年9日2日就任）.  .  . 久原 躬弦. ［事務取扱］. （明治45年5月8日就任）. 第2代. 岡田 良平. （明治40年10月16日就任）.  .  . 岡田 良平. ［兼任］※. （明治41年7月21日就任）. 第1代. 木下 廣次. （明治30年6月28日就任）.  .  . 久原 躬弦. ［事務取扱］. （明治40年7月1日就任）. ※文部次官を兼任. ※※東大総長を兼任. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 研究・産官学連携 研究上の倫理・安全 研究の情報公開（オプトアウト等）. 研究の情報公開（オプトアウト等）. サイドメニュー. 研究活動を知る 研究成果の公開ポリシー 産官学連携（共同研究・知的財産・ベンチャー支援） 京都大学に寄附する 京都大学で研究したい 研究支援体制について 研究者が受けられる研究費・研究支援 研究上の倫理・安全 公的研究費の適正管理 公正な研究活動の推進 京都大学における軍事研究に関する基本方針 安全保障輸出管理 利益相反マネジメント ライフサイエンス研究等に係る倫理の保持と安全の確保 研究の情報公開（オプトアウト等） 本学の研究者が他の研究機関への既存試料・情報の提供のみを行うものに関する情報公開一覧 安全衛生管理 教職員に兼業を依頼したい 「人を対象とする医学系研究に関する倫理指針」、「ヒトゲノム・遺伝子解析研究に関する倫理指針」に基づき、本学で実施中または、本学の既存試料や情報を学外に提供して実施されている、人を対象とする医学系研究に関する情報を以下の通り公開します。研究対象者から直接同意を得ている場合は、以下に含まれません。. 1．本学の研究者が既存試料・情報を用いて実施する研究に関する情報公開. 本学で実施中の人を対象とする医学系研究に関する詳しい情報は、以下のリンクをクリックしてご確認ください。ご自身の情報や試料を研究に使われたくないとお考えの方は、情報公開文書に記載されている問い合わせ先までご連絡ください。そのような場合も診療で不利益を受けることはありません。. 医学部附属病院 iPS細胞研究所 ウイルス・再生医科学研究所 2．本学の研究者が他の研究機関への既存試料・情報の提供のみを行うものに関する情報公開. 本学は研究を行わず、本学に保管されている試料や情報を他の研究機関が行う研究に提供をする場合は、以下に情報を掲載しています。. 本学の研究者が他の研究機関への既存試料・情報の提供のみを行うものに関する情報公開一覧 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"wQsTS9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/social\/hc-day","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":42,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","social"],"url_file_name_s":"hc-day","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["京都大学ホームカミングデイ | 京都大学"],"description_txt":["京都大学ホームカミングデイとは\/About Kyoto University Homecoming Day\n\n京都大学ホームカミングデイは、年に一度の京都大学に関わる全ての方々との交流イベントです。卒業生の方には母校でなつかしい同窓生や恩師との交流を深めていただき、 一般の方々には本学についてより広く知っていただく機会としています。\n\n第17回の開催は、実地とオンラインのハイブリッドを予定しています。"],"last_modified":"Wed Jun 08 14:29:30 UTC 2022","dates_in_content_count_i":4,"dates_in_content_dts":["Sat Nov 05 00:00:00 UTC 2022","Sun Nov 05 00:00:00 UTC 2023","Wed Nov 30 00:00:00 UTC 2022","Thu Nov 30 00:00:00 UTC 2023"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-7261321675105249831,"exact_signature_unique_b":true,"fuzzy_signature_l":-9195792297665113992,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["京都大学ホームカミングデイ"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","京都大学ホームカミングデイとは\/About Kyoto University Homecoming Day","開催実績\/Past events","問い合わせ先\/Inquiries","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":9,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jasocialhc_dayimageshcd.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button sites default files embed jasocialhc dayimageshcd jpg facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":745,"text_t":"京都大学ホームカミングデイ | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 社会連携 京都大学ホームカミングデイ. 京都大学ホームカミングデイ. サイドメニュー. 公開講座・講演会等 京大の研究・教育を知る 社会人向け教育 教育・研究に関する情報・資料公開 産官学連携（共同研究・知的財産・ベンチャー支援） 京大とのつながりを深める 125周年記念事業 京都大学同窓会 ホームカミングデイ 京都大学に寄附する 社会に開かれた施設等 終了した全学的なイベント 京都大学ホームカミングデイとは\/About Kyoto University Homecoming Day. 京都大学ホームカミングデイは、年に一度の京都大学に関わる全ての方々との交流イベントです。卒業生の方には母校でなつかしい同窓生や恩師との交流を深めていただき、 一般の方々には本学についてより広く知っていただく機会としています。. 第17回の開催は、実地とオンラインのハイブリッドを予定しています。. 実地開催は、2022年11月5日（土曜日）の1日の開催です。. オンラインの公開期間は、2022年11月5日（土曜日）10時00分から2022年11月30日（水曜日）17時00分までの予定です。. ※ 新型コロナウイルス感染症拡大状況等により、内容や日程を変更しての開催となる場合があります。. Homecoming Day is an annual event for the general public and the entire Kyoto University community. For alumni, it is an opportunity to deepen their bonds with former classmates and teachers, and for members of the general public, a chance to get to know the many facets of the university. The 17th Kyoto University Homecoming Day will be a hybrid of online and onsite events. . Onsite events will be held for one day only: November 5 (Sat.), 2022. Videos of the online events can be viewed on-demand during the following period:　. 10:00 on November 5 (Sat.) until 17:00 on November 30 (Wed.), 2022. *Please note that the content and schedule of Homecoming Day may change depending on the development of the novel coronavirus situation. 開催実績\/Past events. 第16回（2021年）. 開催案内  | . 開催の様子 第15回（2020年）. 開催案内  |. 開催の様子 第14回（2019年）. 開催案内 |. 開催の様子 第13回（2018年）. 開催案内 |. 開催の様子 第12回（2017年）. 開催案内 |. 開催の様子 第11回（2016年）. 開催案内 |. 開催の様子 第10回（2015年）. 開催案内 |. 開催の様子 第9回（2014年）. 開催案内 |. 開催の様子 第8回（2013年）. 開催案内 |. 開催の様子 第7回（2012年）. 開催案内 |. 開催の様子 第6回（2011年）. 開催案内 |. 開催の様子 第5回（2010年）. 開催案内 |. 開催の様子 第4回（2009年）. 開催案内 |. 開催の様子 第3回（2008年）. 開催案内 |. 開催の様子 第2回（2007年）. 開催案内 |. 開催の様子 第1回（2006年）. 開催案内 |. 開催の様子 問い合わせ先\/Inquiries. 〒606-8501 京都市左京区吉田本町. 京都大学同窓会事務局. Tel： 075-753-2029／5563. Fax： 075-753-2286. E-mail： alumni＊www.adm.kyoto-u.ac.jp （＊を@に変えてください）. Kyoto University Alumni Secretariat. Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501 JAPAN. Phone: 075-753-2029\/5563. Fax: 075-753-2286. Email: alumni*www.adm.kyoto-u.ac.jp (replace * with @). 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 キャンパスライフ支援 学生生活で注意してほしいこと. 学生生活で注意してほしいこと. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生生活で注意してほしいこと 健康管理 在学中の保険加入 住まい・食と購買 アルバイト 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. 京大生としての自覚と責任 SNS利用上の注意 マイカー通学の禁止 自転車運転マナー等について 盗難・置き引きに注意 政治セクト（過激派）、カルト団体などに注意 法令の遵守について 立て看板等の設置について 飲酒に関する注意 危険ドラッグについて 本学からのアルバイト料、旅費の受給について 「悪徳商法」にだまされないために 海外旅行へ行く前に安全性の確認を クレジットカードの利用について 国民年金へ加入しましょう 災害時には自主的な安否確認登録を 地震発生時の対応について 選挙に関する注意 キャンパスライフに潜んでいる様々な危険・あやうさをよく知り、自身がその危険に陥らないように注意することが大切です。その主なものを以下に示します。. 京大生としての自覚と責任. 本学の特徴の一つとして、「自由の学風」があります。ただし、「自由」とは、自分勝手に何をしてもよいということではありません。我々は集団の中で生活しており、生活する基盤となる社会には、法令以外にも規範やスタンダードがあります。社会人、いわゆる大人としての行動の自由とは、社会的規範の遵守が前提です。例えば、差別に関する問題もそのひとつです。民族や地域、障害や性別などの差別は根絶されるべき、ということは皆さんも十分理解しているでしょうし、意図的に差別をする人はいないでしょう。自分ではそのつもりはなくても、相手にとって、差別や不快と感じるような発言や行動もあります。人を思いやる気持ちで接するのはなかなか難しいものです。. 人とのかかわり方や意見交換の方法をしっかりと身につけるとともに、自らの言動に責任を持ち、他人を尊重することにも心がけるようにしてください。. SNS利用上の注意. FacebookやTwitter、LINEなどのSNS（ソーシャル・ネットワーキング・サービス）は、「仲間内だけのもの」と思いがちですが、そこでの発言が他人を傷つけたり、炎上して自分が傷つくこともあります。また、仲間内だけの空間であるという安心感を利用して詐欺（ワンクリック詐欺やフィッシング詐欺など）やウイルスの配布を行う事例も急増していますので、SNSの利用にあたっては、以下の点に注意してください。. むやみに個人に関する情報を公開しない. 写真の掲載は、意図しない位置情報流出の危険性がある. プライバシー、著作権や肖像権の侵害に注意. 発信内容は慎重に、どこで誰に対して公言しても良いことだけを書き込むべき. 偽アカウントやスパムアプリに注意. マイカー通学の禁止. 本学のキャンパスでは、教育・研究の場にふさわしい環境保持および緊急災害時の通路確保、歩行者の安全確保、騒音の防止のため交通規制を実施しており、身体障害者等特別な事情のあるものを除き、マイカー通学は原則として禁止となっています。通学に当たっては、公共交通機関を利用するようにしてください。. 自転車運転マナー等について. 大学構内や周辺道路において、自転車による接触事故や衝突事故が多発しています。また、大学生の自転車運転マナーが悪いといった苦情が絶えません。自転車に乗る場合は、常に安全運転を心掛け、周囲に配慮した運転を怠らないようにしてください。. 1.違反と罰則. 自転車安全利用五則を守りましょう。. 自転車は、車道が原則、歩道は例外. 車道は左側を通行. 歩道は歩行者優先で、車道寄りを徐行. 安全ルールを守る. 子供はヘルメットを着用. 京都府道路交通規則の一部改正に伴い、自転車運転中の携帯電話等の使用やイヤホン、ヘッドフォン等で音楽等を聞くことに対し、罰則規程（5万円以下の罰金）が適用されることとなりました。また、改正道路交通法の施行に伴い、自転車運転中に危険なルール違反（信号無視、一時不停止、酒酔い運転、ブレーキ不良自転車運転等）を繰り返すと、「自転車運転者講習」の受講を受けることになります。. 自転車運転者講習の対象となる危険行為（PDF） 2.自転車保険の加入義務化. 2018年4月1日から、京都市内・京都府内で自転車に乗る場合、自転車損害賠償保険等（以下「保険」という）の加入が義務づけられました。. 最近の自転車事故による賠償事例では、約9500万円の支払いが命ぜられた事例があります。自転車を利用中の事故により、他人にけがをさせてしまった場合などに備え、相手の生命または身体の損害を補償できる保険に加入しましょう。. 以下のWebサイトに保険加入義務化の詳細や、保険加入チェックシートがありますので、参照してください。. 自転車保険 | 京都市サイクルサイト 3.交通事故相談. 交通事故にあったとき、無料で相談できる窓口が京都府に設置されています。損害賠償・示談・保険請求など専門の相談員がアドバイスし、また必要により弁護士にも無料で相談できます。. 相談・問い合わせ先. 京都府交通事故相談所（上京区下立売通新町西入ル（京都府庁旧本館1階））. Tel： 075-414-4274. 面接相談時間. 9時00分～11時30分、13時00分～16時30分（土日曜・祝日・年末年始は休みです。）. 4.駐輪場. 自転車、バイクは、歩行者の安全・避難経路確保等のため、駐輪場が指定されていますので、必ず所定の場所に置くようにしてください。なお、長期間の放置自転車は強制的に撤去されることがあります。. 盗難・置き引きに注意. 本学では、体育館、部室、グラウンド、講義室、研究室など、様々な場所で盗難・置き引きが発生しています。大学は、学生や教職員だけでなく、外部の人も多数出入りしています。貴重品等は常に身に付け、わずかな時間であっても自分の持ち物から目を離さないよう十分注意してください。. また、自転車は「鍵をかけていない」または「鍵が1つしかない」ものが特に狙われています。盗難防止のために備え付けの鍵だけでなく、ワイヤー錠等を併用し、2個以上の鍵（ツーロック）をかけましょう。短い時間であっても必ず施錠し、万が一に備え、防犯登録を必ず行いましょう。. 政治セクト（過激派）、カルト団体などに注意. 思想、信教の自由は憲法で保障されています。しかし、世の中にはそのことを逆手に取り、嘘や違法行為を勝手な解釈で正当化する反社会的な政治セクト（過激派）やカルト団体も存在します。. 政治セクト（過激派）による勧誘. 不法行為も辞さない政治セクト（過激派）が、大学にも潜んでいます。彼らは、学習会系サークルや学生自治会の名を騙って勧誘することがあります。学習会への誘いや、クラス討論、さまざまな署名集めなどであなたに近づき、個人情報を集めたりすることもあります。彼らはかつて力ずくで反対意見を屈服させていた「過激派」の正体を今は隠し、学生運動や市民運動の体裁をとって「戦争反対」などを掲げ、若者の正義感に訴えてきます。しかし、反社会的な政治セクト（過激派）であるというその本質に今も変わりはありません。こういった輩に騙されないよう注意してください。. カルト団体による勧誘. また、カルト団体による違法な勧誘、脅迫、献金強要等のトラブルも発生しています。その手口は、呼吸法、自己啓発セミナー、ボランティア、国際交流などのサークル活動への勧誘やアンケート調査などと言って声を掛け、世間話や趣味などの話題から親しくなり、住所や電話番号、SNSのアカウントといった個人情報を聞き出し、セミナーや合宿等に参加するようにしつこく勧めるというケースが多く見られます。. いったんこういった団体に入ってしまうと、その団体のさまざまな活動にかり出され、時間と労力を浪費し、人間関係が崩壊し、授業にも一切出られず大学を除籍となるなど健全な学生生活を送ることができなくなります。. 被害にあわないために. ひとりでいる時に声をかけられるケースが多発しています。トラブルに巻き込まれないためには、その人が何のために近づいてきたのかを確認してください。名前を言わなかったり、目的を言わなかったり、曖昧にぼかす時は注意してください。また、初対面の人には絶対に個人情報を教えないこと、安易にSNSでつながったりしてはいけません。しつこく勧誘されてもきっぱりと断る勇気が大切です。勧誘時の団体名や活動説明と実際の団体名や活動実態が異なる団体は特に注意してください。おかしいと思ったら、すぐに友人や家族、大学に相談しましょう。社会情報がみな誤りであり、この団体の言うことだけが正しいなどと情報操作・情報規制をされたらすぐ逃げてください。. なお、不審な勧誘を見かけたり、自分が勧誘を受けた時は、すぐに教育推進・学生支援部厚生課に相談してください。. 法令の遵守について. 近年、本学においては、性犯罪や薬物乱用等により逮捕者が出るなど、学生の不祥事が連続して発生しています。これらの犯罪行為は、法律により厳罰に処せられるとともに、大学においても放学または停学といった厳しい懲戒処分を行うことになります。これらの犯罪行為は、被害者やその家族はもとより、皆さん自身の今後の人生にも大きな影響を及ぼすことになります。学生の皆さんは、日常のさまざまな行動の中で、人権の尊重や法令遵守の重要性を自覚し、良識ある行動をとるようにしてください。. 京都大学通則. 第32条　学生の本分を守らない者があるときは、総長は懲戒する。. 前項に規定する懲戒の必要がない学生についても、当該学生の所属する学部長が必要と認めたときは、当該学部長が、厳重注意その他の教育的措置を行うことができる。. 懲戒に関し必要な事項は、別に定める。. 第33条　懲戒の種類は、次のとおりとする。. 譴責. 停学. 放学. 立て看板等の設置について. 京都市より、本学の外構周辺に公衆に表示される形で設置されている立て看板等について、屋外広告物に該当し、「京都市屋外広告物等に関する条例」（以下「条例」という。）に抵触しているほか、当該看板等が道路にはみ出すと不法占用になり、さらに、強風等による倒壊も危惧され、通行に危険を及ぼす. ことにもなりかねない旨の通知がありました。. 条例では、擁壁に屋外広告物を表示してはならないこと（第5条）、屋外広告物を表示しようとする者は市長の許可を受けなければならないこと（第9条）等が規定されていますが、本学の外構周辺に設置されている立て看板等はそれらの規定に違反しているとのことです。. 京都市では、京都らしい景観の保全・継承のため、多くの市民・事業者の理解・協力の下、条例に基づいて屋外広告物の適正化が進められています。また、本学周辺の住民の方々からも、キャンパス周辺の立て看板は市の景観政策に反している、歩行者に危険である等のご指摘を受けています。. つきましては、上記のことを踏まえ、学生各位において、条例等の法令を遵守するようお願いします。. 飲酒に関する注意. 新入生歓迎の行事やコンパなどの集まりで、アルコールが出される機会がありますが、未成年者の飲酒は法律違反であり、未成年者は勧められても飲まないこと、未成年者へ飲酒を勧めないことを厳守してください。. また、成年に達した学生についても、急性アルコール中毒に注意してください。短時間に大量のアルコールを摂取すると、自力で立てないほどの運動障害を起こしたり、昏睡状態になることがあります。最悪の場合は呼吸停止や急性心不全が起き、死につながったり、蘇生しても重篤な後遺症が残ること. があります。また、こうした症状に至らなくても、足下がふらついた状態で転倒したり、嘔吐により窒息死した例もみられます。. 他大学では、クラブ・サークル等の飲み会で、急性アルコール中毒により、学生が死亡するという事故が発生しています。本学においても、急性アルコール中毒等により重篤状態となり、病院に搬送され一歩誤れば死に至る危険な状態となる事件が発生しています。. 飲酒に当たっては、次の点を厳守してください。. 未成年者の飲酒は法律違反であり、勧めない、勧められても飲まない。. 成年者でも、イッキ飲み等の危険な飲酒はしない、他人にさせない。. 体質的にアルコールを受け付けない人に飲酒を勧めない。. 飲酒をしたら、自動車・バイク・自転車の運転をしない。. 万が一、酔いつぶれた者が出た場合は、一人にしないで責任をもって介抱（衣服を緩め、横向きに寝かす等）すること。名前を呼んだり身体をゆすっても反応せず、大きないびきや呼吸を時々しかしないなど、おかしいと思ったら、直ちに救急車を呼ぶこと。. 危険ドラッグについて. 薬物使用は、以下のように、精神と身体に悪影響を及ぼし、人間関係の崩壊など、本人だけでなく社会全体に害悪をもたらします。. 近年では、危険ドラッグの使用が引き金となった事件や死亡事故が多発するなど極めて憂慮すべき状況です。この危険ドラッグは、強力な依存性があり、また強い精神毒性があるため、使用すると急性症状による暴力事件や交通事故を引き起こしたり、慢性的な精神疾患にかかることがあります。. 安易に使用した結果、本人だけでなく周囲にも大きな影響を及ぼすことがあるため、一時的な興味本位で使用することがないように十分注意してください。. 危険ドラッグとは、「ハーブ」、「お香」、「芳香剤」などと用途を偽装したり、「合法ドラッグ」、「合法ハーブ」などと称して販売されている薬物です。. 本人の精神や身体への悪影響. 自分の意志では止められない. 幻想や妄想が表れ、重大犯罪を引き起こす. 友人・家族等の人間関係の崩壊. 法律で禁止されており、厳罰をうけること. 薬物乱用については下記のWebサイト等に詳しく掲載されています。. 麻薬・覚せい剤乱用防止センター 危険ドラッグについて （厚生労働省パンフレット）. マンガ「危険ドラッグゼロの世界」 （京都府発行）. 本学からのアルバイト料、旅費の受給について. 本学からのアルバイト料（OA・TA・RA給与. （注）. 、謝金）、旅費（交通費等）の受給にあたって、以下の行為は. 不正受給に該当します。. （注）OA： Office Assistant、TA： Teaching Assistant、RA： Research Assistant. 架空アルバイト・カラ出張. 実際に勤務・従事していない業務に対してアルバイト料を受け取ることや、実際に行っていない出張に対して旅費を受け取ること。なお自身の学修のために指導教員等のフィールド活動に同行するようなケースでは、アルバイト料は支給されません。. 事実と異なるアルバイト料・旅費. 勤務日数や時間数は同じでも、勤務した日が異なるなど、勤務・業務に従事した事実と異なる勤務表を提出し、アルバイト料を受け取ることや、出張先は同じだが、出張日が異なるなど、出張した事実と異なる旅費報告書を提出し、旅費を受け取ること。. 水増し請求. 安価な交通手段を利用して出張したにもかかわらず、実際よりも高額な旅費を受け取ること。. 研究室への還流行為. 大学から支給されるアルバイト料、旅費の全部又は一部を、研究室における他の用途に使用することや、他の学生に再分配することなどを目的に、教員の指示による回収に応じること。. 大学から支給されるアルバイト料、旅費は、原則として学生本人の銀行口座に振り込まれます。研究室等がこのような還流行為をすることは禁止されています。. 上記の行為を求められた場合や、事務手続きが分からない場合は、まずは、所属する学部・研究科等の事務室・事務職員にご相談ください。. 不正経理の事実を知った場合等の本学の通報窓口. 学内の窓口. 京都大学公正調査監査室 学外の窓口. 京都大学コンプライアンスホットライン窓口（はばたき綜合法律事務所 本学顧問弁護士） 通報に関する詳細については、各URLを参照ください。. 通報したことを理由として、不利益な取り扱いを受けることはありません。. 本学からアルバイト料、旅費を受給される学生のみなさまへ（PDF） 「悪徳商法」にだまされないために. 学生をねらった悪徳商法が多発しています。これらの悪徳商法は、学生の社会的経験の少なさなどにつけこみ、「楽して儲かる」といった気持ちを起こさせ、時には脅迫まがいの方法で引き込んだりします。次にあげるような悪徳商法の他にも巧妙な新しい手口もでてきていますので、くれぐれも注意してください。. キャッチセールス. 街で「アンケートに答えてください……」などと呼び止められ、営業所に連れて行かれて、高価な化粧品や会員権などの契約をさせられます。. あいまいな態度をとらず、はっきり断ること！. アポイントメントセールス. 「格安で海外旅行ができ、レジャー施設も安く利用できる」などと電話で営業所に呼び出され、実際には別の商品（ビデオ教材等）とのセット販売で結局高額な商品を買わされることになります。. 見知らぬ人からのうまい話に要注意！. マルチ商法. 「人を紹介するだけで、どんどん収入がふえる」などのうまい話で誘われます。商品を買って会員になり、知人や友人を紹介して商品を買ってもらうと、リベートがはいり、さらに孫・ひ孫からのリベートがはいるというものです。手軽にできるアルバイトと思って契約したものの、結局残ったのは買い込んだ商品と借金だけということにもなりかねません。. うまい話はありません。もうけ話には注意しましょう！. 振込め詐欺. 電話で「オレだよ、オレ」と言って息子や孫を装い家族に交通事故示談名目やサラ金等借金返済名目で現金を騙し取る手口や、警察官や弁護士などを名乗り示談金を騙し取る、いわゆる「振込め詐欺」が多発しています。本当の出来事かどうか、振込む前に家族と学生は相互に確認をしてください。. すぐに振り込まない。一人で振り込まない！. 架空請求詐欺. 実際には利用していない有料サイトの利用料金等の名目で金銭をだまし取る架空請求詐欺事件が増加しています。学生の皆さんがこうした被害にあわないよう、次のようなことに心掛けてください。. 身に覚えのない請求は無視する。（請求のはがきやメールは保管しておく）. 指定された連絡先には絶対に連絡しない。. 迷惑メールの受信拒否設定をする。. 一人で判断せずに警察や家族、周囲の人に相談する。. 若年層が陥りやすい消費者トラブル事例（経済産業省）（PDF） 海外旅行へ行く前に安全性の確認を. 夏季休業等を利用して海外旅行に行く機会もありますが、特定の国・地域によっては治安の悪化等により、渡航を自粛したり、特別の注意が必要な場合があります。海外旅行へ行く前に旅行先の安全性を詳しく調べるようにしましょう。また、万一の場合に備え、保険に加入することを勧めます。. これらの安全情報は、外務省から提供されていますので活用してください。また旅行社でも確認するようにしてください。. 外務省 海外安全ホームページ 海外渡航に関する安全対策手引き なお、海外旅行へ行く場合は、事前に所属の学部・研究科等の教務担当掛へ「海外渡航届」を必ず提出してください。. クレジットカードの利用について. 学生証ですぐ借りることができる学生ローン、また、サインするだけで手軽にショッピングやレストラン等の利用ができるクレジットカードを安易に利用すると、その返済に追われ学生生活の継続が危ぶまれることになります。. 本学では、「小口短期貸付金」という無利子の短期貸付制度がありますので、病気、不慮の事故、送金の延着等により、急に出費が必要となった場合は、教育推進・学生支援部学生課奨学掛で相談してください。. 国民年金へ加入しましょう. 国民年金は、高齢や不慮の事故などによって私たちの生活が損なわれることのないよう、前もってみんなで保険料を出し合い、経済的にお互いを支え合う制度で、日本に住む20歳から60歳までのすべての人が加入することになっています。本学の学生諸君も20歳になれば必ず国民年金に加入してください。. なお、加入手続きは、住民票を登録している市区町村の国民年金担当窓口で行ってください。. また、収入がない学生のために「学生納付特例制度」があります。詳しくは、同じく住民票を登録している市区町村の国民年金担当窓口に問い合わせてください。. 災害時には自主的な安否確認登録を. 地震や台風などの大規模な自然災害、パンデミック発生などの緊急時に、皆さんの安否を把握することは、授業や試験の再開、被災した皆さんの生活・修学支援、災害対応方針の検討等極めて重要なことです。. 皆さんは、災害等が発生すれば、自分のスマートフォンやパソコン、タブレット端末から「安否確認システム」にアクセスして、自主的に自身の安否状況を登録してください。なお、システムにより登録がない場合、所属部局が、直接、電話やメール、友人等への問合せなどの方法により、安否を確認することになります。. 地震発生時の対応について. 下記に「地震対応マニュアル学生用」を掲載しています。地震の知識や発生時の対応、自宅での備え、家族等との安否確認方法等を記載していますので、参考にしてください。地震を感じたり、緊急地震速報を受信したときは、まず、自分自身の身の安全を確保し、火の始末や家族の安全確認等を行った後、周囲の方と協力して消火、救出、救護、避難活動を実施してください。そして、落ち着いてから自主的に「安否確認システム」を使用して自身の安否状況の登録を行ってください。. 地震対応マニュアル（PDF） Earthquake Safety Manual （PDF） 選挙に関する注意. 公職選挙法等の一部を改正する法律が平成28年6月19日から施行され、選挙権年齢等の満18歳以上への引下げが実施されました。皆さんの中には、投票だけでなく、選挙運動にも積極的に関わっていきたいと考えている方もいるかもしれません。ただし、注意が必要です。公職選挙法では、選挙運動についてさまざまな制限があり、違反した場合、罰則等もあります。法令を遵守し、適切な行動をとるには、ルールの正しい理解が不可欠です。. 1.「選挙運動」とは？. 選挙運動とは「特定の選挙について、特定の候補者の当選を目的として、投票を得または得させるために直接または間接に必要かつ有利な行為」と解されています。. 選挙運動は、選挙ごとに決められた選挙運動期間（選挙の公示日または告示日に候補者が立候補の届出をした時から投票日の前日までの間）内にしか行うことができません。. 候補者への投票を呼びかけるチラシ（選挙運動用ビラ）を配ることは、他の者から指示されたとおりに機械的に行ったとしても一般的には選挙運動になります。また、配れる選挙も限られ、配れる場所も演説会場内や街頭演説の場所等に限られるため、例えば、チラシを選挙人の家のポストに入れるような配り方はできませんので、注意が必要です。. また、満18歳未満の者は選挙運動を行うことはできず、誰であっても、満18歳未満の者を使用して選挙運動をすることはできません。. 2.選挙運動は基本的に「ボランティア」. 選挙応援のためのアルバイトには特に注意が必要です。例えば、チラシを配る者は、報酬を受け取ることはできません。公職選挙法では、「選挙運動は原則として自発的に無報酬で行うものである」とされており、選挙運動に従事する者に対する報酬は、選挙運動に関する事務に従事する者、選挙運動用自動車での車上運動員や手話通訳者に対するものを除き、買収罪に当たることとなります。. なお、公職選挙法に規定されている範囲内で交通費などの実費を支払うことはできるため、こうしたものを受け取ることは可能です。. 3.制度をきちんと理解しよう. 上記のほか、インターネットの利用や飲食物の提供を受けることの可否など注意すべきことはたくさんあります。わからないことがあるときは、みなさん自身がインターネット等を利用して調べるなど、制度をただしく理解しようとする努力が必要です。. 【参考】. 総務省｜高校生向け副教材「私たちが拓く日本の未来」 1. と2. は総務省Webサイトに掲載の教材「私たちが拓く日本の未来」を一部抜粋して作成しました。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム アクセス 食堂、レストラン、ショップなど. 食堂、レストラン、ショップなど. 2階層目ページ用メニュー. 採用情報 アクセス 見学にいらっしゃる方 各キャンパスについて（吉田・宇治・桂） 各学部・研究科、センター等の場所について 食堂、レストラン、ショップなど ATMコーナー キャンパスマップ（ダウンロード用） キャンパス間の移動（学内者向け） 連絡バスの運行 本学にある、食堂やレストラン、ショップなどを紹介しています。. ※ 営業時間が急遽変更されることがあります。ご利用の際は、事前に各店舗の営業状況をWebサイトにてご確認いただくようお願いします。. ※ 大学休暇期間中は営業時間が短縮される場合があります。. 吉田キャンパス 本部・西部構内 吉田南構内 医学部構内 病院構内 北部構内 宇治キャンパス 桂キャンパス 吉田キャンパス. 本部・西部構内. 店舗名. 概要. 場所. 営業時間・電話番号 等. フレンチレストラン ラ・トゥール（La Tour）.  . 落ち着いた雰囲気の中でゆっくりとくつろげるフレンチレストランです. 吉田キャンパス本部構内. 百周年時計台記念館 1階. 本部・西部構内マップ ［3］. 「ラ・トゥール」Webサイト Tel：075-753-7623. カフェレストラン カンフォーラ. 百周年時計台記念館の正面にあり、「楠」の学名にちなんだカフェレストランです. 吉田キャンパス本部構内 正門前. 本部・西部構内マップ ［2］. 営業時間：. 京都大学生協Webサイト をご覧ください. 《カンフォーラ》. Tel：075-753-7628. 《京大ショップ》. Tel：075-753-7630. 《時計台生協ショップ》. Tel：075-753-7630. 《中央食堂》. Tel：075-752-0832. 《西部食堂》. カフェテリア・ルネ. Tel：075-752-9271. ショップ・ルネ. 各コーナーごとに電話番号は. こちらのページ をご覧ください。. 京大ショップ. 京都大学に関する書籍やオリジナルグッズの販売コーナーです. 吉田キャンパス本部構内. 百周年時計台記念館 1階. 本部・西部構内マップ ［3］. 時計台生協ショップ. 学生や教職員の日々の生活を支える各種サービスを提供しています. 吉田キャンパス本部構内. 百周年時計台記念館 地階. 本部・西部構内マップ ［3］. 中央食堂. 本部構内にある食堂で、日々たくさんの学生たちが利用しています。. 吉田キャンパス本部構内. 総合研究8号館. 本部・西部構内マップ ［59］. 西部生協食堂. （ショップ、カフェテリアルネ）. ショップコーナーは、書籍やパソコンなどを取り扱っています. 吉田キャンパス 西部構内. 本部・西部構内マップ ［79］. タリーズコーヒー京都大学時計台店. ゆっくりとくつろげる憩いの学内カフェです. 吉田キャンパス本部構内. 百周年時計台記念館 地階北側. 本部・西部構内マップ ［3］. 営業時間：詳しくは. 店舗のWebサイト をご覧ください. ATMコーナー. 本部構内にある、三井住友銀行のATMです. 吉田キャンパス本部構内. 百周年時計台記念館地階ピロティ内 （タリーズコーヒーとコピールームの間）. 本部・西部構内マップ ［3］. 営業時間：詳しくは「. ATMコーナー 」のページをご覧ください. 吉田南構内. 店舗名. 概要. 場所. 営業時間・電話番号 等. 吉田食堂. 昼休みには、学部１回生を中心に非常に多くの人が利用する食堂です. 吉田南構内マップ 　吉田グラウンドの南. 営業時間：. 京都大学生協Webサイト をご覧ください. 《吉田食堂》. Tel：075-761-9557. 《吉田ショップ》. Tel：075-752-1587. 吉田ショップ. 食品、日用品、書籍を店内に展開販売しています. 吉田南構内マップ ［89］の南（生協店舗）. 医学部構内. 店舗名. 概要. 場所. 営業時間・電話番号 等. 南部喫茶 ブリュッケ.  . ドリアやオムライスの軽食メニューや日替わりランチを提供しています. 吉田キャンパス 医学部構内. 医学部構内マップ ［食堂・生協店舗］. 営業時間：. 京都大学生協Webサイト をご覧ください. 《ブリュッケ・南部食堂》. Tel：075-753-7635. 《南部購買》. Tel：075-752-1586. 南部食堂. 煮物、炒め物、揚げ物、焼き物など家庭的なメニューを中心にお好きな単品メニューを取り合わせてのカフェテリアスタイルです. 南部購買. 学生や教職員の日々の生活を支える各種サービスを提供しています. 病院構内. 医学部附属病院内のレストランおよび売店等については、以下のページをご覧ください。. レストラン・売店 等｜院内マップ（京都大学医学部附属病院Webサイト） 北部構内. 店舗名. 概要. 場所. 営業時間・電話番号 等. 北部食堂. 北部食堂では注文を受けてその場で調理するオーダーコーナーが名物となっています. 北部構内マップ ［食堂・生協店舗］. 営業時間：. 京都大学生協Webサイト をご覧ください. 《北部食堂》. Tel：075-753-7633. 《北部食堂》. Tel：075-722-0706. 北部購買. お弁当・お菓子、文具、日用品に加え、PC・ソフトや実験機器・用品も扱っています. 宇治キャンパス. 店舗名. 概要. 場所. 営業時間・電話番号 等. 生協食堂. 宇治キャンパスの食生活を支える食堂です. 宇治キャンパス構内マップ ［76］. 営業時間：. 京都大学生協Webサイト をご覧ください. 《宇治食堂》. Tel：0774-38-4385. 《宇治購買》. Tel：0774-38-4388. 宇治購買. カフェレストランきはだ. 懐かしの洋食レストラン。ランチには新鮮野菜たっぷりのサラダブッフェもあります. 宇治キャンパス構内マップ ［26］. 「カフェレストランきはだWebサイト Tel：0774-31-7111. 桂キャンパス. 店舗名. 概要. 場所. 営業時間・電話番号 等. カフェテリアSelene（セレネ）. バランスの取れた健康的な食生活をサポートするカフェテリアです。展望デッキもご利用いただけます。. Bクラスター 福利棟2階. 桂キャンパス構内マップ ［16］. 営業時間：. 京都大学生協Webサイト をご覧ください. 《セレネ、アルテ》. Tel：075-383-7302. 《生協ショップ（A～Cショップ共通）》. Tel：075-383-7300. カフェArte（アルテ）. お手軽ランチやカフェメニューなどがあるカフェテリアです. 生協ショップ. AおよびCショップは食料品中心、Bショップでは吉田地区の生協各店舗で提供している様々な商品・サービスの大半網羅しています. 【Aショップ】. AクラスターA2棟. 桂キャンパス構内マップ ［2］. 【Bショップ】. Bクラスター 福利・保健管理棟. 桂キャンパス構内マップ ［16］. 【Cショップ】. Cクラスター C3棟. 桂キャンパス構内マップ ［21］. カフェテリア ソレイユ. こだわりスパイスカレーとふわとろ卵のオムライスの店です. Cクラスター C3棟1階. 桂キャンパス構内マップ  ［24］. 営業時間. 平日： 11時00分～14時00分、17時00分～20時00分. 土曜日： 11時00分～14時00分. 定休日： 日曜日・祝日・お盆・年末年始. 自販機（飲料、パン、軽食、お菓子）は無休. ATMコーナー. 桂キャンパス構内にある三井住友銀行ATMです. Bクラスター 福利・保健管理棟2階. 桂キャンパス構内マップ  ［16］. 営業時間：詳しくは「. ATMコーナー 」のページをご覧ください. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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from the President","Operations and future plans","History","Facts and Figures","Awards and honors","University profile","Living in Kyoto","Public relations","Employment","Human rights","Health, safety, and environment","Study at KyotoU","Education and admissions","Career paths","Visa\/tuition\/scholarships\/housing","University facilities","FAQ","Research &amp; Collaborate","Research news and departments","Policies","Collaboration","Research support and scholarships","Visa\/Housing\/Daily Life","About","Research compliance &amp; ethics","Global KyotoU","International strategy","Global profile","Global partners","International risk &amp; crisis management","Global University Social Responsibility","KyotoU International Concierge and information resources","Current Students &amp; Staff","Alumni","Directions","Contact","日本語 ホーム","中文简体 首页","中文繁体 首頁","한국어 홈페이지","NEWS","Global Engagement","Education","日本語","Current Students &amp; Staff","Wild &amp; Wise Collaborative Learning 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Twitter","","","","","Safety Confirmation System"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:44:33 UTC 2022","fresh_date_dt":"Wed Jun 08 17:24:09 UTC 2022","referrer_id_s":"_sUDfCU8j7Mo","language_s":"en","size_i":69018,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085704377008128,"text_t":"\"Short-term Intern Program in Primatology and Wildlife Science\" held online (1-9 March 2022) | KYOTO UNIVERSITY. Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home NEWS \"Short-term Intern Program in Primatology and Wildlife Science\" held online (1-9 March 2022). Global Engagement Education \"Short-term Intern Program in Primatology and Wildlife Science\" held online (1-9 March 2022). In other languages. 日本語 ターゲット. Current Students & Staff Published on. 2022\/04\/13. From 1 to 9 March 2022, the Primate Research Institute (PRI) and the Wild Research Center (WRC) jointly held the \"Short-term Intern Program in Primatology and Wildlife Science\" online as a Kyoto University Wild & Wise Collaborative Learning Program. The ten participants were short-term exchange students from nine countries: Singapore, India, the United States, Germany, the Netherlands, Australia, Malaysia, Chile, and China. The program introduced cutting-edge research in primatology and wildlife science through videos and virtual tours of laboratories and observation sites in locations such as the Arashiyama district of Kyoto and Yakushima. Activities included participation in the CICASP Seminar in Science Communication, organized by PRI's Center for International Collaboration and Advanced Studies in Primatology (CICASP). Students, together with peers from Kyoto University, listened to research presentations spanning a wide range of fields, took part in debates on primatology and wildlife science topics, and received science communication training focused on these same topics. The final day featured the 17th International Symposium on Primatology and Wildlife Science, where the interns delivered presentations. There were also virtual culture tours of Kyoto and Inuyama, the home of PRI, where the participants learned about local cultural events and historical buildings, and experienced day-to-day life in these cities. The program received highly positive reviews afterwards from many of the participants. Some commented that, in addition to enriching their knowledge of research activities and of the actual studies being conducted, the program enabled them to interact with other students with similar academic interests and to consider their own future from an expanded perspective. First day of the program. Orientation. Lecture by PRI Associate Professor Michael Huffman (currently affiliated with the Wildlife Research Center)   . Another scene from Associate Professor Huffman's presentation. Related link. Wild & Wise Collaborative Learning Programs Related departments & centers. Center for the Evolutionary Origins of Human Behavior (EHUB) Wildlife Research Center (WRC) Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. Published on. 2022\/06\/03. KIZUNA hosts \"Manga and Anime Presentation Contest\" online (24 May 2022). Published on. 2022\/04\/22. Swiss President Ignazio Cassis visits Kyoto University to Deliver Lecture. Published on. 2022\/04\/22. 2021 Kyoto iUP Preparatory Course concludes with ceremony. Published on. 2022\/04\/18. \"AY2021 IAfP Short-term online exchange programs – Addis Ababa University – \/ AY2021 Online lecture series on lifelong sciences\". More news 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home About Public relations. Public relations. サイドメニュー（英語）. Message from the President Operations and future plans History Facts and Figures Awards and honors University profile Living in Kyoto Public relations Kyoto University Introductory Brochure Kyoto University Annual Report Admissions Guide for International Applicants Handbook for International Students Handbook for International Researchers Housing Guidebook for International Researchers and Students MANGA Kyoto University (external link) Social Media Terms and Conditions Kyoto U Research News Safe Travel Abroad (leaflet for students, faculty, and staff traveling abroad) Safety Guide for Students Engaged in Overseas Fieldwork Videos Employment Human rights Gender equality Health, safety, and environment 3階層目以降ページ用メニュー（英語）. Kyoto University Introductory Brochure. Kyoto University Annual Report. Admissions Guide for International Applicants. Handbook for International Students. Handbook for International Researchers. Housing Guidebook for International Researchers and Students. MANGA Kyoto University (external link). Social Media Terms and Conditions. Kyoto U Research News. Safe Travel Abroad (leaflet for students, faculty, and staff traveling abroad). Safety Guide for Students Engaged in Overseas Fieldwork. Videos. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"yE1Ll9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/about\/procurance\/sikaku","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":52,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["ja","about","procurance"],"url_file_name_s":"sikaku","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["入札への参加資格について | 京都大学"],"description_txt":["1．物品の製造・販売および役務の提供等\n\n国立大学法人京都大学の所管する物品の製造・販売および役務の提供等について契約を締結する場合、一般競争参加資格を得ようとする者の申請方法等については、次のとおりです。\n\n本学においては国の一般競争参加資格制度について利用する方法を採用していますので、最寄りの国の機関において競争参加資格（全省庁統一資格）の申請を行ってください。申請後認定された者は、国立大学法人京都大学において認定されたものとして取り扱います。\n\nまた、以下のホームページからも申請が行えます。\n\n統一資格審査申請・調達情報検索サイト\n\n問い合わせ先\n\n〒606-8501 京都市左京区吉田本町\n京都大学 財務部 経理課 経理総括掛\nTel： 075-753-2131\n\n2．建設工事・建設コンサルタント業務等\n\n物品の製造・販売および役務の提供等とは別に取り扱いますので、以下へお問い合わせください。"],"last_modified":"Wed Jun 08 16:44:39 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":7161995791492234395,"exact_signature_unique_b":true,"fuzzy_signature_l":-5523957750594464146,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["入札への参加資格について"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","1．物品の製造・販売および役務の提供等","2．建設工事・建設コンサルタント業務等","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["問い合わせ先"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1866,"text_t":"入札への参加資格について | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 調達情報 入札への参加資格について. 入札への参加資格について. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 物品・役務等の調達情報について 建設・工事等 資産の有効活用（不動産の売却・貸付等） 契約に係る情報等の公表について 調達に関する本学の取り組みについて 【取引先の皆様へ】本学への取引先登録、誓約書の提出および消費税の適格請求書発行事業者の登録申請について 本学における発注・検収・検査について 入札への参加資格について 電子メールによる振込通知サービスについて 入札情報（文部科学省共済組合京都大学支部） 取引金融機関の選定（メインバンク契約）について 法人としての取組 男女共同参画について 公表事項 情報公開 1．物品の製造・販売および役務の提供等. 国立大学法人京都大学の所管する物品の製造・販売および役務の提供等について契約を締結する場合、一般競争参加資格を得ようとする者の申請方法等については、次のとおりです。. 本学においては国の一般競争参加資格制度について利用する方法を採用していますので、最寄りの国の機関において競争参加資格（全省庁統一資格）の申請を行ってください。申請後認定された者は、国立大学法人京都大学において認定されたものとして取り扱います。. また、以下のホームページからも申請が行えます。. 統一資格審査申請・調達情報検索サイト 問い合わせ先. 〒606-8501 京都市左京区吉田本町. 京都大学 財務部 経理課 経理総括掛. Tel： 075-753-2131. 2．建設工事・建設コンサルタント業務等. 物品の製造・販売および役務の提供等とは別に取り扱いますので、以下へお問い合わせください。. 〒606-8501 京都市左京区吉田本町. 京都大学 施設部 施設企画課 施設契約掛. Tel： 075-753-2308. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 「アジアと世界の持続可能な発展に関する国際合同大学院生セミナー」を実施しました. 国際交流 教育 「アジアと世界の持続可能な発展に関する国際合同大学院生セミナー」を実施しました. 公開日. 2022年04月01日. 　経済学研究科では、2022年2月28日～3月12日の約2週間、部局間交流協定校であるタイのタマサート大学（経済学部、地域研究研究所、政治学部）、カセサート大学（経済学部）、チェンマイ大学（経済学部）とインドネシアのガジャマダ大学（経済経営学部）から短期交流学生計17名を受け入れる「Kyoto Winter School～アジアと世界の持続可能な発展に関する国際合同大学院生セミナー」をオンラインで行いました。このプログラムはワイルド&ワイズ共学教育受入れプログラム事業の一環として実施されました。. 　本プログラムに参加した留学生は、東アジア持続的経済発展研究コースを中心とする本研究科の学生とともに本学および協定校の教員の講義に参加し、経済発展と持続可能性について活発な議論を行い、学習を深めました。また、プログラム中盤では日本および各国の都市の様子、伝統行事、食文化等、多岐に渡るテーマの映像や写真を用いての異文化交流が行われました。最終日には大学院生による合同のワークショップが開催され、盛況のうちに終了しました。. 　講義とディスカッション、異文化交流、ワークショップを通じて得た知識や経験が、本学の学生および留学生の相互の学びに繋がりました。. 小林舞 学際融合教育研究推進センター特定助教による講義. 学生によるワークショップでの集合写真. 関連リンク. ワイルド&ワイズ共学教育受入れプログラム事業 関連部局. 経済学部・経済学研究科 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home About History. History. サイドメニュー（英語）. Message from the President Operations and future plans History Chronological table Facts and Figures Awards and honors University profile Living in Kyoto Public relations Employment Human rights Gender equality Health, safety, and environment Historical sketch. Kyoto Imperial University was founded by imperial ordinance on 18 June 1897, the second university to be established in Japan. Within ten years of the founding of the University, the Colleges of Science and Engineering, Law, Medicine, and Letters were opened. In July 1914 the College of Science and Engineering was divided into the College of Science and the College of Engineering, giving the University five Colleges. In accordance with the promulgation of the Imperial University Law, the Colleges were reorganized in February 1919 to form the Faculties of Law, Medicine, Engineering, Letters, and Science, and in the following May the Faculty of Economics was established. In the early period of the University, presidents were chosen by the Ministry of Education. However, faculty members increasingly desired more autonomy. In 1915, opinions from the faculties were considered in the selection of the new president for the first time, and eventually, in 1919, a system for the election of presidents by the faculty members themselves was introduced. In 1923 the Faculty of Agriculture was established. As the demand for education increased, so did the number of students, and a succession of research institutes attached to the University's faculties were founded together with various other establishments. The prewar and war years were a difficult period for the University, but with the end of the war a liberal atmosphere was restored, and in March 1947 the School Education Law was enacted, bringing widespread reforms to the Japanese education system. The aims of this reform were to affirm the principles of equal educational opportunity, and to expand all levels of education, including higher education. In October 1947 Kyoto Imperial University was renamed Kyoto University. In May 1949 the National School Establishment Law was enacted and Kyoto University was reorganized as a four-year instead of a three-year university. The 8th faculty, the Faculty of Education, was also established that year. In May 1949 the Third High School (. Dai San Kou. ) was affiliated to the University. The High School had been providing education in the liberal arts since 1894, and traced its origin to the Seimi-Kyoku, founded in Osaka in 1869, one of the first institutions in Japan to teach western sciences. In September 1949, the Third High School was reorganized to provide general education and renamed the College of Liberal Arts. In April 1953, in accordance with a new national educational regulation, the Kyoto University graduate school system was founded to provide a more systematic postgraduate education in the Graduate Schools of Letters, Education, Law, Economics, Science, Pharmaceutical Sciences, Engineering, and Agriculture. In April 1954 the College of Liberal Arts and Sciences was established, and in July 1955 the Graduate School of Medicine was founded following an amendment of the Education Law. In April 1960, the Faculty of Medicine was divided into two new institutions. One retained the name of the Faculty of Medicine, and the second was named the Faculty of Pharmaceutical Sciences. More recently, in October 1992, the College of Liberal Arts and Sciences was reorganized as the university’s 10th Faculty and named the Faculty of Integrated Human Studies. In recent years, Kyoto University increasingly emphasized graduate-level activities, and has established new graduate schools to tackle emerging problems anticipated to be critical in this century: the Graduate Schools of Human and Environmental Studies, Energy Science, Asian and African Area Studies, Informatics, Biostudies, and Global Environmental Studies. Kyoto University currently has fifteen graduate schools, and traditional academic domains are being reorganized into new disciplines to meet the needs of the coming age. As of 2017, Kyoto University has ten Faculties, eighteen Graduate Schools, thirteen Research Institutes, and twenty-two Research and Educational Centers. Chronological table 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 栄誉等 栄誉 各賞の受賞 叙位・叙勲. 叙位・叙勲. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 京都大学名誉博士 京都大学名誉フェロー 栄誉 各賞の受賞 ノーベル賞 フィールズ賞 ガウス賞 ラスカー賞 チャーン賞 京都賞 日本国際賞 文化勲章 文化功労者 日本学士院会員 恩賜賞・日本学士院賞・日本学士院エジンバラ公賞 紫綬褒章 叙位・叙勲 令和4年度閣議決定分 その他 各賞の授与 歴代総長 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 令和4年度閣議決定分 令和3年度閣議決定分 令和2年度閣議決定分 令和元年度閣議決定分 平成30年度閣議決定分 平成29年度閣議決定分 平成28年度閣議決定分 平成27年度閣議決定分 平成26年度閣議決定分 平成25年度閣議決定分 平成24年度閣議決定分 平成23年度閣議決定分 平成22年度閣議決定分 平成21年度閣議決定分 平成20年度閣議決定分 平成19年度官報掲載分 平成18年度官報掲載分 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム アクセス キャンパスマップ（ダウンロード用）. キャンパスマップ（ダウンロード用）. 2階層目ページ用メニュー. 採用情報 アクセス 見学にいらっしゃる方 各キャンパスについて（吉田・宇治・桂） 各学部・研究科、センター等の場所について 食堂、レストラン、ショップなど キャンパスマップ（ダウンロード用） キャンパス間の移動（学内者向け） 連絡バスの運行 Contents. キャンパスマップ フリーアクセスマップ 散策マップ キャンパスマップ. 2022年4月現在. キャンパス 全体図／吉田キャンパス配置図 吉田キャンパス. 北部構内 本部・西部・吉田南構内 医学部・病院・薬学部構内 宇治キャンパス 桂キャンパス 一般の方にご利用いただける施設（吉田キャンパス）／アクセス 学内の教職員で、マップの利用を希望される方. 教職員ポータルサイト内「ファイル管理」に、キャンパスマップ最新版データが保存されていますので、そちらをご覧ください。. （注意）ここに掲載されている情報の無断転載を禁じます。. 過去に掲載したファイル. こちらからご覧ください。 フリーアクセスマップ. 各キャンパスのバリアフリーマップです。以下のページよりダウンロードください。. フリーアクセスマップ | DRC Kyoto Univ. 散策マップ. 吉田キャンパス. 本学を訪れる方へ多彩な散策コースを提案しています。. 日本語版. PDF（4.96MB） 英語版. PDF（4.78MB） スマートフォンアプリもご利用いただけます。 「ambula map」（株式会社コギト）. iOS版 過去に掲載したファイル. こちらからご覧ください 桂キャンパス. 日本語・英語 併記. 桂キャンパス／Katsura Campus  （1.5MB）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 特色入試 出願状況と選考結果. 出願状況と選考結果. サイドメニュー. 一般選抜 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 出願状況と選考結果 よくある質問 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 京都大学特色入試の出願状況と選考結果は下記のとおりです。. 募集要項に記載の日程に基づいて、随時更新しています。. 令和4年度. 2022年3月22日現在. 学部・学科・コース等. 募集人員. 志願者数. 倍率. 第1次選考. 合格者数. 第2次選考. 合格者数. 最終選考. 合格者数. 男. 女. 合計. 総合人間学部. 5. 35. 7.0. 35. －. 3. 2. 5. 文学部. 10. 60. 6.0. 60. －. 3. 6. 9. 教育学部. 6. 23. 3.8. 12. 6. 2. 3. 5. 法学部. 20. 360. 18.0. 301. －. 14. 8. 22. 経済学部. 25. 67. 2.7. 56. －. 9. 6. 15. 理学部. （数理科学入試）. 5. 69. 13.8. 69. 5. 4. 0. 4. （生物科学入試）. 5. 19. 3.8. 15. －. 2. 4. 6. 医学部. 医学科. 5. 8. 1.6. 6. －. 0. 1. 1. 人間健康科学科. 先端看護科学コース. 20. 45. 2.3. 40. 36. 2. 13. 15. 人間健康科学科. 先端リハビリテーション科学コース. （先端理学療法学講座）. 5. 17. 3.4. 16. 10. 3. 1. 4. 人間健康科学科. 先端リハビリテーション科学コース. （先端作業療法学講座）. 5. 13. 2.6. 13. 11. 0. 3. 3. 薬学部. 薬科学科. 3. 3. 1.0. 2. －. 0. 1. 1. 薬学科. 3. 3. 1.0. 3. －. 1. 0. 1. 工学部. 地球工学科. 3. 11. 3.7. －. －. 1. 0. 1. 建築学科. 2. 3. 1.5. －. －. 0. 1. 1. 物理工学科. 5. 19. 3.8. －. －. 1. 4. 5. 電気電子工学科. 7. 9. 1.3. －. －. 2. 2. 4. 情報学科. 3. 10. 3.3. 4. 2. 1. 1. 2. 工業化学科. 7. 5. 0.7. －. －. 3. 0. 3. 農学部. 資源生物科学科. 3. 20. 6.7. 6. 4. 1. 0. 1. 応用生命科学科. 4. 6. 1.5. 5. 3. 1. 0. 1. 地域環境工学科. 3. 9. 3.0. 6. 4. 1. 2. 3. 食料・環境経済学科. 3. 14. 4.7. 14. 3. 0. 2. 2. 森林科学科. 5. 23. 4.6. 12. 7. 1. 2. 3. 食品生物科学科. 3. 3. 1.0. 3. 0. 0. 0. 0. 計. 165. 854. 5.2. 678. 91. 55. 62. 117. 過去の選考結果. 令和4年度特色入試の結果 令和3年度特色入試の結果 令和2年度特色入試の結果 平成31年度特色入試の結果 平成30年度特色入試の結果 平成29年度特色入試の結果 平成28年度特色入試の結果 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"yjurj9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/education-campus\/student-3\/purpose","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":63,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["ja","education-campus","student-3"],"url_file_name_s":"purpose","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["留学の目的・種類 | 京都大学"],"description_txt":["Contents\n\n\n\t留学の目的 \n\t留学の種類 \n\n\n\n  留学の目的\n\n何事をするにも目的というものがあります。留学は学生時代における大きな出来事となります。何のために留学するのか、それをよく考えておく必要があります。海外へ行ってみたい、というのは大事な気持ちですが、せっかく留学するのであれば、それをどう生かすかを考えることこそが大切です。そこで、準備と並行して、自分が何故海外へ行きたいのか、何を成し遂げたいのかをよく考えておきましょう。\n\n留学には様々な目的があり、その目的にそって留学の仕方や期間が変わってきます。ここでは代表的な目的をいくつか考えてみましょう。"],"last_modified":"Wed Jun 08 15:18:57 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-4752346417511393540,"exact_signature_unique_b":true,"fuzzy_signature_l":4852047231302838136,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["留学の目的・種類"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","留学の目的","留学の種類","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["1. 大学院レベルの専門教育を受ける","2. 学部レベルの教育を受ける","3. 海外の大学や社会を体験する","4. その国の言葉を勉強する","5. 海外で社会貢献をする","6. 海外で職業体験をする","交換留学","短期留学プログラム","私費留学"],"h4_txt":["語学留学","大学院留学","インターンシップ留学","国際ボランティア"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1446,"text_t":"留学の目的・種類 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 海外留学を希望する京大生へ 留学の目的・種類. 留学の目的・種類. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 留学の目的・種類 京大の留学プログラム 留学に関する予備知識 海外へ留学する京大生向け奨学金 海外留学のリスク管理 京都で学べる英語の授業 海外留学のための相談窓口 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. 留学の目的 留学の種類 留学の目的. 何事をするにも目的というものがあります。留学は学生時代における大きな出来事となります。何のために留学するのか、それをよく考えておく必要があります。海外へ行ってみたい、というのは大事な気持ちですが、せっかく留学するのであれば、それをどう生かすかを考えることこそが大切です。そこで、準備と並行して、自分が何故海外へ行きたいのか、何を成し遂げたいのかをよく考えておきましょう。. 留学には様々な目的があり、その目的にそって留学の仕方や期間が変わってきます。ここでは代表的な目的をいくつか考えてみましょう。. 1. 大学院レベルの専門教育を受ける. 大学院生には、より研究志向の留学を目指す人が多いでしょう。海外での優れた研究者や組織の中に入って、先端的な研究に触れることは、その後の進路に大きな影響を与える貴重な経験となるでしょう。. 留学の仕方として、本学での学位取得を目的とした交換留学、海外の大学院で学位を取得する. 大学院留学 などが考えられます。. 交換留学 2. 学部レベルの教育を受ける. 学部生の中でも、海外で専攻分野を勉強し、帰国した際に単位認定を受けたいと考える人が多くいます。また、海外の大学院への進学を念頭において、その一段階前の準備として学部留学をする人もいるでしょう。. このような目的には、交換留学が理想的な制度といえるでしょう。. 交換留学 3. 海外の大学や社会を体験する. 大学の勉強以外にも、海外での大学生活そのものや、その国の言葉・文化を経験することも大事な目的となるでしょう。海外での生活は、広い視野と複眼的な価値観を持って物を見ることを可能にします。. 渡航先の国にもよりますが、大学附属の語学学校に語学留学すれば、その大学の正規授業を聴講できる場合がありますので、希望する大学に問い合わせてみましょう。. 4. その国の言葉を勉強する. 日本での学習に加えて、実際にその国でその言葉を使ってみる、あるいは授業を受けることも、海外留学の大事な目標でありえます。言葉の学習は単に字面を追うだけのものではないはずです。. このような目的には、語学留学が適しているといえるでしょう。語学留学については、京都大学生協でも取り扱っています。. 5. 海外で社会貢献をする. 海外で社会貢献をしたい、何か行動を起こしたいという気持ちも、留学の意義深い目標でありえます。有名なものでは、青年海外協力隊（JOCV）や国連ボランティア計画（UNV）などがあります。. このような目的のためには、. 国際ボランティア が適しているでしょう。. 6. 海外で職業体験をする. 職業研修として海外の企業や団体で一定期間働き、職業経験と海外の社会経験を積みたいと考える人もいます。このような目的のために、. インターンシップ留学 があります。. 留学の種類. 海外留学は、京都大学の交換留学制度を利用する「交換留学」、春休みや夏休みを利用する「短期留学プログラム」、そして「私費留学」に大別されます。. 交換留学. 休学せずに、京都大学の協定校に１学期以上、１年以内の期間、派遣先大学の授業料は不徴収で教育を受け、単位を取得または研究指導を受ける留学です。. 多くの場合、実際の出発の１年以上前に学内募集に応募する必要がありますので、周到な準備が必要です。. 交換留学 短期留学プログラム. 春休みや夏休みを利用する、京都大学実施・推奨の3ヵ月未満の短期留学プログラムがあります。. 単位を取得できる場合もありますが、留学先大学および京都大学の所属学部・研究科のシステムによるので、確認が必要です。. 短期留学プログラム 私費留学. 留学希望先機関に、直接または留学斡旋（あっせん）業者を通して、申込をして留学します。. 語学留学. 専門の語学学校や大学等の教育機関が実施するものなどがあり、期間は数週間から１年以上のものまであります。. 大学院留学. 海外の大学院で学位を取得する場合、学部生とは異なった十分な準備が必要です。また高い語学力やGPAも求められます。. 学位を取得する目的で大学院留学を志望する京都大学学生は、JASSO海外留学支援制度（大学院学位取得型）に、京都大学を通じて応募することができます。所属学部・研究科の教務担当掛または国際教育交流課に問い合わせてください。. JASSO海外留学支援制度（大学院学位取得型）について インターンシップ留学. 海外の企業や団体で一定期間働き、職業経験と海外での社会経験を積む留学です。斡旋団体は、団体の実績や費用等を他団体と比較するなどして、十分な調査を行った上で決定してください。主に文系の学生のためのAIESEC（アイセック）、主に理系の学生のためのIAESTE（イアエステ）という学生支援団体があります。また、日欧産業協力センターによる、日本の理工系学生を対象とした1年間の奨学金付プログラム（ヴルカヌス・ イン・ヨーロッパ）などもあります。. アイセック・ジャパン IAESTE京都 日欧産業協力センター JICAインターンシップ・プログラム 青年海外協力隊 地球の歩き方の旅. 国際ボランティア. 社会貢献などのための留学です。斡旋団体の主旨、歴史、実績などを十分に考慮して選びましょう。. 国連ボランティア計画（UNV） CIEE国際ボランティアプロジェクト 日本ワーキング・ホリデー協会 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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In other languages. 日本語 교토대학은 고도 정보통신 사회에서 개인정보 보호의 중요성을 인식하고, 교토대학 홈페이지( http:\/\/www.kyoto-u.ac.jp\/ )에서 이를 취급함에 있어 법령을 준수하며. 교토대학 개인정보 보호 관련 규정 (이하 '규정') 과 함께 아래 방침에 의거해 개인정보 보호를 위해 노력하겠습니다. 1. 개인정보의 수집과 보유. 교토대학은 적법하고 공정한 수단으로 개인정보를 수집합니다. 교토대학은 본교 업무 수행상 필요한 경우이며 동시에 그 목적을 가능한 한 특정하고 그 특정된 이용 목적 달성에 필요한 범위를 넘어서 개인정보를 보유하지 않습니다. 개인정보를 수집할 경우에는 수집할 정보의 이용 목적, 수집한 정보의 관리자를 표시합니다. 2. 개인정보의 이용. 교토대학은 법령에 의거한 경우를 제외하고 개인정보를 수집 시 제시한 이용 목적 외의 목적을 위해 이용하거나 제3자에게 제공하지 않습니다. 교토대학은 법령에 정해진 경우를 제외하고 개인정보를 본인의 사전 동의 없이 제3자에게 제공하지 않습니다. 교토대학은 법령에 의거해 개인정보를 제3자에게 제공하거나 개인정보 취급 관련 업무를 제3자에게 위탁할 경우는 해당 제3자의 해당 개인정보 안전 확보, 관리 능력 등에 대해 엄정한 조사를 실시하고 개인정보 취급과 관련해 적정한 감독을 실시합니다. 3. 개인정보의 관리. 교토대학은 보유 중인 개인정보의 정확성을 유지함과 동시에 그 분실, 파괴, 변조 및 유출 등을 방지하기 위해 부정 접속, 컴퓨터 바이러스 등에 대한 적정한 정보 보안 대책을 강구합니다. 4. 개인정보의 공개, 정정, 이용 정지, 삭제. 교토대학은 본인이 자기 개인 정보에 대해 공개, 정정, 이용 정지, 삭제 등을 요구할 권리를 가짐을 확인하고, 이들 청구에 대해서는 '규정'이 정하는 바에 따라 신속히 처리합니다. 교토대학의 개인정보 취급과 관련해 의견 및 질문 등이 있으실 경우 교토대학 개인정보 관련 공개 청구 창구(총무부 법무실) 830jyohokoukai * mail2.adm.kyoto-u.ac.jp (*를 @로 바꿔 주세요)로 연락 부탁드립니다. 공개 청구 방법 5. 개인정보보호규정의 수립, 실시, 유지, 개선. 이 방침에 정한 바 외에 교토대학의 개인정보 보호와 관련해서는 '규정'이 정하는 바에 따라 적절히 실시합니다. 6. 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{"id":"x_Mn_NU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/en\/education-campus\/education-and-admissions","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":70,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["en","education-campus"],"url_file_name_s":"education-and-admissions","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Education and admissions | KYOTO UNIVERSITY"],"last_modified":"Wed Jun 08 16:41:27 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":6002324580831898617,"exact_signature_unique_b":true,"fuzzy_signature_l":3461770233944815233,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["Education and admissions"],"h2_txt":["グローバルナビゲーション（英語）","ヘッダー プライマリーリンク（英語）","ヘッダー セカンダリーリンク（日本語以外）","サイドメニュー（英語）","3階層目以降ページ用メニュー（英語）","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":44,"text_t":"Education and admissions | KYOTO UNIVERSITY. Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Study at KyotoU Education and admissions. Education and admissions. サイドメニュー（英語）. Education and admissions Kyoto University degree system Admission schedule Undergraduate degree programs Graduate degree programs English-taught degree programs Double\/joint degree programs Non-degree programs Japanese language education Online learning opportunities and resources Admissions policy, curriculum policy, and diploma policy Academic calendar Resources for prospective international students Career paths Visa\/tuition\/scholarships\/housing University facilities FAQ 3階層目以降ページ用メニュー（英語）. Kyoto University degree system. Admission schedule. Undergraduate degree programs. Graduate degree programs. English-taught degree programs. Double\/joint degree programs. Non-degree programs. Japanese language education. Online learning opportunities and resources. Admissions policy, curriculum policy, and diploma policy. Academic calendar. Resources for prospective international students. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（韓国語）. 교토대학 소개. 대학소개. 총장인사. 안전. 교육・학생 지원. 유학・국제교육. 입시. 학사일정. 학생생활. 연구・산관학 연계. Security Export Control. 국제교류. 연구지원체제. 젊은층 연구자 지원. 학부・대학원. ヘッダー セカンダリーリンク（日本語以外）. 교통편 안내 문의처 기부 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 이동 경로. Home 教職員の方へ. 教職員の方へ. 教育 研究 国際交流 会議･委員会 採用・公募 イベント情報 総長選考関係（学内限定） その他通知・手続き 学内の問い合わせ先等 教職員が利用できる施設 京都大学教職員グループウェア 教育. 大学教育支援プログラム FD（ファカルティ・ディベロップメント）（FD専門委員会） 京都大学のFD 研究. 外部資金 競争的資金等の適正管理 研究倫理・生命倫理 研究成果発表（記者発表の手続き） 国際交流. 国際交流事務手続き（学内限定） 会議・委員会. 会議報告・委員会報告 採用・公募. 教職員公募一覧 イベント情報. EVENTS （イベントカレンダー）. 総長選考関係. 総長選考関係 その他通知・手続き. 監査. 内部監査 公益通報 総務. 諸規程 法務（学内限定）. 「コンプライアンスガイドブック（日本語）」および「Compliance Guidebook（英語）」について. 教職員ポータル内の以下の場所に掲載しています。. 京都大学教職員ポータル 日本語：ファイル管理＞文書共有＞総務部＞05法務室＞03コンプライアンスガイドブック＞コンプライアンスガイドブック. 英語：ファイル管理＞文書共有＞総務部＞05法務室＞03コンプライアンスガイドブック＞Compliance Guidebook. （2020年10月）担当部署名の変更、法令や学内規程等の改正に伴う部分的な改訂を行いました。（日本語／英語）. 広報. 刊行物等の請求方法 記者発表 広報用データ（学内限定）. ビジュアル・アイデンティティ マップ等 ホームページ・京大広報への情報掲載. 掲載に関する手続きは、「京都大学教職員グループウェア」の「ファイル管理」内に掲載しています。. 京都大学教職員ポータル ファイル管理＞文書共有＞総務部＞02広報課＞京大広報・ホームページ等への情報掲載手続き. 職員・人事・共済. 就業規則一覧 人事に関する情報（学内限定） その他 （人事制度・異動関係のお知らせ、人事異動、職員研修・自己啓発支援、兼業、障害者雇用等） ハラスメントの防止と対応 叙位・叙勲 育児支援制度について 財務. 京都大学における発注・検収について 環境安全衛生（学内限定）. 環境管理 安全衛生管理 情報環境. 情報セキュリティ 学内の問い合わせ先等. 公益通報 ハラスメント 学生総合支援センター 研究活動上の不正行為 競争的資金等適正管理 人事（学内限定） 教職員が利用できる施設. 学内施設 学内で使用できる実験室・会議室等 京都大学教職員グループウェア. 役職者内線番号一覧. 本部座席表. 資産の供用公募. などについては、. 京都大学教職員ポータル からご覧いただけます。. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 간행물 채용 교통편 안내 フッター セカンダリーリンク. 사이트 맵 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 男女共同参画について 次世代育成支援対策推進. 次世代育成支援対策推進. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について アクションプラン 女性活躍推進法について 意識・実態調査 改革 次世代育成支援対策推進 育児・介護支援制度 公表事項 情報公開 次世代育成支援推進について. 次世代育成支援対策推進法が一部改正され、（平成21年4月1日施行）行動計画の公表および従業員への周知が義務化されました。. 本学では平成21年3月10日に決定された、. 男女共同参画推進アクション・プラン を公表し、中期計画なども視野に入れて、. 一般事業主行動計画 を策定しました。. 時限立法であった次世代育成支援対策推進法が10年間延長されることに伴い、平成26年度に新たな. 一般事業主行動計画（平成26年4月1日から平成31年3月31日までの5年間） を策定し実施してきましたが、本計画期間の終了に伴い、新たな. 一般事業主計画（平成31年4月1日から平成36年3月31日） を策定しました。. 次世代育成支援対策推進法に基づく基準適合一般事業主に認定について. 次世代育成支援対策推進法第13条の規定に基づき認定申請を行った結果、京都労働局長から、認定通知がありました。. 次世代法に基づく基準適合一般事業主の認定は雇用環境の整備について適切な一般事業主行動計画を策定したこと、その行動計画に定めた目標を達成したことなどの一定要件を満たす場合は都道府県労働局長の認定をうけることができます。. 認定をうけた事業主はその旨を示す表示（次世代認定マーク愛称：くるみん）を求人広告や名刺、封筒などにつけることができます。このマークには「職場「ぐるみ」・社会「ぐるみ」で両立支援に取り組む」という意味が込められているそうです。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 キャンパスライフ支援 アルバイト. アルバイト. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生生活で注意してほしいこと 健康管理 在学中の保険加入 住まい・食と購買 アルバイト 祭礼行列員等のアルバイト紹介 官公庁からのアルバイト紹介 学内部局からのアルバイト紹介 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. アルバイトについての諸注意 アルバイトの紹介 京大生のアルバイトをお探しの皆様へ 問い合わせ先 アルバイトについての諸注意. アルバイトをする場合は、日常の勉学に支障のないよう心掛けるとともに、危険を伴う作業や健康を害する恐れのある職種は避けてください。. 留学生がアルバイトをする場合には、アルバイトを始める前に「資格外活動許可書」を取得しなければなりません。詳しくは、「. 資格外活動（アルバイト）許可 」を参照してください。. 最近、京大生を名乗り一般家庭へ電話し、執拗に家庭教師の勧誘を行うといった悪質な手口が多発していますので、十分にご注意ください。なお、本学では「京都大学生活協同組合（京大生協）」へ家庭教師を含むアルバイト紹介業務を委託していますので、紹介を希望する際は、「. アルバイト先の紹介（学生）（京都大学生活協同組合 ）」をご覧ください。. アルバイトにおけるトラブルを防止するため、「. 確かめようアルバイトの労働条件（厚生労働省） 」をご確認ください。. アルバイトの紹介. アルバイトを探すためのサイトを紹介します。本学が直接に紹介しているのは下記の「祭礼行列員などの紹介」のみです。. 祭礼行列員等のアルバイト紹介 官公庁等からのアルバイト紹介 学内部局からのアルバイト紹介 株式会社学生情報センター（外部リンク） 京都大学生活協同組合（外部リンク） 京都府ブラックバイト相談窓口（外部リンク） 京大生のアルバイトをお探しの企業の皆さまへ. 京都大学生活協同組合で求人を受け付けています。. 詳しくは、コンベンション・サービスセンター（アルバイト紹介係）へお問い合わせください。. Tel： 075-753-7655. Fax： 075-752-0368. 平日： 10時00分～17時00分 （土曜日、日曜日、祝日、年末年始は休み）. 問い合わせ先. 教育推進・学生支援部 厚生課. Tel： 075-753-2539. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"1alE39U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/about\/facilities\/campus\/kizuna","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":59,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["ja","about","facilities","campus"],"url_file_name_s":"kizuna","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["留学生ラウンジ きずな | 京都大学"],"description_txt":[""],"last_modified":"Wed Jun 08 16:44:20 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-388788946971845594,"exact_signature_unique_b":true,"fuzzy_signature_l":-8684402686327114060,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["留学生ラウンジ きずな"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","留学生ラウンジ「きずな」","最新情報","Twitter","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"underline_txt":["「きずな」のメーリングリストにご登録ください。"],"imagescount_i":12,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jaaboutfacilitiescampuskizunaimagesKIZUNA_eyecatch.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jaaboutfacilitiescampuskizunaimagesStudentLoungeKIZUNA_Poster01.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jaaboutfacilitiescampuskizunaimages2020online_language-exchange_flyer.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jaaboutfacilitiescampuskizunaimages2020online_lounge_flyer_s.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","","","","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,180,180,180,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button sites default files embed jaaboutfacilitiescampuskizunaimagesKIZUNA eyecatch jpg jaaboutfacilitiescampuskizunaimages KIZUNA jaaboutfacilitiescampuskizunaimagesStudentLoungeKIZUNA Poster01 Student Lounge Poster 01 jaaboutfacilitiescampuskizunaimages2020online language exchange flyer 2020online lounge s facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1460,"text_t":"留学生ラウンジ きずな | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について キャンパス・施設案内 大学の施設 留学生ラウンジ きずな. 留学生ラウンジ きずな. サイドメニュー. 世界とつながる京都大学 京都大学への留学を希望する方へ 京都大学で学ぶ 留学生へのキャリア支援 留学生ラウンジ「きずな」 在留資格・学費・奨学金・住まい 学部・大学院研究科等事務室一覧 FAQ 便利な情報 留学に関する情報 国際化を支える学内組織 外国人研究者へのサポート 海外留学を希望する京大生へ In other languages. English 留学生ラウンジ「きずな」. 留学生ラウンジ「きずな」の施設案内 きずな月例イベント ML登録＆連絡先 きずなオンライン言語交換 留学生のためのアドバイジング・指導 「きずな」ポスター（PDF） 最新情報. 新型コロナウイルス感染症拡大防止のため、全面休館としていましたが、2021年10月1日（金曜日）から読書室を開室し、以下のとおりサービスを再開しています。. 図書貸出・閲覧. 外国人留学生のためのアドバイジング. （. ※）. 留学生ラウンジ「きずな」では予約なしで相談を受け付けます。. 相談員が在室する時間内に直接来室してください。. 留学生相談室でのカウンセリングを希望する場合は、まずはメールにてご連絡ください。. E-mail： advising＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. なお、留学生ラウンジ「きずな」では、オンラインでのさまざまな国際交流の場を提供しています。. ご関心のある方は. 「きずな」のメーリングリストにご登録ください。. オンライン言語交換（PDF） オンラインラウンジ（PDF） Twitter. 教育推進・学生支援部運営の京都大学公式アカウントからも「きずな」の最新情報が閲覧できます。. Campus Life【京都大学公式】. @CLI_KU ハッシュタグ. #留学生ラウンジきずな 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求 Kyoto U Research News. Kyoto U Research News. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） Kyoto U Research News 入手方法（KYOTO U Research News） アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 Webサイト. Kyoto U Research News 2021 Winter. 9号[2021 Winter] PDF：5.29MB Contents. 03. Backstage at the lab. Crossroads of the world and the mind. 08. KyotoU moving forward. 10. In Covid’s wake. 11. Cutting edge. A special elemental magic. How do birds understand ‘foreign’ calls? 12. From bacteria to you: the biological reactions that sustain our rhythms. 13. Two for the price of one. Flaring, massively. 14. Two for the price of one. Order from chaos. 15. Changing ties that naturally bind. 16. Reducing the high social cost of death. Detect with PKAch. 17. Terahertz zaps alter gene activity in stem cells. 18. How does the spider spin its self-assembled silk? 19. KyotoU today. Science with industry: The forest and the trees: science supporting ASEAN. 20. New strategic partnerships established. 21. News from overseas centers. 22. Student voices. 23. Eternal aesthetic. 2020 Spring. 8号[2020 Spring] PDF：4.90MB Contents. 03. Backstage at the lab. Into the human unknown. 10. Cutting edge. Bridging the scales of the brain. An overactive cerebellum causes issues across the brain. 11. Why fruit flies eat practically anything. 12. Lend me a flipper. Using a chip to find better cancer fighting drugs. 13. The little auks that lived in the Pacific. 14. Reconstructing the clock of human development. 15. KyotoU today. Science with industry: KyotoU venture bringing light to patients’ eyes. 16. News from overseas centers. 18. Student voices. 19. Eternal aesthetic. 2019 Autumn. 7号[2019 Autumn] PDF：4.43MB Contents. 03. Backstage at the lab. An Asian Humanities, for the future. 10. Cutting edge. A petrifying virus key to evolution. Let there be light-activated genes. 11. Infer tility’s roots in DNA packaging. 12. Prince Charming’s kiss unlocking brain’s regenerative potential? Chimps caught crabbing. 13. A study in scarlet Japanese macaques. 14. Testing corneal cell quality? Apply physics. The worms that roared. 15. KyotoU today. So tell us... 16. News from overseas centers. 17. On-Site Laboratory network established. 18. Student voices. 19. Eternal aesthetic. 2019 Spring. 6号[2019 Spring] PDF：5.38MB Contents. 03. Nobel Prize 2018. Congratulations, Honjo-sensei! 04. Defining ‘Kyotoness’ in the research environment. 08. Driven by curiosity. 10. Cutting edge. New gears in your sleep clock. What does the koala genome tell us about the taste of eucalyptus? 11. Meet ERICA. She wants to listen to you. 12. Giant, recently extinct seabird also inhabited Japan. Japanese in Canada, with mixed feelings. 13. Finding success through failure. Using microcredit to increase rice yield in Bangladesh. 14. Making an eye for you. 15. KyotoU today. Science with industry: Microbes are us. But how can we see them? 16. News from overseas centers. 18. Student voices. 19. Eternal aesthetic. 2018 Autumn. 5号[2018 Autumn] PDF： 4.76MB Contents. 03. Backstage at the lab. The high-tech astronomy of Seimei. 08. Personnel profiles. Understanding the machines, Kyoto style. 12. Cutting edge. High achievement not always based on high student initiative. How does HIV escape cellular booby traps? 13. Visualizing danger from songbird warning calls. 14. What a handsome schnoz! Electro-mechano-optical NMR detection. 15. Twisting graphene into spirals. 16. Studying the human impact on sea urchin abundance. Capturing the rare Yanbaru whiskered bat. 17. Using geometry and qubits to build real universes. 18. KyotoU today. KURNe: research news in motion. 19. News from overseas centers. 20. Science with industry: Mining answers from the data mountain. 21. So tell us... what do you do and how did you get to Japan? 22. Student voices. 23. Eternal aesthetic. 2018 Spring. 4号[2018 Spring] PDF： 2.18MB Contents. 03. Backstage at the lab. Humanities 101 : A Kyoto approach to history, society, and the study of humanity. 12. Cutting edge. Special delivery : macromolecules via spider’s ‘bite’. Cells pumping iron to prevent anemia. 13. Black hole pair born inside a dying star? How am I feeling? Ask my house. 14. Lightning, with a chance of antimatter. 15. Winds blowing off a dying star. The secret lives of ancient land plants. 16. What grosses out a chimpanzee? 17. Cells rockin’ in their DNA. 18. News from overseas centers. 20. Student voices. 22. Global outreach. 23. Eternal aesthetic. 2017 Autumn. 3号[2017 Autumn] PDF： 2.93MB Contents. 03. Backstage at the lab. Earth, sky, space, and everything in between. 08. Personnel Profiles. ARN:student perspectives. 12. Cutting edge. Energizing immune cells to combat cancer. Visualizing nuclear radiation. Choosing a simpler path to drug discovery. Charting the skies of history. Born to love superheroes. High stakes, high risk, and a bad bet. You don’t see what I see? Disentangling chloroplast genetics. Untangling the knots in cell stress. Take a look, and you’ll see, into your imagination. Nicking in new nucleotides. 18. International activities. 19. News from overseas centers. 20. Student voices. 22. Global outreach. 23. Eternal aesthetic. 2017 Spring. 2号[2017 Spring] PDF： 2.43MB Contents. 03. Backstage at the lab. Kyoto into Africa. 08. Personnel Profiles. The genetics of wildlife breeding and conservation. 12. Cutting edge. Smiling baby monkeys and the roots of laughter. Homosexual termite regicide. Is it your second cousin? Cotton swabs may tell you. Drinking green tea to prevent artery explosion. Great apes can “read minds”. Mt Aso could erupt much sooner, scientists warn. Aging bonobos in the wild could use reading glasses too. A big nano boost for solar cells. Proteins at the movies. The sea roils and life returns. Safety in darkness. 18. International activities. 20. Student voices. 22. Global outreach. 23. Eternal aesthetic. 2016 Autumn. 1号[2016 Autumn] PDF： 2.57MB Contents. 03. from the President. 04. On the cover: Backstage at the lab. Reading beyond time and across disciplines. 08. Personnel Profiles. Working in CiRA labs. 11. News from overseas centers. 12. Cutting edge. The search for happiness: using MRI to find where happiness happens. Fish oil helps burn fat by transforming fat-storage cells into fat-burning cells. \"Seeing\" black holes with home-use telescopes. To hear a pit ter patter from afar: catching hear tbeats with milimeter-wave radar. Genetically modified E.coli pump out morphine precursor. \"Popular girls\" have less lice . in the monkey world. Australopithecus fossils found east of the Great Rif t Valley. 'Slow' NZ seabed quake sheds light on tsunami-earthquake mechanism. Cancer cell immunity in the crosshairs: wor th the expense? Flipping a protein switch to illuminate brain functions. \"Big mama\" bonobos help younger females stand up for themselves. Did the LIGO gravitational waves originate from primordial black holes? 18. International activities. 20. Student voices. 22. Global outreach. 23. Eternal aesthetic. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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{"id":"zC1609U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/research\/research-policy\/kanrikoukai","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":65,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["ja","research","research-policy"],"url_file_name_s":"kanrikoukai","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["研究データ管理・公開ポリシー | 京都大学"],"description_txt":["京都大学研究データ管理・公開ポリシー\n\n京都大学は、「研究の自由と自主を基礎に、高い倫理性を備えた研究活動により、世界的に卓越した知の創造を行う」とともに、「世界に開かれた大学として、自由と調和に基づく知を社会に伝える」ことを基本理念として掲げている。研究活動の過程で生み出される研究データを適切に管理・保存し、また公開により利活用を促進することは、研究データの価値を守り高めることであり、それによりこれらの理念を具現化し、学術研究の広範囲にわたる発展と地球社会の調和ある共存に大きく貢献することができる。\n\nそこで、京都大学で行われる研究が多様であることを踏まえ、研究データの管理・保存・公開に関して、それぞれの研究分野における法的および倫理的要件に従うことを認め、それらを最大限尊重した上で、学術研究の発展と京都大学における将来の研究を守るため、指針として京都大学研究データ管理・公開ポリシーとして以下の原則を定める。"],"last_modified":"Wed Jun 08 15:51:23 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":1030623025150353706,"exact_signature_unique_b":true,"fuzzy_signature_l":1496851468262947624,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["研究データ管理・公開ポリシー"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","京都大学研究データ管理・公開ポリシー","関連リンク","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":668,"text_t":"研究データ管理・公開ポリシー | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 研究・産官学連携 研究成果の公開ポリシー 研究データ管理・公開ポリシー. 研究データ管理・公開ポリシー. サイドメニュー. 研究活動を知る 研究成果の公開ポリシー オープンアクセス方針 研究データ管理・公開ポリシー 産官学連携（共同研究・知的財産・ベンチャー支援） 京都大学に寄附する 京都大学で研究したい 研究支援体制について 研究者が受けられる研究費・研究支援 研究上の倫理・安全 教職員に兼業を依頼したい In other languages. English 京都大学研究データ管理・公開ポリシー. 京都大学は、「研究の自由と自主を基礎に、高い倫理性を備えた研究活動により、世界的に卓越した知の創造を行う」とともに、「世界に開かれた大学として、自由と調和に基づく知を社会に伝える」ことを基本理念として掲げている。研究活動の過程で生み出される研究データを適切に管理・保存し、また公開により利活用を促進することは、研究データの価値を守り高めることであり、それによりこれらの理念を具現化し、学術研究の広範囲にわたる発展と地球社会の調和ある共存に大きく貢献することができる。. そこで、京都大学で行われる研究が多様であることを踏まえ、研究データの管理・保存・公開に関して、それぞれの研究分野における法的および倫理的要件に従うことを認め、それらを最大限尊重した上で、学術研究の発展と京都大学における将来の研究を守るため、指針として京都大学研究データ管理・公開ポリシーとして以下の原則を定める。. 本ポリシーにおける「研究データ」とは、本学の研究活動の過程で研究者によって収集または生成された情報を指し、デジタル・非デジタルを問わない。. 京都大学において、研究者は研究データを適切に管理・保存すること、すなわち研究データ管理は、優れた研究を行う上で必要不可欠であると認識する。. 京都大学は、原則として、研究データを収集または生成した研究者が、研究データ管理を行う権利と責務を有していることを認め、研究者は、研究データの価値を守るため、それぞれの研究分野における法的および倫理的要件に従って研究データ管理を実施する。. 京都大学は、研究データが、論文などと同様に、今後の学術や社会の発展に貢献する知の基盤の一つであるとの認識に基づき、特段の定めがある場合を除き、可能な限り社会に公開し、その利活用を促進する。. 京都大学は、研究データ管理および公開を支援する環境を整える責務がある。. なお、社会や学術状況の変化に応じて、適宜本ポリシーの見直しを行うものとする。. 2020年3月19日 研究者情報整備委員会 承認. 京都大学研究データ管理・公開ポリシー ［日本語版］. Kyoto University Policy on Research Data Management and Sharing ［英語版］. ポリシーについての解説・補足 関連リンク. 図書館機構「研究データの公開支援」 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 研究・産官学連携 研究支援体制について 研究戦略タスクフォース. 研究戦略タスクフォース. サイドメニュー. 研究活動を知る 研究成果の公開ポリシー 産官学連携（共同研究・知的財産・ベンチャー支援） 京都大学に寄附する 京都大学で研究したい 研究支援体制について 研究戦略タスクフォース 過去の情報 学術研究支援室（KURA） 研究活動と研究者生活を支える組織 研究者が受けられる研究費・研究支援 研究上の倫理・安全 教職員に兼業を依頼したい 　国の厳しい財政事情を背景に、国立大学法人への運営費交付金が年々削減されるなど、国立大学を取り巻く環境が年々厳しくなる中、外部の競争的資金等を獲得することの重要性がますます高まっております。. 　本学は、このような状況を踏まえ、研究担当理事の下、研究資金獲得を支援するための体制を整備しています。. 研究戦略タスクフォース. プログラムディレクター. 氏名. 役職. 松浦 健二. 理事補（研究・評価・産官学連携担当）、農学研究科 教授. 田中 勝久. 理事補（研究・評価・産官学連携担当）、工学研究科 教授. 上杉 志成. 理事補（研究・評価・産官学連携担当）、化学研究所 教授. 赤松 明彦. 白眉センター長. 石川 冬木. 学術研究支援室長. プログラムオフィサー. 氏名. 役職. 小野澤 透. 文学研究科 教授. 北川 宏. 理学研究科 教授. 土居 雅夫. 薬学研究科 教授. 江利口 浩二. 工学研究科 教授. 船曳 康子. 人間・環境学研究科 教授. 生田 宏一. 医生物学研究所 教授. 大村 善治. 生存圏研究所 教授. 過去に掲載した情報は. こちらをご覧ください。 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 経済支援 急に困った時に. 急に困った時に. サイドメニュー. 教育の体制と内容 経済支援 高等教育の修学支援新制度について 授業料の免除／入学料の免除と徴収猶予 さまざまな奨学金 急に困った時に 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. 緊急支援一時金 小口短期貸付 JASSO災害支援金 問い合わせ先 学資負担者が亡くなられた場合や災害により被災した場合などの緊急時に一時金を給付する制度や、急な出費の場合に無利子の貸付を行う制度を紹介します。. 緊急支援一時金. 学資を主として負担する方（学資負担者）が亡くなられたり、学生または学資負担者が災害により被災した等経済的な緊急時に、一時金（25万円）を給付し、修学・生活を支援します（「京都大学基金緊急支援一時金」）。. 該当事由が学資負担者の死亡または被災である場合は当該事由発生日より3か月以内（ただし、卒業・修了予定者については、卒業・修了予定日が属する月の前月末まで）に出願する必要がありますので、希望する場合は事由発生後速やかに教育推進・学生支援部学生課奨学掛にご相談ください。. 申請様式：「. 京都大学基金緊急支援一時金申請書 」（証明書類の添付が必要）. 【参考】京都大学基金緊急支援一時金. 出願事由：本学に在籍する学部学生および大学院学生（外国学生、委託生、科目等履修生、聴講生、特別聴講学生、特別研究学生、特別交流学生等および研究生を除く）のうち、次の各号に掲げる事由のある者。ただし、休学中または懲戒処分中の者は出願できない。. 学生の学資を主として負担する者が死亡したとき。. 学生または学資負担者が災害（風水害、地震または火災をいう）により被災したとき。なお、被災による出願事由は学生または学資負担者の自宅の全壊・半壊・全焼・半焼・床上浸水に限られる。. 小口短期貸付. 学生援助会が病気、不慮の事故、送金の延着、その他急な出費の場合に、無利子の貸与融資を行っています。この貸付金は、本学関係者の寄附によるものです。. 貸付額. 1人 1万円～5万円（1万円単位）. なお、貸付には学生証および本人印（スタンプ印不可）が必要です。ただし、2万円以上は貸付時に. 債務保証書 （4万円以上は2名分）の提出も必要です。. 返済方法. 1万円の場合は、1ヶ月以内に一括返済. 2万円以上の場合は、3ヶ月以内に一括返済. ※ 2015年10月より、2万円以上の貸付金の返済期限が6ヶ月以内から3ヶ月以内へと変更となりました。. JASSO災害支援金. JASSO災害支援金とは、日本学生支援機構（JASSO）が行う寄附金事業です。自然災害等により本人またはその生計維持者が居住する住宅に半壊以上等の被害を受けたことで、学生生活の継続に支障をきたした学生が、一日も早く通常の学生生活に復帰し学業を継続するための支援をする制度です。. 事由発生月の翌月から6か月以内かつ本人が在学中に申請する必要がありますので、希望する場合は事由発生後、速やかに教育推進・学生支援部学生課奨学掛にご相談ください。. 支給額： 10万円（返還不要）. JASSO災害支援金について 問い合わせ. 教育推進・学生支援部学生課奨学掛. 〒606-8501 京都市左京区吉田本町. 吉田キャンパス本部構内 総合研究10号館1階. Tel： 075-753-2480. Fax： 075-753-2563. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 価値創造人材育成プログラム「Kyoto Creative Assemblage」の募集について. 産官学連携 価値創造人材育成プログラム「Kyoto Creative Assemblage」の募集について. ターゲット. 卒業生の方 一般・地域の方 企業・研究者の方 公開日. 2022年05月30日. 　京都大学、京都市立芸術大学、京都工芸繊維大学の3大学を中心に、時代の最前線で活躍するデザイナー、アーティスト、起業家らが結集した、文部科学省「大学等における価値創造人材育成拠点の形成事業」採択プログラム、「Kyoto Creative Assemblage」を2022年度より開講します。. 　本プログラムは社会人を対象にした創造性育成プログラムです。 人文社会学、デザイン、アート分野で京都を代表する3大学の講師と創造的実践の第一線で活躍する実務家が集結し、6ヶ月の課程を通して社会を読み解き、新しい時代を表現するための考え方や方法を学びます。. 　デザイン思考などの従来の方法の限界を乗り越える、新しいイノベーション手法を学び、特に、潜在ニーズを満たすという従来の枠組みではなく、人文社会学的視点で社会をよく見て、現在を捉え直し、新しい時代を表現することにより、歴史をつくるイノベーションを生み出すことを目指します。芸術大学でのアート教育の体験や新しく表現された世界を実体験するフィールドワークなども実施します。. 　この度、本プログラムの受講募集を開始します。これからの時代を切り開く実践を日々模索している方々の受講をお待ちしています。. 募集要項. 受講者像の一例. 企業や事業の今後の展望や戦略を考える糸口を掴めず困っている経営企画部や事業部の方. 問題意識ややりたいことはあるが、どのように表現していいかわからない次世代経営者. 既存のユーザー調査とは異なる視点から社会を読み解く力を身につけたいマーケティング部、デザイン本部の方. イノベーションの方法論を指導、探求する立場から、新たな視座の必要性を感じているデザインリサーチャー・ストラテジスト. 開講期間. 2022年9月〜2023年3月. 履修時間. 約90時間. 定員. 20〜30名. 場所. オンライン、京都大学、京都市立芸術大学、京都工芸繊維大学 ほか。. 受講料. 200,000円 (税別). 応募方法、詳細等. 応募、その他詳細な情報は以下のホームページよりご確認ください。. Kyoto Creative Assemblage 応募締め切り. 2022年6月17日（金曜日）23時59分. 選考結果. 2022年7月8日（金曜日）にメールでご連絡します（必要に応じて面接を実施）。. 問い合わせ. 京都大学経営管理研究部（経営管理大学院）. 606-8501 京都市左京区吉田本町 総合研究14号館103. E-mail： info＊assemblage.kyoto （＊を@に変えてください）. 関連部局. 経営管理研究部\/経営管理教育部（経営管理大学院） Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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2022","referrer_id_s":"9Y1wL9U8j7Mo","language_s":"ja","size_i":113550,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085782035595264,"text_t":"学生意見箱 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. 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PandAの不具合について 2022年5月23日. 2022年5月26日. 学習室24について 2022年5月16日. 2022年5月26日. 検索エンジンでKULASISを検索したときの結果表示について 2022年4月14日. 2022年5月26日. 学習室24について 2022年5月10日. 2022年5月24日. 放置自転車について 2022年5月8日. 2022年5月24日. 健康診断について 2022年4月21日. 2022年5月24日. 性別について 2022年3月26日. 2022年5月19日. 「他大生の課外活動参加禁止の具体的理由について」の補足希望 2022年4月6日. 2022年5月18日. 尿検査の再提出日について 2022年5月11日. 2022年5月16日. 教職課程に関する要望 2022年3月28日. 2022年5月16日. 講義形態について 2022年4月14日. 2022年5月10日. バイク駐車場の範囲拡大と明確化提案 2022年4月12日. 2022年5月10日. 法経第四・十一教室前の自転車の駐輪について 2022年4月11日. 2022年5月10日. 弓道場横の不法投棄について 2022年3月25日. 2022年5月10日. 「ながら運転」について 2022年4月19日. 2022年4月28日. 学習室24について 2022年4月10日. 2022年4月27日. 研究室での業務・アルバイトについて（２） 2022年4月5日. 2022年4月27日. キャンパス間連絡バスのダイヤについて 2022年4月5日. 2022年4月27日. サークルのビラの不適切な内容、および貼り方について 2022年3月21日. 2022年4月25日. 卒業式の来賓席の対応について（2） 2022年4月4日. 2022年4月13日. 証明書自動発行機に関して 2022年4月2日. 2022年4月13日. 大学院学生証の配布日（特に専攻事務と異なるキャンパスの場合）等について 2022年4月1日. 2022年4月13日. E1科目の登録について 2022年4月1日. 2022年4月5日. 卒業式の来賓席の対応について（1） 2022年3月24日. 2022年4月4日. 附属図書館内の学習室24について 2022年3月18日. 2022年4月4日. 研究室での業務・アルバイトについて 2022年2月24日. 2022年4月4日. 他大生の課外活動参加禁止の具体的理由について 2022年3月7日. 2022年3月25日. BOX棟の多目的トイレの不正使用について 2022年3月4日. 2022年3月25日. 課外活動について 2022年3月2日. 2022年3月25日. 弓道場横の不法投棄について 2022年2月21日. 2022年3月22日. 京都大学による新型コロナワクチン3回目接種の実施予定の有無について 2022年3月2日. 2022年3月18日. シラバスについて 2022年3月2日. 2022年3月15日. BOX棟の多目的トイレの閉鎖について 2022年2月27日. 2022年3月2日. バンドクラブの活動について 2021年11月25日. 2022年3月2日. 課外活動の制限について 2022年2月6日. 2022年3月1日. 農学部の成績発表について 2022年2月16日. 2022年2月18日. 法経第四・十一教室前の自転車の駐輪について 2022年2月2日. 2022年2月18日. 新型コロナウイルス感染症の対応について 2022年1月18日. 2022年2月9日. 附属図書館の運営が民間に委託されることについて 2022年1月24日. 2022年2月8日. 附属図書館に関する業務の外部委託について 2022年1月24日. 2022年2月8日. 2021年度後期期末試験の実施について 2022年1月21日～. 2022年1月28日. 2022年2月8日. コロナの感染状況とオンライン授業に関して 2022年1月18日. 2022年2月8日. KULASISのシラバスについて 2022年1月23日. 2022年1月31日. 図書館におけるグミやガムの摂取について 2022年1月14日. 2022年1月31日. PandAの不具合について 2022年1月9日. 2022年1月31日. 課外活動の制限について 2022年1月8日. 2022年1月31日. 宇治キャンパス生協食堂について 2021年12月1日. 2022年1月20日. 教室でのビラ配布について 2021年12月10日. 2022年1月19日. 「対応レベル」基準について 2021年11月25日. 2022年1月19日. Office365について 2021年12月1日. 2022年1月11日. ソフトウェアについて 2021年11月11日. 2022年1月11日. 署名について 2021年12月18日. 2021年12月27日. 保健診療所の閉鎖について 2021年12月17日. 2021年12月27日. 保健診療所の廃止について 2021年12月9日. 2021年12月27日. 保健診療所の継続について 2021年12月8日. 2021年12月27日. 保健診療所について 2021年12月1日. 2021年12月27日. 京都大学保健診療所（吉田・宇治・桂）の診療終了について 2 2021年12月1日～. 2021年12月6日. 2021年12月27日. 京都大学保健診療所（吉田・宇治・桂）の診療終了について 1 2021年12月1日～. 2021年12月5日. 2021年12月27日. PandAの不具合について 2021年12月21日. 2021年12月24日. 第二外国語について 2021年12月16日. 2021年12月24日. 課外活動等の実施における感染拡大予防マニュアルについて 2021年12月9日. 2021年12月24日. 「他大生とのクラブ活動再開の願い」の回答について 2021年12月8日. 2021年12月24日. 学生と大学との対話、情報の開示方法について 2021年12月8日. 2021年12月17日. 課外活動について 2021年12月2日. 2021年12月17日. 西部課外活動棟の使用について 2021年12月1日. 2021年12月17日. 総長の選考過程について 2021年11月22日. 2021年12月17日. 意見箱について 2021年12月1日. 2021年12月6日. 大学構内の電灯について 2021年12月1日. 2021年12月6日. 他大生とのクラブ活動再開の願い 2021年11月25日. 2021年12月6日. 京都大学大学院教育支援機構プログラムの応募条件について 2021年11月24日. 2021年12月6日. 京都大学大学院教育支援機構プログラムについて 2021年11月22日. 2021年12月6日. デジタル学生証について 2021年11月12日. 2021年11月18日. 学習室24の再開について 2021年11月3日. 2021年11月18日. 附属図書館の開館時間・学習室24の開室時間延長および「京大サロン」開室についての要望について 2021年10月29日. 2021年11月18日. 不審者対策について 2021年11月9日. 2021年11月17日. 吉田南構内におけるバンドクラブの活動について 2021年11月9日. 2021年11月17日. 桂キャンパスの駐輪場について 2021年10月21日. 2021年11月17日. 共北1階の多目的トイレの鍵の故障について 2021年11月6日. 2021年11月12日. 生理を持つ人が授業を受ける際の配慮に関する意見について 2021年10月31日. 2021年11月12日. おもろチャレンジの再開について 2021年11月2日. 2021年11月8日. 期間が九か月以下の派遣留学の規制緩和のお願い 2021年11月1日. 2021年11月8日. Web版KULASISの時間割について 2021年10月22日. 2021年10月25日. 霊長研「解体」の報道について 2021年10月15日. 2021年10月25日. 教員から重要メールが転送されないことと対策について 2021年9月27日. 2021年10月25日. PandAの仕様について 2021年10月10日. 2021年10月14日. KULASISのシラバス検索機能について 2021年10月6日. 2021年10月13日. 授業実施形態の決定について 2021年10月4日. 2021年10月13日. 他学部聴講のPandAシステムの改善について 2021年10月4日. 2021年10月13日. 後期をオンライン授業で開始することについて 2021年9月17日. 2021年10月13日. 自転車での通行にあたっての提案 2021年7月30日. 2021年10月11日. 体育館施設の使用状況について 2021年9月21日. 2021年9月30日. 課外活動について 2021年9月21日. 2021年9月30日. トレーニングルームの使用について 2021年9月19日. 2021年9月30日. Adobe Illustrator等有料ソフトの利用について 2021年9月5日. 2021年9月30日. Microsoft Office 365 包括契約について 2021年8月10日. 2021年9月30日. 就職活動交通費の助成について 2021年9月22日. 2021年9月28日. 全学共通科目の採点基準に関して 2021年9月21日. 2021年9月28日. 法科大学院授業の実施方法について 2021年9月18日. 2021年9月28日. 転研究科制度について 2021年9月15日. 2021年9月28日. 「新型コロナウイルス感染症ワクチン接種状況調査協力のお願い」について 2021年9月10日. 2021年9月16日. キャンパス間の移動連絡バスの乗車条件について 2021年9月4日. 2021年9月16日. 新型コロナウイルスの対応について 2021年8月12日. 2021年9月8日. 学内のトイレのハンドドライヤーについて 2021年6月18日. 2021年9月8日. 保健診療所の受付について 2021年8月26日. 2021年8月31日. 新型コロナウイルス感染拡大防止に伴う活動制限のガイドラインにおける対応レベルの引き上げについて 2021年8月18日. 2021年8月31日. 課外活動における他大学学生の参加制限について 2021年7月31日. 2021年8月31日. Student suggestions about school and campus（学校やキャンパスに関する提案について） 2021年6月22日. 2021年8月31日. 附属図書館のウォータークーラーについて 2021年7月19日. 2021年8月27日. ルネの前の放置自転車について 2021年7月24日. 2021年8月11日. 全学共通科目学生窓口の対応について 2021年7月15日. 2021年7月26日. 京都大学桂キャンパスの生垣の刈込について 2021年7月6日. 2021年7月26日. 6月22日回答の「告示について（2）」について 2021年6月22日. 2021年7月26日. 自転車での通行について 2021年6月16日. 2021年7月26日. 「2021年6月16日掲示のお知らせ『【注意喚起】大学内における違法な行為について』について」について 2021年7月3日. 2021年7月14日. 対面再開について 2021年6月22日. 2021年7月14日. 今後の授業計画と新型コロナウイルスの広報について 2021年6月19日. 2021年7月14日. 本部構内の自転車駐輪について 2021年6月26日. 2021年7月8日. 学生意見箱に対する提案 2021年6月24日. 2021年7月8日. 京都大学の契約しているライセンスについて 2021年6月23日. 2021年7月8日. 新型コロナウイルスワクチン接種に関する教員の発言 2021年6月29日. 2021年7月2日. 新型コロナウイルスのワクチン接種と海外渡航について 2021年6月22日. 2021年7月2日. 大学間交換留学について 2021年6月15日. 2021年7月2日. オンライン授業による金銭的補償について 2021年5月31日. 2021年7月2日. 6月20日以降の授業形態について 2021年6月17日. 2021年6月30日. オンライン授業に対する意見 2021年6月15日. 2021年6月30日. KULASISの機能と健康診断の案内サイトについて 2021年6月15日. 2021年6月22日. 大規模健康診断を緊急事態宣言下に実施することの是非について 2021年6月13日. 2021年6月22日. クラシスの文字化けについて 2021年6月12日. 2021年6月22日. 京都大学の英語版Twitterについて 2021年6月11日. 2021年6月22日. 告示について（2） 2021年6月10日. 2021年6月22日. 新型コロナウイルスワクチンの職域接種について 2021年6月10日～. 2021年6月15日. 2021年6月17日. 「【注意喚起】大学内における違法な行為について」について 2021年6月16日～. 2021年6月17日. 2021年6月21日. 西部構内の駐輪場について 2021年6月2日. 2021年6月10日. 6月1日以降の授業形態について 2021年5月24日～. 2021年5月31日. 2021年6月10日. 5月18日回答「授業のオンライン化について」に対する意見 2021年5月20日. 2021年6月10日. 学生の大学構内での行動について 2021年6月2日. 2021年6月9日. 京都大学の住所について 2021年5月29日. 2021年6月9日. 桂キャンパスの電子錠について 2021年5月28日. 2021年6月9日. 授業実施形式の度重なる変更について 2021年4月26日. 2021年6月3日. これまでのご意見・ご要望. 各カテゴリー名をクリックすると投稿内容・回答内容の一覧ページが閲覧できます。. 正課・カリキュラム 課外活動 経済支援 施設・設備・環境 大学からの情報発信 その他 留意事項. ご意見・ご要望の投稿は「学生メール（KUMOI）」から送信されたものに限ります。. メールには投稿者の氏名と学籍番号の二つを記してください。いずれか一方でも未記入のメールは受け付けられません。. セキュリティ保持のために、ご意見・ご要望内容を添付ファイルにして送信したり、メール本文に学外のサイトのURLの参照を求めるなどの記載は行わないでください。. ご意見・ご要望には、内容に応じて関係部局等と協力して、回答します。内容によっては回答までに多少時間がかかる場合がありますので、予めご了承ください。. 大学からの回答は、原則的には、以下の二つの方法で行います。. 投稿者への返信. 本サイト内の「これまでのご意見・ご要望」ページへの掲載. 上記 5.-(2) の掲載にあたっては、投稿者の氏名・学籍番号などの個人情報は厳に秘匿し、いただいた個人情報は本学の学生支援の充実の目的以外には使用しません。. 上記 5.-(2) の掲載を希望しない場合は、投稿メール本文にその旨を明記してください。. いただいたご意見・ご要望に回答できない場合があります。その場合には回答できない旨を上記 5. の方法でお知らせします。. 投稿内容の秘密は厳守します。. その他、利用にあたっては以下の「学生意見箱 運用方針」を必ずお読みいただき、了解の上投稿してください。この学生意見箱については、下記の問い合わせ先までお願いします。. 　. 学生意見箱 運用方針（PDF） 問い合わせ先. 教育推進・学生支援部厚生課厚生掛. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 法人としての取組 新型コロナウイルス感染症への対応 学生の皆さんへ. 学生の皆さんへ. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 新型コロナウイルス感染症への対応 学生の皆さんへ 教職員の皆さんへ 総長メッセージ 本学における新型コロナウイルス感染症罹患者の発生について 武田／モデルナ社製 新型コロナワクチンへの異物混入の報道について（お知らせ） 熊本地震への対応 環境への取組 東日本大震災への対応 人権 障害に関すること 障害者雇用促進への取組 安全への取組 情報環境の整備 公益通報 監査 その他 男女共同参画について 公表事項 情報公開 In other languages. English 学生の皆さんへ. 新型コロナウイルスの集団感染を防ぐために、「換気が悪く、人が密に集まって過ごすような空間」に「集団で集まる」ことは避けるようにしてください。. （2022年3月18日）. 新入生勧誘活動について （2022年2月10日）. 令和4年度前期開講科⽬履修における要配慮申請について（通知） 　. 別紙様式  . （2022年2月8日）. 令和4年度前期の授業について （2021年12月10日）. 対面授業時の換気の徹底について（通知） （2021年11月9日）. 令和4年度授業時間割について （2021年10月22日）. 新型コロナウイルス感染症にかかる基本的な感染防止対策の徹底のお願い （2021年10月12日）. 対面授業の再開について （2021年10月1日）. 課外活動等の実施における感染拡大予防マニュアル（第10版） （2021年10月1日）. 課外活動の自粛要請について（第10版） （2021年9月17日）. 令和3年度後期の授業実施について （2021年8月11日）. 新型コロナウイルスの影響により、昨季と比べ収入が著しく減少した学生へ （2021年8月10日）. 京都大学修学支援基金緊急給付型奨学金の申請について （2021年8月2日）. 令和3年度後期開講科目履修における要配慮申請について（通知） 　　　　　　　　 . 申請書 （2021年6月21日）. 6月21日以降の以降の授業実施⽅法について（通知） （2021年6月15日）. 新型コロナウイルスワクチンの学内接種について （2021年5月31日）. 6月1日以降の授業の実施方法について（通知） （2021年5月10日）. 5月13日以降の授業の実施方法について（通知） （2021年4月19日）. 「新型コロナウイルス感染拡大防止に伴う活動制限のガイドラインにおける対応レベル」の変更に伴う授業の実施方法について（通知） （2021年4月12日）. 飲食を伴う懇親会・歓迎会などの会合等について （2021年3月10日）. 令和3年度授業時間割の変更について （2021年2月12日）. 令和3年度前期開講科目履修における要配慮申請について（通知） 　　　　　　　　. 　　　　　　　　　 . 申請書 （2021年2月12日）. 令和3年度の授業方針について （2020年12月21日）. 年末年始の忘年会・新年会・成人式等及び帰省の留意事項について （2020年10月8日）. 令和2年度後期の授業実施について（その2） （2020年8月31日）. 令和2年度後期の授業実施について  （2020年8月13日）. 百周年時計台記念館国際交流ホール「臨時自習室」の設置について （2020年7月31日）. 新型コロナウイルス感染症の予防のために（注意喚起）（第3版） （2020年7月24日）. 新型コロナウイルス感染症罹患者の発生を受けての要請について （2020年7月21日）. 課外活動及び施設使用の停止について （2020年6月25日）. 課外活動自粛要請の継続について （2020年6月1日）. 「学びの継続」のための『学生支援緊急給付金』の申請について （2020年5月20日）. 新型コロナウイルス感染症拡大の影響により困窮する学生に対する緊急支援プラン（緊急学生支援プラン）の実施について （2020年5月12日）. 【重要】令和2年度前期授業料納付期限の延期について（詳細のお知らせ） （2020年4月30日）. 【重要】新型コロナウイルス感染症の影響により授業料納付が困難になった方へ （2020年4月27日）. オンライン授業に係るネットワーク環境の支援について（大学院学生） （2020年4月24日）. オンライン授業に係るネットワーク環境の支援について（学部学生） （2020年3月31日）. 課外活動の自粛及び施設使用の一時停止について 海外渡航者向け. （2021年9月29日）. 新型コロナウイルス感染症に関する水際対策措置等について（水際対策上特に対応すべき変異株等に対する新たな指定国・地域について等） 外国人留学生向け. （2021年9月29日）. 新型コロナウイルス感染症に関する水際対策措置等について（水際対策上特に対応すべき変異株等に対する新たな指定国・地域について等） （2020年11月4日）. 再入国関連書類の変更について （2020年10月21日）. 新型コロナウイルス感染症の影響により帰国が困難な外国人留学生の在留資格について （2020年6月16日）. 在留資格諸申請に関する取扱い等について（上陸拒否の措置に関し、特段の事情があるものとして再入国を許可する具体的事例について、再入国許可期限経過による在留資格認定証明書交付申請について） （2020年4月3日）. 新型コロナウイルスへの対応について（第3報） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 令和5年度 特色入試入学者選抜方法における変更について（農学部森林科学科、食品生物科学科）（予告）. 入試 令和5年度 特色入試入学者選抜方法における変更について（農学部森林科学科、食品生物科学科）（予告）. ターゲット. 受験生の方 公開日. 2022年05月31日. 　農学部森林科学科、食品生物科学科では、令和5年度 特色入試入学者選抜方法について、次のとおり変更する予定です。. 　詳細は、7月頃に公表する特色入試学生募集要項や本Webサイトを必ずご確認ください。. （1）募集人員の増. 農学部森林科学科. 5名→7名. （2）求める人物像の変更. 農学部食品生物科学科. 最先端の生命科学研究に取り組むことに興味を持ち、大学卒業後に大学院修士・博士後期課程に進学して博士の学位を取得したうえで、国内外のアカデミックなポジション（大学、国の研究機関等）で、世界トップレベルの食と健康に関する基礎研究を推進する人材を求めています。. 関連部局. 農学部・農学研究科 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム EVENTS. EVENTS. 条件で絞り込む. 日付で絞り込む. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. 要申し込み. - すべて -. 必要. 不要. 開催地. - すべて -. 吉田キャンパス. 宇治キャンパス. 桂キャンパス. 東京オフィス・京都アカデミアフォーラム. その他の地域. オンライン. 開催日. 前月. 1月. 2月. 3月. 4月. 5月. 6月. 7月. 8月. 9月. 10月. 11月. 12月. 翌月. 日. 月. 火. 水. 木. 金. 土. 絞り込み条件をリセット. Events. タグ. 入試. 令和5（2023）年度入学者京都大学大学院総合生存学館（思修館）博士課程学生募集入試説明会. 開催日. 2022年06月10日、他. 開催地. 吉田キャンパス. 桂キャンパス. その他の地域. Events. タグ. 125周年. 公開講座. 京都大学創立125周年記念 市民講座「物理と宇宙」. 開催日. 2022年06月11日、他. 開催地. オンライン. Events. タグ. 研究. 2022年度 総合生存学ミニワークショップ［春］. 開催日. 2022年06月13日、他. 開催地. 吉田キャンパス. オンライン. Events. タグ. 教育. 公開講座. オンラインコース「教育評価の基礎講座」2022. 開催日. 2022年06月15日、他. 開催地. オンライン. Events. 創立125周年記念 アカデミックマルシェ. 開催日. 2022年06月18日. 開催地. その他の地域. Events. 体育会学生主催イベント 「京大体育会、まるごと魅せます」. 開催日. 2022年06月19日. 開催地. その他の地域. オンライン. Events. タグ. 教育. 公開講座. 教育学研究科 E.FORUM 2022 全国スクールリーダー育成研修「スクールリーダー育成のためのオンライン・リレー講座」. 開催日. 2022年06月19日、他. 開催地. オンライン. Events. タグ. 公開講座. 京都大学金融セミナー「『金融リテラシーが未来を拓く』2022 ～シン・金融教育元年を迎えて～」. 開催日. 2022年06月20日. 開催地. オンライン. Events. タグ. 入試. 2023年度 教育学研究科・教育学部学士入学 入試説明会. 開催日. 2022年06月25日. 開催地. オンライン. Events. タグ. 公開講座. 第121回 京都大学丸の内セミナー「雄と雌の対立が作り出す植物と花の多様な性」. 開催日. 2022年07月01日. 開催地. 東京オフィス・京都アカデミアフォーラム. Events. タグ. 公開講座. 第122回 京都大学丸の内セミナー「利己性の進化が維持する生物多様性」. 開催日. 2022年08月05日. 開催地. 東京オフィス・京都アカデミアフォーラム. Events. タグ. 公開講座. 第123回 京都大学丸の内セミナー「量子計算と量子暗号」. 開催日. 2022年09月09日. 開催地. 東京オフィス・京都アカデミアフォーラム. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム EVENTS EVENTS. EVENTS. 条件で絞り込む. 日付で絞り込む. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. 要申し込み. - すべて -. 必要. 不要. 開催地. - すべて -. 吉田キャンパス. 宇治キャンパス. 桂キャンパス. 東京オフィス・京都アカデミアフォーラム. その他の地域. オンライン. 開催日. 前月. 1月. 2月. 3月. 4月. 5月. 6月. 7月. 8月. 9月. 10月. 11月. 12月. 翌月. 日. 月. 火. 水. 木. 金. 土. 絞り込み条件をリセット. アイキャッチ画像. 京都大学創立125周年記念 附属図書館所蔵貴重資料展示. 開催日. 2022年06月01日〜12月22日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 産官学連携. 京大MBA×日経ビジネススクール共同講座「サービス経営のフロンティアとセオリー」. 開催日. 2022年06月09日、他. 開催地. オンライン. Events. タグ. 入試. 令和5（2023）年度入学者京都大学大学院総合生存学館（思修館）博士課程学生募集入試説明会. 開催日. 2022年06月10日、他. 開催地. 吉田キャンパス. 桂キャンパス. その他の地域. Events. タグ. 125周年. 公開講座. 京都大学創立125周年記念 市民講座「物理と宇宙」. 開催日. 2022年06月11日、他. 開催地. オンライン. Events. タグ. 125周年. 研究. 京都大学創立125周年記念 研究の魅力を見出す”鏡”プロジェクト. 開催日. 2022年06月11日. 開催地. オンライン. Events. タグ. 公開講座. 人文研アカデミー2022「ヴィリエ・ド・リラダンとフランス象徴主義 －『残酷物語』と『未来のイヴ』が現代に語りかけてくるもの」. 開催日. 2022年06月11日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 研究. 2022年度 総合生存学ミニワークショップ［春］. 開催日. 2022年06月13日、他. 開催地. 吉田キャンパス. オンライン. Events. タグ. 公開講座. 多様性を多様な視点で捉え、行動する ～京都大学と考えるD&Iセミナー～. 開催日. 2022年06月14日. 開催地. オンライン. Events. タグ. 教育. 公開講座. オンラインコース「教育評価の基礎講座」2022. 開催日. 2022年06月15日、他. 開催地. オンライン. Events. タグ. 研究. 第256回アフリカ地域研究会「南アフリカの社会保障制度改革とケアの政治」. 開催日. 2022年06月16日. 開催地. その他の地域. オンライン. Events. タグ. 入試. オンライン個別進学相談（2022年6月開催）. 開催日. 2022年06月17日、他. 開催地. オンライン. Events. タグ. 教育. グローバル教育展開オフィス主催：2022年度レクチャーシリーズ「ダイバシティー時代の「日本型教育」の行方 」第1回. 開催日. 2022年06月17日. 開催地. オンライン. Events. 創立125周年記念 アカデミックマルシェ. 開催日. 2022年06月18日. 開催地. その他の地域. Events. 体育会学生主催イベント 「京大体育会、まるごと魅せます」. 開催日. 2022年06月19日. 開催地. その他の地域. オンライン. Events. タグ. 教育. 公開講座. 教育学研究科 E.FORUM 2022 全国スクールリーダー育成研修「スクールリーダー育成のためのオンライン・リレー講座」. 開催日. 2022年06月19日、他. 開催地. オンライン. Events. タグ. 公開講座. 京都大学金融セミナー「『金融リテラシーが未来を拓く』2022 ～シン・金融教育元年を迎えて～」. 開催日. 2022年06月20日. 開催地. オンライン. Events. タグ. 研究. 学術情報メディアセンターセミナー「教育DXと音声言語処理」. 開催日. 2022年06月21日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 入試. 2023年度 教育学研究科・教育学部学士入学 入試説明会. 開催日. 2022年06月25日. 開催地. オンライン. Events. タグ. 入試. 高校生のための進学ガイダンス2022（岐阜）. 開催日. 2022年06月26日. 開催地. その他の地域. Events. タグ. 研究. ヒト生物学高等研究拠点 研究室紹介「ヒトは、いかにしてヒトとなりえたのか？」. 開催日. 2022年06月27日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 公開講座. プラットフォーム学連続セミナーVol. 12「モビリティとプラットフォーム学」～シェアリング・MaaS・数理最適化から考える「これからの移動のかたち」～. 開催日. 2022年06月30日. 開催地. オンライン. ページ送り. カレントページ. 1 ページ. 2 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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Tel：03-5252-7070　Fax：03-5252-7075. E-mail： t-office＊www.adm.kyoto-u.ac.jp （＊を@に変えてください）. お知らせ. 【重要】入館方法が変わります。（2022年3月25日より）. 京都大学東京オフィスでは2022年3月25日より、さらなる利便性向上を図るとともに、安心してご利用いただけるよう検温・マスク着用確認機能を有した顔認証による入退館システムを導入します。利用者の皆様におかれましては、本システム導入の趣旨をご理解いただき、来館の際にはぜひ利用者登録にご協力いただきますようお願いします。. 顔認証を使用した入退館方法. 来館時に顔認証機器の前にお立ちください。. 検温、消毒後、入館ボタンを押下し、利用目的を選択したら受付までお越しください。. 初回来館時のみIC教職員証・IC学生証・名誉教授証を受付にご提示ください。. 「入館記録票兼利用者登録票」および「個人情報保護方針」をお渡しします。. 在館中に「入館記録票兼利用者登録票」全ての欄に記入し、退館時受付にお渡しください。. 退館の際は、再度顔認証機器前にお立ちいただき、退館ボタンを押下ください。. 利用者登録（任意）. お渡しした「個人情報保護方針」をご一読ください。. 同意いただける場合は、「入館記録票兼利用者登録票」内、「利用者登録」欄の「利用者登録する」にチェックを入れ、受付までお持ちください。. 「利用者登録」には、顔写真の撮影等も含め、数分程度の時間がかかります。. 登録後は「入館記録票兼利用者登録票」の記入は不要となり、顔認証を使用した入退館方法の1．2の手続きのみで利用いただけます。. 【重要】施設利用上のお願い（2022年5月23日より）. 東京都のリバウンド警戒期間は終了しましたが、東京都より引き続き新型コロナウイルス感染症の拡大防止のため基本的な感染防止対策の徹底、感染を広げないための行動の要請、協力依頼がありますので、当施設利用の際は、以下記載のお願い事項をご一読いただき、ご理解、ご協力のほどよろしくお願いします。. お願い事項. 1．入館時・在館中. 入館時、手指の消毒. をお願いします。. 館内では、他の利用者と2メートルの間隔をあけることが難しいため、. 必ずマスクを着用願います. 。. ※マスクを着用いただけない場合、入館をお断りします。. 受付は、間隔を空けてお待ちください。. 2．会議室利用.  各会議室定員までご利用可能です。. 大会議室A： スクール形式 24名. 大会議室B： スクール形式 24名. 大会議室AB合同： スクール形式 48名、シアター形式 90名. 中会議室： 14名. 小会議室： 6名. 主催者側で参加者名簿を作成. いただき、万一の際の公共機関等からの問い合わせに対応できるようお願いします。. 会議室内での飲食は可能ですが、⼤⼈数や⻑時間におよぶ飲⾷・飲酒など、感染リスクの⾼い場⾯を避けるため、. 酒類持込・飲酒につきましてはご遠慮いただきますようお願いします。. 東京都から飲食、イベント、県をまたいだ移動等の活動に際して、参加者のワクチン接種歴及び陰性の検査結果をご確認いただくことが推奨されています。. 3．ラウンジ利用. ご利用人数は、座席数（カウンター10名・ソファ8名・テーブル4名）を上限とさせていただきます。. ラウンジが混みあってきた場合、最大2時間のご利用とさせていただくことがありますので、ご了承願います。. ラウンジ、館内での大きな声での会話、電話、Web会議の利用はご遠慮願います. 。. 教職員の方は、できるだけ小会議室（有料）をご予約願います。. ご利用者の皆さまにはご不便をおかけしますが、ご理解・ご協力のほどよろしくお願いします。. 新丸の内ビルディングへの入館方法に関するお知らせ. 新丸の内ビルディングにおける新型コロナウイルスの感染予防策を強化すべく「ビル入館対応」が以下の通りとなります。. ビル入館対応方法の変更内容. 1. 入館ルートの制限. 新丸の内ビルディングへの入館は. 1階は正面入口のみ. と. 地下1階入口（スターバックス前）. となります。. 2. 健康確認および検温の実施. 1階正面入口を入って右手にて、. 健康確認カードを記入. し、. 健康確認. および. 検温. 後、. 健康確認カードをエレベーターホール手前のビルスタッフに提出. の上での入館となります。. ただし、. 以下に該当する場合は入館を断られる. ことがあります。. 37.5℃以上の発熱がある場合. 強いだるさ（倦怠感）や息苦しさ（呼吸困難）がある場合. 検温に協力いただけない場合. お願い. 趣旨ご理解の上、. ビル側の対策にご協力いただくようお願いします. 。. ビル入館にあたり、. 上記理由等で入館を断られた場合はすみやかに退館をお願いします. 。. なお、入館の必要性がある場合は、各受付に電話連絡をお願いします。. （入館の目的等を確認のうえ、入館を判断させていただきます。）. 会議等の参加者に. 関しましては、. 予め主催者側の連絡先（電話番号）を通知願います. 。. （参加者の方が上記理由で入館を断られた場合は、主催者にて判断いただきます。）. 会議室のご予約について. 東京オフィス会議室の予約方法は、原則、ウェブ予約システムからとさせていただきます。. ウェブ予約システム （あらかじめ利用者登録が必要です。）. 会議室等予約方法 イベントのご案内. 現在ご案内はありません。. 主要なセミナーについて 東京オフィスについて. ご挨拶 アクセスマップ フロアマップ（施設のご案内） 利用案内. 利用案内 会議室等予約方法 （あらかじめ利用者登録が必要です。）. よくあるご質問 会議室等予約状況  パンフレット 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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2022","referrer_id_s":"3hN-L9U8j7Mo","language_s":"ja","size_i":80901,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085764188831744,"text_t":"任期中の基本方針 ―世界に輝く研究大学を目指して― | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 総長からのメッセージ 任期中の基本方針 ―世界に輝く研究大学を目指して―. 任期中の基本方針 ―世界に輝く研究大学を目指して―. サイドメニュー. 総長からのメッセージ 湊 長博総長からのメッセージ 任期中の基本方針 ―世界に輝く研究大学を目指して― 湊長博 総長 略歴 総長の動き 式辞・挨拶等 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 京都大学総長. 湊 長博. 　2001年に制定した京都大学の基本理念、2015年に策定した「京都大学の改革と将来構想（WINDOW構想）」、そして2017年に指定を受けた指定国立大学法人構想を踏まえ、私の総長任期中に新たに注力する取り組みを中心に、基本方針をまとめました。2020年10月の総長就任以降、執行部内で議論を重ね、まず2021年3月に学内において報告し、このたび、より多くの皆様にご覧いただきたくここに公開します。. 　世界に伍する研究大学を目指し、教育・研究支援体制の再構築、人材多様性の確保、財政基盤の強化という3つのビジョンのもと、教育と学生支援、教員の研究活動支援、業務運営体制の改善、施設、組織運営、基金活動の6つの課題について、それぞれ方向性を示しています。. 　今後、全教職員一丸となって、この方針に基づく具体的な取り組みを着実に実行していきます。また、大学を取り巻く状況の変化に応じて、本基本方針を改定していきます。皆様のご理解とご支援を賜りますようお願いします。. 任期中の基本方針 ―世界に輝く研究大学を目指して―. 　本学では、2017年における指定国立大学法人の指定に当たり、全学での議論を経て世界に伍する研究大学を目指すための中長期的基本戦略を策定した。この基本戦略自体は、第4期中期目標・中期計画期間においても維持されるべきものと考える。他方、総長就任以来、理事を含む執行部内での意見交換を踏まえて第4期に向けた職務遂行方針を検討してきたが、その中で従来の取り組みを変更すべき点や新たに必要となる取り組みが想定されるため、それらを中心に任期中の職務遂行の基本方針として本文書をとりまとめた。本文書に記した事項については、できる限り第４期中期目標・中期計画案に反映したいと考えている。. 　なお、中期目標・中期計画においては、従前からの計画の変更や新規計画だけでなく、継続すべき計画も含まれるので、本文書の内容がそのまま中期目標・中期計画案を意味するわけではない。本基本方針については、今後、状況の変化や新たな必要性に応じ、全学的議論を充分に踏まえた上で、逐次改定していくつもりである。　. 基本的な考え方. 　本学の基本理念に従った職務遂行を進める。その中でもとりわけ、自由の学風の下で独創的な研究を推進するため、多様で有為の人材が世界中から本学へ集うことのできる魅力ある教育・研究環境の整備と、自律的な運営を可能とする基盤の強化に注力したい。. 　魅力ある教育・研究環境を実現するための方策として、学生の修学環境や教育内容・体制の改善とともに、教員の教育・研究活動を支援する体制の整備や処遇の改善に取り組む。特に、学生・教員の多様性の確保は重要であり、優秀な海外留学生の増加や若手・女性教員の増員・育成に充分配慮する。また、自律的な運営を行う基盤の強化のため、社会への発信力の強化、産学連携活動や基金活動の一層の推進に積極的に取り組む。これらにより、本学の社会における存在感を高めるとともに、社会からのより大きな支援獲得へと繋げたい。. 　以上の基本的な考え方を具体化するための取り組みについて、主に従前からの変更点や新たな取り組みを中心に以下に述べる。. 具体的な取り組み. I.　教育と学生支援. 　学部・大学院において、熱意に溢れ適性に優れた多様な入学者の確保とその育成のため、学生の生活支援と福利厚生の向上、学生の希望と時代のニーズに対応した教育内容・体制の改善を進める。. （1）学部教育. 入学者選抜等；女子学生の比率向上に向け、中・高生や保護者への働きかけを強化するとともに、Kyoto iUPを拡充して優秀な海外留学生の増加を図る。入試については、現在の制度を当面維持するが、特色入試については、その趣旨と入学後のカリキュラムの整合性や同入学者の修学状況及び学部卒業者の追跡調査に基づいて適宜検証を進める。. 学部入学定員と転学部・転学科；学部の入学定員については、中長期的な社会の要請や学術領域の変化への対応を含めて、引き続き検討する。また、入学後の学生の進路志望変化などに適切・柔軟に対応できるよう、転学部・転学科制度の再検討を進める。. 学部カリキュラム改革；各学部の要卒単位数が適切であるか再検討を行う。特に専門科目の増加に伴い、教養・共通教育については、一部で学生が外形的な単位数収集のみに終始してしまうという懸念がある。教養・共通教育がより効果的・魅力的なものとなるようカリキュラム改革を進めるとともに、要卒単位数や必修科目などについて再検討を進める。. 短期海外留学；海外留学促進のため、各学部の要卒単位数の再検討、海外大学との単位互換の拡張などを積極的に進める。語学留学の単位化を推進し、単位認定を伴わない短期留学及び受入れについては再検討を行う。. （2）大学院・専門職大学院教育. 給付型奨学金制度の拡充と整理；従来の日本学術振興会特別研究員やTA／RA制度に加え、新たに文部科学省による博士後期課程への給付型奨学金の拡充が予定されているが、並行して大学の独自基金による奨学金制度の導入により、全体としてバランスのとれた大学院生の生活支援を拡充し、優秀な学生の大学院進学を促進する。複雑化する各種奨学金制度の趣旨と内容（生活支援、研究支援など）を整理し、その情報をわかりやすく学生に開示・説明できる体制を作る。. 大学院定員と専攻組織の改革；大学院定員の管理柔軟化を引き続き文部科学省に要求していくが、現行定員が社会的要請や学術動向の変化に適切に対応できているかについての検証を進める。専攻・講座の統合・集約化（大講座制など）等の組織改革とそれに応じた大学院課程のカリキュラム改革を進め、全体として効果的な研究者育成プログラムを構築していく必要がある。. 優秀な留学生の獲得；海外の諸拠点も活用しつつ、大学院研究科全体としての統一的な学生リクルート活動を展開し、受け身ではなく積極的な海外の優秀な大学院生の獲得を推進する。そのため、次に述べる大学院教育支援機構（仮称）の内部にリクルーティング・オフィスを設置し、全学大学院統一ポータルサイトの一層の充実化を図るとともに、現行の留学生の授業料・入学料免除や奨学金システムを再検討して、優秀な留学生の確保につながる仕組みへ変更する。. 大学院教育支援機構（仮称）の設置；上記の諸活動に加え、大学院共通科目の充実、各種大学院学位プログラム（卓越大学院、ジョイントディグリー、ダブルディグリーなど）の統括などを一元的に進めるために、全学組織としてGraduate Divisionに相当する大学院教育支援機構(仮称）を新たに設置する。これにより、各研究科の多方面での教育・研究活動における連携協力を推進する。. （3）学生生活. 学生相談窓口の強化；学部・大学院を問わず、学生の生活、心身の健康、修学状況などにかかる相談のニーズは急速に増加しており、学生総合支援センターでの対応は限界を超えてきている。少なくとも各キャンパスに、学生の相談・支援全般に対応しうる窓口体制を整備する。. 学生サービス環境の改善；老朽化している福利厚生施設の更新など、学生サービス環境の改善を図る。. II.　教員の研究活動支援. 　国内外の若手・女性を含む多様で卓越した研究者人材の獲得、及びそのために、教員が充分に教育・研究活動に専念できる環境の整備が最重要課題であると考える。. （1）研究支援体制の再構築. 研究遂行の支援にかかる要員（事務系秘書、技術系職員、URAなど）を増員・拡充し、また、その雇用と運用については全学的に一元化された体制を整備して、これにかかる教員の負担を軽減する。. 全学の諸会議・委員会などの設置、構成、運用方法の見直し､事務諸手続きの電子化を進め、教員の教育・研究エフォート率の実質的増加を促進する。. 部局教員の基礎的研究経費配分の増加を検討するとともに、URAによる競争的研究費獲得支援、競争的研究費応募にかかるセーフティーネットのための学内研究費の仕組みなどの強化を進める。. （2）教員の処遇改善. 人事給与制度を改正し、民間の研究職の給与額も参考にしつつ、教員の教育・研究における業績が透明性の高い評価に基づき適正に処遇に反映される制度を導入する。. 現在極めて多数に及ぶ身分の不安定な有期雇用教員（特定教員）については、可能な限りテニュア・トラック制への移行を図り、適正な業績評価に基づき正規教員へのキャリアパスの拡大を図る。. 保育室など、若手教員の研究支援環境の整備を進める。. （3）教員の多様性の確保. 本学の現状では、女性教員、若手教員、外国人教員の割合はいずれも全国主要国立大学で最低のレベルにあり、大学の社会的信頼性や将来性において大きな懸念材料となっている。外部資金の導入、教員・職員間の定員枠の柔軟化、教員定員再配置の仕組みの検討など様々な方法により、教員の多様化を推進する。. 各学系において現況を把握し、その状況に応じて適切な若手・女性教員比率の目標設定を行う。. （4）教育研究組織の改革. 教育研究組織の多様性が本学の特徴であるが、各組織の規模に非常に大きな開きがあることが事務の効率性やコンプライアンスなどの点でしばしば問題となっている。学術領域の変化や社会的要請を踏まえて各教育研究組織の設置目的と現況評価を改めて検証し、必要に応じて組織再編や統合を促す。. 学域・学系制の趣旨を最大限に活用しつつ、より柔軟で機能的な教育研究組織の改編を進める。. （5）研究活動の国際化と成果の発信. 海外拠点の役割・機能を検証し、教育・研究の国際展開に資する効果的な体制に整備する。. On-site Laboratoryについては、自律的運営を確立し継続的な国際共同研究の拠点化を目指す。. 海外重点研究機関との戦略的パートナーシップにより、形式的な大学間学術交流協定から、より実質的で恒常的な国際共同研究の強化へ移行する。. 人文・社会科学分野の教育・研究成果に関する社会的発信力を高めるため、ハブとなる組織の整備などを行う。. 図書館機構による研究成果の自律的なオープンアクセス方針を推進する。. （6）産学共同・連携活動の推進. 産官学連携本部の機能の充実・強化と透明性の確保を進め、特に全学の教育研究組織・研究者とのコミュニケーションを促進する。学内研究者との情報交換の窓口を拡大し、大学発ベンチャーを含め学内シーズの発掘・支援機能を高める。. 民間からの博士後期課程学生の在籍受け入れなど、大学院教育を基礎にした民間との共同研究の連携支援を拡大する。. III.　業務運営体制の改善. 　コンプライアンスに最大限の配慮をしつつ効率的で機能的な業務運営を行うために、事務組織や雇用体系の抜本的改善と事務職員の政策立案能力向上に向けての取り組みを進め、全体として非効率的経費の削減に努める。. （1）本部と部局の事務改革. 事務本部については、縦割り意識の解消、部門間の連携と情報共有の強化により、効率的な業務運営ができるよう組織再編を進める。. 事務本部の事務組織再編に応じて各部の学内予算再配分機能を見直し、適正で効果的な予算配分と執行体制を整備する。. 現行の共通事務部のあり方を再検証し、事務本部及び部局事務部との効率的な役割分担と協力体制をつくる。. 新規に構築する研究支援体制と各事務部との連携を強化し、両者のシームレスな関係を構築する。. （2）全学機能組織の見直し. 全学機能組織の必要性、役割、適正な規模を再検証し、必要に応じて統廃合を進める。. 各機構を支える事務組織の役割についても明確にし、必要な規模に見直す。. （3）財務改革. 各事務部への慣例的、固定的な予算配分方式や単年度予算消化体制などについては、抜本的な見直しを進める。. 総長裁量経費、全学経費、アクションプランなどの本部組織の基盤的経費については、慣例的配分を抜本的に見直し、役割や運用の評価に基づき適正な配分を行う。. 部局に対し各種経費（機能強化経費、施設維持経費など）で措置される予算や、新たに基盤化された予算についても、事業内容や実績の評価を踏まえ必要に応じて運用の見直しを進める。. （4）職員の人事給与制度の改善. 職員の昇給・昇格については、一律の年功序列的運用によることなく、職責を明確化し、その業績評価に基づく運用を基本とする。これにより、職階と給与の逆転現象など職員の労働意欲の阻害につながるような弊害を防ぐ。. 職員のキャリアパスとして、一般事務職と総合職のコースを作り、職員の事情によりコース変更も可能な制度を検討する。. 特定職員（URA、法務職、技術職など）について、本人の希望と実績に基づき一般事務職と並立の正規職員としてのポストを創設する。. （5）医学部附属病院の管理運営. 先端医療研究開発機構（. iACT. ）を拠点として、企業治験を含む臨床試験の遂行と支援機能を充実強化し、世界的に遅れを取る我が国の先端臨床研究のリーダーシップをとる。. 産官学連携本部と密に連携して大学出資事業や企業連携などを最大限に活用し、医療情報や機器開発などの事業展開を積極的に進めて、医学部附属病院の財務基盤強化を進める。. 教員兼任制による病院管理運営と事業推進には限界があり、専任担当者の採用などを含めて今後の体制についての抜本的な検討を始める。. IV.　施設. 　大学の施設整備計画について、その資金計画を含めて検討を進め確定していく。. （1）キャンパスマスタープラン. コロナ禍での教訓とポストコロナ時代の想定を踏まえ、施設の安全性、機能性の向上に向け各キャンパスの既存マスタープランを再検討する。特に、最大の学生数を収容する吉田南キャンパスの整備が喫緊の課題である。. （2）創立125周年記念事業. （1）の事情を勘案し、当初の事業案を見直し、国際的にも魅力あるキャンパス環境に資する学生の福利厚生施設など明らかに必要性の高い施設への変更案を速やかに検討する。. V.　組織運営. 　各レベルでの組織運営の責任の所在の明確化と、大学運営を担う次世代の人材育成の仕組み作りを進める。. （1）部局長の職務. 部局長が部局運営の一義的責任者であることを明示し、部局長の裁量性を拡大するとともに必要な全学的情報が充分に提供される仕組みを整備する。また、これに対応して、各部局には部局長の任期についての検討を促す。. （2）総長・役員会. 総長・役員会は教育・研究活動にかかる全学的環境整備に責任を負うが、特に全学機能組織の整備と運用、全学教員部における人事と人員配置に一義的責任を持つ。. プロボストは、次世代の大学運営人材の養成に充分に配慮して、戦略調整会議の構成員の選定と運営を進める。. VI.　基金活動、その他. 　大学の財務基盤強化に向けて、基金活動、同窓会活動のための事務体制を充実強化する。特に今後は、法人に加え、個人を対象とする基金活動に重点を置き、海外を含む地域同窓会、保護者会やファンクラブの組織化と広報活動を強く推進する。.  . 令和3年3月5日. 京都大学総長. 湊 長博.  . 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 法人としての取組 新型コロナウイルス感染症への対応 本学における新型コロナウイルス感染症罹患者の発生について. 本学における新型コロナウイルス感染症罹患者の発生について. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 新型コロナウイルス感染症への対応 学生の皆さんへ 教職員の皆さんへ 総長メッセージ 本学における新型コロナウイルス感染症罹患者の発生について 本学における新型コロナウイルス感染症罹患者の発生について（2022年2月2日～2022年3月30日） 本学における新型コロナウイルス感染症罹患者の発生について（2021年10月1日～2022年1月26日） 武田／モデルナ社製 新型コロナワクチンへの異物混入の報道について（お知らせ） 熊本地震への対応 環境への取組 東日本大震災への対応 人権 障害に関すること 障害者雇用促進への取組 安全への取組 情報環境の整備 公益通報 監査 その他 男女共同参画について 公表事項 情報公開 本学における新型コロナウイルス感染症罹患者の発生について. 2022年6月8日（水曜日）. 　5月18日（水）から6月3日（金）に学生21名及び教職員4名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在ＰＣＲ検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年6月1日（水曜日）. 　5月19日（木）から5月31日（火）に学生34名及び教職員10名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年5月25日（水曜日）. 　5月9日（月）から5月23日（月）に学生49名及び教職員5名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年5月18日（水曜日）. 　4月19日（火）から5月13日（金）に学生44名及び教職員10名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年5月11日（水曜日）. 　4月11日（月）から5月8日（日）に学生43名及び教職員5名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年4月27日（水曜日）. 　4月14日（木）から4月25日（月）に学生25名及び教職員9名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年4月20日（水曜日）. 　4月6日（水）から4月18日（月）に学生22名及び教職員14名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年4月13日（水曜日）. 　3月28日（月）から4月11日（月）に学生25名及び教職員13名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。. 2022年4月6日（水曜日）. 　3月16日（水）から4月5日（火）に学生23名及び教職員12名が新型コロナウイルスに感染していることが判明しました。. 　確認された濃厚接触者については. 、現在PCR検査や自宅待機を実施しております。. 　なお、感染者が立ち入った場所については、消毒を実施するなどして感染拡大防止の対応を行っております。.  . 京都大学の取組.  . 　本学では、危機対策本部において、「新型コロナウイルスに対する本学の方針について」を定め、学生及び教職員にも学内の掲示板やメール、本学ホームページ等で周知に努めております。. 　引き続き、国、京都府、京都市、所轄保健所等関係機関の対応に鑑み、それらの情報も踏まえ新型コロナウイルス感染症のまん延防止のために必要な対策を講じ、感染予防・拡大防止等に努めてまいります。. 　なお、感染症法の趣旨を踏まえ、感染者やそのご家族・関係者等が特定されることがないよう人権の尊重・個人情報保護にご留意いただきますようご理解とご配慮をお願いします。. ・. 参考：本学における新型コロナウイルスへの対応について 上記以前の罹患者の発生状況. 本学における新型コロナウイルス感染症罹患者の発生について（2022年2月2日～2022年3月30日） 本学における新型コロナウイルス感染症罹患者の発生について（2021年10月1日～2022年1月26日） 本学における新型コロナウイルス感染症罹患者の発生について（2021年9月7日～28日） 本学における新型コロナウイルス感染症罹患者の発生について（2021年8月4日～27日） 本学における新型コロナウイルス感染症罹患者の発生について（2021年7月9日～30日） 本学における新型コロナウイルス感染症罹患者の発生について（2021年6月4日～6月29日） 本学における新型コロナウイルス感染症罹患者の発生について（2021年5月6日～5月26日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年4月14日～4月28日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年4月13日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年4月12日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年4月9日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年4月7日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年4月5日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年3月1日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年2月22日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年2月8日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年2月3日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年1月26日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年1月22日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年1月14日）. 本学における新型コロナウイルス感染症罹患者の発生について（2021年1月4日）. 本学における新型コロナウイルス感染症罹患者の発生について（2020年12月28日）. 本学における新型コロナウイルス感染症罹患者の発生について（2020年12月24日）. 本学における新型コロナウイルス感染症罹患者の発生について（第6報）（2020年8月29日）. 本学における新型コロナウイルス感染症罹患者の発生について（第5報）（2020年8月24日）. 本学における新型コロナウイルス感染症罹患者の発生について（第4報）（2020年7月24日）. 本学における新型コロナウイルス感染症罹患者の発生について（第3報）（2020年7月22日）. 本学における新型コロナウイルス感染症罹患者の発生について（第2報）（2020年7月21日）. 本学における新型コロナウイルス感染症罹患者の発生について（2020年4月9日）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション（繁体中国語）. 京都大學概況. 校長的話. 大學介紹. 校內安全措施. 教育與學生支援服務. 學生生活. 留學・國際教育. 研究・產官學合作. 支援研究體制. Security Export Control. 國際交流. 學部與院係. ヘッダー セカンダリーリンク（日本語以外）. 交通資訊 聯絡我們 捐贈 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 導航連結. Home NEWS 舉行了2021學年度畢業典禮。（2022年3月24日）. University News 舉行了2021學年度畢業典禮。（2022年3月24日）. In other languages. 日本語 ターゲット. Current Students & Staff Visitors 發表日期. 2022\/03\/29. 　 2022年3月24日，2021學年度畢業典禮於京都市勸業館Miyako Messe舉行，井村裕夫前校長、各理事與副校長、各部門領導出席了儀式。. 畢業生人數 湊長博校長的致辭. 2021年學年度畢業典禮 致辭  . Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 發表日期. 2022\/04\/11. 舉行了2022學年度研究所入學典禮。（2022年4月7日）. 發表日期. 2022\/04\/08. 舉行了2022學年度大學入學典禮。（2022年4月7日）. 發表日期. 2022\/04\/05. 舉行了2021學年度研究所學位授予儀式。（2021年3月24日）. 發表日期. 2022\/03\/29. 舉行了2021學年度畢業典禮。（2022年3月24日）. 最新新聞 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通資訊 フッター セカンダリーリンク. 站點地圖 Privacy policy Site Policy フッター スペシャルリンク. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS. 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News. 公開日. 2022年02月18日. タグ. 産官学連携. 本学とNTT西日本による産学共同「オープンラボ（名称：Platform Initiative Lab）」を設立しました. News. 公開日. 2022年02月18日. タグ. 大学の動き. 令和3年度国際危機管理・国際法務講習会を実施しました. News. 公開日. 2022年02月18日. タグ. 大学の動き. 次期高等教育研究開発推進センター長に飯吉教授を再選しました. News. 公開日. 2022年02月18日. タグ. 教育. 京都大学のフィールド実習をオンライン体験できる映像教材を公開しました. News. 公開日. 2022年02月18日. タグ. 教育. 国際交流. 2021年度冬季アジア未来リーダー育成奨学金プログラム（AFLSP）成果報告会を開催しました. News. 公開日. 2022年02月16日. タグ. 教育. 国際交流. 京都大学化学学生発掘事業を実施しました. News. 公開日. 2022年02月15日. 村上章 理事・副学長、谷誠 名誉教授および小﨑隆 名誉教授が、2022年度日本農学賞および読売農学賞を受賞しました. News. 公開日. 2022年02月15日. タグ. 受賞. 第14回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式の開催について. ページ送り. 先頭ページ. chevron_left. chevron_left. 前ページ. chevron_left. &hellip;. ページ. 2 ページ. 3 ページ. 4 ページ. 5 カレントページ. 6 ページ. 7 ページ. 8 ページ. 9 ページ. 10 &hellip;. 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 高大連携 高大連携・入試広報事業. 高大連携・入試広報事業. サイドメニュー. 一般選抜 特色入試 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 高大連携・入試広報事業 学びコーディネーター事業 ELCAS（エルキャス） 高大接続・入試センター 関連情報 事業概要. 本学では「国際展開を担えるグローバル人材」、「自ら課題を発見し、チャレンジする人材」を求めています。そのためには、特に入試広報と高大連携を一体化させた取り組みがますます重要となっており、高大連携・入試広報事業に力を入れています。. 年間を通じて高等学校および予備校等との連携による計画的・効率的な入試広報業務や、高等学校および教育委員会との連携協定に基づく連携事業を行っています。. 最新のお知らせやイベントについては、以下のぺージのNews、Event欄をご覧ください。. 入試・高大連携 社会連携 2021年度. 令和3年度京都大学高大連携事業 高校生による研究発表大会を開催しました（2022年3月12日） 兵庫県教育委員会との高大連携協定に基づく「令和3年度高大連携課題研究合同発表会 at 京都大学」を実施しました（2021年11月3日） 京都大学サマープログラム2021を開催しました（2021年9月26日、10月24日、10月25日～10月29日） 2021年度 京都大学 高大連携事業 京都大学スーパーレッスンの実施について（2021年07月02日） 2020年度. 「京都大学サイエンスフェスティバル2020」を開催しました。（2021年3月13日、14日） 高校生による研究発表大会「京都大学ポスターセッション2020」をオンライン開催しました。（2021年3月20日、21日） 高大連携・入試広報ポータルサイト「LEOPARD」マスコットキャラクターの愛称が決まりました。（2021年1月12日） 「京都大学チャレンジセミナー2020 in TOKYO」を開催しました。（2020年10月31日） 京都大学サマープログラム2020「暑い夏の1週間、京大生になろう！」をオンライン開催しました。（2020年8月15日～21日） 2019年度. 兵庫県教育委員会との連携協定に基づく高大連携事業を実施しました。（2019年11月4日） 学びコーディネーターによる課題探求活動支援型授業を実施しました。（2019年10月30日） 高大連携事業「京大発見ゼミ2019」を開催しました。（2019年10月25日） 「京都大学 学びのフロンティア2019」を開催しました。（2019年9月21日） 京都大学サマースクール2019「暑い夏の1日、京大生になろう！」を開催しました。（2019年8月17日） 「京都大学チャレンジセミナー2019」を実施しました。（2019年8月8日） 2018年度. 「京都大学サイエンスフェスティバル2018 －科学の頭脳戦－」を実施しました。（2019年3月16日） 「京都大学ポスターセッション2018」を実施しました。（2019年3月16日） 「京都大学　くすのきセミナー2018in福岡」を開催しました。（2018年12月22日) 大阪府教育委員会との連携協定に基づく高大連携事業を実施しました。（2018年11月4日） 「京都大学 学びのフロンティア2018」を開催しました。（2018年10月20日） 本学主催の高大連携事業「京大発見ゼミ2018」を開催しました。（2018年9月13日） 京都大学サマースクール2018「暑い夏の1日、京大生になろう！」を開催しました。（2018年8月18日） 2017年度以前. 「京都大学サイエンスフェスティバル2017 －科学の頭脳戦－」を実施しました。（2018年3月17日） 滋賀県教育委員会との連携協定に基づく高大連携事業を実施しました。（2017年12月17日） 東京都教育委員会との連携協定に基づく高大連携事業を実施しました。（2017年11月18日） 奈良県との連携協定に基づく高大連携事業を実施しました。（2017年9月23日） 京都大学サマースクール2017「暑い夏の1日、京大生になろう！」を開催しました。（2017年8月11日） 京都市教育委員会との連携協定に基づく高大連携事業を実施しました。（2017年7月29日） 「京都大学サイエンスフェスティバル2016 －科学の頭脳戦－」を実施しました。（2017年3月18日） 大阪府教育委員会との連携協定に基づく高大連携事業を実施しました。（2016年11月23日） 京都大学サマースクール2016「暑い夏の1日、京大生になろう！」を開催しました。（2016年8月19日） 京都大学高校生フォーラム in Tokyo感想文コンクール表彰式を開催しました。（2015年1月28日） 各教育委員会との連携協定に基づく高大連携事業を実施しました。（2014年12月13日・18日・20日） 京都大学高校生フォーラム in TOKYOを開催しました。（2014年11月7日） 京都大学は福井県教育委員会との連携に関する協定を締結しました。（2014年8月18日） 京都大学サマースクール2014「暑い夏の1日、京大生になろう！」を開催しました。（2014年8月19日） 京都大学は徳島県教育委員会および徳島市教育委員会との連携に関する協定を締結しました。（2014年8月5日） 京都大学と石川県教育委員会との連携に関する協定を締結しました。（2014年7月23日） 京都大学と東京都教育委員会との連携に関する協定を締結しました。（2014年7月16日） 京とーく2014（京都大学・立命館大学合同イベント・広島会場）を実施しました。（2014年7月13日） 本学と三重県教育委員会との連携に関する協定を締結しました。（2014年5月19日） 本学と奈良県との教育分野の連携に関する協定を締結しました。（2014年4月1日） 本学と和歌山県教育委員会との連携に関する協定を締結しました。（2014年3月25日） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 総長からのメッセージ 湊長博 総長 略歴. 湊長博 総長 略歴. サイドメニュー. 総長からのメッセージ 湊 長博総長からのメッセージ 任期中の基本方針 ―世界に輝く研究大学を目指して― 湊長博 総長 略歴 総長の動き 式辞・挨拶等 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 生年月日： 1951年1月 生. 学位： 京都大学医学博士（1983年7月）. 略歴. 1975（昭和50）年3月. 京都大学医学部卒業. 1975（昭和50）年7月. 京都大学結核胸部疾患研究所附属病院医員（研修医）. 1977（昭和52）年7月. 京都大学結核胸部疾患研究所附属病院医員. 1977（昭和52）年10月. 米国アルバートアインシュタイン医科大学微生物免疫学教室客員研究員. 1980（昭和55）年9月. 自治医科大学助手. 1982（昭和57）年10月. 自治医科大学講師. 1990（平成2）年8月. 自治医科大学助教授. 1992（平成4）年6月. 京都大学医学部附属免疫研究施設教授. 1993（平成5）年4月. 京都大学医学部教授. 1995（平成7）年4月. 京都大学大学院医学研究科教授. 1999（平成11）年4月. 京都大学大学院生命科学研究科教授. 2002（平成14）年4月. 京都大学評議員（2004（平成16）年3月まで）. 2007（平成19）年4月. 京都大学大学院医学研究科教授（2016（平成28）年3月まで）. 2007（平成19）年10月. 京都大学大学院医学研究科附属ゲノム医学センター長（2008（平成20）年10月まで）. 2010（平成22）年10月. 京都大学大学院医学研究科長・医学部長（2014（平成26）年9月まで）. 2014（平成26）年10月. 京都大学理事・副学長（2020（令和2）年9月まで）. 2017（平成29）年10月. 京都大学プロボスト（2020（令和2）年9月まで）. 2020（令和2）年10月. 京都大学総長.  . 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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(2022년 4월 7일) | 교토대학. 주요 콘텐츠로 건너뛰기 사이트 검색. 연구원. 찾기. close. CLOSE. search. menu. グローバルナビゲーション（韓国語）. 교토대학 소개. 대학소개. 총장인사. 안전. 교육・학생 지원. 유학・국제교육. 입시. 학사일정. 학생생활. 연구・산관학 연계. Security Export Control. 국제교류. 연구지원체제. 젊은층 연구자 지원. 학부・대학원. ヘッダー セカンダリーリンク（日本語以外）. 교통편 안내 문의처 기부 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 이동 경로. Home NEWS 2022년도 학부 입학식이 거행되었습니다. (2022년 4월 7일). University News 2022년도 학부 입학식이 거행되었습니다. (2022년 4월 7일). In other languages. 日本語 ターゲット. Current Students & Staff Visitors 발행일. 2022\/04\/08. 　2022년 4월 7일(목요일), 교토시 권업관 ‘미야코멧세’에서, 마쓰모토 히로시(松本紘) 전 총장, 야마기와 주이치(山極壽一) 전 총장, 각 이사 및 부학장, 각 부국장이 참석한 가운데 2022년도 학부 입학식이 거행되었습니다. 입학식에서는 미나토 나가히로(湊長博) 총장이. 축사 를 전했습니다. 금년도 학부 입학생은 2,938명입니다. 학부 입학생 수 미나토 나가히로 총장 축사. 2022년도 학부 입학식 축사(2022년 4월 7일) Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 발행일. 2022\/04\/11. 2022년도 대학원 입학식이 거행되었습니다. (2022년 4월 7일). 발행일. 2022\/04\/08. 2022년도 학부 입학식이 거행되었습니다. (2022년 4월 7일). 발행일. 2022\/04\/05. 2021년도 대학원 학위 수여식이 거행되었습니다. (2022년 3월 24일). 발행일. 2022\/03\/29. 2021년도 졸업식이 거행되었습니다. (2022년 3월 24일). 더 많은 뉴스 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 간행물 채용 교통편 안내 フッター セカンダリーリンク. 사이트 맵 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"mtg6s8U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/sites\/default\/files\/inline-files\/1-kengyo-r205-9b14ef97be7215f41c102371088fceb7.pdf","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":109,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["sites","default","files","inline-files"],"url_file_name_s":"1-kengyo-r205-9b14ef97be7215f41c102371088fceb7","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Microsoft Word - 1-2【見消】兼業指針.docx"],"author":"170307d","author_sxt":["170307d"],"last_modified":"Fri May 22 02:56:47 UTC 2020","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":6210115191197182593,"exact_signature_unique_b":true,"fuzzy_signature_l":-6702198101866101071,"fuzzy_signature_unique_b":true,"wordcount_i":625,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:45:52 UTC 2022","fresh_date_dt":"Sun Jan 22 11:48:11 UTC 2023","referrer_id_s":"jpjC89U8j7Mo","publisher_t":"Acrobat Distiller 17.0 (Windows)","language_s":"ja","size_i":180902,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085787960049664,"text_t":"1 \n国立大学法人京都大学に勤務する教職員の兼業に関する指針 \n \n平成１６年 ４月 １日 \n    \n第１章 総則 \n（目的） \n第１条 この指針は、国立大学法人京都大学（以下「大学」という。）に勤務する教職員（以下\n「教職員」という。）の兼業の取扱いに関する事項を定める。 \n（定義） \n第２条 この指針において「兼業」とは、報酬の有無にかかわらず、継続的又は定期的に次に掲げ\nることを行う場合をいう。 \n(1) 商業、工業又は金融業その他営利を目的とする会社その他の団体（株式の全てを大学が保有\nするものを除く。以下「営利企業」という。）の役員、顧問又は評議員の職を兼ねること。\n（以下「営利企業の役員等兼業」という。）  \n(2) 自ら営利企業を営むこと。（名義人が他人であっても本人が営利企業を営むものと客観的に\n判断される場合を含む。）（以下「自営の兼業」という。） \n (3) 営利企業の事業に直接関与しない職を兼ねること。（以下「営利企業の事業に直接関与しな\nい兼業」という。） \n (4) 営利企業以外の各種法人及び法人格を有しない団体（以下「営利企業以外の法人等」とい\nう。）の役員の職又はその事業の職を兼ねること。（以下「営利企業以外の法人等の兼業」と\nいう。） \n (5) 公立、私立の学校、専修学校、各種学校等の教育施設等で教育に関する事業又は事務の職を\n兼ねること。（以下「教育に関する兼業」という。） \n (6) 国又は地方公共団体の行政機関（附置された機関及び施設並びに病院等を含み、前号の「教\n育施設等」を除く。以下「国等の行政機関」という。）に設置されている審議会等の非常勤の\n職、研究機関若しくは研究施設の非常勤の職、非常勤医師又はその他必要に応じて置かれてい\nる職を兼ねること。（以下「国等の行政機関の兼業」という。） \n (7) 独立行政法人通則法（平成１１年法律第１０３号）第２条第１項の規定に基づき、個別法に\nより設置された法人及び地方独立行政法人法（平成１５年法律第１１８号）に基づき設置され\nた法人の職を兼ねること並びに国立大学法人法（平成１５年法律第１１２号）に基づき設置さ\nれた国立大学法人又は大学共同利用機関法人（国立大学法人又は大学共同利用機関法人により\n設置された国立大学及び大学共同利用機関の職を含む。）の職を兼ねること（以下「独立行政\n法人等の兼業」という。） \n２ 前項に規定する「継続的又は定期的」とは、次の各号のいずれかに該当するもの以外のものを\nいう。 \n (1) １日限りのもの \n (2) ２日以上６日以内の場合で総時間数が１０時間未満のもの \n  \n   第２章 営利企業の役員等兼業 \n    第１節 営利企業の役員等兼業 \n（許可できる営利企業の役員等兼業） \n第３条 営利企業の役員等兼業は、原則として許可しない。ただし、次に掲げる兼業で、当該各兼\n業の許可基準のいずれにも該当する場合には、許可することができるものとする。 \n (1) 技術移転事業者の役員（会計参与、監査役及び社外取締役を除き、取締役、執行役、業務を\n執行する社員、理事、支配人、発起人及び清算人その他これらに準ずる者をいう。）、顧問又\nは評議員（以下「役員等」という。）を兼ねる場合 \n (2) 研究成果活用企業の役員等を兼ねる場合 \n (3) 株式会社（会社法（平成１７年法律第８６号）第二編に定める会社をいう。会社法の施行に\n伴う関係法律の整備等に関する法律（平成１７年法律第８７号）第３条に規定する特例有限会\n社を含む。以下同じ。）の監査役又は信用金庫（信用金庫法（昭和２６年法律第２３８号）に\nより設立される法人をいう。以下同じ。）の監事を兼ねる場合 \n (4) 株式会社の社外取締役又は信用金庫の理事のうち株式会社の社外取締役に相当するもの（以\n2 \n \n下「社外取締役等」という。）を兼ねる場合 \n    第２節 技術移転事業者の役員等の兼業 \n（許可権者） \n第４条 教員（教授、准教授、講師、助教及び助手の職にあるものに限る。以下同じ。）が前条第\n１号に規定する技術移転事業者の役員等の職を兼ねる場合（以下「技術移転兼業」という。）に\nは、総長の許可を受けなければならない。 \n（技術移転事業者） \n第５条 技術移転事業者とは、営利企業を営むことを目的とする会社その他の団体であって、次の\nいずれかの事業を実施するものをいう。 \n (1) 大学等における技術に関する研究成果の民間事業者への移転の促進に関する法律（平成１０\n年法律第５２号。以下「大学等技術移転促進法」という。）第２条第１項に規定する特定大学\n技術移転事業（大学等技術移転促進法第５条第２項に規定する承認計画に係るものに限る。次\n条第１項第２号において「承認事業」という。） \n (2) 大学等技術移転促進法第１１条第１項の認定に係る事業又は特許法（昭和３４年法律第１２\n１号）第１０９条の２第３項第５号の事業（次条第１項第２号において「大学認定事業等」と\nいう。） \n（技術移転兼業の許可基準） \n第６条 総長は、教員から技術移転兼業の申請があった場合には、当該兼業が次に掲げる基準のい\nずれにも適合すると認めるときは、これを許可するものとする。 \n (1) 技術移転兼業を行おうとする教員が、技術に関する研究成果又はその移転について、技術移\n転事業者の役員等としての職務に従事するために必要な知見を有していること。 \n (2) 教員が就こうとする役員等としての職務の内容が、主として承認事業又は大学認定事業等に\n関係するものであること。 \n (3) 教員と申請に係る技術移転事業者（当該技術移転事業者が会社法第２条第３号に規定する子\n会社である場合にあっては、同条第４号に規定する親会社を含む。第１１条第１項第３号の研\n究成果活用企業及び第１６条第１項第２号の株式会社に同じ。）との間に、物品購入等の契約\n関係その他の特別な利害関係又はその発生のおそれがないこと。 \n(4) 兼業の申請前２年以内に、教員が当該申請に係る技術移転事業者との間に、物品購入等の契\n約関係その他の特別な利害関係のある職を占めていた期間がないこと。 \n (5) 教員としての職務の遂行に支障が生じないこと。 \n (6) その他職務の公正性及び信頼性の確保に支障が生じないこと。 \n２ 前項の許可は、役員等の任期等を考慮して定める期限を付して行うものとする。 \n（技術移転兼業の許可の取消し） \n第７条 総長は、技術移転兼業が前条第１項各号の許可の基準に適合しなくなったと認めるときは、\nその許可を取り消すものとする。 \n（技術移転兼業終了後の業務の制限） \n第８条 部局又は学系若しくは全学教員部の長（以下「部局等の長」という。）は、技術移転兼業\nの終了の日から２年間、当該技術移転兼業を行った教員を、技術移転兼業に係る技術移転事業者\nとの間に、物品購入等の契約関係その他の特別な利害関係のある業務に従事させないようにしな\nければならない。 \n \n第３節 研究成果活用企業の役員等の兼業 \n（許可権者） \n第９条 教員が第３条第２号に規定する研究成果活用企業の役員等の職を兼ねる場合（以下「研究\n成果活用兼業」という。）には、総長の許可を受けなければならない。 \n（研究成果活用企業）  \n第１０条 研究成果活用企業とは、営利企業を営むことを目的とする会社その他の団体であって、\n教員の研究成果を活用する事業（次条第１項第２号及び第５号において「研究成果活用事業」と\nいう。）を実施するものをいう。 \n（研究成果活用兼業の許可基準） \n第１１条 総長は、教員から研究成果活用兼業の申請があった場合には、当該兼業が次に掲げる基\n準のいずれにも適合すると認めるときは、これを許可するものとする。 \n3 \n \n (1) 研究成果活用兼業を行おうとする教員が、当該申請に係る研究成果活用企業の事業において\n活用される研究成果を自ら創出していること。 \n (2) 教員が就こうとする役員等としての職務の内容が、主として研究成果活用事業に関するもの\nであること。 \n (3) 教員が申請に係る研究成果活用企業との間に、物品購入契約等の契約関係その他の特別な利\n害関係又はその発生のおそれがないこと。 \n  (4) 兼業の申請前２年以内に、教員が当該申請に係る研究成果活用企業との間に、物品購入契約\n等の契約関係その他の特別な利害関係のある職を占めていた期間がないこと。 \n (5) 教員が就こうとする役員等としての職務内容に、大学に対する契約の締結又は検定、検査等\nの申請に係る折衝の業務（研究成果活用事業に関係する業務を除く。）が含まれていないこと。 \n (6) 教員としての職務の遂行に支障が生じないこと。 \n (7) その他職務の公正性及び信頼性の確保に支障が生じないこと。 \n２ 前項の許可は、役員等の任期等を考慮して定める期限を付して行うものとする。  \n（休職） \n第１２条 総長は、職員が許可を受けて従事している研究成果活用企業の役員等の職務に、主とし\nて従事する必要があり、職員としての職務に従事することができないと認めるときは、就業規則\n第１５条第１項第５号に基づき休職とすることができる。 \n（研究成果活用兼業の許可の取消し） \n第１３条 総長は、研究成果活用兼業が第１１条第１項各号の許可の基準に適合しなくなったと認\nめるときは、その許可を取り消すものとする。 \n（研究成果活用兼業終了後の業務の制限） \n第１４条 部局等の長は、研究成果活用兼業の終了の日から２年間、当該研究成果活用兼業を行っ\nた教員を、研究成果活用兼業に係る研究成果活用企業との間に、物品購入等の契約関係その他の\n特別な利害関係のある業務に従事させないようにしなければならない。 \n    第４節 株式会社の監査役等の兼業 \n（許可権者） \n第１５条 教員が第３条第３号に規定する株式会社の監査役又は信用金庫の監事（以下「監査役等」\nという。）の職を兼ねる場合（以下「監査役等兼業」という。）には、総長の許可を受けなけれ\nばならない。 \n（監査役等兼業の許可基準） \n第１６条 総長は、教員から監査役等兼業の申請があった場合には、当該監査役等兼業が次に掲げ\nる基準のいずれにも適合すると認めるときは、これを許可するものとする。 \n (1) 監査役等兼業を行おうとする教員が、当該申請に係る株式会社又は信用金庫（以下「株式会\n社等」という。）における監査役等の職務に従事するために必要な知見を教員の職務に関連し\nて有していること。 \n (2) 教員が申請に係る株式会社等との間に、物品購入等の契約関係その他の特別な利害関係又は\nその発生のおそれがないこと。 \n  (3) 兼業の申請前２年以内に、教員が当該申請に係る株式会社等との間に、物品購入等の契約関\n係その他の特別な利害関係のある職を占めていた期間がないこと。 \n (4) 教員としての職務の遂行に支障が生じないこと。 \n (5) その他職務の公正性及び信頼性の確保に支障が生じないこと。 \n２ 前項の許可は、監査役等の任期等を考慮して定める期限を付して行うものとする。  \n（監査役等兼業の許可の取消し） \n第１７条 総長は、監査役等兼業が前条第１項各号の許可の基準に適合しなくなったと認めるとき\nは、その許可を取り消すものとする。 \n（監査役等兼業終了後の業務の制限） \n第１８条 部局等の長は、監査役等兼業の終了の日から２年間、当該監査役等兼業を行った教員を\n監査役等兼業に係る株式会社等との間に、物品購入等の契約関係その他の特別な利害関係のある\n業務に従事させないようにしなければならない。 \n    第５節 株式会社の社外取締役等の兼業 \n（許可権者） \n第１８条の２ 教員が第３条第４号に規定する株式会社の社外取締役又は信用金庫の理事のうち株\n4 \n \n式会社の社外取締役に相当するもの（以下「社外取締役等」という。）の職を兼ねる場合（以下\n「社外取締役等兼業」という。）には、総長の許可を受けなければならない。 \n （社外取締役等兼業の許可基準） \n第１８条の３ 総長は、教員から社外取締役等兼業の申請があった場合には、当該社外取締役等兼\n業が次に掲げる基準のいずれにも適合すると認めるときは、これを許可するものとする。 \n (1) 社外取締役等兼業を行おうとする教員が、当該申請に係る株式会社等における社外取締役等\nの職務に従事するために必要な知見を教員の職務に関連して有していること。 \n (2) 教員が就こうとする社外取締役等としての職務の内容が、主として、取締役会又は理事会等\nの監督機能強化を目的とするものであること。 \n (3) 教員が申請に係る株式会社等との間に、物品購入等の契約関係その他の特別な利害関係又は\nその発生のおそれがないこと。 \n  (4) 兼業の申請前２年以内に、教員が当該申請に係る株式会社等との間に、物品購入等の契約関\n係その他の特別な利害関係のある職を占めていた期間がないこと。 \n (5) 教員としての職務の遂行に支障が生じないこと。 \n (6) その他職務の公正性及び信頼性の確保に支障が生じないこと。 \n２ 前項の許可は、社外取締役等の任期等を考慮して定める期限を付して行うものとする。  \n（社外取締役等兼業の許可の取消し） \n第１８条の４ 総長は、社外取締役等兼業が前条第１項各号の許可の基準に適合しなくなったと認\nめるときは、その許可を取り消すものとする。 \n（社外取締役等兼業終了後の業務の制限） \n第１８条の５ 部局等の長は、社外取締役等兼業の終了の日から２年間、当該社外取締役等兼業を\n行った教員を社外取締役等兼業に係る株式会社等との間に、物品購入等の契約関係その他の特別\nな利害関係のある業務に従事させないようにしなければならない。 \n \n   第３章 自営の兼業 \n（許可権者） \n第１９条 教職員が不動産若しくは駐車場の賃貸に係る自営の兼業又は不動産若しくは駐車場の賃\n貸以外の自営の兼業を行おうとする場合には、総長の許可を受けなければならない。 \n（自営の定義） \n第２０条 第２条第１項第２号に規定する自営の兼業で、農業、牧畜、酪農、果樹栽培、養鶏等に\nあっては大規模に経営され客観的に営利を主目的とする企業と判断されるとき、不動産又は駐車\n場の賃貸にあっては次のいずれかに該当するときは、自営に当たるものとして取り扱う。 \n (1) 不動産の賃貸が次のいずれかに該当する場合 \n  イ 独立家屋の賃貸については、独立家屋の数が５棟以上であること。 \n  ロ 独立家屋以外の建物の賃貸については、貸与することができる独立的に区画された一の部\n分の数が１０室以上であること。 \n  ハ 土地の賃貸については、賃貸契約の件数が１０件以上であること。 \n  ニ 賃貸に係る不動産が劇場、映画館、ゴルフ練習場等の娯楽集会、遊技等のための設備を設\nけたものであること。 \n  ホ 賃貸に係る建物が旅館、ホテル等特定の業務の用に供するものであること。 \n (2) 駐車場の賃貸が次のいずれかに該当する場合 \n  イ 建築物である駐車場又は機械設備を設けた駐車場であること。 \n  ロ 駐車台数が１０台以上であること。 \n (3) 不動産又は駐車場の賃貸に係る賃貸料収入の額（これらを併せて行っている場合には、これ\nらの賃貸に係る賃貸料収入の合計額）が年額５００万円以上である場合 \n (4) 第１号又は第２号に掲げる不動産等の賃貸と同様の事情にあると認められる場合 \n（自営の兼業の許可基準） \n第２１条 総長は、教職員から自営の兼業の申請があった場合には、当該兼業が次に掲げる基準の\nいずれにも適合すると認めるときは、これを許可するものとする。 \n (1) 不動産又は駐車場の賃貸を行う場合   \n    イ 教職員と申請に係る不動産又は駐車場の賃貸との間に特別な利害関係又はその発生のおそ\nれがないこと。 \n5 \n \n  ロ 入居者の募集、賃貸料の集金、不動産の維持管理等の不動産又は駐車場の賃貸に係る管理\n業務を事業者に委ねること等（親族による管理も含む。）により教職員の職務の遂行に支障\nが生じないことが明らかであること。 \n  ハ その他職務の公正性及び信頼性の確保に支障が生じないこと。 \n (2) 不動産又は駐車場の賃貸以外の事業を行う場合 \n  イ 教職員と当該事業との間に特別な利害関係又はその発生のおそれがないこと。 \n  ロ 教職員以外の者を当該事業の業務の遂行のための責任者としていること等により教職員の\n職務の遂行に支障が生じないことが明らかであること。 \n  ハ 当該事業が相続、遺贈等により家業を継承したものであること。 \n  ニ その他職務の公正性及び信頼性の確保に支障が生じないこと。 \n（自営の兼業の再許可） \n第２２条 自営の許可を受けた教職員に就業規則第１１条から第１３条の昇任、降任、配置換・出\n向等（以下この条において「昇任等」という。）があった場合並びに許可に係る自営の内容に変\n更があった場合には、当該昇任等又は自営の内容の変更の後１月以内に改めて許可を受けなけれ\nばならない。 \n（自営の兼業の許可の取消し） \n第２３条 総長は、自営の兼業が第２１条の許可の基準に適合しなくなったと認めるときは、その\n許可を取り消すものとする。 \n \n \n第４章 営利企業の役員等兼業以外の兼業 \n（許可権者） \n第２４条 第２条第１項第３号から第７号までに掲げる兼業を行う場合には、部局等の長（部局等\nの長にあっては総長）の許可を受けなければならない。 \n（営利企業の事業に直接関与しない兼業） \n第２５条 営利企業の事業に関与する場合は、原則として許可しない。ただし、次に掲げる第２条\n第１項第３号の営利企業の事業に直接関与しない兼業に該当する場合には、許可することができ\nるものとする。 \n (1) 公的な要素が強く、兼業内容が営利企業付設の診療所等の非常勤医師など営利企業の営業に\n直接関与するものでない場合 \n (2) 大学が管理する特許（出願中のものを含む。）の実施のための契約に基づく実施企業に対す\nる技術指導である場合 \n (3) 営利企業付設の教育施設、研修所及び研修会等又は文化講座等の非常勤講師で従業員教育又\nは社会教育の一環と考えられる場合 \n (4) 営利企業における研究開発（基礎研究、応用研究及び開発研究をいい、技術の開発を含む。\nこの号において同じ。）に従事し、又は研究開発に関する技術指導に従事する場合 \n (5) 公益性が強く法令（条例を含む。）で学識経験者から意見聴取を行うことが義務づけられて\nいる場合 \n (6) 大学等技術移転促進法第２条第１項に規定する特定大学技術移転事業及び同法第１１条第１\n項の認定に係る事業並びに特許法第１０９条の２第３項第５号の事業を実施するもの（次号に\nおいて「技術移転事業者」という。）が行う他の企業に対する技術指導に従事する場合 \n (7) 前号の技術移転事業者が行う技術に関する研究成果の発掘、評価、選別に関する業務に従事\nする場合 \n (8) 営利企業の経営及び法務に関する助言を行う場合 \n（営利企業以外の法人等の兼業） \n第２６条 第２条第１項第４号の営利企業以外の法人等の兼業で、次に掲げるものに該当する場合\nは、原則として許可しない。 \n (1) 医療法人及び社会福祉法人の理事長、理事、監事、顧問及び評議員並びに病院長（医療、療\n養機関の長を含む。）を兼ねる場合 \n (2) 学校法人及び放送大学学園の役員（理事長、理事、監事）及び学校長並びに専修学校、各種\n学校又は幼稚園の設置者若しくはこれらを設置する団体の役員（理事長、理事、監事）及び学\n校（園）長を兼ねる場合 \n6 \n \n (3) 前各号に掲げるものを除く営利企業以外の法人等の会長、理事長、理事、監事、顧問、評議\n員、その他これらに準ずる者（次項において「会長等」という。）を兼ねる場合 \n２ 前項第３号の規定にかかわらず、次に掲げる営利企業以外の法人等の会長等を兼ねる場合は、\n許可することができる。 \n (1) 国際交流を図ることを目的とする営利企業以外の法人等 \n (2) 学会等学術研究上有益であると認められ、当該教職員の研究分野と密接な関係がある営利企\n業以外の法人等 \n (3) 学内に活動範囲が限られた営利企業以外の法人等及びこれに類するものの営利企業以外の法\n人等 \n  (4) 育英奨学に関する営利企業以外の法人等 \n (5) 産学の連携・協力を図ることを目的とする営利企業以外の法人等 \n (6) その他、教育、学術、文化、スポーツの振興を図ることを目的とする営利企業以外の法人等\nで、著しく公益性が高いと認められるもの \n（教育に関する事業） \n第２７条 第２条第１項第５号の教育に関する兼業における教育に関する事業又は事務の職の範囲\nは、次に掲げるものをいう。 \n (1) 公立、私立の学校、専修学校、各種学校又は放送大学学園の設置する大学の長及びこれらの\n教育施設の職員のうち、教育を担当し、又は教育事務（庶務又は会計の事務に係るものを除く。\nこの項において同じ。）に従事する者の職を兼ねる場合 \n  (2) 公立又は私立の図書館、博物館、公民館、青年の家その他の社会教育施設の長及びこれらの\n施設の職員のうち、教育を担当し、又は教育事務に従事する者の職 \n  (3) 教育委員会の委員、指導主事、社会教育主事その他教育委員会の職員のうちもっぱら教育事\n務に従事するもの及び地方公共団体におかれる審議会等で教育に関する事項を所掌するものの\n構成員の職 \n (4) 学校法人、放送大学学園及び社会教育関係団体（文化財保護又はユネスコ活動を主たる目的\nとする団体を含む。）のうち、教育の事業を主たる目的とするものの役員、顧問、参与又は評\n議員の職及びこれらの法人又は団体の職員のうち、もっぱら教育を担当し、又は教育事務に従\n事する者の職 \n (5) 国会、裁判所、防衛省、公共企業体又は地方公共団体に附置された教育関係機関又は教育関\n係施設の長及びこれらの機関又は施設の職員のうち、もっぱら教育を担当し、又は教育事務に\n従事する者の職 \n２ 前項各号に該当する職を兼ねる場合で次に掲げるものは、原則として許可しない。 \n  (1) 公立、私立の学校、専修学校、各種学校又は放送大学学園の設置する大学の長を兼ねる場合 \n (2) 公立又は私立の図書館等の社会教育施設の長を兼ねる場合 \n (3) 部局等の長が教育委員会の委員を兼ねる場合 \n (4) 学校法人、放送大学学園及び社会教育関係団体の理事長及びその他の役員の職を兼ねる場合 \n (5) 国会、裁判所、防衛省、公共企業体又は地方公共団体に附置された教育関係機関又は教育関\n係施設の長を兼ねる場合 \n（国等の行政機関及び独立行政法人等の兼業）  \n第２８条 第２条第１項第６号の国等の行政機関の兼業及び第７号の独立行政法人等の兼業で、次\nに掲げるものに該当する場合は、原則として許可しない。 \n (1) 部局等の長が地方公共団体の執行機関の委員を兼ねる場合 \n (2) 国、地方公共団体、独立行政法人、地方独立行政法人、国立大学法人、大学共同利用機関法\n人等の常勤の職を兼ねる場合（ただし、在籍出向により常勤の職に就く場合を除く。） \n（営利企業の役員等兼業以外の兼業の許可基準） \n第２９条 総長又は部局等の長は、第２条第１項第３号から第７号までに掲げる兼業の申請があっ\nた場合であって、当該兼業が次に掲げる各号の一に該当するときは、第２５条から前条までに定\nめるもののほか、この場合においても原則として許可しない。 \n (1) 教職員としての職務の遂行に支障が生ずると認められる場合 \n (2) 兼業による心身の著しい疲労のため、職務遂行上その能率に悪影響を与えると認められる場\n合 \n (3) 教職員が申請に係る兼業先との間に、物品購入等の契約関係その他の特別な利害関係がある\n7 \n \n場合又はその発生のおそれがある場合 \n (4) 兼業する事業の経営上の責任者となる場合 \n (5) 兼業することが、大学教職員としての信用を傷つけ、又は大学全体の不名誉となるおそれが\nあると認められる場合 \n (6) 兼ねる職が常勤の職である場合（第２条第１項第３号から第５号までに掲げる兼業） \n (7) 大学等の入学試験の準備を目的として設置又は開講されている予備校又はこれに類する教室、\n塾、講座等の講師を行う場合  \n（営利企業の役員等兼業以外の兼業の許可の取消し） \n第３０条 総長又は部局等の長は、第２条第１項第３号から第７号までに掲げる兼業が第２５条か\nら前条までに規定する許可の基準に適合しなくなったと認めるときは、その許可を取り消すもの\nとする。 \n \n第５章 兼業の期間 \n（許可する兼業） \n第３１条 兼業を許可する期間は、技術移転兼業、研究成果活用兼業、自営の兼業、監査役等兼業\n及び社外取締役等兼業を除き、原則として３年以内とする。 \n２ 前項の規定にかかわらず、法令等に任期の定めのある職につく場合は、６年を限度として許可\nすることができる。 \n \n第６章 勤務時間 \n（勤務時間の取扱い） \n第３２条 兼業に従事する時間は、原則として勤務時間外とする。 \n２ 前項の規定にかかわらず、総長又は部局等の長が必要と認めたときは、勤務時間をさいて兼業\nに従事することができる。ただし、勤務時間をさいて兼業に従事した時間については、給与を減\n額する。 \n（勤務時間内の従事） \n第３３条 前条の規定にかかわらず、次の各号の一に該当するもので、無報酬の場合は、勤務時間\n内に職務として従事することができる。 \n (1) 第２条第１項第６号に規定する国等の行政機関の兼業を行う場合 \n (2) 大学の関連団体及び第２条第１項第７号に規定する独立行政法人等の兼業を行う場合 \n  (3) 学会の兼業を行う場合 \n (4) 部局等の長が、教育、学術、文化、スポーツの振興を図ることを目的とする営利企業以外の\n法人等の各種委員等を兼ねる場合で、特に公益性が高いと認められるとき \n (5) 部局等の長が、第２条第１項第５号に掲げる教育に関する兼業に従事する場合 \n \n   第７章 雑則 \n（その他） \n第３４条 この指針に定めるもののほか、教職員の兼業の取扱いに関し必要な事項は、総長又は部\n局等の長が別に定める。 \n \n附 則 \n１ この指針は、平成１６年４月１日から施行する。 \n２ この指針の施行日の前日において、既に許可等を受けている兼業については、施行日以後新た\nにこの指針による許可を要しない。 \n   附 則（平成１６年８月６日 人職職第１７号） \n１ この指針は、平成１６年８月６日から施行する。 \n２ 大学院法学研究科法政実務交流センター法科大学院準備部門の教授、准教授又は講師として任\n期を付して任用された法実務又は行政実務の経験者の兼業については、この指針に定めるものの\nほか、大学院法学研究科が定める別紙の許可基準によるものとする。 \n \n   附 則（平成２１年１０月５日 総職労第４３号） \n１ この指針は、平成２１年１１月１日から施行する。 \n8 \n \n２ この指針の施行日の前日において、既に許可等を受けている兼業については、改正後の指針に\nより許可されたものとみなす。 \n   附 則（平成２４年１２月１２日 総人服第２２０号） \n１ この指針は、平成２５年１月１日から施行する。 \n２ この指針の施行日の前日において、既に許可等を受けている兼業については、改正後の指針に\nかかわらず、なお従前の例による。 \n   附 則（平成２７年６月９日 総人職第１９号） \n この指針は、平成２７年６月９日から施行する。 \n   附 則（平成２９年３月２９日 総人職第６４号） \n この指針は、平成２９年４月１日から施行する。 \n   附 則（平成３０年５月１５日 総人職第２号） \n この指針は、平成３０年６月１日から施行する。 \n   附 則（令和元年５月１７日 総人職第１２号） \n この指針は、令和元年５月１７日から施行する。 \n   附 則（令和２年５月２２日 総人職第１０号） \n この指針は、令和２年５月２２日から施行する。 \n \n"}
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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Study at KyotoU Career guidance for students. Career guidance for students. サイドメニュー（英語）. Education and admissions Career paths Visa\/tuition\/scholarships\/housing University facilities FAQ In other languages. 日本語 Business Japanese classes by the Education Center for Japanese Language and Culture of the Institute for Liberal Arts and Sciences of Kyoto University. Students seeking employment in Japan may learn Japanese and business etiquette through the classes offered by the Education Center for Japanese Language and Culture of the Institute for Liberal Arts and Sciences of Kyoto University. The first (spring) semester is from April to July and the second (fall) semester is from October to January. The course catalogue and registration are available online twice a year, around late March and late September (at the website below). Course catalogues are also available at your faculty\/graduate school office and the International Education and Student Mobility Division. Please note that the application is accepted only through the special website below. Japanese language support classes not included in the regular curriculum . Kyoto University General Student Support Center Career Support Office. The office provides information and suggestions on career development and job searching for Kyoto University students. Kyoto University General Student Support Center Career Support Office Open: Monday - Friday 9:00-17:00 (except on holidays). Off-campus career information centers. Employment Service Center for Foreigners. Osaka Employment Service Center for Foreigners Tokyo Employment Service Center for Foreigners Japan Student Services Organization (JASSO). Japan Student Services Organization (JASSO) Kyoto Job Park (in Japanese language only). Kyoto Job Park Ministry of Health, Labor, and Welfare. Ministry of Health, Labor, and Welfare Changing status of residence to work after graduation. International students who continue to stay in Japan after graduation must change their Status of Residence. Anyone engaging in an activity other than those permitted by his\/her Status of Residence may be subject to punitive measures. International students are responsible for undertaking all of the above procedures by themselves at the immigration office. It may be necessary to obtain various documents from the university, work place, etc. Please inquire about the documents required to complete the process at. the Immigration Bureau . If employed in Japan after graduating. If employed in Japan after graduating, it is necessary to make the appropriate change in your status of residence. Depending on the type of employment, the status may change to \"Specialist in Humanities\/International Services\", \"Skilled Labor\", etc. The documents required vary depending on which status of residence is applied for. When seeking employment in Japan after graduating. When seeking employment in Japan after graduating, it is necessary to change your status of residence to \"Designated Activities\" status. International students with this status are allowed to stay in Japan for six months (extendable up to one year) in order to seek employment in Japan. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP EN JP KR CN BOOK NOW ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP JP. EN. KR. CN. ABOUT. 「神戸のふつう」に出会う旅を、. ここから。. 古くからさまざまな国の人が行き交い、文化がまじりあってできてきた港町・神戸。. その中心、三宮からほんのすこし離れただけで、ふつうの人びとが豊かに暮らす町の日常があります。. そんな「神戸のふつう」を感じてほしくて、ゲストハウス萬家は灘区にオープンしました。. 見上げれば緑まぶしい山のながめ。. ふもとには長いアーケードの活気ある商店街。. 親しみやすく、気どらないお店や人びと。. 「地元にいちばん近い旅」をわたしたちはお手伝いします。. たとえば──. ケーブルカーや山歩きで摩耶山に登り、町と海を一望する。. 王子動物園や美術館をゆったり散策し、レトロとモダンに出会う。. 町の台所である水道筋商店街や市場をめぐって買い物を楽しむ。. 歩き疲れたら、お気に入りの喫茶店でお茶を飲みながらおしゃべりをして、一日の終わりに町の銭湯で汗を流す。. おふろ上がりには、仕事帰りの人びとでにぎわうお店に寄り道して、おいしい料理とお酒をいただく。. そんなふうにして神戸のこと、山や海のこと、お店や食べ物のこと、日々の暮らしのこと、さまざまな表情にふれてもらいたいのです。. 「地元にいちばん近いゲストハウス」だから知っている町の魅力、ご案内できる旅のかたちがここにあります。. あなたの旅のページに. たくさんの「神戸のふつう」が書き加えられますように。. [Join the local experience in Nada!]. NEWS. 2022年5月26日. 《ひょうご旅しようキャンペーン＋》　兵庫県民割6月まで延長になります。. ゲストハウス萬家は「ひょうご旅しようキャンペーン」対象施設です。 「ふるさと応援！ひょうごを旅しよう[…]. read more 2022年4月17日. 価格改定のお知らせ. 価格改定のお知らせ 平素は格別のお引き立て、ご愛顧を賜り厚く御礼を申し上げます。 このたびゲストハウ[…]. read more 2022年3月11日. ”旧摩耶観光ホテル”見学付き宿泊プラン始めます！. かつて摩耶山は、天上寺への参拝客をはじめ、多くの人が訪れる場所として、食事処や宿泊施設等などで賑わい[…]. read more 2021年8月23日. 【8\/25(水)インスタライブのお知らせ】. 韓国文学オンライントーク\/K-BOOK TALK #3【インスタライブのお知らせ】K-BOOK TA[…]. read more ROOM. 地元から愛される医院として使われていた一軒家を地元の人と一緒にリノベーションしました。木を基調に、スタイリッシュさと可愛さのある空間に仕上げました。窓や屋上からは摩耶山、阪急電車、王子公園の観覧車をご覧頂けます。. 男女混合6人ドミトリー. 6 Mixed Dormitory. 木のぬくもりが優しい、形がユニークな二段ベッドのお部屋です。. ・読書灯、コンセント、ベッドカーテン、鍵付き収納ボックス完備. ・バス、トイレ共用. ・部屋の入口ダイヤル施錠可能. 女性専用6人ドミトリー. 6 Female Dormitory. 山小屋のような屋根がついた可愛らしい二段ベッドのお部屋です。. パウダースペースもご用意しております。. ・読書灯、コンセント、ベッドカーテン、鍵付き収納ボックス完備. ・バス、トイレ共用. ・部屋の入口ダイヤル施錠可能. シングルルーム. Single Room. 広々としたベッドを独り占め。ゆったりとお休み頂けます。. ・読書灯完備. ・バス、トイレ共用. マルチルーム(2~4名). Multi Room. 木の柱が印象的な、シングルベッド二台と二段ベッドのお部屋です。. ・読書灯完備. ・バス、トイレ共用. ダブルルーム. Double Room. 落ち着いた雰囲気の、ダブルベッドのお部屋です。. プライベートな時間をゆっくりとお過ごし頂けます。. ・読書灯完備. ・バス、トイレ共用. ※料金や空室状況は『BOOK NOW』をクリックしてご確認下さいませ。. FACILITIES. ラウンジ. Lounge. 食事に、交流に、リラックスタイムに、萬家を訪れた人々が自由に使えるスペースです。スパイラルベンチやUFO型スピーカーなどアイコニックなインテリアが皆様をお迎えします。. キッチン. Kitchen. IHクッキングヒーター、電子レンジ、オーブントースター、電気ケトル、冷凍・冷蔵庫をご用意。フリーティーなどもお楽しみ頂けます。. バスルーム. Bath Room. 各部屋には洗面、シャワーの設備がありませんので 共用のものをご利用ください。シャワー内にはシャンプーとボディソープが常備されており、24 時間ご利用可能です。ドライヤーは深夜帯のご利用を制限させて頂いております。. 無料サービス. Wifi、ヘアドライヤー、シャンプー&ボディソープ、お茶、耳栓、目覚まし時計. 有料サービス. ランドリー、レンタルタオル、歯ブラシ、電源アダプター、自転車、アイロン、パジャマ、荷物預かり. INFORMATION. Check in／16:00～21:00. Check out／10:30. ※上記の時間以外のチェックイン・チェックアウトもできる限り対応致します。事前にご相談ください。. ※チェックイン後の門限はありません。. ※チェックイン前、後のお荷物のお預かりも可能です。（チェックイン日、チェックアウト日は無料、それ以外は300円\/泊）. ・お支払い方法. お支払いはチェックイン時にお願い致します。※一万円以上のお支払いよりクレジットカードのご利用が可能です。(VISA・MasterCard・AMERICAN EXPESS). ・ご予約について. ・ご予約は4ヶ月先までの受付となります。. ・キャンセル料は宿泊日の3日前から発生致します。. 　キャンセル規定\/3日前：30%, 2日前 : 50%, 前日：80%, 当日・無連絡不泊：100％. ・当宿は小さな宿ですので、申し訳ありませんが5名様以上のご予約は受付致しかねます。予めご了承ください。. ・未就学のお子様のご利用はお受けしておりません。. ・安全上、階段をお一人で上るのが大変な方のご予約をお断りする場合がございます。. ・セキュリティー. 正面エントランスと各部屋に鍵を常備しており、セキュリティー対策は万全です。. 営業時間外はエントランスを施錠し、夜間もスタッフが常駐しておりますので、安心してご利用いただけます。. ・女性ゲストへの配慮. 女性スタッフもおり、女性専用のシャワーとトイレもご用意しております。治安も良い地域ですので、女性の一人旅でも安心してご利用いただけます。. LOCAL EXPERIENCE. 神戸のふつうの1日をご紹介. たとえば、朝食は近くの小さなパン屋さん、昼は商店街や酒蔵や美術館など色んな観光を楽しみ、夜は摩耶山から見る夜景やちょっとディープでお洒落な飲み屋を堪能。是非、神戸のふつうを満喫しにきてください。. Things to enjoy around our guesthouse. In The Morning　#1. 神戸のパン屋さん. Details In The Morning　#2. 神戸の喫茶店とコーヒー. Details In The Morning　#3. 朝5時から営業の灘温泉. Details In The Late Morning. 海の見えるなぎさ公園までお散歩. Details In The Afternoon #1. ザ・ローカルな水道筋商店街. Details In The Afternoon #2. 王道を行くなら、王子動物園. Details In The Afternoon #3. 雨の日には、美術館へ. Details In The Afternoon #4. 灘の新しい見所“灘高架下エリア”. Details The Afternoon #5. お酒が好きなら、酒蔵巡り. Details In The Late Afternoon. いよいよ摩耶山へ. Details In The Evening. 灘の美味しくて面白いお店. Details KOBE BEEF. 神戸と言えば、神戸牛. Details ACCESS. JR灘駅から. 改札を出て左方面にある東北口を出ます。東北口を出て左にある横断歩道を渡り、王子動物園に向かって北に進みます。1分くらい歩くと電車の高架が見えます。その高架に添って右に進みます。30秒くらい歩くと右側に「ワールドエンズガーデン」という古本屋が見えます。そのままさらに30秒ほど歩くとゲストハウス萬家に到着します。. 阪急電鉄　王子公園から. 王子公園駅の西口を出て、左に見える横断歩道を渡るとセブンイレブンがあります。セブンイレブンと高架下の間の道を２分ほど歩くと左側にゲストハウス萬家が見えます。. 阪神電鉄　岩屋駅から. 改札を出て外に出ると右にセブンイレブンが見えます。そこを右に曲がるとＪＲ灘駅が見えます。灘駅の2階に上って東北口に出てください。（ここからは灘駅からゲストハウスまでのアクセスと同じ）. ゲストハウス萬家(MAYA). 〒657-0836. 兵庫県神戸市灘区城内通4丁目4-10. 予約・問い合わせ：090-9856-8280（受付時間 8:00-22:00）. TEL: 078-940-2809（受付時間 8:00-22:00）. e-mail: info@kobemaya.com. Copyright © GUEST HOUSE MAYA KOBE JAPAN All Rights Reserved."}
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ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP EN JP KR CN BOOK NOW ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP JP. EN. KR. CN. [Join The Local experience11] - In The Evening. よりディープな街の魅力に出会いたいなら夜は寝ていられません。下町情緒溢れる居酒屋やお洒落なバー、地場のものを使ったおいしいご飯が楽しめるレストランなど、灘には美味しくて面白いお店がたくさんあるんです。地元の人しか知らないようなお店で食事やお酒を楽しみながら、地元の人たちとの会話も楽しんだりすると、旅がより深いものになる のではないでしょうか。さて、今夜はどこに飲みに出掛けましょう。. You don’t have a time to sleep if you want to find a deeper spot in Nada. A local izakaya, stylish bar and good restaurant. Many interesting and good restaurants here. When you have dinner and sake at a restaurant where only local people go ,have a good conversation with them and laugh together, the experience make your trip better and deeper. So, where shall we go tonight? #神戸 #神戸ゲストハウス萬家 #摩耶 #王子公園 #灘区 #神戸ゲストハウス #ゲストハウス #guesthouse #kobe #MAYA #マヤ #ゲストハウスマヤ #guesthousemaya #maya #localexperience #suidousuji #dinner #水道筋 #水道筋商店街. #夜ご飯. ゲストハウス萬家(MAYA). 〒657-0836. 兵庫県神戸市灘区城内通4丁目4-10. 予約・問い合わせ：090-9856-8280（受付時間 8:00-22:00）. TEL: 078-940-2809（受付時間 8:00-22:00）. e-mail: info@kobemaya.com. 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ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP EN JP KR CN BOOK NOW ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP JP. EN. KR. CN. [Join The Local experience04] - In The Late Morning. お昼まで少し時間があるならば、なぎさ公園までお散歩。JR灘駅から海側に向かって歩いていくと兵庫県立美術館に着きます。. ここを右にぐるっと回るとシンボルのサンシスターが見えてきます。. そこまで来るともう目の前は海。そこをのんびり潮風を感じながら歩いていくとなぎさ公園へ。芝生もあり、ほっと一息に最適. If you have a time in the morning, how about taking a walk to the Nagisa park? When you walk to the seaside from JR Nada station, you find Hyogo Prefectural Museum of Art. Then you walk around to the right, you can meet Sun sister. Here is the Seto inland sea and the Nagisa park. Feel the ocean and take a break. #神戸 #神戸ゲストハウス萬家 #摩耶 #王子公園 #灘区 #神戸ゲストハウス #ゲストハウス #guesthouse #kobe #MAYA #マヤ #ゲストハウスマヤ #guesthousemaya #maya #localexperience #nagisapark #なぎさ公園. ゲストハウス萬家(MAYA). 〒657-0836. 兵庫県神戸市灘区城内通4丁目4-10. 予約・問い合わせ：090-9856-8280（受付時間 8:00-22:00）. TEL: 078-940-2809（受付時間 8:00-22:00）. e-mail: info@kobemaya.com. 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このページはJavaScriptを使用しています。JavaScriptを有効にしてご覧ください。. 本文へジャンプ ナビゲーションへジャンプ サイトマップへジャンプ ヘッダーの始まり. ホーム 教育. 教育. 学び合い教え合い、. ともに成長する「半学半教」の気風. 教育. 本文の始まり. 学び合い教え合い、. ともに成長する「半学半教」の気風. 小学校から大学院までを擁する総合学塾である慶應義塾は、問題の発見、検証、解決というプロセスを身につけ、自分の頭で考えることのできる人材を育成します。. 半学半教. 教える者と学ぶ者との師弟の分を定めず、先に学んだ者が後で学ぼうとする者を教える。教員と学生も半分は教えて、半分は学び続ける存在という、草創期からの精神です。. 学部. 多様な領域をカバーする10学部が、. 建学以来の“実学の精神”を共有しつつ、. それぞれ自由な発想に立って. 個性と特色ある教育・研究活動を進めています。. →. 詳しく見る. 大学院. 慶應義塾大学の大学院は、その活動の目的を. “学理及びその応用を教授研究し、. 学術の深奥を究めて文化の進展に寄与すること”. と定め、社会への貢献を使命としています。. →. 詳しく見る. 教育分野. 留学・. 国際プログラム. アカデミック. カレンダー. 図書館・. 情報環境. 一貫教育校. （小・中・高等学校）. 生涯学習. 特色ある教育の取り組み. 慶應義塾は、総合大学の強みを生かし、グローバル社会を見すえた新しい教育や環境づくりに取り組んでいます。その一例をご紹介します。. PEARL（経済学部）. Programme in Economics Alliances, Research and Leadership（PEARL）では、世界で活躍する経済人を目指し、4年間、英語のみで経済学を学びます。. →. PEARL（経済学部）. 外部サイトへリンク. Global Information and Governance Academic (GIGA) Program. 総合政策学部・環境情報学部で開講している、英語で提供される授業の履修のみで学部卒業が可能なプログラムです。講義形式の授業に加え、学生が能動的に参加するICT関連プロジェクトや海外フィールドワーク、インターンシップなど多彩な取り組みが実践されています。. →. GIGA. 外部サイトへリンク. Global Interdisciplinary Courses (GIC). 英語（またはその他の外国語）による授業を一定単位取得した学生に対して修了証を与えるプログラムです（全ての学部学生が受講可能）。. →. GIC. 外部サイトへリンク. 医療系3学部合同教育プログラム. 医学部、看護医療学部、薬学部の3学部を擁する大学として、グループアプローチによる患者中心の医療の実践を追求するプログラムです。. →. 医療系3学部合同教育プログラム. 外部サイトへリンク. リーディング大学院. 優秀な学生を俯瞰力と独創力を備えグローバルに活躍するリーダーへと導くための２つのプログラムです。. →. リーディング大学院. 未来創造塾（湘南藤沢キャンパス）. 開設25周年を迎えたSFC（湘南藤沢キャンパス）に、地球視点の課題に取り組むグローバルゲートウェイとして創られる、大規模滞在型教育研究施設です。. →. 未来創造塾. 外部サイトへリンク. 未来先導基金. 未来の先導者にふさわしい、「独立して生きる力」と「協力して生きる力」を兼ね備え、国際的な体験と視野をもって社会をリードする人間を育成することを目的に創設されました。. →. 未来先導基金. 外部サイトへリンク. 慶應義塾一貫教育校派遣留学制度. 2013年、慶應義塾は「慶應義塾一貫教育校派遣留学制度」を新たに設立しました。一貫教育校（高校段階）各校の枠を越えて選抜した生徒を国外の名門ボーディングスクール（米国 Ten Schools、英国 The Nine）に約1年間派遣しています。. →. 慶應義塾一貫教育校派遣留学制度. 外部サイトへリンク. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. 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本文へジャンプ ナビゲーションへジャンプ サイトマップへジャンプ ヘッダーの始まり. ホーム 慶應義塾について もっと知る 卒業生向けメールマガジン. 卒業生向けメールマガジン. 本文の始まり. 慶應義塾メールマガジンは、慶應オンライン登録者でメールマガジンの配信を希望される卒業生の方に、毎月15日頃配信しています （休刊月あり）。. 【内容】. ・Keio Top News & Event Calendar. ・義塾の最新動向／特集. ・塾員の皆様へのお知らせ　他. 定期購読されるには. 慶應オンライン に登録をお願いいたします。（登録は卒業生の方のみ可能です。一般の方は以下よりご覧いただけます）. バックナンバー. No. 配信日. 特集. 232. 2022\/05\/16. 上智大学×慶應義塾大学「ウクライナ平和シンポジウム」を開催 231. 2022\/04\/15. 春の式典開催 230. 2022\/03\/15. 矢上キャンパス移転から50年 229. 2022\/02\/15. 『学問のすゝめ』出版150年 228. 2022\/01\/20. 福澤先生誕生記念会での塾長年頭挨拶 227. 2021\/12\/15. 塾歌発表から80年 226. 2021\/11\/15. 30年ぶりの東京六大学野球リーグ戦春秋連覇 225. 2021\/10\/15. 「2020年度入学生の集い」と秋の式典 224. 2021\/09\/15. 東京2020オリンピック・パラリンピック：義塾社中の活躍 223. 2021\/07\/15. 福澤諭吉記念慶應義塾史展示館が開館 222. 2021\/06\/15. 新塾長・常任理事の紹介 221. 2021\/05\/17. 小泉信三と慶應義塾 220. 2021\/04\/15. 慶應義塾ミュージアム・コモンズ（KeMCo）開館 219. 2021\/03\/15. 慶應義塾三田移転150年 218. 2021\/02\/15. Apple Watchを利用した臨床研究を開始 217. 2021\/01\/20. 福澤先生誕生記念会での塾長年頭挨拶 216. 2020\/12\/15. \"世界で影響力のある研究者（Highly Cited Researchers 2020）\" 215. 2020\/11\/16. まもなく開催　第62回三田祭はオンラインで 214. 2020\/10\/15. 湘南藤沢キャンパス開設30年 213. 2020\/09\/15. オンライン学園祭 212. 2020\/07\/15. 遠藤周作の未発表小説が『三田文學』に掲載される 211. 2020\/06\/15. 新型コロナウイルス感染症への対応について 210. 2020\/05\/15. 新型コロナウイルス感染症への対応について 209. 2020\/04\/16. 新型コロナウイルス感染症への対応について 208. 2020\/03\/17. 新・日吉記念館が竣工 207. 2020\/02\/17. 東京オリンピック・パラリンピックを目指す塾生・塾員 206. 2020\/01\/21. 福澤先生誕生記念会での塾長年頭挨拶 205. 2019\/12\/16. 145年続く慶應義塾の伝統　三田演説会 204. 2019\/11\/15. 東京六大学野球3季ぶりの優勝・パレード・三田山上祝賀会 203. 2019\/10\/15. 9月学位授与式・卒業式・入学式 202. 2019\/09\/17. 日本ラグビーのルーツ 201. 2019\/07\/16. 理工学部創立80年 200. 2019\/06\/17. メールマガジン200号を振り返る 199. 2019\/05\/15. 2020年度 慶應義塾ミュージアム・コモンズ開設に向けて 198. 2019\/04\/15. 横浜初等部が初めての卒業式を挙行 197. 2019\/03\/15. 世界で初めてiPS細胞を用いた脊髄損傷治療の臨床研究へ 196. 2019\/02\/15. 社会人向け新講座「慶應義塾 三田オープンカレッジ」開講 195. 2019\/01\/21. 福澤先生誕生記念会での塾長年頭挨拶 194. 2018\/12\/17. 慶應義塾と「アート」 193. 2018\/11\/15. 開設10年を迎えた学部・研究科 192. 2018\/10\/15. 慶應医学賞－ノーベル賞受賞者8人輩出－ 191. 2018\/09\/19. 第18回アジア競技大会で塾生・塾員が活躍 190. 2018\/07\/17. 「慶應大阪シティキャンパス開設10周年記念式典」を開催 189. 2018\/06\/15. 体育会野球部、東京六大学野球で27年ぶりの連覇 188. 2018\/05\/15. 慶應義塾大学病院1号館（新病院棟）がオープン 187. 2018\/04\/16. 元住吉国際学生寮・綱島SST国際学生寮がオープン 186. 2018\/03\/15. 医学部 健康医療ベンチャー大賞 185. 2018\/02\/15. 高等学校野球部　春の甲子園に出場決定 184. 2018\/01\/18. 福澤先生誕生記念会での塾長年頭挨拶 183. 2017\/12\/15. BSジャパン建築特番「地震から守れ！ニッポンの建築遺産」 182. 2017\/11\/15. 体育会野球部、東京六大学野球で7季ぶりの優勝 181. 2017\/10\/16. 慶應義塾での最新研究成果紹介 180. 2017\/09\/15. 図書館旧館改修工事の進捗状況 179. 2017\/07\/18. 福澤諭吉・慶應義塾史 新収資料展開催中（7\/3-8\/5） 178. 2017\/06\/15. 新塾長・常任理事の紹介 177. 2017\/05\/15. 医学部開設100年記念事業－新病院棟建設－ 176. 2017\/04\/20. 体育会創立125年記念事業 175. 2017\/03\/15. 慶應義塾大学国際学生寮の新設 174. 2017\/02\/15. グローバルなオンライン講座「FutureLearn」での新コース 173. 2017\/01\/17. 福澤先生誕生記念会での塾長年頭挨拶 172. 2016\/12\/15. 大学入試風景の移り変わり 171. 2016\/11\/15. サイバー脅威に対応する国際連携組織が設立 170. 2016\/10\/17. 山形県鶴岡の先端生命科学研究所が15周年 169. 2016\/09\/15. アジアの最も革新的な大学ランキング10位・司法試験合格者数1位 168. 2016\/07\/15. リオ五輪：慶應義塾大学学生・卒業生の出場決定者 167. 2016\/06\/15. 慶應義塾ウェブサイトを全面リニューアル 166. 2016\/05\/16. 伊勢志摩にゆかりのある義塾関係者たち 165. 2016\/04\/15. 慶應義塾の2015年度：プレスリリースで振り返る 164. 2016\/03\/15. 慶應義塾のベンチャーキャピタルとベンチャー支援 163. 2016\/02\/16. 日吉キャンパスが東京五輪における英国事前キャンプ地に決定 162. 2016\/01\/19. 福澤先生誕生記念会での塾長年頭挨拶 161. 2015\/12\/15. 福澤先生誕生記念会 160. 2015\/11\/17. 慶應義塾　大学部開設125年 記念式典・講演会　開催 159. 2015\/10\/15. 湘南藤沢キャンパス（SFC） 開設25周年記念式典開催 158. 2015\/09\/15. 慶應義塾大学病院　新病院棟I期棟 オープニングセレモニー開催 157. 2015\/07\/15. 「慶應義塾と戦争III 慶應義塾の昭和二十年」展 156. 2015\/06\/16. 「Keio Research Highlights」更新中 155. 2015\/05\/15. 平成26年度大学卒業式・ 平成27年度大学入学式の動画を公開 154. 2015\/04\/15. 大学部開設125年記念 ハーバード大学訪問団 153. 2015\/03\/17. 図書館(三田)展示室 「アインシュタインの見た日本」（3\/25-5\/23） 152. 2015\/02\/17. 第180回福澤先生誕生記念会 年頭の挨拶「平等の慶應社中」 151. 2015\/01\/15. 塾長から新年のご挨拶 サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. 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入学案内. ↓. 教育. ↓. 研究. ↓. 学生生活. ↓. 訪問者別ページ. ↓. 更新情報. ↓. その他. ↓. 言語別ページ. ホーム 慶應義塾について 塾長メッセージ →. 塾長プロフィール. →. 塾長室だより一覧. →. 塾長スピーチ一覧. →. 歴代の塾長. 理念 →. 創立者 福澤諭吉. 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス →. 三田キャンパス. →. 日吉キャンパス. →. 矢上キャンパス. →. 信濃町キャンパス. →. 湘南藤沢キャンパス. →. 芝共立キャンパス. 数字で見る慶應義塾 もっと知る 情報公開 →. 常任理事一覧. →. 理事・監事一覧. →. 評議員一覧. →. 大学学部長・研究科委員長一覧. →. 一貫教育校長一覧. →. 2011（平成23）年度. 点検・評価報告書. →. 2004（平成16）年度. 点検・評価報告書. 社会貢献 →. 社会・地域連携. →. 環境への取り組み. →. 東日本大震災への対応. →. 熊本地震への対応. 協生環境 歴史ある取り組み →. 三田演説会. →. ウェーランド講演会. →. 小泉信三賞全国高校生. 小論文コンテスト. →. 審査結果. →. 過去の入選作品. →. よくあるご質問（FAQ）. →. 慶應義塾の名誉博士称号. 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット →. [塾] バックナンバー一覧. ロゴマーク使用 →. シンボルマーク. →. よくあるご質問（FAQ）. →. 使用の手順. ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 →. 【学部】学費. →. 【大学院】学費. →. 【専門職大学院（法務研究科）】学費. →. 【一貫教育校】学費. →. 学費納入について. （在学生・保護者の方へ）. 教育 教育分野 学部 大学院 →. リーディング大学院. 留学・国際プログラム →. 海外留学プログラム. →. ダブルディグリープログラム. →. 英語のみで学位が取得できる. プログラム. アカデミックカレンダー 図書館・情報環境 一貫教育校 （小・中・高等学校） 生涯学習 研究 研究者情報・業績 →. 教員プロフィール. →. 研究活動による受賞. 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 慶應義塾のオリンピック・ パラリンピック選手 訪問者別ページ. 受験生 在学生 卒業生 寄付 外部サイトへリンク. 更新情報. Keio Times（特集） ニュース プレスリリース イベントカレンダー オリンピック・パラリンピックと慶應義塾 その他. アクセス →. 三田キャンパス. →. 日吉キャンパス. →. 矢上キャンパス. →. 信濃町キャンパス. →. 湘南藤沢キャンパス. →. 芝共立キャンパス. →. キャンパス見学. 塾内サイト 教職員採用情報 このサイトについて 個人情報の取り扱い 取材情報 お問い合わせ →. その他のお問い合わせ・ご意見. 外部サイトへリンク. →. 研究活動に関する不正申し立て. 外部サイトへリンク. →. 部署別お問い合わせ（三田）. →. 部署別お問い合わせ（日吉）. →. 部署別お問い合わせ（矢上）. →. 部署別お問い合わせ（湘南藤沢）. →. 部署別お問い合わせ（芝共立）. →. 海外オフィス. 言語別ページ. English Chinese Korean 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. 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to the content","Home","Event Calendar","Hiyoshi Commemorative Hall, Hiyoshi Campus","here","About Keio","Global Engagement","Top 5 Reasons to Choose Keio","Keio by the Numbers","Office of the President","Philosophy","History","Alumni Network","Campus Information","Facts and Figures","Publications","Conferment of Honorary Degree of Doctor","Projects Supported by the Ishii-Ishibashi Fund","Keio Logos","Social Media &amp; Videos","e-Newsletter","Admissions","Undergraduate Admissions","Graduate Admissions","Study Abroad Information Fairs","Academic Fees","FAQ","Academics","Fields of Study","Undergraduate","Graduate","Study Abroad and International Programs","Degree Programs Offered in English","Double Degree Programs","Short-term Programs","Academic Calendar","Libraries &amp; Information Technology Services","Affiliated Schools","Research","Find a Researcher","Research Centers &amp; Institutes","Research Development &amp; Collaboration","Research Hubs Abroad","Research by Campus","Student 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Links to external sites","","","","","","Keio Research Highlights Links to external sites"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 13:37:45 UTC 2022","fresh_date_dt":"Thu Jun 23 03:23:17 UTC 2022","referrer_id_s":"_p_9TRRb7flo","language_s":"en","size_i":45838,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735073952877772800,"text_t":"AY 2020 Graduate Commencement Ceremony: Keio University. This page uses JavaScript. Please enable JavaScript in your browser. Jump to the content Jump to the navigation Header start. Home Event Calendar Event. Event. Tweet AY 2020 Graduate Commencement Ceremony. Updated on:. March 18. , 2021. December 25, 2020. The AY 2020 Graduate Commencement Ceremony will be held for graduates, degree awardees, and related parties only, after giving maximum consideration to the health and safety of graduates and degree awardees and preventing the spread of the COVID-19 (coronavirus disease.). While the utmost efforts to facilitate the participation of families were made, we request your kind understanding regarding this decision, reached with consideration to the circumstances at Hiyoshi Campus, which is implementing access management for entry to all buildings to prevent infections, and the impact on the areas surrounding the campus. While we believe that all family members were sincerely looking forward to attending the ceremony, we thank you for your kind understanding. No venue will be made available for the use of family members, etc., and we as such request that they refrain from accompanying ceremony attendees on campus. On the day, a video of the ceremony for those unable to attend will be streamed online which we encourage you to watch. *Please consult with us if for various reasons a graduate or degree awardee requires a chaperone. (See telephone number below). *The method for holding the ceremonies may be subject to change according to the ongoing situation regarding infections with COVID-19. If there are any changes, we will inform you via this web page. Please be sure to check the latest information. *We will notify you via this webpage in the case of any changes to the schedule such as the start time of the ceremony due to any unforeseen circumstances on the day. Date. March 26 (Fri.), 2021. Venue opens: 9:00. Arrive by: 9:40. Commencement ceremony: 10:00–11:00 (planned). Please be sure to enter the venue by 9:40. Please note that you will not be permitted to enter or leave the venue once the ceremony has begun. Venue. Hiyoshi Commemorative Hall, Hiyoshi Campus *Building \"17\" as indicated at the above link. Participating Graduate Schools. All graduate schools. Advance Registration. Not required. Notes. [About the video stream]. For those who cannot attend, a video of the ceremony will be made available on the official Keio University website on the below date and time. Please watch the ceremony from here. URL:. https:\/\/youtu.be\/HLUir3LPMAE Start of broadcast: March 26 (Fri.), 9:45 (planned). Language: Japanese. [To all visitors]. ・ Please use public transportation on the day to come to the venue. Your cooperation is requested in wearing a mask at all times when using public transportation, and taking steps to prevent the spread of infections such as refraining from conversation, etc. ・ You are requested to take your temperature on the day of the ceremony and refrain from attending if you have a fever of 37.5 degrees or above, or are feeling unwell. In addition, temperature checks will be performed at the reception desk of the venue. If you have a fever, you will not be permitted to enter. ・Please wear a mask at all times on campus and refrain from talking as much as possible. Please do not remove masks when taking commemorative photos. ・ Note that there will be no campus signboards provided with a view to preventing congestion. Instead, we plan to prepare a photo frame that can be used on smartphones. Please use the photo frame from. here . ・ Please refrain from remaining on campus after the conclusion of the ceremony. ・ Because communal dining involving multiple people gathering to converse and eat and drink together results in the participants removing their masks at close distance while also engaging in conversation, they carry an extremely high risk of infection through droplets, and this has actually been a major source of the spread of infections. Please refrain from communal dining before and after the ceremony, even outside the campus. [To family members]. ・ Family members are kindly requested to refrain from accompanying attendees to the campus. [To all graduates]. ・ Information on the conferral of degrees will be given on the. keio.jp portal. For inquiries on the ceremony: +81(0)3-5427-1517. Footer start. About Keio Global Engagement Top 5 Reasons to Choose Keio Keio by the Numbers Office of the President Philosophy History Alumni Network Campus Information Facts and Figures Publications Conferment of Honorary Degree of Doctor Projects Supported by the Ishii-Ishibashi Fund Keio Logos Social Media & Videos e-Newsletter Admissions Undergraduate Admissions Graduate Admissions Study Abroad Information Fairs Academic Fees FAQ Academics Fields of Study Undergraduate Graduate Study Abroad and International Programs Degree Programs Offered in English Double Degree Programs Short-term Programs Academic Calendar Libraries & Information Technology Services Affiliated Schools Research Find a Researcher Research Centers & Institutes Research Development & Collaboration Research Hubs Abroad Research by Campus Keio Research Highlights Links to external sites. Student Life Student Voices Student Dormitories Scholarships Career Services Athletics & Recreation Health & Well-being Latest News. Keio Times News Press Releases Event Calendar The Olympics \/ Paralympics and Keio University Jobs About This Site Site Map Privacy Policy Contact Us Copyright © Keio University. All rights reserved. ↑. Page Top. Navigation start. Search. Maps. Language. Maps. Language. ↓. Japanese. English. Chinese. Korean. About Keio. ↓. Global Engagement. Top 5 Reasons to Choose Keio. Keio by the Numbers. Office of the President. Philosophy. History. Alumni Network. Campus Information. Continue Exploring. Admissions. ↓. Undergraduate Admissions. Graduate Admissions. Study Abroad Information Fairs. Academic Fees. FAQ. Academics. ↓. Fields of Study. Undergraduate. Graduate. Study Abroad and International Programs. Double Degree Programs. Degree Programs Offered in English. Short-term Programs. Academic Calendar. Libraries & Information Technology Services. Affiliated Schools. Research. ↓. Find a Researcher. Research Centers and Institutes. Research Development and Collaboration. Research Hubs Abroad. Research by Campus. Keio Research Highlights. Links to external sites. Student Life. ↓. Student Voices. Student Dormitories. Scholarships. Career Services. Athletics & Recreation. Health & Well-being. Visit. Apply. Students. Alumni. Internal Links. More. ↓. Internal Links. News. Press Releases. Event Calendar. Keio Times. Contact Us. Visit. Apply. Students. Alumni."}
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慶應義塾の長い歴史の中で行われてきた、演説会や講演会などの取り組みについて紹介します。. →. 三田演説会. 演説や討論という方法を日本に紹介したのは、福澤諭吉と初期の慶應義塾入門生でした。明治7年には三田演説会を組織し、以降、毎年春秋の2回、日本最初の演説会堂、三田演説館において、三田演説会を開催しています。. →. 小泉信三賞全国高校生小論文コンテスト. 小泉信三元塾長の没後10年を記念し、1976年、青少年の文章表現能力の向上に寄与することを目的として始まったコンテストです。. →. ウェーランド講演会. 慶應義塾では、5月15日を「福澤先生ウェーランド経済書講述記念日」として毎年記念講演会を開催しています。. →. 慶應義塾の名誉博士称号. 慶應義塾は、義塾内外を問わず学問・文化の向上に対し功績があると認められた方に名誉博士称号を授与しています。1953年に義塾初の名誉博士称号が授与されました。. 関連コンテンツ. →. 小泉信三記念講座. 外部サイトへリンク. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. 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Life  you were born with...\" Links to external sites","KGRI2040\/CCRC Joint Symposium: Implication of Ukraine-Russia Digital  Conflict on Democracy (Hybrid Format) Links to external sites","Drive My Car: A Symposium on Hamaguchi's Cross-Media Vehicle (Hybrid Format) Links to external sites","Keio Academy of New York  Information Session (Osaka) Links to external sites","KGRI 2040 Archive Online Symposium: Can Media Build the \"Trust\" in the Digital  Society: Thinking through the \"Trust Indicators\"（Video to be released in  June 2022） Links to external sites","Free Online Courses FutureLearn","Keio Research Highlights Links to external sites","","","","","","Keio Research Highlights Links to external sites"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 13:37:38 UTC 2022","fresh_date_dt":"Wed Jun 08 15:55:18 UTC 2022","language_s":"en","size_i":61834,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735073945281888256,"text_t":"Keio University. 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ミュージアム. ミュージアム. 本文の始まり. 慶應義塾は、160年を越える歴史の中で、多様な領域にわたる文化財コレクションを形成してきました。複数の重要文化財も含むこれらのコレクションは、慶應義塾のさまざまな場所に収蔵され、活用されています。キャンパスのミュージアムや展示スペースでは、専門性を活かしたユニークな展覧会が開催されているほか、文化財を使った授業も行われています。. 慶應義塾ミュージアム・コモンズは、塾のコレクションを相互に結びつけ、コレクションに関わる人々の交流を生み出す大学ミュージアムです。また、福澤諭吉記念慶應義塾史展示館では、創立者である福澤の生涯と160年にわたる慶應義塾の歴史が展示されています。. ↓. 慶應義塾ミュージアム・コモンズ. ↓. 福澤諭吉記念慶應義塾史展示館. ↓. 慶應義塾大学アート・スペース. 慶應義塾ミュージアム・コモンズ. →. 慶應義塾ミュージアム・コモンズ. 外部サイトへリンク. 慶應義塾ミュージアム・コモンズ（Keio Museum Commons　略称：KeMCo）は、三田キャンパス東別館に設置されており、義塾が蓄積してきた文化財や学術資料に加え、一般財団法人センチュリー文化財団から寄贈された美術作品などを収蔵しています。KeMCoは新たな時代のミュージアムとして、アナログとデジタルが融け合う環境を構築し、展覧会、イベント、講義、調査・研究やデジタル・アーカイヴなどの活動を通じて、慶應義塾のアートとカルチャーを社会に展開する拠点としての役割を担っていきます。. 福澤諭吉記念慶應義塾史展示館. →. 福澤諭吉記念慶應義塾史展示館. 外部サイトへリンク. 福澤諭吉記念慶應義塾史展示館は、三田キャンパス図書館旧館２階に設置されています。創立者である福澤の生涯と160年にわたる慶應義塾の歴史が展示されます。常設展示は「颯々の章－福澤諭吉の出発」「智勇の章－文明の創造と学問の力」「独立自尊の章－私立の矜持と苦悩」「人間交際の章－男女・家族・義塾・社会」の4章構成で、実物の歴史資料の展示を大切にし、塾史を通して広く日本の近現代史を学べる施設です。. 慶應義塾大学アート・スペース. →. 慶應義塾大学アート・スペース. 外部サイトへリンク. 慶應義塾大学アート・スペースは、三田キャンパス南別館1階に設置されています。博物館相当施設に指定された慶應義塾大学アート・センターが運営するギャラリーで、塾ゆかりの芸術家や、日本の戦後芸術に関するコレクションを所蔵しています。年間を通じて、現代美術、建築、パフォーミング・アーツなど、多岐にわたる企画展を開催しています。. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. 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天は人の上に人を造らず、. 人の下に人を造らずと云えり”. 明治初期、『学問のすゝめ』で人間の自由・平等・権利の尊さを説き、新しい時代の先導者となった福澤諭吉。その教育理念は、福澤が創立した慶應義塾に今も脈々と受け継がれています。ひとりの人間としての福澤は、儀礼的な慣習や常識などにとらわれることを嫌い、飾らず気さくで、また「独立自尊」を身をもって体現する行動的な人物でした。. →. 創立者 福澤諭吉. 「慶應義塾」と命名し. 教育の理念を確立. 幕末の大阪、蘭学者の緒方洪庵の適塾に学んだ福澤諭吉は故郷である中津藩の命を受け、江戸に下り1858（安政5）年、蘭学塾を創始。敢為な精神をもって欧米諸国を見聞した福澤は、帰国後、古いしきたりや慣習にとらわれない教育を実践しました。「独立」や「実学」など、現在も義塾が大切にしている理念は、このときすでに福澤教育の基礎となっています。. 福澤は、1868（慶応4）年、塾舎を芝に移転するとともに学塾の礎を構築。あわせて時の年号にちなみ、学塾に「慶應義塾」という名称を冠したのです。. 福澤が標した道を、その理念を受け継ぐ塾生たちが、新たな時代へ繋ごうとしています。. 慶應義塾の目的. “. 気品の泉源、智徳の模範. ・・・以て全社会の先導者たらんことを欲するものなり”. 福澤諭吉は学問を修める過程で、「智徳」とともに「気品」を重視し、社会の先導者にふさわしい人格形成を志しました。福澤が門下生たちにその志を託した「慶應義塾の目的」と呼ばれる一文があります。. 「慶應義塾は単に一所の学塾として自から甘んずるを得ず。其目的は我日本国中に於ける気品の泉源、智徳の模範たらんことを期し、之を実際にしては居家、処世、立国の本旨を明にして、之を口に言ふのみにあらず、躬行実践、以て全社会の先導者たらんことを欲するものなり」. これは慶應義塾の真に目的とするところを最も簡明に言い表した一文として知られています。. 福澤諭吉と慶應義塾の精神. ぎじゅく. 【義塾】. 義塾. ぎじゅく. もともと漢籍に義塾の用例はありますが福澤は英語の“パブリック・スクール”を参考に、この言葉に新しい知識のための学塾という意味を込めました。. どくりつじそん. 【独立自尊】. 独立自尊. どくりつじそん. 「心身の独立を全うし、自らのその身を尊重して人たるの品位を辱めざるもの、之を独立自尊の人と云う」。自他の尊厳を守り、何事も自分の判断・責任のもとに行うことを意味する、慶應義塾の基本精神です。. じつがく. 【実学】. 実学. じつがく. 福澤がいう実学はすぐに役立つ学問ではなく、「科学（サイヤンス）」を指します。実証的に真理を解明し問題を解決していく科学的な姿勢が義塾伝統の「実学の精神」です。. きひんのせんげん. 【気品の泉源】. 気品の泉源. きひんのせんげん. 福澤は学問を修得していく過程で、「智徳」とともに「気品」を重視していました。人格を備えた社会の先導者となることが、義塾における教育の目標の一つです。. はんがくはんきょう. 【半学半教】. 半学半教. はんがくはんきょう. 教える者と学ぶ者との師弟の分を定めず、先に学んだ者が後で学ぼうとする者を教える。教員と学生も半分は教えて、半分は学び続ける存在という、草創期からの精神です。. じがさっこ. 【自我作古】. 自我作古. じがさっこ. 「我より古を作す（われよりいにしえをなす）」と訓み、前人未踏の新しい分野に挑戦し、たとえ困難や試練が待ち受けていても、それに耐えて開拓に当たるという、勇気と使命感を表した言葉で、義塾の信条となっています。. しゃちゅうきょうりょく. 【社中協力】. 社中協力. しゃちゅうきょうりょく. 社中は、学生・卒業生・教職員など、すべての義塾関係者の総称。塾の運営を経済的に支えている「慶應義塾維持会」など、社中の協力は義塾の誇るべき伝統です。. 関連コンテンツ. →. 創立者 福澤諭吉. →. 塾長メッセージ. →. 歴史. →. 卒業生ネットワーク. →. キャンパス. →. もっと知る. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン 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Engage with Keio from around the globe. 慶應義塾大学は、創立者福澤諭吉の「実学の精神」を基盤とし、総合大学としての強みである分野横断的アプローチにより、教育・研究・医療を通じた社会へのさらなる貢献を目指しています。. 3つの研究クラスター. 長寿クラスター. 豊かな長寿社会の実現. さらに詳しく. 安全クラスター. 安全で安心な社会の構築. さらに詳しく. 創造クラスター. 新たな価値の創造. さらに詳しく. 現在世界が直面する重要課題は数多くありますが、先進国、中でも日本が真っ先に向き合わなければならない課題には、たとえば、少子高齢化や、経済、地域、日常生活に関するセキュリティ、新しい価値の創造をめぐる問題などが挙げられます。慶應義塾大学は、これらの課題解決に向けて、学際的な3つの研究クラスター、「長寿 Longevity」「安全 Security」「創造 Creativity」を軸とした分野横断的な研究を進めています。2014年度に採択された文部科学省スーパーグローバル大学創成支援事業では「実学（サイエンス）により地球社会の持続可能性を高める」を構想テーマに掲げ、各クラスターにおける国際共同研究や教育プログラム、学術交流などを通じて国際的に活躍できる人材の育成にも積極的に取り組んでいます。より良い未来を築くために、国際社会との連携を強化し、学際的かつ国際的な最先端の研究を推進することにより、グローバル社会の発展に貢献していきます。. →. 慶應義塾大学スーパーグローバル大学創成支援事業. Links to external sites. 慶應義塾大学におけるSDGs. 持続可能な開発目標（SDGs, Sustainable Development Goals）とは、2015年9月の国連サミットで採択された17の国際目標（ゴール）です。2030年までに、発展途上国と先進国双方が持続可能でよりよい世界を目指すための普遍的な目標で、「地球上の誰一人取り残さない」を理念として掲げ、貧困、不平等、気候変動、環境問題、平和と繁栄、正義などに関する17の目標と、169のターゲットから構成されています。慶應義塾大学も、大学病院を有する教育・研究機関として、世界と地域社会に貢献する研究大学として、SDGsに関連するさまざまな取り組みを行っています。. →. 慶應義塾大学におけるSDGs. Links to external sites. グローバル・パートナーシップ. 慶應義塾大学は、世界の300以上の大学・高等教育機関等と協定を締結し、学生交換、学術交流、研究連携を展開しています。グローバル社会でのリーダーに必要なスキルや専門性を磨く機会として、日本の大学では最多規模の31（2021年10月現在）の海外の大学とのダブルディグリープログラムを有しています。また、学内では、約1000の英語による授業を提供しています。さらに、. 世界経済フォーラム（ダボス会議）の世界学長会議（GULF） 、. U7+Alliance 、. APRU（環太平洋大学協会） 、ASEAN+3UNet などの大学コンソーシアムの一員として、海外の大学と連携し、世界や地域に貢献する大学として、積極的に活動しています。. →. 海外協定校一覧. Links to external sites. →. 数字で見る慶應義塾. →. Keio University Facts and Figures. PDF. →. The Penmark (e-news letter). →. お問い合わせ：. Keio University Global Engagement Office e-Mail. →. Visit Keio. Keio Hyperlapse Tour. 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塾員（卒業生）ホームページ. 外部サイトへリンク. 塾員（卒業生）向けコンテンツ. 住所変更・証明書. →. 住所変更・改姓. 外部サイトへリンク. →. 証明書. 外部サイトへリンク. 塾員サービス・三田会. →. 塾員（卒業生）ホームページ. 外部サイトへリンク. →. 社中交歡 萬來舍. 外部サイトへリンク. →. 慶應連合三田会. 外部サイトへリンク. →. 食堂・ラウンジの利用. 外部サイトへリンク. ご寄付・ご支援. →. 慶應義塾へのご支援. 外部サイトへリンク. →. 慶應カード. 外部サイトへリンク. →. 維持会. 外部サイトへリンク. 奨学金・就職関連. →. 日本学生支援機構奨学金の貸与を受けていた塾員の皆様へ. 外部サイトへリンク. →. 塾員の皆様への就職支援. 外部サイトへリンク. 慶應義塾について. →. 慶應義塾について. →. 慶應義塾三田オープンカレッジ. 外部サイトへリンク. →. 卒業生ネットワーク. →. 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他大学との連携. 連携窓口・問い合わせ先. [研究機関・企業等の方へ] 産官学連携の窓口となる部門と連絡先は以下のとおりです。. 担当部門. 連絡先（メール）. 連絡先（電話）. 連携全般. 研究連携推進本部. 学術研究支援部総務担当 お問い合わせフォーム 03-5427-1678. 大型連携／インキュベーション. イノベーション推進本部. イノベーション推進本部 お問い合わせフォーム 03-5427-1023. 人文・社会科学分野の研究. 人文・社会科学（三田キャンパス）. 学術研究支援部学術研究支援三田担当 jutaku-mita@adst.keio.ac.jp 03-5427-1756. 理工学分野の研究. 先端科学技術研究センター（理工学部\/理工学研究科）（矢上キャンパス）. 先端科学技術研究センター事務室 kll-staff@adst.keio.ac.jp 045-566-1794. 産官学地域連携（新川崎タウンキャンパス）. 新川崎先端研究教育連携スクエア事務室 k2-tc@adst.keio.ac.jp 044-580-1580. 医学・薬学・生命科学分野の研究. 医学部\/医学研究科（信濃町キャンパス）. 信濃町キャンパス学術研究支援課. ras-shinanomachi@adst.keio.ac.jp 03-5363-3879. 薬学部\/薬学研究科（芝共立キャンパス）. 芝共立キャンパス学術研究支援課. skc-shien@adst.keio.ac.jp 03-5400-2481. 先端生命科学研究所（鶴岡タウンキャンパス）. 鶴岡先端研究教育連携スクエア事務室 office@ttck.keio.ac.jp 0235-29-0800. ウェルビーイングリサーチセンター（殿町タウンキャンパス）. 殿町先端研究教育連携スクエア事務室 contact@tonomachi-wb.jp 044-201-7466. 学際・融合分野の研究. SFC研究所. （総合政策学部\/環境情報学部\/看護医療学部\/政策・メディア研究科\/健康マネジメント研究科）. （湘南藤沢キャンパス）. 湘南藤沢事務室 学術研究支援担当 info-kri@sfc.keio.ac.jp 0466-49-3436. 人文・社会・自然科学、経営管理研究科\/メディアデザイン研究科\/システムデザイン・マネジメント研究科. （日吉キャンパス）. 日吉学術研究支援課 kras-hiyoshi@adst.keio.ac.jp 045-564-2512. 研究成果の発表・展示. 研究成果を広く発信し、共同研究や技術移転など産官学連携のきっかけとなる出会いの場を提供するために、以下のようなイベントを開催しています。. →. 慶應科学技術展（KEIO TECHNO-MALL）. 外部サイトへリンク. →. SFC Open Research Forum. 外部サイトへリンク. →. Hiyoshi Research Portfolio. 外部サイトへリンク. →. KMD Forum. 外部サイトへリンク. 知的資産. 研究連携推進本部の知的資産部門による技術移転活動を通じ、社会貢献度の高い研究成果の発信を総合的かつ戦略的にプロデュースしています。. →. 知的資産部門. 外部サイトへリンク. →. 慶應イノベーション・イニシアティブ（KII）. 外部サイトへリンク. 2015年12月に慶應義塾の関連会社である慶應学術事業会と野村ホールディングスが共同で設立したベンチャーキャピタルです。慶應義塾大学の研究成果を活用したベンチャー企業の育成を今まで以上に推進し、新産業の創造を通じて社会に貢献していくことを目指しています。. 連絡先. （電話）03-6435-0945. （メール）. pr@keio-innovation.co.jp 研究機関との連携（海外）. 海外の大学や学術研究機関と協定等を結び、個別の研究プログラムにおいても積極的に共同研究などの連携を図ることで、学術研究の高度化に取り組んでいます。. →. 海外研究連携拠点. 海外大学との連携. →. 海外協定校一覧. 外部サイトへリンク. →. ダブルディグリープログラム. 研究機関との連携（国内）. 企業との共同研究や人的交流等は、各キャンパスで活発に行われていますが、慶應義塾全体としての包括的な連携協定により、分野横断的な研究や、組織としての多面的協力体制が強化されています。. 2018年12月現在の包括連携協定締結機関（全学対象）は以下のとおりです。. 日本航空株式会社. 持続可能な開発目標（SDGs）に関する共同研究の実施などをめざした連携協力。. →. JALと連携協力協定を締結（2018\/09\/21）. 国立研究開発法人 理化学研究所. 人間知性の解明研究など学部・組織を横断した研究の実施、国際的人材育成のハブ機能の構築などに関しての包括的な連携協力。. 日本電信電話株式会社. 情報通信分野、特にブロードバンド通信、ユビキタス、IPネットワークなどの分野を中心とした連携協力。. 株式会社日立製作所. 国際的かつ分野融合的な技術の発信をめざした連携協力。. 他大学との連携（国内）. 教育・研究体制をさらに強化するために、他大学・大学院との連携にも力を入れています。各学部・研究科単位で多くの連携が行われていますが、全学的な連携についてその一例をご紹介します。. 東京工業大学. 学生交流に関する協定を締結。文理の枠組み、国立私立の壁を越える人材育成を推進。. →. 東京工業大学と学生交流に関する協定を締結（2010\/12\/06）. 慶應義塾大学、早稲田大学、東京工業大学、東京大学「4大学ナノ・マイクロファブリケーションコンソーシアム」. ナノ･マイクロファブリケーションに関する大学間連携コンソーシアムを設立。教育研究体制の整備と産学連携を推進。. →. 慶應義塾大学、早稲田大学、東京工業大学、東京大学による「4大学ナノ・マイクロファブリケーションコンソーシアム」を設立、連携・協力を開始（2009\/02\/18）. 一橋大学. 教育・研究上の連携・協力により研究や実務の世界で活躍するグローバルリーダーの育成を推進。. →. 慶應義塾大学と一橋大学が教育・研究上の連携・協力に関する協定を締結（2008\/10\/29）. 京都大学、東京大学、早稲田大学. 大学院教育における大学間学生交流に関する協定を締結。修士課程または博士課程に在籍する大学院生を対象に、多様な教育・研究指導を受ける機会を提供し、学術の発展と有為な人材を育成。. サイトマップの始まり. 慶應義塾について 塾長メッセージ 理念 歴史 ミュージアム グローバル・エンゲージメント 卒業生ネットワーク キャンパス 数字で見る慶應義塾 情報公開 社会貢献 協生環境 歴史ある取り組み 石井・石橋基金による支援事業 塾歌・カレッジソング 広報誌・パンフレット ロゴマーク使用 ソーシャルメディア・動画 卒業生向けメールマガジン 慶應義塾公式グッズ 外部サイトへリンク. 入学案内 学部入学案内 入試制度 イベント 資料請求 動画ギャラリー ウェブマガジン アプリ・メールマガジン FAQ・お問い合わせ 大学院入学案内 修士課程入学試験 博士課程入学試験 留学生入学試験 FAQ・お問い合わせ 留学生向け入学案内 学費 教育 教育分野 学部 大学院 留学・国際プログラム アカデミックカレンダー 図書館・情報環境 一貫教育校(小・中・高等学校) 生涯学習 研究 研究者情報・業績 研究紹介 研究所・センター 産官学連携 海外研究連携拠点 研究推進・支援 学生生活 奨学金 学生寮 就職・進路 体育会・サークル 学園祭 学生の活躍 オリンピック・パラリンピック選手 更新情報. Keio Times ニュース プレスリリース 取材情報 イベントカレンダー オリンピック・パラリンピックと慶應義塾 受験生 在学生 卒業生 寄付 外部サイトへリンク. 塾内サイト アクセス 情報公開 教職員採用情報 このサイトについて サイトマップ 個人情報の取り扱い お問い合わせ Copyright © Keio University. All rights reserved. ↑. Page Top. ナビゲーションの始まり. 検索. アクセス. Language. アクセス. Language. ↓. Japanese. English. Chinese. Korean. 慶應義塾について. About Keio. ↓. 塾長メッセージ. 理念. 歴史. ミュージアム. グローバル・. エンゲージメント. 卒業生ネットワーク. キャンパス. 数字で見る慶應義塾. もっと知る. 入学案内. Admissions. ↓. 学部入学案内. 入試制度. イベント. 資料請求. 動画ギャラリー. ウェブマガジン. アプリ・メールマガジン. FAQ・お問い合わせ. 大学院入学案内. 修士課程入学試験. 博士課程入学試験. 留学生入学試験. FAQ・お問い合わせ. 留学生向け入学案内. 学費. 教育. Academics. ↓. 教育分野. 学部. 大学院. 留学・国際プログラム. アカデミックカレンダー. 図書館・情報環境. 一貫教育校（小・中・高等学校）. 生涯学習. 研究. Research. ↓. 研究者情報・業績. 研究紹介. 研究所・センター. 産官学連携. 海外研究連携拠点. 研究推進・支援. 学生生活. Student Life. ↓. 奨学金. 学生寮. 就職・進路. 体育会・サークル. 学園祭. 学生の活躍. オリンピック・パラリンピック選手. More. ↓. ニュース. プレスリリース. イベントカレンダー. Keio Times. お問い合わせ. 受験生. 在学生. 卒業生. 寄付. 外部サイトへリンク. 塾内サイト."}
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ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP EN JP KR CN BOOK NOW ABOUT NEWS\/EVENT ROOM\/FACILITIES LOCAL EXPERIENCE ACCESS CONTACT SHOP JP. EN. KR. CN. ABOUT. 「神戸のふつう」に出会う旅を、. ここから。. 古くからさまざまな国の人が行き交い、文化がまじりあってできてきた港町・神戸。. その中心、三宮からほんのすこし離れただけで、ふつうの人びとが豊かに暮らす町の日常があります。. そんな「神戸のふつう」を感じてほしくて、ゲストハウス萬家は灘区にオープンしました。. 見上げれば緑まぶしい山のながめ。. ふもとには長いアーケードの活気ある商店街。. 親しみやすく、気どらないお店や人びと。. 「地元にいちばん近い旅」をわたしたちはお手伝いします。. たとえば──. ケーブルカーや山歩きで摩耶山に登り、町と海を一望する。. 王子動物園や美術館をゆったり散策し、レトロとモダンに出会う。. 町の台所である水道筋商店街や市場をめぐって買い物を楽しむ。. 歩き疲れたら、お気に入りの喫茶店でお茶を飲みながらおしゃべりをして、一日の終わりに町の銭湯で汗を流す。. おふろ上がりには、仕事帰りの人びとでにぎわうお店に寄り道して、おいしい料理とお酒をいただく。. そんなふうにして神戸のこと、山や海のこと、お店や食べ物のこと、日々の暮らしのこと、さまざまな表情にふれてもらいたいのです。. 「地元にいちばん近いゲストハウス」だから知っている町の魅力、ご案内できる旅のかたちがここにあります。. あなたの旅のページに. たくさんの「神戸のふつう」が書き加えられますように。. [Join the local experience in Nada!]. NEWS. 2022年5月26日. 《ひょうご旅しようキャンペーン＋》　兵庫県民割6月まで延長になります。. ゲストハウス萬家は「ひょうご旅しようキャンペーン」対象施設です。 「ふるさと応援！ひょうごを旅しよう[…]. read more 2022年4月17日. 価格改定のお知らせ. 価格改定のお知らせ 平素は格別のお引き立て、ご愛顧を賜り厚く御礼を申し上げます。 このたびゲストハウ[…]. read more 2022年3月11日. ”旧摩耶観光ホテル”見学付き宿泊プラン始めます！. かつて摩耶山は、天上寺への参拝客をはじめ、多くの人が訪れる場所として、食事処や宿泊施設等などで賑わい[…]. read more 2021年8月23日. 【8\/25(水)インスタライブのお知らせ】. 韓国文学オンライントーク\/K-BOOK TALK #3【インスタライブのお知らせ】K-BOOK TA[…]. read more ROOM. 地元から愛される医院として使われていた一軒家を地元の人と一緒にリノベーションしました。木を基調に、スタイリッシュさと可愛さのある空間に仕上げました。窓や屋上からは摩耶山、阪急電車、王子公園の観覧車をご覧頂けます。. 男女混合6人ドミトリー. 6 Mixed Dormitory. 木のぬくもりが優しい、形がユニークな二段ベッドのお部屋です。. ・読書灯、コンセント、ベッドカーテン、鍵付き収納ボックス完備. ・バス、トイレ共用. ・部屋の入口ダイヤル施錠可能. 女性専用6人ドミトリー. 6 Female Dormitory. 山小屋のような屋根がついた可愛らしい二段ベッドのお部屋です。. パウダースペースもご用意しております。. ・読書灯、コンセント、ベッドカーテン、鍵付き収納ボックス完備. ・バス、トイレ共用. ・部屋の入口ダイヤル施錠可能. シングルルーム. Single Room. 広々としたベッドを独り占め。ゆったりとお休み頂けます。. ・読書灯完備. ・バス、トイレ共用. マルチルーム(2~4名). Multi Room. 木の柱が印象的な、シングルベッド二台と二段ベッドのお部屋です。. ・読書灯完備. ・バス、トイレ共用. ダブルルーム. Double Room. 落ち着いた雰囲気の、ダブルベッドのお部屋です。. プライベートな時間をゆっくりとお過ごし頂けます。. ・読書灯完備. ・バス、トイレ共用. ※料金や空室状況は『BOOK NOW』をクリックしてご確認下さいませ。. FACILITIES. ラウンジ. Lounge. 食事に、交流に、リラックスタイムに、萬家を訪れた人々が自由に使えるスペースです。スパイラルベンチやUFO型スピーカーなどアイコニックなインテリアが皆様をお迎えします。. キッチン. Kitchen. IHクッキングヒーター、電子レンジ、オーブントースター、電気ケトル、冷凍・冷蔵庫をご用意。フリーティーなどもお楽しみ頂けます。. バスルーム. Bath Room. 各部屋には洗面、シャワーの設備がありませんので 共用のものをご利用ください。シャワー内にはシャンプーとボディソープが常備されており、24 時間ご利用可能です。ドライヤーは深夜帯のご利用を制限させて頂いております。. 無料サービス. Wifi、ヘアドライヤー、シャンプー&ボディソープ、お茶、耳栓、目覚まし時計. 有料サービス. ランドリー、レンタルタオル、歯ブラシ、電源アダプター、自転車、アイロン、パジャマ、荷物預かり. INFORMATION. Check in／16:00～21:00. Check out／10:30. ※上記の時間以外のチェックイン・チェックアウトもできる限り対応致します。事前にご相談ください。. ※チェックイン後の門限はありません。. ※チェックイン前、後のお荷物のお預かりも可能です。（チェックイン日、チェックアウト日は無料、それ以外は300円\/泊）. ・お支払い方法. お支払いはチェックイン時にお願い致します。※一万円以上のお支払いよりクレジットカードのご利用が可能です。(VISA・MasterCard・AMERICAN EXPESS). ・ご予約について. ・ご予約は4ヶ月先までの受付となります。. ・キャンセル料は宿泊日の3日前から発生致します。. 　キャンセル規定\/3日前：30%, 2日前 : 50%, 前日：80%, 当日・無連絡不泊：100％. ・当宿は小さな宿ですので、申し訳ありませんが5名様以上のご予約は受付致しかねます。予めご了承ください。. ・未就学のお子様のご利用はお受けしておりません。. ・安全上、階段をお一人で上るのが大変な方のご予約をお断りする場合がございます。. ・セキュリティー. 正面エントランスと各部屋に鍵を常備しており、セキュリティー対策は万全です。. 営業時間外はエントランスを施錠し、夜間もスタッフが常駐しておりますので、安心してご利用いただけます。. ・女性ゲストへの配慮. 女性スタッフもおり、女性専用のシャワーとトイレもご用意しております。治安も良い地域ですので、女性の一人旅でも安心してご利用いただけます。. LOCAL EXPERIENCE. 神戸のふつうの1日をご紹介. たとえば、朝食は近くの小さなパン屋さん、昼は商店街や酒蔵や美術館など色んな観光を楽しみ、夜は摩耶山から見る夜景やちょっとディープでお洒落な飲み屋を堪能。是非、神戸のふつうを満喫しにきてください。. Things to enjoy around our guesthouse. In The Morning　#1. 神戸のパン屋さん. Details In The Morning　#2. 神戸の喫茶店とコーヒー. Details In The Morning　#3. 朝5時から営業の灘温泉. Details In The Late Morning. 海の見えるなぎさ公園までお散歩. Details In The Afternoon #1. ザ・ローカルな水道筋商店街. Details In The Afternoon #2. 王道を行くなら、王子動物園. Details In The Afternoon #3. 雨の日には、美術館へ. Details In The Afternoon #4. 灘の新しい見所“灘高架下エリア”. Details The Afternoon #5. お酒が好きなら、酒蔵巡り. Details In The Late Afternoon. いよいよ摩耶山へ. Details In The Evening. 灘の美味しくて面白いお店. Details KOBE BEEF. 神戸と言えば、神戸牛. Details ACCESS. JR灘駅から. 改札を出て左方面にある東北口を出ます。東北口を出て左にある横断歩道を渡り、王子動物園に向かって北に進みます。1分くらい歩くと電車の高架が見えます。その高架に添って右に進みます。30秒くらい歩くと右側に「ワールドエンズガーデン」という古本屋が見えます。そのままさらに30秒ほど歩くとゲストハウス萬家に到着します。. 阪急電鉄　王子公園から. 王子公園駅の西口を出て、左に見える横断歩道を渡るとセブンイレブンがあります。セブンイレブンと高架下の間の道を２分ほど歩くと左側にゲストハウス萬家が見えます。. 阪神電鉄　岩屋駅から. 改札を出て外に出ると右にセブンイレブンが見えます。そこを右に曲がるとＪＲ灘駅が見えます。灘駅の2階に上って東北口に出てください。（ここからは灘駅からゲストハウスまでのアクセスと同じ）. ゲストハウス萬家(MAYA). 〒657-0836. 兵庫県神戸市灘区城内通4丁目4-10. 予約・問い合わせ：090-9856-8280（受付時間 8:00-22:00）. TEL: 078-940-2809（受付時間 8:00-22:00）. e-mail: info@kobemaya.com. Copyright © GUEST HOUSE MAYA KOBE JAPAN All Rights Reserved."}
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Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. 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We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Sorbonne Université. Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home University Rental of Conference Facilities Spaces offered by the Faculty of Medicine. Spaces offered by the Faculty of Medicine. Spaces offered by the Faculty of Medicine . Publié le 24\/12\/2019 - Mis à jour le 2\/06\/2020. Spaces offered by the Faculty of Medicine. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Education Ensure successful studies Tools and Resources. Tools and Resources. Sorbonne University provides its students with all the resources necessary for their success, on site or remotely, and to benefit from innovative teaching methods. Students have access to the entire network of Sorbonne University libraries, which offer co-working and modular workspaces, as well as online resources. Computer rooms and a Wifi zone are available. A digital workspace (ENT) is accessible to all. Sorbonne University is also committed to the development of student entrepreneurship. . Publié le 7\/12\/2019 - Mis à jour le 4\/05\/2022. Menu. Tools and Resources. SIAL, a language learning service ENT. The digital workspace (ENT) brings together all the services and information for Sorbonne University students, including: administrative files, certificate of attendance, exam dates, transcripts, faculty news (room changes, absence of a professor, brochures), and messaging. . This site is only accessible to Sorbonne University students. It gives access to the portal to obtain course documents, schedules, and e-learning. Students also have access to their online courses on Moodle via their NDE. Libraries. Solo or co-working areas. Solo or in groups, work spaces are available to students:. 50 group work rooms and 40 individual cubicles on the Pierre and Marie Curie campus. Bookable on the Burésa site. 20 group workrooms on the Faculty of Arts and Humanities sites. Bookable on the Affluences app. 1 coworking area with free access in graduate library on the Pierre and Marie Curie campus. 1 coworking space, 6 project development rooms and 1 meeting room for project development on the Pierre and Marie Curie campus. By reservation at the Student Life Space.  . Online Resources. The Sorbonne University Library provides access to a host of electronic resources:. Browse through the list of electronic resources, then log in using your regular institutional account;. Discover ebooks and online journals in all fields of knowledge taught at the University. To search for them, go to the AZ Publication Finder. If you have a specific question about searching for online resources, a subscription or an access problem... contact. the BSU team who will answer you as soon as possible. Multimedia rooms. A student card enables self-service access to specialised databases so students can carry out their dissertation and documentary research. Clignancourt Campus - Room E06. Self-service according to the schedule available on the ENT from the end of September. Malesherbes Campus. Self-service according to the schedule available on the ENT from the end of September. Self-service research house according to the schedule available on the ENT from the end of September. Campus Pitié-Salpêtrière 91 bd de l’hôpital, salle 218. Monday to Friday 9 am - 6 pm. Campus Pierre et Marie Curie. Atrium, CAPSULE, 1er étage, salle 113. self-service, Monday to Friday 8am-7pm. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. Let's build the network together Join a club or association Alumni portraits Contact us Education Education. Study at Sorbonne University Study at Sorbonne University. Our Academic Programs Degree-seeking Students Exchange Students Continuing Education Ensure successful studies Ensure successful studies. Office of the Registrar Tutoring Student Services Student internships Tools and Resources Reorientation and caesura Student life Student life. Grants and financial assistance Health and Handicap Services Sports and Cultural Activities Student associations Work Spaces Student Life and Cultural Spaces Cafeteria and Food Service Housing Prepare Your Professional Goals Prepare Your Professional Goals. Employability Facts and Figures Student Entrepreneruial Training Sorbonne University Librairies Sorbonne University Librairies. 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Sorbonne University, an attractive environment for doctoral candidates Doctoral College Application and admission Doctoral Course Career plans and professional development Milestones Administrative procedures Culture and outreach Culture and outreach. Cultural Events Noteworthy Places, Collections and Bequests Noteworthy Places, Collections and Bequests. Architecture and works of art Outstanding collections and bequests An Interdisciplinary Approach to Benefit Society An Interdisciplinary Approach to Benefit Society. Multidisciplinary thematic institutes University for All Ages University for All Ages. Summer University The Inter-Age University Community Outreach Community Outreach. The Science House The Patients’ University The Participatory Science Portal My Life as a Researcher The Teddy Bear Hospital Bookshop Bookshop. The Sorbonne University Press Other publications Foundation Foundation. \"Welcome to the Future\" \"Welcome to the Future\". Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Research and innovation The HRS4R Label. The HRS4R Label. The HRS4R label. In order to build and reinforce the European Research Area, the European Union is deploying a policy to promote the profession of research, with the goal, among other things, of encouraging mobility and career opportunities. A. European Charter for Researchers and a Code of Conduct for Recruitment define the rights, responsibilities and roles of researchers, as well as those of their employers. The Charter and Code aim to ensure the appeal of careers and to improve the recruitment and working conditions of researchers in Europe. These objectives are fundamentally in line with Sorbonne University's policy of enhancing the attractiveness and international visibility of the institution and its active participation in the European Research Area. Sorbonne University has therefore. engaged in the European. Human Resources Strategy for Researchers. (HRS4R) labelling process. This label, delivered by the European Commission, aims to support institutions and organisations working in the field of research in improving their practices in terms of working conditions, recruitment and ethics. Once awarded, it is a. guarantee of the attractiveness. of research careers and of the university to researchers throughout the European continent. It ensures equal treatment in the different institutions and facilitates. access to European research funds. Again, these objectives are fully in line with the Sorbonne University's policy in terms of support for research and its actors. . The labelling process. The process of obtaining the label began with a. self-evaluation of practices at Sorbonne University , encompassing all research personnel (doctoral researchers, doctoral researchers, post-doctoral researchers, teacher-researchers). It focused on four main areas: human resources, working conditions, professional training and ethics. On this basis, an. action plan has been defined to enable the improvement of Sorbonne University’s practices in accordance with the 40 principles defined in the European Charter for Researchers and the Code of Conduct for Recruitment. This action plan followed a. survey carried out among 400 researchers. of all types: doctoral researchers, post-doctoral researchers, assistant professors and professors. With 43% of complete responses, the responses obtained are representative of the diversity of our community. . HRS4R Documents of Sorbonne University:. Self-assessment Action plan  . Open, transparent and merit-based recruitment policy  Sorbonne University charters and engagements :. European Charter for Researchers - Code of conduct for the recruitment of researchers National Charter of Ethics for Research Professions Charter for Equality between Women and Men  Charter for Open Acces to Publications PhD researchers  . Missions and support structures:. Research ethics committee Ethics and whistleblowing officer Sorbonne declaration on research data rights Commitment to open science    Mission for scientific integrity  Mission for gender equality Mission to combat racism, antisemitism, and anti-LGBT discrimination External unit for victims of sexual and gender-based violence Contact. Contacts. Catherine Achard and  . Lucie Salvaudon , head of the HRS4R labelling project. Steering and Supervising Committees. Steering committee. Catherine Achard - Coordinator of the Sorbonne University certification project and Academic Researcher. Lucie Salvaudon - Coordinator of the Sorbonne University certification project. Stéphanie Bonneau - Vice President Personnel, Professional Equality and Diversity and Academic Researcher. Nathalie Drach-Temam - Vice President Research, Innovation and Open Science and Academic Researcher. Véronique Perdereau - Vice President Europe and Academic Researcher. 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Read\/write 25XX NOR flash via SPI protocol. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Do not use CH341A! Identify which flash type you have SOIC8 SOIC16 WSON8 DIP8 Software configuration BeagleBone Black (BBB) Caution about BBB Rasberry Pi (RPi) Caution about RPi Install flashrom How to use flashrom Reading Writing Hardware configuration WARNINGS MISO\/MOSI\/CS\/CLK lines ISP programming and VCC diode GPIO pins on BeagleBone Black (BBB) GPIO pins on Raspberry Pi (RPi) 40 Pin GPIO pins on Raspberry Pi (RPi) 26 Pin SOIC8\/DIP8\/WSON8 wiring diagram SOIC16 wiring diagram (Raspberry Pi) Pull-up resistors and decoupling capacitors SOIC8\/WSON8\/DIP8\/SOIC16 not mounted to a mainboard SOIC8\/SOIC16 soldered to a mainboard DIP8 soldered to the mainboard Replace WSON8 IC with SOIC8 LICENSING Return to index This guide will teach you how to use various tools for externally reprogramming a 25xx NOR flash via SPI protocol. This is the most common type of flash IC for computers that coreboot runs on. Almost every system currently supported by Libreboot uses this type of boot flash; the only exception is ASUS KFSN4-DRE, which uses LPC flash in a PLCC32 socket, which you can simply hot-swap after booting the vendor firmware, and then flash internally. Simple! We will be using the. flashrom software which is written to dump, erase and rewrite these flash chips. Libreboot currently documents how to use these SPI programmers:. Raspberry Pi (RPi). BeagleBone Black (BBB). Many other SPI programmers exist. More of them will be documented on this page, at a date in the future. Most systems in Libreboot have to be re-flashed externally, using instructions on this and similar guides, the first time you flash. However, on all currently supported systems, it’s possible that you can re-flash. internally. when Libreboot is running. Internal. flashing means that the host CPU on your system can re-program the SPI flash, using an on-board SPI programmer (which all boards have). You do this from GNU+Linux, with flashrom. This. guide that you’re reading now is for using an. external. programmer. It is called. external. because it’s not the. internal. one on your mainboard. Do not use CH341A! [link] NOR flashes on Libreboot systems run on 3.3V DC, and this includes data lines. CH341A has 5V logic levels on data lines, which will damage your SPI flash and also the southbridge that it’s connected to, plus anything else that it’s connected to. These ch341a programmers are unfortunately very popular. DO NOT use it unless you have fixed the issue. You CAN fix it so that the data lines are 3.3v, if you follow the notes here:. https:\/\/www.eevblog.com\/forum\/repair\/ch341a-serial-memory-programmer-power-supply-fix\/ In practise, most people will not fix their ch341a and instead just risk it, so no documentation will be provided for ch341a on this website. It is best to discourage use of that device. Not covered on that eevblog page: the WP\/HOLD pins (pins 3 and 7) must be held high via pull-up resistors, but on CH341A dongles, they are directly connected to 3.3V DC (continuity with pin 8). It is advisable to cut these two connections, to the WP and HOLD pins, and jump the cuts using pull-up resistors instead. A value between 1k to 10k (ohms) should be fine. In case it’s not clear:. Please do not buy the ch341a! It is incorrectly engineered for the purpose of ROM flashing on systems with 3.3v SPI (which is most coreboot systems). DO NOT USE IT! This issue still isn’t fixed by the manufacturer, and it doesn’t look like they will ever fix it. If you see someone talking about CH341A, please direct them to this page and tell them why the CH341A is bad. Identify which flash type you have. [link] In all of them, a dot or marking shows pin 1 (in the case of WSON8, pad 1). Use the following photos and then look at your board. When you’ve figured out what type of chip you have, use that knowledge and the rest of this guide, to accomplish your goal, which is to read from and\/or write to the boot flash. SOIC8. [link] SOIC16. [link] SOIC8 and SOIC16 are the most common types, but there are others:. WSON8. [link] It will be like this on an X200S or X200 Tablet:. On T400S, it is in this location near the RAM:. NOTE: in this photo, the chip has been replaced with SOIC8. DIP8. [link] Software configuration. [link] BeagleBone Black (BBB). [link] SSH into your BeagleBone Black. It is assumed that you are running Debian 9 on your BBB. You will run. flashrom. from your BBB. NOTE: This section is out of date, because it is written for Debian 9 (running on the BBB). Run the following commands as root to enable spidev:. config-pin P9.17 spi_cs config-pin P9.18 spi config-pin P9.21 spi config-pin P9.22 spi_sclk. Verify that the spidev devices now exist:. ls \/dev\/spidev*. Output:. \/dev\/spidev1.0 \/dev\/spidev1.1 \/dev\/spidev2.0 \/dev\/spidev2.1. Now the BBB is ready to be used for flashing. The following systemd service file can optionally be enabled to make this persistent across reboots. [Unit] Description=Enable SPI function on pins [Service] Type=oneshot ExecStart=config-pin P9.17 spi_cs ExecStart=config-pin P9.18 spi ExecStart=config-pin P9.21 spi ExecStart=config-pin P9.22 spi_sclk RemainAfterExit=yes [Install] WantedBy=multi-user.target. Now test flashrom:. .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512. It is important to use. spispeed=512. or a lower number such as 256 or 128, because otherwise the BBB will be quite unstable. Example output:. Calibrating delay loop... OK. No EEPROM\/flash device found. Note: flashrom can never write if the flash chip isn't found automatically. This means that it’s working (the clip isn’t connected to any flash chip, so the error is fine). Caution about BBB. [link] BeagleBone Black is not recommended, because it’s very slow and unstable for SPI flashing, and nowadays much better options exist. We used to mainly recommend the BBB, because of the fact that it can be used with entirely Free Software on it, but nowadays there are superior options. TODO: document other SPI flashers. Rasberry Pi (RPi). [link] SSH into your Raspberry Pi. You will run. flashrom. from your Raspberry Pi. You must configure. spidev. on your Raspberry Pi. This is a special driver in the Linux kernel; technically, the driver name is. spi-bcm2835. . This page has info:. https:\/\/www.raspberrypi.org\/documentation\/hardware\/raspberrypi\/spi\/README.md In your Raspberry Pi, which we assume you’re running the latest Raspbian version on, do this:. sudo raspi-config. Under the Interface section, you can enable SPI. The device for communicating via SPI as at. \/dev\/spidev0.0. Caution about RPi. [link] Basically, the Raspbian project, now called Raspberry Pi OS, put in their repo an update that added a new “trusted” repository, which just so happened to be a Microsoft software repository. They seem to have done this for VS Code, but the problem here is that it gave Microsoft free reign to define whatever dependencies they liked (as per apt-get rules), and every time you updated, you would be pinging Microsoft servers. Do you think that is strange? Microsoft shouldn’t have. any. access to your GNU+Linux system! This was the commit that Raspbian added to their distro, which added this what should rightly be called a security vulnerability, intentionally:. https:\/\/github.com\/RPi-Distro\/raspberrypi-sys-mods\/commit\/655cad5aee6457b94fc2336b1ff3c1104ccb4351 They then removed it, after a public backlash, via the following commits:. https:\/\/github.com\/RPi-Distro\/raspberrypi-sys-mods\/commit\/ed96790e6de281bc393b575c38aa8071ce39b555 https:\/\/github.com\/RPi-Distro\/raspberrypi-sys-mods\/commit\/4d1afece91008f3787495b520ac03b53fef754c6 For now, Raspbian \/ Raspberry Pi OS (which is based on Debian) should be safe, but this whole episode proves that the distro can no longer be trusted to respect its users. Therefore, it’s now on the. tasks page a TODO entry for recommending and documenting alternative GNU+Linux distros on the Raspberry Pi, for the purposes of SPI flashing. Install flashrom. [link] If you’re using a BBB or RPi, you will do this while SSH’d into those. Flashrom is the software that you will use, for dumping, erasing and rewriting the contents of your NOR flash. In the Libreboot build system, from the Git repository, you can download and install flashrom. Do this after downloading the. lbmk Git repository :. cd lbmk sudo .\/build dependencies ubuntu2004. NOTE: debian, arch or void can be written instead of ubuntu2004. the debian script is also applicable to newer ubuntu versions. .\/download flashrom .\/build module flashrom. If the. ubuntu2004. script complains about missing dependencies, just modify the script and remove those dependencies. The script is located at. resources\/scripts\/build\/dependencies\/ubuntu2004. and it is written for Ubuntu 20.04, but it should work fine in other GNU+Linux distributions that use the. apt-get. package manager. A. flashrom\/. directory will be present, with a. flashrom. executable inside of it. If you got an error about missing package when running the dependencies command above, tweak. resources\/scripts\/build\/dependencies\/ubuntu2004. . That script downloads and installs build dependencies in apt-get and it is intended for use on x86-64 systems running Ubuntu 20.04, but it should work in Raspbian on the Raspberry Pi. Alternatively, you may download flashrom directly from upstream at. https:\/\/flashrom.org\/Flashrom If you’re flashing a Macronix flashchip on a ThinkPad X200, you will want to use a special patched version of flashrom, which you can download here:. https:\/\/vimuser.org\/hackrom.tar.xz - patched source code is available, and a binary is also available that you can simply run. Pass the. --workaround-mx. argument in flashrom. This mitigates stability issues. If you downloaded the flashrom source code directly, you can go into the directory and simply type. make. . In the Libreboot build system, build dependencies are documented in script located at. resources\/scripts\/build\/dependencies\/. which you can install using the. apt-get. software. How to use flashrom. [link] Read past these sections, further down this page, to learn about specific chip types and how to wire them. Reading. [link] Before flashing a new ROM image, it is highly advisable that you dump the current chip contents to a file. Run this command to see if 25xx flash is detected, with your RPi properly wired. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev0.0,spispeed=32768. For BBB, you must use a lower speed and a different device path:. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512. On BBB, never use a speed higher than. spispeed=512. . In some cases, you may even need to go as low as. spispeed=128. . The BBB is highly unstable and unreliable for SPI flashing. When you’re reading, take multiple dumps and verify that the checksums match, before you flash. You may have to flash your chip several times! NOTE: On some systems, higher speeds will be unstable. On those systems, try lower speed like. spispeed=4096. or even. spispeed=2048. which should, in most cases, work just fine but it will obviously be slower. The. spispeed=32768. setting works just fine on most setups if you use short wires, within 10cm. If flash chip is detected you may try reading (dumping) the flash contents now, or you can try flashing it with a new ROM. Dump it like so (RPi):. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev0.0,spispeed=32768 -r dump.bin. For BBB, do this:. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512 -r dump.bin. It is advisable to take a. 2nd. dump, e.g. . dump2.bin. , and then check sha1sum:. sha1sum dump*.bin. If the checksums match, it indicates that you have a good dump. If they do not, check your wiring. Wires should be within 10cm length for best stability, and they should all be the same length (3.3V VCC and GND wires can be longer). This advice is. especially. applicable to the BBB, which is highly unreliable. Writing. [link] Next, run this command (RPi):. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev0.0,spispeed=32768 -w \/path\/to\/libreboot.rom. If using BBB:. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512 -w \/path\/to\/libreboot.rom. If using BBB, you may have to use a lower speed than 512. You may also have to re-flash several times before it works fully. Again, use a lower. spispeed. value if you need to, for stability. Once that command outputs the following, the flash has completed successfully. If not, just flash again. Reading old flash chip contents... done. Erasing and writing flash chip... Erase\/write done. Verifying flash... VERIFIED. If it says “VERIFIED” or says that the chip contents are identical to the requested image, then the chip is properly flashed. Hardware configuration. [link] Refer to the above guidance about software configuration. The following advice will teach you how to wire each type of flash chip. WARNINGS. [link] Do not connect a 3.3V DC power source until your chip is otherwise properly wired. For instance, do not connect a test clip that has 3.3V DC attached. Do not. disconnect. your chip from the flasher until you’ve disconnected or turned off the 3.3V DC power source. BE CAREFUL that you are indeed supplying 3.3V DC to the chip. All SPI flashes on all currently supported Libreboot hardware run on 3.3V DC and logic at that level too. It is important to CHECK that you are running on the correct voltage, when you do anything with these chips. Lower than 3.3V won’t damage anything, but higher will fry your chip (on most chips, the tolerated voltage range is between 2.7V and 3.6V, but 3.3V is the most ideal level). DO NOT connect more than 1 DC power source to your flash chip either! Mixing voltages like that can easily cause damage to your equipment, and to your chip\/mainboard. MISO\/MOSI\/CS\/CLK lines. [link] You may want to add 47ohm series resistors on these lines, when flashing the chips. Only do it on those lines (NOT the 3.3V or GND lines). This provides some protection from over-current. On Intel platforms, the SPI flash is usually connected via such resistors, directly to the Southbridge chipset. ISP programming and VCC diode. [link] ISP means in-system programming. It’s when you flash a chip that is already mounted to the mainboard of your computer that you wish to install Libreboot on. It may be beneficial to modify the mainboard so that the SPI flash is powered (on the VCC pin) through a diode, but please note: a diode will cause a voltage drop. The tolerated range for VCC on these chips is 2.7V to 3.6V DC. If you do this, please also ensure that the WP and HOLD pins are still held to a high logic state; each via their own resistor (1K to 10K ohms) connected to the. same. 3.3V DC source going through the diode. The reason is simple: on most systems, the flash shares a common 3.3V DC rail with many other components on the board, which draw a lot of current. Further, if you accidentally provide too much voltage or cause an overcurrent, you could fry those other components but if there is diode protection, you’ll only fry the boot flash (and it is very easy to replace, if you have good soldering skills). When you’ve placed the diode, ensure that VCC on the chip is isolated from all other components on that board, which share the same 3.3V DC rail. Further, ensure that the pull-up resistors for WP\/HOLD are. only. connected to the side of the diode that has continuity with the VCC pin (this is important because if they’re not, they won’t be held high while doing ISP flashing, even if they’re still held high when the mainboard is fully powered on). Furthermore: ensure that the SPI flash is operating at 2.7 to 3.6V when fully powered on, after installing the diode. If it’s a desktop\/workstation\/server board (not a laptop), you could de-solder the SOIC8\/WSON8 if it uses that, and replace with an IC socket (for SOIC8, WSON8 or DIP8, whatever you want), because then you could easily just insert the flash into a breadboard when flashing. TODO: Make a page on libreboot.org, showing how to do this on all mainboards supported by Libreboot. GPIO pins on BeagleBone Black (BBB). [link] Use this image for reference when connecting the pomona to the BBB:. https:\/\/beagleboard.org\/Support\/bone101#headers (D0 = MISO or connects to MISO). On that page, look at the. P9 header. . It is what you will use to wire up your chip for flashing. GPIO pins on Raspberry Pi (RPi) 40 Pin. [link] This diagram shows the pinout for most modern Pi’s and Pi derivatives. The diagram shows the pins of an RPi on the left and the two SOIC clips on the left. GPIO pins on Raspberry Pi (RPi) 26 Pin. [link] Diagram of the 26 GPIO Pins of the Raspberry Pi Model B (for the Model B+ with 40 pins, start counting from the right and leave 14 pins):. Use this as a reference for the other sections in this page, seen below:. SOIC8\/DIP8\/WSON8 wiring diagram. [link] Refer to this diagram:. Pin #. 25xx signal. RPi(GPIO). BBB(P9 header). 1. CS. 24. 17. 2. MISO. 21. 21. 3. not used. not used. not used. 4. GND. 25. 1. 5. MOSI. 19. 18. 6. CLK. 23. 22. 7. not used. not used. not used. 8. 3.3V. 1. 3. On your SOIC8, there will be a dot in one of the corners. The dot is pin 1. NOTE: pins 3 and 7 are WP\/HOLD pins. If flashing a chip on a breadboard, please use pull-up resistors on those (see notes below), and decoupling capacitor on pin 8 (VCC). NOTE: On X60\/T60 thinkpads, don’t connect pin 8. Instead, plug in your the PSU to the charging port on your mainboard, but do not power on the mainboard. This will provide a stable 3.3V voltage, with adequate current levels. On those laptops, this is necessary because the flash shares a common 3.3V DC rail with many other ICs that all draw quite a lot of current. SOIC16 wiring diagram (Raspberry Pi). [link] RPi GPIO header:. BBB P9 header:. https:\/\/beagleboard.org\/static\/images\/cape-headers.png Refer to this diagram:. Pin #. 25xx signal. RPi(GPIO). BBB(P9 header). 1. not used. not used. not used. 2. 3.3V. 1. 3. 3. not used. not used. not used. 4. not used. not used. not used. 5. not used. not used. not used. 6. not used. not used. not used. 7. CS#. 24. 17. 8. MISO. 21. 21. 9. not used. not used. not used. 10. GND. 25. 1. 11. not used. not used. not used. 12. not used. not used. not used. 13. not used. not used. not used. 14. not used. not used. not used. 15. MOSI. 19. 18. 16. SCLK. 23. 22. Refer to the RPi GPIO guide above, on this page. On your SOIC16, there will be a dot in one of the corners. The dot is pin 1. NOTE: pins 1 and 9 are WP\/HOLD pins. If flashing a chip on a breadboard, please use pull-up resistors on those (see notes below), and decoupling capacitor on pin 2 (VCC). Pull-up resistors and decoupling capacitors. [link] Do this for chips mounted to a breadboard. Ignore this section if you’re flashing a chip that is already soldered to a mainboard. This section is only relevant if you’re flashing a new chip that is not yet mounted to a mainboard. You need pull-up resistors on the WP and HOLD pins, and decoupling capacitors on the VCC pin. If the chip is already mounted to a board, whether soldered or in a socket, these capacitors and resistors will probably already exist on the board and you can just flash it without pulling WP\/HOLD high, and without capacitors(just connect your external 3.3V DC power source). The best way is as follows:. Insert the DIP8 IC into a breadboard (2.54mm holes), if it’s DIP8. Insert WSON8 into a WSON8 socket and put on a breadboard, if WSON8. Insert SOIC8 into a SOIC8 socket and put on a broadboard, if SOIN8. Wire an SPI flasher, using 2.54mm dupont leads, to the breadboard, using the correct wiring (see link to SPI flashing guides below). SOIC8\/WSON8\/DIP8: pin 3 and 7 must be held to a high logic state, which means that each pin has its own pull-up resistor to VCC (from the 3.3v voltage plane that pin 8 connects to); anything from 1Kohm to 10Kohm will do. When you’re flashing a chip that’s already on a laptop\/desktop\/server mainboard, pin 3 and 7 are likely already held high, so you don’t need to bother. SOIC8\/WSON8\/DIP8: pin 8, which is 3.3v VCC, will already have decoupling capacitors on it if the chip is on a mainboard, but lone chip flashing means that these capacitors do not exist. A capacitor passes AC but blocks DC. Due to electromagnetic indunctance, and RF noise from high-speed switching ICs, a DC voltage line isn’t actually straight (when viewed on an oscilloscope), but actually has low voltage AC mixed in; on a particularly noisy line under high load, noise of around 300mV or more is common. To smooth out that noise, you wire capacitors from the DC line to ground, with the side of the capacitor on VCC as close to the IC’s VCC pin as possible. We recommend that you use ceramic capacitors for this purpose. The recommended capacitors for this are: 100nF and 4.7uF ceramic capacitors. Electrolytic capacitors are inferior for this, because they have higher ESR (ceramic capacitors have super low ESR, which is very good for decoupling). The result of using a decoupling capacitor is that some of the noise on the DC line is filtered to ground, making the DC signal much cleaner\/straighter (when seen on an oscilloscope). SOIC16: same as above, but use a SOIC16 socked on a breadboard. On SOIC16, WP\/HOLD are not pin 3\/7 like above, but instead pins 1 and 9, so wire your pull-up resistors on those. 3.3v VCC on SOIC16 is pin 2, so wire your decoupling capacitors up on that. SOIC8\/WSON8\/DIP8\/SOIC16 not mounted to a mainboard. [link] If your system has lower capacity SPI flash, you can upgrade. On. most. systems, SPI flash is memory mapped and the maximum (in practise) that you can use is a 16MiB chip. For example, KGPE-D16 and KCMA-D8 mainboards in Libreboot have 2MiB flash by default, but you can easily upgrade these. Another example is the ThinkPad X200S, X200 Tablet and T400S, all of which have WSON8 where the best course of action is to replace it with a SOIC8 flash chip. In all such cases, flashing a new chip should be done using a breadboard, not a test clip. You will use 2.54mm dupont leads to connect your Raspberry Pi. For data lines, make sure that all wires are the same length, and about 10cm in length (don’t use longer lengths than this). Some advice:. DIP8: Strong choice is Winbond W25Q128FVIQ. It is a direct drop-in replacement. SOIC8 is possible: Winbond W25Q128FVSIG is a strong choice. DIP8 using adapter and SOIC8 is also possible. Use a 208-mil 1.27mm SOP8\/SOIC8 to DIP8 adapter PCB with a 2.54mm 4-pin header on each side (square pins), then you can slot that in as though it were a normal P-DIP 8 IC. This page shows a perfect example:. https:\/\/coolcomponents.co.uk\/products\/soic-to-dip-adapter-8-pin The above SOP8\/DIP8 adapter is actually what we recommend, if you’re going that route. It’s made by Sparkfun and widely available; you don’t have to buy from that particular website. The part number is: BOB-13655. If you use a SOP\/DIP adapter with a SOIC8 IC, you’ll have to solder it obviously. K tip is a nice choice for soldering ICs like these. Use good flux and 60\/40 leaded solder (or 63\/37), none of that Rohs lead-free crap. If you go for a SOIC8, mounted it to the SOP to DIP adapter (208mil 1.27mm one) and solder 2.54mm headers to it. You could put the 2.54mm pins in a breadboard, then solder the chip to the adapter PCB and mount that to the pins on the breadboard, to keep it aligned, and solder that. Whith the PCB on the pins, and the pins in the breadboard, push the pins inwards a little bit. This is for a new SOIC8 chip, but you can get sockets similar to the one in the video, but for WSON8. Sometimes they are called DFN8 or QFN8 sockets. Get one that is 1.27mm pitch. If you’re flashing\/dumping a lone WSON8, get a WSON8\/QFN8\/DFN8 socket (1.27mm pitch) and mount it to a breadboard for flashing. If your mainboard’s landing pads for the flash IC can take a SOIC8, we recommend that you use a SOIC8 instead because a test clip is possible later on when you wish to re-flash it, however you may be dealing with a board where replacing existing WSON8 with SOIC8 is desirable; in that case, you might still want to dump the contents of the original WSON8. Here is a SOIC8 in a socket, mounted to a breadboard, for flashing:. Here is a photo of a DIP8 IC:. Here is a photo of a SOIC8 in 1.27mm 208mil SOP to DIP adapter:. NOTE: DIP8 and WSON8-in-socket, and SOIC16-in-socket, are basically the same, just adapt accordingly. If you’re replacing a DIP8 but using SOIC8 on an adapter, solder it to the adapter first, then insert 2.54mm headers (square pins) into a breadboard to keep them aligned. Put the SOIC8 on the PCB, onto the pins, and push the pins inwards a little bit, and solder that. Alternatively to the breadboard, you can just put the 2.54mm pins directly in the DIP8 socket and mount the SOIC8 + adapter onto that, and solder that. Use quality rosin flux (not acid based) and good 60\/40 or 63\/37 leaded solder (don’t use lead-free):. SOIC8\/SOIC16 soldered to a mainboard. [link] This is an example of. in-system programming. or. ISP. for short. SOIC8:. Pomona 5250 is a SOIC8 test clip. There are others available, but this is the best one. Use that. Use the SOIC8 diagram (see above) to wire up your Raspberry Pi. Your mainboard likely already pulls WP\/HOLD (pins 3 and 7) high, so don’t connect these. 3.3v VCC on SOIC8’s pin 8 probably already has decoupling capacitors on the mainboard, so just hook that up without using a capacitor. SOIC16:. Pomona 5252 is a SOIC16 test clip. There are others available, but this is the best one. Use that. Use the SOIC16 diagram (see above) to wire up your Raspberry Pi. WP\/HOLD pins are pins 1 and 9, and likely already held high, so no pull-up resistors needed. You do not need a decoupling capacitor for pin 2 (VCC) either because the mainboard will already have one. Here is an example of a test clip connected for SOIC16:. And here is an example photo for SOIC8:. DIP8 soldered to the mainboard. [link] It is extremely cursed for DIP8 to be soldered directly to the mainboard. It is usually mounted to a socket. The pins are large enough that you can just use test hooks to wire up your chip for flashing. You might want to de-solder the chip, using a solder vacuum (extractor) tool, and then you can install a socket in its place. You can then insert the DIP8 IC into the socket. In the Libreboot project, we have never heard of a board where the DIP8 is directly soldered. It is almost always mounted in a socket. Your DIP8 IC has the same pinout as a SOIC8 IC. Replace WSON8 IC with SOIC8. [link] You. can. connect a SOIC8 test clip, but you will struggle to get good connections and it will be extremely unreliable. DO NOT solder to the pads of the WSON8 directly; some people do this, but you shouldn’t do it, because you can easily damage the pads that way. WSON8 has the same pinout as SOIC8, but it’s a ball mounted QFN (quad flat pack, no leads). There are no clips for it. Sometimes referred to as QFN8. On all currently supported Libreboot hardware, boards that have WSON8 can also have a SOIC8 because the pads are long enough to accomodate either type of chip. A good choice of soldering iron would be a T12-D08 or T12-K tip, on a T12 soldering station. KSGER makes nice soldering stations:. https:\/\/vid.puffyan.us\/watch?v=w0nZCK7B-0U The case on that KSGER station is not grounded by default, so you should modify it to ground the case, in case of an electrical fault. This is for your safety. This video shows how to do it:. https:\/\/vid.puffyan.us\/watch?v=-6IZ_sBgw8I Use quality 60\/40 or 63\/37 lead+tin solder. Do not use lead-free! Lead-free is not suitable for hobbyist use such as this. Use quality. rosin. flux. Fluxes with an acid base should never be used. Amtech and MG Chemicals make good flux pastes. Use it in a dispenser tube. Some of these fluxes will contain adapic acid which has a low pH level, and it is simply used as a mild activator. So long as you clean the flux afterwards, you should be fine. Make sure to have a copper wire brush and a wet sponge handy. You wipe the iron on the wire brush and tap it on the wet sponge(to remove oxides) to keep it clean. Always clean your tip constantly. Also, after cleaning it, always re-tin the tip with fresh solder, to prevent the tip from oxidizing! Make sure to buy 99.9% isopropyl alcohol. Don’t buy weaker solutions because they contain water, and don’t use other chemicals because most other chemicals are corrosive. You use the isopropyl to clean the area you’re soldering, before soldering it, and then soak up the wet alcohol with a cloth. You will also use it to clean off any flux that you used. Use of flux is very important, to get a good solder joint, because it removes oxides and prevents further oxidation during soldering, ensuring that the solder flows properly, otherwise the solder will ball up and you won’t get a good joint. In case you’re not comfortable with soldering, we have some excellent videos linked on the. FAQ page which you can watch. WSON8 IC:. Surround a large area around the chip with layers of kapton tape, and then aluminium foil. This will act as a heat shield, to reduce the risk of re-flowing other solder joints (which can make them turn into cold joints, and you risk knocking them off of the board):. Notice that the kapton+foil does not cover the chip itself, or the solder pads. It’s important that these are exposed to the heat. Use a hot air rework station, set to about 330-340C. The reason for the higher temperature is because air doesn’t conduct heat as efficiently as an iron, so you must use a higher temperature. You should put lots of rosin flux above the IC. Do not hold the nozel too close to the board. The diameter of the nozel should be slightly higher than the length of the chip. Apply even heat, at high air flow. While blasting the chip with hot air, hold the chip with tweezers but do not use any real force. Do not try to forcefully pry off the chip. Simply hold the chip with your tweezers, gently nudging it until it feels like the chip can move freely. While in this state, the solder is fully melted and the chip can be lifted off with ease. If you’re doing it correctly, the chip will come off within 1 minute, like so:. Add fresh solder to the pads, including the thermal pad:. Now wick it out using a copper braid, dunked in rosin flux:. Ensure that all of the solder is removed:. You will notice that one of the pads doesn’t have all of the solder removed. The pad on the top-left in this photo. This is intentional, to show you a comparison for reference. The other pads are free of solder. You. can. simply solder the chip unflashed, and flash it using a test clip. Alternatively, you can put the SOIC8 in a socket on a breadboard, and flash it before soldering it. If you wish to dump the contents of the WSON8, you can put the removed WSON8 in a socket on a breadboard and dump it using your SPI flasher. Align the new SOIC8, and tack it in the corner pins. Then solder it fully. Use lots of flux! A T12-D08 tip is being used in this photo, but a mini chisel, mini hoof or knife (e.g. T12-K) tip would be ideal. Ensure that all the joints are perfect. A good solder joint is shiny, and with concave fillets where the solder has flowed. Observe:. After you’re done, use a soft bristle brush and 99.9% isopropyl alcohol to break up the remaining flux, then soak up the flux using a cloth, while the alcohol is still wet. 99.9% isopropyl is the best liquid to use, because it evaporates quickly and it does not leave a corrosive residue. LICENSING. [link] This page is released under different copyright terms than most other pages on this website. This page and the photos on it are available under. CC BY SA 4.0 Check the Git repository for history of who owns what part of the document. Photos showing a BeagleBone Black are under the normal GNU Free Documentation license like other pages and images on this website, or you can use them under the CC-BY-SA 4.0 license if you wish (I, Leah Rowe, own all BBB photos shown on this page, except for the one on the beaglebone website, and that one is merely linked here, instead of being hosted on the av.libreboot.org server). This version of the page is hosted in the. lbwww. git repository, with images for it hosted in the. lbwww-img. repository. Images and this page were both forked from the. old. Libreboot git repository, which was available here:. https:\/\/notabug.org\/libreboot\/libreboot\/ (you can still download it but this repository is no longer worked on. You can find both the website and images under the. www\/. and\/or. docs\/. directory, in that repository). Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/spi.md Site map This HTML page was generated by the. untitled static site generator ."}
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Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction ich9utils ICH9 show utility ICH9 gen utility Write-protecting the flash chip ICH9 deblob utility demefactory utility Early notes Flash chips Early development notes GBE (gigabit ethernet) region in SPI flash GBE region: change MAC address Flash descriptor region platform data partition in boot flash (factory.rom \/ lenovo bios) Return to index Introduction. [link] The. ich9utils. utility in Libreboot is used to manipulate Intel Flash Descriptors for ICH9M on laptops such as ThinkPad X200 or T400. Specifically, the. ich9gen. utility can generate 12KiB descriptor+GbE files for inserting into the start of a ROM, where everything after that is the BIOS region. These are special descriptors with the Intel ME region disabled, and Intel ME itself fully disabled. ich9utils is handled by the. lbmk. (libreboot-make) build system, but the code itself is hosted in a separate repository. You can check the Git repositories linked on. ..\/..\/git.html if you wish to download and use it. It is very. uncommon. , on GM45\/ICH9M systems, to have an Intel Flash Descriptor and GbE but. without. an Intel ME. On. most. of these systems (without Libreboot, Libreboot or coreboot), there is either descriptor+GbE+ME+BIOS or just BIOS, where on systems with just the BIOS region an Intel GbE NIC is not present. In Libreboot (and Libreboot), we provide descriptor+GbE images with Intel ME disabled and not present in the ROM; this enables the Intel GbE NIC to be used, while not having an Intel ME present. A consequence of this is that the malicious features of ME (such as AMT) are not present, however the Intel ME also handles TPM which is therefore disabled in this setup. NOTE: If you accidentally flash a ROM. without. descriptor+GbE, it will still work but the Intel GbE NIC will be dysfunctional. If you do that, just boot up and correct the problem (and you can use a USB\/cardbus\/expresscard NIC or WiFi for internet if necessary). That is the. main reason. why. ich9utils. was written in the first place; it was already very possible to boot without an Intel ME by simply not having a descriptor or anything in ROM, just coreboot. The purpose of. ich9gen. specifically is to get the Intel GbE NIC working but without the Intel ME being enabled! ICH9 based systems were the last generation that could be booted. without. an Intel ME. Future platforms (such as Sandybridge and Ivybridge) require an Intel ME since the ME on those platforms also handles power management and some minor initialization functions. On ICH9 based systems (such as X200 or T400) the Intel ME only handles AMT and TPM, and there’s no 30 minute timer (if you boot later platforms without an Intel ME and descriptor, or invalid Intel ME firmware, the system will either not boot or will turn off after 30 minutes per a watchdog reset timer). More information about the ME can be found at. http:\/\/www.coreboot.org\/Intel_Management_Engine and. http:\/\/me.bios.io\/Main_Page . Another project:. http:\/\/io.netgarage.org\/me\/ ich9utils. [link] You can find. ich9utils. on the. Git page or you can download. lbmk. from the same page and run the following command in there:. .\/build module ich9utils. You can also find it in the source code tar archives, on Libreboot releases. In. lbmk. , you can use the following command to generate descriptors:. .\/build descriptors ich9m. The Libreboot build system will use the descriptors under. descriptors\/ich9m. when building ROM images for these machines. Alternatively, you can just clone. ich9utils. directly and run. make. in the directory, and run the. ich9gen. program directly. ICH9 show utility. [link] The. ich9show. utility outputs the entire contents of the descriptor and GbE regions in a given ROM image as supplied by the user. Output is in Markdown format (Pandoc variant) so that it can be converted easily into various formats. It could even be piped. directly. into pandoc! ICH9 gen utility. [link] When you simply run. ich9gen. without any arguments, it generates descriptor+GbE images with a default MAC address in the GbE region. If you wish to use a custom macaddress, you can supply an argument like so:. ich9gen --macaddress 00:1f:16:80:80:80. The above MAC address is just an example. It is recommended that you use the MAC address officially assigned to your NIC. Three new files will be created:. ich9fdgbe_4m.bin. : this is for GM45 laptops with the 4MB flash chip. ich9fdgbe_8m.bin. : this is for GM45 laptops with the 8MB flash chip. ich9fdgbe_16m.bin. : this is for GM45 laptops with the 16MB flash chip. These files contain the descriptor+GbE region and are suitable for systems that have an Intel GbE NIC present. The flash regions (as defined by the Intel Flash Descriptor) are set. read-write. which means that you can also re-flash using. flashrom -p internal. in your operating system running on that machine. This is the default setup used when Libreboot’s build system compiles ROM images. Alternative versions of these files are also created, which have. ro. in the filename. If you use. those. versions, all flash regions (as defined by the Intel Flash Descriptor) will be set to. read only. . This can be useful, for security purposes, if you wish to ensure that malicious software in your operating system cannot simply re-flash new firmware. The region setup created by these descriptors is as follows:. First 4KiB of flash is: Intel Flash Descriptor. Next 8KiB after Descriptor: Intel GbE region. Rest of the flash, after GbE: BIOS region (BIOS region will have Libreboot). The GbE region contains configuration data for your Intel GbE NIC. You can find information about this in Intel datasheets, and it is very well described in the. ich9utils. source code. Assuming that your Libreboot image is named. libreboot.rom. , copy the file to where. libreboot.rom. is located and then insert the descriptor+gbe file into the ROM image. For 16MiB flash chips:. dd if=ich9fdgbe_16m.bin of=libreboot.rom bs=12k count=1 conv=notrunc. For 8MiB flash chips:. dd if=ich9fdgbe_8m.bin of=libreboot.rom bs=12k count=1 conv=notrunc. For 4MiB flash chips:. dd if=ich9fdgbe_4m.bin of=libreboot.rom bs=12k count=1 conv=notrunc. If you wish to have read-only flash (write protected flash), substitute the above examples with descriptor+GbE images that have. ro. in the filename. RO here means. read only. , not. Romania. ! The above commands assume that in coreboot you have specified the CBFS size as no more than the size of the flash, minus 12KiB. NOTE:. ich9gen. also generates descriptors without a GbE region, where in those descriptors the Intel GbE is not specified. Those are highly experimental, and. theoretical. since no such system exists in the wild where ICH9 is used, no Intel GbE NIC present. and. descriptor present; on such systems, the vendor will just supply a descriptor-less setup. Those GbE-less descriptor images created by. ich9gen. are only 4KiB in size, and should. never be used. except for fun, like, basically shits and\/or giggles. For shits and giggles, R500 ROM images in Libreboot use these no-GbE descriptor images generated by ich9gen. However, a descriptorless setup would also work just fine. ThinkPad R500 doesn’t have an Intel PHY in it, and it instead uses a Broadcom NIC for ethernet. In descriptorless mode, ICH9M works very similarly to older ICH7 chipsets. Your libreboot.rom image is now ready to be flashed on the system. Refer back to. ..\/install\/#flashrom for how to flash it. Write-protecting the flash chip. [link] The. ich9gen. utility (see below) generates two types of descriptor+GbE setup:. read-write. read-only. Read on for more information. Use the. ro. files mentioned below, and your flash will be read-only in software (you can still externally re-flash and read the contents of flash). For ease of use, Libreboot provides ROMs that are read-write by default. In practise, you can boot a Linux kernel with access to lower memory disabled which will make software re-flashing impossible (unless you reboot with such memory protections disabled, e.g. . iomem=relaxed. kernel parameter). ICH9 deblob utility. [link] This was the tool originally used to disable the ME on X200 (later adapted for other systems that use the GM45 chipset). ich9gen now supersedes it; ich9gen is better because it does not rely on dumping the factory.rom image (whereas, ich9deblob does). Simply speaking,. ich9deblob. takes an original dump of the boot flash, where that boot flash contains a descriptor that defines the existence of Intel ME, and modifies it. The Intel Flash Descriptor is modified to disable the ME region. It disables the ME itself aswell. The GbE region is moved to the location just after the descriptor. The BIOS region is specified as being after the descriptor+GbE regions, filling the rest of the boot flash. The GbE region is largely unedited when using this utility. Run it like so, with. factory.rom. in the same directory:. .\/ich9deblob. The. factory.rom. file is your dump of the vendor boot flash. Older versions of this utility have this file name hardcoded, and for compatibility reasons it will still work in this manner. However, you can now specify your own file name. For example:. .\/ich9deblob lenovo.rom. A 12kiB file named. deblobbed_descriptor.bin. will now appear. Keep this and the factory.rom stored in a safe location! The first 4KiB contains the descriptor data region for your system, and the next 8KiB contains the gbe region (config data for your gigabit NIC). These 2 regions could actually be separate files, but they are joined into 1 file in this case. A 4KiB file named. deblobbed_4kdescriptor.bin. will alternatively appear, if no GbE region was detected inside the ROM image. This is usually the case, when a discrete NIC is used (eg Broadcom) instead of Intel. Only the Intel NICs need a GbE region in the flash chip. Assuming that your Libreboot image is named. libreboot.rom. , copy the. deblobbed_descriptor.bin. file to where. libreboot.rom. is located and then run:. dd if=deblobbed_descriptor.bin of=libreboot.rom bs=12k count=1 conv=notrunc. Alternatively, if you got a the. deblobbed_4kdescriptor.bin. file (no GbE defined), do this:. dd if=deblobbed_4kdescriptor.bin of=libreboot.rom bs=4k count=1 conv=notrunc. (it’s very unlikely that you would ever see this. Descriptor without GbE is very rare, probably non-existant, but theoretically possible and this functionality is implemented based on Intel datasheets). The utility will also generate 4 additional files:. mkdescriptor.c. mkdescriptor.h. mkgbe.c. mkgbe.h. These are. self-written. by. ich9deblob. . The. ich9gen. utility was created, based on this very functionality, with some tweaks made afterwards. These are C source files that can re-generate the very same Gbe and Descriptor structs (from ich9deblob\/ich9gen). To use these, place them in src\/ich9gen\/ in ich9deblob, then re-build. The newly build. ich9gen. executable will be able to re-create the very same 12KiB file from scratch, based on the C structs, this time. without. the need for a. factory.rom. dump! You should now have a. libreboot.rom. image containing the correct 4K descriptor and 8K gbe regions, which will then be safe to flash. Refer back to. index.html\/#gm45 for how to flash it. demefactory utility. [link] This utility has never been tested, officially, but it. should. work. This takes a. factory.rom. dump and disables the ME\/TPM, but leaves the region intact. It also sets all regions read-write. Simply put, this means that you can use the original factory firmware but without the Intel ME enabled. The ME interferes with flash read\/write in flashrom, and the default descriptor locks some regions. The idea is that doing this will remove all of those restrictions. Simply run (with. factory.rom. in the same directory):. .\/demefactory. It will generate a 4KiB descriptor file (only the descriptor, no GbE). Insert that into a factory.rom image (NOTE: do this on a copy of it. Keep the original factory.rom stored safely somewhere):. dd if=demefactory_4kdescriptor.bin of=factory_nome.rom bs=4k count=1 conv=notrunc. Use-case: a factory.rom image modified in this way would theoretically have no flash protections whatsoever, making it easy to quickly switch between factory\/Libreboot in software, without ever having to disassemble and re-flash externally unless you brick the device. The sections below are adapted from (mostly) IRC logs related to early development getting the ME removed on GM45. They are useful for background information. This could not have been done without sgsit’s help. Early notes. [link] http:\/\/www.intel.co.uk\/content\/dam\/doc\/datasheet\/io-controller-hub-10-family-datasheet.pdf page 230 mentions about descriptor and non-descriptor mode (which wipes out gbe and ME\/AMT). See reference to HDA_SDO (disable descriptor security). strap connected GPIO33 pin is it on ICH9-M (X200). HDA_SDO applies to later chipsets (series 6 or higher). Disabling descriptor security also disables the ethernet according to sgsit. sgsit’s method involves use of ‘soft straps’ (see IRC logs below) instead of disabling the descriptor. and the location of GPIO33 on the x200s: (was an external link. Putting it here instead). https:\/\/av.libreboot.org\/x200\/gpio33_location.jpg - it’s above the number 7 on TP37 (which is above the big intel chip at the bottom). The ME datasheet may not be for the mobile chipsets but it doesn’t vary that much. This one gives some detail and covers QM67 which is what the X201 uses:. http:\/\/www.intel.co.uk\/content\/dam\/www\/public\/us\/en\/documents\/datasheets\/6-chipset-c200-chipset-datasheet.pdf Flash chips. [link] X200 laptop (Mocha-1): ICH9-M overrides ifd permissions with a strap connected to GPIO33 pin (see IRC notes below). The X200 can be found with any of the following flash chips:. ATMEL AT26DF321-SU 72.26321.A01 - this is a 32Mb (4MiB) chip. MXIC (Macronix?) MX25L3205DM2I-12G 72.25325.A01 - another 32Mb (4MiB) chip. MXIC (Macronix?) MX25L6405DMI-12G 41R0820AA - this is a 64Mb (8MiB) chip. Winbond W25X64VSFIG 41R0820BA - another 64Mb (8MiB) chip. sgsit says that the X200s (Pecan-1) with the 64Mb flash chips are (probably) the ones with AMT (alongside the ME), whereas the 32Mb chips contain only the ME. Early development notes. [link] Start (hex) End (hex) Length (hex) Area Name ----------- --------- ------------ --------- 00000000 003FFFFF 00400000 Flash Image 00000000 00000FFF 00001000 Descriptor Region 00000004 0000000F 0000000C Descriptor Map 00000010 0000001B 0000000C Component Section 00000040 0000004F 00000010 Region Section 00000060 0000006B 0000000C Master Access Section 00000060 00000063 00000004 CPU\/BIOS 00000064 00000067 00000004 Manageability Engine (ME) 00000068 0000006B 00000004 GbE LAN 00000100 00000103 00000004 ICH Strap 0 00000104 00000107 00000004 ICH Strap 1 00000200 00000203 00000004 MCH Strap 0 00000EFC 00000EFF 00000004 Descriptor Map 2 00000ED0 00000EF7 00000028 ME VSCC Table 00000ED0 00000ED7 00000008 Flash device 1 00000ED8 00000EDF 00000008 Flash device 2 00000EE0 00000EE7 00000008 Flash device 3 00000EE8 00000EEF 00000008 Flash device 4 00000EF0 00000EF7 00000008 Flash device 5 00000F00 00000FFF 00000100 OEM Section 00001000 001F5FFF 001F5000 ME Region 001F6000 001F7FFF 00002000 GbE Region 001F8000 001FFFFF 00008000 PDR Region 00200000 003FFFFF 00200000 BIOS Region Start (hex) End (hex) Length (hex) Area Name ----------- --------- ------------ --------- 00000000 003FFFFF 00400000 Flash Image 00000000 00000FFF 00001000 Descriptor Region 00000004 0000000F 0000000C Descriptor Map 00000010 0000001B 0000000C Component Section 00000040 0000004F 00000010 Region Section 00000060 0000006B 0000000C Master Access Section 00000060 00000063 00000004 CPU\/BIOS 00000064 00000067 00000004 Manageability Engine (ME) 00000068 0000006B 00000004 GbE LAN 00000100 00000103 00000004 ICH Strap 0 00000104 00000107 00000004 ICH Strap 1 00000200 00000203 00000004 MCH Strap 0 00000ED0 00000EF7 00000028 ME VSCC Table 00000ED0 00000ED7 00000008 Flash device 1 00000ED8 00000EDF 00000008 Flash device 2 00000EE0 00000EE7 00000008 Flash device 3 00000EE8 00000EEF 00000008 Flash device 4 00000EF0 00000EF7 00000008 Flash device 5 00000EFC 00000EFF 00000004 Descriptor Map 2 00000F00 00000FFF 00000100 OEM Section 00001000 00002FFF 00002000 GbE Region 00003000 00202FFF 00200000 BIOS Region Build Settings -------------- Flash Erase Size = 0x1000. It’s a utility called ‘Flash Image Tool’ for ME 4.x that was used for this. You drag a complete image into in and the utility decomposes the various components, allowing you to set soft straps. This tool is proprietary, for Windows only, but was used to deblob the X200. End justified means, and the utility is no longer needed since the ich9deblob utility (documented on this page) can now be used to create deblobbed descriptors. GBE (gigabit ethernet) region in SPI flash. [link] Of the 8K, about 95% is 0xFF. The data is the gbe region is fully documented in this public datasheet:. http:\/\/www.intel.co.uk\/content\/dam\/doc\/application-note\/i-o-controller-hub-9m-82567lf-lm-v-nvm-map-appl-note.pdf The only actual content found was:. 00 1F 1F 1F 1F 1F 00 08 FF FF 83 10 FF FF FF FF 08 10 FF FF C3 10 EE 20 AA 17 F5 10 86 80 00 00 01 0D 00 00 00 00 05 06 20 30 00 0A 00 00 8B 8D 02 06 40 2B 43 00 00 00 F5 10 AD BA F5 10 BF 10 AD BA CB 10 AD BA AD BA 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 00 40 28 12 07 40 FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF D9 F0 20 60 1F 00 02 00 13 00 00 80 1D 00 FF 00 16 00 DD CC 18 00 11 20 17 00 DD DD 18 00 12 20 17 00 00 80 1D 00 00 00 1F. The first part is the MAC address set to all 0x1F. It’s repeated haly way through the 8K area, and the rest is all 0xFF. This is all documented in the datasheet. The GBe region starts at 0x20A000 bytes from the *end* of a factory image and is 0x2000 bytes long. In Libreboot (deblobbed) the descriptor is set to put gbe directly after the initial 4K flash descriptor. So the first 4K of the ROM is the descriptor, and then the next 8K is the gbe region. GBE region: change MAC address. [link] According to the datasheet, it’s supposed to add up to 0xBABA but can actually be others on the X200. https:\/\/web.archive.org\/web\/20150912070329\/https:\/\/communities.intel.com\/community\/wired\/blog\/2010\/10\/14\/how-to-basic-eeprom-checksums “One of those engineers loves classic rock music, so they selected 0xBABA”. In honour of the song. Baba O’Reilly. by. The Who. apparently. We’re not making this stuff up…. 0x3ABA, 0x34BA, 0x40BA and more have been observed in the main Gbe regions on the X200 factory.rom dumps. The checksums of the backup regions match BABA, however. We think. 0xBABA. is the only correct checksum, because those other, similar checksums were only ever found in the “backup” GbE regions on factory ROM dumps. In Libreboot, we simply use. 0xBABA. and ensure that both 4KiB regions in GbE NVM have that checksum. By default, the X200 (as shipped by Lenovo) actually has an invalid main gbe checksum. The backup gbe region is correct, and is what these systems default to. Basically, you should do what you need on the *backup* gbe region, and then correct the main one by copying from the backup. Look at. ich9deblob.c. in ich9utils. Add the first 0x3F 16bit numbers (unsigned) of the GBe descriptor together (this includes the checksum value) and that has to add up to 0xBABA. In other words, the checksum is 0xBABA minus the total of the first 0x3E 16bit numbers (unsigned), ignoring any overflow. Flash descriptor region. [link] http:\/\/www.intel.co.uk\/content\/dam\/doc\/datasheet\/io-controller-hub-9-datasheet.pdf from page 850 onwards. This explains everything that is in the flash descriptor, which can be used to understand what Libreboot is doing about modifying it. How to deblob:. patch the number of regions present in the descriptor from 5 - 3. originally descriptor + bios + me + gbe + platform. modified = descriptor + bios + gbe. the next stage is to patch the part of the descriptor which defines the start and end point of each section. then cut out the gbe region and insert it just after the region. all this can be substantiated with public docs (ICH9 datasheet). the final part is flipping 2 bits. Halting the ME via 1 MCH soft strap and 1 ICH soft strap. the part of the descriptor described there gives the base address and length of each region (bits 12:24 of each address). to disable a region, you set the base address to 0xFFF and the length to 0. and you change the number of regions from 4 (zero based) to 2. There’s an interesting parameter called ‘ME Alternate disable’, which allows the ME to only handle hardware errata in the southbridge, but disables any other functionality. This is similar to the ‘ignition’ in the 5 series and higher but using the standard firmware instead of a small 128K version. Useless for Libreboot, though. To deblob GM45, you chop out the platform and ME regions and correct the addresses in flReg1-4. Then you set meDisable to 1 in ICHSTRAP0 and MCHSTRAP0. How to patch the descriptor from the factory.rom dump. map the first 4k into the struct (minus the gbe region). set NR in FLMAP0 to 2 (from 4). adjust BASE and LIMIT in flReg1,2,3,4 to reflect the new location of each region (or remove them in the case of Platform and ME). set meDisable to 1\/true in ICHSTRAP0 and MCHSTRAP0. extract the 8k GBe region and append that to the end of the 4k descriptor. output the 12k concatenated chunk. Then it can be dd’d into the first 12K part of a coreboot image. the GBe region always starts 0x20A000 bytes from the end of the ROM. This means that Libreboot’s descriptor region will simply define the following regions:. descriptor (4K). gbe (8K). bios (rest of flash chip. CBFS also set to occupy this whole size). The data in the descriptor region is little endian, and it represents bits 24:12 of the address (bits 12-24, written this way since bit 24 is nearer to left than bit 12 in the binary representation). So,. x. If it’s in descriptor mode, then the first 4 bytes will be 5A A5 F0 0F. platform data partition in boot flash (factory.rom \/ lenovo bios). [link] Basically useless for Libreboot, since it appears to be a blob. Removing it didn’t cause any issues in Libreboot. We think it’s just random data that the manufacturer can put there, to use in their firmware. Intel datasheets seem to suggest that the platform region serves no specific function except to provide a region in flash for the hardware manufacturer to use, for whatever purpose (probably just to store other configuration data, to be used by software running from the BIOS region as per region layout specified in the descriptor). This is a 32K region from the factory image. It could be data (non-functional) that the original Lenovo BIOS used, but we don’t know. It has only a 448 byte fragment different from 0x00 or 0xFF, on the X200 thinkpads that were tested. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/ich9utils.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"FiS532Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/t500.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":45,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"t500","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – ThinkPad T500"],"last_modified":"Sat Apr 02 12:48:37 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":6488308648850646036,"exact_signature_unique_b":true,"fuzzy_signature_l":-2435182554187394577,"fuzzy_signature_unique_b":true,"h1_txt":["ThinkPad T500","Navigate this page:","Introduction","EC update"],"h2_txt":["Compatibility (without blobs)"],"h3_txt":["Hardware virtualization (vt-x)"],"bold_txt":["kvm_intel","kvm","The T500 laptops come with the ME (and sometimes AMT in addition) before flashing libreboot. Libreboot disables and removes it by using a modified descriptor: see ..\/install\/ich9utils.html","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":380,"text_t":"Libreboot – ThinkPad T500. ThinkPad T500. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction EC update Compatibility (without blobs) Hardware virtualization (vt-x) Return to index Introduction. [link] It is believed that all or most T500 laptops are compatible. See notes about. CPU compatibility for potential incompatibilities. W500 is also compatible, and mostly the same design as T500. There are two possible flash chip sizes for the T500: 4MiB (32Mbit) or 8MiB (64Mbit). This can be identified by the type of flash chip below the palmrest: 4MiB is SOIC-8, 8MiB is SOIC-16. The T500 laptops come with the ME (and sometimes AMT in addition) before flashing libreboot. Libreboot disables and removes it by using a modified descriptor: see. ..\/install\/ich9utils.html (contains notes, plus instructions). Flashing instructions can be found at. ..\/install\/#flashrom EC update. [link] It is recommended that you update to the latest EC firmware version. The. EC firmware is separate from libreboot, so we don’t actually provide that, but if you still have Lenovo BIOS then you can just run the Lenovo BIOS update utility, which will update both the BIOS and EC version. See:. ..\/install\/#flashrom http:\/\/www.thinkwiki.org\/wiki\/BIOS_update_without_optical_disk NOTE: this can only be done when you are using Lenovo BIOS. How to update the EC firmware while running libreboot is unknown. Libreboot only replaces the BIOS firmware, not EC. Updated EC firmware has several advantages e.g. bettery battery handling. Compatibility (without blobs). [link] Hardware virtualization (vt-x). [link] The T500, when run without CPU microcode updates in coreboot, currently kernel panics if running QEMU with vt-x enabled on 2 cores for the guest. With a single core enabled for the guest, the guest panics (but the host is fine). Working around this in QEMU might be possible; if not, software virtualization should work fine (it’s just slower). On GM45 hardware (with libreboot), make sure that the. kvm. and. kvm_intel. kernel modules are not loaded, when using QEMU. The following errata datasheet from Intel might help with investigation:. http:\/\/download.intel.com\/design\/mobile\/specupdt\/320121.pdf The T500 is almost identical to the X200, code-wise. See. x200.html . Binary blob policy Edit this page Who develops Libreboot? 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{"id":"12AUH_Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/news\/libreboot20140605.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["news"],"url_file_name_s":"libreboot20140605","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Libreboot 20140605 release"],"author":"Leah Rowe","author_sxt":["Leah Rowe"],"last_modified":"Sat Apr 02 12:48:46 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Thu Jun 05 00:00:00 UTC 2014"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-6861695812326173171,"exact_signature_unique_b":true,"fuzzy_signature_l":-6259970166569510702,"fuzzy_signature_unique_b":true,"h1_txt":["Libreboot 20140605 release","Navigate this page:"],"h2_txt":["Revision notes (5th June 2014):"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":372,"text_t":"Libreboot – Libreboot 20140605 release. Libreboot 20140605 release. Leah Rowe. 5 June 2014. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Revision notes (5th June 2014): Return to index Article published by: Leah Rowe. Date of publication: 5 June 2014. Revision notes (5th June 2014):. [link] added backlight support (Fn+Home and Fn+End) on X60. fixed broken\/unstable 3D when using kernel 3.12 or higher. (see ‘BACKPORT’ file). Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/news\/libreboot20140605.md Site map This HTML page was generated by the. untitled static site generator .","wordcount_i":107,"linkscount_i":21,"linksnofollowcount_i":0,"inboundlinkscount_i":17,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/news\/libreboot20140605.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/news\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/news\/libreboot20140605.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Revision notes (5th June 2014):","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:01 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:09 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":4107,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078561998438400}
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{"id":"XeC3e2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/gnulinux\/grub_hardening.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":55,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","gnulinux"],"url_file_name_s":"grub_hardening","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Hardening GNU GRUB"],"last_modified":"Sat Apr 02 12:48:35 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-957267216854456980,"exact_signature_unique_b":true,"fuzzy_signature_l":-3944609538340795398,"fuzzy_signature_unique_b":true,"h1_txt":["Hardening GNU GRUB","Navigate this page:","GRUB secure boot with GPG","GRUB Password","GPG keys"],"bold_txt":["actually","strongly","trust","more secure","not","easy of use","random pattern","every single time you boot your computer","do not use the hash that you see above!","obviously","words","diceware method","plain text","This should be different than your LUKS passphrase and user password.","compromised","restricted","my.rom","controlled conditions","operational","someone else","very same development setup","any","expert only","many","most","bare metal","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":424,"wordcount_i":1944,"linkscount_i":28,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":10,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/gnulinux\/grub_hardening.html","libreboot.org\/docs\/gnulinux\/","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/gnulinux\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/gnulinux\/grub_hardening.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","GRUB secure boot with GPG","Return to index","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","007-https","008-https","009-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","gnupg.org\/","reproducible-builds.org\/","www.gnu.org\/software\/grub\/manual\/html_node\/Security.html","git.savannah.gnu.org\/cgit\/grub.git\/tree\/docs\/grub.texi","www.coreboot.org\/GRUB2","en.wikipedia.org\/wiki\/Pbkdf2","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","GnuPG","","GRUB manual","GRUB info pages","Coreboot GRUB security howto","PBKDF2","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:01 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:15 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":17707,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078562389557248,"text_t":"Libreboot – Hardening GNU GRUB. Hardening GNU GRUB. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. GRUB secure boot with GPG GRUB Password GPG keys Return to index This article only applies to those people who use the GNU GRUB bootloader as their default payload (options besides GNU GRUB are also available in Libreboot). Whenever this article refers to GNU GRUB, or configuration files used in GNU GRUB, it is referring exclusively to those files hosted in CBFS (coreboot file system) in the Libreboot ROM image. In this configuration, GNU GRUB is running on. bare metal. as a coreboot payload (instead of relying on BIOS or UEFI services, like it does on. most. x86 based configurations). This guide deals with various ways in which you can harden your GNU GRUB configuration, for security purposes. These steps are optional, but. strongly. recommended by the Libreboot project. GNU GRUB provides. many. advanced security features, which most people don’t know about but are fully documented on the Libreboot website. Read on! This article doesn’t cover how to dump your ROM, or flash a new one. Please read other sections in the Libreboot documentation if you don’t know how to do that. As such, this is an. expert only. guide. There is a great possibility for bricking your system if you follow this guide incorrectly, or otherwise don’t know what you’re doing. GRUB secure boot with GPG. [link] GNU GRUB contains code, based on. GnuPG , that can verify PGP signatures on. any. type of file, on any storage medium supported by GNU GRUB (it supports basically everything, including CBFS which is short for coreboot file system and it is what we will focus on in this article). We will be using this functionality to verify the signature of a Linux kernel, at boot time. In conjunction with reproducible builds (both Libreboot and your Linux kernel), this can greatly improve system security; Debian is an excellent example of a GNU+Linux distribution that is fully reproducible nowadays (in stable releases). For your reference: a reproducible build is one where, given a precise (and well documented) development setup, the exact same binary can be produced each time the source code is compiled when that. very same development setup. is replicated by another person. In other words, the file checksum (e.g. SHA512 hash) will be exactly the same at all times. In practise, this means that metadata such as time stamps are not included in the binary, or if they are, they are constant (in many scenarios, it’s based on the date of a Git commit ID that the build is based on, if the software is built from a Git repository). More information about reproducible builds can be found here:. https:\/\/reproducible-builds.org\/ Reproducibility is a key goal of the Libreboot project, though it has not yet achieved that goal. However, it is an important part of any secure system. We suggest that, when securing your Libreboot system as instructed by this guide, you should also use a reproducible GNU+Linux distribution (because checking GPG signatures on a non-reproducible binary, such as a Linux kernel, is meaningless if that binary can be compromised as a result of literally not being able to verify that the source code. actually. corresponds to the provided binary, which is exactly what reproducible builds allow). If. someone else. compiles an executable for you, and that executable is non-reproducible, you have no way to verify that the source code they provided. actually. corresponds to the binary they gave you. Based on these facts, we can observe that checking GPG signatures will improve your. operational. security, but only in specific circumstances under. controlled conditions. . This tutorial assumes you have a Libreboot image (ROM) that you wish to modify, which from now on we will refer to simply as. my.rom. . It should go without saying that this ROM uses the GNU GRUB bootloader as payload. This page shows how to modify grubtest.cfg, which means that signing and password protection will work after switching to it in the main boot menu and bricking due to incorrect configuration will be impossible. After you are satisfied with the new setup, you should transfer the new settings to grub.cfg to make your machine truly secure. First, extract the old grubtest.cfg and remove it from the Libreboot image:. cbfstool my.rom extract -n grubtest.cfg -f my.grubtest.cfg cbfstool my.rom remove -n grubtest.cfg. You can build. cbfstool. in the Libreboot build system. Run this command:. .\/build module cbutils. This assumes that you already downloaded coreboot:. .\/download coreboot. This, in turn, assumes that you have installed the build dependencies for Libreboot. On Ubuntu 20.04 and other apt-get distros, you can do this:. .\/build dependencies ubuntu2004. The. cbfstool. executables will be under each coreboot directory, under each. coreboot\/boardname\/. directory for each board. Just pick one, presumably from the coreboot directory for your board. Libreboot creates multiple coreboot archives for different board revisions, on different boards. References:. GRUB manual GRUB info pages SATA connected storage considered dangerous. Coreboot GRUB security howto GRUB Password. [link] The security of this setup depends on a good GRUB password as GPG signature checking can be disabled through the interactive console:. set check_signatures=no. This is useful because it allows you to occasionally boot unsigned live CD\/USB media and such. You might consider supplying signatures on a USB stick, but the signature checking code currently looks for. \/path\/to\/filename.sig. when verifying. \/path\/to\/filename. and, as such, it will be impossible to supply signatures in any other location (unless the software is modified accordingly). It’s worth noting that this is not your LUKS password but, rather, a password that you must enter in order to use. restricted. functionality (such as the GNU GRUB terminal for executing commands). This behaviour protects your system from an attacker simply booting a live USB key (e.g. live GNU+Linux distribution) for the purpose of flashing modified boot firmware, which from your perspective is. compromised. boot firmware. This should be different than your LUKS passphrase and user password. GNU GRUB supports storing salted, hashed passwords in the configuration file. This is a far more secure configuration, because an attacker cannot simply read your password as. plain text. inside said file. Use of the. diceware method. is. strongly. recommended, for generating secure passphrases (as opposed to passwords). The diceware method involves rolling dice to generate random numbers, which are then used as an index to pick a random word from a large dictionary of words. You can use any language (e.g. English, German). Look it up on a search engine. Diceware method is a way to generate secure passphrases that are very hard (almost impossible, with enough words) to crack, while being easy enough to remember. On the other hand, most kinds of secure passwords are hard to remember and easier to crack. Diceware passphrases are harder to crack because of far higher entropy (there are many words available to use, but only about 50 commonly used symbols in pass. words. ). This high level of entropy is precisely what makes such pass phrases secure, even if an attacker knows exactly which dictionary you used! The GRUB password can be stored in one of two ways:. plaintext. protected with. PBKDF2 We will. obviously. use the latter method. Generating the PBKDF2 derived key is done using the. grub-mkpasswd-pbkdf2. utility. You can get it by installing GRUB version 2. Generate a key by giving it a password:. NOTE: This utility is included under the. grub\/. directory, when you build GRUB using the Libreboot build system. Run the following commands (assuming you have the correct build dependencies installed) to build GNU GRUB, from the Libreboot Git repository:. .\/download grub .\/build module grub. The following executable will then be available under the. grub\/. directory:. grub-mkpasswd-pbkdf2. Its output will be a string of the following form:. grub.pbkdf2.sha512.10000.HEXDIGITS.MOREHEXDIGITS. Now open my.grubtest.cfg and put the following before the menu entries (prefered above the functions and after other directives). Of course use the pbdkf string that you had generated yourself:. set superusers=\"root\" password_pbkdf2 root grub.pbkdf2.sha512.10000.711F186347156BC105CD83A2ED7AF1EB971AA2B1EB2640172F34B0DEFFC97E654AF48E5F0C3B7622502B76458DA494270CC0EA6504411D676E6752FD1651E749.8DD11178EB8D1F633308FD8FCC64D0B243F949B9B99CCEADE2ECA11657A757D22025986B0FA116F1D5191E0A22677674C994EDBFADE62240E9D161688266A711. Obviously, replace it with the correct hash that you actually obtained for the password you entered. In other words,. do not use the hash that you see above! With this configuration in place, you must now enter the passphrase. every single time you boot your computer. . This completely restricts an attacker in such a way that they cannot simply boot an arbitrary operating system on your computer. NOTE: An attacker could still open your system and re-flash new firmware externally. You should implement some detection mechanism, such as epoxy applied in a. random pattern. on every screw; this slows down the attack and means that you will know someone tampered with it because they cannot easily re-produce the exact same blob of epoxy in the same pattern (when you apply it, swirl it around a bit for a few minutes while it cures. The purpose is not to prevent disassembly, but to slow it down and make it detectable when it has occured). Another good thing to do, if we chose to load signed on-disk GRUB configurations, is to remove (or comment out). unset superusers. in function try_user_config:. function try_user_config { set root=\"${1}\" for dir in boot grub grub2 boot\/grub boot\/grub2; do for name in '' autoboot_ libreboot_ coreboot_; do if [ -f \/\"${dir}\"\/\"${name}\"grub.cfg ]; then #unset superusers configfile \/\"${dir}\"\/\"${name}\"grub.cfg fi done done }. The. unset superusers. command disables password authentication, which will allow the attacker to boot an arbitrary operating system, regardless of signature checking. The default Libreboot configuration is tweaked for. easy of use. by end users, and it is. not. done with security in mind (though security is preferred). Thus, Libreboot is less restrictive by default. What you are doing, per this article, is making your system. more secure. but at the expense of user-friendliness. That just about covers it, where password setup is concerned! GPG keys. [link] First, generate a GPG keypair to use for signing. Option RSA (sign only) is ok. WARNING: GRUB does not read ASCII armored keys. When attempting to trust … a key filename it will print. error: bad signature. on the screen. mkdir --mode 0700 keys gpg --homedir keys --gen-key gpg --homedir keys --export-secret-keys --armor boot.secret.key # backup gpg --homedir keys --export boot.key. Now that we have a key, we can sign some files with it. We must sign:. a kernel. (if we have one) an initramfs. (if we wish to transfer control to it) an on-disk. grub.cfg. grubtest.cfg. (so that you can go back to. grubtest.cfg. after signature checking is enforced. You can always get back to. grub.cfg. by pressing ESC, but, afterwards,. grubtest.cfg. is not signed and it will not load. Suppose that we have a pair of. my.kernel. and. my.initramfs. and an on-disk. libreboot_grub.cfg. . We will sign them by running the following commands:. gpg --homedir keys --detach-sign my.initramfs gpg --homedir keys --detach-sign my.kernel gpg --homedir keys --detach-sign libreboot_grub.cfg gpg --homedir keys --detach-sign my.grubtest.cfg. Of course, some further modifications to my.grubtest.cfg will be required. We need to. trust. the key and enable signature enforcement (put this before menu entries):. trust (cbfsdisk)\/boot.key set check_signatures=enforce. What remains now is to include the modifications into the Libreboot image (ROM):. cbfstool my.rom add -n boot.key -f boot.key -t raw cbfstool my.rom add -n grubtest.cfg -f my.grubtest.cfg -t raw cbfstool my.rom add -n grubtest.cfg.sig -f my.grubtest.cfg.sig -t raw. Now, flash it. If it works, copy it over to. grub.cfg. in CBFS. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/gnulinux\/grub_hardening.md Site map This HTML page was generated by the. untitled static site generator ."}
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Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Automated freedom Libreboot blob policy osbmk What is lbmk? AUTOMATED automation COPYING Makefile README.html build download modify projectname resources\/coreboot\/ resources\/coreboot\/BOARDNAME\/ resources\/coreboot\/BOARDNAME\/board.cfg resources\/coreboot\/BOARDNAME\/config\/* COREBOOT build system resources\/coreboot\/DEFAULT\/blobs.list resources\/coreboot\/BOARDNAME\/patches\/* resources\/grub\/background\/ resources\/grub\/config\/grub.cfg resources\/grub\/config\/grub_memdisk.cfg resources\/grub\/keymap\/ resources\/grub\/modules.list resources\/grub\/patches\/ resources\/memtest86plus\/patch\/ resources\/scripts\/build\/boot\/roms resources\/scripts\/build\/boot\/roms_helper resources\/scripts\/build\/clean\/cbutils resources\/scripts\/build\/clean\/crossgcc resources\/scripts\/build\/clean\/flashrom resources\/scripts\/build\/clean\/grub resources\/scripts\/build\/clean\/ich9utils resources\/scripts\/build\/clean\/memtest86plus resources\/scripts\/build\/clean\/payloads resources\/scripts\/build\/clean\/rom_images resources\/scripts\/build\/clean\/seabios resources\/scripts\/build\/dependencies\/arch resources\/scripts\/build\/dependencies\/debian resources\/scripts\/build\/dependencies\/fedora35 resources\/scripts\/build\/dependencies\/ubuntu2004 resources\/scripts\/build\/dependencies\/void resources\/scripts\/build\/descriptors\/ich9m resources\/scripts\/build\/module\/cbutils resources\/scripts\/build\/module\/flashrom resources\/scripts\/build\/module\/grub resources\/scripts\/build\/module\/ich9utils resources\/scripts\/build\/module\/memtest86plus resources\/scripts\/build\/payload\/grub resources\/scripts\/build\/payload\/seabios resources\/scripts\/build\/release\/roms resources\/scripts\/build\/release\/src resources\/scripts\/build\/release\/u-boot-libre resources\/scripts\/download\/coreboot resources\/scripts\/download\/flashrom resources\/scripts\/download\/grub resources\/scripts\/download\/ich9utils resources\/scripts\/download\/memtest86plus resources\/scripts\/download\/seabios resources\/scripts\/misc\/versioncheck resources\/scripts\/modify\/coreboot\/configs resources\/scripts\/modify\/seabios\/configs resources\/scripts\/update\/coreboot\/configs resources\/scripts\/update\/seabios\/configs resources\/seabios\/config\/libgfxinit resources\/seabios\/config\/vgarom update Return to previous index Automated freedom. [link] This manual describes the nature of. lbmk. (LibreBoot MaKe), the automated build system used to produce Libreboot releases. It is intended as a reference for. libreboot development. . If you simply wish to compile Libreboot from source, you should instead refer to the. build instructions This documentation covers. modern. Libreboot; version 20160907 and below use a much older, less polished version, from before it was actually called. lbmk. (it was simply called “the libreboot build system” back then). Information about those build systems are provided in the documentation provided with those releases. Generally speaking,. testing. releases of Libreboot do not come with documentation; if you’re using. old. testing releases, it is prudent to check the. lbwww.git. repository on a revision from around the same time as those releases. Future stable releases of Libreboot will come with a snapshot of the. lbwww.git. repository, for documentation pertaining to such releases. One way to do this, for releases from 2021 onwards, is to simply run. git log. on the. news\/. section of. lbwww.git. and find the revision that added the. announcement. for a given release (again, 2021 onwards), and then you can just reset to that revision. As such, you should always refer to the. live. version of this page, on libreboot.org, when working on the. lbmk.git. repository; the live version is intended for development on the Git repository! Libreboot blob policy. [link] Libreboot has a strict policy of. excluding. non-free software. It is to only distribute. free software. . Please keep this in mind, when you work on the Libreboot build system, if you will later submit patches to the project. Learn more about Libreboot’s policies osbmk. [link] Another project, named osboot, is also maintained by Leah Rowe, forked from Libreboot:. https:\/\/osboot.org\/ - this project is just the same as Libreboot, with the same build system, except for some tweaks:. all. blobs are allowed, and CPU microcode updates are enabled by default, and the build system is modified accordingly, but mostly the same. The libreboot build system is. lbmk. . The osboot build system is. osbmk. . These two build systems are almost identical, except for a few differences. They are both actively maintained, in parallel, and lead\/founded by Leah Rowe. Why bring up osboot? Because it’s relevant for licensing and compliance, in case of audit in the future. The. osboot. project was forked from Libreboot 20160907’s build system (which did not have a name back then), but with Libreboot documentation from December in 2020. It was then expanded upon, fixing many flaws in the 20160907 build system. At that time, a failed experimental re-write of Libreboot’s build system had to either be revived and succeed (but that build system was very badly designed), or scrapped; the latter was decided, and osboot-libre was born, which was used to create. lbmk. . With this act, the Libreboot project, formerly a. dead project. for all intents and purposes, was completely restored. All of these acts took place during the early months of the year 2021; the failed Libreboot re-write took place after the Libreboot 20160907 release, and was scrapped during March of 2021, in favour of lbmk which is a fork of osbmk. (and lbmk is now, as of 2 January 2021, being re-forked to bring osbmk\/lbmk back into feature parity. so you can fork your forks of your forks, and maybe fork the fork of your fork of your fork). Libreboot and osboot are two sides of a coin; neither should exist alone. The osboot project scrapts Libreboot’s. zero blobs policy and it is targeted at those who don’t want to (or can’t) use Libreboot, but who still want some freedoms compared to otherwise fully non-free vendor firmware. What is lbmk? [link] In the same way that Trisquel and Debian are GNU+Linux distributions, Libreboot is a. coreboot distribution. . The. lbmk. build system. is. that distribution, providing the glue necessary to integrate coreboot plus anything else that’s needed, unifying everything in a completely automated and pre-configured fashion, so as to provide a distribution that is ease to install and use by non-technical users. In the past, installation of coreboot. required. extensive amounts of configuration by the user, because there was no automation available. It was a problem, and one that. lbmk. has. solved. ; it is a problem, because most users simply want to. install. coreboot without giving it much thought. The. lbmk. build system is written for. those. people, while also providing some flexibility for those who do want to tinker and get their hands dirty. The. lbmk. build system is designed to be simple. Each part of it is its own separate program, which is to run independently. Write one program that does one thing well. . Technically,. lbmk. isn’t necessarily a build system, but rather, a handful of small scripts that run other scripts, or even C programs if you wish. What makes. lbmk. be. lbmk. is what each individual script does, and how scripts interact with or call each other to produce working ROM images. It takes a. light touch. approach, providing only the most minimal glue necessary to build working ROM images that the user can install, with sane defaults, while also providing some ability to customize the firmware, with documentation describing how to do just that. User-friendly documentation is provided, with simple installation steps, automating as much of it as possible. This. document is different. The document you’re reading right now is written for. technical. users who want to know how Libreboot is put together. The lbmk design also helps to ease copyright licensing and compliance, because each part of lbmk is literally its own separate program. With this design, it means that most scripts do not directly link\/embed\/include each other. Because of this, it’s much easier to have different licenses in use for different files. Generally speaking, lbmk is GNU GPLv3+, but it’s perfectly OK, for example, to add files that are GPLv2 or other licenses. By comparison, if you were to have a C program under GPLv3, you could not #include C libraries that are GPLv2, at least not directly, or there would be many pitfalls to avoid at the very least. With lbmk’s design, you can think of it as like when you have many programs running in your operating system, and not all of those programs are under the same license, and most of those different licenses are not compatible with each other; this is perfectly OK there, and it’s OK here too. The purpose of this document is to (hopefully) cause you to understand the entire build system in Libreboot, so that you can contribute patches or otherwise make whatever changes you like. As such, this is a reference guide for Libreboot development. Libreboot is a. coreboot distro. , focusing on integration. As such, direct development on software such as coreboot, GNU GRUB, SeaBIOS etc should ideally be done upstream, or if it’s a project hosted by Libreboot (such as ich9utils) developed in the corresponding separate repository. This document is written for developers and power users alike, or otherwise for anyone who is curious enough to learn more about what. makes. Libreboot! A major planned addition to lbmk in the future is: use it to implement a small busybox+linux distribution, with musl libc, plus u-root, and implement a linux-based bootloader setup similar to Heads, but do it. lbmk-style. . The lbmk build system is designed for absolute simplicity and modularity, making it easy to understand and maintain. It intentionally avoids use of rather complicated programs such as GNU Autoconf; the Makefile in lbmk is just bolted on but it not required. The. lbmk. build system is a. non-design. ; it evolved over time, into what it is today. Its modularity and simplicity of non-design allows you to easily rewrite large parts of it, whenever you want to do so. lbmk is largely written in GNU BASH, and this is unlikely to change in the future. However, lbmk integrates several projects such as coreboot, GNU GRUB or SeaBIOS, and these all have. their. own build systems aswell. The. lbmk. build system is the glue that puts all of these together to produce ROM images for users, in a completely automated fashion. The purpose of. lbmk. is to provide an. unattended. build process, with as little user interaction as possible. Thus,. lbmk. is an. automated build system. . It says on the Libreboot home page that Libreboot is a. coreboot distribution. in much the same way that Trisquel is a. GNU+Linux distribution. , and. lbmk. is what implements that! Continue reading, and you will learn of each file contained in. lbmk. . This document largely pertains to the version of. lbmk. as hosted in. lbmk.git. , but this manual also covers source code archives containing the full downloaded set of modules such as coreboot and GRUB. In general, it is advisable to open. every. file in lbmk, after you downloaded it (from the Git repository), and study the logic in great detail. This manual attempts to explain all of it, and provide a general idea, but nothing beats simply. studying. the logic directly. AUTOMATED automation. [link] Every part of lbmk checks if the prerequisite steps are done, and performs them automatically if not. The. roms_helper. script is no different; for example, it automatically downloads coreboot if not present, aswell as GRUB and everything else. You can run each and every part of lbmk without having to worry about running something before it, because it is handled automatically; if that is ever not the case, it’s a bug that should be fixed immediately (in Libreboot 20160907, such fine tuned automation did not exist and you did have to run specific parts of the build system manually, in a precise order, but this is no longer the case in. lbmk. for Libreboot from 2021 onwards). Another example: if you run. .\/build payload grub. but. .\/build module grub. is not completed, it will automatically run that first, to produce the. grub-mkstandalone. binary which is then used by. .\/build payload grub. Another example: if you run. .\/build boot roms. and crossgcc isn’t yet built for the revision used on each given board, it will automatically compile that version of it, using. that. coreboot tree’s own build system to do it. This level of automation means that. lbmk. from 2021 onwards is much easier to use, compared to the build system present in Libreboot 20160907. Massive improvements to that build system were made, during most of 2021, when implementing both the. osbmk. and. lbmk. build systems. All sections below pertain to actual files in lbmk:. COPYING. [link] This file contains a copy of the GNU General Public License, version 3.0. It is the license that most parts of. lbmk. are released under. Makefile. [link] For use with GNU Make, this is a frontend to. lbmk. , which can be used to run various commands in. lbmk. . Use of this file is purely optional, and largely beneficial if you simply want to build all of. lbmk. (just run. make. when the current work directory is the root directory of. lbmk. ). README.html. [link] This file contains a brief description of Libreboot, along with information about the project. build. [link] This is the main BASH script, part of. lbmk. , used for running most. lbmk. commands. You could say that this file. is. lbmk. . Run. .\/build help. for usage instructions. It calls scripts in. resources\/scripts\/build\/. . For example, the command. .\/build boot roms. will execute. resources\/scripts\/build\/boot\/roms. . When running such commands, additional parameters can be given, which will be passed along to the corresponding script. For example, try:. .\/build boot roms x60 x200_8mb w500_16mb. This will run:. .\/resources\/scripts\/build\/boot\/roms x60 x200_8mb w500_16mB. The. list. function is very helpful. For example:. .\/build boot list. At the time of writing this section, this would have outputted:. Available options for mode 'boot': roms roms_helper. Another use of. list. would be:. .\/build boot roms list. However, the. roms. script merely happens to implement a. list. command. For example,. .\/build payload grub list. does nothing differently than. .\/build payload grub. . You may also refer to the. build instructions download. [link] This is the main BASH script for downloading various components used by. lbmk. . For example, this script downloads coreboot. Scripts called by. download. may also apply patches and such, to the corresponding project; for example, it will apply custom patches to GNU GRUB. This runs scripts in. resources\/scripts\/download. . For example:. .\/download coreboot. This would run:. .\/resources\/scripts\/download\/coreboot. Additional parameters can be given, for example:. .\/download coreboot default. This would run:. .\/resources\/scripts\/download\/coreboot default. For a full list of all. download. commands, run:. .\/download help. modify. [link] This can be used to modify SeaBIOS and coreboot configs. It calls scripts in. resources\/scripts\/modify\/. , for example:. .\/modify coreboot configs. This runs:. .\/resources\/scripts\/modify\/coreboot\/configs. Additional parameters can be given, for example:. .\/modify coreboot configs x200_8mb x60. This would run:. .\/resources\/scripts\/modify\/coreboot\/configs x200_8mb x60. projectname. [link] This file contains a single line of text, with the string “libreboot”. This file exists because of. osboot existing, which uses a modified version of. lbmk. . Leah Rowe, the founder of Libreboot, is also the founder of osboot and actively maintains both projects. A lot of scripts in. lbmk. make use of the. projectname. file. If you were to fork Libreboot, you could very easily just modify this file, so as to rename your fork in a largely automated way. resources\/coreboot\/. [link] This directory contains configuration, patches and so on, for each mainboard supported in the. lbmk. build system. These directories contain such configuration, so that. lbmk. can build working ROM images. The scripts in. resources\/scripts\/build\/boot\/. make heavy use of this directory. resources\/coreboot\/BOARDNAME\/. [link] Each. BOARDNAME. directory defines configuration for a corresponding mainboard. It doesn’t actually have to be for a board; it can also be used to just define a coreboot revision, with patches and so on. resources\/coreboot\/BOARDNAME\/board.cfg. [link] This file can contain several configuration lines, each being a string, such as:. cbtree=\"default\". (example entry). romtype=\"normal\". (example entry). cbrevision=\"ad983eeec76ecdb2aff4fb47baeee95ade012225\". (example entry). arch=\"x86_64\". (example entry). payload_grub=\"y\". (example entry). payload_grub_withseabios=\"y\". (example entry). payload_seabios=\"y\". (example entry). payload_memtest=\"y\". (example entry). payload_seabios_withgrub=\"y\". (example entry). grub_scan_disk=\"ata\". More information about these and other variables will be provided throughout this document. The. cbtree. entry is actually a link, where its value is a directory name under. resources\/coreboot. . For example,. cbtree=\"default\". would refer to. resources\/coreboot\/default. and the corresponding coreboot source tree created (when running. .\/download coreboot. , which makes use of. board.cfg. ) would be. coreboot\/default\/. . In other words: a. board.cfg. file in. resources\/coreboot\/foo. might refer to. resources\/coreboot\/bar. by specifying. cbtree=\"bar\". , and the created coreboot source tree would be. coreboot\/bar\/. . ALSO:. FUN FACT: such references are infinitely checked until resolved. For example,. foo. can refer to. bar. and. bar. can refer to. baz. but if there is an infinite loop, this is detected and handled by. lbmk. . For example, if. bar. refers to. foo. which refers back to. bar. , this is not permitted and will throw an error in. lbmk. . The. romtype. entry largely defines what. .\/build boot roms. does once the ROM is built; for example,. romtype=\"4MiB ICH9 IFD NOR flash\". would specify that an Intel Flash Descriptor for ICH9M, generated by. ich9gen. , would have to be inserted. The. cbrevision. entry defines which coreboot revision to use, from the coreboot Git repository. At present, lbmk only supports use of the official repository from the upstream coreboot project. . The. arch. entry specifies which CPU architecture is to be used: currently recognized entries are. x86_32. ,. x86_64. and. ARMv7. . At present, setting it to ARMv7 only means that crossgcc-arm will be compiled, but no support for actually building ROMs exists in lbmk exists yet, except for 32-bit and 64-bit x86 machines. The. payload_grub. entry specifies whether or not GNU GRUB is to be included in ROM images. The. payload_grub_withseabios. entry specifies whether or not SeaBIOS is to be included. with. GRUB, in ROM images. Turning this on also turns on. payload_seabios_withgrub. , unless that option is explicitly turned off. The. payload_seabios. entry specifies whether or not SeaBIOS is to be included in ROM images. This option is. automatically. enabled if. payload_grub_withseabios. and\/or. payload_seabios_withgrub. are also turned on. The. payload_memtest. entry specifies whether or not MemTest86+ is to be included in ROM images; it will only be included in ROM images for. text mode. startup, on x86 machines. The. grub_scan_disk. option specifies can be. ahci. ,. ata. or. both. , and it determines which types of disks are to be scanned, when the. grub.cfg. file in GRUB payloads tries to automatically find other. grub.cfg. files supplied by your GNU+Linux distribution. On some machines, setting it to. ata. or. ahci. can improve boot speed by reducing delays; for example, trying to scan. ata0. on a ThinkPad X60 with the optical drive may cause GRUB to hang, so on that machine it is advisable to set this option to. ahci. (becuse the default HDD slot is AHCI). resources\/coreboot\/BOARDNAME\/config\/*. [link] Files in this directory are. coreboot. configuration files. Configuration file names can be as follows:. libgfxinit_corebootfb. libgfxinit_txtmode. vgarom_vesafb. vgarom_txtmode. normal. Information pertaining to this can be found on the. installation manual In. lbmk. , a board-specific directory under. resources\/coreboot\/. should never specify a coreboot revision. Rather, a directory. without. coreboot configs should be created, specifying a coreboot revision. For example, the directory. resources\/coreboot\/default\/. specifies a coreboot revision. In the board-specific directory, your. board.cfg. could then specify. cbtree=\"default\". but without specifying a coreboot revision (this is specified by. resources\/coreboot\/default\/board.cfg. ). When you create a coreboot configuration, you should set the payload to. none. because. lbmk. itself will assume that is the case, and insert payloads itself. Configurations with. libgfxinit. will use coreboot’s native graphics init code if available on that board. If the file name has. txtmode. in it, coreboot will be configured to start in. text mode. , when setting up the display. If the file name has. corebootfb. in it, coreboot will be configured to set up a high resolution. frame buffer. , when initializing the display. NOTE: If the configuration file is. libgfxinit_txtmode. , the SeaBIOS payload can still run. external. VGA option ROMs on graphics cards, and this is the recommended setup (SeaBIOS in text mode) if you have a board with both onboard and an add-on graphics card (e.g. PCI express slot) installed. Configuration files with. vgarom. in the name have coreboot itself configured to run VGA option ROMs (and perhaps other option ROMs). This. setup is not strictly recommended for. SeaBIOS. , and it is recommended that you only run GRUB in this setup. As such, if you wish for a board to have coreboot initialize the VGA ROM (on an add-on graphics card, as opposed to onboard chipset), you should have a. separate. directory just for that, under. resources\/coreboot\/. ; another directory for that board will have configs with. libgfxinit. . HOWEVER:. It. is. supported in lbmk to have SeaBIOS used, on either setup. In the directory. resources\/seabios\/. there are SeaBIOS configs for both; the vgarom one sets VGA hardware type to. none. while the libgfxinit one sets it to. coreboot linear framebuffer. . However, if you use SeaBIOS on a setup with coreboot also doing option ROM initialization, such initialization is being performed. twice. . As such, if you want to use an add-on graphics card in SeaBIOS, but the board has libgfxinit, it is recommended that you do it from a. libgfxinit. ROM. HOWEVER: there’s no hard and fast rule. For example, you could make a vgarom configuration, on a board in lbmk, but in its coreboot configuration, don’t enable native init. or. oproms, and do SeaBIOS-only on that board. On. vgarom. setups, coreboot can be configured to start with a high resolution VESA frame buffer (NOT to be confused with the coreboot frame buffer), or just normal text mode. Text mode startup is always recommended, and in that setup, GRUB (including coreboot GRUB, but also PC GRUB) can use VGA modes. The name. libgfxinit. is simply what. .\/build boot roms. uses, but it may be that a board uses the old-school native video init code written in C. On some platforms, coreboot implemented a 3rd party library called. libgfxinit. , which is written in Ada and handles video initialization. In this setup, coreboot. itself. should. never. be configured to run any option ROMs, whether you start in text mode or with the coreboot framebuffer initialization. The. normal. config type is for desktop boards that lack onboard graphics chipsets, where you would always use an add-on graphics card (or. no. graphics card, which would be perfectly OK on servers). Even if your board doesn’t actually use. libgfxinit. , the config for it should still be named as such. From a user’s perspective, it really makes no difference. COREBOOT build system. [link] If you wish to know about coreboot, refer here:. https:\/\/doc.coreboot.org\/tutorial\/part1.html This and other documents from coreboot shall help you to understand. coreboot. . You create a config, for. resources\/coreboot\/BOARDNAME\/configs. , by running the. make menuconfig. command in the. coreboot. build system. You should do this after running. .\/download coreboot. in lbmk. You can simply clone coreboot upstream, add whatever patches you want, and then you can make your config. It will appear afterwards in a file named. .config. which is your config for inside. resources\/coreboot\/BOARDNAME\/. . You can then use. git format-patch -nX. where. X. is however many patches you added to that coreboot tree. You can put them in the patches directory under. resources\/coreboot\/BOARDNAME. . The. base. revision, upon which any custom patches you wrote are applied, shall be the. cbrevision. entry. REMINDER: Do not enable a payload in coreboot’s build system. Set it to. none. , and enable whatever payload you want in lbmk. If a payload is not supported in lbmk, patches are very much welcome! It is the policy of Libreboot, to only ever use the. coreboot. build system inside coreboot, but not use any of. coreboot’s. own integration for payloads. It is far more flexible and. robust. to handle payloads externally, relative to the coreboot build system. Scripts exist in. lbmk. for automating the modification\/updating of. existing. configs, but not for adding them. Adding them is to be done manually, based on the above guidance. ALSO:. If the option exists, for a given board, please configure coreboot to clear all DRAM upon boot. This is for security reasons. An exception is made when such functionality is not available, on the specific board\/revision that you’re configuring in coreboot. resources\/coreboot\/DEFAULT\/blobs.list. [link] For directories in. resources\/coreboot\/. that specify. cbrevision. , a. blobs.list. file can be included. When running. .\/download coreboot. , lbmk will delete whatever files are listed in. blobs.list. for that coreboot tree. When downloading coreboot, lbmk checks out coreboot 3rdparty submodules, but only does. git submodule update --init. ; on coreboot’s side, it is set up so that this doesn’t download most of the non-free software that coreboot distributes (for that, you run. git submodule --init --checkout. (you’ll note that the. --checkout. option is included). However, some binary blobs still remain even when only doing. --init. . These are discovered, whenever a new coreboot revision is added to lbmk, by running the. linux-libre. deblob script on the coreboot source tree,. after. doing the. git submodule update --init. command. See. deblob-check. from the fsfla website:. https:\/\/www.fsfla.org\/ikiwiki\/selibre\/linux-libre\/ The. deblob-check. script in fact. does. work quite well on the coreboot source tree! However, coreboot is far simpler than the Linux kernel, and much more conservative in its general scope, that the script was never actually forked specifically for Libreboot. Simply speaking, the way deblobbing is handled in Libreboot is as follows:. Copy the. blobs.list. from the. last. deblobbed coreboot revision. Run. deblob-check. on the. new. coreboot revision. Run. deblob-check. on the. last. deblobbed coreboot revision. Diff the results. Any file that was deleted on coreboot side, remove from the new. blobs.list. Any new files get added to the new. blobs.list. Doing it manually, and in such a crude fashion as this, is perfectly acceptable because coreboot makes a good habit of always separating binary code blobs into completely. separate. files. There is some nuance in exactly how Libreboot handles binary blobs. As far as Libreboot is concerned, only. software. is deleted if a blob. Non-software blobs are retained, so long as they are in a well-understood format or are otherwise trivial. Of course, such data must not be under a non-free license! On the other hand, blobs such as CPU microcode are always to be deleted. For example: DDR training data is retained. These are data patterns used for memory controller initialization, specifically during. training. (bruteforcing the precise timings required at boot time). More nuance: lbmk does. not. disable any code for. loading. blobs, but rather, it only deletes the actual blobs. For example, you can still add CPU microcode updates to Libreboot ROM images, and libreboot’s version of coreboot will still use them, if present. This has always been the case. Libreboot will never try to prevent you from running blobs; it merely does not. include. them. This is for the sake of efficiency, because deblobbing is actually only a very minor aspect of what Libreboot is, and time is better spent on other areas of development. Deblobbing is done in the most low-effort way possible, just so as to comply with the. GNU Free System Distribution Guidelines. . Of course, deleting blobs from coreboot. breaks. coreboot, in situations where you actually want to build for a board where those blobs are used, but since those boards are not to be supported in lbmk anyway, it’s moot (the boards that lbmk does support will all boot just fine, because all of the required files exist, and are free). resources\/coreboot\/BOARDNAME\/patches\/*. [link] In cases where. cbrevision. is specified, where the given directory under. resources\/coreboot\/. does in fact define a version of coreboot to download, you can add custom. patches. on top of that revision. When you run the command. .\/download coreboot. , those patches will be applied chronologically in alphanumerical order as per patch file names. The patch files should be named with. .patch. file extensions. All other files will be ignored. By having. lbmk. do it this way, you could add a. README. file for instance, and. lbmk. will not erroneously try to apply. README. as though it were a patch file. This might be useful if you have a. lot. of patches, and you want to provide some explanations about specific files. FUN FACT: If you run. NODELETE= .\/download coreboot. , lbmk will. skip. deleting blobs, and also skip deleting the. .git. files and directories in those coreboot clones. By default, the Git history is deleted, because it contains blobs. However, you may want to make changes and then create a patch using. git format-patch. , and you can do just that! Afterwards, you would simply delete the blobs manually and delete the Git history (or you could just run. .\/download coreboot. again, without. NODELETE. ). resources\/grub\/background\/. [link] Splash screen images applied duing startup when using the GNU GRUB payload. resources\/grub\/config\/grub.cfg. [link] This is a configuration file. It is used to program GRUB’s shell. This is inserted (as. grub.cfg. ) into the root of CBFS, in the ROM image. It contains a lot of logic in it, for booting various system configurations, when the GRUB payload is in use. resources\/grub\/config\/grub_memdisk.cfg. [link] This is a configuration file. It is used to program GRUB’s shell. This file is inserted (as. grub.cfg. ) into the GRUB. memdisk. , when building the GRUB payload (for coreboot), using GRUB’s. grub-mkstandalone. utility. It simply loads the. grub.cfg. file from CBFS (see above). resources\/grub\/keymap\/. [link] This directory contains keymaps for GRUB. They allow for different keyboard layouts to be used. The. lbmk. build system uses these to produce ROM images with various keyboard layouts used by default, when the GRUB payload is to be used. They are stored here, directly in GRUB’s own. .gkb. file format, which is a binary format defining which scancodes correspond to which character input. This binary format is documented by GRUB; the code for it is easy to understand. Please read. grub-core\/commands\/keylayouts.c. in the GRUB source code. resources\/grub\/modules.list. [link] This file defines all modules that are to be included in builds of GNU GRUB. They are standalone builds, created using the. grub-mkstandalone. utility. resources\/grub\/patches\/. [link] This directory contains custom patches for GNU GRUB. resources\/memtest86plus\/patch\/. [link] This directory contains custom patches for Memtest86+. resources\/scripts\/build\/boot\/roms. [link] This script builds coreboot ROM images. It is largely a shim, which calls the. roms_helper. script, which does most of the legwork. Command:. .\/build boot roms. Additional parameters can be provided. This lists all boards available:. .\/build boot roms list. Pass several board names if you wish to build only for specific targets. For example:. .\/build boot roms x60 x200_8mb. resources\/scripts\/build\/boot\/roms_helper. [link] This script builds coreboot ROM images. It is not to be executed directory; user interaction must be done via the main. roms. script. It heavily makes use of the. board.cfg. file, for a given board. This script will. only. operate on a single target, from a directory in. resources\/coreboot\/. . If. grub_scan_disk. is set, it sets that in the. grub.cfg. file that is to be inserted into a ROM image, when. payload_grub. is turned on. It automatically detects if. crossgcc. is to be compiled, on a given coreboot tree (in cases where it has not yet been compiled), and compiles it for a target based on the. arch. entry in. board.cfg. . It creates ROM images with GRUB and\/or SeaBIOS, optionally with Memtest86+ also included, in various separate configurations in many different ROM images for user installation. The. romtype. entry in. board.cfg. tells this script what to do with the ROM, after it has been built. Currently, it operates based on these possible values for. romtype. :. 4MiB IFD BIOS region. will cause only the upper 4MB section of the ROM to be included in a release. This option is largely deprecated, a hangover from osboot, which also no longer uses this option on any boards, and it is thus subject for removal. d8d16sas. will cause. fake. (empty) files named. pci1000,0072.rom. and. pci1000,3050.rom. to be inserted in CBFS. This prevents SeaBIOS from loading or executing the option ROM stored on PIKE2008 modules, present on certain configurations with the ASUS KCMA-D8 or KGPE-D16 mainboards. Those option ROMs cause the system to hang, so they should never be executed (this means however that booting Linux kernels from SAS devices is impossible on those boards, unless a Linux payload is used; Linux can use those SAS drives, without relying on the PIKE2008 option ROMs). When SeaBIOS runs, it will default to loading the corresponding option ROM from CBFS, if it exists, for a given PCI device, overriding whatever option ROM is present on the device itself, but if the option ROM is invalid\/empty, SeaBIOS will not attempt to load another one, until the empty\/invalid one (in CBFS) is deleted. 4MiB ICH9 IFD NOR flash. : the. ich9gen. program will be used to insert an Intel Flash Descriptor and Gigabit Ethernet Non-volatile memory file into the ROM image. This is used on GM45\/ICH9M based laptops, such as: ThinkPad X200, T400, T500, W500, X200 Tablet, X200S, T400S, X301. 8MiB ICH9 IFD NOR flash. : Same as the 4MB one as described above, but for ROM images with 8MB (64Mbit) of boot flash. The one above is for systems with 4MB (32Mbit) of flash. 16MiB ICH9 IFD NOR flash. : ditto, but for 16MB (128Mbit) flash. In this and the other two cases as described above, the first 4KB is the Intel Flash Descriptor, the next 8KB is GbE NVM and the rest is BIOS (for the coreboot part). In all cases, the default. ME. (Intel Management Engine) region is disabled, as is the ME itself, based on bits set to disable it in the Intel Flash Descriptor. The descriptor is used in such a setup, because on all such boards in Libreboot, GbE NVM is needed to get gigabit ethernet working correctly; it is the sole reason. ich9gen. was written, because it is otherwise possible to boot these machines in a. descriptorless. setup, where ICH9M behaves similarly to ICH7: all one region of flash, for the boot firmware (coreboot), but it results in a non-functional gigabit enternet device. 4MiB ICH9 IFD NOGBE NOR flash. : Intel Flash Descriptor. on its own. , without ME or GbE NVM. Just IFD and BIOS. This is used on the ThinkPad R500. 8MiB ICH9 IFD NOGBE NOR flash. : Same as above, but for 8MB (64Mbit) ROMs. 16MiB ICH9 IFD NOGBE NOR flash. : Same as above, but for 16MB (128Mbit) ROMs. i945 laptop. : in this configuration, the upper 64KB section of the ROM is copied into the 64KB section below that. This results in there being two bootblocks in the ROM, and you can decide which one is used by setting. bucts. If no payload is defined in. board.cfg. , the. roms_helper. script will exit with error status. If SeaBIOS is to be used, on. libgfxinit. setups, SeaVGABIOS will also be inserted. This provides a minimal VGA compatibility layer on top of the coreboot framebuffer, but does not allow for. switching. the VGA mode. It is currently most useful for directly executing ISOLINUX\/SYSLINUX bootloaders, and certain OS software (some Windows setups might work, poorly, depending on the board configuration, but don’t hold your breath; it is far from complete). If SeaBIOS is to be used, in. vgarom. setups or. normal. setups, SeaVGABIOS is not inserted and you rely on either coreboot and\/or SeaBIOS to execute VGA option ROMs. In all cases, this script automatically inserts several SeaBIOS runtime configurations, such as:. etc\/ps2-keyboard-spinup. set to 3000 (PS\/2 spinup wait time),. etc\/pci-optionrom-exec. set to 2 (despite that already being the default anyway) to enable. all. option ROMs, unless. vgarom. setups are used, in which case the option is set to. 0. (disabled) because coreboot is then expected to handle option ROMs, and SeaBIOS should not do it. Essentially, the. roms_helper. script makes use of each and every part of lbmk. It is the heart of Libreboot. When the ROM is finished compiling, it will appear under a directory in. bin\/. resources\/scripts\/build\/clean\/cbutils. [link] This simply runs. make clean. on various utilities from coreboot, which lbmk makes use of. Command:. .\/build clean cbutils. resources\/scripts\/build\/clean\/crossgcc. [link] This runs. make crossgcc-clean. on all of the coreboot revisions present in lbmk. Command:. .\/build clean crossgcc. resources\/scripts\/build\/clean\/flashrom. [link] This runs. make clean. in the. flashrom\/. directory. Command:. .\/build clean flashrom. resources\/scripts\/build\/clean\/grub. [link] This runs. make clean. in the. grub\/. directory. It does not delete anything in. payload\/grub\/. . Command:. .\/build clean grub. resources\/scripts\/build\/clean\/ich9utils. [link] This runs. make clean. in the. ich9utils\/. directory. Command:. .\/build clean ich9utils. resources\/scripts\/build\/clean\/memtest86plus. [link] This runs. make clean. in the. memtest86plus\/. directory. Command:. .\/build clean memtest86plus. resources\/scripts\/build\/clean\/payloads. [link] This deletes the. payload\/. directory. Command:. .\/build clean payloads. resources\/scripts\/build\/clean\/rom_images. [link] This deletes the. bin\/. directory. Command:. .\/build clean rom_images. resources\/scripts\/build\/clean\/seabios. [link] This runs. make clean. in the. seabios\/. directory. Command:. .\/build clean seabios. resources\/scripts\/build\/dependencies\/arch. [link] Using. pacman. , this installs build dependencies in Arch. It may also work on similar distros like Manjaro or Artix. Command:. .\/build dependencies arch. resources\/scripts\/build\/dependencies\/debian. [link] Using. apt-get. , this installs build dependencies in Debian. It may work on other. apt-get. distros. Command:. .\/build dependencies debian. resources\/scripts\/build\/dependencies\/fedora35. [link] Using. dnf. , this installs build dependencies in Fedora 35. Command:. .\/build dependencies fedora35. resources\/scripts\/build\/dependencies\/ubuntu2004. [link] Using. apt-get. , this installs build dependencies for Ubuntu 20.04 (for later versions, you might use the Debian script). This script should also work with Trisquel 9 and 10. Command:. .\/build dependencies ubuntu2004. resources\/scripts\/build\/dependencies\/void. [link] Using. xbps. , this installs build dependencies for Void. Command:. .\/build dependencies void. resources\/scripts\/build\/descriptors\/ich9m. [link] This runs. ich9gen. to generate descriptors for ICH9M platforms. These are then stored in. descriptors\/ich9m\/. Command:. .\/build descriptors ich9m. resources\/scripts\/build\/module\/cbutils. [link] This compiles various coreboot utilities (such as cbfstool). Command:. .\/build module cbutils. resources\/scripts\/build\/module\/flashrom. [link] This compiles flashrom. Command. .\/build module flashrom. resources\/scripts\/build\/module\/grub. [link] This compiles GRUB utilities. It does not build the actual payloads. Command:. .\/build module grub. resources\/scripts\/build\/module\/ich9utils. [link] This compiles. ich9utils. , which includes the. ich9gen. utility. Command:. .\/build module ich9utils. resources\/scripts\/build\/module\/memtest86plus. [link] This compiles Memtest86+. Command:. .\/build module memtest86plus. resources\/scripts\/build\/payload\/grub. [link] This builds the GRUB payloads. Command:. .\/build payload grub. resources\/scripts\/build\/payload\/seabios. [link] This builds the SeaBIOS payloads. Command:. .\/build payload seabios. resources\/scripts\/build\/release\/roms. [link] This builds release archives, containing ROM images. You must only run this after you’ve built all of the ROM images that you wish to release. Command:. .\/build release roms. resources\/scripts\/build\/release\/src. [link] This builds source archives. You must only run this after compiling crossgcc on all coreboot source trees. Command:. .\/build release src. resources\/scripts\/build\/release\/u-boot-libre. [link] This builds a release of u-boot-libre. Once released officially, the files produced by the command below are then expected to be used by FSDG distributions packages. While this isn’t useful as-is for Libreboot, it enables to share the deblobbing code between various FSDG distributions and also enables Libreboot to reuse that deblobbing process to support boards with u-boot in the future. Command:. .\/build release u-boot-libre. resources\/scripts\/download\/coreboot. [link] This downloads, patches and deblobs coreboot, as per. board.cfg. files in. resources\/coreboot\/. . Command:. .\/download coreboot. resources\/scripts\/download\/flashrom. [link] This downloads and patches flashrom. Command:. .\/download flashrom. resources\/scripts\/download\/grub. [link] This downloads and patches GNU GRUB. Command:. .\/download grub. resources\/scripts\/download\/ich9utils. [link] This downloads. ich9utils. , which includes. ich9gen. . Command:. .\/download ich9utils. resources\/scripts\/download\/memtest86plus. [link] This downloads and patches Memtest86+. Command:. .\/download memtest86plus. resources\/scripts\/download\/seabios. [link] This downloads and patches SeaBIOS. Command:. .\/download seabios. resources\/scripts\/misc\/versioncheck. [link] This updates the text file containing version information. It is used by many other build scripts. It also updates the files containing the version date. You need not run this yourself, directly. resources\/scripts\/modify\/coreboot\/configs. [link] Loads coreboot configs into coreboot trees, and runs. make menuconfig. , so that you can easily modify them in an ncurses interface. Additional parameters are accepted, for example:. .\/modify coreboot configs x60 x200_8mb. With no additional parameters given, it simply cycles through all configs under. resources\/coreboot\/. . Command:. .\/modify coreboot configs. resources\/scripts\/modify\/seabios\/configs. [link] This lets you modify SeaBIOS configs. Command:. .\/modify seabios configs. resources\/scripts\/update\/coreboot\/configs. [link] This runs. make oldconfig. on coreboot configs under. resources\/coreboot\/. . It is most useful when updating a coreboot revision, per. board.cfg. . It allows additional parameters, for example:. .\/update coreboot configs x60 x200_8mb. With no additional parameters given, it simply cycles through all configs under. resources\/coreboot\/. . Command:. .\/update coreboot configs. resources\/scripts\/update\/seabios\/configs. [link] This runs. make oldconfig. on SeaBIOS configs. It is most useful when updating the version of SeaBIOS used by lbmk. Command:. .\/update seabios configs. resources\/seabios\/config\/libgfxinit. [link] SeaBIOS configuration file, when. libgfxinit. is to be used. It enables the. coreboot linear framebuffer. option in the SeaBIOS. make menuconfig. configuration interface. resources\/seabios\/config\/vgarom. [link] This version is for normal SeaBIOS configurations, where. libgfxinit. is not to be used. update. [link] This can be used to update SeaBIOS and coreboot configs. It calls scripts in. resources\/scripts\/update\/. , for example:. .\/update coreboot configs. This runs:. .\/resources\/scripts\/update\/coreboot\/configs. Additional parameters can be given, for example:. .\/update coreboot configs x200_8mb x60. This would run:. .\/resources\/scripts\/update\/coreboot\/configs x200_8mb x60. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/maintain\/index.md Site map This HTML page was generated by the. untitled static site generator ."}
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ASUS Chromebook C201. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Google’s intent with CrOS devices Considerations about ChromeOS and free operating systems Debian GNU+Linux Devuan GNU+Linux Parabola GNU+Linux Caution: Video acceleration requires a non-free blob, software rendering can be used instead. Caution: WiFi requires a non-free blob, a USB dongle can be used instead. EC firmware is free software! No microcode! Depthcharge payload Flash chip write protection: the screw Return to index NOTE: support for this machine is dropped in recent Libreboot releases. It will be re-added at a later date. For now, please use Libreboot 20160907 on this machine. NOTE: much of this page is outdated. for instance, it references cafe beverage who later revealed herself to be Alyssa Rosenzweig, who then launched the Panfrost project. This is a Chromebook, using the Rockchip RK3288 SoC. It uses an ARM CPU, and has free EC firmware (unlike some other laptops). More RK3288-based laptops will be added to libreboot at a later date. Flashing instructions can be found at. ..\/install\/#flashrom Google’s intent with CrOS devices. [link] CrOS (Chromium OS\/Chrome OS) devices, such as Chromebooks, were not designed with the intent of bringing more freedom to users. However, they run with a lot of free software at the boot software and embedded controller levels, since free software gives Google enough flexibility to optimize various aspects such as boot time and most importantly, to implement the CrOS security system, that involves various aspects of the software. Google does hire a lot of Coreboot developers, who are generally friendly to the free software movement and try to be good members of the free software community, by contributing code back. CrOS devices are designed (from the factory) to actually coax the user into using proprietary web services (SaaSS) that invade the user’s privacy (ChromeOS is literally just the Google Chrome browser when you boot up, itself proprietary and comes with proprietary add-ons like flash. It’s only intended for SaaSS, not actual, real computing). Google is even a member of the. PRISM. program, as outlined by Edward Snowden. See notes about ChromeOS below. The libreboot project recommends that the user replace the default. ChromeOS. with a distribution that can be used in freedom, without invading the user’s privacy. We also use a similar argument for the MacBook and the ThinkPads that are supported in libreboot. Those laptops are supported, in spite of Apple and Lenovo, companies which are actually. hostile. to the free software movement. Considerations about ChromeOS and free operating systems. [link] This laptop comes preinstalled (from the factory) with Google ChromeOS. This is a GNU+Linux distribution, but it’s not general purpose and it comes with proprietary software. It’s designed for SaaSS. Libreboot recommends that users of this laptop replace it with another distribution. Debian GNU+Linux. [link] https:\/\/wiki.debian.org\/InstallingDebianOn\/Asus\/C201 shows how to install Debian. Devuan GNU+Linux. [link] https:\/\/notabug.org\/dimkr\/devsus produces bootable and installable Devuan images. Parabola GNU+Linux. [link] See:. https:\/\/lists.gnu.org\/archive\/html\/libreboot\/2015-12\/msg00026.html In this discussion thread (on the old GNU Libreboot mailing lists), there are instructions for installing Parabola on C201 and other rockchip chromebooks supported by Libreboot. Caution: Video acceleration requires a non-free blob, software rendering can be used instead. [link] The C201 has a Mali T GPU, which requires a non-free blob. A driver, Tamil, was written, but its source code has not been released. The developer has so-far. withheld it . Use software rendering to avoid the blob instead. Most tasks can still be performed without video acceleration, without any noticeable performance penalty. In practise, this means that certain things like games, blender and GNOME shell (or other fancy desktops) won’t work well. The libreboot project recommends a lightweight desktop which does not need video acceleration, such as. XFCE. or. LXDE. . As it is unlikely that Tamil will be released, the. chai project is writing a driver as well. Ask on IRC if you think you can contribute. Caution: WiFi requires a non-free blob, a USB dongle can be used instead. [link] These laptops have non-removeable (soldered on) M.2 Type 1216 card with WiFi+Bluetooth, which requires non-free firmware to be loaded by the Linux kernel in order to work. The libreboot project recommends using an external USB wifi dongle that works with free software. See. #recommended_wifi . There are 2 companies (endorsed by Free Software Foundation, under their. Respects your Freedom. guidelines), that sell USB WiFi dongles guaranteed to work with free software (i.e. linux-libre kernel):. ThinkPenguin sells them (company based in USA). Tehnoetic sells them (company based in Europe). These wifi dongles use the AR9271 (atheros) chipset, supported by the free. ath9k_htc. driver in the Linux kernel. They work in. linux-libre. too. EC firmware is free software! [link] It’s free software. Google provides the source. Build scripts will be added later, with EC sources provided in libreboot, and builds of the EC firmware. This is unlike the other current libreboot laptops (Intel based). In practise, you can (if you do without the video\/wifi blobs, and replace ChromeOS with a distribution that respects your freedom) be more free when using one of these laptops. The libreboot FAQ briefly describes what an. EC. is:. ..\/..\/faq.html#firmware-ec No microcode! [link] Unlike x86 (e.g. Intel\/AMD) CPUs, ARM CPUs do not use microcode, not even built in. On the Intel\/AMD based libreboot systems, there is still microcode in the CPU (not considered problematic by the FSF, provided that it is reasonably trusted to not be malicious, since it’s part of the hardware and read-only), but we exclude microcode updates (volatile updates which are uploaded at boot time by the boot firmware, if present), which are proprietary software. On ARM CPUs, the instruction set is implemented in circuitry, without microcode. Depthcharge payload. [link] These systems do not use the GRUB payload. Instead, they use a payload called depthcharge, which is common on CrOS devices. This is free software, maintained by Google. Flash chip write protection: the screw. [link] It’s next to the flash chip. Unscrew it, and the flash chip is read-write. Screw it back in, and the flash chip is read-only. It’s called the screw. The screw. is accessible by removing other screws and gently prying off the upper shell, where the flash chip and the screw are then directly accessible. User flashing from software is possible, without having to externally re-flash, but the flash chip is SPI (SOIC-8 form factor) so you can also externally re-flash if you want to. In practise, you only need to externally re-flash if you brick the laptop; read. ..\/install\/spi.html for an example of how to set up an SPI programmer. Write protection is useful, because it prevents the firmware from being re-flashed by any malicious software that might become executed on your GNU+Linux system, as root. In other words, it can prevent a firmware-level. evil maid. attack. It’s possible to write protect on all current libreboot systems, but CrOS devices make it easy. The screw is such a stupidly simple idea, which all designs should implement. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/c201.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"I1egv2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/bsd\/openbsd.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":43,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","bsd"],"url_file_name_s":"openbsd","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – How to install OpenBSD on x86 GNU GRUB payload"],"last_modified":"Sat Apr 02 12:48:34 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":3771442756473652111,"exact_signature_unique_b":true,"fuzzy_signature_l":-6836515499363626884,"fuzzy_signature_unique_b":true,"h1_txt":["How to install OpenBSD on x86 GNU GRUB payload","Navigate this page:","Troubleshooting"],"h2_txt":["Prepare the USB drive (in LibertyBSD or OpenBSD)","Prepare the USB drive (in NetBSD)","Prepare the USB drive (in FreeBSD)","Prepare the USB drive (in GNU+Linux)","Installing OpenBSD without full disk encryption","Installing OpenBSD with full disk encryption","Booting","Configuring Grub","Won’t boot…something about file not found"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":404,"wordcount_i":862,"linkscount_i":25,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":7,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/bsd\/openbsd.html","libreboot.org\/docs\/bsd\/","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/bsd\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/bsd\/openbsd.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Prepare the USB drive (in LibertyBSD or OpenBSD)","Return to index","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","www.openbsd.org\/faq\/faq4.html","wiki.netbsd.org\/tutorials\/how_to_install_netbsd_from_an_usb_memory_stick\/","www.freebsd.org\/doc\/handbook\/bsdinstall-pre.md","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","","This page","This page","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:03 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:17 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":11241,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078563676160000,"text_t":"Libreboot – How to install OpenBSD on x86 GNU GRUB payload. How to install OpenBSD on x86 GNU GRUB payload. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Prepare the USB drive (in LibertyBSD or OpenBSD) Prepare the USB drive (in NetBSD) Prepare the USB drive (in FreeBSD) Prepare the USB drive (in GNU+Linux) Installing OpenBSD without full disk encryption Installing OpenBSD with full disk encryption Booting Configuring Grub Troubleshooting Won’t boot…something about file not found Return to index This guide is written for OpenBSD 6.1, but it can be adapted easily for other versions (of OpenBSD). If you want an encrypted install, use SeaBIOS instead of GRUB and go here:. https:\/\/www.openbsd.org\/faq\/faq4.html (official installation guide, which says how to use encryption). GRUB supports booting OpenBSD kernels directly. However, you’re better off simply using the SeaBIOS payload; BSD works well with BIOS or UEFI setups. GRUB is acceptable for booting unencrypted BSD installations. However, encrypted BSD installations will probably require the use of SeaBIOS\/Tianocore. install61.fs is the installation image for OpenBSD 6.1. Adapt the filename accordingly, for a different OpenBSD version or LibertyBSD. Prepare the USB drive (in LibertyBSD or OpenBSD). [link] If you downloaded your ISO on a LibertyBSD or OpenBSD system, here is how to create the bootable LibertyBSD\/OpenBSD USB drive:. Connect the USB drive and check the system message buffer:. dmesg | tail. Check to confirm which drive it is, for example, if you think it’s. sd3. :. disklabel sd3. Check that it wasn’t automatically mounted. If it was, unmount it:. doas umount \/dev\/sd3i. Now write the OpenBSD installer to the drive with. dd. :. doas dd if=install60.fs of=\/dev\/rsdXc bs=1M; sync. You should now be able to boot the installer from your USB drive. Continue reading, for information about how to do that. Prepare the USB drive (in NetBSD). [link] This page on the NetBSD website shows how to create a NetBSD bootable USB drive from within NetBSD itself. You should use the. dd. method documented there. This will also work with the OpenBSD image. Prepare the USB drive (in FreeBSD). [link] This page on the FreeBSD website shows how to create a bootable USB drive for installing FreeBSD. Use the. dd. on that page. You can also use the same instructions with a OpenBSD ISO image. Prepare the USB drive (in GNU+Linux). [link] If you downloaded your ISO on a GNU+Linux system, here is how to create the bootable OpenBSD USB drive:. Connect the USB drive. Check dmesg:. dmesg. Check lsblk to confirm which drive it is:. lsblk. Check that it wasn’t automatically mounted. If it was, unmount it. For example:. sudo umount \/dev\/sdX\\* umount \/dev\/sdX\\*. dmesg told you what device it is. Overwrite the drive, writing your distro ISO to it with dd. For example:. sudo dd if=install61.fs of=\/dev\/sdX bs=8M; sync dd if=install61.fs of=\/dev\/sdX bs=8M; sync. You should now be able to boot the installer from your USB drive. Continue reading, for information about how to do that. Installing OpenBSD without full disk encryption. [link] Press C in GRUB to access the command line:. grub kopenbsd (usb0,openbsd1)\/6.1\/amd64\/bsd.rd grub boot. It will start booting into the OpenBSD installer. Follow the normal process for installing OpenBSD. Installing OpenBSD with full disk encryption. [link] If you are using SeaBIOS, OpenBSD’s bootloader will be chainloaded automatically and everything will just work. Alternatively, it would be good to port OpenBSD either natively as a coreboot payload, or port it to libpayload (payload library in coreboot; it has a basic C library and a few functions for certain operations e.g. text\/bitmap). Booting. [link] Press C in GRUB to access the command line:. grub kopenbsd -r sd0a (ahci0,openbsd1)\/bsd grub boot. OpenBSD will start booting. Yay! Configuring Grub. [link] If you don’t want to drop to the GRUB command line and type in a command to boot OpenBSD every time, you can create a GRUB configuration that’s aware of your OpenBSD installation and that will automatically be used by Libreboot. On your OpenBSD root partition, create the. \/grub. directory and add the file. libreboot_grub.cfg. to it. Inside the. libreboot_grub.cfg. add these lines:. default=0 timeout=3 menuentry \"OpenBSD\" { kopenbsd -r sd0a (ahci0,openbsd1)\/bsd }. If your OpenBSD installation uses a GPT scheme, use the. gpt4. partition instead of. openbsd1. . The next time you boot, you’ll see the old Grub menu for a few seconds, then you’ll see the a new menu with only OpenBSD on the list. After 3 seconds OpenBSD will boot, or you can hit enter to boot. Troubleshooting. [link] Most of these issues occur when using Libreboot with coreboot’s ‘text mode’ instead of the coreboot framebuffer. This mode is useful for booting payloads like memtest86+ which expect text-mode, but for OpenBSD it can be problematic when they are trying to switch to a framebuffer because it doesn’t exist. Won’t boot…something about file not found. [link] Your device names (i.e. usb0, usb1, sd0, sd1, wd0, ahci0, hd0, etc) and numbers may differ. Use TAB completion. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/bsd\/openbsd.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"4HMUY_Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/news\/libreboot20140716.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["news"],"url_file_name_s":"libreboot20140716","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Libreboot 20140716 release"],"author":"Leah Rowe","author_sxt":["Leah Rowe"],"last_modified":"Sat Apr 02 12:48:46 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Wed Jul 16 00:00:00 UTC 2014"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":8400851627559610447,"exact_signature_unique_b":true,"fuzzy_signature_l":-6015109145662162377,"fuzzy_signature_unique_b":true,"h1_txt":["Libreboot 20140716 release","Navigate this page:"],"h2_txt":["Revisions for r20140716 (2nd beta) (16th July 2014)"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":354,"text_t":"Libreboot – Libreboot 20140716 release. Libreboot 20140716 release. Leah Rowe. 16 July 2014. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Revisions for r20140716 (2nd beta) (16th July 2014) Return to index Article published by: Leah Rowe. Date of publication: 16 July 2014. Revisions for r20140716 (2nd beta) (16th July 2014). [link] Deleted all git-related files from the coreboot directory. This was necessary because with those it is possible to run ‘git diff’ which shows the changes made in the form of a patch (diff format); this includes the blobs that were deleted during deblobbing. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/news\/libreboot20140716.md Site map This HTML page was generated by the. untitled static site generator .","wordcount_i":136,"linkscount_i":21,"linksnofollowcount_i":0,"inboundlinkscount_i":17,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/news\/libreboot20140716.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/news\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/news\/libreboot20140716.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Revisions for r20140716 (2nd beta) (16th July 2014)","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:03 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:11 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":4314,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078563875389440}
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{"id":"Fcq7K2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/install\/c201.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":44,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","install"],"url_file_name_s":"c201","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – ASUS Chromebook C201 installation guide"],"last_modified":"Sat Apr 02 12:48:41 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":6748086473062700431,"exact_signature_unique_b":true,"fuzzy_signature_l":-2102480848407111488,"fuzzy_signature_unique_b":true,"h1_txt":["ASUS Chromebook C201 installation guide","Navigate this page:","Installing Libreboot to the SPI flash"],"h2_txt":["Gaining root privileges on ChromeOS","Configuring verified boot parameters","Removing the write protect screw","Installing Libreboot internally, from the device","Installing Libreboot externally, with a SPI flash programmer","Installing Debian"],"bold_txt":["Note: it might be a good idea to copy the produced flash.img file at this point and store it outside of the device for backup purposes.","read-only","Google’s modified version of flashrom","Beware: there are cables attached to it!","enabled","disabled","Next","Ctrl + Alt + Next","Enable","Proceed","Enable debugging features","Ctrl + D","veyron speedy","This documentation is retained from Libreboot 20160907, but it may also be prudent to check documentation from Libreboot 20160907 itself. It is included in the source code archive, for that release.","Libreboot 20160907 only. This board was dropped in subsequent releases, but may be added again in the future.","Buy preinstalled","Donate"],"imagescount_i":5,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico","av.libreboot.org\/c201\/screws.jpg","av.libreboot.org\/c201\/wp-screw.jpg","av.libreboot.org\/c201\/spi-flash-layout.jpg","av.libreboot.org\/c201\/battery-connector.jpg"],"images_alt_sxt":["Libreboot logo","Screws","WP screw","SPI flash layout","Battery connector"],"images_height_val":[-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1],"images_text_t":"libreboot org favicon ico Libreboot logo av c201 screws jpg c 201 Screws wp screw WP screw spi flash layout SPI flash layout battery connector Battery connector","responsetime_i":390,"wordcount_i":1371,"linkscount_i":26,"linksnofollowcount_i":0,"inboundlinkscount_i":20,"outboundlinkscount_i":6,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https","018-https","019-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/install\/c201.html","libreboot.org\/docs\/misc\/codenames.html","libreboot.org\/docs\/depthcharge\/","libreboot.org\/docs\/hardware\/c201.md","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/install\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/install\/c201.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Gaining root privileges on ChromeOS","veyron speedy","recovery mode screen","","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","chromium.googlesource.com\/chromiumos\/third_party\/flashrom\/","av.libreboot.org\/c201\/screws.jpg","av.libreboot.org\/c201\/wp-screw.jpg","av.libreboot.org\/c201\/spi-flash-layout.jpg","av.libreboot.org\/c201\/battery-connector.jpg","wiki.debian.org\/InstallingDebianOn\/Asus\/C201","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","flashrom","","","","","","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:03 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:14 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":15005,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078564231905280,"text_t":"Libreboot – ASUS Chromebook C201 installation guide. ASUS Chromebook C201 installation guide. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Gaining root privileges on ChromeOS Configuring verified boot parameters Removing the write protect screw Installing Libreboot to the SPI flash Installing Libreboot internally, from the device Installing Libreboot externally, with a SPI flash programmer Installing Debian Return to index Libreboot 20160907 only. This board was dropped in subsequent releases, but may be added again in the future. This documentation is retained from Libreboot 20160907, but it may also be prudent to check documentation from Libreboot 20160907 itself. It is included in the source code archive, for that release. The version of Depthcharge used here is very old, from 2016, and support for this laptop was dropped in recent releases. It will be re-added at a later date. If you wish to use Libreboot on this board, right now you must install Libreboot 20160907. This page has been retained on libreboot.org for now, but you should refer to the documentation provided in the Libreboot 20160907 release if you want to learn more. NOTE: in that release, the documentation is written in raw HTML, because the Markdown-based static site generator used on libreboot.org had not yet been written at that point. These instructions are for installing Libreboot to the ASUS Chromebook C201 (more known under a name. veyron speedy. ). Since the device ships with Coreboot, the installation instructions are the same before and after flashing Libreboot for the first time. Look at the. list of ROM images to see which image is compatible with your device. Libreboot can be installed internally from the device, with sufficient privileges. The installation process requires using. Google’s modified version of flashrom. , that has support for reflashing the Chromebook’s SPI flash. Otherwise, flashing externally will work with the upstream flashrom version. Google’s modified version of flashrom. is free software and its source code is made available by Google:. flashrom . It is not distributed along with Libreboot yet. However, it is preinstalled on the device, with ChromeOS. Installing Libreboot internally requires sufficient privileges on the system installed on the device. When the device has ChromeOS installed (as it does initially), it is necessary to gain root privileges in ChromeOS, to be able to access a root shell. Gaining root privileges on ChromeOS. [link] In order to gain root privileges on ChromeOS, developer mode has to be enabled from the recovery mode screen and debugging features have to be enabled in ChromeOS. Instructions to access the. recovery mode screen and. enabling developer mode are available on the page dedicated to. depthcharge . Once developer mode is enabled, the device will boot to the. developer mode screen . ChromeOS can be booted by waiting for 30 seconds (the delay is shortened in Libreboot) or by pressing. Ctrl + D. After the system has booted, root access can be enabled by clicking on the. Enable debugging features. link. A confirmation dialog will ask whether to proceed. After confirming by clicking. Proceed. , the device will reboot and ask for the root password to set. Finally, the operation has to be confirmed by clicking. Enable. . After setting the root password, it becomes possible to log-in as root. A tty prompt can be obtained by pressing. Ctrl + Alt + Next. . The. Next. key is the one on the top left of the keyboard. Preparing the device for the installation Before installing Libreboot on the device, both its software and hardware has to be prepared to allow the installation procedure and to ensure that security features don’t get in the way. Configuring verified boot parameters. [link] It is recommended to have access to the. developer mode screen and to. configure the following verified boot parameters :. Kernels signature verification:. disabled. External media boot:. enabled. Those changes can be reverted later, when the device is known to be in a working state. Removing the write protect screw. [link] Since part of the SPI flash is write-protected by a screw, it is necessary to remove the screw to remove the write protection and allow writing Libreboot to the. read-only. part of the flash. To access the screw, the device has to be opened. There are 8 screws to remove from the bottom of the device, as shown on the picture below. Two are hidden under the top pads. After removing the screws, the keyboard plastic part can be carefully detached from the rest. Beware: there are cables attached to it! It is advised to flip the keyboard plastic part over, as shown on the picture below. The write protect screw is located next to the SPI flash chip, circled in red in the picture below. It has to be removed. The write protect screw can be put back in place later, when the device is known to be in a working state. Installing Libreboot to the SPI flash. [link] The SPI flash (that holds Libreboot) is divided into various partitions that are used to implement parts of the CrOS security system. Libreboot is installed in the. read-only. coreboot partition, that becomes writable after removing the write-protect screw. Installing Libreboot internally, from the device. [link] Before installing Libreboot to the SPI flash internally, the device has to be reassembled. All the files from the. veyron_speedy. release (or build) have to be transferred to the device. The following operations have to be executed with root privileges on the device (e.g. using the. root. account). In addition, the. cros-flash-replace. script has to be made executable:. chmod a+x cros-flash-replace. The SPI flash has to be read first:. flashrom -p host -r flash.img. Note: it might be a good idea to copy the produced flash.img file at this point and store it outside of the device for backup purposes. Then, the. cros-flash-replace. script has to be executed as such:. sudo bash .\/cros-flash-replace flash.img coreboot ro-frid. If any error is shown, it is definitely a bad idea to go further than this point. The resulting flash image can then be flashed back:. flashrom -p host -w flash.img. You should also see within the output the following:. Verifying flash... VERIFIED. Shut down. The device will now boot to Libreboot. Installing Libreboot externally, with a SPI flash programmer. [link] Before installing Libreboot to the SPI flash internally, the device has to be opened. The SPI flash is located next to the write protect screw. Its layout is indicated in the picture below. Note that it is not necessary to connect. WP#. since after removing the screw it is pulled up weakly to 3v3. Before writing to the chip externally, the battery has to be unplugged. Battery connector is located under the heat spreader, that has to be unscrewed from the rest of the case. It is located on the right and has colorful cables, as shown on the picture below. All the files from the. veyron_speedy. release (or build) have to be transferred to the host. The following operations have to be executed with root privileges on the host (e.g. using the. root. account). In addition, the. cros-flash-replace. script has to be made executable:. chmod a+x cros-flash-replace. The SPI flash has to be read first (using the right spi programmer):. flashrom -p *programmer* -r flash.img. Note: it might be a good idea to copy the produced flash.img file at this point and store it outside of the device for backup purposes. Then, the. cros-flash-replace. script has to be executed as such:. .\/cros-flash-replace flash.img coreboot ro-frid. If any error is shown, it is definitely a bad idea to go further than this point. The resulting flash image can then be flashed back (using the right spi programmer):. flashrom -p *programmer* -w flash.img. You should also see within the output the following:. Verifying flash... VERIFIED. The device will now boot to Libreboot. Installing Debian. [link] Debian is recommended for this device (which is on that list. See. https:\/\/wiki.debian.org\/InstallingDebianOn\/Asus\/C201 . Also look at the HCL entry for operating systems (Debian, Devuan, Parabola):. https:\/\/libreboot.org\/docs\/hardware\/c201.md Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/c201.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"LiawK2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/r400.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":45,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"r400","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – ThinkPad R400"],"last_modified":"Sat Apr 02 12:48:37 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2370427274555947914,"exact_signature_unique_b":true,"fuzzy_signature_l":-1361154136924221561,"fuzzy_signature_unique_b":true,"h1_txt":["ThinkPad R400","Navigate this page:","EC update"],"h2_txt":["Compatibility (without blobs)"],"h3_txt":["Hardware virtualization (vt-x)"],"bold_txt":["kvm_intel","kvm","The R400 laptops come with the ME (and sometimes AMT in addition) before flashing libreboot. Libreboot disables and removes it by using a modified descriptor: see ..\/install\/ich9utils.html","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":370,"text_t":"Libreboot – ThinkPad R400. ThinkPad R400. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. EC update Compatibility (without blobs) Hardware virtualization (vt-x) Return to index It is believed that all or most R400 laptops are compatible. See notes about. CPU compatibility for potential incompatibilities. There are two possible flash chip sizes for the R400: 4MiB (32Mbit) or 8MiB (64Mbit). This can be identified by the type of flash chip below the palmrest: 4MiB is SOIC-8, 8MiB is SOIC-16. The R400 laptops come with the ME (and sometimes AMT in addition) before flashing libreboot. Libreboot disables and removes it by using a modified descriptor: see. ..\/install\/ich9utils.html (contains notes, plus instructions). Flashing instructions can be found at. ..\/install\/#flashrom EC update. [link] It is recommended that you update to the latest EC firmware version. The. EC firmware is separate from libreboot, so we don’t actually provide that, but if you still have Lenovo BIOS then you can just run the Lenovo BIOS update utility, which will update both the BIOS and EC version. See:. ..\/install\/#flashrom http:\/\/www.thinkwiki.org\/wiki\/BIOS_update_without_optical_disk NOTE: this can only be done when you are using Lenovo BIOS. How to update the EC firmware while running libreboot is unknown. Libreboot only replaces the BIOS firmware, not EC. Updated EC firmware has several advantages e.g. bettery battery handling. Compatibility (without blobs). [link] Hardware virtualization (vt-x). [link] The R400, when run without CPU microcode updates in coreboot, currently kernel panics if running QEMU with vt-x enabled on 2 cores for the guest. With a single core enabled for the guest, the guest panics (but the host is fine). Working around this in QEMU might be possible; if not, software virtualization should work fine (it’s just slower). On GM45 hardware (with libreboot), make sure that the. kvm. and. kvm_intel. kernel modules are not loaded, when using QEMU. The following errata datasheet from Intel might help with investigation:. http:\/\/download.intel.com\/design\/mobile\/specupdt\/320121.pdf The R400 is almost identical to the X200, code-wise. See. x200.html . TODO: put hardware register logs here like on the. X200 and. T400 page. Binary blob policy Edit this page Who develops Libreboot? 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2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":13897,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078564748853248,"text_t":"Libreboot – ThinkPad X60 Recovery guide. ThinkPad X60 Recovery guide. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Brick type 1: bucts not reset. Brick type 2: bad ROM image Return to index This section documents how to recover from a bad flash that prevents your ThinkPad X60 from booting. ROM images for this machine are well-tested in Libreboot, so bricks are rare. The most common cause of a brick is operator error, when flashing a ROM image. In. most. cases, the cause will be that there is no bootblock, or an invalid one. Brick type 1: bucts not reset. [link] You still have Lenovo BIOS, or you had libreboot running and you flashed another ROM; and you had bucts 1 set and the ROM wasn’t dd’d.* or if Lenovo BIOS was present and libreboot wasn’t flashed. There are. 2. 64KiB bootblocks possible, in the upper part of the ROM image. By default (bucts set to 0), the top one is used. If bucts is set to 1, the lower one (the one before the top one) is used. This bootblock is the first code that executes, during. romstage. as per coreboot hardware initialization. BUC is short for. Backup Control. and TS is short for. Top Swap. . This is a special register on Intel platforms. Lenovo BIOS sets PRx registers, preventing software re-flashing, but there is a bug in the protection, allowing everything. except. the upper 64KiB from being flashed. By default, coreboot only puts a bootblock in the upper region. If you flash such a ROM, while bucts is set to 1, the system won’t boot because there’s not a valid bootblock; this is common if you’re re-flashing when coreboot is already installed, and you didn’t set bucts back to 0. When you install on X60\/T60 the first time, you set this bucts bit to 1, then you re-flash a second time and set it back to 0. In this case, unbricking is easy: reset BUC.TS to 0 by removing that yellow cmos coin (it’s a battery) and putting it back after a minute or two:. *Those dd commands should be applied to all newly compiled X60 ROM images (the ROM images in libreboot binary archives already have this applied!):. dd if=coreboot.rom of=top64k.bin bs=1 skip=$[$(stat -c %s coreboot.rom) - 0x10000] count=64k dd if=coreboot.rom bs=1 skip=$[$(stat -c %s coreboot.rom) - 0x20000] count=64k | hexdump dd if=top64k.bin of=coreboot.rom bs=1 seek=$[$(stat -c %s coreboot.rom) - 0x20000] count=64k conv=notrunc. (doing this makes the ROM suitable for use when flashing a system that still has Lenovo BIOS running, using those instructions:. http:\/\/www.coreboot.org\/Board:lenovo\/x60\/Installation . Brick type 2: bad ROM image. [link] In this instance, you might have flashed a ROM without the top bootblock copied to the lower 64KiB section in the ROM, and you flashed the ROM for the first time (from Lenovo BIOS), in which case there is not a valid bootblock. In this scenario, you compiled a ROM that had an incorrect configuration, or there is an actual bug preventing your system from booting. Or, maybe, you set BUC.TS to 0 and shut down after first flash while Lenovo BIOS was running. In any case, your system is bricked and will not boot at all. “Unbricking” means flashing a known-good (working) ROM. The problem: you can’t boot the system, making this difficult. In this situation, external hardware (see hardware requirements above) is needed which can flash the SPI chip (where libreboot resides). Remove those screws:. Push the keyboard forward (carefully):. Lift the keyboard up and disconnect it from the board:. Grab the right-hand side of the chassis and force it off (gently) and pry up the rest of the chassis:. You should now have this:. Disconnect the wifi antenna cables, the modem cable and the speaker:. Unroute the cables along their path, carefully lifting the tape that holds them in place. Then, disconnect the modem cable (other end) and power connection and unroute all the cables so that they dangle by the monitor hinge on the right-hand side:. Disconnect the monitor from the motherboard, and unroute the grey antenna cable, carefully lifting the tape that holds it into place:. Carefully lift the remaining tape and unroute the left antenna cable so that it is loose:. Remove the screw that is highlighted (do NOT remove the other one; it holds part of the heatsink (other side) into place):. Remove those screws:. Carefully remove the plate, like so:. Remove the SATA connector:. Now remove the motherboard (gently) and cast the lcd\/chassis aside:. Lift back that tape and hold it with something. Highlighted is the SPI flash chip:. Here is another photo, with the numbers of the pins written:. This photo shows an SPI flasher used, with SOIC8 test clip:. Refer to the following guide:. Externally rewrite 25xx NOR flash via SPI protocol NOTE: Do not use the 3.3v rail from your raspberry pi. Leave that disconnected. For 3.3v, plug your charger into the mainboard (but do not power on the mainboard) when the clip is connected. Before removing the clip, disconnect the charger. This will provide adequate 3.3v DC at correct current levels. The SPI flash on an X60 shares a common 3.3V rail with many other components on the mainboard, which all draw a lot of current, more than your programmer can provide. Example RPi command:. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev0.0,spispeed=4096 -w libreboot.rom -V. If flashrom complains about multiple flash chips detected, just pass the. -c. option as it suggests, and pick any of the chips it lists. spispeed=4096. or lower (e.g. . spispeed=512. ) is recommended on this board. The flashing becomes unstable, on this machine, when you use higher speeds. Reverse the steps to re-assemble your system, after you’ve flashed the chip. It should be. Verifying flash... VERIFIED. at the end. If flashrom complains about multiple flash chip definitions detected, then choose one of them following the instructions in the output. Remove the programmer and put it away somewhere. Put back the tape and press firmly over it:. Your empty chassis:. Put the motherboard back in:. Reconnect SATA:. Put the plate back and re-insert those screws:. Re-route that antenna cable around the fan and apply the tape:. Route the cable here and then (not shown, due to error on my part) reconnect the monitor cable to the motherboard and re-insert the screws:. Re-insert that screw:. Route the black antenna cable like so:. Tuck it in neatly like so:. Route the modem cable like so:. Connect modem cable to board and tuck it in neatly like so:. Route the power connection and connect it to the board like so:. Route the antenna and modem cables neatly like so:. Connect the wifi antenna cables. At the start of the tutorial, this system had an Intel wifi chip. Here you see I’ve replaced it with an Atheros AR5B95 (supports 802.11n and can be used without blobs):. Connect the modem cable:. Connect the speaker:. You should now have this:. Re-connect the upper chassis:. Re-connect the keyboard:. Re-insert the screws that you removed earlier:. Power on! Operating system:. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/x60_unbrick.md Site map This HTML page was generated by the. untitled static site generator ."}
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Libreboot 20150208 release. Leah Rowe. 8 February 2015. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Machines supported in this release: Revisions for r20150208 (relative to r20150126) Return to index Article published by: Leah Rowe. Date of publication: 8 February 2015. Machines supported in this release:. [link] Lenovo ThinkPad X60\/X60s. You can also remove the motherboard from an X61\/X61s and replace it with an X60\/X60s motherboard. An X60 Tablet motherboard will also fit inside an X60\/X60s. Lenovo ThinkPad X60 Tablet. (1024x768 and 1400x1050) with digitizer support. See. hardware\/#supported_x60t_list. for list of supported LCD panels. It is unknown whether an X61 Tablet can have it’s mainboard replaced with an X60 Tablet motherboard. Lenovo ThinkPad T60. (Intel GPU) (there are issuesinstall\/x200_external.html; see below):. See notes below for exceptions, and. hardware\/#supported_t60_list. for known working LCD panels. It is unknown whether a T61 can have it’s mainboard replaced with a T60 motherboard. See. future\/#t60_cpu_microcode. . T60p (and T60 laptops with ATI GPU) will likely never be supported:. hardware\/#t60_ati_intel. Lenovo ThinkPad X200. X200S and X200 Tablet are also supported, conditionally; see. hardware\/x200.html#x200s. ME\/AMT. : libreboot removes this, permanently. hardware\/gm45_remove_me.html. Lenovo ThinkPad R400. (r20150208 and later, only). ME\/AMT. : libreboot removes this, permanently. hardware\/gm45_remove_me.html. Apple MacBook1,1. (MA255LL\/A, MA254LL\/A, MA472LL\/A). See. hardware\/#macbook11. . Apple MacBook2,1. (MA699LL\/A, MA701LL\/A, MB061LL\/A, MA700LL\/A, MB063LL\/A, MB062LL\/A). See. hardware\/#macbook21. . Revisions for r20150208 (relative to r20150126). [link] This is a maintenance release (polishing) based on r20150126. Users who installed r20150126 don’t really need to update to this release. buildrom-withgrub: use gnulove.jpg background on 16:10 laptops (MacBook2,1 and X200). build-release: include grub-background script in libreboot_bin. grub-background (new): lets user change GRUB background image. grub-assemble: Add link to original utility. buildrom-withgrub: Put background.jpg in CBFS, not GRUB memdisk. grub-assemble: merge scripts into a single script gen.sh. Documentation: implement theme, drastically improve readability. docs\/hardware\/: update list of compatible T60 LCD panels. docs\/: more clarification of libreboot’s stated purpose. build-release: include the commitid file in the release archives. docs\/: Further emphasize the GNU+Linux requirement. lenovobios_firstflash: fix BASH errors. lenovobios_secondflash: fix BASH errors. docs\/install\/x200_external.html: Tell user to switch MAC address. docs\/git\/: Add to the list of x86_64 compatible hosts. docs\/install\/: Remove old (obsolete) information. docs\/git\/: Say that the build dependencies are for src (and not nedeed for libreboot_bin). build: re-factor the descriptor\/gbe generating loop for GM45\/ICH9M. X60, X60S and X60 Tablet now the same ROM images. Add QEMU (q35\/ich9) support to libreboot. Add QEMU (i440fx\/piix4) support to libreboot. docs\/: Re-write the description of what libreboot is. docs\/release.html: Add notes about how to use GPG. build-release: delete the commitid file from release archives. build-release: create file named commitid after build-release. Binary blob policy Edit this page Who develops Libreboot? 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Libreboot 20140911 release. Leah Rowe. 11 September 2014. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. 6th release (pre-release, 7th beta) Machines still supported (compared to previous release): New systems supported in this release: Machines no longer supported (compared to previous release): Revisions for r20140911 (7th beta) (11th September 2014) Return to index Article published by: Leah Rowe. Date of publication: 11 September 2014. 6th release (pre-release, 7th beta). [link] Released 11th July 2014 (pre-release) 1st beta. Revised (pre-release, 2nd beta) 16th July 2014. Revised (pre-release, 3rd beta) 20th July 2014. Revised (pre-release, 4th beta) 29th July 2014. Revised (pre-release, 5th beta) 11th August 2014 (corrected 11th August 2014). Revised (pre-release, 6th beta) 3rd September 2014. Revised (pre-release, 7th beta) 11th September 2014. Machines still supported (compared to previous release):. [link] Lenovo ThinkPad X60\/X60s. You can also remove the motherboard from an X61\/X61s and replace it with an X60\/X60s motherboard. New systems supported in this release:. [link] Lenovo ThinkPad X60 Tablet. (1024x768 and 1400x1050) with digitizer support. See. hardware\/#supported_x60t_list. for list of supported LCD panels. It is unknown whether an X61 Tablet can have its mainboard replaced with an X60 Tablet motherboard. Lenovo ThinkPad T60. (Intel GPU) (there are issues; see below). See notes below for exceptions, and. hardware\/#supported_t60_list. for known working LCD panels. It is unknown whether a T61 can have its mainboard replaced with a T60 motherboard. T60p (and T60 variants with ATI GPU) will likely never be supported:. hardware\/#t60_ati_intel. Apple MacBook1,1. (MA255LL\/A, MA254LL\/A, MA472LL\/A). See. hardware\/#macbook11. . Apple MacBook2,1. (MA699LL\/A, MA701LL\/A, MB061LL\/A, MA700LL\/A, MB063LL\/A, MB062LL\/A). See. hardware\/#macbook21. . Machines no longer supported (compared to previous release):. [link] All previous systems still supported! Revisions for r20140911 (7th beta) (11th September 2014). [link] The changes below were made in a git repository, unlike in previous releases. Descriptions below are copied from ‘git log’. Update .gitignore for new dependencies. Use a submodule for i945-pwm. Don’t clean packages that fail or don’t need cleaning. Don’t clean i945-pwm, it’s not needed. Regression fix: Parabola live ISO boot issues. Re-enable background images in ISOLINUX\/SYSLINUX GRUB parser menus. Regression fix: Re-add CD-ROM (ata0) in GRUB. Documentation: add notes about performance penalty when using ecryptfs. Documentation: Fixed spelling and grammatical errors. Documentation: macbook21: add new system as tested. Documentation: macbook21: add info about improving touchpad sensitivity. Documentation: X60 Tablet: add more information about finger input. Documentation: release.html: Add information about recently merged commit in coreboot. Binary blob policy Edit this page Who develops Libreboot? 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{"id":"tma5o_Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/news\/libreboot20140622.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["news"],"url_file_name_s":"libreboot20140622","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Libreboot 20140622 release"],"author":"Leah Rowe","author_sxt":["Leah Rowe"],"last_modified":"Sat Apr 02 12:48:46 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Sun Jun 22 00:00:00 UTC 2014"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":8482164125724832774,"exact_signature_unique_b":true,"fuzzy_signature_l":-4782922516565809872,"fuzzy_signature_unique_b":true,"h1_txt":["Libreboot 20140622 release","Navigate this page:","Release 20140622 (5th release)"],"h2_txt":["Officially supported","Revision (22nd June 2014 - extra)","Revision (22nd June 2014 - extra)","Revision notes (22nd June 2014)"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":427,"text_t":"Libreboot – Libreboot 20140622 release. Libreboot 20140622 release. Leah Rowe. 22 June 2014. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Release 20140622 (5th release) Officially supported Revision (22nd June 2014 - extra) Revision (22nd June 2014 - extra) Revision notes (22nd June 2014) Return to index Article published by: Leah Rowe. Date of publication: 22 June 2014. Release 20140622 (5th release). [link] 7th March 2014. revised 22nd June 2014. Officially supported. [link] ThinkPad X60. ThinkPad X60s. Revision (22nd June 2014 - extra). [link] Documentation: added X60 Unbricking tutorial. Documentation: added info about enabling or disabling wifi. Documentation: added info about enabling or disabling trackpoint. Revision (22nd June 2014 - extra). [link] Documentation: Improved the instructions for using flashrom. Documentation: Improved the instructions for using cbfstool (to change the default GRUB menu). Documentation: Numerous small fixes. Revision notes (22nd June 2014). [link] updated GRUB (git 4b8b9135f1676924a8458da528d264bbc7bbb301, 20th April 2014). Made “DeJavu Sans Mono” the default font in GRUB (fixes border corruption). re-added background image in GRUB (meditating GNU). added 6 more images:. coreboot_ukqwerty.rom (UK Qwerty keyboard layout in GRUB). coreboot_serial_ukqwerty.rom (UK Qwerty keyboard layout in GRUB). coreboot_dvorak.rom (US Dvorak keyboard layout in GRUB). coreboot_serial_dvorak.rom (US Dvorak keyboard layout in GRUB). coreboot_ukdvorak.rom (UK Dvorak keyboard layout in GRUB). coreboot_serial_ukdvorak.rom (UK Dvorak keyboard layout in GRUB). 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{"id":"35fzJ_Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/news\/libreboot20140729.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["news"],"url_file_name_s":"libreboot20140729","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Libreboot 20140729 release"],"author":"Leah Rowe","author_sxt":["Leah Rowe"],"last_modified":"Sat Apr 02 12:48:47 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Jul 29 00:00:00 UTC 2014"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":202481354506572699,"exact_signature_unique_b":true,"fuzzy_signature_l":7421228210315736025,"fuzzy_signature_unique_b":true,"h1_txt":["Libreboot 20140729 release","Navigate this page:"],"h2_txt":["Revisions for r20140729 (4th beta) (29th July 2014)"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":414,"text_t":"Libreboot – Libreboot 20140729 release. Libreboot 20140729 release. Leah Rowe. 29 July 2014. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Revisions for r20140729 (4th beta) (29th July 2014) Return to index Article published by: Leah Rowe. Date of publication: 29 July 2014. Revisions for r20140729 (4th beta) (29th July 2014). [link] Documentation: improved (more explanations, background info) in docs\/security\/x60_security.html (courtesy of Denis Carikli). MacBook2,1 tested (confirmed). macbook21: Added script ‘macbook21_firstflash’ for flashing libreboot while Apple EFI firmware is running. Documentation: macbook21: added software-based flashing instructions for flashing libreboot while Apple EFI firmware is running. Reduced size of libreboot_src.tar.gz:. Removed .git and .gitignore from grub directory (libreboot_src); not needed. Removing them reduces the size of the archive (by a lot). GRUB development should be upstream. Removed .git and .gitignore from bucts directory (libreboot_src); not needed. Removing them reduces the size of the archive. bucts development should be upstream. Removed .svn from flashrom directory (libreboot_src); not needed. Removing it reduces the size of the archive. flashrom development should be upstream. Added ROMs with Qwerty (Italian) layout in GRUB (libreboot*itqwerty.rom). Added resources\/utilities\/i945gpu\/intel-regs.py for debugging issues related to LCD panel compatibility on X60 Tablet and T60. (courtesy of. Michał Masłowski ). Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/news\/libreboot20140729.md Site map This HTML page was generated by the. untitled static site generator .","wordcount_i":239,"linkscount_i":22,"linksnofollowcount_i":0,"inboundlinkscount_i":17,"outboundlinkscount_i":5,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/news\/libreboot20140729.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/news\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/news\/libreboot20140729.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Revisions for r20140729 (4th beta) (29th July 2014)","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","004-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","mtjm.eu\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","Michał Masłowski","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:05 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:14 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":5325,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078565782749184}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Foundation Our Organization Foundation team. Foundation team. A team dedicated to seeking private funding to contribute to the development and promotion of Sorbonne University. Publié le 30\/11\/2019 - Mis à jour le 17\/11\/2021. Foundation team. Mélina Mercier, Executive Director of the Foundation. Mélina Mercier has a background in neurosciences from the Faculty of Science and Engineering at Sorbonne University. She worked for 5 years for two private foundations in England, before joining the Foundation in 2009. 01 44 27 80 33 Contact by email. Anne-Lydie Ardiet, Head of Corporate Sponsorship and Foundations. After creating and developing a communications department in the subsidiary of a major industrial group for almost 7 years, Anne-Lydie joined CELSA, her former school, as head of communications and then corporate relations. In September 2019, she joined the Sorbonne University Foundation as Head of Sponsorship for Foundations and Companies. . 01 44 27 26 11 Contact by email. Audrey Cotrez, Administrative and Financial Manager. Audrey worked as a project manager in the Public Contracts Office of the Directorate General of Research and Technology Transfer at Sorbonne University for 4 years before joining the Foundation in April 2015. 01 44 27 68 12 Contact by email. Marine Basset, communication and events manager. Marine Basset worked for 12 years in higher education as a communications manager, notably in the sponsorship sector at Sciences Po Paris and the Bordeaux University Foundation. She joined the foundation in May 2016. . 01 44 27 50 74 Contact by email. Charlotte Guiot, prospect researcher. Charlotte Guiot holds a Masters degree in International Relations. She joined the Sorbonne University Foundation in May 2016. . 01 44 27 61 40 Contact by email. Thierry Pfistner, project manager. Thierry has worked for 20 years in the private sector in the commercial field and in the management of profit centres. . In 2006, he joined the Institut de la Vision as head of employment contract oversight, participated in the oversight of research projects and was then appointed head of general services. . In 2017, he joined Polytech Sorbonne as administrative manager before joining the Foundation as project manager. . 01 44 27 61 78 Contact by email. Véronique Lévy, Executive Assistant. Véronique Lévy began working in the private sector in very different fields (commercial, legal, medical, business creation, wealth management) for 15 years. In 2000 she joined the Versailles Rectorate and since 2005 has been working in higher education in various positions. She joined the Foundation in April 2019. 01 44 27 21 58 Contact by email. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Inter-Age University","SHOP","","","Faculty of medicine","Editorial","Management","Regulatory acts","Code of ethics","Divisions","Senior management","Health Courses Division","Human Resources Department","Procurement and Finance Division","Research and Development Division","Rental of conference facilites","Courses","Medical studies","Degree and inter-university degree catalogue","PASS and L.AS","Second cycle [years 2-6 of medical studies]","Third cycle [year 7 onwards of medical studies]","Midwifery studies","Paramedical studies","Speech and language therapy","Orthoptics","Psychomotricity","Research","Research units","MU 75 - Paris Brain Institute","MU 80 - Institut de la Vision","MRSU 933 - Laboratory of childhood genetic diseases","MRSU 938 – Saint-Antoine Research Center","MRSU 959- Immunology-Immunopathology-Immunotherapy","MRSU 974 – Center of Research in Myology","MRSU 1135 - Center for Immunology and Microbial Infections","MRSU 1135 - Centre for Immunology and Infectious 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Culture and outreach University for All Ages The Inter-Age University. The Inter-Age University. A lecture series, in French, open to the general public. Publié le 17\/01\/2020 - Mis à jour le 25\/10\/2021. The Inter-Age University. Sorbonne University offers a series of lectures that address different themes and disciplinary fields, providing the general public with the opportunity to cultivate their knowledge and understand current issues. These lectures are in French, and are open to all without any conditions pertaining to age or educational level. The lecture series does not, however, lead to the awarding of any diploma or certificate, and does not offer preparation for any examination. Useful links. The Inter-Age University. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Student Entrepreneurial Training. The Pépite Student Innovation Center The FabLabs Support for Student Entrepreneurs Your student entrepreneurship contacts. Pépite Sorbonne University. Contact by email. Pépite Faculty of Arts. Contact by email. Pépite Faculty of Science and Engineering. Contact by email. Pépite National Muséum of Natural History. Contact by email. Download the Pépite brochure. Plaquette Pépite 2021 - pdf - 864.92 KB. A lire aussi. Entrepreneuriat Programme d’accompagnement et d’accélération des étudiants entrepreneurs Entrepreneuriat Prix Pépite 2021 - Tremplin pour l’entrepreneuriat étudiant PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. 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{"id":"SyaYu2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/mac_address.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":52,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"mac_address","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Changing the MAC address"],"last_modified":"Sat Apr 02 12:48:37 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Jan 16 00:00:00 UTC 2018"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-4049390981086298549,"exact_signature_unique_b":true,"fuzzy_signature_l":9064994094518930414,"fuzzy_signature_unique_b":true,"h1_txt":["Changing the MAC address","Navigate this page:","Introduction (GM45+e1000)","Obtaining the existing MAC address","Changing the MAC address in the operating system","Changing the MAC address on X200\/T400\/T500\/W500"],"bold_txt":["ethXXX","enpXXX","Buy preinstalled","Donate"],"imagescount_i":2,"images_protocol_sxt":["000-https","001-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico","av.libreboot.org\/t400\/macaddress1.jpg"],"images_alt_sxt":["Libreboot logo",""],"images_height_val":[-1,-1],"images_width_val":[-1,-1],"images_text_t":"libreboot org favicon ico Libreboot logo av t400 macaddress1 jpg t 400 macaddress 1","responsetime_i":407,"text_t":"Libreboot – Changing the MAC address. Changing the MAC address. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction (GM45+e1000) Obtaining the existing MAC address Changing the MAC address in the operating system Changing the MAC address on X200\/T400\/T500\/W500 Return to index Introduction (GM45+e1000). [link] This section is applicable to all Libreboot-supported laptops with the mobile 4 series chipset (as shown in. $ lspci. ) that use the e1000 ethernet controller (e.g. T400, X200). The R500 is an exception to this as it does not use the built-in e1000. On all these laptops, the. MAC address for the built-in gigabit ethernet controller is stored inside the flash chip, along with Libreboot and other configuration data. Therefore, installing Libreboot will overwrite it. Thus, for these laptops, prebuilt Libreboot already contains a generic MAC address in the configuration section. This address is. 00:f5:f0:40:71:fe in builds before 2018-01-16 and. 00:4c:69:62:72:65` (see the ascii character set) afterwards. Unless you change it, your computer will boot and use it. This can lead to network problems if you have more than one Libreboot computer on the same layer2 network (e.g. on the same network switch). The switch (postman) will simply not know who to deliver to as the MAC (house) addresses will be the same. To prevent these address clashes, you can either modify prebuilt Libreboot to use an address of your own choosing or you can change the address in your operating system’s boot scripts. In either case, it is a good idea to write down the address that your computer originally had. Obtaining the existing MAC address. [link] The existing MAC address may be obtained by the following methods:. Run. ip link. or. ifconfig. in a terminal\/console\/shell; find your ethernet device (e.g.,. enpXXX. or. ethXXX. ), and look for a set of 12 colon-delimited. hexadecimal digits . For example:. 00:f3:f0:45:91:fe. . `$ ip link. … link\/ether ??:??:??:??:??:?? brd …. Alternatively:. ifconfig. … ether ??:??:??:??:??:?? txqueuelen …. Otherwise you can read the white label that is often found on the motherboard under the memory sticks:. The MAC address is usually listed on the laptop chassis as well. This one will be incorrect if the motherboard was changed and the stickers were not updated. Changing the MAC address in the operating system. [link] There are three portable ways of doing so:. Using the new iproute2 package:. ip link set. down. ip link set dev. address 00:4c:69:62:72:65. ip link set. up. Using the old. ifconfig. command:. ifconfig. hw ether 00:4c:69:62:72:65. Using the macchanger package. You can use use of these three methods in your operating system’s init scripts or you can use your operating system’s own networking configuration. Refer to your operating system’s documentation for how to do this. Changing the MAC address on X200\/T400\/T500\/W500. [link] On GM45 laptops with ICH9M southbridge and Intel PHY module, the MAC address is hardcoded in boot flash, which means it can be changed if you re-flash. See. ich9utils documentation Binary blob policy Edit this page Who develops Libreboot? 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Flashing the T500 with a BeagleBone Black. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Libreboot T400 Serial port A note about CPUs Quad-core CPUs A note about GPUs CPU paste required Flash chip size MAC address Initial BBB configuration The procedure Thermal paste (IMPORTANT) Wifi WWAN Memory Boot it! Return to index Initial flashing instructions for T500. This guide is for those who want libreboot on their ThinkPad T500 while they still have the original Lenovo BIOS present. This guide can also be followed (adapted) if you brick your T500, to know how to recover. W500 is also mostly compatible with this guide. Libreboot T400. [link] You may also be interested in the smaller, more portable. Libreboot T400 . Serial port. [link] EHCI debug might not be needed. It has been reported that the docking station for this laptop has a serial port, so it might be possible to use that instead. A note about CPUs. [link] ThinkWiki has a list of CPUs for this system. The Core 2 Duo P8400, P8600 and P8700 are believed to work in libreboot. The T9600 was also tested on the T400 and confirmed working. T9550 and T9900 was tested by a user, and is compatible as reported in the IRC channel. T9500 and T9400 may also work, but YMMV. Quad-core CPUs. [link] Very likely to be compatible, but requires hardware modification. Based on info from German forum post about installing Core Quad CPU on T500 found in coreboot mailing list. Currently work in progress and no guide available. Q9100 is compatible and confirmed working (after hw mod), as reported by users in the IRC channel. Coreboot mailing list post German forum post about install Core Quad on T500 This video from FrostKiwi (Libera IRC) does a nice illustration and explains everything nicely:. https:\/\/vid.puffyan.us\/watch?v=Fs4GjDiOie8 A note about GPUs. [link] Some models have an Intel GPU, while others have both an ATI and an Intel GPU; this is referred to as “switchable graphics”. In the. BIOS setup. program for lenovobios, you can specify that the system will use one or the other (but not both). Libreboot is known to work on systems with only the Intel GPU, using native graphics initialization. On systems with switchable graphics, the Intel GPU is used and the ATI GPU is disabled, so native graphics initialization works all the same. CPU paste required. [link] See. #paste . Flash chip size. [link] Use this to find out:. flashrom -p internal. MAC address. [link] Refer to. mac_address.html . Initial BBB configuration. [link] Refer to. spi.html as a guide for external re-flashing. The following shows how to connect clip to the BBB (on the P9 header), for SOIC-16 (clip: Pomona 5252):. POMONA 5252 (correlate with the BBB guide) === ethernet jack and VGA port ==== NC - - 21 1 - - 17 NC - - NC NC - - NC NC - - NC NC - - NC 18 - - 3.3V (PSU) 22 - - NC - this is pin 1 on the flash chip === SATA port === This is how you will connect. Numbers refer to pin numbers on the BBB, on the plugs near the DC jack. The following shows how to connect clip to the BBB (on the P9 header), for SOIC-8 (clip: Pomona 5250):. POMONA 5250 (correlate with the BBB guide) === RAM slots ==== 18 - - 1 22 - - NC NC - - 21 3.3V (PSU) - - 17 - this is pin 1 on the flash chip === slot where the AC jack is connected === This is how you will connect. Numbers refer to pin numbers on the BBB, on the plugs near the DC jack. The procedure. [link] Remove all screws:. It is also advisable to, throughout the disassembly, place any screws and\/or components that you removed in the same layout or arrangement. The follow photos demonstrate this:. Remove the HDD\/SSD and optical drive:. Remove the palm rest:. Remove the keyboard and rear bezel:. If you have a WWAN\/3G card and\/or sim card reader, remove them permanently. The WWAN-3G card has proprietary firmware inside; the technology is identical to what is used in mobile phones, so it can also track your movements:. Remove this frame, and then remove the wifi chip:. Remove the speakers:. Remove the NVRAM battery (already removed in this photo):. When you re-assemble, you will be replacing the wifi chip with another. These two screws don’t hold anything together, but they are included in your system because the screw holes for half-height cards are a different size, so use these if you will be installing a half-height card:. Unroute the antenna wires:. Disconnect the LCD cable from the motherboard:. Remove the LCD assembly hinge screws, and then remove the LCD assembly:. Remove the fan and heatsink:. Remove this screw:. Remove these cables, keeping note of how and in what arrangement they are connected:. Disconnect the power jack:. Remove the motherboard and cage from the base (the marked hole is where those cables were routed through):. Remove all screws, arranging them in the same layout when placing the screws on a surface and marking each screw hole (this is to reduce the possibility of putting them back in the wrong holes):. Also remove this:. Separate the motherboard from the cage:. The flash chip is next to the memory slots. On this system, it was a SOIC-8 (4MiB or 32Mb) flash chip:. Connect your programmer, then connect GND and 3.3V. A dedicated 3.3V PSU was used to create this guide, but at ATX PSU is also fine:. Of course, make sure to turn on your PSU:. Now, you should be ready to install libreboot. Flashrom binaries for ARM (tested on a BBB) are distributed in libreboot_util. Alternatively, libreboot also distributes flashrom source code which can be built. Log in as root on your BBB, using the instructions in. bbb_setup.html#bbb_access . Test that flashrom works:. .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512. In this case, the output was:. flashrom v0.9.7-r1854 on Linux 3.8.13-bone47 (armv7l) flashrom is free software, get the source code at http:\/\/www.flashrom.org Calibrating delay loop... OK. Found Macronix flash chip \"MX25L6405(D)\" (8192 kB, SPI) on linux_spi. Found Macronix flash chip \"MX25L6406E\/MX25L6436E\" (8192 kB, SPI) on linux_spi. Found Macronix flash chip \"MX25L6445E\/MX25L6473E\" (8192 kB, SPI) on linux_spi. Multiple flash chip definitions match the detected chip(s): \"MX25L6405(D)\", \"MX25L6406E\/MX25L6436E\", \"MX25L6445E\/MX25L6473E\" Please specify which chip definition to use with the -c option. How to backup factory.rom (change the -c option as neeed, for your flash chip):. .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512 -r factory.rom .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512 -r factory1.rom .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512 -r factory2.rom. Note: the. -c. option is not required in libreboot’s patched flashrom, because the redundant flash chip definitions in. flashchips.c. have been removed. Now compare the 3 images:. sha512sum factory\\*.rom. If the hashes match, then just copy one of them (the factory.rom) to a safe place (on a drive connected to another system, not the BBB). This is useful for reverse engineering work, if there is a desirable behaviour in the original firmware that could be replicated in coreboot and libreboot. While there is a default MAC address inside the gbe region of flash image, it is not one you want to use. Make sure to change the MAC address to the one that is correct for your system, for. later internal flash. , but always remember to. flash unmodfied txtmode image first. as it is known to work and only this variant provides memtest. You can follow instructions at. ich9utils.html#ich9gen to change the MAC address inside the libreboot image. Now flash it:. .\/flashrom -p linux_spi:dev=\/dev\/spidev1.0,spispeed=512 -w path\/to\/libreboot\/rom\/image.rom -V. You might see errors, but if it says. Verifying flash... VERIFIED. at the end, then it’s flashed and should boot. If you see errors, try again (and again, and again); the message. Chip content is identical to the requested image. is also an indication of a successful installation. Example output from running the command (see above):. flashrom v0.9.7-r1854 on Linux 3.8.13-bone47 (armv7l) flashrom is free software, get the source code at http:\/\/www.flashrom.org Calibrating delay loop... OK. Found Macronix flash chip \"MX25L6405(D)\" (8192 kB, SPI) on linux_spi. Reading old flash chip contents... done. Erasing and writing flash chip... FAILED at 0x00001000! Expected=0xff, Found=0x00, failed byte count from 0x00000000-0x0000ffff: 0xd716 ERASE FAILED! Reading current flash chip contents... done. Looking for another erase function. Erase\/write done. Verifying flash... VERIFIED. Thermal paste (IMPORTANT). [link] Because part of this procedure involved removing the heatsink, you will need to apply new paste. Arctic MX-4 is ok. You will also need isopropyl alcohol and an anti-static cloth to clean with. When re-installing the heatsink, you must first clean off all old paste with the alcohol\/cloth. Then apply new paste. Arctic MX-4 is also much better than the default paste used on these systems. NOTE: the photo above is for illustration purposes only, and does not show how to properly apply the thermal paste. Other guides online detail the proper application procedure. Wifi. [link] The T500 typically comes with an Intel wifi chipset, which does not work without proprietary software. For a list of wifi chipsets that work without proprietary software, see. ..\/hardware\/#recommended_wifi . Some T500 laptops might come with an Atheros chipset, but this is 802.11g only. It is recommended that you install a new wifi chipset. This can only be done after installing libreboot, because the original firmware has a whitelist of approved chips, and it will refuse to boot if you use an ‘unauthorized’ wifi card. The following photos show an Atheros AR5B95 being installed, to replace the Intel chip that this T500 came with:. WWAN. [link] If you have a WWAN\/3G card and\/or sim card reader, remove them permanently. The WWAN-3G card has DMA, and proprietary firmware inside; the technology is identical to what is used in mobile phones, so it can also track your movements. Not to be confused with wifi (wifi is fine). Memory. [link] In DDR3 machines with Cantiga (GM45\/GS45\/PM45), northbridge requires sticks that will work as PC3-8500 (faster PC3\/PC3L sticks can work as PC3-8500). Non-matching pairs may not work. Single module (meaning, one of the slots will be empty) will currently only work in slot 0. NOTE: according to users reports, non matching pairs (e.g. 1+2 GiB) might work in some cases. Make sure that the RAM you buy is the 2Rx8 configuration when buying 4GiB sticks (In other words: maximum of 2GiB per rank, 2 ranks per card). This page might be useful for RAM compatibility info (note: coreboot raminit is different, so this page might be BS). The following photo shows 8GiB (2x4GiB) of RAM installed:. Boot it! [link] You should see something like this:. Now. install GNU+Linux . Binary blob policy Edit this page Who develops Libreboot? 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GRUB payload. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. GRUB keyboard layouts Custom keyboard layout Return to previous index TODO: this guide should be reviewed and updated. Some info might be out of date. GNU GRUB already has excellent documentation, but there are aspects of Libreboot that deserve special treatment. Libreboot provides the option to boot GNU GRUB directly, running on bare metal (instead of using BIOS or UEFI services). The GNU+Linux section also has Libreboot-specific guides for dealing with GNU+Linux distributions when using GNU GRUB directly, in this setup. A similar section exists for BSD operating systems GRUB keyboard layouts. [link] It is possible to use. any. keymap in GNU GRUB. Custom keyboard layout. [link] Keymaps are stored in. resources\/grub\/keymap\/. You can use the. ckbcomp. program to generate a keymap, based on Xorg keymap files:. ckbcomp fr frazerty. When you build GRUB from source, you can use the. grub-mklayout. program to create a special keymap file for GRUB. Learn how to build GRUB When you’ve built GRUB, using. lbmk. (libreboot build system), take your kepmap file (generated by ckbcomp) and run it through. grub-mklayout. like so:. cat frazerty | .\/grub\/grub-mklayout -o frazerty.gkb. Place the newly created. .gkb. file under. resources\/grub\/keymap. in lbmk. When you build Libreboot, a ROM image with GRUB payload and your newly created keymap will be available under the. bin\/. directory. Learn how to build Libreboot ROM images Many keymaps exist in the Libreboot build system, but sometimes you must manually tweak the file created by. ckbcomp. , adjusting the scan codes in that file, before converting to a GRUB keymap file. Therefore, it would be unwise to automatically add all keymaps in GRUB. If you’ve added a keymap to lbmk, and it works,. please submit a patch! Binary blob policy Edit this page Who develops Libreboot? 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2022","fresh_date_dt":"Tue Jul 12 03:52:05 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":14909,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078557021896704,"text_t":"Libreboot – Flashing the ThinkPad T400 externally. Flashing the ThinkPad T400 externally. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Serial port A note about CPUs Quad-core CPUs A note about GPUs CPU paste required Flash chip size MAC address How to flash externally The procedure Thermal paste (IMPORTANT) Memory Boot it! Return to index Initial flashing instructions for T400. This guide is for those who want libreboot on their ThinkPad T400 while they still have the original Lenovo BIOS present. This guide can also be followed (adapted) if you brick your T400, to know how to recover. An. “HMM” (Hardware Maintenance Manual) detailing the process of [dis]assembly is available for this model. Be careful when reassembling the laptop as the screws on page 114 (with title “1130 Keyboard bezel”) are swapped and if you follow the HMM you will punch a hole through the bezel in the upper right corner. Serial port. [link] EHCI debug might not be needed. It has been reported that the docking station for this laptop has a serial port, so it might be possible to use that instead. A note about CPUs. [link] ThinkWiki has a list of CPUs for this system. The Core 2 Duo P8400, P8600 and P8700 are believed to work in libreboot. T9600, T9500, T9550 and T9900 are all compatible, as reported by users. Quad-core CPUs. [link] Very likely to be compatible, but requires hardware modification. Based on info from German forum post about installing Core Quad CPU on T500 found in coreboot mailing list. Currently work in progress and no guide available. Coreboot mailing list post German forum post about install Core Quad on T500 A note about GPUs. [link] Some models have an Intel GPU, while others have both an ATI and an Intel GPU; this is referred to as “switchable graphics”. In the. BIOS setup. program for lenovobios, you can specify that the system will use one or the other (but not both). Libreboot is known to work on systems with only the Intel GPU, using native graphics initialization. On systems with switchable graphics, the Intel GPU is used and the ATI GPU is disabled, so native graphics initialization works all the same. CPU paste required. [link] See. #paste . Flash chip size. [link] Use this to find out:. flashrom -p internal. MAC address. [link] Refer to. mac_address.html . How to flash externally. [link] Refer to. spi.html as a guide for external re-flashing. The procedure. [link] Remove. all. screws, placing them in the order that you removed them:. Remove those three screws then remove the rear bezel:. Remove the speakers:. Remove the wifi:. Remove this cable:. Unroute those antenna wires:. Remove the LCD assembly:. Disconnect the NVRAM battery:. Disconnect the fan:. Unscrew these:. Unscrew the heatsink, then lift it off:. Disconnect the power jack:. Loosen this:. Remove this:. Unscrew these:. Remove this:. Unscrew this:. Remove the motherboard (the cage is still attached) from the right hand side, then lift it out:. Remove these screws, placing the screws in the same layout and marking each screw hole (so that you know what ones to put the screws back into later):. Separate the motherboard:. Connect your programmer, then connect GND and 3.3V. A dedicated 3.3V PSU was used to create this guide, but at ATX PSU is also fine:. Of course, make sure to turn on your PSU:. Now, you should be ready to install libreboot. Refer to the external flashing instructions. here , and when you’re done, re-assemble your laptop. Thermal paste (IMPORTANT). [link] Because part of this procedure involved removing the heatsink, you will need to apply new paste. Arctic MX-4 is ok. You will also need isopropyl alcohol and an anti-static cloth to clean with. When re-installing the heatsink, you must first clean off all old paste with the alcohol\/cloth. Then apply new paste. Arctic MX-4 is also much better than the default paste used on these systems. NOTE: the photo above is for illustration purposes only, and does not show how to properly apply the thermal paste. Other guides online detail the proper application procedure. Memory. [link] In DDR3 machines with Cantiga (GM45\/GS45\/PM45), northbridge requires sticks that will work as PC3-8500 (faster PC3\/PC3L sticks can work as PC3-8500). Non-matching pairs may not work. Single module (meaning, one of the slots will be empty) will currently only work in slot 0. NOTE: according to users reports, non matching pairs (e.g. 1+2 GiB) might work in some cases. Make sure that the RAM you buy is the 2Rx8 configuration when buying 4GiB sticks (In other words: maximum of 2GiB per rank, 2 ranks per card). This page might be useful for RAM compatibility info (note: coreboot raminit is different, so this page might be BS). The following photo shows 8GiB (2x4GiB) of RAM installed:. Boot it! [link] You should see something like this:. Now. install GNU+Linux . Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/t400_external.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"ZBgQ-2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/d945gclf.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"d945gclf","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Intel D945GCLF desktop board"],"last_modified":"Sat Apr 02 12:48:38 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":1190415547909166215,"exact_signature_unique_b":true,"fuzzy_signature_l":598665201054717739,"fuzzy_signature_unique_b":true,"h1_txt":["Intel D945GCLF desktop board","Navigate this page:","Introduction"],"h2_txt":["Remarks about vendor bios:","How to replace thermal paste and fan"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":13,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico","av.libreboot.org\/d945gclf\/d945gclf.jpg","av.libreboot.org\/d945gclf\/20160923_141521.jpg","av.libreboot.org\/d945gclf\/20160923_141550.jpg","av.libreboot.org\/d945gclf\/20160923_141620.jpg","av.libreboot.org\/d945gclf\/20160923_141614.jpg","av.libreboot.org\/d945gclf\/20160923_142618.jpg","av.libreboot.org\/d945gclf\/20160923_141738.jpg","av.libreboot.org\/d945gclf\/20160923_141814.jpg","av.libreboot.org\/d945gclf\/20160923_141601.jpg","av.libreboot.org\/d945gclf\/20160923_142031.jpg","av.libreboot.org\/d945gclf\/20160923_141659.jpg","av.libreboot.org\/d945gclf\/20160923_141635.jpg"],"images_alt_sxt":["Libreboot logo","","","","","","","","","","","",""],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"libreboot org favicon ico Libreboot logo av d945gclf jpg d 945gclf 20160923 141521 141550 141620 141614 142618 141738 141814 141601 142031 141659 141635","responsetime_i":372,"text_t":"Libreboot – Intel D945GCLF desktop board. Intel D945GCLF desktop board. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction Remarks about vendor bios: How to replace thermal paste and fan Return to index If you just want flashing instructions, go to. ..\/install\/d945gclf.html D945GCLF2D also reported working by a user. Introduction. [link] This board is a mini-itx desktop board for 2008. It uses an atom 230, which is a singe core CPU but it is hyperthreaded so it appears to have 2 thread to the OS. The flash chip is very small, 512KiB, so grub2 does not fit, which is why libreboot has to use seabios on this target. Full disk encryption like on other supported targets will not be possible, so plan accordingly. This board has a 945gc chipset which is the desktop equivalent of 945gm which can be found in the Lenovo x60\/t60 or macbook2,1. This chipset features an ICH7 southbridge. It has 1 DIMM slot that can accommodate up to 2G of DDR2 RAM. Connectivity-wise it has 1 PCI slot, a 10\/100 ethernet port, 4 usb slot and 4 usb ports, with one internal header and 2 SATA ports. The D945GCLF2 is an upgraded version of this board. The differences are: 1 more USB header, 10\/100\/1000 ethernet and a dual core cpu (also hyperthreaded). Since the board is almost identical (and coreboot code seem to indicate that it works, since MAX_CPU=4 is set), it is believed that it should also work but this is untested. Remarks about vendor bios:. [link] Without coreboot\/libreboot this board is utery useless, since the vendor bios is very bad. It cannot boot from any HDD whether it is connected to the SATA port or USB. With libreboot it works just fine. The vendor bios write protects the flash so it requires external flashing to install libreboot on this device. Once libreboot is flashed there is no problem to update the firmware internally. Here is an image of the board:. Here is an image of the D945GCLF2 board:. And SPI SOIC8 flash chip. How to replace thermal paste and fan. [link] This board comes with very crappy disposable loud fan, that one has no bearings, which can not be repaired or oiled properly, do not waste your time trying to fix it, just buy one chinese same size fan. Make sure that new one has same wiring. This is a new one, with bearing and maintenable. Now remove the both coolers rotating them a bit, slowly, then clean both silicons and both coolers (removing cmos battery first is recommended). Put a little bit of non conductive thermal paste on both silicons (only cpu silicon iis shown on that image). Before assembling new fan, some need new longer screws, make sure having these (on the left is original one, too short for new fan). After that, assemble your new fan into CPU cooler. Finally assemle both coolers on both chips, do not forget put in the CPU fan connector back, and you are done. Binary blob policy Edit this page Who develops Libreboot? 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{"id":"FFf1L2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/install\/kgpe-d16.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":48,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","install"],"url_file_name_s":"kgpe-d16","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – KGPE-D16 external flashing instructions"],"last_modified":"Sat Apr 02 12:48:39 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-5062351551607097348,"exact_signature_unique_b":true,"fuzzy_signature_l":3385536559368432260,"fuzzy_signature_unique_b":true,"h1_txt":["KGPE-D16 external flashing instructions","Navigate this page:","External programmer"],"bold_txt":["DO NOT","Memory initialization is still problematic, for some modules. We recommend avoiding Kingston modules.","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":399,"text_t":"Libreboot – KGPE-D16 external flashing instructions. KGPE-D16 external flashing instructions. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. External programmer Return to index Initial flashing instructions for KGPE-D16. This guide is for those who want libreboot on their ASUS KGPE-D16 motherboard, while they still have the proprietary ASUS BIOS present. This guide can also be followed (adapted) if you brick you board, to know how to recover. Memory initialization is still problematic, for some modules. We recommend avoiding Kingston modules. For more general information about this board, refer to. ..\/hardware\/kgpe-d16.html . TODO: show photos here, and other info. External programmer. [link] Refer to. spi.html for a guide on how to re-flash externally. The flash chip is in a PDIP 8 socket (SPI flash chip) on the motherboard, which you take out and then re-flash with libreboot, using the programmer. DO NOT. remove the chip with your hands. Use a chip extractor tool. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/kgpe-d16.md Site map This HTML page was generated by the. untitled static site generator .","wordcount_i":193,"linkscount_i":23,"linksnofollowcount_i":0,"inboundlinkscount_i":19,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https","018-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/install\/kgpe-d16.html","libreboot.org\/docs\/hardware\/kgpe-d16.html","libreboot.org\/docs\/install\/spi.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/install\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/install\/kgpe-d16.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","External programmer","..\/hardware\/kgpe-d16.html","spi.html","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:57 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:06 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":4623,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078557792600064}
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{"id":"eeFZa2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/bsd\/freebsd.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":43,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","bsd"],"url_file_name_s":"freebsd","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – How to install FreeBSD on x86 GNU GRUB payload"],"last_modified":"Sat Apr 02 12:48:33 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-2986717210430262344,"exact_signature_unique_b":true,"fuzzy_signature_l":8150616520437372229,"fuzzy_signature_unique_b":true,"h1_txt":["How to install FreeBSD on x86 GNU GRUB payload","Navigate this page:","Troubleshooting"],"h2_txt":["Prepare the USB drive (in FreeBSD)","Prepare the USB drive (in NetBSD)","Prepare the USB drive (in LibertyBSD or OpenBSD)","Prepare the USB drive (in GNU+Linux)","Installing FreeBSD without full disk encryption","Installing FreeBSD with full disk encryption","Booting","Configuring Grub","won’t boot…something about file not found"],"bold_txt":["dd","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":353,"wordcount_i":658,"linkscount_i":24,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":6,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/bsd\/freebsd.html","libreboot.org\/docs\/bsd\/","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/bsd\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/bsd\/freebsd.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Prepare the USB drive (in FreeBSD)","Return to index","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","www.freebsd.org\/doc\/handbook\/bsdinstall-pre.html","wiki.netbsd.org\/tutorials\/how_to_install_netbsd_from_an_usb_memory_stick\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","This page","This page","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:58 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:11 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":9701,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078559103320064,"text_t":"Libreboot – How to install FreeBSD on x86 GNU GRUB payload. How to install FreeBSD on x86 GNU GRUB payload. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Prepare the USB drive (in FreeBSD) Prepare the USB drive (in NetBSD) Prepare the USB drive (in LibertyBSD or OpenBSD) Prepare the USB drive (in GNU+Linux) Installing FreeBSD without full disk encryption Installing FreeBSD with full disk encryption Booting Configuring Grub Troubleshooting won’t boot…something about file not found Return to index FreeBSD might show graphical corruption during bootup. You can fix this by altering the order in which kernel modules\/drivers are loaded. First, try moving video to an earlier stage on the boot process, or try moving it to a later stage instead. With this, you should be able to get a working display. freebsd.img is the installation image for FreeBSD. Adapt the filename accordingly, for whatever FreeBSD version you use. Prepare the USB drive (in FreeBSD). [link] This page on the FreeBSD website shows how to create a bootable USB drive for installing FreeBSD. Use the. dd. on that page. Prepare the USB drive (in NetBSD). [link] This page on the NetBSD website shows how to create a NetBSD bootable USB drive from within NetBSD itself. You should use the. dd. method documented there; you can use this with any ISO, including FreeBSD. Prepare the USB drive (in LibertyBSD or OpenBSD). [link] If you downloaded your ISO on a LibertyBSD or OpenBSD system, here is how to create the bootable FreeBSD USB drive:. Connect the USB drive. Check dmesg:. dmesg | tail. Check to confirm which drive it is, for example, if you think its sd3:. disklabel sd3. Check that it wasn’t automatically mounted. If it was, unmount it. For example:. doas umount \/dev\/sd3i. dmesg told you what device it is. Overwrite the drive, writing the FreeBSD installer to it with dd. For example:. doas dd if=freebsd.img of=\/dev\/rsdXc bs=1M; sync. You should now be able to boot the installer from your USB drive. Continue reading, for information about how to do that. Prepare the USB drive (in GNU+Linux). [link] If you downloaded your ISO on a GNU+Linux system, here is how to create the bootable FreeBSD USB drive:. Connect the USB drive. Check dmesg:. dmesg. Check lsblk to confirm which drive it is:. lsblk. Check that it wasn’t automatically mounted. If it was, unmount it. For example:. sudo umount \/dev\/sdX\\* umount \/dev\/sdX\\*. dmesg told you what device it is. Overwrite the drive, writing your distro ISO to it with dd. For example:. sudo dd if=freebsd.img of=\/dev\/sdX bs=8M; sync dd if=freebsd.img of=\/dev\/sdX bs=8M; sync. You should now be able to boot the installer from your USB drive. Continue reading, for information about how to do that. Installing FreeBSD without full disk encryption. [link] Press C in GRUB to access the command line:. grub kfreebsd (usb0,gpt3)\/boot\/kernel\/kernel grub set FreeBSD.vfs.mountfrom=ufs:\/dev\/da1p3\\ grub boot. It will start booting into the FreeBSD installer. Follow the normal process for installing FreeBSD. Installing FreeBSD with full disk encryption. [link] TODO. Booting. [link] TODO. Configuring Grub. [link] TODO. Troubleshooting. [link] Most of these issues occur when using libreboot with coreboot’s ‘text mode’ instead of the coreboot framebuffer. This mode is useful for booting payloads like memtest86+ which expect text-mode, but for FreeBSD it can be problematic when they are trying to switch to a framebuffer because it doesn’t exist. In most cases, you should use the corebootfb ROM images. There ROM images have. corebootfb. in the file name, and they start in a high resolution frame buffer, provided by coreboot’s. libgfxinit. library. won’t boot…something about file not found. [link] Your device names (i.e. usb0, usb1, sd0, sd1, wd0, ahci0, hd0, etc) and numbers may differ. Use TAB completion. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/bsd\/freebsd.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"5MvOK2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/depthcharge\/","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":39,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","depthcharge"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Depthcharge payload"],"last_modified":"Sat Apr 02 12:48:34 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":5112375421779456248,"exact_signature_unique_b":true,"fuzzy_signature_l":2071483689638342863,"fuzzy_signature_unique_b":true,"h1_txt":["Depthcharge payload","Navigate this page:","CrOS security model","Developer mode screen","Recovery mode screen","Configuring verified boot parameters"],"h2_txt":["Holding the developer mode screen","Booting normally","Booting from different mediums","Showing device information","Warnings","Recovering from a bad state","Enabling developer mode"],"bold_txt":["Ctrl + D","weakening the security of the device","Escape + Refresh + Power","Tab","Ctrl + I","Ctrl + L","Ctrl + U","3 seconds","Ctrl + H","This documentation is retained from Libreboot 20160907, but it may also be prudent to check documentation from Libreboot 20160907 itself. It is included in the source code archive, for that release.","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":392,"wordcount_i":1034,"linkscount_i":24,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":6,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/depthcharge\/","libreboot.org\/docs\/fdl-1.3.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/depthcharge\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/depthcharge\/index.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","CrOS security model","..\/fdl-1.3.html","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","chromium.googlesource.com\/chromiumos\/platform\/vboot_reference\/","chromium.googlesource.com\/chromiumos\/platform\/mosys\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","crossystem","mosys","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:58 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:11 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":12611,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078559124291584,"text_t":"Libreboot – Depthcharge payload. Depthcharge payload. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. CrOS security model Developer mode screen Holding the developer mode screen Booting normally Booting from different mediums Showing device information Warnings Recovery mode screen Recovering from a bad state Enabling developer mode Configuring verified boot parameters Return to previous index This documentation is retained from Libreboot 20160907, but it may also be prudent to check documentation from Libreboot 20160907 itself. It is included in the source code archive, for that release. This section relates to the depthcharge payload used in libreboot. CrOS security model. [link] CrOS (Chromium OS\/Chrome OS) devices such as Chromebooks implement a strict security model to ensure that these devices do not become compromised, that is implemented as the verified boot (vboot) reference, most of which is executed within depthcharge. A detailed overview of the CrOS security model is available on the dedicated page. In spite of the CrOS security model, depthcharge won’t allow booting kernels without verifying their signature and booting from external media or legacy payload unless explicitly allowed: see. configuring verified boot parameters . Developer mode screen. [link] The developer mode screen can be accessed in depthcharge when developer mode is enabled. Developer mode can be enabled from the recovery mode screen. It allows booting normally, booting from internal storage, booting from external media (when enabled), booting from legacy payload (when enabled), showing information about the device and disabling developer mode. Holding the developer mode screen. [link] As instructed on the developer mode screen, the screen can be held by pressing. Ctrl + H. in the first 3 seconds after the screen is shown. After that delay, depthcharge will resume booting normally. Booting normally. [link] As instructed on the developer mode screen, a regular boot will happen after. 3 seconds. (if developer mode screen is not held). The default boot medium (internal storage, external media, legacy payload) is shown on screen. Booting from different mediums. [link] Depthcharge allows booting from different mediums, when they are allowed (see. configuring verified boot parameters to enable or disable boot mediums). As instructed on the developer mode screen, booting from various mediums can be triggered by pressing various key combinations:. Internal storage:. Ctrl + D. External media:. Ctrl + U. (when enabled). Legacy payload:. Ctrl + L. (when enabled). Showing device information. [link] As instructed on the developer mode screen, showing device information can be triggered by pressing. Ctrl + I. or. Tab. . Various information is shown, including vboot non-volatile data, TPM status, GBB flags and key hashes. Warnings. [link] The developer mode screen will show warnings when:. Booting kernels without verifying their signature is enabled. Booting from external media is enabled. Booting legacy payloads is enabled. Recovery mode screen. [link] The recovery mode screen can be accessed in depthcharge, by pressing. Escape + Refresh + Power. when the device is off. It allows recovering the device from a bad state by booting from a trusted recovery media. When accessed with the device in a good state, it also allows enabling developer mode. Recovering from a bad state. [link] When the device fails to verify the signature of a piece of the boot software or when an error occurs, it is considered to be in a bad state and will instruct the user to reboot to recovery mode. Recovery mode boots using only software located in write-protected memory, that is considered to be trusted and safe. Recovery mode then allows recovering the device by booting from a trusted recovery media, that is automatically detected when recovery mode starts. When no external media is found or when the recovery media is invalid, instructions are shown on screen. Trusted recovery media are external media (USB drives, SD cards, etc) that hold a kernel signed with the recovery key. Google provides images of such recovery media for Chrome OS (which are not advised to users as they contain proprietary software). They are signed with Google’s recovery keys, that are pre-installed on the device when it ships. When replacing the full flash of the device, the pre-installed keys are replaced. When the recovery private key is available (e.g. when using self-generated keys), it can be used to sign a kernel for recovery purposes. Enabling developer mode. [link] As instructed on the recovery mode screen, developer mode can be enabled by pressing. Ctrl + D. . Instructions to confirm enabling developer mode are then shown on screen. Configuring verified boot parameters. [link] Depthcharge’s behavior relies on the verified boot (vboot) reference implementation, that can be configured with parameters stored in the verified boot non-volatile storage. These parameters can be modified with the. crossystem. tool, that requires sufficient privileges to access the verified boot non-volatile storage. crossystem. relies on. mosys. , that is used to access the verified boot non-volatile storage on some devices. crossystem. and. mosys. are both free software and their source code is made available by Google:. crossystem . mosys . These tools are not distributed along with Libreboot yet. However, they are preinstalled on the device, with ChromeOS. Some of these parameters have the potential of. weakening the security of the device. . In particular, disabling kernels signature verification, external media boot and legacy payload boot can weaken the security of the device. The following parameters can be configured:. Kernels signature verification:. crossystem dev_boot_signed_only=1 # enable crossystem dev_boot_signed_only=0 # disable. External media boot:. crossystem dev_boot_usb=1 # enable crossystem dev_boot_usb=0 # disable. Legacy payload boot:. crossystem dev_boot_legacy=1 # enable crossystem dev_boot_legacy=0 # disable. Default boot medium:. crossystem dev_default_boot=disk # internal storage crossystem dev_default_boot=usb # external media crossystem dev_default_boot=legacy # legacy payload. Copyright © 2015 Paul Kocialkowski. contact@paulk.fr Permission is granted to copy, distribute and\/or modify this document under the terms of the GNU Free Documentation License Version 1.3 or any later version published by the Free Software Foundation with no Invariant Sections, no Front Cover Texts, and no Back Cover Texts. A copy of this license is found in. ..\/fdl-1.3.html Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/depthcharge\/index.md Site map This HTML page was generated by the. untitled static site generator ."}
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2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":13884,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078559654871040,"text_t":"Libreboot – Flashing the ThinkPad R400. Flashing the ThinkPad R400. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Libreboot T400 Serial port A note about CPUs Quad-core CPUs A note about GPUs CPU paste required Flash chip size MAC address External flashing Disassembly Thermal paste (IMPORTANT) Memory Boot it! Return to index Initial flashing instructions for R400. This guide is for those who want libreboot on their ThinkPad R400 while they still have the original Lenovo BIOS present. This guide can also be followed (adapted) if you brick your R400, to know how to recover. Before following this section, please make sure to setup your libreboot ROM properly first. Although ROM images are provided pre-built in libreboot, there are some modifications that you need to make to the one you chose before flashing. (instructions referenced later in this guide). Libreboot T400. [link] You may also be interested in the smaller, more portable. Libreboot T400 . Serial port. [link] EHCI debug might not be needed. It has been reported that the docking station for this laptop has a serial port, so it might be possible to use that instead. A note about CPUs. [link] ThinkWiki has a list of CPUs for this system. The Core 2 Duo P8400 and P8600 are believed to work in libreboot. The Core 2 Duo T9600 was confirmed to work, so the T9400 probably also works. The Core 2 Duo T5870\/5670 and Celeron M 575\/585 are untested! Quad-core CPUs. [link] Incompatible. Do not use. A note about GPUs. [link] Some models have an Intel GPU, while others have both an ATI and an Intel GPU; this is referred to as “Dual Graphics” (previously “switchable graphics”). In the. BIOS setup. program for lenovobios, you can specify that the system will use one or the other (but not both). Libreboot is known to work on systems with only the Intel GPU, using native graphics initialization. On systems with switchable graphics, the Intel GPU is used and the ATI GPU is disabled, so native graphics initialization works all the same. CPU paste required. [link] See. #paste . Flash chip size. [link] Use this to find out:. flashrom -p internal. MAC address. [link] Refer to. mac_address.html . External flashing. [link] Refer to. spi.html as a guide for external re-flashing. Disassembly. [link] Remove all screws:. Remove the HDD and optical drive:. Remove the hinge screws:. Remove the palm rest and keyboard:. Remove these screws, and then remove the bezel:. Remove the speaker screws, but don’t remove the speakers yet (just set them loose):. Remove these screws, and then remove the metal plate:. Remove the antennas from the wifi card, and then start unrouting them:. Disconnect the LCD cable from the motherboard:. Remove the hinge screws, and then remove the LCD panel:. Remove this:. Remove this long cable (there are 3 connections):. Disconnect the speaker cable, and remove the speakers:. Remove the heatsink screws, remove the fan and then remove the heatsink\/fan:. Remove the NVRAM battery:. Remove this screw:. Disconnect the AC jack:. Remove this screw and then remove what is under it:. Remove this:. Lift the motherboard (which is still inside the cage) from the side on the right, removing it completely:. Remove all screws, marking each hole so that you know where to re-insert them. You should place the screws in a layout corresponding to the order that they were in before removal:. Remove the motherboard from the cage, and the SPI flash chip will be next to the memory slots:. Now, you should be ready to install libreboot. Read. this article to learn how you may flash the chip, which is near to the RAM. Thermal paste (IMPORTANT). [link] Because part of this procedure involved removing the heatsink, you will need to apply new paste. Arctic MX-4 is ok. You will also need isopropyl alcohol and an anti-static cloth to clean with. When re-installing the heatsink, you must first clean off all old paste with the alcohol\/cloth. Then apply new paste. Arctic MX-4 is also much better than the default paste used on these systems. NOTE: the photo above is for illustration purposes only, and does not show how to properly apply the thermal paste. Other guides online detail the proper application procedure. Memory. [link] In DDR3 machines with Cantiga (GM45\/GS45\/PM45), northbridge requires sticks that will work as PC3-8500 (faster PC3\/PC3L sticks can work as PC3-8500). Non-matching pairs may not work. Single module (meaning, one of the slots will be empty) will currently only work in slot 0. NOTE: according to users reports, non matching pairs (e.g. 1+2 GiB) might work in some cases. Make sure that the RAM you buy is the 2Rx8 configuration when buying 4GiB sticks (In other words: maximum of 2GiB per rank, 2 ranks per card). This page might be useful for RAM compatibility info (note: coreboot raminit is different, so this page might be BS). The following photo shows 8GiB (2x4GiB) of RAM installed:. Boot it! [link] You should see something like this:. Now. install GNU+Linux . Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/r400_external.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"T21I22Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/ga-g41m-es2l.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":53,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"ga-g41m-es2l","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Gigabyte GA-G41M-ES2L desktop board"],"last_modified":"Sat Apr 02 12:48:38 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":7660893004825900673,"exact_signature_unique_b":true,"fuzzy_signature_l":3820853176214709878,"fuzzy_signature_unique_b":true,"h1_txt":["Gigabyte GA-G41M-ES2L desktop board"],"h2_txt":["RAM"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":378,"text_t":"Libreboot – Gigabyte GA-G41M-ES2L desktop board. Gigabyte GA-G41M-ES2L desktop board. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to index This is a desktop board using intel hardware (circa ~2009, ICH7 southbridge, similar performance-wise to the Libreboot X200. It can make for quite a nifty desktop. Powered by libreboot. IDE on the board is untested, but it might be possible to use a SATA HDD using an IDE SATA adapter. The SATA ports do work. You need to set a custom MAC address in GNU+Linux for the NIC to work. In \/etc\/network\/interfaces on debian-based systems like Debian or Devuan, this would be in the entry for your NIC:. hwaddress ether macaddressgoeshere. Alternatively:. cbfstool libreboot.rom extract -n rt8168-macaddress -f rt8168-macaddress. Modify the MAC address in the file. rt8168-macaddress. and then:. cbfstool libreboot.rom remove -n rt8168-macaddress cbfstool libreboot.rom add -f rt8168-macaddress -n rt8168-macaddress -t raw. Now you have a different MAC address hardcoded. In the above example, the ROM image is named. libreboot.rom. for your board. You can find cbfstool under. coreboot\/default\/util\/cbfstool\/. after running the following command in the build system:. .\/build module cbutils. You can learn more about using the build system, lbmk, here:. Libreboot build instructions Flashing instructions can be found at. ..\/install\/ RAM. [link] Kingston 8 GiB Kit KVR800D2N6\/8G with Elpida Chips E2108ABSE-8G-E. this is a 2x4GB setup and these work quite well, according to a user on IRC. Many other modules will probably work just fine, but raminit is very picky on this board. Binary blob policy Edit this page Who develops Libreboot? 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Modifying grub.cfg in CBFS. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Compile flashrom and cbfstool Dump the boot flash Extract grub.cfg Insert new grub.cfg Flash the modified ROM image Return to index Before you follow this guide, it is advisable that you have the ability to flash externally, just in case something goes wrong. This guide assumes that you use the GNU GRUB bootloader as your default payload. In this configuration, GNU GRUB is flashed alongside coreboot and runs on. bare metal. as a native coreboot payload and does. not. use BIOS or UEFI services (but it. can. load and execute SeaBIOS, in addition to any other coreboot payload, by chainloading it). In most circumstances, this guide will not benefit you. Libreboot’s default GRUB configuration file contains scripting logic within it that intelligently searches for GRUB partitions installed onto a partition on your SSD, HDD or USB drive installed on your computer. If such a file is found, libreboot’s default GRUB configuration is configured to switch automatically to that configuration. While not perfect, the logic. does. work with most configurations. Therefore, you should only follow. this. guide if the automation (described above) does not work. It goes without saying that modifying the default GRUB configuration is risky, because a misconfiguration could create what’s called a. soft brick. where your machine is effectively useless and, in that scenario, may or may not require external flashing equipment for restoring the machine to a known state. Compile flashrom and cbfstool. [link] Libreboot does not currently distribute utilities pre-compiled. It only provides ROM images pre-compiled, where feasible. Therefore, you have to build the utilities from source. As for the ROM, there are mainly three methods for obtaining a libreboot ROM image:. Dump the contents of the the main. boot flash. on your system, which already has libreboot installed (with GNU GRUB as the default payload). Extract the GRUB configuration from. that. ROM image. Extract it from a libreboot ROM image supplied by the libreboot project, on the libreboot website or mirrors of the libreboot website. Build the ROM yourself, using the libreboot build system. Instructions for how to do this are covered in the following article:. How to build libreboot from source In either case, you will use the. cbfstool. supplied in the Libreboot build system. This can be found under. coreboot\/*\/util\/cbfstool\/. as source code, where. *. can be any coreboot source code directory for a given mainboard. The directory named. default. should suffice. Install the build dependencies. For Ubuntu 20.04 and similar, you can run the following command in the libreboot build system, from the root directory of the libreboot Git repository. .\/build dependencies ubuntu2004. Then, download coreboot:. .\/download coreboot. Finally, compile the. cbutils. module:. .\/build module cbutils. Among other things, this will produce a. cbfstool. executable under any of the subdirectories in. coreboot\/. under `util\/cbfstool\/cbfstool. For example:. coreboot\/default\/util\/cbfstool\/cbfstool. The. cbfstool. utility is what you shall use. It is used to manipulate CBFS (coreboot file system) which is a file system contained within the coreboot ROM image; as a. coreboot distribution. , libreboot inherits this technology. You will also want to build. flashrom. which libreboot recommends for reading from and\/or writing to the boot flash. In the libreboot build system, you can build it by running this command:. .\/build module flashrom. An executable will be available at. flashrom\/flashrom. after you have done this. Dump the boot flash. [link] If you wish to modify your. existing. libreboot ROM, which was installed on your computer, you can use. flashrom. to acquire it. Simply run the following, after using libreboot’s build system to compile flashrom:. sudo .\/flashrom\/flashrom -p internal -r dump.bin. If flashrom complains about multiple flash chip definitions, do what it says to rectify your command and run it again. You may want to use the following, instead of. -p internal. :. -p internal:laptop=force_I_want_a_brick,boardmismatch=force. Do not let the word. brick. fools you. This merely disables the safety checks in flashrom, which is sometimes necessary depending on what ROM was already flashed, versus the new ROM image. The. internal. option assumes that internal read\/write is possible; this is when you read from and\/or write to the boot flash from an operating systems (usually GNU+Linux) that is. running on. the target system. In other cases, you may need to connect an SPI programmer externally (with the machine powered down) and read the contents of the boot flash. Learn how to externally reprogram these chips Extract grub.cfg. [link] Libreboot images that use the GNU GRUB bootloader will have. two. configuration files in CBFS:. grub.cfg. grubtest.cfg. We recommend that you modify. grubtest.cfg. first, and boot. Select the boot menu option for loading. grubtest.cfg. and verify that your new config works correctly. If it doesn’t, keep modifying. grubtest.cfg. until it does work. When that it done, copy the changes over to `grub.cfg. You can use the following commands to modify the contents of CBFS, where GRUB’s configuration file is concerned (dump.bin is the ROM that you dumped, or it could refer to the libreboot ROM image that you compiled or otherwise acquired). Show the contents of CBFS, in your ROM:. cbfstool dump.bin print. Extract. grub.cfg. (substitude with. grubtest.cfg. as desired):. cbfstool dump.bin extract -n grub.cfg -f grub.cfg. You will now have a file named. grub.cfg. . Make your desired modifications. You should then delete the old. grub.cfg. from your ROM image. Insert new grub.cfg. [link] Remove the old. grub.cfg. (substitute with. grubtest.cfg. as desired):. cbfstool dump.bin remove -n grub.cfg. Add your modified. grub.cfg. (substitute with. grubtest.cfg. as desired):. cbfstool dump.bin add -f grub.cfg -n grub.cfg -t raw. Flash the modified ROM image. [link] Your modified. dump.bin. or other modified libreboot ROM can then be re-flashed using:. sudo .\/flashrom -p internal -w dump.bin. If a. -c. option is required, use it and specify a flash chip name. This is only useful when. flashrom. complains about multiple flash chips being detected. If flashrom complains about wrong chip\/board, make sure that your ROM is for the correct system. If you’re sure, you can disable the safety checks by running this instead:. sudo .\/flashrom -p internal:laptop=force_I_want_a_brick,boardmismatch=force -w dump.bin. If you need to use external flashing equipment, see the link above to the Raspberry Pi page. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/gnulinux\/grub_cbfs.md Site map This HTML page was generated by the. untitled static site generator ."}
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Libreboot 20160818 release. Leah Rowe. 18 August 2016. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Machines supported in this release: Changes for this release of Libreboot, relative to Libreboot version 20150518 (earliest changes are shown last and the most recent changes are shown first first) Return to index Article published by: Leah Rowe. Date of publication: 18 August 2016. This is in comparison to the Libreboot 20150518 release. Installation instructions can be found at. docs\/install\/. . Building instructions (for source code) can be found at. docs\/git\/\\#build. . Machines supported in this release:. [link] ASUS Chromebook C201. Check notes in. docs\/hardware\/c201.html. Gigabyte GA-G41M-ES2L desktop motherboard. Check notes in. docs\/hardware\/ga-g41m-es2l.html. Intel D510MO desktop motherboard. Check notes in. docs\/hardware\/d510mo.html. Intel D945GCLF desktop motherboard. Check notes in. docs\/hardware\/d945gclf.html. Apple iMac 5,2. Check notes in. docs\/hardware\/imac52.html. ASUS KFSN4-DRE server board. PCB revision 1.05G is the best version (can use 6-core CPUs). Check notes in. docs\/hardware\/kfsn4-dre.html. ASUS KGPE-D16 server board. Check notes in. docs\/hardware\/kgpe-d16.html. ASUS KCMA-D8 desktop\/workstation board. Check notes in. docs\/hardware\/kcma-d8.html. ThinkPad X60\/X60s. You can also remove the motherboard from an X61\/X61s and replace it with an X60\/X60s motherboard. An X60 Tablet motherboard will also fit inside an X60\/X60s. ThinkPad X60 Tablet. (1024x768 and 1400x1050) with digitizer support. See. docs\/hardware\/#supported_x60t_list. for list of supported LCD panels. It is unknown whether an X61 Tablet can have it’s mainboard replaced with an X60 Tablet motherboard. ThinkPad T60. (Intel GPU) (there are issues; see below):. See notes below for exceptions, and. docs\/hardware\/#supported_t60_list. for known working LCD panels. It is unknown whether a T61 can have it’s mainboard replaced with a T60 motherboard. See. docs\/future\/#t60_cpu_microcode. . T60p (and T60 laptops with ATI GPU) will likely never be supported:. docs\/hardware\/#t60_ati_intel. ThinkPad X200. X200S and X200 Tablet are also supported, conditionally; see. docs\/hardware\/x200.html#x200s. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. ThinkPad R400. See. docs\/hardware\/r400.html. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. ThinkPad T400. See. docs\/hardware\/t400.html. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. ThinkPad T500. See. docs\/hardware\/t500.html. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. Apple MacBook1,1. (MA255LL\/A, MA254LL\/A, MA472LL\/A). See. docs\/hardware\/#macbook11. . Apple MacBook2,1. (MA699LL\/A, MA701LL\/A, MB061LL\/A, MA700LL\/A, MB063LL\/A, MB062LL\/A). See. docs\/hardware\/#macbook21. . Changes for this release of Libreboot, relative to Libreboot version 20150518 (earliest changes are shown last and the most recent changes are shown first first). [link] NEW BOARDS ADDED:. ASUS Chromebook C201 (ARM laptop) (thanks to Paul Kocialkowski). Gigabyte GA-G41M-ES2L motherboard (desktop) (thanks to Damien Zammit). Intel D510MO motherboard (desktop) (thanks to Damien Zammit). ASUS KCMA-D8 motherboard (desktop) (thanks to Timothy Pearson). ASUS KFSN4-DRE motherboard (server) (thanks to Timothy Pearson). ASUS KGPE-D16 motherboard (server) (thanks to Timothy Pearson). For details development history on these boards, refer to the git log and documentation. For boards previously supported, many fixes from upstream have been merged. Other changes (compared to libreboot 20150518), for libreboot in general or for previously supported systems: (this is a summary. For more detailed change list, refer to the git log). 256MiB VRAM allocated on GM45 (X200, T400, T500, R400) instead of 32MiB. This is an improvement over both Lenovo BIOS and Libreboot 20150518, allowing video decoding at 1080p to be smoother. (thanks Arthur Heymans) To clarify, GM45 video performance in libreboot 20160818 is better than on the original BIOS and the previous libreboot release. 64MiB VRAM on i945 (X60, T60, MacBook2,1) now supported in coreboot-libre, and used by default (in the previous release, it was 8MiB allocated). Thanks to Arthur Heymans. Higher battery life on GM45 (X200, T400, T500, R400) due to higher cstates now being supported (thanks Arthur Heymans). C4 power states also supported. Higher battery life on i945 (X60, T60, MacBook2,1) due to better CPU C-state settings. (Deep C4, Dynamic L2 shrinking, C2E). Text mode on GM45 (X200, T400, T500, R400) now works, making it possible to use MemTest86+ comfortably. (thanks to Nick High from coreboot). Dual channel LVDS displays on GM45 (T400, T500) are now automatically detected in coreboot-libre. (thanks Vladimir Serbinenko from coreboot). Partial fix in coreboot-libre for GRUB display on GM45, for dual channel LVDS higher resolution LCD panels (T400, T500). (thanks Arthur Heymans). Massively improved GRUB configuration, making it easier to boot more encrypted systems automatically, and generally a more useful menu for booting the system (thanks go to Klemens Nanni of the autoboot project). Libreboot now uses the grub.cfg provided by the installed GNU+Linux distribution automatically, if present, switching to that configuration. This is done across many partitions, where libreboot actively searches for a configuration file (also on LVM volumes and encrypted volumes). This should make libreboot more easy to use for non-technical users, without having to modify the GRUB configuration used in libreboot. Utilities archives is now source only. You will need to compile the packages in there (build scripts included, and a script for installing build dependencies). (binary utility archives are planned again in the next release, when the new build system is merged). SeaGRUB is now the default payload on all x86 boards. (SeaBIOS configured to load a compressed GRUB payload from CBFS immediately, without providing an interface in SeaBIOS. This way, GRUB is still used but now BIOS services are available, so you get the best of both worlds). Thanks go to Timothy Pearson of coreboot for this idea. crossgcc is now downloaded and built as a separate module to coreboot-libre, with a universal revision used to build all boards. Individual boards now have their own coreboot revision and patches, independently of each other board. This makes maintenance easier. Updated all utilities, and modules (coreboot, GRUB, etc) to newer versions, with various bugfixes and improvements upstream. RTC century byte issue now fixed on GM45 in coreboot-libre, so the date should now be correctly displayed when running the latest linux kernel, instead of seeing 1970-01-01 when you boot (thanks to Alexander Couzens from coreboot). Build system now uses multiple CPU cores when building, speeding up building for some people. Manually specifying how many cores are needed is also possible, for those using the build system in a chroot environment. (thanks go to Timothy Pearson from coreboot). In the build system (git repository), https:\/\/ is now used when cloning coreboot. http:\/\/ is used as a fallback for GRUB, if git:\/\/ fails. New payload, the depthcharge bootloader (free bootloader maintained by Google) for use on the ASUS Chromebook C201. (thanks go to Paul Kocialkowski). Various fixes to the ich9gen utility (e.g. flash component density is now set correctly in the descriptor, gbe-less descriptors now supported). Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/news\/libreboot20160818.md Site map This HTML page was generated by the. untitled static site generator ."}
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This can be identified by the type of flash chip below the palmrest: 4MiB is SOIC-8, 8MiB is SOIC-16. The T400 laptops come with the ME (and sometimes AMT in addition) before flashing libreboot. Libreboot disables and removes it by using a modified descriptor: see. ..\/install\/ich9utils.html (contains notes, plus instructions). Flashing instructions can be found at. ..\/install\/#flashrom EC update. [link] It is recommended that you update to the latest EC firmware version. The. EC firmware is separate from libreboot, so we don’t actually provide that, but if you still have Lenovo BIOS then you can just run the Lenovo BIOS update utility, which will update both the BIOS and EC version. See:. ..\/install\/#flashrom http:\/\/www.thinkwiki.org\/wiki\/BIOS_update_without_optical_disk NOTE: this can only be done when you are using Lenovo BIOS. How to update the EC firmware while running libreboot is unknown. Libreboot only replaces the BIOS firmware, not EC. Updated EC firmware has several advantages e.g. bettery battery handling. Compatibility (without blobs). [link] Hardware virtualization (vt-x). [link] The T400, when run without CPU microcode updates in coreboot, currently kernel panics if running QEMU with vt-x enabled on 2 cores for the guest. With a single core enabled for the guest, the guest panics (but the host is fine). Working around this in QEMU might be possible; if not, software virtualization should work fine (it’s just slower). On GM45 hardware (with libreboot), make sure that the. kvm. and. kvm_intel. kernel modules are not loaded, when using QEMU. The following errata datasheet from Intel might help with investigation:. http:\/\/download.intel.com\/design\/mobile\/specupdt\/320121.pdf The T400 is almost identical to the X200, code-wise. See. x200.html . Binary blob policy Edit this page Who develops Libreboot? 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{"id":"OSLLE2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/macbook21.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":50,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"macbook21","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – MacBook2,1 and MacBook1,1"],"last_modified":"Sat Apr 02 12:48:38 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-198374910105737091,"exact_signature_unique_b":true,"fuzzy_signature_l":1066629522199453255,"fuzzy_signature_unique_b":true,"h1_txt":["MacBook2,1 and MacBook1,1","Navigate this page:","Compatibility","Internal flashing","External flashing","GNU+Linux on Apple EFI firmware","Coreboot wiki page","Issues and solutions\/workarounds"],"h2_txt":["Models","Make it overheat less","Enable AltGr","Enable 3-finger tap","Make touchpad more responsive"],"bold_txt":["(tested - working)","resources\/utils\/macbook21-three-finger-tap","This section is less relevant for Libreboot 20211122 and newer, because cstate level 3 support was added, thanks to vitali64 on IRC. This means that the CPU temperature is much lower most of the time, as is power consumption. However, you might still benefit from the steps below, just not as much as you would have previously benefited.","The MacBook2,1 comes with a webcam which does not work with free software. Webcams are a privacy and security risk; cover it up! Or remove it.","Buy preinstalled","Donate"],"imagescount_i":2,"images_protocol_sxt":["000-https","001-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico","av.libreboot.org\/chip\/soic8.jpg"],"images_alt_sxt":["Libreboot logo",""],"images_height_val":[-1,-1],"images_width_val":[-1,-1],"images_text_t":"libreboot org favicon ico Libreboot logo av chip soic8 jpg soic 8","responsetime_i":446,"wordcount_i":1289,"linkscount_i":29,"linksnofollowcount_i":0,"inboundlinkscount_i":19,"outboundlinkscount_i":10,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https","018-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/hardware\/macbook21.html","libreboot.org\/docs\/hardware\/","libreboot.org\/docs\/install\/spi.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/hardware\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/hardware\/macbook21.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Compatibility","Return to index","Externally rewrite 25xx NOR flash via SPI protocol","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","005-https","006-https","007-https","009-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","www.everymac.com\/ultimate-mac-lookup\/?search_keywords=MacBook1,1","www.everymac.com\/ultimate-mac-lookup\/?search_keywords=MacBook2,1","www.ifixit.com\/Device\/MacBook_Core_2_Duo","www.ifixit.com\/Guide\/MacBook+Core+2+Duo+PRAM+Battery+Replacement\/529","www.coreboot.org\/Board:apple\/macbook21","cweiske.de\/tagebuch\/tuxbook.htm","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","","","This page shows disassembly guides","How to remove the motherboard","","cover it up","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:06 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:20 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":15079,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078567197278208,"text_t":"Libreboot – MacBook2,1 and MacBook1,1. MacBook2,1 and MacBook1,1. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Compatibility Models Internal flashing External flashing GNU+Linux on Apple EFI firmware Coreboot wiki page Issues and solutions\/workarounds Make it overheat less Enable AltGr Enable 3-finger tap Make touchpad more responsive Return to index There is an Apple laptop called the macbook1,1 from 2006 which uses the same i945 chipset as the ThinkPad X60\/T60. A developer (Mono Moosbart) ported the Macbook2,1 to coreboot, working alongside Vladimir Serbinenko. The ROM images also work on the macbook1,1. Libreboot’s support and documentation for this is based on the Libreboot project, which also supports macbook2,1. Some macbook2,1 models are late 2006, others are early 2007. You do not need to use external flashing equipment when flashing the MacBook2,1 but the MacBook1,1 requires external flashing equipment while running Apple EFI firmware. Macbook2,1 laptops come with Core 2 Duo processors which support 64-bit operating systems (and 32-bit). The MacBook1,1 uses Core Duo processors (supports 32-bit OS but not 64-bit), and it is believed that this is the only difference. Compatibility. [link] The following pages list many models of MacBook1,1 and MacBook2,1:. http:\/\/www.everymac.com\/ultimate-mac-lookup\/?search_keywords=MacBook1,1 http:\/\/www.everymac.com\/ultimate-mac-lookup\/?search_keywords=MacBook2,1 Models. [link] Specifically (Order No. \/ Model No. \/ CPU) for macbook 1,1:. MA255LL\/A \/ A1181 (EMC 2092) \/ Core Duo T2500. (tested - working). MA254LL\/A \/ A1181 (EMC 2092) \/ Core Duo T2400. (tested - working). MA472LL\/A \/ A1181 (EMC 2092) \/ Core Duo T2500 (untested). For macbook 2,1:. MA699LL\/A \/ A1181 (EMC 2121) \/ Intel Core 2 Duo T5600. (tested - working). MA701LL\/A \/ A1181 (EMC 2121) \/ Intel Core 2 Duo T7200. (tested - working). MB061LL\/A \/ A1181 (EMC 2139) \/ Intel Core 2 Duo T7200 (untested). MA700LL\/A \/ A1181 (EMC 2121) \/ Intel Core 2 Duo T7200. (tested - working). MB063LL\/A \/ A1181 (EMC 2139) \/ Intel Core 2 Duo T7400 (works). MB062LL\/A \/ A1181 (EMC 2139) \/ Intel Core 2 Duo T7400. (tested - working). Internal flashing. [link] Macbook2,1 can always be flashed internally, even if running Apple firmware:. sudo flashrom -p internal:laptop=force_I_want_a_brick,boardmismatch=force -w your.rom. Macbook1,1 same as above, but if running Apple firmware see below for external flashing. External flashing. [link] macbook1,1 requires external flashing, if running the default Apple firmware. macbook2,1 can be flased internally, regardless. If running coreboot, libreboot or Libreboot, you can already internally re-flash. This page shows disassembly guides Locate the flash. It’ll be a SOIC8, which looks like this:. motherboard. How to remove the motherboard . Refer to the following guide:. Externally rewrite 25xx NOR flash via SPI protocol You need to replace OS X with GNU+Linux before flashing Libreboot. (OSX won’t run at all in Libreboot), if you wish to internally flash on a macbook21. Libreboot won’t boot OSX either (well, maybe with Tianocore it would, but that’s untested and OSX is inferior to GNU+Linux). In general you should think of your Macbook like a regular laptop, for the purposes of anything coreboot. If it’s a macbook2,1 with the core2duo processors, you can run a 64-bit distro. For macbook 1,1 the CPU probably only has 32-bit support. GNU+Linux on Apple EFI firmware. [link] How to boot an ISO: burn it to a CD (like you would normally) and hold down the Alt\/Control key while booting. The bootloader will detect the GNU+Linux CD as ‘Windows’ (because Apple doesn’t think GNU+Linux exists). Install it like you normally would. When you boot up again, hold Alt\/Control once more. The installation (on the HDD) will once again be seen as ‘Windows’. (it’s not actually Windows, but Apple likes to think that Apple and Microsoft are all that exist.). Coreboot wiki page. [link] The following page has some information:. https:\/\/www.coreboot.org\/Board:apple\/macbook21 Issues and solutions\/workarounds. [link] There is one mouse button only, however multiple finger tapping works. Battery life is poor compared to X60\/T60. The Apple logo on the back is a hole, exposing the backlight, which means that it glows. You should. cover it up . The MacBook2,1 consumes more power with libreboot than with the Apple EFI firmware, which means it overheats a lot. The MacBook2,1 comes with a webcam which does not work with free software. Webcams are a privacy and security risk; cover it up! Or remove it. Make it overheat less. [link] This section is less relevant for Libreboot 20211122 and newer, because cstate level 3 support was added, thanks to vitali64 on IRC. This means that the CPU temperature is much lower most of the time, as is power consumption. However, you might still benefit from the steps below, just not as much as you would have previously benefited. The MacBook2,1 overheats a lot with libreboot, we still don’t know why but a simple workaround is to install macfanctld. Macfanctld is available on the default repos of many distributions. For example, to install macfanctld on an Arch-based distro (Parabola, …), you would run as root. pacman -S macfanctld. and don’t forget to enable it by using. systemctl. or by a script that will run macfanctld if using runit. Then, you want to install powertop and tlp. And then, run the following on battery. sudo tlp start && sudo powertop --calibrate. Then, after quitting powertop, run :. sudo powertop --auto-tune. Now, configure tlp, edit the. \/etc\/tlp.conf. and uncomment\/add\/modify the following:. CPU_BOOST_ON_AC=1 CPU_BOOST_ON_BAT=0 SCHED_POWERSAVE_ON_AC=0 SCHED_POWERSAVE_ON_BAT=1 PLATFORM_PROFILE_ON_AC=performance PLATFORM_PROFILE_ON_BAT=low-power. The MacBook will still overheat, just less. Enable AltGr. [link] The keyboard has a keypad enter instead of an AltGr. The first key on the right side of the spacebar is the Apple “command” key. On its right is the keypad enter. We can make it act as an AltGr. If your operating system is Trisquel or other dpkg-based distribution, there is an easy solution. Under root (or sudo) run. dpkg-reconfigure keyboard-configuration. and select the option “apple laptop”, leave other settings as their defaults until you are given the option “Use Keypad Enter as AltGr”. Select this. The keypad enter key will then act as an AltGr everywhere. For Parabola or other systemd-based distributions you can enable AltGr manually. Simply add the line. KEYMAP_TOGGLE=lv3:enter_switch. to the file \/etc\/vconsole.conf and then restart the computer. Enable 3-finger tap. [link] A user submitted a utility to enable 3-finger tap on this laptop. It’s available at. resources\/utils\/macbook21-three-finger-tap. in the Libreboot git repository. Make touchpad more responsive. [link] Linux kernels of version 3.15 or lower might make the touchpad extremely sluggish. A user reported that they could get better response from the touchpad with the following in their xorg.conf:. Section \"InputClass\" Identifier \"Synaptics Touchpad\" Driver \"synaptics\" MatchIsTouchpad \"on\" MatchDevicePath \"\/dev\/input\/event*\" Driver \"synaptics\" The next two values determine how much pressure one needs for tapping, moving the cursor and other events. Option \"FingerLow\" \"10\" Option \"FingerHigh\" \"15\" Do not emulate mouse buttons in the touchpad corners. Option \"RTCornerButton\" \"0\" Option \"RBCornerButton\" \"0\" Option \"LTCornerButton\" \"0\" Option \"LBCornerButton\" \"0\" One finger tap = left-click Option \"TapButton1\" \"1\" Two fingers tap = right-click Option \"TapButton2\" \"3\" Three fingers tap = middle-mouse Option \"TapButton3\" \"2\" Try to not count the palm of the hand landing on the touchpad as a tap. Not sure if helps. Option \"PalmDetect\" \"1\" The following modifies how long and how fast scrolling continues after lifting the finger when scrolling Option \"CoastingSpeed\" \"20\" Option \"CoastingFriction\" \"200\" Smaller number means that the finger has to travel less distance for it to count as cursor movement. Larger number prevents cursor shaking. Option \"HorizHysteresis\" \"10\" Option \"VertHysteresis\" \"10\" Prevent two-finger scrolling. Very jerky movement Option \"HorizTwoFingerScroll\" \"0\" Option \"VertTwoFingerScroll\" \"0\" Use edge scrolling Option \"HorizEdgeScroll\" \"1\" Option \"VertEdgeScroll\" \"1\" EndSection. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/macbook21.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"PqUoO_Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/news\/libreboot20140720.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["news"],"url_file_name_s":"libreboot20140720","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Libreboot 20140720 release"],"author":"Leah Rowe","author_sxt":["Leah Rowe"],"last_modified":"Sat Apr 02 12:48:47 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Sun Jul 20 00:00:00 UTC 2014"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2197992472470899771,"exact_signature_unique_b":true,"fuzzy_signature_l":4309277905215095304,"fuzzy_signature_unique_b":true,"h1_txt":["Libreboot 20140720 release","Navigate this page:"],"h2_txt":["Revisions for r20140720 (3rd beta) (20th July 2014)"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":439,"text_t":"Libreboot – Libreboot 20140720 release. Libreboot 20140720 release. Leah Rowe. 20 July 2014. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Revisions for r20140720 (3rd beta) (20th July 2014) Return to index Article published by: Leah Rowe. Date of publication: 20 July 2014. Revisions for r20140720 (3rd beta) (20th July 2014). [link] Fixed typo that existed in 2nd beta where the release date of the 2nd beta was listed as being in year 2016, when in actual fact it was 2014. Documentation: added (preliminary) details about (rare) buggy CPUs on the ThinkPad T60 that were found to fail (instability, kernel panics, etc) without the microcode updates. Documentation: added docs\/hardware\/x60_heatsink.html for showing how to change the heatsink on the Thinkpad X60. Added ROM images for Azerty (French) keyboard layout in GRUB (courtesy of Olivier Mondoloni). Tidied up some scripts:. Re-factored those scripts (made easier to read\/maintain): build-x60, build-x60t, build-t60, build-macbook21. Reduced the number of grub configs to 2 (or 1, for macbook21), the build scripts now generate the other configs at build time. Deleted build-x60, build-x60t, build-t60, build-macbook21 and replaced with intelligent (generic) buildrom-withgrub script. Updated build to use buildrom-withgrub script for building the ROM images. coreboot.rom and coreboot_serial.rom renamed to coreboot_usqwerty.rom and coreboot_serial_usqwerty.rom. coreboot_dvorak and coreboot_serial_dvorak.rom renamed to coreboot_usdvorak.rom and coreboot_serial_usdvorak.rom. Renamed coreboot*rom to libreboot*rom. Made flash, lenovobios_firstflash and lenovobios_secondflash scripts fail if the specified file does not exist. Updated all relevant parts of the documentation to reflect the above. Replaced background.png with background.jpg. added gnulove.jpg. (resources\/grub\/background\/). Updated buildrom-withgrub to use background.jpg instead of background.png. Updated buildrom-withgrub to use gnulove.jpg aswell. Updated resources\/grub\/config\/macbook21\/grub*cfg to use gnulove.jpg background. Updated resources\/grub\/config\/{x60,t60,x60t}\/grub*cfg to use background.jpg background. Documentation: updated docs\/#grub_custom_keyboard to be more generally useful. nvramtool:. Updated builddeps-coreboot script to build it. Updated build script to include it in libreboot_bin. Documentation: added docs\/security\/x60_security.html (security hardening for X60). Binary blob policy Edit this page Who develops Libreboot? 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{"id":"hW3qM2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/gnulinux\/guix.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":45,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","gnulinux"],"url_file_name_s":"guix","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Encrypted Guix GNU+Linux"],"last_modified":"Sat Apr 02 12:48:35 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2559859293611837266,"exact_signature_unique_b":true,"fuzzy_signature_l":-9146506490249532520,"fuzzy_signature_unique_b":true,"h1_txt":["Encrypted Guix GNU+Linux","Navigate this page:","Objective","Scope","Process","Conclusion","References","Acknowledgements"],"h2_txt":["Preparation","Pre-Installation","Installation","Post-Installation"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":373,"wordcount_i":1152,"linkscount_i":25,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":7,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/gnulinux\/guix.html","libreboot.org\/docs\/gnulinux\/","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/gnulinux\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/gnulinux\/guix.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Objective","Return to index","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","006-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","ftp.gnu.org\/gnu\/guix\/guix-system-install-a.b.c.sss-linux.iso.xz","ftp.gnu.org\/gnu\/guix\/guix-system-install-a.b.c.sss-linux.iso.xz.sig","guix.gnu.org\/manual\/en\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","","","","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:07 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:23 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":15259,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078567894581248,"text_t":"Libreboot – Encrypted Guix GNU+Linux. Encrypted Guix GNU+Linux. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Objective Scope Process Preparation Pre-Installation Installation Post-Installation Conclusion References Acknowledgements Return to index Objective. [link] To provide step-by-step guide for setting up guix system (stand-alone guix) with full disk encryption (including \/boot) on devices powered by Libreboot. Scope. [link] Any users, for their generalised use cases, need not stumble away from this guide to accomplish the setup. Advanced users, for deviant use cases, will have to explore outside this guide for customisation; although this guide provides information that is of paramount use. Process. [link] Preparation. [link] In your current GNU+Linux System, open terminal as root user. Insert USB drive and get the USB device name \/dev\/sdX, where “X” is the variable to make a note of. lsblk. Unmount the USB drive just in case if it’s auto-mounted. umount \/dev\/sdX. Download the latest (a.b.c) Guix System ISO Installer Package (sss) and it’s GPG Signature; where “a.b.c” is the variable for version number and “sss” is the variable for system architecture. wget https:\/\/ftp.gnu.org\/gnu\/guix\/guix-system-install-a.b.c.sss-linux.iso.xz wget https:\/\/ftp.gnu.org\/gnu\/guix\/guix-system-install-a.b.c.sss-linux.iso.xz.sig. Import required public key. gpg --keyserver pool.sks-keyservers.net --recv-keys 3CE464558A84FDC69DB40CFB090B11993D9AEBB5. Verify the GPG Signature of the downloaded package. gpg --verify guix-system-install-a.b.c.sss-linux.iso.xz.sig. Extract the ISO Image from the downloaded package. xz --decompress guix-system-install-a.b.c.sss-linux.iso.xz. Write the extracted ISO Image to the USB drive. dd if=guix-system-install-a.b.c.sss-linux.iso of=\/dev\/sdX; sync. Reboot the device. reboot. Pre-Installation. [link] On reboot, as soon as you see the Libreboot Graphic Art, press arrow keys to change the menu entry. Choose “Search for GRUB2 configuration on external media [s]” and wait for the Guix System from USB drive to load. Set your keyboard layout lo, where “lo” is the two-letter keyboard layout code (example: us or uk). loadkeys lo. Unblock network interfaces (if any). rfkill unblock all. Get the names of your network interfaces. ifconfig -a. Bring your required network interface nwif (wired or wireless) up, where “nwif” is the variable for interface name. For wired connections, this should be enough. ifconfig nwif up. For wireless connection, create a configuration file using text editor, where “fname” is the variable for any desired filename. nano fname.conf. Choose, type and save ONE of the following snippets, where ‘nm’ is the name of the network you want to connect, ‘pw’ is the corresponding network’s password or passphrase and ‘un’ is user identity. For most private networks:. network={ ssid=\"nm\" key_mgmt=WPA-PSK psk=\"pw\" }. (or). For most public networks:. network={ ssid=\"nm\" key_mgmt=NONE }. (or). For most organisational networks:. network={ ssid=\"nm\" scan_ssid=1 key_mgmt=WPA-EAP identity=\"un\" password=\"pw\" eap=PEAP phase1=\"peaplabel=0\" phase2=\"auth=MSCHAPV2\" }. Connect to the configured network, where “fname” is the filename and “nwif” is the network interface name. wpa_supplicant -c fname.conf -i nwif -B. Assign an IP address to your network interface, where “nwif” is the network interface name. dhclient -v nwif. If your Guix installation image doesn’t have support for LVM, do the following. guix pull --branch=master && guix install lvm2. Obtain the device name \/dev\/sdX in which you would like to deploy and install Guix System, where “X” is the variable to make a note of. lsblk. Wipe the respective device. Wait for the command operation to finish. shred --random-source=\/dev\/urandom \/dev\/sdX. Load device-mapper module in the current kernel. modprobe dm_mod. Partition the respective device. Just do, GPT – New – Write – Quit; defaults will be set. cfdisk \/dev\/sdX. Encrypt the respective partition. cryptsetup --verbose --hash whirlpool --cipher serpent-xts-plain64 --verify-passphrase --use-random --key-size 512 --iter-time 500 luksFormat \/dev\/sdX1. Obtain and note down the “LUKS UUID”. cryptsetup luksUUID \/dev\/sdX1. Open the respective encrypted partition and map it as ‘fde’. cryptsetup luksOpen \/dev\/sdX1 fde. Create a physical volume in the partition. pvcreate \/dev\/mapper\/fde. Create a volume group in the physical volume, named ‘matrix’. vgcreate matrix \/dev\/mapper\/fde. Create a logical volume of 2GiB for swap, named ‘swapvol’. lvcreate --size 2G matrix --name swapvol. Create a logical volume of rest of free-space for root, named ‘rootvol’. lvcreate --extents 100%FREE matrix --name rootvol. Create swap space in the logical volume ‘swapvol’, labeled ‘swap’. mkswap --label swap \/dev\/matrix\/swapvol. Create filesystem in the logical volume ‘rootvol’, labeled ‘root’. mkfs.btrfs --metadata dup --label root \/dev\/matrix\/rootvol. Mount the root filesystem under the current system. mount --label root --target \/mnt --types btrfs. Installation. [link] Make the installation packages to be written on the respective mounted filesystem. herd start cow-store \/mnt. Create the required directory. mkdir \/mnt\/etc. Create, edit and save the configuration file by typing the following code snippet. WATCH-OUT for variables in the code snippet and replace them with your relevant values. nano \/mnt\/etc\/config.scm. Snippet:. (use-modules (gnu) (gnu system nss)) (use-package-modules certs gnome linux) (use-service-modules desktop xorg) (operating-system (kernel linux-libre-lts) (bootloader (bootloader-configuration (bootloader (bootloader (inherit grub-bootloader) (installer #~(const #t)))) (keyboard-layout keyboard-layout))) (keyboard-layout (keyboard-layout \"xy\" \"altgr-intl\")) (host-name \"hostname\") (mapped-devices (list (mapped-device (source (uuid \"luks-uuid\")) (target \"fde\") (type luks-device-mapping)) (mapped-device (source \"matrix\") (targets (list \"matrix-rootvol\" \"matrix-swapvol\")) (type lvm-device-mapping)))) (file-systems (append (list (file-system (type \"btrfs\") (mount-point \"\/\") (device (file-system-label \"root\")) (flags '(no-atime)) (options \"space_cache=v2\") (needed-for-boot? #t) (dependencies mapped-devices))) %base-file-systems)) (swap-devices (list (file-system-label \"swap\"))) (users (append (list (user-account (name \"username\") (comment \"Full Name\") (group \"users\") (supplementary-groups '(\"audio\" \"cdrom\" \"kvm\" \"lp\" \"netdev\" \"tape\" \"video\" \"wheel\")))) %base-user-accounts)) (packages (append (list nss-certs) %base-packages)) (timezone \"Zone\/SubZone\") (locale \"ab_XY.1234\") (name-service-switch %mdns-host-lookup-nss) (services (append (list (service gnome-desktop-service-type)) %desktop-services))). Initialise new Guix System. guix system init \/mnt\/etc\/config.scm \/mnt. Reboot the device. reboot. Post-Installation. [link] On reboot, as soon as you see the Libreboot Graphic Art, choose the option ‘Load Operating System [o]’. Enter LUKS Key, for Libreboot’s grub, as prompted. You may have to go through warning prompts by repeatedly pressing the “enter\/return” key. You will now see guix’s grub menu from which you can go with the default option. Enter LUKS Key again, for kernel, as prompted. Upon login screen, login as “root” with password field empty. Open terminal from the GNOME Dash. Set passkey for “root” user. Follow the prompts. passwd root. Set passkey for “username” user. Follow the prompts. passwd username. Update the guix distribution. Wait for the process to finish. guix pull. Update the guix system. Wait for the process to finish. guix system reconfigure \/etc\/config.scm. Reboot the device. reboot. Conclusion. [link] Everything should be stream-lined from now. You can follow your regular boot steps without requiring manual intervention. You can start logging in as regular user with the respective “username”. You will have to periodically (at your convenient time) login as root and do the update\/upgrade part of post-installation section, to keep your guix distribution and guix system updated. That is it! You have now setup guix system with full-disk encryption on your device powered by Libreboot. Enjoy! References. [link] [1] Guix Manual (http:\/\/guix.gnu.org\/manual\/en\/). Acknowledgements. [link] [1] Thanks to Guix Developer, Clement Lassieur (clement@lassieur.org), for helping me with the Guile Scheme Code for the Bootloader Configuration. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/gnulinux\/guix.md Site map This HTML page was generated by the. untitled static site generator ."}
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YOUR HELP IS NEEDED!","Navigate this page:","Introduction","What’s happening in New Hampshire?","HOW TO HELP"],"h2_txt":["TESTIFY IN SUPPORT OF THE BILL","More states\/countries will follow","OUR ENEMIES WILL BE THERE"],"bold_txt":["is","proprietary","the modern internet, where ISPs run free software almost exclusively!","money","You","Matrix","you","your","their","“Free software isn’t controlled, and is unknown.”","are","distro","Distributions","“Free software comes from potentially untrustworthy sources”","free as in zero price","free as in freedom","“Free software is unreliable \/ costly to maintain”","“Developers should get recognition for their work”","defined","shareholders","it is fun to do so and you make a lot of friends too!","paid","“Software authors deserve to be paid!”","more secure","hacker","“Free software is used by criminal hackers”","“Free software is insecure \/ you will get hacked”","against","must","YOU NEED TO GO TO NEW HAMPSHIRE IN PERSON!","Please read both threads very carefully!","nitter","Eric Gallager","Location of hearing: Legislative Office Building in Concord, New Hampshire: https:\/\/en.wikipedia.org\/wiki\/New_Hampshire_Legislative_Office_Building","The reading of the bill is happening on 11 January 2022. 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YOUR HELP IS NEEDED! New Hampshire (USA) may soon enshrine Software Freedom into law. YOUR HELP IS NEEDED! Leah Rowe. 8 January 2022. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction What’s happening in New Hampshire? HOW TO HELP TESTIFY IN SUPPORT OF THE BILL More states\/countries will follow OUR ENEMIES WILL BE THERE Return to index Article published by: Leah Rowe. Date of publication: 8 January 2022. Introduction. [link] This event of such global importance to. Free Software projects, and the Free Software movement as a whole, has made me decide to write an article. The events in question, covered by this article, will occur on 11 January 2022. This is just three days away from today, 8 January 2022 when this article was written, so if you make a decision, you should make it now, today, and prepare. Please continue reading. If you live in New Hampshire or in one of the neighbouring states, especially Massachusetts, please listen up! If you are further away and unable to reach New Hampshire all that easily, please spread the following news anyway. It’s important. As alien as it may seem to many of my readers, I’m actually writing parts of this article as though someone who has never heard of Free Software is reading it, because I expect precisely that such people. will. read this particular article. You will see the term. Free Software. used in this article, but some people call it Open Source Software. However, you should call it Free Software. The word “free” refers to freedom, not price, though the software is usually also free as in gratis \/ zero price. The opposite of Free Software is called. proprietary software. , or. non-free software. . Proponents of Open Source sometimes call non-free software. Closed Source. , but you should call it. non-free. or proprietary, to highlight the fact that it isn’t free. What’s happening in New Hampshire? [link] An important bill is being proposed in New Hampshire, which would enshrine much of what we know as Free Software. into law. . Here is the proposed bill, technically named “HB1273”:. https:\/\/gencourt.state.nh.us\/bill_status\/legacy\/bs2016\/billText.aspx?sy=2022&id=1363&txtFormat=html You can read it for yourself, but here is a paraphrasing of what it proposes:. Specifically. bans state-run websites from serving non-free javascript to clients. Creates a commission to provide oversight, watching the use of Free Software by state agencies. Bans state agencies from using proprietary software - maybe this could include schools, in the future! If a person is tried in a criminal case, they have the right to audit the source code of any proprietary software that collects evidence against them. Encourages data portability (able to transfer data from one program to another). Bans certain non-compete clauses and NDAs (non-disclosure agreements) pertaining to Free Software projects. Bans state\/local law enforcement from assisting with the enforcement of copyright claims against Free Software projects. Bans state agencies from purchasing non-free software if free software exists, for a given task. However, this is only a short summary. You are advised to read the bill in detail. It’s not very long. At first glance, it may not seem that the bill affects individuals, but don’t be fooled; this is a hugely positive step forward for everyone! If the state is using Free Software, that most likely means it’ll be used in education aswell. Although perhaps not immediately and readily apparent, this is a stake in the heart of proprietary software’s current dominance, because it would remove one key element of its attack against us; its abuse of education services. If education services are using Free Software, that means they’ll probably have children (the ones being educated) using it too. This is a. huge. step, and it will result in more Free Software developers in the future. Free Software will become more and more mainstream to the masses, which can surely only be a good thing! Freedom is always superior. The more people that have it, the better off we all are, because freedom is also collective; it relies on others around us also having it, so that we can defend each other. If more people have it, especially if it results in more Free Software developers in the future, that’s one thing, but imagine if. more. states like what they see and start to copy the new legislation. Now imagine that countries besides the US start doing it, inspired by the US’s success (and I think it will be a resounding success). Imagine a world where. Free Software , free as in freedom, is the default everywhere. Imagine a world where. Free Software licensing is required reading material in schools. Imagine a world where any five year old can install a free operating system such as GNU+Linux, and Computer Science is mandatory in schools from a young age. Imagine filing your tax returns with Free Software, exclusively. Imagine not even thinking about that, because it became the norm. Imagine a world where proprietary software doesn’t exist, because it is obsolete; entire generations of people are taught to value freedom, and to staunchly defend it, helping each other learn and grow (and produce better software in the process, with less bugs, because people are now free to do that, without relying on some evil company). Imagine a world where you’re no longer being spied on because NSA, Apple and Microsoft no longer have backdoor access to your computer. Imagine having the ability to say no, because that’s what freedom is. Try to imagine it! Free Software is a revolution that we in the Free Software movement have rigorously upheld and fought for, over many years, but we still face an uphill battle because children are not taught in schools about free computing, nor are they encouraged to learn; they are taught to view computers as. products. to throw away every 1-2 years, that they can run a few. apps. on but otherwise are not allowed to do anything with. The. concept. of a. general purpose, fully reprogrammable computer. is heavily suppressed in mainstream culture. Most. people in the world do not run a free operating system; the idea of a computer being a mere. appliance. is normalized (as opposed to the idea of it being a highly liberating tool for development and the expansion of human knowledge). This. is what we in the Free Software movement have fought for over the years. We believe that knowledge is a human right, that the ability to share, study, learn, adapt and modify the software is an inalienable right that everyone must have. The four freedoms are absolute. One of our biggest problem has been simply that schools and governments do not teach people about free computing. The right to learn, the right to read and the right to hack. Our governments are made up of human beings just like you or me, and they can be bought\/corrupted; Microsoft, Apple and many others (such as IBM) have done this for years, having the national infrastructures governing us run on their proprietary systems, instead of systems that respect freedom; it is essential that these systems run free software, because a free and democratic society should expect nothing less. Those companies buy influence. and they own your politicians. . All of this could change very soon. Something is happening in New Hampshire, which could redefine our movement and give. free software. real power instead. HOW TO HELP. [link] TESTIFY IN SUPPORT OF THE BILL. [link] The reading of the bill is happening on 11 January 2022. This is when you should go to New Hampshire. Location of hearing: Legislative Office Building in Concord, New Hampshire:. https:\/\/en.wikipedia.org\/wiki\/New_Hampshire_Legislative_Office_Building The organizer of the proposed bill,. Eric Gallager. , has left instructions on Twitter. The following is a. nitter. link, which lets you view the relevant Twitter thread without running non-free Javascript in your browser:. https:\/\/nitter.net\/cooljeanius\/status\/1479663133207764992 The original Twitter URL is:. https:\/\/twitter.com\/cooljeanius\/status\/1479663133207764992 Further instructions for what room to go to, when you get there:. See Nitter link:. https:\/\/nitter.net\/cooljeanius\/status\/1479062316532604930 (original twitter link:. https:\/\/twitter.com\/cooljeanius\/status\/1479062316532604930 ). Please read both threads very carefully! YOU NEED TO GO TO NEW HAMPSHIRE IN PERSON! If you’re able to go to New Hampshire to attend the reading of the bill, please do so! Voice your support of the bill, and say why you think it’s important. Tell the lawmakers that you demand freedom! This thread on Twitter is where Eric announced that the reading of the bill is to proceed (original Twitter URL):. https:\/\/twitter.com\/cooljeanius\/status\/1479555737223413760 More states\/countries will follow. [link] If this bill is passed in New Hampshire, more states will likely follow. It will lead to a massively renewed drive to liberate all computer users, and US laws tend to be copied\/pasted around the world too. This bill, if passed, will have a hugely positive impact on Free Software at a global level. You. must. support this bill. If you want to see it pass, please go to New Hampshire on 11 January 2022 to make sure your voice is heard. OUR ENEMIES WILL BE THERE. [link] The. proprietary. software companies like Microsoft and Apple will also be there, trying to argue the case. against. the use of Free Software. There is already precedent; please watch this video, which shows how Microsoft (for example) might behave in the reading of the bill. This video is from a discussion within the European Union, several years ago:. https:\/\/vid.puffyan.us\/watch?v=W_S0k1sx8EM (invidious link. works without javascript enabled, if you wish). They will try to trick the law makers by claiming things such as:. “Free software is insecure \/ you will get hacked”. - nothing could be further from the truth! Free operating systems such as GNU+Linux, FreeBSD and especially OpenBSD, are among the most secure operating systems available. “Free software is used by criminal hackers”. - here, they use the term. hacker. to describe someone who illegally gains access to someone elses computer. Don’t fall for it. Maintainers of free operating systems like GNU+Linux distros or the BSDs are actively working to make the internet and computers in general. more secure. “Software authors deserve to be paid!”. - In fact, many free software devs are. paid. to work on Free Software! Many companies, including big ones, work on it. There are also hobbyists or otherwise unpaid people, who might work on Free Software for a number of reasons (wanting to make the world a better place, wanting the glory of recognition for solving a major problem, and more often than not, simply because. it is fun to do so and you make a lot of friends too! ) - No, these companies (e.g. Microsoft) are only arguing in reality for the ability to pay their. shareholders. , and they control the software exclusively. In fact, free software has repeatedly and consistently over the years. defined. the computing industry, creating all kinds of new employment opportunities; for example, docker is widely used today and it is free software, used by millions of companies for commercial gain, and the apache web server revolutionized the web back in the day, enabling lots of ISPs to easily host websites - many of the common protocols that we depend upon today, that businesses depend upon (and get paid to maintain or provide services\/support for) are in fact free as in freedom! “Developers should get recognition for their work”. - in free software, you can easily make a name for yourself with relatively few resources except your own computer and an internet connection, plus some cheap hosting. When most developers work on. proprietary. software such as Windows, they don’t get recognition; their copyright is assigned to their employer (e.g. Microsoft) who will take all the credit! “Free software is unreliable \/ costly to maintain”. - actually, it has been well known for years that free software is generally more stable and reliable than proprietary. In cases where it isn’t, it is quickly improved, and in complete freedom. Free software has a lower cost to maintain and service, and you have a free market where you can choose who you hire to write\/maintain it for you (if you won’t do that yourself); meanwhile, proprietary software such as Windows is often full of bugs, crashes often and there is only one provider of support most of the time, who will charge a heavy price, while also charging a lot of money for the software itself - free software is. free as in freedom. , but also usually. free as in zero price. . “Free software comes from potentially untrustworthy sources”. - This is pure nonsense, because the very freedoms provided by free software (access to source code, ability to work on it yourself, and see what others did) means that people generally do not add malware to public software sources, because they’d be discovered instantly. Distributions. of GNU+Linux and other free operating systems are often maintained by many people, who verify the safety of each software package that they provide; they are also usually provided by each. distro. , in a central repository unlike with, say, Windows where you really. are. randomly executing binaries from all kinds of locations (often even without checking the cryptographic checksums of those files, to verify their integrity). It’s very hard to become infected with malware on a free system, precisely because security is handled much better; the design of unix-like operating systems in particular is also naturally more secure, due to better separation of root\/user privileges. “Free software isn’t controlled, and is unknown.”. - this is completely false. These non-free software companies are only talking about. their. control, and it’s quite telling that they completely disregard yours, in this very sentence. In fact, Free Software. is. controlled, but it’s not controlled by some external entity;. your. installation of free software is controlled by. you. . If you’re familiar with the. Matrix. films, proprietary operating systems like Windows\/MacOS are basically like the Matrix; bland, no individuality, no independent thought, everything tightly controlled. By contrast, free operating systems (such as GNU+Linux distributions or the BSDs) are like zion\/io; vibrant, full of life, buzzing with activity, everything loose and free, and everyone is different (a highly diverse culture of people from all walks of life, acting in common cause but nonetheless individuals). Meanwhile, Windows is known to have backdoors. Microsoft actively informs the NSA about how to exploit them, so that it can break into people’s computers and steal private data. Proprietary software companies are evil, and must be opposed. They know that if this bill passes, their days are numbered. Defend freedom! Don’t listen to any of the arguments against it by proprietary software companies; they don’t care about you, and instead only care about profit. They fundamentally do not want you to have any sort of freedom over your own computer, and they actively pursue tactics (such as DRM) to thwart you. Microsoft and Apple are not your friends. There is no such thing as the Windows community. When you use proprietary systems, you are isolated from everyone around you, and so are they. You. are the product, for the non-free software to exploit at the behest of their developers who only care about. money. . However, there. is. such a thing as the Free Software community. It is a vibrant community, consisting of millions of people collectively all over the world, and they are all free to work with each other infinitely. It gave us most of the technology that we take for granted today, including. the modern internet, where ISPs run free software almost exclusively! Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/news\/usa-libre.md Site map This HTML page was generated by the. untitled static site generator ."}
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ASUS KFSN4-DRE server\/workstation board. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Form factor Flash chips Native graphics initialization Memory Hex-core CPUs Board configurations Current issues Other information Return to index This is a server board using AMD hardware (Fam10h). It can also be used for building a high-powered workstation. Powered by libreboot. Flashing instructions can be found at. ..\/install\/#flashrom Form factor. [link] These boards use the SSI EEB 3.61 form factor; make sure that your case supports this. This form factor is similar to E-ATX in that the size is identical, but the position of the screws are different. Flash chips. [link] These boards use LPC flash (not SPI), in a PLCC socket. The default flash size 1MiB (8Mbits), and can be upgraded to 2MiB (16Mbits). SST49LF080A is the default that the board uses. SST49LF016C is an example of a 2MiB (16Mbits) chip, which might work. It is believed that 2MiB (16Mbits) is the maximum size available for the flash chip. DO NOT hot-swap the chip with your bare hands. Use a PLCC chip extractor. These can be found online. See. http:\/\/www.coreboot.org\/Developer_Manual\/Tools#Chip_removal_tools Native graphics initialization. [link] Native graphics initialization exists (XGI Z9s) for this board. Framebuffer- and text-mode both work. A serial port is also available. Memory. [link] DDR2 533\/667 Registered ECC. 16 slots. Total capacity up to 64GiB. Hex-core CPUs. [link] PCB revision 1.05G is the latest version of this board and the best one (the revision number is be printed on the board), if you want to use dual hex-core CPUs (Opteron 2400\/8400 series), though only two board configurations are believed to support them. Other revisions are believed to only support dual quad-core CPUs. To be sure your board supports a CPU check the official ASUS website here:. https:\/\/www.asus.com\/support\/cpu_support . Note: not all CPUs are listed. If you are running a Hex-Core CPU on any board version, please contact us. Board configurations. [link] There are 7 different configurations of this board: “standard”, 2S, iKVM, iKVM\/IST, SAS, SAS\/iKVM and SAS\/iKVM\/IST. The 2S boards have two PCI-E slots with the numbers of lanes shared, making each slot have 8 lanes. The iKVM boards are so called because they offer a remote real-time access to the machine through a removable PCI management card, their hardware is the same as the non-iKVM ones. The SAS versions have a 4-port SAS controller and a four 7-pin SAS connectors instead of the PCI-E 8x slot which is present in all the other board configurations. Note: the SAS functionality is. not supported. by libreboot. The IST versions with PCB revision 1.05G are the ones who are believed to support the six core Opteron Istanbul processors (2400 and 8400 series). Current issues. [link] There seems to be a 30 second bootblock delay (observed by tpearson); the system otherwise boots and works as expected. See. text\/kfsn4-dre\/bootlog.txt - this uses the ‘simple’ bootblock, while tpearson uses the ‘normal’ bootblock, which tpearson suspects may be a possible cause. This person says that they will look into it. This config doesn’t have the issue. Text-mode is jittery and it may not be usable, so it’s recommended to flash the BIOS with the coreboot frame-buffer image (kfsn4-dre_corebootfb.rom). The jitter disappears if using KMS once the kernel starts, but it will remain, if booting the kernel in text-mode. Booting from USB mass storage devices is not possible; neither GRUB nor SeaBIOS detect USB drives when present. USB keyboards function under both GRUB and SeaBIOS, albeit slowly under GRUB (several seconds per character typed). To install an operating system you will need a hard disk with a pre-installed OS otherwise you have to plug in another hard disk or a CD\/DVD reader in order to boot a copy of the installer of your OS, since the USB booting doesn’t work. Other information. [link] specifications Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/kfsn4-dre.md Site map This HTML page was generated by the. untitled static site generator ."}
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If power usage is a concern, then you should not use this. (we’re also not sure whether this workaround is appropriate)","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":374,"wordcount_i":1682,"linkscount_i":28,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":10,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/misc\/","libreboot.org\/docs\/gnulinux\/grub_cbfs.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/misc\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/misc\/index.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","High Pitched Whining Noise on Idle in Debian or Devuan","this guide","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","009-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","en.wikibooks.org\/wiki\/Grsecurity\/Appendix\/Grsecurity_and_PaX_Configuration_Options","help.ubuntu.com\/community\/SerialConsoleHowto","01.org\/sites\/default\/files\/documentation\/g45_vol_3_register_0_0.pdf","review.coreboot.org\/","www.mouser.com\/pdfdocs\/945gmedatasheet.pdf","web.archive.org\/web\/20210416010634\/https:\/notabug.org\/libreboot\/libreboot\/issues\/23","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","Note: using a grsec enabled kernel will disable the powertop function.","","","","","this page","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:07 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:20 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":17935,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078568691499008,"text_t":"Libreboot – Miscellaneous. Miscellaneous. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. High Pitched Whining Noise on Idle in Debian or Devuan High Pitched Whining Noise on Idle in Parabola Note: using a grsec enabled kernel will disable the powertop function. Finetune backlight control on intel gpu’s Power Management Beeps on Thinkpads Get EDID: Find out the name (model) of your LCD panel e1000e driver trouble shooting (Intel NICs) USB keyboard wakeup on GM45 laptops Return to previous index TODO: this page is very old, and could do with an update. High Pitched Whining Noise on Idle in Debian or Devuan. [link] Start powertop automatically at boot time. Included with libreboot is a script called ‘powertop.debian’. Run this as root and it will setup powertop to run with –auto-tune at boot time. Load the file in your text editor to see how it does that. sudo .\/resources\/scripts\/misc\/powertop.debian. Might want to run with –calibrate first. If powertop doesn’t work, another way (reduces battery life slightly) is to add. processor.max_cstate=2. to the. linux. line in grub.cfg, using. this guide . High Pitched Whining Noise on Idle in Parabola. [link] The following removes most of the noise. It reduces what is a high frequency whine (that not everyone can hear) to a slight buzz (which most people can’t hear or doesn’t bother most people). This is not perfect! The full solution is still not discovered but this is a step towards that. Also, in some instances you will need to run ‘sudo powertop –auto-tune’ again. This needs to be implemented properly in coreboot itself! On the X60 with coreboot or libreboot, there is a high pitched sound when idle. So far we have use processor.max_cstate=2 or idle=halt in GRUB. These consume power. Stop using them! Be root. su -. Installed powertop:. pacman -S powertop. and added the following to \/etc\/systemd\/system\/powertop.service :. [Unit] Description=Powertop tunings [Service] Type=oneshot RemainAfterExit=no ExecStart=\/usr\/bin\/powertop --auto-tune \"powertop --auto-tune\" still needs a terminal for some reason. Possibly a bug? Environment=\"TERM=xterm\" [Install] WantedBy=multi-user.target. Finally, as root do that:. systemctl enable powertop systemctl start powertop. The next time you boot the system, the buzz will be gone. Might want to run with –calibrate first. If powertop doesn’t work, another way (reduces battery life slightly) is to add. processor.max_cstate=2. to the. linux. line in grub.cfg, using. this guide . X60\/T60: Serial port - how to use (for dock owners). Note: using a grsec enabled kernel will disable the powertop function. [link] For the Thinkpad X60 you can use the “UltraBase X6” dock (for the X60 Tablet it is called X6 Tablet UltraBase). For the ThinkPad T60, you can use the “Advanced Mini Dock”. If you are using one of the ROM images with ‘serial’ in the name, then you have serial port enabled in libreboot and you have memtest86+ included inside the ROM. Connect your null modem cable to the serial port on the dock and connect the other end to a 2nd system using your USB Serial adapter. On the 2nd system, you can try this (using GNU Screen): sudo screen \/dev\/ttyUSB0 115200 How to quit GNU Screen: Ctrl+A then release and press K, and then press Y. There are also others like Minicom but I like GNU Screen By doing this before booting the X60\/T60, you will see console output from libreboot. You will also see GRUB displaying on the serial output, and you will be able to see MemTest86+ on the serial output aswell. You can also configure your distro so that a terminal (TTY) is accessible from the serial console. The following guide is for Ubuntu, but it should work in Debian and Devuan, to enable a serial console using GeTTY: https:\/\/help.ubuntu.com\/community\/SerialConsoleHowto (we DO NOT recommend Ubuntu, because it contains non-free software in the default repos. Use Debian or Devuan) Note: part of the tutorial above requires changing your grub.cfg. Just change the linux line to add instructions for enabling getty. See ..\/gnulinux\/grub_cbfs.html . Finetune backlight control on intel gpu’s. [link] Sometimes the backlight control value (BLC_PWM_CTL) set by libreboot is not ideal. The result is either flicker, which could cause nausea or epilepsy or an uneven backlight and\/or coil whine coming from the display. To fix this a different value for the gpu reg BLC_PWM_CTL needs to be set. See p94 of https:\/\/01.org\/sites\/default\/files\/documentation\/g45_vol_3_register_0_0.pdf for more information on this reg. The tool for setting registry values on intel gpu’s is included in intel-gpu-tools. Install intel-gpu-tools: sudo apt-get install intel-gpu-tools You can set values: sudo intel_reg write 0x00061254 your_value_in_C_hex_format NOTE: on older versions of this utility, use intel_reg_write instead. The value set has the following structure: bits [31:16] is PWM divider. PWM \/ PWM_divider = frequency bits [15:0] is the duty cycle and determines the portion of the backlight modulation frequency. A value of 0 will mean that the display is off. A value equal to the backlight modulation frequency means full on. The value should not be larger than the backlight modulation frequency. On displays with a CCFL backlight start from: 0x60016001 To verify if all modes work as desired use: xbacklight -set 10 and gradually increase until 100. Displays with an LED backlight need a lower backlight modulation. Do the same thing but start from 0x01290129 . Try setting different values until you have found a value which presents no issue. It is important to know that there are four failure modes: flickering very fast, which could cause epilepsy (frequency is too low. decrease divider). randomly flickering in random intervals - driver IC cannot keep up with toggling the mosfet (frequency too fast, increase divider). frequency is in audible range and causes coils to whine (frequency is too high. increase divider). backlight is uneven. (CCFL specific, frequency is too high. increase divider). To check for flickering try moving your laptop while looking at it. A higher frequency equals higher power consumption. You want to find the highest good working value. Next this value should be set at boot: either add intel_reg write 0x00061254 &ltyour_ideal_value NOTE: on older versions of this utility, use intel_reg_write instead. before exit 0 in \/etc\/rc.local or create a systemd service file \/etc\/systemd\/system\/backlight.service: [Unit] Description=Set BLC_PWM_CTL to a good value [Service] Type=oneshot RemainAfterExit=no ExecStart=\/usr\/bin\/intel_reg write 0x00061254 &ltyour_value [Install] WantedBy=multi-user.target Now start and enable it: sudo systemctl start backlight && sudo systemctl enable backlight Special note on i945: i945 behaves differently. Bit 16 needs to be 1 and the duty cycle is not updated when backlight is changed. There are no available datasheets on this target so why it behaves in such a way is not known. So to find a working value BLC_PWM_CTL set bit 16 to 1 and make sure PWM divider == duty cycle. see https:\/\/review.coreboot.org\/#\/c\/10624\/ on bit 16. The cause of this issue is that i945, in contrast with to GM45, is set to work in BLM Legacy Mode. This makes backlight more complicated since the duty cycle is derived from 3 instead of 2 registers using the following formula: if(BPC[7:0] xFF) then BPCR[15:0] * BPC[7:0] Else BPCR[15:0] BPC is LBB - PCI Backlight Control Register, described on http:\/\/www.mouser.com\/pdfdocs\/945gmedatasheet.pdf on page 315. BPCR is BLC_PWM_CTL described in https:\/\/01.org\/sites\/default\/files\/documentation\/g45_vol_3_register_0_0.pdf on page 94. More research needs to be done on this target so proceed with care. Power Management Beeps on Thinkpads. [link] When disconnecting or connecting the charger, a beep occurs. When the battery goes to a critically low charge level, a beep occurs. Nvramtool is included in libreboot, and can be used to enable or disable this behaviour. You need to write changes in a libreboot rom image, and flash it, in order to apply them. You can either use a pre-compiled rom image, or create an image from the current one in your computer. See here https:\/\/libreboot.org\/docs\/gnulinux\/grub_cbfs.html#get-the-rom-image for more information on how to do that. Once you have a libreboot rom image, say ‘libreboot.rom’, you can write changes on the image with the following commands. Disable or enable beeps when removing\/adding the charger: sudo .\/nvramtool -C libreboot.rom -w power_management_beeps=Enable sudo .\/nvramtool -C libreboot.rom -w power_management_beeps=Disable Disable or enable beeps when battery is low: sudo .\/nvramtool -C libreboot.rom -w low_battery_beep=Enable sudo .\/nvramtool -C libreboot.rom -w low_battery_beep=Disable You can check that the parameters are set in the image with : sudo .\/nvramtool -C libreboot.rom -a Finally, you need to flash the rom with this new image. See here https:\/\/libreboot.org\/docs\/gnulinux\/grub_cbfs.html#with-re-flashing-the-rom for a detailed explanation. Get EDID: Find out the name (model) of your LCD panel. [link] Get the panel name: sudo get-edid | strings Or look in \/sys\/class\/drm\/card0-LVDS-1\/edid Alternatively you can use i2cdump. In Debian and Devuan, this is in the package i2c-tools. sudo modprobe i2c-dev sudo i2cdump -y 5 0x50 (you might have to change the value for -y) sudo rmmod i2c-dev You’ll see the panel name in the output (from the EDID dump). If neither of these options work (or they are unavailable), physically removing the LCD panel is an option. Usually, there will be information printed on the back. e1000e driver trouble shooting (Intel NICs). [link] Example error, ¿may happen on weird and complex routing schemes(citation needed for cause): e1000e 0000:00:19.0 enp0s25: Detected Hardware Unit Hang Possible workaround, tested by Nazara: Disable C-STATES. NOTE: this also disables power management, because disabling C-States means that your CPU will now be running at full capacity (and therefore using more power) non-stop, which will drain battery life if this is a laptop. If power usage is a concern, then you should not use this. (we’re also not sure whether this workaround is appropriate). To disable c-states, do this in GNU+Linux: for i in \/sys\/devices\/system\/cpu\/cpu\/cpuidle\/state\/disable; do echo 1 $i; done You can reproduce this issue more easily by sending lots of traffic across subnets on the same interface (NIC). More information, including logs, can be found on this page . USB keyboard wakeup on GM45 laptops. [link] Look at resources\/scripts\/helpers\/misc\/libreboot_usb_bugfix Put this script in \/etc\/init.d\/ on debian-based systems. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page: https:\/\/libreboot.org\/docs\/misc\/index.md Site map This HTML page was generated by the untitled static site generator ."}
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ThinkPad X60 Tablet Recovery guide. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Brick type 1: bucts not reset. Brick type 2: bad ROM image Return to index This section documents how to recover from a bad flash that prevents your ThinkPad X60 Tablet from booting. ROM images for this machine are well-tested in Libreboot, so bricks are rare. The most common cause of a brick is operator error, when flashing a ROM image. In. most. cases, the cause will be that there is no bootblock, or an invalid one. Brick type 1: bucts not reset. [link] You still have Lenovo BIOS, or you had libreboot running and you flashed another ROM; and you had bucts 1 set and the ROM wasn’t dd’d.* or if Lenovo BIOS was present and libreboot wasn’t flashed. There are. 2. 64KiB bootblocks possible, in the upper part of the ROM image. By default (bucts set to 0), the top one is used. If bucts is set to 1, the lower one (the one before the top one) is used. This bootblock is the first code that executes, during. romstage. as per coreboot hardware initialization. BUC is short for. Backup Control. and TS is short for. Top Swap. . This is a special register on Intel platforms. Lenovo BIOS sets PRx registers, preventing software re-flashing, but there is a bug in the protection, allowing everything. except. the upper 64KiB from being flashed. By default, coreboot only puts a bootblock in the upper region. If you flash such a ROM, while bucts is set to 1, the system won’t boot because there’s not a valid bootblock; this is common if you’re re-flashing when coreboot is already installed, and you didn’t set bucts back to 0. When you install on X60\/T60 the first time, you set this bucts bit to 1, then you re-flash a second time and set it back to 0. In this case, unbricking is easy: reset BUC.TS to 0 by removing that yellow cmos coin (it’s a battery) and putting it back after a minute or two:. *Those dd commands should be applied to all newly compiled X60 ROM images (the ROM images in libreboot binary archives already have this applied!):. dd if=coreboot.rom of=top64k.bin bs=1 skip=$[$(stat -c %s coreboot.rom) - 0x10000] count=64k dd if=coreboot.rom bs=1 skip=$[$(stat -c %s coreboot.rom) - 0x20000] count=64k | hexdump dd if=top64k.bin of=coreboot.rom bs=1 seek=$[$(stat -c %s coreboot.rom) - 0x20000] count=64k conv=notrunc. (doing this makes the ROM suitable for use when flashing a system that still has Lenovo BIOS running, using those instructions:. http:\/\/www.coreboot.org\/Board:lenovo\/x60\/Installation . Brick type 2: bad ROM image. [link] In this instance, you might have flashed a ROM without the top bootblock copied to the lower 64KiB section in the ROM, and you flashed the ROM for the first time (from Lenovo BIOS), in which case there is not a valid bootblock. In this scenario, you compiled a ROM that had an incorrect configuration, or there is an actual bug preventing your system from booting. Or, maybe, you set BUC.TS to 0 and shut down after first flash while Lenovo BIOS was running. In any case, your system is bricked and will not boot at all. “Unbricking” means flashing a known-good (working) ROM. The problem: you can’t boot the system, making this difficult. In this situation, external hardware (see hardware requirements above) is needed which can flash the SPI chip (where libreboot resides). Remove those screws:. Remove the HDD:. Push keyboard forward to loosen it:. Lift:. Remove those:. Also remove that (marked) and unroute the antenna cables:. For some X60T laptops, you have to unroute those too:. Remove the LCD extend board screws. Also remove those screws (see blue marks) and remove\/unroute the cables and remove the metal plate:. Remove that screw and then remove the board:. This photo shows the flash location:. This photo shows an SPI flasher used, with SOIC8 test clip:. Refer to the external flashing guide:. Externally rewrite 25xx NOR flash via SPI protocol NOTE: Do not use the 3.3v rail from your SPI programmer. Leave that disconnected. For 3.3v, plug your charger into the mainboard (but do not power on the mainboard) when the clip is connected. Before removing the clip, disconnect the charger. This will provide adequate 3.3v DC at correct current levels. The SPI flash on an X60 Tablet shares a common 3.3V rail with many other components on the mainboard, which all draw a lot of current, more than most flashers can provide. Example command:. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev0.0,spispeed=4096 -w libreboot.rom -V. If flashrom complains about multiple flash chips detected, just pass the. -c. option as it suggests, and pick any of the chips it lists. spispeed=4096. or lower (e.g. . spispeed=512. ) is recommended on this board. The flashing becomes unstable, on this machine, when you use higher speeds. Reverse the steps to re-assemble your system, after you’ve flashed the chip. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/x60tablet_unbrick.md Site map This HTML page was generated by the. untitled static site generator ."}
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This is important, if you want 64-bit booting on your system. 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Installing GNU+Linux. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction Prepare the USB Drive in GNU+Linux Prepare the USB drive in NetBSD Prepare the USB drive in FreeBSD Prepare the USB drive in LibertyBSD or OpenBSD Debian or Devuan net install Booting ISOLINUX Images (Automatic Method) Booting ISOLINUX Images (Manual Method) Troubleshooting debian-installer Graphical Corruption in Text-Mode (Debian and Devuan) Return to index Introduction. [link] This guide assumes that you are using the GNU GRUB bootloader directly. If you’re using SeaBIOS, it’s quite intuitive and works similarly to other BIOS software; refer to the documentation on. https:\/\/seabios.org\/SeaBIOS . This guide explains how to prepare a bootable USB for Libreboot systems that can be used to install several GNU+Linux distributions. For this guide, you will only need a USB flash drive and the. dd. utility (it’s installed into all GNU+Linux distributions, by default). These instructions are intended to be generic, applicable to just about any GNU+Linux distribution. Prepare the USB Drive in GNU+Linux. [link] If you downloaded your ISO while on an existing GNU+Linux system, here is how to create the bootable GNU+Linux USB drive:. Connect the USB drive. Check. lsblk. , to confirm its device name (e.g.,. \/dev\/sdX. ):. lsblk. For this example, let’s assume that our drive’s name is. sdb. . Make sure that it’s not mounted:. sudo umount \/dev\/sdb. Overwrite the drive, writing your distro ISO to it with. dd. . For example, if we are installing. Foobarbaz. GNU+Linux, and it’s located in our Downloads folder, this is the command we would run:. sudo dd if=~\/Downloads\/foobarbaz.iso of=\/dev\/sdb bs=8M; sync. That’s it! You should now be able to boot the installer from your USB drive (the instructions for doing so will be given later). Prepare the USB drive in NetBSD. [link] This page on the NetBSD website shows how to create a NetBSD bootable USB drive, from within NetBSD itself. You should the. dd. method documented there. This will work with any GNU+Linux ISO image. Prepare the USB drive in FreeBSD. [link] This page on the FreeBSD website shows how to create a bootable USB drive for installing FreeBSD. Use the. dd. method documented. This will work with any GNU+Linux ISO image. Prepare the USB drive in LibertyBSD or OpenBSD. [link] If you downloaded your ISO on a LibertyBSD or OpenBSD system, here is how to create the bootable GNU+Linux USB drive:. Connect the USB drive. Run. lsblk. to determine which drive it is:. lsblk. To confirm that you have the correct drive, use. disklabel. . For example, if you thought the correct drive were. sd3. , run this command:. disklabel sd3. Make sure that the device isn’t mounted, with. doas. ; if it is, this command will unmount it:. doas umount \/dev\/sd3i. The. lsblk. command told you what device it is. Overwrite the drive, writing the OpenBSD installer to it with. dd. . Here’s an example:. doas dd if=gnulinux.iso of=\/dev\/rsdXc bs=1M; sync. That’s it! You should now be able to boot the installer from your USB drive (the instructions for doing so will be given later). Debian or Devuan net install. [link] Download the Debian or Devuan net installer. You can download the Debian ISO from. the Debian homepage , or the Devuan ISO from. the Devuan homepage . Secondly, create a bootable USB drive using the commands in. #prepare-the-usb-drive-in-gnulinux . Thirdly, boot the USB and enter these commands in the GRUB terminal (for 64-bit Intel or AMD):. set root='usb0' linux \/install.amd\/vmlinuz initrd \/install.amd\/initrd.gz boot. If you are on a 32-bit system (e.g. some Thinkpad X60’s) then you will need to use these commands (this is also true for 32-bit running on 64-bit machines):. set root='usb0' linux \/install.386\/vmlinuz initrd \/install.386\/initrd.gz boot. Booting ISOLINUX Images (Automatic Method). [link] Boot it in GRUB using the. Parse ISOLINUX config (USB). option. A new menu should appear in GRUB, showing the boot options for that distro; this is a GRUB menu, converted from the usual ISOLINUX menu provided by that distro. Booting ISOLINUX Images (Manual Method). [link] These are generic instructions. They may or may not be correct for your distribution. You must adapt them appropriately, for whatever GNU+Linux distribution it is that you are trying to install. If the. ISOLINUX parser. or. Search for GRUB configuration. options won’t work, then press. C. in GRUB to access the command line, then run the. ls. command:. ls. Get the device name from the above output (e.g.,. usb0. ). Here’s an example:. cat (usb0)\/isolinux\/isolinux.cfg. Either the output of this command will be the ISOLINUX menuentries for that ISO, or link to other. .cfg. files (e.g,. \/isolinux\/foo.cfg. ). For example, if the file found were. foo.cfg. , you would use this command:. cat (usb0)\/isolinux\/foo.cg`. And so on, until you find the correct menuentries for ISOLINUX. For Debian-based distros (e.g., Ubuntu, Devuan), there are typically menuentries listed in. \/isolinux\/txt.cfg. or. \/isolinux\/gtk.cfg. . For dual-architecture ISO images (i686 and x86_64), there may be separate files directories for each architecture. Just keep searching through the image, until you find the correct ISOLINUX configuration file. NOTE: Debian 8.6 ISO only lists 32-bit boot options in txt.cfg. This is important, if you want 64-bit booting on your system. Devuan versions based on Debian 8.x may also have the same issue. Now, look at the ISOLINUX menuentry; it’ll look like this:. kernel \/path\/to\/kernel append PARAMETERS initrd=\/path\/to\/initrd ... GRUB works similarly; here are some example GRUB commands:. set root='usb0' linux \/path\/to\/kernel PARAMETERS MAYBE_MORE_PARAMETERS initrd \/path\/to\/initrd boot. Note:. usb0. may be incorrect. Check the output of the. ls. command (in GRUB), to see a list of USB devices\/partitions. Of course, this will vary from distro to distro. If you did all of that correctly, then it should now be booting your USB drive in the way that you specified. Troubleshooting. [link] Most of these issues occur when using Libreboot with coreboot’s. text-mode. with libgfxinit for video initialization. This mode is useful for text mode payloads, like. MemTest86+. , which expect. text-mode. , but for GNU+Linux distributions it can be problematic when they are trying to switch to a framebuffer, because no mode switching support is present (Linux\/BSD kernels do Kernel Mode Setting, so they are able to initialize a frame buffer in bare metal regardless of whatever coreboot is doing). debian-installer Graphical Corruption in Text-Mode (Debian and Devuan). [link] When using the ROM images that use Coreboot’s. text mode. , instead of the coreboot framebuffer, while using libgfxinit, booting the Debian or Devuan net installer results in graphical corruption, because it is trying to switch to a framebuffer while no mode switching support is present. Use this kernel parameter on the. linux. line, when booting it:. fb=false. This forces debian-installer to start in. text-mode. , instead of trying to switch to a framebuffer. If selecting. text-mode. from a GRUB menu created using the ISOLINUX parser, you can press. E. on the menu entry to add this. Or, if you are booting manually (from GRUB terminal), then just add the parameters. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/gnulinux\/grub_boot_installer.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"Ukxoo2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/kgpe-d16.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"kgpe-d16","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – ASUS KGPE-D16 server\/workstation board"],"last_modified":"Sat Apr 02 12:48:38 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-2539364165999199172,"exact_signature_unique_b":true,"fuzzy_signature_l":8568848225042008047,"fuzzy_signature_unique_b":true,"h1_txt":["ASUS KGPE-D16 server\/workstation board","Navigate this page:","Introduction","CPU compatibility","Board status (compatibility)","Form factor","IPMI iKVM module add-on","Flash chips","Native graphics initialization","Current issues"],"h2_txt":["Hardware specifications"],"h3_txt":["Processor \/ system bus","Core logic","Memory compatibility (with libreboot)","Expansion slots","Form factor","ASUS features","Storage","Networking","Graphics","On board I\/O","Back I\/O ports","Environment","Monitoring","Note:"],"bold_txt":["Non operation humidity:","Non operation temperature:","Operation temperature:","SAS\/SATA Controller:","SATA controller:","Additional Slot 1:","Slot Location 6:","Slot Location 5:","Slot Location 4:","Slot Location 3:","Slot Location 2:","Slot Location 1:","Total slot:","Compatible sizes per memory module:","Memory Type that is compatible:","Capacity:","Total Slots:","AVOID LIKE THE PLAGUE - virtualization is broken without microcode updates.","highly recommended - fast, and works well without microcode updates, including virtualization","Not","DO NOT hot-swap the chip with your bare hands. Use a P-DIP 8 chip extractor. These can be found online. See http:\/\/www.coreboot.org\/Developer_Manual\/Tools#Chip_removal_tools","Use Opteron 6200 series (works without microcode updates, including hw virt).","For working configurations see https:\/\/www.coreboot.org\/Board:asus\/kgpe-d16 .","Memory initialization is still problematic, for some modules. We recommend avoiding Kingston modules.","and Fam15h","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":379,"wordcount_i":1178,"linkscount_i":30,"linksnofollowcount_i":0,"inboundlinkscount_i":19,"outboundlinkscount_i":11,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https","018-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/hardware\/kgpe-d16.html","libreboot.org\/docs\/hardware\/","libreboot.org\/docs\/install\/spi.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/hardware\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/hardware\/kgpe-d16.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Introduction","Return to index","25xx NOR flashing guide","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","007-https","010-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","www.coreboot.org\/Board:asus\/kgpe-d16","raptorengineeringinc.com\/coreboot\/kgpe-d16-status.php","www.coreboot.org\/Developer_Manual\/Tools","web.archive.org\/web\/20210416011941\/https:\/notabug.org\/libreboot\/libreboot\/issues\/736","github.com\/facebook\/openbmc","support.amd.com\/TechDocs\/41322_10h_Rev_Gd.pdf","support.amd.com\/TechDocs\/48063_15h_Mod_00h-0Fh_Rev_Guide.pdf","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","","","","Notabug issue tracker","This project","","","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:08 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:24 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":15487,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078569739026432,"text_t":"Libreboot – ASUS KGPE-D16 server\/workstation board. ASUS KGPE-D16 server\/workstation board. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction CPU compatibility Board status (compatibility) Form factor IPMI iKVM module add-on Flash chips Native graphics initialization Current issues Hardware specifications Processor \/ system bus Core logic Memory compatibility (with libreboot) Expansion slots Form factor ASUS features Storage Networking Graphics On board I\/O Back I\/O ports Environment Monitoring Note: Return to index Introduction. [link] This is a server board using AMD hardware (Fam10h. and Fam15h. CPUs available). It can also be used for building a high-powered workstation. Powered by libreboot. The coreboot port was done by Timothy Pearson of Raptor Engineering Inc. and, working with them (and sponsoring the work), merged into libreboot. Memory initialization is still problematic, for some modules. We recommend avoiding Kingston modules. For working configurations see. https:\/\/www.coreboot.org\/Board:asus\/kgpe-d16 . Flashing instructions can be found at. ..\/install\/#flashrom - note that external flashing is required (e.g. BBB), if the proprietary (ASUS) firmware is currently installed. If you already have libreboot, by default it is possible to re-flash using software running in GNU+Linux on the KGPE-D16, without using external hardware. CPU compatibility. [link] Use Opteron 6200 series (works without microcode updates, including hw virt). 6300 series needs microcode updates, so avoid those CPUs. 6100 series is too old, and mostly untested. Board status (compatibility). [link] See. https:\/\/raptorengineeringinc.com\/coreboot\/kgpe-d16-status.php . Form factor. [link] These boards use the SSI EEB 3.61 form factor; make sure that your case supports this. This form factor is similar to E-ATX in that the size is identical, but the position of the screws are different. IPMI iKVM module add-on. [link] Don’t use it. It uses proprietary firmware and adds a backdoor (remote out-of-band management chip, similar to the. Intel Management Engine . Fortunately, the firmware is unsigned (possibly to replace) and physically separate from the mainboard since it’s on the add-on module, which you don’t have to install. Flash chips. [link] 2MiB flash chips are included by default, on these boards. It’s on a P-DIP 8 slot (SPI chip). The flash chip can be upgraded to higher sizes: 4MiB, 8MiB or 16MiB. With at least 8MiB, you could feasibly fit a compressed linux+initramfs image (BusyBox+Linux system) into CBFS and boot that, loading it into memory. Libreboot has configs for 2, 4, 8 and 16 MiB flash chip sizes (default flash chip is 2MiB). DO NOT hot-swap the chip with your bare hands. Use a P-DIP 8 chip extractor. These can be found online. See. http:\/\/www.coreboot.org\/Developer_Manual\/Tools#Chip_removal_tools This guide shows how to flash the chip:. 25xx NOR flashing guide Native graphics initialization. [link] Only text-mode is known to work, but linux(kernel) can initialize the framebuffer display (if it has KMS - kernel mode setting). Current issues. [link] LRDIMM memory modules are currently incompatible. SAS (via PIKE 2008 module) requires non-free option ROM (and SeaBIOS) to boot from it (theoretically possible to replace, but you can put a kernel in CBFS or on SATA and use that to boot GNU, which can be on a SAS drive. The linux kernel can use those SAS drives (via PIKE module) without an option ROM). SeaBIOS lacked serial console support out-of-the-box in release 20160907 and as such a workaround using SGABIOS is necessary. You can find instructions on how to do this on the. Notabug issue tracker IPMI iKVM module (optional add-on card) uses proprietary firmware. Since it’s for remote out-of-band management, it’s theoretically a backdoor similar to the Intel Management Engine. Fortunately, unlike the ME, this firmware is unsigned which means that a free replacement is theoretically possible. For now, the libreboot project recommends not installing the module. This project might be interesting to derive from, for those who want to work on a free replacement. In practise, out-of-band management isn’t very useful anyway (or at the very least, it’s not a major inconvenience to not have it). Graphics: only text-mode works. See. #graphics Hardware specifications. [link] The information here is adapted, from the ASUS website. Processor \/ system bus. [link] 2 CPU sockets (G34 compatible). HyperTransport™ Technology 3.0. CPUs supported:. AMD Opteron 6100 series (Fam10h. No IOMMU support. Not. recommended - old. View errata datasheet here:. http:\/\/support.amd.com\/TechDocs\/41322_10h_Rev_Gd.pdf ). AMD Opteron 6200 series (Fam15h, with full IOMMU support in libreboot -. highly recommended - fast, and works well without microcode updates, including virtualization. ). AMD Opteron 6300 series (Fam15h, with full IOMMU support in libreboot. AVOID LIKE THE PLAGUE - virtualization is broken without microcode updates. NOTE: 6300 series CPUs have buggy microcode built-in, and libreboot recommends avoiding the updates. The 6200 series CPUs have more reliable microcode. Look at this errata datasheet:. http:\/\/support.amd.com\/TechDocs\/48063_15h_Mod_00h-0Fh_Rev_Guide.pdf (see Errata 734 - this is what kills the 6300 series). 6.4 GT\/s per link (triple link). Core logic. [link] AMD SR5690. AMD SP5100. Memory compatibility (with libreboot). [link] Total Slots:. 16 (4-channel per CPU, 8 DIMM per CPU), ECC. Capacity:. Maximum up to 256GB RDIMM (Tested max 128GB). Memory Type that is compatible:. DDR3 1600\/1333\/1066\/800 UDIMM*. DDR3 1600\/1333\/1066\/800 RDIMM*. Compatible sizes per memory module:. 16GB, 8GB, 4GB, 3GB, 2GB, 1GB RDIMM. 8GB, 4GB, 2GB, 1GB UDIMM. Expansion slots. [link] Total slot:. 6. Slot Location 1:. PCI 32bit\/33MHz. Slot Location 2:. PCI-E x16 (Gen2 X8 Link). Slot Location 3:. PCI-E x16 (Gen2 X16 Link), Auto switch to x8 link if slot 2 is occupied. Slot Location 4:. PCI-E x8 (Gen2 X4 Link). Slot Location 5:. PCI-E x16 (Gen2 X16 Link). Slot Location 6:. PCI-E x16 (Gen2 X16 Link), Auto turn off if slot 5 is occupied, For 1U FH\/FL Card, MIO supported. Additional Slot 1:. PIKE slot (for SAS drives. See notes above). Follow SSI Location#. Form factor. [link] SSI EEB 3.61 (12“x13”). ASUS features. [link] Fan Speed Control. Rack Ready (Rack and Pedestal dual use). Storage. [link] SATA controller:. AMD SP5100. 6 x SATA2 300MB\/s. SAS\/SATA Controller:. ASUS PIKE2008 3Gbps 8-port SAS card included. Networking. [link] 2 x Intel® 82574L + 1 x Mgmt LAN. Graphics. [link] Aspeed AST2050 with 8MB VRAM. On board I\/O. [link] 1 x PSU Power Connector (24-pin SSI power connector + 8-pin SSI 12V + 8-pin SSI 12V power connector). 1 x Management Connector , Onboard socket for management card. 3 x USB pin header , Up to 6 Devices. 1 x Internal A Type USB Port. 8 x Fan Header , 4pin (3pin\/4pin fan dual support). 2 x SMBus. 1 x Serial Port Header. 1 x TPM header. 1 x PS\/2 KB\/MS port. Back I\/O ports. [link] 1 x External Serial Port. 2 x External USB Port. 1 x VGA Port. 2 x RJ-45. 1 x PS\/2 KB\/Mouse. Environment. [link] Operation temperature:. 10C ~ 35C. Non operation temperature:. -40C ~ 70C. Non operation humidity:. 20% ~ 90% ( Non condensing). Monitoring. [link] CPU temperatures. Fan speed (RPM). Note:. [link] * DDR3 1600 can only be supported with AMD Opteron 6300\/6200 series processor. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/kgpe-d16.md Site map This HTML page was generated by the. untitled static site generator ."}
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Libreboot 20140309 release. Leah Rowe. 9 March 2014. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Revision notes (9th March 2014): Development notes Return to index Article published by: Leah Rowe. Date of publication: 9 March 2014. Revision notes (9th March 2014):. [link] recreated coreboot config from scratch. GRUB loads even faster now (less than 2 seconds). Total boot time reduced by further ~5 seconds. Added crypto and cryptodisk modules to GRUB. cbfstool now included in the binary archives. Development notes. [link] Binary archive now have 2 images:. With serial output enabled and memtest86+ included (debug level 8 in coreboot). With serial output disabled and memtest86+ excluded (faster boot speeds) (debugging disabled). Reduced impact on battery life:. ‘processor.max_cstate=2’ instead of ‘idle=halt’ for booting default kernel. coreboot.rom (faster boot speeds, debugging disabled):. Disabled coreboot serial output (Console- in “make menuconfig”). Set coreboot debug level to 0 instead of 8 (Console- in “make menuconfig”). Changed GRUB timeout to 1 second instead of 2 (in grub.cfg. Removed background image in GRUB. Removed memtest86+ payload (since it relies on serial output). coreboot_serial.rom (slower boot speeds, debugging enabled):. Boot time still reduced, but only by ~2 seconds. has the memtest86+ payload included in the ROM. has serial port enabled. How this is achieved (from X60_source): Turn on debugging level to 8, and enable serial output. (in Console- in coreboot “make menuconfig”). (and build with grub_serial.cfg and grub_memdisk_serial.cfg). Binary blob policy Edit this page Who develops Libreboot? 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{"id":"ltoPr2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/bsd\/netbsd.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":42,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","bsd"],"url_file_name_s":"netbsd","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – How to install NetBSD on x86 GNU GRUB payload"],"last_modified":"Sat Apr 02 12:48:34 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-8745963424650595344,"exact_signature_unique_b":true,"fuzzy_signature_l":-2622267021171219651,"fuzzy_signature_unique_b":true,"h1_txt":["How to install NetBSD on x86 GNU GRUB payload","Navigate this page:","Troubleshooting"],"h2_txt":["Prepare the USB drive (in NetBSD)","Prepare the USB drive (in FreeBSD)","Prepare the USB drive (in OpenBSD or NetBSD)","Prepare the USB drive (in GNU+Linux)","Installing NetBSD without full disk encryption","Installing NetBSD with full disk encryption","Booting","Configuring Grub","won’t boot…something about file not found"],"bold_txt":["dd","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":378,"wordcount_i":741,"linkscount_i":24,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":6,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/bsd\/netbsd.html","libreboot.org\/docs\/bsd\/","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/bsd\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/bsd\/netbsd.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Prepare the USB drive (in NetBSD)","Return to index","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","wiki.netbsd.org\/tutorials\/how_to_install_netbsd_from_an_usb_memory_stick\/","www.freebsd.org\/doc\/handbook\/bsdinstall-pre.html","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","This page","This page","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:09 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:27 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":10227,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078570269605888,"text_t":"Libreboot – How to install NetBSD on x86 GNU GRUB payload. How to install NetBSD on x86 GNU GRUB payload. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Prepare the USB drive (in NetBSD) Prepare the USB drive (in FreeBSD) Prepare the USB drive (in OpenBSD or NetBSD) Prepare the USB drive (in GNU+Linux) Installing NetBSD without full disk encryption Installing NetBSD with full disk encryption Booting Configuring Grub Troubleshooting won’t boot…something about file not found Return to index GRUB supports booting NetBSD kernels directly. However, you’re better off simply using the SeaBIOS payload; BSD works well with BIOS or UEFI setups. GRUB is acceptable for booting unencrypted BSD installations. However, encrypted BSD installations will probably require the use of SeaBIOS\/Tianocore. Prepare the USB drive (in NetBSD). [link] This page on the NetBSD website shows how to create a NetBSD bootable USB drive from within NetBSD itself. You should use the. dd. method documented there. Prepare the USB drive (in FreeBSD). [link] This page on the FreeBSD website shows how to create a bootable USB drive for installing FreeBSD. Use the. dd. on that page. You can also use the same instructions with a NetBSD ISO image. Prepare the USB drive (in OpenBSD or NetBSD). [link] If you downloaded your ISO on a OpenBSD or NetBSD system, here is how to create the bootable NetBSD USB drive:. Connect the USB drive. Check dmesg:. dmesg | tail. Check to confirm which drive it is, for example, if you think its sd3:. disklabel sd3. Check that it wasn’t automatically mounted. If it was, unmount it. For example:. doas umount \/dev\/sd3i. dmesg told you what device it is. Overwrite the drive, writing the NetBSD installer to it with dd. For example:. doas netbsd.iso of=\/dev\/rsdXc bs=1M; sync. You should now be able to boot the installer from your USB drive. Continue reading, for information about how to do that. Prepare the USB drive (in GNU+Linux). [link] If you downloaded your ISO on a GNU+Linux system, here is how to create the bootable NetBSD USB drive:. Connect the USB drive. Check dmesg:. dmesg. Check lsblk to confirm which drive it is:. lsblk. Check that it wasn’t automatically mounted. If it was, unmount it. For example:. sudo umount \/dev\/sdX\\* umount \/dev\/sdX\\*. dmesg told you what device it is. Overwrite the drive, writing your distro ISO to it with dd. For example:. sudo dd if=install60.fs of=\/dev\/sdX bs=8M; sync dd if=netbsd.iso of=\/dev\/sdX bs=8M; sync. You should now be able to boot the installer from your USB drive. Continue reading, for information about how to do that. Installing NetBSD without full disk encryption. [link] You might have to use an external USB keyboard during the installation. Press C to access the GRUB terminal. grub knetbsd -r sd0a (usb0,netbsd1)\/netbsd grub boot. It will start booting into the NetBSD installer. Follow the normal process for installing NetBSD. Installing NetBSD with full disk encryption. [link] TODO. Booting. [link] Press C in GRUB to access the command line:. grub knetbsd -r wd0a (ahci0,netbsd1)\/netbsd grub boot. NetBSD will start booting. Yay! Configuring Grub. [link] If you don’t want to drop to the GRUB command line and type in a command to boot NetBSD every time, you can create a GRUB configuration that’s aware of your NetBSD installation and that will automatically be used by libreboot. On your NetBSD root partition, create the. \/grub. directory and add the file. libreboot_grub.cfg. to it. Inside the. libreboot_grub.cfg. add these lines:. default=0 timeout=3 menuentry \"NetBSD\" { knetbsd -r wd0a (ahci0,netbsd1)\/netbsd }. The next time you boot, you’ll see the old Grub menu for a few seconds, then you’ll see the a new menu with only NetBSD on the list. After 3 seconds NetBSD will boot, or you can hit enter to boot. Troubleshooting. [link] Most of these issues occur when using Libreboot with coreboot’s ‘text mode’ instead of the coreboot framebuffer. This mode is useful for booting payloads like memtest86+ which expect text-mode, but for NetBSD it can be problematic when they are trying to switch to a framebuffer because it doesn’t exist. won’t boot…something about file not found. [link] Your device names (i.e. usb0, usb1, sd0, sd1, wd0, ahci0, hd0, etc) and numbers may differ. Use TAB completion. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/bsd\/netbsd.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"3PDeq2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/install\/x200_external.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":53,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","install"],"url_file_name_s":"x200_external","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – First-time ThinkPad X200 flashing"],"last_modified":"Sat Apr 02 12:48:40 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":8547451180479756292,"exact_signature_unique_b":true,"fuzzy_signature_l":-7554200842115204304,"fuzzy_signature_unique_b":true,"h1_txt":["First-time ThinkPad X200 flashing","Navigate this page:","Flash chip size","MAC address","Memory"],"h2_txt":["The procedure","Boot it!","X200S and X200 Tablet users: GPIO33 trick will not work."],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":12,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico","av.libreboot.org\/x200\/disassembly\/0003.jpg","av.libreboot.org\/x200\/disassembly\/0004.jpg","av.libreboot.org\/x200\/disassembly\/0005.jpg","av.libreboot.org\/x200\/disassembly\/0006.1.jpg","av.libreboot.org\/x200\/disassembly\/0006.jpg","av.libreboot.org\/x200\/x200_soic16.jpg","av.libreboot.org\/x200\/x200_soic8.jpg","av.libreboot.org\/x200\/disassembly\/0008.jpg","av.libreboot.org\/x200\/disassembly\/0018.jpg","av.libreboot.org\/x200\/disassembly\/0019.jpg","av.libreboot.org\/x200\/gpio33_location.jpg"],"images_alt_sxt":["Libreboot logo","","","","","","","","","","",""],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"libreboot org favicon ico Libreboot logo av x200 disassembly 0003 jpg x 200 0004 0005 0006 1 soic16 soic 16 soic8 8 0008 0018 0019 gpio33 location gpio 33","responsetime_i":374,"wordcount_i":940,"linkscount_i":26,"linksnofollowcount_i":0,"inboundlinkscount_i":21,"outboundlinkscount_i":5,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https","018-https","019-https","020-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/install\/x200_external.html","libreboot.org\/docs\/hardware\/mac_address.html","libreboot.org\/docs\/install\/spi.html","libreboot.org\/docs\/gnulinux\/","libreboot.org\/docs\/install\/ich9utils.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/install\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/install\/x200_external.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Flash chip size","mac_address.html","SPI programming instructions","install GNU+Linux","ich9utils.html#demefactory","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","004-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","www.coreboot.org\/Board:lenovo\/x200\/internal_flashing_research","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:09 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:24 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":10999,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078570513924096,"text_t":"Libreboot – First-time ThinkPad X200 flashing. First-time ThinkPad X200 flashing. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Flash chip size MAC address The procedure Memory Boot it! X200S and X200 Tablet users: GPIO33 trick will not work. Return to index This guide is for those who want libreboot on their ThinkPad X200 while they still have the original Lenovo BIOS present. This guide can also be followed (adapted) if you brick your X200, to know how to recover. If you have the original Lenovo firmware running, you will need to take the keyboard and palmrest off so that you can access the flash chip, which is just underneath the palm rest. You will then connect an external SPI programmer, to re-flash the chip externally while it is powered off with the battery removed. NOTE: This guide only applies to the regular X200. For X200S and X200 Tablet flashing, please read other guides available on libreboot.org. Flash chip size. [link] Run this command on x200 to find out flash chip model and its size:. flashrom -p internal. MAC address. [link] Refer to. mac_address.html . The procedure. [link] This section is for the X200. This does not apply to the X200S or X200 Tablet (for those systems, you have to remove the motherboard completely, since the flash chip is on the other side of the board). Remove these screws:. Gently push the keyboard towards the screen, then lift it off, and optionally disconnect it from the board:. Disconnect the cable of the fingerpring reader, and then pull up the palm rest, lifting up the left and right side of it:. This shows the location of the flash chip, for both SOIC-8 and SOIC-16:. Lift back the tape that covers a part of the flash chip, and then connect the clip:. Now, you should be ready to install libreboot. Refer to the. SPI programming instructions . When you’re done, put the system back together. If it doesn’t boot, try other RAM modules because raminit is very unreliable on this platform (in coreboot). Memory. [link] In DDR3 machines with Cantiga (GM45\/GS45\/PM45), northbridge requires sticks that will work as PC3-8500 (faster PC3\/PC3L sticks can work as PC3-8500). Non-matching pairs may not work. Single module (meaning, one of the slots will be empty) will currently only work in slot 0. NOTE: according to users reports, non matching pairs (e.g. 1+2 GiB) might work in some cases. Make sure that the RAM you buy is the 2Rx8 configuration when buying 4GiB sticks (In other words: maximum of 2GiB per rank, 2 ranks per card). In this photo, 8GiB of RAM (2x4GiB) is installed:. Boot it! [link] You should see something like this:. Now. install GNU+Linux . X200S and X200 Tablet users: GPIO33 trick will not work. [link] sgsit found out about a pin called GPIO33, which can be grounded to disable the flashing protections by the descriptor and stop the ME from starting (which itself interferes with flashing attempts). The theory was proven correct; however, it is still useless in practise. Look just above the 7 in TP37 (that’s GPIO33):. By default we would see this in lenovobios, when trying flashrom -p internal -w rom.rom:. FREG0: Warning: Flash Descriptor region (0x00000000-0x00000fff) is read-only. FREG2: Warning: Management Engine region (0x00001000-0x005f5fff) is locked. With GPIO33 grounded during boot, this disabled the flash protections as set by descriptor, and stopped the ME from starting. The output changed to:. The Flash Descriptor Override Strap-Pin is set. Restrictions implied by the Master Section of the flash descriptor are NOT in effect. Please note that Protected Range (PR) restrictions still apply. The part in bold is what got us. This was still observed:. PR0: Warning: 0x007e0000-0x01ffffff is read-only. PR4: Warning: 0x005f8000-0x005fffff is locked. It is actually possible to disable these protections. Lenovobios does, when updating the BIOS (proprietary one). One possible way to go about this would be to debug the BIOS update utility from Lenovo, to find out how it’s disabling these protections. Some more research is available here:. http:\/\/www.coreboot.org\/Board:lenovo\/x200\/internal_flashing_research Of course, it’s likely that the Lenovo BIOS is checking for some bit in memory that tells it not to disable flashing, and then it won’t set PRx registers. The way the Lenovo BIOS updater works is, it is executed in Windows first and then a reboot happens, triggering the re-flashing to happen during early boot. It is probably setting something in memory and loading the ROM, plus a payload program that does the flashing; Lenovo BIOS then probably sees that and runs that, instead of setting PRx and going for normal boot. It is theoretically possible that we could discover how this works, by debugging the Lenovo BIOS update utility (in Windows), and then replicate what it is doing, with some tool for GNU+Linux, then load a flashrom binary into memory and the ROM to flash (for the BIOS region). You would do this with GPIO33 grounded, and the payload program would actually flash the entire chip, with just a normal Libreboot image. It’s possible. The above is likely the only way that the Lenovo BIOS updater program works. So if we discover precisely how to do that, then you could just connect some pogo pins to ground GPIO33, then boot up, run some software (which would have to be written) that does the above. On a related note, libreboot has a utility that could help with investigating this:. ich9utils.html#demefactory Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/x200_external.md Site map This HTML page was generated by the. untitled static site generator ."}
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To see which panel type you have, see #led_howtotell . If you need the inverter\/cable, here are part numbers: 44C9909 for CCFL LVDS cable with bluetooth and camera connections, and 42W8009 or 42W8010 for the inverter.","resources\/utilities\/x200t-screen-rotation","kvm_intel","kvm","The X200 laptops come with the ME (and sometimes AMT in addition) before flashing libreboot. 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ThinkPad X200. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction EC update Battery Recall Compatibility (without blobs) Hardware virtualization (vt-x) LCD compatibility list AFFS\/IPS panels X200 X200S How to tell if it has an LED or CCFL? Hardware register dumps Return to index Introduction. [link] It is believed that all X200 laptops are compatible. X200S and X200 Tablet will also work,. depending on the configuration . It may be possible to put an X200 motherboard in an X201 chassis, though this is currently untested by the libreboot project. The same may also apply between X200S and X201S; again, this is untested. It’s most likely true. There are two possible flash chip sizes for the X200: 4MiB (32Mbit) or 8MiB (64Mbit). This can be identified by the type of flash chip below the palmrest: 4MiB is SOIC-8, 8MiB is SOIC-16. The X200 laptops come with the ME (and sometimes AMT in addition) before flashing libreboot. Libreboot disables and removes it by using a modified descriptor: see. ..\/install\/ich9utils.html (contains notes, plus instructions). Flashing instructions can be found at. ..\/install\/#flashrom EC update. [link] It is recommended that you update to the latest EC firmware version. The. EC firmware is separate from libreboot, so we don’t actually provide that, but if you still have Lenovo BIOS then you can just run the Lenovo BIOS update utility, which will update both the BIOS and EC version. See:. ..\/install\/#flashrom http:\/\/www.thinkwiki.org\/wiki\/BIOS_update_without_optical_disk X200, X200s, X200si BIOS Update X200t BIOS Update NOTE: this can only be done when you are using Lenovo BIOS. How to update the EC firmware while running libreboot is unknown. Libreboot only replaces the BIOS firmware, not EC. Updated EC firmware has several advantages e.g. better battery handling. Battery Recall. [link] On 21 April 2015, Lenovo expanded a recall on Lenovo batteries found in some ThinkPad models, which includes the X200 and X200S. To find out if you are affected, use. this Lenovo tool. Lenovo advises that owners of the recalled models “should turn off the system, remove the battery, and only power your ThinkPad by plugging in the AC adapter and power cord.” Upon battery verification, Lenovo will replace recalled batteries free of charge. Battery replacement instructions for the X200\/X200s are found. on this page. Compatibility (without blobs). [link] Hardware virtualization (vt-x). [link] The X200, when run without CPU microcode updates in coreboot, currently kernel panics if running QEMU with vt-x enabled on 2 cores for the guest. With a single core enabled for the guest, the guest panics (but the host is fine). Working around this in QEMU might be possible; if not, software virtualization should work fine (it’s just slower). On GM45 hardware (with libreboot), make sure that the. kvm. and. kvm_intel. kernel modules are not loaded, when using QEMU. The following errata datasheet from Intel might help with investigation:. https:\/\/web.archive.org\/web\/20150110195728\/https:\/\/download.intel.com\/design\/mobile\/specupdt\/320121.pdf Anecdotal reports from at least 1 user suggests that some models with CPU microcode 1067a (on the CPU itself) might work with vt-x in libreboot. LCD compatibility list. [link] LCD panel list (X200 panels listed there):. http:\/\/www.thinkwiki.org\/wiki\/TFT_display All LCD panels for the X200, X200S and X200 Tablet are known to work. The X200 Tablet has a screen rotation button on its bezel. Depending on the operating system it might or might not rotate the screen, the digitizer (stylus), or the trackpoint accordingly. Utilities are provided to fix this at. resources\/utilities\/x200t-screen-rotation. in the libreboot git repository. AFFS\/IPS panels. [link] X200. Adapted from. https:\/\/github.com\/bibanon\/Coreboot-ThinkPads\/wiki\/ThinkPad-X200 Look at wikipedia for difference between TN and IPS panels. IPS have much better colour\/contrast than a regular TN, and will typically have good viewing angles. These seem to be from the X200 tablet. You need to find one without the glass touchscreen protection on it (might be able to remove it, though). It also must not have a digitizer on it (again, might be possible to just simply remove the digitizer). BOE-Hydis HV121WX4-120, HV121WX4-110 or HV121WX4-100 - cheap-ish, might be hard to find. Samsung LTN121AP02-001 - common to find, cheap. If your X200 has an LED backlit panel in it, then you also need to get an inverter and harness cable that is compatible with the CCFL panels. To see which panel type you have, see. #led_howtotell . If you need the inverter\/cable, here are part numbers: 44C9909 for CCFL LVDS cable with bluetooth and camera connections, and 42W8009 or 42W8010 for the inverter. There are glossy and matte versions of these. Matte means anti-glare, which is what you want (in this authors opinion). Refer to the HMM (hardware maintenance manual) for how to replace the screen. Sources:. ThinkPad Forums - Matte AFFS Panel on X200 ThinkPad Forums - Parts for X200 AFFS Mod ThinkWiki.de - X200 Displayumbau X200S. [link] http:\/\/forum.thinkpads.com\/viewtopic.php?p=618928#p618928 explains that the X200S screens\/assemblies are thinner. You need to replace the whole lid with one from a normal X200\/X201. How to tell if it has an LED or CCFL? [link] Some X200s have a CCFL backlight and some have an LED backlight, in their LCD panel. This also means that the inverters will vary, so you must be careful if ever replacing either the panel and\/or inverter. (a CCFL inverter is high-voltage and will destroy an LED backlit panel). CCFLs contain mercury. An X200 with a CCFL backlight will (unless it has been changed to an LED, with the correct inverter. Check with your supplier!) say the following:. “This product contains Lithium Ion Battery, Lithium Battery and a lamp which contains mercury; dispose according to local, state or federal laws”. (one with an LED backlit panel will say something different). Hardware register dumps. [link] The coreboot wiki. shows how to collect various logs useful in porting to new boards. Following are outputs from the X200:. BIOS 3.15, EC 1.06. hwdumps\/x200\/ Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/x200.md Site map This HTML page was generated by the. untitled static site generator ."}
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Libreboot 20150126 release. Leah Rowe. 26 January 2015. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Machines supported in this release: Revisions for r20150126 (relative to r20150124) Return to index Article published by: Leah Rowe. Date of publication: 26 January 2015. Machines supported in this release:. [link] Lenovo ThinkPad X60\/X60s. You can also remove the motherboard from an X61\/X61s and replace it with an X60\/X60s motherboard. An X60 Tablet motherboard will also fit inside an X60\/X60s. Lenovo ThinkPad X60 Tablet. (1024x768 and 1400x1050) with digitizer support. See. hardware\/#supported_x60t_list. for list of supported LCD panels. It is unknown whether an X61 Tablet can have it’s mainboard replaced with an X60 Tablet motherboard. Lenovo ThinkPad T60. (Intel GPU) (there are issuesinstall\/x200_external.html; see below):. See notes below for exceptions, and. hardware\/#supported_t60_list. for known working LCD panels. It is unknown whether a T61 can have it’s mainboard replaced with a T60 motherboard. See. future\/#t60_cpu_microcode. . T60p (and T60 laptops with ATI GPU) will likely never be supported:. hardware\/#t60_ati_intel. Lenovo ThinkPad X200. X200S and X200 Tablet are also supported, conditionally; see. hardware\/x200.html#x200s. ME\/AMT. : libreboot removes this, permanently. hardware\/gm45_remove_me.html. Lenovo ThinkPad R400. (r20150208 and later, only). ME\/AMT. : libreboot removes this, permanently. hardware\/gm45_remove_me.html. Apple MacBook1,1. (MA255LL\/A, MA254LL\/A, MA472LL\/A). See. hardware\/#macbook11. . Apple MacBook2,1. (MA699LL\/A, MA701LL\/A, MB061LL\/A, MA700LL\/A, MB063LL\/A, MB062LL\/A). See. hardware\/#macbook21. . Revisions for r20150126 (relative to r20150124). [link] This is a bug fix release based on r20150124. It contains a few small changes:. grub.cfg: hardcode the list of partitions to search (speeds up booting considerably. GRUB regexp isn’t very well optimized). Docs (x200.html hcl): Remove incorrect information. Documentation (bbb_setup.html): Fix typos. build-release: delete ich9fdgbe_{4m,8m}.bin files from ich9gen. These were accidentically included in the r20150124 release. They are generated from ich9gen so it’s ok, but they don’t need to be in the archive. Documentation (grub_cbfs.html): Looping in libreboot_grub.cfg (Add notes about it if the user copied from grub.cfg in CBFS.). Binary blob policy Edit this page Who develops Libreboot? 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{"id":"Z0HAS2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/r500.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":45,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"r500","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – ThinkPad R500"],"last_modified":"Sat Apr 02 12:48:37 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2299599894490965735,"exact_signature_unique_b":true,"fuzzy_signature_l":2464243823248600879,"fuzzy_signature_unique_b":true,"h1_txt":["ThinkPad R500"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":366,"text_t":"Libreboot – ThinkPad R500. ThinkPad R500. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to index This board as basically identical to the T500, and has very similar disassembly. You must take it apart and flash the chip externally. The chip is 4MiB NOR flash (SPI protocol) is SOIC8 form factory. Refer to the following guide:. Externally rewrite 25xx NOR flash via SPI protocol Unlike other GM45+ICH9M thinkpads in Libreboot, the R500 doesn’t have an Intel PHY (for Gigabit Ethernet). However, Libreboot still includes an Intel flash descriptor, but with just the descriptor and BIOS region. The. ich9gen. program supports this fully. Therefore, you do not have to worry about the MAC address. The onboard NIC for ethernet is made by Broadcom (and works in linux-libre). Refer to T500 disassembly guide. The R500 disassembly procedure is almost identical. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/r500.md Site map This HTML page was generated by the. untitled static site generator .","wordcount_i":177,"linkscount_i":22,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/hardware\/","libreboot.org\/docs\/install\/spi.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/hardware\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/hardware\/r500.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Return to index","Externally rewrite 25xx NOR flash via SPI protocol","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:05 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:19 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":4119,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078566253559808}
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{"id":"V7U6Y_Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/news\/libreboot20150518.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["news"],"url_file_name_s":"libreboot20150518","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Libreboot 20150518 release"],"author":"Leah Rowe","author_sxt":["Leah Rowe"],"last_modified":"Sat Apr 02 12:48:49 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Mon May 18 00:00:00 UTC 2015"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":1778190126324981088,"exact_signature_unique_b":true,"fuzzy_signature_l":-5018561483528614303,"fuzzy_signature_unique_b":true,"h1_txt":["Libreboot 20150518 release","Navigate this page:"],"h2_txt":["Machines supported in this release:","Changes for this release, relative to r20150208 (earliest changes last, recent changes first)"],"bold_txt":["docs\/hardware\/gm45_remove_me.html","ME\/AMT","New board","docs\/hardware\/#macbook21","docs\/hardware\/#macbook11","docs\/hardware\/t500.html","docs\/hardware\/t400.html","docs\/hardware\/r400.html","docs\/hardware\/x200.html#x200s","docs\/hardware\/#t60_ati_intel","docs\/future\/#t60_cpu_microcode","docs\/hardware\/#supported_t60_list","docs\/hardware\/#supported_x60t_list","docs\/git\/#build","docs\/install\/","New board:","download","getall","DEBLOB","build","part_bsd","bsd","coreboot-libre: backport patches for X200 Tablet digitizer support","Apple MacBook2,1","Apple MacBook1,1","ThinkPad T500","ThinkPad T400","ThinkPad R400","ThinkPad X200","ThinkPad T60","ThinkPad X60 Tablet","ThinkPad X60\/X60s","Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":405,"wordcount_i":1406,"linkscount_i":21,"linksnofollowcount_i":0,"inboundlinkscount_i":17,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/news\/libreboot20150518.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/news\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/news\/libreboot20150518.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Machines supported in this release:","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:05 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:14 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":17244,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078566601687040,"text_t":"Libreboot – Libreboot 20150518 release. Libreboot 20150518 release. Leah Rowe. 18 May 2015. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Machines supported in this release: Changes for this release, relative to r20150208 (earliest changes last, recent changes first) Return to index Article published by: Leah Rowe. Date of publication: 18 May 2015. Installation instructions can be found at. docs\/install\/. . Building instructions (for source code) can be found at. docs\/git\/#build. . Machines supported in this release:. [link] ThinkPad X60\/X60s. You can also remove the motherboard from an X61\/X61s and replace it with an X60\/X60s motherboard. An X60 Tablet motherboard will also fit inside an X60\/X60s. ThinkPad X60 Tablet. (1024x768 and 1400x1050) with digitizer support. See. docs\/hardware\/#supported_x60t_list. for list of supported LCD panels. It is unknown whether an X61 Tablet can have it’s mainboard replaced with an X60 Tablet motherboard. ThinkPad T60. (Intel GPU) (there are issues; see below):. See notes below for exceptions, and. docs\/hardware\/#supported_t60_list. for known working LCD panels. It is unknown whether a T61 can have it’s mainboard replaced with a T60 motherboard. See. docs\/future\/#t60_cpu_microcode. . T60p (and T60 laptops with ATI GPU) will likely never be supported:. docs\/hardware\/#t60_ati_intel. ThinkPad X200. X200S and X200 Tablet are also supported, conditionally; see. docs\/hardware\/x200.html#x200s. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. ThinkPad R400. See. docs\/hardware\/r400.html. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. ThinkPad T400. See. docs\/hardware\/t400.html. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. ThinkPad T500. See. docs\/hardware\/t500.html. ME\/AMT. : libreboot removes this, permanently. docs\/hardware\/gm45_remove_me.html. Apple MacBook1,1. (MA255LL\/A, MA254LL\/A, MA472LL\/A). See. docs\/hardware\/#macbook11. . Apple MacBook2,1. (MA699LL\/A, MA701LL\/A, MB061LL\/A, MA700LL\/A, MB063LL\/A, MB062LL\/A). See. docs\/hardware\/#macbook21. . Changes for this release, relative to r20150208 (earliest changes last, recent changes first). [link] Add a whitelist entry to board_enable.c in flashrom, for the ThinkPad R400, T400 and T500. Updated flashrom (to SVN revision 1889). X200 whitelist patch removed (merged upstream). X200 whitelist modified to include X200S and X200 Tablet. libreboot_util: don’t include cmos layout files (not needed anymore). coreboot-libre: backport patches for X200 Tablet digitizer support. build\/release\/archives: create SHA512 sum manifest file of the release archives. build\/release\/archives: separate crossgcc into a new archive. disabled generation of txtmode ROM images for now (they will be back again in the next release). coreboot-libre: delete unused code (reduce size of src archive). Flashing guides: make them more friendly to colourblind people. docs\/gnulinux\/encrypted_*.html: Remove mention of password length - it was arbitrary and pointless. docs\/maintain\/: Finish the guide. scripts\/download\/coreboot: use diffs included in libreboot, not external gerrit cherry-picks - review.coreboot.org (gerrit) being down no longer kills libreboot (backup mirrors of the master repository exist). docs\/install\/bbb_setup.html: Add info about wp\/hold and pinouts. docs\/: improve the description of libreboot. docs\/hardware\/gm45_remove_me.html: notes about the demefactory utility. docs\/install\/bbb_setup.html: EHCI debug: recommend linux-libre. docs\/install\/bbb_setup.html: EHCI Debug logging setup guide. docs\/hardware\/t500.html: Add screen compatibility report (TODO: fix incompatible screens). Update coreboot(again) + merge GM45 hybrid GPU patches - means that T400\/T500 with the ATI+Intel hybrid GPU setup will work (ATI disabled, Intel permanently enabled). power_on_after_fail nvram option added to all GM45 boards, defaulting to No, so that plugging it AC doesn’t boot up the system against the users will. Net20DC is now the default debug dongle on all boards (compatible with BBB). demefactory (new utility): create GM45 factory.rom without the ME. ich9deblob: re-factor descriptor.c functions. docs\/hardware\/t500.html: add hardware logs. docs\/gnulinux\/encrypted_*.html: No password for default entry. docs\/git\/: Add more details about BUC.TS. grub.cfg: Also scan for grub2\/grub.cfg, not just grub\/grub.cfg. docs\/maintain\/ (new section. WIP!): Maintaining libreboot. docs\/gnulinux\/grub_boot_installer.html: Fix hazardous instruction. docs\/tasks.html: Better categorization between intel\/amd\/arm. docs\/install\/bbb_setup.html: notes about SPI flashing stability. docs\/install\/bbb_setup.html: more names for the 0.1\" cables. docs\/install\/*_external.html: add disclaimer about thermal paste. docs\/install\/bbb_setup.html: Fix broken links. docs\/install\/bbb_setup.html: preliminary notes about EHCI debug. docs\/hardware\/gm45_remove_me.html: Link to websites talking about the ME. docs\/install\/{t400,t500,r400}_external.html: Notes about CPU compatibility. Delete the ich9macchange script. It’s useless, and confuses people. docs\/hardware\/gm45_remove_me.html: prioritize ich9gen executable path. docs\/hardware\/gm45_remove_me.html: prioritize changing mac address. docs\/hardware\/gm45_remove_me.html: less confusing notes about ich9gen. build\/dependencies\/parabola: Add dependencies for x86_64. scripts\/dependencies\/paraboladependencies: build dependencies (32-bit Parabola). New board. : ThinkPad T500. Add diffs for descriptor\/gbe differences between T500 and X200. coreboot-libre: provide better blob categorization. docs\/hardware\/gm45_remove_me.html: add notes about flash write protect. New board. : ThinkPad T400. GRUB: add partial vesamenu.c32 support (fixes tails ISOLINUX menu). Update GRUB (to revision fa07d919d1ff868b18d8a42276d094b63a58e299). Update coreboot (to revision 83b05eb0a85d7b7ac0837cece67afabbdb46ea65). Intel CPU microcode (most of it) no longer deleted, because it was deleted upstream (moved to a 3rd party repository). MacBook2,1 cstate patch is no longer cherry picked (merged upstream). Patch to disable use of timestamps in coreboot no longer included (merged upstream). coreboot-libre: don’t list vortex86ex kbd firmware as microcode (list it separately). coreboot-libre: don’t rm *\/early_setup_ss.h (these are not blobs). coreboot-libre: add GPLv3 license to the findblobs script. coreboot-libreboot: don’t rm raminit_tables (nahelem\/sandybridge) (they are not blobs). coreboot-libre: don’t delete the .spd.hex files (they are not blobs). build\/release\/archives: don’t put rmodtool in libreboot_util. docs\/install\/x200_external.html: recommend installing GNU+Linux at the end. docs\/install\/x200_external.html: add more photos, improve instructions. build\/clean\/grub: use distclean instead of clean. grub-assemble: Add the. bsd. and. part_bsd. modules. build\/roms\/withgrub: Only run ich9gen if gm45\/gs45 images exist. docs\/git\/: Add notes about building for specific boards. build\/roms\/withgrub: Allow building for a custom range of boards. grub-assemble: Disable verbose output. Add documentation on how to unlock root encrypted fs with key in initramfs in Parabola Linux. docs\/gnulinux\/grub_cbfs.html: Improve structure (easier to use). grub.cfg: Disable the beep on startup. docs\/install\/bbb_setup.html: Make the guide easier to use. docs\/gnulinux\/grub_cbfs.html: Remove redundant instructions. docs\/install\/x200_external.html: Mark pins in the images. docs\/install\/bbb_setup.html: Replace 3.3V PSU photo with ATX PSU. docs\/hardware\/x200.html: Add dumps from 4-MiB X200 with Lenovo BIOS 3.22. docs\/hardware\/x200.html: Add dumps from 4-MiB X200 with Lenovo BIOS 3.18. grub.cfg: add syslinux_configfile menuentry for ahci0. grub.cfg: Add more paths for syslinux_configfile. docs\/future.html: T60: Add EDID dump from LG-Philips LP150E05-A2K1. docs\/install\/bbb_setup.html: Further clarify which clip is needed. bash scripts: Make script output more user-friendly in general. bash scripts: Only enable verbose output if DEBUG= is used. build: Support multiple extra options - now possible to build multiple images for arbitrary boards (configs), but without building the entire collection. Deleted the signing archive key - the finger print and ID is given instead, so that the user can download it from a key server. scripts\/helpers\/build\/release: Move docs to separate archive - reduces the size of the other archives considerably. Move DEBLOB to resources\/utilities\/coreboot-libre\/deblob. scripts\/helpers\/build\/release: Delete DEBLOB from libreboot_src\/ - not needed in libreboot_src (release archive) because it contains a coreboot revision that has already been deblobbed. flash (script): Use. build. instead of. DEBLOB. to know if in src. docs\/install\/r400_external.html: Show images, don’t link. docs\/install\/x200_external.html: Show images, don’t link. docs\/install\/bbb_setup.html: Show images, instead of linking. Documentation: optimize all images (reduce file sizes). Remove download links from the release page (and the archive page) - release archives are hosted differently following this release, which means that the old methods are no longer viable. Moved ich9macchange to resources\/scripts\/misc\/ich9macchange. ich9macchange: assume that the script is being run from _util (act only on one ROM image, defined by a user-provided path). Move grub-background to resources\/scripts\/misc\/grub-background. grub-background: assume that it is being run from libreboot_util. grub-background: change only one ROM image, specified by path. build (release archives): Add the commitid file to release\/. build-release: Move the release archives to release\/. Merge all build scripts into a single generic script, with helpers in resources\/scripts\/helpers\/build\/. Replace. getall. with. download. , which takes as input an argument specifying which program the user wants to download. Moved the get scripts to resources\/scripts\/helpers\/download\/. build-release: Remove the powertop entries. Documentation: general improvements to the flashing instructions. Merged all flashing scripts into a single script. Updated GRUB. bucts: Make it build without git. Moved dejavu-fonts-ttf-2.34\/AUTHORS to resources\/grub\/font\/. Deleted GRUB Invaders from libreboot. Deleted SeaBIOS from libreboot. build-release: optimize use of tar (reduced file sizes). grub.cfg: add another SYSLINUX config location (\/syslinux\/syslinux.cfg). build-release: remove the bin\/ directory from libreboot_util. cleandeps: delete the bin\/ directory. buildrom-withgrub: create the bin directory if it does not exist. coreboot-libre: don’t use git for version timestamp. i945-pwm: add clean command to Makefile. i945-pwm: add -lz to Makefile. docs\/install\/x200_external: Mention GPIO33 non-descriptor mode. docs\/hardware\/: Remove redundant links. ich9macchange: Add R400. build-release: Separate ROM images into individual archives. build-release: rename libreboot_bin to libreboot_util. New board:. ThinkPad R400 support added to libreboot. bbb_setup.html: tell user to use libreboot’s own flashrom. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/news\/libreboot20150518.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"uYaok2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/d510mo.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":47,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"d510mo","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Intel D510MO and D410PT desktop boards"],"last_modified":"Sat Apr 02 12:48:36 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":7724567657873906747,"exact_signature_unique_b":true,"fuzzy_signature_l":9221990997405355037,"fuzzy_signature_unique_b":true,"h1_txt":["Intel D510MO and D410PT desktop boards"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":374,"text_t":"Libreboot – Intel D510MO and D410PT desktop boards. Intel D510MO and D410PT desktop boards. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to index This is a desktop board using intel hardware (circa ~2009, ICH7 southbridge, similar performance-wise to the Libreboot X200. It can make for quite a nifty desktop. Powered by libreboot. NOTE: D410PT is another name and it’s the same board. Flash the exact same ROM and it should work. NOTE: This board has a working framebuffer in Grub, but in GNU+Linux in native resolution the display is unusable due to some raminit issues. This board can however be used for building a headless server. Flashing instructions can be found at. ..\/install\/d510mo.html Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/d510mo.md Site map This HTML page was generated by the. untitled static site generator .","wordcount_i":153,"linkscount_i":22,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":4,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/hardware\/","libreboot.org\/docs\/install\/d510mo.html","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/hardware\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/hardware\/d510mo.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Return to index","..\/install\/d510mo.html","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:51:06 UTC 2022","fresh_date_dt":"Tue Jul 12 03:52:21 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":4039,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078566784139264}
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{"id":"wDJja2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/install\/t60_unbrick.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":51,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","install"],"url_file_name_s":"t60_unbrick","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – ThinkPad T60 Recovery guide"],"last_modified":"Sat Apr 02 12:48:40 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":6990287183853272274,"exact_signature_unique_b":true,"fuzzy_signature_l":-901742491940058257,"fuzzy_signature_unique_b":true,"h1_txt":["ThinkPad T60 Recovery guide","Navigate this page:","Brick type 1: bucts not reset.","Brick type 2: bad ROM image"],"bold_txt":["except","Top Swap","Backup Control","romstage","2","most","Buy 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2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":14047,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078548089077760,"text_t":"Libreboot – ThinkPad T60 Recovery guide. ThinkPad T60 Recovery guide. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Brick type 1: bucts not reset. Brick type 2: bad ROM image Return to index This section documents how to recover from a bad flash that prevents your ThinkPad T60 from booting. This section documents how to recover from a bad flash that prevents your ThinkPad X60 from booting. ROM images for this machine are well-tested in Libreboot, so bricks are rare. The most common cause of a brick is operator error, when flashing a ROM image. In. most. cases, the cause will be that there is no bootblock, or an invalid one. Brick type 1: bucts not reset. [link] You still have Lenovo BIOS, or you had libreboot running and you flashed another ROM; and you had bucts 1 set and the ROM wasn’t dd’d.* or if Lenovo BIOS was present and libreboot wasn’t flashed. There are. 2. 64KiB bootblocks possible, in the upper part of the ROM image. By default (bucts set to 0), the top one is used. If bucts is set to 1, the lower one (the one before the top one) is used. This bootblock is the first code that executes, during. romstage. as per coreboot hardware initialization. BUC is short for. Backup Control. and TS is short for. Top Swap. . This is a special register on Intel platforms. Lenovo BIOS sets PRx registers, preventing software re-flashing, but there is a bug in the protection, allowing everything. except. the upper 64KiB from being flashed. By default, coreboot only puts a bootblock in the upper region. If you flash such a ROM, while bucts is set to 1, the system won’t boot because there’s not a valid bootblock; this is common if you’re re-flashing when coreboot is already installed, and you didn’t set bucts back to 0. When you install on X60\/T60 the first time, you set this bucts bit to 1, then you re-flash a second time and set it back to 0. In this case, unbricking is easy: reset BUC.TS to 0 by removing that yellow cmos coin (it’s a battery) and putting it back after a minute or two:. *Those dd commands should be applied to all newly compiled T60 ROM images (the ROM images in libreboot binary archives already have this applied!):. dd if=coreboot.rom of=top64k.bin bs=1 skip=$[$(stat -c %s coreboot.rom) - 0x10000] count=64k dd if=coreboot.rom bs=1 skip=$[$(stat -c %s coreboot.rom) - 0x20000] count=64k | hexdump dd if=top64k.bin of=coreboot.rom bs=1 seek=$[$(stat -c %s coreboot.rom) - 0x20000] count=64k conv=notrunc. (doing this makes the ROM suitable for use when flashing a system that still has Lenovo BIOS running, using those instructions:. http:\/\/www.coreboot.org\/Board:lenovo\/x60\/Installation . (it says x60, but instructions for t60 are identical). Brick type 2: bad ROM image. [link] In this instance, you might have flashed a ROM without the top bootblock copied to the lower 64KiB section in the ROM, and you flashed the ROM for the first time (from Lenovo BIOS), in which case there is not a valid bootblock. In this scenario, you compiled a ROM that had an incorrect configuration, or there is an actual bug preventing your system from booting. Or, maybe, you set BUC.TS to 0 and shut down after first flash while Lenovo BIOS was running. In any case, your system is bricked and will not boot at all. “Unbricking” means flashing a known-good (working) ROM. The problem: you can’t boot the system, making this difficult. In this situation, external hardware (see hardware requirements above) is needed which can flash the SPI chip (where libreboot resides). Remove those screws and remove the HDD:. Lift off the palm rest:. Lift up the keyboard, pull it back a bit, flip it over like that and then disconnect it from the board:. Gently wedge both sides loose:. Remove that cable from the position:. Now remove that bezel. Remove wifi, nvram battery and speaker connector (also remove 56k modem, on the left of wifi):. Remove those screws:. Disconnect the power jack:. Remove nvram battery:. Disconnect cable (for 56k modem) and disconnect the other cable:. Disconnect speaker cable:. Disconnect the other end of the 56k modem cable:. Make sure you removed it:. Unscrew those:. Make sure you removed those:. Disconnect LCD cable from board:. Remove those screws then remove the LCD assembly:. Once again, make sure you removed those:. Remove the shielding containing the motherboard, then flip it over. Remove these screws, placing them on a steady surface in the same layout as they were in before you removed them. Also, you should mark each screw hole after removing the screw (a permanent marker pen will do), this is so that you have a point of reference when re-assembling the system:. This photo shows the flash chip, near the RAM, with numbers of pins written:. Refer to the external flashing guide:. Externally rewrite 25xx NOR flash via SPI protocol NOTE: Do not use the 3.3v rail from your SPI programmer. Leave that disconnected. For 3.3v, plug your charger into the mainboard (but do not power on the mainboard) when the clip is connected. Before removing the clip, disconnect the charger. This will provide adequate 3.3v DC at correct current levels. The SPI flash on an X60 shares a common 3.3V rail with many other components on the mainboard, which all draw a lot of current, more than your flasher can provide. Example command:. sudo .\/flashrom -p linux_spi:dev=\/dev\/spidev0.0,spispeed=4096 -w libreboot.rom -V. If flashrom complains about multiple flash chips detected, just pass the. -c. option as it suggests, and pick any of the chips it lists. spispeed=4096. or lower (e.g. . spispeed=512. ) is recommended on this board. The flashing becomes unstable, on this machine, when you use higher speeds. Reverse the steps to re-assemble your system, after you’ve flashed the chip. It should be. Verifying flash... VERIFIED. at the end. If flashrom complains about multiple flash chip definitions detected, then choose one of them following the instructions in the output. Put those screws back:. Put it back into lower chassis:. Attach LCD and insert screws (also, attach the lcd cable to the board):. Insert those screws:. On the CPU (and there is another chip south-east to it, sorry forgot to take pic) clean off the old thermal paste (with the alcohol) and apply new (Artic Silver 5 is good, others are good too) you should also clean the heatsink the same way. Attach the heatsink and install the screws (also, make sure to install the AC jack as highlighted):. Reinstall that upper bezel:. Do that:. Re-attach modem, wifi, (wwan?), and all necessary cables. Sorry, forgot to take pics. Look at previous removal steps to see where they go back to. Attach keyboard and install nvram battery:. Place keyboard and (sorry, forgot to take pics) reinstall the palmrest and insert screws on the underside:. It lives! Always stress test (‘stress -c 2’ and xsensors. below 90C is ok) when replacing cpu paste\/heatsink:. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/install\/t60_unbrick.md Site map This HTML page was generated by the. untitled static site generator ."}
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GA-G41M-ES2L flashing tutorial. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Return to index This guide is for those who want libreboot on their Intel GA-G41M-ES2L motherboard while they still have the original BIOS present. Flash chip size. [link] Use this to find out:. flashrom -p internal. Flashing instructions. [link] Refer to. spi.html for how to set up an SPI programmer for external flashing. You can only externally reprogram one of the chips at a time, and you need to disable the chip that you’re not flashing, by connecting 3v3 to \/CS of that chip, so you will actually need second test clip or IC pin mini grabber. NOTE: on GA-G41M-ES2L, the flash shares a common voltage plane with the southbridge, which draws a lot of current. This will cause under-voltage on most SPI flashers, so do not use the 3.3V rail from your flasher. Do not connect +3.3V to the chip. Instead, turn the board on and then turn it off by holding the power button. With the board powered down, but plugged in, there will be a 3.3V supply from the ATX PSU. You can then flash, but DO NOT connect the +3.3V supply from your SPI flasher! NOTE: You should use a resistor in series, between 1K to 10K ohms, for the 3.3v connection to the CS pin. This is to protect from over-current. Here is an image of the flash chip:. Internal flashing is possible. Boot with the proprietary BIOS and GNU+Linux. There are 2 flash chips (one is backup). Flash the first chip:. .\/flashrom -p internal:dualbiosindex=0 -w libreboot.rom. Flash the second chip:. .\/flashrom -p internal:dualbiosindex=1 -w libreboot.rom. NOTE: you can still boot the system with just the main flash chip connected, after desoldering the backup chip. This has been tested while libreboot was already installed onto the main chip. NOTE: If you don’t flash both chips, the recovery program from the default factory BIOS will kick in and your board will be soft bricked. Make sure that you flash both chips! NOTE: You need the latest flashrom. Just get it on flashrom.org from their SVN or Git repos. NOTE: due to a bug in the hardware, the MAC address is hardcoded in coreboot-libre. Therefore, you must set your own MAC address in your operating system. Binary blob policy Edit this page Who develops Libreboot? 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{"id":"6Ha2_2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/hardware\/kcma-d8.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":48,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","hardware"],"url_file_name_s":"kcma-d8","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – ASUS KCMA-D8 desktop\/workstation board"],"last_modified":"Sat Apr 02 12:48:36 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Thu Mar 18 00:00:00 UTC 2021"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-4915613430135416648,"exact_signature_unique_b":true,"fuzzy_signature_l":-936524811602716810,"fuzzy_signature_unique_b":true,"h1_txt":["ASUS KCMA-D8 desktop\/workstation board","Navigate this page:","Introduction","Flashing","PCI option ROMs","CPU coolers","CPU compatibility","Board status (compatibility)","Form factor","IPMI iKVM module add-on","Flash chips","Native graphics initialization","Current issues"],"h2_txt":["Hardware specifications"],"bold_txt":["also","can","DO NOT hot-swap the chip with your bare hands. Use a P-DIP 8 chip extractor. These can be found online. 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ASUS KCMA-D8 desktop\/workstation board. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction Flashing PCI option ROMs CPU coolers CPU compatibility Board status (compatibility) Form factor IPMI iKVM module add-on Flash chips Native graphics initialization Current issues Hardware specifications Return to index Introduction. [link] Specifications available here:. https:\/\/www.asus.com\/uk\/Commercial-Servers-Workstations\/KCMAD8\/ Quite a nice board; can have up to 16 Opteron 4200\/4300 CPU cores, with up to 64GiB of RAM. It holds its own against more modern machines, especially when compiling large source trees (for compilers, what you want is high RAM and more CPU cores). This is a desktop board using AMD hardware (Fam10h. and Fam15h. CPUs available). It can also be used for building a high-powered workstation. Libreboot also supports it. The coreboot port was done by Timothy Pearson of Raptor Engineering Inc. and, working with them, merged into libreboot many years ago. Note that not all boards are compatible. See. board status below to determine compatibility with your board. Flashing instructions can be found at. ..\/install\/ - note that external flashing is required (e.g. RPi), if the proprietary (ASUS) firmware is currently installed. If you already have libreboot\/osboot\/coreboot, by default it is possible to re-flash using software running in GNU+Linux on the kcma-d8, without using external hardware. If you currently have the ASUS firmware, please ignore the above link and instead refer to the section below:. Flashing. [link] The default ASUS firmware write-protects the flash, so you have to remove the chip and re-flash it using external hardware. It has a 25XX NOR flash (SPI protocol) in a P-DIP 8 socket, which looks like this:. The default chip is a 2MiB one, but we recommend upgrading it to a 16MiB chip. NOTE: If you’re already running Libreboot, you probably don’t need to re-flash externally. Refer instead to the generic instructions on this page:. ..\/install\/ Refer to the following guide:. Externally rewrite 25xx NOR flash via SPI protocol PCI option ROMs. [link] Unlike Libreboot 20160907, Libreboot in newer releases now supports finding and loading PCI option ROMs automatically, both in GRUB and SeaBIOS on this machine. So for example, if you wish to use an add-on graphics card, you can! It’s no problem, and should work just fine. CPU coolers. [link] With some creativity, standard AM3+ coolers will work fine. 2 x Socket C32 (LGA1207) available, so you can use 2 CPUs. (up to 32GiB per CPU). CPU compatibility. [link] Opteron 4100 series: Incompatible. Opteron 4200 series: Compatible, does not require microcode updates. Opteron 4300 series: Compatible, requires microcode updates (nonfree!). “Requires” means needed for stability. Opteron 4200 series CPUs work very well, even without microcode updates. Because the updates are non-free, Libreboot does not include them. In particular, the 4200 series works well with hardware virtualization even without the microcode updates; 4300 series on the other hand is more dependent upon these microcode updates. Microcode configures the logic gate arrays inside your CPU, to implement the instruction set architecture, which in turn is what enables. instructions. to execute on your CPU. What do you think will happen if you don’t have these updates? The answer is: bugs are possible, and the updates fix those bugs. AMD CPUs are generally better engineered than Intel ones, and work much nicer without updates compared to Intel CPUs, but CPU manufacturers design their chips to accept updates for a reason! Board status (compatibility). [link] There are two ways to identify a supported KCMA-D8 board:. Serial number (sticker attached to the 24-pin ATX power connector). BIOS version (sticker next to CPU slot 1, last four digits). Supported boards begin with a serial number of. B9S2xxxxxxxx. or above where the first character refers to the year of manufacture (A = 2010, B = 2011, etc.) and the following character the month in hexadecimal (1…9, A, B, C). Thus, any board produced September 2011. or later. are compatible with Libreboot. Boards originally shipped with BIOS version. 2001. or higher are also compatible. For help locating these identifying markers, see. ASUS documentation for determining Opteron 4200 series compatibility For more detailed information regarding the coreboot port, see. https:\/\/raptorengineeringinc.com\/coreboot\/kcma-d8-status.php Form factor. [link] This board is ATX form factor. While the. ATX standard, version 2.2 specifies board dimensions 305mm x 244mm, this board measures 305mm x 253mm; please ensure that your case supports this extra ~cm in width. IPMI iKVM module add-on. [link] Don’t use it. It uses proprietary firmware and adds a backdoor (remote out-of-band management chip, similar to the. Intel Management Engine . Fortunately, the firmware is unsigned (possible to replace) and physically separate from the mainboard since it’s on the add-on module, which you don’t have to install. Flash chips. [link] 2MiB flash chips are included by default, on these boards. It’s on a P-DIP 8 slot (SPI chip). The flash chip can be upgraded to higher sizes: 4MiB, 8MiB or 16MiB. With at least 8MiB, you could feasibly fit a compressed linux+initramfs image (BusyBox+Linux system) into CBFS and boot that, loading it into memory (and nowadays there is LinuxBoot, for which we would recommend a 16MiB boot flash). DO NOT hot-swap the chip with your bare hands. Use a P-DIP 8 chip extractor. These can be found online. See. http:\/\/www.coreboot.org\/Developer_Manual\/Tools#Chip_removal_tools Ideally, you should not hot-swap. Only remove the IC when the system is powered down and disconnected from mains. Native graphics initialization. [link] Only text-mode is known to work, but linux(kernel) can initialize the framebuffer display (if it has KMS - kernel mode setting). NOTE: This section relates to the onboard ASpeed GPU. You. can. use an add-on PCI-E GPU in one of the available slots on the mainboard. Nvidia GTX 780 cards are what Libreboot recommends; it has excellent support in Nouveau (free Linux kernel \/ mesa driver for Nvidia cards) and generally works well; however, the performance won’t be as high in Nouveau, compared to the non-free Nvidia driver because the Nouveau driver can’t increase the GPU clock (it doesn’t know how, as of 18 March 2021). Current issues. [link] Opteron 4100 series CPUs are currently incompatible. LRDIMM memory modules are currently incompatible (use UDIMMs please). Memory initialization is still problematic for some modules. We recommend avoiding Kingston and Super Talent modules for this reason. The coreboot wiki has some information about RAM compatibility. The wiki is deprecated but the info on it is still correct for this board. Some other considerations:. Booting from USB mass storage devices is currently broken under GRUB. Consequently, the textmode ROM with SeaBIOS is recommended otherwise in order to install an operating system you will need a hard disk with a pre-installed OS or will have to plug in another HDD or CD\/DVD reader in order to boot OS installation media. SeaBIOS lacked serial console support out-of-the-box in release 20160907 and as such a workaround using SGABIOS is necessary. You can find instructions on how to do this on the. Notabug issue tracker TODO: test whether this is still the case in Libreboot, which uses a newer version of coreboot nowadays). SAS (via PIKE 2008 module) requires non-free option ROM (and SeaBIOS) to boot from it (theoretically possible to replace, but you can put a kernel in CBFS or on SATA and use that to boot GNU, which can be on a SAS drive. The linux kernel can use those SAS drives (via PIKE module) without an option ROM). NOTE: SeaBIOS can load PCI-E option ROMs, and by default it will do so in Libreboot, so you could use it. However, you could. also. simply install 16MiB NOR flash with linuxboot payload in it, and use linuxboot which has the Linux kernel, which can use SAS drives without needing that option ROM; then it can kexec another linux kernel, which in turn also can can use those drives. Or just put a standard linux kernel and initramfs in cbfs and chainload that from GRUB, with the right parameters. IPMI iKVM module (optional add-on card) uses proprietary firmware. Since it’s for remote out-of-band management, it’s theoretically a backdoor similar to the Intel Management Engine. Fortunately, unlike the ME, this firmware is unsigned which means that a free replacement is theoretically possible. For now, the Libreboot project recommends not installing the module. This project might be interesting to derive from, for those who want to work on a free replacement. In practise, out-of-band management isn’t very useful anyway (or at the very least, it’s not a major inconvenience to not have it). Graphics: only text-mode works. See. #graphics Hardware specifications. [link] Check the ASUS website. Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/hardware\/kcma-d8.md Site map This HTML page was generated by the. untitled static site generator ."}
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{"id":"d6wBR2Q1CANx","host_id_s":"Q1CANx","sku":"https:\/\/libreboot.org\/docs\/misc\/codenames.html","host_s":"libreboot.org","host_organization_s":"libreboot","url_chars_i":46,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["docs","misc"],"url_file_name_s":"codenames","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Libreboot – Product Codenames"],"last_modified":"Sat Apr 02 12:48:43 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":6658090950445999531,"exact_signature_unique_b":true,"fuzzy_signature_l":3963193167644371472,"fuzzy_signature_unique_b":true,"h1_txt":["Product Codenames","Navigate this page:","Introduction","List of models and codenames","See also"],"h3_txt":["A note on GPUs","Codenames","Miscellaneous"],"bold_txt":["Buy preinstalled","Donate"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["libreboot.org\/favicon.ico"],"images_alt_sxt":["Libreboot logo"],"images_height_val":[-1],"images_width_val":[-1],"images_text_t":"libreboot org favicon ico Libreboot logo","responsetime_i":401,"wordcount_i":638,"linkscount_i":26,"linksnofollowcount_i":0,"inboundlinkscount_i":18,"outboundlinkscount_i":8,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https"],"inboundlinks_urlstub_sxt":["libreboot.org\/","libreboot.org\/faq.html","libreboot.org\/download.html","libreboot.org\/docs\/install\/","libreboot.org\/docs\/","libreboot.org\/news\/","libreboot.org\/git.html","libreboot.org\/contact.html","libreboot.org\/docs\/misc\/codenames.html","libreboot.org\/docs\/misc\/","libreboot.org\/news\/policy.html","libreboot.org\/who.html","libreboot.org\/license.html","libreboot.org\/template-license.html","libreboot.org\/docs\/misc\/logo-license.html","libreboot.org\/contrib.html","libreboot.org\/docs\/misc\/codenames.md","libreboot.org\/sitemap.html"],"inboundlinks_anchortext_txt":["Home","FAQ","Download","Install","Docs","News","Send patch","Contact","Introduction","Return to index","Binary blob policy","Who develops Libreboot?","License","Template","Logo","Authors","","Site map"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"outboundlinks_urlstub_sxt":["notabug.org\/libreboot\/lbmk\/issues","www.patreon.com\/libreleah","minifree.org\/","ark.intel.com\/products\/codename\/5950\/Calistoga","ark.intel.com\/products\/codename\/26552\/Cantiga","en.wikipedia.org\/wiki\/List_of_Intel_codenames","review.coreboot.org\/cgit\/coreboot.git\/tree\/Documentation\/thinkpad\/codenames.csv","untitled.vimuser.org\/"],"outboundlinks_anchortext_txt":["Bugs","Donate","Buy preinstalled","Calistoga","Cantiga","Wikipedia","","untitled static site generator"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 14:50:49 UTC 2022","fresh_date_dt":"Tue Jul 12 03:51:52 UTC 2022","referrer_id_s":"e18XSsQ1CANx","language_s":"en","size_i":9075,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735078549395603456,"text_t":"Libreboot – Product Codenames. Product Codenames. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. Introduction A note on GPUs List of models and codenames Codenames Miscellaneous See also Return to index TODO: this page could do with an update. More info, about more boards. Introduction. [link] This document lists product codenames for some hardware. Please note that just because a certain device is listed here does NOT mean that it is supported in Libreboot. For supported devices refer to the installation documentation. A note on GPUs. [link] Some laptops come with and without a discrete GPU (dGPU). Whether the motherboard includes one or not can be determined by (in descending order of reliability):. often thorough disassembly and searching for the actual chip. looking at white PCB markings near RAM slots \/ under keyboard and comparing with some known codenames (if not available FRU ID sticker) listed below. sometimes by looking at heatsink grills: on discrete GPU laptops these will look orange and on intergrated ones they will appear silver. List of models and codenames. [link] Codenames. [link] Asus Chromebook C201PA: speedy_rk3288, veyron-speedy. ThinkPad X60: KS Note (Sumo). ThinkPad X60s (slim): KS Note-2 \/ KS-2 (Kabuki). ThinkPad X60 Tablet: Dali (Same PCB as KS Note-2, different EC firmware). ThinkPad X200: Mocha-1. ThinkPad X200s (slim): Pecan-1. ThinkPad X200 Tablet: Caramel-1. ThinkPad R400\/T400: Malibu-3. with discrete GPU (dGPU), at board revision 0: \"MLB3D-0. with only integrated GPU (iGPU), at board revision 0: “MLB3I-0”. ThinkPad T500\/W500: Coronado-5. with dGPU (radeon): “COR5D-0” (last number is the board revision). with only iGPU: “COR5I-0”. ThinkPad T400s (slim): Shinai-MV. ThinkPad R500: Waikiki-3. T6x (whole family): Davinci. They don’t have codename label in silkscreen so you need to use FRU label of the board, which is placed under RAM sticks. T60:. with dGPU (radeon): Magi-0 (last number is the board revision). with iGPU: Lisa-0. R60(e): RP-1, RP-2 - Rockwell \/ Picasso. With ThinkPads on Intel platforms newer than Montevina (Except T410), the codenames become more consistent. All boards have the following appended based on the type of graphics they have:. with dGPU: SWG (SWitchable Graphics). with only iGPU: UMA (Unified Memory Access). *Note that Intel platforms newer than Montevina are not supported by libreboot yet!. Currently only Calistoga and Montevina platforms are supported. These are the known model codenames:. ThinkPad T410: NOZOMI-1 # EXT\/INT. ThinkPad T410s: SHINAI-2 # SWG\/UMA. ThinkPad T420: NOZOMI-3 # SWG\/UMA. ThinkPad T420s: SHINAI-3 # SWG\/UMA. ThinkPad T430: NOZOMI-4 # SWG\/UMA. ThinkPad T430s: SHINAI-4 # SWG\/UMA. ThinkPad T520: KENDO-1. ThinkPad W520: KENDO-1 WS. ThinkPad T520: KENDO-3. ThinkPad W520: KENDO-3 WS. ThinkPad T530: KENDO-4. ThinkPad W530: KENDO-4 WS. ThinkPad T-series dock codenames:. Advanced Mini Dock - Donatello. Advanced Mini Dock (lacking SuperIO) - Donatello-lite. Advanced Dock - Rodin. Dock II - Aswan. Mini Dock - Seville. Port Replicator II - Seville-lite. Miscellaneous. [link] Calistoga : 945GM\/945PM chipset family name. Napa: calistoga based platform. Cantiga : GM45\/GS45\/PM45 chipset family name. This is the chipset used in T400,X200 and similar. Montevina: cantiga based platform. PMH: the Power Management Hub is a gate array for managing the power up\/down sequence. It is additionally tasked with extending EC’s I\/O. Its later version was called “Thinker-1”, and eventually it was merged with PMIC (Rinkan) as ThinkEngine (Do not confuse it with EC chip which is also has ThinkEngine logo on ThinkPad boards). Kozak, Tsurumai, Rinkan: These are successive versions of power management ICs for Notebook PCs. Tsurumai chip marking is “TB62501F” and datasheet of it fully describes its operation. See also. [link] Many more Intel codenames can be found at. Wikipedia . For ThinkPads see [Documentation\/thinkpad\/codenames.csv @ Coreboot] (https:\/\/review.coreboot.org\/cgit\/coreboot.git\/tree\/Documentation\/thinkpad\/codenames.csv). Binary blob policy Edit this page Who develops Libreboot? License Template Logo Authors Markdown file for this page:. https:\/\/libreboot.org\/docs\/misc\/codenames.md Site map This HTML page was generated by the. untitled static site generator ."}
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Encrypted Debian GNU+Linux. Home FAQ Download Install Docs News Bugs Send patch Donate Contact Buy preinstalled Navigate this page:. If booting in text mode Boot the installer Partitioning Further partitioning Kernel Tasksel Postfix configuration Install the GRUB boot loader to the master boot record Clock UTC Booting your system ecryptfs LUKSv2 Generate distro’s grub.cfg Troubleshooting Return to index This guide is written for the Debian distribution, but it should also work for Devuan with the net installer. Other Debian based GNU+Linux distributions may also work, using these instructions. This guide assumes that you are using the GNU GRUB bootloader as a coreboot payload. In this configuration, GNU GRUB runs on. bare metal. instead of relying on BIOS or UEFI. GNU GRUB has support for directly reading LUKS volumes and it can directly boot your Linux kernel this way. With GRUB already in the boot flash, this means that your. \/boot\/. directory (containing your Linux kernel) can be fully encrypted. The same cannot be said for most other systems, and no other coreboot payload provides this functionality. If booting in text mode. [link] Libreboot ROM images are provided, which will either boot the system in classic text mode, or with a framebuffer implemented by coreboot for video display initialization (not to be confused with int10h VGA modes). Text mode. is the default video mode on. most. x86 platforms, using. INT 10H. functions. It’s an interrupt service that text-mode applications use, a hangover from the days of CS\/M and DOS. In this mode, no framebuffer exists and Libreboot currently does not implement VGA modes. The Debian net installer will attempt to use VGA modes that most implementations of INT 10H provide. Therefore, you must force Debian’s installation program to operate in text mode. To boot the Debian net installer, make sure to specify. fb=false. on the linux kernel parameters in GRUB. This will boot the installer in text mode instead of using a framebuffer. By default, the netinstaller will try to switch to a high resolution framebuffer. Due to lack of INT10H video BIOS services and mode switching support in. libgfxinit. , this will fail. In. some. setups, you don’t need this. For example, if you’re using an add-on PCIe GPU on a desktop\/server board (e.g. ASUS KGPE-D16\/KCMA-D8, Gigabyte GA-G41M-ES2L), you would use SeaBIOS payload with text-mode startup, but the Video BIOS. option ROM. in your add-on graphics card would be executed, and it would presumably handle INT10H VGA modes. Boot the installer. [link] Libreboot on x86 can use the GNU GRUB bootloader as a bare metal coreboot. payload if you wish, which means that the GRUB configuration file (where your GRUB menu comes from) is stored directly alongside Libreboot and its GRUB payload executable, inside the flash chip. In context, this means that installing distributions and managing them is handled slightly differently compared to traditional BIOS or UEFI systems. On most systems, the. \/boot\/. partition has to be left unencrypted while the others are encrypted. This is so that GRUB, and therefore the kernel, can be loaded and executed since the firmware can’t open a LUKS volume. Not so with Libreboot! Since GRUB is already included directly as a payload, even. \/boot\/. can be encrypted. This protects \/boot from tampering by someone with physical access to the system. This guide is written for Debian net installer. You can download the ISO from the homepage on. debian.org . Use this on the GRUB terminal to boot it from USB (for 64-bit Intel or AMD):. set root='usb0' linux \/install.amd\/vmlinuz initrd \/install.amd\/initrd.gz boot. If you are on a 32-bit system (e.g. X60):. set root='usb0' linux \/install.386\/vmlinuz initrd \/install.386\/initrd.gz boot. This guide shows how to create a boot USB drive with the Debian ISO image. This guide is only for the GRUB payload. If you use the depthcharge payload, ignore this section entirely. Note: on some thinkpads, a faulty DVD drive can cause the cryptomount -a step during boot to fail. If this happens to you, try removing the drive. Set a strong user password (lots of lowercase\/uppercase, numbers and symbols). Use of the. diceware method. is recommended, for generating secure passphrases (instead of passwords). When the installer asks you to set up encryption (ecryptfs) for your home directory, select ‘Yes’ if you want to:. LUKS is already secure and performs well. Having ecryptfs on top of it will add noticeable performance penalty, for little security gain in most use cases. This is therefore optional, and not recommended. Choose ‘no’. Your user password should be different from the LUKS password which you will set later on. Your LUKS password should, like the user password, be secure. Partitioning. [link] Choose ‘Manual’ partitioning:. Select drive and create new partition table. Single large partition, but not with all the free space, let more than 512MB left (prevent an installer loop on Debian 11 netinst with the bootloader question on the same encrypted device). The following are mostly defaults:. Use as: physical volume for encryption. Encryption: aes. key size: whatever default is given to you. IV algorithm: whatever default is given to you. Encryption key: passphrase. erase data: Yes (only choose ‘No’ if it’s a new drive that doesn’t contain your private data). Select ‘configure encrypted volumes’. Create encrypted volumes. Select your partition. Finish. Really erase: Yes. (erase will take a long time. be patient). (if your old system was encrypted, just let this run for about a minute to make sure that the LUKS header is wiped out). Select encrypted space:. use as: physical volume for LVM. Choose ‘done setting up the partition’. Configure the logical volume manager:. Keep settings: Yes. Create volume group:. Name:. matrix. (use this exact name). Select crypto partition. Create logical volume. select. matrix. (use this exact name). name:. rootvol. (use this exact name). size: default, minus 2048 MB. Create logical volume. select. matrix. (use this exact name). name:. swap. (user this exact name). size: press enter. Further partitioning. [link] Now you are back at the main partitioning screen. You will simply set mountpoints and filesystems to use. LVM LV rootvol. use as: btrfs. mount point: \/. done setting up partition. LVM LV swap. use as: swap area. done setting up partition. Now you select ‘Finished partitioning and write changes to disk’. Kernel. [link] Installation will ask what kernel you want to use. linux-generic is fine, but you can choose whatever you want here. Tasksel. [link] For Debian, use the. MATE. option, or one of the others if you want. The Libreboot project recommends MATE, unless you’re saavy enough to choose something else. If you want debian-testing, then you should only select barebones options here and change the entries in \/etc\/apt\/sources.list after install to point to the new distro, and then run. apt-get update. and. apt-get dist-upgrade. as root, then reboot and run. tasksel. as root. This is to avoid downloading large packages twice. NOTE: If you want the latest up to date version of the Linux kernel, Debian’s kernel is sometimes outdated, even in the testing distro. You might consider using. this repository instead, which contains the most up to date versions of the Linux kernel. These kernels are also deblobbed, like Debian’s kernels, so you can be sure that no binary blobs are present. Postfix configuration. [link] If asked, choose. No Configuration. here (or maybe you want to select something else. It’s up to you.). Install the GRUB boot loader to the master boot record. [link] Choose. No. , and then it will still ask you what HDD to install GRUB on. Select your HDD\/SSD from the automatically generated list. The installer will provide GRUB on your HDD\/SSD, but not try to install it to an MBR section. However, the. \/boot\/grub\/grub.cfg. on your system will be maintained automatically by. apt-get. when handling kernel packages. Clock UTC. [link] Just say ‘Yes’. At this point, your Debian system is installed. Shut down when the installer tells you to. Booting your system. [link] If you didn’t install GRUB during the net installation process, don’t worry. You can boot your installed system manually, using the. terminal. in GRUB on your boot flash (the version that Libreboot gives you). At this point, you will have finished the installation. At your GRUB payload, press C to get to reach the GRUB terminal and enter these commands:. cryptomount -a set root='lvm\/matrix-rootvol' linux \/vmlinuz root=\/dev\/mapper\/matrix-rootvol cryptdevice=\/dev\/mapper\/matrix-rootvol:root initrd \/initrd.img boot. If you did install GRUB, ignore the above. Just select the default. Load Operating System. menu option and it should fully boot into your system. When you type your encryption passphrase in GRUB, it will seem like the process has stalled. The same will be true when you load your linux kernel in Debian. Just be patient and it will boot. If you see errors, just press enter to skip them until you see the Debian GRUB menu. ecryptfs. [link] If you didn’t encrypt your home directory, then you can safely ignore this section. Immediately after logging in, do that:. sudo ecryptfs-unwrap-passphrase. This will be needed in the future if you ever need to recover your home directory from another system, so write it down and keep the note somewhere secret. Ideally, you should memorize it and then burn the note (or not even write it down, and memorize it still). LUKSv2. [link] LUKSv2 is fully supported nowadays, in recent Libreboot releases. The old Libreboot release, version 20160907 (and earlier releases), did not support LUKSv2 in GNU GRUB. By default, modern Debian distributions will use LUKSv2. You do not need to downgrade LUKSv2 to v1, but you shouldn’t use any of the special features that LUKSv2 offers. Basically, the partitioning should be done exactly the same way as with LUKSv1 (but with newer encryption\/hashing algorithms used by LUKSv2 partitions). This is because of limitations in the implementation of LUKSv2 in GNU GRUB. GRUB uses its own custom implementation, instead of directly adapting the Linux kernel implementation. At the moment it is. only the PBKDF2 key derivation function supported. Argon2i, is not yet supported. That’s the point, you must convert it from Argon2i to PBKDF2, if you wish to use LUKSv2. Therefor you can use any live distribution with the package, that include dm-crypt. If the installation is finished, boot with a live CD and change it with:. cryptsetup luksConvertKey --pbkdf pbkdf2 \/dev\/sdX. Generate distro’s grub.cfg. [link] If. \/boot\/grub\/grub.cfg. already exists, ignore this step. Now you need to set it up so that the system will automatically boot, without having to type a bunch of commands. Install grub-coreboot if not already installed:. apt-get install grub-coreboot. Modify or add following lines to \/etc\/default\/grub. GRUB_CMDLINE_LINUX=\"cryptdevice=\/dev\/mapper\/matrix-rootvol:root\" GRUB_ENABLE_CRYPTODISK=y. Copy fonts\/backgrounds to \/boot\/grub and generate grub.cfg using following command:. grub-install --target=i386-coreboot. Refer to this guide for further guidance on hardening your GRUB configuration, for security purposes. Troubleshooting. [link] A user reported issues when booting with a docking station attached on an X200, while decrypting the disk in GRUB. The error. AHCI transfer timed out. was observed. The workaround was to remove the docking station or remove the CD\/DVD drive. Here is the information on that DVD drive, which said user had:. \"sudo wodim -prcap\" shows information about the drive: Device was not specified. Trying to find an appropriate drive... Detected CD-R drive: \/dev\/sr0 Using \/dev\/cdrom of unknown capabilities Device type : Removable CD-ROM Version : 5 Response Format: 2 Capabilities : Vendor_info : 'HL-DT-ST' Identification : 'DVDRAM GU10N ' Revision : 'MX05' Device seems to be: Generic mmc2 DVD-R\/DVD-RW. Drive capabilities, per MMC-3 page 2A: Does read CD-R media Does write CD-R media Does read CD-RW media Does write CD-RW media Does read DVD-ROM media Does read DVD-R media Does write DVD-R media Does read DVD-RAM media Does write DVD-RAM media Does support test writing Does read Mode 2 Form 1 blocks Does read Mode 2 Form 2 blocks Does read digital audio blocks Does restart non-streamed digital audio reads accurately Does support Buffer-Underrun-Free recording Does read multi-session CDs Does read fixed-packet CD media using Method 2 Does not read CD bar code Does not read R-W subcode information Does read raw P-W subcode data from lead in Does return CD media catalog number Does return CD ISRC information Does support C2 error pointers Does not deliver composite A\/V data Does play audio CDs Number of volume control levels: 256 Does support individual volume control setting for each channel Does support independent mute setting for each channel Does not support digital output on port 1 Does not support digital output on port 2 Loading mechanism type: tray Does support ejection of CD via START\/STOP command Does not lock media on power up via prevent jumper Does allow media to be locked in the drive via PREVENT\/ALLOW command Is not currently in a media-locked state Does not support changing side of disk Does not have load-empty-slot-in-changer feature Does not support Individual Disk Present feature Maximum read speed: 4234 kB\/s (CD 24x, DVD 3x) Current read speed: 4234 kB\/s (CD 24x, DVD 3x) Maximum write speed: 4234 kB\/s (CD 24x, DVD 3x) Current write speed: 4234 kB\/s (CD 24x, DVD 3x) Rotational control selected: CLV\/PCAV Buffer size in KB: 1024 Copy management revision supported: 1 Number of supported write speeds: 4 Write speed # 0: 4234 kB\/s CLV\/PCAV (CD 24x, DVD 3x) Write speed # 1: 2822 kB\/s CLV\/PCAV (CD 16x, DVD 2x) Write speed # 2: 1764 kB\/s CLV\/PCAV (CD 10x, DVD 1x) Write speed # 3: 706 kB\/s CLV\/PCAV (CD 4x, DVD 0x) Supported CD-RW media types according to MMC-4 feature 0x37: Does write multi speed CD-RW media Does write high speed CD-RW media Does write ultra high speed CD-RW media Does not write ultra high speed+ CD-RW media. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. Let's build the network together Join a club or association Alumni portraits Contact us Education Education. Study at Sorbonne University Study at Sorbonne University. Our Academic Programs Degree-seeking Students Exchange Students Continuing Education Ensure successful studies Ensure successful studies. Office of the Registrar Tutoring Student Services Student internships Tools and Resources Reorientation and caesura Student life Student life. Grants and financial assistance Health and Handicap Services Sports and Cultural Activities Student associations Work Spaces Student Life and Cultural Spaces Cafeteria and Food Service Housing Prepare Your Professional Goals Prepare Your Professional Goals. Employability Facts and Figures Student Entrepreneruial Training Sorbonne University Librairies Sorbonne University Librairies. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Donate to our projects. Donate to our projects. Whether you are a professional or a private individual, you can to support the project of your choice. Publié le 17\/02\/2020 - Mis à jour le 4\/12\/2020. The Sorbonne University Foundation is mobilising to finance current and future research into the coronavirus and similar viruses. In the framework of the urgency of the SARS-Cov2 epidemic, research teams at Sorbonne University have been working tirelessly for several weeks to ensure that their work makes effective progress. An exceptional budget of €500,000 has been made available, but researchers need more money to go faster and be more efficient. We are facing an unprecedented epidemic. We must act quickly. Research teams are beyond mobilised. . Now is the time to give! Five main priorities were chosen by the working group set up by Professor Bruno Riou, Dean of the Faculty of Medicine of Sorbonne University and Medical Director of Crisis Medical Director on COVID-19 for the PA-HP. . Epidemiological studies. Translational Studies. Pathophysiological studies. Clinical research and infection prevention, care and control measures in health care and community settings. Ethics\/Human and Social Sciences. Crisis and Attack Centre (CASC). Participate in the creation of a unique simulation site for everyone: knowing how to protect yourself in order to take good care of oneself. Encourage all stakeholders to interact, including pre-hospital and hospital teams, firefighters, armed forces, police, the gendarmerie and their respective special forces, as well as crisis psychologists. In addition, the general public and students will be trained. Our fundraising objective:. 1 million for the 2nd phase of construction. After the refurbishment of the premises (financing acquired) construction of the infrastructures: immersive room, control room, observation and evaluation stations. . Make a donation Sorbonne Centre for Artificial Intelligence (SCAI). Contribute to the creation of a think tank for research and training in Artificial Intelligence (AI): setting up chairs of excellence, creating doctoral programs, supporting and hosting interdisciplinary projects and responding to invitations for tenders. Some twenty industrial partnerships have been agreed on (from startups to large corporations) which will make it possible to finance 2 million euros per year until the end of 2022. Sorbonne University will provide €1 million per year.  . Fundraising Objectives: . The creation of chairs: from 1.5 to 2 million euros. The Image Wall (free use for researchers and students, with a 3D projection room inside): €246,000. . The AI Library: €6,000 . Make a donation The Institute of Hispanic Studies (IEH). Help renovate the Institute of Hispanic Studies, rethink its use, improve facilities for students, academic researchers and guest delegations, make it accessible to people with disabilities, restore its identity as a prestigious building and make it a true international crossroads of Iberian and Latin American culture. . Fundraising Objectives:. Construction of the auditorium: €550,000. Construction and fitting out of the garden and the terrace-cafeteria: €695,000. Restoration of the mural fresco by Francisco Mateos: €55,000. Make a donation Passeports Pluri’Elles. Finance 4-year scholarships, from the second year of the Bachelor’s to a completed Master’s. These scholarships are awarded to deserving first-year undergraduate women enrolled in demanding courses (dual or dual-disciplinary courses). Indeed, despite their higher success rate than men in the first year, women are a minority among Sorbonne University students in scientific fields (41% of students, and only 27% in the math-IT stream). . Fundraising objective:. €5,000 a year, i.e. €20,000 in total for a bursary. . Make a donation Passports for the Olympic Games (PPJO). Support annual scholarships to accompany students in competitive sports. In addition to the existing facilities at Sorbonne University, students preparing for the Olympics will be able to finance internships, competitions, physical therapy or training. The University has 40 high-level sporting students. Fundraising objective:. €3,000 a year and per top-level student athlete . Make a donation Mobility Passports. Offer deserving but financially challenged students the opportunity to experience a new culture and language in an international setting. Fundraising objective:. €5,000 per student for one semester. . Make a donation Passports to the Master (PPM). Contribute to 5-year scholarships, from a Bachelor’s to a completed Master’s, awarded to deserving high-school students with financial difficulties. The financial support is accompanied by tutoring by a mentor from the sponsoring company. . Fund-raising objective:. €10,000 a year, i.e. €50,000 in total for one fellow. Make a donation Research and education in Astrophysics - Institut Astrophysique de Paris. Support research and education in astrophysics and enable scientific advances (black holes and exoplanets, for example) that are becoming more and more regular, alongside the development of optics, large-capacity telescopes and the increasing power of calculation and modelling machines. The aim is to attract the most brilliant doctoral students internationally (in string theory, modelling, data analysis) by providing them with material support and hosting them in the best possible conditions. . Fundraising target:. €100,000 . Make a donation I would like to leave my donation to Sorbonne University’s discretion. Make a donation PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Goals The campaign committee Five scientific priorities Commit Alumni Chapters Projects to Fund Projects to Fund. Expanding Interdisciplinarity Cultivating Talent Building an Inspiring Environment Community of Donors Community of Donors. Corporate Donors Individual Donor Page Former Staff Group Our Organization Our Organization. Missions Governance Foundation team Activity Reports The Foundation’s non-disclosure policy Giving fr en You are. Future international student International student Doctoral Candidate Company Alumni Giving Close. Search. Home Culture and outreach An Interdisciplinary Approach to Benefit Society Multidisciplinary institutes and initiatives. Multidisciplinary institutes and initiatives. The deepening of knowledge enables us to extend the scope of its utilization and to ensure its presence in interdisciplinary research. Rather than making a distinction between a classical and disciplinary approach to research versus a contemporary and interdisciplinary one, Sorbonne University has learned that successful interdisciplinary research results in more innovative disciplinary research. Therefore, for the challenges facing today’s society that do not fall within the remit of any particular science, Sorbonne University has set up various institutes. Publié le 2\/12\/2019 - Mis à jour le 5\/05\/2022. Multidisciplinary institutes and initiatives. The Sorbonne University Alliance brings together six higher education and research institutions covering all disciplinary fields of arts, humanities, medicine, science, engineering, technology and management. The diversity of its members promotes a comprehensive approach to teaching and research. It supports access to knowledge for all and develops numerous joint programs in lifelong learning across all disciplines, in research and innovation, and in the use and dissemination of their results. . The Sorbonne University Alliance currently boasts several multidisciplinary thematic institutes and initiatives. Bringing together several institutions, they seek to make a range of disciplines work together, freeing them from traditional divisions and thus providing researchers from different backgrounds with the opportunity to compare their knowledge and experience. . The institutes are financed partly by the Idex (the French Excellence Initiative) and partly by external project funding (ANR, European Commission, industrial partners, etc.). Institutes. Sorbonne Center for Artificial Intelligence. SCAI, the Sorbonne Center for Artificial Intelligence, brings together a strategic range of modern artificial intelligence disciplines in a unique location in the heart of the Latin Quarter. SCAI's ambition is to make a significant contribution to the excellence of interdisciplinary research and education in artificial intelligence by promoting exchanges between researchers, academics, students and industrialists. Its aim is to showcase the Sorbonne University Alliance's know-how in the field of artificial intelligence and to be a gateway for academic and industrial partners. This project is supported by numerous academic researchers from the three faculties of Sorbonne University and the institutions of the Sorbonne University Alliance, who have been mobilised to ensure its success. Director: Gérard Biau. The SCAI website  Collegium Musicæ. The Collegium Musicæ is an institute that brings together musicians, researchers and academics around the creation, research, conservation and practice of music. It provides a unifying, fertile space, conducive to innovative projects combining humanities, social sciences and exact sciences. It enhances synergies between research, musical creation, education, heritage and musical practice and thus allows a cross-discipline approach to music. Director: Benoît Fabre. The Collegium Musicæ website The Institute of Computing and Data Sciences. The Institute of Computing and Data Sciences (ISCD) is a research, expertise and educational centre in scientific computing, based on high-performance computing and visualisation resources. Dedicated to research activities in digital simulation, it is a structure hosting multidisciplinary teams of the Sorbonne University Alliance who are engaged in a collaborative project in scientific computing. Its mission and objectives are to accompany, support, structure and stimulate projects from the community of researchers, engineers and students of the. Sorbonne University Alliance involved in scientific computing and simulation. Director: Pascal Frey. The ISCD website The University Institute for Health Engineering. The mission of the University Institute for Health Engineering (IUIS) is to encourage educational, scientific and technological innovation in health by providing a structure to host multidisciplinary teams grouped around three themes to be approached in an interactive way: engineering, health, humanities and social sciences. . The IUIS thus makes it possible to federate, coordinate and develop education, research and development, innovation and transfer activities on a larger scale. Directors: Pierre Mozer - Marie-Aude Vitrani . The IUIS website The Institute for Environmental Transition. The Sorbonne University Institute for Environmental Transition (SU-ITE) aims to create innovative education and research dynamics in major issues of the environment, sustainable development and the construction of a post-COP 21 society. To this end, it relies on its ability to implement interdisciplinary and inter-sectoral approaches to contribute to the environmental transition of societies. It encourages interactions between research and education teams and the stakeholders of society (political, economic, social). The Institute's work in education and research is structured around three main scientific areas: controlling climate change and its consequences, creating the conditions for responsible and sustainable use of resources and energies, and governing biodiversity and drawing inspiration from nature. Director: Luc Abbadie. The SU-ITE website The Heritage Observatory. The Heritage Observatory (OPUS) promotes a broad, innovative and operational field through a synergy of scientific, pedagogical and heritage resources of the Sorbonne University Alliance. The Observatory, using different approaches and tools, links objects, specialities, temporalities, to the concept of heritage (cultural and natural). The complementary nature of the Sorbonne University Alliance's scientific expertise in these heritage issues, as well as the extraordinary range of expertise and technical resources available to it, is a major asset in terms of research, the dissemination of knowledge, and the management, conservation and promotion of collections. . It has identified four main areas of research: to deal with heritage objects in all their diversity; to cultivate an interdisciplinary approach, to keep an active watch on the evolution of the concept of heritage and the phenomena of the preservation of heritage; and to affirm the social usefulness of the Alliance's work. Director: Nathalie Ginoux. The OPUS website The Materials Science Institute. An interdisciplinary institute to meet the major scientific, industrial and societal challenges of materials science. Materials are at the heart of important sectors of the economy and ubiquitous in society. The Sorbonne University Materials Science Institute encourages innovation, an interdisciplinary approach and both basic and applied research on materials to help meet the major societal and economic challenges. The Sorbonne University Alliance brings together major players in the field of materials. The Institute’s missions consist in strengthening their collaborations by supporting ambitious research projects, by promoting new interdisciplinary education and by developing international exchanges with academic and industrial circles, thus participating actively in the international influence of the Alliance. . The Institute's field of activity covers current and prospective research in materials science, at the interface of several disciplines: chemistry, physics, earth sciences and engineering. In the field of energy and sustainable development, themes can include the production and storage of energy or the life cycle of materials. Bio-assisted synthesis, optimized or targeted for specific properties and the functionalization of surfaces are relevant examples for the industrial and health fields. The institute is also interested in the new fundamental properties of hybrid, functional, or reduced dimensional materials and the devices that can emerge from them. Finally, developments in materials engineering and innovative study techniques and methods naturally constitute the Institute's themes. Director: Abhay Shukla. The Ocean Institute. Overview:. The Sorbonne University Alliance’s Ocean Institute, in association with the French Navy and naval school, brings together marine and maritime sciences. . Challenges and impacts:. Marine and maritime sciences play an important role in a range of fronts of knowledge and economic and societal issues: sustainable development, food, energy, geostrategy... Transdisciplinary aspects:. Relying on a large number of laboratories and the Alliance’s five marine stations, the Institut de la Mer assembles a strategic set of disciplinary fields and skills, conducive to the emergence of new concepts, innovative projects and unexpected synergies. Scientific themes:. The institute studies the sea as part of the history of the Earth, life and society. This approach is undertaken for analyses of the sea at different time scales and periods (geological, prehistoric, historical, current) and is a firmly original aspect of the institute, made possible in large part thanks to the diverse components that make up the Alliance. The Ocean Institute also centers on the specificities of marine and coastal environments in terms of risks and adaptations to global changes. This subject is at once topical and near inescapable in many disciplines, which require a transdisciplinary approach that takes into account the oceanic and coastal specificities of impacts as well as the notion of marine biomes, maritime geopolitics and marine traffic.This will involve studying the expansion of marine and maritime traffic in all its forms—ships, people and cultures, goods, pandemics, species, etc. in the Mediterranean region— and their ecological and geopolitical implications, such as the issues of appropriation and exploitation of the oceans, which may become the source of tomorrow's conflicts. Director Christophe Prazuck. Assistant directors: François Lallier, Olivier Chahine, Sylvie Dufour. Quantum Information Centre Sorbonne. Missions and objectives. The Quantum Information Centre Sorbonne (QICS) is an institute gathering the knowledge and skills in the Alliance Sorbonne Université on quantum information, mainly in computer science and physics, but also in mathematics and human sciences. Its scientific scope is Quantum Information — quantum computing and quantum communications —, as well as its impact in other fields. Born in the 1980’s, quantum information has shown that basing computers and communications on quantum physics could dramatically improve their performance or achieve result previously thought to be impossible. With many industries around the world possibly impacted by even limited quantum machines, a race has started for research results and for talents both for software and hardware. The QICS, opening in 2020, builds on the unique set of knowledge and skills from fundamental computer science to quantum physics, experimental demonstrations of quantum advantage, social, economic and philosophical studies of digital technologies present within the Alliance Sorbonne Université. The QICS is dedicated to intensifying experimental – theoretical research and engineering interactions along with exploring societal implications of this new way to treat information. This includes increased teaching efforts in this field and raise of awareness and collaboration within the innovation and industrial ecosystems. . General objectives of the Centre can be summarized as:. Trigger research excellence and collaborative works among Alliance Sorbonne Université. Develop a consistent teaching curriculum, from undergraduate to doctoral studies and continuing education.  . Outreach toward the general public, other scientific areas, and industries. To do so, QICS identified 3 main actions:. Animate the QI community . hosting two doctoral contract per year . organizing predoctoral schools, seminars, workshop. inviting professors. answering to calls\/hosting grants. To be involved in the teaching programs. constituting a teaching commission. creating a Quantum information PhD student community. integrating QI related teaching everywhere in SU. To build bridges between academia, the general public and the business world. supporting technology transfer and link with industry. developing continuing education. and any other action that would be relevant for our community. Scientific areas. hardware and software development of quantum computing and simulation . hardware and software development of quantum communications. theoretical investigation of the foundations of the quantum advantage. impact of Quantum Information in other fields, from classical cryptography to humanities and economic implications. Directors : Nicolas Treps & Fréderic Grosshans. About QICS Initiatives. i-Bio. Initiative for Biology: i-Bio, for interdisciplinary research in Biology. Biological research has evolved considerably over the past decade in its practices, discoveries and implications in and for society. Research now covers all scales of time and space, and aims to understand living things in their entirety and history. Biology is becoming nanometric, quantitative but also integrative. It is now a question of covering the changes in scale, and of understanding the emerging properties of the constituent elements of living organisms via their interactions with each other and the environment. The stakes are particularly high in the fields of evolution and biodiversity of organisms, the stochasticity and robustness of biological processes, the phenomena of adaptation and plasticity in the face of a changing environment, and the emergence and dynamics of biological networks. To meet these challenges, biology requires input from other disciplines. The i-Bio initiative is part of this research dynamic. It aims to promote the exploration of fundamental biological questions through the application of approaches and\/or concepts from other disciplines, such as mathematics, physics, chemistry, engineering and computer science. Teams from the fundamental biology group within the. Paris-Seine Biology Institute and the. Fer à Moulin Institute are leading this initiative. Its four-year program includes doctoral contracts, support for transdisciplinary research projects, the recruitment of new teams and active scientific animation through seminars, colloquia and summer school programs. By promoting new collaborations and attracting new talents, i-Bio will help make the fundamental biology center of the Pierre-et-Marie Curie campus of Sorbonne University a leading site for the study of biological systems. Email i-Bio Director: Catherine Jessus. Visit i-Bio’s. website IBEES. The Biodiversity, Evolution, Ecology and Society Initiative (IBEES) aims to stimulate and strengthen the organization of fundamental research activities and academic programs for the scientific community. Highly interdisciplinary, IBEES is made up of a thousand scientists working in approximately twenty research units in the fields of evolutionary science, ecology, earth sciences and human-environment relations, all considerably diverse and diachronic. IBEES contributes to a truly interdisciplinary spirit by using their unique research approach to connect natural sciences with humanities and social sciences. Through the reinforcement of the Bachelor's and Master's minors it already offers, and by relying on doctoral and postdoctoral programs, IBEES wishes to illuminate the blind spots of current research, at the multiscale interfaces between various units and disciplines. In order to do so, it relies on help from existing foundations such as the CNRS Ecology and Environment Partnership Schemes of the *MNHN  and Sorbonne University (DIPEE) and from the experience of the BCDiv  *LabEx, until recently restricted to the MNHN site but now open to all interested teams in the Sorbonne University Alliance. Director: Anthony Herrel. *MNHN: the French National Museum of Natural History. *Laboratoire d’Excellence, a prestigious status that indicates selection in the French Government’s Future Investments program. The Theater Initiative. Overview and objectives:. The Theater Initiative is a transdisciplinary initiative that arranges and links together activities related to theater and live performance within the Sorbonne University Alliance. . Challenges and impacts:. The Initiative promotes actions and events that help people to think about theater and stagecraft and their place in society. The Theater Initiative’s role is to develop and maintain partnerships with academic and theatrical institutions dedicated to the performing arts, and to organize the cultural dissemination of the project's themes in the form of symposiums, conferences, exhibitions, installations, performances, and publications. Transdisciplinary aspects:. Coordinated by Elisabeth Angel-Perez, professor of English literature (theater), and Andrea Fabiano, professor of Italian literature and culture, this initiative, in partnership with the Interdisciplinary Research Program on Theater and Staging Practices (PRITEPS), brings together researchers specializing in theater and the performing arts (literature experts, linguists, comparatists, musicologists and historians). The initiative is interested in both contemporary theatrical innovations and the exploration of ancient dramaturgy. Scientific themes:. The Theater Initiative will develop structured methods of reflection around theater and stagecraft and the professions associated with them (acting, staging, lighting, musical composition, translation, etc.). Central to the scientific project is research on reflections about theater and the city, which is divided into three focal points: . Theater and memory (heritage, transmission, translation.). Performing arts and political engagement (theater and community, actors\/actresses and representativeness, theater venues.). Theater and music (rituals, entertainment, musicals, opera.). Directors: Elisabeth Angel-Perez and Andrea Fabiano. Health Economics. Overview and objectives:. The Health Economics initiative will develop synergies between several Sorbonne University communities, in particular the Faculty of Medicine, the Pierre Louis Institute of Epidemiology and Public Health (iPLESP), the Compiègne University of Technology (UTC) and the European Business Institute (INSEAD). . Challenges and impacts:. The study of the intersections between economic and medical sciences can address major societal challenges. For example, it can contribute to the establishment of a sustainable health system by studying the mechanisms around drug pricing regulations (price adjustments when a new therapeutic use is identified for treatments already on the market, for example.) It can also lead to the improvement of treatments via the integration of new approaches, notably from behavioral economics (for example, by integrating neurobiology and behavioral economics models to better understand mechanisms of self-control and improve the effectiveness of weight loss operations.). Transdisciplinary aspects:. This initiative will make it possible to structure research around health-related issues (pathologies, treatments, prevention) that have an economic, financial or managerial aspect. For example, the initiative could facilitate collaboration between iPLESP and INSEAD on drug pricing or between the Brain Institute and INSEAD on dietary behavior. . One of the objectives of the initiative is to create new fields of knowledge out of collaborations between disciplines that until now have remained compartmentalized despite shared topics. . Director: Alexandra Roulet. The Gender Initiative. Overview and objectives:. The Gender Initiative, backed by PHILOMEL—an interdisciplinary program on gender, will contribute, from within Sorbonne University to the development of educational and scientific initiatives related to gender issues. The objectives of the Gender Initiative center on communication, education and research. Transdisciplinary aspects:. The Gender Initiative contributes to ensuring links between its members and highlighting the various initiatives related to gender (available courses, research programs, research teams, potential directors, seminars, cultural events, conferences, etc.). Its goal is to promote, transdisciplinary ‘gender’ courses across all three faculties starting from the second year of the bachelor’s, transdisciplinary exchanges between master’s students, and an environment for transdisciplinary doctoral research on gender by establishing, among others, a Doctoral Program on gender and other specific doctoral contracts. The Gender Initiative is also designed to develop an interdisciplinary dynamic around gender issues by supporting and encouraging the emergence of transdisciplinary research projects on gender at the Sorbonne University level. Scientific axes:. Several transdisciplinary projects are already in operation, with others under development. These include: . ‘A sociological approach to the place of female students in physics and biology studies’ (transdisciplinary science and humanities.). ‘Gender and authority’ (transdisciplinary both within the humanities and between the humanities and medicine.). ‘Trans identities in the Middle Ages’ (transdisciplinary in litterature, history and the arts.). ‘Heritage and transmission’ (developed, among others, by ‘Les Jaseuses,’ a multidisciplinary group of doctoral students and young researchers.). For further information: . • Join the. mailing list • The Gender Initiative. website (in French). Directors: Anne Tomiche, Frédéric Regard. Biomedical Humanities. Transdisciplinary aspects:. The Biomedical Humanities Initiative offers an interdisciplinary approach to issues related to medicine and health. It forms part of a long tradition of exchange between, on the one hand, disciplines relating to the humanities (philosophy, literature, sociology, cultural anthropology, history and art history) and, on the other, medicine, understood here as \"art at the crossroads of several sciences\" (G. Canguilhem)— anatomy, physiology, biology, or even chemistry. The initiative intends to strengthen this dialogue through work that balances both historical and critical perspectives:. By integrating the contributions of disciplines such as law, economics, information and communication sciences. By mobilizing Sorbonne University’s heritage resources (Dupuytren collection, Charcot Archives etc.). By prioritizing interdisciplinary research at the national and international levels (via partnerships with Stanford University and their Medicine and the Muse center, the University of Montreal, and the University of Geneva.). Scientific themes:. Four areas of research are currently in development: . On the history of representations of the body, health, illness, and physician and patient relationships using resources from history, literature, philosophy and art history. The aim here is to question the evolution of care practices, and to develop projects dedicated to observing practices (sociology, anthropology.). On the evolution of tools and techniques, such as questioning the conservation techniques for anatomical remains or data collection related to health and the body (biomarkers, databases,) and researching the challenges of introducing artificial intelligence and digital technologies in the health sector. On the ethics of biomedical research, including doctor-patient ethics, the challenges of conducting surveys on the quality of life of patients, the issue of access to care, the relationships between different social environments, the state of health (physical and mental) of individuals and populations in the workplace, and justice theories as applied to health. On new methods of disseminating and communicating information on diseases and health, their role in the prevention of diseases and in ‘therapeutic education’ of patients (such as the ‘expert patient’) and the growing feeling of mistrust of medicine. Directors: Claire Crignon, Alexandre Escargueil. Read. our interview with Claire Crignon. Europe and Interculturality Initiative. Overview and objectives:. The principal pillars of this initiative have been explored over a long period time that is neither limited to the contemporary period nor European construction. The initiative will highlight foundational concepts on interculturality, in particular everything related to circulation, networks and transfers. . Transdisciplinary aspects:. This initiative was established in close collaboration with the Encyclopedia of Digital History of Europe (EHNE), which is part of the LabEx of the same name. A bilingual French\/English encyclopedia, EHNE attracts more than 1,500 visitors every day and brings together 70 international researcher-authors who have written more than 400 pieces on the following themes: Material Civilization, Epistemology of Politics, European Humanism, Europe, Europeans and the World, Wars and Traces of War, Gender and Europe, Art in Europe, Education, Teaching and Training, Ecology and the Environment, and Migration in Europe. . Director: Olivier Dard. Mastering Safe and Sustainable Technology Systems. Overview and objectives:. The initiative on ‘Mastering safe and sustainable technology systems' is designed to build and animate a community around research on systems created by humans and intended for our own use. Its scientific objectives center on the understanding of the functioning of technological systems, considered in all their complexity, but also their production systems, their designs, controls, diagnostics and lifespans. Challenges and impacts:. The goal of this research is to contribute to the improvement or creation of economically and socially acceptable technology systems, capable of protecting against voluntary acts that seek to harm them and respectful of the environment and of people. This is in terms of their design, realization, use and end of life. For this, it requires the development of interdisciplinary and transdisciplinary approaches based on the hybridization of knowledge and practices. Transdisciplinary aspects:. The concepts of safety and sustainability will be central to the work of the initiative, and must be understood in a broad sense, with interdisciplinary integration. For example, safety can identify and address the need for a technology system to be reliable, maintainable, available and secure, as well as the need to ensure that it cannot adversely affect humans. For its part, sustainability can refer to the maintenance, over time, of the functional character of the technology system, as well as to the minimization of human and environmental costs over its entire life cycle, with particular concern for the preservation of natural resources. Director: Jérôme Favergeon. Physics of the Infinite. Overview and objectives:. The Physics of the infinite initiative has the goal to organise and promote top educational programs for Sorbonne University students at the intersection between two flagship fields bringing together ten laboratories: high-energy physics (particle physics, cosmology) and plasma physics. Challenges and impacts:. From understanding elementary physical processes to the laws governing how complex systems function; from fundamental research (such as astrophysical objects or interactions between particles) to applied research with a strong societal impact (for example the environment or medicine), the initiative will provide a better understanding of phenomena whose properties range from the infinitely small (particle physics) to the infinitely large (physics of the Universe.). Transdisciplinary aspects:. The initiative will capitalize on the diversity of approaches of the laboratories involved (such as theory, modeling, development of nanosatellites, experiments on large instruments including terrestrial and astronomical observatories, particle accelerators, high-intensity or electron lasers and ion beams) to provide research programs ranging from master's to the post-doctoral level. In particular, the initiative will rely on scientific excellence and on the interdisciplinary appraoch of its research laboratories to develop a unifying and ambitious doctoral and post-doctoral program, in conjunction with doctoral schools. It will also carry out scientific dissemination actions (the development of MOOCs) and communication aimed at the general public (conference cycles). Directors: Christophe Balland, Thierry Dufour. Visit the initiative’s. website Sciences of Antiquity. Overview and objectives:. This initiative assembles research and education strengths within the Sorbonne University Alliance that pertain to Sciences of Antiquity. It aims to provide students with the opportunity, given the richness of the pedagogical program, to follow a path that corresponds to their interests and professional projects. Innovative educational methods will introduce them to digital humanities and modern sciences as applied to archaeology and the study of ancient documents, while providing them with a solid foundation of knowledge in source languages, history, literature, philosophy and the history of art. In a university where teachers and researchers are geographically dispersed, it is fundamental to create an intellectual community by establishing bridges between research structures, doctoral schools and university departments. The initiative therefore also aims to bring together strengths dedicated to the study of Antiquity with a view to opening up transdisciplinary approaches. This will enable them to access funding for education or research projects; for invitations to foreign colleagues, for doctoral and post-doctoral contracts, and for new forms of publication. The objective of the initiative is also to increase the employability of our students, by providing them with training that better corresponds to today's needs in terms of research and preservation of knowledge and heritage, both intangible and tangible. Transdisciplinary aspects:. Knowledge of the works of antiquity to which we have access and of the people who created them requires us to take into account a series of insights and discoveries that have developed over the last few decades, developing into disciplines in their own right. Anyone working in the science of antiquity must possess a certain familiarity with a large number of methodological and analytical tools in order to reconcile the specialization that is required for high-level research with an understanding of ancient societies as a whole. . This necessary openness is based on an obvious transdisciplinary approach: despite the separations imposed by university structures and curricula, the study of Antiquity extends through history, literature, art history, history of science, archaeology and philosophy as well as epigraphy, papyrology, paleography or numismatics, which are often considered as auxiliary sciences, ignored by non-specialists. Only an inclusive approach will be able to show the whole of our student population, and more generally our contemporary society, what we can still learn from Antiquity. Director: Alessandro Garcea. The Molecular Science and Engineering Initiative. The molecular scale finds itself at the center of many disciplines: chemistry, physics, biology, medicine, geosciences and others. Our goal is to strengthen the impact of Sorbonne University Alliance research by strengthening resources across a selection of projects that involve multiple molecular science and engineering skills. Challenges and impacts:. The molecule corresponds to the elementary and organisational scale of matter, from the atom to the nano-object, and is therefore at the crossroads of many scientific concerns and primordial societal issues such as health, energy, the environment, information, all in the context of a circular economy. Indeed, all major experimental scientific disciplines rely heavily on molecular properties to analyze, understand, reproduce and manipulate the complexity of matter. Transdisciplinary aspects:. A multidisciplinary approach at the molecular level is essential to meeting major scientific challenges around fundamental or applied questions such as biomimicry, artificial photosynthesis, the cosmic origins of the molecular bricks of life; artificial life, storage and transmission, molecular and supramolecular information, catalysis, theoretical modelling of complex molecular assemblies, molecular machines and more. Director: Matthieu Sollogoubz. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Rental Spaces in the Zamansky Tower. Capacity. The Zamansky Tower offers 2 panoramic spaces:. 1 reception area: Panoramic room (120-140 people for receptions). 1 meeting room: Panorama room (40 people in meetings). Availability. Monday to Friday, 7:30 am to 6 pm . In the evening, until 10 pm - On request. Weekends - On request. Address and contact. 4 place Jussieu. 75005 Paris. Contact by email. Presentation of the Zamansky Tower (pdf) (in French). Booking the Zamansky Tower (pdf) (in French). Reception Hall. Panorama room. Sorbonne Université - Faculté des Sciences et Ingénierie. Campus Pierre et Marie Curie. 4 place Jussieu 75005 Paris. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? 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Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. 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Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Romane Dicko, a second-year undergraduate student in Mathematics, has now been crowned European Judo Champion twice: in Tel Aviv in 2018 and in Prague in 2020. Having already become the French champion at 16 years old, she is considered one of the most promising athletes in French judo. The twice champion is supported by the Sorbonne University Foundation’s \"Passport for the Olympics\" program. Supported by Crédit Agricole d'Ile-de-France, the program provides student athletes with greater financial independence, allowing them to devote themselves fully to both their studies and training. What does this European Championship title mean to you?  . Romane Dicko: First of all, it's the result of a lot of work. I had a difficult season due to several injuries that led me to redouble my efforts in order to return to the best level. Secondly, for me, this victory represents the start of a new stage and proves that I am capable of having a great career. . How did you discover judo? . R. D.: I started judo quite late, at the age of 13. It was while watching the London Olympics in 2012 with my family that I discovered Audrey Tcheuméo, the bronze medalist. I watched a report on her and it made me want to try it. My father supported the idea and I joined a club. Why did you choose to join Sorbonne University?  . R. D.: I wanted to become an aeronautical engineer and reconcile this goal with high-level sport. I had to find an institution that would allow me to take a course in engineering while continuing my training. Sorbonne University, being in partnership with the national institute of sport, expertise and performance (Insep), was the ideal solution. . How does Sorbonne University help you balance high-level sport and your studies on a daily basis? R. D.: Having the status of a high-level sports student offered by the university allows me to adapt the course of study according to my sporting objectives. This means I can spread out the duration of my degree over several years in order to succeed in both my educational ambitions and my sporting goals. . I also benefit from the fact that in MIPI (the Mathematics, Information technology Physics and Engineering core first year) can be done directly at the National Sports, Expertise and Performance Institute, (Insep), which allows me to save time and efficiency. I have two training sessions of two hours per day, one in the morning and one in the afternoon, and about three hours of classes in one day. . The professors at Sorbonne University come directly to Insep. They are very attentive to our needs, always available to answer our questions and accompany us throughout our double project.   . What are your next goals? R. D.: I want to succeed in improving on two levels, both athletic and academic. On the sports side, I will participate in the world championships at the beginning of September. As for my professional project, I still want to become an aeronautical engineer and join the engineering school of Sorbonne University. . My double career path brings me a lot because through high-level sport, I meet new and different people, I learn how to push through psychological milestones and overcome obstacles. This helps me on a daily basis in my studies and enriches my university experience. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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companies | Sorbonne Université. Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Thus, Sorbonne Université has about ten laboratories shared with companies ranging from large groups such as Thales, Pierre Fabre, ESSILOR or SBM Company to SMEs such as Bleckwen, Genious, Microphyt or Fluorotechnique.   . Sorbonne Université also includes 4 Carnot structures, out of the 40 or so accredited by the Ministry of Higher Education, Research and Innovation for their scientific and technological skills and their ability to meet the research and innovation needs of businesses. . About a hundred people work in the research departments of Sorbonne Université and its faculties to support researchers in the formalisation of their relations with companies.    . Contact the Research and Faculty Development departments:. in Literature, languages, humanities and social sciences in Medicine in Science and engineering PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. Contact us Find us! Site réalisé par. Faculty of medicine Faculty of medicine. Editorial Management Regulatory acts Regulatory acts. Code of ethics Divisions Divisions. Senior management Health Courses Division Human Resources Department Procurement and Finance Division Research and Development Division Rental of conference facilites Courses Courses. Medical studies Medical studies. Degree and inter-university degree catalogue PASS and L.AS Second cycle [years 2-6 of medical studies] Third cycle [year 7 onwards of medical studies] Midwifery studies Paramedical studies Paramedical studies. Speech and language therapy Orthoptics Psychomotricity Research Research. Research units Research units. MU 75 - Paris Brain Institute MU 80 - Institut de la Vision MRSU 933 - Laboratory of childhood genetic diseases MRSU 938 – Saint-Antoine Research Center MRSU 959- Immunology-Immunopathology-Immunotherapy MRSU 974 – Center of Research in Myology MRSU 1135 - Center for Immunology and Microbial Infections MRSU 1135 - Centre for Immunology and Infectious Diseases MRSU 1136 – Pierre Louis Institute of Epidemiology and Public Health MRSU 1155 – Common and Rare Kidney Diseases MRSU 1158 - Relations between the nervous system and the respiratory system MRSU 1166 - Research Unit on Cardiovascular Diseases MRSU 1269 - Nutrition and obesities : systemic approaches Autonomia -Real-life Research Center Clinical Research Groups Clinical Research Groups. CRG 1 - GREEN CRG 4 - Theranostic Biomarkers of Non-Small Cell Bronchial Cancers CRG 5 - Predictive Onco-Urology Research Group CRG 6 - Expert Center for Endometriosis CRG 14 - BIOSFAST PSYDEV CRG 16 - TT CRG 23 - NeurON CRG 25 - DREPS CRG 28 : GRAASU CRG 29 - ARPE CRG 30 - RESPIRE CRG 32 - TTIC GRC 33 : RIC Mixed service units Mixed service units. UMS 028 PASS Institute for Health Engineering International International. Incoming Incoming. Enroll in 1st year of medical studies Incoming in exchange student Internship for medicine students from partner universities Research and teaching stay Practical information 4EU+ Alliance 4EU+ Alliance. 4EU+ Against Cancer SUMMER SCHOOL 4EU+PharmacoEpi: an integrated 4EU+ Pharmacoepidemiology educational course Donate fr en Donate Close. Search. Welcome. to the faculty of medicine. Direct access. University and inter-university degrees. Registrations. Erasmus+. Rental of conferences facilities. Library. Explore. Courses. News. Research. The Patients' University: the Inauguration of the Skills and Vulnerabilities Chair 07.04.2022. Education. Immersion in the simulation department 10.09.2020. Lionel Naccache receives two prestigious awards for his work 04.01.2022. Vittoria Colizza - Irène Joliot-Curie price 2021 06.01.2022. All the news. The events. Online Conference. Distal myopathies in Japan On March 28th. Our events. Courses. Discover our courses catalog. Medicine. The Faculty of Medicine teaches the 3 cycles of medical studies: from PASS (integrated into the faculty) to the 3rd cycle including DES, DESC, DU and DIU. The lessons are given mainly on two sites: Pitié-Salpêtrière and Saint-Antoine. The faculty also provides paramedical education: speech therapy, psychomotricity and orthoptics. The Saint-Antoine site includes a midwifery school. University and inter-university degrees catalog Courses. Study. at the faculty of medicine. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students and offers them a wide range of training as well as support adapted to their profile and their project. David Klatzmann. Immunologiste et codécouvreur du VIH. Dominique Costagliola. Une scientifique sur le front des maladies infectieuses. Simon Pouzet. Associate dean student of the faculty of Medicine. Maha Dahawi. Doctorante en génétique. Our portraits. Associative life. One of our riches is the diversity of students and their backgrounds. Sorbonne University is committed to the success of each of its students. 21 393. users. 17 527. students. 715. Doctors in medecine and research. 12. Research centers. Incoming PRESS Intranet RENTAL OF CONFERENCE FACILITIES Legal notice SITE MAP Pitié-Salpétrière site. 91-105, Bd de l'Hôpital - 75013 Paris. Saint-Antoine site. 27, rue Chaligny - 75012 Paris. Contact us Find us! © Sorbonne Université 2021. Site réalisé par. Study Study. Overview Why Choose Us ? Admissions Timeline Our Programmes The International Welcome Desk Research Research. Overview Research Areas Research Areas. Chemistry Computer Science Earth, Environmental & Climate Sciences Electronics Life Sciences Mathematics Mechanics Physics Research Units & Doctoral Schools Societal & Scientific Challenges Societal & Scientific Challenges. AI, Computing & Data Engineering and Healthcare Environmental Transition Heritage and culture Partnerships & Innovation Partnerships & Innovation. Overview Partnership-Oriented Research Innovation & Entrepreneurship Research Support Services Economic Impact & Contributions Campus Life Campus Life. Overview The Faculty's Campuses Libraries & Student Venues Sport, Art & Societies Sport, Art & Societies. Sport & Fitness Art & Culture Societies & Clubs Health & Wellness Health & Wellness. Health Disability Resources Housing & Dining About About. Overview Vision & Values Our People, Our Strength Our People, Our Strength. Our Portraits The Faculty in Numbers Schools, Observatories & Institutes Venues, Sites & History Join Our Team Foundation fr en Foundation Close. Search. Apply Now: New European Master's in High Performance Computing. Discover EUMaster4HPC, an exciting new master's program in High Performance Computing (HPC), and be a part of its first-ever cohort. Direct access. ACCES 1. ACCES 2. ACCES 3. ACCES 4. ACCES 5. ACCES 6. Explore. Why Choose Us? What's Happening at the faculty. University. Sorbonne University Moves Up in the QS World University Rankings 11.04.2022. Open position for Director of the new Institute of Biology 03.03.2022. How do termite queens and kings stay healthy for decades? 26.01.2022. Research. Henri de Lacaze-Duthiers, pioneering scientist 26.11.2021. All News. Our Upcoming Events. The EPS Forum From. June 2nd. to. June 3rd. Conférence. LCPMR | REXS 2022 Conference in Paris On June 28th. Apply Now for New 4EU+ MICROBS Course: \"Biodiversity, sustainability in the agro-food system\" From. April 14th. to. April 29th. All Events. A unique combination of courses and expertise. Our international study programmes are organised according to the major disciplinary areas of the faculty. They represent the graduate study programmes that are not strictly conducted in French or that could be suitable (in part) for non French-speaking students. They also reflect part of the diversity of the disciplines involved and the bi- and inter-disciplinary aspects of many of the courses we offer.  . Explore Our International Study Programmes by Discipline. Research & Innovation. We rely on disciplinary skills and on interdisciplinary approaches to renew concepts, methods and research subjects and to focus on some of the crucial issues faced by our societies: transformations affecting the very construction of knowledge (data, AI), the treatment of complex objects (the environment, marine and ocean sciences, cultural heritage) or our contributions to addressing societal challenges (climate change, healthcare). Study at. The Faculty of Science & Engineering. Our campuses offer different and unique experiences to our students, visitors and staff. Resources and support services are also available to ensure an equal chance at success to all. Joana Desbordes. Student in biology-geosciences-chemistry and top-level swimmer. Thomas Matagne. Graduate - Master of Science and Environmental Policy. Our People, Our Strength. Our portraits. Campus Life. In addition to the cultural activities and events organised throughout the year by our clubs and societies, th. e. Parismus society. organises numerous events, evenings and cultural visits that will help. you discover France and its parisian life. 23. ACADEMIC DEPARTMENTS & INSTITUTE. 72. RESEARCH UNITS. 22,000. STUDENTS. Why Choose Us Campuses Intranet Join Our Team Faculty of Science & Engineering. Sorbonne University. 4, place Jussieu. 75005 Paris. Contact Find us! © Sorbonne Université 2020. Site réalisé par. Envoyer cette page à un ami. ×. Your email address. Your name. Send to. Enter the email address of the recipient. Your personal message. HTML is not allowed in this field."}
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Sorbonne Université. Skip to main navigation. Skip to main content. Skip to search. A+. A. A-. fr en Université. Arts and Humanities. Medicine. Science. and Engineering. University University. About us About us. The great adventure of knowledge The Strategic Plan Governance and organization Governance and organization. The Governance Team Central Councils Faculties Sorbonne University, Globally Engaged Sorbonne University, Globally Engaged. Sorbonne University in Europe Sorbonne University Worldwide History and Heritage History and Heritage. History Noteworthy Places, Collections and Bequests SorbonNum Archives Rental of Conference Facilities Rental of Conference Facilities. The International Conference Centre Rental Spaces in the Zamansky Tower Rental Spaces on the Cordeliers campus At the Faculty of Arts and Humanities At the Faculty of Medicine At the Faculty of Science and Engineering Alumni Alumni. 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Login with a Sorbonne University student login and password. Accommodation based on social criteria. To obtain a CROUS accommodation, students can create a. DSE  (Demande de dossier social étudiant\/ Request for a social file) between 15 January and 31 May of the year prior to signing up to Sorbonne University. The Cité internationale universitaire de Paris (CIUP). Starting at the master's degree, the. CIUP welcomes 12,000 people every year in its 40 houses. Personalised housing assistance (APL). For students to benefit from the APL, they must apply to the. CAF , once they have signed a rental contract. The ViSALE housing deposit. ViSALE is entirely free, simple and digital, and is aimed at people under 30 years of age (subject to conditions beyond 30 years of age). Check with the related faculty. PUBLIC PROCUREMENT LEGAL NOTICE INTERNATIONAL PRESS RELATIONS CREDITS RENTAL OF CONFERENCE FACILITIES SHOP Sorbonne University. 21, rue de l'école de médecine. 75006 Paris. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求. 刊行物・資料請求. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 以下の刊行物は、PDFやデジタルパンフレットにより閲覧が可能です。また、入手方法のリンクのある刊行物は、資料請求していただくこともできます（一部を除く）。. 大学案内. 京都大学概要 入手方法 京大英文概要 広報誌. 紅萠（くれなゐもゆる） 入手方法 「紅萠」特設サイト アンケート KYOTO U Research News 入手方法 KURN 特設サイト 入試関連. 京都大学一般選抜学生募集要項 （ダウンロード版（PDF）のみ）. 京都大学一般選抜入学者選抜要項 （ダウンロード版（PDF）のみ）. 知と自由への誘い （受験生向け大学案内）. 入手方法 京都大学特色入試学生募集要項 （ダウンロード版（PDF）のみ）. 学部案内冊子の請求 京都大学への留学案内 （外国人入学希望者向け）. 入手方法 各種報告書. 環境報告書 アニュアルレポート 将来構想、中期目標・中期計画. 京都大学の改革と将来構想（WINDOW構想） WINDOW構想特設サイト 第4期中期目標・中期計画リーフレット 学内向け. 京大広報 Campus Life Information（旧学生便覧） 留学生ハンドブック TUTOR GUIDEBOOK  . 就職のしおり 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ 海外渡航に関する安全対策手引き 外国人研究者ハンドブック 学生の海外でのフィールドワーク実施に関する安全手引き ハウジングガイドブック  . その他. 京都大学 第26代総長 山極壽一 活動報告書 －おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」 －尾池和夫第24代京都大学総長アクションレポート 共通教育通信  京都大学百年史 ※上記の刊行物・資料のほかに、. 京都大学メールマガジン を発行しています。. 過去の刊行物. こちらからご覧ください。 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 社会連携. 社会連携. 公開講座や一般公開している施設の情報など、広く一般・地域の方に向けた情報を掲載しています。. メインナビゲーション. 公開講座・講演会等 京都大学春秋講義 京都大学未来フォーラム 京大の知 京都大学地域講演会 京都大学 － 稲盛財団合同京都賞シンポジウム 公開講座お知らせメール受付 京大の研究・教育を知る 京都大学白浜水族館　展示 京都大学総合博物館　展示 京都大学大学文書館　歴史展示室 京都大学アカデミックデイ 京大ウィークス オープンキャンパス 京都大学オープンコースウェア（OCW） KoALA（Kyoto University Online for Augmented Learning Activities） edX 京都大学（KyotoUx） 社会人向け教育 リカレント教育（生涯教育等） 社会人のための大学院コース 社会人・一般向け講座 京都大学エグゼクティブ・リーダーシップ・プログラム（京都大学ELP） 教育・研究に関する情報・資料公開 京都大学研究資源アーカイブ（総合博物館） 京都大学貴重資料デジタルアーカイブ（図書館機構） 京都大学学術情報リポジトリKURENAI 京都大学大学文書館 京都大学教育研究活動データベース K.U. RESERCH 京大新刊情報ポータル 京大先生シアター 産官学連携（共同研究・知的財産・ベンチャー支援） 京大とのつながりを深める 125周年記念事業 京都大学同窓会 ホームカミングデイ 京都大学に寄附する 社会に開かれた施設等 京都大学医学部附属病院 京都大学大学文書館 京都大学図書館機構 京都大学総合博物館 京都大学白浜水族館 京都大学大学院教育学研究科附属臨床教育実践研究センター「心理教育相談室」 終了した全学的なイベント ジュニアキャンパス NEWS. 公開日. 2022年05月30日. タグ. 同窓会. 中国同窓会（京京会、京仙会）と京都大学同窓会の懇談会が開催されました. 公開日. 2022年05月27日. タグ. 社会連携. 「文部科学省 情報ひろば」旧庁舎3階 企画展示室「バイオ3Dプリンタから作製した神経導管による末梢神経再生医療」の開催について. 公開日. 2022年05月26日. タグ. 社会連携. 2022年度 クロックタワーコンサートを開催しました. 公開日. 2022年05月19日. タグ. 同窓会. 第24回「技術士を目指そう」説明会が開催されました. 最新ニュース一覧を見る EVENTS. タグ. 125周年. 公開講座. 京都大学創立125周年記念 市民講座「物理と宇宙」. 開催日. 2022年06月11日、他. 開催地. オンライン. タグ. 公開講座. 人文研アカデミー2022「ヴィリエ・ド・リラダンとフランス象徴主義 －『残酷物語』と『未来のイヴ』が現代に語りかけてくるもの」. 開催日. 2022年06月11日. 開催地. 吉田キャンパス. オンライン. タグ. 公開講座. 多様性を多様な視点で捉え、行動する ～京都大学と考えるD&Iセミナー～. 開催日. 2022年06月14日. 開催地. オンライン. タグ. 教育. 公開講座. オンラインコース「教育評価の基礎講座」2022. 開催日. 2022年06月15日、他. 開催地. オンライン. 最新イベント一覧を見る 関連タグ. 社会連携 公開講座 同窓会 Features. 大学創立125周年記念事業特設サイト. 京都大学へのご支援. 京都大学同窓生向けサービス KUON. 公開講座. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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{"id":"aOr4o9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/news\/2022-04-06-2","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":46,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","news"],"url_file_name_s":"2022-04-06-2","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["京都大学におけるウクライナ学生受入へのご支援のお願い | 京都大学"],"description_txt":["京都大学を支えてくださる皆様へ\n\n　皆様には、日頃より京都大学にご支援いただき、誠に有難うございます。\n\n　京都大学は、この度のロシア軍のウクライナ侵攻に伴い、学業や研究活動を継続することが困難となった大学生および大学院生を、特別枠として受入れることを決定いたしました。\n\n　本学にはウクライナとの間に大学間学術交流協定を結んでいる大学があり、これまでもウクライナから留学生を迎えておりましたが、今戦火の只中にあるウクライナでは、多くの学生が国外避難を余儀なくされ、やむなく国内に留まっている学生も、大学で学べる環境には全くありません。本学は、そのような学生達を一人でも多く受入れ、その学びと研究を継続できる環境を提供したいと考えています。"],"last_modified":"Wed Jun 08 14:38:08 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":3508794328143874842,"exact_signature_unique_b":true,"fuzzy_signature_l":4920653386805980180,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["京都大学におけるウクライナ学生受入へのご支援のお願い"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","問い合わせ","関連リンク","Latest news","2023年度 農学研究科修士課程学生募集について","マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて","令和4年6月1日付告示第4号","留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Wed Apr 06 12:00:00 UTC 2022","Fri Jun 03 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1739,"text_t":"京都大学におけるウクライナ学生受入へのご支援のお願い | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 京都大学におけるウクライナ学生受入へのご支援のお願い. 大学の動き 学生支援 京都大学におけるウクライナ学生受入へのご支援のお願い. ターゲット. 一般・地域の方 公開日. 2022年04月06日. 京都大学を支えてくださる皆様へ. 　皆様には、日頃より京都大学にご支援いただき、誠に有難うございます。. 　京都大学は、この度のロシア軍のウクライナ侵攻に伴い、学業や研究活動を継続することが困難となった大学生および大学院生を、特別枠として受入れることを決定いたしました。. 　本学にはウクライナとの間に大学間学術交流協定を結んでいる大学があり、これまでもウクライナから留学生を迎えておりましたが、今戦火の只中にあるウクライナでは、多くの学生が国外避難を余儀なくされ、やむなく国内に留まっている学生も、大学で学べる環境には全くありません。本学は、そのような学生達を一人でも多く受入れ、その学びと研究を継続できる環境を提供したいと考えています。. 　受け入れに当たっては、入学料・授業料等を全て免除するとともに、住居についても、自治体とも連携し無償で提供できるよう確保を進めますし、渡航費用や生活支援については、政府にも速やかに働きかけ、京都大学として取り得る最大限の支援を考えています。. 　毎日のようにニュースで流れるウクライナ国内の悲惨な状況を目の当たりにするにつけ、若い学生たちの学びや研究への意欲が、無残に断ち切られている事態は耐えがたいことです。京都大学として、希望する学生の最大限の受入れを進めるつもりですが、家族を伴っての来日を希望する学生のニーズもあり、日本滞在中の経済的支援については、その全てを京都大学のみでサポートするのは容易ではない状況です。. 　皆様には、何卒本学の意志をご理解いただき、そのためのご協力とご支援を、心からお願い申し上げる次第です。. 令和4年4月. 京都大学総長. 　湊 長博. 問い合わせ. 【基金に関すること】. 　京都大学基金事務局. 　Tel： 075-753-2210. 関連リンク. 　. ウクライナ情勢に係る本学の方針 　. ウクライナからの学生受入れについて 　. 京都大学基金「ウクライナ危機支援基金」  .  . Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教職員の方へ. 教職員の方へ. 教育 研究 国際交流 会議･委員会 採用・公募 イベント情報 総長選考関係（学内限定） その他通知・手続き 学内の問い合わせ先等 教職員が利用できる施設 京都大学教職員グループウェア 教育. 大学教育支援プログラム FD（ファカルティ・ディベロップメント）（FD専門委員会） 京都大学のFD 研究. 外部資金 競争的資金等の適正管理 研究倫理・生命倫理 研究成果発表（記者発表の手続き） 国際交流. 国際交流事務手続き（学内限定） 会議・委員会. 会議報告・委員会報告 採用・公募. 教職員公募一覧 イベント情報. EVENTS （イベントカレンダー）. 総長選考関係. 総長選考関係 その他通知・手続き. 監査. 内部監査 公益通報 総務. 諸規程 法務（学内限定）. 「コンプライアンスガイドブック（日本語）」および「Compliance Guidebook（英語）」について. 教職員ポータル内の以下の場所に掲載しています。. 京都大学教職員ポータル 日本語：ファイル管理＞文書共有＞総務部＞05法務室＞03コンプライアンスガイドブック＞コンプライアンスガイドブック. 英語：ファイル管理＞文書共有＞総務部＞05法務室＞03コンプライアンスガイドブック＞Compliance Guidebook. （2020年10月）担当部署名の変更、法令や学内規程等の改正に伴う部分的な改訂を行いました。（日本語／英語）. 広報. 刊行物等の請求方法 記者発表 広報用データ（学内限定）. ビジュアル・アイデンティティ マップ等 ホームページ・京大広報への情報掲載. 掲載に関する手続きは、「京都大学教職員グループウェア」の「ファイル管理」内に掲載しています。. 京都大学教職員ポータル ファイル管理＞文書共有＞総務部＞02広報課＞京大広報・ホームページ等への情報掲載手続き. 職員・人事・共済. 就業規則一覧 人事に関する情報（学内限定） その他 （人事制度・異動関係のお知らせ、人事異動、職員研修・自己啓発支援、兼業、障害者雇用等） ハラスメントの防止と対応 叙位・叙勲 育児支援制度について 財務. 京都大学における発注・検収について 環境安全衛生（学内限定）. 環境管理 安全衛生管理 情報環境. 情報セキュリティ 学内の問い合わせ先等. 公益通報 ハラスメント 学生総合支援センター 研究活動上の不正行為 競争的資金等適正管理 人事（学内限定） 教職員が利用できる施設. 学内施設 学内で使用できる実験室・会議室等 京都大学教職員グループウェア. 役職者内線番号一覧. 本部座席表. 資産の供用公募. などについては、. 京都大学教職員ポータル からご覧いただけます。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 大学評価. 大学評価. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 自己点検・評価 認証評価 国立大学法人評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 自己点検・評価. 学校教育法第109条第1項の定めにより、大学は教育研究水準の向上に資するため、教育研究・組織運営・施設設備の総合的な状況について自ら点検及び評価を行い、その結果を公表することとなっています。. 自己点検・評価 認証評価. 大学機関別認証評価. 学校教育法第109条第2項の定めにより、大学は、文部科学大臣が認証した機関（認証評価機関）を評価者として、認証評価機関が定める評価基準に基づき、教育研究・組織運営・施設設備の総合的な状況が、評価基準に適合しているか否かの認定を受けることとなっています。. 専門職大学院別認証評価. 学校教育法第109条第3項の定めにより、専門職大学院は文部科学大臣が認証した機関（認証評価機関）を評価者として、認証評価機関が定める専門職大学院別に特化した評価基準に基づき、教育研究・組織運営・施設設備の総合的な状況が、評価基準に適合しているか否かの認定を受けることとなっています。. 認証評価 国立大学法人評価. 国立大学法人法第31条の2の定めにより、文部科学省の国立大学法人評価委員会が国立大学法人の中期目標・中期計画・年度計画に対する教育研究活動や業務運営、財務内容等の総合的な達成状況について評価することとなっています。評価結果は次期の中期目標・中期計画の内容や運営費交付金等の算定に反映されます。. 国立大学法人評価 問い合わせ. 企画部企画課IR推進室大学評価掛. E-mail： hyouka880＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 令和4年6月1日付告示第4号. 告示 令和4年6月1日付告示第4号. ターゲット. 在学生の方 公開日. 2022年06月01日. 告示第4号. 　現在、「同学会全学処分対策委員会」、「全学処分対策委員会」を名乗り、「6\/3京大学生処分阻止・撤回全学集会」、「6\/3大集会」と称する集会を呼びかけている団体がある。. 　京都大学は、学生の自主的な活動について、直接関与するものではない。. 　しかし、本学構内において学内ルールを無視して共用スペースを無断で占有する行為や拡声器等を使用して大音量を発する行為などの迷惑行為により、本学の平穏な教育研究環境を妨害することは、断じて許されるものではない。. 　京都大学はこれらの行為には法的措置も含め厳正に対処する。. 　上記の団体は、本学ウェブサイトにおいて、「吉田南1号館の封鎖について（2015年10月28日）」として掲載している吉田南1号館の封鎖に関わった、中核派系全学連（全日本学生自治会総連合）と関係する団体と考えられる。. 　本学学生には、この集会に関わらないとともに、車道・歩道の通行の妨げとならないように注意を促すものである。. 　特に、新型コロナウイルス感染症の感染の再拡大を防ぐため、多数の人間が集まる状況は回避すべきであり、学生諸君には賢明な判断を求める。. 　　　　　　　　　　　　　　　　　　　　　　 令和4年6月1日. 　　　　　　　　　　　　　　　　　　　　　　　　　京都大学. Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS ウクライナからの学生受入れについて. 大学の動き ウクライナからの学生受入れについて. ターゲット. 一般・地域の方 公開日. 2022年04月06日. 　京都大学では、3月8日付け「ウクライナ情勢に係る本学の方針」において、ロシアによるウクライナ侵攻で影響を受けている学生や大学・学術機関関係者への支援、特にウクライナからの学生の受入れについて発表したところですが、ウクライナの危機的状況が続くなかで検討を行い、このたび、まずはウクライナにおける本学の学術交流協定校（キーウ（キエフ）⼯科⼤学及びタラス・シェフチェンコ記念キーウ（キエフ）国⽴⼤学）に対して、学生受入れを表明し、本学において一時的に受け入れて、学習及び研究環境を提供したいと考えております。. 　併せて、本学では、ウクライナからの学生に対する一時的受入れを可能とする特別措置（授業料免除等を含む）を実施するとともに、必要な生活支援については、ウクライナからの避難者に対する積極的な支援を発表している京都府及び京都市と連携しながら実施していく予定です。加えて、更に必要な経済的支援を実施するために、特別基金「ウクライナ危機支援基金」を立ち上げ、寄附募集を開始したところです。. 　本学でのこうした取組みが、ウクライナの大学で学ぶ学生の未来に繋がる有用な学習及び研究の機会となり、本邦におけるウクライナ支援の一助となれば幸いです。. 京都大学総長. 湊 長博. 関連リンク. 　. ウクライナ情勢に係る本学の方針 　. 京都大学におけるウクライナ学生受入へのご支援のお願い 　. 京都大学基金「ウクライナ危機支援基金」 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"Lfu_Y9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/admissions\/fees-exemption","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":54,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","admissions"],"url_file_name_s":"fees-exemption","content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["入学検定料の免除について | 京都大学"],"description_txt":["本学では、下記「対象となる災害」に該当する災害により被災した入学志望者に対して、当該入学志望者の主たる家計支持者が下記「要件」に該当する場合は、希望者からの申請により入学者選抜試験（学部一般選抜・学部特色入試・学部の特別選抜試験・大学院および専門職大学院入学者選抜試験）入学検定料を免除することがあります。"],"last_modified":"Wed Jun 08 16:42:57 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":4930216995855970748,"exact_signature_unique_b":true,"fuzzy_signature_l":3243169600734349402,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["入学検定料の免除について"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","対象となる災害","要件","連絡先等","関連リンク","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["学部一般選抜・学部特色入試の入学検定料免除について","その他の入学者選抜試験の入学検定料免除について"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1403,"text_t":"入学検定料の免除について | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 入学検定料の免除について. 入学検定料の免除について. サイドメニュー. 一般選抜 特色入試 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 本学では、下記「対象となる災害」に該当する災害により被災した入学志望者に対して、当該入学志望者の主たる家計支持者が下記「要件」に該当する場合は、希望者からの申請により入学者選抜試験（学部一般選抜・学部特色入試・学部の特別選抜試験・大学院および専門職大学院入学者選抜試験）入学検定料を免除することがあります。. 対象となる災害. 平成23年3月 東日本大震災. 平成28年4月 熊本地震. 平成30年7月 豪雨. 平成30年9月 北海道胆振東部地震. 令和元年10月 台風第19号. 令和2年7月 豪雨. 要件. 総長が指定する災害（上記「対象となる災害」に掲げるもの）に係る災害救助法適用地域（以下「災害救助法適用地域」という）において被災し死亡した。. 災害救助法適用地域に準ずると総長が認めた地域において被災し死亡した。. 災害救助法適用地域にある住居が、全壊、半壊、一部損壊、全焼、半焼または床上浸水の損害を受けた。. 災害救助法適用地域に準ずると総長が認めた地域にある住居が、全壊、半壊、全焼、半焼または床上浸水の損害を受けた。. 災害救助法適用地域にある、勤務先（事業所、店舗、田畑、漁業船舶等を含む。次号において同じ）が被災し、収入が見込めなくなった。. 都道府県知事等から指示が出されている区域に居住しており避難等を行ったためまたは、勤務先が当該区域にあるため、就労困難となり収入が見込めなくなった。. 申請には、罹災証明書等が必要です。. 連絡先等. 検定料免除を希望する方は、学部一般選抜・学部特色入試は入試企画課まで、その他の入学者選抜試験は各学部・大学院等の窓口までご連絡ください。. 学部一般選抜・学部特色入試の入学検定料免除について. 教育推進・学生支援部入試企画課. Tel： 075-753-2521、2522. その他の入学者選抜試験の入学検定料免除について. 各学部・大学院等の窓口までご連絡ください。. 各学部・研究科、教養・共通教育の窓口 検定料免除申請書はこちら 関連リンク. 授業料の免除／入学料の免除と徴収猶予 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home NEWS. NEWS. 适合对象. - 所有 -. Prospective Students. Current Students & Staff. Alumni. Visitors. Corporations & Researchers. 标签. - 所有 -. Admissions. Educational Outreach. Community Engagement. Public Lectures. Alumni. University News. Global Engagement. Notifications. Research. Industry-Academia-Government Collaboration. Education. Support Student. Student Activities. Awards. Announcements. 125周年. 适合对象. アイキャッチ画像. 发表日期. 2022\/02\/18. タグ. Global Engagement. Education. 举办了留学生社交室KI-ZU-NA“在线活动‘通过折纸来制作圣诞装饰吧！’”. アイキャッチ画像. 发表日期. 2022\/02\/18. タグ. Global Engagement. Education. 实施了全球环境人才培养项目. News. 发表日期. 2022\/04\/11. タグ. University News. 举行了2022学年研究生院入学典礼。（2022年4月7日）. News. 发表日期. 2022\/04\/08. タグ. University News. 举行了2022学年本科入学典礼。（2022年4月7日）. News. 发表日期. 2022\/04\/05. タグ. University News. 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. News. 发表日期. 2022\/03\/29. タグ. University News. 举行了2021学年毕业典礼。（2022年3月24日）. News. 发表日期. 2022\/02\/18. 法国驻京都总领事一行造访了数理分析研究所. News. 发表日期. 2022\/02\/18. タグ. University News. 举办了京都大学校长特别荣誉奖颁奖仪式. News. 发表日期. 2022\/02\/18. タグ. Global Engagement. Education. 实施了“旨在实现可持续性的森里海连环”. News. 发表日期. 2022\/02\/18. タグ. Global Engagement. 举办了加强全球开展能力项目“第1届非洲实务组织与大学交流会”. News. 发表日期. 2022\/01\/25. タグ. Student Activities. Awards. 京都大学的学生队伍“iGEM Kyoto”在合成生物学国际大赛“iGEM2021”上荣获了金奖. News. 发表日期. 2021\/12\/09. タグ. Awards. 本校5人入选“Clarivate Highly Cited Researchers 2021”. News. 发表日期. 2021\/10\/19. タグ. Awards. 数理分析研究所望月拓郎教授荣获数学突破奖. News. 发表日期. 2021\/10\/19. タグ. Global Engagement. Education. 实施了“Interview@KIZUNA”. News. 发表日期. 2021\/10\/14. タグ. University News. 2021学年研究生院秋季入学典礼贺辞已发送. News. 发表日期. 2021\/10\/04. タグ. University News. 举行2021学年研究生院秋季学位授予仪式。（2021年9月24日）. News. 发表日期. 2021\/04\/19. タグ. University News. 举行了2020学年入学典礼。（2021年4月7日）. News. 发表日期. 2021\/04\/05. 举行了2020学年毕业典礼。（2021年3月24日）. News. 发表日期. 2021\/04\/05. 举行了2020学年研究生院学位授予仪式。（2021年3月24日）. News. 发表日期. 2020\/11\/02. Call for applications: 13th Tachibana Award. News. 发表日期. 2020\/10\/15. KIZUNA hosts online rakugo show featuring Sayohime (16 September 2020). News. 发表日期. 2020\/10\/13. Clock Tower to be closed Sunday 25 October 2020. 分页. 当前页. 1 页面. 2 页面. 3 页面. 4 页面. 5 页面. 6 页面. 7 页面. 8 页面. 9 &hellip;. 下一页. chevron_right. 末页. chevron_right. chevron_right. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"xrS-zkU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/zh-cn\/site-policy","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":43,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["zh-cn"],"url_file_name_s":"site-policy","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["网站政策 | 京都大学"],"description_txt":["京都大学网站　运用指针\n\n目的\n\n我们为了实现京都大学网站（以下称“网站”）的顺利运用而规定了本指针。\n\n定义\n\n本指针中的网站指的是拥有“kyoto-u.ac.jp”域名的下述网站、内容及页面。\n\n\n\t京都大学官方网站（http:\/\/www.kyoto-u.ac.jp\/）（日文\/英文\/中文繁体\/中文简体\/韩文页面）及其下属的各种内容、页面、以及链接的由京都大学（包括部门及其他设置于京都大学的组织）管理运营的特设网站 。\n\t由设置于京都大学的部门等运营的网站（以下等同于“部门等网站”） 。\n\n\n管理负责人\n\n京都大学官方网站由京都大学公共关系委员会网站企划专门部会（以下称“专门部会”）进行管理。部门等网站由相关部门等进行管理。但是，从全校角度来看有必要时，专门部会可以就部门等网站发表意见。\n\n运用负责人\n\n京都大学官方网站由总务部公共关系课进行运用。关于部门等网站，各相关部门应设置1名运用负责人，在进行运用的同时，负责与专门部会的联系协调。"],"last_modified":"Wed Jun 08 16:43:08 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-1507689879659085203,"exact_signature_unique_b":true,"fuzzy_signature_l":-4336660526002848903,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["网站政策"],"h2_txt":["グローバルナビゲーション（簡体中国語）","ヘッダー セカンダリーリンク（日本語以外）","京都大学网站　运用指针","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["目的","定义","管理负责人","运用负责人","刊登请求者","违反公共秩序或善良风俗的信息","改善劝告","其他规定"],"italic_txt":["menu","search","close"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":932,"text_t":"网站政策 | 京都大学. 跳转到主要内容 网站内搜索. 教师搜索. 搜索. close. CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home 网站政策. 网站政策. In other languages. 日本語 京都大学网站　运用指针. 目的. 我们为了实现京都大学网站（以下称“网站”）的顺利运用而规定了本指针。. 定义. 本指针中的网站指的是拥有“kyoto-u.ac.jp”域名的下述网站、内容及页面。. 京都大学官方网站（http:\/\/www.kyoto-u.ac.jp\/）（日文\/英文\/中文繁体\/中文简体\/韩文页面）及其下属的各种内容、页面、以及链接的由京都大学（包括部门及其他设置于京都大学的组织）管理运营的特设网站 。. 由设置于京都大学的部门等运营的网站（以下等同于“部门等网站”） 。. 管理负责人. 京都大学官方网站由京都大学公共关系委员会网站企划专门部会（以下称“专门部会”）进行管理。部门等网站由相关部门等进行管理。但是，从全校角度来看有必要时，专门部会可以就部门等网站发表意见。. 运用负责人. 京都大学官方网站由总务部公共关系课进行运用。关于部门等网站，各相关部门应设置1名运用负责人，在进行运用的同时，负责与专门部会的联系协调。. 此外，此处所说的由总务部公共关系课进行的运用指的是仔细检查运用负责人自主刊登的信息、或者由后述规定的刊登请求者要求刊登的信息后发布在京都大学官方网站上。. 刊登请求者. 网站的刊登请求者（以下称“请求者”）将鉴于网站会向不特定的多数人发布伴有速报性的信息、以及积累信息等特性，将网站理解为是正确传达本校应发布事项或快速且有效地传达本校魅力的手段，事先对应刊登信息的内容彻底进行仔细检查。此外，在信息刊登后也将努力使相关内容始终保持最新且正确的状态。. 违反公共秩序或善良风俗的信息. 网站上刊登的信息应是与京都大学进行的教育、研究、社会贡献相关的信息，例如，不得刊登违反公共秩序或善良风俗的信息、以商业行为、政治\/宗教活动等为目的的信息。. 改善劝告. 专门部会判断网站上刊登的报道内容违反前款规定时，专门部会会向该报道的刊登者及该网站的运用负责人提出改善劝告。此外，对于链接至违反公共秩序或善良风俗的网站的行为，也将采取相同的措施。. 其他规定. 其他与网站运用相关的必要事项由专门部会规定。. （1997年10月27日 网站专门委员会制定）. （2015年2月24日 公共关系委员会网站企划专门部会改正）. （2007年6月15日 京都大学公共关系委员会网站企划专门部会改正）. （2015年4月 1日 公共关系委员会网站企划专门部会改正）. （2017年4月 1日 公共关系委员会网站企划专门部会改正）. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"YJmbfkU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/zh-cn\/news\/2022-03-29","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":47,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["zh-cn","news"],"url_file_name_s":"2022-03-29","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["举行了2021学年毕业典礼。（2022年3月24日） | 京都大学"],"description_txt":["　 2022年3月24日，2021学年毕业典礼在京都市劝业馆Miyako Messe举行，井村裕夫前校长、各理事和副校长、以及各部门领导出席了典礼。\n\n毕业生人数"],"last_modified":"Wed Jun 08 16:43:08 UTC 2022","dates_in_content_count_i":4,"dates_in_content_dts":["Tue Mar 29 00:00:00 UTC 2022","Mon Apr 11 00:00:00 UTC 2022","Fri Apr 08 00:00:00 UTC 2022","Tue Apr 05 00:00:00 UTC 2022"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":6321900780987337277,"exact_signature_unique_b":true,"fuzzy_signature_l":-3914327296468076740,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["举行了2021学年毕业典礼。（2022年3月24日）"],"h2_txt":["グローバルナビゲーション（簡体中国語）","ヘッダー セカンダリーリンク（日本語以外）","凑长博校长的致辞","Latest news","举行了2022学年研究生院入学典礼。（2022年4月7日）","举行了2022学年本科入学典礼。（2022年4月7日）","举行了2021学年研究生院学位授予仪式。（2022年3月24日）","举行了2021学年毕业典礼。（2022年3月24日）","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Tue Mar 29 12:00:00 UTC 2022","Mon Apr 11 12:00:00 UTC 2022","Fri Apr 08 12:00:00 UTC 2022","Tue Apr 05 12:00:00 UTC 2022","Tue Mar 29 12:00:00 UTC 2022"],"imagescount_i":11,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakubu-1-cd9bbcf8ee8d3acae2b113b38bbd7574.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakubu-2-b3b92460466fc2035d8961ecb00034cd.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakubu-3-f8ac4784776f0f47bd9141924cc07d2b.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakubu-4-6685ebe7f747e78e52fa64d805a1a8b4.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","京都大学交響楽団","湊総長","学位授与代表者","会場の様子","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal sites default files inline 20220324 gakubu 1 cd9bbcf8ee8d3acae2b113b38bbd7574 jpg cd 9bbcf 8ee 8d 3acae 2b 113b 38bbd 7574 京都大学交響楽団 2 b3b92460466fc2035d8961ecb00034cd b 3b 92460466fc 2035d 8961ecb 00034cd 湊総長 3 f8ac4784776f0f47bd9141924cc07d2b f 8ac 4784776f 0f 47bd 9141924cc 07d 学位授与代表者 4 6685ebe7f747e78e52fa64d805a1a8b4 6685ebe 7f 747e 78e 52fa 64d 805a 1a 8b 会場の様子 icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":942,"text_t":"举行了2021学年毕业典礼。（2022年3月24日） | 京都大学. 跳转到主要内容 网站内搜索. 教师搜索. 搜索. close. CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home NEWS 举行了2021学年毕业典礼。（2022年3月24日）. University News 举行了2021学年毕业典礼。（2022年3月24日）. In other languages. 日本語 ターゲット. Current Students & Staff Visitors 发表日期. 2022\/03\/29. 　 2022年3月24日，2021学年毕业典礼在京都市劝业馆Miyako Messe举行，井村裕夫前校长、各理事和副校长、以及各部门领导出席了典礼。. 毕业生人数 凑长博校长的致辞. 2021学年毕业典礼 致辞 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 发表日期. 2022\/04\/11. 举行了2022学年研究生院入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/08. 举行了2022学年本科入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/05. 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. 发表日期. 2022\/03\/29. 举行了2021学年毕业典礼。（2022年3月24日）. 最新新闻 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home 交通. 交通. 2階層目ページ用メニュー（簡体中国語）. 招聘信息 交通资讯 In other languages. 日本語 京都大学的位置. 广域交通图　. 放大图点击此处 从机场出发的交通图　. 放大图点击此处 从机场到京都的交通. 从机场到JR京都站，可以乘坐铁路、公交车、出租车等交通工具。所需时间及费用请查看各机场的网站。. 从关西国际机场出发. 详情请查看. 关西国际机场官网的“机场交通”页 。. 从大阪国际机场 （伊丹机场）出发. 详情请查看. 大阪国际机场（伊丹机场）官网的“机场交通”页 。. 从成田国际机场出发. 详情请查看. 成田国际机场官网的“交通信息”页 。. 京都大学的3个校区. 放大地图点击此处 京都大学大致分为三个校区，即吉田校区、宇治校区、桂校区。. 吉田校区 宇治校区 桂校区 各校区的详情请查看下述页面。. 关于各校区（吉田、宇治、桂） 研究所与附属研究设施等. 本校的研究所及附属研究设施等散布于日本全国。关于研究所及附属研究设施等的详情请查看下述页面。. 日本国内的研究所与附属研究设施等 京都府内的设施. 舞鹤水产实验所（野外调查科学教育研究中心） 芦生研究林 花山天文台（研究生院理学研究科附属） 宇治校区内. 防灾研究所 宇治川开放实验室 研究生院农学研究科附属牧场 研究生院农学研究科附属农场 栽培植物起源学研究室（研究生院农学研究科） 共8家设施，截至2020年4月. 全日本的设施. 北海道研究林［標茶区］（野外调查科学教育研究中心） 北海道研究林［白糠区］（野外调查科学教育研究中心） 大潟波浪观测所（防灾研究所附属流域灾害研究中心） 上宝观测所（防灾研究所附属地震预知研究中心） 穗高砂防观测所（防灾研究所附属流域灾害研究中心） 飞驒天文台（研究生院理学研究科附属） 北陆观测所（防灾研究所附属地震预知研究中心） 灵长类研究所 生态学研究中心 流域圈综合环境质研究中心（研究生院工学研究科附属） 生存圈研究所信乐MU观测所 逢坂山观测所（防灾研究所附属地震预知研究中心） 阿武山观测所（防灾研究所附属地震预知研究中心） 复合原子力科学研究所 屯鹤峰观测所（防灾研究所附属地震预知研究中心） 纪伊大岛实验所（野外调查科学教育研究中心） 潮岬风力实验所（防灾研究所附属流域灾害研究中心） 濑户临海实验所（野外调查科学教育研究中心） 白浜海洋现象观测所（防灾研究所附属流域灾害研究中心） 和歌山研究林（野外调查科学教育研究中心） 鸟取观测所（防灾研究所附属地震预知研究中心） 冈山天文台（研究生院理学研究科附属） 德山试验地（野外调查科学教育研究中心） 德山试验地（防灾研究所附属地震预知研究中心） 德岛山崩观测所（防灾研究所附属斜面灾害研究中心） 附属地球热学研究设施（研究生院理学研究科） 火山研究中心（研究生院理学研究科附属地球热学研究设施） 野生动物研究中心 熊本庇护所 宫崎观测所（防灾研究所附属地震预知研究中心） 幸岛观察站（幸岛观察所） 樱岛火山观测所（防灾研究所附属火山活动研究中心） 生存圈研究所生活与森林圈模拟场 屋久岛观察站（屋久岛观察所） 共33家设施，截至2020年4月. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"bwF6xkU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/zh-cn\/news\/2022-04-05","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":47,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["zh-cn","news"],"url_file_name_s":"2022-04-05","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["举行了2021学年研究生院学位授予仪式。（2022年3月24日） | 京都大学"],"description_txt":["　2022年3月24日，2021学年研究生院学位授予仪式在京都市劝业馆Miyako Messe举行， 各理事和副校长、以及各部门领导等出席了仪式。\n\n　本次获得学位的包括硕士2,211名、（专业）硕士176名、（专业）法务博士129名、以及分别在2021年11月24日、2022年1月24日和3月23日获得学位的课程博士共510名、论文博士共44名。\n\n　12名同学完成了始于2012年的博士课程教育领导课程。\n\n　12名同学完成了始于2018年的卓越研究生院课程。\n\n　凑长博校长向学生代表颁发了学位证书并进行了校长致辞。"],"last_modified":"Wed Jun 08 16:43:09 UTC 2022","dates_in_content_count_i":4,"dates_in_content_dts":["Tue Apr 05 00:00:00 UTC 2022","Mon Apr 11 00:00:00 UTC 2022","Fri Apr 08 00:00:00 UTC 2022","Tue Mar 29 00:00:00 UTC 2022"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":8476884593700931984,"exact_signature_unique_b":true,"fuzzy_signature_l":6062561263262572873,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["举行了2021学年研究生院学位授予仪式。（2022年3月24日）"],"h2_txt":["グローバルナビゲーション（簡体中国語）","ヘッダー セカンダリーリンク（日本語以外）","凑长博校长的致辞","Latest news","举行了2022学年研究生院入学典礼。（2022年4月7日）","举行了2022学年本科入学典礼。（2022年4月7日）","举行了2021学年研究生院学位授予仪式。（2022年3月24日）","举行了2021学年毕业典礼。（2022年3月24日）","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Tue Apr 05 12:00:00 UTC 2022","Mon Apr 11 12:00:00 UTC 2022","Fri Apr 08 12:00:00 UTC 2022","Tue Apr 05 12:00:00 UTC 2022","Tue Mar 29 12:00:00 UTC 2022"],"imagescount_i":11,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakuijyuyo-1-5b5bc3bf11c39fa7257ddd33344a8cf5.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakuijyuyo-2-ca2803e63585a713fb618f828752b3b0.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakuijyuyo-3-47ba5c4af611677b4a1706f466feffd6.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/20220324-gakuijyuyo-4-a06ffe97596cde8558c2b0e5aac42839.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","学位授与","学位授与","会場の様子","湊総長","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal sites default files inline 20220324 gakuijyuyo 1 5b5bc3bf11c39fa7257ddd33344a8cf5 jpg 5b 5bc 3bf 11c 39fa 7257ddd 33344a 8cf 5 学位授与 2 ca2803e63585a713fb618f828752b3b0 ca 2803e 63585a 713fb 618f 828752b 3b 0 3 47ba5c4af611677b4a1706f466feffd6 47ba 5c 4af 611677b 4a 1706f 466feffd 6 会場の様子 4 a06ffe97596cde8558c2b0e5aac42839 a 06ffe 97596cde 8558c 2b 0e 5aac 42839 湊総長 icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":803,"text_t":"举行了2021学年研究生院学位授予仪式。（2022年3月24日） | 京都大学. 跳转到主要内容 网站内搜索. 教师搜索. 搜索. close. CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home NEWS 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. University News 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. In other languages. 日本語 ターゲット. Current Students & Staff Visitors 发表日期. 2022\/04\/05. 　2022年3月24日，2021学年研究生院学位授予仪式在京都市劝业馆Miyako Messe举行， 各理事和副校长、以及各部门领导等出席了仪式。. 　本次获得学位的包括硕士2,211名、（专业）硕士176名、（专业）法务博士129名、以及分别在2021年11月24日、2022年1月24日和3月23日获得学位的课程博士共510名、论文博士共44名。. 　12名同学完成了始于2012年的博士课程教育领导课程。. 　12名同学完成了始于2018年的卓越研究生院课程。. 　凑长博校长向学生代表颁发了学位证书并进行了. 校长致辞 。. 博士学位获得者人数一览（2021年11月24日、2022年1月24日、2022年3月23日） 硕士学位、（专业）硕士学位、（专业）法务博士学位获得者人数一览 凑长博校长的致辞. 2021学年学位授予仪式 致辞 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 发表日期. 2022\/04\/11. 举行了2022学年研究生院入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/08. 举行了2022学年本科入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/05. 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. 发表日期. 2022\/03\/29. 举行了2021学年毕业典礼。（2022年3月24日）. 最新新闻 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"fwjeMkU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/zh-cn\/privacy-policy","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":46,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["zh-cn"],"url_file_name_s":"privacy-policy","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["隐私政策 | 京都大学"],"description_txt":["京都大学知道个人信息保护在高度信息通信社会中的重要性，在京都大学网站（http:\/\/www.kyoto-u.ac.jp\/）中处理相关信息时，将遵守法令，并根据 《京都大学有关个人信息保护的规程》（以下称“规程”）及下述方针努力保护个人信息。"],"last_modified":"Wed Jun 08 16:43:10 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":2497490045155049949,"exact_signature_unique_b":true,"fuzzy_signature_l":1967472654301805620,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["隐私政策"],"h2_txt":["グローバルナビゲーション（簡体中国語）","ヘッダー セカンダリーリンク（日本語以外）","1. 关于个人信息的取得与持有","2. 关于个人信息的利用","3. 关于个人信息的管理","4. 关于个人信息的披露、订正、停用及删除","5. 个人信息保护规程的制定、实施、维持及改善","6. 关于Google Analytics服务的利用","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":836,"text_t":"隐私政策 | 京都大学. 跳转到主要内容 网站内搜索. 教师搜索. 搜索. close. CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 面包屑. Home 隐私政策. 隐私政策. In other languages. 日本語 京都大学知道个人信息保护在高度信息通信社会中的重要性，在京都大学网站（. http:\/\/www.kyoto-u.ac.jp\/ ）中处理相关信息时，将遵守法令，并根据. 《京都大学有关个人信息保护的规程》 （以下称“规程”）及下述方针努力保护个人信息。. 1. 关于个人信息的取得与持有. 京都大学将采取合法且公正的方式取得个人信息。. 京都大学将在执行本校业务的必要情况下尽可能特别指定其目的，不会持有超出实现该指定利用目的之必要范围的个人信息。. 在取得个人信息时，我们将表明需取得信息的利用目的及已取得信息的管理员。. 2. 关于个人信息的利用. 除了法令规定的情况外，京都大学不会将个人信息用于取得时表明的利用目的之外的目的或提供给第三方。. 除了法令规定的情况外，京都大学不会在未事先征得本人同意的情况下将个人信息提供给第三方。. 京都大学根据法令将个人信息提供给第三方、或者将涉及个人信息处理的业务委托给第三方时，将就该个人信息的安全确保、管理能力等对该第三方进行严格调查，并在让其处理个人信息时进行合理的监督。. 3. 关于个人信息的管理. 京都大学将对非法访问和计算机病毒等采取合理的信息安全对策，从而确保持有的个人信息的正确性并防止其遗失、破坏、篡改及泄露等。. 4. 关于个人信息的披露、订正、停用及删除. 京都大学承认本人有权对自己的个人信息提出披露、订正、停用、删除等要求，对于这些要求，我们将根据“规程”的规定内容及时处理。. 若对京都大学的个人信息处理存在意见或疑问，请联系京都大学与个人信息相关的披露请求窗口（总务部法务室） 830jyohokoukai * mail2.adm.kyoto-u.ac.jp （请将*改为@）。. 关于请求披露方法 5. 个人信息保护规程的制定、实施、维持及改善. 除了本方针规定的内容外，我们还将根据《规程》的规定内容适当地实施京都大学的个人信息保护。. 6. 关于Google Analytics服务的利用. 本网站利用Google Analytics服务，以便统计性地掌握用户的浏览情况，并将其作为今后组织运营及网站改善的参考。在收集该服务所需的流量数据时使用了第一方cookie。关于Google Analytics中的隐私数据处理内容，请查看Google公司的该服务的. 概要说明页 。. 2005年3月29日制定. 京都大学. （2005年3月29日 京都大学网站专门部会制定）. （2016年11月7日 京都大学网站企划专门部会改正）. （2017年 4月1日 京都大学网站企划专门部会改正）. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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(2022년 4월 7일)","2022년도 학부 입학식이 거행되었습니다. (2022년 4월 7일)","2021년도 대학원 학위 수여식이 거행되었습니다. (2022년 3월 24일)","2021년도 졸업식이 거행되었습니다. (2022년 3월 24일)","2021년도 대학원 후기 입학식 영상 축사를 안내 드립니다.","2021년도 대학원 후기 학위 수여식이 거행되었습니다.","2021년도 학부 입학식이 거행되었습니다. (2021년 4월 7일)","2021년도 대학원 입학식이 거행되었습니다. 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(2022년 3월 24일). 説明. 2022년 3월 24일, 교토시 권업관 미야코멧세에서 이무라 히로오(井村裕夫) 전 총장, 각 이사 및 부학장, 각 부국장이 참석한 가운데 졸업식이 거행되었습니다. 画像. 2021년도 대학원 후기 입학식 영상 축사를 안내 드립니다. 説明. 2021년도 대학원 후기 입학식은 신종 코로나바이러스 감염증의 영향으로 현장 행사는 개최하지 않고 미나토 나가히로(湊長博) 총장의 영상 축사를 보내 드리게 되었습니다. 画像. 2021년도 대학원 후기 학위 수여식이 거행되었습니다. 説明. 2021년 9월 24일(금요일), 백주년 시계탑 기념관에서 미나토 나가히로(湊長博) 총장, 이사 및 부학장을 비롯해 각 연구과장, 학관장, 학사장, 교육부장, WISE Program(Doctoral Program for World-leading Innovative & Smart Education, 문부과학성 선정) 코디네이터 등이 참석한 가운데 대학원 후기 학위 수여식이 거행되었습니다. 画像. 2021년도 학부 입학식이 거행되었습니다. (2021년 4월 7일). 説明. 2021년 4월 7일(수요일), 교토시 권업관 ‘미야코멧세’에서 이무라 히로오(井村裕夫) 전 총장, 마쓰모토 히로시(松本紘) 전 총장, 야마기와 주이치(山極壽一) 전 총장, 각 이사 및 부학장, 각 부국장이 참석한 가운데 2021년도 학부 입학식이 거행되었습니다. 画像. 2021년도 대학원 입학식이 거행되었습니다. (2021년 4월 7일). 説明. 2021년 4월 7일(수요일), 교토시 권업관 ‘미야코멧세’에서 마쓰모토 히로시(松本紘) 전 총장, 야마기와 주이치(山極壽一) 전 총장, 각 이사 및 부학장, 각 부국장이 참석한 가운데 2021년도 대학원 입학식이 거행되었습니다. chevron_left. play_arrow. pause. chevron_right. Announcements. Statement. Statement regarding the Situation in Ukraine. COVID-19. Regarding the novel coronavirus: Kyoto University’s Response to the Novel Coronavirus (COVID-19). Latest research news (available in English). 발행일. 2022\/06\/03. COVID-19 Research Using Bronchial Organoids and Drug Discovery Applications. 발행일. 2022\/05\/31. Invention of technology to create mature skeletal muscle cells from iPS cells without genetic manipulation. 발행일. 2022\/05\/30. Breathtaking solution for a breathless problem. 발행일. 2022\/05\/27. Some monkeys might pull a poker face. More research news KyotoU News. アイキャッチ画像. 발행일. 2022\/02\/18. タグ. Global Engagement. Education. 유학생 라운지 ‘기즈나’ 온라인 이벤트 ‘종이접기로 크리스마스 장식을 만들자!’를 개최했습니다. アイキャッチ画像. 발행일. 2022\/02\/18. タグ. Global Engagement. Education. 글로벌 환경 인재 양성 프로그램을 실시했습니다. 발행일. 2022\/04\/11. タグ. University News. 2022년도 대학원 입학식이 거행되었습니다. (2022년 4월 7일). 발행일. 2022\/04\/08. タグ. University News. 2022년도 학부 입학식이 거행되었습니다. (2022년 4월 7일). 발행일. 2022\/04\/05. タグ. University News. 2021년도 대학원 학위 수여식이 거행되었습니다. (2022년 3월 24일). 발행일. 2022\/03\/29. タグ. University News. 2021년도 졸업식이 거행되었습니다. (2022년 3월 24일). 더 많은 뉴스 Upcoming Events. 더 많은 이벤트 Features. Kyoto University's125th Anniversary. edX 공개 온라인 수업. 교토대학 오픈코스웨어. 대학소개. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 간행물 채용 교통편 안내 フッター セカンダリーリンク. 사이트 맵 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 課外活動支援. 課外活動支援. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 課外活動学生団体の紹介 課外活動のための物品貸出など 学生が利用できる本学の遠隔地施設 近隣の文化施設の優待制度 「学生ボランティア」学校サポート事業 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 3階層目以降ページ用メニュー. 課外活動学生団体の紹介. 京都大学には多くの課外活動学生団体が熱心な活動をしています。ここでは、非公認の団体を除いて、公認団体を「体育会所属団体」、「体育系サークル」、「文化系サークル」に区分して示しています。それぞれの団体の活動内容については紹介していませんので、関連のホームページなどをご覧ください。 課外活動のための物品貸出など. 物品貸出など課外活動をサポートする取組を紹介します。 学生が利用できる本学の遠隔地施設. 本学には以下の3つの遠隔地の施設があり、学生諸君が課外活動などで利用できます。 近隣の文化施設の優待制度. 本学学生が近隣の博物館、美術館や歴史的文化施設等において多様な文化芸術に触れるための優遇制度です。是非活用してください。 「学生ボランティア」学校サポート事業. 本学学生が教育現場におけるボランティア活動によって自己の資質向上に取り組むための事業です。 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション（簡体中国語）. 京都大学概况. 校长致辞. 大学介绍. 有关安全的措施. 教育・学生的支援. 学生生活. 留学・国际教育. 学部与院系. 研究・行政・民间合作. 支援研究体制. Security Export Control. 国际交流. ヘッダー セカンダリーリンク（日本語以外）. 交通资讯 联系我们 捐赠 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 画像. Resources for prospective international students. 説明. 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Announcements. Statement. Statement regarding the Situation in Ukraine. COVID-19. Regarding the novel coronavirus: Kyoto University’s Response to the Novel Coronavirus (COVID-19). Latest research news (available in English). 发表日期. 2022\/06\/03. COVID-19 Research Using Bronchial Organoids and Drug Discovery Applications. 发表日期. 2022\/05\/31. Invention of technology to create mature skeletal muscle cells from iPS cells without genetic manipulation. 发表日期. 2022\/05\/30. Breathtaking solution for a breathless problem. 发表日期. 2022\/05\/27. Some monkeys might pull a poker face. More research news KyotoU News. アイキャッチ画像. 发表日期. 2022\/02\/18. タグ. Global Engagement. Education. 举办了留学生社交室KI-ZU-NA“在线活动‘通过折纸来制作圣诞装饰吧！’”. アイキャッチ画像. 发表日期. 2022\/02\/18. タグ. Global Engagement. Education. 实施了全球环境人才培养项目. 发表日期. 2022\/04\/11. タグ. University News. 举行了2022学年研究生院入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/08. タグ. University News. 举行了2022学年本科入学典礼。（2022年4月7日）. 发表日期. 2022\/04\/05. タグ. University News. 举行了2021学年研究生院学位授予仪式。（2022年3月24日）. 发表日期. 2022\/03\/29. タグ. University News. 举行了2021学年毕业典礼。（2022年3月24日）. 最新新闻 Upcoming Events. 最新活动 Features. Kyoto University's125th Anniversary. 网上公布课程. 京都大学开放课程讲座影片内容. 大学介绍. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通资讯 フッター セカンダリーリンク. 站点地图 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n附属図書館　図書館企画課 図書館企画掛 〒 606-8501 京都市左京区吉田本町 075-753-2616 soumu660*mail2.adm.kyoto-u.ac.jp\n医学研究科総務企画課企画広報掛 〒 606-8501 京都市左京区吉田近衛町 075-753-4305、4669 060k-kouhou*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n生態学研究センター総務掛 〒 520-2113 滋賀県大津市平野2丁目509-3 077-549-8200 620groupA*mail2.adm.kyoto-u.ac.jp\n化学研究所\nエネルギー理工学研究所\n生存圏研究所\n防災研究所\n生態学研究センター\n医学研究科\n工学研究科\n学術情報メディアセンター\n附属図書館\nｉＰＳ細胞研究所\n東南アジア地域研究研究所\n霊長類研究所\n複合原子力科学研究所\n数理解析研究所\n経済研究所\n基礎物理学研究所\nウイルス・再生医科学研究所\n人文科学研究所\n経営管理研究部\n公共政策連携研究部\n地球環境学堂\n理学研究科\n経済学研究科\n法学研究科\n教育学研究科\n文学研究科\n総合生存学館\n生命科学研究科\n情報学研究科\nアジア・アフリカ地域研究研究科\nエネルギー科学研究科\n人間・環境学研究科\n農学研究科\n薬学研究科\n医学部附属病院\n担当部署名\n住所\n担当部署連絡先\n部局名\n部局別の担当部署連絡先\n【留意事項】\n・部局名をクリックすると、当該部局のHPにリンクします。（部局HPがある場合のみ）\n・依頼に係る手続きについては、予め各部局HP等を参照のうえ、各担当部署にご確認ください。\n・兼業許可にあたっては会議附議等、学内手続きに時間を要しますので、予め余裕をもってご依頼ください。\n特に、研究科長等の部局の長が行うものや営利企業の役員の職にかかるものについては、通常よりも\n審査に時間を要しますので、ご留意ください。\n【依頼時期の目安】\n・部局の長が行うもの、営利企業の職にかかるもの…兼業開始日の２～３か月前\n・上記以外のもの…兼業開始日の１～２か月前\n※依頼日から兼業開始日までの期間が短い場合、開始日までに許可できないことがあります。\nTEL E-mail(*を@に変更してください)\n担当部署名\n住所\n担当部署連絡先\n部局名\n野生動物研究センター事務掛（総務担当） 〒 606-8203 京都府京都市左京区田中関田町2-24 075-771-4393 soumu_yasei*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n北部構内総務課人事掛（農学、基礎研、フィールド等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-6024 a60somu2*mail2.adm.kyoto-u.ac.jp\n福井謙一記念研究センター事務室 〒 606-8103 京都市左京区高野西開町34-4 075-711-7708 090akinen*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n企画・情報部情報推進課総務掛 〒 606-8501 京都市左京区吉田本町　学術情報メディアセンター北館 075-753-7400、7402 740jyohokikaku-soumu*mail2.adm.kyoto-u.ac.jp\n研究推進部産官学連携課産官学連携企画掛 〒 606-8501\n京都市左京区吉田本町京都大学\n国際科学イノベーション棟 東館5階\n075-753-5536 sanren-jinji*mail2.adm.kyoto-u.ac.jp\n研究推進部産官学連携課オープンイノベーション機構支援室 〒 606-8501\n京都市左京区吉田本町\n京都大学国際科学イノベーション棟\n075-753-7767 oi-shienjimu*mail2.adm.kyoto-u.ac.jp\n企画・情報部国際交流課国際企画掛 〒 606-8501 京都市左京区吉田本町 075-753-2045 kokusa850*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n大学文書館事務室 〒 606-8305 京都市左京区吉田河原町15-9 075-753-2651 archives*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n学術研究支援室事務部門 〒 606-8501 京都市左京区吉田本町　学術研究支援棟 075-753-5108 shienshitu-jimu*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n次世代研究創成ユニット 本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\nアジア研究教育ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\nスーパーグローバルコース人文社会科学系ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n人社未来形発信ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n地域連携教育研究推進ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n実験と理論計算科学のインタープレイによる\n触媒・電池の元素戦略研究拠点ユニット\n桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\n構造材料元素戦略研究拠点ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\nインフラシステムマネジメント研究拠点ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\nレジリエンス実践ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\nスーパーグローバルコース化学系ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\n光量子センシング研究拠点 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\n宇宙総合学研究ユニット 北部構内総務課人事掛（理学、数研等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-3606 a60somu1*mail2.adm.kyoto-u.ac.jp\n森里海連環学教育研究ユニット 北部構内総務課人事掛（農学、基礎研、フィールド等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-6024 a60somu2*mail2.adm.kyoto-u.ac.jp\n企画・情報部企画課総務掛 〒 606-8501 京都市左京区吉田本町 075-753-2600 kikakuka-soumu*mail2.adm.kyoto-u.ac.jp\n教育推進・学生支援部学生課総務掛 〒 606-8501 京都市左京区吉田本町 075-753-2506 840gakusei-soumu*mail2.adm.kyoto-u.ac.jp\n研究推進部研究推進課総務掛 〒 606-8501 京都市左京区吉田本町 075-753-2041 kensui.jinji*mail2.adm.kyoto-u.ac.jp研究推進部\n教育推進・学生支援部\n企画・情報部\n福井謙一記念研究センター\n野生動物研究センター\n白眉センター\nアフリカ地域研究資料センター\n学術研究支援室\n高等研究院\n大学文書館\n大学文書館\n学際融合教育研究推進センター\nこころの未来研究センター\n国際高等教育院\nフィールド科学研究センター\n総合博物館\n高等教育研究開発推進センター\n学生総合支援センター\n国際戦略本部\nオープンイノベーション機構\n産官学連携本部\n情報環境機構\n男女共同参画推進センター\n"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 特色入試 特色入試の過去問題. 特色入試の過去問題. サイドメニュー. 一般選抜 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 令和3年度 試験問題 令和4年度 試験問題 試験問題等の利用について 出願状況と選考結果 よくある質問 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 3階層目以降ページ用メニュー. 令和3年度 試験問題. 令和4年度 試験問題. 試験問題等の利用について. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Research & Collaborate Research news and departments. Research news and departments. サイドメニュー（英語）. Research news and departments Research news Departments Policies Collaboration Research support and scholarships Careers Visa\/Housing\/Daily Life About Research compliance & ethics Overview: To wander with purpose. A tradition of non-traditional thinking has defined Kyoto University over its 120-year history. The freedom to think unorthodox ideas allows cutting-edge science and unique ideas to flourish at Kyoto. President. Juichi Yamagiwa 's ambition is to keep Kyoto University a safe haven for free thinking: free from societal pressures, free from prejudice, and free from existing paradigms. \"In present-day Japan, people tend to be treated harshly when they think outside the box,\" Yamagiwa observes. \"Universities should protect students and researchers from that. Free thinking arising from the liberal atmosphere here helps pioneer the future.\". Many come to Kyoto for the freedom to conceive and explore radical ideas. They include physicist. Hideki Yukawa , the first Japanese Nobel laureate, and Kitaro Nishida, who founded the Kyoto school of philosophy. Yamagiwa is also a pioneer, spending years in the African jungle living with gorillas and studying human evolution. Ultimately, Yamagiwa plans to make Kyoto an incubator for 'frontier science' -- progressive research based on interdisciplinary communication. \"Frontier science should transcend national borders and also the differences between the arts and the sciences,\" he says, reaffirming his intention to spread Kyoto's distinctive studies to the rest of the world. \"Our researchers don't just follow the cutting edge. They are the pioneers that plough the field for others to follow,\" he emphasizes. Animals and technology. And plowing through the field of great ape behavior is. Fumihiro Kano , who applies the latest technologies, including eye tracking and thermal sensing, to his research. In. a recent study , Kano used eye-tracking technology to examine how well apes recall one-time events stored in their long-term memories. Kano's team recorded where the apes looked while viewing video clips starring a colleague dressed as a 'bad' gorilla. The results demonstrated that apes can remember events they had seen or experienced only once. \"Studying animal behavior is no longer just about direct observation and long-term training,\" Kano explains. Meanwhile, Kano's colleague. Masaki Tomonaga has been testing the visual perception of ponies using touch screens. In. his study , ponies discriminate between different shapes by nuzzling a touch screen. Comparison of these results with those of studies involving other mammals reveals that ponies see the world as humans do, he says. \"New technology is developing quickly. It opens up so many options.\". Peering into germ-cell development. Behavioral scientists are not the only ones finding new uses for the latest advances. Kotaro Sasaki is breaking new ground with induced pluripotent stem (iPS) cells, whose discovery led to Kyoto University's. Shinya Yamanaka being awarded the 2012 Nobel Prize in Physiology or Medicine. Pluripotent stem cells have been used to generate just about every cell type in the body except germ cells, which include sperm and eggs. Unlike other stem-cell-based therapies, which affect only the patient, those that induce germ cells could also benefit the patient's offspring. Germ cells are difficult to study because of the ethical and technical issues involved in experimentation. \"When I read about his work, I knew I had to come back to Japan,\" says Sasaki, referring to lab leader. Mitinori Saitou 's work in the field. Sasaki, who was an established pathologist in the United States, returned to join Saitou's team in Kyoto. Sasaki spent a month at the. Center for iPS Research and Application learning how to work with human iPS cells and about the developmental process of human germ cells. In addition to illustrating key transcription interactions and signaling events, Sasaki gained insight into how epigenetic marks -- traits that are inherited without changes to the DNA sequence -- are 'erased' at the beginning of germ-cell development. The team's model , still in its early stages, may form a foundation for continuing studies on germ cell lineage. \"By further reconstituting human germ-cell development. in vitro. ,\" explains Saitou, \"we may be able to elucidate the mechanisms throughout the developmental process, from embryo to adult.\". 'Seeing' black holes with visible light. In the field of astronomy,. Mariko Kimura , a master's student, put Kyoto University's international network to full use when she witnessed a major event unfold in front of her eyes. Kimura called affiliated astronomers when the binary black-hole system V404 Cygni generated an outburst, an event that occurs only a few times a century. Within minutes, she had international collaborators supporting her with unprecedented amounts of observational data. \"We now know that we can make observations based on optical rays -- visible light, in other words -- and that black holes can be observed without high-spec X-ray or gamma-ray telescopes,\" Kimura says. Her study indicates that the main factor triggering repetitive activity around black holes is not the quantity of mass accretion, but rather the length of orbital periods. \"The study would not have come to fruition,\" says. Daisaku Nogami , leader of the research unit, \"were it not for the readily available network Kyoto astronomers have established through years of effort with a view to advancing mutual support in observation.\". \"Stars can only be observed after dark, and there are only so many hours each night, but by making observations from different locations around the globe we're able to gather data more comprehensively,\" elaborates Nogami. Searching for where happiness happens. In Yamagiwa's safe haven for free thinking, other researchers delve into a question left unanswered for millennia -- our own well-being. Throughout human history, eminent scholars such as Aristotle have contemplated happiness, says. Wataru Sato . He is determined to find answers with his own twist and with the help of functional magnetic resonance imaging (fMRI). \"Emotions arise from our subconscious mind and spirit,\" says Sato. \"People bond through emotions, but we can't access our subconscious minds directly. Being able to understand that part of ourselves could help us solve problems associated with the mind and spirit -- including how to be happy.\". At the. Kokoro Research Center , scholars like Sato explore the mind and spirit, with some examining the topic from a philosophical perspective, and others, like Sato, integrating neurological insights in their studies. Using fMRI, Sato has identified the part of the brain that 'feels' happiness and has clarified what factors constitute happiness. According to. his study , happiness is a combination of happy emotions and satisfaction with life coming together in the precuneus, a region in the medial parietal lobe that becomes active when experiencing consciousness. For Yamagiwa, this is only the beginning. Even in the face of the uncertainty of this globalized era, Yamagiwa believes that Kyoto can lead the world with its creativity. And he is not merely talking about edgy science. \"We wander in academic freedom with a purpose,\" Yamagiwa says. \"These innovations will lead to solutions for a harmonious, sustainable, global society.\". Related links. Research activities Graduate degree programs Undergraduate degree programs Non-degree programs Faculties, Graduate Schools, Research Institutes, and other centers 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 学生表彰など 京都大学久能賞. 京都大学久能賞. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 総長賞（すべての学生個人・団体） たちばな賞（若手女性研究者・学生） 久能賞（理系の女子学生） SPEC（京大生チャレンジコンテスト） 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 京都大学久能賞とは. 　京都大学久能賞は、京都大学OGの久能和子氏（工学部昭和50年卒）、祐子氏（同52年卒）のお母様である久能悠子氏からのご寄附により設立されました。悠子氏は、ご自身の学生時代に、科学の道へ進みたいとの思いを抱いておりましたが、その夢は戦争によって叶いませんでした。久能賞には、悠子氏のお二人のご令嬢が京都大学の女子学生として素晴らしい教育を受けることが出来たという感謝の思いと、悠子氏の叶えられなかった夢を今の若い世代の方々に託したいという二つの思いが込められています。. 　本賞は、同氏からのご寄附の趣意を踏まえ、21世紀における地球規模の課題を解決し、よりよい世界を目指し、社会に貢献したいという高い志を持ち、科学・技術分野において自ら定めた独創的な夢を持つ意欲のある女子学生を支援することを目的としています。. 2021年度 京都大学久能賞の募集について. 2021年度 京都大学久能賞の募集は終了しました。. 2021年度募集要項等. 募集要項・申請書. 募集要項（PDF） 願書（Excel） 課題エッセイ「私の夢と志」（Word） 申請期間. 2021年9月24日（金曜日）～2021年10月22日（金曜日）. 二次選考（面接審査）. 2021年12月の予定. ※ 二次選考（面接審査）実施形態および実施予定日は後日告知します。. 授賞式. 2022年1月中旬頃の予定. ※ 授賞式の実施形態および実施予定日は後日告知します。. ポスター（PDF） 2021年度 京都大学久能賞受賞者について. 2021年度 京都大学久能賞受賞者3名を決定しました。. 受賞者. 童 友さん（生命科学研究科 1年生）. 合屋 智尋さん（医学部 2年生）. 齊藤 良佳さん（医学部 4年生）. 過去の情報. 2021年度京都大学久能賞受賞者が決定しました（2022年2月8日） 2021年度 京都大学久能賞の募集について（2021年09月08日） 2020年度京都大学久能賞受賞者が決定しました（2021年2月3日） 2020年度 京都大学久能賞の募集について（2020年9月10日） 第4回京都大学久能賞授賞式・報告会を開催しました。（2020年1月15日） 2019年度 京都大学久能賞の募集について（2019年9月20日） 第3回京都大学久能賞授賞式・報告会を開催しました。（2019年1月17日） 平成30年度 京都大学久能賞の募集について （2018年7月20日） 第2回京都大学久能賞授賞式・報告会を開催しました。（2018年1月23日） 平成29年度 京都大学久能賞の募集について （2017年7月24日） 第1回京都大学久能賞授賞式を開催しました。（2016年12月22日） 平成28年度 京都大学久能賞の募集について （2016年9月26日） 問い合わせ. 教育推進・学生支援部学生課奨学掛（総合研究10号館1階）. E-mail： 840scholarship＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. Tel： 075-753-2481. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 教育の体制と内容 中学・高校の教員免許. 中学・高校の教員免許. サイドメニュー. 教育の体制と内容 京都大学の基本理念と方針 学部新入生の方へ（国際高等教育院のサイトへ） アカデミックカレンダー 教養・共通教育と学部専門教育 大学院の教育 成績評価とGPA制度 補助金による教育プログラム 中学・高校の教員免許 リカレント教育（生涯教育等） 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. 京都大学がめざす教師像 教育職員免許状の取得について 教職課程ポートフォリオ 教職課程の体制 教員の養成に係る教育の質の向上に係る取組 教員免許の取得と就職の状況 教員免許更新制について 問い合わせ窓口 京都大学がめざす教師像. 京都大学は、研究型総合大学としての特質を活かし、「学問する」教師の育成を目指しています。. 学問と教育、理論と実践を統一することはもちろん大事ですが、それらの間にズレがあることも事実です。理論と実践の間を往還しながら、異なる文化の狭間で思考し、それらを実践場面において融合していくことが、即戦力に止まらない真の実践的指導力につながるのだと考えます。. たとえば、カリキュラム・デザインや教材開発の力量形成を通して、長期的な見通しの中で授業を構想する視点を得ることにより、「適応的熟達者」としての教師に成長する。また、実践経験を分析・一般化して、教師自身の言葉と論理（「現場の教育学」）を創ったり再構成したりする力を高めることで、問題把握の枠組み自体をも問い直しつつ、より広い視野に立って問題解決に当たっていく「問題探求的省察」の力を身につける。その基盤として、教科専門と教育学、両方の学問的素養が必要だと考えます。. 教職の高度化. 教育職員免許状の取得について. 以下に掲載する内容は、平成31年度以降入学者（平成28年改正教育職員免許法【新法】適用者）が対象です。. ※. 平成30年度以前入学者（平成10年改正教育職員免許法【旧法】が適用されない者を除く）に関する情報については、本学在籍者はKULASISの「全学向け共通掲示板」で確認してください。在籍者以外は課程を持つ学部・研究科等の教務窓口にお問い合わせください。. 1. 教職課程カリキュラムについて. 教育職員免許法及び教育職員免許法施行規則の改正により、平成31年度入学者より改正後の新法に基づく教職課程のカリキュラムが適用されます。なお、改正法の施行に伴う経過措置により旧法が適用される場合があります。. 2. 教育職員免許状について. 大学を除くすべての国公立、私立学校の教員となるためには教育職員免許状が必要です。このうち本学で取得できるのは、高等学校教諭、中学校教諭及び特別支援学校教諭の免許状です。. 高等学校及び中学校の教員免許状は教科《国語・社会・地理歴史・公民・理科・数学・英語など》別になっており、教育職員免許法に定められた所要の単位を修得すれば、学部・学科等の専攻分野に対応した教科の免許状が取得できます。. 取得希望者は4月に開催する教職課程オリエンテーションに必ず参加してください。また、中学校の教員免許状取得には介護等体験が必要です。詳細は、「4. 教育実習及び介護等体験について」を参照してください。. ※. 本学在籍者への教員免許状取得に関する情報については、KULASIS「全学生向け共通掲示板」ページの左下「教員免許関係情報はこちらをClick」をクリックしたページに掲示しますので、見落としのないよう注意してください。）. 取得することができる免許状の種類と教科 3. 「教科及び教職に関する科目」の単位修得について. 免許状を取得するには、教育職員免許法に定められた所要の単位を修得する必要があります。単位は、「教科及び教科の指導法に関する科目」、「教育の基礎的理解に関する科目」、「道徳、総合的な学習の時間等の指導法及び生徒指導、教育相談等に関する科目」、「教育実践に関する科目」、及び「大学が独自に設定する科目」に区分され、それぞれ必要な単位を修得しなければなりません。. ※. 教員免許状取得に必要な単位については、所属学部の教務掛で配布している「履修カルテ（単位修得状況）」で確認してください。. 「教科及び教職に関する科目」の単位修得について 「教科及び教科の指導法に関する科目」のうち、教科に関する専門的事項を含む科目は、所属の教務担当に確認してください。. 上記以外の「教科及び教科の指導法に関する科目」、「教育の基礎的理解に関する科目」、「道徳、総合的な学習の時間等の指導法及び生徒指導、教育相談等に関する科目」、「教育実践に関する科目」については、下記のファイルを確認してください。. 平成31年度以降の入学者（平成28年改正教育職員免許法【新法】適用者）対象 シラバス集はこちら 4. 教育実習及び介護等体験について. 教育実習は「教育実践に関する科目」として必修で、実習に係る事前及び事後指導（いずれも講義）並びに中・高等学校で行う実習（中学校の教員免許状は4週間・高等学校の教員免許状は2週間）からなっています。. また、中学校の教員免許状取得希望者については、平成10年度入学者から、特別支援学校で2日間と社会福祉施設等（保育所を除く）で5日間、合計7日間の介護等体験を行うことが義務づけられています。. ※. 本学在籍者への実施日程、申請方法等、教育学部からの情報はKULASIS「全学向け共通掲示板」に掲載します。. 教育実習日程及び参加申込資格 介護等体験申請方法等 教育実習及び介護等体験実施に当たっての共通の注意事項. 教育実習及び介護等体験への参加には、健康診断証明書が必要です。当該年度に実施される学生一般定期健康診断を、胸部X線検査も含めて必ず受検しなければなりません。また、事故対策として、「学生教育研究災害傷害保険」（学研災）と「学研災付帯賠償責任保険」（付帯賠責）に加入していない場合は、教育実習や介護等体験に参加できません。. 保険に関する問い合わせ窓口. 教育推進・学生支援部厚生課厚生掛. Tel： 075-753-2539. 5.教育職員免許状の授与申請について. 京都府教育委員会への一括申請手続きについては、所属学部・研究科等で行います。例年9～10月頃に各学部・研究科等から通知されるので、見落としのないよう注意してください。なお、課程認定の関係で個人申請になる場合もあります。. 6. 教育職員免許状取得までの道筋について. 一般的事項として、学部学生で1回生から教育職員免許状の取得を目指しているケースの例を掲載します。教育職員免許状は、学部の2回生以上から、または大学院学生でも、所定の単位を修得することでその取得を目指すことが可能です。. 教育職員免許状取得までの道筋 教職課程ポートフォリオ. 本学では、教職課程を履修する学生に、教職課程ポートフォリオ（学生の成果資料を体系的に整理して記録するファイル）の作成と活用を求めています。. 本学では、教師として求められる力量を5つの柱（A．教職に求められる教養、B．生徒理解と人間関係構築力、C．教科内容に関する知識・技能、D．教科等の授業づくりの力量、E．課題探求力）で明確化するとともに、それを「履修カルテ」において具体化しています。. 学生はこれらの5つの柱を意識しながら、教職課程の学びのプロセスや成果物を教職課程ポートフォリオに蓄積していきます。. そして、教職課程の総まとめとして、教職課程ポートフォリオに基づいた自己評価・相互評価と学生の自主的活動を軸に、教職実践演習を実施し、教員としての資質能力の向上をはかっています。. 教師に求められる力量の5つの柱. 京都大学教職課程ポートフォリオ（様式） 教職課程の体制. 本学の教職教育に関する検討、実施、および調査は全学の教職教育委員会等が行っています。. また、教育学研究科が教職課程の実際の運用（教職科目の企画・提供、教育実習・介護等体験などの受付や指導）を担当しています。. 必要に応じて、教職に関する責任部局である教育学研究科の教職担当教員が中心となり、教職教育の実施に関し学部・研究科等相互間の連絡調整を行う体制を整えています。. 組織図 職員数 教員研究活動データベース 教員の養成に係る教育の質の向上に係る取組. 履修カルテや教職課程ポートフォリオは、教員の側のヴィジョンの共有や実践事例の相互交流の土台にもなっています。. また、京都市教育委員会や教職課程を設けている京都地区の国公私立大学（短期大学含む）が加盟する「京都地区大学教職課程協議会」に本学も加盟しており、教職教育委員会の委員が定期総会や小委員会に参加しています。定期総会で教員養成や教職教育のあり方について議論されたことは、参加した委員より教職教育委員会に報告し、本学の教職課程教育に反映しています。. 卒業者の教員免許状の取得および教員への就職の状況. 令和2年度卒業者教員免許状取得状況・学校教員就職状況 令和元年度卒業者教員免許状取得状況・学校教員就職状況 平成30年度卒業者教員免許状取得状況・学校教員就職状況 平成29年度卒業者教員免許状取得状況・学校教員就職状況 平成28年度卒業者教員免許状取得状況・学校教員就職状況 平成27年度卒業者教員免許状取得状況・学校教員就職状況 平成26年度卒業者教員免許状取得状況・学校教員就職状況 平成25年度卒業者教員免許状取得状況・学校教員就職状況 教員免許更新制について. 平成21年（2009年）4月1日から教員免許更新制が導入され、有効期間を更新して免許状の有効性を維持するには免許状更新講習の受講・修了が必要となっています。. 本学では免許状更新講習を開設していませんので、免許状所持者は、各自で他大学等の更新講習を受講・修了の上、有効期間の更新又は更新講習修了確認のための申請を行ってください。. なお、制度の概要や手続きについては、文部科学省や各都道府県教育委員会のホームページ等で確認してください。. 教員免許更新制 （文部科学省ホームページ）. 教員免許状の有効期間確認ツール （文部科学省ホームページ）. 教員免許更新制について （京都府教育委員会ホームページ）. 問い合わせ窓口. 所属学部・研究科等の教務担当. 「教科及び教職に関する科目」のうち、教科に関する専門的事項を含んだ科目及び教育職員免許状の授与申請に関すること. 教育学部教職教務掛. 上記以外の「教科及び教職に関する科目」及び介護等体験を含めた教員免許状全般に関すること. 所属研究科等の教務担当. 専修免許状に関すること. 関連リンク. 各学部・研究科、教養・共通教育の窓口 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Contacts. Contacts. Mailing address:. Kyoto University. Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501 JAPAN. Phone: +81-75-753-7531. Admissions, student exchange programs, tuition, fees, financial aid, etc Visa applications Kyoto University International House and accommodation information Executive visits Faculties and Graduate Schools Research Institutes, Centers, and Organizations Researcher database Employment and other inquiries Admissions, student exchange programs, tuition, fees, financial aid, etc. Make sure to check the following links before contacting the office specified below. How to apply for undergraduate programs. How to apply for graduate programs. Research students. Non-degree programs. Tuition and fees. Scholarships. Admissions Guide for International Applicants. International Education and Student Mobility Division. Email: studyku*mail2.adm.kyoto-u.ac.jp (replace * with @). Admissions procedures and requirements vary by faculty\/graduate school. Visit the website of each faculty\/graduate school to read its application guidelines, and if you have additional questions, contact the appropriate administrative office. Visa applications. International Service Office. Phone: +81-75-753-2245, 5412. Email: kuiso*mail2.adm.kyoto-u.ac.jp (replace * with @). Website:. https:\/\/kuiso.oc.kyoto-u.ac.jp\/ Kyoto University International House and accommodation information. International Service Office. Phone: +81-75-753-7614. Email: kuiso*mail2.adm.kyoto-u.ac.jp (replace * with @). Website:. https:\/\/kuiso.oc.kyoto-u.ac.jp\/ Executive visits. International Affairs Division. Email: ipromo*mail2.adm.kyoto-u.ac.jp (replace * with @). Website:. http:\/\/www.oc.kyoto- u.ac.jp\/en\/ Faculties and Graduate Schools. To contact specific Faculties or Graduate Schools, see:. Links to Faculty and Graduate School websites Administrative offices of Faculties and Graduate Schools Research Institutes, Centers, and Organizations. Research institutes Centers Organizations Researcher database. If you wish to find information on researchers or details of their work, conduct a search within the. Activity Database on Education and Research Employment and other inquiries. Employment opportunities General inquiries. Email: ku-info*mail2.adm.kyoto-u.ac.jp (replace * with @). 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 特色入試 特色入試における選抜方法等の変更. 特色入試における選抜方法等の変更. サイドメニュー. 一般選抜 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 出願状況と選考結果 よくある質問 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 　本学では、特色入試における選抜方法等を以下のとおり変更します。. 令和5年度入試. 令和5年度 特色入試入学者選抜方法における変更について（農学部森林科学科、食品生物科学科）（予告） （2022年5月31日付）. 令和5年度 特色入試入学者選抜方法における変更について（予告） （2022年3月29日付）. 令和5年度 特色入試入学者選抜方法における変更について（経済学部）（予告） （2021年5月13日付）. 令和5年度 特色入試入学者選抜方法における変更について（工学部建築学科、物理工学科）（予告） （2021年5月13日付）. 令和6年度入試. 令和6年度 特色入試入学者選抜方法における変更について（経済学部）（予告） （2022年3月9日付）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 同窓会. タグ:. 同窓会. すべて. Events. News. News. 公開日. 2022年05月30日. タグ. 同窓会. 中国同窓会（京京会、京仙会）と京都大学同窓会の懇談会が開催されました. News. 公開日. 2022年05月19日. タグ. 同窓会. 第24回「技術士を目指そう」説明会が開催されました. News. 公開日. 2022年04月22日. タグ. 同窓会. 2021年度 応援団OB・OG会「いぶき会」役員会が開催されました. News. 公開日. 2022年04月07日. タグ. 同窓会. 第23回「技術士を目指そう」説明会が開催されました. News. 公開日. 2022年04月07日. タグ. 同窓会. 第1回京大以文会東京支部茶話会を開催しました. News. 公開日. 2022年03月29日. タグ. 同窓会. 2022年 京都大学地球物理学教室同窓会「京大知球会」講演会・総会を開催しました. News. 公開日. 2022年03月25日. タグ. 同窓会. 京都大学技術士会 第14回講演会（イブニングセミナー東京）が開催されました. News. 公開日. 2022年01月18日. タグ. 同窓会. 京都大学土木会「2021 椿の会 学生・OGオンライン交流会」が開催されました. News. 公開日. 2021年12月23日. タグ. 同窓会. 第16回京都大学ホームカミングデイをオンラインで開催しました. News. 公開日. 2021年12月17日. タグ. 同窓会. 第20回京都大学同窓会若手会年次総会・講演会を開催しました. News. 公開日. 2021年12月13日. タグ. 同窓会. 第8回（令和3年度）京都大学福岡講演会が開催されました. News. 公開日. 2021年12月08日. タグ. 同窓会. 京都大学技術士会 第9回大会・特別講演会が開催されました. News. 公開日. 2021年12月08日. タグ. 同窓会. 第22回「技術士を目指そう」説明会が開催されました. News. 公開日. 2021年10月07日. タグ. 同窓会. 京都大学工学部建築学教室創立100周年記念 京大建築会総会・国際ワークショップ・シンポジウムを開催しました. News. 公開日. 2021年08月27日. タグ. 同窓会. 第5回総合生存学館同窓会「遊聞会」が開催されました. News. 公開日. 2021年08月24日. タグ. 同窓会. 京都大学技術士会 第13回講演会（イブニングセミナー東京）が開催されました. ページ送り. カレントページ. 1 ページ. 2 ページ. 3 ページ. 4 ページ. 5 ページ. 6 ページ. 7 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 研究・産官学連携 京都大学に寄附する 寄附講座・寄附研究部門. 寄附講座・寄附研究部門. サイドメニュー. 研究活動を知る 研究成果の公開ポリシー 産官学連携（共同研究・知的財産・ベンチャー支援） 京都大学に寄附する 京都大学基金 寄附講座・寄附研究部門 寄附講座等設置状況 その他のご寄附 京都大学で研究したい 研究支援体制について 研究者が受けられる研究費・研究支援 研究上の倫理・安全 教職員に兼業を依頼したい 企業等からの寄附金をもとに、寄附者の意向に沿いながら、本学が主体的に新しい講座や研究部門を設置します。. 名称. 寄附講座・寄附研究部門には、その講座等における教育研究の内容を示す名称を付けますが、寄附者から申出があったときには、寄附者が明らかになるような字句（いわゆる冠）を付けることができます。. 期間. 寄附講座・寄附研究部門の設置期間は、原則として3年以上5年以下ですが、更新することができます。更新の手続は、設置の手続きに準じて行うこととなります。. 寄附講座・寄附研究部門. 構成. 寄附講座・寄附研究部門には、少なくとも教授又は准教授に相当する者一人及び准教授又は助教に相当する者一人の教員を置く必要があります。. 教員の名称等. 寄附講座を担当する教員の名称は、「寄附講座教員」、寄附研究部門を担当する教員の名称は、「寄附研究部門教員」とします。なお、京都大学客員教授及び客員准教授選考基準により、「客員教授」又は「客員准教授」を称することができます。. 採用、選考. 寄附講座教員・寄附研究部門教員は、特定教員、有期雇用教員又は時間雇用教職員として採用されます。選考は、部局の教員選考基準及び選考方法に準じて行うことになります。. .職務. 寄附講座教員は当該寄附講座における教育研究に、寄附研究部門教員は当該寄附研究部門における研究に従事するほか、当該寄附講座等における業務の遂行に支障のない範囲内で、その他の授業又は研究指導を担当することができます。. 手続き方法. 寄附申込書 及び. 寄附講座 又は. 寄附研究部門の概要 を各部局の受け入れ担当窓口までお送りください。. 設置の決定は部局長が決定します。この場合、あらかじめ部局教授会等の意見を聞いて決定します。. 設置を決定しましたら、学内周知のほか、Webサイトやマスコミ発表により社会へ広報します。. 寄附講座等に係る経費の寄附は、寄附講座等の存続期間に係る総額を一括して受け入れることを原則とします。ただし、継続して受け入れることが確実であるときは、年度毎に分割して受け入れることができます。. 寄附講座等設置状況. 寄附講座・寄附研究部門設置状況 関連規定. 京都大学寄附講座及び寄附研究部門規程 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム EVENTS. EVENTS. 条件で絞り込む. 日付で絞り込む. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. 要申し込み. - すべて -. 必要. 不要. 開催地. - すべて -. 吉田キャンパス. 宇治キャンパス. 桂キャンパス. 東京オフィス・京都アカデミアフォーラム. その他の地域. オンライン. 開催日. 前月. 1月. 2月. 3月. 4月. 5月. 6月. 7月. 8月. 9月. 10月. 11月. 12月. 翌月. 日. 月. 火. 水. 木. 金. 土. 絞り込み条件をリセット. アイキャッチ画像. 京都大学創立125周年記念 附属図書館所蔵貴重資料展示. 開催日. 2022年06月01日〜12月22日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 入試. 令和5（2023）年度入学者京都大学大学院総合生存学館（思修館）博士課程学生募集入試説明会. 開催日. 2022年06月10日、他. 開催地. 吉田キャンパス. 桂キャンパス. その他の地域. Events. タグ. 125周年. 公開講座. 京都大学創立125周年記念 市民講座「物理と宇宙」. 開催日. 2022年06月11日、他. 開催地. オンライン. Events. タグ. 研究. 2022年度 総合生存学ミニワークショップ［春］. 開催日. 2022年06月13日、他. 開催地. 吉田キャンパス. オンライン. Events. タグ. 研究. 第256回アフリカ地域研究会「南アフリカの社会保障制度改革とケアの政治」. 開催日. 2022年06月16日. 開催地. その他の地域. オンライン. Events. タグ. 教育. グローバル教育展開オフィス主催：2022年度レクチャーシリーズ「ダイバシティー時代の「日本型教育」の行方 」第1回. 開催日. 2022年06月17日. 開催地. オンライン. Events. 創立125周年記念 アカデミックマルシェ. 開催日. 2022年06月18日. 開催地. その他の地域. Events. 体育会学生主催イベント 「京大体育会、まるごと魅せます」. 開催日. 2022年06月19日. 開催地. その他の地域. オンライン. Events. タグ. 公開講座. 京都大学金融セミナー「『金融リテラシーが未来を拓く』2022 ～シン・金融教育元年を迎えて～」. 開催日. 2022年06月20日. 開催地. オンライン. Events. タグ. 研究. 学術情報メディアセンターセミナー「教育DXと音声言語処理」. 開催日. 2022年06月21日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 入試. 2023年度 教育学研究科・教育学部学士入学 入試説明会. 開催日. 2022年06月25日. 開催地. オンライン. Events. タグ. 研究. ヒト生物学高等研究拠点 研究室紹介「ヒトは、いかにしてヒトとなりえたのか？」. 開催日. 2022年06月27日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 公開講座. プラットフォーム学連続セミナーVol. 12「モビリティとプラットフォーム学」～シェアリング・MaaS・数理最適化から考える「これからの移動のかたち」～. 開催日. 2022年06月30日. 開催地. オンライン. Events. タグ. 研究. 京都大学医生物学研究所 第16回公開講演会「医生物学研究所の船出」. 開催日. 2022年07月09日. 開催地. 吉田キャンパス. Events. タグ. 公開講座. 数理解析研究所 第43回数学入門公開講座. 開催日. 2022年08月01日〜08月04日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 国際交流. 国際ワークショップ「ブロックチェーン会議 2022」. 開催日. 2022年08月04日〜08月05日. 開催地. その他の地域. オンライン. Events. タグ. 公開講座. 臨床倫理学 入門コース 2022. 開催日. 2022年08月19日、他. 開催地. 吉田キャンパス. オンライン. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 調達情報 本学における発注・検収・検査について. 本学における発注・検収・検査について. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 物品・役務等の調達情報について 建設・工事等 資産の有効活用（不動産の売却・貸付等） 契約に係る情報等の公表について 調達に関する本学の取り組みについて 【取引先の皆様へ】本学への取引先登録、誓約書の提出および消費税の適格請求書発行事業者の登録申請について 本学における発注・検収・検査について 入札への参加資格について 電子メールによる振込通知サービスについて 入札情報（文部科学省共済組合京都大学支部） 取引金融機関の選定（メインバンク契約）について 法人としての取組 男女共同参画について 公表事項 情報公開 業者の皆さまへ. 2020年10月21日. 本学における発注・検収・検査制度は以下のとおりとなっていますので、ご理解の上、ご協力をお願いします。. 京都大学における発注・検収・検査（PDF） 検収所を設置する構内. 各構内の設置場所については以下を参照ください。ただし熊取構内、犬山構内については事務室内に設置していますので、設置場所図の掲載を省略しています。. 北部構内 本部構内 医学部構内 病院西構内 宇治構内 桂構内 熊取構内. 犬山構内. 注意. 検収所を設置していない構内（吉田南構内・西部構内・薬学部構内・病院東構内）へ納品される際には、最寄りの検収所にて、検収を受けてください。. 納品先が所在する構内の検収所に限らず、どの構内の検収所でも本学に対する納入物品の検収を行います。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 国際交流 京都大学への留学を希望する方へ 京都大学で学ぶ 学部で学位をとる 学部入学案内. 学部入学案内. サイドメニュー. 一般選抜 特色入試 その他の学部入学試験 留学生のための入試 学部で学位をとる 学部入学案内 大学院で学位をとる 大学院入学案内 交換留学・短期留学・研究生等 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 日本政府（文部科学省）奨学金留学生として入学を希望する場合は、詳細を. こちら から確認してください。. 出願資格. 出願者は、各学部により設定された出願資格に該当しなければなりません。以下は、出願資格からの主な抜粋です。また、出願書類を提出する前に、出願資格の審査が必要になる場合があります。詳細については、入学を希望する学部へ問い合わせてください。. 外国において、学校教育における12年の課程を修了した者またはこれに準ずる者で、文部科学大臣の指定した者（2021年3月31日までに修了見込みの者を含む）. 文部科学大臣の指定した者（国際バカロレア資格を有する者等）. 選抜方法等. 原則として日本人と同じ選抜方法（一般入試等）により入学者を決定していますが、薬学部・工学部について は、私費外国人留学生特別選抜試験を実施しています。入学の時期は4月です。. 1.私費外国人留学生特別選抜試験. 私費外国人留学生特別選抜試験は、薬学部・工学部で実施しています。. 私費外国人留学生特別選抜試験を受験するためには、日本学生支援機構（JASSO）が実施する日本留学試験（EJU）に加え、TOEFL、各学部が設定する出願資格、学力試験、面接試験等の基準を満たさなければなりません。また、 出願書類を提出する前に、出願資格の審査が必要になる場合があります。. 新年度の日程は変更の可能性があります。詳細は、各学部の募集要項を確認してください。不明な点などがある場合は、当該 の学部事務室に問い合せてください。. 問い合わせ先（学部・大学院研究科等事務室一覧） 私費外国人留学生特別選抜試験スケジュール. 学部. 担当掛. 電話. 所在地. 出願期間. 試験日. 入学時期. 薬学部 教務掛. 075-753-4514. 〒606-8501 京都市左京区下阿達町. 11月上旬. 一次：11月下旬. 二次：12月中旬. 4月. 工学部 留学生掛. 075-383-2051. 〒615-8530 京都市西京区京都大学桂. 1月初旬. 2月下旬. 4月. 2. 英語で学位を取得するコース. 京都大学では、海外からの学生を受け入れるための方策の一環として、留学生のためのさまざまなコースを展開しています。ここでご紹介するのは、授業や評価がすべて英語で行われるコースであり、入学試験や学生サポートなども英語で行われます。日本語ができない学生にも、世界レベルの研究施設の利用や指導を受ける機会が開かれています。詳細はこちらを参照してください。. 英語で学位を取得するコース 3.Kyoto University International Undergraduate Program （Kyoto iUP）. Kyoto iUPは、6ヶ月の予備教育と4年の学士課程からなる4.5年のプログラムであり、プログラム開始段階での日本語能力は不問です。プログラム開始後の充実した日本語教育と、1・2年次の英語による教養・共通教育を経て、概ね3年次以降から、各自の選んだ専門分野を日本語で学びます。 詳細はこちらを参照してください。. Kyoto iUP (Kyoto University International Undergraduate Program) 4.外国学校出身者のための選考. 法学部・経済学部では、日本以外の国の学校を卒業した人のための特別試験を実施しています。日本人と外国人の志願者からの申し込みを受け付けています。詳細については、「. 入学者選抜要項 」をご参照ください。. 外国学校出身者のための選考 5.特色入試. 2015年度より、特色入試を実施しています。一部の学部においては、留学生も出願可能です。なお、法学部特色入試につい ては、後期日程で実施しています。詳細は、教育推進・学生支援部入試企画課に確認してください。. 特色入試 6.一般入試. 一般入学試験では、大学入試センターの実施する大学入試センター試験に加えて、各学部が実施する個別学力検査を受験しなければなりません。. 大学入試センター試験は、例年1月に実施されます。詳細については、. 大学入試センター（外部リンク） へ照会してください。. 例年、個別学力検査の出願期間は1月下旬で、試験は2月下旬に実施されます。個別学力検査の詳細については、. 京都大学学務部入試企画課 に確認してください。. 7.第3年次編入学試験. 教育学部・経済学部・医学部人間健康科学科で実施しています。出願資格等の詳細については、当該の学部事務室へ問い合わせてください。. 問い合わせ先（編入学試験・学士入学試験） 8.学士入学試験. 文学部は、大学を卒業した者および卒業見込みの者を対象に、総合人間学部・理学部・工学部は京都大学を卒業した者および卒業見込みの者を対象に実施しています。詳細については、当該の学部事務室へ問い合わせてください。. 問い合わせ先（編入学試験・学士入学試験） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 法人としての取組 人権 京都大学におけるハラスメントの防止と対応について. 京都大学におけるハラスメントの防止と対応について. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 新型コロナウイルス感染症への対応 熊本地震への対応 環境への取組 東日本大震災への対応 人権 人権委員会 京都大学におけるハラスメントの防止と対応について その他の相談機関（リンク） ハラスメント相談に関する心得 ハラスメント窓口相談員のための研修会 障害に関すること 障害者雇用促進への取組 安全への取組 情報環境の整備 公益通報 監査 その他 男女共同参画について 公表事項 情報公開 In other languages. English English (Harassment Prevention and Countermeasures) English (Booklet) 教育・研究・医療を使命とする京都大学では、多くの人間がさまざまな関係を形成しています。そこでの修学・就労は、これに関わるすべての者にとって快適で有意義なものでなければなりません。しかしながら、指導する者と指導される者、指示する者と指示される者という関係は、時として、不適切な権力関係と化し、修学・就労を妨害するだけでなく、被害者の人格を著しく傷つけることがあります。また、性的な欲求や関心に基づく行動が、相手の修学・就労の環境を害し、あるいは、その人格を著しく傷つけることもあります。. 京都大学は、このようなハラスメント行為が生じないよう適切な措置を講じるとともに、また、万が一そのような事態が生じた場合には、迅速に事態が改善されるよう、最大限の努力をするものです。京都大学に関わるすべての人が、一個の人間として尊重され、それぞれの立場から教育・研究・医療の活動に打ち込めるようにしたいと考えています。. ハラスメントを受けた場合. 一人で我慢せずに、誰か周囲の信頼できる人に相談しましょう。. ことばと態度で、嫌だ、不快だ、という気持ちをはっきりと相手に伝えましょう。無視したり、受け流したりしているだけでは、状況は少しも改善されません。相手が嫌がっていることに、行為者が気づいていない場合もあります。. 記録をつけておきましょう。見ている人がいたら、その人にも確認しておきましょう。. 窓口に相談しましょう。. ハラスメントに第三者として気づいた場合. ハラスメントがなされていることを知ったときには、第三者であっても被害者の力になってあげてください。. 不快な場面を目撃したら、すぐ行為者に注意をしましょう。. 必要なら、証人になってあげましょう。. 被害者の相談にのって精神的に支えてあげましょう。被害者を責めてはいけません。. 相談窓口へ行くようにすすめ、同行してあげましょう。. ハラスメントはその性質上、被害者自身が事実を公然と訴えることは容易ではありません。相談窓口は、被害に気づいた第三者からの相談に対しても開かれています。. 関連リンク. 京都大学におけるハラスメントの防止と対応について（冊子） （ハラスメントの防止や排除のための方策、ハラスメントに起因する問題が生じた場合の対応方策などについて、規程等の内容をわかり易く示したものです）. 相談窓口 京都大学におけるハラスメントの防止等に関する規程（京都大学規程集へ） 京都大学におけるハラスメントの防止等に関する規程の運用について（通知） 学域・学系制導入に伴う各部局における懲戒の流れについて（教員： ハラスメントの場合） 妊娠・出産・育児休業等に関するハラスメントとなり得る言動例について （参考資料）. ハラスメント関係規程等 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム アクセス 各キャンパスについて（吉田・宇治・桂）. 各キャンパスについて（吉田・宇治・桂）. 2階層目ページ用メニュー. 採用情報 アクセス 見学にいらっしゃる方 各キャンパスについて（吉田・宇治・桂） 吉田キャンパス マップ一覧 宇治キャンパス構内マップ 桂キャンパス構内マップ 各学部・研究科、センター等の場所について 食堂、レストラン、ショップなど キャンパスマップ（ダウンロード用） キャンパス間の移動（学内者向け） 連絡バスの運行 京都大学には、大きく分けて吉田キャンパス、宇治キャンパス、桂キャンパスの三つのキャンパスがあります。. 各キャンパス相関図はこちら 各キャンパスについて. 吉田キャンパス. 所在地： 〒606-8501 京都市左京区吉田本町. 京都大学のメインキャンパスで、七つの構内に分かれています。. 本学を象徴する建物である時計台や、エンブレムに描かれている楠も吉田キャンパス内にあります。. 周辺地図（吉田キャンパス相関図）. キャンパスマップ. 北部構内マップ 本部・西部構内マップ 吉田南構内マップ フリーアクセスマップ （各キャンパスのバリアフリーマップ）. 医学部構内マップ 薬学部構内マップ 病院構内マップ 宇治キャンパス. 所在地： 〒611-0011 京都府宇治市五ケ庄. 1949年に旧陸軍の敷地を譲り受けて発足したキャンパスです。現在は主に自然科学・エネルギー系の研究所が置かれています。最新鋭のラボが集まるテクノロジー開発の最先端地域です。. 周辺地図. キャンパスマップ. 宇治キャンパス構内マップ フリーアクセスマップ （各キャンパスのバリアフリーマップ）. 桂キャンパス. 所在地： 〒615-8530 京都市西京区京都大学桂. 2003年10月にオープンした京都大学3番目のキャンパスです。テクノロジーとサイエンスが融合する新しい研究教育の場「テクノサイエンスヒル」の形成を目指しています。. 周辺地図. キャンパスマップ. 桂キャンパス構内マップ フリーアクセスマップ （各キャンパスのバリアフリーマップ）. ダウンロード用マップ. 以下のぺージに、ダウンロード用のPDFファイルを載せています。. キャンパスマップ（ダウンロード用） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CDC42-C末端異常症に於ける炎症病態を解明―ゴルジ体への異常蓄積がパイリンインフラマソーム形成を過剰促進―. 研究成果. 公開日. 2022年04月28日. DNA光回復酵素のFAD光還元反応をXFEL時分割構造解析にて解明. 研究成果. 公開日. 2022年04月28日. マウス卵子形成不全の遺伝子治療の成功―女性不妊治療への応用に期待―. 研究成果. 公開日. 2022年04月28日. ゲノム編集による内在性レトロウイルス排除―より安全なワクチン製造を目指して―. 研究成果. 公開日. 2022年04月27日. 現状維持バイアスの重い腰を上げさせる！東日本大震災後の変動型電気料金に関する横浜市でのフィールド実験. 研究成果. 公開日. 2022年04月27日. 糖鎖の違いで糖タンパク質を見分ける新技術―抗体分離、ウイルス変異検出の高度化―. 研究成果. 公開日. 2022年04月27日. グレビーシマウマiPS細胞の樹立に成功. 研究成果. 公開日. 2022年04月27日. 核磁気共鳴のレーザー検出器の小型化に成功―量子技術による化学分析の道を開拓―. ページ送り. 先頭ページ. chevron_left. chevron_left. 前ページ. chevron_left. ページ. 1 ページ. 2 カレントページ. 3 ページ. 4 ページ. 5 ページ. 6 ページ. 7 ページ. 8 ページ. 9 &hellip;. 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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Tel： 075-753-7531（代表番号）. 情報環境機構. 吉［南］ 93. 京都市左京区吉田二本松町. 企画・情報部 情報学研究科. 吉[本部・西部] 59. 京都市左京区吉田本町. 情報学研究科事務部 情報センター（工学研究科附属）. 桂 14（事務管理棟2階）. 京都市西京区京都大学桂. 桂地区（工学研究科）事務部 人文科学研究所. 吉[本部・西部] 38. 京都市左京区吉田本町. 人文科学研究所事務部 す. 数理解析研究所. 吉［北部］ 7. 京都市左京区北白川追分町. 数理解析研究所事務部 せ. 生存圏研究所. 宇治 27 他. 宇治市五ケ庄. 宇治地区事務部 生態学研究センター. 大津 －. 大津市平野2-509-3. 理学研究科事務部 （生態学研究センター担当） 生命科学研究科. 吉［医学］ 18. 京都市左京区吉田近衛町. 生命科学研究科事務部 そ. 総合生存学館（思修館）. 東一条館 －. 京都市左京区吉田中阿達町1. 総合生存学館事務部 総合人間学部. 吉［南］ 84. 京都市左京区吉田二本松町. 人間・環境学研究科事務部 総合博物館. 吉[本部・西部] 22. 京都市左京区吉田本町. 総合博物館事務室 た行.  . 名称. 場所. 住所. 担当事務部. （連絡先については、リンク先ページをご確認ください。）. マップ名. 番号. た. 大学文書館. 吉［医学］ 29. 京都市左京区吉田河原町15-9. 大学文書館事務室 （総務部総務課）. 男女共同参画推進センター. （男女共同参画推進本部）. 吉［医学］ 28. 京都市左京区吉田橘町. 人事部 職員育成課 ダイバーシティ推進室 ち. 地球環境学堂・学舎. 吉[本部・西部] 40. 京都市左京区吉田本町. 地球環境学堂･学舎事務部 と. 東南アジア地域研究研究所. 吉［薬学］ 61,64. 京都市左京区吉田下阿達町46. 地域研究事務部 な行.  . 名称. 場所. 住所. 担当事務部. （連絡先については、リンク先ページをご確認ください。）. マップ名. 番号. に. 人間・環境学研究科. 吉［南］ 89. 京都市左京区吉田二本松町. 人間・環境学研究科事務部 の. 農学部・農学研究科. 吉［北部］ 8,18. 京都市左京区北白川追分町. 農学研究科等事務部 農学研究科附属農場（本場）. 木津川市 －. 京都府木津川市城山台4-2-1. 農学研究科等事務部 は行.  . 名称. 場所. 住所. 担当事務部. （連絡先については、リンク先ページをご確認ください。）. マップ名. 番号. は. 白眉センター. 吉[本部・西部] 61. 京都市左京区吉田本町. 研究推進部 研究推進課 ひ. 光・電子理工学教育研究センター（工学研究科附属）. 桂 1. 京都市西京区京都大学桂. 桂地区（工学研究科）事務部 東アジア人文情報学研究センター（人文科学研究所附属）. 吉［北部］ 26. 京都市左京区北白川東小倉町47. 人文科学研究所事務部 ヒト行動進化研究センター. 犬山 －. 犬山市官林41-2. ヒト行動進化研究センター事務部 人と社会の未来研究院. 吉［薬学］ 64. 京都市左京区吉田下阿達町46. 人と社会の未来研究院事務室 ふ. フィールド科学教育研究センター. 吉［北部］ 8. 京都市左京区北白川追分町. 農学研究科等事務部 福井謙一記念研究センター. 吉［北部］ （近隣施設）. －. 京都市左京区高野西開町34-4. 桂地区（工学研究科）事務部 複合原子力科学研究所. 熊取 －. 大阪府泉南郡熊取町朝代西二丁目1010. 複合原子力科学研究所 附属図書館（図書館機構）. 吉[本部・西部] 18. 京都市左京区吉田本町. 附属図書館事務部 物質－細胞統合システム拠点. 吉[本部・西部] 31～33. 京都市左京区吉田牛ノ宮町. 高等研究院等事務部 文化財総合研究センター. 吉[本部・西部] 34. 京都市左京区吉田本町. 文学研究科事務部 文学部・文学研究科. 吉[本部・西部] 8. 京都市左京区吉田本町. 文学研究科事務部 ほ. 法学部・法学研究科. 吉[本部・西部] 4～6. 京都市左京区吉田本町. 法学研究科事務部 放射線生物研究センター. 吉［医学］ 9. 京都市左京区吉田近衛町. 医学研究科事務部 防災研究所. 宇治 27 他. 宇治市五ケ庄. 宇治地区事務部 や行.  . 名称. 場所. 住所. 担当事務部. （連絡先については、リンク先ページをご確認ください。）. マップ名. 番号. や. 薬学部・薬学研究科. 吉［薬学］ 66,67. 京都市左京区吉田下阿達町46-29. 薬学研究科事務部 野生動物研究センター. 関田南研究棟 －. 京都市左京区田中関田町2-24.  . ゆ. 湯川記念館. 吉［北部］ 11. 京都市左京区北白川追分町.  . ら行.  . 名称. 場所. 住所. 担当事務部. （連絡先については、リンク先ページをご確認ください。）. マップ名. 番号. り. 理学部・理学研究科. 吉［北部］ 2～5 他. 京都市左京区北白川追分町. 理学研究科事務部 留学生ラウンジ「きずな」. 吉[本部・西部] 13. 京都市左京区吉田本町. 国際・共通教育推進部. 留学生支援課. （. 連絡先 ）. 流域圏総合環境質研究センター（工学研究科附属）. 大津 －. 大津市由美浜1-2. 桂地区（工学研究科）事務部 量子理工学教育研究センター（工学研究科附属）. 宇治 31. 宇治市五ケ庄. 桂地区（工学研究科）事務部 目的別連絡先について. 教育・学生支援関係. 学生支援窓口 各学部・研究科の窓口 留学関係. 国際化を支える学内組織 学部・大学院研究科等事務室一覧 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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{"id":"3_R9Q9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/education-campus\/students2\/intl-scholors","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":69,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["ja","education-campus","students2"],"url_file_name_s":"intl-scholors","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["留学生のための奨学金 | 京都大学"],"description_txt":["留学生のための奨学金情報を提供しています。"],"last_modified":"Wed Jun 08 14:41:09 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-7268527148486726770,"exact_signature_unique_b":true,"fuzzy_signature_l":9015787474292570547,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["留学生のための奨学金"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","3階層目以降ページ用メニュー","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":757,"text_t":"留学生のための奨学金 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. 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Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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International Education and Student Mobility Division, Kyoto University. 〒606-8501　京都市左京区吉田本町. Yoshida Honmachi, Sakyo-ku, Kyoto, Japan, 606-8501. E-mail： studyku＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） 紅萠アンケート 紅萠（くれなゐもゆる） 21号～30号 紅萠（くれなゐもゆる） 11号～20号 紅萠（くれなゐもゆる） 1号～10号 Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 「 紅萠（くれなゐもゆる）」は「京都大学の現在」の姿をわかりやすい形で広く社会にお知らせすることを目的として2002年3月に創刊された広報誌です。本学の面白い授業や研究、学生の活躍、卒業生からのメッセージなど京大の魅力が満載です。. 以下のとおり、PDF版を掲載しておりますので是非ご覧ください。. また、「紅萠」は誌面をWeb化した特設サイトもありますので、こちらもご覧ください。. 京都大学広報誌『紅萠』 ご覧になられた後は以下のリンクからアンケートにお答えください。. 回答いただいた方の中から抽選で「プティ・ゴーフル」をプレゼントします。. 紅萠アンケート 41号（2022年3月）. 41号（2022年3月） PDF：10.8MB 表紙. モノ語る京大の歴史. 百周年時計台記念館（時計塔とその内部）. 1925年に誕生した京都大学のシンボル。2003年12月に、創立百周年記念事業の一環として、最新の免震構法を取り入れて改修。外装や内装は、誕生当時の雰囲気を残しながら、京都大学と社会とのつながりを作る場として生まれ変わった。. シンポジウムやイベントの開催などに利用できる「百周年記念ホール」、「国際交流ホール」のほか、レストランや「京大ショップ」など、一般の方も利用できる施設を備える。詳細は12ページ「施設探訪」に。. 1．巻頭座談会. 九重に 花ぞ匂へる. 自主の気風が耕す土壌に. 芽吹く花こそ麗しい. 湊 長博+ 木村里子+ 彭 永成+ 北田せり. 2．研究室でねほりはほり. 記念品とあなどるなかれ。. 歴史家の眼で編む大学沿革史. 西山 伸. 3．施設探訪. 百周年時計台記念館. 4．授業に潜入！ おもしろ学問. 間違いを恥じることこそ間違い！. 仮説実験授業で出会う物理の世界. 舟橋春彦. 5．輝け！ 京大スピリット. 囲碁部／アーチェリー部. 6．私を変えたあの人、あの言葉. ドラマティックなことなんてない。. でも、意味は確かにあったのだ. 山西利和. 7．京都大学基金事務局より／京都大学同窓会だより. 8．京都大学創立125 周年 特別企画. クイズ　京大博士をめざせ！. 40号（2021年9月）. 40号（2021年9月） PDF：8.96MB 表紙. モノ語る京大の歴史. 阿武山地震観測所 歴代地震計. 1930年の設立当時の超大型地震計から最新の地震計まで、多様な地震計を所蔵する阿武山観測所。. サイエンスミュージアムとして整備され、週1回程度、見学会を実施する。. 佐々式大震計は、2代目所長の佐々憲三が開発した世界にたった1台の地震計。. 近地地震の完全な波形をはじめて捉えることに成功した。. 1．巻頭鼎談. 京都大学創立125周年記念企画. 革新してこそ伝統は続く. 笹岡隆甫＋鈴鹿可奈子＋西平 直　. 2．追憶の京大逍遥. 自由の学風にあこがれて. 原点となった京大での学生生活. 上田輝久. 3．授業に潜入！　おもしろ学問. 憶良は子煩悩？ それとも社会派歌人？. 時代性から読み解く萬葉の世界. 佐野 宏. 4．恩師を語る. 啓蒙時代の哲学者と相対し、. 己の「実存」を問い続けた大学者 中川久定. 多賀 茂. 5．京都大学をささえる人びと. 「どこに言うても恥ずかしくない」。. 極め続けるトマト栽培の最前線. 西川浩次. 6．萌芽のきらめき・結実のとき. 哲学と統計学を対話させ、揺らぐ〈正しさ〉の在り方を問う. 大塚 淳. 「もっと速く、正確に」を追求。緊急地震速報システムの立役者. 山田真澄. 7．輝け！ 京大スピリット. 体育会ボート部／11月祭事務局／村山陽奈子. 8．まなび遊山. 京都大学の景観を織りなす樹木 宇治キャンパス編. 9．京都大学基金事務局より／京都大学同窓会だより. 10．触発ギャラリー. いろ＋おと＋ことば. 39号（2021年3月）. 39号（2021年3月） PDF：9.65MB 表紙. モノ語る京大の歴史. 『しほやきぶんしょう』（附属図書館 所蔵）. 『文正草子』の通称で知られ、『御伽文庫』の巻頭を飾る物語の一本。. 貧しい庶民が、身の才と娘の徳により立身出世を果たすという祝儀物。. 内容のめでたさから、女子の正月の読み物にふさわしいと、嫁入り道具にも加えられ、豪華な奈良絵本や絵巻物が多数作られた。. 表紙の画像は、物語の挿絵を加工、合成して構成したもの。. 1．巻頭鼎談. 大学の磁場が育てる独創的な個性と研究. 湊 長博＋曽我部真裕＋深澤愛子　. 2．私を変えたあの人、あの言葉. 京大との出会いは、まるで運命の悪戯のようだった. 水原碧衣. 3．研究室でねほり・はほり. 遅咲きの研究者が挑むがん免疫療法の新たな切り札. 河本宏. 4．授業に潜入！　おもしろ学問. 「やかんが沸騰した」はなぜ伝わる？比喩から考える言葉の不思議. 谷口一美. 5．フィールド科学教育研究センター. 瀬戸臨海実験所　白浜水族館. 下村通誉＋原田桂太. 6．京都大学をささえる人びと. 原子力科学の飛躍は徹底した安全管理があってこそ. 藤原靖幸. 7．輝け！ 京大スピリット. 鳥人間チーム／体育会フィールドホッケー部／レスキューロボット開発・運用チームSHINOBI. 有朋自遠方来 ジョ・ソウ. 8．京都大学基金事務局より／京都大学同窓会だより. 9．触発ギャラリー. いろ＋おと＋ことば. 38号（2020年11月）. 38号（2020年11月） PDF：5.68MB 表紙. モノ語る京大の歴史. 平塚貝類コレクション（総合博物館）. 明治・大正時代の商人で貝類研究の創始者、平瀬與一郎が日本各地から収集した貝類の標本。岡崎公園前にあった平瀬貝類博物館の展示品も含め、京都大学の理学部動物学教室、地質学鉱物学教室を経て、総合博物館に約9,000点が保存されている。. この入手には、貝類博物館に勤務し、その後京都帝国大学理学部助手に就職した黒田徳米が関与していると推定されている。. 貝類が保管されている箱には、それぞれ手書きで学名と和名、産地の記載が残る。おそらく黒田が作成したものと考えられる。. 1．巻頭鼎談. 〈自学自習〉の教えのもとに自身の手で描くコロナ禍後の未来. 出席者　松下佳代＋諸富 徹＋田中祐理子. 2．追憶の京大逍遥. モラトリアムの時間、闇鍋のごとし地下空間. 福原伸治. 3．授業に潜入！おもしろ学問. 私たちをとり巻く目に見えない原子・分子たちの化学反応. 中村敏浩. 4．恩師を語る. 日本の発生生物学を愛し、育てた名物教授　岡田節人. ケチはアカン。. 余裕のあるところにしかサイエンスは生まれへん. 高橋淑子. 5．京都大学をささえる人びと. 難題であるほど燃えてくる。μg単位で分析値を求める日々. 坂田文恵. 6．萌芽のきらめき・結実のとき. 脳の動きを追跡し複雑で多面的な人間の本性を描き出す. 阿部修士. 生命科学の最先端から進歩の先のある社会を描く. 澤井 努. 7．輝け！ 京大スピリット. 体育会居合道部／永田理奈／山仕事サークルすぎりょうたろう. 8．まなび遊山. 京都大学の景観を作り出す樹木. 本部・吉田南構内編. 9．京都大学基金事務局より／京都大学同窓会だより. 10．触発ギャラリー. いろ＋おと＋ことば. 37号（2020年3月）. 37号（2020年3月） PDF：6.84MB 表紙. モノ語る京大の歴史. 本部構内で出土した京都帝国大学時代の陶器土瓶と磁器染付椀（大学院文学研究科附属文化遺産学・人文知連携センター 所蔵）. 土瓶の体部外面と染付椀の底部内面に描かれているのは、「大学」の円形意匠。染付椀の底部外面には、表紙に掲載の「物理」のほか、「機械」といった学科名、部局名などが記されている。こうした出土文字資料は、遺跡の性格を明らかにするうえで、大きな役割を果たす場合が少なくないといえる。. 1．巻頭対談. 壁にぶつかり乗り越えるたびゴールは近づく. 出席者： 吉野 彰＋山極壽一. 2．私を変えたあの人、あの言葉. 〈伝統〉に浮かびあがる、先人の生きた証に惹きつけられて. 大塚真帆. 3．研究室でねほり・はほり. 粘土をこね、土器を形作る手から〈個人〉を見つめて. 金子守恵. 4．授業に潜入！おもしろ学問. 統計学を駆使する秘訣は、データに潜む「ほつれ」の理解. 市村賢士郎. センター長に聞く. データ科学を学ぶ意義. 山本章博. 5．施設探訪. 大学院理学研究科 附属天文台 岡山天文台. 木野 勝. 6．京都大学をささえる人びと. ユーザー目線で思考し、安全で便利な情報環境を築く. 石橋由子. 7．輝け！ 京大スピリット. 劇団ケッペキ／体育会ゴルフ部／ガオ・ジエ. 有明自遠方来 カルラ・ラバット・デ・オス. 8．京都大学基金事務局より／京都大学同窓会だより. 9．触発ギャラリー. いろ＋おと＋ことば. 36号（2019年10月）. 36号（2019年10月） PDF：6.84MB 表紙. モノ語る京大の歴史. 総合博物館は京都帝国大学時代から集められた2万点以上の鉱物標本を収蔵。その中でも、1万点を超える「比企鉱物標本」は、現代では入手できない日本最高峰の鉱物コレクション。工学部採鉱冶金学科の教授であった比企忠（ひき・ただす、1866年-1927年）が蒐集した。表紙の石英は、二つの石英結晶（水晶）が接合した双晶の一つ「日本式双晶」。大型で、左右の結晶のバランスが秀逸な一品。背景に使用した比企が記したノートは、標本の保管方法や、入手経緯が細かく記された「バイブル」である。. （石英（日本式双晶）総合博物館蔵）. 1．巻頭対談. 遊んで、悩んで、ことばに沈む. 現実と虚構に生きる作家という仕事とその日々. 出席者： いしいしんじ＋青羽 悠. 進行： 廣野由美子. 2．追憶の京大逍遥. 京大ならではの、ポップな頭脳集団だった. 川下大洋. 3．授業に潜入！ おもしろ学問. リーダーの条件は自分と他者の〈こころ〉を知ること. 河合江理子＋河合美宏. 4．恩師を語る. 戦後日本を鋭く見つめた先覚者 高坂正堯　潜れども潜れども叡智は深く. 中西 寛. 5．京都大学をささえる人びと. 24時間365日、集まり続ける地震データを整理. 澤田麻沙代. 6．萌芽のきらめき・結実のとき. 「怪異」の跋扈する社会のしたたかな人びと. 高谷知佳. 100年後の人類に役立つ可能性を秘めた、厖大な標本を次世代へ. 西川完途. 7．輝け！ 京大スピリット. 男子ラクロス部／民族舞踊研究会／村津 蘭. 8．まなび遊山 もの言わぬモノが語りだす. 「秘めたる美学」の物語. 9．京都大学基金事務局より／京都大学同窓会だより. 10.触発ギャラリー. いろ＋おと＋ことば. 35号（2019年3月）. 35号（2019年3月） PDF：10.5MB 表紙. 本庶佑特別教授ノーベル賞授賞式に出席. 本庶 佑高等研究院副院長・特別教授が、ジェームズ・アリソン博士とともに、「免疫抑制の阻害によるがん治療法の発見」により生理学・医学賞を受賞。2018年12月10日の授賞式に出席し、スウェーデン王国カール16世グスタフ国王から、メダルと賞状が授与された。. 1．巻頭鼎談. 「探究心」という強烈なエンジンこそが、新たな道を拓く. 出席者　本庶 佑＋山中 伸弥＋湊 長博. 2．研究室でねほり・はほり. 好奇心に火をつけて 骨が宿す物語に目をこらす. 松岡 廣繁. 3．授業に潜入！ おもしろ学問. 粒であり波でもある 量子のふるまいを見破る切り札. 中家 剛. 4．施設探訪. 生存圏研究所 材鑑調査室. 5．私を変えたあの人、あの言葉. 名もなき建築を追いかけた ユーラシア大陸横断の旅. 森田 一弥. 6．京都大学をささえる人びと. キャンパスの電気を見守り40年. 足立 融正. 7．輝け！ 京大スピリット. 探検部／高橋唯基／フィギュアスケート部. Var Elif Berna. 8．京都大学基金事務局より／京都大学同窓会だより. 9．触発ギャラリー. いろ＋おと＋ことば. 34号（2018年9月）. 34号（2018年9月） PDF：8.75MB 表紙. マダイのゲノム編集. ゲノム編集は、特定の遺伝子を改変し、その働き方を変える技術。農学研究科の木下政人助教は、マダイの筋肉細胞の増加や成長を止めるブレーキとなるミオスタチンの遺伝子の機能を欠損させ、筋肉量を増やした肉厚マダイをつくりだした。マダイの絵は、理学部動物学教室所蔵のシーボルトらが編集した「日本動物誌」から抜粋し、処理により合成した。. （画像提供は理学研究科・理学部生物科学専攻図書室）. 1．巻頭対談. 京大発、「肉厚マダイ」参上　食に革命を起こすゲノム編集と安全. 出席者 木下政人＋高橋拓児. 2．追憶の京大逍遥. 新築の熊野寮で受けた洗礼. 村上憲郎. 3．授業に潜入！ おもしろ学問. 多角的な視点で古典に新しい価値を. 道坂昭廣. 4．恩師を語る. 日本の動物行動学の始祖 日高敏隆　「なぜ」のこころが出発点. 幸島司郎. 5．京都大学をささえる人びと. 図書館は知の宝物庫　扉は君にも開かれている. 赤澤久弥. 6．萌芽のきらめき・結実のとき. 学びへの意欲を呼び覚ます「京大らしい」授業とは. 田口真奈. 未知なる世界へいざなう、小さな救世主. 亀井謙一郎. 7．輝け！ 京大スピリット. 心茶会／藤井 駿／ヨット部. 8．まなび遊山. 建築の細部に宿るこだわり. 9．触発ギャラリー. いろ＋おと＋ことば. 33号（2018年3月）. 33号（2018年3月） PDF：12.2MB 表紙. 源氏物語. 室町時代後期に近衛稙家（このえたねいえ）があつらえたと伝えられる源氏物語の写本。稙家が当時の高名な書家を動員して書写させたと考えられる。のちに、毛利元康の手元に渡り、元康自身や連歌師の里村紹巴（さとむらじょうは）が欠本を補写した。巻ごとに表紙や用紙が異なる。冊子が納められる箱には蒔絵が施され、引き出し、鍵に至るまで細かな神経がゆき届いた造形となっている。文学研究科図書館所蔵。. （関連項目は3ページ「巻頭特集」に。画像提供：京都大学附属図書館）. 1．触発ギャラリー. いろ＋おと＋ことば. 2．特集. おそれずに、苦手科目に飛びこんでみよう. カードの一面だけを見ていませんか. 3．研究室でねほり・はほり. 人類学の視点から東南アジアの未来を探る. マリオ・ロペズ. 4．授業に潜入！ おもしろ学問. 景観に潜在する歴史のレイヤーをあぶりだす. 増井正哉. 5．施設探訪. 京都大学農学部・農学研究科. 附属農場. 6．京都大学をささえる人びと. 守備範囲は日本全土、マネジャーは飛び回る. 高橋秀典. 7．輝け！ 京大スピリット. 体操部／落語研究会／久保田結子. 有朋自遠方来 Ei. 8．私を変えたあの人・あの言葉. 多様な出会いと交流のなかで. 千松信也. 32号（2017年9月）. 32号（2017年9月） PDF：10.4MB 表紙. Space Jungle. 制作年度2014年。宇宙のどこかにあるジャングルに咲く花、なにかが起こったあとの静けさ、花びらが霧の中に消えていき、空間自体に幽遠な雰囲気が漂う。. （土佐尚子情報環境機構教授・総合生存学館兼任教授）. 1．触発ギャラリー. いろ＋おと＋ことば. 2．巻頭対談. 人間とAIとがせめぎあい共存する時代. AIの無限の可能性に人間社会の未来を賭ける. 出席者　金出武雄＋土佐尚子. 3．授業に潜入！ おもしろ学問. どうして地球には、これほど多くの酸素があるのだろう. 酒井 敏. 4．恩師を語る. 臨床心理学の礎を築いた河合隼雄. 〈なにもしない〉ことに全力を注げ 生きている〈あなた〉一人ひとりを見つめつづけた臨床家. 皆藤 章. 5．. 京都大学をささえる人びと. 勉強も研究も仕事も、すべてはカラダが資本. 雜賀真梨. 6．萌芽のきらめき・結実のとき. 幸せと生きづらさは背中あわせ 現役世代に届く政策をさぐる. 柴田 悠. 季節のない熱帯林で出会う、 したたかで多彩な生きものたち. 酒井章子. 7．輝け！ 京大スピリット. 柔道部／大木暁登／陶芸部「天山窯」. 8．まなび遊山. 「掘り出しもの」で古今往来！ 対岸にみやこの栄華を眺めつつ……. 千葉 豊. 9．追憶の京大逍遥. 私の血に流れる自由な批判精神. 山口 繁. 31号（2017年4月）. 31号（2017年4月） PDF：11.0MB 表紙. 京都大学吉田キャンパス周辺地図. （左）1889（明治22）年測量 1：20000仮製地形図「京都」の一部を拡大。（右）1996 （平成8）年修正 1：25000地形図「京都東北部」の一部を拡大。（国土地理院発行の地図を使用）. 明治22年は京都大学の前身である第三高等中学校が京都に移転した年であり、敷地（現本部構内）には数棟の建物しかない。東大路通や今出川通のような大正時代の都市計画でつくられた道路もまだない。京都大学の周辺は一面の水田と農村であり、鴨川以西の洛中は建物が密集する市街地であるとわかる。いっぽう、平成図の京都大学周辺は、洛中と同等の住宅密集地となっている。. （関連項目は4ページ「巻頭特集」に）. 1．触発ギャラリー. いろ＋おと＋ことば. 2．. 特集〈教養〉の大地を耕して好奇心の種を育てよう. 地図にのこされた歴史の痕跡をたどる. 山村亜希. 物理学は恐るるに足らず。 厳密な実験の先に待っている楽しみ. Roger Wendell ／ Anthony Beaucamp. 少人数授業で「ゼミ」スタイルの醍醐味を体験. 語りつくせぬ鳥類の魅力。 学生たちと分かち合いたい. Craig Barnett. 学術世界に羽ばたくきみに、 「英語力」という翼を. 金丸敏幸／ John Rylander. 教養教育は、人生の果実を豊かに 実らせるための「土づくり」です. 村中孝史. 3．. 研究室でねほり・はほり. 野生動物の「個性」に遺伝子からせまる. 村山美穂. 4．. 施設探訪. 宇治川オープンラボラトリー. 5．. 京都大学をささえる人びと. 小さな疑問に、等身大のことばで寄り添いたい. 中内彩香. 6．輝け！ 京大スピリット. フットサル部ARI ／小木曽友輔／ 京都着物企画. 有朋自遠方来 Nalaka GEEKIYANAGE. 7．. 私を変えたあの人・あの言葉. 自分の頭で考えるということ. 平野啓一郎. バックナンバー. 30号以前のバックナンバーは、以下のぺージよりご覧ください。. 紅萠21号～30号 紅萠11号～20号 紅萠1号～10号 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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{"id":"86UPD8U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jainternationalorg2documents01.pdf","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":86,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["sites","default","files","embed"],"url_file_name_s":"jainternationalorg2documents01","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["jainternationalorg2documents01.pdf"],"author":"i_strategy","author_sxt":["i_strategy"],"last_modified":"Thu Jul 11 05:46:37 UTC 2019","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Mar 22 00:00:00 UTC 2016"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-8421927373772883270,"exact_signature_unique_b":true,"fuzzy_signature_l":-1623454919915430134,"fuzzy_signature_unique_b":true,"wordcount_i":11471,"linkscount_i":3,"linksnofollowcount_i":0,"inboundlinkscount_i":2,"outboundlinkscount_i":1,"inboundlinks_urlstub_sxt":["www.kyoto-u.ac.jp\/en\/global\/international_strategy\/international-strategy"],"inboundlinks_anchortext_txt":[""],"outboundlinks_protocol_sxt":["000-https","001-https"],"outboundlinks_urlstub_sxt":["www.oc.kyoto-u.ac.jp\/overseas-centers\/en\/","u.kyoto-u.jp\/ex8y9"],"outboundlinks_anchortext_txt":["",""],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:43:50 UTC 2022","fresh_date_dt":"Sat Mar 25 06:54:46 UTC 2023","referrer_id_s":"qVq_w9U8j7Mo","publisher_t":"Microsoft® Word 2019","language_s":"en","size_i":1664548,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085659381563392,"text_t":" \n \n0 \n \n \n   \n \n \n \n \nAnnual report  \nacademic year 2018 \n \nInternational Strategy Office \nKyoto University \n(iSO-KU) \n \n \n1 \n \nContents \n \n1. Greeting from the Director ................................................................................................... 2 \n2. Introduction .......................................................................................................................... 4 \n2.1. Background to the establishment of the iSO-KU ........................................................... 4 \n2.2. Overview of the iSO-KU’s mission ................................................................................. 6 \n2.3. Future internationalization at Kyoto University .............................................................. 6 \n3. iSO-KU: Mission and functions ............................................................................................ 8 \n4. Orchestrating internationalization & diplomatic response .................................................. 10 \n4.1. Internal connect ........................................................................................................... 11 \n4.2. External connect.......................................................................................................... 15 \n4.3. Broader interactions .................................................................................................... 16 \n5. iSO-KU at work .................................................................................................................. 17 \n5.1. Kyoto University Basic Concept for Internationalization .............................................. 17 \n5.2. Work and achievements in Connect, Boost, Promote, and Synergize ........................ 19 \n5.2.1. Connect ................................................................................................................. 19 \n5.2.2. Boost ..................................................................................................................... 20 \n5.2.3. Promote ................................................................................................................ 22 \n5.2.4. Synergize .............................................................................................................. 25 \n6. iSO-KU academic role ....................................................................................................... 27 \n6.1. Academic contribution ................................................................................................. 27 \n6.1.1. International presentations .................................................................................... 27 \n6.1.2. International presentations in previous years ........................................................ 28 \n6.1.3. Presentations in Kyoto University ......................................................................... 28 \n6.2. Contribution to staff development in Kyoto University ................................................. 29 \n7. iSO-KU future plans: Looking ahead ................................................................................. 30 \n8. Meet the iSO-KU people .................................................................................................... 35 \nDirector ............................................................................................................................... 35 \nVice-Directors ..................................................................................................................... 35 \nProgram-Specific Senior Lecturers .................................................................................... 35 \nDirectors of Kyoto University Overseas Centers ................................................................ 36 \n \n \n \n2 \n \n1. Greeting from the Director \nWelcome to the International Strategy Office \nof Kyoto University or iSO-KU! \n \n \n \nThe Annual Report offers an overview of the \nwork of iSO-KU in the 2018 academic year; \ndescribing activities, contributions and \nachievements to promote internationalization, \nand a vision to the future.  \nIn October 2018, iSO-KU reached university-\nwide consensus on the Kyoto University Basic \nConcept for Internationalization, a document \nthat gives a sense of shared direction on the \nefforts of the University’s to remain open to the \nworld. Inspired by the principles set there, the \nOffice (iSO-KU) promotes the global impact of \nthe University, in education, research and \nsocial contribution by producing and enhancing \npolicies and measures, as well as \norganizational transformation for an effective \ninternationalization. \nIn terms of research, a major achievement \niSO-KU had was to support the establishment \nof On-site Laboratories, accomplishing one of \nthe goals set in the Designated National \nUniversity Initiative. The Office became a focal \npoint providing smooth support for the \nimplementation and operation of the project, \nhence increasing international cooperation and \nmobility of researchers.  \nSimilarly, iSO-KU supports the “Strategic \nPartnership Development Project to \nStrengthen the University’s Research \nCapabilities” to enhance collaboration with \nleading universities around the globe and to \npromote further cooperation and exchange.  \nIn education, a major accomplishment iSO-KU \nhad was the establishment of the International \nAdmissions Assistance Office (IAAO), in April \nof 2019 under the iSO-KU umbrella. Also \nenvisioned in the Designated National \nUniversity Initiative, this project aims to \nfacilitate the admission process for new \ninternational students throughout the \nUniversity.  \niSO-KU supported the management of the \nKingfisher Global Leadership Program, a short-\nterm package dispatching students on \nexchange to the USA. This year the Office \nstarted managing three short-term outbound \nprograms formerly managed by International \nEducation Administration (IEA) Office. \nRegarding social contribution, iSO-KU \nparticipated actively in activities organized for \nand with the USRNetwork together with the \nEducation Planning and Promotion Office; \nsimilarly, in cooperation with Kyoto University \nResearch Administration Office (KURA), the \nOffice planned “Kyoto University IPCC Weeks \n2019”, a side event of the IPCC \n(Intergovernmental Panel on Climate Change) \nheld in Kyoto in May 2019. \nProf. Yasuyuki Kono \nVice-President for Int’l Strategy \nDirector of the International Strategy Office  \n \n \n3 \n \nDuring 2018 iSO-KU created a new site in the \nEnglish Website (Global KyotoU) that brings \ntogether international information of the \nUniversity under one single entry point to \nfacilitate access to information for foreign \nprofessors, researchers and students within \nand outside the University.  \nIn the strive to consolidate an international and \ndiverse campus, iSO-KU continues to foster an \ninclusive environment, by creating and \nmaintaining mechanisms to connect and bring \nthe views of foreign researchers and students \ninto the policy-making channels of the \nUniversity.  \nRegarding organizational management, the \nUniversity's President appointed a Vice-\nPresident for International Strategy to \nstrengthen the governance of the University; \nwhile simultaneously serving as the Director of \nthe iSO-KU, to enhance discussions in \ninstances like the iSO-KU Core Meeting -a \nforum specially created to promote strategic \nreforms for the internationalization of the \nUniversity, composed of Deputy Executive \nVice Presidents- and other key players in the \nUniversity.  \nSimilarly, iSO-KU International Advisory Board, \ncomposed of international professors and \nresearchers, was established as a mechanism \nto capitalize the experience and views from \ndifferent perspectives to foster the University’s \ninternationalization.  \nRegarding the University’s Overseas Centers, \niSO-KU, in cooperation with the International \nStrategy Section at the Planning and \nInformation Department, set the North \nAmerican Center, a longstanding issue in the \ninternational affairs of the University. As a \nresult, a comprehensive system of university-\nwide Overseas Centers in Europe, North \nAmerica, and ASEAN has been put in place. In \nearly 2019, the San Diego Liaison Office was \nre-structured to become a part of the North \nAmerican Center.  \niSO-KU has been able to carry out these \nactivities thanks to the cooperation and support \nfrom relevant departments and sections both in \nheadquarters and throughout the University. \nAs a team, the iSO-KU members would like to \nexpress to them all a deep sense of gratitude. \nWhen considering the future of iSO-KU, a key \nchallenge is to design and enact necessary \nmeasures to promote internationalization in \nline with the Basic Concept, and to foster its \nimplementation in the University; focusing \nmainly on the promotion of international \nmobility of students; the acquisition of larger \nnumbers of qualified foreign students; and \ncontinuing to develop a more internationally \ncompatible environment on campus.  \nFor these goals to be effectively and efficiently \nachieved, iSO-KU and the University will \ncontinue to strengthen the functions of \ninternational relations organizations through \nthe reorganization and reform of the University \nas a whole.  \nThe iSO-KU team and myself look forward to \nyour continued support and cooperation. \n \n \n \n4 \n \n2. Introduction \n2.1. Background to the establishment of the iSO-KU \n Until March 2016 the \nOrganization for the \nPromotion of International \nRelations (OPIR) handled \nand promoted the University’s international \nmatters and exchanges, including \nuniversity-wide international student \nmobility, and supporting the departments in \ntheir international affairs. \nWithin OPIR, the International Relations \nCommittee served as a platform for \ndeliberation and information share, \nchanneling and coordinating interactions \nthroughout the University; including the \ndepartments, the central administration and \nthe management.  \nAiming to boost the University’s international \nprojection, in April 2016 the International \nDevelopment Committee and the \nInternational Education Committee were \ncreated respectively to enhance academic \nand research internationalization on the one \nhand, while promoting student mobility and \ninternational education on the other.  \nAt that time, the International Strategy Office \nof Kyoto University (iSO-KU) was created, \ninheriting OPIR’s functions, as the \nmechanism in charge of planning and \ncoordinating internationalization of the \nUniversity as a whole.  \nKyoto University has a long international \nvocation, rooted on its very foundation, \npromoting the creation of knowledge and \nacademic excellence built on a strong \ncommitment to the principle of “Academic \nFreedom”. \nFor Kyoto University being open to the world \nmeans promoting research beyond borders \nand nurturing high intellectual curiosity. To \ndo so, internationalization represents a key \npillar to how the University relates to society \nand materializes its mission: \"Kyoto \nUniversity will deepen its international \nexchanges as a university open to the world \naiming to contribute to the harmonious \ncoexistence of the global society\". \nSystematic internationalization at the \nUniversity began with the establishment of \nthe International Relations Committee in \n1977, which brought order and coordination \nto the international activities that were, until \nthen, carried out mostly at the department \nlevel and on individual bases, especially \namong researchers. \nThe International Relation Committee’s role \nwas to enhance university-wide information \nshare, towards a more integrated interest in \npromoting academic agreements that \ngradually expanded leading to the setting up \nof the International Student Center.  \n \n \n5 \n \nThese reforms evidence the University’s \ncommitment to diversify opportunities; a \nshift from quantity to quality-oriented vision \nof how internationalization should happen; \nand a transition in the way the university \ninteracted with the world, from a passive \nand receptive mode to a more transmissive \nand synergic mode, hence overcoming the \nindividualistic response of the past towards \na systematic and proactive approach. \nToday, in addition to professional \nknowledge, universities are expected to \ndevelop human resources who can carry out \ndiverse international activities and who are \nready to address changing global demands.  \nAs the world changes, the way people and \ninstitutions interact change too; in the same \nway international relations evolve and in so \ndoing the nature of those interactions and \nthe broad implications they have also \nchange. For example, as partner \norganizations change in their composition, \ntheir institutional interests and capabilities \nover time, the way university react to \naddresses those changes and adapt to the \nnew circumstance evolve too.  \nThis constant new conditions in the \ninternational relations have deep impacts in \nthe academic fields, the way research is \ndone, and on what and how the education \nuniversities provide is delivered. Being \nready to address these changes is a \nchallenge all universities share.   \nFor Kyoto University, as a research-oriented \ninstitution, fostering and expanding its \ninternational outreach has always been \nclosely related to the internationalization of \nresearch, which, in turn works as propeller \nfor the University as a whole.  \nGlobalization demands that universities \ncontinue to internationalize, to address the \ndouble challenge of providing internationally \ncompetitive education and producing \ninnovative research, to do it, they need to \nreorganize and strengthen their functions \nfrom within. Likewise, they need to enhance \nthe support they provide to activities at the \ndepartment level while maintaining a flexible \nand balanced approach so as to expand \nand maximize their global interactions. \nFor Kyoto University the main purpose of \ninternationalizing is to promote broader \nacademic cooperation with meaningful \nsocial impacts. The setting up of iSO-KU \naims to make efficient use of available \nresources, boosting effective management \nand integrating all efforts in the area of \ninternationalization. \niSO-KU’s mission is to steer necessary \nchanges in the University so as to maximize \npositive impacts that having an international \nenvironment at home imply, while \nmaintaining the fundamental spirit of the \nUniversity. Opportunities for growth are \nendless, yet, one needs to be realistic about \nthe changes the office can actually bring. \n \n \n6 \n \n2.2. Overview of the iSO-KU’s mission \n iSO-KU’s main role is to \norchestrate, promote and \nsupport the process of \ninternationalization for the \nUniversity as a whole. The Office \ncoordinates the global interactions of the \nUniversity integrating transversally the \nefforts of the central administration and the \ndepartments towards internationalization \nwhile respecting autonomy and promoting \nleadership and innovation.  \nBased on the regulations upon which the \noffice (Notification of its establishment, \nMarch 22, 2016) was created, its main \nresponsibilities are: \n1. Plan and coordinate strategies and \nmeasures to internationalize the \nuniversity. \n2. Support for the University’s global \ninteractions. \n3. Support the establishment and \noperation of the University’s Overseas \nCenters. \n4. Steer other necessary activities to \npromote the internationalization of the \nUniversity. \nAlong with these responsibilities, iSO-KU \ncontributes to the accomplishment of the \nUniversity’s mission working in four broad \nfunctions, broadly described as: Connect, \nBoost, Promote and Synergize, to provide \nsupport to activities aimed at creating an \nincreasingly inclusive campus; referring to: \nConnect from within and outside the \nUniversity.  \nBoost effective planning, coordination and \nsupport for international education, \nresearch, global contributions. \nPromote further internationalization of the \nUniversity’s campus and environments. \nSynergize the work of different \norganizations in the University, to maximize \nthe use of internal resources in the process \nof internationalization. \n \n2.3. Future internationalization at Kyoto University \niSO-KU achieved university-\nwide consensus regarding the \nfuture of internationalization of \nthe University by producing \nthe Kyoto University Basic Concept for \nInternationalization; setting an agreed \nvision on why and how internationalization \nshall be promoted in the University.  \nAlthough not strictly a strategy, the Basic \nConcept for internationalization aims to \ninspire coordinated and effective action in \nregards to internationalization of education, \n \n \n7 \n \ndevelopment of innovative research and \ncontribution from the University to \nharmonious coexistence on the planet. As \nsuch, it sets lines to inspire engagement \nand action from all corners of the University \ntowards proactive relation with society.  \niSO-KU functions as a transversal bridge \nthat connects the efforts of administrative \nsections and offices in the University; \ntogether, they work to internationalize the \nUniversity as a whole and comprise the \nPlanning and Information Department, the \nPlanning Division, the International Affairs \nDivision, the International Exchanges \nDivision, the Education and Promotion \nStudent Support Division, the International \nEducation Exchange Division (as of March \n2013), and Kyoto University Research \nAdministrators. Coordinating such a \ncomplex array of stakeholders is \nchallenging yet much needed.  \nOrganizational reforms require time and \neffort; and accomplishing a smooth and \neffective support system that promotes \ninternationalization requires a well-prepared \nand highly professional administration. \nThese reforms need to be planned in such \na way that they respect institutional culture, \nfoster effective communication and efficient \ncoordination of functions.  \nAlthough challenges remain ahead, the \ninternationalization road is open to new \nopportunities. iSO-KU’s role is to find and \npromote ways to turn those opportunities \ninto reality. This is not a small task; \nhowever, it is achievable and the following \npages present evidence of how important \nstructural changes have been achieved.  \nIn addition to planning and implementing \nactions to promote internationalization of \nthe University, iSO-KU also plays a key role \nin supporting the role of the University’s \nHeadquarters as a synergic factor, from the \nstanding point of the management and the \nadministration, to enhance not only \ngovernance of the University as a whole but \nownership in the strategies developed to \ncontinue the process of internationalization. \n  \n \n \n8 \n \n3. iSO-KU: Mission and functions \n \nGuided by the spirit of \nself-reliance and self-\nrespect since its \nfoundation in 1897, \nKyoto University has \nestablished a strong tradition of \nacademic freedom, pioneering in the \ncultivation of leaders and the creation of \nknowledge through outstanding research. \nAs the world faces unprecedented \nchallenges, internationalization is \nessential for Kyoto University to enhance \nits contributions to the global community. \nKyoto University is a world-leading \ninstitution that fosters dynamic \nadvancement of international initiatives in \neducation and research under a robust \nadministrative support system inspired by \nhigh standards of expertise and \nprofessionalism.  \niSO-KU was created to bring better \ncoordination throughout the University, \nserving as a support mechanism and as \na promoter of good practices in terms of \nhow internationalization is to happen. \nAccording to the statutory rules set in the \norganizational basis of Kyoto University1 \niSO-KU’ mission’s and functions are: \n                                               \n \n1 Statutory establishment of iSO-KU: March \n22, 2016 Notice No. 18 Establishment. \n• Planning and coordination of \nmeasures related to the international \nstrategies \n• Support for international exchange \n• Establishment and operation of \noverseas centers \n• Other matters necessary for \npromoting internationalization \nThe broad areas of activities carried out \nby the iSO-KU reflects the versatile \nnature of the office, at the time of \nimplementing its work, be it by \ncooperating or supporting other offices or \ndepartments, or be it based on its own \ninitiatives.  \nThe implementation of these tasks falls in \nthe broad areas of functions described as \nConnect internal and external networks, \nBoost international visibility, Promote \nmulticultural and inclusive campus and \nSynergize institutional processes and \nuse of internal resources.  \nConnect encompasses activities that \nbridge offices and departments to \nenhance internal coordination, \ninformation share and informed decision \nmaking about the global interactions of \nthe University; specifically, this refers to:  \n \n \n9 \n \na) Planning and coordinating \nheadquarters-led international \nactivities \nb) Coordinating and supporting \ndepartment-led international activities \nc) Research on issues of \ninternationalization \nBoost international visibility of the \nUniversity and provide support towards \neffective planning and coordination to \nfurther internationalize education, \nresearch, global contributions; \nspecifically, this refers to: \na) Strengthening branding and a \nuniversity-wide strategy for \ninternational visibility \nb) Strengthening the University’s \ninternational networks \nc) Assisting the University’s \nHeadquarters and departments in \ntheir diplomatic responses \nPromote further internationalization of \nthe University fostering an inclusive \ncampus with internationally friendly \nenvironments; specifically, this refers to: \na) Designing, implementing, and \nfollowing-up measures for \ninternationalization \nb) Improving access and distribution of \ninformation throughout the University \nc) Nurturing the University’s \nadministrative staff \nd) Promoting participation and \nnetworking in international activities \ne) Fostering Risk prevention and crisis \nmanagement \nSynergize the work of university-wide \nadministration, management, and the \ndepartments, to improve planning and \nmaximize the use of resources for a \nmore integrated and effective process of \ninternationalization; specifically, this \nrefers to: \na) Strengthening governance to \npromote internationalization by \nre-adjusting administrative offices \nand functions \nb) Strengthening cooperation among \nthe University’s management and \nadministrative offices in relation to \ninternational matters \nc) Supporting offices and \ndepartments in the \nimplementation of university-wide \ninitiatives for internationalization \nAs a whole, iSO-KU has a strategic \ncommunication role that enhances \ncollaboration with and among internal \nand external partners in order to \ncontribute to the core mission of Kyoto \nUniversity in education, research, and \nrelation with the global society.\n  \n \n \n10 \n \n4. Orchestrating internationalization & \ndiplomatic response\nEffective promotion of internationalization \nrequires timely information share, wide \ncoordination and informed decisions. iSO-\nKU is an institutional hub, bridging the \nUniversity from within and with the world \non matters related to internationalization to \nshare information, create synergies and \nmake efficient use of existing resources.  \nThe orchestration of the process of \ninternationalization of the University \nhappens through iSO-KU’s interactions, \nengagements and collaboration. \nWithin the University, these interactions \ninclude iSO-KU-led activities, support for \nother offices or departments; externally, \nwhile interactions with stakeholders and \nexternal partners fall under diplomatic \nresponse.  \niSO-KU has an academic role representing \nthe University in conferences, organizing \nseminars and staff development; while it is \nincreasingly engaging in research on \ninternationalization.  \nInteractions happening under this \norchestration are classified as follows: \n• Internal Connect (IC) includes \nHeadquarters coordination (IC.HQC), \nHeadquarters Advice & Support \n(IC.HQAS), Department Coordination \n(IC.BKC), Department Advise & Support \n(IC.DAS), Staff Development (IC.SD), \nand Meeting with Key Internationalizers \n(IC.MKI). \n• External Connect (EC) refers to Inbound \nDiplomatic Response (EC.IDR), and \nOutbound Diplomatic Response \n(EC.ODR).  \n• Academic Contribution (AC) comprises \nResearch activities (AC.Re), Educational \nactivities (AC.Ed), and staff development \nactivities (AC.SD).  \nDuring 2018 iSO-KU networking \ndistribution totaled more than 350 \ninstances for cooperation, including 229 IC \nmeetings, 111 EC meetings, and 13 \ninstances of AC. \nInternal connect \n65%\nExternal connect\n31%\nAcademic \ncontribution 4%\niSO-KU Networking distribution \n2018\n \n \n11 \n \n4.1. Internal connect \n4.1.1. Mechanisms for internal communication on international matters \n \niSO-KU steers \nand manages \nplatforms that \nbring together \nkey stakeholders \nin the University to share information and \npromote internal cooperation; the \ncombined outcomes from the decisions \nmade and their implementation materialize \nas University-wide policy for \ninternationalization. These mechanisms \ninclude: \nThe iSO-KU Management Council is the \nOffice’s highest decision-making stance, \nincluding 4 University’s EVPs and high-\nprofile professors to set the broad lines of \naction of the Office.  \nThe iSO-KU Core Meeting brings \ntogether high-profile academic and non-\nacademic staff and it represents a key \nadvisory body on international matters of \nthe university. Meetings fare organized in \ncooperation with International Strategy \nSection; in 2018 15 encounters took place \naddressing a wide array of topics \nThe Specialists Meetings (University-\nlevel partnership, Housing for Int’l \nStudents & Researchers, and Kyoto \nUniversity Overseas Centers). Each \nfocusing on areas promoting \ninternationalization. These meetings are \norganized in cooperation with the \nInternational Affairs Section; during 2018 \n12 meetings took place resulting in \nimportant decisions pertaining to the \nimplementation the Strategic Partnerships \ninitiative, the management of the \ndormitories for international students, and \na list of suggested translations for official \nnames in the University. \nThe Regular Meeting of Kyoto \nUniversity Overseas Centers brings \ntogether the directors of the centers, and \nother representatives and relevant \nstakeholders working both in Japan and \noverseas to share advancements at the \ncenters in their activities in their respective \nareas. These organized bimonthly in \ncooperation with International Strategy \nSection; in 2018 5 topic-specific meetings \nand 5 general meetings took place. \nThe ASEAN Center Network brings \ntogether representatives from \ndepartments carrying out research and \neducational activities in Southeast Asia. \nAs a transdisciplinary platform it promotes \nnot only information share but also in-\nfocus decision making for interactions with \nthe region. These meetings are organized \nin cooperation with International Strategy \n \n \n12 \n \nSection and KURA, in 2018 5 meetings \nwere held in Kyoto, added to the JASTIP \nsteering meeting in November in Bangkok. \nThe European Center Advisor Meeting \nbrings together representatives from \ndepartments and headquarters working on \nissues related to the relation of the \nUniversity to Europe, its presence and \nvisibility there.  \nThe Internationalization Promotion \nMeeting (IPM) gathers academic and \nadministrative staff working on \ninternational issues from departments and \nHeadquarters. Organized in cooperation \nwith International Affairs Section; in 2018 \n7 meetings were held steering information \nshare and cooperation on crosscutting \nissues of concern for the university.  \nThe International Cooperation Platform \n(PF) bridges across administrative offices \nand sections at headquarters; to share \ninformation transversally. Organized in \ncooperation with International Strategy \nSection; in 2018 12 regular meetings were \nheld resulting in a more coordinated \nadministrative approach and more \ninformed and integrated approach to \ninternationalization.  \nThe Advisory Board to iSO-KU was \nlaunched in October of 2018; this is formal \nmechanism provides advice and \nrecommendations to iSO-KU on policy to \npromote internationalization of the \nUniversity including the point of view of \ninternational academic staff from \ndepartments and Headquarters. \nCooperation spurring from these meeting \nare expected to increase the influence of \ninternational staff in the decision-making \nprocess of the University as well as \nhelping bridge needs and initiatives from \nthe departments with possible response \nfrom the Headquarters. In cooperation \nwith International Strategy Section, a kick \noff meeting and a two follow up meetings \nhave been held.  \nThe Kyoto University international \nAcademics Network (KUIAN) is a regular \nyet informal platform for international \nacademic and administrative staff to share \nviews on internationalization. The network \nis closely related to the role of Advisory \nBoard providing a closer connection to the \nreality of international staff throughout the \nuniversity and the policy-making system. \nThis is an iSO-KU initiative whose efforts \nare bringing important information for \ndecision making to the Headquarters.  \nThe Meeting with Key Internationalizers \nare ad hoc institutional consultations with \nleading stakeholders in the university on \ninternationalization policy. These meetings \nare held as informal encounters they \nprovide deep insights on matters that \nrequire more in-focus attention. This is an \niSO-KU initiative, during 2018 19 meetings \nwere held.  \nThe International Crises Management is \na topic-specific working group focusing on \npolicy for travel safety, hence risk and \n \n \n13 \n \ncrises prevention and management for \npersonnel and students engaged in \ninternational travel. The team has \nproduced a travel safe check list and an \ninformation flow chart for emergencies. \nCommunication took place online or over \ne-mail regularly in cooperation with the \nInternational Strategy Section. \n \n4.1.2. iSO-KU Support for administrative offices \n \niSO-KU has a key \nrole in supporting the \ninternationalization \nwork of other \nadministrative offices \nin the University. In 2018 iSO-KU provided \nadvice and recommendations mainly to: \n• To the International Affairs Division, \nregarding support for the University’s \nparticipation in the Rectors meetings in \nOctober in Hiroshima University, \nproducing the University’s report on \nSocial Contribution for the Presidents \nMeeting to be held in June 2019 in \nHamburg University; providing content \nand editorial advice towards the yearly \npublication of the Kyoto University \nIntroductory Brochure. iSO-KU regularly \nplays the role of MC in international \nevents such as the President’s Welcome \nDinner for Researchers; iSO-Ku offers \nsupport in issues related to contents and \nEnglish language in presentations or \nseminars organized internal and \nexternally.  \n• To the International Strategy Line \nSection, iSO-KU supports and provides \nrecommendations regarding the setting \nand functioning of the University’s \nOverseas Centers. \n• To the Office of Global Communications \nand Public Relations as a member of the \nKyoto University Research News \nEditorial Board providing contents on \nOverseas Centers, editorial support and \ndissemination of the magazine both \ninternally and externally.  \n• To the International Education \nAdministrators (IEA) providing advice and \nnetworking support towards MOUs and \nSEAs with partners or future partner \ninstitutions.  \n• To the International Student Exchange \nOffice, through support in the selection \ncommittee for programs like Kingfisher \nand other outbound opportunities for \nstudents. As well as in providing advice \nand support regarding policies towards \nan inclusive campus (e.g. raising \nawareness on existing praying room and \nother good practices)  \n• To the Web Strategy Office as a core \nmember providing advice on matters of \nUniversity Identity, integration, contents \nand design for both Japanese and \n \n \n14 \n \nEnglish home pages. \n• To the Office of Educational Planning in \nmatters related to participation in the \nUniversity Social Responsibility Network \nas well as other dissemination of SDGs \nrelated activities.  \n• To administrators of programs such as \nELCAS or Kyoto University-Bhutan \nFriendship by proving support for the \ndissemination of their activities and \nboosting international visibility. \n• To Office of Society-Academia \nCollaboration for Innovation (SACI) \nthrough regular meetings and supporting \ninternational response, for example \nfostering cooperation with Universidad \nNacional de Cordoba, Argentina.  \n• To the Institutional Research Office (IR) \nproviding support and data on the \nindicators for Social Impact for Times \nHigher Education. \n \n4.1.3. iSO-KU Support for activities by faculties and departments \n \niSO-KU supports the \ninternationalization \nwork of faculties and \ndepartments \nproviding advice and recommendations \nincluding but not limited to initiatives:  \n• Support for Kyoto University Museum to \norganize and disseminate the Open Day \nas well as Children’s Museum. \n• Bridging research interests (Engineering) \nDepartment on Urban Planning with the \nNanjing University, China.  \n• Bridging the Center for Advanced Study \nof International Development at Michigan \nState University with Kyoto University’s \nDepartment of Education by organizing \nan academic event in cooperation with \nKURA, the Center for the Promotion of \nExcellence in Higher Education and \nGlobal Communications. The visit has \nresulted in a Department level MOU with \nthe Department of Education in Kyoto \nUniversity. \n• Support for Graduate School of Global \nEnvironmental Studies in boosting \ninternational visibility to programs such \nas Ecole de Kyodai and activities like the \nJikatsu project, the SPIRITS Initiative \nand other SDGs related initiatives. \n• To the Graduate School of Agriculture in \nits participation in the ASEAN+3 \nConference in Manila on International \nCrisis Management and risk prevention. \n• Support to the Center for the Promotion \nof Excellence in Higher Education during \nthe seminar for newly appointed \nprofessors at the university as well as in \nthe development and implementation of \njoint MOOC with USRN on University \nSocial Responsibility. \n \n \n15 \n \n4.2. External connect \niSO-KU has an important role in leading \nand supporting the work of other offices \nand departments in the interactions of the \nUniversity with the external stakeholders. \nAs inbound diplomatic response, iSO-KU \norganizes and receives courtesy visits, it \nalso carries out outbound diplomatic \nresponse through participating in \ninternational events as well as dispatching \npersonnel for external visits and \nnetworking. \n \n4.2.1. Inbound diplomatic response: receiving courtesy visits \n \niSO-KU is a key player in \nreceiving delegations \nfrom external partners. In \n2018 iSO-KU welcomed \nrepresentatives from: \n• The Multimedia Delegation from the \nNational University of Malaysia (UKM) \non internationalization and academic \ncooperation for research at CSEAS.  \n• The networking session organized with \nKURA with delegates from the Ministry \nof Research Technology and Higher \nEducation of Indonesia on innovative \nresearch opportunities.  \n• Representative from Ort University, \nUruguay, on academic cooperation for \nstudent mobility and research in the field \nof architectural sciences.\n \n4.2.2. Outbound interactions: delegations sent abroad \n \nIn 2018 iSO-KU had an \nactive international \nparticipation through \npersonnel dispatched \nfor networking purposes \nas well as in especial academic events.  \n• Representing Kyoto University, iSO-KU \nparticipated in the steering committees \nof USRN organized at the University of \nSao Paulo, Brazil in April, in Haifa \nUniversity in December and in Hong \nKong Polytechnic in February 2019. \nGiving high visibility to the University and \nits leading role in the network as well as \nin the creation of a joint MOOC on USR.     \n• In November iSO-KU’s director led the \n4th Steering Committee of Japan \nASEAN Science Technology and \nInnovation Program (JASTP), serving as \nmid-term information share and \n \n \n16 \n \nevaluation of ongoing activities in \nSoutheast Asia, as well as in stating the \ndesign of the next phase of the program.  \n• iSO-KU represented Kyoto University in \nAPAIE 2019, Kuala Lumpur, networking \nwith current and potential partners on \nmobility and research cooperation. \n \n4.3. Broader interactions \niSO-KU participated in \nnetworking events in \ncooperation with \noffices of departments \nwith representatives \nfrom organizations, including government \nagencies as JICA Kasai on the initiative for \nan SDG related portal.  \nTogether with the University’s Students \nAffairs EVP, iSO-KU promoted cooperation \nwith the Asian Development Bank (ADB) in \nthe organization of a joint seminar aimed at \nboosting inbound students and \nresearchers. \n \n \n  \n \n \n17 \n \n5. iSO-KU at work \nAchievements in internationalization \n \nThe following pages describe iSO-KU’s \nachievements, firstly by introducing its \nmost important achievement since its \nestablishment -the Kyoto University Basic \nConcept for Internationalization- and then \ndescribing the work \nand achievements in the respective four \nfunctions -Connect, Boost, Promote, and \nSynergize-. \n \n5.1. Kyoto University Basic Concept for \nInternationalization \nIn September 2018 iSO-\nKU achieved university-\nwide consensus on the \nfuture internationalization \nof the University by \nproducing a widely agreed instrument, the \nKyoto University Basic Concept for \nInternationalization; an inspirational \ndocument that describes how Kyoto \nUniversity will continue its work to remain a \nleading university open to the world.  \nFaithful to the University’s traditions and \ndesigned to sustain its role as a prominent \nglobal knowledge hub, the Kyoto \nUniversity Basic Concept presents the \nvisions through which the University aims \nto contribute to a sustainable future for \nglobal society.  \nThe document sets an agreed vision on \nhow internationalization shall be promoted \nin the University; while it leaves wide \nfreedom for actual implementation to those \nin charge of their respective areas and \nfields both in the administration as well as \nin the departments.  \nThe Basic Concept is not strictly a \nstrategy; however, it gives a sense of \nshared direction about how efforts and \naccomplishments shall lead to the \nattainment of a common vision regarding: \n• Education that cultivates international \noutlooks and competencies \n• Global development of innovative \nresearch  \n• Contribution to harmonious coexistence \non the planet \nThe document is the product of a long \nconsensus building process that started in \nearly 2017 and it contains views from \n \n \n18 \n \nmajor stakeholders throughout the \nuniversity. In order to ensure agreement \nand to promote ownership, iSO-KU visited \nand consulted with the departments and \nrelevant offices in the University at the time \nof drafting and revising it.  \nOnce agreement on the text was achieved, \nthe University’s Vice-President for \nInternational Strategy and iSO-KU’s \nDirector, Professor Yasukuki Kono made \nthe Basic Concept public and produced a \ncomplementary statement offering details \non the significance of the document, the \npurpose of its creation and how it should \nbe used. \nThe Basic Concept becomes a reality \nthrough the University’s comprehensive \nsupport system, contributing to the \nattainment of the University’s mission in  \nfostering well-being around the world, \nraising awareness of global challenges \nand human rights among members of its \ncommunity and beyond.  \nThe Basic Concept sets general lines that \ninspire action from all corners of the \nUniversity towards proactive engagements \nwith society aiming to contribute to social \ndevelopment through systematic and \nconstant internationalization of education \nand research.  \nBased on already existing good practices \nand lessons learned on \ninternationalization, the text encourages \nthe consolidation, expansion and \nreplication of initiatives that have proven \neffective at the central management and \nadministration level.  \nThis includes implementing the plan set by \nthe Designated National University Status, \nas well as other initiatives designed and \nexecuted by the departments. In the same \nline, the Basic Concept aims to motivate \nnew initiatives, such as developing \nattractive programs to further enhance \ninternational student mobility. \n \nFull version https:\/\/u.kyoto-u.jp\/ex8y9  \n \n \n19 \n \n5.2. Work and achievements in Connect, Boost, Promote, and \nSynergize \n5.2.1. Connect \n \nThe Connect function consist of \nbridging the University internally, \nand enhancing global \ninteractions to increase \ninternational presence and \nacademic development. The \noffice’s main tasks and achievements were:  \n \nA) Planning, coordinating headquarters-led \ninternational activities \n- Supporting the implementation of mid-\nterm goals and plans.  \n- Strengthening collaboration with overseas \npartners, supporting the matching process \nfor research collaboration, or participating \nin international events.  \n- Supporting the planning\/screening \nprocess of short-term programs to expand \nstudent mobility opportunities; including \ninternship programs, and the creation of \nnew programs.  \nSpecific contributions \n→ iSO-KU hosted a delegation of students \nand researchers from Michigan State \nUniversity researching on \ninternationalization of higher education \n→ iSO-KU hosted a delegation of students \nand researchers from National University of \nMalaysia (UKM) and connecting them with \nCSEAS at Kyoto University \n→ iSO-KU promoted interactions with SACI in \nthe search for new initiatives \n→ iSO-KU supported offices to foster \ncooperation with international \norganizations: with OECD to expand \ninternship schemes; with ADB to expand \nresearch scheme \n→ iSO-KU supported activities for Wild & \nWise initiative, MEXT Special Program, the \nUniversity’s participation in the Kingfisher \nProgram \n→ iSO-KU supported Alumni Support Office to \nboost interactions with international alumni \n(Japanese graduates and former foreign \nstudents in Kyoto University) \n- Supporting the Overseas Centers, while \npromoting their visibility and connectivity  \n \nSpecific contributions \n→ iSO-KU supported policy-making for the \ncenters and their mechanisms (ASEAN \nCenter Network, Education Working Group, \nEuropean Center Advisor Meeting)  \n→ iSO-KU supported the contents about the \ncenters for KURN \n→ iSO-KU supported and contributed to \nstrengthening the ASEAN Center Network \n→ iSO-KU steered and managed the Special \nCommittee for Overseas Centers \n→ iSO-KU promoted networking within Kyoto \nUniversity towards a more integrated \napproach to SDGs, E.g. the Graduate \nSchool of Global Environmental Studies \n→ iSO-KU gathered and shared information \nabout the centers in Global KyotoU site  \n→ iSO-KU supported procedures for NGO \nstatus of ASEAN Center \n→ iSO-KU supported visit from delegates from \nMinistry of Research, Technology, and \nHigher Education (Indonesia) and \nnetworking with Kyoto University’s \nresearchers.  \n→ iSO-KU contributed to relations with \nregional networks: SEAMEO and AUN \n \nB) Coordinating, supporting department-\nled international activities \n \n \n20 \n \n - Promoting internationalization of education \nby the International Board of Education, \nplanning innovative activities, offering support \nin networking efforts, as well as for short term \nprograms and events. \n \nSpecific contributions \n→ iSO-KU supported Kyoto University \nMuseum activities: bridging with ILAS and \nStudent Exchange Division for Kodomo-no-\nhakubutsukan; promoting Museum’s Open \nDay in IPM and organization of the event  \n→ iSO-KU supported the establishment of \ndouble degree at the Management \nDepartment with Cornell University \n \nSpecific contributions \n→ iSO-KU promoted participation of students \nand faculty in activities organized with \npartner universities in USRNetwork such \nas: SERVE initiative on post disaster \nrecovery; and the Research Attachment \nprogram at Kyoto University  \n \n- Having supported the University’s On-site \nLabs establishment and management  \n \nSpecific contributions \n→ iSO-KU drafted guidelines to enhance \ncommon understanding of how labs are set \nand function, in cooperation with \nInternational Strategy Section \n→ iSO-KU set mechanisms for coordination \nand communication among departments \nand between departments and \nheadquarters \n→ iSO-KU set a system to support the lab’s \noperation and management \n \nC) Producing research and publications \n- Analyzing and sharing good practices on \ninternationalization, with special focus on \nsharing in-house lessons learned and good \npractices.  \n \nSpecific contributions \n→ iSO-KU collected evidence base data from \nwithin the University to enhance and \nreplicate good practices and lessons \nlearned \n→ iSO-KU initiated considerations to create a \nreport with policy suggestions and \nobservations with views from international \nfaculty members and staff \n→ iSO-KU started considerations to create an \nelectronic newsletter from the Office to \nshare information about internationalization \ndevelopments for the University \n \n5.2.2. Boost \n \nThe Boost function aims to \nincrease international visibility \nof the University, supporting, \nplanning and coordinating \ninitiatives to internationalize \neducation, research, and \nexpanding global contributions. \nThe office’s main tasks and achievements \nwere: \n \nA) Strengthening branding and a \nuniversity-wide strategy for \ninternational visibility \n- Providing advice for international branding \nand University Identity  \n \nSpecific contributions \n→ iSO-KU collected and shared data from the \nUniversity to enhance and replicate good \npractices and lessons learned \n→ iSO-KU supported Web Strategy Office and \nPublic Relations Office on international \nbranding policy.  \n→ iSO-KU contributed to publishing of the \nUniversity’s “Visual Identity Guidebook”, \nproduced by the Public Relations Office \n→ iSO-KU supported IR Office regarding THE \nranking on social impact  \n \n- Supporting the production of materials for \ninternational promotion  \n \n \n \n21 \n \nSpecific contributions \n→ iSO-KU contributed to the development \nand update of the University’s websites, \nbrochures and other ICT resources \n→ iSO-KU supported renewal of the Japanese \nWebsite with coherent titles, contents, \nstructure and overall international look \n→ iSO-KU offered advice for the Japanese \nwebsite especially in Education, Research, \nand Admission related to consistency, \ncontents, and inter-page navigation  \n \n- Developing and maintaining the Global \nKyotoU site that brings together relevant \ninformation of the University in regards to \ninternationalization \n \nSpecific contributions \n→ iSO-KU developed the concept and \nstructure of the site to enhance \naccessibility to information, visibility of good \npractices and navigability \n→ iSO-KU developed contents for Global \nKyotoU and bridged with key stakeholders \n→ As of April 2019, 5 of 12 sections have \nbeen uploaded: International strategy, \nGlobal profile, Global partners, \nInternational Risk & Crisis Management, \nand Global USR \n→ iSO-KU made Kyoto University is the first \nNational University in Japan to disclose \ninformation on its University Social \nResponsibility \n \n- Provided support to other offices in the \nproduction of printed materials \n \nSpecific contributions \n→ As a member of the editorial board to \nKURN, iSO-KU supported the Global \nCommunications Section, Public Relations \nOffice and KURA.  \n→ iSO-KU supported the editorial work for \nKyoto University Introductory Brochure \nproduced by the International Affairs Office. \n→ iSO-KU supported the production and \ndistribution of the Kyoto University Risk \nManagement framework and Safe travels \ncheck list \n→ iSO-KU drafted the Kyoto University North \nAmerican Center introductory brochure  \n \n- Boosting the visibility of the University’s \nOverseas Centers, internally and externally \n \nSpecific contributions \n→ iSO-KU supported the production, revision, \nedition and distribution of the Center’s \nrespective newsletters, leaflets, events, etc.  \n→ iSO-KU distributed information from and \nabout the Centers, both throughout and \nbeyond the University \n \nB) Strengthening the University’s \ninternational networks \n- Supporting interactions within \ninternational networks to boost visibility  \n \nSpecific contributions \n→ iSO-KU supported interactions with \nHeKKSaGOn, RENKEI and ASEAN+3, \ncarried out by the International Affairs \nOffice  \n→ iSO-KU contributed to the strengthening of \nalumni collaboration & support. Resulting in \nhigh participation in ASEAN Network Fora \nthrough Southeast Asia \n→ iSO-KU contributed in the organization and \nmanagement of the Universities \nAdministrators Workshop (UAW) including \nstaff support for presentations of reps from \nInternational Affairs and other offices to \nworkshops abroad (Taiwan, Hong Kong) \n→ iSO-KU supported staff participation in \ninternational events (presentation skills, \ncontents development, capabilities to meet \nglobal standards language and delivery)  \n→ iSO-KU engaged with ASEAN+3 \nConference Risk Management (Manila)  \n→ iSO-KU participated in USRNetwork \n \n \n22 \n \nsteering committee, contributing to raising \nawareness on the network and activities of \nthe University in the Network; supporting \nthe development of a joint MOOC on USR \nand representing Kyoto University in a joint \npresentation on SDGs in GEEF 2019 in \nYonsei University, Seoul \n \n- Fostering collaboration with overseas \npartners in industry-government-academia,  \n \nSpecific contributions \n→ iSO-KU supported offices in setting, \ndrafting and implementing MOUs, SEAs, \nand other agreements boosting the \ninternational visibility of Kyoto University. \n→ iSO-KU offered advisory support to the \nInternational Education Administration \nOffice for decision making regarding \npartnerships \n→ iSO-KU supported International Affairs \nOffice in the implementation of the \ninternship program and staff exchange \nprogram with UCDavis \n→ iSO-KU connected ASAFAS to University \nof Bayreuth, and then to Hankuk University \nof Foreign Studies. As a result of the \nconnection made during the visit by \nGerman Universities’ rectors.   \n→ iSO-KU supported SACI and departments \nto strengthen relations with local industry in \norder to create short-term internship \nprograms \n \nC) Assisting Headquarters in its \ndiplomatic response \n- Hosting visits and sending Office’s staff \nas representatives to international events \n \nSpecific contributions \n→ iSO-KU cooperated with SACI’s work with \npartners on commitment for SDGs; \nstrengthening access and improved \nvisibility of SACI Global KyotoU site  \n→ iSO-KU provided advice on diplomatic \nresponse for regional approaches \nregarding the Americas and Oceania. \nAddressing visits from Australian \nUniversities consortium, U. Guanajuato \n(Mexico), U. Concordia (Canada), CJJC \n(Canada-Japan) \n→ iSO-KU contributed to the promotion of \nbilateral relationships by delivering \naddresses at joint symposia or other \nevents: Japan-Chile, Japan-Austria \n→ iSO-KU attended and supported high-\nprofile events: Japan-Taiwan Presidents’ \nForum; Japan-UK Dialogue on Exchange in \nHigher Education & Research; Japan-\nFrance Symposium in Higher Education \n \n- Other international ad hoc activities \n \nSpecific contributions \n→ iSO-KU drafted, revised and edited high \nprofile speeches. etc. \n→ iSO-KU supported the preparation for \npresentations by other offices externally \n→ iSO-KU supported the draft of \ndocumentation for projects of international \nrelevance: proposal to MEXT for the \nDesignated National University, Canada-\nJapan Committee \n \n5.2.3. Promote \n \nThe Promote function refers to \nhow iSO-KU encourages \ninternationalization fostering an \ninclusive campus with \ninternationally friendly \nenvironments. The office’s \nmain tasks and achievements were:  \n \nA) Designing, implementing, and \nfollowing-up measures for \ninternationalization \n- Achieving university-wide consensus on a \ncomprehensive scheme: Kyoto University \nBasic Concept for Internationalization \n \nSpecific contributions \n \n \n23 \n \n→ The Basic Concept represents a major \nachievement that gives the University an \nagreed the direction on how it should foster \nits endeavors to internationalize \n→ Even if Basic Concept encompasses \nuniversity-wide activities, iSO-KU has taken \ninitiative in number of specific actions:  \n→ In cooperation with the International \nStrategy Section, iSO-KU organized a joint \nworkshop on internationalization with \nrepresentatives from Ritsumeikan \nUniversity, Doshisha University, Kyushu \nUniversity and Osaka University.  \n→ iSO-KU has contributed to gather and \nconsolidate internal information on \ninternationalization in the University. Based \non this iSO-KU is revising the \nimplementation section of the Basic \nconcept (tactics and actions)  \n→ iSO-KU started mechanisms for planning, \nbrining better understanding and boosting \ninternal communication through KUIAN and \niSO-KU Advisory Board  \n→ In 2018 seven KUIAN meetings were held; \nthe network currently has more than 40 \nscholars. So far five blockages were spot: \npromoting more inclusion, information \nshare and access in English and support \nfor scholars and families. \n \n- Implementation, monitoring, reporting of \nactivities contained in the Basic Concept \n \nSpecific contributions \n→ iSO-KU started collection and analysis of \ndata on international education, research, \nsocially impactful activities, management \nand administration (reflected in the \nImplementation section of the Basic \nConcept: tactics and actions) \n→ iSO-KU publicized the Basic Concept and \nraising awareness on its relevance \n→ iSO-KU started the design of a university-\nwide questionnaire for international staff \norganized with cooperation from academic \nstaff at KUIAN to deepen understanding of \nthe needs and expectations for foreign staff \nin the University \n→ iSO-KU started the collection of data to \ndevelop an evidence-based report about \nsituation of international faculty members \nand staff to boost informed decision making \n \n- Starting implementation of the Basic \nConcept on management, planning, \nparticipation and domestic connection \ntowards effective internationalization \n \nSpecific contributions: \n→ iSO-KU set up mechanisms for better \ninformation share and decision making: \nrestructuring iSO-KU as a broad network  \n→ iSO-KU refurbished the decision-making \nprocess, through empowering iSO-KU Core \nMeeting, and building stronger \ncommunication tools with international \nacademic staff through the iSO-KU \nAdvisory Board and KUIAN \n \nB) Improving access and distribution of \ninformation throughout the \nUniversity  \n- Building and maintaining mechanisms for \ninformation share \n \nSpecific contributions \n→ iSO-KU designed, developed and \nmaintained Global KyotoU sites in the \nEnglish Homepage of the University to \nfacilitate access and navigability \n→ iSO-KU started integration of information \non international matters in a single portal  \n→ Currently developing the “Kyoto University \nInternational Concierge” webpage to \nprovide a strong information management \nservice to serve as a traffic controller \n→ iSO-KU started works to make GAROON \nmore understandable and user friendly, for \nthose who do not read Japanese \n→ iSO-KU supported the developments for \nthe University’s Researchers Database to \nfacilitate access to institutional and \npersonal contact information \n \n \n24 \n \n- Fostering a culture of bilingual \ncommunication for official documents  \n \nSpecific contributions \n→ iSO-KU supported offices and departments’ \nefforts towards multilingual and \nmulticultural measures: creation and  \nconsolidation of guidelines or glossaries \n→ iSO-KU supported International Affairs \nOffice in written documentation, as well as \nthe work of its Translation Section. \n→ iSO-KU promoted the use of bilingual \ndocuments on campus: for those official \nand for publication including presentations, \nspeeches and the like  \n→ iSO-KU advocated for Englization of sings \nand information displayed in English: in \nKyoto University Museum, iUP flier, English \nHomepage, Guideline for University Identity  \n \nC) Nurturing the University’s \nadministrative staff  \n- Promoting the skill up of human resources \nregarding internationalization through staff \ndevelopment programs and activities \n \nSpecific contributions \n→ iSO-KU promoted multidisciplinary \ncooperation in the administration. E.g. \nactivities organized during the University \nAdministrators Workshop on institutional \nlearning \n→ iSO-KU made a presentation in the training \nworkshop for newly appointed scholars \norganized by CPEHE \n→ iSO-KU provided support for administrative \nstaff joining the John Mung Program for \nmobility in the Overseas Centers \n→ iSO-KU organized biweekly English Lunch \nwith staff from Planning and Information \nDepartment as well as with the \nInternational Strategy Section \n \nD) Promote and facilitate \nparticipation\/networking in \ninternational activities \n- Enhancing networking capabilities with \ninternational researchers and students \nwithin and outside the university \n \nSpecific contributions \n→ iSO-KU supported and made a \npresentation during the “From scholars to \nscholars” seminar organized by MSA \n(MEXT Scholars Association) on career \npaths in Japan \n→ iSO-KU brought the students’ voice to \ninformation share mechanisms such as the \nIPM and PF by inviting them to share good \npractices on risk management, etc. \n→ iSO-KU supported students’ activities such \nas the organization of TED talks \n \n- Promoting networking of international \nresearchers within university \n \nSpecific contributions \n→ iSO-KU created Kyoto University \nInternational Academics Network (KUIAN) \nand the iSO-KU Advisory Board, both \nplatforms offer potential for cooperation \n→ iSO-KU promoted connections between \ndepartments. Advocating among \ndepartments so as to offer Japanese \nclasses for foreign researches  \n→ iSO-KU has been working to enhance the \nconnections between the Kyoto University \nMuseum and ILAS in order to boost \nexchange students’ interests in the \nactivities of the Museum  \n→ iSO-KU promoted the activities of the \nGSGES regarding SDGs projects in the \nASEAN Center Network \n→ Based on the data gathered through the \nVisits to Key Internationalizers (Guerrilla \nmeetings) iSO-KU promoted internal \ncooperation among researchers \n \nE) Risk prevention and crisis \nmanagement \n \n \n25 \n \n- Creating a framework for risk prevention \n& management for international travel \n \nSpecific contributions \n→ In cooperation with International Strategy \nSection, iSO-KU gathered broad internal \nand external data on risk management as \nreference for the framework to set and \ndesigning the international travel check list, \ncrisis information flow and guideline \n→ In Cooperation with International Strategy \nSection, iSO-KU created an emergency \nflow chart, easy to read and to use for \ninformation share and prevention measures \nfor in and outbound students and \nresearchers \n→ Also, in cooperation with International \nStrategy Section iSO-KU set the Kyoto \nUniversity International travel check list, \ncrisis management information flow and its \nguideline, which are currently being used \nby departments \n→ In cooperation with International Strategy \nSection, having held a “Consideration \nmeeting” with the working group to further \nimprove the framework \n→ iSO-KU started gathering of information on \ngood practices from leading departments \n(13 so far) to share information on existing \nsafety guidelines for fieldwork at the \ndepartment level \n→ As of May 2019, information on safety \nguidelines is being compiled and analyzed \nto create a university-wide framework for \ninternational research safety as a guideline \nto be included in the framework of Kyoto \nUniversity regulations on fieldwork \n \n5.2.4. Synergize \n \nThe synergize function portrays \nefforts to coordinate university-\nwide organizations, offices, and \ndepartments, to improve \nplanning and maximize the use \nof resources towards a more \nintegrated and effective internationalization. \nThe office’s tasks and achievements were:  \n \nA) Strengthen governance to promote \ninternationalization by re-adjusting \nadministrative offices & functions \n \n- Providing advice in the planning of \nuniversity-wide offices with international \nscope, their functions and organization of \ntasks.  \n \nSpecific contributions \n→ iSO-KU supported the Provost Office at the \ntime of designing the proposal for the \nDesignated National University: \nconsistency of texts and English)  \n→ iSO-KU supported iUP team at the start of \nthe project regarding implementation of the \nproject, university identity in the fliers and \nwebsite; also, in recruiting activities \n→ iSO-KU supported the University Council in \nthe design and implementation of the \nInternational Admissions Assistance Office \n→ iSO-KU started to create and consolidate a  \nlist of admission information and \nrequirements from all the graduate schools \nin the University towards an integrated \nprocedure \n→ iSO-KU collected information on admission \ninformation and requirements from leading \nuniversities, to learn from existing good \npractices \n→ iSO-KU made necessary arrangements \nand preparations towards the launch of the \nnew office, including appointment of the \nstaff, regulations, etc. \n \n- Strengthening cooperation among the \nUniversity’s management and \nadministrative offices in relation to \ninternational matters \n \nSpecific contributions \n→ iSO-KU supported the work of the \nInternational Exchange Division, the \nInternational Education Administration \nSection and the International Strategy \nSection of the Planning Division; while \n \n \n26 \n \ncontributing to increased harmonizing on \nefforts with offices like KURA and SACI \n→ iSO-KU collaborated with KURA on the \ndesign of posters and development of \ncontents for the website for IPCC 2019, \nincluding organization of events and \ntranslation of documents \n \n- Managing and steering mechanisms for \ninformation share on international matters  \n \nSpecific contributions \n→ iSO-KU set effective means of \ncommunication as part of the institutional \nstructure of the University: \nInternationalization Promotion Meeting to \nbridge administrative offices and \ndepartments; and International Cooperation \nPlatform to bridge administrative offices \nthrough the use of regular meetings and \nmailing lists, among others.  \n→ iSO-KU made these mechanisms part of \nthe structure of the University making them \nsystematic and stable; which facilitates \ncommunication, setting of agenda, follow \nup mechanisms as well opening the way \nfor new issues.  \n→ In cooperation with KURA and International \nStrategy Section, iSO-KU overviewed and \ncoordinated the work of the University’s \nOverseas Offices with regional focus. iSO-\nKU was reformed in 2019 to enhance the \nregional work of the university assigning \nand redistributing responsibilities:  \n- For Europe, Africa and the Middle East \nthrough the EU Overseas Center \nEuropean Advisor Meeting, and the \nAfrica Unite Meeting \n- For Southeast Asia and East Asia, \nthrough the ASEAN Center and the \nASEAN Center Network Meeting \n- For the Americas and Oceania through \nthe North American Center and the San \nDiego Liaison Office \n→ The above resignation of responsibilities \nimplies providing support for activities of \nthe three Overseas Centers, the Africa Unit \n→ For activities of the ASEAN Center support \nfor: JUNThai, NGO status, flier, website, \nJASTIP, information shared at KURN, \nconnections with departments for ASEAN \nKyoten Network, increasing visibility of \nactivities, etc. \n→ For North American Center, iSO-KU \nsupported in the design and \nimplementation of the project. Contributing \nto development of networks for North \nAmerican Center, in the US and Canada \n→ iSO-KU supported the activities of the San \nDiego Liaison Office and fostering \ncoordination with Medical faculty at On-site \nLaboratory \n→ For the Oceania region, iSO-KU has been \npromoting the University’s presence \nthrough networking with universities in \nAustralia and New Zealand \n \n- Steering roles and relations of high-profile \ncouncils and committees \n \nSpecific contributions \n→ iSO-KU supported the work of \nmanagement councils, special advisory \ncouncils and committees of specialists \n(Strategic partnerships, international \nenvironment, the Kyoto University Library; \nand working groups such as the \nResearcher’s information database and the \nInternational Crisis prevention and \nManagement) \n \n- Supporting relevant offices and \ndepartments in the implementation of \nuniversity-wide initiatives such as the Top \nGlobal Universities or the Designated \nNational Universities \n \n \n \n27 \n \n6. iSO-KU academic role\niSO-KU’s academic \nmission becomes a \nreality by participating \nin academic events like \nconferences or \nseminars. \nIn 2018 iSO-KU members offered \npresentations on internationalization, SDGs \nresponse, university social responsibility, \nthematic partnerships and more, both \nabroad and also in Kyoto University. \niSO-KU has continued to contribute to the \nUniversity’s staff development through \nregular activities such as the English Lunch \ninitiative for the Planning and Information \nDivision as a biweekly informal session for \ninformation share and language skill up.  \nThe following paragraphs describe iSO-KU \ncontributions as presentations and staff \ndevelopment activities during 2018. \n \n6.1. Academic contribution \n6.1.1. International presentations \n \nUSR through culture: the power of \nmultidisciplinarity. Kyoto University. \nSao Paulo University. USRN Staff \nDevelopment. Sao Paulo. Apr 2018 \n \nExploring new horizon of academic \nexchange (Japan & Thailand). \nCeremony: Granting of NGO Status and \nSigning of the MOU between the NSTDA \nand Kyoto University. Bangkok. Sep 2018 \n \nSDGs and roles of JASTIP. 5th \nJASTIP Symposium. Disaster Risk \nReduction & Environmental Sustainability \nfor Social Resilience. KL. Oct 2018 \nRegion-based collaboration for \nSDGs: Challenges of JASTIP. 6th \nJASTIP Symposium. Biodiversity, Genetic \nResources and Innovative Bioresource \nTechnology. Roles in ASEAN’s Sustainable \nDevelopment. Tangerang. Nov 2018 \n \nRole of regional collaboration in \nscience and technology research \nfor achieving SDGs: Perspectives from the \nJASTIP experience. The 18th Science \nCouncil of Asia Conference. Role of Science \nfor Society: Strategies towards SDGs in \nAsia. Science Council of Japan. Dec 2018 \n \n \n28 \n \nUniversity Social Contribution & \nresponsibility. What’s behind \nlanguage. Universities’ Social Responsibility \nin Shaping a Future of Critical Hope. Haifa \nUniversity. Dec. 2018 \n \nUSRN Summit. Jaifa. Israel.  Dec. 2018  \nKyotoU x SDGs: Comprehensive & \nmultilayer approach. Some KyotoU \ncontributions to SDGs. Yonsei University. \nGlobal Engagement and Empowerment \nForum for Sustainable Development 2019. \nSeoul. Feb 2019 \n \nThematic Partnership: exploring \nsynergies in multilateral \ncollaboration. Joint presentation with \nNational Taiwan University and Hamburg \nUniversity. APAIE 2019. KL. Mar 2019 \n \n6.1.2. International presentations in previous years \n \nInstitutional learning. Do \ninstitutions learn or is it only the \npeople? University Administrators \nWorkshop. Kyoto University. Feb 2017  \nLessons learned on student mobility: \nTAG-AIMS. Trans-ASEAN Global \nAgenda for the AIMS Program at University \nof Tsukuba. JUNThai. Bangkok. Dec 2016 \n6.1.3. Presentations in Kyoto University  \n \nComparative internationalization. \nMichigan State University & Kyoto \nUniversity. Visit from Michigan State \nUniversity Department of Higher Education. \nKyoto University. May 2018 \n \nStudents Comparative discussion on \ninternationalization. 1) student \ndemographics; 2) student characteristics; 3) \nstudent services; and 4) teaching\/learning \nissues of students. Hosting the visit from \nMichigan State University. Kyoto University. \nCPEHE. May 2018.  \nMSU visit to KyotoU. CPEHE. May 2018  \n \n \n29 \n \nFrom MEXT scholarship cocoon to \nreal life careers in Japan. Seminar \non Job Hunting and career development. \nMEXT Scholars Association. Kyoto \nUniversity. Jul 2018   \n \nIntroducing Kyoto University’s \ninternationalization Plan and the \nwork of iSO-KU. Hosting visit delegation \nfrom National University of Malaysia (UKM) \nunder the Multimedia Delegation to Japan. \nKyoto University. Jan 2019 \n \nIntroducing Kyoto University \nCampus-life environment and \nnetworking at CSEAS. Hosting visit from \nNational University of Malaysia (UKM) under \nthe Multimedia Delegation to Japan. Kyoto \nUniversity, iSO-KU. Jan 2019 \n \n6.2. Staff development in Kyoto University \nIntroducing iSO-KU: Strategic \npromotion of internationalization in \nKyoto University. Seminar for newly \nappointed professors organized by the \nCenter for the Promotion of Excellence in \nHigher Education(新任教員教育セミナー\n2018). Kyoto University. September 2018.  \n \n第 2回 国際系業務講習会：本学\nの国際交流業務に関する概要、お\nよび各種手続き等について『国際戦略本部\nの体制と国際化推進について－京都大学が\n世界に輝く知の拠点であり続けるために\n－』Kyoto University. September 2018 \n \n第 3回 国際系業務講習会:本学の\n国際戦略と国際化に向けて『国際\n戦略本部の体制と国際化推進について－京都\n大学が世界に輝く知の拠点であり続けるため\nに－』Kyoto University. September 2018 \n \nDefining University Social \nResponsibility for Kyoto \nUniversity. 1st Workshop on University Social \nResponsibility. Kyoto University \nHeadquarters. March 2019 \n \nEnglish Lunch for Staff. Biweekly \ninformal lunch organized in \nEnglish to discuss issues of common \ninterest with staff from the Planning and \nInformation Department and John Mung \noutbound candidates. Since January 2016. \n \n \n \n30 \n \n7. iSO-KU future plans: Looking ahead \n \nIn academic year \n2018, iSO-KU had \nan active role on \nestablishing and \npromoting the \nrelations to boost \ninternationalization \nof Kyoto University \nand its collaborations for academic \ndevelopment and research.  \nBased on these experiences and \ncontributions, iSO-KU looks into the future \naiming to continue strengthening Kyoto \nUniversity’s internal networks; to create \nstronger synergies; to lead and support the \ncoordination of internationalization \nprograms and activities, to elaborate and \nshare academic materials about good \npractices in internationalization in higher \neducation, as well as to facilitate the \noperation of the University’s Overseas \nCenters and On-site-Labs.  \nIn order to continue accomplishing its \nmission of serving as a support mechanism \nand as a promoter of good practices for \ninternationalization iSO-KU will further \nconsolidate its work in the core functions \ndescribed as “Connect, Boost, Promote, \nand Synergize”, while exploring new \npotential areas to be identified and \ndeveloped in the implementation of Kyoto \nUniversity Basic Concept for \nInternationalization for Academic Year \n2019. \nBased on the current achievements iSO-KU \nplans to focus its efforts as follows:  \n \n7.1 Connect  \n \niSO-KU will continue \nto connect the \nUniversity from \nwithin and with the \nworld, by bridging \noffices and \ndepartments to enhance coordination, \ninformation share and effective decision \nmaking about the global interactions of the \nUniversity.   \na. Planning and coordinating \nheadquarters-led international activities \nb. Coordinating and supporting \ndepartment-led international activities \nc. Research on issues of \ninternationalization \nIn 2018, iSO-KU worked intensely on \n \n \n31 \n \nconnecting internal and external \norganizations consolidating internal \nnetworks leading towards an increased \ncapacity and coordination of headquarters \nand the departments to respond to the \nchallenges of internationalization at the \nUniversity.  \nFor academic year 2019, iSO-KU plans to \nadd more effort for promoting information \nshare, supporting the Overseas Centers, \nthe On-site Labs and supporting other \nheadquarters-led international activities \nand events. Additionally, iSO-KU will keep \non facilitating international liaison with the \ndepartments while attending diplomatic \nresponse. \nA major achievement iSO-KU had in 2018 \nwas the establishment of Kyoto University \nNorth American Center. For 2019 iSO-KU \nwill continue to nurture the relations with \ncurrent and potential partners towards the \nincrease and dynamization of academic \nmobility, promoting the collaboration with \nother departments and offices such as \nILAS, SACI, KURA and more.  \nAnother potential area for development is \nthe reception and exchange of remarkable \nresearchers and academics during their \nsabbatical periods in overseas universities \nand the implementation of fellowships and \nspecial programs for this purpose. \nFurthermore, it is expected that in the \nfollowing months iSO-KU will take lead on \nheadquarters-led tailored short-term \nspecific programs, while supporting \nnecessary development to expand \ninternship opportunities and international \nfieldwork.  \nThese programs will operate in \ncoordination with the International Mobility \nSupport Office, the Overseas Centers as \nwell as the University’s departments. \n \n7.2. Boost  \n \niSO-KU will \ncontinue to foster \nactions to boost \ninternational \nrecognition of Kyoto \nUniversity by \nsupporting policies and measures to \nenhance worldwide visibility; while providing \nsupport towards effective planning and \ncoordination to further internationalize the \neducation provided, the research carried, \nand contributions towards a harmonious \nglobal existence. \na. Strengthening branding and a \nuniversity-wide strategy for international \nvisibility \nb. Strengthening the University’s \ninternational networks \n \n \n32 \n \nc. Assisting the University’s \nheadquarters and departments in their \ndiplomatic responses \nFor academic year 2019, iSO-KU aims to \ncontinue its proactive participation in the \ndevelopment of contents and visual identity \nof Kyoto University in collaboration with \nPublic Relations Office and Web Strategy \nOffice.  \nAn important challenge ahead for iSO-KU \nrelates to developing the contents and \nupdates for the University’s English website, \nconcretely, developing the remaining \nsections and maintaining the Global KyotoU \nsite with updated information.  \nIn order to boost visibility of the activities of \nKyoto University, iSO-KU is considering \nways to enhance access to information of \ninternational activities carried out by \nfaculties, graduate schools, institutes and \nresearch centers, and create an umbrella \nmechanism to share them in the community \nas good practices for internationalization.  \nAnother area under consideration is how to \nextend and deepen more systematic internal \ncooperation among internal organizations \nworking on international matters; promoting \nheadquarters-led initiatives with external \npartners, while boosting the participation \nand closer coordination of the departments \naccording to their education and research \ninterest.  \niSO-KU also aims to support the Alumni \nnetworks to boosting ownership and \nengagement, including foreign alumni in \nJapan and overseas through the activities \ncarried out by the Overseas Centers and the \nExternal Affairs Division. \n \n7.3. Promote  \n \niSO-KU will continue \nto promote the \ninternationalization of \nthe University by \nsupporting the efforts \nof different \nstakeholders in the University towards the \nconsolidation of an increasingly inclusive \ncampus with internationally friendly \nenvironments. \na. Designing, implementing, and \nfollowing-up measures for \ninternationalization \nb. Improving access and distribution of \ninformation throughout the University \nc. Nurturing the University’s \nadministrative staff \nd. Promoting participation and \nnetworking in international activities \ne. Fostering Risk prevention and crisis \nmanagement \n \n \n33 \n \nOne of iSO-KU’s functions is to promote \ninternationalization throughout the different \nenvironments on campus by encouraging \npractices that promote multidisciplinary \neducation and research, while contributing \ntowards a harmonious coexistence of \ncultural diversity. In this regard, a wide \nnumber of potential activities are at sight. \n For example, the organization of workshops \nand seminars in topics related to good \npractices in internationalization of higher \neducation and research with participation of \nother institutions such as local and \ninternational universities, workshops on \ninternationalization in Kyoto University.   \nIn order to deepen the Office’s academic \ncontribution, while continuing with its current \ncommitment to presentations abroad and in \nKyoto University; iSO-KU aims to develop \npublications about best practices in \ninternationalization. iSO-KU also expects to \ngather, consolidate and publicize data about \non-going strategies inside Kyoto University \nto promote replicability of good practices \nand experiences.  \nFor this purpose, the iSO-KU aims to \ndevelop cooperation with the departments to \nachieve a mechanism to systematize \ninformation share about how Kyoto \nUniversity’s practices contribute to the \nenhancement of internationalization of \neducation in Japan.  \nRegarding the design of more inclusive \npolicy-making process within the University, \niSO-KU is working to deepen and \nconsolidate the role of the iSO-KU \nInternational Advisory Board as a formal \nmechanism that connects international staff \nand the policy-making bodies of University.  \nSimilarly, iSO-KU has steered the activities \nof the KUIAN network, which is an informal \nmechanism designed to incorporate the \nperspectives of international faculty, to \nbetter connect headquarters and \ndepartments, by including more staff \nmembers in the policy-making system of \nKyoto University.  \nFor academic year 2019, iSO-KU is \nplanning to organize meetings to integrate \nthe community members based at Katsura \nand Uji campuses to connect with them and \ninclude their opinions on the compilation of \nevidence-based needs in KU towards \npolicy-making and reform. \nIn regards to the English website of the \nUniversity, iSO-KU is in charge of Global \nKyotoU section that provides useful \ninformation for the international members of \nthe community and beyond as a single-entry \npoint to Kyoto University international \naffairs.  \nFor academic year 2019 iSO-KU plans to \ndevelop the remaining sections prepared to \ndisseminate and articulate information in \nEnglish.\n \n \n \n34 \n \n7.4. Synergize  \niSO-KU will \ncontinue to \npromote \nsynergies and \nteamwork of the \nmanagement, the \nadministration, \nand the departments, towards a more \nintegrated and effective \ninternationalization. \na. Strengthening governance to \npromote internationalization by re-\nadjusting administrative offices and \nfunctions \nb. Strengthening cooperation among \nthe University’s management and \nadministrative offices in relation to \ninternational matters \nc. Supporting offices and \ndepartments in the implementation of \nuniversity-wide initiatives for \ninternationalization \nFor academic year 2019 iSO-KU aims to \ncontinue promoting effective \ncommunication and information-share \nthroughout the University as a means to \nmake available resources visible and \nusable, contributing to more effective \ndecision-making processes and \nenhancing ownership on the efforts to be \nmade towards further internationalization.  \nAmong the potential activities iSO-KU is \nconsidering for academic year 2019, the \nfollowing are considered priorities: \ncoordinating headquarters-led short-term \ninbound and outbound programs, \npromoting opportunities like internships \nwith local actors and incorporating the \nparticipation of enterprises in Japan and \nabroad in close cooperation with the \nUniversity’s Overseas Centers.  \nFor this purpose, iSO-KU aims to extend \nthe coordination with the faculties and \nexchange offices to make existing MOUs \nand SEAs more dynamic, and in so doing \nto increase the international exchange \nand cooperation activities.  \n  \n \n \n35 \n \n8. Meet the iSO-KU people \nDirector \nProf. Yasuyuki Kono \nProf. Yasuyuki Kono is \nthe Vice-President for \nInternational Strategy, \nDirector of iSO-KU, Director of European \nCenter, former Director of the Center for \nSoutheast Asian Studies (CSEAS, 2014–\n2018), and professor of CSEAS. He \nsupervises the activities of iSO-KU by \ncoordinating them with departments in the \nuniversity and external stakeholders. \n \nVice-Directors \nSpecially appointed \nProf. Yukari Mitsuhashi \nProf. Mitsuhashi is the \nVice-Director of iSO-KU. \nHer responsibilities include enhancing the \nUniversity's international profile through \nplanning, coordinating, and implementing \nits strategy for internationalization — both \nat home and abroad — and providing \nsupport in the University's diplomatic \nresponse. She is in charge of overseeing \nthe University's international initiatives, \nespecially those related to Europe, Africa, \nthe Middle East, and Americas.  \n \nDr. Junji Yamamoto \nDr. Yamamoto is iSO-\nKU's Administrative \nDirector; he is mainly \nresponsible for international affairs \nmanagement, strategy, and other matters \nrelated to the International Strategy Office. \nHe also is in charge of overseeing the \nUniversity's activities, above all those \nrelated to East Asian and African countries \non the strategic plan. \n \nProgram-Specific Senior Lecturers \nDr. Fernando Palacio \nDr. Palacio is a Program-\nSpecific Senior Lecturer \nat iSO-KU. His main \nresponsibilities relate to the design and \nimplementation of university-wide policy on \ninternationalization, and support for iSO-\nKU's efforts to boost an inclusive \nuniversity. He is in charge of overseeing \nthe University's activities in Southeast Asia. \nHis tasks also relate to the promotion of \nUniversity Social Responsibility. \n \nDr. Paola Sanoni \nDr. Sanoni is a Program-\nSpecific Senior Lecturer \nat iSO-KU. Her main \nresponsibilities relate to the design and \nimplementation of measures for promoting \nthe University's internationalization, \nattending diplomatic response, and \nsupporting activities to promote an \ninclusive university. She is in charge of \ndomestic and international liaison affairs \nand the University's institutional relations in \nthe Americas and Oceania. \n \n \n36 \n \n \nMs. Mariko Adachi \nMs. Adachi is a Program-\nSpecific Senior Lecturer \nat iSO-KU. Her main \nresponsibilities relate to \nthe development of the internationalization \nof Kyoto University in multilateral \ncooperation. She is in charge of promoting \nrisk prevention and management of \ninternational fieldwork, and summarizing \nthe contents of activities at the Faculties \nand Graduate Schools for publication on \nthe web. She also has a role in supporting \nthe oversight of the University's Overseas \nCenters and contributing to the formation \nof an inclusive campus. \n \nDirectors of Kyoto University \nOverseas Centers  \nASEAN Center \nProf. Emeritus Mamoru \nShibayama \nProf. Shibayama’s \nresearch focused on Area \nInformatics in Area \nStudies. He established the ASEAN Center \nin May of 2014 and served as the Center \nDirector since June 2014 to March 2019. \nDuring this period, he contributed to the \nnetworking capabilities of the University in \nthe ASEAN region, boosting its research \nand education outreach as well as the \nattainment of the official NGO Status for \nthe Center in Thailand.  \n \nProf. Eiji NAWATA \nProf. NAWATA is the \nDirector of ASEAN \nCenter since May 2019. \nProfessor of the \nGraduate School of Agriculture, his field is \ntropical agriculture, covering diverse areas \nin agriculture and agricultural environment \nin the tropics, including ecophysiology of \ntropical crops, cropping system analysis, \ncrop evolution. He is committed to work \ntowards the constructive and sustainable \nrelationships with universities and \nacademic institutions in ASEAN region, \nand is doing his best to support various \nactivities of students and colleagues. \n \nNorth American Center \nProf. Nathan Badenoch \nProf Badenoch worked for \n9 years at the Center for \nSoutheast Asian Studies \nbefore assuming the \nposition of North American Center Director. \nHe has spent more than 25 year living in \nAsia, but is now happily resettled on the \nEast Coast of the U.S., where he grew up. \nNathan is a graduate of Graduate School \nof Asian and African Studies. His research \ninterests include documentation of \nendangered languages and knowledge \nsystems, multilingualism and ideophones.  \n \n \n \n37 \n \n \n \n \n \n \n \n \n \n \nInternational Strategy Office \nKyoto University \n(iSO-KU) \n \n \n606-8501 Kyoto University \nYoshida-Honmachi, Sakyo-ku, Kyoto, JAPAN \n \nTel: +81-75-753-5062 \nFax: +81-75-753-2089 \n \nEmail: contact.iso-ku[@]mail2.adm.kyoto-u.ac.jp \nwww.kyoto-u.ac.jp\/en\/global\/international_strategy\/international-strategy \n \n \n \n \n \nAnnual Report - Academic year 2018  \nProduced by International Strategy Office \nJune 2019. Kyoto University. Japan \n"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について キャンパス・施設案内 大学の施設 楽友会館. 楽友会館. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 キャンパスの概要 国内の研究所・附属研究施設等 海外の拠点 On-site Laboratory 大学の施設 百周年時計台記念館 楽友会館 宇治おうばくプラザ 国際科学イノベーション棟 ローム記念館 船井哲良記念講堂・船井交流センター 稲盛財団記念館 京都大学 東京オフィス 京都アカデミアフォーラム 医学部附属病院 白浜水族館 図書館機構（附属図書館を含む） 学術情報メディアセンター・情報環境機構 総合博物館 留学生ラウンジ きずな 国際交流会館 教職員が利用できる施設 福利厚生施設・課外活動施設 写真・動画で見る京都大学 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 　京都大学楽友会館は、1925（大正14）年に京都大学創立25周年を記念して、当時京都大学工学部建築科助教授森田慶一氏の設計により立てられました。鉄筋コンクリート2階建の瓦葺でスパニッシュ・ミッション様式を基調とした外装と東京高等工芸学校教授森谷延雄氏による室内装飾は大正建築の特徴をよく伝えており、1998（平成10）年に国の登録有形文化財に指定されています。. 　2010（平成22）年9月に改装を終え、当時の趣きを残して会議室や喫茶食堂を備えた会館として再生しました。. 　本館は、京都大学の教職員の方々の学術交流の場として使用することができます。. 会議室のご利用について 会議室等予約状況 会館へのアクセス 館内マップ 施設概要 お知らせ. 京都大学楽友会館規程を一部改正しました。（2022年4月1日）. 楽友会館使用申請書の様式を変更しました。（2022年4月1日）. レストラン「近衛Latin」は2021（令和3）年3月27日をもって営業を終了しました。. 利用時間. 火曜日～土曜日 9時00分～17時30分 （会議室利用は9時00分～17時00分）. 休館日は日曜日・月曜日・祝日、12月28日～1月3日、6月18日（創立記念日）、8月第3週の月曜日、火曜日および水曜日（夏期一斉休業日）です。. 連絡先. 京都大学楽友会館事務室 （火曜日～土曜日 9時00分～17時00分）. 〒606-8501 京都市左京区吉田二本松町. Tel： 075-753-7603. Fax： 075-753-2608. E-mail： rakuyuukaikan.office＊gmail.com（＊を@に変えてください）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"8ln2-9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/news\/2022-04-25","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":44,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","news"],"url_file_name_s":"2022-04-25","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["三菱マテリアル株式会社の支援による寄附講座「非鉄製錬学講座」を更新しました | 京都大学"],"description_txt":["　本学と三菱マテリアル株式会社は、2017年4月より、三菱マテリアル株式会社の支援による寄附講座として「非鉄製錬学講座」を設立し、工学研究科材料工学専攻にて開講していましたが、2022年4月以降も第2期として同講座を更新することとなりました。\n\n　2022年4月15日、百周年時計台記念館にて本更新に伴う記者発表を行いました。同発表では、椹木哲夫 工学研究科長による挨拶のあと、伊左治勝義 三菱マテリアル株式会社 執行役常務 金属事業カンパニープレジデントより本講座の主旨と第2期への期待について、本寄附講座の教授を兼任する宇田哲也 工学研究科教授より第1期の活動成果についての概要説明がありました。"],"last_modified":"Wed Jun 08 16:43:46 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-4248254310652031674,"exact_signature_unique_b":true,"fuzzy_signature_l":-8836606142261445675,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["三菱マテリアル株式会社の支援による寄附講座「非鉄製錬学講座」を更新しました"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","関連リンク","Latest news","2023年度 農学研究科修士課程学生募集について","マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて","令和4年6月1日付告示第4号","留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Mon Apr 25 12:00:00 UTC 2022","Fri Jun 03 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022"],"imagescount_i":9,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/220415-news-01-5b8a1c89e4495ba034dddbc2eda3d87c.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/220415-news-02-40f743c2348be64d85e791eabd18e35d.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","文章を入れてください","文章を入れてください","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal sites default files inline 220415 news 01 5b8a1c89e4495ba034dddbc2eda3d87c jpg 5b 8a 1c 89e 4495ba 034dddbc 2eda 3d 87c 文章を入れてください 02 40f743c2348be64d85e791eabd18e35d 40f 743c 2348be 64d 85e 791eabd 18e 35d icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":2402,"text_t":"三菱マテリアル株式会社の支援による寄附講座「非鉄製錬学講座」を更新しました | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 三菱マテリアル株式会社の支援による寄附講座「非鉄製錬学講座」を更新しました. 産官学連携 三菱マテリアル株式会社の支援による寄附講座「非鉄製錬学講座」を更新しました. ターゲット. 企業・研究者の方 公開日. 2022年04月25日. 　本学と三菱マテリアル株式会社は、. 2017. 年. 4. 月より、三菱マテリアル株式会社の支援による寄附講座として「非鉄製錬学講座」を設立し、工学研究科材料工学専攻にて開講していましたが、. 2022. 年. 4. 月以降も第. 2. 期として同講座を更新することとなりました。. 　. 2022. 年. 4. 月. 15. 日、百周年時計台記念館にて本更新に伴う記者発表を行いました。同発表では、椹木哲夫. 工学研究科長による挨拶のあと、伊左治勝義. 三菱マテリアル株式会社. 執行役常務. 金属事業カンパニープレジデントより本講座の主旨と第. 2. 期への期待について、本寄附講座の教授を兼任する宇田哲也. 工学研究科教授より第. 1. 期の活動成果についての概要説明がありました。. 　本寄附講座では、高校生から社会人までの幅広い世代に対する教育活動、ならびに高効率な製錬プロセスやリサイクルプロセスを実現するための基礎研究を実施することで、未来の非鉄製錬業界を牽引・革新する人材を育成するとともに、持続型社会を支える基幹産業のさらなる発展、地球環境の保全に貢献します。. 左から、伊左治執行役常務、椹木研究科長. 非鉄製錬学講座の活動内容. 関連リンク. プレスリリース資料  関連部局. 工学部・工学研究科 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 特色入試 合格者発表・入学手続. 合格者発表・入学手続. サイドメニュー. 一般選抜 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 出願状況と選考結果 よくある質問 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 お知らせ. 現在公開中のものは、令和4年度特色入試の情報です。. このページでは、特色入試（法学部以外）の合格発表と入学手続について記載しています。. ※ 法学部の特色入試（後期日程）については、一般選抜学生募集要項をご確認ください。. 合格者発表. 最終合格者の発表は、「合否照会システム」により行います。. なお、第2次選考結果（教育・理（数理科学入試）・医（人間健康科学科）・工（情報学科）・農学部）については、「合否照会システム」により発表を行います。. ※ 詳細は、令和4年度特色入試学生募集要項の「. 11. 合格者発表 」をご確認ください。. 令和4年度特色入試学生募集要項 入学手続. 入学手続は、「入学手続に必要な書類」をすべて同封し、本ページから印刷した「. 特色入試入学手続書類送付用宛名ラベル 」を貼り付けた市販の封筒により、必ず. 書留速達. で郵送してください。. ※ 詳細は、令和4年度特色入試学生募集要項「. 12. 入学手続 」をご確認ください。.   . 令和4年度特色入試学生募集要項 郵送締め切り日時. 令和4年2月21日（月曜日）17時00分必着. ※ 上記締め切り日時までに手続ができない場合は、合格した学部の教務担当へ締め切り日時までに連絡のうえ、指示に従ってください。. ※ 入学料免除・徴収猶予を申請する場合は、以下のページをご参照ください。. 　. 授業料の免除／入学料の免除と徴収猶予 入学手続に必要な書類. 提出書類. 書式など. 入学料振込金受付証明書. 入学料振込時に入手した原本. ※ 受験票送付時に同封する「振込金受付証明書貼付用台紙」に貼付して提出してください。. 大学入学共通テスト受験票. 原本. 本学特色入試受験票. 原本. 学籍登録シート. 入学者の写真を貼付し、氏名などの修正の有無を記入して提出してください。. ※ 受験票送付時に送付します。. ※ 学籍登録シートに貼付する写真の規格についてもご確認ください。. 特色入試入学手続書類. 送付用宛名ラベル. PDF ※ ここからダウンロードして印刷してください。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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and future plans","History","Facts and Figures","Awards and honors","University profile","Living in Kyoto","Public relations","Employment","Human rights","Health, safety, and environment","Study at KyotoU","Education and admissions","Career paths","Visa\/tuition\/scholarships\/housing","University facilities","FAQ","Research &amp; Collaborate","Research news and departments","Policies","Collaboration","Research support and scholarships","Visa\/Housing\/Daily Life","About","Research compliance &amp; ethics","Global KyotoU","International strategy","Global profile","Global partners","International risk &amp; crisis management","Global University Social Responsibility","KyotoU International Concierge and information resources","Current Students &amp; Staff","Alumni","Directions","Contact","日本語 ホーム","中文简体 首页","中文繁体 首頁","한국어 홈페이지","Key Presidential Policies: Operating as a World-Leading Research Institution","Profile of President Nagahiro Minato","Speeches","List of Kyoto University 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2022","referrer_id_s":"_sUDfCU8j7Mo","language_s":"en","size_i":49887,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":3,"references_internal_i":3,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085666539143168,"text_t":"Message from the President | KYOTO UNIVERSITY. Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home About Message from the President. Message from the President. サイドメニュー（英語）. Message from the President Key Presidential Policies: Operating as a World-Leading Research Institution Profile of President Nagahiro Minato Speeches List of Kyoto University Presidents Operations and future plans History Facts and Figures Awards and honors University profile Living in Kyoto Public relations Employment Human rights Gender equality Health, safety, and environment In other languages. 日本語 Kyoto University was established in 1897 as Japan's second imperial university. At that time, the role of science and technology in society was rapidly expanding, and momentum was being gained by the idea that universities should not only be responsible for education, but should also pursue research. Since its establishment, Kyoto University has pursued its core mission of contributing to harmonious coexistence within the global community by promoting original and creative research, creating new intellectual values, and fostering capable human resources who can generate such new values. In its pursuit of that mission, the University has made significant achievements as one of Japan's leading research-oriented universities. Those achievements are testified by the fact that Kyoto University has produced more Nobel Prize- and Fields Medal-winning research than any other university in Asia. The University has also been exploring new avenues of development since its appointment by the Japanese government as a Designated National University in 2017. Kyoto University's educational undertaking is significant, involving approximately 5,500 faculty and staff members and 23,000 undergraduate and graduate students. The University has a tradition of providing practical education that actively involves students in creative research, promoting a unique form of \"self-reliant learning\". Students are encouraged to independently identify and address issues to foster a broad range of abilities and a creative spirit founded on robust scientific knowledge. Therefore, all students enrolled at the University are required to take liberal and general education courses provided by the Institute for Liberal Arts and Sciences (ILAS) regardless of their future field of specialization, an approach which promotes their development as healthy and intelligent citizens. Creativity is born from intellectual curiosity and inquisitiveness, and I hope that the University's tradition of \"teaching through research\" will foster a high degree of expertise in our students, and that they will become the next generation of creative researchers. Currently, multiple complex issues threaten the health and lives of all people on earth. Such issues include the unexpectedly rapid worsening of climate change, large-scale natural disasters, and environmental degradation, as well as the escalation of international conflicts, widening disparity, and the international spread of infectious diseases, such as the novel coronavirus. We must now reaffirm our commitment to Kyoto University's mission — we must earnestly and determinedly seek solutions to the multifaceted and complex issues being faced by global society, and consistently share our progress with society. 2022 will mark the 125th anniversary of Kyoto University's founding. Throughout its history the University's pioneering spirit of creativity and diversity has borne much fruit, but to maintain the production of such nourishing fruit in the future, we must plant seedlings and cultivate them properly. By firmly linking the University's chain of achievements throughout the past 125 years to its future, we will continue to make the utmost efforts to contribute to society through the creation of new intellectual value and the cultivation of outstanding human resources. October 2020. Nagahiro Minato. President. Kyoto University. Profile of President Nagahiro Minato List of Kyoto University Presidents 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 研究・産官学連携 研究者が受けられる研究費・研究支援 研究資金・賞に関する公募等 京都大学たちばな賞（優秀女性研究者賞）. 京都大学たちばな賞（優秀女性研究者賞）. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 総長賞（すべての学生個人・団体） たちばな賞（若手女性研究者・学生） 久能賞（理系の女子学生） SPEC（京大生チャレンジコンテスト） 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 京都大学たちばな賞（優秀女性研究者賞）. 本賞は、京都大学における若手の女性研究者の優れた成果を讃える制度として、人文・社会科学または自然科学の分野において、優れた研究成果を挙げた若手の女性研究者を顕彰し、当該若手女性研究者およびこれに続く若手女性研究者の研究意欲を高め、もって、本学、さらには我が国の学術研究の将来を担う優れた女性研究者の育成等に資することを目的に、平成20年度に創設しました。. 令和3年度. 第14回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催しました（2022年3月3日） 第14回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式の開催について 第14回京都大学たちばな賞（優秀女性研究者賞）の募集について 令和2年度. 第13回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催しました。（2021年3月3日） 第13回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式について 第13回京都大学たちばな賞（優秀女性研究者賞）の募集について 令和元年度. 第12回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催しました。（2020年3月3日） 第12回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式について 第12回京都大学たちばな賞（優秀女性研究者賞）の募集について 平成30年度. 第11回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催しました。（2019年3月4日） 第11回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式について 第11回京都大学たちばな賞（優秀女性研究者賞）の募集について 平成29年度. 第10回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催しました。（2018年3月2日） 第10回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式について 第10回京都大学たちばな賞（優秀女性研究者賞）の募集について 平成28年度. 第9回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催 第9回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式について 第9回京都大学たちばな賞（優秀女性研究者賞）の募集について 平成27年度. 第8回京都大学たちばな賞 （優秀女性研究者賞）の受賞者決定および表彰式について 第8回京都大学たちばな賞（優秀女性研究者賞）の募集について 平成26年度. 第7回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催 第7回京都大学たちばな賞 （優秀女性研究者賞）の受賞者決定および表彰式について 第7回京都大学たちばな賞（優秀女性研究者賞）の募集について 平成25年度. 第6回京都大学たちばな賞（優秀女性研究者賞）表彰式を開催 第6回京都大学たちばな賞（優秀女性研究者賞）の受賞者決定および表彰式について 第6回京都大学たちばな賞（優秀女性研究者賞）の募集について 平成24年度. 第5回京都大学優秀女性研究者賞（たちばな賞）の表彰式を開催 第5回京都大学優秀女性研究者賞（たちばな賞）の受賞者決定および表彰式について 第5回京都大学優秀女性研究者賞（たちばな賞）の募集について 平成23年度. 第4回京都大学優秀女性研究者賞（たちばな賞）の表彰式を開催 第4回京都大学優秀女性研究者賞（たちばな賞）の受賞者決定および表彰式について 第4回京都大学優秀女性研究者賞（たちばな賞）の募集について 平成22年度. 第3回京都大学優秀女性研究者賞（たちばな賞）の表彰式を開催 第3回京都大学優秀女性研究者賞（たちばな賞）の受賞者決定および表彰式について 第3回京都大学優秀女性研究者賞（たちばな賞）の募集について 平成21年度. 第2回京都大学優秀女性研究者賞（たちばな賞）の表彰式を開催 第2回京都大学優秀女性研究者賞（たちばな賞）の受賞者決定および表彰式について 第2回京都大学優秀女性研究者賞（たちばな賞）の募集について 平成20年度. 第1回京都大学優秀女性研究者賞（たちばな賞）の表彰式を開催 第1回京都大学優秀女性研究者賞（たちばな賞）の表彰式および研究発表について 第1回京都大学優秀女性研究者賞（たちばな賞）の受賞者決定について 第1回京都大学優秀女性研究者賞（たちばな賞）の募集について 女性研究者支援モデル育成事業. 京都大学では、平成18年度の科学技術振興調整費：女性研究者支援モデル育成事業「女性研究者の包括的支援『京都大学モデル』」の採択を受けて、平成18年9月に「女性研究者支援センター」を設立しました。本事業では、女性研究者の抱える問題を解決し研究環境を改善することにより、更に卓越した女性研究者を生み出していく包括的な支援体制を整備すべく、10個のプログラムを実施してきました。平成20年度にて科学技術振興調整費による支援は終了しましたが、大学経費においてそれぞれの支援事業を推進しています。. 京都大学男女共同参画推進センターホームページ（外部リンク） 女性研究者養成システム改革加速事業. 本事業は、多様な人材の養成・確保および男女共同参画の推進の観点から、特に女性研究者の採用割合等が低い分野である、理学系、工学系、農学系の研究を行う優れた女性研究者の養成を加速する必要性に基づきスタートした、文部科学省科学技術振興調整費事業の一つです。本事業を実施し、機関におけるシステム改革に効果的な分野・規模で当該女性研究者の採用を行うことにより、人材の多様化、研究の活性化および男女共同参画意識の醸成、さらには、機関として本来取り組まなければならない柔軟な組織編成や環境整備等を同時に促進し、総合的なシステム改革の構築を目指すものです。. 平成21年度に、京都大学が提案した「京大式女性研究者養成コーディネートプラン」を含む5課題が採択され、本学では以下の支援を実施しました。. なお、科学技術振興調整費による支援は平成23年度をもって終了しました。. 概要. 京都大学は、世界でも有数の知の拠点であり、多様な人材の育成という観点からも、女性研究者の活躍促進は喫緊の課題です。それを踏まえ本課題では、女性研究者の採用・育成のコーディネート等を担う「女性研究者養成システム改革推進室」の設置を皮切りに、新規養成女性研究者へのワークライフバランス等に関するソフトな支援から、研究活動に根ざした段階的支援を実施し、育成を図ります。本システム改革自身は、女性研究者の参画促進等の広告塔となることで、学内の意識啓発や次世代を担う女性研究者の裾野開拓・拡大、その結果、更なる女性研究者の輩出という一連の好連鎖を生み出し、アカデミア全体の知力の底上げに寄与することが期待されます。. 実施体制（PDF）（93KB） 実施内容（PDF）（271KB） 女性研究者養成システム改革推進室. 京都大学における「女性研究者養成システム改革加速プログラム」を円滑に実施し、これに関連する部局において女性研究者の養成に関し、連絡調整およびその推進に係る企画立案等を行うため、研究担当理事の下に、女性研究者養成システム改革推進室（以下、推進室という。）を設置しました。. また推進室に、女性研究者養成の推進等に関し必要な事項を審議するため、女性研究者養成システム改革推進委員会を設け、さらに推進委員会に理学系・工学系・農学系ワーキンググループを置いて、事業の推進に取り組みました。. 段階的研究支援. 平成23年度 研究スキルアップ経費の募集について（学内限定） 平成22年度 研究スキルアップ経費の募集について（学内限定） 平成21年度 研究スキルアップ経費の募集について（学内限定） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 最新の研究成果を知る. 最新の研究成果を知る. 研究成果. 公開日. 2022年04月13日. スズを含むペロブスカイト太陽電池：23.6%の世界最高効率を達成―ペロブスカイト薄膜の上下表面構造修飾法を開発―. 研究成果. 公開日. 2022年04月13日. 代謝を通じて免疫応答を制御する新たなタンパク質の発見―マクロファージに発現するCyclin Jを介した新たながんや感染制御機序―. 研究成果. 公開日. 2022年04月13日. ヒトは小さく生まれて大きく育つ―その秘密は鎖骨にあり―. 研究成果. 公開日. 2022年04月12日. 子どもの健康と環境に関する全国調査（エコチル調査）＜妊婦の血中重金属濃度と生まれた子どもの口唇口蓋裂との関連について―エコチル調査＞. 研究成果. 公開日. 2022年04月11日. 遺伝暗号開始コドンの修飾がmRNAの性能を変える 〜修飾塩基によりmRNAからタンパク質の作られやすさが変わる新しいメカニズムを発見！〜. 研究成果. 公開日. 2022年04月08日. 深い海の底からアシタラズのウミナナフシの新種を発見. 研究成果. 公開日. 2022年04月07日. ヒトiPS細胞から作製した膵島様細胞に混入する目的外細胞の特性評価にもとづく除去. 研究成果. 公開日. 2022年04月07日. みんな電力との共同研究「電力カラーリング」で特許を取得ー世界初の試みー. 研究成果. 公開日. 2022年04月07日. 京大式サイエンスの創り方～狙ってもできないことがある～ ー京都大学理学研究科が取り組んだ型破りな教育プログラムの活動報告ー. 研究成果. 公開日. 2022年04月04日. 珪藻の光化学系II-集光性色素タンパク質超分子複合体の立体構造解明 －集光性色素タンパク質の進化を紐解く糸口に－. 研究成果. 公開日. 2022年03月31日. 小笠原諸島の固有植物の起源を解明 －ヤロード属の2種は異なる祖先種から種分化した－. 研究成果. 公開日. 2022年03月30日. ヒトiPS細胞からの視細胞直接誘導法 －視細胞を迅速かつ簡便に分化誘導することが可能に－. 研究成果. 公開日. 2022年03月30日. 植物の枝分かれ調節ホルモンの合成メカニズムを解明 －植物ホルモンを活性化する酵素タンパク質を発見－. 研究成果. 公開日. 2022年03月30日. 産官学民コンソーシアム“PEGASAS”が考えるHealthy Smart Cityの構造モデル －「健康・医療・介護」の視点から超少子･超高齢社会の新しいあり方を提案－. 研究成果. 公開日. 2022年03月28日. 細胞が作り出すモザイクパターンのパズル －その仕組みを生物学と数学の連携で読み解く－. 研究成果. 公開日. 2022年03月28日. 歴史的観測から蘇る1957～1958年のオーロラ観測の全貌：過去4世紀最大の太陽活動極大期に起きた太陽嵐の痕跡. ページ送り. 先頭ページ. chevron_left. chevron_left. 前ページ. chevron_left. ページ. 1 ページ. 2 ページ. 3 ページ. 4 カレントページ. 5 ページ. 6 ページ. 7 ページ. 8 ページ. 9 &hellip;. 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 法人としての取組 障害者雇用促進への取組. 障害者雇用促進への取組. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 新型コロナウイルス感染症への対応 熊本地震への対応 環境への取組 東日本大震災への対応 人権 障害に関すること 障害者雇用促進への取組 京都大学業務支援室 安全への取組 情報環境の整備 公益通報 監査 その他 男女共同参画について 公表事項 情報公開 3階層目以降ページ用メニュー. 京都大学業務支援室. WINDOW構想（Diverse and Dynamic）におけるダイバーシティ推進のもと、障害者雇用の促進の一環として2018年4月1日に設置されました。業務を特定せず、学内のあらゆる部局の教職員から、さまざまな依頼に対応しています。 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 学生支援の窓口・証明書発行など 入学料・授業料などの納付. 入学料・授業料などの納付. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援窓口のご案内 入学料・授業料などの納付 学納金の額一覧 各種証明書や学生証などの発行 学生関係諸規程一覧 事例別相談窓口一覧 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. 入学料・授業料納付方法 その他の入学時納付金 授業料滞納による除籍等 問い合わせ 入学料・授業料納付方法. 入学料・授業料の納付方法は次のとおりです。. なお、授業料などの金額は、「. 授業料等の額について 」に別途掲載されていますのでご確認ください。. 1.入学料. 本学所定の振込依頼書により、本学指定の銀行口座にお振り込みください。. 2.授業料（学部学生・大学院生）. 原則として、口座振替により納入いただきます（手数料は不要です）。. 振替日. 前期：. 5月27日. 後期：. 11月27日. （ただし、振替日が土曜・日曜・祝日の場合は、翌営業日となります。振替日の前日までに入金くださいますようお願いします。）. 口座振替に関する申し込み（開始および変更）は、以下のインターネット口座振替申込ページから行ってください。. （印鑑不要、夜間・休日も手続き可能です。）. 一旦登録した口座振替の登録口座を変更される場合も、再度申し込みいただくことで変更が可能です。. ※. ボタンを押すと申し込みページに移動します。. セキュリティ強化のため、 Windows XP SP2以前のOSはご利用いただけません。. また、京都大学内の共有PC、無線LANネットワークおよび一部の携帯端末からはご利用いただけない場合があります。. 申し込み期限. 前期開始：. 4月末日. 後期開始：. 10月末日. インターネットによる申し込みが困難な場合等、書面による申し込みを希望される場合は、所属学部等の教務担当掛または財務部経理課資金管理掛にて口座振替の申し込み用紙をお渡しします。. 書面での申し込み期限. 前期開始：. 3月末日. 後期開始：. 9月末日. 口座振替の事前通知について. 口座振替の事前通知は、メールでのみ行います。口座振替実施前に、KULASIS. ※. 上の「授業料に関する書類の送付先および連絡先の設定」で選択された連絡先メールアドレスへご連絡します。. ※KULASIS（クラシスー京都大学教務情報システム）. 連絡先メールアドレスの変更は、KULASISにより変更の手続きを行ってください。. 連絡先メールアドレスを届け出されていない方には通知されませんので、ご注意ください。. その他. 口座振替のお申し込みが間に合わないなどの事情により、口座振替による納入ができない学生にのみ、振替依頼書を前期分は5月20日頃、後期分は11月20日頃に送付します。（送付先はKULASIS上の「授業料に関する書類の送付先および連絡先の設定」で選択された住所です。）. 授業料の納入は原則、口座振替となります。上記の期限に間に合うよう、申し込みを行ってください。. 振り込みにあたっての注意事項. 学納金の振り込みは、必ず本学所定の振込依頼書を使用し、金融機関の窓口での現金振り込みにより行ってください。. 入金の確認が困難になるおそれがありますので、ATM・インターネット等からの振り込みは行わないでください。. やむをえず本学発行の振込依頼書を利用できない場合は、事前に下記. 問い合わせ先 へご相談ください。. 3.授業料（研究生・聴講生・科目等履修生・研修員）. 本学所定の振込依頼書により、本学指定の銀行口座にお振り込みください。. その他の入学時納付金. 大学内の団体の会員または組合員となるために必要な諸会費として、次のようなものがあります。. 1.学生教育研究災害傷害保険（学研災）・学研災付帯賠償責任保険. 保険料等については「. 学生教育研究災害傷害保険 」をご覧ください。. 2. 大学生活協同組合費出資金. 20,000円以上。出資金は卒業の際に返還されます。. 3. 体育会費. 医学部以外の学部学生：. 10,000円（入会費 2,000円、会費4年分 8,000円）. 医学部学生：. 14,000円（入会費 2,000円、会費6年分 12,000円）. なお、学部によっては上記のほかに必要な会費等があります。新入生にはこれらについて別途所属学部から通知されます。. 授業料滞納者の取扱いについて. 1．概要. 平成29年4月1日から授業料を滞納. *1. している学生（以下「授業料滞納者」という。）の取扱いが次のとおり変更となりました。. 授業料滞納者の除籍に係る期間を4期から2期とする。（経過措置あり. *2. ）. 授業料滞納者の休学および授業料免除の申請を認める。. これまで、授業料滞納者については、休学や授業料免除の申請が認められておらず、授業料を4期滞納することで除籍となっていましたが、今回の変更により、滞納による除籍が抑制されるとともに、滞納額の累積が抑えられることで除籍となった場合でも従前より再入学. *3 . しやすくなりました。詳細は、以下をご覧ください。. 授業料滞納とは、授業料の一部または全部を納入期限（前期分は5月中、後期分は11月中）までに納付せず（未納状態）、その期の末日（前期は9月30日、後期は3月31日）を迎えることをいいます。. 平成28年度から引き続いて授業料の滞納がある場合は従前通り4期滞納で除籍となります。. 授業料滞納により除籍された者が、滞納した授業料を納付後、再入学を願い出た時は、除籍された日から3年以内に限り、再入学を許可することがあります。. 2．授業料滞納者の取扱いに関するQ＆A. Q1．. 今回、授業料滞納者の取扱いが変更された趣旨を教えてください。 Q2．. 授業料2期滞納で除籍となるのは、平成29年度以降の新入生のみでしょうか。 Q3．. 平成28年度後期分1期のみを滞納している者が、平成29年度前期分を未納状態のまま、平成29年度前期終了前に平成28年度後期分を納付した場合、4期滞納または2期滞納いずれにより除籍となるのでしょうか。 Q4．. 平成29年度前期分の授業料を滞納し、同年後期分の授業料も納付しないまま11月1日から休学に入った場合の取り扱いはどうなりますか。 Q5．. 授業料1期滞納後、2期目の中途で休学した場合、2期目の終了日までに滞納している授業料の少ない方の期（2期目）の分を納付することは可能でしょうか。 Q6．. 授業料滞納者でも休学申請および授業料免除申請ができるようになったことについて、これは平成29年前期分以降の滞納者のみが対象となるのでしょうか。 Q7．. 休学している者が翌期初日から復学する場合の期にかかる授業料免除の申請を休学中にすることは可能でしょうか。 Q8．. 授業料滞納により除籍された学生が再入学した場合、在学中ではあるが授業料を滞納していた期間に取得した単位は再入学後にどのように取り扱われますか。 3．授業料滞納者の休学および授業料免除に関する取扱い. 授業料を滞納している方でも、. 平成29年4月から「. 授業料免除 」および「. 休学 」. の申請が出来るようになりました。. 授業料免除について. 学部および大学院正規生で、経済的に困難な状況にあり、学業成績が優秀な学生に対しては、授業料の全額または半額を免除する制度があります。詳細は「. 授業料免除 」をご覧ください。. 休学について. 休学期間の授業料は、前期または後期の初日から休学する場合にはその期の授業料は免除されます。期の途中からの休学にかかる授業料や休学の詳細については「. 休学 」を参照してください。. 授業料滞納者の取扱いに関するQ＆A. Q1．. 今回、授業料滞納者の取扱いが変更された趣旨を教えてください。. A1．. これまで、授業料滞納による除籍までの期間を4期としており、授業料を滞納している学生には休学および授業料免除の申請を認めていませんでした。そのため、授業料の滞納が経済的困窮を理由とする場合、家計が好転しなければ滞納の解消が困難であり、結果として、除籍に至るケースが少なからずありました。. また、授業料滞納を事由とする除籍の場合、除籍後3年以内に滞納した授業料を支払えば再入学できますが、この取扱いを決定した昭和30年当時の授業料は6,000円であったものが、現在は535,800円まで上昇しており、再入学に際して100万円を超える額の支払いを要することになります。そのため、現行の授業料水準のもとでは、4期という条件は再入学を困難にしていると考えられます。. これらのことから、授業料を納付することが困難である学生に係る授業料債務の累積を抑止するとともに、万が一除籍となる場合にあっても再入学を困難とする要因を緩和するよう取扱いを変更することとなりました。. Q2．. 授業料2期滞納で除籍となるのは、平成29年度以降の新入生のみでしょうか。. A2．. 授業料2期滞納で除籍となるのは入学年度に関係なく、平成29年度以降に新たに授業料を滞納した方が対象となります。. 平成28年度から引き続いて授業料の滞納がある場合は当該滞納分を完納するまでは従前通り4期滞納で除籍となります。また、平成29年度以降に平成28年度以前の期から継続する授業料滞納分が完済した後に新たに授業料を滞納した場合は、授業料2期滞納で除籍となります。. Q3．. 平成28年度後期分1期のみを滞納している者が、平成29年度前期分を未納状態のまま、平成29年度前期終了前に平成28年度後期分を納付した場合、4期滞納または2期滞納いずれにより除籍となるのでしょうか。. A3．. 2期滞納で除籍となります。. 平成28年度以前から引き続く滞納分が全て完済された場合、完済した後に新たに滞納した授業料は新規扱いとなり2期滞納で除籍となります。. また、平成28年度以前から引き続く滞納分が完済されていない場合は、4期滞納で除籍となります。（以下参考例）. Q4．. 平成29年度前期分の授業料を滞納し、同年後期分の授業料も納付しないまま11月1日から休学に入った場合の取り扱いはどうなりますか。. A4．. この場合、11月分から授業料が免除されることとなります. ※. が、10月分のひと月分の滞納でもその期の授業料を滞納したことになるので、後期の末日（3月31日）までに当該授業料が納付されなければ、前期分と併せて2期滞納していることになり、除籍となります。. ※前期または後期の初日からその期の末まで休学する場合にはその期の授業料は免除されます。期の途中（授業料納入期限後）から休学する場合には、その期の授業料を全額納付しなければなりません。ただし、4月2日～5月1日または10月2日～11月1日を始期とする休学をする場合は、1ヶ月分の授業料となります。（残り5ヶ月分の授業料は免除されます）. Q5．. 授業料1期滞納後、2期目の中途で休学した場合、2期目の終了日までに滞納している授業料の少ない方の期（2期目）の分を納付することは可能でしょうか。. A5．. 可能です。. 通常であれば、古い滞納分から充当されることとなりますが、滞納額の少ない方の期（2期目）の分の授業料を納付することが可能です。. ※期の途中からの休学にかかる授業料はQ4を参照してください。. Q6．. 授業料滞納者でも休学申請および授業料免除申請ができるようになったことについて、これは平成29年前期分以降の滞納者のみが対象となるのでしょうか。. A6．. 平成28年度から引き続いて滞納している者も申請可能です。滞納の始期がいつであるかに関わらず、滞納者は平成29年4月以降からの休学または授業料免除の申請が可能です。. Q7．. 休学している者が翌期初日から復学する場合の期にかかる授業料免除の申請を休学中にすることは可能でしょうか。. A7．. 可能です。. ただし、申請期間中に所定の手続きを行う必要があります。詳細は「. 授業料免除 」のページをご覧ください。. Q8．. 授業料滞納により除籍された学生が再入学した場合、在学中ではあるが授業料 を滞納していた期間に修得した単位は再入学後にどのように取り扱われますか。. A8．. 再入学後は、滞納期間中に修得した単位であっても、修得したものとして取り扱われます。なお、滞納による除籍後から滞納した授業料を納付して再入学するまでの間は、滞納期間中に修得した単位は、修得単位として証明されませんのでご注意ください。. 問い合わせ. 入学料・授業料納付方法について. 京都市左京区吉田本町. 京都大学財務部経理課資金管理掛. Tel：  075-753-2141（直通）. E-mail： 820gakunoukin＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. その他の入学時納付金. 学生教育研究災害傷害保険（学研災）・学研災付帯賠償責任保険. 京都市左京区吉田本町. 京都大学教育推進・学生支援部厚生課. Tel：  075-753-2539（直通）. 大学生活協同組合費出資金. 京都大学生活協同組合 体育会費. 京都大学体育館事務室. Tel：  075-753-2574（直通）. 授業料滞納による除籍等について. 休学等に関すること. 各学部・研究科等の教務担当窓口 授業料免除に関すること. 京都市左京区吉田本町. 京都大学教育推進・学生支援部学生課奨学掛. Tel：  075-753-2532（直通）. E-mail： 840menjo＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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Jun 08 16:43:45 UTC 2022","fresh_date_dt":"Wed Aug 10 22:32:16 UTC 2022","referrer_id_s":"z3uZu9U8j7Mo","publisher_t":"Adobe PDF Library 17.11.238","language_s":"ja","size_i":291649,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085654300164096,"text_t":"1 \n学生・教職員の新型コロナウイルス感染確認時等における対応について（第８版） \n（R2.4.22 一部改正） \n（R2.8.7 一部改正） \n（R2.9.30 一部改正） \n（R2.11.13 一部改正） \n（R3.2.22 一部改正） \n（R3.9.7 一部改正） \n（R4.1.18 一部改正） \n（R4.2.2 一部改正） \n \n（１）学生・教職員の感染が確認された場合 \n \n感染した者が行うこと \n①速やかに所属部局に報告 \n②保健所の指示に従い、入院又は宿泊施設もしくは自宅にて療養（可能な範囲で体\n調の経過観察表を記載） \n③退院又は宿泊（自宅）療養解除時に受ける主治医や保健所の指示・指導に従い、そ\nの内容を所属部局に報告 \n※退院等後、登校禁止、就業禁止が解除されるまでの間は、引き続き、体調の経過観\n察を行う \n※無症状患者の場合、検体採取日から１０日間の体調の経過観察を行う \n \n該当部局が行うこと \n①速やかに感染者の情報を危機対策本部へ報告、なお、当該感染者が教職員の場合\nは、就業の禁止を決定・通知 \n②保健所及び危機対策本部の指示により、活動・業務を停止し、関係者へ自宅待機\nの指示 \n③保健所及び危機対策本部と連携し、感染者が発病する２日前以降の行動及び濃厚\n接触した者を特定 \n④保健所の指導のもと、危機対策本部と連携し、感染者の勤務等していた建物や部\n屋などの消毒作業（必要な場合は一時閉鎖） \n⑤危機対策本部の指示により、活動・業務の再開（感染者が担当していた業務等の\n支援体制の整備） \n⑥感染者より、退院又は宿泊（自宅）療養解除時に受けた主治医や保健所からの指\n示・指導の内容や職場復帰の目安（※）に基づき、当該感染者が学生の場合は登校禁\n止の解除を決定し、併せて危機対策本部に報告、教職員の場合は、主治医や保健所\nからの指示・指導の内容及び職場復帰の目安に基づき就業の禁止の解除を決定・通\n知のうえ、人事部人事企画課に報告 \n（※）職場復帰の目安は、次の１）及び２）の両方の条件を満たすこと。無症状患者\nの場合は、次の３）の条件を満たすこと \n１）発症後に少なくとも１０日が経過している \n２）薬剤(*) を服用していない状態で、解熱後に少なくとも７２時間が経過してお\nり、発熱以外の症状(**) が改善傾向(***) である \n (*) 解熱剤を含む症状を緩和させる薬剤 \n (**) 咳・咽頭痛・息切れ・全身倦怠感・下痢 \n (***) 各症状を４段階（なし：新型コロナウイルス感染症罹患による症状出現前と同程度、\n2 \n軽度：何かに集中すると忘れる程度、中等度：常に不良を感じる程度、重度：日常\n生活に支障をきたす程度）で評価し、すべて「なし」または「軽度」の状態が３日\n連続している \n３）無症状患者の場合は、検体採取日から７日が経過している \n \n危機対策本部が行うこと \n①保健所より感染者情報を入手した場合、速やかに該当部局に報告 \n②感染者の発生状況について、京大病院クラスター班に報告 \n③必要に応じ、京大病院クラスター班と連携し、以下の初動対応を実施 \n１）保健所の指導のもと、該当部局の活動・業務の停止を指示 \n２）保健所及び該当部局と連携し、濃厚接触者の特定 \n３）保健所の指導のもと、該当部局と連携し消毒作業 \n④保健所の指導のもと、該当部局の活動・業務の再開の指示 \n⑤学内外への広報 \n⑥業務等の支援体制への協力 \n \n（２）学生・教職員が濃厚接触者となった場合 \n \n濃厚接触者となった者が行うこと \n①速やかに所属部局に報告 \n②保健所及び本学の指示・指導に従い、感染者と接触した最後の日から７日間は自\n宅待機（在宅勤務、自宅学習など）及び１０日間は体調の経過観察 \n※当該期間については、自治体の変異株の流行状況等によって取り扱いが異なる場\n合があるため、各自治体保健所の指示・指導に従う。また、医療従事者については、\n別に定めることが出来る（以下同じ） \n③発熱・咳等の症状が出れば、医療機関には直接行かず、相談窓口又は管轄の保健\n所に相談 \n④相談窓口又は管轄の保健所から指定された医療機関を受診し、新型コロナウイル\nス陽性と診断された場合は、上記（１）により対応 \n⑤７日間の体調の経過観察後、体調に問題が無ければ、所属部局へ経過観察の結果\nを報告。引き続き１０日間を経過するまで体調の経過観察を行う \n※濃厚接触者と判断されなかった場合、発熱・咳等の症状がなければ、登校、就業は\n可能 \nただし、必ず７日間は体調の経過観察を行い、所属部局へ経過観察の結果を報告 \n \n該当部局が行うこと \n①速やかに濃厚接触者の情報を危機対策本部へ報告 \n②保健所及び危機対策本部の指示・指導のもと、濃厚接触した者に対して、感染者\nと接触した最後の日から７日間の自宅待機（在宅勤務、自宅学習など）及び１０日\n間の体調の経過観察を指示 \n③濃厚接触者が担当していた業務等の支援体制の整備 \n④報告を受けた７日間の体調の経過観察の結果を危機対策本部に報告後、自宅待機\n（在宅勤務、自宅学習など）の解除を指示。引き続き１０日間を経過するまで体調\nの経過観察を指示 \n⑤濃厚接触者が、医療機関を受診し新型コロナウイルス陽性と診断された場合は、\n上記（１）により対応 \n3 \n※濃厚接触者と判断されなかった者より報告のあった経過観察の結果を危機対策本\n部へ報告 \n \n危機対策本部が行うこと \n①保健所より濃厚接触者情報を入手した場合は、速やかに該当部局に報告 \n②保健所の指導のもと、該当部局へ濃厚接触した者に対して、感染者と接触した最\n後の日から７日間の自宅待機（在宅勤務、自宅学習など）及び１０日間の体調の経過観\n察を指示 \n③業務等の支援体制への協力 \n \n（３）同居者が濃厚接触者となった場合 \n \n学生・教職員が行うこと \n①速やかに状況を所属部局に報告 \n②学生・教職員は、登校や出勤に制限を設けないが、同居者に症状が出た場合は、自\n宅待機（在宅勤務、自宅学習など）。登校や出勤時には必ずマスクの着用、手洗い、\n密の回避等感染予防対策の徹底を図る \n③同居者がＰＣＲ検査又は抗原検査を受け、「陽性」となった場合、所属部局に状況\nを報告し、保健所及び危機対策本部等の指示・指導のもと、上記（２）により対応 \n \n（厚生労働省：家庭内でご注意いただきたいこと～８つのポイント～） \nhttps:\/\/www.mhlw.go.jp\/content\/10900000\/000601721.pdf \n（日本環境感染学会：新型コロナウイルスの感染が疑われる人がいる場合の家庭内での注\n意事項） \nhttps:\/\/www.mhlw.go.jp\/stf\/seisakunitsuite\/newpage_00009.html \n \n該当部局が行うこと \n①速やかに状況を危機対策本部へ報告 \n②同居している濃厚接触者が、医療機関を受診し新型コロナウイルス陽性と診断さ\nれた場合は、上記（２）により対応 \n \n危機対策本部が行うこと \n①該当者に関する情報を部局と共有 \n②必要に応じ、保健所等との連絡調整 \n③濃厚接触者となった場合は、上記（２）により対応 \n \n（４）学生・教職員が新型コロナウイルス感染類似症状を呈した場合 \n※類似症状とは、咳・喉頭痛・息切れ・全身倦怠感・下痢・高熱などを指す。また、それ以外の疾患の確定\n診断が付いている場合は除外する。 \n \n学生・教職員が行うこと \n①類似症状を呈した場合は、医療機関又は保健所に相談（ＰＣＲ検査等を受けるこ\nとも含め）のうえ、自宅待機（業務命令、自宅学習など）及び体調の経過観察 \n※京都府内の場合は、まず、かかりつけ医に相談（休日、夜間、かかりつけ医がい\nない場合などは、「きょうと新型コロナ医療相談センター」（075-414-5487）に相談） \n②速やかに状況を所属部局に報告 \n4 \n※ＰＣＲ検査を受けることとなった場合にも所属部局に報告 \n※新型コロナウイルス陽性と診断された場合は、上記（１）により対応 \n③以下２点の両方を満たした場合、自宅待機は解除 \n・発症後、８日以上が経過している（起算日：発症日を０日目） \n・ 薬剤(a) を服用していない状態で、解熱後に７２時間以上が経過しており、発熱\n以外の症状(b) が改善傾向である（起算日：解熱日・症状消失日を０日目） \n(a) 解熱剤を含む症状を緩和させる薬剤 \n(b) 咳・咽頭痛・息切れ・全身倦怠感・下痢 \n④自宅待機の解除後１週間は、体調の経過観察を行う \n \n該当部局が行うこと \n①以下２点の両方を満たした場合、自宅待機の解除 \n・発症後、８日以上が経過している（起算日：発症日を０日目） \n・ 薬剤(a) を服用していない状態で、解熱後に７２時間以上が経過しており、発熱\n以外の症状(b) が改善傾向である（起算日：解熱日・症状消失日を０日目） \n(a) 解熱剤を含む症状を緩和させる薬剤 \n(b) 咳・咽頭痛・息切れ・全身倦怠感・下痢 \n \n（５）学外の感染者が学内に立ち入った場合 \n \n該当部局が行うこと \n①危機対策本部より、濃厚接触した者がいたと連絡があった場合は、上記（２）によ\nり対応 \n \n危機対策本部が行うこと \n①保健所等の指示・指導のもと、学内で濃厚接触した者を把握 \n②濃厚接触した者がいた場合は、保健所の指示・指導のもと、所属部局に報告し、上\n記（２）により対応 \n③保健所の指導のもと、該当部局と連携し、消毒作業 \n \n（参 考） \n \n◎ 「濃厚接触者」とは、「患者（確定例）」が感染可能期間に接触した者のうち、次の範\n囲に該当する者を指す。 \n・患者（確定例）と同居あるいは長時間の接触（車内、航空機内等を含む）があった者 \n・適切な感染防護無しに患者（確定例）を診察、看護若しくは介護していた者  \n・患者（確定例）の気道分泌液もしくは体液等の汚染物質に直接触れた可能性が高い者 \n・手で触れることの出来る距離（目安として１メートル）で、必要な感染予防策なしで、\n「患者（確定例）」と１５分以上の接触があった者（周辺の環境や接触の状況等個々の状\n況から患者の感染性を総合的に判断する） \n※ 京都府ＨＰ（濃厚接触者Ｑ＆Ａ） https:\/\/www.pref.kyoto.jp\/kentai\/corona\/qa.html \n \n◎ 「患者（確定例）」とは、「新型コロナウイルス感染症の臨床的特徴を有し、かつ検査\nにより新型コロナウイルス感染症と診断された者」を指す。 \n \n◎ 「感染可能期間」とは、発熱及び咳・呼吸困難などの急性の呼吸器症状を含めた新型コ\n5 \n（問い合わせ先） \n総務部企画管理主幹付リスク管理掛 \nTel.075-753-2226 \nE-mail: 830riskkanrikakari@mail2.adm.kyoto-u.ac.jp \n \n※体調の経過観察の提出先 \n施設部環境安全保健課保健衛生掛 \nTel.075-753-2400 \nE-mail: hoken-corona@hoken.kyoto-u.ac.jp \nロナウイルス感染症を疑う症状（以下参照）を呈した２日前から退院又は宿泊（自宅）療\n養の解除の基準を満たすまでの期間とする。 \n※ 発熱、咳、呼吸困難、全身倦怠感、咽頭痛、鼻汁・鼻閉、頭痛、関節・筋肉痛、下\n痢、嘔気・嘔吐など \n \n◎ 登校禁止、就業禁止及び自宅待機（在宅勤務、業務命令及び自宅学習）の解除に際し、\n医療機関に対して「治癒証明書」や「陰性証明書」の発行を求めない。 \nhttps:\/\/www.mhlw.go.jp\/content\/000758785.pdf \n \n◎ 保健所等の行政機関との連絡調整は、危機対策本部が担当する。 \n \n◎ 濃厚接触者の判断、消毒作業について \n・保健所が感染者の行動を確認し、現地調査などに基づき事業所と協議の上、濃厚接触者\nや消毒場所を決定することになる。 \n・濃厚接触者の判断については、同じ事務室内で業務を行なっていたとしても、明らかな\n接触がないことなどにより、濃厚接触者とならない場合もある。 \n・消毒作業に関しては、小さな事務室であれば１日程度で終了し、翌日から業務を再開し\nた例もある。 \n \n◎ 消毒作業については、保健所及び危機対策本部の指示・指導のもと、専門業者に依頼す\nることを基本とする。 \nただし、緊急を要し、清掃業者を手配できない場合、危機対策本部において当該部局と\n連携し、消毒作業を実施する。 \n \n本学教職員が消毒作業を行う場合は、感染者が利用した区域（部屋、エレベータ、廊下、\nトイレ等）のうち手指が頻回に接触する箇所（ドアノブ、スイッチ類、手すり、洗面、便\n座、流水レバー等）を中心に、「（厚生労働省）新型コロナウイルスの消毒・除菌方法につ\nいて」※を参考に実施する。 \n※ https:\/\/www.mhlw.go.jp\/stf\/seisakunitsuite\/bunya\/syoudoku_00001.html \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n※ 体調の経過観察については、別添様式をご使用ください。 \n"}
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{"id":"9Ney09U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/education-campus\/students2\/campus\/career","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":69,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["ja","education-campus","students2","campus"],"url_file_name_s":"career","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["留学生へのキャリア支援 | 京都大学"],"description_txt":["京都大学国際高等教育院附属日本語・日本文化教育センターのビジネス日本語講座\n\n日本での就職を希望する留学生を対象に、日本の会社で働くために必要な日本語のスキルやビジネスマナー等の修得を目指します。\n\n前期（春季）講義は4月から7月、後期（秋季）講義は10月から1月まで行われます。授業案内と受講申請は、年2回、3月と9月の下旬に、「課外の日本語学習支援講座」サイトで閲覧・登録できます。授業案内は学部・研究科等事務室と国際教育交流課でも配付します。申請は上記の日本語・日本文化教育センターのサイト上でのみ受け付けとなりますので、注意してください。\n\n京都大学学生総合支援センターキャリアサポートルーム\n\n全学の学生のキャリア・就職支援として、各種情報提供や相談対応などを行っています。\n\n京都大学学生総合支援センターキャリアサポートルーム"],"last_modified":"Wed Jun 08 16:43:43 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":4280722877940911611,"exact_signature_unique_b":true,"fuzzy_signature_l":-1974568277858211582,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["留学生へのキャリア支援"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","京都大学国際高等教育院附属日本語・日本文化教育センターのビジネス日本語講座","京都大学学生総合支援センターキャリアサポートルーム","学外の就職相談機関一覧","卒業後の就労に関する在留資格変更手続","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["利用時間","外国人雇用サービスセンター","日本学生支援機構（JASSO）","京都ジョブパーク","厚生労働省職業安定局外国人雇用対策課","留学生就職支援ネットワーク","卒業後日本において就職するとき","卒業後日本において就職活動を行うとき"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1542,"text_t":"留学生へのキャリア支援 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 京都大学に在籍する留学生の方へ 日本での学生生活 留学生へのキャリア支援. 留学生へのキャリア支援. サイドメニュー. 世界とつながる京都大学 京都大学への留学を希望する方へ 京都大学で学ぶ 留学生へのキャリア支援 留学生ラウンジ「きずな」 在留資格・学費・奨学金・住まい 学部・大学院研究科等事務室一覧 FAQ 便利な情報 留学に関する情報 国際化を支える学内組織 外国人研究者へのサポート 海外留学を希望する京大生へ In other languages. English 京都大学国際高等教育院附属日本語・日本文化教育センターのビジネス日本語講座. 日本での就職を希望する留学生を対象に、日本の会社で働くために必要な日本語のスキルやビジネスマナー等の修得を目指します。. 前期（春季）講義は4月から7月、後期（秋季）講義は10月から1月まで行われます。授業案内と受講申請は、年2回、3月と9月の下旬に、「. 課外の日本語学習支援講座 」サイトで閲覧・登録できます。授業案内は学部・研究科等事務室と国際教育交流課でも配付します。申請は上記の日本語・日本文化教育センターのサイト上でのみ受け付けとなりますので、注意してください。. 京都大学学生総合支援センターキャリアサポートルーム. 全学の学生のキャリア・就職支援として、各種情報提供や相談対応などを行っています。. 京都大学学生総合支援センターキャリアサポートルーム 利用時間. 月曜日～金曜日　9時00分～17時00分（祝日を除く）. 学外の就職相談機関一覧. 外国人雇用サービスセンター. 外国人雇用サービスセンター（大阪） 外国人雇用サービスセンター（東京） 日本学生支援機構（JASSO）. 日本学生支援機構（JASSO） 京都ジョブパーク. 京都ジョブパーク 厚生労働省職業安定局外国人雇用対策課. ｢日本で就職しようとする留学生の皆さんへ｣ 留学生就職支援ネットワーク. 「日本で就職を希望する留学生の皆さんへ」 卒業後の就労に関する在留資格変更手続. 留学生が卒業後も引き続き日本に在住する場合は、すみやかに在留資格を変更してください。在留資格以外の活動を行う者は処罰の対象となります。. 「留学」から各種の在留資格に変更する場合、手続きは原則として本人が最寄りの入国管理局へ出向いて行う必要があります。変更に際して必要な書類は、自分で用意するもの、大学または就職先からもらうものなどがあります。在留資格に関わる必要書類などの情報は. 法務省のWebサイト で確認し、詳しい情報は. 入国管理局 に問い合わせてください。. 卒業後日本において就職するとき. 現在の「留学」の在留資格を「人文知識・国際業務」、「技術」など就労可能な在留資格へ変更することが必要です。変更する在留資格によって、必要書類は異なります。. 卒業後日本において就職活動を行うとき. 「特定活動」へ在留資格を変更する必要があります。この在留資格で、6ヶ月間、または更新手続きにより最長1年間は日本に滞在し、就職活動を行うことができます。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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English 　詳細情報は随時更新します。. 令和４年度　大学院秋季学位授与式・入学式. 令和4年度 大学院秋季学位授与式. 日時. 令和4年9月26日（月曜日）10時00分開始　11時10分終了予定. （入場開始時刻（予定）：9時20分）. 場所. 百周年時計台記念館　百周年記念ホール. 本部・西部構内マップ［3］ 対象者. 令和4年5月23日付け、7月25日付け、9月26日付け博士学位授与者. 令和4年9月26日付け修士・専門職学位授与者. その他. 式典にふさわしい服装を着用してください。. 各研究科等の問い合わせ先 令和4年度 大学院秋季入学式. 日時. 令和4年10月1日（土曜日）10時00分開始　10時30分終了予定. （入場開始時刻（予定）：9時20分）. 場所. 百周年時計台記念館　百周年記念ホール. 本部・西部構内マップ［3］ 対象者. 令和4年10月1日付け大学院入学者・進学者. その他. 式典にふさわしい服装を着用してください。. 各研究科等の問い合わせ先 令和４年度　大学院学位授与式・卒業式. 令和4年度 大学院学位授与式. 日時. 令和5年3月23日（木曜日）14時00分開始　15時10分終了予定. （入場開始時刻（予定）：13時10分）. 場所. 京都市勧業館みやこめっせ　第3展示場（3階）. アクセス 対象者. 令和4年11月24日付け、令和5年1月23日付け、3月23日付け博士学位授与者. 令和5年3月23日付け修士・専門職学位授与者. その他. 式典にふさわしい服装を着用してください。. 各研究科等の問い合わせ先 令和4年度 卒業式. 日時. 令和5年3月24日（金曜日）10時00分開始　10時40分終了予定. （入場開始時刻（予定）：9時10分）. 場所. 京都市勧業館みやこめっせ　第3展示場（3階）. アクセス 対象者. 令和5年3月10日付け、3月24日付け学部卒業者. その他. 式典にふさわしい服装を着用してください。. 各学部の問い合わせ先 担当部署. 教育推進・学生支援部 教務企画課教務掛. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 男女共同参画について 改革. 改革. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について アクションプラン 女性活躍推進法について 意識・実態調査 改革 次世代育成支援対策推進 育児・介護支援制度 公表事項 情報公開 1. 委員会の設置. 平成17年7月. 役員懇談会において、本学の男女共同参画の推進に向けた具体的な体制・施策等の検討のため、担当理事の下にプロジェクトチームを設置することを了承. 平成17年9月. 部局長会議に男女共同参画推進体制等検討プロジェクトチームの設置検討状況等を報告. 平成17年10月. 部局長会議に男女共同参画推進体制等検討プロジェクトチームから報告「京都大学における男女共同参画推進体制等について」（概要）本学における男女共同参画の現状や施策について検討するため、担当理事の下に、「京都大学男女共同参画企画推進委員会」を設置するべき等. 同. 京都大学男女共同参画企画推進委員会設置（役員会決定）. 2. 京都大学男女共同参画企画推進委員会. 役割. 男女共同参画推進に関する調査・分析. 男女共同参画推進に関する政策提言. 提言に基づく施策の実施状況調査・検証・評価等. これまでの活動. 本学の男女共同参画に関する予備的な調査を実施（平成18年1～2月）. 「京都大学における男女共同参画」（基本理念・基本方針）の原案策定（平成18年3月部局長会議決定）. 男女共同参画に関する全学調査の実施（平成18年12月～1月）報告書の作成. 「京都大学男女共同参画推進アクション・プランに向けて（提言）」を報告（平成19年10月部局長会議報告）. 平成20年1月. 役員懇談会において、男女共同参画企画推進委員会を発展的に解消し男女共同参画推進室を設置することを了承。. 役員会において「男女共同参画企画推進委員会要項」の廃止の決議。. 3. 男女共同参画推進室. 平成20年1月15日. 部局長会議において男女共同参画推進室要項の審議、了承. 平成20年1月31日. 第1回 男女共同参画推進室会議. 議事 京都大学男女共同参画推進アクション・プランについて. ※ アクション・プラン作成に向けた意見交換. 平成20年3月4日. 第2回 男女共同参画推進室会議. 議事 男女共同参画推進のための緊急重点プロジェクトについて. ※ 21年度までに取り組むべき事業について原案の検討. ※ 「男女共同参画の推進にむけて －第1期中期計画期間中の事業－」提言の（案）了承. 平成20年4月15日. 部局長会議に. 「男女共同参画の推進にむけて －第1期中期計画期間中の事業－」 について報告. 平成20年9月26日. 第3回 男女共同参画推進室会議. 議事 男女共同参画推進の推進にむけて. ※ アクション・プラン作成WGの設置の了承。部局長会議に「男女共同参画の推進にむけて －第1期中期計画期間中の事業－」について報告. 平成20年12月16日. 第4回 男女共同参画推進室会議議事. アクション・プラン策定について. ※ WGの進捗状況の報告と今後の進め方の了承. 平成21年2月23日. 第5回 男女共同参画推進室会議. 議事 アクションプラン策定について. ※ 京都大学男女共同参画アクションプラン（案）の審議、了承. 平成21年3月10日. 部局長会議において「京都大学男女共同参画推進アクション・プラン」の審議、了承. 平成21年4月24日. 第6回 男女共同参画推進室会議. 【議題】. 京都大学男女共同参画推進室要項の一部改正について. ※ 室員の任期を定める（案）の審議・了承. 男女共同参画推進アクション・プランのフォローアップの進め方ついて. ※ 年度ごとに計画・実績の報告を受け公表することの審議・了承. 次世代育成支援対策推進法の改正に伴う一般事業主行動計画の届出および認定の申請 について. ※ 計画期間を5年として策定する（案）了承. 平成21年5月12日. 部局長会議に「男女共同参画推進アクション・プラン」のフォローアップの進め方について報告. 平成21年5月28日. 第7回 男女共同参画推進室会議. 【議題】. 「京都大学重点事業アクションプラン2006～2009」にかかる事業計画について. ※ 平成20年度に計画された平成21年度計画の審議・了承. 男女共同参画推進アクション・プランのフォローアップの進め方について. ※ 事前アンケートにより、部局選定について検討. 平成21年7月14日. 部局長会議に「男女共同参画推進アクション・プランの進捗状況」および平成21年5月1日現在の「平成21年度教員部局別・職別女性比率」を報告. 平成21年9月8日. 部局長会議に「男女共同参画推進アクション・プラン1-（1）・（2）」の推進にむけ男女共同参画推進室メンバーとアンケートにより選定した各部局との懇談会（7月8日～7月24日の間に行った）の報告. 平成21年11月11日. 第8回 男女共同参画推進室会議. 【議題】. 男女共同参画推進室主催講演会の開催について. ※ 審議・了承. 男女共同参画推進アクション・プラン推進計画と実績の様式による担当部局からの計画報告の公表について. ※ 21年度計画・実績、22年度計画の報告をまとめ22年度に公表する事を審議・了承. 平成20年度教職員の育児・介護制度の利用状況の公表について. ※ 部局長会議にて報告し21年度から公表する事の審議・了承. 平成21年11月17日. 部局長会議に「男女共同参画推進アクション・プラン」の進捗状況について報告. 各部局における平成21年度の計画と実績の報告を依頼しその結果を平成22年度にまとめ部局長会議に報告し公表する。. アクション・プラン5-（2）に基づき調査し平成21年度分から公表する。. アクション・プラン1-（2）関連を調査し平成21年度分から公表する。. 平成22年3月4日. 第9回 男女共同参画推進室会議. 【議題】. 「男女共同参画推進アクション・プラン」推進室担当部分の記載（案）と平成22年度の推進室重点取り組みテーマについて. ※ 審議・了承. 男女共同参画推進アクション・プラン1-（2）・5-（2）の推進に向けた平成21年度実績の調査について. ※ 継続審議とする。. 男女共同参画推進室のホームページについて. ※ 改定に向け審議・了承. 「日本女性会議2010きょうと」開催にかかる協力依頼について. ※ 審議、広報面について協力をすることで了承. 平成22年5月13日. 第1回 男女共同参画推進室会議. 【議題】. 「男女共同参画推進アクション・プラン」平成22年度の推進室重点取り組みテーマについて. ※ 22年度重点取り組みテーマを審議、了承。また、昨年度の重点取り組みテーマを継続重点取り組みテーマとした。. 男女共同参画推進アクション・プラン1-（2）・5-（2）の推進に向けた平成21年度の調査について. ※ 調査書の内容を精査、審議・了承. 平成22年11月19日. 第2回 男女共同参画推進室会議. 【報告】. 「男女共同参画推進アクション・プラン」. 平成21年度のフォローアップと調査結果の報告. 平成23年5月23日. 第1回 男女共同参画推進室会議. 【議題】. 「男女共同参画推進アクション・プラン」. 平成23年度の推進室重点取り組みテーマについて. ※ 23年度取り組みテーマを審議、了承. 「1．男女共同参画の視点に立った教育・研究および就業の確立」を引き続き今年度の重点テーマとする。. 男女共同参画推進アクションプラン1-（2）・5-（2）の推進に向けた平成22年度実績の調査について. ※ 調査書の内容を精査、審議・了承. 【報告】. 平成22年度のアクション・プランフォローアップの調査を進める旨を報告. 平成24年3月2日. 第2回 男女共同参画推進室会議. 【議題】. 平成22年度および平成23年度「男女共同参画推進アクション・プラン」男女共同参画推進室担当部分の記載（案）について. ※ 記載案について審議・了承され、平成22年度実績について部局長会議にて報告後、全学に公表することとなった。. 男女共同参画推進室　平成24年度重点テーマの設定について. ※ 引き続き、「1. 男女共同参画の視点に立った教育・研究および就業の確立」および「5. 教職員・学生への啓発活動の推進」を重点テーマとする。. 【報告】. 「男女共同参画推進アクション・プラン推進計画と実績」平成23年度の計画と平成22年度の実績の報告について. 「男女共同参画アクション・プラン」1-（2）、5-（2）関連の平成22年度の調査結果について. ※ 平成23年度計画と平成22年度の実績、調査報告について、部局長会議にて報告後、全学に公表することとなった。. 平成25年3月29日. 平成24年度 男女共同参画推進室会議. 【議事】. 平成24年度「男女共同参画推進アクション・プラン」男女共同参画推進室担当部分の記載（案）について. 男女共同参画推進室平成25年度重点テーマの設定および部局訪問について. 【報告】. 「男女共同参画アクション・プラン推進計画と実績」平成23年度の実績等の報告について. 「男女共同参画推進アクション・プラン」1-（2）、5-（2）関連の平成23年度の調査結果について. 平成25年7月5日. 平成25年度 第1回男女共同参画推進室会議. 【議事】. 男女共同参画推進アクション・プラン（平成21年～平成25年度）まとめについて. 平成26年度以降の男女共同参画推進について. こども未来財団 ベビーシッター育児支援事業への申し込みについて. 【報告】. 京都大学教員・研究員の生活時間に関するアンケート調査報告. 今年度の部局訪問について. 4．男女共同参画推進本部. 平成26年4月1日. 男女共同参画推進体制の整備・再編により、新たに男女共同参画推進本部を設置. 平成26年7月9日. 平成26年度 第1回男女共同参画推進本部会議. 【議事】. 一般事業主行動計画策定届の提出について. ※ 審議・了承. 平成26年11月7日. 平成26年度 第2回男女共同参画推進本部会議. 【議事】. 平成27年度～31年度のアクションプランについて. ※ アクション・プラン作成WGの設置の了承。. 平成26年12月26日. 平成26年度 第3回男女共同参画推進本部会議. 【議事】. 京都大学男女共同参画推進アクション・プラン実施報告書について. ※ 平成21年～25年のアクション・プラン実施報告書の作成について、審議・了承. 平成27年1月22日. 平成26年度 第4回男女共同参画推進本部会議. 【議事】. 京都大学男女共同参画推進アクション・プラン2015～2024（案）について. 平成27年3月16日. 平成26年度 第5回男女共同参画推進本部会議. 【議事】. 「京都大学男女共同参画推進に関する意識・実態調査」について. ※ 調査の実施について、審議・了承. 「日仏討論会 ～出生率の上昇と男女間の格差との闘い： 迷路から抜けるためのイニシアティブとは？ ～」について. ※ 開催について、審議・了承. 平成27年5月15日. 平成27年度 第1回男女共同参画推進本部会議. 【議事】. 「京都大学男女共同参画推進に関する意識・実態調査」の質問項目について. 平成27年6月22日. 平成27年度 第2回男女共同参画推進本部会議. 【議事】. 「京都大学男女共同参画推進に関する意識・実態調査」報告書作成WGの設置について. ベビーシッター派遣事業について. 平成28年3月10日. 平成27年度 第3回男女共同参画推進本部会議. 【議事】. 女性活躍推進法施行に伴う行動計画の策定について. 「京都大学男女共同参画推進に関する意識・実態調査」報告書について. 平成28年7月4日. 平成28年度 第1回男女共同参画推進本部会議. 【議事】. 女性研究者の出産に伴う雇用経費支援制度の創設について. 平成29年10月6日. 平成29年度 第1回男女共同参画推進本部会議. 【議事】. 女性研究者の出産に伴う雇用経費支援制度について. 【報告】. 今年度前半期（平成29年4月1日～平成29年9月30日）の本学男女共同参画推進の主な活動報告について. 今後の予定について. 平成30年6月22日. 平成30年度 第1回男女共同参画推進本部会議. 【議事】. 「京都大学男女共同参画推進アクション・プラン」について. 【報告】. 平成29年度の男女共同参画推進関係の主な活動について. 平成30年7月26日. 平成30年度 第2回男女共同参画推進本部会議. 【議事】. 「京都大学男女共同参画推進アクション・プラン」について. 【報告】. 女子学生を増やすための取組みについて. 「女性教員比率の向上に関する意見交換」に係る部局訪問の中間報告について. 平成30年9月12日. 平成30年度 第3回男女共同参画推進本部会議. 【議事】. 女性教員比率の向上に関するインセンティブ案について. ​​​​【報告】. 「女性教員比率の向上に関する意見交換」に係る部局訪問の最終報告について. 平成30年10月4日. 平成30年度 第4回男女共同参画推進本部会議. 【議事】. 「女性教員比率の向上に関するインセンティブ案」の修正案について. 【報告】. 女子高生応援大使について問の最終報告について. 平成30年11月20日. 平成30年度 第5回男女共同参画推進本部会議. 【議事】. 「女性教員比率の向上に関するインセンティブ案」の修正案について. 平成31年2月4日. 平成30年度 第6回男女共同参画推進本部会議. 【議事】. 「男女共同参画支援たちばな基金」の設立及び申請について. 平成31年2月22日. 平成30年度 第7回男女共同参画推進本部会議. 【議事】. 女性教員採用に関する支援策について. 平成31年3月12日. 平成30年度 第8回男女共同参画推進本部会議. 【議事】. 「一般事業主行動計画策定届」について. 令和元年10月3日. 令和元年度 第1回男女共同参画推進本部会議. 【報告】. 他大学における男女共同参画にかかる取組について. 「女性教員登用等支援事業」採択一覧について. 令和2年3月18日. 令和元年度 第2回男女共同参画推進本部会議. 【議事】. 「京都大学一般事業主行動計画策定届（女性活躍促進法）」について. 「女性教員登用等支援事業」について. 令和2年5月25日. 令和2年度 第1回男女共同参画推進本部会議. 【議事】. 「女性教員登用等支援事業」について. 男女共同参画推進アクション・プランのフォローアップ等について. 令和2年12月21日. 令和2年度 第2回男女共同参画推進本部会議. 【議事】. 京都大学男女共同参画推進アクション・プランについて. 令和3年1月18日. 令和2年度 第3回男女共同参画推進本部会議. 【議事】. 京都大学男女共同参画推進アクション・プランについて. 令和3年3月1日. 令和2年度 第4回男女共同参画推進本部会議. 【議事】. 「女性教員登用等支援事業」について. 男女共同参画推進本部保育事業タスクフォース要項の制定について. 【報告】. 国立大学法人における男女共同参画推進について‐アクションプラン（2021年度～2025年度）‐. 令和3年3月23日. 令和2年度 第5回男女共同参画推進本部会議. 【議事】. 男女共同参画推進本部保育事業タスクフォース要項の改正について. 【報告】. 京都大学男女共同参画推進本部要項、京都大学男女共同参画推進本部内規及び京都大学男女共同参画推進センター内規の改正について. 令和3年9月2日. 令和3年度 第1回男女共同参画推進本部会議. 【議事】. 京都大学男女共同参画推進アクションプラン（2015年度～2021年度）について. 京都大学男女共同参画推進アクションプラン（2022年度～2027年度）について. 令和4年2月28日. 令和3年度 第2回男女共同参画推進本部会議. 【議事】. 男女共同参画推進体制について. 女性教員登用等支援事業の継続について. 令和4年3月4日. 令和3年度 第3回男女共同参画推進本部会議. 【議事】. 女性教員登用等支援事業の継続について.  . 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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{"id":"tUy6zNU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/zh-tw\/research\/export","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":47,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["zh-tw","research"],"url_file_name_s":"export","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Security Export Control | 京都大學"],"description_txt":["What is \"Security Export Control\"? In order to maintain both national and international peace and security, the Government of Japan controls military sensitive goods and technologies, including relevant dual-use goods and technologies. Based on the Foreign Exchange and Foreign Trade Act (1949) and its relevant legislations, exporters, including universities in Japan, are requested by the Ministry of Economy, Trade and Industry (METI) to apply for an export license in case:  - Items or technologies are listed in the control list; or"],"last_modified":"Wed Jun 08 16:45:06 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-3366578873490749698,"exact_signature_unique_b":true,"fuzzy_signature_l":2571137475231250656,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["Security Export Control"],"h2_txt":["グローバルナビゲーション（繁体中国語）","ヘッダー セカンダリーリンク（日本語以外）","サイドメニュー（繁体中国語）","What is \"Security Export Control\"?","Further information","Contact","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1011,"text_t":"Security Export Control | 京都大學. 移至主內容 網站內搜索. 教師搜索. 搜索. close. CLOSE. search. menu. グローバルナビゲーション（繁体中国語）. 京都大學概況. 校長的話. 大學介紹. 校內安全措施. 教育與學生支援服務. 學生生活. 留學・國際教育. 研究・產官學合作. 支援研究體制. Security Export Control. 國際交流. 學部與院係. ヘッダー セカンダリーリンク（日本語以外）. 交通資訊 聯絡我們 捐贈 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 導航連結. Home 研究・產官學合作 Security Export Control. Security Export Control. サイドメニュー（繁体中国語）. 支援研究體制 Security Export Control 國際交流 In other languages. 日本語 What is \"Security Export Control\"? In order to maintain both national and international peace and security, the Government of Japan controls military sensitive goods and technologies, including relevant dual-use goods and technologies. Based on the Foreign Exchange and Foreign Trade Act (1949) and its relevant legislations, exporters, including universities in Japan, are requested by the Ministry of Economy, Trade and Industry (METI) to apply for an export license in case:. - Items or technologies are listed in the control list; or. - The export is under the provision of end-use control such as 'catch-all control.'. The export of goods and technology designated in international export control agreements without permission from METI is subject to criminal and legal penalties. Such penalties may be applied not only to individual(s) but also to institutions involved. In the course of the various research and educational activities undertaken by universities, certain activities, such as the exportation or carrying of research equipment or materials abroad, collaborative research with foreign researchers or institutions, the hosting and education of foreign researchers or students and the exchange of sensitive information with foreign researchers, could be subject to security export control. Furthermore, the export control legislation of the US, such as the Export Administration Regulations (EAR), must also be observed when research activities involve the receipt of materials or technologies from US researchers, institutions, industries or government. \"Security Export Control Regulations of Kyoto University\" (internal) (PDF) \"Complying The Security Export Control Laws\" (internal) (PDF) \"Security Trade Control Guidelines for Academic Exchange Agreements at Kyoto University\" (internal) (PDF) Further information. Security Export Control Policy Division, Trade Control Department,. The Ministry of Economy, Trade and Industry (English Homepage) (external link). http:\/\/www.meti.go.jp\/policy\/anpo\/englishpage.html Guidance for the Control of Sensitive Technologies for Academic Institutions February 2010 (PDF) Export Administration Regulations (US) (external link). https:\/\/www.bis.doc.gov\/index.php\/regulations\/export-administration-regulations-ear Contact. Research Promotion Division, Research and International Affairs Department. (Security Export Control Section). Phone: ext. 2297, 2298. Email: info-yusyutu*mail2.adm.kyoto-u.ac.jp (replace * with @). 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通資訊 フッター セカンダリーリンク. 站點地圖 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 キャンパスライフ支援 住まい・食と購買. 住まい・食と購買. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生生活で注意してほしいこと 健康管理 在学中の保険加入 住まい・食と購買 「吉田寮生の安全確保についての基本方針」の実施に関するFAQ（第1版） 学生寄宿舎（女子寮）の入居について アルバイト 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. 学生寄宿舎 民間アパート等の紹介 食と購買 キャンパスライフの基礎となる住まい・食・書籍や物品の購入については、本学は主として京都大学生活協同組合に業務委託しています。. お知らせ. 学生寄宿舎（女子寮）の入居について 学生寄宿舎. 本学では学生寄宿舎として次の各寮を設置しており、それぞれの寄宿舎の運営は各寮自治会が主体となって行っています。なお、各学生寄宿舎は、居室形態、寄宿料、施設・設備の内容等が異なるほか、寄宿舎によっては建築後相当の年数が経過しており老朽化している建物もありますので、入寮を希望する方には、それらの状況を事前に確認することをお勧めします。. 1．各学生寄宿舎の状況. 1．学生寄宿舎の概要. 学生寄宿舎名. 吉田寮. 熊野寮. 室町寮. 女子寮. 所在地. 平成30年1月以降、吉田寮への新規入寮は認めておりません。. 関連情報：. 吉田寮生の安全確保についての基本方針 「吉田寮生の安全確保についての基本方針」の策定・実施について 「吉田寮生の安全確保についての基本方針」にかかる補足説明 「吉田寮生の安全確保についての基本方針」の実施に関するFAQ（第1版） 「吉田寮生の安全確保についての基本方針」の実施に関するFAQ（第1版更新版） 「吉田寮生の安全確保についての基本方針」の実施に関するFAQ（第2版） 「吉田寮生の安全確保についての基本方針」の実施に関するFAQ（第3版） 「吉田寮生の安全確保についての基本方針」の実施に関するFAQ（第4版） 「吉田寮生の安全確保についての基本方針」の実施状況について 吉田寮の退舎期限経過後の状況について 吉田寮の今後のあり方について 現在吉田寮に居住する者へ 〒606-8393. 左京区東竹屋町. 〒602-0001. 上京区竹園町. 下記、教育推進・学生支援部厚生課厚生掛へ問い合わせください。. 電話番号. 075-751-4050,4051. 075-431-8888. 対象学生. 本学学生. 本学大学院学生. 本学女子学生. 収容定員. 422名. 19名. 65名. 建築年. 昭和39・40年. 昭和17年. 平成31年. 建築構造. 鉄筋コンクリート4階建（3棟）. 木造2階建（1棟）. 鉄筋コンクリート4階建. 居室様式. 洋室30㎡：84室（4人部屋）. 洋室15㎡：43室（2人部屋）. 和室8畳：1室. 和室6畳：6室. 和室4畳：12室. （いずれも1人部屋）. 居室64室（約12㎡・個室・バス・トイレなし）. バリアフリー室1室（約19㎡・個室・バス・トイレあり）. 付属施設. 食堂、談話室、図書室、会議室、音楽室、シャワー室. 談話室、自炊室、シャワー室. 集会所、学習室、リビングダイニング、キッチン、シャワー室、浴室. 寄宿料（月額）. 700円. 400円. 25,000円. 光熱水料. 月額1,500～2,500円（各寮によって異なります）. 実費. 吉田キャンパスまでの通学時間. 徒歩約15分. 市バス約20分. 徒歩約7分. 特記事項. 熊野寮の一部は、中核派系全学連の関係先のひとつとされ、警察による強制捜査（直近では令和3年6月24日）が行われたことがあります。. なし. なし. 2．各学生寄宿舎の連絡先. 吉田寮. 下記の問い合わせ先までお問い合わせください。. 熊野寮. 熊野寮Webサイト（熊野寮自治会） （学外リンク）. 室町寮. 室町寮Webサイト（室町寮自治会） （学外リンク）. 女子寮. 学生寄宿舎（女子寮）の入居について 2．学生寄宿舎に関する問い合わせ先. 教育推進・学生支援部 厚生課 厚生掛. Tel： 075-753-2540. 民間アパート等の紹介. 京都大学生活協同組合が行っていますので、以下のサイトをご覧ください。. 京大生協「住まい探し」のページ 食と購買. 京大生協のページ 生協以外のレストランなどの店舗については以下をご覧ください。. 食堂、レストラン、ショップなど 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 公表事項 業務方法書. 業務方法書. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 国立大学法人ガバナンス・コード 自己点検・評価（教員評価を含む） 認証評価 業務方法書 会議報告 国立大学法人法等による公表事項 教育情報の公表 贈与報告書 資金運用について 労働施策総合推進法に基づく中途採用比率の公表 情報公開 国立大学法人京都大学業務方法書. 平成16年5月24日. 平成23年3月31日. 平成26年9月1日. 平成27年4月1日. 平成30年7月11日. 令和4年3月25日. 文部科学大臣認可. 第1章　目的. （目的）. 第1条　この業務方法書は、国立大学法人法（平成15年法律第112号）第35条において準用する独立行政法人通則法（平成11年法律第103号）第28条第1項の規定に基づき、国立大学法人京都大学（以下「法人」という。）の業務方法書について、国立大学法人法施行規則（平成15年文部科学省令第57号）第11条に規定する事項を定め、その業務の適正な運営に資することを目的とする。. 第2章　役員（監事を除く。）の職務の執行が国立大学法人法又は他の法令に適合することを確保するための体制その他業務の適正を確保するための体制の整備に関する事項. （内部統制に関する基本事項）. 第2条　法人は、役員（監事を除く。）の職務の執行が国立大学法人法又は他の法令に適合することを確保するための体制その他業務の適正を確保するための体制（以下「内部統制システム」という。）を整備し、継続的にその見直しを図るとともに、役員及び職員（以下「役職員」という。）への周知や研修の実施、必要な情報システムの更新に努めるものとする。. 第3条　法人は、内部統制システムに関する事務を統括する役職員その他の内部統制システムの整備の推進のための体制について決定するものとする。. 法人は、前項の体制に基づき、モニタリングを行うために必要な規程を整備するものとする。. 内部統制システムに関する事務を統括する役職員は、定期的な連絡の機会を設け、内部統制システムに関する事務を統括する役員に対し、必要な報告が定期的に行われることを確保するものとする。. 第4条　法人は、職員の職務の執行にあたり、国立大学法人法又は他の法令、本学の定める規程に違反する事由が発生した場合における、違反した職員に対する懲戒に関する規程その他の対応の指針をあらかじめ定めるものとする。. 法人は、前項に規定する事由が発生した場合には、速やかな是正措置をとり、あわせて再発防止を図るものとする。. 法人は、職員の人事異動に係る基本方針を策定するほか、長期在職者の把握等により業務の適正確保に努めるものとする。. 第5条　法人は、学長から役職員への意思の伝達や、職員から役員への危機管理、内部統制に係る情報その他の必要な情報の伝達が確実に行われるよう、必要な措置を講ずるものとする。. （法人運営に関する基本的事項）. 第6条　法人は、法人の運営に係る基本理念を定め、これを公表するものとする。. 法人は、役職員の倫理指針を定めるものとする。. 第7条　法人は、業務執行に係る決裁及び経費支出の承認に係る手順を明らかにするとともに、役職員は、その過程における確認機能を着実に果たすものとする。. 法人は、業務の適正かつ効率的な実施にあたり必要とされるマニュアルの整備及び効率的な業務運営を可能とするための情報システムの整備を行うものとする。. （理事の分掌に関する事項）. 第8条　法人は、理事の分掌を決定し、これを公表するものとする。. （中期計画の策定に関する事項）. 第9条　法人は、中期計画について、役員会、経営協議会及び教育研究評議会の関与その他中期計画の策定の過程を整備するものとする。. （中期計画に係る評価及び評価に基づく予算の適正な配分に関する事項）. 第10条　法人は、中期計画の進捗管理及び中期計画に基づき実施する業務の評価（以下「評価活動」という。）を定期的に実施するものとし、役員会、経営協議会及び教育研究評議会その他の評価活動のために必要な体制について整備を行うとともに、評価活動の結果を踏まえ、国立大学法人法第31条の2第2項に規定する報告書の作成を適切に行うものとする。. 評価活動については、あらかじめ定める手順に沿った適正な実施を確保するとともに、恣意的とならない評価の実施に努めるものとする。また、評価活動を通じ、法人の業務執行が、必要とされる業務の手順を踏まえたものとなっているかの確認を行うものとする。. 法人は、予算の配分が適正に実施されることを確保するための体制を整備するものとする。その中において、評価活動の結果を予算の配分に活用する仕組みの構築を行うものとする。. （リスク評価と対応に関する事項）. 第11条　法人は、業務実施の障害となる要因を事前にリスクとして識別、分析及び評価し、当該リスクへの適切な対応を可能とするため、業務の整理並びに業務実施におけるリスク及びその発生原因の分析並びに必要な規程の整備に努めるとともに、以下の取組を行うものとする。. リスク管理に係る事務を統括する部署の設置. 把握したリスクを低減するための検討. 把握したリスクに対する評価の定期的かつ継続的な見直し. 把握したリスクに関する広報の体制及び広報における留意事項の整理. 第12条　法人は、事故、災害その他の緊急時における業務の継続のための計画を策定するものとする。当該計画には、以下に係る事項を定めなければならない。. 計画に基づく訓練等の実施. 緊急事態発生時における対策本部の設置及び当該本部の構成員. 緊急事態発生時における初動体制. 緊急事態発生時における情報収集の迅速な実施. 法人は、反社会的勢力への対応の在り方についての方針を整備するものとする。. 第13条　法人は、施設の定期的な点検及び必要な補修の実施を行うものとする。. 第14条　法人は、情報システムに係るリスクへの対策として必要な取組を行うものとし、その状況について、定期的な点検を行うものとする。. （入札・契約に関する事項）. 第15条　法人は、契約事務の適切な実施及び契約事務における相互牽制の確立を確保するため、以下の取組を行うものとする。. 契約の適正な履行に関する審査を行うための委員会の活用. 談合情報がある場合の対応方針の整備. （研究の適切な実施に関する事項）. 第16条　法人は、研究活動について、以下の事項を確保するための規程を整備するものとする。. 内部牽制機能による研究費の適正経理. 研究不正の防止. 知的財産の保護. 法人は、特に厳格な規律を要すると考えられる研究を実施する際の適切性の確保に努めるものとする。. （情報の適切な管理に関する事項）. 第17条　法人は、情報セキュリティの確保に関する規程の整備その他情報漏えいの防止に係る取組を推進するものとする。. 2 法人は、個人情報の保護に関する規程を整備し、個人情報の適切な管理にあたり必要とされる取組を着実に実施するとともに、取組の実施状況に関する点検を定期的に行うものとする。. 第18条　法人は、法人の意思決定に係る文書が適切に管理されることを担保するために、文書の適切な保存管理及び文書情報公開に関する規程を整備するものとする。. 第19条　法人は、所有する情報について、閲覧権限を整理するとともに、閲覧権限を有する者が、効率的に情報を検索できるよう、体系的な情報の保存及びそれを可能とする情報システムの整備を行うものとする。. （監事及び監事監査に関する事項）. 第20条　法人は、監事及び監事監査に関する規程を整備するものとする。同規程には、以下に係る事項を定めなければならない。. 監事が有する権限. 監査の結果に係る学長への報告. 監査の結果の業務への適切な反映. 監査の結果に対する改善状況の監事への報告. 役職員の不正及び違法行為並びに著しい不当事実がある場合の監事への報告義務. 法人の意思決定に係る文書の閲覧. 第21条　法人は、監事監査の円滑かつ適切な実施のため、以下の事項が確保されるよう、適切な措置を講じるものとする。. 役職員による監事及び監査に関する業務の支援に従事する職員への協力. 監事による役職員への文書提出や説明の要請権限. 監事の重要な会議への出席. 監事及び会計監査人の連携. 監事及び内部監査担当部署との連携. 監査に関する業務の支援に従事する職員の独立性. 監事による国立大学法人法第11条第5項及び第7項に基づく法人及び子法人の財産の状況の調査権限. 監事による国立大学法人法第11条第6項に規定する文部科学省令で定める書類の調査. 第22条　法人は、第20条に定める監事及び監事監査に関する規程を定め、又はこれを変更する場合には、監事の意見を聴かなければならない。. 第23条　法人は、学長、監事及び会計監査人の意思疎通を確保できるよう、定期的な連絡の機会を設けるなど、必要な体制の整備を行うものとする。. （内部監査に関する事項）. 第24条　法人は、内部監査を担当する組織を設置し、内部監査を実施するとともに、内部監査の結果及びそれに対する改善措置状況を学長に報告するものとする。. （内部通報・外部通報に関する事項）. 第25条　法人は、内部通報及び外部通報に係る通報窓口の設置及び運営等に関する規程を整備するものとする。. 第3章 出資の方法に関する基本的事項. （出資の方法に関する基本的事項）. 第26条　法人は、技術に関する研究の成果の活用を促進することが十分に期待できる場合、国立大学法人法第22条第1項第6号から第9号まで及び国立大学法人法施行令（平成15年政令第478号）第3条の規定に基づき、教育研究に係る施設、設備又は知的基盤の管理・利用促進に係る事業を実施する者、研究成果を活用する事業を実施する者、研究の成果の活用を促進する事業を実施する者又は認定特定研究成果活用支援事業者に出資することができる。. 第26条の2　法人は、国立大学法人法第34条の4第1項の規定により指定国立大学法人の指定を受けている場合、同法第34条の5第1項の規定に基づき、大学における技術に関する研究の成果の提供を受けて商品を開発し、若しくは生産し、又は役務を開発し、若しくは提供する事業を実施する者に出資することができる。. 第27条　法人は、出資に関し、国立大学法人法第22条第2項及び第34条の5第2項に規定する認可を申請しようとするときは、経営協議会の審議を経た上で役員会の議を経るものとする。. 2 前項の経営協議会及び役員会については、議事録を作成し、出資の認可の申請に係る議事の内容を明瞭に記載するものとする。. 第4章 委託の基準. （業務の委託）. 第28条　法人は、国立大学法人法第22条第1項第1号、第2号、第4号、第5号及び第10号に規定する業務の一部を法人以外の者に委託することにより効率的にその業務を遂行することができると認められ、かつ、委託することにより優れた成果を得られることが十分期待できる場合、業務の一部を委託することができる。. （委託契約）. 第29条　法人は、前条の規定により業務を委託するときは、受託者との間に業務に関する委託契約を締結するものとする。. 第5章 競争入札その他契約に関する基本的事項. （競争入札その他契約に関する基本事項）. 第30条　法人は、売買、貸借、請負その他の契約を締結する場合においては、すべて公告して申込みをさせることにより競争に付するものとする。ただし、契約の性質又は目的が競争を許さない場合その他規程で定める場合は、指名競争又は随意契約によることができるものとする。. 2 政府調達に関する協定（平成7年条約第23号）、政府調達に関する協定を改正する議定書（平成26年条約第4号）によって改正された協定その他の国際約束の適用を受ける契約については、国際約束に定められた調達手続きによるものとする。. 附則. この業務方法書は、文部科学大臣の認可のあった日から施行する。. 附則. この業務方法書は、文部科学大臣の認可のあった日から施行する。. 附則. この業務方法書は、文部科学大臣の認可のあった日から施行する。ただし、改正後の第2条及び第5条の規定は平成26年4月1日から、改正後の第4条第2項の規定は平成26年4月16日から適用する。. 附則. この業務方法書は、文部科学大臣の認可のあった日から施行する。. 附則. この業務方法書は、文部科学大臣の認可のあった日から施行する。. 附則. この業務方法書は、令和4年4月1日から施行する。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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{"id":"tE7dN9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/news\/2022-03-08","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":44,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","news"],"url_file_name_s":"2022-03-08","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["ウクライナ情勢に係る本学の方針 | 京都大学"],"description_txt":["　京都大学は、現在のウクライナ情勢に関し、平和的な解決を強く望むとともに、本学の学術交流協定校を含む多くのウクライナの大学・学術機関の関係者に対して、学生の受入れ等最大限の便宜を図ります。\n\n　また、現在本学に在籍するウクライナ及びロシアからの留学生に関して、今回の事態により本人の教育・研究活動に支障が生じることがないよう、精神的なケアを含めた様々な支援を行います。\n\n京都大学総長\n湊 長博\n\n "],"last_modified":"Wed Jun 08 16:33:12 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":1003022442932798366,"exact_signature_unique_b":true,"fuzzy_signature_l":-4169981323003423405,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["ウクライナ情勢に係る本学の方針"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","Latest news","2023年度 農学研究科修士課程学生募集について","マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて","令和4年6月1日付告示第4号","留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Tue Mar 08 12:00:00 UTC 2022","Fri Jun 03 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":682,"text_t":"ウクライナ情勢に係る本学の方針 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS ウクライナ情勢に係る本学の方針. ウクライナ情勢に係る本学の方針. 公開日. 2022年03月08日. 　京都大学は、現在のウクライナ情勢に関し、平和的な解決を強く望むとともに、本学の学術交流協定校を含む多くのウクライナの大学・学術機関の関係者に対して、学生の受入れ等最大限の便宜を図ります。. 　また、現在本学に在籍するウクライナ及びロシアからの留学生に関して、今回の事態により本人の教育・研究活動に支障が生じることがないよう、精神的なケアを含めた様々な支援を行います。. 京都大学総長. 湊 長博.  . Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 最新の研究成果を知る. 最新の研究成果を知る. 研究成果. 公開日. 2014年04月24日. ヒトの体細胞は原条の細胞に似た状態を経て初期化される －初期化メカニズムの一端を解明－. 研究成果. 公開日. 2014年04月23日. ブラックホールを記述する新理論をコンピュータで検証. 研究成果. 公開日. 2014年04月22日. スーパーコンピュータ「京」による計算で超新星爆発のニュートリノ加熱説が有望に. 研究成果. 公開日. 2014年04月22日. 陸上植物の季節に依存して花を咲かせる仕組みの起源を紐解く. 研究成果. 公開日. 2014年04月22日. 体に本来備わっているガン予防のしくみに迫る. 研究成果. 公開日. 2014年04月22日. 世界で初めて、溶液反応の超高速時間・角度分解光電子分光に成功 －溶液化学反応の機構解明に前進－. 研究成果. 公開日. 2014年04月18日. 新奇層状ペロブスカイト圧電体材料の発見 －層状構造と酸素八面体回転により破られる中心対称性－. 研究成果. 公開日. 2014年04月15日. イノベーションは創造するだけでは十分ではなく、それを普及させるためのコミュニケーション活動も重要であることを示唆. 研究成果. 公開日. 2014年04月14日. ワット級高出力フォトニック結晶レーザーを世界に先駆けて実現 －世界初、面発光型レーザーにより高ビーム品質でワット級の高出力化を達成－. 研究成果. 公開日. 2014年04月11日. 刺激性皮膚炎（おむつかぶれ）のメカニズムを解明. 研究成果. 公開日. 2014年04月08日. 超極細ナノ炭素細線の画期的高効率合成法を開発 －高効率太陽電池への応用に期待－. 研究成果. 公開日. 2014年04月04日. なぜ筋肉と脂肪のハイブリッド細胞はできないのか？ 筋分化と脂肪分化が排他的に起こる仕組みの解明 －肥満、再生医療などの治療応用に期待－. 研究成果. 公開日. 2014年04月02日. 海洋天然物ヘロナミド類の抗真菌作用メカニズムを解明 －新しい創薬シーズと創薬標的の提案に期待－. 研究成果. 公開日. 2014年03月24日. 濃い液体が秘める新機能を発見、新世代の電解液へ －電池の充電時間が1／3以下に－. 研究成果. 公開日. 2014年03月18日. 透明紙の開発に成功. ページ送り. 先頭ページ. chevron_left. chevron_left. 前ページ. chevron_left. &hellip;. ページ. 122 ページ. 123 ページ. 124 ページ. 125 ページ. 126 ページ. 127 ページ. 128 ページ. 129 カレントページ. 130 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 「日本と南仏におけるワイン醸造学と食文化比較論」を実施しました. 教育 国際交流 「日本と南仏におけるワイン醸造学と食文化比較論」を実施しました. In other languages. English ターゲット. 在学生の方 公開日. 2022年04月11日. 　農学研究科では、ワイルド&ワイズ共学教育受入れプログラム事業の一環として、「日本と南仏におけるワイン醸造学と食文化比較論」を、2022年3月7日から3月10日の4日間にわたり実施しました。本プログラムは、日本と南フランスにおけるワイン醸造学および世界の国において関連する食文化に関する知識や交流を深めることを目的としています。今回は、フランスのInstitute Agroより、モンペリエ校（スパグロ）から34名、ディジョン校から40名、レンヌ校から61名の計135名、日本からは京都大学、山梨大学、秋田県立大学や名城大学から40名の学生が参加し、オンラインによる講義とフィールドトリップを実施しました。. 　1日目は、「Fermentation Science」をテーマとし、Ingrid Collombel Institute Agro准教授およびYves Wache 同准教授よる講演を実施し、発酵食品とヒトの健康や人類の文化について学修しました。. 　2日目は、「Production & Distribution System」をテーマとし、 Louis-Antoine Saisset Institute Agro准教授と伊庭治彦 農学研究科准教授がコーディネーターとなり、日本とフランスから各2名の学生が講演を行いました。また、ワインや酒類のマーケティングや持続可能性について議論しました。. 　3日目は、玉置尚德 鹿児島大学教授とAurélie Roland Institute Agro准教授による講演の実施、日本酒とワインの発酵原理や製造工程の違い、ワインの香りとヒトの健康についてを学修しました。その後、「Beverages & Cuisine in the World」をテーマとし、3つのルームに分かれ、世界の飲料と料理の関係について交流討論会を行いました。. 　4日目は、「Viticulture & Wines in Japan」をテーマとし、Laurent Torregrossa Institute Agro教授と奥田徹 山梨大学ワイン研究所長による講演を行いました。その後、3つのワイナリーおよび生産者とともに、日本のワイナリーでの実地研修の可能性について意見交換を行いました。. アジアにおける発酵食品の解説. ワインの日本市場開拓における戦略. ブルゴーニュ地方の肉料理とワイン. 伝統的な日本酒と焼酎. マテ茶とは. 日本におけるぶどうの品種. 関連リンク . ワイルド&ワイズ共学教育受入れプログラム事業 関連部局. 農学部・農学研究科 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 社会連携 公開講座・講演会等 京都大学未来フォーラム. 京都大学未来フォーラム. サイドメニュー. 公開講座・講演会等 京都大学春秋講義 京都大学未来フォーラム 過去の開催記録（2007年度・第28回～2011年度・第52回） 過去の開催記録（2004年度・第1回～2006年度・第27回） 京大の知 京都大学地域講演会 京都大学 － 稲盛財団合同京都賞シンポジウム 公開講座お知らせメール受付 京大の研究・教育を知る 社会人向け教育 教育・研究に関する情報・資料公開 産官学連携（共同研究・知的財産・ベンチャー支援） 京大とのつながりを深める 社会に開かれた施設等 終了した全学的なイベント 「京都大学未来フォーラム」とは？. 京都大学で過ごした学生時代の経験を活かして、様々な分野で活躍する卒業生が、学生の皆様や今の社会を生きる皆様にとって、素晴らしい未来を志向するヒントとなるようなメッセージを発信します。. 次回ご案内. 回数・開催日. 第79回・2021年12月10日（金曜日）. テーマ. 「働く人生のアップデート」 講師. 株式会社堀場製作所 理事／HORIBA COLLEGE学長兼CSR担当. 野崎 治子 氏. 演題：人生やって無駄なことはない！. ウォンテッドリー株式会社 代表取締役CEO. 仲 暁子 氏. 演題：シゴトでココロおどろう. 開催報告. 当日の様子 過去の開催記録. 回数・開催日. 第78回・2021年6月11日（金曜日）. テーマ. 「わからないものとの戦い ～「気分」と「イノベーション」の狭間で～」 講師. 株式会社スチールヘッド代表取締役. テレビプロデューサー・演出家. 小松 純也 氏. 開催報告. 当日の様子  . 動画 （YouTube）. 回数・配信期間. 【オンライン開催】第77回・2020年11月2日（月曜日）9時00分～2021年3月26日（金曜日）17時00分まで. テーマ. 「森見登美彦さんと語る小説の世界」 出演者. 作家　森見 登美彦 氏. 京都大学 人文科学研究所　藤原 辰史 准教授. 開催報告. 当日の様子 第1部 ｜. 第2部 ｜. 第3部 　動画（YouTube）. 回数・開催日. 第76回・2019年12月11日（水曜日）. テーマ. 「どうすれば幸福になれるか」 講師. 哲学者. 岸見 一郎 氏. 開催報告. 当日の様子 回数・開催日. 第75回・2019年10月18日（金曜日）. テーマ. 「農林水産業と食と地域のくらしを支えるリーディング・バンクを目指して－農林中央金庫初のはえ抜き理事長の46年間の歩み－」 講師. 農林中央金庫 顧問. 河野 良雄 氏. 開催報告. 当日の様子 回数・開催日. 第74回・2019年6月3日（月曜日）. テーマ. 「笑いの花咲く国へ」 講師. 関西大学 人間健康学部 教授. 森下 伸也 氏. 開催報告. 当日の様子 回数・開催日. 第73回・2018年11月20日（火曜日）. テーマ. 「企業力の源泉と産業界が予想する未来“Society 5.0”」 講師. 三菱電機株式会社特別顧問. 山西 健一郎 氏. 開催報告. 当日の様子 回数・開催日. 第72回・2018年10月23日（火曜日）. テーマ. 「パンとサーカスの危ない時代に」 講師. 宝塚歌劇団演出家. 上田 久美子 氏. 開催報告. 当日の様子 回数・開催日. 第71回・2018年6月15日（金曜日）. テーマ. 「多様化する現代コミュニティと建築空間－明日の地域社会のあり方を考える」 講師. 京都精華大学長. ウスビ・サコ 氏. 開催報告. 当日の様子 回数・開催日. 第70回・2018年1月18日（木曜日）. テーマ. 「現代人はなぜ不幸なのか －医学モデルから愛着モデルへ」 講師. 岡田クリニック院長. 岡田 尊司 氏. 開催報告. 当日の様子 回数・開催日. 第69回・2017年12月1日（金曜日）. テーマ. 「新しい価値の創造への挑戦 ～京大女子の冒険～」 講師. 株式会社サキコーポレーション代表取締役社長. 秋山 咲恵 氏. 開催報告. 当日の様子 回数・開催日. 第68回・2017年10月3日（火曜日）. テーマ. 「本物の起業家とは ～Get it done! ～」 講師. 株式会社レノバ 代表取締役会長. 千本 倖生 氏. 開催報告. 当日の様子 回数・開催日. 第67回・2017年2月21日（火曜日）. テーマ. 「京都の文化遺産、明日はどうなっているだろう」 講師. 立命館大学歴史都市防災研究所教授／京都大学 名誉教授. 土岐 憲三 氏. 開催報告. 当日の様子 回数・開催日. 第66回・2016年12月1日（木曜日）. テーマ. 「激変する情報通信市場と新たなチャレンジ」 講師. 西日本電信電話株式会社 代表取締役社長 村尾 和俊 氏. 開催報告. 当日の様子 回数・開催日. 第65回・2016年6月16日（木曜日）. テーマ. 「国際機関で働く」 講師. 公益財団法人フォーリン・プレスセンター理事長／元国連事務次長. 赤阪 清隆 氏. 開催報告. 当日の様子 回数・開催日. 第64回・2015年12月2日（水曜日）. テーマ. 「世界をリードする日本の鉄鋼産業 －もの造りの現場第一線から－」 講師. 新日鐵住金株式会社相談役. 友野 宏 氏. 開催報告. 当日の様子 回数・開催日. 第63回・2015年6月22日（月曜日）. テーマ. 「TVドラマの作りかた －プロデューサーの現場から－」 講師. NHKドラマ番組部チーフ・プロデューサー. 訓覇 圭 氏. 開催報告. 当日の様子 回数・開催日. 第62回・2015年5月25日（月曜日）. テーマ. 「揺れてどこまで行くのやら」 講師. 医師・高雄病院地域医療室室長. 川島 実 氏. 開催報告. 当日の様子 回数・開催日. 第61回・2014年12月2日（火曜日）. テーマ. 「変化の時代」 講師. ボストンコンサルティンググループ日本代表. 御立 尚資 氏. 開催報告. 当日の様子 回数・開催日. 第60回・2014年10月24日（金曜日）. テーマ. 「呼吸する小説」 講師. 作家 いしいしんじ 氏. 開催報告. 当日の様子 回数・開催日. 第59回・2014年7月4日（金曜日）. テーマ. 「「国難」研究の最前線 －南海トラフ巨大地震と首都直下地震の被害の定量化－」 講師. 関西大学社会安全研究センター長・教授／京都大学名誉教授／阪神・淡路大震災記念 人と防災未来センター長. 河田 惠昭 氏. 開催報告. 当日の様子 回数・開催日. 第58回・2013年12月3日（火曜日）. テーマ. 「あしたを、ちがう「まいにち」に。～社会から必要とされる存在であり続けるために～」 講師. TOTO株式会社代表取締役会長兼取締役会議長. 木瀬 照雄 氏. 開催報告. 当日の様子 回数・開催日. 第57回・2013年11月7日（木曜日）. テーマ. 「ジェンダー研究のススメ」 講師. 社会学者・東京大学名誉教授. 上野 千鶴子 氏. 開催報告. 当日の様子 回数・開催日. 第56回・2013年7月2日（火曜日）. テーマ. 「歌声合成システムVOCALOID－新しい楽器と新しい音楽」 講師. ヤマハ株式会社VOCALOIDプロジェクトリーダー（VOCALOID開発者）. 剣持 秀紀 氏. 開催報告. 当日の様子 回数・開催日. 第55回・2012年12月5日（水曜日）. テーマ. 「M＆Aの本質と企業価値向上－組織と個人の最適な関係－」 講師. GCAサヴィアングループ株式会社取締役／一橋大学大学院 国際企業戦略研究科教授／インテグラル株式会社代表取締役. 佐山 展生 氏. 開催報告. 当日の様子 回数・開催日. 第54回・2012年9月26日（水曜日）. テーマ. 「なぜエンターテインメントに残虐な表現が必要なのか」 講師. 作家 貴志 祐介 氏. 開催報告. 当日の様子 回数・開催日. 第53回・2012年7月9日（月曜日）. テーマ. 「アヴァンギャルド京都と「美術京都」の未来」 講師. 京都市美術館長・元京都市立芸術大学長. 潮江 宏三氏. 開催報告. 当日の様子 過去の開催記録（2007年度・第28回～2011年度・第52回） 過去の開催記録（2004年度・第1回～2006年度・第27回） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\n北部構内総務課人事掛（農学、基礎研、フィールド等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-6024 a60hokubujinji*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n宇治地区人事掛 〒 611-0011 宇治市五ケ庄 0774-38-3337 uji.jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n宇治地区人事掛 〒 611-0011 宇治市五ケ庄 0774-38-3337 uji.jinji*mail2.adm.kyoto-u.ac.jp\n宇治地区人事掛 〒 611-0011 宇治市五ケ庄 0774-38-3337 uji.jinji*mail2.adm.kyoto-u.ac.jp\n宇治地区人事掛 〒 611-0011 宇治市五ケ庄 0774-38-3337 uji.jinji*mail2.adm.kyoto-u.ac.jp\n北部構内総務課人事掛（農学、基礎研、フィールド等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-6024 a60hokubujinji*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n北部構内総務課人事掛（理学、数研等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-3606 a60hokubujinji*mail2.adm.kyoto-u.ac.jp\n複合原子力科学研究所総務掛 〒 590-0494 大阪府泉南郡熊取町朝代西2丁目 072-451-2310 soumu2*rri.kyoto-u.ac.jp\n北部構内総務課霊長類研究所総務掛 〒 484-8506 愛知県犬山市官林41-2 0568-63-0567 soumu_reichou*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n附属図書館　図書館企画課 図書館企画掛 〒 606-8501 京都市左京区吉田本町 075-753-2616 soumu660*mail2.adm.kyoto-u.ac.jp\n医学研究科総務企画課企画広報掛 〒 606-8501 京都市左京区吉田近衛町 075-753-4305、4669 060k-kouhou*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n生態学研究センター総務掛 〒 520-2113 滋賀県大津市平野2丁目509-3 077-549-8200 620groupA*mail2.adm.kyoto-u.ac.jp\n人文科学研究所\n数理解析研究所\n経済研究所\n基礎物理学研究所\nウイルス・再生医科学研究所\n学術情報メディアセンター\n附属図書館\nｉＰＳ細胞研究所\n東南アジア地域研究研究所\n霊長類研究所\n経営管理研究部\n公共政策連携研究部\n地球環境学堂\n化学研究所\nエネルギー理工学研究所\n生存圏研究所\n防災研究所\n生態学研究センター\n複合原子力科学研究所\n部局別の担当部署連絡先\n【留意事項】\n・部局名をクリックすると、当該部局のHPにリンクします。（部局HPがある場合のみ）\n・依頼に係る手続きについては、予め各部局HP等を参照のうえ、各担当部署にご確認ください。\n・兼業許可にあたっては会議附議等、学内手続きに時間を要しますので、予め余裕をもってご依頼ください。\n　特に、研究科長等の部局の長が行うものや営利企業の役員の職にかかるものについては、通常よりも\n　審査に時間を要しますので、ご留意ください。\n【依頼時期の目安】\n・部局の長が行うもの、営利企業の職にかかるもの…兼業開始日の２～３か月前\n・上記以外のもの…兼業開始日の１～２か月前\n　※依頼日から兼業開始日までの期間が短い場合、開始日までに許可できないことがあります。\nエネルギー科学研究科\n人間・環境学研究科\n農学研究科\n薬学研究科\n理学研究科\n経済学研究科\n法学研究科\n教育学研究科\n文学研究科\n医学研究科\n工学研究科\n担当部署名\n住所\n担当部署連絡先\n部局名\n医学部附属病院\n総合生存学館\n生命科学研究科\n情報学研究科\nアジア・アフリカ地域研究研究科\nTEL E-mail(*を@に変更してください)\n担当部署名\n住所\n担当部署連絡先\n部局名\n野生動物研究センター事務掛（総務担当） 〒 606-8203 京都府京都市左京区田中関田町2-24 075-771-4393 soumu_yasei*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n北部構内総務課人事掛（農学、基礎研、フィールド等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-6024 a60hokubujinji*mail2.adm.kyoto-u.ac.jp\n福井謙一記念研究センター事務室 〒 606-8103 京都市左京区高野西開町34-4 075-711-7708 090akinen*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n情報部情報推進課総務掛 〒 606-8501 京都市左京区吉田本町　学術情報メディアセンター北館 075-753-7400、7402 740jyohokikaku-soumu*mail2.adm.kyoto-u.ac.jp\n研究推進部産官学連携課産官学連携企画掛 〒 606-8501\n京都市左京区吉田本町京都大学\n国際科学イノベーション棟 東館5階\n075-753-5536 sanren-kikaku*mail2.adm.kyoto-u.ac.jp\n研究推進部産官学連携課オープンイノベーション機構支援室 〒 606-8501\n京都市左京区吉田本町\n京都大学国際科学イノベーション棟\n075-753-7767 oi-shienjimu*mail2.adm.kyoto-u.ac.jp\n企画部国際交流課国際企画掛 〒 606-8501 京都市左京区吉田本町 075-753-2045 kokusai-jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n大学文書館事務室 〒 606-8305 京都市左京区吉田河原町15-9 075-753-2651 archives*mail2.adm.kyoto-u.ac.jp\n吉田南構内共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田二本松町 075-753-6582 A30jinji-kyuyo*mail2.adm.kyoto-u.ac.jp\n学術研究支援室事務部門 〒 606-8501 京都市左京区吉田本町　学術研究支援棟 075-753-5108 shienshitu-jimu*mail2.adm.kyoto-u.ac.jp\n南西地区共通事務部総務課人事掛 〒 606-8507 京都市左京区聖護院川原町53 075-366-7107 A50jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\n次世代研究創成ユニット 本部構内（理系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町３６番地１ 075-753-3587 A20jinji*mail2.adm.kyoto-u.ac.jp\nアジア研究教育ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\nスーパーグローバルコース人文社会科学系ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n人社未来形発信ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n地域連携教育研究推進ユニット 本部構内（文系）共通事務部総務課人事掛 〒 606-8501 京都市左京区吉田本町 075-753-3008 A10jinji-kyousai*mail2.adm.kyoto-u.ac.jp\n実験と理論計算科学のインタープレイによる\n触媒・電池の元素戦略研究拠点ユニット\n桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\n構造材料元素戦略研究拠点ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\nインフラシステムマネジメント研究拠点ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\nレジリエンス実践ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\nスーパーグローバルコース化学系ユニット 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\n光量子センシング研究拠点 桂地区（工学研究科）総務課人事掛 〒 615-8530 京都市西京区京都大学桂　 075-383-2009 090sjinji*mail2.adm.kyoto-u.ac.jp\n宇宙総合学研究ユニット 北部構内総務課人事掛（理学、数研等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-3606 a60hokubujinji*mail2.adm.kyoto-u.ac.jp\n森里海連環学教育研究ユニット 北部構内総務課人事掛（農学、基礎研、フィールド等担当） 〒 606-8502 京都府京都市左京区北白川追分町 075-753-6024 a60hokubujinji*mail2.adm.kyoto-u.ac.jp\n企画部企画課総務掛 〒 606-8501 京都市左京区吉田本町 075-753-2600 kikakuka-soumu*mail2.adm.kyoto-u.ac.jp\n教育推進・学生支援部学生課総務掛 〒 606-8501 京都市左京区吉田本町 075-753-2506 840gakusei-soumu*mail2.adm.kyoto-u.ac.jp\n研究推進部研究推進課総務掛 〒 606-8501 京都市左京区吉田本町 075-753-2041 kensui.jinji*mail2.adm.kyoto-u.ac.jp\n施設部環境安全保健課機構事務掛 〒 606-8501 京都市左京区吉田本町 075-753-2387 810kikochosei*mail2.adm.kyoto-u.ac.jp\n学術研究支援室\n高等研究院\n大学文書館\n大学文書館\n高等教育研究開発推進センター\n学生総合支援センター\n環境安全保健機構\n研究推進部\n企画部\n福井謙一記念研究センター\n野生動物研究センター\n白眉センター\nアフリカ地域研究資料センター\nフィールド科学研究センター\n国際戦略本部\nオープンイノベーション機構\n産官学連携本部\n情報環境機構\n総合博物館\n教育推進・学生支援部\n男女共同参画推進センター\n学際融合教育研究推進センター\nこころの未来研究センター\n国際高等教育院\n"}
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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求 京大英文概要. 京大英文概要. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 バックナンバー 京大英文概要 多言語版 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 京都大学英文概要 Kyoto University 2021-2022. 全ページPDF：9.26 MB Front Cover (PDF) (1.47 MB) Contents (PDF) (1.64 MB) Mission Statement (PDF) (1.64 MB) Message from the President (PDF) (421 KB) Basic Concept for Internationalization (PDF) (421 KB) History of Kyoto University (PDF) (804 KB) Award-Winning Research (PDF) (804 KB) Kyoto University at a Glance (PDF) (1.063 MB) Current Initiatives at Kyoto University (PDF) (2.91 KB) Global Engagement (PDF) (3.36 MB) General Information (PDF) (6.66 MB) Kyoto: Japan's Cultural Heartland (PDF) (3.89 MB) For More Information (PDF) (467 KB) Back Cover (PDF) (1.47 MB) 資料編　Kyoto University Facts and Figures. 全ページ. 2021 (PDF)(958 KB) バックナンバー. 京都大学英文概要. Kyoto University 2020-2021 Kyoto University 2019-2020 Kyoto University 2018-2019 Kyoto University 2017-2018 Kyoto University at a Glance 2016 2nd Edition Kyoto University at a Glance 2016 Kyoto University at a Glance 2015 Kyoto University at a Glance 2014 Kyoto University Profile 2013 Kyoto University Profile 2012 Kyoto University Profile 2011 Kyoto University Profile 2010 Kyoto University Profile 2008\/2009（PDF） Kyoto University Profile 2007\/2008（PDF） Kyoto University Profile 2006\/2007（PDF） Kyoto University Profile 2004\/2005（PDF） 資料編　Kyoto University Facts and Figures. 2020(PDF) 2019 (PDF) 2018 (PDF) 2017 (PDF) 2016 2nd（PDF） 2016（PDF） 2015（PDF） 2014（PDF） 2013（PDF） 2012（PDF） 2011（PDF） 2010（PDF） 2008\/2009（PDF） 2007\/2008（PDF） 2006\/2007（PDF） 2004\/2005（PDF） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム EVENTS. EVENTS. 条件で絞り込む. 日付で絞り込む. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. 要申し込み. - すべて -. 必要. 不要. 開催地. - すべて -. 吉田キャンパス. 宇治キャンパス. 桂キャンパス. 東京オフィス・京都アカデミアフォーラム. その他の地域. オンライン. 開催日. 前月. 1月. 2月. 3月. 4月. 5月. 6月. 7月. 8月. 9月. 10月. 11月. 12月. 翌月. 日. 月. 火. 水. 木. 金. 土. 絞り込み条件をリセット. アイキャッチ画像. 京都大学創立125周年記念 附属図書館所蔵貴重資料展示. 開催日. 2022年06月01日〜12月22日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 産官学連携. 京大MBA×日経ビジネススクール共同講座「サービス経営のフロンティアとセオリー」. 開催日. 2022年06月09日、他. 開催地. オンライン. Events. タグ. 125周年. 公開講座. 京都大学創立125周年記念 市民講座「物理と宇宙」. 開催日. 2022年06月11日、他. 開催地. オンライン. Events. タグ. 125周年. 研究. 京都大学創立125周年記念 研究の魅力を見出す”鏡”プロジェクト. 開催日. 2022年06月11日. 開催地. オンライン. Events. タグ. 公開講座. 人文研アカデミー2022「ヴィリエ・ド・リラダンとフランス象徴主義 －『残酷物語』と『未来のイヴ』が現代に語りかけてくるもの」. 開催日. 2022年06月11日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 教育. 公開講座. オンラインコース「教育評価の基礎講座」2022. 開催日. 2022年06月15日、他. 開催地. オンライン. Events. タグ. 研究. 第256回アフリカ地域研究会「南アフリカの社会保障制度改革とケアの政治」. 開催日. 2022年06月16日. 開催地. その他の地域. オンライン. Events. タグ. 教育. グローバル教育展開オフィス主催：2022年度レクチャーシリーズ「ダイバシティー時代の「日本型教育」の行方 」第1回. 開催日. 2022年06月17日. 開催地. オンライン. Events. 創立125周年記念 アカデミックマルシェ. 開催日. 2022年06月18日. 開催地. その他の地域. Events. 体育会学生主催イベント 「京大体育会、まるごと魅せます」. 開催日. 2022年06月19日. 開催地. その他の地域. オンライン. Events. タグ. 教育. 公開講座. 教育学研究科 E.FORUM 2022 全国スクールリーダー育成研修「スクールリーダー育成のためのオンライン・リレー講座」. 開催日. 2022年06月19日、他. 開催地. オンライン. Events. タグ. 研究. 学術情報メディアセンターセミナー「教育DXと音声言語処理」. 開催日. 2022年06月21日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 公開講座. プラットフォーム学連続セミナーVol. 12「モビリティとプラットフォーム学」～シェアリング・MaaS・数理最適化から考える「これからの移動のかたち」～. 開催日. 2022年06月30日. 開催地. オンライン. Events. タグ. 公開講座. 第121回 京都大学丸の内セミナー「雄と雌の対立が作り出す植物と花の多様な性」. 開催日. 2022年07月01日. 開催地. 東京オフィス・京都アカデミアフォーラム. Events. タグ. 公開講座. 親子理科実験教室 第1回（全4回）「DNAって何だろう？～観察したあなたのDNAでネックレスを作ろう！～」. 開催日. 2022年07月03日. 開催地. 吉田キャンパス. Events. タグ. 研究. 京都大学医生物学研究所 第16回公開講演会「医生物学研究所の船出」. 開催日. 2022年07月09日. 開催地. 吉田キャンパス. Events. タグ. 公開講座. 京大MBA 2022 短期集中講座「企業価値評価とファイナンス －財務モデル・ESG・サステナビリティ－」. 開催日. 2022年07月21日、他. 開催地. 吉田キャンパス. 東京オフィス・京都アカデミアフォーラム. オンライン. Events. タグ. 125周年. 公開講座. 京都大学創立125周年記念 工学部公開講座・オープンセミナー. 開催日. 2022年07月30日. 開催地. オンライン. Events. タグ. 公開講座. 数理解析研究所 第43回数学入門公開講座. 開催日. 2022年08月01日〜08月04日. 開催地. 吉田キャンパス. オンライン. Events. タグ. 国際交流. 国際ワークショップ「ブロックチェーン会議 2022」. 開催日. 2022年08月04日〜08月05日. 開催地. その他の地域. オンライン. Events. タグ. 公開講座. 第122回 京都大学丸の内セミナー「利己性の進化が維持する生物多様性」. 開催日. 2022年08月05日. 開催地. 東京オフィス・京都アカデミアフォーラム. ページ送り. カレントページ. 1 ページ. 2 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 海外留学を希望する京大生へ 海外留学のリスク管理. 海外留学のリスク管理. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 留学の目的・種類 京大の留学プログラム 留学に関する予備知識 海外へ留学する京大生向け奨学金 海外留学のリスク管理 海外留学に伴うリスク リスクへの対応 海外留学と保険 海外安全情報 海外渡航に関する安全対策手引き 京都で学べる英語の授業 海外留学のための相談窓口 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 海外留学に伴うリスク. 海外留学には安全面、身体面、精神面、そして加害者になってしまうリスクなど、さまざまなリスクが伴います。ここでは、具体的にどのようなリスクがあるかを説明します。. 詳細はこちら リスクへの対応. ここでは、上記「海外留学に伴うリスク」に対してどのように対応するのが良いかを説明します。. ここでは一般的な事項について触れていますが、外務省の海外安全情報のページでは国ごとの安全対策について詳細な記述があります。留学前に、渡航先の安全情報を必ず確認し、それを念頭において行動してください。. 詳細はこちら 外務省 海外安全ホームページ 海外留学と保険. 留学には「海外留学に伴うリスク」で述べたようなリスクがあるので、十分な保険をかけていく必要があります。具体的なことは保険会社と相談の必要がありますが、ここでは保険の種類およびその内容について説明します。. 詳細はこちら 海外安全情報. 外務省が海外に旅行したり海外に滞在する人のために提供している情報等について説明します。. 詳細はこちら 渡航安全対策. 海外に渡航する際に、特定の国・地域によっては治安の悪化等により、渡航を自粛する等、特別の注意が必要な場合があります。渡航前に渡航計画や安全策をしっかり検討しましょう。. 海外に渡航する際の安全対策に関する情報を紹介しています。新型コロナウイルス感染症の世界的な感染拡大を踏まえ、「国際的に懸念される感染症への対策」を含めています。. 海外渡航に関する安全対策手引き（日・英） 海外でフィールドワークを実施する際は、上記「海外渡航に関する安全対策手引き」に加え、こちらの手引きを確認してください。留意事項を紹介しています。. 学生の海外でのフィールドワーク実施に関する安全手引き（日・英） 海外渡航にかかる可否判断基準. 大学では、渡航の可否判断にあたってのガイドラインとして、外務省の危険情報に準じ、「海外渡航にかかる可否判断基準」を制定しています。大学が主催するプログラムにおける渡航の可否は、渡航地の安全状況および渡航者にかかる諸事情を考慮のうえ、渡航許可にかかる権限者が判断します。. 研究旅行、観光旅行、帰省等の私事渡航においても、渡航先の安全情報を確認の上、本基準を参考にして渡航の判断をするよう推奨します。. 海外渡航にかかる可否判断基準 現在、大学では、新型コロナウイルスの世界的な感染拡大を受け、コロナ禍における海外渡航に関する方針を定めていますので、渡航に際しては以下の通知を確認してください。. 新型コロナウイルスに対する本学の方針について（第14版修正1）－海外渡航等を中心とした対応について－（日・英） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"ty0Ek9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/admissions\/undergrad\/jisshijokyo","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":61,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["ja","admissions","undergrad"],"url_file_name_s":"jisshijokyo","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["入学者選抜実施状況 | 京都大学"],"description_txt":["令和4年度（PDF） ［1.07MB］\n\t令和3年度の入学試験実施結果［246KB］"],"last_modified":"Wed Jun 08 16:45:03 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-1021594108634452731,"exact_signature_unique_b":true,"fuzzy_signature_l":-2384034514134969908,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["入学者選抜実施状況"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1592,"text_t":"入学者選抜実施状況 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. 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Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 関連情報 資料請求. 資料請求. サイドメニュー. 一般選抜 特色入試 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 資料請求 受験生サポート 新入生の方へ（国際高等教育院Webサイト内）（外部リンク） 入学前後のスケジュール（国際高等教育院Webサイト内）（外部リンク） 本学の選抜要項・大学案内等の請求方法には、１.～3．の3つの方法があります。. 1. テレメールで請求する方法. テレメール（インターネット）. 二次元バーコード（対応する携帯電話で読み取れます）. 料金は、. こちらのテレメール（インターネット）サイト でご確認ください。. 料金は、お届けした資料に同封されている支払方法に従い表示料金をお支払いください。（支払いに際しても手数料が別途必要になります。）. 大学案内は本Webサイトにも掲載しています。. 大学案内冊子「知と自由への誘い 」 総合人間学部案内、工学部案内、農学部案内につきましては、各学部Webサイトにて、電子媒体での公開を行っています。. 総合人間学部Webサイト（外部リンク） 工学部Webサイト（外部リンク） 農学部Webサイト（外部リンク） テレメールでの請求についての問い合わせ先. テレメールカスタマーセンター. IP電話： 050-8601-0102 （9時30分～18時00分）. 2. 「モバっちょ」で請求する方法. 携帯電話を利用する場合. 二次元バーコード. ※ 料金は、. こちらのDJC資料請求サイト「モバっちょ」 でご確認ください。. 特色入試選抜要項. 特色入試選抜要項・大学案内「知と自由への誘い 」. 大学案内「知と自由への誘い 」. ※ 携帯電話の機種、携帯電話会社との契約状況によって、通話料金と一緒にお支払いできない場合があります。その場合は、コンビニ後払いを選択してください。. パソコンの場合（クレジットカード決済のみとなりますが同じ料金で請求できます）. 「モバっちょ」での請求についての問い合わせ先. モバっちょカスタマーセンター IP電話： 050-3540-5005（10時00分～18時00分）. 3. 京都大学で直接入手する方法. 教育推進・学生支援部入試企画課. 9時00分～17時00分（月曜日～金曜日〈祝休日・年末年始・創立記念日および夏季休業日は除く〉）. ※ 大学案内「知と自由への誘い」は、教育推進・学生支援部入試企画課でのみ配付します。. 一般選抜入学者選抜要項・一般選抜学生募集要項および特色入試学生募集要項について. 一般選抜入学者選抜要項・一般選抜学生募集要項および特色入試学生募集要項については、冊子の配付は行いませんので、以下のリンクより、ダウンロード・閲覧・印刷してご利用ください。 . 一般選抜入学者選抜要項 一般選抜学生募集要項 特色入試学生募集要項 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 教育・学生支援 キャンパスライフ支援 健康管理. 健康管理. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生生活で注意してほしいこと 健康管理 健康科学センター 在学中の保険加入 住まい・食と購買 アルバイト 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ Contents. 学生健康診断 保健診療所 桂エクササイズ・ヒーリングルーム 本学には学生の健康のために「健康科学センター」が設置され、「保健診療所」で診療を行っています。また、すべての学生に対して健康診断を実施しています。. 学生健康診断. 健康科学センターでは、学校保健法ならびに学内規程に基づき、学生の健康管理のため以下の健康診断を実施しています。. 健康科学センター 学生定期健康診断（新学期に実施）. 毎年、学年始めに定期健康診断が行われます。これは全学生が必ず受けなければならないもので、これを無届けで受けない場合は当該年度の試験を受けることができませんので、特に留意してください。. 1. 定期健康診断欠席届. 指定された日に病気、その他やむを得ない事情により定期健康診断を受けられない場合は、所属学部等へ理由書（病気の場合は診断書）を添えて欠席届を提出してください。. 欠席届を提出した場合は、後日健康診断を受けてください。. 2. 健康診断結果通知書・健康診断証明書・健康診断書. 就職や大学院受験等で健康診断結果通知書、健康診断証明書が必要なときは、証明書自動発行機で受け取ってください。（学年始めの定期健康診断を受けていない者には交付できません。）. なお、健康診断書が必要なときは、保健診療所へ申し出てください。. 期間外学生健康診断. 定期健康診断終了後の入学者・留学者が対象、所属長の依頼書に従って実施しています。. 学生特別定期健康診断（血液検査）. 化学薬品等有害物質、組み換えDNA実験、腎症候性出血熱（HFRS）などに関連する作業を行う学生が対象です。. その他健康診断. その他、体育実習健診、学生団体健診などを随時行っています。. 保健診療所. 保健診療所では、下記各科の専門医が、学生および教職員の傷病診療と健康相談・精神衛生相談を行っています。. 診療科名. 内科、神経科. 診療日および時間. 詳しいことは保健診療所の受付掲示板を見てください。. 休診日. 土曜日、日曜日、国民の祝日および年末年始（12月28日～1月3日）は全日休診です。. なお、臨時休診日（定期健康診断実施日等）は、その都度受付の掲示板に掲示します。. 料金. 学生の健康相談・精神衛生相談および正課中における傷害についての初診時の料金は無料ですが、傷病診療（薬価、検査科、処置料その他の経費）は実費となります。. 詳細は. 環境安全保健機構 健康管理部門／健康科学センターのホームページ へ（外部リンク）. 桂エクササイズ・ヒーリングルーム. 健康科学センター桂分室では、京都大学の学生、研究生、職員、研究者の方々の心と体の健康維持･増進を目的として、エクササイズとヒーリングルーム（マッサージ・アロマテラピー・ミュージックテラピー）を設けています。無料でご利用いただけます。. 利用時間. 月曜日～金曜日 9時00分～13時00分、14時00分～17時00分. ご利用にあたって. ご利用当日に電話にてご予約ください。. ご予約の際は　桂分室（075-383-7308）までご連絡ください。. ルーム種別、希望時間、氏名、所属、電話番号をお伺いします。. アロマオイルご希望の方はお申し付けください。. 開室時間は診療カレンダーでご確認ください。（. 健康科学センターWebサイト ）. これまで通り、空室がある場合は予約なしでもご利用いただけます。. エクササイズルーム. サイクルエルゴメーター. ダンベル. トレッドミル. ヒーリングルーム. ミュージックテラピー. マッサージ（マッサージを使用しない場合はベッドとして利用いただけます。）. シャワールーム. エクササイズ利用の方のみの使用となります。. ※ シャワーのみのご利用はご遠慮願います。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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Singapore Statement on Research Integrity （研究公正に関するシンガポール宣言 - 2010）. 日本語訳 その他. 日本学術会議　科学研究における健全性の向上に関する検討委員会・研究健全性問題検討分科会. 「科学研究における健全性の向上について（回答）」 平成27年3月6日. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"sNzad8U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/sites\/default\/files\/inline-files\/5-yoshiki9-1-r303-2e26b4affb312a5b92ba32abe671b70a.docx","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":114,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["sites","default","files","inline-files"],"url_file_name_s":"5-yoshiki9-1-r303-2e26b4affb312a5b92ba32abe671b70a","url_file_ext_s":"docx","content_type":["application\/vnd.openxmlformats-officedocument.wordprocessingml.document"],"crawldepth_i":2,"collection_sxt":["user"],"title":["阪大総人第３４０号"],"author":"宮園  基彦","author_sxt":["宮園  基彦"],"last_modified":"Thu Mar 04 06:47:00 UTC 2021","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":7901136002889693330,"exact_signature_unique_b":true,"fuzzy_signature_l":-3384283232118648865,"fuzzy_signature_unique_b":true,"text_t":"（様式９－１：社外取締役等兼業依頼状）\n\n令和　　年　　月　　日\n\n　国立大学法人京都大学総長　殿\n\n                                                  \n　　　　　　　　　　　　　　　　　　　　　　　　　　　企業名\n                                                      代表者名　　　　　　　　 \n\n\n社外取締役の委嘱について（依頼）\n\n　このたび、貴学教員を当社社外取締役として下記のとおり委嘱したいので、ご同意くださるようお願いいたします。\n\n記\n\n　１．従事予定者：\n\n\n  ２．役員等の職名：\n２－２．（※）会社法に定める社外取締役要件を満たすこと：　□　確認済\n２－３．（※）業務執行を担当しないこと：　□　確認済\n（※）信用金庫の理事のうち株式会社の社外取締役に相当する職を依頼する場合のみ記入\n\n  ３．職務内容：\n\n　４．報酬の有無：　有（予定年額　　　　 円） ・ 無\n\n　５．職務への予定従事時間：\n　（出席予定会議名、曜日、時間数、時間帯、従事回数等の勤務形態を記入）\n\n　６．任　期：　令和　　年　　月　　日　から　令和　　年　　月　　日　まで\n\n　７．株主総会における選任予定日：　令和　　年　　月　　日\n\n　８．従事予定者の親族による経営への強い影響力の有無：　有 ・ 無\n\n　９．事業内容：\n\n１０．親会社の有無：　有 ・ 無\n　　　　名　称：\n　　　　所在地：\n　　　　事業内容：\n\n１１．事業者の所在地・担当者連絡先：\n　　　〒\n\n１２．その他：\n\n-----------------------------------------\n【回答について】\n本学では、兼業許可にあたっての文書回答を原則省略しております。\n文書回答が必要な場合は、ご依頼の際、その旨お申出のうえ、以下、ご留意ください。\n・文書回答を行う場合であっても、メール添付による回答（押印なし）を原則といたします。\n・やむを得ず、押印入りの文書回答を必要とする場合は、その旨記載のうえ、宛先が明記された返信用封筒を必ず同封し、郵送にてこの様式をお送りください。\n　・所定様式による回答が必要な場合にはご依頼の際、併せて、その様式をお送りください。\n\n（注意事項）\n１　この様式は、本学教員に株式会社の社外取締役又は信用金庫の理事のうち株式会社の社外取締役に相当する職を依頼する場合に使用してください。\n　　なお、信用金庫の理事のうち株式会社の社外取締役に相当する職を依頼する場合には依頼状題名及び依頼文の「社外取締役」を「非常勤理事」等、依頼する職名に修正してください。\n\n２　添付資料として、定款・組織図・事業報告書等を提出してください。\n\n３　「１．従事予定者」は、本学の所属部局及び職名も記入してください。\n\n４　「２－２．会社法に定める社外取締役要件を満たすこと」については、「信用金庫の理事のうち株式会社の社外取締役に相当する職」を依頼する場合のみ「確認済」の□にチェックしてください。\nなお、会社法に定める社外取締役要件とは以下のとおりであり、「株式会社」を「信用金庫」に、「取締役」や「監査役」を「非常勤理事」や「監事」等の信用金庫での役職に読み替えて確認してください。　\n24257011938000\n会社法第２条（抜粋）\n十五　社外取締役　株式会社の取締役であって、次に掲げる要件のいずれにも該当するもの\nをいう。\nイ　当該株式会社又はその子会社の業務執行取締役若しくは執行役又は支配人その他の使用人（以下「業務執行取締役等」という。）でなく、かつ、その就任の前十年間当該株式会社又はその子会社の業務執行取締役等であったことがないこと。\nロ　その就任の前十年内のいずれかの時において当該株式会社又はその子会社の取締役、会計参与（会計参与が法人であるときは、その職務を行うべき社員）又は監査役であったことがある者（業務執行取締役等であったことがあるものを除く。）にあっては、当該取締役、会計参与又は監査役への就任の前十年間当該株式会社又はその子会社の業務執行取締役等であったことがないこと。\nハ　当該株式会社の親会社等（自然人であるものに限る。）又は親会社等の取締役若しくは執行役若しくは支配人その他の使用人でないこと。\nニ　当該株式会社の親会社等の子会社等（当該株式会社及びその子会社を除く。）の業務執行取締役等でないこと。\nホ　当該株式会社の取締役若しくは執行役若しくは支配人その他の重要な使用人又は親会社等（自然人であるものに限る。）の配偶者又は二親等内の親族でないこと。\n\n\n５　「３．職務内容」については、具体的に記入してください。\n\n６　「４．報酬の有無」、「８．従事予定者の親族による株式会社の経営への強い影響力の有無」、及び「１０．親会社の有無」については、「有」又は「無」に○を附してください。\n　　「４．報酬の有無」が「有」の場合、括弧内に金額を記入してください。\n　　「１０．親会社の有無」が「有」の場合、名称、所在地及び事業内容を記入してください。\n\n７　「９．事業内容」については、貴社の事業内容を詳細に記入してください。\nなお、記入欄が不足する場合は、別紙に記入してください。別添の定款等に記載されている場合は、参照箇所を記入してください。\n\n８　「１２．その他」については、従事予定者を社外取締役候補者とする理由等を詳細に記入してください。\n\n９　この様式によるほか、必要事項が記入されていれば、他の様式で作成していただいても結構です。\n\n１０　この様式は押印不要です。メール添付による送付も受け付けます。\n（様式９－１：社外取締役等兼業依頼状）　　　　　　　　　　　　　　　　　　記　入　例\n\n令和　　年　　月　　日\n\n　国立大学法人京都大学総長　殿\n\n                                                  \n　　　　　　　　　　　　　　　　　　　　　　　　　　　○○○○株式会社\n                                                      代表取締役社長　○○　○○\n\n\n社外取締役の委嘱について（依頼）\n\n　このたび、貴学教員を当社社外取締役として下記のとおり委嘱したいので、ご同意くださるようお願いいたします。\n\n記\n\n　１．従事予定者：\n　　　　京都大学大学院○○研究科教授　○○　○○\n\n  ２．役員等の職名：　社外取締役\n\n  ３．職務内容：（教員が就こうとする社外取締役としての職務の内容が、主として取締役会等\nの監督機能強化を目的とするものであることを具体的に記載）\n126682513589000\n　４．報酬の有無：　有（予定年額　3,600,000 円） ・ 無\n\n　５．職務への予定従事時間：\n　（出席予定会議名、曜日、時間数、時間帯、従事回数等の勤務形態を記入）\n\n　　　　取締役会　第２土曜日　午後１時から午後３時（月１回　１回２時間）\n\n　６．任　期：　令和○○年○○月○○日　から　令和○○年○○月○○日　まで\n\n７．株主総会における選任予定日：　令和○○年○○月○○日\n473392511938000\n　８．従事予定者の親族による株式会社の経営への強い影響力の有無：　有 ・ 無\n\n９．事業内容：（具体的に記載）\n180022511938000\n１０．親会社の有無：　有 ・ 無\n　　　　名　称：\n　　　　所在地：\n　　　　事業内容：\n\n１１．事業者の所在地・担当者連絡先：\n　　　〒○○○－○○○○　○○市○○区○番○号\n○○○○株式会社○○部○○課　　担当：○○\n電話番号：○○○○○　ＦＡＸ：○○○○○\n\n１２．その他：\n　　　従事予定者を社外取締役候補者とする理由など\n","wordcount_i":232,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:45:13 UTC 2022","fresh_date_dt":"Sun Jan 22 11:46:56 UTC 2023","referrer_id_s":"jpjC89U8j7Mo","language_s":"ja","size_i":30284,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085746878939136}
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In view of the ongoing crisis, the university has decided to first offer a temporary hosting program to students from its partner universities in Ukraine, and ensure that they are provided with study and research facilities."],"last_modified":"Wed Jun 08 15:25:23 UTC 2022","dates_in_content_count_i":4,"dates_in_content_dts":["Wed Apr 06 00:00:00 UTC 2022","Fri Jun 03 00:00:00 UTC 2022","Fri Apr 22 00:00:00 UTC 2022","Mon Apr 18 00:00:00 UTC 2022"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-7907473104460631977,"exact_signature_unique_b":true,"fuzzy_signature_l":270263409336971013,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["Statement regarding the Hosting of Students from Ukraine"],"h2_txt":["グローバルナビゲーション（英語）","ヘッダー プライマリーリンク（英語）","ヘッダー セカンダリーリンク（日本語以外）","Related link","Latest news","KIZUNA hosts \"Manga and Anime Presentation Contest\" online (24 May 2022)","Swiss President Ignazio Cassis visits Kyoto University to Deliver Lecture","2021 Kyoto iUP Preparatory Course concludes with ceremony","\"AY2021 IAfP Short-term online exchange programs – Addis Ababa University – \/ AY2021 Online lecture series on lifelong sciences\"","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Wed Apr 06 12:00:00 UTC 2022","Fri Jun 03 12:00:00 UTC 2022","Fri Apr 22 12:00:00 UTC 2022","Fri Apr 22 12:00:00 UTC 2022","Mon Apr 18 12:00:00 UTC 2022"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":3056,"text_t":"Statement regarding the Hosting of Students from Ukraine | KYOTO UNIVERSITY. Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home NEWS Statement regarding the Hosting of Students from Ukraine. University News Statement regarding the Hosting of Students from Ukraine. ターゲット. Alumni Visitors Published on. 2022\/04\/06. On March 8, Kyoto University issued a Statement regarding the Situation in Ukraine. The statement announced the university’s intention to provide the maximum assistance to students, universities, and academic institutions affected by the Russian invasion of Ukraine, and specifically to make every effort to host Ukrainian students. In view of the ongoing crisis, the university has decided to first offer a temporary hosting program to students from its partner universities in Ukraine, and ensure that they are provided with study and research facilities. Kyoto University’s partner institutions in Ukraine are the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” and Taras Shevchenko National University of Kyiv. The university will implement special measures (including tuition fee waivers, etc.) to facilitate the temporary hosting of students from Ukraine, and will also provide the necessary living support in cooperation with the local governments of Kyoto Prefecture and Kyoto City, which have declared their intention to aid Ukrainian evacuees. In addition, a special Ukraine Crisis Relief Fund has been launched to help provide the necessary financial support, and the university has commenced seeking donations. We hope that Kyoto University’s efforts will help to secure a study and research environment for Ukrainian students, and contribute to Japan’s efforts to aid Ukraine. Nagahiro Minato. President. Related link. 　. Statement regarding the Situation in Ukraine 　. 　. Kyoto University Ukraine Crisis Relief Fund Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. Published on. 2022\/06\/03. KIZUNA hosts \"Manga and Anime Presentation Contest\" online (24 May 2022). Published on. 2022\/04\/22. Swiss President Ignazio Cassis visits Kyoto University to Deliver Lecture. Published on. 2022\/04\/22. 2021 Kyoto iUP Preparatory Course concludes with ceremony. Published on. 2022\/04\/18. \"AY2021 IAfP Short-term online exchange programs – Addis Ababa University – \/ AY2021 Online lecture series on lifelong sciences\". More news 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"r6Jb39U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/news\/2022-05-30-1","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":46,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","news"],"url_file_name_s":"2022-05-30-1","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["2022年度 本部構内定期入構許可証交付申請の受付（学外者・追加）について | 京都大学"],"description_txt":["ポスター（PDF）"],"last_modified":"Wed Jun 08 15:45:24 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":705264885684183022,"exact_signature_unique_b":true,"fuzzy_signature_l":-3559935596284355256,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["2022年度 本部構内定期入構許可証交付申請の受付（学外者・追加）について"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","受付期間","申請書","問い合わせ先","Latest news","2023年度 農学研究科修士課程学生募集について","マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて","令和4年6月1日付告示第4号","留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Mon May 30 12:00:00 UTC 2022","Fri Jun 03 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/220530-news-poster-319d56e99d722457c1a89fa899fe8770.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","””","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,300,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal sites default files inline 220530 news poster 319d56e99d722457c1a89fa899fe8770 jpg 319d 56e 99d 722457c 1a 89fa 899fe 8770 ”” icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1775,"text_t":"2022年度 本部構内定期入構許可証交付申請の受付（学外者・追加）について | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS 2022年度 本部構内定期入構許可証交付申請の受付（学外者・追加）について. 事務連絡 2022年度 本部構内定期入構許可証交付申請の受付（学外者・追加）について. ターゲット. 企業・研究者の方 公開日. 2022年05月30日. ポスター（PDF） 　本学では、頻繁に出入構を要する方に対して、事業年度ごとに定期入構許可を行っています。今回、学外の方を対象に、追加で本部構内の定期入構許可証を発行します。構内交通量規制のため、駐車場台数内の許可となり、定期入構整理料は月額3,000円（税込）です。. 　今回受け付けるのは2022年7月1日からの許可で、定期入構許可証交付申請受付期間は以下のとおりです。. 　※ 吉田南構内は別の扱いとなります。（窓口：吉田南構内共通事務部　環境安全室）. 受付期間. 2022年5月30日（月曜日）～6月10日（金曜日）. 申請書. こちらからダウンロードできます。. 定期入構許可証交付申請書（学外者用） 問い合わせ先. 施設部プロパティ運用課キャンパス安全掛. Tel ： 075-753-5305. E-mail： 810campusanzen＊mail2.adm.kyoto-u.ac.jp（＊を＠に変えてください）. Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS. NEWS. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 絞り込み条件をリセット. News. 公開日. 2022年05月30日. タグ. 産官学連携. 価値創造人材育成プログラム「Kyoto Creative Assemblage」の募集について. News. 公開日. 2022年05月30日. タグ. 同窓会. 中国同窓会（京京会、京仙会）と京都大学同窓会の懇談会が開催されました. News. 公開日. 2022年05月19日. タグ. 同窓会. 第24回「技術士を目指そう」説明会が開催されました. News. 公開日. 2022年05月13日. タグ. 産官学連携. 「サステイナブル・ガストロノミー起業塾 －食と農のミライを考える 2022」受講生の募集について. News. 公開日. 2022年04月22日. タグ. 同窓会. 2021年度 応援団OB・OG会「いぶき会」役員会が開催されました. News. 公開日. 2022年04月08日. タグ. 教育. 理化学研究所数理創造プログラムと東京大学教養学部および京都大学理学部が、オンライン授業を用いた新しい大学間連携講義を開設します. News. 公開日. 2022年04月07日. タグ. 同窓会. 第23回「技術士を目指そう」説明会が開催されました. News. 公開日. 2022年04月07日. タグ. 同窓会. 第1回京大以文会東京支部茶話会を開催しました. News. 公開日. 2022年04月06日. タグ. 研究. 日本電産株式会社の支援による寄附講座「優しい地球環境を実現する先端電気機器工学」の継続について発表しました. News. 公開日. 2022年04月06日. タグ. 研究. 京都大学貴重資料デジタルアーカイブ基金により附属図書館所蔵『奇兵隊日記』付属絵図を電子化・公開しました. News. 公開日. 2022年04月01日. タグ. 大学の動き. 第4期中期目標期間が始まりました. News. 公開日. 2022年03月29日. タグ. 同窓会. 2022年 京都大学地球物理学教室同窓会「京大知球会」講演会・総会を開催しました. News. 公開日. 2022年03月25日. タグ. 国際交流. 「第5回マヒドン大学オンサイトラボラトリーワークショップ」をオンライン開催しました. News. 公開日. 2022年03月25日. タグ. 同窓会. 京都大学技術士会 第14回講演会（イブニングセミナー東京）が開催されました. News. 公開日. 2022年03月24日. タグ. 入試. 令和5（2023）年度総合生存学館（思修館）博士課程学生募集について. News. 公開日. 2022年03月10日. タグ. 社会連携. 第17回京都大学附置研究所・センターシンポジウム／京都大学松山講演会を開催しました. News. 公開日. 2022年02月14日. Kyoto iUP留学生支援のためクラウドファンディングを開始しました. News. 公開日. 2022年02月09日. タグ. 教育. 第6回思修館の集いを開催しました. News. 公開日. 2022年02月08日. タグ. 受賞. 第15回湯川記念財団・木村利栄理論物理学賞授賞式を開催しました. News. 公開日. 2022年02月03日. タグ. 大学の動き. 医学部附属病院で、コロナ禍における新たな医療を生み出す取組への寄附を募集しています. ページ送り. カレントページ. 1 ページ. 2 ページ. 3 ページ. 4 ページ. 5 ページ. 6 ページ. 7 ページ. 8 ページ. 9 &hellip;. 次ページ. chevron_right. 最終ページ. chevron_right. chevron_right. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"r1Vtu8U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/sites\/default\/files\/inline-files\/guideline-v3-210419-9076967217f5952bcf62418b89dbd0c5.pdf","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":115,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["sites","default","files","inline-files"],"url_file_name_s":"guideline-v3-210419-9076967217f5952bcf62418b89dbd0c5","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["<4D6963726F736F667420576F7264202D208A8893AE90A78CC082CC834B83438368838983438393816991E6825294C5816A28323032312E342E313929>"],"author":"SOURIS001","author_sxt":["SOURIS001"],"last_modified":"Mon Apr 19 08:39:06 UTC 2021","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-339932464313834123,"exact_signature_unique_b":true,"fuzzy_signature_l":-6995462965536428182,"fuzzy_signature_unique_b":true,"text_t":"2021.4.19 \n新型コロナウイルス感染拡大防止に伴う活動制限のガイドラインについて \n（第３版） \n \n新型コロナウイルス感染拡大防止に対し、各部局において必要な対策が講じられるよう、危機対\n策本部において、下記のとおり活動制限のガイドラインを示します。 \n \n【 Category１： 授業（講義、演習、実験、実習）・課外活動】 \n \nレベル 授業活動等 課外活動 \nレベル０ \n（制限なし） \n通常 通常 \nレベル１ \n（最小限の制限） \n感染拡大の防止に最大限の配慮をした\n上で、可能なものは通常の形式で実施す\nる。 \n実施可能な感染対策を講じたとしても対\n面授業を実施することが困難である場\n合、対面とオンラインの併用又はオンライ\nンのみにより授業を実施する。 \n※別途定める感染拡大予防マニュアル\nに定める配慮を行う。 \n屋外における活動及び感染拡大\nの予防に関して十分な安全対策\nが確認された屋内施設における活\n動などを除き、課外活動を自粛す\nる。 \n \n \n※課外活動の実施にあたっては\n別途通知による制約あり。 \nレベル２（－） \n（対面授業の停止 \n・課外活動の制限） \n対面授業は原則停止し、オンライン授業\nを中心に実施する。やむを得ず対面授\n業を実施する場合には、感染拡大予防\nマニュアルを踏まえ、感染拡大の予防に\n十分留意しつつ、必要な安全対策を確\n認した上で実施する。 \n \n感染拡大の予防に関して十分な安\n全対策が確認された屋外における\n活動など一部を除き課外活動を\n自粛する。 \n \n \n※課外活動の実施にあたっては\n別途通知による留意事項あり。 \nレベル２ \n（対面授業の停止 \n・課外活動の自粛） \n対面授業は原則停止し、オンライン授業\nを中心に実施する。やむを得ず対面で\n実施する場合には感染拡大の防止に最\n大限の配慮をして行う。特に演習、実\n験、実習を対面で行う場合には、密集、\n密閉、密接を厳密に排除できることを確\n認した上で実施する。 \n全ての課外活動を自粛する｡ \n \nレベル３ \n（対面授業の停止 \n・課外活動の自粛） \n同上 同上 \nレベル４ \n（最小限の活動） \nオンライン授業のみ実施する。 全ての課外活動を停止する。 \nレベル５ \n（全学休講） \n全ての授業を停止する。 同上 \n \n※ オンライン授業の実施については、情報環境が整わない学生に対する配慮のために \n学内 WiFi 等を利用させる場合は、感染防止対策の措置が確実に講じられること \n※ 活動の自粛あるいは禁止中、やむを得ず一時的に部室等に立ち入らざるを得ない場 \n合には、感染防止対策の措置を講じ、用事が済み次第、速やかに退出すること \n  \n2021.4.19 \n新型コロナウイルス感染拡大防止に伴う活動制限のガイドラインについて \n（第３版） \n \n新型コロナウイルス感染拡大防止に対し、各部局において必要な対策が講じられるよう、危機対\n策本部において、下記のとおり活動制限のガイドラインを示します。 \n \n【 Category２： 学内会議の実施・職員の勤怠】 \n \nレベル 学内会議の実施 職員の勤怠 \nレベル０ \n（制限なし） \n通常 通常 \nレベル１ \n（最小限の制限） \n感染拡大の防止に最大限の配慮をし\nた上で、可能なものは通常の形式で\n実施する。 \n実施可能な感染対策を講じたとして\nも対面会議を実施することが困難で\nある場合、対面とオンラインの併用又\nはオンラインのみにより会議を実施\nする。 \n※別途定める感染拡大予防マニュ\nアルの授業の実施上の配慮に準じ\nた配慮を行う。 \n感染拡大の防止に最大限の配慮を\nした上で、可能なものは通常の勤務\n形態で実施する。 \n実施可能な感染対策を講じたとして\nも対面での勤務を実施することが困\n難な者については、在宅での勤務を\n実施する。 \n通勤時の混雑を回避しつつ、時差出\n勤を推奨する。 \n \nレベル２（－） \n（対面の制限） \n※レベル２と同じ 通常の業務量の維持に努めつつ、\n在宅で可能な業務は在宅勤務を推\n奨する。 \n時差出勤を推奨する。 \nレベル２ \n（対面の制限） \n感染拡大の防止に最大限の配慮を\nしたうえで、対面会議を実施する場\n合には、オンライン参加を推奨す\nる。 \n執務室における人の密度を抑制す\nるため、必要な業務の見直しを行\nいつつ、在宅で可能な業務は在宅\n勤務を推奨する。 \nレベル３ \n（業務の制限） \n原則、オンライン会議で実施する。 \n \n運営上必要な業務を絞り、執務の\n体制を分割し、出勤と在宅勤務と\n交代で実施する。 \nレベル４ \n（大幅な業務の制限） \n同上 非常に優先度の高い最小限の業務\nに従事する職員のみ出勤し、他は、\n原則として、在宅勤務とする。 \nレベル５ \n（緊急業務に限定） \n緊急に必要な会議のみオンライン\nで実施する。 \n \n緊急に出勤を要する最小限の要員\n以外、原則として、全ての職員の出\n勤を禁止する。 \n \n※ 活動制限の期間中は、緊急性の高くない会議の開催を延期や、案件の絞り込み等に \nよる所要時間の短縮、メールによる書面審議の実施など、運営上の配慮を講じること \n \n※ レベルによらず総じて、公共交通機関を利用する職員は、優先的に在宅勤務となるよう \n配慮し、用務等が終了次第速やかに帰宅すること \n  \n2021.4.19 \n新型コロナウイルス感染拡大防止に伴う活動制限のガイドラインについて \n（第３版） \n \n新型コロナウイルス感染拡大防止に対し、各部局において必要な対策が講じられるよう、危機対\n策本部において、下記のとおり活動制限のガイドラインを示します。 \n \n【 Category３：研究活動】 \n \nレベル 制限・活動等 \nレベル０ \n（制限なし） \n通常 \nレベル１ \n（最小限の制限） \n感染拡大の防止に最大限の配慮をしたうえで、研究業務は通常通り継続\nする。 \nレベル２（－） \n（研究室活動の制限） \n研究室内の換気、各員の手洗い、マスク着用の徹底、及び接触や不要不\n急な滞在時間の削減など、感染防止に充分な注意を払いつつ、通常の研\n究活動への復帰をめざす。        \nレベル２ \n（研究室活動の制限） \n感染拡大の防止に最大限の配慮をしたうえで、可能な限り研究室における\n研究作業時間を削減し、可能な作業は自宅で行うように努める。 \n研究室メンバー（学生を含む）の歓送迎会など､研究室外での行事や催し\nは禁止する。 \nレベル３ \n（研究従事員の制限） \n現在実施中の実験等および研究室の運営に必要な最低限の研究室メンバ\nーのみ出勤し、研究活動に従事する。 \nあらたに実験を開始することは原則禁止する。 \nレベル４ \n（最小限の研究活動） \n以下に示す非常に優先度が高い研究や作業に従事する研究室メンバー \nのみ、出勤する。 \n 容易に中断できない長期間に亘る実験等に従事している者 \n 上記以外の実験等を終了または中断のための作業に従事する者 \n 以下に従事する最低限の研究室構成員 \n機器装置・試薬類等の維持管理、実験動植物等の維持、寒剤の補充\nや研究室内設備等の維持管理など \n・サーバーメンテナンスを行う者 \nレベル５ \n（研究室閉鎖） \n全ての研究活動を停止し、停止期間終了後の研究室再開に不可欠な以下\nの作業に従事する最低限の者以外の出勤を禁止する。 \n・重要な機器装置・試薬類等の維持管理、不可欠で代替不能な実験動植\n物等の維持、寒剤の補充や研究室設備の緊急時対応など \n・サーバーメンテナンスを行う者 \n \n※ レベルによらず総じて、出勤する研究室構成員は、可能な限り公共交通機関の利用を避 \nけ、用務等が終了次第速やかに帰宅すること \n \n※ 実験動物施設等については、各レベルに対応する具体的な活動制限内容が定められてい\nるので、関係部局は、別途施設へ問い合わせること \n \n","wordcount_i":164,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:45:14 UTC 2022","fresh_date_dt":"Mon Jan 02 08:32:47 UTC 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 刊行物・資料請求 学生の海外でのフィールドワーク実施に関する安全手引き. 学生の海外でのフィールドワーク実施に関する安全手引き. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 京都大学概要 京大英文概要 京都大学の改革と将来構想 学部案内冊子の請求 京都大学への留学案内 紅萠（くれなゐもゆる） Kyoto U Research News アニュアルレポート／Annual Report 京大広報 留学生ハンドブック 海外留学の手引き 留学支度 －大学のプログラムで行く短期留学－ ハウジングガイドブック 学生の海外でのフィールドワーク実施に関する安全手引き 京都大学 第26代総長 山極壽一 活動報告書－おもろい大学の歩み 京都大学 松本紘 第二十五代総長の挑戦 －魅力・活力・実力ある大学を目指して－ 「京都大学法人化の前後」－ 尾池和夫第24代京都大学総長アクションレポート メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 　海外でフィールドワークを実施する際の一般的な留意事項を紹介しています。渡航前、渡航中、渡航後の安全対策に役立ててください。. 学生の海外でのフィールドワーク実施に関する安全手引き. 学生の海外でのフィールドワーク実施に関する安全手引き （769KB）. 英語版. Safety Guide for Students Engaged in Overseas Fieldwork （556KB）. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"rjOcTNU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/en\/global\/global-profile","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":50,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["en","global"],"url_file_name_s":"global-profile","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Global profile | KYOTO UNIVERSITY"],"description_txt":["The year 2022 will mark the   125th anniversary of the foundation of Kyoto University   . 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Skip to main content Site search. Researcher. Search. close. CLOSE. search. menu. グローバルナビゲーション（英語）. About. Message from the President. Operations and future plans. History. Facts and Figures. Awards and honors. University profile. Living in Kyoto. Public relations. Employment. Human rights. Gender equality. Health, safety, and environment. Study at KyotoU. Education and admissions. Career paths. Visa\/tuition\/scholarships\/housing. University facilities. FAQ. Research & Collaborate. Research news and departments. Policies. Collaboration. Research support and scholarships. Careers. Visa\/Housing\/Daily Life. About. Research compliance & ethics. Global KyotoU. International strategy. Global profile. Global partners. International risk & crisis management. Global University Social Responsibility. KyotoU International Concierge and information resources. ヘッダー プライマリーリンク（英語）. Current Students & Staff Alumni ヘッダー セカンダリーリンク（日本語以外）. Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Global KyotoU Global profile. Global profile. サイドメニュー（英語）. International strategy Global profile International facts & figures Overseas offices, international facilities, and research bases in Japan Research facilities run by Departments Kyoto University Offices in Tokyo Global partners International risk & crisis management Global University Social Responsibility KyotoU International Concierge and information resources The year 2022 will mark the. 125th anniversary of the foundation of Kyoto University . This represents both a historical moment to commemorate the University’s academic contribution to Japan and the world, and it also represents a new beginning, a new opportunity to look both in and outwards and to reinvent itself into the future. This represents a new bright start, guided by the motto 京大力、新輝点 (\". Kyodai-ryoku, shin kiten. \", or roughly translated as \"Kyoto University’s power, a new shining point\") from where Kyoto University aims to continue fostering its well-established global interactions. Kyoto University is located at the cultural heart of Japan, and its international vocation spurs out of its commitment to academic freedom and diversity. The University serves as a bridge between Eastern and Western philosophies; it also connects ancient traditions with most advanced human developments, and academic expansion. It is through wide exchange and interaction with the world that innovative research is created, that education becomes dynamic and inclusive, and that the value of the University overflows and engages with society at large. As a university opened to the world, Kyoto University actively engages and expands its frontiers. The figures in its international facts, its connections and networking capabilities through Overseas Offices, its worldwide network of. On-site Laboratories. , the setting of. strategic partnerships with other institutions of excellence, together with other international facilities, provide a strong ground upon which it continues bridging with the global society. International facts & figures Overseas offices and international facilities Kyoto University Offices in Tokyo 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション（韓国語）. 교토대학 소개. 대학소개. 총장인사. 안전. 교육・학생 지원. 유학・국제교육. 입시. 학사일정. 학생생활. 연구・산관학 연계. Security Export Control. 국제교류. 연구지원체제. 젊은층 연구자 지원. 학부・대학원. ヘッダー セカンダリーリンク（日本語以外）. 교통편 안내 문의처 기부 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 이동 경로. Home 학부・대학원. 학부・대학원. 학부. 종합인간학부 문학부 교육학부 (사범학부) 법학부 경제학부 이학부 (자연과학학부) 의학부 약학부 공학부 농학부 대학원. 문학연구과 교육학연구과 법학연구과 경제학연구과 이학연구과 의학연구과 약학연구과 공학연구과 농학연구과 인간·환경학연구과 에너지과학연구과 아시아·아프리카지역연구연구과 정보학연구과 생명과학연구과 종합생존학관(思修館) 지구환경학사 전문직대학원. 공공정책대학원 경영관리대학원 법과대학원 (Japanese) 의학연구과사회건강의학계전공 (Japanese) Program for Leading Graduate Schools. Graduate School of Advanced Leadership Studies Inter-Graduate School Program for Sustainable Development and Survaivable Societies Training Program of Leaders for Integrated Medical System Collaborative Graduate Program in Design Leading Graduate Program in Primatology and Wildlife Science 연구소. 화학연구소 인문과학연구소 재생의과학연구소 에너지이공학연구소 생존권연구소 방재연구소 기초물리학연구소 바이러스연구소 경제연구소 수리해석연구소 원자로실험소 영장류연구소 동남아시아연구소 iPS 세포연구소 Kyoto University Institute for Advanced Study (KUIAS). 물질-세포통합시스템거점 Institute for the Advanced Study of Human Biology (ASHBi) 교육 연구 시설. 학술정보미디어센터 방사선생물연구센터 생태학연구센터 지역연구통합정보센터 야생동물연구센터 고등교육연구개발추진센터 종합박물관 저온물질과학연구센터 (Japanese) 필드과학교육연구센터 후쿠이켄이치기념연구센터 마음의미래연구센터 문화재종합연구센터 학생종합지원센터 대학문서관 아프리카지역연구자료센터 하쿠비(白眉)센터 대학간융합교육연구추진센터 (Japanese) 기구. 국제고등교육원 (Japanese) 환경안전보건기구 (Japanese) 국제교류추진기구 정보환경기구 도서관기구 산관학연대본부 Features. Kyoto University's 125th Anniversary. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 간행물 채용 교통편 안내 フッター セカンダリーリンク. 사이트 맵 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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2022","referrer_id_s":"6vYqZ9U8j7Mo","language_s":"ja","size_i":111783,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":5,"references_internal_i":5,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085762570878976,"text_t":"京大の知 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 社会連携 公開講座・講演会等 京大の知. 京大の知. サイドメニュー. 公開講座・講演会等 京都大学春秋講義 京都大学未来フォーラム 京大の知 京都大学地域講演会 京都大学 － 稲盛財団合同京都賞シンポジウム 公開講座お知らせメール受付 京大の研究・教育を知る 社会人向け教育 教育・研究に関する情報・資料公開 産官学連携（共同研究・知的財産・ベンチャー支援） 京大とのつながりを深める 社会に開かれた施設等 終了した全学的なイベント 「京大の知」とは？. 「京大の知」は、全国各地を巡回して京都大学の研究成果を発信することを目的とする連続講演会です。毎回定められたテーマに沿って、京都大学の教員が講演を行います。無料の講演会で、どなたでもご参加いただけます。. ※ 「京大の知」は、2019年度までは「東京で学ぶ　京大の知」として東京地区において実施していました。. 今後の開催予定. 開催日. ※中止となりました。（2022年1月20日）. 2022年02月6日 日曜日. テーマ. 大学の使命と科学する心 総長 湊 長博. 場所. ANAクラウンプラザホテル富山. （富山市大手町2番3号）. 過去の開催記録. シリーズ33 「京都大学の女性リーダーたち」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2019年11月5日. 師弟関係という文化. 教育学研究科 稲垣 恭子 研究科長・教授. 当日の様子 2019年11月11日. 野生動物保全ラボの挑戦. 野生動物研究センター. 村山 美穂 センター長・教授. 当日の様子 2019年11月18日. タイで生きる・老いる・ケアすること －フィールドワークから考える. 東南アジア地域研究研究所. 速水 洋子 所長・教授. 当日の様子 2019年11月25日. 免疫のしくみ －生体の恒常性維持－. 稲葉 カヨ 京都大学理事. 当日の様子 シリーズ32「再生医療－現状と展望－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2019年9月4日. 幹細胞の制御 －再生医療の産業化へ向けて－. 高等研究院 物質-細胞統合システム拠点. 長谷川 光一 特定拠点講師. 当日の様子 2019年9月11日. iPS細胞を用いたパーキンソン病治療. iPS細胞研究所 高橋 淳 教授. 当日の様子 2019年9月18日. 再生医療とリハビリテーション. 医学研究科 青山 朋樹 教授. 当日の様子 2019年9月25日. 社会で考えるiPS細胞研究をめぐる倫理. iPS細胞研究所 藤田 みさお 特定教授. 当日の様子 シリーズ31「地球環境と私たちの暮らし－環境を考える－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2019年5月23日. 再エネで地球環境に貢献し、地域を豊かにする－全国各地の試みから－. 地球環境学堂 諸富 徹 教授. 当日の様子 2019年5月29日. マイクロプラスチックによる水環境汚染と私たちの生活. 地球環境学堂 田中 周平 准教授. 当日の様子 2019年6月6日. きれいな土と汚れた土を分けるもの. 地球環境学堂 勝見 武 教授. 当日の様子 2019年6月13日. 環境の汚染とリスク. 工学研究科 島田 洋子 准教授. 当日の様子 シリーズ30「明治150年－明治の歩みを考える－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2019年1月31日. 記憶としての明治維新－「明治50年」と「明治150年」のあいだで－. 法学研究科 奈良岡 聰智 教授. 当日の様子 2019年2月7日. 国産初の長距離海底ケーブル、日本－台湾－中国大陸を結ぶ. 東南アジア地域研究研究所 貴志 俊彦 教授. 当日の様子 2019年2月12日. 日本の近代美術－西洋とのであい－. 人文科学研究所 高階 絵里加 准教授. 当日の様子 2019年2月21日. 「極東」から「極西」へ－「日本哲学」への序章としての明治－. 文学研究科 杉村 靖彦 教授. 当日の様子 シリーズ29「未来につながる情報化社会－展望と課題－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2018年11月22日. ソーシャルメディアの光と影. 人間・環境学研究科 吉田 純 教授. 当日の様子 2018年11月27日. 脳から心を読む技術. 情報学研究科 神谷 之康 教授. 当日の様子 2018年12月4日. 情報化社会の働き方と労働法の未来. 法学研究科 島田 裕子 准教授. 当日の様子 2018年12月11日. プラットフォームの経済学. －なぜ独り勝ちが起きるのか？GoogleとUberのビジネスの秘密を考える－. 経済学研究科 依田 高典 教授. 当日の様子 シリーズ28「未来につながる情報化社会－日常生活の中の技術革新－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2018年5月15日. 子どもたちと機械翻訳が世界を繋ぐ. 情報学研究科 石田 亨 教授. 当日の様子 2018年5月22日. コンピュータ技術は医療をどう変えようとしているのか. 医学部附属病院 黒田 知宏 教授. 当日の様子 2018年5月29日. 情報化の進展と都市交通の変革. 工学研究科 宇野 伸宏 教授. 当日の様子 2018年6月5日. 防災・減災における情報化. －先端的な技術はどのように使われるのか－. 防災研究所 畑山 満則 教授. 当日の様子 シリーズ27「国際社会の中の日本－日本の果たす役割－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2018年2月8日. トランプ時代の暮らしと経済. 地球環境学堂 宇佐美 誠 教授. 当日の様子 2018年2月15日. インドネシアにおける荒廃乾燥泥炭地の回復への挑戦. 東南アジア地域研究研究所. 水野 広祐 教授. 当日の様子 2018年2月22日. グローバル人材育成と開発援助－総合生存学館の試み－. 総合生存学館（思修館）. 木邨 洗一 特定教授. 当日の様子 2018年3月1日. 国連の平和維持活動（PKO）と日本. 法学研究科 酒井 啓亘 教授. 当日の様子 シリーズ26「国際社会の中の日本－世界との関係・日本の現状－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2017年11月22日. 介護の担い手をめぐるグローバルな政策と実態. 文学研究科 安里 和晃 准教授. 当日の様子 2017年11月28日. 世界と日本の食料・農業. 経済学研究科 久野 秀二 教授. 当日の様子 2017年12月5日. 国際貿易と国際政治の交錯. 公共政策大学院. 鈴木 基史 教授. 当日の様子 2017年12月12日. 核軍縮と日本. 法学研究科 淺田 正彦 教授. 当日の様子 シリーズ25「時を考える」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2017年5月18日. 経済学における時間の捉え方. 経済学研究科 若井 克俊 教授. 当日の様子 2017年5月25日. 被災地の時間／未被災地の時間. 経済学研究科 岡田 知弘 教授. 当日の様子 2017年6月1日. インド仏教と時間. 文学研究科 宮崎 泉 教授. 当日の様子 2017年6月8日. 日を数える. 人文科学研究所 浅原 達郎 教授. 当日の様子 シリーズ24「時を知る」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2017年1月18日. 植物は体のどこで時間を測っているのか. 生命科学研究科 遠藤 求 准教授. 当日の様子 2017年1月25日. 体の時計は今何時？. 薬学研究科 岡村 均 教授. 当日の様子 2017年2月1日. ヒトの一生を科学する. 生命科学研究科 石川 冬木 教授. 当日の様子 2017年2月8日. 物理学における時間. 理学研究科 佐々 真一 教授. 当日の様子 シリーズ23「人工知能と人間社会」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2016年10月25日. 人工知能さん、あなたは一体誰？. 情報学研究科 西田 豊明 教授. 当日の様子 2016年11月1日. 話がわかるコンピュータ －音声認識と会話ロボットの最前線－. 情報学研究科 河原 達也 教授. 当日の様子 2016年11月8日. ビッグデータと「人工知能」 －社会へのインパクト－. 情報学研究科 田中 克己 教授. 当日の様子 2016年11月15日. AIブームは本物か？ －米国の場合、日本の場合－. 文学研究科 林 晋 教授. 当日の様子 シリーズ22「世界とつながる日本の問題 －憲法・格差・環境・食－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2016年5月12日. 立憲民主主義の国際化と日本国憲法  －公布70年の今－. 法学研究科 毛利 透 教授. 当日の様子 2016年5月19日. 所得格差はなぜ拡大しているのか？ －日本とアメリカのちがい－. 公共政策連携研究部. 宇仁 宏幸 教授. 当日の様子 2016年5月26日. 日本と世界の水環境問題 －日本の経験は生かせるか？. 地球環境学堂 藤井 滋穂 教授. 当日の様子 2016年6月2日. 食品価格とフードシステムの持続 －安ければいいのか？－. 農学研究科 新山 陽子 教授. 当日の様子 シリーズ21「大地震、何が起きる？どう備える？」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2016年3月9日. 巨大地震に備えた都市・建築物の地震対策. 工学研究科 林康裕 教授. 当日の様子 2016年3月16日. 巨大地震の予測と地震科学の限界. 防災研究所 橋本学 教授. 当日の様子 2016年3月23日. コンピュータ地震なまずの飼育法. 理学研究科 平原和朗 教授. 当日の様子 2016年3月30日. 復興から考える、地震で起きること・備えること. 防災研究所 牧紀男 教授. 当日の様子 シリーズ20「「コミュニケーション」を考える」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2016年1月13日. 心を通わせる能力はどうして進化したのか？. 霊長類研究所 正高信男 教授. 当日の様子 2016年1月20日. 民意の読み書き能力. 教育学研究科 佐藤卓己 教授. 当日の様子 2016年1月27日. アフリカの声の世界とインターネット・コミュニケーション. アジア・アフリカ地域研究研究科 木村大治 教授. 当日の様子 2016年2月3日. 人間言語の特質とその起源・進化. 人間・環境学研究科 藤田耕司 教授. 当日の様子 シリーズ19「「家族」はどこへ行く？」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2015年10月14日. 教育家族のゆくえ －愛護され、教育される子どもたち－. 国際高等教育院 小山静子 教授. 当日の様子 2015年10月21日. あれも「家族」、これも「家族」 －多様化する「家族」と家族政策－. 文学研究科 伊藤公雄 教授. 当日の様子 2015年10月28日. 縮小する日本らしさ －ライフコースと家族の40年－. 教育学研究科 岩井八郎 教授. 当日の様子 2015年11月4日. ワーク・ライフ・バランス －家族の問題、それとも社会の問題？ －. 公共政策連携研究部 久本憲夫 教授. 当日の様子 シリーズ18「生命・いのち －誕生から死まで－」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2015年7月9日. 寿命と老化の仕組みを探る. 生命科学研究科 西田英介 教授. 当日の様子 2015年7月16日. 「いのち」と向き合う. 医学研究科 田村恵子 教授. 当日の様子 2015年7月23日. 「生老病死」に活きる日本人の経験智. こころの未来研究センター. カール・ベッカー 教授. 当日の様子 2015年7月30日. 細胞培養から見えてきた生命像. 医学研究科 篠原隆司 教授. 当日の様子 シリーズ17「変動する社会と激変する自然災害」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2015年3月5日. 豪雨と斜面災害 －雨の降り方、地質、地形から地域の減災を考える－. 防災研究所 松四雄騎 准教授. 当日の様子 2015年3月12日. 災害リスク・コミュニケーションの新しいかたち －想定を活かすために－. 防災研究所 矢守克也 教授. 当日の様子 2015年3月19日. 火山活動の予測 －できて当然の火山・ハードルの高い火山－. 理学研究科 鍵山恒臣 教授. 当日の様子 2015年3月26日. 都市水害 －地下と車に要注意－. 経営管理研究部 戸田圭一 教授. 当日の様子 シリーズ16「社会に浸透する情報技術」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2014年9月18日. オープンエデュケーションの世界 －21世紀の「知の革命」と教育の未来－. 高等教育研究開発推進センター長. 飯吉 透 教授. 当日の様子 2014年9月22日. ゲノム情報のコンピュータ解析 －高等数学+αによる先端的解析手法－. 化学研究所附属バイオインフォマティクスセンター長 阿久津達也 教授. 当日の様子 2014年10月1日. 安全・安心な社会の構築 －情報技術を活用したこれからの社会基盤整備と維持管理－. 工学研究科 杉浦邦征 教授. 当日の様子 2014年10月6日. 情報技術の社会への展開 －ビッグデータの検索と分析－. 情報学研究科 田中克己 教授. 台風の影響により、中止しました。. シリーズ15「こころの未来－私たちのこころは何を求めているのか」. 詳細はこちら 開催日. テーマ. 講演者. 開催報告. 2014年5月28日. 日本文化における主体性とは何か －日本人の意識、感情、関係性からの考察. こころの未来研究センター 内田由紀子 特定准教授. 当日の様子 2014年6月4日. 自分の意思で決めるとはどういうことか？ －心理学と脳科学の視点から. こころの未来研究センター上廣こころ学研究部門 阿部修士 特定准教授. 当日の様子 2014年6月11日. 「求めるべき幸福とは －ブータンの国民総幸福とその根底に横たわる精神性」. こころの未来研究センター上廣こころ学研究部門 熊谷誠慈 特定准教授. 当日の様子 2014年6月25日. 「主体性は超えられるのか？ －心理療法における揺らぎと超越」. こころの未来研究センター 河合俊雄 教授. 当日の様子 シリーズ14「美術研究最前線」. 開催日. テーマ. 講演者. 開催報告. 2014年2月20日. 運慶研究の最前線. 文学研究科 根立研介 教授. 当日の様子 2014年2月27日. 若い画家たちの師としてのレンブラント. 文学研究科 中村俊春 教授. 当日の様子 2014年3月6日. 16世紀のヨーロッパにおける絵画形態の多様化現象をめぐって. 文学研究科 平川佳世 准教授. 当日の様子 2014年3月13日. アヴァンギャルド・アートを考える. 文学研究科 吉岡 洋 教授. 当日の様子 シリーズ13「宇宙に新たな知を求めて」. 開催日. テーマ. 講演者. 開催報告. 2013年11月27日. 宇宙と人文社会科学 －宇宙総合学研究ユニットの試み－. 学際融合教育研究推進センター 磯部洋明 特任准教授. 当日の様子 2013年12月4日. 探査機・人工衛星・ロケットの飛ばし方の極意. 生存圏研究所 山川 宏 教授. 当日の様子 2013年12月11日. 宇宙論の進展. 基礎物理学研究所 田中貴浩 教授. 当日の様子 2013年12月18日. 赤外線で、銀河系の中心部まで見とおし、系外惑星をさぐる. 理学研究科 長田哲也 教授. 当日の様子 シリーズ12「人間と宗教」. 開催日. テーマ. 講演者. 開催報告. 2013年8月8日. 西田幾多郎「善の研究」における＜宗教的要求＞とは何か. 文学研究科 氣多雅子 教授. 当日の様子 2013年8月20日. アウグスティヌス：邂逅の生涯とキリスト教. 文学研究科 川添信介 教授. 当日の様子 2013年9月5日. 「倫理と宗教」という観点から見た「人間と宗教」の問題. 総合生存学館 藤田正勝 教授. 当日の様子 2013年9月17日. マハトマ・ガンディーと宗教. 赤松明彦 理事・副学長. 当日の様子 シリーズ11「人はなぜハマる？」. 開催日. テーマ. 講演者. 開催報告. 2013年5月28日. 熱中と依存の境界線：精神医学の視点から. 医学研究科 村井俊哉 教授. 当日の様子 2013年6月4日. 社会科学で依存症をどうとらえるか？. 国立社会保障・人口問題研究所 西村周三 所長. 当日の様子 2013年6月11日. やめる脳、やめられない脳. こころの未来研究センター 船橋新太郎 教授. 当日の様子 2013年6月25日. 依存の脳画像. 医学研究科 福山秀直 教授. 当日の様子 シリーズ10「教育を考える」. 開催日. テーマ. 講演者. 開催報告. 2013年2月6日. 大学入試を生涯学習という観点から考える. 教育学研究科長 前平泰志 教授. 当日の様子 2013年2月20日. 臨床の知－心理療法の実践から. 教育学研究科 皆藤章 教授. 当日の様子 2013年2月27日. 三つ子の魂、どんな魂？－幼児期の心の発達を探る. 教育学研究科 子安増生 教授. 当日の様子 2013年3月6日. 才を見極める－教育哲学の観点から. 教育学研究科 鈴木晶子 教授. 当日の様子 シリーズ9「身近なナノテクノロジーの世界」. 開催日. テーマ. 講演者. 開催報告. 2012年11月29日. 生物の力を借りたバイオマスの電気への変換. 農学研究科 加納健司 教授. 当日の様子 2012年12月6日. 自然界に学ぶナノテクノロジー. 工学研究科 平尾一之 教授. 当日の様子 2012年12月13日. 燃料電池の実用化のためのナノテクノロジーの世界. 人間・環境学研究科 内本喜晴 教授. 当日の様子 2012年12月20日. 小さな機械が創る大きな機会. 工学研究科 田畑修 教授. 当日の様子 シリーズ8「農学部、京の食を語る」. 開催日. テーマ. 講演者. 開催報告. 2012年7月24日. うまさの科学 京の出汁（ダシ）. 農学研究科 伏木亨 教授. 当日の様子 2012年7月25日. 京で野菜を作りながら世界の食料を語る. 農学研究科 間藤徹 教授. 当日の様子 2012年8月2日. しもふり肉 和牛改良の最前線. 農学研究科 祝前博明 教授. 当日の様子 2012年8月20日. 京の魚 健康を支える力の秘密. 農学研究科 平田孝 教授. 当日の様子 シリーズ7「新しい社会、そのための経済政策」. 開催日. テーマ. 講演者. 開催報告. 2012年5月16日. 世界の成長の極、新興市場経済の行方. 経済研究所長 溝端佐登史 教授. 当日の様子（PDF） 2012年5月23日. 日本のデフレとマクロ経済政策の役割. 経済研究所附属先端政策分析研究センター 中澤正彦 准教授. 当日の様子（PDF） 2012年5月30日. イノベーションと被災地復興. 経済研究所附属先端政策分析研究センター 佐分利応貴 准教授. 当日の様子（PDF） 2012年6月13日. 討論：新しい社会の経済政策. 討論者：経済研究所附属先端政策分析研究センター長 矢野誠 教授、財務総合政策研究所長 稲垣光隆、独立行政法人 経済産業研究所理事長 中島厚志. 当日の様子（PDF） シリーズ6「中国学研究最前線」. 開催日. テーマ. 講演者. 開催報告. 2012年2月1日. 電子時代の漢字研究. 人間・環境学研究科 阿辻哲次 教授. 当日の様子 （PDF） 2012年2月8日. 国宝「王勃（おうぼつ）詩巻」続編の発見. 人間・環境学研究科 道坂昭廣 准教授. 当日の様子 （PDF） 2012年2月22日. 中国芸術史研究への展望. 文学研究科 宇佐美文理 准教授. 当日の様子 （PDF） 2012年2月29日. 辛亥革命100周年に際して. 人間・環境学研究科 江田憲治 教授. 当日の様子 （PDF） シリーズ5「人間とその進化の隣人たち」. 開催日. テーマ. 講演者. 開催報告. 2011年10月1日. 人間とは何か、想像するちから. 霊長類研究所長 松沢哲郎 教授. 当日の様子 （PDF） 2011年10月8日. ゴリラの社会に探る人間家族の起源. 理学研究科長 山極壽一 教授. 当日の様子 （PDF） 2011年10月15日. オラウータンとヒト、形態から考える. 野生動物研究センター 幸島司郎 教授. 当日の様子 （PDF） 2011年10月22日. チンパンジーとボノボから探るヒト社会の成立. 野生動物研究センター長 伊谷原一 教授. 当日の様子 （PDF） シリーズ4「女性として、研究リーダーとして」. 開催日. テーマ. 講演者. 開催報告. 2011年7月26日. 人はなぜ人間を分類するのか. 人文科学研究所 竹沢泰子 教授. 当日の様子 （PDF） 2011年7月30日. 「わたし」から始まる社会学. 文学研究科 落合恵美子 教授. 当日の様子 （PDF） 2011年8月2日. 免疫学の面白さ. 生命科学研究科 稲葉カヨ 教授. 当日の様子 （PDF） 2011年8月6日. 植物の生き方に魅せられて. 理学研究科 西村いくこ 教授. 当日の様子 （PDF） シリーズ3「安心安全生活まちづくり」. 開催日. テーマ. 講演者. 開催報告. 2011年5月9日. 交通・物流とまちづくり. 工学研究科 谷口栄一 教授. 当日の様子 （PDF） 2011年5月16日. 地域医療とまちづくり. 医学研究科 野本慎一 教授. 当日の様子 （PDF） 2011年5月23日. 景観とまちづくり. 工学研究科 川崎雅史 教授. 当日の様子 （PDF） 2011年5月30日. 人間の感性とまちづくり. 医学研究科 精山明敏 教授. 当日の様子 （PDF） シリーズ2「生きものの多様な世界」. 開催日. テーマ. 講演者. 開催報告. 2011年1月15日. 人類の夢、不老不死のベニクラゲの神秘. フィールド科学教育研究センター 瀬戸臨海実験所. 久保田信 准教授. 当日の様子 （PDF） 2011年1月29日. 海洋生物の行動学. フィールド科学教育研究センター 舞鶴水産実験所. 益田玲爾 准教授. 当日の様子 （PDF ）. 2011年2月5日. マツ枯れナラ枯れ. フィールド科学教育研究センター 上賀茂試験地長. 安藤信 准教授. 当日の様子 （PDF） 2011年2月12日. 海洋生物の生物多様性. フィールド科学教育研究センター長・瀬戸臨海実験所長. 白山義久 教授. 当日の様子 （PDF） 当日の様子 シリーズ1「王朝文学の世界」. 開催日. テーマ. 講演者. 開催報告. 2010年11月10日. 王朝文学、理解のカギは漢文学. 文学研究科 大谷雅夫 教授. 当日の様子 （PDF） 2010年11月17日. 物語の金字塔「源氏物語」. 文学研究科 金光桂子 准教授. 当日の様子 （PDF） 2010年11月24日. カタカナとひらがな. 文学研究科 大槻信 准教授. 当日の様子 （PDF） 2010年12月1日. 「話すように書かれた」源氏物語. 文学研究科 木田章義 教授. 当日の様子 （PDF） 当日の様子 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Global KyotoU KyotoU International Concierge and information resources. KyotoU International Concierge and information resources. サイドメニュー（英語）. International strategy Global profile Global partners International risk & crisis management Global University Social Responsibility KyotoU International Concierge and information resources Students Researchers & professors Find your academic partner at KyotoU General References, Access & Maps, Navigating Garoon (internal portal) Others, Alumni, Donate Welcome to KyotoU’s International Concierge & information resource management site. This site is meant to facilitate access to information relevant to the reader’s experience in, with, or in relation to Kyoto University, guiding the reader to find the \"entry points\" to different kinds of information according to the his or her needs and standpoint. The International Concierge offers access to a wide range of information, covering issues of interest to students such as procedures for registration, kinds of programs available, forms of academic and financial support, life as a student, and more. Content for researchers and professors covers job opportunities, working rules and procedures, benefits, human resources issues, life in Japan, and more. Students Prospective, current, graduating\/ed Researchers \/ professors Prospective, current, leaving KyotoU Find your academic institutions, partners, and contacts at Kyoto University General References Access & Maps Navigating Garoon (internal portal) Others Alumni Donate Note: Some of the information shared through International Concierge is for internal use at KyotoU; access requires logging into the University’s internal network on campus (KUINS Air) or providing a KyotoU ID (ECS-ID or SPS-ID) and password. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"u8vz9NU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/en\/research\/collaboration","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":51,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["en","research"],"url_file_name_s":"collaboration","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Collaboration | KYOTO UNIVERSITY"],"description_txt":["The threefold mission of the Office of Society-Academia Collaboration for Innovation is (1) to promote collaborative research among academia, industries and the government, (2) to support business start-ups by researchers or students, and (3) to manage and utilize the university’s intellectual properties."],"last_modified":"Wed Jun 08 15:30:14 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-9150946667954255395,"exact_signature_unique_b":true,"fuzzy_signature_l":-2118169394898234259,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["Collaboration"],"h2_txt":["グローバルナビゲーション（英語）","ヘッダー プライマリーリンク（英語）","ヘッダー セカンダリーリンク（日本語以外）","サイドメニュー（英語）","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":760,"text_t":"Collaboration | KYOTO UNIVERSITY. 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Directions Contact Make a gift Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 Breadcrumb. Home Research & Collaborate Collaboration. Collaboration. サイドメニュー（英語）. Research news and departments Policies Collaboration Research support and scholarships Careers Visa\/Housing\/Daily Life About Research compliance & ethics The threefold mission of the Office of Society-Academia Collaboration for Innovation is (1) to promote collaborative research among academia, industries and the government, (2) to support business start-ups by researchers or students, and (3) to manage and utilize the university’s intellectual properties. Collaborative Research Courses\/Laboratories, created by SACI in April 2010, specialize in collaborative research contracts and are established to improve the efficiency of collaborative research with Kyoto University. These courses are supervised by the deans and directors of the university and are supported by the various faculties to help ensure the successful progress of collaborative research. Comprehensive Collaboration Agreements are entered into between SACI and major global companies to identify common research interests from Kyoto University’s many distinguished academic research. SACI combines academic research and business applications into an educational foundation that encourages entrepreneurial and business-oriented thinking. SACI also maintains a close collaborative relationship with the Kyoto University Venture Fund (NVCC #1), promoting spin-out venture companies based on the university’s research discoveries. The Notable R&D for Technology Transfer website is managed by SACI as a portal to information on up-to-date technologies in their various stages of development: not only on intellectual properties such as patents, but also on technologies which are currently being researched. http:\/\/www.saci.kyoto-u.ac.jp\/en\/ 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. Publications Careers Directions フッター セカンダリーリンク. Site map Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"uzV7q9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/research\/rule\/export","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":49,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["ja","research","rule"],"url_file_name_s":"export","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["安全保障輸出管理 | 京都大学"],"description_txt":["Contents\n\n\n\tお知らせ \n\t安全保障輸出管理とは \n\t関係規程・通知等 \n\t本学の体制と取り組み \n\t啓発・教育 \n\t参考 \n\t安全保障輸出管理についての問い合わせ先 \n\n\n\n  お知らせ\n\n\n\t「外国ユーザーリストの改正情報」  が掲載されました。"],"last_modified":"Wed Jun 08 16:44:56 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-8103148066280488129,"exact_signature_unique_b":true,"fuzzy_signature_l":-5185740927129324889,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["安全保障輸出管理"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","お知らせ","安全保障輸出管理とは","関係規程・通知等","本学の体制と取り組み","啓発・教育","参考","安全保障輸出管理についての問い合わせ先","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"h3_txt":["学内説明会・e-Learning","その他説明会・セミナー等"],"italic_txt":["menu","search","close"],"imagescount_i":8,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1462,"text_t":"安全保障輸出管理 | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 研究・産官学連携 研究上の倫理・安全 安全保障輸出管理. 安全保障輸出管理. サイドメニュー. 研究活動を知る 研究成果の公開ポリシー 産官学連携（共同研究・知的財産・ベンチャー支援） 京都大学に寄附する 京都大学で研究したい 研究支援体制について 研究者が受けられる研究費・研究支援 研究上の倫理・安全 公的研究費の適正管理 公正な研究活動の推進 京都大学における軍事研究に関する基本方針 安全保障輸出管理 安全保障輸出管理の概要 関係規程・通知等 本学の取り組み 安全保障輸出管理についての問い合わせ先 利益相反マネジメント ライフサイエンス研究等に係る倫理の保持と安全の確保 研究の情報公開（オプトアウト等） 安全衛生管理 教職員に兼業を依頼したい Contents. お知らせ 安全保障輸出管理とは 関係規程・通知等 本学の体制と取り組み 啓発・教育 参考 安全保障輸出管理についての問い合わせ先 お知らせ. 「外国ユーザーリストの改正情報」 が掲載されました。（2022年3月10日）. ウクライナ情勢に関する措置 が講じられました。（2022年2月26日～）. みなし輸出管理制度が明確化 されました。（2021年11月18日公布（2022年5月1日施行））. 規制対象品目リストが改正 されました。（2021年10月15日（12月15日施行））. 安全保障輸出管理とは. 「学内限定」資料に関しては、教職員ポータル内の以下の場所に掲載しています。. 教職員ポータル ＞ ファイル管理 ＞文書共有 ＞ 研究推進部 ＞ 研究規範マネジメント室＞ 安全保障輸出管理. 安全保障輸出管理の概要 パンフレット「研究が兵器等に転用されないために～安全保障輸出管理の基本～」（学内限定）. 「安全保障輸出管理法令遵守について（日英対応）」（学内限定）. 関係規程・通知等. 「学内限定」資料に関しては、教職員ポータル内の以下の場所に掲載しています。. 教職員ポータル ＞ ファイル管理 ＞文書共有 ＞ 研究推進部 ＞ 研究規範マネジメント室＞ 安全保障輸出管理. 京都大学における安全保障輸出管理に関する規程 京都大学安全保障輸出管理規則（PDF）（学内限定）. 学術交流協定締結にかかるガイドライン（PDF）（学内限定）. 関係法令・改正情報 本学の体制と取り組み. 京都大学の安全保障輸出管理体制図 本学の取り組み（用語解説を含む） リスト規制 （全地域対象）. リスト規制対象貨物一覧表（輸出貿易管理令別表第1概要） 貨物・技術のマトリクス表（輸出貿易管理令別表第1、外国為替令別表の詳細）（経済産業省） キャッチオール規制 （. グループAの国（ホワイト国）  以外対象）. キャッチオール規制の詳細（経済産業省） 外国ユーザーリスト掲載大学 外国ユーザーリスト（経済産業省） 少額特例等の円表示金額換算方法及び換算率について（経済産業省） Q＆A（経済産業省） 啓発・教育. 学内説明会・e-Learning. 「学内限定」資料に関しては、教職員ポータル内の以下の場所に掲載しています。. 教職員ポータル ＞ ファイル管理 ＞文書共有 ＞ 研究推進部 ＞ 研究規範マネジメント室＞ 安全保障輸出管理. 学内説明会 実施通知、学内説明会資料（学内限定）. e-Learning研修（安全保障輸出管理） その他説明会・セミナー等. 安全保障貿易に係る大学・研究機関の教職員向けe-ラーニング（経済産業省）. ［日本語版］ ［英語版］ CISTEC e-Learning研修（ 安全保障輸出管理実務能力認定試験（STC）対策コース） （一般財団法人 安全保障貿易情報センター）. 安全保障貿易管理説明会資料（経済産業省） 参考. 経済産業省 安全保障貿易管理 「安全保障貿易に係る機微技術管理ガイダンス（大学・研究機関用）第四版」  . 特定非営利法人 産学連携学会. 「研究者のための安全保障貿易管理ガイドライン（改訂第2版）」  財団法人 安全保障貿易情報センター（CISTEC） EAR（米国輸出管理規則） 安全保障輸出管理についての問い合わせ先. 研究推進部研究規範マネジメント室（安全保障輸出管理担当）. Tel： 075-753-2298、5602. E-mail： info-yusyutu＊mail2.adm.kyoto-u.ac.jp （＊を@に変えてください）. 部局担当者（部局問い合わせ先）一覧（学内限定）については、教職員ポータル内の以下の場所に掲載しています。. ※ 教職員ポータル ＞ ファイル管理 ＞文書共有 ＞研究推進部 ＞ 研究規範マネジメント室＞ 安全保障輸出管理. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"upj8O9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/about\/facilities\/campus\/kizuna\/event","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":65,"url_protocol_s":"https","url_paths_count_i":5,"url_paths_sxt":["ja","about","facilities","campus","kizuna"],"url_file_name_s":"event","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["きずな月例イベント | 京都大学"],"description_txt":["最新イベント情報\n\n\n\t5月イベント（オンライン）：お気に入りマンガ・アニメ プレゼ"],"last_modified":"Wed Jun 08 16:44:58 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":2231480579870867734,"exact_signature_unique_b":true,"fuzzy_signature_l":5458739025418676196,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["きずな月例イベント"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","サイドメニュー","最新イベント情報","Interview @KIZUNA","過去のイベント","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"bold_txt":["きずなでは、今までに数々のイベントを行いました。"],"italic_txt":["menu","search","close"],"imagescount_i":10,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-sidebar-button.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jaaboutfacilitiescampuskizunaeventimagesKIZUNA_Event_eyecatch.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/embed\/jaaboutfacilitiescampuskizunaeventimagesKIZUNA_Event03.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","","","","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,250,255,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,900,900,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon sidebar button sites default files embed jaaboutfacilitiescampuskizunaeventimagesKIZUNA Event eyecatch jpg jaaboutfacilitiescampuskizunaeventimages KIZUNA Event03 03 facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":1841,"text_t":"きずな月例イベント | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について キャンパス・施設案内 大学の施設 留学生ラウンジ きずな きずな月例イベント. きずな月例イベント. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 キャンパスの概要 国内の研究所・附属研究施設等 海外の拠点 On-site Laboratory 大学の施設 百周年時計台記念館 楽友会館 宇治おうばくプラザ 国際科学イノベーション棟 ローム記念館 船井哲良記念講堂・船井交流センター 稲盛財団記念館 京都大学 東京オフィス 京都アカデミアフォーラム 医学部附属病院 白浜水族館 図書館機構（附属図書館を含む） 学術情報メディアセンター・情報環境機構 総合博物館 留学生ラウンジ きずな ML登録＆連絡先 きずな月例イベント きずなイベント（2022年） きずなイベント（2021年） きずなイベント（2020年） きずなイベント（2019年） 過去のイベント（2018年） 過去のイベント（2017年） 過去のイベント（2016年） 過去のイベント（2015年） 過去のイベント（2014年） 留学生ラウンジ「きずな」の施設案内 国際交流会館 教職員が利用できる施設 福利厚生施設・課外活動施設 写真・動画で見る京都大学 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 In other languages. English 最新イベント情報. 5月イベント（オンライン）：お気に入りマンガ・アニメ プレゼン大会／Manga and Anime Presentation Contest Interview @KIZUNA. 第1回 TALA BAKHTIARYさん（2021年8月21日） 第2回 Eko Heru Prasetyoさん（2021年9月17日） 第3回 SAPRA SHRESHTHさん（2021年11月15日） 第4回 江 欣樺さん（2022年1月31日） 過去のイベント. きずなでは、今までに数々のイベントを行いました。. 2022年 2021年 2020年 2019年 2018年 2017年 2016年 2015年 2014年 2013年 2012年 2011年 2010年 2009年 2008年 2007年 2006年 2005年 2004年 2003年 2002年 留学生ラウンジ「きずな」の施設案内 留学生ラウンジ「きずな」(トップページ) ML登録＆連絡先 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 入試・高大連携 特色入試 よくある質問. よくある質問. サイドメニュー. 一般選抜 特色入試 特色入試における選抜方法等の変更 特色入試学生募集要項 提出書類様式 特色入試インターネット出願 合格者発表・入学手続 特色入試に係る情報開示について 特色入試の過去問題 出願状況と選考結果 よくある質問 その他の学部入学試験 留学生のための入試 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 お知らせ. 現在公開中のものは、令和4年度特色入試の情報です。. 出願について 募集人員について 選抜日程について 出願（推薦）要件について 提出書類について. 調査書について 学業活動報告書・推薦書について 学びの設計書について 学びの報告書について 大学入学共通テストについて 医学部医学科の特色入試について その他 出願について. Q.1. インターネットによる出願の登録で、「出願資格情報」の「出身校」の記載がよくわからない。. A.1. 高等学校・中等教育学校の出身の方. 出身の学校名を先頭から入力ください。入力に従い、学校の候補がプルダウン形式で表示されますので、正しいものを選択してください。「都道府県立」や「学校法人名」などは不要です。. 入力後に候補が表示されない場合は、[出身校がみつからない場合はこちらから検索できます]のボタンをクリックし、出身の学校を検索してください。. 高等学校・中等教育学校以外の学校の出身などの方. [出身校がみつからない場合はこちらから検索できます]のボタンをクリックし、[その他の学校・高卒認定試験]を選択のうえ、該当する資格を選択してください。. 詳しくは下記をご確認ください。. 出願資格情報の入力例一覧 Q.2. インターネットによる出願の登録で、「出願資格情報」の「出願資格」でエラーが生じる。. A.2. 該当する資格をプルダウンメニューから選択してください。学部・学科により、出願資格は異なります。出願資格は、学部・学科のページをご確認ください。. なお、各学部・学科が指定する出願資格を満たさない資格を選択された場合は、エラーとなります。正しいものを選択してください。. Q.3. インターネットによる出願の登録で、「出願資格情報」の「資格取得年月」はいつになるのか。卒業月でよいのか。. A.3. 令和4年（2022年）3月に高等学校または中等教育学校を卒業見込みの方. 卒業式開催月にかかわらず、資格取得年月を「. 2022 . 」年「. ３ . 」月とプルダウンメニューから選択してください。. 医学部医学科志望の方（下記該当者のみ）. 国際科学オリンピック（数学、物理、化学、生物）日本代表で世界大会に出場し、高等学校、中等教育学校、又は国内の国際バカロレア認定校を令和5（2023）年3月卒業見込みの方は、資格取得年月を「. 2023 . 」年「 . 3 . 」月とプルダウンメニューから選択してください。. 募集人員について. Q.1. 学部間の併願は可能か。. A.1. 後期日程で実施する法学部を除き、学校推薦型・総合型選抜の別なく学部・学科・コース間の併願はできません。なお、理学部は、「数理科学入試」および「生物科学入試」に分けて募集し、両方に出願することはできません。. Q.2. 特色入試と一般選抜の併願は可能か。. A.2. 特色入試は、学部によって総合型選抜、学校推薦型選抜、後期日程試験と種別が異なっています。. 総合型選抜、学校推薦型選抜として実施する学部を受験する場合は、本学又は他大学における一般選抜の前・後期日程と併願は可能です。. 法学部は後期日程で実施しますので、本学又は他大学における一般選抜の前期日程と併願は可能ですが、他大学の後期日程との併願はできません。. Q.3. 募集人員どおりの合格者となるのか。. A.3. 募集人員数を目安に合格者を出す予定です。ただし、多少の増減はあり得ます。法学部を除き、選考の結果によって合格者数が募集人員に満たない場合は、その欠員分は一般選抜（前期日程）の募集人員に加えます。. 選抜日程について. Q.1. 10月出願の学部（教育学部・理学部・医学部人間健康科学科）の場合、2学期制の高校では高校3年生の成績を記載した調査書が出せないが、どうすればよいか。. A.1. 2学期制の高校は、出願の時点で高校3年生前期中間試験までの成績を記載し、提出してください。その後、期末試験まで入った高校3年生前期の成績を指定の期日（今年度（令和4年度）入試では11月1日（月曜日）午後3時必着）までに出願した学部（教育学部・理学部・医学部人間健康科学科）の教務掛へ書留速達で郵送してください。. 各学部教務掛の宛先は、特色入試学生募集要項に記載の問い合わせ先をご確認ください。. 出願（推薦）要件について. Q.1. 外国の学校を卒業（見込みを含む）したが、出願は可能か。. A.1. 外国の学校を卒業（見込みを含む）した人が出願できるのは法学部のみです。他の学部・学科では、日本の高等学校および在外教育施設を卒業（見込みを含む）した人のみ出願が可能です。. 提出書類：調査書について. Q.1. 高等学校卒業程度認定試験合格者で法学部を志望する場合、提出書類はどのようにすればよいか。. A.1. 調査書の代わりに高等学校卒業程度認定試験の「合格成績証明書」を提出してください。なお、高等学校卒業程度認定試験合格者が出願できるのは法学部のみです。. 提出書類：学業活動報告書・推薦書について. Q.1. 既卒生の学業活動報告書は、担任がいない場合は責任を持って記載できないが、どうすればよいか。. A.1. 学校に保存されている指導要録等における記載事項や、可能であれば志願者への聞き取りによって、箇条書きで事実のみを書いてください。特に、理学部の数理科学入試においては、数学および数理科学に関する特記事項、生物科学入試においては、理科もしくは数学に関する特記事項を、指導要録等に記載されている事項や志願者への聞き取りによって、箇条書きで事実のみを書いてください。. Q.2. 学年の順位を出していない場合どうするのか。. A.2. お手数ですが、学年（学科・コース）の順位を改めて出していただくようお願いします。. Q.3. 志願者の学年における順位は、成績評定平均値を小数点第1位までしか出していないので、同点者が多くなるがそれでもよいか。. A.3. 同点者が多くなっても構わないので、順位をつけてください。. Q.4. 理学部の学業活動報告書について、SSH校でない高校では、数学や理科に特化した行事は少なく、求められているような実績や成果を伴う活動内容を書くことは困難である。このような場合、学業活動報告書にどのようなことを書けばよいのか。未記入でもよいのか。. A.4. 研究発表で優秀賞を取ったなどの実績がなくても、生徒の勉学態度や生徒との会話などから、本人の数学もしくは生物学に関する能力や勉学意欲を示すようなエピソードがあれば具体的に書いてください。以下に例を記載します。. 数学や理科の教科で、強く興味を示した項目・題材. 授業中または授業外で、どのような質問をしたか. 授業中または授業外で、解いたり考えたりした問題と生徒の解答. 授業中に他の生徒とは違う発想で発言することがある. 自分自身で設定して粘り強く取り組んだ課題（授業中でも授業外の活動でも良い）. 理学に対する強い夢、熱意がある、そのような話をしていた　など. 未記入ではなく、受賞歴や教科の成績と直結しないエピソードでも結構ですので、生徒の能力・意欲を伺える事柄を書いてください。. 提出書類：学びの設計書について. Q.1. 学びの設計書は志望理由書と同じようなものと考えてよいか。. A.1. 本学への志望理由だけでなく、高等学校での学習から卒業後の進路まで含んだ、高等学校、大学、社会人までをつなぐ設計書として考えてください。. 提出書類：学びの報告書について. Q.1. 教育学部の「学びの報告書」の資料にはどのようなものを添付すればよいか。. A.1. 添付する資料、記載できる資格や検定の例としては、次のようなものが考えられます。ただし、これらに限定するものではありません。. 各教科で執筆したレポート、論文など. 総合的な学習の時間、SSH・SGH等課題研究などで探求・研究した成果をまとめたレポート、ポスター、論文、パワーポイント（印刷されたもの）など（以下省略）. 大学入学共通テストについて. Q.1. 各学部・学科の大学入学共通テストの成績基準は、どのように定められているのか。. A.1. 大学入試センター試験の成績を参考に、各学部・学科が入学者に求める大学入学共通テストの成績を「選抜方法及び基準」に定めています。ただし、大学入学共通テスト導入初期は、試験問題の傾向変化を考慮し、各学部・学科が定める得点（率）に「概ね」が付記されている場合、合否判定の際にその成績基準に配慮を行うことがあります。. 医学部医学科の特色入試について. Q.1. （医学部医学科）大学入学共通テストの成績評価提出は推薦要件となるのか。. A.1. 医学部医学科では、高大接続の観点から入学後の教育指導の参考とするため、高等学校の教育課程における学修成果の一つとして、大学入学共通テストの成績評価の提出を求めます。なお、大学入学共通テストの結果を合否判定に利用はいたしません。. Q.2. （医学部医学科）大学入学共通テストの指定科目をすべて受験しなくてはいけないのか。. A.2. 高大接続の観点から入学後の教育指導の参考とするために、高等学校の教育課程の教科・科目の習得による基礎学力の判断の一つとして、合否判定には利用しませんが、科目の指定は行います。. Q.3. （医学部医学科）大学入学共通テストについて、体調不良等による欠席や間違った科目を受験した場合は、不合格となるのか。. A.3. 大学入学共通テストの受験については、推薦要件としています。高等学校長がこのことを確認のうえで推薦いただくことになります。合格発表の後に万が一、欠席による不受験や間違った科目を受験しているケース等が判明した場合には、理由を確認させていただき、その結果、入学を認めない場合があります。. Q.4. （医学部医学科）出願資格2,3,4で受験する場合は、大学入学共通テストの取り扱いはどうするのか。. A.4. 出願資格3（飛び級）については、大学入学共通テストの出願資格に該当しないことからそもそも受験できません。ただし自然科学の領域において傑出した能力を有していることとします。. 出願資格2,4（IB）については、大学入学共通テストの成績の代わりに、出願時に国際バカロレア（IB）のディプロマスコア（予想点の場合には最終スコアを後日提出）を参考とします。. 出願資格については、下記をご確認ください。. 高等学校又は中等教育学校を令和4年3月卒業見込みの者. 国内の国際バカロレア認定校（DP）を令和3年4月から令和4年3月までに卒業又は卒業見込みの者. 国際科学オリンピック（数学、物理、化学、生物）日本代表として世界大会に出場した者で、高等学校又は中等教育学校を令和5年3月卒業見込みの者. 国際科学オリンピック（数学、物理、化学、生物）日本代表として世界大会に出場した者で、国内の国際バカロレア認定校（DP）を令和4年4月から令和5年3月までに卒業見込みの者. 文部科学大臣が高等学校の課程と同等の課程又は相当する課程を有するものとして認定又は指定した在外教育施設の当該課程を令和3年4月から令和4年3月までに修了又は修了見込みの者. Q.5. （医学部医学科）口頭試問の出題範囲はどうなるのか。. A.5. 医学部医学科の令和4年度特色入試では、口頭試問を第2次選考に課します。. 口頭試問では、与えられた課題に基づき口頭試問を行うことで、思考過程を評価することが効果的であると考えています。そのことから出題範囲については、高等学校での物理・化学・生物の3科目の範囲からの出題を予定していますが、これらの科目の複合的な問題として出題する可能性もあります。. 高等学校での履修状況によって不利にならないように複数の問題を用意しますので、受験生の方は、上記の3科目のうち、2科目について十分な準備をしていただく必要があります。. Q.6. （医学部医学科）口頭試問と面接試験の違いはなにか。. A.6. 口頭試問では、資料を読んでレポートを作成し、それに基づく口頭試問を行い、論理的思考力、文章構成力などについて評価します。面接試験では、京都大学が望む医学研究者、医師としての適性、社会的能力、科学的能力などについて評価します。. その他. Q.1. 受験時の宿泊先の斡旋は行っているか。. A.1. 本学では斡旋を行っていません。宿泊先の手配は、京都大学生活協同組合や旅行会社をご活用ください。. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. Scroll to top"}
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TUTOR GUIDEBOOK. サイドメニュー. 教育の体制と内容 経済支援 海外留学を希望する京大生へ 大学行事 課外活動支援 学生表彰など 学生相談・障害学生支援・就職 キャンパスライフ支援 学生支援の窓口・証明書発行など 学生支援の情報発信 京都大学に在籍する留学生の方へ 日本での学生生活 来日に際して 住まい・生活 健康・医療・保険 チューター制度 TUTOR GUIDEBOOK 留学生のための就職情報 留学生のためのアドバイジング・指導 留学生ラウンジ「きずな」 Prayer Room 防犯・防災 留学生の家族サポート 帰国に際して 留学生ハンドブック 留学生のための奨学金 在留資格変更・在留期限更新・再入国・休学・復学・退学等 手続担当窓口早見表 日本語教育 チューターについて説明しています。. TUTOR GUIDEBOOK （. 2.51 MB. ）. バックナンバー. チューターの手引き（2016） チューターの手引き（2012） チューターの手引き（2008） 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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{"id":"wJgaNNU8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/zh-tw\/privacy-policy","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":46,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["zh-tw"],"url_file_name_s":"privacy-policy","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["隱私權政策 | 京都大學"],"description_txt":["京都大學知道保護個人資料在進步的資通訊社會的重要性，於京都大學網站（ http:\/\/www.kyoto-u.ac.jp\/ ）處理相關資訊時，將遵守法令，並根據《京都大學個人資料保護相關章程》（以下稱「章程」）及下述方針致力於個人資料的保護。"],"last_modified":"Wed Jun 08 16:44:51 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-8968718899713983116,"exact_signature_unique_b":true,"fuzzy_signature_l":5347180211490271714,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["隱私權政策"],"h2_txt":["グローバルナビゲーション（繁体中国語）","ヘッダー セカンダリーリンク（日本語以外）","1. 關於個人資料的取得與保存","2. 關於個人資料的利用","3. 關於個人資料的管理","4. 關於個人資料的揭露、訂正、停用及刪除","5. 個人資料保護章程的制定、實施、維護及改善","6. 關於Google Analytics服務的利用","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":999,"text_t":"隱私權政策 | 京都大學. 移至主內容 網站內搜索. 教師搜索. 搜索. close. CLOSE. search. menu. グローバルナビゲーション（繁体中国語）. 京都大學概況. 校長的話. 大學介紹. 校內安全措施. 教育與學生支援服務. 學生生活. 留學・國際教育. 研究・產官學合作. 支援研究體制. Security Export Control. 國際交流. 學部與院係. ヘッダー セカンダリーリンク（日本語以外）. 交通資訊 聯絡我們 捐贈 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 導航連結. Home 隱私權政策. 隱私權政策. In other languages. 日本語 京都大學知道保護個人資料在進步的資通訊社會的重要性，於京都大學網站（. http:\/\/www.kyoto-u.ac.jp\/ ）處理相關資訊時，將遵守法令，並根據. 《京都大學個人資料保護相關章程》 （以下稱「章程」）及下述方針致力於個人資料的保護。. 1. 關於個人資料的取得與保存. 京都大學以合法且公正的方式取得個人資料。. 京都大學因執行本校業務必須取得個人資訊時，會盡可能特別指定其目的，不會保有超出實現該特定使用目的之必要範圍的個人資訊。. 在取得個人資料時 ，將揭露取得資訊的使用目的以及取得資訊的管理者。. 2. 關於個人資料的利用. 除了法律規定的情形外，京都大學不會將個人資料用在取得時所揭示的使用目的以外的目的或提供給第三方。. 除了法律規定的情形外，京都大學不會在未事先徵得本人同意的情況下，將個人資料提供給第三方。. 京都大學依法將個人資料提供給第三方，或將涉及個人資料處理的相關業務委託給第三方時，將就該個人資料的安全確保、管理能力等，對該第三方進行嚴格調查，並在讓其處理個人資料時進行適當監督。. 3. 關於個人資料的管理. 京都大學要針對駭客行為、電腦病毒等採取適當的資訊安全對策，以確保所保有的個人資料的正確性，並防止其遺失、遭到破壞、竄改及洩漏等。. 4. 關於個人資料的揭露、訂正、停用及刪除. 京都大學確認本人有就自己的個人資料提出揭露、訂正、停用、刪除等要求，對於這些要求，將依「章程」規定及時處理。. 若對京都大學的個人資料處理有任何意見或疑問，請聯繫京都大學個人資料相關揭露請求負責窗口（總務部法務室） 830jyohokoukai ＊ mail2.adm.kyoto-u.ac.jp （請將＊改為@）. 關於請求揭露方法 5. 個人資料保護章程的制定、實施、維護及改善. 除了本方針規定的內容外，並將根據「章程」的規定內容妥善實施京都大學的個人資料保護。. 6. 關於Google Analytics服務的利用. 本網站利用Google Analytics服務，統計掌握用戶的瀏覽情況，作為今後組織營運及網站改善的參考。在蒐集該服務所需的流量數據時使用了第一方cookie。關於Google Analytics的隱私數據處理內容，請查看Google公司該服務的. 概要說明頁 。. 2005年3月29日制定. 京都大學. （2005年3月29日 京都大學網站專門部會制定）. （2016年11月7日京都大學網站企劃專門部會修訂）. （2017年4月1日京都大學網站企劃專門部會修訂）. 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 出版物 招聘信息 交通資訊 フッター セカンダリーリンク. 站點地圖 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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{"id":"v_igQ9U8j7Mo","host_id_s":"U8j7Mo","sku":"https:\/\/www.kyoto-u.ac.jp\/ja\/news\/2022-04-06","host_s":"www.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":44,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","news"],"url_file_name_s":"2022-04-06","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["オンライン短期交流プログラム「霊長類学・ワイルドライフサイエンス短期インターンプログラム」を実施しました | 京都大学"],"description_txt":["　霊長類研究所と野生動物研究センターでは、ワイルド&ワイズ共学教育受入れプログラム事業の一環として「霊長類学・ワイルドライフサイエンス短期インターンプログラム」を、2022年3月1日～9日にオンラインで実施しました。シンガポール、インド、アメリカ、ドイツ、オランダ、オーストラリア、マレーシア、チリ、中国の大学より、外国人学部生および修士課程院生10名を短期交流学生として受け入れました。\n\n　プログラムでは、京都嵐山や屋久島にある野生霊長類の調査地や実験室での様子など、研究の現場を動画やバーチャルツアーを通して体験することで、最先端の「霊長類学・ワイルドライフサイエンス」研究を紹介しました。\n\n　他にも霊長類研究所 国際共同先端研究センター（国際センター）が主催する「CICASP Seminar in Science Communication」にも参加し、幅広い分野の研究者による研究紹介、「霊長類学・ワイルドライフサイエンス」に関連するトピックを扱った学生間のディベートやそれらを聴衆に伝えるサイエンスコミュニケーションのトレーニングなどを本学の学生とともに体験しました。"],"last_modified":"Wed Jun 08 15:28:54 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-1798490170671084010,"exact_signature_unique_b":true,"fuzzy_signature_l":7042906356193203914,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/favicon.ico","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/apple-touch-icon.png"],"icons_rel_sxt":["icon","apple-touch-icon"],"icons_sizes_sxt":["",""],"h1_txt":["オンライン短期交流プログラム「霊長類学・ワイルドライフサイエンス短期インターンプログラム」を実施しました"],"h2_txt":["グローバルナビゲーション","ヘッダー プライマリーリンク","ヘッダー セカンダリーリンク","関連リンク ","Latest news","2023年度 農学研究科修士課程学生募集について","マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて","令和4年6月1日付告示第4号","留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました","フッター プライマリーリンク","フッター セカンダリーリンク","フッター スペシャルリンク"],"italic_txt":["menu","search","close"],"startDates_dts":["Wed Apr 06 12:00:00 UTC 2022","Fri Jun 03 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022","Wed Jun 01 12:00:00 UTC 2022"],"imagescount_i":11,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https"],"images_urlstub_sxt":["www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-vertical.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/header-logo-horizontal.svg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/220309-news-01-ce61166225529a32a9aae2d237d3a8dd.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/220309-news-02-c33d79b920ff61fd3601063b9c00e528.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/220309-news-03-8ce8bbb10139a05bc13323e795081df3.jpg","www.kyoto-u.ac.jp\/sites\/default\/files\/inline-images\/220309-news-04-7881d5ebc0da6aa37018c74c0fa56277.jpg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-facebook.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-twitter.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-youtube.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/icon-instagram.svg","www.kyoto-u.ac.jp\/themes\/custom\/camphor\/images\/logo.svg"],"images_alt_sxt":["Kyoto University","Kyoto University","文章を入れてください","文章を入れてください","文章を入れてください","文章を入れてください","Facebook","Twitter","YouTube","Instagram","Kyoto University"],"images_height_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_width_val":[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],"images_text_t":"www kyoto u ac jp themes custom camphor images header logo vertical svg Kyoto University horizontal sites default files inline 220309 news 01 ce61166225529a32a9aae2d237d3a8dd jpg ce 61166225529a 32a 9aae 2d 237d 3a 8dd 文章を入れてください 02 c33d79b920ff61fd3601063b9c00e528 c 33d 79b 920ff 61fd 3601063b 9c 00e 528 03 8ce8bbb10139a05bc13323e795081df3 8ce 8bbb 10139a 05bc 13323e 795081df 3 04 7881d5ebc0da6aa37018c74c0fa56277 7881d 5ebc 0da 6aa 37018c 74c 0fa 56277 icon facebook Facebook twitter Twitter youtube YouTube instagram Instagram","responsetime_i":755,"text_t":"オンライン短期交流プログラム「霊長類学・ワイルドライフサイエンス短期インターンプログラム」を実施しました | 京都大学. メインコンテンツに移動 サイト内検索. 教員検索. 検索. close. CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム NEWS オンライン短期交流プログラム「霊長類学・ワイルドライフサイエンス短期インターンプログラム」を実施しました. 教育 国際交流 オンライン短期交流プログラム「霊長類学・ワイルドライフサイエンス短期インターンプログラム」を実施しました. 公開日. 2022年04月06日. 　霊長類研究所と野生動物研究センターでは、ワイルド&ワイズ共学教育受入れプログラム事業の一環として「霊長類学・ワイルドライフサイエンス短期インターンプログラム」を、2022年3月1日～9日にオンラインで実施しました。シンガポール、インド、アメリカ、ドイツ、オランダ、オーストラリア、マレーシア、チリ、中国の大学より、外国人学部生および修士課程院生10名を短期交流学生として受け入れました。. 　プログラムでは、京都嵐山や屋久島にある野生霊長類の調査地や実験室での様子など、研究の現場を動画やバーチャルツアーを通して体験することで、最先端の「霊長類学・ワイルドライフサイエンス」研究を紹介しました。. 　他にも霊長類研究所 国際共同先端研究センター（国際センター）が主催する「CICASP Seminar in Science Communication」にも参加し、幅広い分野の研究者による研究紹介、「霊長類学・ワイルドライフサイエンス」に関連するトピックを扱った学生間のディベートやそれらを聴衆に伝えるサイエンスコミュニケーションのトレーニングなどを本学の学生とともに体験しました。. 　最終日には「The 17th International Symposium on Primatology and Wildlife Science」の国際シンポジウムにも参加し、発表も行いました。さらに、京都および犬山の文化イベントや歴史建造物等のバーチャルツアーを行い、日本文化の紹介、京都や霊長類研究所の所在地である愛知県犬山市での生活をオンラインで体験しました。. 　参加学生からは、研究の現場や実際に行われている研究を知るだけでなく、自分と同様の分野に興味がある他の学生とも交流でき、自分の将来を考える上でとても有意義なプログラムだった、などの感想が寄せられました。. プログラム初日の様子. オリエンテーションの様子. Michael Huffman 霊長類研究所准教授（現 野生動物研究センター准教授）による講義1. Huffman准教授による講義2. 関連リンク . ワイルド&ワイズ共学教育受入れプログラム事業 関連部局. ヒト行動進化研究センター 野生動物研究センター Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 公開日. 2022年06月03日. 2023年度 農学研究科修士課程学生募集について. 公開日. 2022年06月01日. マルウェア感染が原因と思われる京都大学工学研究科メールアドレスを悪用したメール送信のお詫びについて. 公開日. 2022年06月01日. 令和4年6月1日付告示第4号. 公開日. 2022年06月01日. 留学生ラウンジきずな「オンラインイベント『お気に入りマンガ・アニメ プレゼン大会』」を開催しました. 最新のニュース一覧を見る 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について キャンパス・施設案内 大学の施設 百周年時計台記念館. 百周年時計台記念館. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 キャンパスの概要 国内の研究所・附属研究施設等 海外の拠点 On-site Laboratory 大学の施設 百周年時計台記念館 新型コロナウイルスへの対応について 来館者の方へ 主催者の方へ 時計台地震計 楽友会館 宇治おうばくプラザ 国際科学イノベーション棟 ローム記念館 船井哲良記念講堂・船井交流センター 稲盛財団記念館 京都大学 東京オフィス 京都アカデミアフォーラム 医学部附属病院 白浜水族館 図書館機構（附属図書館を含む） 学術情報メディアセンター・情報環境機構 総合博物館 留学生ラウンジ きずな 国際交流会館 教職員が利用できる施設 福利厚生施設・課外活動施設 写真・動画で見る京都大学 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 supervisor_account. 来館者の方へ. supervisor_account. 主催者の方へ. menu_book. アクセスマップ. menu_book. 予約状況. 　1925（大正14）年に誕生した時計台は、80年近くにわたって京都大学のシンボルとして親しまれ続けてきましたが、2003（平成15）年12月、創立百周年記念事業の一環として最新の免震構法を取り入れた改修工事を終え、外観や内装の雰囲気はそのままに、「百周年記念ホール」や「国際交流ホール」などを備えた学術交流の場へ、さらには京都大学から社会への情報発信の場へと再生しました。. 　京都大学百周年時計台記念館は、京都大学における学術の交流および社会との連携を図り、本学における研究教育ならびに学術および文化の発展に寄与することを目的として使用することができます。. 開館日. 年末年始（12月28日～翌1月3日）を除く毎日. 開館時間. 9時00分～18時00分. 場所. 京都大学本部構内正門正面. 連絡先. 京都大学 百周年時計台記念館事務室. 〒606-8501 京都市左京区吉田本町. Tel： 075-753-2285. Fax： 075-753-2107. E-mail： kinenkan＊mail.adm.kyoto-u.ac.jp （＊を@に変えてください）. ※ . 新型コロナウイルスへの対応について お知らせ. 「京都大学百周年時計台記念館使用規則」が改正になりました （2022年4月1日）. 百周年時計台記念館の開館時間の変更、国際交流ホールの施設使用料の改定について　. 開館時間の変更. 開館時間. 9時00分～18時00分. 18時00分以降に百周年記念ホール等の施設使用がある場合は、終了するまで、開館時間を延長します。. 百周年記念ホール、国際交流ホール等の使用について. 百周年記念ホール、国際交流ホール等の施設は、9時00分～21時30分まで使用することができます。. 国際交流ホールの料金の改定.  . 使用料（1時間）. 国際交流ホール 1室の利用. 8,600円. 国際交流ホール 2室の利用. 17,200円. 国際交流ホール 3室の利用. 25,800円. その他のお知らせ. 京大サロンの利用時間の変更について（2022年4月1日）. 百周年時計台記念館の開館時間変更に伴い、10時00分～18時00分に変更になりました。. 利用については、新型コロナウイルス感染拡大防止のための制限を設けています。. 「京大サロンの利用について」 をご確認のうえ、ご協力をお願いします。. レストラン「ラトゥール」の営業再開について. レストラン「ラトゥール」は、2022年3月24日（木曜日）より営業を再開することになりました。. 「百周年時計台記念館の使用における遵守事項（新型コロナウイルス感染拡大予防対応分）」 を変更しました。（2022年3月3日）. 百周年時計台記念館事務室への参加者名簿の提出は不要となりました。. 新型コロナウイルス感染リスク低減の為、行事を開催する場合は. 「行事を主催する方へのお願い」 をご確認ください。. 「百周年時計台記念館使用申請書様式」 （様式1-3、様式1-4-1、様式1-4-2）を変更しました。（2021年4月1日）. 「百周年時計台記念館使用申請書様式」  を変更しました。（2020年7月1日）. その他. 公開講座お知らせメール受付 春秋講義・市民講座・未来フォーラム等を開催ごとにメールでお知らせします。. 京都大学メールマガジン登録 時計台で開催されますさまざまな催しものも含めて京都大学の最新情報をお届けします。. 時計台地震計. 時計台記念館には、基礎地面（基盤）、建物底（塔基礎）、および時計台塔（塔頂）の3カ所に研究用の地震計が設置されています。. 詳細 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 栄誉等 京都大学名誉博士. 京都大学名誉博士. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 京都大学名誉博士 京都大学名誉フェロー 栄誉 歴代総長 キャンパス・施設案内 広報活動 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 京都大学名誉博士の称号授与について. 平成15年4月1日に称号授与の取扱いが変更され、それまでは本学における学術研究に寄与した功績が特に顕著である者に限定していましたが、変更後は学術研究はもとより、教育や学術文化への寄与が加えられ、授与対象の範囲が拡げられました。また、称号も分野毎に区分されていたものが「京都大学名誉博士」の一つとなりました。. 第8号. 氏名. 吉野 彰 Akira Yoshino. 国籍. 日本. 授与時の年齢. 72歳. 授与年月日. 令和2年5月15日. 授与時の所属等. 旭化成名誉フェロー、国立研究開発法人産業技術総合研究所フェロー兼ゼロエミッション国際共同研究センター長、技術研究組合リチウムイオン電池材料評価研究センター（LIBTEC）理事長、九州大学栄誉教授・グリーンテクノロジー研究教育センター訪問教授、名城大学特別栄誉教授・大学院理工学研究科教授. 研究業績等. リチウム含有遷移金属酸化物正極材料に対する負極材料の研究開発に取り組まれ、様々な炭素材料の中で非晶質炭素材料を用いることにより、世界で初めて4V級リチウムイオン二次電池の開発に成功された。. 同博士の研究は、小型携帯機器用電源に始まり、環境負荷低減に資する電気自動車並びに再生可能エネルギーのための蓄電池として不可欠なリチウムイオン二次電池を実現されたものであり、高く評価されている。. また、1972年に旭化成工業株式会社（現・旭化成株式会社）に勤務されて以来、現在に至るまで実に40年以上の長きに亘り、蓄電池開発の指導を行い、多くの研究者を育成された。さらに2011年4月より2年間、京都大学工学研究科の特命教授として、本学の教育研究に寄与した功績が特に顕著である。. 関連リンク. 京都大学名誉博士の称号を吉野彰 先生に授与しました。（2020年5月15日） 第7号. 氏名. 大隅 良典 Yoshinori Ohsumi. 国籍. 日本. 授与時の年齢. 72歳. 授与年月日. 平成29年7月13日. 授与時の所属等. 東京工業大学栄誉教授、東京工業大学科学技術創成研究院特任教授. 研究業績等. 細胞の恒常性維持にとって極めて重要な機構であるオートファジーの研究に取り組み、酵母遺伝学を駆使して関連遺伝子を次々に同定するとともに、それら遺伝子機能を解析することで、それまで全く未解明であったオートファジーの分子機構を明らかにし、その生理的な意義をも解明した。加えて、我が国の基礎科学研究の振興と発展に尽力するとともに、次世代を担う理系人材の育成に対する貢献も顕著である。. 同博士の研究は、オートファジーが、細胞がその恒常性を維持するために欠くことのできない極めて基本的な機能の一つであることを明らかにしたものであり、その功績は国際的にも高く評価されている。また、オートファジーと疾患の関係についても多くの先導的な研究成果を挙げており、医学分野における貢献も大きい。. 関連リンク. 大隅良典 先生への京都大学名誉博士称号贈呈式および記念講演会を行いました。（2017年7月13日） 第6号. 氏名. アウンサンスーチー Aung San Suu Kyi. 国籍. ミャンマー連邦共和国. 授与時の年齢. 71歳. 授与年月日. 平成28年11月3日. 授与時の所属等. ミャンマー連邦共和国国家最高顧問、外務大臣、大統領府大臣. 研究業績等. ミャンマーの自由と民主主義のために、軍事政権による弾圧に屈することなく非暴力的に戦い続け、同国の民政移管以降の政治経済発展と、その後の平和的政権交代に貢献された。. その影響力はミャンマーにとどまらず、自由と民主主義を求める非暴力闘争の世界的シンボルになっている。. 学術文化の礎ともいえる自由と民主主義という普遍的価値の発展に寄与した功績が特に顕著であり、本学の基本理念である「自由と調和」を根本的に支える貢献をされた。. 関連リンク. アウン・サン・スー・チー氏（ミャンマー連邦共和国国家最高顧問）名誉博士称号贈呈式および学生との対話（OCW） アウン・サン・スー・チールーム 第5号. 氏名. 赤﨑 勇 Isamu Akasaki. 国籍. 日本. 授与時の年齢. 86歳. 授与年月日. 平成27年5月15日. 授与時の所属等. 名城大学教授（終身）、名古屋大学特別教授・名誉教授. 研究業績等. 従来、物性制御が困難であると信じられてきた窒化ガリウム（GaN）の研究に取り組まれ、高品質GaNエピタキシャル結晶成長とp型およびn型の電気伝導度制御を実現し、世界で初めてGaN pn接合青色\/紫外LEDの開発に成功された。また、化合物半導体の結晶成長、基礎物性解明や光電子素子に関連する分野の研究者育成にも大きく貢献された。同博士の研究は、光の3原色の構成要素として不可欠な青色発光を実現されたものであり、フルカラーディスプレイの根幹をなすものである。また、青色LEDと蛍光体の組み合わせによる白色LEDは、地球環境に優しい明りとして高く評価されている。. 関連リンク. ノーベル物理学賞受賞者・赤﨑勇博士と京都大学 －大学時代に育まれた研究者の芽－（インタビュー記事） 京都大学 赤﨑勇先生 名誉博士称号授与記念講演会（OCW） 第4号. 氏名. アラン・ケイ Alan Kay. 国籍. 米国. 授与時の年齢. 68歳. 授与年月日. 平成21年1月20日. 授与時の所属等. Viewpoints Research Institute, Inc. （ビューポイントリサーチ社）　社長. 研究業績等. 現在のパーソナルコンピュータやその基本ソフトウェアの原型を世界に先駆けて構想するとともに実証を通じてこれらの機能およびその有用性を明らかにした。さらに、プログラミング言語のみならず、情報システム分析・設計、データベース設計などに対して、オブジェクト指向という計算機科学における革新的なパラダイム変革を起こす概念を提唱し、現在広く普及しているプログラミング言語などに大きな影響を与えた。パーソナルコンピューティングおよびオブジェクト指向言語の概念は、20世紀の計算機科学における最も重要な発明であるといえる。. 第3号. 氏名. ジェーン・グドール Jane Goodall. 国籍. 英国. 授与時の年齢. 73歳. 授与年月日. 平成19年11月12日. 授与時の所属等. 1960年7月～1967年6月タンザニア・ゴンベ国立公園研究員. 1967年7月～2003年6月タンザニア・ゴンベストリーム研究センター所長. 研究業績等. 称号授与の趣旨 第2号. 氏名. 利根川 進 Susumu Tonegawa. 国籍. 日本. 授与時の年齢. 64歳. 授与年月日. 平成16年4月12日. 授与時の所属等. アメリカ合衆国マサチューセッツ工科大学細胞神経科学ピコワー教授. 研究業績等. B 細胞が合成放出する免疫グロブリン（抗体）の遺伝子構造を遺伝子工学を駆使して解析し、高等動物の遺伝子が発生・分化の過程で組み換えられる再構成の仕組みを明らかにした。これは遺伝子組換え機序に基づいて新しい機能遺伝子が体細胞レベルで組み立てられることを示したものであり、それまでの生物学の常識を覆す物学上20世紀最大級の発見として高く評価されている。. 第1号. 氏名. ライオネル マッケンジー Lionel W. McKenzie. 国籍. アメリカ合衆国. 授与時の年齢. 85歳. 授与年月日. 平成16年2月3日. 授与時の所属等. アメリカ合衆国ロチェスター大学名誉教授. 研究業績等. 数理経済学なかでも一般均衡理論、国際貿易モデル、経済成長モデルの理論的研究に関する研究業績. 内外の数理経済学者への育成に貢献. 本学経済研究所への蔵書寄贈により教員や学生の研究に永続的な指針となる基盤形成. 第8号（名誉農学博士）. 氏名. プテン－クリシュ ラーマチャンドラン ナイヤー Puthen-Krish Ramachandran Nair. 国籍. アメリカ合衆国. 授与時の年齢. 59歳. 授与年月日. 平成14年2月5日. 授与時の所属等. インド国農学アカデミー会員 アメリカ合衆国フロリダ大学教授. 研究業績等. 林業と農業・畜産業・水産業を組み合わせて、土地あたりの総生産力を増加させる持続的な土地利用システムであるアグロフォレストリーの基礎研究と応用・実践の結合に関する研究業績. 第7号（名誉工学博士）. 氏名. ロバート バイロン バード  Robert Byron Bird. 国籍. アメリカ合衆国. 授与時の年齢. 71歳. 授与年月日. 平成7年12月12日. 授与時の所属等. 米国国立科学アカデミー会員 アメリカ合衆国ウィスコンシン大学バイラス名誉教授. 研究業績等. 気体と液体の輸送係数の分子論的研究、物質の輸送現象の現象論的研究、高分子を含む液体の輸送現象の分子論的研究等の移動現象に関する研究業績. 第6号（名誉医学博士）. 氏名. クリスチャン シュトレッファー Christian Streffer. 国籍. ドイツ連邦共和国. 授与時の年齢. 60歳. 授与年月日. 平成6年12月6日. 授与時の所属等. ドイツ連邦共和国エッセン大学教授. 研究業績等. 放射線障害の防禦、特に癌の温熱療法、放射線増感剤、中性子捕捉療法に関する研究業績. 第5号（名誉農学博士）. 氏名. ターゲ リカルド エリクソン Tage Rickard Eriksson. 国籍. スウェーデン王国. 授与時の年齢. 66歳. 授与年月日. 平成4年11月24日. 授与時の所属等. スウェーデン王国ウプサラ大学名誉教授 スウェーデン王立科学アカデミー会員. 研究業績等. 植物組織の培養法、特に細胞融合法に関する研究業績. 第4号（名誉工学博士）. 氏名. ルドルフ エミール カルマン Rudolf Emil Kalman. 国籍. アメリカ合衆国. 授与時の年齢. 60歳. 授与年月日. 平成2年7月10日. 授与時の所属等. アメリカ合衆国フロリダ大学大学院教授 スイス国立工科大学教授. 研究業績等. 自動制御理論、特にシステム理論に関する研究業績. 第3号（名誉医学博士）. 氏名. ハンス ノイラートHans Neurath. 国籍. アメリカ合衆国. 授与時の年齢. 80歳. 授与年月日. 平成2年1月23日. 授与時の所属等. アメリカ合衆国ワシントン大学名誉教授. 研究業績等. 生物物理化学、特に蛋白質の構造と機能に関する研究業績. 第2号（名誉工学博士）. 氏名. ジョージ ガーフィールド ホール George Garfield Hall. 国籍. 連合王国. 授与時の年齢. 64歳. 授与年月日. 平成元年4月18日. 授与時の所属等. 国際量子分子科学アカデミー会員. 研究業績等. 理論化学、特に分子の電子理論、化学反応論、数学教育論に関する研究業績. 第1号（名誉理学博士）. 氏名. イズライル モイセーヴィチ ゲルファント Izrail Moiseevich Gelfand. 国籍. ソビエト社会主義共和国連邦. 授与時の年齢. 76歳. 授与年月日. 平成元年2月7日. 授与時の所属等. ソビエト科学アカデミー会員 ソビエト社会主義共和連邦国立モスクワ大学教授. 研究業績等. 数学、特に函数解析学の分野における研究業績. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 京都大学について 広報活動 京都大学ソーシャルメディア公式アカウントに係るポリシー. 京都大学ソーシャルメディア公式アカウントに係るポリシー. サイドメニュー. 総長からのメッセージ 理念・沿革・運営・将来構想 大学評価 データ集（学生数、入学状況等） 役員等・組織・諸規程 栄誉等 キャンパス・施設案内 広報活動 刊行物・資料請求 メールマガジン 記者発表の手続き ビジュアル・アイデンティティ 京都大学ソーシャルメディア公式アカウントに係るポリシー 調達情報 法人としての取組 男女共同参画について 公表事項 情報公開 令和3年11月1日. 広報委員会ホームページ企画専門部会制定. 　国立大学法人京都大学（以下「本学」という。）総務部広報課（以下「広報課」という。）が管理するソーシャルメディアの公式アカウント（以下「当アカウント」という。）における運用および情報発信に関する方針を以下のように定めています。. 1. 利用目的. 　本学の取組や教育研究活動に関するニュース、イベント情報、研究成果等の情報を発信し、利用者が本学に対する理解を深め、本学への関心を高めることを目的とします。. 2. 公式アカウント一覧. 　広報課では、現在以下の本学公式アカウント（以下「本アカウント」という。）を管理および運用しています。. ソーシャルメディア名. 言語. アカウント名. Twitter. 日本語. 京都大学 Twitter. 英語. Kyoto University Facebook. 日本語. 京都大学（Kyoto University） Facebook. 英語. Kyoto University Instagram. 日本語. 京都大学 Instagram. 英語. Kyoto University YouTube. 日本語・英語. Kyoto University  . 3. 本アカウントの運用. (1)    投稿内容について. 　本アカウントには、本学に関する情報を掲載します。その際、プライバシー権、名誉権、肖像権、著作権、商標権など他者の権利を不当に侵害することのないよう細心の注意を払い、関連する法令等を遵守します。. (2)    返信・コメントについて. 　本アカウントは情報発信専用のアカウントです。利用者からのコメントやダイレクトメッセージ等には、返信やコメントを行いません。また、本アカウントが発信する情報に無関係または不適切と思われるコメントやリプライは、利用者の承諾を得ずに非表示または削除する場合があります。なお、無関係または不適切と思われるコメントやリプライが続いた場合、当該利用者のアカウントをブロックする場合があります。. 4. 知的財産権. 　本アカウントに掲載されている文章、写真、イラスト、音声、および動画の知的財産権は本学または正当な権利を有する権利者に帰属します。. 5. 免責事項. 　本アカウントが発信する情報の正確性、完全性および正当性の確保に努めますが、これらを完全に保証するものではありません。. 　利用者が本アカウントの掲載情報を用いて行う一切の行為について、本学は何ら責任を負うものではありません。. 　利用者により投稿されたコメントおよびリプライや、本アカウントに関連して利用者間または利用者と第三者間でトラブルや紛争が発生した場合であっても、本学は何ら責任を負うものではありません。本学は、ソーシャルメディアを利用したこと、または利用出来なかったことによって被った損害について一切責任を負いません。. 　システム障害、保守などにより事前に通知することなく本アカウントの運用を停止する場合があります。また、本学の事情により、事前に通知することなく本アカウントを削除する場合があります。. 6. その他. 　本学の正式な発表に関しては本学公式ホームページおよびプレスリリースを通じて情報発信します。. 　本ポリシーは予告なく変更する場合があります。.  . 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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(2022년 3월 24일) | 교토대학. 주요 콘텐츠로 건너뛰기 사이트 검색. 연구원. 찾기. close. CLOSE. search. menu. グローバルナビゲーション（韓国語）. 교토대학 소개. 대학소개. 총장인사. 안전. 교육・학생 지원. 유학・국제교육. 입시. 학사일정. 학생생활. 연구・산관학 연계. Security Export Control. 국제교류. 연구지원체제. 젊은층 연구자 지원. 학부・대학원. ヘッダー セカンダリーリンク（日本語以外）. 교통편 안내 문의처 기부 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 이동 경로. Home NEWS 2021년도 대학원 학위 수여식이 거행되었습니다. (2022년 3월 24일). University News 2021년도 대학원 학위 수여식이 거행되었습니다. (2022년 3월 24일). In other languages. 日本語 ターゲット. Current Students & Staff Visitors 발행일. 2022\/04\/05. 　2022년 3월 24일, 교토시 미야코메세에서 각 이사, 부학장, 각 부국장 등이 참석한 가운데 대학원 학위수여식이 거행되었습니다. 　이번에 학위를 수여받은 인원은 석사 2,211명, 석사(전문직) 176명, 법무박사(전문직) 129명, 그리고 2021년 11월 24일자,  2022년 1월 24일자, 동 3월 23일자 학위를 합해 과정박사는 총 510명, 논문박사는 총 44명입니다. 　2012년부터 시작된 박사 과정 교육 리딩 프로그램을 수료한 수강생은 12명이었습니다. 　2018년부터 시작된 WISE Program(Doctoral Program for World-leading Innovative & Smart Education, 문부과학성 선정)을 수료한 수강생은 12명이었습니다. 　미나토 나가히로(湊長博) 총장이 대표자에게 학위기를 전달한 후. 축사 를 전했습니다. 박사 학위 수여자 수 목록(2021년 11월 24일, 2022년 1월 24일, 2022년 3월 23일) 석사 학위\/석사(전문직) 학위\/법무박사(전문직) 학위 수여자 수 목록 미나토 나가히로 총장 축사. 2021년도 학위 수여식 축사 Share. Share on Facebook. Share on Twitter. Share via E-mail. Latest news. 발행일. 2022\/04\/11. 2022년도 대학원 입학식이 거행되었습니다. (2022년 4월 7일). 발행일. 2022\/04\/08. 2022년도 학부 입학식이 거행되었습니다. (2022년 4월 7일). 발행일. 2022\/04\/05. 2021년도 대학원 학위 수여식이 거행되었습니다. (2022년 3월 24일). 발행일. 2022\/03\/29. 2021년도 졸업식이 거행되었습니다. (2022년 3월 24일). 더 많은 뉴스 〒. 606-8501. Yoshida-honmachi, Sakyo-ku. ,. Kyoto-shi. ,. Kyoto. JAPAN. Tel:. 075-753-7531. フッター プライマリーリンク. 간행물 채용 교통편 안내 フッター セカンダリーリンク. 사이트 맵 Privacy policy Site Policy フッター スペシャルリンク. Safety Confirmation System Staff portal (in Japanese) Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム 国際交流 京都大学への留学を希望する方へ 京都大学で学ぶ 大学院で学位をとる 大学院入学案内. 大学院入学案内. サイドメニュー. 一般選抜 特色入試 その他の学部入学試験 留学生のための入試 学部で学位をとる 学部入学案内 大学院で学位をとる 大学院入学案内 交換留学・短期留学・研究生等 大学院入試 入学検定料の免除について 京都大学受験生ナビゲーション 京大の『学び』を知る 高大連携 関連情報 In other languages. English 日本政府（文部科学省）奨学金留学生として入学を希望する場合は、詳細を. こちら から確認してください。. 出願に関して. 出願者は、各研究科等により設定された出願資格に該当しなければなりません。以下は、出願資格からの主な抜粋です。ま た、出願書類を提出する前に、出願資格の審査が必要になる場合があります。詳細は、研究科等のWebサイトに掲載されている募集要項などで確認してください。不明な点などがある場合は、入学を希望する研究科等事務室へ問い合せてください。. 研究科等事務室一覧 外国の大学を卒業した方. 外国の大学を卒業 し、研究生または大学院生として京都大学へ入学を希望する志願者は、入学を希望する研究科等への出願手続きの前にアドミッション支援オフィス （Admissions Assistance Office: AAO）での照会手続を行う必要があります。詳しくは以下のWebサイトをご覧ください。. 外国の大学を卒業した方へ 出願資格. 1．修士課程、専門職学位課程、アジア・アフリカ地域研究研究科および総合生存学館（思修館）5年一貫制博士課程. 日本において、大学を卒業した者. 外国において、学校教育における16年の課程を修了した者*. 当該研究科が大学を卒業した者と同等以上の学力があると認めた者で、22歳に達した者. 2．博士後期課程. 日本において、修士の学位または修士（専門職）もしくは法務博士（専門職）の学位を取得した者. 外国において、本学研究科の修士課程、または専門職学位課程に相当する課程を修了した者. 当該研究科が修士の学位または修士（専門職）もしくは法務博士（専門職）の学位を有する者と同等以上の学力があると認めた者で、24歳に達した者. 3．医学研究科医学専攻および薬学研究科薬学専攻の4年制博士課程. 日本において、大学における修業年限が6年の課程（医学、歯学、薬学または獣医学を履修する課程に限る）を卒業した者. 外国において、学校教育における18年の課程を修了した者*. 当該研究科が大学（医学、歯学、薬学または獣医学を履修する修業年限が6年の課程）を卒業した者と同等以上の学力があると認めた者で、24歳に達した者. * 外国の学校教育における16年（4年制博士課程の場合は18年）に満たない課程を修了した者でも、一定の要件を満たした 場合には出願することが可能です。詳細は、研究科等のWebサイトに掲載されている募集要項などで確認してください。不明な点などがある場合は、入学を希望する研究科等事務室へ問い合せてください。. 選抜方法等. 大学院の入学者選考試験は、研究科毎に出願時期や試験日程等が異なります。出願書類を提出する前に、出願資格の審査が必 要になる場合があります。選考試験は、おおむね専門科目・外国語科目および口述試験により実施されます。入学の時期は4月ですが、一部の研究科では10月入学も実施しています。. 詳細は、研究科等のWebサイトに掲載されている募集要項などで確認してください。不明な点などがある場合は、入学を希望する. 研究科等事務室 へ問い合せてください。. 関連リンク. 大学院一覧 大学院募集要項（外国人留学生特別入試を含む） 英語で学位を取得するコース 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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CLOSE. search. menu. グローバルナビゲーション. 京大について. 総長からのメッセージ. 理念・沿革・運営・将来構想. 大学評価. データ集（学生数、入学状況等）. 役員等・組織・諸規程. 栄誉等. キャンパス・施設案内. 広報活動. 調達情報. 法人としての取組. 男女共同参画について. 公表事項. 情報公開. 入試・高大連携. 一般選抜. 特色入試. その他の学部入学試験. 留学生のための入試. 大学院入試. 入学検定料の免除について. 京都大学受験生ナビゲーション. 京大の『学び』を知る. 高大連携. 関連情報. 教育・学生支援. 教育の体制と内容. 経済支援. 海外留学を希望する京大生へ. 大学行事. 課外活動支援. 学生表彰など. 学生相談・障害学生支援・就職. キャンパスライフ支援. 学生支援の窓口・証明書発行など. 学生支援の情報発信. 京都大学に在籍する留学生の方へ. 研究・産官学連携. 研究活動を知る. 研究成果の公開ポリシー. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京都大学に寄附する. 京都大学で研究したい. 研究支援体制について. 研究者が受けられる研究費・研究支援. 研究上の倫理・安全. 教職員に兼業を依頼したい. 国際交流. 世界とつながる京都大学. 京都大学への留学を希望する方へ. 国際化を支える学内組織. 外国人研究者へのサポート. 海外留学を希望する京大生へ. 社会連携. 公開講座・講演会等. 京大の研究・教育を知る. 社会人向け教育. 教育・研究に関する情報・資料公開. 産官学連携（共同研究・知的財産・ベンチャー支援）. 京大とのつながりを深める. 社会に開かれた施設等. 終了した全学的なイベント. 学部・大学院等. ヘッダー プライマリーリンク. 受験生 在学生 卒業生 一般・地域 企業・研究者 ヘッダー セカンダリーリンク. アクセス 大学施設案内 刊行物・資料請求 寄附 Language. 日本語 ホーム English Home 中文简体 首页 中文繁体 首頁 한국어 홈페이지 パンくず. ホーム EVENTS. EVENTS. 条件で絞り込む. 日付で絞り込む. タグ. - すべて -. 入試. 高大連携. 社会連携. 公開講座. 同窓会. 大学の動き. 国際交流. 事務連絡. 研究. 産官学連携. 教育. 学生支援. 学生の活動. 受賞. 告示. 125周年. 対象者. - すべて -. 受験生の方. 在学生の方. 卒業生の方. 一般・地域の方. 企業・研究者の方. 要申し込み. - すべて -. 必要. 不要. 開催地. - すべて -. 吉田キャンパス. 宇治キャンパス. 桂キャンパス. 東京オフィス・京都アカデミアフォーラム. その他の地域. オンライン. 開催日. 前月. 1月. 2月. 3月. 4月. 5月. 6月. 7月. 8月. 9月. 10月. 11月. 12月. 翌月. 日. 月. 火. 水. 木. 金. 土. 絞り込み条件をリセット. Events. タグ. 産官学連携. 京大MBA×日経ビジネススクール共同講座「サービス経営のフロンティアとセオリー」. 開催日. 2022年06月09日、他. 開催地. オンライン. Events. タグ. 125周年. 研究. 京都大学創立125周年記念 研究の魅力を見出す”鏡”プロジェクト. 開催日. 2022年06月11日. 開催地. オンライン. Events. タグ. 公開講座. 多様性を多様な視点で捉え、行動する ～京都大学と考えるD&Iセミナー～. 開催日. 2022年06月14日. 開催地. オンライン. Events. タグ. 教育. 公開講座. オンラインコース「教育評価の基礎講座」2022. 開催日. 2022年06月15日、他. 開催地. オンライン. Events. タグ. 研究. 第256回アフリカ地域研究会「南アフリカの社会保障制度改革とケアの政治」. 開催日. 2022年06月16日. 開催地. その他の地域. オンライン. Events. タグ. 教育. グローバル教育展開オフィス主催：2022年度レクチャーシリーズ「ダイバシティー時代の「日本型教育」の行方 」第1回. 開催日. 2022年06月17日. 開催地. オンライン. Events. タグ. 教育. 公開講座. 教育学研究科 E.FORUM 2022 全国スクールリーダー育成研修「スクールリーダー育成のためのオンライン・リレー講座」. 開催日. 2022年06月19日、他. 開催地. オンライン. Events. タグ. 公開講座. 京都大学金融セミナー「『金融リテラシーが未来を拓く』2022 ～シン・金融教育元年を迎えて～」. 開催日. 2022年06月20日. 開催地. オンライン. Events. タグ. 公開講座. プラットフォーム学連続セミナーVol. 12「モビリティとプラットフォーム学」～シェアリング・MaaS・数理最適化から考える「これからの移動のかたち」～. 開催日. 2022年06月30日. 開催地. オンライン. Events. タグ. 公開講座. 第121回 京都大学丸の内セミナー「雄と雌の対立が作り出す植物と花の多様な性」. 開催日. 2022年07月01日. 開催地. 東京オフィス・京都アカデミアフォーラム. Events. タグ. 研究. 京都大学医生物学研究所 第16回公開講演会「医生物学研究所の船出」. 開催日. 2022年07月09日. 開催地. 吉田キャンパス. Events. タグ. 公開講座. 京大MBA 2022 短期集中講座「企業価値評価とファイナンス －財務モデル・ESG・サステナビリティ－」. 開催日. 2022年07月21日、他. 開催地. 吉田キャンパス. 東京オフィス・京都アカデミアフォーラム. オンライン. Events. タグ. 国際交流. 国際ワークショップ「ブロックチェーン会議 2022」. 開催日. 2022年08月04日〜08月05日. 開催地. その他の地域. オンライン. Events. タグ. 公開講座. 第122回 京都大学丸の内セミナー「利己性の進化が維持する生物多様性」. 開催日. 2022年08月05日. 開催地. 東京オフィス・京都アカデミアフォーラム. Events. タグ. 公開講座. 臨床倫理学 入門コース 2022. 開催日. 2022年08月19日、他. 開催地. 吉田キャンパス. オンライン. Events. タグ. 公開講座. 「人を知る」人工知能講座 2022 オンライン. 開催日. 2022年08月25日〜08月26日、他. 開催地. オンライン. Events. タグ. 公開講座. 第123回 京都大学丸の内セミナー「量子計算と量子暗号」. 開催日. 2022年09月09日. 開催地. 東京オフィス・京都アカデミアフォーラム. 京都大学. 〒. 606-8501. 京都府. 京都市. 左京区吉田本町. Tel:. 075-753-7531. フッター プライマリーリンク. 刊行物・資料請求 採用情報 アクセス フッター セカンダリーリンク. サイトマップ プライバシーポリシー サイトポリシー フッター スペシャルリンク. 安否確認システム 教職員の方へ Copyright © Kyoto University. All Rights Reserved. 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Arithmetic 　　. Elliptic Curves... 」によく似ている）をするので、久しぶりの「モード. 　　の切り替え」に少し戸惑っているところである。. 2011年05月02日. 　・. 2011-03-03のコメント で懸案となっていた付値論的な問題が簡単に解決できる. 　　ことに気付いた。これにより、Pop-Stixのp進セクション予想関連の仕事で. 　　得られた主定理が、（玉川氏による「非特異性の解消」のような難しい. 　　数論的な結果を必要としない）ごく初等的なグラフ論と可換環論による考察. 　　から導くことが可能であることを示したことになる。詳しくは. こちら へ。. 2011年03月03日. 　・以前の. コメント で説明したように、Pop-Stixのp進セクション予想関連の. 　　仕事を用いることによって、副有限p進セクション予想はその「tempered. 　　版」に帰着させることが可能である（＝Y. Andre氏による指摘）。一方、. 　　この帰着だけなら、（玉川氏による「非特異性の解消」のような難しい. 　　数論的な結果を必要としない）ごく初等的なグラフ論的考察からも導く. 　　ことが可能であることに最近気付いた。詳しくは. こちら へ。. 2010年11月18日. 　・IUTeich理論の連続論文の進捗状況について報告する。現在、4篇からなる. 　　連続論文全体の3分の2強程書き終わっており、IUTeich IIIの最後の節の. 　　執筆に向けての準備に入ったところである。この節では、IUTeich理論の. 　　. 「主定理」. （詳しくは先月の講演の. OHPシート や. アブストラクト を参照）. 　　を定式化し、証明する予定である。作業が順調に進めば、IUTeich IVの. 　　. 「体積計算」. 等の執筆は、来年度早々から開始できる見込みである。. 2010年11月11日. 　・先日のPop-Stixの定理に関するコメントを少し修正した。詳しくは、. 　　. こちら をご参照下さい。. 2010年11月02日. 　・「組合論的セクション予想」と、Pop-Stixのp進セクション予想関連の仕事. 　　の関係について、もっと一般的な体や、素数集合Σに対しても同様な議論. 　　ができることに気付いたので、2010-10-29の議論を少し修正した。詳しく. 　　は、. こちら をご参照下さい。. 2010年10月29日. 　・Pop-Stixのp進セクション予想関連の仕事が、数年前の論文. 「Semi-graphs 　　. of anabelioids」 で証明した「組合論的セクション予想」と関係している. 　　ことが分かった。詳しくは、. こちら をご参照下さい。. 2010年04月12日. 　・IUTeich理論の連続論文の進捗状況について報告する。最近、IUTeich II. 　　を（イントロを除いて）一通り書き終えたので、（2008年7月に始めた）. 　　このプロジェクト（＝つまり、連続論文の執筆）全体の約半分が完成した. 　　ことになる。本当は、IUTeich IIIとIVは、IとIIより短くなる予定なので、. 　　2年弱で既に「半分強」書き終わったことになる。一方、最近は研究以外. 　　の義務が以前より増えているので、残りの「半分弱」には、2年強掛かっ. 　　ても可笑しくない感じがしている。つまり、現時点での（イントロを除い. 　　た本文の執筆の）終了時期の予測は、これまでと変わらず、2012年の夏頃. 　　ということになる。　　. 　　IUTeich IとIIの理論は、p進Teich理論との類似でいうと、「曲線の標準的. 　　な持ち上げ」と「付随するガロア表現」の構成に対応している。既に発表. 　　している論文のうち、この理論において一番本質的に使うのは、. 「Semi- 　　. graphs of anabelioids」 と. 「Etale theta」 という２篇の論文である。. 　　p進Teich理論との類似からも分かるように、既に書き終わっている理論だ. 　　けでも、「既にかなり出来ている」という気分になる。一方、残りの. 　　IUTeich IIIとIVは、（当然ながら）決して「自明」ではなく、まだ暫く. 　　の時間を要しそうである。p進Teich理論との類似でいうと、IUTeich III. 　　は「フロベニウス持ち上げ」の構成に対応していて、IUTeich IVは. 　　「Hasse不変量」（＝「フロベニウス持ち上げの微分」）の計算に対応し　. 　　ている。また、IUTeich I, II, IIIは、既に発表している様々な論文の中. 　　で用意した「部品」を用いて「ある装置を組み立てる」作業に対応してい. 　　るという意味において一つの「自然な単位」を成していると見ることがで. 　　きる。つまり、その装置の組み立ては、既に3分の2強程、完成していると. 　　いうことになる。一方、IUTeich IVは、組み立てた装置に関する計算である。. 2009年10月15日. 　・現在執筆中のIUTeich理論の論文の進捗状況について報告する。前回（＝. 　　2009年02月）の報告では、理論を3篇の論文に分けて執筆中であると書いた. 　　が、その後3篇が次の通り4篇に増えた。. 　　　IUTeich I: Construction of Hodge Theaters. 　　　IUTeich II: Hodge-Arakelov-theoretic Evaluation. 　　　IUTeich III: Canonical Splittings of the Log-Theta-Lattice. 　　　IUTeich IV: An Analogue of the Hasse Invariant. 　　これは、別に数学的内容が増えたことによって起こったことではなく、む. 　　しろ理論をきちんと定式化することによって理解がより精密になり、それ. 　　によって理論を前と比べてある程度簡略化することができているが、理論. 　　を詳しく解説しながら、1篇の長さを100ページ余りに抑えようとすると、. 　　さらなる分割が必要になることが判明しただけのことである。. 　　現在は、IUTeich II をほぼ半分書き終えたところである。つまり、この. 　　プロジェクト全体が「4.0」だとすると、そのうちの「1.5」位まで書き. 　　進んでいることになる。IUTeich I は、2月頃一旦書き終えたが、その後. 　　ある部分を書き直したり、また新たな節を追加したりしたことによって. 　　全体のペースが少し落ちてしまった。現在のところ、大体「1篇に1年. 　　（弱？）位」掛かるというペースなので、書き終わるのは2012年の夏頃. 　　という計算になる。（もちろん保障はできないが!）. 　　より精密な理解に到達した課題は無数にあるが、中でも恐らく最も特筆. 　　すべきなのは、以前の定式化の最後辺りに出てくる. 　　　　　　　　　「複雑な極限」の理論を完全に削除できた. 　　ことであろう。p進Teich理論との類似でいうと、これによって「p進体ま. 　　での標準的持ち上げ」ではなく、「mod p^2までの標準的持ち上げ」で. 　　十分であることが分かったということである。この簡略化によって、. 　　理論全体の論理構造がより透明になり、（. 以前から主張 していたように！）. 　　. Etale Theta や. Absolute Topics III の、「遠アーベル系」理論こそが. 　　IUTeich理論全体の中でも最も中心的かつ本質的な部分であることがより. 　　明示的になった。. 2009年02月11日. 　・IUTeichの論文を昨年の7月から執筆しているが、最近の進捗状況について. 　　報告する。まず、2008-03-25の報告（. 過去と現在の研究 を参照）では、. 　　この理論を二篇の論文に分けて書く予定であると書いたが、この半年. 　　余りの間、（論文一篇の長さが100ページを大幅に超過しないように）. 　　理論を三篇の論文に分割して書くことに方針を変更した。現時点で. 　　考えている題名は次の通りである：. 　　　IUTeich I: Construction of Hodge Theaters. 　　　IUTeich II: Hodge-Arakelov-theoretic Evaluation. 　　　IUTeich III: Canonical Splittings. 　　このうち、IUTeich I は（イントロを除いて）一通り書き終わっていて、. 　　IUTeich IIを書き始めているところである。これまでのペースで作業が. 　　進めば、（2008-03-25の報告で予定した通り）2010年末までに一通り. 　　書き終わる見通しであるが、もちろんこれについては現時点では何も. 　　保障できない。. 　　IUTeich I では、. 　　　　　　　　　　　　　(a). Frobenioid I ,. II の理論. 　　の他、. 　　　　　　　　　　　　　(b). Etale Theta の理論. 　　や. 　　　　　　　　　　　　　(c). Absolute Topics III の理論. 　　. 　　の、非自明ながら比較的表面的な部分を、本質的な形で利用したが、. 　　IUTeich II では、(b)の最も深い部分を使う予定である。一方、. 　　IUTeich III では、(c)の最も深い部分を適用する予定である。. 2008年06月11日. 　・組合せ論的カスプ化（前回04月09日の報告を参照）の論文が完成した. 　　（. 論文 を参照）。この論文では、properな双曲的曲線の場合、配置空間. 　　の次元が2から1に下がるときの単射性は証明されていないが、論文が. 　　完成した後で、星裕一郎氏との共同研究でこの単射性を証明することが. 　　できそうになった。この共同研究が完成すると、松本氏の定理のproper. 　　な場合への拡張ができたことになる。この展開で特に面白いと思うのは、. 　　スキーム論の枠組に留まる限りとてもできそうな感じがしなかったproper. 　　な場合が、スキーム論に「パターンのヒント」を得ながらスキーム論の. 　　枠組の外にある組合せ論的な理論を適用することによってすんなり解決. 　　できたこと。即ちこの展開は、正に「IU幾何の精神」の有効性のよい例に. 　　なったと思う。. 　　組合せ論的カスプ化の論文では、GT（＝Grothendieck-Teichmuller群）. 　　に含まれる「対称性」が、（次元が下がったときの配置空間の幾何的. 　　基本群の外部自己同型群の）全射性の証明では重要な役割を果たす。. 　　最近、興味深いことに、このGT的対称性を使うことによって、p進局所体. 　　上の絶対遠アーベル幾何において、初となる副pのGC（＝Grothendieck. 　　予想）型の定理を証明できることに気付いた。簡単な議論だが、そろそろ. 　　IUTeichの論文の執筆を再開したいと思うので、いつ書くことになるか. 　　分からない。. 2008年04月09日. 　・「combGC」（＝「Grothendieck予想の組合せ論版」--- 2007年の論文を. 　　参照）を適用することによって、松本眞氏の有名な「単射性定理」（＝. 　　1996年のCrelleの論文のTheorem 2.2）の「組合せ論版」ができそう。. 　　これは２つの意味において興味深い展開だと思う。まず、第一に、. 　　「Grothendieck予想型」の定理の＊応用＊になっているところが面白い。. 　　第二に、証明では、「combGC」は一種の＊「canonicalな分裂」＊を. 　　構成するのに使うのだが、IUTeichにおいても、遠アーベル幾何は正に一. 　　種の「canonicalな分裂」を構成するのに使うことを連想させるところ. 　　がある。（最近の「. 過去と現在の研究の報告 」を参照。）特に、この. 　　「canonicalな分裂」が、松本さんの議論における「スキーム論から. 　　生じる」という性質の「代役」を果たしているところが、IUTeichとの. 　　類似性を更に感じさせるものである。因みに、この「GCのようなもの. 　　がもたらす分裂＝半単純性」という現象の原型は、「center-freeな. 　　群Gと任意の群Hに対して、HのGによる拡大と、HによるGへの外作用は. 　　同値である」という事実だと思う。. 2006年10月31日. 　・「対称化された配置空間」に関する訂正：先日、数理研で開かれたガロア. 　　理論の研究集会で玉川さんの（配置空間の代数および遠アーベル幾何に関. 　　する）講演の後、私と玉川さんの共著論文で得られている結果が、「対称. 　　化された配置空間」（＝配置空間を対称群で割って得られる空間）に対し. 　　て拡張可能かどうか、（伊原康隆先生とT. Szamuely先生より）質問が. 　　あった。そこで、私は、「早とちり」で、共著論文で得られている二種類. 　　の定理のうち、一種類については、拡張可能であると答えたが、これは私. 　　の誤解によるものであった。この場をお借りしてお詫びしたい。. 　　講演から帰った後、改めて考えてみたところ、次の結論に達した：対称化. 　　配置空間への拡張は、「pro-l、かつ l 配置空間の次元」という条件の. 　　下では、（単に最大pro-l 部分群を考えることによって）共著論文の二. 　　種類の定理に対して可能だが、それ以外の場合（＝例えば、一般の. 　　profiniteの場合等）は、共著論文で得られている結果から直ちに従う　. 　　帰結はなさそうだ。つまり、対称化配置空間への一般的な拡張は、今のと. 　　ころ、面白い＊未解決問題＊と言わざるを得ない。また、配置空間の基本. 　　群の同型によって各々の対角因子に付随する分解群たちが保たれるかどう. 　　か（これも伊原先生による質問だが）という問題も、現時点では＊未解決＊　. 　　なままである。　. 2006年08月25日. 　・pro-(p,l)のabs pGCに関する補足：06月24日の報告の「点論性から幾何性」. 　　が従うという議論はまだ大丈夫だと思うが、「できそうだ」と言っていた. 　　「点論性」の議論にギャップが見付かり、今のところ修復の目処が立って. 　　いない。. 2006年06月24日. 　・pro-(p,l)のabs pGCに関する補足：05月17日の時点ではまだ出来て. 　　いなかった部分（＝Green自明化に関係する部分）があったが、これは無事. 　　解決できたと思う。ただし、この「pro-(p,l)のabs pGCが出来た」という. 　　話は全部「点論的」（＝「Cuspidalization」の論文の「point-theore-. 　　tic」）という仮定の下での話。一方、「点論性」については、学生の. 　　星氏との共同研究によってそう遠くないうちにできそうだ。すると、現在. 　　の認識では、「p進遠アーベル幾何のもっとも重要な未解決問題」は、. 　　pro-pのabs pGCかな。これまで出来ていることを考慮すると、この問題は、. 　　pro-p abs的な設定において曲線のspecial fiberの（閉点や既約成分の）. 　　幾何を復元することと事実上同値である。. 　・双曲的曲線の配置空間の遠アーベル幾何に関する玉川さんとの共同研究は. 　　順調に進んでいて、そう遠くないうちに論文も公表できそうだ。. 2006年05月17日. 　・pro-(p,l)のabs pGC （p進局所体上の絶対グロタンディーク予想）. 　　が出来そうな気がしてきた（まだ完全にチェックしたわけではないが）。. 　　これを受けて、「pro-p Green 自明化がこのようなabs pGCの設定で. 　　保たれる」ことを示すことが、p進遠アーベル幾何のもっとも重要な. 　　未解決問題であると改めて認識させられた。（Green自明化については、. 　　「Cuspidalization」の論文を参照。）."}
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Anabelioids. 2022年05月01日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2022年04月24日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2022年03月20日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を掲載（修正箇所は、Introductionの冒頭の部分を中心に、. 　　表現等の軽微なものが数件。）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 　・（. 論文 ）修正版を掲載（修正箇所は、表現等の軽微なものが多数。）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves III: Tripods and Tempered Fundamental Groups. 　・（. 論文 ）コメントを更新：. 　　A Theory of Ordinary p-adic Curves. 　　Foundations of p-adic Teichmuller Theory. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 　・（. 論文 ）コメントを掲載：. 　　Introduction to p-adic Teichmuller Theory. 2022年01月14日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年12月14日. 　・（. 出張・講演 ）RIMS共同研究（公開型）「代数的整数論とその周辺」での. 　　宇宙際タイヒミューラー理論に関する連続講演の関連資料を掲載。. 2021年12月08日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Explicit Estimates in Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年11月17日. 　・（. 過去と現在の研究 ）世界の数学者に向けた、今年度の. 宇宙際タイヒミューラー. 　　. 理論関連集会のビデオ閲覧. の. 招待状 を掲載。. 2021年11月08日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Group-theoreticity of numerical invariants and distinguished subgroups. 　　　　of configuration space groups. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Explicit Estimates in Inter-universal Teichmuller Theory. 2021年09月12日. 　・（. 出張・講演 ）今年度の訪問滞在型研究「宇宙際タイヒミューラー理論の. 　　拡がり」の残りの2件の（宇宙際タイヒミューラー理論関連の）集会の. 　　関連資料を掲載。. 2021年08月30日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年07月12日. 　・（. 出張・講演 ）今年度の訪問滞在型研究「宇宙際タイヒミューラー理論の. 　　拡がり」の最初の2件の集会の関連資料を掲載。. 　・（. 論文 ）修正版を掲載（修正箇所は表現等の軽微なものが多数。定理等の. 　　番号は変更なし。）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves IV: Discreteness and Sections. 2021年06月18日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Explicit Estimates in Inter-universal Teichmuller Theory. 2021年06月04日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年05月27日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年04月25日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年04月18日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年04月15日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年03月06日. 　・（. 論文 ）宇宙際タイヒミューラー理論に関する論文4篇の出版を記念して、. 　　新論文を掲載：. 　　On the Essential Logical Structure of Inter-universal Teichmuller. 　　　Theory in Terms of Logical AND \"∧\"\/Logical OR \"∨\" Relations:. 　　　Report on the Occasion of the Publication of the Four Main Papers. 　　　on Inter-universal Teichmuller Theory. 2021年01月15日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Combinatorial Construction of the Absolute Galois Group of the Field of. 　　　　Rational Numbers. 2020年12月23日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2020年12月12日. 　・（. 論文 ）新論文を掲載：. 　　Combinatorial Construction of the Absolute Galois Group of the Field of. 　　　　Rational Numbers. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　・（. 過去と現在の研究 ）2021年度に開催予定の訪問滞在型研究. 　　. 「宇宙際タイヒミューラー理論の拡がり」 のウェブページを掲載。. 2020年11月30日. 　・（. 論文 ）新論文を掲載：. 　　Explicit Estimates in Inter-universal Teichmuller Theory. 2020年10月31日. 　・（. 出張・講演 ）11月6日（金・日本時間）に予定されているBerkeley Colloquium. 　　のオンライン講演の. スライド を掲載。. 2020年08月04日. 　・（. 論文 ）修正版を掲載（修正箇所は、Section 4を中心に、表現等の軽微な. 　　ものが多数。またSection 4の最後に、Corollary 4.18, Lemma 4.19を追加。. 　　一方、定理等の番号は変更なし。）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 2020年05月18日. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 　　A Combinatorial Version of the Grothendieck Conjecture. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2020年04月25日. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）コメントを掲載：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 2020年04月22日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2020年04月18日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2020年04月12日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2020年04月04日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2020年03月22日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2020年03月10日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2020年02月05日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2020年01月24日. 　・（. 過去と現在の研究 ）第二の. IUTeichアニメーション. 「Computation of the 　　. log-volume of the q-pilot via the multiradial representation」 を公開。. 2019年12月30日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2019年12月21日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2019年12月10日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2019年11月30日. 　・（. 論文 ）修正版を掲載（修正箇所は表現等の軽微なものが多数。定理. 　　等の番号は変更なし。）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 2019年11月15日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2019年11月08日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2019年11月04日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2019年10月31日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2019年07月20日. 　・（. 論文 ）コメントを更新：. 　　On the Combinatorial Cuspidalization of Hyperbolic Curves. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 2019年06月28日. 　・（. 論文 ）コメントを更新：. 　　The Local Pro-p Anabelian Geometry of Curves. 　　A Combinatorial Version of the Grothendieck Conjecture. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2019年06月07日. 　・（. 論文 ）コメントを更新：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2019年05月18日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2019年05月14日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2019年05月12日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2019年05月01日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2019年04月22日. 　・（. 過去と現在の研究 ）2020年度に開催予定の訪問滞在型研究. 　　. 「宇宙際タイヒミューラー理論の拡がり」 のウェブページを掲載。. 2019年04月03日. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 2019年03月22日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2019年03月18日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2019年03月10日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2019年02月10日. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2019年02月01日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 過去と現在の研究 ）2018年3月、数理研で行なわれたIUTeich. 　　に関する議論を纏めた報告書＋関連文書の. ウェブページ を更新。. 2019年01月19日. 　・（. 論文 ）コメントを更新：. 　　Topics in Absolute Anabelian Geometry III. 2019年01月15日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2018年12月28日. 　・（. 過去と現在の研究 ）2018年3月、数理研で行なわれたIUTeich. 　　に関する議論を纏めた報告書＋関連文書の. ウェブページ を更新。. 2018年11月26日. 　・（. 過去と現在の研究 ）2018年3月、数理研で行なわれたIUTeich. 　　に関する議論を纏めた報告書＋関連文書の. ウェブページ を更新。. 2018年11月08日. 　・（. 過去と現在の研究 ）2018年3月、数理研で行なわれたIUTeich. 　　に関する議論を纏めた報告書＋関連文書の. ウェブページ を更新。. 　・（. 過去と現在の研究 ）南出新氏による、. IUTeichにおける明示的な 　　. 不等式に関する講演のスライド を掲載。. 2018年10月06日. 　・（. 過去と現在の研究 ）2018年3月、数理研で行なわれたIUTeich. 　　に関する議論を纏めた報告書＋関連文書の. ウェブページ を更新。. 2018年09月28日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2018年09月21日. 　・（. 過去と現在の研究 ）2018年3月、数理研で行なわれたIUTeich. 　　に関する議論を纏めた. 報告書 （および. 関連文書 ）を掲載。. 　・（. 過去と現在の研究 ）山下剛氏による. 「IUTeichに関するFAQ」 を掲載。. 2018年06月28日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2018年04月03日. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　・（. 論文 ）コメントを更新：. 　　Foundations of p-adic Teichmuller Theory. 2018年02月22日. 　・（. 過去と現在の研究 ）譚福成氏による、. IUTeichに関する講演のスライド を掲載。. 　・（. 過去と現在の研究 ）加藤文元氏による、. IUTeichに関する講演ビデオ へのリンク. 　　を掲載。. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2017年12月30日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）コメントを更新：. 　　Topics in Absolute Anabelian Geometry I. 　・（. 論文 ）コメントを掲載：. 　　The Local Pro-p Anabelian Geometry of Curves. 2017年12月14日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）コメントを更新：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）コメントを更新：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 2017年11月01日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2017年08月25日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2017年08月18日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 論文 ）コメントを掲載：. 　　The Geometry of the Compactification of the Hurwitz Scheme. 2017年07月20日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2017年07月12日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）コメントを更新：. 　　Topics in Absolute Anabelian Geometry III. 2017年06月30日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2017年05月20日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2017年04月24日. 　・「. 数理研近辺の桜2017 」を掲載。. 2017年04月08日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Group-theoreticity of numerical invariants and distinguished subgroups. 　　　　of configuration space groups. 2017年04月01日. 　・（. 過去と現在の研究 ）南出新氏による、. 組合せ論的遠アーベル幾何. の入門講義. 　　の. スライド を掲載。. 　・（. 論文 ）新論文を掲載：. 　　Group-theoreticity of numerical invariants and distinguished subgroups. 　　　　of configuration space groups. 2017年03月04日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2017年01月28日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2016年12月27日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2016年12月16日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2016年11月12日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2016年10月24日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves III: Tripods and Tempered Fundamental Groups. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves IV: Discreteness and Sections. 2016年10月01日. 　・（. 過去と現在の研究 ）Ivan Fesenko氏による、IUTeichに関する. 解説記事 へのリンク. 　　を掲載。. 　・（. 過去と現在の研究 ）南出新氏による、. 配置空間群の数値的不変量. に関する講演. 　　の写真を纏めたPDFファイル（＝　. 1 2 3 4 ）を掲載。. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 　　Topics in Absolute Anabelian Geometry I. 2016年09月04日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2016年08月18日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2016年08月08日. 　・（. トップページ ）本サイト（リンク先を含む）のコンテンツの使用に関する. 　　メッセージを追加した。. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2016年08月01日. 　・（. 過去と現在の研究 ）今年7月に数理研で開催されたIUTeichに関する. 研究集会 　　に関する項目を更新し、. 写真 の共有フォルダーへのリンクを追加した。. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2016年07月16日. 　・（. 論文 ）コメントを更新・掲載：. 　　Correspondences on Hyperbolic Curves. 　　Noncritical Belyi Maps. 　　Arithmetic Elliptic Curves in General Position. 2016年07月12日. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2016年07月02日. 　・（. 論文 ）新論文を掲載：. 　　The Mathematics of Mutually Alien Copies: from Gaussian Integrals to. 　　　　Inter-universal Teichmuller Theory. 2016年06月21日. 　・（. 論文 ）コメントを更新・掲載：. 　　The Absolute Anabelian Geometry of Hyperbolic Curves. 　　A Combinatorial Version of the Grothendieck Conjecture. 　　Topics in Absolute Anabelian Geometry I. 　　Topics in Absolute Anabelian Geometry II. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2016年05月26日. 　・（. 過去と現在の研究 ）山下剛氏による、多重ゼータ値入門講義（ただし、最後の. 　　4節を除く）の写真を纏めたPDFファイル（＝　. 1 2 3 4 5 6 ）を掲載。. 2016年04月30日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　・（. 論文 ）コメントを掲載：. 　　Topics in Absolute Anabelian Geometry III. 2016年04月14日. 　・「. 数理研近辺の桜2016 」を掲載。. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 　・（. 過去と現在の研究 ）南出新氏による、. 組合せ論的遠アーベル幾何. の入門講義. 　　（続編）の写真を纏めたPDFファイル（＝　. 1 2 3 4 ）を掲載。. 2016年02月25日. 　・（. 過去と現在の研究 ）今年7月に数理研で開催予定の、IUTeichに関する. 研究集会 　　のホームページへのリンクを追加。. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 過去と現在の研究 ）南出新氏による、. 組合せ論的遠アーベル幾何. の入門講義. 　　の写真を纏めたPDFファイル（＝　. 1 2 3 4 ）を掲載。. 2016年01月12日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2015年12月24日. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2015年12月18日. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2015年12月15日. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2015年12月06日. 　・（. 論文 ）コメントを更新・掲載：. 　　The Absolute Anabelian Geometry of Hyperbolic Curves. 　　Semi-graphs of Anabelioids. 　　The Geometry of Frobenioids I: The General Theory. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 　　Arithmetic Elliptic Curves in General Position. 　　Conformal and Quasiconformal Categorical Representation of. 　　　　Hyperbolic Riemann Surfaces. 　・（. 過去と現在の研究 ）星裕一郎氏による、IUTeichに関する. サーベイ と. 講演スライド 　　へのリンクを追加。. 　・（. 過去と現在の研究 ）山下剛氏による、IUTeichに関する. 講演スライド へのリンクを. 　　追加。. 　・（. 過去と現在の研究 ）Frobenioidの理論に関する. 問答集 を修正（. 修正箇所のリスト ）。. 2015年11月11日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を更新（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）コメントを更新：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 2015年10月29日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2015年10月25日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2015年10月16日. 　・（. 過去と現在の研究 ）. Frobenioid. の理論に関する. 問答集 を公開。. 2015年10月12日. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2015年10月04日. 　・（. 過去と現在の研究 ）IUTeichに関するアニメーション（＝[IUTchIII], Theorem A. 　　の内容に対応）の新しいヴァージョン（. 「石碑版」. ）を公開。. 2015年09月29日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 　・（. 過去と現在の研究 ）. IUTeichに関するアニメーション. （＝[IUTchIII], Theorem A. 　　の内容に対応）を公開。. 2015年09月14日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2015年09月08日. 　・（. 過去と現在の研究 ）用語\"anabelioid\"および\"Frobenioid\"の語形変化論的側面. 　　に関する. 解説文書 （の項目(7.)の最初のディスプレイの前後の文）を少し修正。. 2015年09月01日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 過去と現在の研究 ）用語\"anabelioid\"および\"Frobenioid\"の語形変化論的. 　　側面に関する. 解説文書 を掲載。. 2015年08月08日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）コメントを更新：. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 　・（. 論文 ）コメントを更新：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 2015年07月18日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」（＝若林氏および和田氏に関する. 　　　項目）を更新。. 2015年07月10日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 2015年05月16日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 2015年05月10日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2015年05月05日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 2015年04月26日. 　・（. 過去と現在の研究 ）今年12月に（英）Oxford大学で開催予定の、IUTeich. 　　に関する. 研究集会 のホームページへのリンクを追加。. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新（前の. 　　2015-04版と比べて、pp. 6（下）, 7（上）, 17（下）を少し修正）。. 　・（. 論文 ）コメントを更新：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 　・（. 論文 ）IUTeichとの関係に関する解説のリストを掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2015年04月24日. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新（前の. 　　2015-02版と比べて、p. 17の真ん中辺りを少し修正）。. 2015年04月22日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）コメントを更新：. 　　The Algebraic and Anabelian Geometry of Configuration Spaces. 　・（. 論文 ）コメントを更新：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 2015年04月10日. 　・「. 数理研近辺の桜2015 」を掲載。. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 　・（. 過去と現在の研究 ）3月の研究集会の写真（＝集合写真＋山下氏の講演の. 　　写真）を掲載。. 　・（. 論文 ）新論文を掲載：. 　　Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from. 　　　　the Point of View of Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2015年03月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2015年02月21日. 　・（. 過去と現在の研究 ）3月に開催予定の研究集会での. 星氏の講演のスライド へ. 　　のリンクを追加。. 2015年02月19日. 　・（. 過去と現在の研究 ）Ivan Fesenko氏による、IUTeichに関する. サーベイ への. 　　リンクを追加。. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves IV: Discreteness and Sections. 　・（. 論文 ）コメントを更新：. 　　On the Combinatorial Cuspidalization of Hyperbolic Curves. 　・（. 論文 ）コメントを更新：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 　・（. 出張・講演 ）本日、数理研の数論セミナーで行なわれた講演のスライド. 　　を掲載。. 2015年02月01日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）コメントを更新：. 　　The Algebraic and Anabelian Geometry of Configuration Spaces. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新（前の. 　　2014-12版と比べて、p. 13を少し修正）。. 2014年12月30日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 2014年12月25日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　・（. 過去と現在の研究 ）IUTeichの検証活動に関する. 報告 を掲載。. 2014年12月22日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）コメントを掲載：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新（前の. 　　2014-11版と比べて、pp. 4, 5, 11, 12, 16を少し修正）。. 2014年11月24日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Monomorphisms in Categories of Log Schemes. 2014年11月21日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 過去と現在の研究 ）2015年3月09日〜20日、数理研で開催予定の. IUTeich. 　　に関する研究集会（＝日本語の講演による）の. 案内 を更新。（英語による）. 　　報告集は数理研の「. 講究録別冊 」として刊行する予定である。. 2014年11月16日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Monomorphisms in Categories of Log Schemes. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新（前の. 　　2014-11版と比べて、p. 2（上）を少し修正）。. 2014年11月12日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新（前の. 　　2014-10版と比べて、p. 2（上）と p. 3（下）を少し修正）。. 2014年10月30日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Monomorphisms in Categories of Log Schemes. 　・（. 論文 ）コメントを更新：. 　　Categorical representation of locally noetherian log schemes. 　・（. 論文 ）コメントを更新：. 　　Categories of log schemes with archimedean structures. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新（前の. 　　2014-10版と比べて、次の頁の上半分を少し修正： pp. 1, 2, 4）。. 　・（. 過去と現在の研究 ）2015年3月09日〜20日、数理研で開催予定の. IUTeich. 　　に関する研究集会（＝日本語の講演による）の. 案内 を掲載。（英語による）. 　　報告集は数理研の「. 講究録別冊 」として刊行する予定である。. 2014年10月04日. 　・（. 過去と現在の研究 ）IUTeichに関する. レクチャーノート を更新. 　　（2014-09版と比べて、p. 4を少し修正）。. 2014年09月20日. 　・（. 論文 ）コメントを更新：. 　　Categorical representation of locally noetherian log schemes. 2014年09月15日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 過去と現在の研究 ）（講演の具体的な予定はないが）IUTeichに関する. 　　. レクチャーノート の最新版を掲載（2014-05の講演版と比べて、pp. 3-4,. 　　p. 9, p. 10, p. 12, p. 13, p.17を少し修正）。. 2014年09月05日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2014年08月29日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 2014年08月24日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）LaTeX版を追加：. 　　Categories of log schemes with archimedean structures. 　・（. 論文 ）コメントを更新：. 　　Categorical representation of locally noetherian log schemes. 　・（. 論文 ）コメントを更新：. 　　Categories of log schemes with archimedean structures. 2014年08月16日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2014年08月09日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）コメントを掲載：. 　　Topics in Absolute Anabelian Geometry II. 　・（. 論文 ）コメントを掲載：. 　　Galois Sections in Absolute Anabelian Geometry. 2014年07月31日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves IV: Discreteness and Sections. 2014年07月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）コメントを更新：. 　　Arithmetic Elliptic Curves in General Position. 2014年07月18日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2014年07月15日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2014年07月03日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　・（. 過去と現在の研究 ）山下剛氏による連続講演. 「宇宙際Teichmuller理論と 　　. そのDiophantus的帰結」 の案内へのリンクを掲載。. 2014年06月30日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2014年06月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2014年06月26日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2014年06月14日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 2014年05月27日. 　・（. 出張・講演 ）2014年05月24日、熊本大学で行なった講演のスライドを少し. 　　修正した（. 修正箇所のリスト ）。. 2014年05月07日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2014年04月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 論文 ）新論文を掲載：. 　　Monomorphisms in Categories of Log Schemes. 　・（. 論文 ）コメントを掲載：. 　　Categorical representation of locally noetherian log schemes. 　・（. 論文 ）コメントを掲載：. 　　Categories of log schemes with archimedean structures. 2014年04月15日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2014年04月12日. 　・「. 数理研近辺の桜2014 」を掲載。. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 2014年03月22日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2014年03月18日. 　・（. 出張・講演 ）2014年05月24日、熊本大学で開催予定の講演のスライド（の. 　　p. 16, 19）を少し修正した。. 2014年03月15日. 　・（. 出張・講演 ）2014年05月24日、熊本大学で開催予定の講演のスライドを. 　　掲載。このスライドは（タイトルを除けば）2014年02年20年の講演のものと. 　　同一である。. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2014年02月20日. 　・（. 出張・講演 ）本日、数理研の数論セミナーで行なわれた講演のスライド. 　　を掲載。. 　・（. 過去と現在の研究 ）2013年12月18日に公開した、IUTeichの検証活動に. 　　関する. 報告 の. 英語版 の表現（＝項目(7)の4行目）を微修正。. 　・（. 論文 ）コメントを掲載：. 　　The Algebraic and Anabelian Geometry of Configuration Spaces. 　・（. 論文 ）コメントを更新：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 2013年12月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves IV: Discreteness and Sections. 2013年12月18日. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 過去と現在の研究 ）IUTeichの検証活動に関する. 報告 を掲載。. 2013年11月22日. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2013年11月21日. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 2013年10月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2013年10月24日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2013年10月08日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry II. 　・（. 出張・講演 ）2013年11月15日開催予定の数理解析研究所５０周年記念講演. 　　のスライドを掲載：. 　　「数論幾何の風景 — 数の加減乗除から対称性の幾何まで」. 2013年09月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry II. 　・（. 論文 ）新論文を掲載：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves IV: Discreteness and Sections. 2013年09月22日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 　・（. 論文 ）コメントを更新：. 　　Arithmetic Elliptic Curves in General Position. 2013年08月20日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 　・（. 論文 ）コメントを更新：. 　　Arithmetic Elliptic Curves in General Position. 2013年07月31日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 2013年07月17日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2013年06月28日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2013年06月08日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2013年05月24日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　・（. 論文 ）コメントを更新：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 　・（. 論文 ）コメントを掲載：. 　　The Absolute Anabelian Geometry of Hyperbolic Curves. 2013年05月12日. 　・（. 論文 ）コメントを更新：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 2013年05月02日. 　・（. 出張・講演 ）2013年06月13日、東京大学で開催予定の講演のスライドを. 　　掲載。このスライドは数ヶ月前に公開したサーベイ. 　　　　A Panoramic Overview of Inter-universal Teichmuller Theory. 　　を基に作成したものである。. 2013年04月24日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　・（. 過去と現在の研究 ）山下剛氏による. 「\"宇宙際\"に関するFAQ」 を掲載。. 2013年04月12日. 　・「. 数理研近辺の桜2013 」を掲載。. 　・（. 論文 ）コメントを更新：. 　　A Combinatorial Version of the Grothendieck Conjecture. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry II. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 2013年03月26日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2013年03月01日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 2013年02月21日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 2013年02月20日. 　・（. 論文 ）新論文（＝. IUTeichのサーベイ. ）を掲載：. 　　A Panoramic Overview of Inter-universal Teichmuller Theory. 　・（. 論文 ）修正版を掲載（＝[Lch]に関する議論を少し修正し、文献リストを. 　　更新した）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves III: Tripods and Tempered Fundamental Groups. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2013年02月04日. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 2013年01月14日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2012年12月26日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 2012年12月20日. 　・（. 論文 ）コメントを更新・掲載：. 　　The Geometry of Frobenioids I: The General Theory. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 　　The Geometry of Anabelioids. 　　Semi-graphs of Anabelioids. 　　The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves. 　　　　over Number Fields. 2012年12月16日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2012年12月02日. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 2012年11月28日. 　・（. 出張・講演 ）2012年12月開催予定の. 研究集会 での講演の. OHPシート を更新。. 2012年11月21日. 　・（. 出張・講演 ）2012年12月開催予定の. 研究集会 での講演の. OHPシート を更新。. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 2012年11月14日. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 2012年11月10日. 　・（. 出張・講演 ）2012年12月開催予定の. 研究集会 での講演の. OHPシート を掲載。　. 　　このOHPシートを基に、. IUTeichに関するサーベイ. を2013年3月末までに書く. 　　予定である。. 2012年11月07日. 　・（. 論文 ）新論文を掲載：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves III: Tripods and Tempered Fundamental Groups. 2012年10月25日. 　・（. 論文 ）コメントを掲載：. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 2012年10月20日. 　・（. 論文 ）コメントを掲載：. 　　A Combinatorial Version of the Grothendieck Conjecture. 2012年10月18日. 　・（. 論文 ）修正版を掲載（. 修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry III. 　・（. 論文 ）コメントを更新：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 　・（. 論文 ）コメントを掲載：. 　　A Survey of the Hodge-Arakelov Theory of Elliptic Curves I. 　　A Survey of the Hodge-Arakelov Theory of Elliptic Curves II. 2012年10月14日. 　・（. 論文 ）次の論文のTheorem 1.10に関する. コメント を掲載：. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2012年10月08日. 　・（. 論文 ）コメントを更新：. 　　Arithmetic Elliptic Curves in General Position. 2012年10月06日. 　・（. 論文 ）コメントを掲載：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 2012年10月02日. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2012年08月30日. 　・（. 論文 ）修正版を掲載（修正は実質的な内容とは無関係な軽微なもの）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 　・（. 論文 ）新論文を掲載：. 　　Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. 　　Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic. 　　　　Evaluation. 　　Inter-universal Teichmuller Theory III: Canonical Splittings of. 　　　　the Log-theta-lattice. 　　Inter-universal Teichmuller Theory IV: Log-volume Computations. 　　　　and Set-theoretic Foundations. 2012年08月10日. 　・（. 論文 ）2012年8月の公開講座の原稿を掲載：. 　　数体と位相曲面に共通する「二次元の群論的幾何」．. 2012年06月18日. 　・（. 論文 ）コメントを掲載：. 　　The Absolute Anabelian Geometry of Canonical Curves. 2012年04月14日. 　・「. 数理研近辺の桜2012 」を掲載。. 2012年04月12日. 　・（. 論文 ）修正版を掲載（Theorem 1.8, (i); Corollary 1.11, (i)と文献. 　　リストを少し修正した）：. 　　Topics in Absolute Anabelian Geometry II. 2012年04月06日. 　・（. 論文 ）修正版を掲載（修正は数学的な内容とは無関係な軽微なもの）：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 2012年04月05日. 　・（. トップページ ）トップページの写真を先日撮ったものに更新。. 2012年04月02日. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 2012年03月14日. 　・（. 論文 ）修正版を掲載（. 主な修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry I. 2012年02月12日. 　・（. 論文 ）修正版を掲載（. 主な修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry I. 　　Topics in Absolute Anabelian Geometry II. 　　Topics in Absolute Anabelian Geometry III. 2012年01月23日. 　・（. 感想・着想 ）IUTeich理論の論文の執筆状況に関する報告を掲載。. 2011年12月22日. 　・（. 論文 ）修正版を掲載（＝Lemma 1.5のミスプリを修正した）：. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 2011年11月25日. 　・（. 論文 ）コメントを更新：. 　　The Geometry of Frobenioids I: The General Theory. 2011年11月08日. 　・（. 論文 ）パソコン環境によってはPDFファイルが開けないことがあるとの. 　　指摘を受け、本サイトのPDFファイルのほとんどを「フォント埋め込み版」. 　　に差し替えることにした。. 　・（. 安否確認情報 ）災害時、非常時等に役に立つことがあると思い、本ページに. 　　「安否確認情報」のボタンを追加した。. 2011年09月08日. 　・（. 論文 ）コメントを更新：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2011年07月15日. 　・（. 論文 ）コメントを掲載：. 　　A Theory of Ordinary p-adic Curves. 　・（. 論文 ）コメントを掲載：. 　　Arithmetic Elliptic Curves in General Position. 2011年06月20日. 　・（. 感想・着想 ）IUTeich理論の論文の執筆状況に関する報告を掲載。. 2011年05月02日. 　・（. 感想・着想 ）. 2011-03-03のコメント で懸案となっていた付値論的な問題が. 　　簡単に解決できることに気付いた。これにより、Pop-Stixのp進セクション　. 　　予想関連の仕事で得られた主定理が、（玉川氏による「非特異性の解消」. 　　のような難しい数論的な結果を必要としない）ごく初等的なグラフ論と可換. 　　環論による考察から導くことが可能であることを示したことになる。詳しくは. 　　. こちら へ。. 2011年04月12日. 　・「. 数理研近辺の桜2011 」を掲載。. 2011年04月01日. 　・（. 論文 ）新論文を掲載：. 　　Topics Surrounding the Combinatorial Anabelian Geometry of. 　　　　Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists. 　・（. 論文 ）コメントを掲載：. 　　On the Combinatorial Cuspidalization of Hyperbolic Curves. 2011年03月03日. 　・（. 感想・着想 ）以前の. コメント で説明したように、Pop-Stixのp進セクション. 　　予想関連の仕事を用いることによって、副有限p進セクション予想はその. 　　「tempered版」に帰着させることが可能である（＝Y. Andre氏による指摘）。. 　　一方、この帰着だけなら、（玉川氏による「非特異性の解消」のような難しい. 　　数論的な結果を必要としない）ごく初等的なグラフ論的考察からも導くことが. 　　可能であることに最近気付いた。詳しくは. こちら へ。. 2010年11月23日. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 2010年11月18日. 　・（. 感想・着想 ）IUTeich理論の論文の執筆状況に関する報告を掲載。. 2010年11月15日. 　・（. 感想・着想 ）先日のPop-Stixの定理に関するコメントをまた少し修正し、　. 　　複数あったリンク先を. 最新版 に統一した。. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 2010年11月11日. 　・（. 感想・着想 ）先日のPop-Stixの定理に関するコメントを少し修正した。. 　　詳しくは、. こちら をご参照下さい。　. 　・（. 論文 ）修正版を掲載（＝高尾氏の仕事に関する記述を少し修正した）：. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 2010年11月02日. 　・（. 感想・着想 ）「組合論的セクション予想」と、Pop-Stixのp進セクション. 　　予想関連の仕事の関係について、もっと一般的な体や、素数集合Σに対して. 　　も同様な議論ができることに気付いたので、2010-10-29の議論を少し修正. 　　した。詳しくは、. こちら をご参照下さい。. 2010年10月29日. 　・（. 感想・着想 ）Pop-Stixのp進セクション予想関連の仕事が、数年前の. 　　論文. 「Semi-graphs of anabelioids」 で証明した「組合論的セクション. 　　予想」と関係していることが分かった。詳しくは、. こちら をご参照下さい。. 2010年10月27日. 　・（. 論文 ）修正版を掲載（Definition 1.9; Corollaries 1.10, 1.11に. 　　おいて、「genus」に関する記述を「type」に関する記述に修正した）：. 　　Topics in Absolute Anabelian Geometry II. 2010年10月13日. 　・（. 出張・講演 ）2010年10月開催予定の. 研究集会 での講演のOHPシートを掲載。　. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 2010年07月16日. 　・（. 過去と現在の研究 ）2010年10月開催予定の. 研究集会 での講演の. 　　. アブストラクト を掲載。. 2010年06月18日. 　・（. 論文 ）修正版を掲載（Propositions 5.7, (ii); 5.8, (vi)を少し修正した）：. 　　Topics in Absolute Anabelian Geometry III. 2010年06月11日. 　・（. 論文 ）修正版を掲載（Definition 5.4, (v)を少し修正した）：. 　　Topics in Absolute Anabelian Geometry III. 2010年06月01日. 　・（. 論文 ）修正版を掲載（Corollary 2.8, (i) と Remark 2.12.1を修正した）：. 　　Topics in Absolute Anabelian Geometry I. 2010年05月06日. 　・（. 論文 ）修正版を掲載（. 主な修正箇所のリスト ）：. 　　Topics in Absolute Anabelian Geometry I. 2010年04月12日. 　・（. 感想・着想 ）IUTeich理論の論文の執筆状況に関する報告を掲載。. 　・. 「数理研近辺の桜2010」 を掲載。. 2009年10月15日. 　・（. 感想・着想 ）IUTeich理論の論文の執筆状況に関する報告を掲載。. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 2009年10月01日. 　・（. 論文 ）修正版を掲載（＝Lemma 1.6, Remark 1.6.1を修正した）：. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 2009年09月13日. 　・（. 論文 ）修正版を掲載（＝Theorem 6.1, Corollary 6.3を修正した）：. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 2009年09月03日. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 　・（. 論文 ）修正版を掲載（＝Remark 2.11.1を少し修正した）：. 　　Topics in Absolute Anabelian Geometry II. 2009年08月19日. 　・（. 論文 ）新論文を掲載：. 　　On the Combinatorial Anabelian Geometry of Nodally Nondegenerate. 　　　　Outer Representations. 2009年07月06日. 　・（. 論文 ）修正版を掲載（＝関連する素数が2のときの「edge-paired」の扱いを. 　　少し修正した）：. 　　Topics in Absolute Anabelian Geometry II. 2009年06月16日. 　・（. 論文 ）コメントを更新：. 　　Semi-graphs of Anabelioids. 2009年05月14日. 　・（. 論文 ）コメントを掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2009年04月21日. 　・（. 論文 ）修正版（＝Remark 2.11.1を追加）を掲載：. 　　Topics in Absolute Anabelian Geometry II. 2009年04月12日. 　・. 「数理研近辺の桜2009」 を掲載。. 2009年04月05日. 　・（. 論文 ）修正版（＝特にlog-shellの定義を修正）を掲載：. 　　Topics in Absolute Anabelian Geometry III. 2009年02月25日. 　・（. 論文 ）修正版を掲載：. 　　On the Combinatorial Cuspidalization of Hyperbolic Curves. 　・（. 論文 ）次の論文の内容に関連したコメントを掲載：. 　　The Algebraic and Anabelian Geometry of Configuration Spaces. 2009年02月17日. 　・（. 論文 ）修正版を掲載：. 　　Arithmetic Elliptic Curves in General Position. 2009年02月11日. 　・（. 感想・着想 ）IUTeich理論の論文の執筆状況に関する報告を掲載。. 2009年01月22日. 　・（. 論文 ）修正版を掲載：. 　　On the Combinatorial Cuspidalization of Hyperbolic Curves. 2008年12月12日. 　・（. 論文 ）論文リストを更新し、修正版を掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2008年11月17日. 　・（. 論文 ）修正版を掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2008年09月05日. 　・（. 論文 ）コメントを掲載：. 　　The Geometry of Frobenioids I: The General Theory. 2008年08月18日. 　・（. 論文 ）修正版を掲載：. 　　Topics in Absolute Anabelian Geometry II. 2008年08月03日. 　・（. 論文 ）修正版を掲載：. 　　Topics in Absolute Anabelian Geometry III. 2008年07月21日. 　・（. 過去と現在の研究 ）. 「IUTeich理論って何？」. 加藤文元氏にご指摘. 　　いただいた「そっくりアニメ」による解説。. 2008年06月17日. 　・（. 論文 ）修正版を掲載：. 　　Arithmetic Elliptic Curves in General Position. 2008年06月11日. 　・（. 論文 ）新論文を掲載：. 　　On the Combinatorial Cuspidalization of Hyperbolic Curves. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 　・（. 感想・着想 ）組合せ論的カスプ化と、絶対副p遠アーベル幾何の最近の. 　　研究に関する報告を掲載。. 2008年05月18日. 　・（. 出張・講演 ）「数論的Teichmuller理論」に関する集中講義と談話会の. 　　講義ノートを掲載。. 　・（. 論文 ）論文の図とコメントを更新：. 　　Semi-graphs of Anabelioids. 2008年04月14日. 　・（. 論文 ）修正版を掲載：. 　　Topics in Absolute Anabelian Geometry I, II, III. 2008年04月09日. 　・. 「数理研近辺の桜2008」 を掲載。. 　・（. 感想・着想 ）松本氏の単射性定理の組合せ論版の研究に関する報告を掲載。. 　・（. 論文 ）コメントを掲載：. 　　Foundations of p-adic Teichmuller Theory. 　・（. 論文 ）コメントを掲載：. 　　Semi-graphs of Anabelioids. 2008年03月25日. 　・（. 論文 ）新論文を掲載：. 　　Topics in Absolute Anabelian Geometry I, II, III. 　・（. 論文 ）新論文を掲載：. 　　Arithmetic Elliptic Curves in General Position. 　・（. 過去と現在の研究 ）これまでの研究に関する報告を更新。. 2008年02月01日. 　・（. 論文 ）コメントを掲載：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 2008年01月05日. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 2007年07月27日. 　・（. 論文 ）修正版を掲載：. 　　The Algebraic and Anabelian Geometry of Configuration Spaces. 2007年06月06日. 　・（. 論文 ）修正版を掲載：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 2007年05月16日. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 　・（. 論文 ）修正版を掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2007年04月12日. 　・. 「数理研近辺の桜2007」 を掲載。. 2007年04月06日. 　・（. 論文 ）修正版を掲載：. 　　A Combinatorial Version of the Grothendieck Conjecture. 2007年01月31日. 　・（. 論文 ）修正版を掲載：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 　・（. 論文 ）修正版を掲載：. 　　A Combinatorial Version of the Grothendieck Conjecture. 2007年01月20日. 　・（. 論文 ）修正版を掲載：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 2006年12月23日. 　・（. 論文 ）修正版を掲載：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 　・（. 論文 ）修正版を掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2006年11月27日. 　・（. 論文 ）修正版を掲載：. 　　The Algebraic and Anabelian Geometry of Configuration Spaces. 2006年11月18日. 　・（. 出張・講演 ）来週の「p進数論幾何」の研究集会での講演のレクチャー・. 　　ノートを掲載。. 2006年10月31日. 　・（. 感想・着想 ）「対称化された配置空間」に関する訂正。. 2006年09月16日. 　・（. 過去と現在の研究 ）ABC予想関連の最近の研究に関する報告を更新。. 2006年09月06日. 　・（. 論文 ）新論文を掲載：. 　　The Algebraic and Anabelian Geometry of Configuration Spaces. 2006年09月01日. 　・（. 論文 ）修正版を掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2006年08月25日. 　・（. 感想・着想 ）前回の報告の補足を掲載。. 　・（. 論文 ）修正版を掲載：. 　　Conformal and Quasiconformal Categorical Representation of. 　　Hyperbolic Riemann Surfaces. 2006年07月14日. 　・（. 論文 ）修正版を掲載：. 　　Global Solvably Closed Anabelian Geometry. 2006年06月24日. 　・（. 感想・着想 ）最近の遠アーベル幾何の研究に関する報告を掲載。. 2006年06月13日. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids I: The General Theory. 2006年06月03日. 　・（. 論文 ）修正版を掲載：. 　　Global Solvably Closed Anabelian Geometry. 2006年06月02日. 　・（. 学生・受験生諸君へ ）「学生・受験生諸君へ」を更新。. 2006年05月22日. 　・（. Curriculum Vitae ）「望月新一」の発音を録音したmp3ファイルを掲載。. 2006年05月21日. 　・（. 論文 ）修正版を掲載：. 　　Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. 2006年04月22日. 　・（. 論文 ）新論文を掲載：. 　　The Etale Theta Function and its Frobenioid-theoretic Manifestations. 2006年04月11日. 　・（. 論文 ）修正版を掲載：. 　　A Combinatorial Version of the Grothendieck Conjecture. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids I: The General Theory. 　・（. 論文 ）修正版を掲載：. 　　The Geometry of Frobenioids II: Poly-Frobenioids. 2006年04月06日. 　・（. 論文 ）修正版を掲載：. 　　Conformal and Quasiconformal Categorical Representation of. 　　Hyperbolic Riemann Surfaces."}
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[1] The Local Pro-p Grothendieck Conjecture　（1997年以前）. PDF [2] Faltingsのp進ホッジ理論、Grothendieck予想、標準的通常曲線の群論的特徴付け　（1997年以前）. PDF [3] 通常p進曲線の理論　（1997年以前）. PDF [4] （玉川安騎男氏による三人の）秋季賞の受賞講演　（東京大学 1997年9月）. PDF [5] 代数曲線に関するGrothendieck予想 - p進幾何の視点から　（津田塾 1998年10月）. PDF [6] 楕円曲線の比較定理とTheta関数　（東京大学 1999年7月）. PDF [7] 楕円曲線のHodge-Arakelov理論　（京都大学数論合同セミナー 1999年12月）. PDF [8] 楕円曲線のHodge-Arakelov理論における遠アーベル幾何、数論的微分とは何か？　（名古屋大学. 　　　2001年11月）. PDF [9] 数論的 log scheme の圏論的表示　（九州大学 2003年7月）. 田口さんのノート [10] 数論的log schemeの圏論的表示から見た楕円曲線の数論　（北海道大学 2003年11月）. PDF [11] 数論的Teichmuller理論入門　（京都大学理学部数学教室 2008年5月）.　　. 月 　. 火 　. 水 　. 木 　. 金 　. 概要 　. 　　　. レポート問題 　. 談話会 　. アブストラクト [12] 宇宙際タイヒミューラー理論への誘（いざな）い　（京都大学数理解析研究所 2012年12月）. PDF [13] 宇宙際タイヒミューラー理論への誘（いざな）い　《拡大版》 （東京大学 2013年06月）. PDF [14] 数論幾何の風景 — 数の加減乗除から対称性の幾何まで　（京都大学2013年11月）. PDF [15] 宇宙際タイヒミューラー理論への誘（いざな）い　《3時間版》 （京都大学数理解析研究所 2014年02月）. 　　　. PDF [16] 宇宙際タイヒミューラー理論への誘（いざな）い　《2+2時間版》 （熊本大学 2014年05月）. PDF [17] 宇宙際タイヒミューラー理論への誘（いざな）い　（2015-02） （京都大学数理解析研究所 2015年02月）. 　　　. PDF [18] 宇宙際タイヒミューラー理論の論理構造 I, II, III　（「代数的整数論とその周辺」 (Zoom) 2021年12月13日〜14日）. 　　　. Lecture notes Lecture notes (marked up version for I, II, III) 　　　. Classical roots of inter-universal Teichmuller theory 　　　. Classical roots of inter-universal Teichmuller theory (marked up version) 　　　　　　　　　　　　　　　　　　　　　　　　　　英語. [1] The Intrinsic Hodge Theory of p-adic Hyperbolic Curves (Oberwolfach 1996年7月). PDF [2] The Intrinsic Hodge Theory of p-adic Hyperbolic Curves (ICM 98). PDF [3] The Intrinsic Hodge Theory of p-adic Hyperbolic Curves (奈良 1998年12月). PDF [4] The Hodge-Arakelov Theory of Elliptic Curves (Berkeley 1999年9月). PDF [5] The Hodge-Arakelov Theory of Elliptic Curves (Utrecht 2000年6月). PDF [6] The Hodge-Arakelov Theory of Elliptic Curves (穂高 2000年7月). PDF [7] A Brief Introduction to Inter-universal Geometry (東京大学 2004年1月). PDF [8] Categorical Representation of Arithmetic Log Schemes – with Applications to the Arithmetic of. 　　　Elliptic Curves (東京大学 2004年2月). PDF [9] Anabelian Geometry from an Inter-universal Point of View (京都大学数理解析研究所 2004年9月). 　　　. PDF [10] Absolute Anabelian Cuspidalizations (京都大学数理解析研究所 2005年2月). PDF [11] Tempered Anabelian Geometry (広島大学 2005年2月). PDF [12] A Brief Survey of the Geometry of Categories (岡山大学 2005年5月). PDF [13] Inter-universal Hodge-Arakelov Theory (京都大学数理解析研究所 2005年12月). PDF [14] A Survey of Absolute p-adic Anabelian Geometry (京都大学数理解析研究所 2006年11月). PDF [15] Inter-universal Teichmuller Theory: a Progress Report (京都大学数理解析研究所 2010年10月). 　　　. Abstract PDF [16] Classical Roots of Inter-universal Teichmuller Theory (Berkeley Colloquium (Zoom) 2020年11月). 　　　. Abstract PDF [17] The Technique of Tripodal Transport in Arithmetic Geometry I, II (Foundations and Perspectives of. 　　　Anabelian Gemetry (Zoom) 2021年06月). Lecture notes (adapted from Section 3.10 of. [Alien] ). 　　　. Lecture Notes (marked up version) Microsoft Whiteboard Image [18] Overview of Combinatorial Anabelian Geometry I, II, III (Combinatorial Anabelian Geometry and. 　　　Related Topics (Zoom) 2021年07月). Lecture notes Lecture notes (marked up version) 　　　. Microsoft Whiteboard Image [24] On the essential logical structure of inter-universal Teichmuller theory I, II, III, IV, V （宇宙際タイヒ. 　　　ミューラー理論への誘い（いざない） (Zoom) 2021年9月 + 宇宙際タイヒミューラー理論サミット２０２１. 　　　(Zoom) 2021年9月）. Lecture notes Lecture notes (marked up version for I, II, III) 　　　. Lecture notes (marked up version for IV, V) Classical roots of inter-universal Teichmuller theory 　　　. Classical roots of inter-universal Teichmuller theory (marked up 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IUTeich. に関する研究集会（＝日本語の講演による）の. 　. 案内 を掲載。（英語による）報告集は数理研の「. 講究録別冊 」として刊行する予定である。. 　. 星氏の講演のスライド は事前に公開しました。. 　山下氏の講演の写真（＝星・望月撮影）：. 　　（星）　　 3月. 09 日. 10 日. 11 日. 12 日. 13 日. 16 日. 17 日. 18 日. 19 日. 20 日. 　　（望月）　3月. 09 日. 10 日. 11 日. 12 日. 13 日. 16 日. 17 日. 18 日. 19 日. 20 日. 　. 集合写真 ・IUTeichの検証活動に関する. 報告 （2014年12月現在）. ・Ivan Fesenko氏による、IUTeichに関する. サーベイ と. 解説記事 ・2015年12月に（英）Oxford大学で開催されたIUTeichに関する. 研究集会のホームページ ・2016年7月に数理研で開催されたIUTeichに関する. 研究集会のホームページ 　（. 写真 ）. ・星裕一郎氏による、IUTeichに関する. サーベイ と. 講演スライド ・山下剛氏による、IUTeichに関する. 講演スライド ・譚福成氏による、. IUTeichに関する講演のスライド ・加藤文元氏による、. IUTeichに関する講演ビデオ ・. IUTeichに関するアニメーション. （＝[IUTchIII], Theorem Aの内容に対応）. 　\"The Multiradial Representation of Inter-universal Teichmuller Theory\"を公開。. 　背景：　　黒　（. avi 　. wmv ）　　黒板　（. avi 　. wmv ）　　和紙　（. avi 　. wmv ）. 　石碑版：　「復元」　フェードアウト版　（. avi wmv ）　　　　　　　背景版　（. avi wmv ）. 　　　　　　　「復元の流れ」　フェードアウト版　（. avi wmv ）　　　背景版　（. avi wmv ）. ・第二の、. IUTeichに関するアニメーション. （＝[IUTchIII], Theorem Bの内容に対応）. 　. \"Computation of the log-volume of the q-pilot via the multiradial representation\" 　を公開。. ・用語\"anabelioid\"および\"Frobenioid\"の語形変化論的側面に関する. 解説文書 ・. Frobenioid. の理論に関する. 問答集 ・南出新氏による、. 組合せ論的遠アーベル幾何. の入門講義の写真やスライドを纏めた. 　PDFファイル（＝　2016-01-25：　. 1 2 3 4 　・　2016-03-15：　. 1 2 3 4 　・　. 　2017-03-28:　. スライド ）. ・南出新氏による、. 配置空間群の数値的不変量. に関する講演の写真を纏めた. 　PDFファイル（＝　2016-09-13：　. 1 2 3 4 ）. ・山下剛氏による、多重ゼータ値入門講義（ただし、最後の4節を除く）の写真を. 　纏めたPDFファイル（＝　. 1 2 3 4 5 6 ）. ・2018年3月、数理研で行なわれたIUTeichに関する議論を纏めた. 報告書 　（および. 関連文書 ）. ・南出新氏による、. IUTeichにおける明示的な不等式に関する講演のスライド を掲載. ・2021年度に開催予定の訪問滞在型研究. 「宇宙際タイヒミューラー理論の拡がり」 の. 　ウェブページを掲載. ・世界の数学者に向けた、今年度の. 宇宙際タイヒミューラー理論関連集会のビデオ閲覧. の. 　. 招待状 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IUTeichの「心」. は、簡単に言うと、次のようなものです：. 　「数論幾何において本質的なのは、環やスキームのような‘具体的’な対象たちではなく、むしろそれら. 　の具体的なスキーム論的な対象たちを統制している、様々な（‘組み合わせ論的アルゴリズム’に近い）. 　. 抽象的なパターン. である。」. このような現象の典型的な例として次のようなものが挙げられます：. (1) log schemeの幾何：詳しくは、私の論文　. Extending Families of Curves over Log Regular Schemes の. 文献リストに出ている加藤和也先生の二つの論文を参照して下さい。簡単にまとめると、. 　「多項式環等、Noether環の構造のある側面の本質は、. モノイド. という組み合わせ論的な対象に集約される」. という内容の理論です。. (2) 遠アーベル幾何：これについては、沢山の論文を書いていますが、入門的な解説では、次の二つが挙げ. られます：. ・. 「代数曲線の基本群に関するGrothendieck予想」 ・. 「代数曲線に関するGrothendieck予想 --- p進幾何の視点から」 簡単にまとめると、「数論的な体」の上で定義された双曲的曲線の構造は、その有限次エタール被覆の自己. 同型群の. 群論的構造. だけで決まるという理論です。. (3) 圏の幾何：これについては、私の論文. ・. Categorical representation of locally noetherian log schemes ・. Categories of log schemes with archimedean structures ・. Conformal and Quasiconformal Categorical Representation of Hyperbolic Riemann Surfaces それから、講演のレクチャーノート. ・. 「A Brief Survey of the Geometry of Categories (岡山大学 2005年5月)」 を参照して下さい。簡単にまとめると、スキーム（または、log schemeやarchimedeanな構造付きのlog. scheme）や双曲的リーマン面の構造は、そのような対象たちが定義する圏（＝‘category'）の. 圏論的構造. だけで決まるという話です。. 因みに、IUTeich関係の話では、p進Teichmuller理論に登場する「標準的なFrobenius持ち上げの微分を. とる」という操作の「抽象的パターン的類似物」が主役です。p進Teichmuller理論の解説としては、. ・. An Introduction to p-adic Teichmuller Theory ・. 「An Introduction to p-adic Teichmuller Theory」 (和文） が挙げられます。. 同じ研究所の. 玉川安騎男教授. も、別の観点から「双曲的代数曲線の数論」の研究に携わっております。. 玉川さんの研究との比較については、. ここ をご参照下さい。. 以上のまとめからも推測されるように、修士課程への入学を希望する学生に対しては次のような予備知識を. 要求しております：. 　(1). 代数位相幾何. の基礎的な知識（＝基本群や特異コホモロジー）. 　(2). リーマン面. の基礎的な知識（＝line bundleやRiemann-Rochの定理）. 　(3). 可換環論やスキーム論. の基礎的な知識（「松村」、「Hartshorne」を参照）. ただし、特に(3)については完全な理解を要求するのではなく、内容に対して一定の「親しみ」さえあれば、. 入学してからセミナーなどで復習することは可能です。. 辻村　昇太. 　（つじむら　しょうた）. 略歴：. 2015年03月　京都大学 理学部 数学科 卒業. 2015年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 2017年03月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了. 　　修士論文：　　　　. 　　Geometric Version of the Grothendieck Conjecture for Universal Curves over. 　　Hurwitz Stacks　. PDF 2017年04月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 進学. 2018年04月〜2020年03月　日本学術振興会 特別研究員（DC2）. 若林　泰央. 　（わかばやし　やすひろ）. 略歴：. 2009年03月　京都大学 理学部 数学科 卒業. 2009年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 2011年03月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了. 　　修士論文：　　　　. 　　On the cuspidalization problem for hyperbolic curves over finite fields　. PDF 2011年04月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 進学. 2012年04月〜2014年03月　日本学術振興会 特別研究員（DC2）. 2014年03月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 修了. 　　学位論文：　　　　. 　　An explicit formula for the generic number of dormant indigenous bundles　. PDF 2014年04月〜07月　京都大学数理解析研究所　研究員. 2014年04月〜2015年03月　大阪市立大学数学研究所　研究所員（兼任）. 2014年08月〜2015年03月. 　　京都大学学際融合教育研究推進センター　特定研究員. 2015年04月〜05月　日本学術振興会　特別研究員（PD）. 2015年06月〜2018年03月　東京大学大学院数理科学研究科　特任助教. 2018年04月〜　東京工業大学理学院数学系　助教. 望月研究室の大学院生. 星　裕一郎 　（ほし　ゆういちろう）. 略歴：. 2004年03月　東京工業大学 理学部 数学科 卒業. 2004年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 2006年03月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了. 　　修士論文：　　　　. 　　Fundamental groups of log configuration spaces and the cuspidalization problem. 　　. PDF 2006年04月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 進学. 2006年04月〜2007年03月　日本学術振興会 特別研究員（DC1）. 2007年04月　京都大学 数理解析研究所 基礎数理研究部門 助教. 2009年07月　京都大学 数理解析研究所 博士学位 (論文博士) 取得. 　　学位論文：　　　　. 　　Absolute anabelian cuspidalizations of configuration spaces of proper hyperbolic. 　　curves over finite fields　. PDF 2011年12月　京都大学 数理解析研究所 無限解析研究部門 講師. 2011年12月　2011年度井上研究奨励賞受賞. 2017年12月　京都大学 数理解析研究所 無限解析研究部門 准教授. なお、仮に修士課程に入学し、私の学生になった場合の、少なくとも最初の一年間の. 「カリキュラム」. は. 大体次のとおりになります：. 　(a) 「松村」、「Hartshorne」の復習. 　(b) 複素多様体や微分多様体の理論の復習. 　(c) エタール・トポス、エタール・コホモロジー、エタール基本群. 　(d) 曲線やアーベル多様体のstable reduction. 　(e) log scheme の幾何. 　(f) エタール基本群のweightの理論. また、これらの基本的なテーマの勉強が済んだら、. 　(i) crystalやcrystalline site, crystalline cohomology. 　(ii) Fontaine氏が定義した様々な「p進周期環」. 　(iii) p-divisible groupsとfiltered Frobenius moduleの関係. 　(iv) Faltingsのp進Hodge理論. 　(v) p進遠アーベル幾何. 　(vi) p進Teichmuller理論. のようなp進的なテーマに進むことなどが考えられます。（(v), (vi)については、本サイトの. 「論文」 、. 「過去と現在の研究」 、または. 「出張・講演」 をご参照下さい。）. 私の研究の主なテーマは、. 「双曲的代数曲線の数論」. です。「双曲的代数曲線」. とは、大雑把に言うと、多項式で定義される幾何学的な対象の中で、. 上半平面. で一意化されるリーマン面. に対応するものです。ただし、複素数体の上でしか. 意味を成さないリーマン面の理論と違って、代数的な対応物を扱うことによっ. て、数体やp進局所体といった「数論的な体」の上で定義されたものの様々な. 興味深い性質を考察することが可能になります。また、双曲的なリーマン面と. 同様に、双曲的代数曲線の研究では、. 基本群. およびその. 基本群へのガロア群の. 作用. が重要な役割を果たします。私の研究に関するもっと詳しい説明について. は本サイトの. 「論文」 、. 「過去と現在の研究」 、または. 「出張・講演」 を. ご参照下さい。. 南出　新. 　（みなみで　あらた）. 略歴：. 2011年03月　京都大学 理学部 数学科 卒業. 2011年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 2013年03月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了. 2013年04月〜09月　京都大学数理解析研究所 研究生. 2013年10月　大阪大学 理学研究科 数学専攻 博士課程 入学. 2014年03月　大阪大学 理学研究科 数学専攻 博士課程 中退. 2014年04月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 入学. 2016年04月〜2017年03月　日本学術振興会 特別研究員（DC2）. 2017年03月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 修了. 　　学位論文：　　　　. 　　Indecomposability of Various Profinite Groups Arising from Hyperbolic Curves　. PDF 2017年04月〜2019年03月　京都大学数理解析研究所　研究員. 2019年04月〜　イギリス・ノッティンガム大学・オックスフォード大学　. 　　Invited Research Visitor. Yang, Yu. 　（やん　ゆー）. 略歴：. 2009年07月　中国国内の大学の修士課程を卒業. 2009年10月〜2011年03月　京都大学数理解析研究所 研究生. 2011年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 2013年03月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了. 2013年04月〜2014年03月　京都大学数理解析研究所 研究生. 2014年04月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 入学. 2016年04月〜2018年03月　日本学術振興会 特別研究員（DC2）. 2017年03月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 修了. 　　学位論文：　　　　. 　　Degeneration of Period Matrices of Stable Curves　. PDF 2018年04月〜　京都大学数理解析研究所　研究員. 河口　. 祐輝. 　（かわぐち　ゆうき）. 略歴：. 2012年03月　東京大学 理学部 数学科 卒業. 2012年04月　東京大学大学院 数理科学研究科 数理科学専攻 修士課程 入学. 2014年03月　東京大学大学院 数理科学研究科 数理科学専攻 修士課程 修了. 　　修士論文：　　　　. 　　Fibered products of Hopf algebras and Seifert-van Kampen theorem for. 　　semi-graphs of tannakian categories　. PDF 2014年04月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 入学. 2019年03月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 修了. 　　学位論文：　　　　. 　　Near Miss abc-Triples in General Number Fields　. PDF 2019年04月〜　私立灘中学校・高等学校　教諭. 和田　. 悠暉. 　（わだ　ゆうき）. 略歴：. 2014年03月　京都大学 理学部 数学科 卒業. 2014年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 2017年03月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了. 　　修士論文：　　　　. 　　Near miss abc-triples in compactly bounded subsets　. PDF 2017年04月　株式会社NTTデータ数理システムに就職. 2019年02月　株式会社NTTデータ数理システムを退職. 2019年03月　富士フィルム株式会社に就職. Ponrod, Pitchayatak. 　（ポンロド　ピッチャヤタク）. 略歴：. 2017年03月　タイ・チュラーロンコーン大学 数学科　修士課程 修了. 2018年04月〜09月　京都大学国際高等教育院　聴講生（文科省 国費留学生）. 2018年10月〜　京都大学数理解析研究所　研究生. 牟　卓群. 　（む　たくぐん　Mou, Zhuoqun）. 略歴：. 2017年07月　中国・中国科学技術大学 数学科 卒業. 2017年10月〜2019年03月　京都大学数理解析研究所　研究生. 2019年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 湯地　智紀. 　（ゆじ　ともき）. 略歴：. 2018年03月　大阪大学 理学部 数学科 卒業. 2018年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. （. 星氏の学生. ）　. 東山　和巳. 　（ひがしやま　かずみ）. 略歴：. 2014年03月　東京工業大学 理学部 数学科 卒業. 2014年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. 2016年03月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了. 　　修士論文：　　　　. 　　Reconstruction of inertia groups associated to log divisors from a configuration. 　　space group equipped with log-full subgroups　. PDF 2016年04月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 進学. 2018年04月〜2020年03月　日本学術振興会 特別研究員（DC2）. 2019年03月　京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 修了. 　　学位論文：　　　　. 　　The semi-absolute anabelian geometry of geometrically pro-p arithmetic. 　　fundamental groups of associated low-dimensional configuration spaces. Tafula Santos, Christian. 　（タフラ・サントス・クリスティアン）. 略歴：. 2016年07月　ブラジル・サンパウロ大学 数学・統計学研究院 卒業. 2017年04月〜09月　京都大学国際高等教育院　聴講生（文科省 国費留学生）. 2017年10月〜2018年03月　京都大学数理解析研究所　研究生. 2018年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学. Sixtel, Ilia. 　（シクステル・イリア）. 略歴：. 2018年09月　イスラエル・エルサレム・ヘブライ大学 アジア学科・数学科 卒業. 2019年04月〜　京都大学数理解析研究所　研究生（文科省 国費留学生）. 孫 澤銘. 　（そん　たくめい　Sun, Zeming）. 略歴：. 2018年07月　中国・中国科学技術大学 数学科 卒業. 2018年10月〜2019年03月　京都大学数理解析研究所　研究生. 2019年04月　京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学."}
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2022","language_s":"en","size_i":20765,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085945631277056,"text_t":"望月新一の論文. 　　　 　　　　　　　　　　　 　　　　　　　　　　　欧文学術雑誌の論文等. 数論幾何一般. [1] The Geometry of the Compactification of the Hurwitz Scheme. PDF Comments NEW !! (2017-08-18). [2] On Semi-Positivity and Filtered Frobenius Crystals. PDF [3] Correspondences on Hyperbolic Curves. PDF Comments NEW !! (2016-07-16). [4] Extending Families of Curves over Log Regular Schemes. PDF [5] Noncritical Belyi Maps. PDF Comments NEW !! (2016-07-16). [6] Arithmetic Elliptic Curves in General Position. PDF NEW !! (2009-02-17). Comments NEW !! (2016-07-16). 内在的Hodge理論. [1] The Intrinsic Hodge Theory of p-adic Hyperbolic Curves (ICM 98). PDF [2] The intrinsic Hodge theory of hyperbolic curves (Seoul 1998). PDF [3] The intrinsic Hodge theory of p-adic hyperbolic curves (Nara 1998). PDF p進Teichmuller理論. [1] A Theory of Ordinary p-adic Curves. PDF Comments NEW !! (2022-03-20). [2] Foundations of p-adic Teichmuller Theory. Comments NEW !! (2022-03-20). [3] An Introduction to p-adic Teichmuller Theory. PDF Comments NEW !! (2022-03-20). 遠アーベル幾何、圏の幾何. [1] The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves over Number Fields. PDF Comments NEW !! (2012-12-20). [2] A Version of the Grothendieck Conjecture for p-adic Local Fields. PDF [3] The Local Pro-p Anabelian Geometry of Curves. PDF Comments NEW !! (2019-06-28). [4] The Grothendieck conjecture on the fundamental groups of algebraic curves. PDF [5] The Absolute Anabelian Geometry of Hyperbolic Curves. PDF Comments NEW!! (2016-06-21). [6] The Absolute Anabelian Geometry of Canonical Curves. PDF Comments NEW !! (2012-06-18). [7] Topics Surrounding the Anabelian Geometry of Hyperbolic Curves. PDF [8] The Geometry of Anabelioids. PDF Comments NEW !! (2022-05-18). [9] Categorical representation of locally noetherian log schemes. PDF Comments NEW !! (2014-10-30). [10] Categories of log schemes with archimedean structures. PDF PDF (LaTeX Version) Comments NEW !! (2014-10-30). [11] Semi-graphs of Anabelioids. PDF Comments NEW !! (2020-05-18). [12] Galois Sections in Absolute Anabelian Geometry. PDF Comments NEW !! (2014-08-09). [13] A Combinatorial Version of the Grothendieck Conjecture. PDF NEW !! (2007-04-06). Comments NEW !! (2020-05-18). [14] Conformal and Quasiconformal Categorical Representation of Hyperbolic Riemann Surfaces. PDF NEW !! 　　　(2006-08-25). Comments NEW !! (2015-12-06). [15] Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves. PDF PDF (LaTeX Version) NEW !! 　　　(2007-06-06). Comments NEW !! (2014-02-20). [16] The Geometry of Frobenioids I: The General Theory. PDF NEW !! (2008-06-11). Comments NEW !! (2020-04-25). [17] The Geometry of Frobenioids II: Poly-Frobenioids. PDF NEW !! (2008-06-11). Comments NEW !! (2019-02-10). [18] Global Solvably Closed Anabelian Geometry. PDF PDF (LaTeX Version) NEW !! (2006-07-14). [19] The Etale Theta Function and its Frobenioid-theoretic Manifestations. PDF NEW !! (2008-12-12). Comments NEW !! 　　　(2022-03-20). Related expositions NEW!! (2015-04-26). Responses to Questions on Frobenioids (cf., especially,. 　　　Questions 3, 8, 23, 24). NEW!! (2015-12-06). [20] The Algebraic and Anabelian Geometry of Configuration Spaces. PDF Comments NEW !! (2015-04-22). 　　　. Comments on strongly torsion-free groups NEW !! (2009-02-25). [21] Topics in Absolute Anabelian Geometry I: Generalities. PDF NEW !! (2012-03-14). Comments NEW !! (2017-12-30). [22] Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms. PDF NEW !! (2013-10-08). 　　　. Comments NEW !! (2016-06-21). [23] Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms. PDF NEW !! (2015-11-11). Comments NEW !! 　　　(2019-06-07). [24] On the Combinatorial Cuspidalization of Hyperbolic Curves. PDF Comments NEW !! (2019-07-20). [25] On the Combinatorial Anabelian Geometry of Nodally Nondegenerate Outer Representations. PDF NEW !! (2011-12-22). 　　　. Comments NEW !! (2019-07-20). [26] Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves I: Inertia Groups and. 　　　Profinite Dehn Twists. PDF NEW !! (2012-08-30). Comments NEW !! (2017-12-14). [27] Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves II: Tripods and. 　　　Combinatorial Cuspidalization. PDF NEW !! (2022-03-20). [28] Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves III: Tripods and. 　　　Tempered Fundamental Groups. PDF NEW !! (2022-03-20). [29] Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves IV: Discreteness and. 　　　Sections. PDF NEW !! (2021-07-12). [30] Monomorphisms in Categories of Log Schemes. PDF NEW !! (2014-11-24). [31] Group-theoreticity of Numerical Invariants and Distinguished Subgroups of Configuration Space Groups. PDF 　　　. NEW !! (2021-11-08). [32] Combinatorial Construction of the Absolute Galois Group of the Field of Rational Numbers. PDF NEW!! (2021-01-15). 楕円曲線のHodge-Arakelov理論. [1] The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories. PDF [2] The Scheme-Theoretic Theta Convolution. PDF [3] Connections and Related Integral Structures on the Universal Extension of an Elliptic Curve. PDF [4] The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve. PDF [5] The Hodge-Arakelov Theory of Elliptic Curves in Positive Characteristic. PDF [6] A Survey of the Hodge-Arakelov Theory of Elliptic Curves I. PDF Comments NEW !! (2012-10-18). [7] A Survey of the Hodge-Arakelov Theory of Elliptic Curves II. PDF Comments NEW !! (2012-10-18). [8] Anabelian Geometry in the Hodge-Arakelov Theory of Elliptic Curves. PDF 宇宙際Teichmuller理論. [1] Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. PDF NEW !! (2020-05-18). [2] Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation. PDF NEW !! (2020-12-23). [3] Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice. PDF NEW !! (2020-05-18). [4] Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations. PDF 　　. NEW !! (2020-04-22). [5] A Panoramic Overview of Inter-universal Teichmuller Theory. PDF NEW !! (2013-08-20). Comments NEW !! (2015-11-11). [6] Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of. 　　Inter-universal Teichmuller Theory. PDF NEW !! (2016-01-12). Comments NEW !! (2020-04-25). [7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF 　　. NEW !! (2020-12-23). [8] Explicit Estimates in Inter-universal Teichmuller Theory. PDF NEW!! (2022-05-18). [9] On the Essential Logical Structure of Inter-universal Teichmuller Theory in Terms of Logical AND \"∧\"\/. 　　Logical OR \"∨\" Relations: Report on the Occasion of the Publication of the Four Main Papers on. 　　Inter-universal Teichmuller Theory. PDF NEW!! (2022-06-01). 　　　　　　　　　　　　　　　　　　　　　　　　　　　和文雑誌の論文. [1] 代数曲線の基本群に関するGrothendieck予想. PDF [2] 基本群に映る双曲的代数曲線の構造. PDF [3] 代数曲線に関するGrothendieck予想 --- p進幾何の視点から. PDF [4] 連載ICM 98、部門別報告：整数論と数論幾何I. PDF [5] モチーフ --- 代数多様体の数論的骨格 ---. PDF [6] 多項式の解の近似がとりもつ数論と幾何の関係 (1), (2), (3), (4). PDF 　　　　　　　　　　　　　　　　　　　　　講演のアブストラクト・レクチャーノート. [1] 実複素多様体のセクション予想と測地線の幾何. PDF [2] p進Teichmuller理論. PDF [3] Anabelioidの幾何学. PDF [4] Anabelioidの幾何学とTeichmuller理論. PDF [5] 離散付値環のalmost etale extensions（学生用のノート）. PDF [6] 数体と位相曲面に共通する「二次元の群論的幾何」（2012年8月の公開講座）. PDF 論文リスト"}
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International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Workshop for Principal Researcher Call for proposals. Call for proposals（RIMS Research Projects） Call for proposals（RIMS Workshops\/Symposia） Call for proposals（RIMS Satellite Seminars） Call for proposals（RIMS Review Seminars） International exchange. TOP 　　 International exchange. Overview. 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Researchers from each field of mathematical science invited from overseas stay for an extended duration through the year and conduct joint research with domestic researchers. Distinguished Visiting Professors. As part of the efforts for assisting the Top Global University Project at Kyoto University, professors with leading international reputations, including Fields Medal winners, are received as distinguished visiting professors. They conduct research and teach students with the staff of Kyoto University, as well as participate in a wide range of education including special lectures. Foreign visitors list RIMS as an international joint research center. As the largest center of international joint research in the field of mathematical science in Japan, prominent researchers from overseas come to stay at this institute. It serves as a place for domestic and overseas researchers to conduct collaborative research. Its activities have earned international acclaim, including a comment in the Notice of the AMS, 2004,. \"RIMS, an Institute for Japan and the World.\" Specific activities include receiving researchers of mathematical science from overseas universities and research institutions as visiting professors or guest scholars, guest research associates, or short-stay visiting research scholars. These guest scholars stay for around a week to half a year and are received in various forms. Three hundred or more researchers are received every year, including overseas researchers coming to conduct joint usage research. The institute also actively accepts international students. Inviting overseas researchers is expected to promote interactions with domestic researchers through international joint research. There is systematic coordination of the stays of overseas researchers with joint usage research such as international symposiums (held around 10 times every year as part of a joint usage research project) in which many overseas researchers participate. RIMS aims at promoting RIMS Research Projects for international joint research and encourages international applications for joint research activities. Furthermore, RIMS is committed to supporting young researchers in pursuing international joint research. Academic Exchange Agreements. As part of the activities of the center of international research, the following academic exchange agreements have been concluded for the purpose of promoting and developing research collaborations in the fields of mathematical science. Number of Visitors from Overseas. Overview Exchange Agreements Number of Visitors from Overseas ←　BACK TO THE TOP About Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Workshop for Principal Researcher Call for proposals. Call for proposals（RIMS Research Projects） Call for proposals（RIMS Workshops\/Symposia） Call for proposals（RIMS Satellite Seminars） Call for proposals（RIMS Review Seminars） Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities. TOP Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities. RIMS recognizes that, from both personal and social perspectives, it is important to ensure that a diverse array of individuals is able to thrive within the mathematics community of Japan. Indeed, not only is there an imperative for talented mathematicians of various circumstances to be provided with an academic environment in which they can fulfil their potential, but also, without such, society risks losing valuable contributions towards future progress. Some of the initiatives that RIMS has introduced in order to support a diverse research community are listed below. Given the extremely low participation of women in mathematics in Japan, these place a particular focus on gender equality and diversity. Code of Conduct. A \"Code of Conduct\" is in place for those involved in joint research activities at RIMS. 　. URL:https:\/\/www.kurims.kyoto-u.ac.jp\/~kenkyubu\/CodeConductVer5.pdf We ask those proposing or participating in a research activity at RIMS to read the above Code of Conduct carefully and understand how it applies to them. \"Promotion of Women's Participation\" in Research Activities. We encourage the participation of female mathematicians in our research community at RIMS. To support this, we have introduced a specific \"Promotion of Women's Participation\" workshop\/symposium program. To be eligible for this, proposals must meet the following three conditions. 　(1) The principal organizer is female. (If there are two, both are female.). 　(2) 50% or more of the speakers (which should be specified in the proposal) are female. 　(3) 50% or more of the expected participants listed in the proposal are female. *Type C is excluded. The Advisory Board will select up to three such proposals. For those accepted, a budget of up to 1.2 million yen will be offered; this is to support travel expenses for the participants, and encourage international participation. Proposals not adopted as this type of project will still hold a chance to be selected as a regular RIMS Workshop\/Symposium. The Technical Committee and the Advisory Board will make the selection and decide on the budget (up to 700,000 yen) for each of the proposals accepted as a regular RIMS Workshop\/Symposium. We encourage the participation of female mathematicians in our research community at RIMS. To support this, we have introduced a specific \"Promotion of Women's Participation\" seminar program. To be eligible for this, proposals must meet the following three conditions. 　(1) The principal organizer is female. (If there are two, both are female.). 　(2) 50% or more of the speakers and all organizing committee members (which should be specified in the proposal) are female. 　(3) 50% or more of the expected participants listed in the proposal are female. The Advisory Board will select up to one such proposal as \"Promotion of Women's Participation\" programs. For those accepted, a budget of up to 1 million yen will be offered; this is to support travel expenses for the participants, and encourage international participation. Proposals not adopted as this type of project will still hold a chance to be selected as a regular RIMS Review Seminar. For more details, please refer to the webpage below. In particular, see the \"Guidelines for Grant Proposal Application Form (for “Promotion of Women's Participation”)\", and \"How to Write the Proposal (for “Promotion of Women's Participation”)\". 　　. RIMS Workshops\/Symposia (Type A,C) URL:https:\/\/www.kurims.kyoto-u.ac.jp\/kyoten\/en\/call_for_proposals-1.html 　　. RIMS Review Seminars URL:https:\/\/www.kurims.kyoto-u.ac.jp\/kyoten\/en\/call_for_proposals-3.html Please note that the application form and the instructions (“How to Write the Proposal”) for the \"Promotion of Women's Participation\" scheme are different from those for the regular RIMS Workshops\/Symposia program. Gender Equality and Fostering Early Career Researchers. Kyoto University supports gender equality and encourages the research activities of early career researchers. RIMS asks that those preparing proposals (for any RIMS program) consider gender diversity and the perspective of fostering early career researchers. We particularly hope to see more female researchers participating as members of organizing committees. We moreover encourage the invitation of female and early career researchers to participate in and present their work at the various Joint Research Activities of RIMS. Finally, we highlight that proposals from female and early career researchers are extremely welcome. Childcare Services during Joint Research Activities. It is possible to set up a temporary nursery on campus during research meetings for the use of researchers with children. For details, please contact the organizer of the meeting you are attending or the Cooperative Research Service Section at RIMS. Data on Women's Participation. For data on female participation in RIMS activities (numbers of participants, speakers and organizers, by year), see the following document. →. Click here Code of Conduct \"Promotion of Women's Participation\" in Research Activities Gender Equality and Fostering Early Career Researchers Childcare Services during Joint Research Activities Data on Women's Participation ←　BACK TO THE TOP About Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Workshop for Principal Researcher Call for proposals. Call for proposals（RIMS Research Projects） Call for proposals（RIMS Workshops\/Symposia） Call for proposals（RIMS Satellite Seminars） Call for proposals（RIMS Review Seminars） Past RIMS Research Projects. TOP Past RIMS Research Projects. Past RIMS Research Projects. 2021. Mathematical Biofluid Mechanics As a part of applied mathematical studies, biofluid mechanics has gathered significant attention from various research communities such as physical and material sciences, engineering, biology and medicine. In particular, novel computational and theoretical techniques, mathematical models and methods are all required to understand complex motions in biological phenomena. In this research project, though a series of workshops, tutorial seminars and symposia, we enthusiastically explore newly-born research topics in collaboration with researchers with various research backgrounds to expand the horizons of fluid mechanics and applied mathematics, in addition to deepening the traditional research topics, aiming at cultivating national and international networks of related researchers. Lectures on patterns and dynamics in biology: From basics to recent advances (RIMS Research Project 2021)【RIMS Review Seminars】 Location: Online via Zoom Period: 2021-06-15〜2021-06-18. Organizer: Kenta Ishimoto(RIMS, Kyoto University). Biofluids Symposium (RIMS Research Project 2021)【RIMS Symposia】 Location: Online via Zoom Period: 2021-06-21〜2021-06-24. Organizer: Makoto Iima(Hiroshima University, Graduate School of Integrated Sciences for Life). Fluid mechanics in ecology and evolution (RIMS Research Project 2021)【RIMS Workshops(Type A)】 Location: Online via Zoom Period: 2021-07-05〜2021-07-07. Organizer: Tsuyoshi Hirashima(Graduate School of Medicine and Faculty of Medicine, Kyoto University). Studies on Mathematical Mechanobiology 2021 (RIMS Research Project 2021)【RIMS Workshops(Type A)】 Location: Online via Zoom Period: 2021-07-14〜2021-07-16. Organizer: YAMAMICHI Masato(The University of Queensland). 　　　　　　　　　　　　　　　　. Biofluid Mehanics of Reproduction (RIMS Research Project 2021)【RIMS Workshops(Type A)】 Location: Online via Zoom Period: 2021-07-29〜2021-07-30. Organizer: Lisa J. Fauci(Tulane University). Mathematical methods for the studies of flow, shape, and dynamics (RIMS Research Project 2021)【RIMS Workshops(Type A)】 Location: Online via Zoom Period: 2021-08-30〜2021-08-31. Organizer: Kenta Ishimoto(RIMS, Kyoto University). 2021. Theory of operator algebras and its applications Operator algebra theory is a branch in functional analysis being studied intensively and extensively with strong ties to ergodic theory, topological dynamical systems, analytic group theory,mathematical physics, quantum information, noncommutative geometry, noncommutative probability, etc. The basic idea of operator algebra theory is to study the algebras of operators (duh), which are noncommutative, as opposed to the algebras of functions. Operator algebras come in two basic varieties: von Neumann algebras and C*-algebras. Von Neumann algebras deal with measure theoretic aspects of the operator algebra theory, while C*-algebras do for topological aspects. The goal of this research project is to promote theory of operator algebras generally and develop the younger generation. For this purpose, we plan to hold three international workshops and conferences, special lecture series by long-term visitors, and a school for younger generation. Workshop on von Neumann algebras and related topics (RIMS Research Project 2021)【RIMS Symposia】. Location: Room 420 Period:. Canceled. Organizer: Narutaka Ozawa(RIMS, Kyoto University). The Second Australia-China-Japan-Singapore-U.S. Index Theory Conference (RIMS Research Project 2021)【RIMS Symposia】. Location: Room 420 Period:. Canceled. Organizer: Yasuyuki Kawahigashi(Graduate School of Mathematical Sciences, The University of Tokyo). 　　　　　　　　　　　　　　　　. Workshop on C*-algebras and related topics (RIMS Research Project 2021)【RIMS Symposia】. Location: Online via Zoom Period: 2021-09-27〜2021-09-28. Organizer: Narutaka Ozawa(RIMS, Kyoto University). Workshop on free probability and related topics (RIMS Research Project 2021)【RIMS Symposia】 Location: Hybrid Meeting Period: 2022-01-13〜2022-01-14. Organizer: Benoit Collins(Graduate School of Science, Kyoto University). 2020→2021. Expanding Horizons of Inter-universal Teichmüller Theory The elucidation of the way in which the additive and multiplicative structure of the integers are intertwined with one another is one of the most important and central themes in number theory. In August 2012, Shinichi Mochizuki (the proposer and chief organizer of the present RIMS Research Project) released preprints of a series of papers concerning \"Inter-universal Teichmüller Theory\", a theory that constitutes an important advance with regard to elucidating this intertwining. Moreover, the proof of the \"ABC Conjecture\", which follows as a consequence of the theory, attracted worldwide attention. In the roughly nine years since the release of these preprints:. ・The four papers were published in the international mathematical journal PRIMS after undergoing a roughly seven and a half year long review. ・A preprint of a joint paper by five authors in which various numerically effective versions of the inequalities that appear in the theory was recently released. ・The number of researchers who have already acquired a thorough understanding of the theory, as well as advanced learners of the theory, has, as a result of the diligent efforts of many of the researchers involved, increased slowly, but steadily. ・Quite a number of surveys and related expositions of the theory (6 of which have been published or accepted for publication; another survey has been released, but remains unpublished) have been written, not only by the author of the theory, but also by researchers who have already acquired a thorough understanding of the theory. ・Although it is difficult to ascertain the precise number, at least on the order of 30 lectures and small-scale workshops on the theory have been conducted all over the world (in Japan, the UK, Russia, the US, China, Germany, and France). ・At least 4 large-scale workshops (of one to two weeks in length) on the theory have been conducted not only within Japan (in Kyoto, March 2015 and July 2016), but also in China (in Beijing, July 2015) and the UK (in Oxford, December 2015). Moreover, a long-term online (Zoom) workshop involving participants mainly from Japan, France, and the UK was conducted during the period September 2020 ～ April 2021. As a result of these activities, a sort of \"inter-universal Teichmüller theory community\", consisting of between ten and twenty researchers, is currently in the process of forming. Moreover, as a result of advances in research, such as combinatorial anabelian geometry, based on ideas closely related to the ideas that underlie inter-universal Teichmüller theory, important links between research on inter-universal Teichmüller theory and research concerning the Grothendieck-Teichmüller group and the absolute Galois group of the rational numbers have begun to form. In light of these developments, the present RIMS Research Project seeks to bring together various researchers not only from the \"inter-universal Teichmüller theory community\", but also researchers interested in various forms of mathematics related to inter-universal Teichmüller theory, and to provide all such researchers an opportunity to engage in lively discussions concerning the various developments discussed above in an environment in which interaction for periods on the order of months is possible, that is to say, unlike the situation in the case of a single workshop (i.e., which typically only lasts for roughly a week). Foundations and Perspectives of Anabelian Geometry (RIMS Research Project 2021) Location: Online via Zoom Period: 2021-06-28〜2021-07-02. Organizer: Ivan FESENKO(University of Nottingham). Combinatorial Anabelian Geometry and Related Topics (RIMS Research Project 2021) Location: Online via Zoom Period: 2021-07-05〜2021-07-09. Organizer: Yuichiro Hoshi(RIMS, Kyoto University). Invitation to inter-universal Teichmüller Theory (RIMS Research Project 2021) Location: Online via Zoom Period: 2021-08-31〜2021-09-03. Organizer: Yuichiro Hoshi(RIMS, Kyoto University). Inter-universal Teichmüller Theory Summit 2021 (RIMS Research Project 2021) Location: Online via Zoom Period: 2021-09-07〜2021-09-10. Organizer: Yuichiro Hoshi(RIMS, Kyoto University). 2020→2021. Differential Geometry and Integrable Systems -Mathematics of Symmetry, Stability and Moduli- Since the 9th MSJ-IRI 2000 “Integrable Systems in Differential Geometry”, (link. http:\/\/www.math.tsukuba.ac.jp\/~moriya\/iri-j2.html ), several scientific activities in the area of differential geometry and integrable systems have been carried out extensively and intensively. There has been remarkable progress in differential geometry based on theory of harmonic maps of Riemann surfaces into symmetric spaces and integrable system methods: The. DPW(Dorfmeister-Pedit-Wu）. method via loop groups and its applications to geometric analysis of surfaces, integrable system approach to the constrained Willmore conjecture; differential geometry of discrete surfaces and discrete geometric analysis; special geometry of minimal submanifolds and their moduli spaces; isoparametric submanifolds of finite and infinite dimensions; Floer homology of Lagragian submanifolds in homogeneous Kähler geometry; special geometry related to infinite integrable systems, Higgs bundles and mirror symmetry; fusion of non-linear PDE methods and integrable system methods based on symmetry of differential equations，and so on. This research project intends to cultivate new areas of“Mathematics of Symmetry, Stability and Moduli”by enhancing and expanding such research fields of differential geometry and integrable systems and encouraging activities of young researchers. Franz Pedit (UMASS Amherst,USA), Chikako Mese(Johns Hopkins U.,USA)，Eric Rains (Caltech,USA)，Fernando Codá Marques (Princeton,USA),Jaigyoung Choe(KIAS,Korea). and others will be invited as mid-term to long-term foreign RIMS visiting professors or international leading researchers. Throughout this academic year we conduct activities such as international workshop，special lectures，joint research，satellite seminar and so on from the viewpoints of geometry of submanifolds and integrable systems, geometric PDE and variational problems，mirror symmetry and its applications to differential geometry. A major internatonal conference “Differential Geometry and Integrable Systems”(MSJ-SI) will be held at the end of this project and we intend to greatly output new research results and to educate young researchers widely. The agreement of academic cooperation between RIMS and OCAMI, which was concluded in 2007, will be also used to promote this project. Applications of Harmonic Maps and Higgs Bundles to Differential Geometry (RIMS Satellite Seminars)(Closed seminar). Location: Seapal Suma Period:. Canceled. Organizer: Franz Pedit(The University of Massachusetts Amherst,USA). Generalized Hitchin Systems,Non-commutative Geometry and Special Functions (RIMS Research Project 2021)【RIMS Review Seminar】 Location: Online via Zoom Period: 2021-11-16〜2021-11-25. Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute). Special Geometry, Mirror Symmetry and Integrable Systems (RIMS Research Project 2021)【RIMS Review Seminars】 Location: Online via Zoom Period: 2021-11-29〜2021-12-02. Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute). Symmetry and Stability in Differential Geometry of Surfaces（RIMS Research Project 2021）【RIMS Review Seminar】(partly open) Location: Sugimoto Campus, Osaka City University Period: 2022-02-14〜2022-02-16. Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute). International Workshop on Geometry of Submanifolds and Integrable Systems (RIMS Research Project 2021)【RIMS Symposia】(Open Symposia) Location: Hybrid Meeting Period: 2022-02-20〜2022-02-23. Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute). Differential Geometry and Integrable Systems (RIMS Research Project 2020)【RIMS Symposia】(Open Symposia) Location: Sugimoto Campus, Osaka City University Period: 2022-03-01〜2022-03-21. Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute). 2019. Cluster Algebras Cluster algebras were original introduced by Fomin and Zelevinsky around 2000 to generalize a class of commutative algebras appearing in Lie theory in view of the Laurent phenomenon. Nowadays they are recognized as a kind of extension of the theory of root systems, and they are actively studied as an underlying algebraic and combinatorial structure ubiquitously appearing in several areas of mathematics. In this research project, we will hold an international workshops series \"Cluster Algebras 2019\" at RIMS in June, 2019, which is the largest comprehensive program on cluster algebras since the semantic program in KIAS, Korea in 2014. We will also hold a mini course on topics in cluster algebras at RIMS in May, 2019 by the visiting professors Bernard Leclerc (Universitè de Caen) and Michael Gekhtman (Notre Dame). Cluster algebras: theory and applications (RIMS Research Project 2019) Location: Rm 420 Period: 2019-06-03--2019-06-07. Organizer: Tomoki Nakanishi(Graduate School of Mathematics, Nagoya University). Participants 115 \/ Overseas 5. Cluster algebras and representation theory (RIMS Research Project 2019) Location: Rm 420 Period: 2019-06-10--2019-06-14. Organizer: Tomoki Nakanishi(Graduate School of Mathematics, Nagoya University). Participants 125 \/ Overseas 60. Cluster algebras, geometry, and mathematical physics (RIMS Research Project 2019) Location: Rm 420 Period: 2019-06-17--2019-06-21. Organizer: Tomoki Nakanishi(Graduate School of Mathematics, Nagoya University). Participants 121 \/ Overseas 53. Photo-1 ／. Photo-2 ／. Photo-3 2019. Discrete Optimization and Related Topics. Discrete optimization occurs frequently in our economic and social activities. The development of discrete optimization in both theory and application has a major impact on our society, since artificial intelligence (AI), machine learning, and big data receive much attention. In this project research, we aim to promote theoretical research on discrete optimization. The project focus not only on classical research but also on the one related to big data, such as sublinear or constant time optimization algorithm. For example, we plan to have the following three international workshops. 1) Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications. 2) International Workshop on Innovative Algorithms for Big Data. 3) International Workshop on Combinatorial Optimization and Algorithmic Game Theory. Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications (RIMS Research Project 2019) Location: Sanjo Conference Hall, The University of Tokyo Period: 2019-05-27--2019-05-30. Organizer: Kazuhisa Makino(RIMS, Kyoto University). Participants 117 \/ Overseas 41. Discrete Structures in Phylogenetic Networks (RIMS Research Project 2019)(closed). Location: Period: 2019-07-14--2019-07-20. Organizer: Kazuhisa Makino(RIMS, Kyoto University). Participants 5 \/ Overseas 2. International Workshop on Innovative Algorithms for Big Data 2019 (RIMS Research Project 2019) Location: Rm 420 Period: 2019-10-30--2019-11-01. Organizer: Kazuhisa Makino(RIMS, Kyoto University). Participants 81 \/ Overseas 8. International Workshop on Combinatorial Optimization and Algorithmic Game Theory (RIMS Research Project 2019) Location: Rm 420 Period: 2020-01-13--2020-01-14. Organizer: Kazuhisa Makino(RIMS, Kyoto University). Participants 41 \/ Overseas 7. Combinatorial Optimization and Algorithms (RIMS Research Project 2019)(closed). Location: Period: 2020-01-15--2020-01-20. Organizer: Kazuhisa Makino(RIMS, Kyoto University). Participants 7 \/ Overseas 4. Past RIMS Research Projects(2018-) ←　BACK TO THE TOP About Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
{"index":{}}
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Public opening：. No.1 〜 No.2212. 京都大学学術情報リポジトリ Kyoto university research information repository. Year 2022 No. 2209-2217 Year 2021 No. 2174-2208 Year 2020 No. 2144-2173 Year 2019 No. 2100-2143 Year 2018 No. 2060-2099 Year 2017 No. 2014-2059 Year 2016 No. 1980-2013 Year 2015 No. 1930-1979 Year 2014 No. 1872-1929 Year 2013 No. 1821-1871 Year 2012 No. 1773-1820 Year 2011 No. 1723-1772 Year 2010 No. 1672-1722 Year 2009 No. 1620-1671 Year 2008 No. 1575-1619 Year 2007 No. 1527-1574 Year 2006 No. 1462-1526 Year 2005 No. 1409-1461 Year 2004 No. 1349-1408 Year 2003 No. 1299-1348 Year 2002 No. 1242-1298 Year 2001 No. 1181-1241 Year 2000 No. 1121-1180 Year 1999 No. 1074-1120 Year 1998 No. 1023-1073 Year 1997 No. 976-1022 Year 1996 No. 934-975 Year 1995 No. 891-933 Year 1994 No. 856-890 Year 1993 No. 818-855 Year 1992 No. 775-817 Year 1991 No. 738-774 Year 1990 No. 712-737 Year 1989 No. 677-711 Year 1988 No. 637-676 Year 1987 No. 603-636 Year 1986 No. 580-602 Year 1985 No. 542-579 Year 1984 No. 509-541 Year 1983 No. 476-508 Year 1982 No. 448-475 Year 1981 No. 411-447 Year 1980 No. 374-410 Year 1979 No. 342-373 Year 1978 No. 317-341 Year 1977 No. 286-316 Year 1976 No. 259-285 Year 1975 No. 225-258 Year 1974 No. 197-224 Year 1973 No. 170-196 Year 1972 No. 133-169 Year 1971 No. 105-132 Year 1970 No. 80-104 Year 1969 No. 63-79 Year 1968 No. 36-62 Year 1967 No. 19-35 Year 1966 No. 8-18 Year 1965 No. 4-7 Year 1964 No. 1-3 RIMS Kôkyûroku published in 2022. No. 2209-2217. No.2217. 角　大輝(Hiroki Sumi). Integrated Research on Random Dynamical Systems and Multi-Valued Dynamical Systems ランダム力学系および多価写像力学系理論の総合的研究 2022.4. 2021\/08\/30--2021\/09\/03. No.2216. 茂木 康平(Kohei Motegi). Combinatorial Representation Theory and Connections with Related Fields 組合せ論的表現論および関連分野との連携 2022.4. 2021\/10\/18--2021\/10\/22. No.2215. 鈴木　政尋(Masahiro Suzuki). Mathematical Analysis in Fluid and Gas Dynamics 流体と気体の数学解析 2022.4. 2021\/07\/07--2021\/07\/09. No.2214. 星野　満博(Mitsuhiro Hoshino). Study on Nonlinear Analysis and Convex Analysis 非線形解析学と凸解析学の研究 2022.4. 2021\/12\/08--2021\/12\/10. No.2213. 嘉田　勝(Masaru Kada). Recent Developments in Set Theory of the Reals 実数の集合論における近年の進展 2022.4. 2021\/10\/12--2021\/10\/15. No.2212. 高坂　良史(Yoshihito Kohsaka). Geometric Aspects of Solutions to Partial Differential Equations 偏微分方程式の解の幾何的様相 2022.2. 2021\/06\/28--2021\/06\/30. No.2211. 高山　茂晴(Shigeharu Takayama). Open problems in complex geometry II 複素幾何学の諸問題II 2022.1. 2021\/09\/06--2021\/09\/09. No.2210. 中田　文憲(Fuminori Nakata). Deepening and merging geometric structures related to submanifold theory 部分多様体論と関連する幾何構造研究の深化と融合 2022.1. 2021\/06\/21--2021\/06\/23. No.2209. 今村　隼人(Hayato Imamura). 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Transformation Groups and Combinatorics","Singularity theory, geometry and topology","Exact WKB Analysis and Microlocal Analysis","New developments in group representation theory and\n   non-commutative harmonic analysis","Mathematical and numerical analysis for interface motion\n   arising in nonlinear phenomena","Functions in Number Theory and Their Probabilistic Aspects","Harmonic Analysis and Nonlinear Partial Differential Equations","Algebraic Number Theory and Related Topics 2010","Far-From-Equilibrium Dynamics","Progress in Mathematics of Integrable Systems","Summer School on the Theory of Uniform Distribution","Infinite Analysis 2010,  Developments in Quantum Integrable Systems","Spectra of Random Operators and Related Topics","Harmonic Analysis and Nonlinear Partial Differential Equations","Algebraic Number Theory and Related Topics 2009","Higher Dimensional Algebraic Geometry","Combinatorial Optimization and Discrete Algorithms","Harmonic Analysis and Nonlinear Partial Differential Equations","Applications of Renormalization Group Methods in Mathematical Sciences","New Viewpoints of Representation Theory and\n　Noncommutative Harmonic Analysis","Algebraic Number Theory and Related Topics 2008","Harmonic Analysis and Nonlinear Partial Differential Equations","Infinite dimensional Teichm&#252;ller spaces and moduli spaces","Spectral and Scattering Theory and Related Topics","Mathematical analysis on the self-organization and self-similarity","Harmonic Analysis and Nonlinear Partial Differential Equations","Expansion of Integrable Systems","Algebraic Number Theory and Related Topics 2007","New Trends in Combinatorial Representation Theory","Differential Equations and Exact WKB Analysis","Higher Dimensional Algebraic Varieties and Vector Bundles","Combinatorial Representation Theory and Related Topics","Representation Theory and Analysis on Homogeneous Spaces","Proceedings of RIMS Workshop on Stochastic Analysis and Applications","Algebraic Analysis and the Exact WKB Analysis\nfor Systems of Differential Equations","Proceedings of the Symposium on Algebraic Number theory and Related Topics","Workshops on\n“Pattern Formation Problems in Dissipative Systems”\nand “Mathematical Modeling and Analysis for Nonlinear Phenomena”","Algebraic, Analytic and Geometric Aspects of Complex Differential\nEquations and their Deformations. Painlev&#233 hierarchies","Kyoto Conference on the Navier-Stokes Equations and their Applications"],"charset_s":"EUC-JP","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:48:33 UTC 2022","fresh_date_dt":"Tue Jul 19 21:21:33 UTC 2022","referrer_id_s":"eOG-td16muho","language_s":"en","size_i":26927,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085956381278208,"text_t":"Bessatsu. Kôkyûroku Bessatsu LaTeX Style Files Kôkyûroku A New and Noteworthy Series of Lecture Notes from RIMS. A new and noteworthy series of Lecture Notes: RIMS Kôkyûroku Bessatsu (ISSN: 1881-6193) is now being published by RIMS. Each volume in this new series will be the proceedings of a conference held at RIMS. All papers are refereed. From the conferences held at RIMS every year, the advisory board of RIMS selects about 10 conferences to publish their proceedings in this new series. Thus the new series will be an up-dated record of the major activities in mathematical sciences in Japan. RIMS Kôkyûroku Bessatsu B88:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. Satoshi Masaki and Hideo Takaoka. December, 2021. RIMS Kôkyûroku Bessatsu B87:. Mathematical structures of integrable systems, its deepening and expansion eds. Takao Suzuki. August, 2021. RIMS Kôkyûroku Bessatsu B86:. Algebraic Number Theory and Related Topics 2018 eds. Takao Yamazaki, Shuji Yamamoto. July, 2021. RIMS Kôkyûroku Bessatsu B85:. Study of the History of Mathematics 2020 eds. Naoki Osada. July, 2021. RIMS Kôkyûroku Bessatsu B84:. Inter-universal Teichmüller Theory Summit 2016 eds. Yuichiro Hoshi, Shinichi Mochizuki. January, 2021. RIMS Kôkyûroku Bessatsu B83:. Algebraic Number Theory and Related Topics 2017 eds. Hiroshi Tsunogai, Takao Yamazaki, Yasushi Mizusawa. October, 2020. RIMS Kôkyûroku Bessatsu B82:. Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations eds. Takayoshi Ogawa, Keiichi Kato, Mishio Kawashita, Masashi Misawa. June, 2020. RIMS Kôkyûroku Bessatsu B81:. Study of the History of Mathematics 2019 ed. Naoki Osada. April, 2020. RIMS Kôkyûroku Bessatsu B80:. Regularity, singularity and long time behavior for partial differential equations with conservation law eds. Keiichi Kato, Mishio Kawashita, Masashi Misawa, Takayoshi Ogawa. April, 2020. RIMS Kôkyûroku Bessatsu B79:. Stochastic Analysis on Large Scale Interacting Systems eds. Ryoki Fukushima, Tadahisa Funaki, Yukio Nagahata, Hirofumi Osada, Kenkichi Tsunoda. April, 2020. RIMS Kôkyûroku Bessatsu B78:. Mathematical structures of integrable systems and their applications ed. Shinsuke Iwao. April, 2020. RIMS Kôkyûroku Bessatsu B77:. Algebraic Number Theory and Related Topics 2016 eds. Yasuo Ohno, Hiroshi Tsunogai, Toshiro Hiranouchi. April, 2020. RIMS Kôkyûroku Bessatsu B76:. On the examination and further development of inter-universal Teichmüller theory ed. S. Mochizuki. August, 2019. RIMS Kôkyûroku Bessatsu B75:. New development of microlocal analysis and singular perturbation theory eds. N. Honda, Y. Okada. June, 2019. RIMS Kôkyûroku Bessatsu B74:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. H. Takaoka, S. Masaki. April, 2019. RIMS Kôkyûroku Bessatsu B73:. The study of the history of mathematics 2018 ed. S. Jochi. April, 2019. RIMS Kôkyûroku Bessatsu B72:. Algebraic Number Theory and Related Topics 2015 eds. H. Takahashi, Y. Ohno, T. Tsushima. December, 2018. RIMS Kôkyûroku Bessatsu B71:. The study of the history of mathematics 2017 ed. S. Jochi. December, 2018. RIMS Kôkyûroku Bessatsu B70:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. H. Takaoka, H. Kubo. April, 2018. RIMS Kôkyûroku Bessatsu B69:. The study of the history of mathematics 2016 ed. S. Jochi. April, 2018. RIMS Kôkyûroku Bessatsu B68:. Various aspects of multiple zeta values ed. K. Ihara. October, 2017. RIMS Kôkyûroku Bessatsu B67:. Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics eds. H. Okamoto, Y. Tsutsumi, N. Ueki, T.Adachi, S. Shimizu. October, 2017. RIMS Kôkyûroku Bessatsu B66:. Geometry and Analysis of Discrete Groups and Hyperbolic Spaces eds. M. Fujii, N. Kawazumi, K. Ohshika. June, 2017. RIMS Kôkyûroku Bessatsu B65:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. H. Kubo, H. Takaoka. May, 2017. RIMS Kôkyûroku Bessatsu B64:. Algebraic Number Theory and Related Topics 2014 eds. T. Tsuji, H. Takahashi, Y. Hoshi. May, 2017. RIMS Kôkyûroku Bessatsu B63:. Regularity and Singularity for Partial Differential Equations with Conservation Laws eds. M. Kawashita, K. Kato, M. Misawa, T. Ogawa. May, 2017. RIMS Kôkyûroku Bessatsu B62:. String theory, integrable systems and representation theory eds. K. Hasegawa, Y. Yamada. May, 2017. RIMS Kôkyûroku Bessatsu B61:. Microlocal Analysis and Singular Perturbation Theory eds. Y. Takei，T. Aoki, N. Honda, K. Kataoka, T. Koike. January, 2017. RIMS Kôkyûroku Bessatsu B60:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. H. Kubo, M. Sugimoto. December, 2016. RIMS Kôkyûroku Bessatsu B59:. Stochastic Analysis on Large Scale Interacting Systems eds. R. Fukushima, T. Funaki, Y. Nagahata, M. Nakashima, H. Osada, Y. Otobe. July, 2016. RIMS Kôkyûroku Bessatsu B58:. Natural extension of arithmetic algorithms and S-adic system ed. S. Akiyama. July, 2016. RIMS Kôkyûroku Bessatsu B57. :. Several aspects of microlocal analysis eds. N. Honda, Y. Okada, S. Yamazaki. May, 2016. RIMS Kôkyûroku Bessatsu B56. :. Harmonic Analysis and Nonlinear Partial Differential Equations eds. H. Kubo, M. Sugimoto. April, 2016. RIMS Kôkyûroku Bessatsu B55. :. Theory of singularities of smooth mappings and around it ed. T. Nishimura. April, 2016. RIMS Kôkyûroku Bessatsu B54. :. Topology optimization theory and applications toward wide fields of natural sciences ed. T. Nakazawa. October, 2015. RIMS Kôkyûroku Bessatsu B53. :. Algebraic Number Theory and Related Topics 2013 eds. T. Ochiai, T. Tsuji, I. Kimura. September, 2015. RIMS Kôkyûroku Bessatsu B52:. Exponential Analysis of Differential Equations and Related Topics ed. Y. Takei. November, 2014. RIMS Kôkyûroku Bessatsu B51:. Algebraic Number Theory and Related Topics 2012 eds. A. Shiho, T. Ochiai, N. Otsubo. October, 2014. RIMS Kôkyûroku Bessatsu B50:. Study of the History of Mathematics ed. T. Ogawa. June, 2014. RIMS Kôkyûroku Bessatsu B49:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. M. Sugimoto, H. Kubo. June, 2014. RIMS Kôkyûroku Bessatsu B48:. Analysis and Geometry of Discrete Groups and Hyperbolic Spaces eds. M. Fujii, N. Kawazumi, K. Ohshika. June, 2014. RIMS Kôkyûroku Bessatsu B47:. Novel Development of Nonlinear Discrete Integrable Systems ed. J. Matsukidaira. June, 2014. RIMS Kôkyûroku Bessatsu B46:. Numeration and Substitution 2012 ed. S. Akiyama. June, 2014. RIMS Kôkyûroku Bessatsu B45:. Spectral and Scattering Theory and Related Topics ed. F. Hiroshima. April, 2014. RIMS Kôkyûroku Bessatsu B44:. Algebraic Number Theory and Related Topics 2011 eds. N. Suwa, A. Shiho, K. Sato. December, 2013. RIMS Kôkyûroku Bessatsu B43:. Potential Theory and its Related Fields ed. K. Hirata. September, 2013. RIMS Kôkyûroku Bessatsu B42:. Harmonic Analysis and Nonlinear Partial Differential Equations ed. M. Sugimoto. August, 2013. RIMS Kôkyûroku Bessatsu B41:. The breadth and depth of nonlinear discrete integrable systems ed. T. Tokihiro. August, 2013. RIMS Kôkyûroku Bessatsu B40:. Recent development of micro-local analysis 　for the theory of asymptotic analysis eds. N. Honda, Y. Okada, M. Uchida. April, 2013. RIMS Kôkyûroku Bessatsu B39:. Geometry of Transformation Groups and Combinatorics ed. M. Masuda. April, 2013. RIMS Kôkyûroku Bessatsu B38:. Singularity theory, geometry and topology ed. K. Saji. April, 2013. RIMS Kôkyûroku Bessatsu B37:. Exact WKB Analysis and Microlocal Analysis ed. T. Koike. March, 2013. RIMS Kôkyûroku Bessatsu B36:. New developments in group representation theory and non-commutative harmonic analysis eds. H. Ishi, T. Hashimoto. December, 2012. RIMS Kôkyûroku Bessatsu B35:. Mathematical and numerical analysis for interface motion arising in nonlinear phenomena eds. K. Nakamura, S. Yazaki, T. Ishiwata. December, 2012. RIMS Kôkyûroku Bessatsu B34:. Functions in Number Theory and Their Probabilistic Aspects eds. K. Matsumoto, S. Akiyama, K. Fukuyama, H. Nakada, H. Sugita, A. Tamagawa. August, 2012. RIMS Kôkyûroku Bessatsu B33:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. M. Sugimoto, T. Ozawa. July, 2012. RIMS Kôkyûroku Bessatsu B32:. Algebraic Number Theory and Related Topics 2010 eds. M. Kida, N. Suwa, S. Kobayashi. July, 2012. RIMS Kôkyûroku Bessatsu B31:. Far-From-Equilibrium Dynamics eds. T. Ogawa, K. Ueda. May, 2012. RIMS Kôkyûroku Bessatsu B30:. Progress in Mathematics of Integrable Systems eds. R. Hirota, D. Takahashi. April, 2012. RIMS Kôkyûroku Bessatsu B29:. Summer School on the Theory of Uniform Distribution ed. S. Akiyama. April, 2012. RIMS Kôkyûroku Bessatsu B28:. Infinite Analysis 2010, Developments in Quantum Integrable Systems eds. A. Kuniba, T. Nakanishi, M. Okado, Y. Takeyama. August, 2011. RIMS Kôkyûroku Bessatsu B27:. Spectra of Random Operators and Related Topics ed. N. Minami. July, 2011. RIMS Kôkyûroku Bessatsu B26:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. T. Ozawa, M. Sugimoto. May, 2011. RIMS Kôkyûroku Bessatsu B25:. Algebraic Number Theory and Related Topics 2009 eds. T. Ichikawa, M. Kida, T. Yamazaki. April, 2011. RIMS Kôkyûroku Bessatsu B24:. Higher Dimensional Algebraic Geometry eds. S. Mukai, N. Nakayama. March, 2011. RIMS Kôkyûroku Bessatsu B23:. Combinatorial Optimization and Discrete Algorithms ed. S. Iwata. December, 2010. RIMS Kôkyûroku Bessatsu B22:. Harmonic Analysis and Nonlinear Partial Differential Equations ed. Y. Tsutsumi. December, 2010. RIMS Kôkyûroku Bessatsu B21:. Applications of Renormalization Group Methods in Mathematical Sciences eds. K. Ito, T. Miyao. December, 2010. RIMS Kôkyûroku Bessatsu B20:. New Viewpoints of Representation Theory and 　Noncommutative Harmonic Analysis eds. M. Itoh, H. Ochiai. October, 2010. RIMS Kôkyûroku Bessatsu B19:. Algebraic Number Theory and Related Topics 2008 eds. H. Nakamura, T. Ichikawa, K. Matsuno. June, 2010. RIMS Kôkyûroku Bessatsu B18:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. T. Ozawa, M. Sugimoto. June, 2010. RIMS Kôkyûroku Bessatsu B17:. Infinite dimensional Teichmüller spaces and moduli spaces ed. E. Fujikawa. June, 2010. RIMS Kôkyûroku Bessatsu B16:. Spectral and Scattering Theory and Related Topics eds. S. Fujiie, T. Umeda. April, 2010. RIMS Kôkyûroku Bessatsu B15:. Mathematical analysis on the self-organization and self-similarity eds. T. Ogawa, M. Kawashita, M. Kurokiba, T. Nakaki, T. Senba. December, 2009. RIMS Kôkyûroku Bessatsu B14:. Harmonic Analysis and Nonlinear Partial Differential Equations eds. T. Ozawa and M. Sugimoto. November, 2009. RIMS Kôkyûroku Bessatsu B13:. Expansion of Integrable Systems ed. A. Nobe. October, 2009. RIMS Kôkyûroku Bessatsu B12:. Algebraic Number Theory and Related Topics 2007 eds. M. Asada, H. Nakamura and H. Takahashi. August, 2009. RIMS Kôkyûroku Bessatsu B11:. New Trends in Combinatorial Representation Theory eds. S. Ariki and M. Okado. April, 2009. RIMS Kôkyûroku Bessatsu B10:. Differential Equations and Exact WKB Analysis ed. Y. Takei. November, 2008. RIMS Kôkyûroku Bessatsu B9:. Higher Dimensional Algebraic Varieties and Vector Bundles ed. S. Mukai. September, 2008. RIMS Kôkyûroku Bessatsu B8:. Combinatorial Representation Theory and Related Topics eds. T. Tanisaki and T. Suzuki. May, 2008. RIMS Kôkyûroku Bessatsu B7:. Representation Theory and Analysis on Homogeneous Spaces ed. H. Sekiguchi. April, 2008. RIMS Kôkyûroku Bessatsu B6:. Proceedings of RIMS Workshop on Stochastic Analysis and Applications eds. M. Fukushima and I. Shigekawa. January, 2008. RIMS Kôkyûroku Bessatsu B5:. Algebraic Analysis and the Exact WKB Analysis for Systems of Differential Equations eds. T. Aoki, N. Kumano-Go and S. Yamazaki. January, 2008. RIMS Kôkyûroku Bessatsu B4:. Proceedings of the Symposium on Algebraic Number theory and Related Topics eds. K. Hashimoto, Y. Nakajima and H. Tsunogai. December, 2007. RIMS Kôkyûroku Bessatsu B3:. Workshops on “Pattern Formation Problems in Dissipative Systems” and “Mathematical Modeling and Analysis for Nonlinear Phenomena” eds. M. Kuwamura, T. Ogawa and M. Nagayama. November, 2007. RIMS Kôkyûroku Bessatsu B2:. Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painlev&#233 hierarchies ed. Y. Takei. March, 2007. RIMS Kôkyûroku Bessatsu B1:. Kyoto Conference on the Navier-Stokes Equations and their Applications eds. Y. Giga, H. Kozono, H. Okamoto and Y. Shibata. March, 2007. Research Institute for Mathematical Sciences , Kyoto University Kyoto, 606-8502 JAPAN FAX: +81-75-753-7272."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Access Joint Usage\/Research. 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Mathematics of quasiperiodic systems and related topics. Location: Hybrid Meeting (RIMS and Zoom) Period: 2020-11-09--2020-11-11. Organizer: YAMAGISHI Yoshikazu(Faculty of Advanced Science and Technology, Ryukoku University). Report ／. Photo-1 ／. Photo-2 ／. Photo-3 List of Past Workshops AY2020 List of Past Workshops AY2019 List of Past Workshops AY2018 List of Past Workshops AY2017- ←　BACK TO THE TOP About Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. 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English Page トップ 論文 講演. 履歴書 予定. 講演タイトル. ,. 研究集会名,. 場所,. 日時. ガウス和に関するいくつかの話題. ,. 全学共通科目 &ndash; 現代の数学と数理解析 基礎概念とその諸科学への広がり &ndash; ,. 京都大学数理解析研究所,. 2022.5.20. アルゴリズム的観点による代数体や混標数局所体の遠アーベル幾何学. ,. 東京工業大学 2022 年度 数学特別講義 A ,. 東京工業大学,. 2022.7.11-2022.7.15. Combinatorial anabelian geometry in the absence of group-theoretic cuspidality. ,. Partial combinatorial cuspidalization for F-admissible outomorphisms. ,. Synchronization of tripods. ,. Glueability of combinatorial cuspidalizations. ,. 第 29 回整数論サマースクール &ldquo;組み合わせ論的遠アーベル幾何学&rdquo; ,. online,. 2022.9.5-2022.9.9（仮）. 講演. 研究発表. (の一部). On the Galois Orbit Version of Inter-universal Teichmüller Theory I: a Progress Report. ,. On the Galois Orbit Version of Inter-universal Teichmüller Theory II: a Progress Report. ,. Inter-universal Teichmüller Theory Summit 2021,. 京都大学数理解析研究所,. 2021.9.7-2021.9.10. A Combinatorial Anabelian Result for Stable Log Curves over Log Points. (. 講演スライド ),. Combinatorial Anabelian Geometry in the Absence of Group-theoretic Cuspidality. (. 講演スライド ),. Synchronization of Tripods and Glueability of Combinatorial Cuspidalizations. (. 講演スライド ),. Applications to the Theory of Tempered Fundamental Groups. (. 講演スライド ),. Combinatorial Anabelian Geometry and Related Topics,. 京都大学数理解析研究所,. 2021.7.6-2021.7.9. The Absolute Anabelian Geometry of Quasi-tripods. (. 講演スライド ),. Foundations and Perspectives of Anabelian Geometry,. 京都大学数理解析研究所,. 2021.6.28-2021.7.2. The Absolute Anabelian Geometry of Quasi-tripods. (. 報告集原稿 ),. Homotopic and Geometric Galois Theory,. The Mathematisches Forschungsinstitut Oberwolfach, online,. 2021.3.7-2021.3.13. Hasse-Witt不変量が非零である次数が小さい正標数代数曲線の巡回被覆の存在. ,. 広島大学代数学セミナー,. 広島大学,. 2019.11.29. An Anabelian Open Basis for a Smooth Variety. ,. Fundamental groups: Geometry and Arithmetic,. University of Science and Technology of China,. 2019.8.18-2019.8.22. An introduction to nilpotent ordinary indigenous bundles via modular curves. ,. Nilpotent indigenous bundles in characteristic three. ,. Witt Vectors, Deformations, and Absolute Geometry,. University of Vermont,. 2018.7.16-2018.7.21. A pro-p group-theoretic criterion for good reduction of ordinary proper hyperbolic curves. ,. The 7th East Asia Number Theory Conference,. National Taiwan University,. 2018.2.5-2018.2.9. On torsion points on a curve with good reduction over an absolutely unramified base. ,. Oberseminar Algebra und Geometrie,. Institut für Mathematik in Johann Wolfgang Goethe-Universität Frankfurt am Main,. 2017.11.16. A simple purely group-theoretic characterization of the Grothendieck-Teichmüller group. ,. Nilpotent Fundamental Groups,. The Banff International Research Station for Mathematical Innovation and Discovery,. 2017.6.18-2017.6.23. A pro-p group-theoretic criterion for good reduction of ordinary proper hyperbolic curves. ,. 2017 East Asian Core Doctorial Forum on Mathematics,. Seoul National University,. 2017.1.6-2017.1.10. Classical p-adic Teichmüller theory in characteristic three. (. 講演スライド ),. AMS Summer Institute in Algebraic Geometry,. University of Utah,. 2015.7.27-2015.7.31. 数体の単遠アーベル的復元. (. 講演スライド ),. 宇宙際タイヒミューラー理論の検証と更なる発展,. 京都大学数理解析研究所,. 2015.3.9-2015.3.20. 数体の単遠アーベル的復元. (. 講演スライド ),. 京都大学 談話会,. 京都大学,. 2014.7.9. 組み合わせ論的遠アーベル幾何学概論. ,. 京都大学 微分トポロジーセミナー,. 京都大学,. 2014.7.8. Reconstruction of a Number Field from the Absolute Galois Group. (. 講演スライド ), (. 絶対 Galois 群による数体の復元 ),. 第 18 回早稲田大学整数論研究集会,. 早稲田大学,. 2014.3.11-2014.3.13. Moderate Points of Hyperbolic Curves. ,. Fundamental groups in arithmetic and algebraic geometry,. Ennio De Giorgi Mathematical Research Center,. 2013.12.16-2013.12.20. 低次元多重双曲曲線に対する Grothendieck 予想. ,. Okayama Anabelful Days,. 岡山大学,. 2013.5.22-2013.5.23. Grothendieck-Teichmüller Groups in the Combinatorial Anabelian Geometry. ,. Grothendieck-Teichmüller theory and Multiple Zeta Values,. Isaac Newton Institute for Mathematical Sciences in University of Cambridge,. 2013.4.8-2013.4.12. Conditional results on the birational section conjecture over small number fields. ,. Galois covers and deformations,. Université de Bordeaux 1,. 2012.6.25-2012.6.29. On the Grothendieck conjecture for hyperbolic polycurves. ,. 数論幾何とその周辺,. 京都大学,. 2012.4.9-2012.4.11. Grothendieck による遠アーベルセクション予想について. (. Grothendieck による遠アーベルセクション予想について ),. 第 56 回代数学シンポジウム,. 岡山大学,. 2011.8.8-2011.8.11. On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves. ,. Séminaire de Théorie des Nombres,. Université de Bordeaux 1,. 2011.6.24. Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero. ,. Séminaire de théorie des nombres de l'IMJ,. Université de Paris 6,. 2011.6.20. Survey on the combinatorial anabelian geometry of hyperbolic curves. (. 講演スライド ),. Development of Galois-Teichmüller theory and anabelian geometry,. 京都大学数理解析研究所,. 2010.10.25-2010.10.30. Monodromic fullness in the anabelian geometry I. ,. Monodromic fullness in the anabelian geometry II. ,. 数論幾何学ワークショップ２０１０,. 沖縄尚学高等学校,. 2010.8.2-2010.8.6. Existence of nongeometric pro-p Galois sections of hyperbolic curves. ,. PIA 2010: The arithmetic of fundamental groups,. MAThematics Center Heidelberg in University of Heidelberg,. 2010.2.8-2010.2.12. Existence of nongeometric pro-p Galois sections of hyperbolic curves. ,. Séminaire d'Arithmétique et de Géométrie Algébrique,. Université Paris-Sud 11,. 2010.2.2. Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero. ,. Sakura Workshop: Torsion of Abelian Schemes and Rational Points on Moduli Spaces,. Université de Bordeaux 1,. 2010.1.25-2010.1.29. On the combinatorial cuspidalizations and the faithfulness of the outer Galois representations of hyperbolic curves (joint work with Shinichi Mochizuki). ,. Anabelian Geometry,. Isaac Newton Institute for Mathematical Sciences in University of Cambridge,. 2009.8.24-2009.8.28. Cuspidalizations of fundamental groups of configuration spaces. ,. Arithmetic Galois Theory and Related Moduli Spaces,. 京都大学数理解析研究所,. 2006.10.23.-2006.10.27; 2006.10.30.-2006.10.31. レクチャ. Multiradial Representations and Log-volume Estimates. (. 講演スライド ),. Inter-universal Teichmüller Theory Summit 2021,. 京都大学数理解析研究所,. 2021.9.7-2021.9.10. Mono-anabelian Transport in Inter-universal Teichmüller Theory. ,. Promenade in Inter-universal Teichmüller Theory,. online,. 2021.3.25. 身近な数学 • 身近でない数学. ,. 京都大学高校生フォーラム in TOKYO,. 東京都教職員研修センター,. 2019.10.26. 混標数局所体に関する単遠アーベル復元アルゴリズム. (全5回; 計10時間;. 目次 ),. 東京大学 平成30年度 集中講義,. 東京大学,. 2018.10.29-2018.11.2. 乗法的情報による有理数の加法の復元. ,. 数理の翼 伊計島セミナー2018,. N 高等学校伊計本校,. 2018.8.23-2018.8.27. 格子点集合の対称性と虚二次数. ,. 膳所高等学校数学講演会,. 膳所高等学校,. 2018.1.12. 有理数・複素数の四則演算に関する復元定理. ,. 数学吉田塾·連続講義,. 京都大学,. 2017.12.18.-2017.12.21. 射影直線による代数幾何学 • 数論幾何学への入門. ,. 第15回吉田塾·数学入門合宿,. あうる京北 (京都府立ゼミナールハウス),. 2017.8.29.-2017.8.31. 〚IUTch-III-IV〛 from the point of view of mono-anabelian transport I. (. 講演スライド ),. 〚IUTch-III-IV〛 from the point of view of mono-anabelian transport II. (. 講演スライド ),. 〚IUTch-III-IV〛 from the point of view of mono-anabelian transport III. (. 講演スライド ),. 〚IUTch-III-IV〛 from the point of view of mono-anabelian transport IV. (. 講演スライド ),. Inter-universal Teichmüller Theory Summit 2016,. 京都大学数理解析研究所,. 2016.7.18-2016.7.27. Introduction to Mono-anabelian Geometry. (全2回; 計4時間;. 目次 ),. Fundamental groups in arithmetic geometry - Paris 2016,. The Institut Henri Poincaré,. 2016.5.26-2016.6.3. An Approximate Statement of the Main Theorem of Inter-universal Teichmüller Theory. (. 講演スライド ),. Mono-anabelian Transport. (. 講演スライド ),. Hodge-Arakelov-theoretic Evaluation I. (. 講演スライド ),. Hodge-Arakelov-theoretic Evaluation II. (. 講演スライド ),. Workshop on IUT theory of Shinichi Mochizuki,. University of Oxford,. 2015.12.7-2015.12.11. 宇宙際 Teichmüller 理論入門 I. (. 講演スライド ;. 付録 ),. 宇宙際 Teichmüller 理論入門 II. (. 講演スライド ),. 宇宙際 Teichmüller 理論入門 III. (. 講演スライド ),. 代数的整数論とその周辺 2015,. 京都大学数理解析研究所,. 2015.11.30-2015.12.4. 宇宙際 Teichmüller 理論入門. ,. 数論幾何学セミナー,. 九州大学,. 2015.6.25. 乗法的情報による加法構造の復元. (全4回; 計5時間;. 乗法的情報による加法構造の復元 ),. 京都大学数理解析研究所 数学入門公開講座,. 京都大学数理解析研究所,. 2014.8.4-2014.8.8. Kummer 忠実体上の代数関数体に対する遠アーベル幾何学. (全5回; 計7.5時間;. 目次 ),. 広島大学 平成26年度 集中講義 数論と基本群,. 広島大学,. 2014.6.23-2014.6.27. 関数体との類似から眺める数体の幾何学. (全7回; 計10.5時間;. 目次 ),. 基礎数学からの展開 B,. 京都大学,. 2013.10-11. p 進局所体に対する遠アーベル幾何学. (全5回; 計7.5時間;. 目次 ),. 東京工業大学 平成25年度 集中講義,. 東京工業大学,. 2013.6.24.-2013.6.28. 遠アーベル幾何学概論. (. 位相曲面に対するある Grothendieck 予想型の結果について ),. 第10回城崎新人セミナー,. 兵庫県豊岡市城崎市民センター,. 2013.2.18-2013.2.22. 代数関数, リーマン面, 現代的代数幾何学. ,. 愛知県立明和高等学校による数理解析研究所訪問・見学,. 京都大学数理解析研究所,. 2012.7.20. Classical Anabelian Geometry. ,. Grothendieck Conjecture over Local Fields &mdash; from &ldquo;Relative&rdquo; to &ldquo;Absolute&rdquo; &mdash;. ,. Automorphic forms and Galois representations,. Durham University,. 2011.7.18-2011.7.28. On &ldquo;cuspidalization&rdquo; and &ldquo;anabelioids&rdquo;. ,. Galois-theoretic Arithmetic Geometry ,. 国際高等研究所,. 2010.10.19.-2010.10.24. ヤコビ和に関するいくつかの話題. (2021.4.23),. ２次体を通じた数論のいくつかの基本的概念への入門. (2020.6.5),. 原始根予想. (2017.4.28),. 平面上の格子と２次体の数論. (2015.7.3),. 正方形や長方形の族による複素数体の復元. (2014.4.25),. 代数関数体の掛け算的情報による足し算の復元. (2013.5.17),. 射影直線の被覆に関する数論幾何学. (2012.6.1),. 射影直線を通しての代数幾何学 • 数論幾何学への入門. (2011.5.20),. 射影直線の幾何学による遠アーベル幾何学の紹介. (2010.4.23),. 全学共通科目 &ndash; 現代の数学と数理解析 基礎概念とその諸科学への広がり &ndash;,. 京都大学数理解析研究所. 遠アーベル幾何学について. ,. 第6回城崎新人セミナー,. 兵庫県豊岡市城崎町大会議館,. 2009.2.16.-2009.2.20. Crystalline cohomology 入門. (全5回; 計15時間),. l進的・p進的手法による幾何学の研究,. 名古屋大学,. 2007.8.23, 2007.9.6, 2007.9.27, 2007.10.17, 2007.11.7."}
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2022","referrer_id_s":"eOG-td16muho","language_s":"ja","size_i":19069,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":0,"references_internal_i":0,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735085951384813568,"text_t":"COVID-19 updates for organizers and participants | RIMS. HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Workshop for Principal Researcher Call for proposals. Call for proposals（RIMS Research Projects） Call for proposals（RIMS Workshops\/Symposia） Call for proposals（RIMS Satellite Seminars） Call for proposals（RIMS Review Seminars） COVID-19 updates for organizers and participants. TOP 　　 COVID-19 updates for organizers and participants. Guidelines for RIMS Joint Research Activities. June 1st, 2022. The Research Institute for Mathematical Sciences (RIMS) has developed the following guidelines for meetings (workshops, symposia and seminars) hosted as part of its Joint Research Activities. Possible formats of each meeting depend on Kyoto University's. Level of Activity Restriction. , the latest status of which can be found here:. https:\/\/www.kyoto-u.ac.jp\/en\/about\/safety\/coronavirus.html *The Current level of activity restriction is Level 1 (October 1st-). i) When the Level of Activity Restriction is. 2&minus; or higher. , the meeting shall be either cancelled, postponed to a later date in the same academic year, or. held entirely online using Zoom . ii) When the Level of Activity Restriction is. 1. , we recommend. an online meeting using Zoom , but if the organizer (principal researcher) sees fit, a (partly) face-to-face meeting is also possible with capacity restrictions and adequate hygiene measures. *Effective one month after Level 1*. *References (For Participants\/ For Organizers) * Detailed information on. hybrid (face-to-face and online) meetings at RIMS iii) When the Level of Activity Restriction is. 0. , the meeting takes place physically as usual. Our staff will ask the organizer of each meeting to decide, at least one month prior to the scheduled start date, whether and in which format the meeting shall take place. Our infection prevention measures. *Effective for one month after Level 1*. ◆For Participants: Measures against COVID-19 at RIMS ◆For Organizers: Measures against COVID-19 at RIMS ** In our buildings **. ・Hand sanitizer is available at the building entrance, at the doors of seminar rooms, and near shared equipment. ・Our staff will clean room equipment (desks, chairs, audio\/video devices, door knobs, etc.) daily. ・The capacities of seminar rooms are reduced so that sufficient distance between participants can be maintained. ・Doors and windows are regularly held open for increased ventilation. ・To further reduce risk, we ask participants to follow some restrictions on the usage of shared equipment and areas within the building. ** Guidelines for RIMS meetings (Joint Research Activities) **. ・Daily temperature checks for participants are conducted, and those in doubt of a symptom are asked not to attend the meeting physically. *Note that under our health policy, persons with some typical symptoms (continued cough, fatigue or temperature 1 °C higher than normal) are not allowed in the building. ・Participants are encouraged to wash hands frequently and wear a mask. ・Participants are asked to keep a distance of two meters from each other. ・No beverage or snacks should be served in meetings. ・Participants should refrain from dining out as a large group during the period of meetings. ・. *Effective for one month after Level 1*. The lobby space on the 1st floor will be closed and the Tea\/Water Dispensers are unavailable. ・We provide larger rooms as much as possible for meetings hosted at RIMS. ・. *Effective for one month after Level 1*. The number of members and participants will be limited each day. *We encourage only selected core members of organizing committees to show up and the rest to participate via Zoom. For the use of Lecture Room 420, the number of attendants should be limited to at most 30. For other rooms, up to 30% or less of its capacity. ・. *Effective for one month after Level 1*. Consider carefully the participation of those who are visiting from regions and countries where cases of COVID-19 are especially notable. Encourage them to participate online as much as possible. As a general rule, we restrict visitors from areas where a state of emergency is declared. ・Participants from regions under mobility restrictions by authorities are asked to refrain from in-person attendance. Guide to organizing an online event using Zoom. Zoom Meeting features: Zoom Pro License, Video Webinar 500, Large Meeting 500, Cloud Recording - 500GB. *. Guidelines for conducting online meetings using Zoom *. Zoom Operation Manual for Hosts *. Zoom Operation Manual for Participants *. Zoom Recording Manual for Speakers Guide to organizing a hybrid (face-to-face and online) meeting at RIMS. *. Guidelines for conducting hybrid meetings using Zoom *. Zoom Operation Manual for Hosts *. Zoom Operation Manual for Participants *. Webinar Operation Manual for Hosts *. Webinar Operation Manual for Participants *. Zoom Recording Manual for Speakers Diagram for Hybrid meeting at Lecture Room 420 Guidelines for RIMS Joint Research Activities Our infection prevention measures Guide to organizing an online event using Zoom Guide to organizing a hybrid (face-to-face and online) meeting at RIMS ←　BACK TO THE TOP Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. 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RIMS aims to develop unique young researchers through graduate education, with its graduate students belonging to the Department of Mathematical Sciences of the Division of Mathematics and Mathematical Sciences in the Graduate School of Science, Kyoto University. In connection with this, the institute implemented. the 21st Century Center of Excellence (COE) Program, Formation of an International COE in the Frontiers of Mathematics and Fostering of Researchers in Future Generations. the Global COE Program, Fostering Top Leaders in Mathematics-Broadening the Core and Exploring New Ground. in the past. Currently, as the Mathematics Unit of the Kyoto University Top Global University Project \"Japan Gateway Program\", it provides graduate students with an international research environment including teaching by top-class overseas researchers. These projects are jointly run by the Division of Mathematics and Mathematical Sciences in the Graduate School of Science and RIMS. Mathematical Sciences Course. Number of Students. The number of students in the Department of Mathematical Sciences (since 2016) is listed below. Fiscal Year. Master's Program. Doctoral Program. Fiscal 2016. 19. 21. Fiscal 2017. 20. 23. Fiscal 2018. 21. 25. Fiscal 2019. 21. 20. Fiscal 2020. 21. 24. Conferment of Degrees. The number of students in the Department of Mathematical Sciences who received a Ph.D. (in science) from Kyoto University (since 1994) is listed below. Doctoral Course. Doctoral Dissertation. Total. 130. 51. 181. About RIMS. Message from the Director Members Organization Graduate Education and Conferment of Degrees ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. 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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Lecture. TOP 　　. Seminars 　　Lecture. Lecture. Speaker. Boris Tsygan (Northwestern University). Date. November 28 (Thu), 14:00--16:00, 2019. Room. RIMS, Rm 110. Title. On noncommutative geometry of algebras in positive characteristic. Abstract. Noncommutative geometry is a theory that takes an associative algebra and assigns to it invariants in such a way that, if the algebra is question is an algebra of functions on a variety, one gets geometric\/topological invariants of this variety. For a variety over complex numbers one can recover much of the usual calculus of differential forms, as well as De Rham cohomology. We will show how to generalize this to algebras over a field of positive characteristic. The talk is an introductory review of recent works of Kaledin, Petrov-Vologodsky, and myself. Comments. Takuro Mochizuki. Speaker. Boris Tsygan (Northwestern University). Date. November 25 (Mon), 14:00--16:00, 2019. Room. RIMS, Rm 110. Title. Microlocal invariants of symplectic manifolds. Abstract. I will discuss how to construct a category starting with a symplectic manifold using microlocal methods, following works of Tamarkin, Nadler-Zaslow, and myself. The talk will be very introductory and concentrate mainly on examples of the plane, cylinder, and 2-torus. Comments. Takuro Mochizuki. Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. 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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Colloquium\/Seminars. TOP 　　. Seminars 　　Low-dimensional Topology Seminar. Low-dimensional Topology Seminar. Date. February 6 (Thu.) 2020, 14:30 -- 15:30. Room. Room 402 RIMS, Kyoto Univ. Speaker. Delphine Moussard (Aix-Marseille University). Title. A splicing formula for the LMO invariant. Abstract. We will see a ``splicing formula'' for the LMO invariant of Le, Murakami and Ohtsuki. Specifically, if a rational homology 3-sphere M is obtained by gluing the exteriors of two framed knots in rational homology 3-spheres, the formula expresses the LMO invariant of M in terms of the Kontsevich-LMO invariants of the two knots. This is a joint work with Gwénaël Massuyeau. Organizers. T. Ohtsuki, K. Habiro. Date. November 14 (Thu.) 2019, 14:30 -- 15:30. Room. Room 402 RIMS, Kyoto Univ. Speaker. Gwenael Massuyeau (University of Burgundy). Title. Generalized Dehn twists on surfaces and surgeries in 3-manifolds. Abstract. (Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the \"Dehn twist\" along C is the homeomorphism of S defined by \"twisting\" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to \"simulate\" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S; another possibility is to view C as a curve on the top boundary of the cylinder S \\times [0,1], to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a new 3-manifold. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a \"symplectic expansion\" of the fundamental group of S. Organizers. T. Ohtsuki, K. Habiro. Date. October 17 (Thu.) 2019, 14:30 -- 15:30. Room. Room 402 RIMS, Kyoto Univ. Speaker. Anderson Vera (RIMS, Kyoto University\/JSPS). Title. Alternative Johnson homomorphisms and the LMO functor. Abstract. One of the main objects associated to a surface S is its mapping class group MCG(S). By considering the action of MCG(S) on the fundamental group of S, it is possible to define different filtrations of MCG(S) together with some homomorphisms on each term of the filtration. In this talk we present our results concerning a filtration of MCG(S), \"the alternative Johnson filtration\", recently introduced by Habiro-Massuyeau and whose definition involves a handlebody bounded by S. We show the relationship between the \"alternative Johnson homomorphisms\" and the functorial extension of the Le-Murakami-Ohtsuki (LMO) invariant of 3-manifolds. Organizers. T. Ohtsuki, K. Habiro. Date. July 9 (Tue) 2019, 14:30 -- 15:30. Room. Room 110 RIMS, Kyoto Univ. Speaker. Fiona Torzewska (University of Leeds). Title. Using model categories to construct topological quantum field theories. Abstract. We are able to use Strom's model category structure on the category of topological spaces to construct a very general class of TQFTs, which can be applied to a variety of source categories. This uses ideas from Yetter's 'TQFTs assoiated to finite groups' and from Quinn's 'Finite total homotopy TQFT'. In this talk I will outline the construction, highlighting why model categories appear to be the right setting for our work. I will also provide plenty of examples and show how explicit calculation is possible. No previous knowledge of model categories will be required. Organizers. T. Ohtsuki, K. Habiro. Date. February 19 (Tue) 2019, 14:30 -- 15:30. Room. Room 110 RIMS, Kyoto Univ. Speaker. Jørgen Ellegaard Andersen 氏 (QGM, Aarhus University). Title. The new formulation of the Teichmüller TQFT. Abstract. A few years ago we define the Teichmüller TQFT, which is based on a deformation of quantum Teichmüller theory. In its original formulation there was a certain homology assumption needed to guarantee finiteness of the involved integrals in the definition of that TQFT. In our new formulation, this assumption is no longer needed. The talk will be focused on our new formulation, its generalisations and if time permits the relation to the old formulation. The talk is based on joint work with Rinat Kashaev. Organizers. T. Ohtsuki, K. Habiro. Date. November 22 (Thur) 2018, 14:30 --. Room. Room 402 RIMS, Kyoto Univ. Speaker. Gaetan Borot (Max Planck Institute for Mathematics). Title. Mapping class group invariants via geometric recursion. Abstract. I will describe the geometric recursion (GR) and its first applications. Its aim is to construct mapping class group invariant quantities for surfaces of all topologies from a small amount of initial data, via excision\/glueing of pairs of pants, and therefore by induction on the Euler characteristic. Although GR is more general, I will focus on the case where it takes values in the space of continuous functions over Teichmuller space of bordered surfaces. The GR functions are designed in such a way that, when integrated against the Weil-Petersson volume form, the result satisfies a topological recursion. For instance, Mirzakhani identities express the constant function 1 on Teichmuller spaces as a result of GR, and after integration one obtains a topological recursion for the Weil-Petersson volumes. I will present a family of generalizations of Mirzakhani identities involving statistics of hyperbolic lengths of simple multicurves, and give other examples of potential applications of GR. Based on a joint work with Jorgen Ellegaard Andersen and Nicolas Orantin. Organizers. T. Ohtsuki, K. Habiro. Date. June 28 (Thur) 2018, 14:30 --. Room. Room 402 RIMS, Kyoto Univ. Speaker. Moussard Delphine (RIMS, Kyoto University \/ JSPS research fellow). Title. 2-knots with factorized Alexander polynomial. Abstract. For knots in the 3-sphere, it is well-known that the Alexander polynomial of a ribbon knot factorizes as f(t)f(1\/t) for some polynomial f(t). For 2-knots, i.e. embeddings of a 2-sphere in the 4-sphere, the Alexander polynomial of a ribbon 2-knot is not even symmetric in general. Via an alternative notion of ribbon 2-knots, we give a topological condition on a 2-knot for recovering the factorization of the Alexander polynomial. This is a joint work with Emmanuel Wagner. Organizers. T. Ohtsuki, K. Habiro. Date. June 21 (Thur) 2018, 14:30 --. Room. Room 402 RIMS, Kyoto Univ. Speaker. Zhongtao Wu (The Chinese University of Hong Kong). Title. An Alexander polynomial for graphs. Abstract. Using Alexander modules, one can define a polynomial invariant for a certain class of graphs with a balanced coloring. We will give different interpretations of this polynomial by Kauffman state formula and MOY relations. Moreover, we propose a potential categorification for the invariant. This is a joint work with Yuanyuan Bao. Organizers. T. Ohtsuki, K. Habiro. Date. 2018年4月26日(木) 14時30分 --. Room. 京都大学 数理解析研究所 402号室. Speaker. 湯淺 亘氏（京都大学 理学研究科）. Title. A_2 colored polynomials of rigid vertex graphs. Abstract. The Kauffman-Vogel polynomials are three variable polynomial invariants of 4-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented 4-valent rigid vertex graphs was given by using the Kauffman bracket. We define an A_2 version of the one-variable Kauffman-Vogel polynomial for 4-valent rigid vertex graphs by using the A_2 bracket. Organizers. T. Ohtsuki, K. Habiro. Date. 2018年4月12日(木) 14時00分 --. Room. 京都大学 数理解析研究所 402号室. Speaker. 辻 俊輔氏（京大数理研 \/ 学振特別研究員PD）. Title. Formulas for the action of Dehn twists on skein modules and their applications. Abstract. We give explicit formulaｓ for the action of the Dehn twist along a simple closed curve of a surface on the completed Kauffman bracket skein modules and the completed HOMFLY-PT type skein modules of the surface. As an application, using this formula, we construct some invariants for an integral homology 3-sphere. Organizers. T. Ohtsuki, K. Habiro. Date. 2018年1月11日(木) 14時30分 --. Room. 京都大学 数理解析研究所 402号室. Speaker. 小鳥居祐香氏（理化学研究所）. Title. On Goussarov-Polyak-Viro's finite type invariants and a virtual $C_n$-equivalence on long virtual knots. Abstract. A $C_n$-move is a family of local moves on knots, which gives a topological characterization of finite type invariants of knots. On the other hand, Goussarov-Polyak-Viro defined finite type invariants and $n$-equivalence on (long) virtual knots. They mentioned that the value of a finite type invariant of degree less than or equal to $n$ depends only on the $n$-equivalence class of (long) virtual knots. In this talk, we extend the $C_n$-move to (long) virtual knots by using the lower central series of the pure virtual braid, and we define a virtual $C_n$-equivalence, which is generated by this extended $C_n$-moves. We then prove that the virtual $C_n$-equivalence coincides with $(n-1)$-equivalence on long virtual knots. Organizers. T. Ohtsuki, K. Habiro. Date. 2017年10月19日(木) 14時30分 --. Room. 京都大学 数理解析研究所 402号室. Speaker. Andrew Kricker 氏 (Nanyang Technological University). Title. What is the l^2-Alexander invariant and how can we approximate it? Abstract. The classical Alexander polynomial marks the beginning of mathematical knot theory. The l^2-Alexander invariant is a recent variation due to Li-Zhang (2006) and Dubois-Friedl-Luck (2014) where you twist the classical construction by the infinite-dimensional representation consisting of square-summable series of elements of the fundamental group. This invariant has remarkable properties - such as the fact that it determines the volume of a hyperbolic knot, determines the Thurston norm, and detects infinitely many knots. In this talk I'll introduce this invariant and survey what is known and unknown about it. It is a very strong invariant, but also is almost impossible to calculate. I'll also talk about some ongoing projects investigating its approximation. Organizers. T. Ohtsuki, K. Habiro. Date. 2016年11月24日(木) 14:30 -- 15:30. Room. 京都大学 数理解析研究所 402号室. Speaker. 高尾和人氏（京大数理研）. Title. Heegaard surfaces and singularities of product maps. Abstract. The classification of Heegaard surfaces is a general problem in 3-manifold topology. Some recent progress on this problem has been made with a technique based on the singularities of the product map of two smooth functions. Further progress seems to depend on a special singularity theory for product maps. In this talk, I would like to survey relevant results, pending problems and my approach. Organizers. T. Ohtsuki, K. Habiro. Date. 2016年10月20日(木) 14:30 -- 15:30. Room. 京都大学 数理解析研究所 402号室. Speaker. Moussard Delphine 氏 (RIMS, Kyoto University \/ JSPS research fellow). Title. Equivariant triple intersections. Abstract. We define an invariant of null-homologous knots in rational homology 3-spheres by means of equivariant triple intersections of surfaces in the infinite cyclic covering of the complement of the knot. The invariant is a map on the triple tensor product of the Alexander module of the knot. We compute the variation of this invariant under null Borromean surgeries, and describe the set of all maps obtained as such invariants. Organizers. T. Ohtsuki, K. Habiro. Date. 2016年6月22日(水) 15:00 -- 16:00. Room. 京都大学 数理解析研究所 402号室. Speaker. 辻 俊輔 氏（東大数理）. Title. カウフマン・ブラケット・スケイン代数による整係数ホモロジー球面の不変量. Abstract. Torelli群から完備化されたカウフマン・ブラケット・スケイン代数への 埋め込みにより、トレリ群とカウフマン・ブラケット・スケイン代数の 新たな関係性を発見することができた。この埋め込みとHeegaard splitting を用いて整係数ホモロジー球面の不変量を構成することができた。この 不変量は Q[[A+1]] に値を持つ。さらにmod (A+1)^n で見たとき 位数 n の有限型不変量になる。とくに A+1 の係数はCasson invariantとなる。 この不変量の構成の仕方と不変量になっている証明の概略を紹介する。. Organizers. T. Ohtsuki, K. Habiro. Date. 2015年12月16日(水) 14:30 -- 15:30. Room. 京都大学 数理解析研究所 110号室. Speaker. Nathan Geer 氏 (Utah State University). Title. Re-normalized Link invariants. Abstract. In the last few years, C. Blanchet, F. Costantino, B. Patureau, N. Reshetikhin, V. Turaev and myself (in various collaborations) have developed a theory of renormalized quantum invariants of links and 3-manifolds which lead to TQFTs. This talk will start out by giving an overview of this work. In the second part of the talk I will discuss the renormalized quantum invariants of links coming from quantized sl(2) at a root of unity. These link invariants contain Kashaev's quantum dilogarithm invariants of knots, the Akutsu-Deguchi-Ohtsuki invariant of links and the multi-variable Alexander Polynomial. Moreover, these re-normalized invariants of knots are meromorphic functions whose residues are closely related to the the colored Jones polynomials. Organizers. T. Ohtsuki, K. Habiro. Date. 2014年3月7日(金) 15:00 --. Room. 京都大学 数理解析研究所 111号室. Speaker. 清水 達郎 氏（東大数理）. Title. An invariant of rational homology 3-spheres via vector fields. Abstract. In this talk, we define an invariant of rational homology 3-spheres with values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using vector fields. The construction of our invariant is a generalization of both that of the Kontsevich-Kuperberg- Thurston invariant $z^{KKT}$ and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$. As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for rational homology 3-spheres. Organizers. T. Ohtsuki, K. Habiro, T. Ito. Date. 2011年2月17日(木) 15:00～. Room. 京都大学 数理解析研究所 402号室. Speaker. Jean-Baptiste Meilhan 氏 (Universite Grenoble 1). Title. Torelli group and equivalence relations for homology cylinders. (joint work with G. Massuyeau). Abstract. Two 3-manifolds are called Y_k-equivalent if one can be obtained from the other by \"twisting\" an embedded surface by an element of the k-th term of the lower central series of its Torelli group. The J_k-equivalence relation is defined similarly, using the Johnson filtration instead of the lower central series. In this talk, we shall consider these equivalence relation among homology cylinders over a given surface S, which are 3-manifolds homologically equivalent to S \\times [0,1]. We classify these equivalence relations, for k \\le 3, using several classical invariants. This provides generalizations of results of W.Pitsch and S.Morita on the structure of integral homology spheres and the Casson invariant. Organizers. T. Ohtsuki, K. Habiro. Date. 2011年 1月25日(火) 15:00～. Room. 京都大学 数理解析研究所 110号室. Speaker. Sergei Duzhin 氏 (Steklov Mathematical Institute). Title. A formula for the HOMFLY polynomial of rational links. (joint work with Mikhail Shkolnikov). Abstract. We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link. Organizers. T. Ohtsuki, K. Habiro. Date. 2010年6月10日(木) 15:00～. Room. 京都大学 数理解析研究所 402号室. Speaker. David Farris 氏 (University of California, Berkeley). Title. The embedded contact homology of circle bundles over Riemann surfaces. Abstract. Embedded contact homology is a topological invariant of three-manifolds which is defined by choosing a contact structure on the manifold and studying pseudoholomorphic curves in the symplectization of the contact 3-manifold. We compute this invariant for circle bundles over Riemann surfaces (prequantization spaces), a case where pseudoholomorphic curves can be concretely understood as meromorphic sections of a holomorphic line bundle. We make use of domain-dependent almost complex structures to achieve transversality for moduli spaces of these curves. Organizers. T. Ohtsuki, K. Habiro. Date. February 10 (Wednesday), 2010, 14:00-. Room. 京都大学 理学部6号館 204講義室. Speaker. Mark Powell 氏（Edinburgh 大学）. Title. Knot concordance and twisted Blanchfield forms. Abstract. In this talk I will recall the notion of knot concordance as defined by Fox and Milnor, which asks whether a knot in S3 bounds a disk in 4 space D4. The work of Casson and Gordon involved a two stage obstruction theory which depends on the intersection form of a 4-manifold. This has been generalised by the work of Cochran-Orr-Teichner. I shall discuss an obstruction theory which is intrinsically 3-dimensional, using Blanchfield linking forms with coefficients twisted using metabelian representations of the knot group. These linking forms obstruct null-concordance. We then describe an algorithm to construct the symmetric chain complex of the universal cover of a knot exterior, and then use this to make calculations of the twisted Blanchfield forms. Organizers. T. Ohtsuki, K. Habiro. Date. December 1 (Tuesday), 11:00-. Room. RIMS Annex (Research Building No.4), Room 307 (総合研究4号館307号室). Speaker. Oliver Dasbach 氏 (Louisiana State University). Title. On the Turaev surface and its applications. Abstract. Turaev constructed for each link diagram an embedded, unknotted surface on which the link projects alternatingly. We will show how the Jones polynomial can be computed from this surface. As a corollary we will get an interesting formula for the determinant of a knot. Furthermore, we will give applications to the study of Khovanov and Ozsvath-Szabo knot homologies. Organizers. T. Ohtsuki, K. Habiro. Date. June 11 (Thur), 15:00-. Room. RIMS, Room 402. Speaker. Ivan Izmestiev 氏 (Technische Universitat Berlin). Title. Hyperbolic cusps with convex polyhedral boundary. Abstract. Consider a hyperbolic cusp bounded by a locally convex piecewise geodesic surface. The intrinsic metric of the boundary is a hyperbolic cone-metric with cone angles less than 2\\pi. We prove the converse: every hyperbolic cone-metric on the torus with cone angles less than 2\\pi can be realized as the boundary of a convex hyperbolic cusp. This theorem is similar to the Alexandrov's characterization of convex polyhedra. In fact, both of them are special cases of a general statement on convex realizations of surfaces with cone-metrics. The proof is based on the variational properties of the discrete Hilbert-Einstein functional. This is a joint work with Francois Fillastre. Organizers. T. Ohtsuki, K. Habiro. Date. 2007年11月6日(木) 15:00 - 16:30. Room. 数理解析研究所402号室. Speaker. Gwénaël Massuyeau (CNRS - Louis Pasteur University, Strasbourg). Title. An infinitesimal version of Morita's homomorphisms. Abstract. Let S be a compact, connected, oriented surface with one boundary component, and let I(S) be the Torelli group of S. For each integer k0, I(S) acts in the natural way on the k-th nilpotent quotient of the fundamental group P(S) of S and, by definition, the kernel of this action is the k-th term of the Johnson filtration of I(S). The Johnson filtration being separated, one can study the Torelli group by approaching P(S) by its successive nilpotent quotients. In this context, Morita defined for each k0 a group homomorphism from the k-th term of the Johnson filtration to the third homology group of the k-th nilpotent quotient of P(S). In this talk, groups will be replaced by their Malcev Lie algebras, which will lead to an ``infinitesimal'' version of Morita's homomorphisms. Although equivalent to their originals, the infinitesimal Morita homomorphisms seem more easy to use. Thus, we will give a diagrammatic description for them and we will explain how they can be computed. Finally, the infinitesimal Morita homomorphisms will be connected to the diagrammatic representation of I(S) derived from the LMO invariant of 3-manifolds. Organizers. T. Ohtsuki, K. Habiro. Date. 2007年8月23日(木), 午後2時～4時(休憩時間30分を含む). Room. 京都大学数理解析研究所 102号室. Speaker. Yoav Rieck 氏 （University of Arkansas, USA）. Title. On the Heegaard genus of knot exteriors. Abstract. We will survey some of the authors' results about the behavior of Heegaard genus of knot exteriors under connected sum operation. As our main result we will prove that given integers g_i 1 (i=1,...,n) there exist infinitely may knots K_i in S^3 so that g(E(K_i)) = g_i and g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). This proves the existence of counterexamples to Morimoto's Conjecture. Organizers. T. Ohtsuki, K. Habiro. Date. 2007年4月12日(木), 14:00～15:30 渡邉氏, 16:00～17:30 石井氏. Room. 京都大学 数理解析研究所 102号室. Speaker. 渡邉 忠之 氏 （京大数理研、機関研究員） \/石井 敦 氏 （京大数理研、日本学術振興会特別研究員）. Title. 配置空間積分の特性類について \/ A bracket polynomial and TQFT for invariants of virtual links. Abstract. Maxim Kontsevichは、Chern--Simons摂動理論の高次元のアナロジーとして、 配置空間積分を用いて奇数次元の枠付きホモロジー球面をファイバーとする ファイバーバンドル(C^\\infty-smooth)の特性類を構成しました。この特性類は、 ファイバーが３次元の場合にはホモロジー３球面の位相不変量になり、 全てのQ値の有限型不変量（大槻不変量）を含む普遍的不変量であることが Kuperberg--Thurstonにより示されています。しかし、ファイバーの次元が ３より大きい場合には、その性質は全く知られていませんでした。 この講演では、３より高い次元におけるKontsevichの特性類の非自明性に関して 得られた結果を紹介する予定です。またそれの系として、Casson不変量の高次元化 と思われる非自明な不変量が得られることなども時間があれば話す予定です。. 仮想結び目理論において、ブラケット多項式を用いた不変量の構成方法と オペレーター不変量の構成方法がどのように働くか（働かないか）を、 宮澤多項式を例にとって、古典的な場合との違いを説明します。. Organizers. T. Ohtsuki, K. Habiro. Date. September 28-29, 各日とも 14:00-17:00, 2006. Room. RIMS Room 402 (Sep. 28), Room 115 (Sep. 29), Kyoto University. Speaker. Sergei Duzhin 氏 \/ (Steklov Institute). Title. Detecting the link orientation \/ Estimation of crossing numbers of knots. Abstract. I will speak about the problem of detecting the orientation of knots and links, i.e. finding the invariants that take distinct values on two links differing only by an inversion. The first result in this direction is a classical theorem of Trotter who proved that the pretzel knot $P_{3,5,7}$ is not equivalent to its inverse. Trotter and some subsequent authors used homomorphisms of the knot group to study the invertibility. It is known that knot polynomials obtained by the Reshetikhin--Turaev procedure do not feel the orientation. Finite type (Vassiliev) knot invariants are strictly stronger than quantum invariants, and there is an important problem if these can tell a knot from its inverse. This problem is open until now. For links with more than one component the corresponding problem is partially solved, namely, a positive answer was obtained for closed links with 6 or more components (X.-S.Lin) and for string links with 2 components (S.Duzhin--M.Karev). I will give a review of the known results on the problem and then speak about an attempt to solve it for closed 2-component links using the invariants with values in the necklace algebra. Another approach that might lead to orientation-detecting invariants is the categorification of the $gl_n$ Lie algebra weight system. Indeed, the weight system with values in the center of the universal enveloping algebra for $gl_n$ is a function on the space of Jacobi diagrams which is given by the alternating sum over a cube of resolutions of the triple points of a diagram. One may therefore try to apply the general scheme of categorification due to Khovanov (through Frobenius algebras) and Bar-Natan (through canopolies) -- in a hope to split the big alternating sum into several smaller sums each of which gives a weight system, whose totality is a stronger invariant than the original weight system. This is an unfinished project, and I will speak about it in a hope to find eventual collaborators among the listeners. Organizers. T. Ohtsuki, K. Habiro. Date. July 13 (Thu), 2006, 15:00-. Room. Room 402, RIMS, Kyoto University. Speaker. Alexander Stoimenow 氏 \/ （京大数理研 COE研究員）. Title. 結び目の交点数の評価 \/ Estimation of crossing numbers of knots. Abstract. 5月10日の談話会で講演した内容の発展として、結び目図式から 結び目の交点数を決定したり評価したりする話題を中心にお話します。 とくに Lickorish-Thistlethwaite により定義された semiadequate 絡み目とその部分クラスに対する交点数の評価を紹介し、 さらに、時間に余裕があった場合は、 それらの Jones 多項式の非自明性についてもお話したいとおもいます。. I intend to talk about problems related to determination and estimation of crossing numbers of semiadequate knots, as defined by Lickorish-Thistlethwaite, and some of their subclasses. If time permits, I will discuss the relation to the non-triviality of their Jones polynomial. Organizers. T. Ohtsuki, K. Habiro. Speaker. 14:30--15:30 Gwenael Massuyeau氏. 16:00--17:00 高瀬 将道氏. Date. April 20 (Thu), 2006, 14:30-17:00. Room. Room 402, RIMS, Kyoto University. Speaker. Gwenael Massuyeau氏 （CNRS - Louis Pasteur University, Strasbourg \/ 日本学術振興会外国人特別研究員、京大数理研）. Title. Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds. Abstract. The Reidemeister-Turaev torsion is an invariant of a closed oriented three-dimensional manifold equipped with an Euler structure, with values in the ring of quotients of the group ring of the first homology group. We will prove that its reductions by powers of the augmentation ideal are finite-type invariants in the sense of M. Goussarov and K. Habiro. For this, we will start off by explaining how their theory of finite-type invariants can be refined to take into account Euler structures (which is a joint work with F. Deloup). Speaker. 高瀬 将道氏 (京大数理研 機関研究員). Title. Homology 3-spheres in codimension three. Abstract. For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers; and besides, gives rise to a complete invariant for homology cobordism classes of all embeddings of homology 3-spheres in the 6-sphere. As a consequence, we show that two embeddings of an oriented integral homology 3-sphere in the 6-sphere are isotopic if and only if they are homology cobordant. We also relate our invariant to the Rohlin invariant and accordingly characterise those embeddings which are compressible into the 5-sphere. Organizers. T. Ohtsuki, K. Habiro, T. Ito. Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Workshop for Principal Researcher Call for proposals. Call for proposals（RIMS Research Projects） Call for proposals（RIMS Workshops\/Symposia） Call for proposals（RIMS Satellite Seminars） Call for proposals（RIMS Review Seminars） Call for proposals (RIMS Research Projects). TOP 　　 Call for proposals (RIMS Research Projects). Type of RIMS Joint Research Activities. RIMS Research Projects RIMS Workshops RIMS Symposia RIMS Satellite Seminars RIMS Review Seminars RIMS Research Projects. An international joint research program implemented in combination with any of the types of joint research activities described below. Leading researchers in their respective fields are invited and stay at RIMS for a medium- to long-term duration, working on specific research subjects for a period of several months to one year. RIMS chooses and implements more than one project under this RIMS Research Project program each academic year. A call for proposals is published around June each academic year for projects to be implemented two years later. Each applicant should organize an organizing committee comprising several members to submit an application. Projects under this RIMS Research Project program are implemented to develop researchers who will lead research projects in the fields of mathematics and mathematical sciences in the future. The applicant may recommend young researchers participate, in cooperation with leading researchers, in planning, drawing and implementing international research projects as \"RIMS Project Fellows.\". List of RIMS Research Projects Past RIMS Reserch Projects Financial support. Expenses directly related to the Research Project (for example, transportation fees between the participants’ home countries and Japan, transportation fees in Japan, accommodation fees, and a per diem for invited overseas researchers) will be covered by RIMS up to a limit of 5,000,000 Japanese yen in accordance with the regulations of Kyoto University. *. Access (domestic travel expenses) Screening process. The Advisory Board of RIMS will select the proposal(s) to be adopted from among the submitted proposals, and approve three to seven organizing committee members in total, including at least one Advisory Board member. Available facilities. Seminar rooms that can be used for joint research activities are introduced on the following page. Seminar rooms and facilities for joint research COVID-19 updates for organizers and participants. Guidelines for RIMS Joint Research Activities Guidelines for Principal Researcher Guidline are available on the following page. Guidelines for Principal Researcher Guidelines and Application form (RIMS Research Projects). AY2024. !! New !! deadline: Aug.31.2022(JST). Web Submission. URL Guidelines for Grant Proposal PDF Application form for proposal PDF WORD Guidelines for Recommendation for RIMS Project Fellow PDF Recommendation form for RIMS Project Fellow PDF WORD AY2023 closed. Web Submission (closed) Guidelines for Grant Proposal PDF Application form for proposal PDF WORD Guidelines for Recommendation for RIMS Project Fellow PDF Recommendation form for RIMS Project Fellow PDF WORD List of Advisory Board menbers RIMS Research Projects RIMS Workshops\/Symposia RIMS Satellite Seminars RIMS Review Seminars ←　BACK TO THE TOP Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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{"id":"2yf3s116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":124,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["ess-lgc-iut.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Fri Jun 03 14:34:07 UTC 2022","dates_in_content_count_i":3,"dates_in_content_dts":["Wed Feb 05 00:00:00 UTC 2020","Fri Apr 03 00:00:00 UTC 2020","Sat Oct 07 00:00:00 UTC 2017"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-2569988016399845793,"exact_signature_unique_b":true,"fuzzy_signature_l":-8793518054336302561,"fuzzy_signature_unique_b":true,"wordcount_i":57015,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:53 UTC 2022","fresh_date_dt":"Sat Jun 11 05:57:41 UTC 2022","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":884726,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086040903843840,"text_t":"ON THE ESSENTIAL LOGICAL STRUCTURE OF\nINTER-UNIVERSAL TEICHMÜLLER THEORY IN TERMS\nOF LOGICAL AND “∧”\/LOGICAL OR “∨” RELATIONS:\nREPORT ON THE OCCASION OF THE\nPUBLICATION OF THE FOUR MAIN PAPERS ON\nINTER-UNIVERSAL TEICHMÜLLER THEORY\nShinichi Mochizuki\nJune 2022\nAbstract. The main goal of the present paper is to\ngive a detailed exposition of the essential logical structure of inter-universal\nTeichmüller theory from the point of view of the Boolean operators — such as the\nlogical AND “∧” and logical OR “∨” operators — of propositional calculus. This\nessential logical structure of inter-universal Teichmüller theory may be summarized\nsymbolically as follows:\nA ∧B = A ∧ (B1 ∨̇ B2 ∨̇ . . . ) =⇒ A ∧ (B1 ∨̇ B2 ∨̇ . . . ∨̇ B′1 ∨̇ B′2 ∨̇ . . . )\n.\n.\n.\n— where\n· the “∨̇” denotes the Boolean operator exclusive-OR, i.e., “XOR”;\n· A, B, B1, B2, B′1, B′2, denote various propositions;\n· the logical AND “∧’s” correspond to the Θ-link of inter-universal Te-\nichmüller theory and are closely related to the multiplicative structures of\nthe rings that appear in the domain and codomain of the Θ-link;\n· the logical XOR “∨̇’s” correspond to various indeterminacies that arise\nmainly from the log-Kummer-correspondence, i.e., from sequences of it-\nerates of the log-link of inter-universal Teichmüller theory, which may be\nthought of as a device for constructing additive log-shells.\nThis sort of concatenation of logical AND “∧’s” and logical XOR “∨̇’s” is reminiscent\nof the well-known description of the “carry-addition” operation on Teichmüller\nrepresentatives of the truncated Witt ring Z\/4Z in terms of Boolean addition “∨̇”\nand Boolean multiplication “∧” in the field F2 and may be regarded as a sort\nof “Boolean intertwining” that mirrors, in a remarkable fashion, the “arith-\nmetic intertwining” between addition and multiplication in number fields and\nlocal fields, which is, in some sense, the main object of study in inter-universal Te-\nichmüller theory. One important topic in this exposition is the issue of “redundant\ncopies”, i.e., the issue of how the arbitrary identification of copies of isomorphic\nmathematical objects that appear in the various constructions of inter-universal Te-\nichmüller theory impacts — and indeed invalidates — the essential logical structure\nof inter-universal Teichmüller theory. This issue has been a focal point of funda-\nmental misunderstandings and entirely unnecessary confusion concerning\ninter-universal Teichmüller theory in certain sectors of the mathematical community.\nThe exposition of the topic of “redundant copies” makes use of many interesting\nelementary examples from the history of mathematics.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\nContents:\nIntroduction\n§1. Summary of non-mathematical aspects for non-specialists\n§1.1. Publication of [IUTchI-IV]\n§1.2. Redundancy assertions of the “redundant copies school” (RCS)\n§1.3. Qualitative assessment of assertions of the RCS\n§1.4. The importance of extensive, long-term interaction\n§1.5. The historical significance of detailed, explicit, accessible records\n§1.6. The importance of further dissemination\n§1.7. The notion of an “expert”\n§1.8. Fabricated versions spawn fabricated dramas\n§1.9. Geographical vs. mathematical proximity\n§1.10. Mathematical intellectual property rights\n§1.11. Social mirroring of mathematical logical structure\n§1.12. Computer verification, mathematical dialogue, and developmental\nreconstruction\n§2. Elementary mathematical aspects of “redundant copies”\n§2.1. The history of limits and integration\n§2.2. Derivatives and integrals\n§2.3. Line segments vs. loops\n§2.4. Logical AND “∧” vs. logical OR “∨”\n§3. The logical structure of inter-universal Teichmüller theory\n§3.1. One-dimensionality via identification of RCS-redundant copies\n§3.2. RCS-redundancy of Frobenius-like\/étale-like versions of objects\n§3.3. RCS-redundant copies in the domain\/codomain of the log-link\n§3.4. RCS-redundant copies in the domain\/codomain of the Θ-link\n§3.5. Gluings, indeterminacies, and pilot discrepancy\n§3.6. Chains of logical AND relations\n§3.7. Poly-morphisms and logical AND relations\n§3.8. Inter-universality and logical AND relations\n§3.9. Passage and descent to underlying structures\n§3.10. Detailed description of the chain of logical AND relations\n§3.11. The central importance of the log-Kummer-correspondence\nList of Examples:\n1.9.1. The insufficiency of geographical proximity\n1.9.2. The remarkable potency of mathematical proximity\n1.10.1. The Pythagorean Theorem\n1.12.1. Explicit parametrization of Pythagorean triples\n2.1.1. False contradiction in the theory of integration\n2.2.1. Symmetry properties of derivatives\n2.3.1. Endpoints of an oriented line segment\n2.3.2. Gluing of adjacent oriented line segments\n2.4.1. “∧” vs. “∨” for adjacent oriented line segments\n2.4.2. Differentials on oriented line segments\n2.4.3. Representation via subgroup indices of “∧” vs. “∨”\n2.4.4. Logical “∧\/∨” vs. “narrative ∧\/∨”\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 3\n2.4.5. Numerical representation of “∧” vs. “∨”\n2.4.6. Carry operations in arithmetic, geometry, and Boolean logic\n2.4.7. The projective line as a gluing of ring schemes along a multiplicative\ngroup scheme\n3.1.1. Elementary models of gluings and intertwinings\n3.2.1. Global multiplicative subspaces and bounds on heights\n3.2.2. Coricity, symmetry, and commutativity properties of the log-theta-lattice\n3.3.1. Classical complex Teichmüller theory\n3.3.2. The Jacobi identity for the classical theta function\n3.3.3. Theta functions and multiplicative structures\n3.5.1. Bounded nature of log-shell automorphism indeterminacies\n3.5.2. Examples of gluings\n3.8.1. Inevitability of inner automorphism indeterminacies\n3.8.2. Inter-universality and the structure of (Θ±ellNF-)Hodge theaters\n3.8.3. Truncated vs. profinite Kummer theory and compatibility with the\np-adic logarithm\n3.8.4. Symmetrizing isomorphisms, truncatibility, and the log-Kummer-\ncorrespondence\n3.9.1. Categories of open subschemes\n3.10.1. Symmetries as a fundamental non-formal aspect of gluings\nIntroduction\nIn the present paper, we give a detailed exposition of the essential logical\nstructure of inter-universal Teichmüller theory in terms of elementary Boolean\noperators such as logical AND “∧” and logical OR “∨”. One important topic in\nthis exposition is the issue of “redundant copies”, i.e., the issue of how the arbi-\ntrary identification of copies of isomorphic mathematical objects that appear in\nthe various constructions of inter-universal Teichmüller theory impacts — and in-\ndeed invalidates — the essential logical structure of inter-universal Teichmüller the-\nory. This issue has been a focal point of fundamental misunderstandings and\nentirely unnecessary confusion concerning inter-universal Teichmüller theory\nin certain sectors of the mathematical community [cf. the discussion of Examples\n2.4.5, 2.4.7].\nWe begin, in §1, by reporting on various non-mathematical aspects of the situa-\ntion surrounding inter-universal Teichmüller theory, such as the issue of “redundant\ncopies”. Perhaps the most central portion of this discussion of non-mathematical as-\npects of the situation surrounding inter-universal Teichmüller theory concerns the\nlong-term, historical importance of producing detailed, explicit, mathemati-\ncally substantive, and readily accessible written documentation of the essential\nlogical structure of the issues under debate [cf. §1.5]. Such written documentation\nof the essential logical structure of the issues under debate is especially important\nin situations such as the situation that has arisen surrounding inter-universal Te-\nichmüller theory, in which the proliferation of logically unrelated fabricated\nversions of the theory has led to fundamental misunderstandings and entirely un-\nnecessary confusion concerning inter-universal Teichmüller theory in certain sectors\nof the mathematical community that are deeply detrimental to the operational\n4 SHINICHI MOCHIZUKI\nnormalcy of the field of mathematics [cf. §1.3, §1.8, §1.10, §1.11, §1.12]. This dis-\ncussion in §1 is supplemented by various interesting historical examples related\nto the Pythagorean theorem [cf. Example 1.10.1] and Pythagorean triples\n[cf. Example 1.12.1].\nWe then proceed, in §2, to discuss elementary aspects of the mathematics\nsurrounding the essential logical structure of inter-universal Teichmüller theory.\nOur discussion of these elementary aspects, which concerns mathematics at the ad-\nvanced undergraduate or beginning graduate level and does not require any advanced\nknowledge of anabelian geometry or inter-universal Teichmüller theory, focuses on\nthe close relationship between\n· integration and differentiation on — i.e., so to speak, the “differential\ngeometry” of — the real line [cf. §2.1, §2.2, as well as Example 2.4.2],\n· the geometry of adjacent closed intervals of the real line and the\nloops that arise by identifying various closed subspaces of such closed\nintervals [cf. §2.3; Example 2.4.1], and\n· Boolean operators such as logical AND “∧” and logical OR “∨” [cf.\n§2.4].\nOne important unifying theme that relates these seemingly disparate topics is the\ntheme of “carry operations”, which appear in the various arithmetic, geomet-\nric [i.e., “gluing”], and Boolean-logical situations discussed in §2 [cf. Example\n2.4.6]. On the other hand, from the point of view of arithmetic geometry, the\ndiscussion of\nthe projective line as a gluing of ring schemes along amultiplicative\ngroup scheme\ngiven in Example 2.4.7 yields a remarkably elementary qualitative model\/analogue\nof the essential logical structure surrounding the gluing given by the Θ-link in\ninter-universal Teichmüller theory. Moreover, over the complex numbers, this ex-\nample of the projective line — i.e., which may be visualized as a sphere — leads\nto an interesting analogy between the well-known [e.g., especially, in a cartographic\ncontext!] metric\/geodesic geometry of the sphere with the multiradial rep-\nresentation of the Θ-pilot in inter-universal Teichmüller theory [cf. Example\n2.4.7, (v)]. Finally, this example of the projective line is also of interest in light of\nthe remarkable parallels between the issue of “redundant copies” in the context\nof inter-universal Teichmüller theory and the well-known 19-th century “algebraic\ntruths” versus “geometric fantasies” dispute between Weierstrass and Riemann\nconcerning approaches to complex function theory [cf. the discussion of §1.5].\nThe preparatory topics of §2 lead naturally to the detailed exposition of the\nessential logical structure of inter-universal Teichmüller theory given in §3. From a\nstrictly rigorous point of view, this exposition assumes a substantial level of knowl-\nedge and understanding of the technicalities of inter-universal Teichmüller theory\n[which are surveyed, for instance, in [Alien]], although the essential mathematical\ncontent of most of the issues discussed may in fact be understood at the level of\nthe elementary considerations discussed in §2. The essential logical structure\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 5\nof inter-universal Teichmüller theory may be represented symbolically as follows:\nA ∧B = A ∧ (B1 ∨̇ B2 ∨̇ . . . )\n=⇒ A ∧ (B1 ∨̇ B2 ∨̇ . . . ∨̇ B′1 ∨̇ B′2 ∨̇ . . . )\n=⇒ A ∧ (B1 ∨̇ B2 ∨̇ . . . ∨̇ B′1 ∨̇ B′2 ∨̇ . . . ∨̇ B′′1 ∨̇ B′′2 ∨̇ . . . )\n...\n— cf. the discussion of (∧(∨̇)-Chn) in §3.10. [Here, “∨̇” denotes the Boolean\noperator exclusive-OR, i.e., “XOR”.] Indeed, §3 is devoted, for the most part, to\ngiving a detailed exposition of various aspects of this symbolic representation, such\nas the following:\n· the logical AND “∧’s” in the above display may be understood as\ncorresponding to the Θ-link of inter-universal Teichmüller theory and are\nclosely related to the multiplicative structures of the rings that appear\nin the domain and codomain of the Θ-link;\n· the logical XOR “∨̇’s” in the above display may be understood as\ncorresponding to various indeterminacies that arise mainly from the log-\nKummer-correspondence, i.e., from sequences of iterates of the log-\nlink of inter-universal Teichmüller theory, which may be thought of as a\ndevice for constructing additive log-shells.\nThis appearance of logical AND “∧’s” and logical XOR “∨̇’s” is of interest in that it\nis reminiscent of the well-known description of the “carry-addition” operation on\nTeichmüller representatives of the truncated Witt ring Z\/4Z in terms of Boolean\naddition “∨̇” and Boolean multiplication “∧” in the field F2 and may be re-\ngarded as a sort of “Boolean intertwining” that mirrors, in a remarkable fashion,\nthe “arithmetic intertwining” between addition and multiplication in num-\nber fields and local fields, which is, in some sense, the main object of study in\ninter-universal Teichmüller theory [cf. the discussion of Example 2.4.6, (iii); the\ndiscussion surrounding (TrHrc) in §3.10]. The above symbolic representation of the\nessential logical structure of inter-universal Teichmüller theory arises naturally from\nconsiderations concerning such key topics as\n· the coricity\/symmetry\/commutativity properties of the log-theta-\nlattice [cf. Example 3.2.2] and the closely related significance of working\nwith both Frobenius-like and étale-like objects [cf. §3.2];\n· the closely intertwined properties of theta functions and [p-adic\/archi-\nmedean] logarithms [cf. Examples 3.3.2, 3.3.3, 3.8.3, 3.8.4] in the con-\ntext of the log-Kummer-correspondence [cf. §3.3; §3.11; Examples\n3.2.1, 3.3.1];\n· generalities concerning gluings [cf. §3.1, §3.4, §3.5, §3.10; Examples\n3.1.1, 3.5.2, 3.10.1];\n· generalities concerning indeterminacies [cf. §3.5, §3.6, §3.7; Example\n3.5.1];\n· generalities concerning inter-universality [cf. §3.8; Examples 3.8.1,\n3.8.2, 3.8.3, 3.8.4];\n· generalities concerning descent to underlying structures [cf. §3.9,\n§3.10, §3.11].\n6 SHINICHI MOCHIZUKI\nAcknowledgements\nThe present paper benefited substantially from numerous discussions with Ben-\njamin Collas, Ivan Fesenko, Yuichiro Hoshi, Fumiharu Kato, Emmanuel Lepage,\nArata Minamide, Wojciech Porowski, Fucheng Tan, and Shota Tsujimura. The au-\nthor would like to express his appreciation to all of these mathematicians for the\ntime and effort that they contributed to these discussions.\nSection 1: Summary of non-mathematical aspects for non-specialists\nWe begin with an overall summary of non-mathematical aspects of the situation\nsurrounding [IUTchI-IV], which may be of interest to both non-mathematicians\nand mathematicians. We also refer to [FsADT], [FKvid], [FsDss], [FsPio] for a\ndiscussion of various aspects of this situation from slightly different points of view.\n§1.1. Publication of [IUTchI-IV]\nThe four main papers [IUTchI-IV] on inter-universal Teichmüller theory (IUT)\nwere accepted for publication in the Publications of the Research Institute for\nMathematical Sciences (PRIMS) on February 5, 2020. This was announced at an\nonline video news conference held at Kyoto University on April 3, 2020. The four\npapers were subsequently published in several special volumes of PRIMS, a leading\ninternational journal in the field of mathematics with a distinguished history dating\nback over half a century.\nThe refereeing for these Special Volumes was overseen by an Editorial Board for\nthe Special Volumes chaired by Professors Masaki Kashiwara and Akio Tamagawa.\n[Needless to say, as the author of these four papers, I was completely excluded\nfrom the activities of this Editorial Board for the Special Volumes.] Professor\nKashiwara, a professor emeritus at RIMS, Kyoto University, is a global leader in\nthe fields of algebraic analysis and representation theory. Professor Tamagawa,\ncurrently a professor at RIMS, Kyoto University, is a leading pioneer in the field\nof anabelian geometry and related research in arithmetic geometry. Here, it should\nbe noted that, to a substantial extent,\ninter-universal Teichmüller theory arose as an extension\/application\n— developed by the author in the highly mathematically stimulating en-\nvironment at RIMS, Kyoto University, over the course of roughly two\ndecades [i.e., 1992 - 2012] — of precisely the sort of anabelian geome-\ntry that was pioneered by Tamagawa.\nIt is for this reason that PRIMS stood out among mathematics journals worldwide\nas the most appropriate — i.e., in the sense of being by far the most [and indeed\nperhaps the only truly] technically qualified — journal for the task of refereeing and\npublishing the four papers [IUTchI-IV] on inter-universal Teichmüller theory.\nBoth Professors Kashiwara and Tamagawa have an outstandingly high inter-\nnational reputation, built up over distinguished careers that span several decades.\nIt is entirely inconceivable that any refereeing process overseen by these mathemati-\ncians might be conducted relative to anything less than the highest mathematical\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 7\nstandards, free of any inappropriate non-mathematical considerations. In an arti-\ncle in the Asahi Shimbun [a major Japanese newspaper] published shortly after the\nannouncement of April 3, 2020, Professor Tamagawa is quoted as saying that he\nhas\n“100 percent confidence in the refereeing”\nthat was done for the four papers [IUTchI-IV].\nIn another article in the Asahi Shimbun [also published shortly after the an-\nnouncement of April 3, 2020], Professors Shigefumi Mori, a professor emeritus at\nRIMS, Kyoto University, and Nobushige Kurokawa, a professor emeritus at the\nTokyo Institute of Technology, express their expectations about the possibility of\napplying inter-universal Teichmüller theory to other unsolved problems in number\ntheory.\nSubsequent to these developments in 2020, a sequel [ExpEst] to the four original\npapers [IUTchI-IV] on inter-universal Teichmüller theory was accepted for publica-\ntion in the Kodai Mathematical Journal in September 2021. This sequel [ExpEst]\nconcerns explicit numerical estimates in inter-universal Teichmüller theory and con-\ntains, in particular, a new proof of Fermat’s Last Theorem.\nIn particular, the results proven in the four original papers [IUTchI-IV] on\ninter-universal Teichmüller theory, as well as the sequel [ExpEst], may now be\nquoted in the mathematical literature as results proven in papers that have been\npublished in leading international journals in the field of mathematics after under-\ngoing, in the case of the four original papers [IUTchI-IV], an exceptionally thorough\n[seven and a half year long] refereeing process.\n§1.2. Redundancy assertions of the “redundant copies school” (RCS)\nUnfortunately, it has been brought to my attention that, despite the develop-\nments discussed in §1.1, fundamental misunderstandings concerning the math-\nematical content of inter-universal Teichmüller theory persist in certain sectors of\nthe mathematical community. These misunderstandings center around a certain\noversimplification — which is patently flawed, i.e., leads to an immediate contradic-\ntion — of inter-universal Teichmüller theory. This oversimplified version of inter-\nuniversal Teichmüller theory is based on assertions of redundancy concerning\nvarious multiple copies of certain mathematical objects that appear in inter-\nuniversal Teichmüller theory. In the present paper, I shall refer to the school of\nthought [i.e., in the sense of a “collection of closely interrelated ideas”] constituted\nby these assertions as\nthe “RCS”, i.e., “redundant copies school [of thought]”.\nOne fundamental reason for the use of this term “RCS” [i.e., “redundant copies\nschool [of thought]”] in the present paper, as opposed to proper names of math-\nematicians, is to emphasize the importance of concentrating on mathematical\ncontent, as opposed to non-mathematical — i.e., such as social, political, or psy-\nchological — aspects or interpretations of the situation.\nThus, in a word, the central assertions of the RCS may be summarized as\nfollows:\n8 SHINICHI MOCHIZUKI\nVariousmultiple copies of certain mathematical objects in inter-universal\nTeichmüller theory are redundant and hence may be identified with one\nanother. On the other hand, once one makes such identifications, one\nobtains an immediate contradiction.\nIn the present paper, I shall refer to redundancy in the sense of the assertions of the\nRCS as “RCS-redundancy”, to the identifications of RCS-redundant copies that\nappear in the assertions of the RCS as “RCS-identifications”, and to the over-\nsimplified version of inter-universal Teichmüller theory obtained by implementing\nthe RCS-identifications as “RCS-IUT”.\nAs discussed in [Rpt2018] [cf., especially, [Rpt2018], §18], there is absolutely\nno doubt that\nRCS-IUT is indeed a meaningless and absurd theory that leads immediately\nto a contradiction.\nA more technical discussion of this contradiction, in the language of inter-universal\nTeichmüller theory, is given in §3.1 below, while a digested version in more elemen-\ntary language of the technical discussion of §3 may be found in Examples 2.4.5,\n2.4.7 below.\nRather, the fundamental misunderstandings underlying the RCS lie in the as-\nsertions of RCS-redundancy. The usual sense of the word “redundant” suggests\nthat there should be some sort of equivalence, or close logical relationship,\nbetween the original version of the theory [i.e., IUT] and the theory obtained [i.e.,\nRCS-IUT] by implementing the RCS-identifications of RCS-redundant objects. In\nfact, however,\nimplementing the RCS-identifications of RCS-redundant objects radi-\ncally alters\/invalidates the essential logical structure of IUT\nin such a fundamental way that it seems entirely unrealistic to verify any sort of\n“close logical relationship” between IUT and RCS-IUT.\nA more technical discussion of the three main types of RCS-redundancy\/RCS-\nidentification — which we refer to as “(RC-FrÉt)”, “(RC-log)”, and “(RC-Θ)” —\nis given, in the language of inter-universal Teichmüller theory, in §3.2, §3.3, §3.4,\nbelow. In fact, however, the essential mathematical content of these three\nmain types of RCS-redundancy\/RCS-identification is entirely elementary and\nlies well within the framework of undergraduate-level mathematics. A discussion of\nthis essentially elementary mathematical content is given in §2.3, §2.4 below [cf.,\nespecially, Examples 2.4.5, 2.4.7].\nOne important consequence of the technical considerations discussed in §3\nbelow is the following:\nfrom the point of view of the logical relationships between various as-\nsertions of the RCS, the most fundamental type of RCS-redundancy is\n(RC-Θ).\nThat is to say, (RC-Θ) may be understood as the logical cornerstone of the\nvarious assertions of the RCS.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 9\n§1.3. Qualitative assessment of assertions of the RCS\nAs discussed in detail in §3.4 below [cf. also §2.3, §2.4],\nimplementing the logical cornerstone RCS-identification of (RC-Θ)\ncompletely invalidates the crucial logical AND “∧” property satis-\nfied by the Θ-link — a property that underlies the entire logical struc-\nture of inter-universal Teichmüller theory.\nIn particular, understanding the issue of how the RCS treats this fundamental\nconflict between the RCS-identification of (RC-Θ) and the crucial ∧-property of the\nΘ-link is central to the issue of assessing the assertions of the RCS.\nIn March 2018, discussions were held at RIMS with two adherents of the RCS\nconcerning, in particular, (RC-Θ) [cf. [Rpt2018], [Dsc2018]]. Subsequent to these\ndiscussions, after a few e-mail exchanges, these two adherents of the RCS informed\nme via e-mail in August 2018 — in response to an e-mail that I sent to them in which\nI stated that I was prepared to continue discussing inter-universal Teichmüller the-\nory with them, but that I had gotten the impression that they were not interested\nin continuing these discussions — that indeed they were not interested in continuing\nthese discussions concerning inter-universal Teichmüller theory. In the same e-mail,\nI also stated that perhaps it might be more productive to continue these discussions\nof inter-universal Teichmüller theory via different participants [i.e., via “represen-\ntatives” of the two sides] and encouraged them to suggest possible candidates for\ndoing this, but they never responded to this portion of my e-mail. [Incidentally,\nit should be understood that I have no objection to making these e-mail messages\npublic, but will refrain from doing so in the absence of explicit permission from the\ntwo recipients of the e-mails.]\nSince March 2018, I have spent a tremendous amount of time discussing the\nfundamental “(RC-Θ) vs. ∧-property” conflict mentioned above with quite\na number of mathematicians. Moreover, during the years following the March 2018\ndiscussions, many mathematicians [including myself!] with whom I have been in\ncontact have devoted a quite substantial amount of time and effort to analyzing\nand discussing certain 10pp. manuscripts written by adherents of the RCS [cf.,\nespecially, the discussion of the final page and a half of the files “[SS2018-05]”,\n“[SS2018-08]” available at the website [Dsc2018]] — indeed to such an extent that\nby now, many of us can cite numerous key passages in these manuscripts by memory.\nMore recently, one mathematician with whom I have been in contact has made a\nquite intensive study of the mathematical content of recent blog posts by adherents\nof the RCS.\nDespite all of these efforts, the only justification for the logical cornerstone\nRCS-identification of (RC-Θ) that we [i.e., I myself, together with the many\nmathematicians with whom I have discussed these issues] could find either in oral\nexplanations during the discussions of March 2018 or in subsequent written records\nproduced by adherents of the RCS [i.e., such as the 10pp. manuscripts referred to\nabove or various blog posts] were statements of the form\n“I don’t see why not”.\n[I continue to find it utterly bizarre that such justifications of the assertions of\nthe RCS appear to be taken seriously by some professional mathematicians.] In\n10 SHINICHI MOCHIZUKI\nparticular, we were unable to find any detailed mathematical discussion by adherents\nof the RCS of the fundamental “(RC-Θ) vs. ∧-property” conflict mentioned\nabove. That is to say, in summary,\nthe mathematical justification for the “redundancy” asserted in the\nlogical cornerstone assertion (RC-Θ) of the RCS remains a complete\nmystery to myself, as well as to all of the mathematicians that I have\nconsulted concerning this issue\n[cf. the discussion of Examples 2.4.5, 2.4.7]. Put another way, the response of all\nof the mathematicians with whom I have had technically meaningful discussions\nconcerning the assertions of the RCS was completely uniform and unanimous, i.e.,\nto the effect that these assertions of the RCS were obviously completely math-\nematically inaccurate\/absurd, and that they had no idea why adherents of the\nRCS continued to make such manifestly absurd assertions. In particular, it should\nbe emphasized that\nI continue to search for a professional mathematician [say, in the field of\narithmetic geometry] who feels that he\/she understands the mathematical\ncontent of the assertions of the RCS and is willing to discuss this math-\nematical content with me or other mathematicians with whom I am in\ncontact\n[cf. the text at the beginning of [Dsc2018]]. It is worth noting that this situation\nalso constitutes a serious violation of article (6.)\nMathematicians should not make public claims of potential new theorems\nor the resolution of particular mathematical problems unless they are able\nto provide full details in a timely manner.\nof the subsection entitled “Responsibilities of authors” of the Code of Practice of\nthe European Mathematical Society (cf. [EMSCOP]).\nIn this context, one important observation that should be kept in mind is the\nfollowing [cf. the discussion of [Rpt2018], §18]:\n(UndIg) There is a fundamental difference between\n(UndIg1) criticism of a mathematical theory that is based on a solid,\ntechnically accurate understanding of the content and logi-\ncal structure of the theory and\n(UndIg2) criticism of a mathematical theory that is based on a funda-\nmental ignorance of the content and logical structure of the\ntheory.\nAn elementary classical example of this sort of difference is discussed in\n§2.1 below.\nIn the case of the RCS, the lack of any thorough mathematical discussion of the\nfundamental “(RC-Θ) vs. ∧-property” conflict mentioned above in the vari-\nous oral\/written explanations set forth by adherents of the RCS demonstrates, in a\ndefinitive way, that none of the adherents of the RCS has a solid, technically accu-\nrate understanding of the logical structure of inter-universal Teichmüller theory in\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 11\nits original form, i.e., in particular, of the central role played in this logical structure\nby the “∧-property” of the Θ-link. Put another way, the only logically consistent\nexplanation of this state of affairs is that the theory “RCS-IUT” that adherents of\nthe RCS have in mind, i.e., the theory that is the object of their criticism, is simply\na completely different — and logically unrelated — theory from the theory\nconstituted by inter-universal Teichmüller theory in its original form.\nFinally, it should be mentioned that although some people have asserted par-\nallels between the assertions of the RCS and the fundamental error in the first\nversion of Wiles’s proof of the Modularity Conjecture in the mid-1990’s, this anal-\nogy is entirely inappropriate for numerous reasons. Indeed, as is well-known,\nnothing even remotely close to the phenomena discussed thus far in the present\n§1.3 occurred in the case of the error in the first version of Wiles’s proof. The fact\nthat there was indeed a fatal error in the first version of Wiles’s proof was never\ndisputed in any way by any of the parties involved; the only issue that arose was\nthe issue of whether or not the proof could be fixed. By contrast, no essential er-\nrors have been found in inter-universal Teichmüller theory, since the four preprints\n[IUTchI-IV] on inter-universal Teichmüller theory were released in August 2012.\nThat is to say, in a word, the assertions of the RCS are nothing more than mean-\ningless, superficial misunderstandings of inter-universal Teichmüller theory on\nthe part of people who are clearly not operating on the basis of a solid, technically\naccurate understanding of the mathematical content and essential logical structure\nof inter-universal Teichmüller theory.\n§1.4. The importance of extensive, long-term interaction\nIn general, the transmission of mathematical ideas between individuals\nwho share a sufficient stock of common mathematical culture may be achieved\nin a relatively efficient way and in a relatively brief amount of time. Typical ex-\namples of this sort of situation in the context of interaction between professional\nmathematicians include\n· one-hour mathematical lectures,\n· week-long mathematical lecture series, and\n· informal mathematical discussions for several days to a week.\nIn the context of mathematical education, typical examples include\n· written or oral mathematical examinations and\n· mathematics competitions.\nThe successful operation of each of these examples relies, in an essential way, on\na common framework of mathematical culture that is shared by the various\nparticipants in the activity under consideration.\nOn the other hand, in the case of a fundamentally new area of research,\nsuch as inter-universal Teichmüller theory, which evolved out of research over the\npast quarter of a century concerning absolute anabelian geometry, certain types\nof categories arising from arithmetic geometry, and certain arithmetic aspects of\ntheta functions, the collection of mathematicians who share such a sufficient stock\nof common mathematical culture tends to be relatively small in number. In par-\nticular, for most mathematicians — even many arithmetic geometers or anabelian\n12 SHINICHI MOCHIZUKI\ngeometers — short-term interaction of the sort that occurs in the various typical\nexamples mentioned above is far from sufficient to achieve an effective trans-\nmission of mathematical ideas. That is to say, no matter how mathematically\ntalented the participants in such platforms of interaction may be, it takes time for\nthe participants to\n· analyze and sort out numerous mutual misunderstandings,\n· develop effective techniques of communication that can transcend such\nmisunderstandings, and\n· digest and absorb new ideas and modes of thought.\nDepending on the mathematical content under consideration, as well as on the\nmathematical talent, mathematical background, and time constraints of the partic-\nipants, this painstaking process of analysis\/development\/digestion\/absorption may\nrequire\npatiently sustained efforts to continue constructive, orderly mathematical\ndiscussions [via e-mail, online video discussions, or face-to-face meetings]\nover a period of months or even years\nto reach fruition. Indeed, my experience in exposing the ideas of inter-universal\nTeichmüller theory to numerous mathematicians over the past decade suggests\nstrongly that, in the case of inter-universal Teichmüller theory, it is difficult to\nexpedite this process to the extent that it can be satisfactorily achieved in less than\nhalf a year or so.\nIn particular, in the case of inter-universal Teichmüller theory, a week-long\nsession of discussions such as the discussions held at RIMS in March 2018 with\ntwo adherents of the RCS [cf. [Rpt2018], [Dsc2018]] is far from sufficient. This is\nsomething that I emphasized, both orally during these discussions and in e-mails\nto these two adherents of the RCS during the summer of 2018 subsequent to these\ndiscussions.\n§1.5. The historical significance of detailed, explicit, accessible records\nAs was discussed in §1.3, I continue to search for a professional mathematician\n[say, in the field of arithmetic geometry] who purports to understand the math-\nematical justification for the RCS-redundancy asserted in the logical corner-\nstone assertion (RC-Θ) — i.e., in particular, who has confronted the mathematical\ncontent of the fundamental “(RC-Θ) vs. ∧-property” conflict mentioned in\n§1.3 — and who is prepared to discuss this mathematical content with me or other\nmathematicians with whom I am in contact. Of course,\na detailed, explicit, mathematically substantive, and readily ac-\ncessible written exposition\n— i.e., as an alternative to direct mathematical discussions [via e-mail, online video\ndiscussions, or face-to-face meetings] — of the mathematical justification for the log-\nical cornerstone assertion (RC-Θ) would also be quite welcome [cf. the discussion\nof [Rpt2014], (7)]. Moreover, in this context, it should be emphasized that such a\ndetailed, explicit, mathematically substantive, and readily accessible written expo-\nsition would be of great value not only for professional mathematicians and graduate\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 13\nstudents who are involved with inter-universal Teichmüller theory at the present\ntime, but also for scholars in the [perhaps distant!] future.\nIn general, it cannot be overemphasized that maintaining such detailed, ex-\nplicit, mathematically substantive, and readily accessible written records is\nof fundamental importance to the development of mathematics.\nIndeed, as was discussed in the final portion of [Rpt2018], §3, from a historical\npoint of view, it is only by maintaining such written records that the field of math-\nematics can avoid the sort of well-known and well-documented confusion that\nlasted for so many centuries concerning “Fermat’s Last Theorem”. Moreover, it\nis fascinating to re-examine, from the point of view of a modern observer, the in-\ntense debates that occurred, during the time of Galileo, concerning the theory of\nheliocentrism or, during the time of Einstein, concerning the theory of relativity.\nPerhaps a more pertinent example [as was pointed out to the author by Fu-\nmiharu Kato], relative to the issue of “redundant copies”, may be found in\nthe well-known dispute — i.e., concerning “algebraic truths” [which corresponds\nto Weierstrass’ approach] versus “geometric fantasies” [which corresponds to Rie-\nmann’s approach] — in the 19-th century between Weierstrass and Riemann con-\ncerning approaches to complex function theory [cf. [Btt]]. That is to say, the\ncriticism by Weierstrass of the “geometric fantasies” approach due to Riemann via\nanalytic continuation on Riemann surfaces, which are obtained by gluing to-\ngether — what are, perhaps to some observers, seemingly “redundant”! — copies\nof open subsets of the complex plane [cf. the discussion of the well-known gluing\nconstruction of the projective line in Example 2.4.7 below; the discussion of ana-\nlytic continuation of the complex logarithm in the discussion surrounding (FxEuc)\nin §3.1 below, as well as in Example 3.10.1 below; the illustrations of [AnCnCv],\n[AnCnLg]!], exhibits remarkable parallels in spirit to the assertions of the RCS.\nHere, it is also of interest to note [as was pointed out to the author by Ben-\njamin Collas] that the issue of analytic continuation of the complex logarithm is\nvery closely related to the long and heated dispute between Leibniz and [Johann]\nBernoulli concerning the appropriate definition of logarithms of negative and more\ngeneral complex numbers — a dispute that was ultimately, in some sense, resolved\nby Euler’s formula and the acceptance of the multi-valued nature of the complex\nlogarithm [cf. the discussion of “Euler’s Formula and Its Consequences” in [AnHst],\nChapter I, §I.5, although it must be kept in mind that in Euler’s time, complex\nfunction theory, i.e., of the sort necessary to treat the complex logarithm as a\nholomorphic function, had not yet been developed]. This multi-valued nature\nof the complex logarithm is closely related not only to the theory of analytic\ncontinuation of the complex logarithm, but also to the theory of coverings [in the\nsense of the classical topological fundamental group], where the indeterminacy in\nvalues may be understood as an indeterminacy in the choice of basepoint, that is\nto say, as a sort of distant — though nonetheless quite direct! — ancestor of\nthe inter-universal indeterminacies that appear in inter-universal Teichmüller\ntheory [cf. the discussion of §3.8, especially Example 3.8.1, below!].\nThe central role occupied by the notion of analytic continuation in these his-\ntorical disputes of Weierstrass\/Riemann and Leibniz\/Bernoulli is also of interest, in\nthe context of inter-universal Teichmüller theory, in light of the central role played,\n14 SHINICHI MOCHIZUKI\nin the history of analytic continuation, by the functional equation [i.e., Jacobi\nidentity] for the theta function, which exhibits numerous remarkable structural\nsimilarities to inter-universal Teichmüller theory [cf. the discussion of Example\n3.3.2 below], and which is also closely related to Riemann’s famous research con-\ncerning the theory of the functional equation\/analytic continuation of theRiemann\nzeta function. Research in the 19-th century concerning analytic continuation\nmay also be regarded as a sort of early precursor of more modern notions such\nas monodromy representations and connections\/crystals — an observation\nwhich is of interest in light of the analogy between connections\/crystals and the\nnotion of multiradiality in inter-universal Teichmüller theory [cf. the discussion of\n[Alien], §3.1, (v), as well as the discussion of §3.5, §3.10 below].\nOf course, from the point of view of a modern observer who is well-versed in\naxiomatic set theory and the theory of [Grothendieck] universes, these bitter\nhistorical disputes of Weierstrass\/Riemann and Leibniz\/Bernoulli seem somewhat\nquaint or even “silly”. One should not, however, in this context, overlook the\nimportance of such bitter historical disputes in motivating the development of such\nmodern tools as axiomatic set theory and the theory of [Grothendieck] universes,\nwhich underlie this privileged viewpoint of the modern observer. Conversely, when\nviewed from this historical perspective, the assertions of the RCS seem all the more\nlike a sort of bizarre anachronism, which has no place in the 21-st century [cf.,\nespecially, the entirely elementary nature of the various examples — such as the\ngluing construction of the projective line discussed in Example 2.4.7 — that appear\nin §2 below]! Moreover, in the case of the Leibniz\/Bernoulli dispute, it is of interest\nto note — especially, in the context of the highly sensationalist nature of the\ncoverage of inter-universal Teichmüller theory in the English-language mass-media\nand internet [cf., e.g., the discussion of §1.8, §1.10, §1.12 below] — that apparently,\nat the time of the dispute, the dispute was kept as secret as possible in order to\navoid “damaging the prestige of pure mathematics as an exact and rigorous science”\n[cf. the discussion of “Euler’s Formula and Its Consequences” in [AnHst], Chapter\nI, §I.5]. At any rate, in the context of this sort of historical discussion, it cannot\nbe overemphasized that\nall of these historical re-examinations [i.e., of the sort that underlie the\ndiscussion of the last few paragraphs!] are technically possible precisely\nbecause of the existence of detailed, explicit, mathematically sub-\nstantive, and readily accessible written expositions of the logical\nstructure underlying the various central assertions that arose in the de-\nbate.\nIn this context, it should be noted that, from a historical point of view, one\npattern that typically underlies the formidable deadlocks that tend to occur\nin such debates is the point of view, on the part of parties opposed to a newly\ndeveloped theory, that\n(CmSn) it is a “matter of course” or “common sense” — i.e., in the lan-\nguage of the above discussion, a matter that is so profoundly self-evident\nthat any “decent, reasonable observer” would undoubtedly find detailed,\nexplicit, mathematically substantive, and readily accessible written expo-\nsitions of its logical structure to be entirely unnecessary — that the\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 15\nissues under consideration can be completely resolved within some exist-\ning, familiar framework of thought without the introduction of the newly\ndeveloped theory, which is regarded as deeply disturbing and unlikely to be\nof use in any substantive mathematical sense.\nIn fact, however,\n(OvDlk) ultimately, the only meaningful technical tool that humanity can apply\nto develop the cultural infrastructure necessary to overcome such\ndeadlocks is precisely the production of detailed, explicit, mathematically\nsubstantive, and readily accessible written expositions of the logical\nstructure underlying points of view that are regarded by certain parties\nas being a “matter of common sense”\n[cf. the discussion of [EMSCOP] in §1.3; the discussion of (UndIg) in §1.3; the\ndiscussion of §1.10 below; §2.1 below; the discussion of (FxRng), (FxEuc), (FxFld)\nin §3.1 below].\n§1.6. The importance of further dissemination\nOne fundamental and frequently discussed theme in the further development\nof inter-universal Teichmüller theory is the issue of increasing the number of profes-\nsional mathematicians who have a solid, technically accurate understanding\nof the details of inter-universal Teichmüller theory. Indeed, this issue is in some\nsense the central topic of [Rpt2013], [Rpt2014]. As discussed in §1.4, in order to\nachieve such a solid, technically accurate understanding of the theory, it is neces-\nsary to devote a substantial amount of time and effort over a period of roughly\nhalf a year to two or three years, depending on various factors. It also requires the\nparticipation of professional mathematicians or graduate students who are\n· sufficiently familiar with numerous more classical theories in arithmetic\ngeometry [cf. the discussion of [Alien], §4.1, (ii); [Alien], §4.4, (ii)],\n· sufficiently well motivated and enthusiastic about studying inter-universal\nTeichmüller theory, and\n· sufficiently mathematically talented, and who have a\n· sufficient amount of time to devote to studying the theory.\nAs a result of quite substantial dissemination efforts not only on my part, but\nalso on the part of many other mathematicians, the number of professional mathe-\nmaticians who have achieved a sufficiently detailed understanding of inter-universal\nTeichmüller theory to make independent, well informed, definitive statements con-\ncerning the theory that may be confirmed by existing experts on the theory [cf. also\nthe discussion of §1.7 below] is roughly on the order of 10. It is worth noting that\nalthough this collection of mathematicians is centered around RIMS, Kyoto Uni-\nversity, it includes mathematicians of many nationalities and of age ranging from\naround 30 to around 60. One recent example demonstrated quite dramatically that\nit is quite possible to achieve a solid mathematical understanding of inter-universal\nTeichmüller theory as a graduate student by studying on one’s own, outside of\nJapan, and with essentially zero contact with RIMS, except for a very brief pe-\nriod of a few months at the final stage of the student’s study of inter-universal\nTeichmüller theory.\n16 SHINICHI MOCHIZUKI\nFinally, we observe, in the context of the discussion [cf. §1.3, §1.4, §1.5] of the\nassertions of the RCS, that another point that should be emphasized is that it is\nalso of fundamental importance to\nincrease the number of professional mathematicians [say, in the field of\narithmetic geometry] who have a solid technical understanding of the\nmathematical content of the assertions of the RCS, and who are pre-\npared to discuss this mathematical content with members of the “IUT\ncommunity”\n[i.e., with mathematicians who are substantially involved in mathematical research\nand\/or dissemination activities concerning inter-universal Teichmüller theory]. Here,\nwe note in passing that such a solid technical understanding of the mathematical\ncontent of the assertions of the RCS is by no means “equivalent” to expressions\nof support for the RCS on the basis of non-mathematical — i.e., such as social,\npolitical, or psychological — reasons. In this context, it should also be emphasized\nand understood [cf. the discussion of [Rpt2014], (7)] that both\n· producing detailed, explicit, mathematically substantive, and readily ac-\ncessible written expositions of the mathematical justification of assertions\nof the RCS [such as (RC-Θ)!], i.e., as discussed in §1.5, and\n· increasing the number of professional mathematicians [say, in the field\nof arithmetic geometry] who have a solid technical understanding of the\nmathematical content of the assertions of the RCS, and who are prepared\nto discuss this mathematical content with members of the IUT community\nare in the interest not only of the IUT community, but of the RCS as well. More-\nover,\nthe process of attaining a solid, technically accurate understanding of the\nprecise logical relationship between RCS-IUT and IUT, i.e., as ex-\nposed, for instance, in the present paper, can serve as a valuable peda-\ngogical tool\n[cf. the discussion of [Rpt2018], §17] for mathematicians currently in the process\nof studying inter-universal Teichmüller theory.\n§1.7. The notion of an “expert”\nOne topic that sometimes arises in the context of discussions of dissemination\nof inter-universal Teichmüller theory [i.e., as in §1.6], is the following issue:\nWhat is the definition of, or criterion for, being an “expert” on inter-\nuniversal Teichmüller theory?\nIn a word, it is very difficult to give a brief, definitive answer, e.g., in the form of a\nstraightforward, easily applicable criterion, to this question. On the other hand, in\nthis context, it should also be pointed out that the difficulties that arise in the case of\ninter-universal Teichmüller theory are, in fact, not so qualitatively different from the\ndifficulties that arise in answering the analogous question for mathematical theories\nother than inter-universal Teichmüller theory. These difficulties arise throughout\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 17\nthe daily life of professional mathematicians in numerous contexts, such as the\nfollowing:\n(Ev1) preparing suitable exercises or examination problems to educate and\nevaluate students,\n(Ev2) evaluating junior mathematicians,\n(Ev3) refereeing\/evaluating mathematical papers for journals.\nFrom my point of view, as the author of [IUTchI-IV], one fundamental criterion\nthat I always keep in mind — not only the in case of [IUTchI-IV], but also in the\ncase of other papers that I have written, as well as when I am involved in the various\ntypes of evaluation procedures (Ev1) ∼ (Ev3) discussed above — is the issue of\nthe extent to which the level of understanding of the mathematician in\nquestion enables the mathematician to “stand on his\/her own two feet”\nwith regard to various assertions concerning the theory, on the basis of\nindependent, logical reasoning, without needing to be “propped up” or cor-\nrected by me or other known experts in the theory.\nI often refer to this criterion as the criterion of autonomy of understanding.\nOf course, from a strictly rigorous point of view, this criterion is, in some sense,\nnot so “well-defined” and, in many contexts, difficult to apply in a straightforward\nfashion. On the other hand, in the past, various mathematicians involved with\ninter-universal Teichmüller theory have demonstrated such an autonomous level of\nunderstanding in the following ways:\n(Atm1) the ability to detect various minor errors\/oversights in [IUTchI-III];\n(Atm2) the ability to propose new, insightful ways of thinking about various\naspects of inter-universal Teichmüller theory;\n(Atm3) the ability to propose ways of modifying inter-universal Teichmüller\ntheory so as to yield stronger or more efficient versions of the theory;\n(Atm4) the ability to produce technically accurate oral or written expositions of\ninter-universal Teichmüller theory;\n(Atm5) the ability to supervise or direct new mathematicians — i.e., by train-\ning\/educating professional mathematicians or graduate students with re-\ngard to inter-universal Teichmüller theory — who, in due time, demon-\nstrate various of the four types of ability (Atm1) ∼ (Atm4) discussed\nabove.\nOf course, just as in the case of other mathematical theories, different experts\ndemonstrate their expertise in different ways. That is to say, experts in inter-\nuniversal Teichmüller theory often demonstrate their expertise with respect to some\nof these five types of ability (Atm1) ∼ (Atm5), but not others.\nIn this context, it should be pointed out that one aspect of inter-universal\nTeichmüller theory that is currently still under development is the analogue for\ninter-universal Teichmüller theory of (Ev1), i.e., preparing suitable exercises for\nmathematicians currently in the process of studying inter-universal Teichmüller\ntheory. This point of view may be seen in the discussion in the final portion of the\nIntroduction to [Alien], as well as in the discussion of “valuable pedagogical tools”\n18 SHINICHI MOCHIZUKI\nin [Rpt2018], §17. Indeed, many of the technical issues discussed in [Rpt2018],\n§15 [or, alternatively, Example 3.2.2 of the present paper], may easily be reformu-\nlated as “exercises” or, alternatively, as “examination problems” for evaluating the\nlevel of understanding of mathematicians in the process of studying inter-universal\nTeichmüller theory.\n§1.8. Fabricated versions spawn fabricated dramas\nAs discussed in §1.6, §1.7, by now there is a substantial number of mathe-\nmaticians who have attained a thorough, accurate, and automous understanding of\ninter-universal Teichmüller theory. In each of the cases of such mathematicians that\nI have observed thus far, such an understanding of the theory was achieved essen-\ntially by means of a thorough study of the original papers [IUTchI-IV], followed by\na period of constructive discussions with one or more existing experts that typically\nlasted roughly from two to six months to sort out and resolve various “bugs” in the\nmathematician’s understanding of the theory that arose when the mathematician\nstudied the original papers on his\/her own [cf. the discussion of §1.4].\nOn the other hand, there is also a growing collection of mathematicians who\nhave a somewhat inaccurate and incomplete — and indeed often quite superficial\n— understanding of certain aspects of the theory. This in and of itself is not\nproblematic — that is to say, so long as the mathematician in question maintains\nan appropriate level of self-awareness of the inaccurate and incomplete nature of\nhis\/her level of understanding of the theory — and indeed is a phenomenon that\noften occurs as abstract mathematical theories are disseminated.\nUnfortunately, however, a certain portion of this collection of mathematicians\n[i.e., whose understanding of the theory is inaccurate and incomplete] have exhibited\na tendency to\nassert\/justify the validity of their inaccurate and incomplete understand-\ning of the theory by means of “reformulations” or “simplifications”\nof the theory, which are in fact substantively different from and have no\ndirectly logical relationship to [e.g., are by no means “equivalent” to!] the\noriginal theory.\nIndeed, the version, referred to in the present paper as “RCS-IUT” [cf. §1.2],\nthat arises from implementing the assertions of the RCS appears to be the most\nfamous of these fabricated versions of inter-universal Teichmüller theory [cf. also the\ndiscussion of Example 2.4.5 below for a more detailed discussion of various closely\nrelated variants of RCS-IUT]. On the other hand, other, less famous fabricated\nversions of inter-universal Teichmüller theory have also come to my attention in\nrecent years.\nHere, before proceeding, we note that, in general, reformulations or simplifica-\ntions of a mathematical theory are not necessarily problematic, i.e., so long as they\nare indeed based on a thorough and accurate understanding of the original theory\nand, moreover, can be shown to have a direct logical relationship to the original\ntheory.\nThe authoring of fabricated versions of inter-universal Teichmüller theory ap-\npears to be motivated, to a substantial extent, by a deep desire to recast inter-\nuniversal Teichmüller theory in a “simplified” form that is much closer to the sort\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 19\nof mathematics with which the author of the fabricated version is already famil-\niar\/feels comfortable. On the other hand, this phenomenon of producing fabricated\nversions also appears to have been\nsubstantially fueled by numerous grotesquely distorted mass media reports\nand comments on the English-language internet that blithely paint inter-\nuniversal Teichmüller theory as a sort of cult religion, fanatical political\nmovement, mystical philosophy, or vague sketch\/proposal for a mathemat-\nical theory.\nMoreover, another unfortunate tendency, of which RCS-IUT is perhaps the most\negregious example, is for fabricated versions of inter-universal Teichmüller theory\nto\nspawn lurid social\/political dramas revolving around the content of\nthe fabricated version, which in fact have essentially nothing to do with\nthe content of inter-universal Teichmüller theory.\nSuch lurid dramas then spawn further grotesquely distorted mass media reports\nand comments on the English-language internet, which then reinforce and enhance\nthe social\/political status of the fabricated version. Here, it should be emphasized\nthat such vicious spirals have little [or nothing] to do with substantive mathe-\nmatical content and indeed serve only to mass-produce unnecessary confusion\nthat is entirely counterproductive, from the point of the view of charting a sound,\nsustainable course in the future development of the field of mathematics [cf. the\ndiscussion of [Alien], §4.4, (iv)].\nIn fact, of course, inter-universal Teichmüller theory is neither a religion, nor\na political movement, nor a mystical philosophy, nor a vague sketch\/proposal for a\nmathematical theory. Rather, it should be emphasized that\ninter-universal Teichmüller theory is a rigorously formulated mathematical\ntheory that has been verified countless times by quite a number of mathe-\nmaticians, has undergone an exceptionally thorough seven and a half year\nlong refereeing process, and was subsequently published in a leading inter-\nnational journal in the field of mathematics.\n[cf. the discussion of §1.1]. In particular, in order to avoid the sort of vicious spirals\nreferred to above, it is of the utmost importance\nto concentrate, in discussions of inter-universal Teichmüller theory, on\nsubstantive mathematical content, as opposed to non-mathematical\n— such as social, political, or psychological — aspects or interpretations\nof the situation.\nAs discussed in §1.2, this is the main reason for the use of the term “RCS” in the\npresent paper.\n§1.9. Geographical vs. mathematical proximity\nHistorically, mathematical interaction between professional mathematicians re-\nlied on physical meetings or the exchange of hardcopy documents. Increasingly,\n20 SHINICHI MOCHIZUKI\nhowever, advances in information technology have made it possible for mathemati-\ncal interaction between professional mathematicians to be conducted electronically,\nby means of e-mail or online video communication. Of course, this does not imply\nthat physical meetings or the exchange of hardcopy documents — especially in sit-\nuations where physical meetings or the exchange of hardcopy documents do indeed\nfunction in a meaningful way, from the point of view of those involved — should\nnecessarily be eschewed.\nOn the other hand, physical meetings between participants who live in dis-\ntant regions requires travel. Moreover, travel, depending on the situations of the\nparticipants, can be a highly taxing enterprise. Indeed, travel, as well as lodging\naccommodations, typically requires the expenditure of a quite substantial amount\nof money, as well as physical and mental effort on the part of those involved.\nThis effort can easily climb to unmanageable [i.e., from the point of view of certain\nof the participants] proportions, especially when substantial cultural — i.e., either in\nmathematical or in non-mathematical culture, or in both — differences are involved.\nThe current situation involving the COVID-19 pandemic adds yet another dimen-\nsion to the reckoning, from the point of view of the participants, of the physical\nand mental effort that must be expended in order to travel. As a result,\nwhen, from the point of view of at least one of the key participants, the\namount of effort, time, and\/or money that must be expended to travel\nclearly exceeds, by a substantial margin — i.e., “≫” — the gain [i.e.,\nrelative to various mathematical or non-mathematical criteria of the key\nparticipant in question] that appears likely to be obtained from the travel\nunder consideration, it is highly probable that the travel under considera-\ntion will end up simply not taking place.\nOne “classical” example of this phenomenon “≫” is the relative scarcity of profes-\nsional mathematicians in Europe or North America who travel to Japan frequently\n[e.g., at least once a year] or for substantial periods of time.\nI have, at various times in my career, been somewhat surprised by assertions\non the part of some mathematicians to the effect that travel should somehow be\nforced on mathematicians, i.e., to the effect that some sort of coercion may somehow\n“override” the fundamental inequality “≫” that exists as a result of the circum-\nstances in which a mathematician finds him\/herself in. In my experience, although\nthis sort of coercion to travel may result in some sort of superficial influence in\nthe very short term, it can never succeed in the long term. That is to say, the\nfundamental circumstances that give rise to the fundamental inequality “≫” can\nnever be altered by means of such coercive measures to travel [cf. the discussion of\n[Rpt2014], (8)].\nIn this context, I was most impressed by the following two concrete examples,\nwhich came to my attention recently. In describing these examples, I have often\nused somewhat indirect expressions, in order to protect the privacy of the people\ninvolved.\nExample 1.9.1: The insufficiency of geographical proximity. This exam-\nple concerns the results obtained in a paper written in the fall of 2019 by a graduate\nstudent (St1) from country (Ct1). This student (St1) showed his paper to a promi-\nnent senior researcher (Pf1) at a university in country (Ct2) in a certain area of\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 21\nnumber theory. The education and career of this researcher (Pf1) was conducted\nentirely at universities in countries (Ct2), (Ct3), and (Ct4). This researcher (Pf1)\ninformed (St1) of his very positive evaluation of the originality of the results ob-\ntained in the paper by (St1). Another prominent senior researcher (Pf2) in a certain\narea of number theory was informed by (Pf1) of the paper by (St1). This researcher\n(Pf2), who works at a university in country (Ct2) in close physical proximity to\n(Pf1), also took a generally positive position with regard to the paper by (St1).\nOn the other hand, several months subsequent to this interaction between (Pf1)\nand (St1), a junior researcher (Pf3), who is originally from country (Ct5), but cur-\nrently works at a university in country (Ct2) in close physical proximity to (Pf1)\nand (Pf2), informed student (St1) [via e-mail contacts between (Pf3) and (St1)’s\nadvisor] that\nthe results of the paper by (St1) are in fact “well-known” and essen-\ntially contained in papers published in the 1990’s by (Pf4), a\nprominent senior researcher in country (Ct6).\n[To be more precise, in fact the results of the paper by (St1) are not entirely\ncontained in the papers by (Pf4) in the sense that the paper by (St1) contains\ncertain numerically explicit estimates that are not contained in the papers by (Pf4).]\nCountry (Ct6) is in close physical proximity to country (Ct3), and in fact, one of\nthe research advisors of researcher (Pf4), when (Pf4) was a graduate student, was a\nprominent researcher (Pf5) who is originally from country (Ct7), but has pursued\nhis career as a mathematician mainly in countries (Ct3) and (Ct6). Here, it should\nbe pointed out that (Pf1), (Pf2), and (Pf4) are very close in age, and that (Pf1)\nreceived his undergraduate education in country (Ct3) at one of the universities\nthat played in prominent role in the career of (Pf5). The paper by student (St1)\nconcerns mathematics that has been studied extensively by — and indeed forms\none of the central themes of the research of — both (Pf1) and (Pf4), but from very\ndifferent points of view, using very different techniques, since (Pf1) and (Pf4) work\nin substantially different areas of number theory. On the other hand, at no time\nduring the initial several months of interaction between (Pf1), (Pf2), and (St1)\nwas the work of (Pf4) mentioned. That is to say, (Pf1) and (Pf2) discussed the\nresults obtained in the paper by (St1) in a way that can only be explained by the\nhypothesis that\n(Pf1) and (Pf2) were, at the time, entirely unaware of the very close\nrelationship between the results obtained in the paper by (St1) and the\npapers in the 1990’s by (Pf4).\n— i.e., despite the numerous opportunities afforded by close physical prox-\nimity, as well as proximity of age, for substantial interaction between (Pf1) and\n(Pf4). The paper by student (St1) is currently submitted for publication to a cer-\ntain mathematical journal. Student (St1) recently received a referee’s report for\nhis paper, which apparently [i.e., judging from the comments made in the referee’s\nreport] was written by a mathematician working in an area of number theory close\nto (Pf1). This referee’s report also makes no mention of the papers in the 1990’s\nby (Pf4) and the fact that the results obtained in the paper by (St1) appear, with\nthe exception of certain numerically explicit estimates, to be essentially contained\nin these papers of (Pf4). Finally, it should be mentioned that each official language\nof each of these countries (Ct1), (Ct2), (Ct3), (Ct4), (Ct5), (Ct6), (Ct7) belongs to\n22 SHINICHI MOCHIZUKI\nthe European branch of the Indo-European family of languages, and that at least\nsix of the ten pairs of countries in the list (Ct2), (Ct3), (Ct4), (Ct6), (Ct7) share a\ncommon official language [i.e., with the other country in the pair].\nExample 1.9.2: The remarkable potency of mathematical proximity. This\nexample concerns the study of inter-universal Teichmüller theory by a graduate stu-\ndent (St2), who is originally from country (Ct8), but was enrolled in the doctoral\nprogram in mathematics at a university in country (Ct9) under the supervision of a\nsenior faculty member (Pf6), who is originally from country (Ct10). This graduate\nstudent (St2) began his study of inter-universal Teichmüller theory as a graduate\nstudent and continued his study during his years as a graduate student with es-\nsentially no mathematical contact with any researchers who are significantly\ninvolved with inter-universal Teichmüller theory, except for his advisor (Pf6) and\none mid-career researcher (Pf7) from country (Ct11). Here, we remark that the\nofficial language of each of these countries (Ct8), (Ct9), (Ct10), (Ct11) belongs\nto the European branch of the Indo-European family of languages. In particular,\nwith the exception of a few very brief e-mail exchanges with me and a brief two-\nweek long stay at RIMS in 2016 to participate in a workshop on IUT, this student\n(St2) had essentially no mathematical contact, prior to the fall of 2019, with\nany researchers at Kyoto University who are involved with inter-universal\nTeichmüller theory. Even in these circumstances,\nthis student was able not only to achieve a very technically sound un-\nderstanding of inter-universal Teichmüller theory on his own, by reading\n[IUTchI-IV] and making use of various resources, activities, and contacts\nwithin country (Ct9), but also to succeed, as a graduate student, in making\nhighly nontrivial original research contributions to a certain mild\ngeneralization of inter-universal Teichmüller theory, as well as to certain\nrelated aspects of anabelian geometry.\nMy first [i.e., with the exception of a few very brief e-mail exchanges prior to this]\nmathematical contact with this student (St2) was in the fall of 2019. Although this\nstudent (St2) initially had some technical questions concerning aspects of inter-\nuniversal Teichmüller theory that he was unable to understand on his own, after a\nfew relatively brief discussions in person with me, he was able to find answers to\nthese technical questions in a relatively short period of time [roughly a month or\ntwo] without much trouble.\n§1.10. Mathematical intellectual property rights\nThe socio-political dynamics generated by the proliferation of logically\nunrelated fabricated versions of inter-universal Teichmüller theory — of which\nRCS-IUT is perhaps the most frequently cited [cf. the discussion of §1.2, as well as\nExamples 2.4.5, 2.4.7 below] — and further fueled by\n· grotesquely distorted mass media coverage and internet comments [cf.\nthe discussion of §1.8], as well as by\n· the conspicuous absence of detailed, explicit, mathematically substantive,\nand readily accessible written expositions of the logical structure underly-\ning the various central assertions of the proponents of such socio-political\ndynamics [cf. the discussion of §1.5],\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 23\nhave had the effect of deeply disrupting the normal process of absorption of\ninter-universal Teichmüller theory by the worldwide mathematical community. Left\nunchecked, this state of affairs threatens to pave the way for a field — i.e., the field\nof mathematics — governed by socio-political dynamics [cf. the discussion\nof (CmSn) in §1.5], rather than by mathematical content.\nFrom a historical point of view, various forms of institutional and concep-\ntual infrastructure — such as the notions of\n· a modern judiciary system;\n· universal, inalienable human rights;\n· the rule of law;\n· due process of law; and\n· burden of proof\n— were gradually developed, precisely with the goal of averting the outbreak of the\nsort of socio-political dynamics that were viewed as detrimental to society. In\nthis context, it is interesting to note the central role played, for instance, in courts\nof law, by the practice of producing\ndetailed, explicit, logically substantive, and readily accessible written doc-\numentation of the logical structure underlying the various central as-\nsertions of the parties involved.\nThis situation is very much reminiscent of the situation in mathematics discussed in\n§1.5 [cf. also the discussion of RCS-IUT in §1.3!], i.e., where we observe that it is not\neven possible to analyze or debate, in any sort of meaningfully definitive way,\nmathematical assertions — such as, for instance, the historically famous assertion\nof Fermat to the effect that he had a proof of “Fermat’s Last Theorem”, but did not\nwrite it down — in the absence of such written documentation of the logical\nstructure of the issues under consideration.\nFrom the point of view of the above discussion, it seems natural, in the case\nof mathematics, to introduce, especially in the context of issues such as the one\ndiscussed above involving logically unrelated fabricated versions of inter-universal\nTeichmüller theory, the notion of mathematical intellectual property rights\n[i.e., “MIPRs”]. As the name suggests, this notion is, in some sense, modeled on\nthe conventional notion of intellectual property rights associated, for instance, with\ntrademarks or brand names of corporations. In the case of this conventional notion,\nintellectual property rights may be understood as a tool for protecting the “reli-\nability” or “creditworthiness” of trademarks or brand names of a corporation\nfrom the sort of severe injury to such trademarks or brand names that may ensue\nfrom the proliferation of shoddy third-party imitations of products produced\nby the corporation. Here, we observe that this “severe injury” often revolves around\nthe creation of severe obstacles to the execution of activities that play a central role\nin the operational normalcy of the corporation.\nUnlike this conventional notion, MIPRs should be understood as being as-\nsociated — not to corporations or individuals for some finite period of time, but\nrather — to mathematical notions and theories and, moreover, are of unlim-\nited duration. The purpose of MIPRs may be understood as the protection of the\n“creditworthiness” of such a mathematical notion or theory from the severe\n24 SHINICHI MOCHIZUKI\ninjury to the operational normalcy of mathematical progress related to no-\ntion\/theory that ensues from the proliferation of logically unrelated fabricated\n“fake” versions of the notion\/theory.\nBefore proceeding, we pause to consider one relatively elementary example of\nthis notion of MIPRs.\nExample 1.10.1: The Pythagorean Theorem.\n(i) Recall the Pythagorean Theorem concerning the length of the hypotenuse\nof a right triangle in the Euclidean plane. Thus, if 0 < x ≤ y < z ∈ R are the\nlengths of the sides of a right triangle in the Euclidean plane, then the Pythagorean\nTheorem states that\nx2 + y2 = z2.\nVarious versions of this result apparently may be found not only in the writings\nof ancient Greece and Rome, but also in Babylonian, ancient Indian, and ancient\nChinese documents. Well-known “elementary proofs” of this result may be ob-\ntained, for instance, by computing, in various equivalent ways, the area of suitable\nplanar regions covered by right triangles or squares that are closely related to the\ngiven right triangle. On the other hand, such “elementary proofs” typically do not\naddress the fundamental issue of how to define such notions as length, angle, and\nrotation, i.e., which are necessary in order to understand the precise content of the\nstatement of the Pythagorean Theorem. Here, we observe that if, for instance, one\ntries to define the notion of the length of a line segment in Euclidean space in the\nconventional way, then the Pythagorean Theorem reduces, in effect, to a meaning-\nless tautology! Moreover, although the notions of length and angle may be defined\nonce one has defined the notion of a rotation, it is by no means clear how to give\na natural definition of the notion of a rotation. For instance, one may attempt\nto define the notion of a rotation of Euclidean space as an element of the group\ngenerated by well-known matrices involving sines and cosines, but it is by no means\nclear that such a definition is “natural” or the “right definition” in some meaningful\nsense. Thus, in summary,\nit is by no means clear that such “elementary proofs” may be regarded as\ngenuine rigorous proofs in the sense of modern mathematics.\n(ii) From a modern point of view, a natural, precise definition of the funda-\nmental notion of a rotation of Euclidean space [from which, as observed in (i),\nnatural definitions of the notions of the notions of length and angle may be easily\nderived] may be given by thinking in terms of invariant tensor forms associ-\nated to compact subgroups of [the topological groups determined by] various\ngeneral linear groups. From this modern point of view, the precise form of the\nPythagorean Theorem [i.e., “x2 + y2 = z2”] — and, in particular, the significance\nof the “2” in the exponent! — may be traced back to the theory of Brauer groups\nand the closely related local class field theory of the archimedean field “R” of\nreal numbers, i.e., in short, to various fundamental properties of the arithmetic of\nthe topological field “R”.\n(iii) Considering the situation discussed in (i), (ii), it is by no means clear, in\nany sort of a priori or naive sense, just why the Pythagorean Theorem should take\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 25\nthe precise form “x2+y2 = z2”. Indeed, since this precise form of the Pythagorean\nTheorem continues to appear utterly mysterious even to numerous modern-day\nhigh school students — i.e., who grow up immersed in an environment replete with\ncountless cultural links to modern mathematics, science, and technology! — it\nseems reasonable to assume that it should have appeared all the more mysterious\nto the individuals who populated the various ancient civilizations mentioned in (i).\nIn particular, it is by no means unnatural to consider the possibility that assertions\nsimilar to the following assertions [stated relative to the notation introduced in (i)]\nmight have been made by some hypothetical individual at some time in human\nhistory:\n(Pyth1) “I don’t understand why the relation in the Pythagorean Theorem is of\nthe form ‘x2 + y2 = z2’, rather than ‘x2 · y2 = z2’.”\n(Pyth2) “I would like to investigate, in the context of the Pythagorean Theorem,\nwhether or not the relation ‘x2 · y2 = z2’ holds.”\n(Pyth3) “I investigated, in the context of the Pythagorean Theorem, whether\nor not the relation ‘x2 · y2 = z2’ holds and discovered that there exist\nexamples that show that this relation does not in fact hold in general.”\n(Pyth4) “The Pythagorean Theorem is false for the following reason: The\nPythagorean Theorem states that ‘x2 · y2 = z2’, but there exist coun-\nterexamples that show that this relation does not hold in general.”\n(iv) From the point of view of the discussion given above of MIPRs, the asser-\ntions (Pyth1), (Pyth2), (Pyth3) do not constitute a violation of the MIPRs of the\nPythagorean Theorem, but rather are precisely the sorts of assertions\/comments\nthat occur naturally in normal, sound research and educational activities in math-\nematics. By contrast,\n(VioMIPR) (Pyth4) may be regarded as a classical example of a violation of the\nMIPRs of the Pythagorean Theorem.\nIt is not difficult to imagine\n(DtrVio) the deeply detrimental effects on the development of mathematics\nthroughout history that would have occurred if violations of MIPRs\nsimilar to (Pyth4) regarding the Pythagorean Theorem arose and were\nleft unchecked.\nMoreover, in this context, it is important to observe that\n(IgNoJst) assertions of ignorance of the technical details that one must un-\nderstand in order to distinguish the modified version of the Pythagorean\nTheorem given in (Pyth4) from the original version of the Pythagorean\nTheorem do not by any means constitute a justification for participating\nin the proliferation\/citation\/dissemination of (Pyth4).\nThat is to say,\n(BurPrf) the burden of proof of establishing any sort of logical relationship\nbetween such a modified version and the original version lies exclusively\nin the hands of the proponents of the modified version\n26 SHINICHI MOCHIZUKI\n[cf. the discussion immediately following the present Example 1.10.1]. Indeed, this\nnotion of burden of proof (BurPrf) constitutes a fundamental pillar underlying\nthe notion of MIPRs and may be readily understood by considering the corre-\nsponding [perhaps more familiar] situation surrounding the conventional notion of\nintellectual property rights as it is typically applied to technological devices such as\ncomputers: that is to say, an assertion of technical ignorance concerning the details\nof the internal technical structure of a computer product manufactured by company\nA [or “A-product” for short] and a computer product manufactured by company X\n[or “X-product” for short] does not by any means [i.e., legal, ethical, or otherwise!]\njustify the sale, by a [say, technically ignorant] computer dealer, of an X-product\nadvertised as an authentic A-product.\n(v) From a historical point of view, it appears, in light of the discussion of (i),\n(ii), (iii), (iv), to be in some sense a sort of miracle that the Pythagorean Theorem\nwas “discovered” in and, moreover, survived throughout the duration of numer-\nous ancient civilizations, i.e., despite the fact that the ancient world was primarily\ndominated by dictatorial, authoritarian political regimes with little regard for such\nmodern notions as a judiciary system [in the modern sense], inalienable human\nrights, the rule of law, due process of law, burden of proof, and so on. Of course, to\na certain extent, this situation may be understood as a consequence of the fact that\nthe Pythagorean Theorem is closely related to the task of direct measurement of\nlengths of various easily accessed [i.e., even in the ancient world!] physical objects.\nFrom this point of view of “direct measurement”, the “Pythagorean Theorem”, as\nunderstood in various ancient civilizations, should perhaps be regarded [cf. the\ndiscussion of (i), (ii)] not so much as a result in mathematics [in the modern sense\nof the term], but rather as a principally empirically substantiated result in physics.\nNevertheless, even when viewed from this point of view, it still seems like some-\nthing of a miracle that this result survived throughout the duration of numerous\nancient civilizations, unaffected by numerous meaningless misunderstandings of the\nsort discussed in (iii), (iv), especially considering that similarly meaningless mis-\nunderstandings of the Pythagorean Theorem continue to plague modern-day high\nschool students!\nPrior to the introduction above of the notion of MIPRs, this notion does not\nappear to have played an important role, at least in any sort of explicit sense, in\ndiscussions or analyses of the development of mathematics. On the other hand,\nExample 1.10.1 [cf., especially, Example 1.10.1, (iv)!] shows how, even at a purely\nimplicit level,\nthis notion of MIPRs has in fact played a fundamentally important\nrole in the development of mathematics throughout history.\nOf course, in the case of violations of the MIPRs of a mathematical notion or\ntheory, conventional courts or judiciary systems are simply not equipped to play a\nmeaningful role in dealing with such violations, since this would require an exten-\nsive technical knowledge and understanding, on the part of the judges or lawyers\ninvolved, of the mathematics under consideration. Indeed, it is useful to recall in\nthis context that\n· traditionally, any detrimental effects arising from such violations of the\nMIPRs of a mathematical notion or theory were typically averted in the\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 27\nfield of mathematics by means of the refereeing systems of various well\nestablished mathematical journals;\n· from the point of view of such traditional refereeing systems of well\nestablished mathematical journals, the burden of proof regarding the\ncorrectness of any novel assertions concerning existing mathematical no-\ntions or theories — such as, for instance, any assertions concerning some\nsort of logical relationship between a modified version of a theory and\nthe original version of the theory— lies [not with the author of the original\nversion of the theory (!), but rather] with the author of the manuscript\ncontaining the novel assertions.\nOn the other hand, as discussed in §1.8 [cf. also the discussion of §1.3], in the\ncase of the quite egregious MIPRs violations constituted by logically unrelated\nfabricated versions of inter-universal Teichmüller theory, numerous mass media re-\nports and internet comments released by individuals who are clearly not operating\non the basis of a solid, technically accurate understanding of the mathematics in-\nvolved are regarded, in certain sectors of the mathematical community, as carrying\nmuch more weight than an exceptionally thorough refereeing process in a well\nestablished mathematical journal by experts on the mathematics under considera-\ntion. This state of affairs is deeply regrettable and should be regarded as a cause\nfor alarm. Perhaps in the long term, new forms of institutional or conceptual\ninfrastructure may be developed for averting the deeply detrimental effects of this\nsort of situation. At the time of writing, however, it appears that\nthe only meaningful technical tool currently available to humanity for deal-\ning with this sort of situation lies in the production of detailed, explicit,\nmathematically substantive, and readily accessible written expositions\nof the logical structure underlying the assertions of the various parties\ninvolved [cf. the discussion surrounding (OvDlk) in §1.5, as well as the\ndiscussion of §1.12 below],\ni.e., even when such assertions are purported to be a “matter of course” or “com-\nmon sense”, that is to say, a matter that is so profoundly self-evident that any\n“decent, reasonable observer” would undoubtedly find such written documentations\nof logical structure to be entirely unnecessary [cf. the discussion surrounding\n(CmSn) in §1.5]. From a historical point of view, such written documentations of\nlogical structure can then serve as a\nvaluable transgenerational or transcultural common core\nof scholarly activity — a point of view that is reminiscent of the logical relator\nAND “∧”, which forms a central theme of the present paper.\n§1.11. Social mirroring of mathematical logical structure\nDiscussions, on the part of some observers, concerning the situation surround-\ning inter-universal Teichmüller theory are often dominated by various mutually\nexclusive and socially divisive\/antagonistic dichotomies [cf. also the related\ndiscussion of §1.8], i.e., such as the following:\n28 SHINICHI MOCHIZUKI\n(ExcDch) Is it the case that adherents of the RCS should be regarded as math-\nematically correct\/reliable\/reasonable mathematicians, while mathemati-\ncians associated with inter-universal Teichmüller theory should not, OR\nis it the other way around?\nHere, the “OR” is typically understood as an “XOR”, i.e., exclusive-or. That is to\nsay, such questions\/dichotomies are typically understood as issues for which it can\nnever be the case that an “AND” relation between the two possible alternatives\nunder consideration holds. In discussions of mutually exclusive dichotomies such\nas (ExcDch), mathematicians associated with inter-universal Teichmüller theory,\nas well as the actual mathematical content of inter-universal Teichmüller theory,\nare often treated as completely and essentially disjoint entities, within the\ninternational mathematical community, from the mathematicians and mathematical\nresearch associated with the RCS.\nIn this context, it is interesting to note that this sort of mutually exclusive\ndichotomy is very much reminiscent of the essential logical structure of RCS-\nIUT, which, as discussed in the latter portion of Example 2.4.5 below, may be\nunderstood as being essentially logically equivalent toOR-IUT, as well as toXOR-\nIUT, i.e., to logically unrelated fabricated versions of inter-universal Teichmüller\ntheory in which the crucial logical AND “∧” relation satisfied by the Θ-link of\ninter-universal Teichmüller theory is replaced by a logical OR “∨” relation or,\nalternatively, by a logical XOR “∨̇” relation.\nIn fact, however,\n(MthCnn) although it is indeed the case that the international mathematical com-\nmunity, as well as the mathematical content of the research performed by\nthe international mathematical community, does not consist [in the lan-\nguage of classical algebraic geometry] of a single irreducible component, it\nis nevertheless undeniably connected.\nIn the case of inter-universal Teichmüller theory, this abundant inter-connectivity\nmay be explicitly witnessed in the following aspects of the theory:\n(IntCnn1) The mathematical content of various aspects of inter-universal Teichmüller\ntheory is closely related to various classical theories such as the following:\n· the invariance of heights of abelian varieties under isogeny\n[cf. the discussion of [Alien], §2.3, §2.4, as well as the discussion\nof Example 3.2.1 below; the discussion of §3.5 below];\n· the classical proof in characteristic zero of the geometric ver-\nsion of the Szpiro inequality via the Kodaira-Spencer morphism,\nphrased in terms of the theory of crystals [cf. the discussion of\n[Alien], §3.1, (v), as well as the discussion of §3.5, §3.10 below];\n· Bogomolov’s proof over the complex numbers of the geomet-\nric version of the Szpiro inequality [cf. the discussion of [Alien],\n§3.10, (vi)];\n· classical complex Teichmüller theory [cf. the discussion\nof Example 3.3.1 in §3.3 below];\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 29\n· the classical theory of the Jacobi identity for the theta func-\ntion [cf. the discussion of Example 3.3.2 in §3.3 below];\n· the classical theory of the computation of the Gaussian inte-\ngral via polar coordinates [cf. [Alien], §3.8].\nWe refer to [Alien], §4, for a more detailed discussion of such relationships\nbetween inter-universal Teichmüller theory and various classical mathe-\nmatical theories.\n(IntCnn2) In the context of the assertions of the RCS, it is important to recall [cf.\nthe discussion of (MthVl) in Example 2.4.5, (viii), below] that [perhaps\nsomewhat surprisingly!]\nin fact there is in some sense no disagreement among any of\nthe parties involved with regard to the mathematical validity\nof the central mathematical assertions of the RCS\n— i.e., so long as one deletes the arbitrary label “inter-universal\nTeichmüller theory” imposed by adherents of the RCS on the logi-\ncally unrelated fabricated versions [i.e., RCS-IUT\/OR-IUT\/XOR-IUT] of\ninter-universal Teichmüller theory that typically appear in discussions of\nadherents of the RCS [cf. the discussion of §1.2, §1.3, §1.8, §1.10].\nThe situation discussed in (InnCnn2) is of particular interest in the context of\nthe present paper since the essential logical structure of this situation discussed in\n(InnCnn2) — i.e.,\n(CmMth) of a common mathematical understanding of the mathematical va-\nlidity of the various assertions under discussion, so long as one keeps track\nof the distinct labels “inter-universal Teichmüller theory” and “RCS-\nIUT\/OR-IUT\/XOR-IUT”\n— is remarkably similar to the essential logical structure of the situation surrounding\nthe central theme of the present paper, namely, the crucial logical AND “∧”\nproperty of the Θ-link in inter-universal Teichmüller theory — cf. the discussion\nof §2.4, §3.4 below. At a more elementary level, (CmMth) may be understood as\nbeing qualitatively essentially the same phenomenon as the phenomenon discussed\nin Example 1.10.1, (iii), i.e., the distinction between (Pyth3) [where the distinct\nlabel “Pythagorean Theorem” is treated properly] and (Pyth4) [where the distinct\nlabel “Pythagorean Theorem” is not treated properly].\n§1.12. Computer verification, mathematical dialogue, and developmental\nreconstruction:\nOne question that is frequently posed, in the context of the entirely unnecessary\nconfusion that results from the plethora of misinformation and false narratives\nconcerning inter-universal Teichmüller theory in the English-language mass media\nand internet [cf. the discussion of §1.8, §1.10], is the following:\n(CmpVer) Why not use computers to verify the mathematical validity of inter-\nuniversal Teichmüller theory?\n30 SHINICHI MOCHIZUKI\nThe implication here is that “computers” may be regarded as an entity inherently\nendowed with a sort of impeccable neutrality\/impartiality with regard to the verifi-\ncation of mathematical assertions.\nThe fundamental problem with (CmpVer) lies in the essentially tautological\nobservation that\n(Algor) no computer verification algorithm for verifying some mathematical as-\nsertion can yield a verification of the validity of the algorithm itself,\ni.e., of the presumed relationship between\n· the mechanical output yielded by the algorithm and\n· the conventional human sense of “mathematical correctness”\n— i.e., a relationship on whose integrity any sort of computer verification\nmust be premised.\nOf course, in situations where the issue raised in (Algor) is not an issue of concern\n— i.e., such as situations involving routine numerical computations or manipulation\nof data in some relatively simple combinatorial framework [such as a finite group, a\nfinite simplicial complex, or a finite chain of Boolean operators] — computer verifi-\ncation can indeed function as a meaningful tool for the verification of mathematical\nassertions.\nOn the other hand, in situations — such as the situation surrounding such\nlogically unrelated fabricated versions of inter-universal Teichmüller theory\nas RCS-IUT — in which the central issue lies [cf. the discussion of §1.2, §1.3,\n§1.10, as well as Example 2.4.5, 2.4.7, below] in the erroneous confusion of such\nlogically unrelated fabricated versions of a theory with the original version of the\ntheory, computer verifications can never yield meaningful or substantive progress,\nsince the erroneous confusion of logically unrelated fabricated versions of a theory\nwith the original version of the theory completely invalidates, in a very essential\nand inevitable way, the “presumed relationship” discussed in (Algor) on which any\nsort of computer verification of a mathematical assertion must be premised. More\nelementary examples of this phenomenon of erroneous confusion of logically unre-\nlated fabricated versions of a theory with the original version of the theory may be\nseen in\n· the situation surrounding the assertion of Example 1.10.1, (iii), (Pyth4);\n· the situation surrounding the [erroneous!] point of view described in\nExample 2.1.1 concerning the classical theory of integration on the real\nline\n— i.e., situations that arise from essentially non-mathematical [e.g., social\/political\/\npsychological] circumstances and, as a result, are clearly not amenable to resolution\nvia computer verification.\nIn the context of (CmpVer), it is perhaps of interest to recall from the In-\ntroduction [cf. also the discussion at the beginning of §3.10] the point of view —\nmotivated by the well-known functional completeness, in the sense of propositional\ncalculus, of the collection of Boolean operators consisting of logical AND “∧”, logical\nOR “∨”, and negation “¬” — that one can, in principle, express\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 31\nthe essential logical structure of any mathematical argument or\ntheory in terms of elementary logical relations, i.e., such as logical\nAND “∧”, logical OR “∨”, and negation “¬”.\nIndeed, it is precisely this point of view that formed the central motivation and\nconceptual starting point of the exposition given in the present paper concerning\nthe essential logical structure of inter-universal Teichmüller theory, which may\nbe represented symbolically as follows:\nA ∧B = A ∧ (B1 ∨̇ B2 ∨̇ . . . )\n=⇒ A ∧ (B1 ∨̇ B2 ∨̇ . . . ∨̇ B′1 ∨̇ B′2 ∨̇ . . . )\n=⇒ A ∧ (B1 ∨̇ B2 ∨̇ . . . ∨̇ B′1 ∨̇ B′2 ∨̇ . . . ∨̇ B′′1 ∨̇ B′′2 ∨̇ . . . )\n...\nThat is to say, in summary:\n(SymIUT) The symbolic representation [cf. the above display!] of the essen-\ntial logical structure of inter-universal Teichmüller theory exposed in the\npresent paper may be understood as being, in some sense, the closest\nrealistic approach to the essential spirit of (CmpVer). Moreover, this\nsymbolic representation [cf. the above display!] is sufficiently simple and\ntransparent that, once it has been properly communicated, it may be veri-\nfied readily by mental computation in a matter of minutes without the use\nof a computer!\nHere, we note that the relative simplicity of this symbolic representation of (SymIUT)\nis obtained as a result of organizing\/compartmentalizing into “blackboxes” vari-\nous “blocks” of inter-universal Teichmüller theory that consist of anabelian geometry\nor the theory of étale theta functions, and whose validity has never arisen as a mat-\nter of discussion.\nUltimately, misunderstandings resulting from logically unrelated fabricated\nversions can only be overcome by studying the original papers [IUTchI-IV] [cf.\nalso [Alien], as well as the present paper!] on inter-universal Teichmüller theory,\nor, if this is not sufficient, by engaging in constructive mathematical dialogue\nwith mathematicians who do have a substantial, accurate understanding of the\ntheory [cf. the discussion of §1.4, §1.5, §1.6]. Indeed, perhaps more to the point,\nthere appears to be a conspicuous tendency, in certain sectors of the mathematical\ncommunity, to\n(RfsDlg) utilize the proliferation\/citation of logically unrelated fabricated\nversions of the theory as a sort of “lame excuse”\/subterfuge to justify\na stance of refusal to engage in such constructive mathematical dialogue\nconcerning the theory\n— cf. the discussion of §1.3, §1.10, especially the discussion of (VioMIPR), (DtrVio),\n(IgNoJst), (BurPrf) in Example 1.10.1, (iv); the discussion at the beginning of Ex-\nample 3.10.1 below. Moreover,\n(DngPrc) the situation described above in (RfsDlg), if left unchecked, constitutes,\nin the long-term, a dangerous precedent from the point of view of\n32 SHINICHI MOCHIZUKI\nmaintaining a state of operational normalcy in the field of mathematics\nin a fashion consistent with such fundamental democratic principles as the\nrule of law, due process of law, and burden of proof\n— cf. the discussion of §1.10, especially the discussion of (VioMIPR), (DtrVio),\n(IgNoJst), (BurPrf) in Example 1.10.1, (iv).\nNevertheless, in this context, it is also of fundamental importance to recall\n(OvrMs) the existence of numerous mathematicians [of many diverse nationali-\nties!] who were indeed successful in overcoming various meaningless\nmisunderstandings concerning inter-universal Teichmüller theory pre-\ncisely by persistently engaging in constructive mathematical dialogue\nconcerning the theory.\nSuch dialogues typically involve the painstaking and time-consuming process of\n(ExplLS) sifting through and analyzing the assertions of the mathematician in\nquestion concerning inter-universal Teichmüller theory in order to make\nprecise and explicit the exact content of the logical structure under-\nlying such assertions [cf. the discussion of §1.5].\nOn the other hand, it is important to emphasize that such efforts, when seen through\nto their conclusion, have always led to a situation in which the mathematician in\nquestion realizes his\/her misunderstandings of the theory and ultimately concedes\nthat, at least so far as he\/she can see,\n(MthVlTh) there is indeed no mathematical reason to deny the mathematical\nvalidity of the theory.\nThat is to say, this chain of events is precisely the chain of events that should be\nexpected from any constructive mathematical dialogue carried out in a suitable,\nsincere, and rational fashion concerning a rigorously formulated mathematical the-\nory.\nThus, in summary, at least in a direct, literal sense, ultimately, the only way\nto overcome meaningless misunderstandings of the theory that arise from logically\nunrelated fabricated versions of the theory is to\n(CfrMth) directly confront the mathematical content of the original theory,\neither\n· by studying the original papers [IUTchI-IV] [cf. also [Alien], as\nwell as the present paper!] on inter-universal Teichmüller theory,\nor, if this is not sufficient,\n· by engaging in constructive mathematical dialogue with mathe-\nmaticians who do have a substantial, accurate understanding of\nthe theory.\n[cf. the discussion of §1.3, §1.10, as well as Example 3.10.1 below].\nConversely,\n(BlkAcc) a stance of systematic and institutionally endorsed refusal to confront\nthe mathematical content of the original theory has the effect of orches-\ntrating the creation of a sort of artificial blackhole relative to the issue of\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 33\nmathematical accountability [cf. the discussion of [EMSCOP] in §1.3]\nand leads to the sort of absurd pathologies and obstructions to the\noperational normalcy of the field of mathematics discussed in detail in\n§1.10 [cf., especially, Example 1.10.1, (iii), (Pyth4)].\nThat is to say, unlike the IUT community, which bears active mathematical\nresponsibility for the mathematical content of inter-universal Teichmüller theory\nin the long-term by maintaining an extensive, sustained infrastructural appa-\nratus of mathematical activities surrounding inter-universal Teichmüller theory —\nsuch as\n· workshops, of one to two weeks in length, concerning inter-universal\nTeichmüller theory [e.g., at RIMS in March 2015, in Oxford in Decem-\nber 2015, at RIMS in July 2016, and at RIMS in September 2021] and\nnumerous lecture series [e.g., in Kumamoto in May 2014, at RIMS in De-\ncember 2015, in Yokohama in November 2018, and at RIMS in December\n2021], which have led to a quite substantial stock of widely and readily ac-\ncessible PDF files of slides and videos of lectures exposing inter-universal\nTeichmüller theory;\n· extensive one-to-one mathematical discussions between mathematicians\nall over the world concerning inter-universal Teichmüller theory via e-mail\nand online video meetings;\n· joint research projects concerning the further development of inter-\nuniversal Teichmüller theory [cf., e.g., [ExpEst]]\n— the author remains entirely unable, despite years of intensive effort [cf. the\ndiscussion of §1.3, §1.5, §1.6], to locate even a single mathematician who is will-\ning to bear active responsibility for the mathematical content of RCS-IUT\nby engaging in similar mathematical activities\/mathematical dialogue. The fun-\ndamental qualitative difference constituted by this egregrious absence of an\ninfrastructural apparatus supporting the assertions of the RCS — that is to say,\nput another way, this sort of “hit-and-run”\/“dead-end” approach to making\nvaguely formulated mathematical assertions that are not supported by detailed\ndocumentation\/exposition apparatuses, i.e., in violation of the [EMSCOP] [cf. the\ndiscussion of the [EMSCOP] in §1.3] — is precisely what is meant by the notion of\na “blackhole” of mathematical accountability discussed in (BlkAcc).\nOn the other hand, from a more long-term, historical point of view, it is perhaps\nof interest to observe that there is another approach to witnessing the validity of a\nmathematical theory, namely,\n(DvpRcn) the approach of developmental reconstruction, i.e., of “reconstructing\nthe validity” of a mathematical theory by witnessing the subsequent\ndevelopments that ensue from the theory.\nThis point of view is particularly of interest in the context of inter-universal Te-\nichmüller theory, given the central role played in inter-universal Teichmüller theory\nby anabelian geometry, i.e., which revolves around the development of recon-\nstruction algorithms that allow one to reconstruct conventional algebraic struc-\ntures [i.e., of the sort that typically appear in algebraic\/arithmetic geometry] from\nmore primitive combinatorial structures such as topological groups.\n34 SHINICHI MOCHIZUKI\nThis approach of developmental reconstruction may be applied, for instance, to\nthe task of evaluating the level of mathematical or scientific development of ancient\ncivilizations, i.e., not via the direct study of detailed theoretical expositions [which\nare typically not readily available — cf. the discussion of §1.5!] of the mathematics\nor science understood by such an ancient civilization, but rather by observing what\nmay be understood as the “fruits” of this mathematics or science, e.g., in the form\nof architectural achievements such as the famous\n· pyramids of Egypt or\n· Nazca lines and mysterious ruins of Puma Punku in South Amer-\nica.\nAnother important [though non-architectural!] example of this sort of phenomenon\nmay be seen in\n· the list of Pythagorean triples in the famous Babylonian tablet\nPlimpton 322\n— i.e., which is particularly notable in that it strongly suggests [that is to say, despite\nthe fact that it is not accompanied by any sort of explicit theoretical exposition!]\nan understanding of algebraic manipulation on a par with the essential content of\nExample 1.12.1 below.\nExample 1.12.1: Explicit parametrization of Pythagorean triples. The\nset of integral solutions — i.e., solutions in the ring of integers Z, also known as\nPythagorean triples— of the equation x2+y2 = z2 may be parametrized by applying\nthe substitutions xz \b→ u, yz \b→ v and considering the set of rational solutions —\ni.e., solutions in the field of rational numbers Q — of the equation u2 + v2 = 1.\nThe solutions of this last equation u2 + v2 = 1 in Q — or, indeed, in any field of\ncharacteristic \n= 2 — may be completely parametrized by the substitutions\nu \b→ t2−1t2+1 , v \b→ 2tt2+1\n— where we observe that\n(\nt2−1\nt2+1\n)2\n+\n(\n2t\nt2+1\n)2\n= 1, t = v\n1−u =\nu+1\nv =\n{(\nt2−1\nt2+1\n)\n+ 1\n}\n·\n(\n2t\nt2+1\n)−1\n,\n2 ·\n(\nt+ 1t\n)−1\n= 2 ·\n(\nu+1\nv +\nv\nu+1\n)−1\n= 2 ·\n(\n(u+1)2+v2\nv(u+1)\n)−1\n= 2 ·\n(\nu2+v2+1+2u\nv(u+1)\n)−1\n= 2 ·\n(\n2(u+1)\nv(u+1)\n)−1\n= v\nThat is to say, this parametrization by t [for t such that t2+1 \n= 0] gives a complete\nlist of all solutions of the equation u2 + v2 = 1 [for u such that u − 1 \n= 0] in\nany field of characteristic \n= 2 [or, indeed, by interpreting “\n= 0” as a condition of\ninvertibility, in any ring in which 2 is invertible].\nOn the other hand, in this context, it is of interest to note that, at least as of\nthe time of writing of the present paper,\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 35\n(AncBas) no ancient civilization has produced evidence of knowledge of the equation\n∑∞\nn=1\n1\nn2 =\nπ2\n6\n— i.e., of the solution of the so-called Basel problem.\nThat is to say, here we note that this observation (AncBas) is valid despite the fact\nthat each of the essential components of this equation — i.e.,\n· the positive integers of “sufficiently large value”,\n· the elementary operations of addition\/multiplication\/division,\n· the notion of the length of the circumference of a circle of radius 1, and\n· the idea of a sum of [real] numbers coming arbitrarily close, up to a very\nsmall margin of error, to another [real] number\n— may be readily expressed in terms understandable to many advanced ancient\ncivilizations. Of course, the discovery of an ancient civilization that produced\nevidence of some sort of knowledge of the equation in the display of (AncBas)\nwould be quite startling since it would strongly suggest [that is to say, even if it\nis not accompanied by any sort of explicit theoretical exposition!] an understanding\nof numerous ideas and theorems not only from elementary differential and integral\ncalculus, but also possibly from Fourier analysis on the circle and complex\nanalysis on the complex plane.\nFinally, we return to our discussion of inter-universal Teichmüller theory. In\nthe case of inter-universal Teichmüller theory, the phenomenon of developmen-\ntal reconstruction may already be seen, albeit in a relatively weak sense, in\nthe numerical results of [ExpEst]. Stronger examples of this phenomenon may be\nseen, however, in various enhanced versions of inter-universal Teichmüller theory\nthat are currently under development, which are expected to give rise to various\nnew types of applications of inter-universal Teichmüller theory. Such enhanced\nversions suggest strongly that the original version of inter-universal Teichmüller\ntheory given in [IUTchI-IV] [cf. also [Alien], as well as the present paper!] should\nperhaps be regarded as only the first example of a much larger collection of exam-\nples of “anabelian adèlic analysis”, i.e., in the spirit of the point of view that\nthe various types of prime-strips that occur in inter-universal Teichmüller theory\nmay be thought of [cf. the discussion at the end of [Alien], §3.3, (iv)] as a sort\nof anabelian\/monoid-theoretic version of the classical notion of adèles\/idèles that\nappears throughout conventional arithmetic geometry and number theory.\nSection 2: Elementary mathematical aspects of “redundant copies”\nThe essence of the central mathematical assertions of the RCS revolves, per-\nhaps somewhat remarkably, around quite elementary considerations that lie well\nwithin the framework of undergraduate-level mathematics. Before examining, in §3,\nthe assertions of the RCS in the technical terminology of inter-universal Teichmüller\ntheory, we pause to give a detailed exposition of these elementary considerations.\n36 SHINICHI MOCHIZUKI\n§2.1. The history of limits and integration\nThe classical notion of integration [e.g., for continuous real-valued functions\non the real line], as well as the more fundamental, but closely related notion of a\nlimit, have a long history, dating back [at least] to the 17-th century. Initially,\nthese notions did not have rigorous definitions, i.e., were not “well-defined”, in the\nsense understood by mathematicians today. The lack of such rigorous definitions\nfrequently led, up until around the end of the 19-th century, to “contradictions”\nor “paradoxes” in mathematical work — such as Grandi’s series\n∞∑\nn=0\n(−1)n\n— concerning integrals or limits.\nUltimately, of course, the theory of limits and integrals evolved, especially\nduring the period starting from around the mid-19-th century and lasting until\naround the early 20-th century, to the extent that such “contradictions\/paradoxes”\ncould be resolved in a definitive way. This process of evolution involved, for instance,\nin the case of integration, first the introduction of the Riemann integral and later the\nintroduction of the Lebesgue integral, which made it possible to integrate functions\n— such as, for instance, the indicator function on the real line of the subset of\nrational numbers — whose Riemann integral is not well-defined.\nHere, it should be noted that at various key points during this evolution of the\nnotions of limits and integration, the central “contradictions\/paradoxes” that, at\ntimes, led to substantial criticism and confusion arose from a solid, technically\naccurate understanding of the content and logical structure of the assertions —\nsuch as, for instance, various possible approaches to computing the value of Grandi’s\nseries — at the center of these “contradictions\/paradoxes”. It is precisely for this\nreason that such criticism and confusion ultimately lead to substantive refinements\nin the theory that were sufficient to resolve the original “contradictions\/paradoxes”\nin a definitive way.\nSuch constructive episodes in the history of mathematics — which may\nbe studied by scholars today precisely because of the existence of detailed,\nexplicit, mathematically substantive, and readily accessible written records!\n[cf. the discussion of §1.5] — stand in stark contrast to [cf. the discussion\nof (UndIg) in §1.3] criticism of a mathematical theory that is based on a\nfundamental ignorance of the content and logical structure of the the-\nory, such as the following “false contradiction” in the theory of integration,\nwhich may be observed in some students who are still in an initial stage\nwith regard to their study of the notion of integration.\nExample 2.1.1: False contradiction in the theory of integration. Consider\nthe following computation of the definite integral of a real-valued function on the\nreal line ∫ 1\n0\nxn dx = 1n+1\nfor n a positive integer. Suppose that one takes the [drastically oversimplified and\nmanifestly absurd, from the point of view of any observer who has an accurate\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 37\nunderstanding of the theory of integration!] point of view that the most general\npossible interpretation of the equation of the above display is one in which the\nfollowing three symbols\n“\n∫ 1\n0\n”, “x”, “dx”\nare allowed to be arbitrary positive real numbers a, b, c. Here, we note that in the\ncase of “dx”, such a substitution “dx \b→ c” could be “justified” by quoting conven-\ntional “\u0002-δ” treatments of the theory of limits and integrals, in which infinitesimals\n— i.e., such as “dx” or “\u0002” — are allowed to be arbitrary positive real numbers,\nwhich are regarded as being “arbitrarily small”, and observing that any positive\nreal number c is indeed much smaller than “most other positive real numbers” [such\nas 1000 · c, etc.]. On the other hand, by substituting the values n = 1, 2, 3, one\nobtains relations\nabc = 1, ab2c = 1\n2\n, ab3c = 1\n3\n.\nThe first two of these relations imply that b = 1\n2\n[so b2 = 1\n4\n], while the first and\nthird relations imply that b2 = 1\n3\n\n= 1\n4\n— a “contradiction”!\n§2.2. Derivatives and integrals\nIn the context of the historical discussion of integration in §2.1, it is interesting\nto recall the fundamental theorem of calculus, i.e., the result to the effect that,\nroughly speaking, the operations of integration and differentiation of functions [i.e.,\nreal-valued functions on the real line satisfying suitable conditions] are inverse to\none another. Thus, from a certain point of view,\nthe “essential information” contained in a function may be understood as\nbeing “essentially equivalent” to the “essential information” contained\nin the derivative of the function\n— that is to say, since one may always go back and forth at will between a function\nand its derivative by integrating and then differentiating. This point of view might\nthen tempt some observers to conclude that\nany mathematical proof that relies, in an essential way, on consideration\nof the derivative of a function must be fundamentally flawed since any\ninformation that might possibly be extracted from the derivative of the\nfunction should already be available [cf. the “essential equivalence” dis-\ncussed above] from the function prior to passing to the derivative, i.e., in\n“contradiction” to the essential dependence of the proof on passing to the\nderivative.\nAlternatively, this point of view may be summarized in the following way:\nthe “essential equivalence” discussed above implies that any usage, in a\nmathematical proof, of the derivative of a function is necessarily inherently\nredundant in nature.\nIn fact, of course, such “pseudo-mathematical reasoning” is itself fundamen-\ntally flawed. Two examples of well-known proofs in arithmetic geometry that\n38 SHINICHI MOCHIZUKI\ndepend, in a essential way, on passing to the derivative will be discussed in the final\nportion of §3.2 below. These examples are in fact closely related to the mathematics\nthat inspired inter-universal Teichmüller theory [cf. the discussion in the final por-\ntion of §3.2 below]. One central aspect of the situations discussed in §3.2 below is\nthe exploitation of properties of [various more abstract analogues of] the derivative\nof a function that differ, in a very substantive, qualitative way, from the prop-\nerties of the original function. One important example of this sort of situation is\nthe validity\/invalidity of various symmetry properties. This phenomenon may\nbe observed in the following elementary example.\nExample 2.2.1: Symmetry properties of derivatives. The real-valued func-\ntion\nf(x) = x\non the real line is not invariant [i.e., not symmetric] with respect to translations\nby an arbitrary constant c ∈ R. That is to say, in general, it is not the case that\nf(x+ c) = f(x). On the other hand, the derivative\nf ′(x) = 1\nof this function is manifestly invariant\/symmetric with respect to such trans-\nlations.\n§2.3. Line segments vs. loops\nBy comparison to the examples given in §2.1, §2.2, the following elementary\ngeometric examples are much more closely technically related to the assertions of\nthe RCS concerning inter-universal Teichmüller theory.\nExample 2.3.1: Endpoints of an oriented line segment.\n(i) Write\nI\ndef\n= [0, 1] ⊆ R\nfor the closed unit interval [i.e., the set of nonnegative real numbers ≤ 1] in the real\nline R. Thus, I is equipped with a natural topology [i.e., induced by the topology\nof R], hence can be regarded as a topological space, indeed more specifically, as a\none-dimensional topological manifold with boundary that is equipped with a natural\norientation [i.e., induced by the usual orientation of R]. Write\nα\ndef\n= {0}, β def= {1}\nfor the topological spaces [consisting of a single point!] determined by the two\nendpoints of I. Thus, α and β are isomorphic as topological spaces. In certain\nsituations that occur in category theory, it is often customary to replace a given\ncategory by a full subcategory called a skeleton, which is equivalent to the given\ncategory, but also satisfies the property that any two isomorphic objects in the\nskeleton are equal. This point of view of working with skeletal categories [i.e.,\ncategories which are their own skeletons] is motivated by the idea that nonequal\nisomorphic objects are “redundant”. Of course, there are indeed various situations\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 39\nin which nonequal isomorphic objects are redundant in the sense that working\nwith skeletal categories, as opposed to arbitrary categories, does not result in any\nsubstantive difference in the mathematics under consideration.\n(ii) On the other hand, if, in the present discussion of I, α, β — which one\nmay visualize as follows\nI\n•——>——•\nα β\n— one identifies α and β, then one obtains a new topological space, that is to say,\nmore specifically, an oriented one-dimensional topological manifold [whose orienta-\ntion is induced by the orientation of I]\nL\ndef\n= I\/〈α ∼ β〉\nthat is homeomorphic to the unit circle, i.e., may be visualized as a loop. Write\nγL ⊆ L for the image of α ⊆ I, or, equivalently, β ⊆ I, in L. As is well-known\nfrom elementary topology, the topological space L is structurally\/qualitatively very\ndifferent from the topological space I. For instance, whereas I has a trivial fun-\ndamental group, L has a nontrivial fundamental group [isomorphic to the additive\ngroup of integers Z]. In particular,\nit is by no means the case that the fact that α and β are isomorphic as\ntopological spaces implies a sort of “redundancy” to the effect that any\nmathematical argument involving I [cf. the above observation concerning\nfundamental groups!] is entirely equivalent to a corresponding math-\nematical argument in which α and β are identified, i.e., in which “I” is\nreplaced by “L”.\n(iii) In this context, we observe that the [one-dimensional oriented topological\nmanifold with boundary] I does not admit any symmetries that switch α and\nβ. Moreover, even if one passes to the quotient I \u0003 L, the [one-dimensional ori-\nented topological manifold] L does not admit any symmetries that reverse the\norientation of L.\nExample 2.3.2: Gluing of adjacent oriented line segments.\n(i) A similar elementary geometric situation to the situation discussed in Ex-\nample 2.3.1, but which is technically a bit more similar to the situation that arises\nin inter-universal Teichmüller theory may be given as follows. We begin with two\ndistinct copies †I, ‡I of I. Thus, †I has endpoints †α, †β [i.e., corresponding re-\nspectively to the endpoints α, β of I]; similarly, ‡I has endpoints ‡α, ‡β. We then\nproceed to form a new topological space J by gluing †I to ‡I via the unique isomor-\nphism of topological spaces †β ∼→ ‡α. Thus, †β and ‡α are identified in J. Let us\nwrite γJ for the one-pointed topological space obtained by identifying\n†β and ‡α.\nThus, J may be visualized as follows:\n†I ‡I\n40 SHINICHI MOCHIZUKI\n•——>——•——>——•\n†α γJ ‡β\n(ii) Observe that the gluing operation that gave rise to J is such that we may\nregard †I and ‡I as subspaces †I ⊆ J, ‡I ⊆ J of J. Since each of these subspaces\n†I, ‡I of J is naturally isomorphic to I, one may take the point of view, as in the\ndiscussion of Example 2.3.1, that these two subspaces are “redundant” and hence\nshould be identified with one another [say, via the natural isomorphisms of †I, †I\nwith I] to form a new topological space\nM\ndef\n= J\/〈†I ∼ ‡I〉\n— where we observe that the natural isomorphisms of †I, †I with I determine a\nnatural isomorphism of topological spaces M\n∼→ L = I\/〈α ∼ β〉, with the loop L\nconsidered in Example 2.3.1. Write γM ⊆ M for the image of γJ ⊆ J in M. Thus,\nthe natural isomorphism M\n∼→ L maps γM isomorphically onto γL. On the other\nhand, just as in the situation discussed in Example 2.3.1,\nit is by no means the case that the fact that †I and ‡I are [in fact, natu-\nrally] isomorphic as topological spaces implies a sort of “redundancy” to\nthe effect that any mathematical argument involving J is entirely equiv-\nalent to a corresponding mathematical argument in which †I and ‡I are\nidentified [say, via the natural isomorphisms of †I, †I with I], i.e., in\nwhich “J” is replaced by “M”.\nIndeed, for instance, one verifies immediately, just as in the situation of Exam-\nple 2.3.1, that the fundamental groups of J and M are not isomorphic. That is\nto say, just as in the situation of Example 2.3.1, the topological space J is struc-\nturally\/qualitatively very different from the topological space M.\n§2.4. Logical AND “∧” vs. logical OR “∨”\nThe essential mathematical content of the elementary geometric examples dis-\ncussed in §2.3 may be reformulated in terms of the symbolic logical relators AND\n“∧” and OR “∨”. This reformulation renders the elementary geometric examples\nof §2.3 in a form that is even more directly technically related to various central\naspects of the assertions of the RCS concerning inter-universal Teichmüller theory.\nExample 2.4.1: “∧” vs. “∨” for adjacent oriented line segments.\n(i) Recall the situation discussed in Example 2.3.2. Thus, J ⊇ †I ⊇ †β = γJ =\n‡α ⊆ ‡I ⊆ J, i.e.,\n(AOL1) the following condition holds:(\nγJ =\n†β ⊆ †I\n)\n∧\n(\nγJ =\n‡α ⊆ ‡I\n)\n.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 41\nOn the other hand, if one identifies †I, ‡I, then one obtains a topological space\nM\n∼→ L, i.e., a loop. Here, “ ∼→ ” denotes the natural isomorphism discussed in\nExample 2.3.2, (ii). Now suppose that we are given a connected subspace\nγI ⊆ I\nwhose image in the quotient I \u0003 L = I\/〈α ∼ β〉 coincides with γL ⊆ L, i.e., with\nthe image of γJ ⊆ J via the composite of the quotient J\u0003 M = J\/〈†I ∼ ‡I〉 with\nthe natural isomorphism M\n∼→ L. Then observe that\n(AOL2) the following condition holds: γI ∈ {α, β}, i.e.,(\nγI = β ⊆ I\n)\n∨\n(\nγI = α ⊆ I\n)\n.\nOf course,\n(AOL3) the essential mathematical content discussed in this condition (AOL2)\nmay be formally described as a condition involving the AND relator “∧”:(\nβ ∈ {α, β}\n)\n∧\n(\nα ∈ {α, β}\n)\n.\nBut the essential mathematical content of the OR relator “∨” statement in (AOL2)\nremains unchanged.\n(ii) On the other hand, [unlike the case with γJ!]\n(AOL4) the following condition does not hold:(\nγI = β ⊆ I\n)\n∧\n(\nγI = α ⊆ I\n)\n.\nThat is to say, in summary, the operation of identifying †I, ‡I — e.g., on the\ngrounds of “redundancy” [cf. the discussion of Example 2.3.2] — has the effect\nof passing from a situation in which\nthe AND relator “∧” holds [cf. (AOL1)]\nto a situation in which\nthe OR relator “∨” holds [cf. (AOL2), (AOL3)], but\nthe AND relator “∧” does not hold [cf. (AOL4)]!\n(iii) It turns out that this phenomenon — i.e., of an identification of “redundant\ncopies” leading to a passage from the validity of an “∧” relation to the validity of\nan “∨” relation coupled with the invalidity of an “∧” relation — forms a very\nprecise model of the situation that arises in the assertions of the RCS concerning\ninter-universal Teichmüller theory [cf. the discussion of §3.2, §3.4 below].\n42 SHINICHI MOCHIZUKI\nExample 2.4.2: Differentials on oriented line segments.\n(i) In the situation of Example 2.4.1, one way to understand the gap between\n(AOL1) and (AOL4) — i.e., the central issue of whether the AND relator “∧” holds\nor does not hold — is to think in terms of the restriction to I ⊆ R of the coordinate\nfunction “x” of Example 2.2.1. Indeed,\n(AOD1) one may interpret (AOL4) as the statement that the coordinate functions\n“x” on the two copies †I, ‡I that constitute J do not glue together to\nform a single, well-defined R-valued function on J [that is to say, since it\nis not clear whether the value of such a function on γJ ⊆ J should be 0 or\n1, i.e., such a function is not well-defined on γJ ⊆ J];\n(AOD2) on the other hand, (AOL1) may be interpreted as the statement that\nsuch a function [i.e., obtained by gluing together the coordinate functions\n“x” on the two copies †I, ‡I that constitute J] can indeed be defined if\none regards its values as being [not in a single copy of R, but rather] in\nthe set †,‡R obtained by gluing together two distinct copies †R, ‡R of R\nby identifying †1 ∈ †R with ‡0 ∈ ‡R.\n(ii) On the other hand, if, instead of considering the coordinate function “x”,\none considers the differential “dx” associated to this coordinate function [cf. the\ndiscussion of Example 2.2.1], then one observes immediately that\n(AOD3) the differentials “dx” on the two copies †I, ‡I that constitute J do\nindeed glue together to form a single, well-defined differential on J that,\nmoreover, is compatible with the quotient J \u0003 M = J\/〈†I ∼ ‡I〉 in the\nsense that, as is easily verified, it arises as the pull-back, via this quotient\nmap J\u0003M, of a [smooth] differential on the [smooth manifold constituted\nby the] loop M.\nNote, moreover, that the gluings and compatibility of (AOD3) may be achieved\nwithout considering functions or differentials valued in some sort of complicated\n[i.e., by comparison to R!] set such as †,‡R.\n(iii) It turns out [cf. the discussion of Example 2.4.1, (iii)] that the phenomenon\ndiscussed in (AOD3) is closely related to the situation that arises in inter-universal\nTeichmüller theory [cf. the discussion of §3.2 below].\nExample 2.4.3: Representation via subgroup indices of “∧” vs. “∨”.\n(i) Let A be an abelian group and B1, B2 ⊆ A subgroups of A such that B1∩B2\nhas finite index in B1 and B2. Then one may define a positive rational number,\nwhich we call the index of B2 relative to B1,\n[B1 : B2]\ndef\n= [B1 : B1 ∩B2]\/[B2 : B1 ∩B2] ∈ Q>0.\nThus, [B1 : B2] · [B2 : B1] = 1; when B2 ⊆ B1, this notion of index coincides with\nthe usual notion of the index of B2 in B1.\n(ii) Let n be a positive integer ≥ 2. Consider the diagram of group homomor-\nphisms\nG1\nn·−→ G2 n·−→ G3\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 43\n— where, for i = 1, 2, 3, Gi denotes a copy of [the additive group of rational\nintegers] Z, and the arrows are given by multiplication by n. For i = 1, 2, 3, write\nGQi\ndef\n= Gi⊗Z Q for the tensor product of Gi over Z with Q. Then observe that this\ndiagram induces a diagram of group isomorphisms\nGQ1\n∼→ GQ2 ∼→ GQ3\n— i.e., in which the arrows are isomorphisms. Let us use these isomorphisms to\nidentify the groups GQi , for i = 1, 2, 3, and denote the resulting group by G\nQ\n∗ .\n(iii) Observe that the first diagram of (ii) is structurally reminiscent of the\nobject J discussed in Examples 2.3.2, 2.4.1, 2.4.2, i.e., if one regards\n· the first arrow of the first diagram of (ii) as corresponding to †I,\n· the second arrow of the first diagram of (ii) as corresponding to ‡I, and\n· G1, G2, and G3 as corresponding to †α, †β = ‡α, and ‡β, respectively.\nHere, we observe that G2 appears simultaneously as the codomain of the arrow\nG1\nn·−→ G2 AND [cf. (AOL1)!] as the domain of the arrow G2 n·−→ G3. Moreover,\nwe may consider indices of G1, G2, and G3 as subgroups of G\nQ\n∗\n[G2 : G1] = [G1 : G2]\n−1 = n; [G3 : G2] = [G2 : G3]−1 = n;\n[G3 : G1] = [G1 : G3]\n−1 = n2\nin a consistent fashion, i.e., in a fashion that does not give rise to any contra-\ndictions.\n(iv) On the other hand, suppose that we delete the “distinct labels” G1, G2, G3\nfrom the copies of Z considered in the first diagram of (ii). This yields a diagram\nZ\nn·−→ Z n·−→ Z\nin which the second arrow may be regarded as a copy of the first arrow. This\nsituation might motivate some observers to conclude that these two arrows are\n“redundant” and hence should be identified with one another — cf. the discus-\nsion of the quotient J\u0003M in Example 2.3.2, (ii) — to form a diagram\nn·\n\u0002 Z\nconsisting of a single copy of Z and the endomorphism of this single copy of Z given\nby multiplication by n. At first glance, this operation of identification may appear\nto give rise to various “contradictions” in the computation of the index, i.e., such\nas\n1 = [G1 : G1] = [Z : Z] = [G2 : G1] = n ≥ 2\nand so on. In fact, however, if one takes into account the OR relator “∨” [but not\nthe AND relator “∧”!] relations that one obtains upon executing the identification\noperation in question [cf. (AOL2), (AOL4)!], then one concludes that [after exe-\ncuting the identification operation in question!] each of the indices [Gi : Gj ], for\ni, j ∈ {1, 2, 3}, may only be computed up to multiplication by an integral power of\nn, i.e., that\n44 SHINICHI MOCHIZUKI\neach index [Gi : Gj ], for i, j ∈ {1, 2, 3}, is only well-defined as “some\nindeterminate element” of nZ\ndef\n= {nm | m ∈ Z} ⊆ Q>0.\nIn particular, in fact there is no contradiction.\nExample 2.4.4: Logical “∧\/∨” vs. “narrative ∧\/∨”. Consider the following\nargument concerning a natural number x ∈ {1, 3}, i.e., a natural number for which\nit holds that (x = 1) ∨ (x = 3):\n(Nar1) Suppose that x = 3. Then it follows that x = 3 > 2. That is to say, we\nconclude that x > 2.\n(Nar2) Since (x = 1) ∨ (x = 3), we may consider the case x = 1. Then, by\napplying the conclusion of (Nar1), we conclude that 1 = x > 2, i.e., that\n1 > 2 — a contradiction!\nOf course, this argument is completely fallacious! On the other hand, it yields\na readily understood concrete example of the absurdity that arises when, as is in\neffect done in (Nar2), logical OR “∨” is confused with logical AND “∧”! In various\ncontexts, this sort of confusion can arise from the ambiguity of various narrative\nexpressions that appear in the discussion of a mathematical argument. This sort\nof ambiguity can lead to a situation in which\na “narrative AND ∧” — i.e., the fact that in a particular narrative\nexposition of an argument, one performs both the task of considering the\ncase “x = 3” [cf. (Nar1)] and the task of considering the case “x = 1” [cf.\n(Nar2)] — is mistakenly construed as a logical AND “∧”.\nIn a similar vein, one may consider situations in which the roles played by “∧”\nand “∨” are reversed, i.e., in which a “narrative OR ∨” — i.e., the fact that in\na particular narrative exposition of an argument, one’s attention is concentrated\neither on the task of considering one situation or on the task of considering another\nsituation — is mistakenly construed as a logical OR “∨”. Indeed, it appears\nthat one fundamental cause, in the context of the essential logical structure of\ninter-universal Teichmüller theory [cf. the discussion of Example 2.4.5 below!], of\nthe confusion on the part of some mathematicians between logical AND “∧” and\nlogical OR “∨” lies precisely in this sort of confusion between “narrative ∧\/∨”\nand logical “∧\/∨”.\nExample 2.4.5: Numerical representation of “∧” vs. “∨”.\n(i) A slightly more sophisticated numerical representation of the difference\nbetween “∧” and “∨” — which in fact mirrors the essential logical structure\nof inter-universal Teichmüller theory in a very direct fashion — may be given as\nfollows [cf. [Alien], Example 3.11.4]. Indeed, the essential logical flow of inter-\nuniversal Teichmüller theory may be summarized as follows:\n· one starts with from the definition of an object called the Θ-link;\n· one then constructs a complicated apparatus that is referred to as the\nmultiradial representation of the Θ-pilot [cf. [IUTchIII], Theorem\n3.11];\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 45\n· finally, one derives a final numerical estimate [cf. [IUTchIII], Corol-\nlary 3.12] in an essentially straightforward fashion from the multiradial\nrepresentation of the Θ-pilot.\n(ii) An elementary model of this essential logical flow may be given by means\nof real numbers A,B ∈ R>0 and \u0002,N ∈ R such that 0 ≤ \u0002 ≤ 1 in the following way:\n· Θ-link: (\nN\ndef\n= −2B\n)\n∧\n(\nN\ndef\n= −A\n)\n;\n· multiradial representation of the Θ-pilot:(\nN = −2A+ \u0002\n)\n∧\n(\nN = −A\n)\n;\n· final numerical estimate:\n−2A+ \u0002 = −A, hence A = \u0002, i.e., A ≤ 1.\nThus, the definition of the Θ-link and the construction of the multiradial represen-\ntation of the Θ-pilot are meaningful\/nontrivial precisely on account of the validity\nof the AND relator “∧”, which is rendered possible, in the definition of the Θ-link,\nprecisely by allowing the real numbers A, B to be [a priori] distinct real numbers\n— cf. (AOL1) vs. (AOL4), where we think in terms of the correspondences\nB ←→ †I, N ←→ γJ, A ←→ ‡I\n−2B ←→ †β, −A ←→ ‡α, −2A ←→ ‡β.\nThe passage from themultiradial representation of the Θ-pilot to the final numerical\nestimate is then immediate\/straightforward\/logically transparent.\n(iii) By contrast, if, in the elementary numerical model of (ii), one replaces\n“∧” by “∨”, then our elementary numerical model of the logical structure of inter-\nuniversal Teichmüller theory takes the following form:\n· “∨” version of Θ-link:(\nN\ndef\n= −2B\n)\n∨\n(\nN\ndef\n= −A\n)\n[cf.\n(\nN\ndef\n= −2A\n)\n∨\n(\nN\ndef\n= −A\n)\n];\n· “∨” version of multiradial representation of the Θ-pilot:(\nN = −2A+ \u0002\n)\n∨\n(\nN = −A\n)\n;\n· final numerical estimate:\n−2A+ \u0002 = −A, hence A = \u0002, i.e., A ≤ 1.\n46 SHINICHI MOCHIZUKI\nThat is to say, the use of distinct real numbers A, B in the definition of the “∨” ver-\nsion of Θ-link seems entirely superfluous [cf. (AOL2), relative to the correspon-\ndences discussed in (ii)]. This motivates one to identify A and B — i.e., to suppose\n“for the sake of simplicity” that A = B — which then has the effect of rendering\nthe definition of the original “∧” version of the Θ-link invalid\/self-contradictory [cf.\n(AOL4), relative to the correspondences discussed in (ii)]. Once one identifies A\nand B, i.e., once one supposes “for the sake of simplicity” that A = B, the pas-\nsage from the “∨” version of Θ-link to the resulting “∨” version of the multiradial\nrepresentation of the Θ-pilot then seems entirely meaningless\/devoid of any\ninteresting content. The passage from the resulting meaningless “∨” version of\nthe multiradial representation of the Θ-pilot to the final numerical estimate then\nseems abrupt\/mysterious\/entirely unjustified, i.e., put another way, looks as\nif\none erroneously replaced the “∨” in the meaningless “∨” version of\nthe multiradial representation of the Θ-pilot by an “∧” without any\nmathematical justification whatsoever.\nIt is precisely this pernicious chain of misunderstandings emanating from the\n“redundancy” assertions of the RCS that has given rise to a substantial amount\nof unnecessary confusion concerning inter-universal Teichmüller theory.\n(iv) Before proceeding, we observe that the sort of confusion discussed in (iii)\nbetween “∧” and “∨” can occur as the result of any of the following phenomena:\n(AOC1) a confusion between “narrative ∧\/∨” and logical “∧\/∨”, as discussed\nin Example 2.4.4;\n(AOC2) thinking in terms of the “fake ∧” of (AOL3), i.e., which, though for-\nmulated as a logical AND “∧” relation, is in fact, substantively speaking,\na logical OR “∨” relation;\n(AOC3) the symptom (Syp2) discussed in §3.6 below, i.e., a desire to see the\n“proof” of some sort of commutative diagram or “compatibility prop-\nerty” to the effect that taking log-volumes of pilot objects in the domain\nand codomain of the Θ-link yields the same real number;\n(AOC4) a fundamental misunderstanding — which is often closely inter-\ntwined with the symptom (Syp2) discussed in (AOC3) — of the meaning\nof the crucial closed loop of §3.10, (Stp7), (Stp8), below [cf. §3.10,\n(Stp7), (Stp8), as well as the following discussion].\n(v) Let us refer to the “∧” version of inter-universal Teichmüller theory dis-\ncussed in (ii) — i.e., the original version of inter-universal Teichmüller theory, in\nwhich one interprets the Θ-link as a logical AND “∧” relation — as AND-IUT.\nThus,\nAND-IUT = IUT is the original version of inter-universal Teichmüller\ntheory.\nLet us refer to the “∨” version of inter-universal Teichmüller theory discussed in\n(iii) — i.e., the version of inter-universal Teichmüller theory that arises if one [mis-\ntakenly!] interprets the Θ-link as a logical OR “∨” relation — as OR-IUT. As\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 47\ndiscussed in (iii), in OR-IUT, one is motivated to implement the RCS-identifications\nof RCS-redundant copies of objects — i.e., in the language of (iii), to “identify A and\nB” — and hence to conclude that OR-IUT =⇒ RCS-IUT, where we recall that\n“RCS-IUT” refers to the version of inter-universal Teichmüller theory obtained by\nimplementing the RCS-identifications of RCS-redundant copies of objects [cf. the\ndiscussion of §1.2]. On the other hand, it is not difficult to see that in RCS-IUT, one\nis forced to work with a (NeuORInd) indeterminacy [cf. the discussion at the end of\n§3.4 below, as well as the discussion of (ΘORInd) in §3.11 below], i.e., to interpret\nthe Θ-link as a logical XOR “∨̇” relation [that is to say, a logical OR “∨”\nrelation such that the corresponding logical AND “∧” relation cannot hold — cf.\nthe discussion of (iii)]. In particular, we conclude that RCS-IUT =⇒ XOR-IUT\n=⇒ OR-IUT [where the second “ =⇒ ” is a consequence of well-known general\nproperties of Boolean operators], i.e., in summary:\n(XOR\/RCS) we have equivalences XOR-IUT ⇐⇒ OR-IUT ⇐⇒ RCS-IUT.\nIn the following, I shall refer to the school of thought [i.e., in the sense of a “collection\nof closely interrelated ideas”] surrounding OR-IUT as ORS, i.e., the “OR school\n[of thought]”, and to the school of thought surrounding XOR-IUT as XORS, i.e.,\nthe “XOR school [of thought]”. Thus, XORS = ORS = RCS.\n(vi) On the other hand, one may also consider yet another version of inter-\nuniversal Teichmüller theory, also motivated by the discussion of (iii), which we\nrefer to as EssOR-IUT, i.e., “essentially OR IUT”. This is the version of inter-\nuniversal Teichmüller theory in which one accepts, at the level of formal definitions,\nthe logical AND “∧” version of the Θ-link as in (ii), i.e., without identifying\nA and B, but [for some unexplained reason!] one then arbitrarily shifts, when\nconsidering the multiradial representation of the Θ-pilot, to the logical OR\n“∨” interpretation of the multiradial representation of the Θ-pilot, i.e., as in (iii).\nThat is to say, as the name “EssOR-IUT” suggests, the fundamental logical AND\n“∧” property of the Θ-link is never actually used in any sort of meaningful way in\nEssOR-IUT. In particular,\n(EssOR\/RCS) although, at a purely formal level, EssOR-IUT rejects RCS-IUT, the\nessential logical structure of EssOR-IUT still nevertheless gives rise to the\nabrupt\/mysterious\/entirely unjustified transition discussed in (iii)\nto the final numerical estimate.\nIt appears that the “arbitrarily shift” referred to above is often precipitated by\nthe various phenomena discussed in (iv) [cf., especially, (AOC3), (AOC4)]. In the\nfollowing, I shall refer to the school of thought [i.e., in the sense of a “collection of\nclosely interrelated ideas”] surrounding EssOR-IUT as EssORS, i.e., the “essen-\ntially OR school [of thought]”.\n(vii) In general, at the level of formalities of Boolean operators, “∧ ⇒ ∨”, but\n“∨ \n⇒ ∧”. In particular, in the context of the transition to the final numerical\nestimate of inter-universal Teichmüller theory,\n(∨ \n⇒ ∧) it appears entirely hopeless\/unrealistic to pass from the “∨” version of\nthe multiradial representation of the Θ-pilot to the “∧” version of the\nmultiradial representation of the Θ-pilot.\n48 SHINICHI MOCHIZUKI\nThis is precisely the “abrupt\/mysterious\/entirely unjustified” transition [to the final\nnumerical estimate] discussed in (iii).\n(viii) The discussion of (v), (vi), (vii) may be summarized as follows [cf. also\nthe discussion of §1.2, §1.3]:\nAssertions of Actual\nvarious schools of thought mathematical content\nRCS = ORS = XORS:\n“IUT ⇔ RCS-IUT” “XOR-IUT ⇔ OR-IUT ⇔ RCS-IUT”\n“∨ \n⇒ ∧”,\nRCS = ORS = XORS: which implies that\n“IUT is logically flawed.” “(AND-IUT =) IUT \n⇒ RCS-IUT”,\n“(AND-IUT =) IUT \n⇒ OR-IUT”,\n“(AND-IUT =) IUT \n⇒ XOR-IUT”\nEssORS:\neither “IUT is logically flawed.” “∨ \n⇒ ∧”,\nor “The logical structure of which implies that\nIUT is infinitely “(AND-IUT =) IUT \n⇒ EssOR-IUT”\ncomplicated\/mysterious.”\n· The fundamental misunderstanding on the part of adherents of the\nRCS = ORS = XORS to the effect that OR-IUT or XOR-IUT is indeed\nthe content of AND-IUT = IUT leads to the mistaken interpretation of\nthe assertion (XOR\/RCS) as an equivalence between AND-IUT = IUT\nand RCS-IUT.\n· The fundamental misunderstanding on the part of adherents of the\nRCS = ORS = XORS to the effect that OR-IUT or XOR-IUT is indeed\nthe content of AND-IUT = IUT leads to the mistaken interpretation of\nthe assertion (∨ \n⇒ ∧) as a logical flaw in AND-IUT = IUT.\n· The fundamental misunderstanding on the part of adherents of the\nEssORS to the effect that EssOR-IUT is indeed the content of AND-IUT =\nIUT leads either to the mistaken interpretation of the assertion (∨ \n⇒ ∧) as\na logical flaw in AND-IUT = IUT or to the mistaken interpretation of the\nassertion (∨ \n⇒ ∧) as an indication the existence of some sort of infinitely\ncomplicated and mysterious argument— i.e., for concluding that “∨ ⇒ ∧”!\n— in inter-universal Teichmüller theory that requires years of concerted\neffort to understand. Thus, as the descriptive “essential” suggests, there\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 49\nis in fact, from the point of view of the essential logical structure under\nconsideration, very little difference between EssORS and XORS = ORS =\nRCS or between EssOR-IUT and XOR-IUT⇐⇒ OR-IUT⇐⇒ RCS-IUT.\n· In fact, the correct interpretation of the assertion (∨ \n⇒ ∧) consists\nof the conclusion that neither XOR-IUT ⇐⇒ OR-IUT ⇐⇒ RCS-IUT nor\nEssOR-IUT has any direct logical relationship to AND-IUT = IUT.\nHere, we observe that the above analysis is in some sense remarkable in that it makes\nexplicit the fact that, if one forgets the arbitrary label “inter-universal Te-\nichmüller theory” placed on XOR-IUT or OR-IUT or EssOR-IUT by adherents\nof the RCS = ORS = XORS or the EssORS, then [perhaps somewhat surprisingly!]\n(MthVl) there is in fact no disagreement among any of the parties involved\nwith regard to themathematical validity of the mathematical assertions\n(XOR\/RCS) and (∨ \n⇒ ∧).\nIndeed, this state of affairs may be understood as in some sense highlighting the es-\nsentially social\/political\/psychological, i.e., in summary, non-mathematical nature\nof the entirely unnecessary confusion that has arisen concerning inter-universal Te-\nichmüller theory [cf. the discussion of §1.8, §1.11]. The above observations are\nsummarized in the “dictionary of assertions” given above.\nExample 2.4.6: Carry operations in arithmetic, geometry, and Boolean\nlogic.\n(i) Observe that if, in the situation of Example 2.4.3, (ii), one focuses one’s\nattention on the subset D4−i ⊆ Gi in the copy of Z denoted by Gi, where i = 1, 2, 3,\ncorresponding to {0, 1, 2, . . . , n− 1} ⊆ Z, then the situation considered in Example\n2.4.3, (ii), closely resembles the situation that arises in elementary arithmetic com-\nputations — such as addition and multiplication — involving base n expansions\nof natural numbers. That is to say, one may think of\n· D1 as the first digit, i.e., when n = 10, the “ones digit”,\n· D2 as the second digit, i.e., when n = 10, the “tens digit”, and\n· D3 as the third digit, i.e., when n = 10, the “hundreds digit”\nof such an expansion. When performing such elementary arithmetic computations\n— such as addition and multiplication — involving base n expansions of natural\nnumbers, recall that it is of fundamental importance to take into account the various\ncarry operations that occur. In particular, we observe that\nthe use of distinct labels for distinct digits plays a fundamental role in\nelementary arithmetic computations involving base n expansions of natural\nnumbers\n— cf. the distinct labels “G1”, “G2”, “G3” in the discussion of Example 2.4.3, (ii),\n(iii). This situation is reminiscent of the important role played by the distinct\nlabels “A”, “B” in the “Θ-link” of Example 2.4.5, (ii). Note, morever, that\ndeletion\/confusion of these distinct labels for distinct digits has the\neffect of completely invalidating, at least in the usual “strict sense”,\n50 SHINICHI MOCHIZUKI\nelementary arithmetic computations involving base n expansions of natural\nnumbers\n— cf. the situation considered in Example 2.4.3, (iv). On the other hand, if one\nrestricts one’s attention to a specific computational algorithm [involving, say,\naddition and multiplication operations], then in fact it is often possible — i.e., de-\npending on the content of the specific computational algorithm under consideration\n— to obtain\nestimates to the effect that applying the algorithm either with or without\nthe use of distinct labels for distinct digits in the base n expansions of\nthe natural numbers involved in fact yields the same result, up to some\nexplicitly bounded discrepancy.\n[For instance, when n = 10, any algorithm that only involves addition and multipli-\ncation operations yields the same result modulo 9 (= 10− 1), regardless of whether\nor not one uses distinct labels for distinct digits in decimal expansions of natural\nnumbers.] Such estimates are reminiscent of the “multiradial representation”\nof Example 2.4.5, (ii).\n(ii) Observe that the discussion of\n· adjacent oriented line segments “†I”, “‡I” and\n· oriented loops “L”, “M”\nin Examples 2.3.1, 2.3.2, 2.4.1, 2.4.2 [cf. also the discussion of Examples 2.4.3, (iii);\n2.4.5, (ii)] may be regarded as a sort of limiting case of the discussion of base n\nexpansions of natural numbers in (i) above, i.e., if one\n· considers the real numbers obtained by dividing the natural numbers\n≤ 2n in the discussion of (i) above by n and then\n· passes to the limit n→ +∞.\nThat is to say, in summary,\nthe adjacency of the oriented line segments “†I”, “‡I” may be under-\nstood as a sort of continuous, geometric representation of the carry\noperation that appears in elementary arithmetic computations involving\nbase n expansions of natural numbers.\n(iii) From the point of view of discussions of the logical structure of math-\nematical arguments represented in terms of Boolean operators such as logical\nAND “∧” and logical OR “∨”, it is of interest to consider the discussion of (i)\nabove in the binary case, i.e., the case n = 2. We begin our discussion of the\nbinary case by recalling the following well-known facts:\n· multiplication in the field F2 = {0, 1}may be regarded as corresponding\nto the Boolean operator AND “∧”;\n· addition in the field F2 = {0, 1} may be regarded as corresponding to\nthe Boolean operator XOR [i.e., “exclusive OR”], which we denote by\n“∨̇”;\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 51\n· “carry-addition” in the truncated ring of Witt vectors F2 × F2 —\ni.e., addition of two elements of F2\n∼→ {0} × F2 ⊆ F2 × F2, regarded as\nTeichmüller representatives in the truncated ring of Witt vectors Z\/4Z,\nthat is to say, an addition operation in which one allows for the carry\noperation [cf. the discussion of (i)!] to the first factor of F2×F2 — may\nbe regarded as corresponding to an operator that we shall refer to as the\n“COR”, or “carry-OR”, operator and denote by “∨̈”; thus, we have\n∨̈ = (∧, ∨̇)\n[so 0 ∨̈ 0 = (0, 0); 1 ∨̈ 0 = 0 ∨̈ 1 = (0, 1); 1 ∨̈ 1 = (1, 0)].\nThese well-known facts case may be summarized as follows:\n(∨̈= ∧ ∨̇) Conventional mixed-characteristic\/“carry” addition in Z consid-\nered modulo 4 — i.e., “∨̈” — may be described in terms of the “splitting”\nof the natural surjection Z \u0003 F2 determined by the “Teichmüller repre-\nsentatives” 0, 1 ∈ Z via the equation\n∨̈ = (∧, ∨̇)\n— i.e., which exhibits “∨̈” as an operation obtained by “stacking” mul-\ntiplication “∧” in F2 on top of addition “∨̇” in F2. Here, we note\nthat this splitting via Teichmüller representatives 0, 1 ∈ Z is compatible\nwith the multiplicative structures in Z and F2, but not with the addi-\ntive structures in Z and F2. Put another way, one may think of the ring\nstructures of Z and F2 as structures that share a common multiplicative\nstructure [cf. “∧”!], but do not share a common additive structure [cf.\n“∨̇”!].\nThese observations will be of fundamental importance in the theory developed in\n§3 [cf., especially, the discussion at the beginning of §3.10].\n(iv) Some readers may object to the comparisons and analogies between inter-\nuniversal Teichmüller theory and the “mathematics of carry operations” dis-\ncussed in (i), (ii), and (iii) as being inappropriate on the grounds that this mathe-\nmatics of carry operations is much too “trivial” to be of any substantive interest.\nOn the other hand, we observe that the mathematics of carry operations discussed\nin (i), (ii), and (iii) is intimately intertwined with numerous important develop-\nments in the history of mathematics. With regard to the content of (i), we recall\nthat the use of place-value decimal numerals, i.e., that make use of notation\nfor zero, appears to date back to Indian texts and inscriptions from the 7-th to\n9-th centuries AD. Such numerals also reached the Arab world during this period,\nbut this Hindu-Arabic numeral system apparently only became widely used in\nEurope during the late middle ages, between the 13-th and 15-th centuries. In\nthis context, one noteworthy development was the book Liber Abaci published by\nthe Italian mathematician Fibonacci in 1202, which promoted the use of the Hindu-\nArabic numeral system in Europe. Here, it is useful to recall that, by comparison to\nearlier numeral systems, such as the Greco-Roman and Babylonian systems, place-\nvalue decimal numerals not only facilitate elementary arithmetic computations —\ni.e., via the systematic use of carry operations, as discussed in (i)! — but also\n52 SHINICHI MOCHIZUKI\nmake it possible to express all — hence, in particular, infinitely many —\nnatural numbers by means of finitely many symbols\n— i.e., unlike earlier numerical systems, in which only finitely many natural num-\nbers could be expressed using finitely many symbols. This revolutionary importance\nof the development of place-value decimal numerals in India was recognized, for in-\nstance, in writings of the 18-th century French mathematician Laplace. In this\ncontext, it is also of interest to observe that the discussion in (ii) of the interpreta-\ntion of the discussion of (i) in terms of line segments is reminiscent of the discussion\nof the “Euclidean algorithm” in Euclid’s Elements, in which numbers are often rep-\nresented as lengths of line segments. Finally, we recall that the Boolean aspects\ndiscussed in (iii) played an important role in the [well-known!] development of\nmodern digital computers in the 20-th century.\nThe gluings of adjacent line segments discussed in Examples 2.3.2, 2.4.1, 2.4.2\nmay in some sense be regarded as a sort of\noptimized elementary geometric\/combinatorial representation of\nthe essential logical “∧\/∨” structure surrounding a gluing\nin a fashion that is qualitatively entirely structually similar to the gluings that occur\nin inter-universal Teichmüller theory, which will be discussed in more detail in §3\nbelow. The somewhat more numerical\/arithmetic situations discussed in Examples\n2.4.3, 2.4.5, 2.4.6 may also be regarded as entirely elementary representations of\nthis essential logical “∧\/∨” structure surrounding a gluing. On the other hand, the\ngluing operation that occurs in the standard construction of the projective line,\nwhile somewhat less elementary than the previously mentioned examples, also con-\nstitutes an important— and, moreover, still relatively elementary! — representation\nof this essential logical “∧\/∨” structure surrounding a gluing. Moreover, this exam-\nple of the projective line is more directly related to scheme-theoretic arithmetic\ngeometry than the previously mentioned examples.\nExample 2.4.7: The projective line as a gluing of ring schemes along a\nmultiplicative group scheme. In the following discussion, we take k to be a field\nand q ∈ k to be an element such that q3 \n= q [i.e., q \n∈ {0, 1,−1}]. Write k× def= k\\{0},\nA1 for the affine line Spec(k[T ]) over k, Gm for the open subscheme Spec(k[T, T\n−1])\nof A1 obtained by removing the origin. Thus, the standard coordinate T on A1,\nGm determines natural bijections A\n1(k)\n∼→ k, Gm(k) ∼→ k× of the respective sets of\nk-rational points of A1, Gm with corresponding subsets of k. Also, we recall that\nA1 is equipped with a well-known natural structure of ring scheme over k, while\nGm is equipped with a well-known natural structure of [multiplicative] group\nscheme over k.\n(i) Write †A1, ‡A1 for the k-ring schemes given by copies of A1 equipped with\nthe respective labels “†”, “‡”. We regard †A1 as being further equipped with the\nk-rational point †q−1 ∈ †A1(k) ( ∼→ k) corresponding to the multiplicative inverse\nof the element q ∈ k and ‡A1 as being further equipped with the k-rational point\n‡q ∈ ‡A1(k) ( ∼→ k) corresponding to the element q ∈ k. Similarly, we write †Gm,\n‡Gm for the [multiplicative] k-group schemes given by copies of Gm, equipped with\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 53\nthe respective labels “†”, “‡”. Thus, †q−1 ∈ †Gm(k) (⊆ †A1(k)), ‡q ∈ ‡Gm(k) (⊆\n‡A1(k)).\n(ii) Relative to the notation of (i), we observe that\n(ii-a) there exists a unique isomorphism of k-ring schemes †A1 ∼→ ‡A1,\nbut that\n(ii-b) the pairs (†A1, †q−1) and (‡A1, ‡q) are not isomorphic, i.e., as pairs\nconsisting of a k-ring scheme equipped with a k-rational point [cf. our\nassumption that q3 \n= q].\nBy contrast,\n(ii-c) there exists a unique isomorphism of pairs (†Gm, †q−1)\n∼→ (‡Gm, ‡q),\ni.e., of pairs consisting of a [multiplicative] k-group scheme equipped\nwith a k-rational point.\nHere, we observe that the isomorphism (†Gm, †q−1)\n∼→ (‡Gm, ‡q) of (ii-c) does not\nextend [cf. (ii-b)!] to an isomorphism (†A1, †q−1) ∼→ (‡A1, ‡q). In particular,\n(ii-d) the isomorphism of [multiplicative] k-group schemes †Gm\n∼→ ‡Gm is\nnot compatible with the k-ring scheme structures of †A1 (⊇ †Gm),\n‡A1 (⊇ ‡Gm).\nNext, we observe that\n(ii-e) the standard construction of the projective line may be regarded as\nthe result of gluing (†A1, †q−1) to (‡A1, ‡q) along the isomorphism\n(†Gm, †q−1)\n∼→ (‡Gm, ‡q)\nof (ii-c); thus, relative to this gluing, †Gm\n∼→ ‡Gm may be regarded si-\nmultaneously as an open subscheme of †A1 AND [cf. “∧”!] as an open\nsubscheme of ‡A1.\nIn particular, (ii-d), (ii-e) may be summarized as follows:\nthe standard construction of the projective line may be regarded as a\ngluing of two ring schemes along an isomorphism of multiplicative\ngroup schemes that is not compatible with the ring scheme struc-\ntures on either side of the gluing.\nMoreover, we note that, relative to this gluing,\n(ii-f) the notion of a regular function on †A1 cannot be expressed di-\nrectly in terms of the notion of a regular function on ‡A1, whereas\n(ii-g) the notion of a rational function on †A1 can be expressed directly\nin terms of — i.e., in essence, coincides, relative to the above gluing, with\n— the notion of a rational function on ‡A1.\nFinally, we observe that\n(ii-h) if, in the gluing of (ii-e), one arbitrarily deletes the distinct labels\n“†”, “‡”, then the resulting “gluing without labels” amounts to a gluing of\n54 SHINICHI MOCHIZUKI\na single copy of A1 to itself that maps the standard coordinate “T” on A1\n[regarded, say, as a rational function on A1] to T−1; that is to say, such a\n“gluing without labels” results in a contradiction [i.e., since T \n= T−1!],\nunless one passes to some sort of quotient of A1 — which amounts, from\na foundational\/logical point of view, to the introduction of some sort of\nindeterminacy, i.e., to the consideration of some sort of collection of\npossibilities [cf. “∨”!].\n(iii) The discussion of the projective line in (ii) is truly remarkable in that it\ncompletely parallels — i.e., relative to the correspondence\n“−1” ←→ “j2”\nbetween the exponent “−1” in the discussion of (ii) and the exponents “j2”, where\nj ranges from 1 to l\u0002, in the discussion of §3.4 — numerous aspects of the Θ-link\nof inter-universal Teichmüller theory, which we shall discuss in more detail in §3\n[cf., especially, §3.4]. Indeed,\n(iii-a) the isomorphism of (ii-a) may be understood as corresponding to the\nfact that the (Θ±ellNF-)Hodge theaters on either side of the Θ-link in\ninter-universal Teichmüller theory are isomorphic, while\n(iii-b) the observation of (ii-b) may be understood as corresponding to the fact\nthat there is no isomorphism of (Θ±ellNF-)Hodge theaters as in (iii-\na) that maps the Θ-pilot in the domain of the Θ-link [which corresponds\nto “†q−1”] to the q-pilot in the codomain of the Θ-link [which corresponds\nto “‡q”].\nOn the other hand,\n(iii-c) the isomorphism of (ii-c) may be understood as corresponding to the full\npoly-isomorphism of [multiplicative!] F\u0002\u0003×μ-prime-strips that\nconstitutes the Θ-link, while\n(iii-d) the observation of (ii-d) may be understood as corresponding to the fact\nthat the full poly-isomorphism of (iii-c) is not compatible with the ring\nstructures determined by the (Θ±ellNF-)Hodge theaters on either side\nof the Θ-link, i.e., in particular, does not arise from a poly-isomorphism\nbetween these (Θ±ellNF-)Hodge theaters on either side of the Θ-link [cf.\n(iii-b)].\nNext, we observe that\n(iii-e) the gluing of (ii-e) may be understood as corresponding to the glu-\ning constituted by the Θ-link between the (Θ±ellNF-)Hodge theaters on\neither side of the Θ-link, i.e., a gluing along [multiplicative!] F\u0002\u0003×μ-\nprime-strips that is not compatible with the ring structures in the\ndomain and codomain of the Θ-link, but which allows one to obtain a\nsingle F\u0002\u0003×μ-prime-strip, up to isomorphism, that may be interpreted\nsimultaneously as the F\u0002\u0003×μ-prime-strip arising from the Θ-pilot in\nthe domain of the Θ-link AND [cf. “∧”!] as the F\u0002\u0003×μ-prime-strip\narising from the q-pilot in the codomain of the Θ-link.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 55\nHere, we recall that this crucial logical AND “∧” property of the Θ-link is the\ncentral theme of the present paper [cf. the discussion of Examples 2.4.1, 2.4.2, 2.4.3,\n2.4.4, 2.4.5, 2.4.6!]. Next, we observe that\n(iii-f) the observation of (ii-f) may be understood as corresponding to the fact\nthat, at least from an a priori point of view, there is no natural way to\nexpress the Θ-pilot of the (Θ±ellNF-)Hodge theater in the domain of the\nΘ-link, relative to the gluing of (iii-e), in terms of the (Θ±ellNF-)Hodge\ntheater in the codomain of the Θ-link, while\n(iii-g) the observation of (ii-g) may be understood as corresponding to the si-\nmultaneous holomorphic expressibility (SHE) property of the mul-\ntiradial representation of the Θ-pilot [cf. [IUTchIII], Remark 3.11.1,\n(iii); [IUTchIII], Remark 3.9.5, (viii), (ix)], which allows one to express\nthe Θ-pilot of the (Θ±ellNF-)Hodge theater in the domain of the Θ-link,\nrelative to the gluing of (iii-e), in terms of the (Θ±ellNF-)Hodge theater\nin the codomain of the Θ-link [cf. also the discussion of (iv), (v), below]..\nFinally, we note that\n(iii-h) the “gluing without labels” discussed in (ii-h) may be understood as cor-\nresponding to the oversimplified version “RCS-IUT” of inter-universal\nTeichmüller theory obtained by implementing the RCS-identifications of\nRCS-redundant copies of objects [cf. the discussion of §1.2, Example\n2.4.5], which leads to an immediate contradiction, unless one introduces\nsome sort of quotient\/indeterminacy, i.e., which amounts to the con-\nsideration of some sort of collection of possibilities [cf. “∨”!].\nIn particular, relative to this remarkably close structural resemblance between\nthe gluing that appears in the standard construction of the projective line and\nthe gluing constituted by the Θ-link of inter-universal Teichmüller theory, the\ncentral assertion\n“IUT ⇔ RCS-IUT”\nof the RCS [cf. the discussion of Example 2.4.5] may be understood as correspond-\ning to the assertion of an “obvious equivalence” [cf. the discussion of §1.3]\nbetween\n· the projective line, on the one hand, and\n· the affine line regarded up to some sort of identification of the standard\ncoordinate “T” on the affine line with its inverse, on the other.\n(iv) From the point of view of the analogy discussed in (iii) between the gluing\nconstruction of the projective line and inter-universal Teichmüller theory, perhaps\nthe closest nontrivial analogue, in the case of the projective line, to the multiradial\nrepresentation of the Θ-pilot in inter-universal Teichmüller theory is the group\nof projective general linear [i.e., “PGL2”] symmetries of the projective line\n[cf. also the discussion of (v) below; the discussion of Example 3.10.1, (i), below].\nThat is to say, although there is also an analogy, discussed in (iii-g), with the\nobservation of (ii-g), the content of (ii-g) is rather formal\/trivial. By contrast, the\nPGL2-symmetries of the projective line are somewhat less trivial, especially from\nthe point of view of the gluing construction of the projective line discussed in (ii)\n[cf. also the discussion of (v) below; the discussion of Example 3.10.1, (i), below].\n56 SHINICHI MOCHIZUKI\n(v) The PGL2-symmetries of the projective line are, in some sense, especially\ninteresting in the case where one takes k to be the field C of complex numbers, and\none restricts to the subgroup\nPU2 ⊆ PGL2(C)\ngiven by the image of the unitary matrices, i.e., the projective unitary group. Thus,\nas is well-known, one may think of PU2 as the group of isometric symmetries of the\nRiemann surface associated to the projective line over C equipped with the Fubini-\nStudy metric. The underlying topological space of this Riemann surface may be\nnaturally identified with the sphere S2. The geodesics associated to the Fubini-Study\nmetric then correspond to great circles on the sphere S2 [cf., e.g., the illustration\nof [GeoSph]]. In particular, the geodesics that pass through the north\/south poles\nof S2 may be thought of as lines of longitude. In the current metrized situation, it\nis natural to think of S2 as being obtained not via a gluing of the complement of\nthe north pole to the complement of the south pole [i.e., as in (ii-e)], but rather as\nbeing obtained via a gluing\nS2 ⊇ H+ ⊇ E def= H+ ∩H− ⊆ H− ⊆ S2\nof the northern hemisphere H+ ⊆ S2 of S2 to the southern hemisphere H− ⊆ S2 of\nS2 along the equator E ⊆ S2. Here, we note that the gluing of (ii-e) — which yields\na single rational function on the projective line that corresponds simultaneously to\nthe standard coordinate “†T” on †A1 AND to the standard coordinate “‡T−1” on\n‡A1 — may be thought of, in the current metrized situation, as corresponding to\nthe following [at first glance, self-contradictory!] phenomenon:\n(OrFlw) an oriented flow along the equator — which may be thought of phys-\nically as a sort of wind current — that flows from east to west appears\nsimultaneously to be flowing, from the point of view of the northern hemi-\nsphere H+ ⊆ S2, in the clockwise direction AND, from the point of view\nof the southern hemisphere H− ⊆ S2, in the counterclockwise direction.\nNext, let us recall that — unlike S2! — both H+ and H− may be thought of as\nclosed discs in the plane. Thus, in summary,\n(GdsFlw) the geodesic geometry of the Fubini-Study metric — i.e., in essence,\nthe (PGL2(C) ⊇) PU2-symmetries of S2 — allow one, by considering\nthe geodesic flow along lines of longitude, to represent, up to a\nrelatively mild distortion, the entirety of S2, i.e., including H− ⊆ S2, as a\nsort of extension\/deformation of the closed disc H+.\nIndeed, (GdsFlw) is precisely the principle that is applied to represent, using lines\nof longitude, the globe [i.e., in the sense of the surface of the planet earth] via\na rectangular, planar, cartesian map [i.e., in the sense of cartography]! Note,\nmoreover, that\n(NoLbDlt) although the approach of (GdsFlw) gives rise to a certain relatively mild\ndegree of distortion in the representation of H− in terms of H+, it does\nnot involve any sort of naive identification of the closed discs H+, H−,\ni.e., any sort of arbitrary label deletion, in the style of (ii-h).\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 57\nThe interpretation discussed in (GdsFlw) and (NoLbDlt) of the (PGL2(C) ⊇) PU2-\nsymmetries of S2 may be understood as strongly suggesting a nontrivial analogy\nbetween these symmetries of S2 and the multiradial representation of the Θ-\npilot in inter-universal Teichmüller theory [cf. the analogy between multiradiality\nand connections\/parallel transport\/crystals discussed in [Alien], §3.1, (iv), (v), as\nwell as §3.5, §3.10, below].\nSection 3: The logical structure of inter-universal Teichmüller theory\nIn the present §3, we give a detailed exposition of the essential logical struc-\nture of inter-universal Teichmüller theory, with a special focus on issues related to\nRCS-redudancy. From a strictly rigorous point of view, this exposition assumes a\nsubstantial level of knowledge and understanding of the technical details of inter-\nuniversal Teichmüller theory [which are surveyed, for instance, in [Alien]]. On the\nother hand, in a certain qualitative sense, the discussion of the present §3 may in\nfact be understood, at a relatively elementary level, via the analogies that we dis-\ncuss with the topics covered in §2. Indeed, in this context, it should be emphasized\nthat, despite the relatively novel nature of the set-up of inter-universal Teichmüller\ntheory,\nthe essential mathematical content that lies at the heart of all of the issues\ncovered in the present §3 concerns entirely well-known mathematics at the\nadvanced undergraduate or beginning graduate level [i.e., the topics covered\nin §2].\n§3.1. One-dimensionality via identification of RCS-redundant copies\nInter-universal Teichmüller theory concerns the explicit description of the re-\nlationship between various possible intertwinings — namely,\nthe “Θ”- and “q-” intertwinings\n— between the two underlying combinatorial\/arithmetic dimensions of a\nring [cf., e.g., [Alien], §2.11; [Alien], §3.11, (v), as well as the discussion of §3.9\nbelow]. There are many different ways of thinking about these two underlying\ncombinatorial\/arithmetic dimensions of a ring; one way to understand these two\ndimensions is to think of them as corresponding, respectively, to the unit group\nand value group of the various local fields that appear as completions of a number\nfield at one of its valuations.\nIn more technical language, this sort of decomposition into unit groups and\nvalue groups may be seen in the F\u0002\u0003×μ-prime-strips that appear in the Θ-link\nof inter-universal Teichmüller theory. Thus, if one thinks in terms of such F\u0002\u0003×μ-\nprime-strips, then inter-universal Teichmüller theory may be summarized as follows:\n(2-Dim) The main content of inter-universal Teichmüller theory is an explicit\ndescription, up to certain relatively mild indeterminacies, of the Θ-\nintertwining on the [two-dimensional!] F\u0002\u0003×μ-prime-strips that ap-\npear in the Θ-link in terms of the q-intertwining on these F\u0002\u0003×μ-prime-\nstrips by means of the log-link and various types of Kummer theory\nthat are used to relate Frobenius-like and étale-like structures.\n58 SHINICHI MOCHIZUKI\nIn particular, the essential mathematical content of inter-universal Teichmüller the-\nory concerns an a priori variable relationship between the two underlying com-\nbinatorial\/arithmetic dimensions of a ring.\nPut another way, if one arbitrarily “crushes” these two dimensions into a single\ndimension — i.e., in more technical language, assumes that\n(1-Dim) there exists a consistent choice of a fixed relationship between these\ntwo dimensions of (2-Dim), so that these two dimensions may, in effect,\nbe regarded as a single dimension\n— then one immediately obtains a superficial contradiction [cf. the discussion of\nExample 3.1.1, (i-b), (ii-b), below]. Indeed, this is one of the central assertions of the\nRCS [cf. the discussion following Example 3.1.1]. This is not a “new” observation,\nbut rather, in some sense, the starting point of inter-universal Teichmüller theory,\ni.e., the initial motivation for regarding the relationship between the two underlying\ncombinatorial\/arithmetic dimensions of a ring as being variable, rather than fixed.\nExample 3.1.1: Elementary models of gluings and intertwinings. In the\nfollowing, we shall write V for the topological group R>0. Let x, y ∈ V be [not\nnecessarily distinct!] elements of V and Y ⊆ V a nonempty subset of V . Let V rl,\nV im be two distinct labeled copies of V , which we think of as corresponding to the\npositive portions of the real and imaginary axes in the complex plane.\n(i) Let †V , ‡V be two not necessarily distinct copies of V . We shall write\n†y ∈ †V , ‡x ∈ ‡V for the respective elements determined by y, x ∈ V .\n(i-a) Suppose that †V , ‡V are distinct copies of V . Write W for the topolog-\nical space obtained by gluing †V , ‡V along the homeomorphic subspaces\n{†y} ⊆ †V , {‡x} ⊆ ‡V . Then observe that this construction of W is well-\ndefined and free of any internal contradictions. Moreover, the existence of\nW does not imply any nontrivial conclusions concerning x and y.\nNote the sharp contrast between the situation discussed in (i-a) and the following\nsituation:\n(i-b) Suppose that †V , ‡V are in fact the same copy of V , i.e., ∗V def= †V = ‡V .\nConsider the assertion that\nthe topological space ∗V is obtained by gluing †V , ‡V along the\nhomeomorphic subspaces {†y} ⊆ †V , {‡x} ⊆ ‡V .\nThen observe that this assertion concerning ∗V is well-defined and free of\ninternal contradictions only in the case where x = y. That is to say, the\nexistence of a topological space ∗V as described in the above assertion\nimplies the nontrivial conclusion that x = y, or, equivalently, a “con-\ntradiction” to the assertion that x \n= y.\nOne may also consider the following variant of (i-b):\n(i-c) One replaces {†y} ⊆ †V in (i-b) by the nonempty subset †Y ⊆ †V [i.e.,\ndetermined by Y ⊆ V ], where one thinks of this subset as a set of “possible\ny’s”. The resulting “assertion” then becomes a corresponding collection\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 59\nof assertions related by logical OR “∨’s”, and the final nontrivial con-\nclusion is that x ∈ Y .\n(ii) The elementary models presented in (i) may be interpreted as essentially\nequivalent representations of various models of “holomorphic structures” [cf.\nthe discussion below of (InfH), as well as Examples 3.3.1, 3.3.2] — i.e., in the ter-\nminology of the discussion preceding the present Example 3.1.1, “intertwinings”\n— between the “real” and “imaginary” dimensions V rl, V im. Here, we think of\n“holomorphic structures”\/“intertwinings” as being defined by assignments\nV rl \u0016 1rl \b→ ? ∈ V im\n[where 1rl ∈ V rl denotes the element determined by 1 ∈ V ], corresponding to “coun-\nterclockwise rotations by 90 degrees”, or, alternatively, “multiplication by\n√−1”. In-\ndeed, let †V rl, ‡V rl be two not necessarily distinct copies of V rl; †V im, ‡V im two\nnot necessarily distinct copies of V im. We shall write †yim ∈ †V im, ‡xim ∈ ‡V im\nfor the respective elements determined by y, x ∈ V . Then the discussion of (i-a)\nmay be translated into a discussion concerning intertwinings by arguing as follows:\n(ii-a) Suppose that †V rl, ‡V rl are distinct copies of V rl; †V im, ‡V im are\ndistinct copies of V im. Here, we think of †V rl, †V im as being equipped\nwith the intertwining given by taking “?” to be †yim ∈ †V im; we think of\n‡V rl, ‡V im as being equipped with the intertwining given by taking “?” to\nbe ‡xim ∈ ‡V im. Then one applies (i-a), relative to the correspondences\n†V im ←→ †V , ‡V im ←→ ‡V . This yields a gluing as in (i-a) that is well-\ndefined and free of any internal contradictions. Moreover, the existence\nof such a gluing does not imply any nontrivial conclusions concerning x\nand y.\nIn a similar vein:\n(ii-b) Suppose that †V rl, ‡V rl are in fact the same copy of V rl, i.e., ∗V rl def=\n†V rl = ‡V rl, and that †V im, ‡V im are in fact the same copy of V im, i.e.,\n∗V im def= †V im = ‡V im. Then one applies (i-b), relative to the correspon-\ndence ∗V im ←→ ∗V . This yields an assertion concerning a gluing as in\n(i-b) — i.e., in the language of the present discussion, concerning a coin-\ncidence of the intertwining on †V rl, †V im with the intertwining on ‡V rl,\n‡V im — that is well-defined and free of internal contradictions only in the\ncase where x = y. That is to say, the existence of such a gluing implies\nthe nontrivial conclusion that x = y, or, equivalently, a “contradiction”\nto the assertion that x \n= y.\n(ii-c) One replaces {†yim} ⊆ †V im in (ii-b) [cf. also the notation of (ii-a)] by\nthe nonempty subset †Y im ⊆ †V im [i.e., the subset determined by Y ⊆ V ],\nwhere one thinks of this subset as a set of “possible y’s”. The resulting\n“assertion” then becomes a corresponding collection of assertions related\nby logical OR “∨’s”, and the final nontrivial conclusion is that x ∈ Y .\n(iii) Relative to the analogy with inter-universal Teichmüller theory, we have\n60 SHINICHI MOCHIZUKI\ncorrespondences with objects that appear in the elementary models of (ii) as follows:\nV rl ←→ the value group portion of an F\u0002\u0003×μ-prime-strip;\nV im ←→ the unit group portion of an F\u0002\u0003×μ-prime-strip;\n†\/‡ ←→ (Θ±ellNF-)Hodge theaters in the domain\/codomain of the Θ-link;\nintertwinings involving “y” ←→ the Θ-intertwinings;\nintertwinings involving “x” ←→ the q-intertwinings;\n[cf. the discussion at the beginning of §3.4]. Here, we note that from the point\nof view of intertwinings, the unit group portion corresponding to “V im” must be\nunderstood as being log-shifted by −1, relative to the value group portion corre-\nsponding to “V rl” [cf. the discussion below of (InfH), as well as Examples 3.3.1,\n3.3.2]. That is to say, if the value group portion corresponding to “V rl” is located\nat the coordinate (n,m) of the log-theta-lattice, then the unit group portion corre-\nsponding to “V im” must be understood as being located at the coordinate (n,m−1)\nof the log-theta-lattice. In particular, the unit group and value group portions cor-\nresponding to a pair “(V rl, V im)” belong to different F\u0002\u0003×μ-prime-strips. From\nthe point of view of the discussion of (1-Dim), the “consistent choice of a fixed\nrelationship” corresponds to the coincidence of intertwinings in (ii-b), while the\nresulting “superficial contradiction” corresponds to the “contradiction” discussed\nin (ii-b). On the other hand, the “explicit description”\/“variable relationship” of\n(2-Dim), which leads naturally to a numerical estimate\/inequality concerning log-\nvolumes [cf. Example 2.4.5, (ii)], corresponds to the situation involving various\npossibilities discussed in (ii-c), which leads to the nontrivial conclusion “x ∈ Y ”\n[cf. the discussion of “closed loops” in (Stp7), (Stp8) of §3.10 below; the discus-\nsion of (DltLb) in §3.11 below; the discussion of [IUTchIII], Remark 3.12.2, (ii)].\n(iv) Finally, we observe in passing that the fixed intertwining of (ii-b) [cf.\nalso the discussion of (ii-b) in (iii), as well as the discussion of (FxRng), (FxEuc),\n(FxFld) below] may be regarded as being analogous to the well-known classical\nholomorphic approach to the theory of moduli of [one-dimensional] complex\ntori, that is to say, in which one works with a copy of the upper half-plane “H”\nwith a fixed holomorphic structure and thinks of the moduli of complex tori as a\n“variation of period matrices” [i.e., the holomorphic parameter “z ∈ H”, which may\nbe taken, in the notation of (ii-b), to be “ix” or “iy”]. By contrast, the situation\ninvolving the set “†Y im ⊆ †V im” discussed in (ii-c) may be regarded as analogous\nto the [real analytic] Teichmüller approach to the theory of moduli of complex\ntori [cf. the discussion of Example 3.3.1], i.e., in which the holomorphic structure\nis subject to Teichmüller dilations [corresponding to various elements in the set\n†Y im], relative to the fixed “real analytic” pair given by †V rl, †V im.\nOne central assertion of the RCS — which appears, for instance, in certain\n10pp. manuscripts written by adherents of the RCS [cf., especially, the discussion\nof the final page and a half of the files “[SS2018-05]”, “[SS2018-08]” available at\nthe website [Dsc2018]] — is to the effect that the existence, as in (1-Dim), of a\nconsistent choice of a fixed relationship between the two dimensions of (2-Dim)\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 61\nmay be derived as a consequence — i.e., in more succinct notation,\n(RC-FrÉt), (RC-log), (RC-Θ) “ =⇒ ” (1-Dim)\n— of certain “redundant copies assertions”, as follows:\n(RC-FrÉt) the Frobenius-like and étale-like versions of objects in inter-universal\nTeichmüller theory are “redundant”, i.e., may be identified with one\nanother without affecting the essential logical structure of the theory;\n(RC-log) the (Θ±ellNF-)Hodge theaters on either side of the log-link in inter-\nuniversal Teichmüller theory are “redundant”, i.e., may be identified\nwith one another without affecting the essential logical structure of the\ntheory;\n(RC-Θ) the (Θ±ellNF-)Hodge theaters on either side of the Θ-link in inter-\nuniversal Teichmüller theory are “redundant”, i.e., may be identified\nwith one another without affecting the essential logical structure of the\ntheory.\nIn the remainder of the present §3 [cf., especially, §3.2, §3.3, §3.4], we discuss in more\ndetail the falsity of each of these “RCS-redundancy” assertions [i.e., (RC-FrÉt),\n(RC-log), (RC-Θ)].\nHere, it should be noted that this falsity of (RC-FrÉt), (RC-log), (RC-Θ) is\nby no means a difficult or subtle issue, but rather a sort of matter of “belaboring\nthe intuitively obvious” from the point of view of mathematicians who are thor-\noughly familiar with inter-universal Teichmüller theory. Nevertheless, as discussed\nin [Rpt2018], §17, it is a pedagogically meaningful exercise to write out and discuss\nthe details surrounding this sort of issue. Moreover, as discussed in §1.5 of the\npresent paper, it is desirable from a historical point of view to produce detailed,\nexplicit, and readily accessible written expositions concerning this sort of issue.\nThis state of affairs prompts the following question:\nWhy do adherents of the RCS continue to insist on asserting the validity\nof these assertions (RC-FrÉt), (RC-log), (RC-Θ)?\nAny sort of complete, definitive answer to this question lies beyond the scope of\nthe present paper. On the other hand, it seems natural to conjecture that one\nfundamental motivation for these assertions of RCS-redundancy may be found in\nthe fact that\n(FxRng) many arithmetic geometers have only experienced working in situations\nwhere all schemes — or, alternatively, rings — that appear in a theory\nare regarded as belonging to a single category that is fixed throughout\nthe theory, hence are related to another via ring homomorphisms, i.e.,\nin such a way that the ring structure of the various rings involved is\nalways respected [cf. the discussion of §1.5, as well as the discussion of\n§3.8 below].\nIt is not difficult to imagine that the heuristics and intuition that result from years\n[or decades!] of immersive experience in such mathematical situations could create\n62 SHINICHI MOCHIZUKI\na mindset that is fertile ground for the RCS-redundancy assertions that will be\ndiscussed in detail in the remainder of the present §3 [cf., especially, §3.2, §3.3,\n§3.4].\nFinally, we observe that this situation is, in certain respects, reminiscent of the\nsituation that occurred in the late 19-th century with regard to such novel [i.e., at\nthe time] notions as the notion of an abstract manifold or an abstract Riemann\nsurface. That is to say,\n(FxEuc) from the point of view of anyone for whom it is a “matter of course”\nor “common sense” that all geometry must take place within some fixed,\nstatic ambient Euclidean space — such as, for instance, the complex\nplane — such abstract geometric notions as the notion of an abstract\nmanifold or abstract Riemann surface might come across as deeply dis-\nturbing and unlikely to be of use in any substantive mathematical sense\n[cf. the discussion of §1.5; the discussion of [IUTchI], §I2].\nIn this context, it is of interest — especially from a historical point of view —\nto recall that, in some sense, the most fundamental classical example of such an\nabstract geometry is the Riemann surface that arises by applying the technique of\nanalytic continuation to the complex logarithm, i.e., which may be regarded\nas a sort of distant ancestor [cf. the discussion of [IUTchI], Remark 5.1.4; [Alien],\n§3.3, (vi)] of the log-link of inter-universal Teichmüller theory. Another [in fact\nclosely related!] fundamental classical example of such an abstract geometry is the\nhyperbolic geometry of the upper half-plane, which may also be regarded as\na sort of distant ancestor of numerous aspects of inter-universal Teichmüller theory\n[cf. (InfH) and Example 3.3.2 in §3.3 below, as well as the discussion of [IUTchI],\nRemark 6.12.3, (iii); [IUTchIII], Remark 2.3.3, (ix), (x)].\nAnother historically important instance of this sort of situation may be seen\nin the introduction, in the early 19-th century, of Galois groups — i.e., of [finite]\nautomorphism groups of abstract fields — as a tool for investigating the roots of\npolynomial equations. That is to say,\n(FxFld) until the advent of Galois groups\/abstract fields, the issue of investigat-\ning the roots of polynomial equations was always regarded — again as a\n“matter of course” or “common sense” — as an issue of investigating var-\nious “exotic numbers” inside some fixed, static ambient field such as\nthe field of complex numbers; moreover, from this more classical “common\nsense” point of view, the idea of working with automorphisms of abstract\nfields — i.e., fields that are not constrained [since such constraints would\nrule out the existence of nontrivial automorphisms!] to be treated as sub-\nsets of some fixed, static ambient field — might come across as deeply\ndisturbing and unlikely to be of use in any substantive mathematical sense\n[cf. the discussion of §1.5].\nOn the other hand, from the point of view of inter-universal Teichmüller theory,\nthis radical transition\nroots as concrete numbers \u0004 Galois groups\/abstract fields\nthat occurred in the early 19-th century may be regarded as a sort of distant ancestor\nof the transition\nGalois groups\/abstract fields \u0004 abstract groups\/anabelian algorithms\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 63\nthat occurs in inter-universal Teichmüller theory [cf. the discussion at the beginning\nof §3.2 below; the discussion of §3.8 below; the discussion of the final portion of\n[Alien], §4.4, (i)].\n§3.2. RCS-redundancy of Frobenius-like\/étale-like versions of objects\nWe begin by recalling that (Θ±ellNF-)Hodge theaters — i.e., lattice points\nin the log-theta-lattice — give rise to both Frobenius-like and étale-like objects.\nWhereas the datum of a Frobenius-like object depends, a priori, on the coordinates\n“(n,m)” of the (Θ±ellNF-)Hodge theater from which it arises, étale-like objects\nsatisfy various [horizontal\/vertical] coricity properties to the effect that they\nmap isomorphically to corresponding objects in a vertically [in the case of vertical\ncoricity] or horizontally [in the case of horizontal coricity] neighboring (Θ±ellNF-\n)Hodge theater of the log-theta-lattice [cf. the discussion of Example 3.2.2, (i),\n(iv), below; [Alien], §2.7, (i), (ii), (iii), (iv); [Alien], §2.8, 2Fr\/ét; [Alien], §3.3, (ii),\n(vi), (vii); [Rpt2018], §15]. Here, we recall that étale-like objects correspond, for\nthe most part, to\narithmetic fundamental groups — such as, for instance, the étale\nfundamental group “π1(X)” of a hyperbolic curve X over a number field\nor mixed characteristic local field\n— or, more generally, to objects that may be reconstructed from such arithmetic\nfundamental groups, so long as the object is regarded as being equipped with auxil-\niary data consisting of the arithmetic fundamental group from which it was recon-\nstructed, together with the reconstruction algorithm that was applied to reconstruct\nthe object. Here, we recall that, in this context, it is of fundamental importance\nthat these arithmetic fundamental groups be treated simply as abstract topological\ngroups [cf. the discussion of §3.8 below for more details]. Étale-like objects also\nsatisfy a\ncrucial symmetry property with respect to permutation of adjacent\nvertical lines of the log-theta-lattice\n[cf. Example 3.2.2, (ii), (iv), below; [Alien], §3.2; the discussion surrounding Fig.\n3.12 in [Alien], §3.6, (i)]. That is to say, in summary,\nthe crucial coricity\/symmetry properties satisfied by étale-like objects\n— which are, in essence, a formal consequence of treating the arithmetic\nfundamental groups that appear as abstract topological groups [cf.\nthe discussion of §3.8 below for more details] — play a central role in\nthe multiradial algorithms of inter-universal Teichmüller theory [i.e.,\n[IUTchIII], Theorem 3.11] and are not satisfied by Frobenius-like ob-\njects\n— cf., the discussion of Example 3.2.2, (i), (ii), (iv), below; [Alien], §2.7, (ii), (iii);\n[Alien], §3.1, (iii); [Alien], Example 3.2.2; [Rpt2018], §15, (LbΘ), (Lblog), (LbMn),\n(EtFr), (EtΘ), (Etlog), (EtMn).\nOn the other hand, once one implements the RCS-identifications discussed\nin (RC-log), (RC-Θ), there is, in effect, “only one” (Θ±ellNF-)Hodge theater in the\nlog-theta-lattice, so all issues of determining relationships between corresponding\n64 SHINICHI MOCHIZUKI\nobjects in (Θ±ellNF-)Hodge theaters at distinct coordinates “(n,m)” of the log-\ntheta-lattice appear, at first glance, to have been “trivially resolved”. Put another\nway,\nonce one implements the RCS-identifications of (RC-log), (RC-Θ), even\nFrobenius-like objects appear, at first glance, to satisfy all possible coric-\nity\/symmetry properties, i.e., at a more symbolic level,\n(RC-log), (RC-Θ) “ =⇒ ” (RC-FrÉt).\nIn particular, the assertions of the RCS discussed in §3.1 and the present §3.2 may\nbe summarized, at a symbolic level, as follows:\n(RC-log), (RC-Θ) “ =⇒ ” (RC-FrÉt), (RC-log), (RC-Θ) “ =⇒ ” (1-Dim).\nIn fact, however, the RCS-identifications of (RC-log), (RC-Θ) do not resolve\nsuch issues [i.e., of relating corresponding objects in (Θ±ellNF-)Hodge theaters at\ndistinct coordinates “(n,m)” of the log-theta-lattice] at all [cf. the discussion of\nsymmetries in Example 2.3.1, (iii)!], but rather merely have the effect of\ntranslating\/reformulating such issues of relating corresponding objects in\n(Θ±ellNF-)Hodge theaters at distinct coordinates “(n,m)” of the log-theta-\nlattice into issues of tracking the effect on objects in (Θ±ellNF-)Hodge the-\naters as one moves along the paths constituted by various composites\nof Θ- and log-links.\nOn the other hand, at a purely formal level,\nthe discussion given above of the falsity of (RC-FrÉt) — i.e., as a conse-\nquence of the crucial coricity\/symmetry properties discussed above —\nis, in some sense, predicated on the falsity of (RC-log), (RC-Θ).\nThis falsity of (RC-log), (RC-Θ) will be discussed in detail in §3.3, §3.4, below.\nIn this context, it is useful to observe that the situation surrounding the Θ-\nlink and (RC-Θ), (RC-FrÉt) (respectively, the log-link and (RC-log), (RC-FrÉt))\nis structurally reminiscent of the object J discussed in Examples 2.3.2, 2.4.1,\n2.4.2 [cf. also the correspondences discussed in Example 2.4.5, (ii); the discussion\nof [IUTchIII], Remark 1.2.2, (vi), (vii)], i.e., if one regards\n(StR1) the domain of the Θ- (respectively, log-) link as corresponding to †I,\n(StR2) the codomain of the Θ- (respectively, log-) link as corresponding to ‡I,\n(StR3) the gluing data — i.e., a certain F\u0002\u0003×μ-prime-strip (respectively, F-\nprime-strip) — that arises from the domain (Θ±ellNF-)Hodge theater of\nthe Θ- (respectively, log-) link as corresponding to †β = γJ,\n(StR4) the gluing data — i.e., a certain F\u0002\u0003×μ-prime-strip (respectively, F-\nprime-strip) — that arises from the codomain (Θ±ellNF-)Hodge theater of\nthe Θ- (respectively, log-) link as corresponding to γJ =\n‡α,\n(StR5) the étale-like objects that are coric with respect to the Θ- (respectively,\nlog-) link as corresponding to the glued differential discussed in (AOD3),\nand\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 65\n(StR6) the RCS-identification of (RC-Θ) (respectively, (RC-log)) as correspond-\ning to the operation of passing to the quotient\nJ \u0003 M = J\/〈†I ∼ ‡I〉 ∼→ L = I\/〈α ∼ β〉.\nThis strong structural similarity will play an important role in the discussion\nof §3.3, §3.4, below.\nFinally, we observe that the portion, i.e., (StR5), of this strong structural\nsimilarity involving the glued differential discussed in (AOD3) is particularly of\ninterest in the context of the discussion of [Alien], §2. That is to say, as discussed in\nthe first paragraph of [Alien], §2.6, étale-like objects in inter-universal Teichmüller\ntheory play an analogous role to the role played by tangent bundles\/sheaves of\ndifferentials in the\n(InvHt) special case of the invariance of the height under isogenies between\nabelian varieties [due to Faltings] discussed in [Alien], §2.3, §2.4 [cf. also\n[Rpt2018], §16, (DiIsm), as well as the discussion of Example 3.2.1 below;\nthe discussion of §3.5 below], as well as in the\n(FrDff) discussion of differentiation of p-adic liftings of the Frobenius mor-\nphism given in [Alien], §2.5.\nThe efficacy of the technique of considering induced maps on differentials in the\nvarious examples discussed in [Alien], §2.3, §2.4, §2.5, may also be observed in\nthe discussion of the fundamental theorem of calculus in §2.2, as well as in the\ncontext of (RC-FrÉt) and (StR5). In light of the central importance of (InvHt),\nas well as the closely related coricity\/symmetry\/commutativity properties of the\nlog-theta-lattice [cf. the discussion at the beginning of the present §3.2, as well\nas the discussion at the beginning §3.3 below], in the essential logical structure of\ninter-universal Teichmüller theory, we pause to give a brief review\/exposition of\n(InvHt) and these closely related coricity\/symmetry\/commutativity properties, in\nthe following Examples 3.2.1, 3.2.2.\nExample 3.2.1: Global multiplicative subspaces and bounds on heights.\n(i) Let p be a prime number, K a finite extension of the field Qp of p-adic\nnumbers, E an elliptic curve over K with bad multiplicative — i.e., in other words,\nnonsmooth semi-stable — reduction over the ring of integers OK of K. Write\nmK ⊆ OK for the maximal ideal of OK . Thus, E is a Tate curve and hence [cf.\nthe theory of [Mumf2]] may be represented, using the theory of formal schemes, as\na sort of quotient\n“Gm\/q\nZ\nE”\nof the multiplicative group scheme Gm over K by the subgroup generated by the\n[nonzero] q-parameter qE ∈ mK of the elliptic curve E. Let l be a prime number.\nThen the subscheme μl of l-torsion points of Gm determines, via the above p-adic\nquotient representation, a canonical exact sequence\n1 −→ μl −→ E[l] −→ Z\/lZ −→ 1\n66 SHINICHI MOCHIZUKI\n— where we write E[l] ⊆ E for the subgroup scheme of l-torsion points of E,\nand we observe that generator “qE” of the group of deck transformations of the\nabove quotient representation determines a canonical generator γl ∈ (E[l]\/μl)(K),\nup to multiplication by ±1, of (E[l]\/μl)(K), i.e., a canonical isomorphism, up to\nmultiplication by ±1, of the quotient group scheme E[l]\/μl with [the group scheme\nover K determined by] Z\/lZ. Thus, in summary,\nthe subgroup scheme E[l] ⊆ E of l-torsion points of E is equipped with\na canonical multiplicative subspace μl (↪→ E[l]), as well as with a\ncanonical generator, up to multiplication by ±1, of the quotient E[l]\/μl.\n(ii) Let (E, 0E) be the pointed Riemann surface determined by an elliptic curve\nover the field of complex numbers C. Write E[∞] def= πtop1 (E) for the [usual topologi-\ncal] fundamental group of E, relative to the basepoint determined by the origin 0E of\nthe given elliptic curve. Thus, E[∞] is a free abelian group on two generators. Let\nM ⊆ E[∞] be a rank one [free abelian] subgroup such that E[∞]\/M is torsion-free.\nThen M corresponds to an infinite covering\nEM → E\nof E such that any point 0EM of EM that lifts 0E determines, up to possible compo-\nsition with the inversion automorphism, an isomorphism EM\n∼→ C× of complex Lie\ngroups, relative to the unique complex Lie group structure on the pointed Riemann\nsurface (EM, 0EM) that lifts the unique complex Lie group structure on the pointed\nRiemann surface (E, 0E). In particular, we conclude that the choice of M [together\nwith a choice of an isomorphism EM\n∼→ C× of the sort just discussed] determines a\ncomplex holomorphic quotient representation\n“C×\/qZE”\nof the given elliptic curve, together with an exact sequence\n1 −→ M −→ E[∞] −→ Z −→ 1\n— where we observe that the rank one free abelian group E[∞]\/M (∼= Z) is equipped\nwith a unique choice of generator γ∞ ∈ E[∞]\/M, up to multiplication by ±1. Thus,\nthe p-adic quotient representation, as well as the associated exact sequence and\ncanonical generator [up to multiplication by ±1], discussed in (i) may be regarded as\np-adic analogues of the complex holomorphic quotient representation and associated\nexact sequence\/canonical generator [up to multiplication by ±1] discussed in the\npresent (ii).\n(iii) Let F be a number field [i.e., a finite extension of the field Q of rational\nnumbers] and E an elliptic curve over F . Write E[l] ⊆ E for the subgroup scheme\nof l-torsion points of E. Then, in general,\nE does not necessarily admit a global multiplicative subspace (GMS)\nM ⊆ E[l] or a global canonical generator (GCG) γ ∈ (E[l]\/M)(F ),\ni.e., a subgroup scheme M ⊆ E[l] or generator γ ∈ (E[l]\/M)(F ) that\nrestricts to the multiplicative subspace or canonical generator discussed in\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 67\n(i) at each nonarchimedean prime of F where E has bad multiplicative\nreduction.\nOn the other hand, let us suppose, for the remainder of the present (iii), that\nE does admit a global multiplicative subspace (GMS) M ⊆ E[l].\nWrite E∗ def= E\/M . Thus, we have an isogeny\nφ : E → E∗\nthat, in light of the discussion of (i), together with the fact that M ⊆ E[l] is a\nGMS, corresponds to the isogeny\n“Gm\/q\nZ\nE” −→ “Gm\/ql·ZE ”\ngiven by raising to the l-th power on Gm at each nonarchimedean prime of F where\nE has bad multiplicative reduction. In particular, at each nonarchimedean prime v\nof F where E has bad multiplicative reduction, the q-parameter qE∗,v of E\n∗ at v is\nthe l-th power of the q-parameter qE,v of E at v, i.e.,\nqE∗,v = q\nl\nE,v.\nIn particular, if we write log(q(−)) ∈ R for the normalized arithmetic degree “deg(−)”\n[cf. the discussion preceding [GenEll], Definition 1.2] of the arithmetic divisor deter-\nmined by the q-parameters at the nonarchimedean primes of an elliptic curve over a\nnumber field where the elliptic curve, which we denote “(−)”, has bad multiplicative\nreduction, then we obtain the relation\nlog(qE∗) = l · log(qE) ∈ R\nbetween log(qE), log(qE∗) ∈ R.\n(iv) We continue to consider the situation discussed in (iii). Write\nMell ⊇ Mell\nfor the compactified moduli stack of elliptic curves — or, equivalently, the moduli\nstack of pointed stable curves of type (1, 1) — over Z and the open substack obtained\nby forming the complement of the divisor at infinity∞Mell ⊆Mell. Write ωMell for\nthe ample line bundle onMell determined by the cotangent space at the origin\nof the tautological family of one-dimensional semi-abelian schemes over Mell [i.e.,\nobtained by forming the complement of the unique node of the tautological pointed\nstable curve of type (1, 1) overMell]. Now recall the discriminant moduli form\nΔMell , which may be thought of as an isomorphism of line bundles\nOMell\n∼→ ω⊗12Mell(−∞Mell)\n— i.e., a section of ω⊗12Mell overMell that has no zeroes or poles except for a zero of\norder 1 at ∞Mell . It follows immediately from the existence of ΔMell [cf., e.g., the\n68 SHINICHI MOCHIZUKI\ndiscussion of [GenEll], §3, for more details] that, if, for the sake of simplicity, we\nignore the contributions at the archimedean primes, then we obtain the relation\nht(−) ≈ 16 log(q(−))\n— where we write ht(−) for the normalized height associated to the ample line\nbundle ω⊗2Mell on Mell of the point determined by an elliptic curve “(−)” defined\nover a number field and “≈” to signify a relationship of bounded discrepancy [i.e.,\nthat the absolute value of the difference between the left- and right-hand sides is\nbounded by some positive real number independently of “(−)”].\n(v) We continue to consider the situation discussed in (iii), (iv). Recall the\nisogeny φ : E → E∗ discussed in (iii). Since this isogeny is of degree l, hence, in\nparticular, étale over nonarchimedean primes of F of residue characteristic \n= l, we\nconclude immediately, via a straightforward computation of the map dφ induced on\ndifferentials by φ [cf. the proof of [GenEll], Lemma 3.5, for more details], that\nhtE − log(l) \u0005 htE∗ \u0005 htE + log(l)\nand hence, by combining the latter relation “\u0005” with the relations of the final\ndisplays of (iii), (iv), that\nhtE \u0005 1l\n(\nhtE + log(l)\n)\n— i.e., that htE \u0005 1l−1 log(l) ≤ 1. That is to say, in summary, if, for the sake of\nsimplicity, we ignore the contributions at the archimedean primes, then\nthe assumption [cf. (iii)] that E admits a GMS implies a bound on the\nheight of E [i.e., htE ] and hence, in particular, that, if one only considers\nnumber fields F of bounded degree over Q, then there are only finitely\npossibilities for the isomorphism class of E.\nThis is precisely the argument given in [GenEll], Lemma 3.5, which may be regarded\nas a special case of the argument given in the original proof [due to Faltings] of the\ninvariance of the height under isogenies of abelian varieties.\n(vi) The argument reviewed in (v) may be understood as consisting of two key\npoints, both of which are closely related to various central aspects of inter-universal\nTeichmüller theory. The first key point is, of course,\n(vi-a) the assumption of the existence of a GMS [cf. (iii)], which implies that\nthe passage E \u0004 E∗ to the quotient of E by the GMS corresponds to a\nrelation\nqE \b→ qlE\nbetween the q-parameters of E and E∗ at each nonarchimedean prime of\nF where E and E∗ have bad multiplicative reduction — i.e., to a relation\nreminiscent of the Frobenius morphism in positive characteristic.\nHere, we observe in passing that, in the absence of this crucial assumption of the\nexistence of a GMS, the passage from E to some arbitrary quotient of E by a\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 69\nfinite subgroup scheme of rank l would give rise to relations “qE \b→ qlE” at some\nnonarchimedean primes of bad multiplicative reduction and to relations “qE \b→ q1\/lE ”\nat other nonarchimedean primes of bad multiplicative reduction — i.e., a situation\nin which the argument reviewed in (v) would break down completely! In inter-\nuniversal Teichmüller theory,\n(vi-b) the fundamental role played by theta functions [cf. the discussion of\nExamples 3.3.2, 3.8.4 below] — i.e., in the multiradial reconstruction\nalgorithms of the Θ-pilot “{qj2\nv\n}” — means that in addition to a GMS,\nit will be necessary to somehow “simulate” [cf. the discussion of Example\n3.8.2, (i), below] the existence of a GCG.\nHere, it is useful to recall that, from the point of view of the classical complex theory\nof theta functions [cf. also Example 3.3.2 below]\n+∞∑\nn=−∞\nqn\n2 · Un\n— where q, U ∈ C and |q| < 1:\n· the “U” may be understood as the standard multiplicative coor-\ndinate on the infinite covering “C× ∼→ EM → E” of (ii), hence is only\ndefined once one has a “multiplicative subspace M ⊆ E[∞]”, i.e., the com-\nplex analogue of the l-torsion multiplicative subspace “μl ⊆ E[l]” of (i), or,\nalternatively, of the coverings “XK”, “CK” of [Alien], §3.3, (i), (iv), (v)\n[cf. also the discussion of Example 3.8.2, (i), below];\n· the “q” may be understood as the complex q-parameter determined by\na generating deck transformation of the finite covering “C× ∼→ EM → E”\nof (ii), hence is only defined once one has a “generator, up to multiplication\nby ±1, of E[∞]\/M”, i.e., the complex analogue of the l-torsion canonical\ngenerator “γl ∈ (E[l]\/μl)(K)”, up to multiplication by ±1, or, alterna-\ntively, the index “j = 1” in the Θ-pilot “{qj2\nv\n}” of [Alien], §3.3, (vii) [cf.\nalso the discussion of Example 3.8.2, (i), below; the discussion of Example\n3.8.4, (vi), below].\n(vii) The second key point of the argument reviewed in (v)\n(vii-a) consists of the computation of dφ discussed at the beginning of (v) —\ni.e., a computation that essentially amounts to the computation of the\nlogarithmic derivative\ndlog(U) = dUU \b→ l · dlog(U)\nof the isogeny φ, written as “U \b→ U l” in terms of the standard multi-\nplicative coordinate “U” on Gm [cf. (i), (iii)] — which, in light of the\nampleness of ωMell [cf. (iv)], implies that\n“ωMell |E ≈ ωMell |E∗”\n70 SHINICHI MOCHIZUKI\n[i.e., the “roughly isomorphic” arithmetic line bundles obtained by restrict-\ning ωMell to E, E\n∗] serves, up to a negligible discrepancy, as a common\ncontainer for the moduli of both E and E∗, i.e., in light of the existence\nof the discriminant modular form “ΔMell” [cf. (iv)], as a common\ncontainer for both “qE” and “q\nl\nE”.\nHere, we observe that this “common container”\/ampleness aspect of ωMell may\nbe understood as corresponding — cf. the analogy between étale-like objects in\ninter-universal Teichmüller theory and tangent bundles\/sheaves of differen-\ntials that was recalled above in the context of (InvHt), (FrDff)! — in inter-universal\nTeichmüller theory, to\n(vii-b) the theory of the log-link\/log-shells and closely relatedmono-anabelian\nreconstruction algorithms in a vertical line of the log-theta-lattice\nthat give rise to the log-Kummer-correspondence of inter-universal\nTeichmüller theory, i.e., which play the fundamental role of furnishing a\nmultiradial container for the Frobenius-like Θ-pilot at the lattice\npoint (0, 0) of the log-theta-lattice [cf. the discussion of §3.6, §3.9, §3.10,\n§3.11, below!].\nFinally, in passing, we note that the discriminant modular form “ΔMell” is also\nreminiscent of the classical complex theta function of Example 3.3.2 below [i.e.,\nwhich may also be regarded as a modular form on the upper half-plane], as well as\nof the discussion of the relationship between the discriminant modular form and\nscheme-theoretic Hodge-Arakelov theory in the final portion of [HASurI],\n§1.2 [cf. also [Alien], Example 2.14.3; [Alien], §3.9, (i), (ii), for a discussion of the\nrelationship between scheme-theoretic Hodge-Arakelov theory and inter-universal\nTeichmüller theory].\n(viii) Thus, one may summarize the discussion of the present Example 3.2.1 as\nfollows:\nAt a very rough, introductory\/expository level, one may think of inter-\nuniversal Teichmüller theory as a sort of generalization of the ap-\nproach of (InvHt) [cf. (v)] to bounding heights of elliptic curves over\nnumber fields to the case of [essentially] arbitrary elliptic curves over\nnumber fields [i.e., which are not assumed to admit a GMS!] by\n· somehow “simulating” a GMS\/GCG and\n· applying the theory of theta functions and mono-anabelian\ngeometry.\nExample 3.2.2: Coricity, symmetry, and commutativity properties of the\nlog-theta-lattice. In the following discussion, we fix notation as follows: Let k\nbe a finite extension of Qp, for some prime number p; k an algebraic closure of k;\nq ∈ k a nonzero element of the maximal ideal mk of the ring of integers Ok of k;\nN ≥ 2 an integer. Write\n· N for the additive monoid of nonnegative integers;\n· Gk def= Gal(k\/k);\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 71\n· Ok for the ring of integers of k, with maximal ideal mk ⊆ Ok;\n· O\u0004\nk\n⊆ Ok for the multiplicative monoid of nonzero elements of Ok;\n· O×\nk\n⊆ O\u0004\nk\nfor the group of invertible elements of O\u0004\nk\n;\n· O×\nk\n\u0003 O×μ\nk\n, k\n× \u0003 k×μ, k \u0003 kμ for the respective quotients of O×\nk\n,\nk\n×μ\n, k\nμ\nby the action of the group μ∞ of torsion elements [i.e., roots of\nunity] of O×\nk\n;\n· Fn def= O×μ\nk\n× (qn)N ⊆ k×μ for the multiplicative submonoid of k×μ\n[equipped with a natural action by Gk] generated by O×μ\nk\nand qn, where\nwe allow n to be an arbitrary positive integer, and, by a slight abuse of\nnotation, we write “q” for the image of q ∈ mk in k×μ;\n· Θk : FN\n∼→ F1 for the isomorphism of Gk-monoids that restricts to the\nidentity isomorphism on O×μ\nk\nand maps qN \b→ q;\n· logk : O×k \u0003 k for the p-adic logarithm on O\n×\nk\n.\nThus, Gk acts naturally on O×μ\nk\n\u0006 O×\nk\n⊆ O\u0004\nk\n⊆ Ok. Let Π\u0003 Gk be a topological\ngroup equipped with a surjection onto Gk, which determines natural actions of Π\non O×μ\nk\n\u0006 O×\nk\n⊆ O\u0004\nk\n⊆ Ok.\n(i) We begin by observing the following properties:\n(i-a) The isomorphism\nΘk : FN\n∼→ F1\nis not compatible with the ring structures in its domain\/codomain in\nthe sense that it does not arise from a Gk-equivariant ring homomorphism\nφ : k ↪→ k, i.e., there does not exist a Gk-equivariant ring homomorphism\nφ : k ↪→ k for which the induced map k×μ → k×μ restricts either to a map\nFN → F1 that coincides with Θk or to a map F1 → FN that coincides\nwith Θ−1\nk\n.\n(i-b) The map\nlogk : O×k → k\nis not compatible with the ring structures in its domain\/codomain in\nthe sense that it does not arise from a Gk-equivariant ring homomorphism\nφ : k ↪→ k, i.e., there does not exist a Gk-equivariant ring homomorphism\nφ : k ↪→ k that restricts to logk.\nIndeed, both (i-a) and (i-b) follow immediately from the easily verified elementary\nfact that any Gk-equivariant ring homomorphism φ : k ↪→ k induces [by passing\nto Gk-invariants and considering the l-divisibility properties of units\/non-units of\nk, for prime numbers l \n= p] an isomorphism of topological fields k ∼→ k, hence an\nisomorphism between the value groups of the copies of k in the domain\/codomain.\n[Here, we recall that, since N ≥ 2, the assignments\nqN \b→ q [cf. (i-a)] and (O×\nk\n\u0016) 1 + p2 \b→ logk(1 + p2) (∈ p · Ok) [cf. (i-b)]\n72 SHINICHI MOCHIZUKI\nyield immediate contradictions to the existence of such an induced isomorphism\nbetween value groups.] By contrast, we observe that\n(i-c) the isomorphism Θk : FN\n∼→ F1 is compatible — in the sense of\nequivariance, relative to the natural actions on the domain\/codomain of\nΘk — with an isomorphism Gk\n∼→ Gk between the copies of Gk in the\ndomain\/codomain of Θk;\n(i-d) the map logk : O×k → k is compatible — in the sense of equivariance,\nrelative to the natural actions on the domain\/codomain of logk — with an\nisomorphism Π\n∼→ Π between the copies of Π in the domain\/codomain\nof logk.\nIndeed, (i-c) and (i-d) follow immediately from the various definitions involved.\nHere, we recall, however, that it is of fundamental importance to observe the fol-\nlowing:\n(i-e) Since Θk and logk are not compatible with the respective ring struc-\ntures in their domains\/codomains [cf. (i-a), (i-b)!], in order to obtain coric\nstructures — i.e., structures that are common to these domains\/codomains,\nhence well-defined in a sense that is independent of any specification of a\nrelationship to these domains\/codomains — it is necessary to regard the\n· isomorphisms\nGk\n∼→ Gk and Π ∼→ Π\nas indeterminate isomorphisms of abstract groups, i.e.,\nnot of Galois groups, that is to say, groups equipped with the\n“Galois-rigidification” constituted by the auxiliary data of some\nsort of action on a field\/ring;\n· the “identity isomorphism” given by restricting Θk to O×μk\nO×μ\nk\n∼→ O×μ\nk\nas an indeterminate isomorphism of topological monoids\nequipped with an action by a topological group [i.e., Gk], as\nwell as with the collection of submonoids given by the images\nin O×μ\nk\nof the intersections O×\nk\n∩ kH [i.e., of O×\nk\nwith the H-\ninvariants k\nH ⊆ k of k], where H ranges over the open subgroups\nof Gk\n— cf. the discussion of §3.8 below. These coricity properties will also play\nan important role in the context of the discussion of (ii-c) below.\nIn the context of (i-e), it is also important to note the following:\n(i-f) The use of the terminology “identity isomorphism” when referring\nto any of the isomorphisms\nGk\n∼→ Gk, Π ∼→ Π, O×μ\nk\n∼→ O×μ\nk\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 73\ndiscussed in (i-e) can be highly misleading and give rise to unnecessary\nconfusion, for the following reason: Strictly speaking, throughout mathe-\nmatics, this terminology “identity isomorphism” is only well-defined when\napplied to an isomorphism from a mathematical object to itself [i.e., not\nto a distinct mathematical object!]. Since, however, the ring structures\nin the domain\/codomain of Θk or logk must be distinguished [i.e., so\nlong as they are related to one another via Θk or logk — cf. (i-a), (i-b),\nas well as the discussion of §3.4 below!], the only possible “well-defined\nsense” in which this terminology “identity isomorphism” may be applied\nis the sense of referring to some sort of “identity isomorphism” between\nweaker underlying structures [i.e., such as “sets”, “abstract topological\ngroups”, “abstract topological monoids”, etc. — cf. the discussion of §3.8\nbelow] that do indeed coincide, hence may indeed be “identified” with one\nanother [cf. the situations discussed in Example 3.5.2 below].\nThe observations of (i-e) and (i-f) play a fundamental role in the essential logical\nstructure of inter-universal Teichmüller theory.\n(ii) Next, we observe the following properties:\n(ii-a) The isomorphism\nΘk : FN\n∼→ F1\nis not symmetric with respect to switching the domain\/codomain in\nthe following sense: there do not exist Gk-equivariant ring homomor-\nphisms φ : k ↪→ k, ψ : k ↪→ k such that the induced maps φ×μ : k×μ →\nk\n×μ\n, ψ×μ : k\n×μ → k×μ fit into a diagram\nk\n×μ ⊇ FN\nΘ\nk−→ F1 ⊆ k×μ⏐⏐\nφ×μ ⏐⏐\nψ×μ\nk\n×μ ⊇ F1\nΘ\nk←− FN ⊆ k×μ\nthat is commutative on the portion of the diagram on which the relevant\ncomposites are defined.\n(ii-b) The isomorphism\nlogk : O×k\n∼→ k\nis not symmetric with respect to switching the domain\/codomain in\nthe following sense: there do not exist Gk-equivariant ring homomor-\nphisms φ : k ↪→ k, ψ : k ↪→ k that fit into a diagram\nk ⊇ O×\nk\nlog\nk−→ k⏐⏐\nφ ⏐⏐\nψ\nk\nlog\nk←− O×\nk\n⊆ k\nthat is commutative on the portion of the diagram on which the relevant\ncomposites are defined.\nIndeed, (ii-a) follows by tracing the image, at the level of value groups, of the qN\nin the upper left-hand corner of the diagram, i.e., which maps [cf. the discussion of\n74 SHINICHI MOCHIZUKI\ninduced isomorphisms of value groups following (i-a), (i-b)] to q via the composite\nof the upper horizontal and right-hand vertical arrows, but to qN\n2\n( \n= qN ) via the\ncomposite of the left-hand vertical and lower horizontal arrows. In a similar vein,\n(ii-b) follows by tracing the image, at the level of value groups, of the element\nu\ndef\n= {exp(p2)}\n1\np2 ∈ O×\nk\n[where “exp(−)” denotes the well-known formal power series of the exponential\nfunction, and the “ 1p2 ” denotes a p\n2-root of the element “{−}”] in the upper left-\nhand corner of the diagram, i.e., which maps to 0 via the composite of the upper\nhorizontal, right-hand vertical, and lower horizontal arrows, but to u ( \n= 0) via the\nleft-hand vertical arrow. By contrast, we observe the following:\n(ii-c) The topological group actions surrounding Θk [cf. (i-c)]\nΠ \u0003 Gk ∼→ Gk \u0006 Π\n\u0002 \u0002\nFN\nΘ\nk−→ F1\nyield a diagram\nΠ \u0003 Gk ∼→ Gk \u0006 Π\nthat is [manifestly!] symmetric with respect to applying the operation of\nreflection of this last diagram around the “\n∼→ ”, where we observe that\nthis symmetry property only holds if the coricity properties discussed in\n(i-e), (i-f) are applied, i.e., if\n· the isomorphism of topological groups Gk ∼→ Gk is regarded\nas an indeterminate isomorphism of abstract topological\ngroups and\n· the arrows “\u0003”\/“\u0006” are regarded as surjections between\nabstract topological groups, i.e., topological groups that are\nonly well-defined up to indeterminate isomorphism.\nIndeed, if one does not apply these coricity properties, then one is in\neffect working with structures [i.e., the various “Π” and “Gk” in “Π \u0003\nGk\n∼→ Gk \u0006 Π”] that depend on Galois-rigidifications that arise from\nstructures specific to the domain or codomain of Θk, hence do not satisfy\nthe desired symmetry property [cf. the discussion of (i-e), (i-f)!].\nThe observation (ii-c), together with the observations (i-e) and (i-f), plays a funda-\nmental role in the essential logical structure of inter-universal Teichmüller theory.\n(iii) Next, we observe the following properties:\n(iii-a) Θk does not commute with logk — i.e., at a purely formal level,\n“Θk ◦ logk \n= logk ◦ Θk” — in the following sense: there do not exist\nGk-equivariant ring homomorphisms φ : k ↪→ k, ψ : k ↪→ k such that the\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 75\ninduced maps φμ : k\nμ → kμ, ψμ : kμ → kμ fit into a diagram\nO×μ\nk\n⊆ FN\nΘ\nk−→ F1 ⊇ O×μ\nk⏐⏐\nlogk ⏐⏐\nlogk\nk k⏐⏐\nφμ|k ⏐⏐\nψμ|k\nk\nμ ⊇ FN\nΘ\nk−→ F1 ⊆ kμ\nthat is commutative on the portion of the diagram on which the relevant\ncomposites are defined.\n(iii-b) Θk is not invariant with respect to logk — i.e., at a purely formal\nlevel, “Θk ◦ logk \n= Θk” — in the following sense: there do not exist Gk-\nequivariant ring homomorphisms φ : k ↪→ k, ψ : k ↪→ k such that the\ninduced maps φμ : k\nμ → kμ, ψμ : kμ → kμ fit into a diagram\nO×μ\nk\n⊆ FN\nΘ\nk−→ F1 ⊇ O×μ\nk⏐⏐\nlogk ⏐⏐\nιk\nk k\nμ⏐⏐\nφμ|k ⏐⏐\nψμ\nk\nμ ⊇ FN\nΘ\nk−→ F1 ⊆ kμ\n— where ιk : O×μk ↪→ k\nμ\ndenotes the natural inclusion — that is commu-\ntative on the portion of the diagram on which the relevant composites are\ndefined.\nIndeed, let m be a positive integer such that pm ·Ok ⊆ logk(O×k ) ⊆ p−m ·Ok, where\nwe write O×k\ndef\n= Ok ∩O×\nk\n. Then (iii-a) follows by tracing the image of the element\nun\ndef\n= exp(pn·N ) ∈ O×\nk\n[where “exp(−)” denotes the well-known formal power series of the exponential\nfunction, and n is any positive integer > 2m [which implies that n < n+(n−2m) =\n2n − 2m ≤ n · N − 2m] such that pn ∈ qN · O×\nk\n] in the upper left-hand portion of\nthe diagram, i.e., which maps [cf. the discussion of induced isomorphisms of value\ngroups following (i-a), (i-b)] to [the image in the value group of k of] some nonzero\nelement ∈ pn·N−2m ·Ok [cf. the discussion of (i-e), as well as of Example 3.5.1 below]\nvia the composite of the upper horizontal and right-hand vertical arrows, but to [the\nimage in the value group of k of] pn ( \n∈ pn·N−2m · Ok) via the left-hand vertical and\nlower horizontal arrows. In a similar vein, (iii-b) follows by tracing the image of the\nsame element un ∈ O×\nk\nin the upper left-hand portion of the diagram, i.e., which\nmaps [cf. the discussion of induced isomorphisms of value groups following (i-a),\n(i-b)] to [the image in the value group of k of] un via the composite of the upper\nhorizontal and right-hand vertical arrows, but to [the image in the value group of k\nof] pn ( \n= un) via the left-hand vertical and lower horizontal arrows.\n76 SHINICHI MOCHIZUKI\n(iv) Finally, we observe that it is now an essentially formal\/routine matter to\ntranslate the various elementary properties discussed above in (i), (ii), (iii) into the\ncorresponding coricity\/symmetry\/commutativity properties of the log-theta-\nlattice, i.e.:\n· the incompatibility with the ring structures in the domain\/codomain\n[cf. the discussion at the beginning of the present §3.2; the discussion at\nthe beginning of §3.8 below] of the Θ-link [cf. (i-a)] and log-link [cf.\n(i-b)];\n· the horizontal [cf. (i-c)] and vertical [cf. (i-d)] coricity [cf. (i-\ne), (i-f)] properties of the étale-like structures that appear in the log-\ntheta-lattice [cf. the discussion at the beginning of the present §3.2; the\ndiscussion at the beginning of §3.8 below];\n· the non-symmetricity with respect to switching the domain\/codomain\n[cf. the discussion at the beginning of the present §3.2] of the Θ-link [cf.\n(ii-a)] and log-link [cf. (ii-b)];\n· the symmetricity with respect to the Θ-link [cf. the discussion at\nthe beginning of the present §3.2] of the étale-like structures of the\nlog-theta-lattice [cf. (ii-c)];\n· the non-commutativity of the log-theta-lattice [cf. (iii-a); the discus-\nsion at the beginning §3.3 below];\n· the non-invariance of the Θ-link with respect to the log-link [cf. (iii-\nb); the discussion at the beginning §3.3 below].\n§3.3. RCS-redundant copies in the domain\/codomain of the log-link\nThe Θ-link of inter-universal Teichmüller theory is defined, in the style of\nclassical complex Teichmüller theory [cf. Example 3.3.1 below; [IUTchI], Remark\n3.9.3], as a deformation of the ring structure in a (Θ±ellNF-)Hodge theater\nthat depends, in an essential way, on the splitting into unit groups and value\ngroups of the various localizations of the number field involved. On the other\nhand, the log-link of inter-universal Teichmüller theory [i.e., in essence, the p-adic\nlogarithm at primes of the number field of residue characteristic p] has the effect of\njuggling\/rotating these unit groups and value groups, e.g., by mapping units to\nnon-units [cf., e.g., the discussion of [Alien], Example 2.12.3, (v)]. In particular,\nthere is no natural way to relate the two Θ-links [i.e., the two horizon-\ntal arrows in the following diagram] that emanate from the domain and\ncodomain of the log-link [i.e., the left-hand vertical arrow in the following\ndiagram]\n• Θ−→ •\u000b⏐⏐log ...??...• Θ−→ •\n— that is to say, there is no natural candidate for “??” [i.e., such as, for instance,\nan isomorphism or the log-link between the two bullets “•” on the right-hand side\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 77\nof the diagram] that makes the diagram commute. Indeed, it is an easy exercise\n[cf. Example 3.2.2, (iii), (iv); [Alien], §3.3, (ii); [Rpt2018], §15, (LbΘ), (Lblog),\n(LbMn)], to show that neither of these candidates for “??” [i.e., an isomorphism\nor the log-link] yields a commutative diagram.\nThus, in summary, any identification of the domain and codomain of the log-\nlink [cf. (RC-log)!] yields a situation in which the local splittings into unit groups\nand value groups of the resulting identified “•’s” are no longer well-defined. In\nparticular,\nany such identification of the domain and codomain of the log-link [cf.\n(RC-log)!] yields a situation in which the Θ-link is not well-defined\n— i.e., a situation in which the apparatus of inter-universal Teichmüller theory\ncompletely ceases to function — cf. the discussion of the definition of the Θ-link in\nthe latter half of [Alien], §3.3, (ii), as well as the discussion of Example 3.3.3, (i),\nbelow. This discussion may be summarized, at a symbolic level, as follows:\ndefinition of the Θ-link =⇒ falsity of (RC-log).\nNext, we observe [cf. the discussion of [IUTchI], Remark 3.9.3, (iii), (iv)] that\nthe non-existence of a solution for “??” in the above diagram [i.e., that makes\nthe diagram commute] amounts, at a structural level, to essentially the same phe-\nnomenon as the incompatibility of the dilations that appear in classical complex\nTeichmüller theory with multiplication by non-real roots of unity [cf. Example\n3.3.1 below]. Write R, C, respectively, for the topological fields of real and complex\nnumbers. Then as observed in the discussion of the latter half of [Alien], §3.3, (ii)\n[cf., especially, the discussion surrounding [Alien], Fig. 3.6]:\n...\n...\u000b⏐⏐log \u000b⏐⏐log\n. . .\nΘ−→ • Θ−→ • Θ−→ . . .\u000b⏐⏐log \u000b⏐⏐log\n. . .\nΘ−→ • Θ−→ • Θ−→ . . .\u000b⏐⏐log \u000b⏐⏐log\n. . .\nΘ−→ • Θ−→ • Θ−→ . . .\u000b⏐⏐log \u000b⏐⏐log\n...\n...\n⊇\n...\n...\u000b⏐⏐log \u000b⏐⏐log\n• •\u000b⏐⏐log \u000b⏐⏐log\n• Θ−→ •\u000b⏐⏐log \u000b⏐⏐log\n• •\u000b⏐⏐log \u000b⏐⏐log\n...\n...\n(InfH) this structural similarity is consistent with the analogy discussed in loc.\ncit. between\n· the “infinite H” portion of the log-theta-lattice consisting\nof the two vertical lines [i.e., of log-links] on either side of a\nhorizontal arrow [i.e., a Θ-link] of the log-theta-lattice and\n78 SHINICHI MOCHIZUKI\n· the elementary theory surrounding the bijection\nC×\\GL+2 (R)\/C× ∼→ [0, 1)(\nλ 0\n0 1\n) \b→ λ−1λ+1\n— where λ ∈ R≥1; GL+2 (R) denotes the group of 2 × 2 real\nmatrices of positive determinant; C× denotes the multiplicative\ngroup of C, which we regard as a subgroup of GL+2 (R) via the\nassignment a+ ib \b→ ( a b−b a), for a, b ∈ R such that (a, b) \n= (0, 0);\nthe domain of the bijection is the set of double cosets.\nThat is to say,\n· the dilation (λ 0\n0 1\n)\n— cf. the dilations that appear in classical\ncomplex Teichmüller theory, i.e., as reviewed in Example 3.3.1\nbelow — corresponds to the Θ-link portion of an “infinite H”\n[cf. Example 3.3.2, (iii), below], while\n· the two copies of the group of toral rotations “C×” [e.g., by\nroots of unity in C×] on either side of “GL+2 (R)” — which may\nbe thought of as a representation of the holomorphic structures in\nthe domain and codomain of the dilation\n(\nλ 0\n0 1\n)\n[cf. the discussion\nof Example 3.3.1 below] — correspond, respectively, to the two\nvertical lines of log-links in the “infinite H” on either side of the\nΘ-link [cf. the discussion of Example 3.3.2, (iv), below].\nExample 3.3.1: Classical complex Teichmüller theory. Let λ ∈ R>1. Re-\ncall the most fundamental deformation of complex structure in classical complex\nTeichmüller theory\nΛ : C → C\nC \u0016 z = x+ iy \b→ ζ = ξ + iη def= λ · x+ iy ∈ C\n— where x, y ∈ R. Let n ≥ 2 be an integer, ω a primitive n-th root of unity. Write\n(ω ∈) μn ⊆ C for the group of n-th roots of unity. Then observe that\nif n ≥ 3, then there does not exist ω′ ∈ μn such that Λ(ω · z) = ω′ · Λ(z)\nfor all z ∈ C.\n[Indeed, this observation follows immediately from the fact that if n ≥ 3, then\nω \n∈ R.] That is to say, in words,\nΛ is not compatible with multiplication by μn unless n = 2 [in which\ncase ω = −1].\nThis incompatibility with “indeterminacies” arising from multiplication by μn,\nfor n ≥ 3, may be understood as one fundamental reason for the special role played\nby square differentials [i.e., as opposed to n-th power differentials, for n ≥ 3] in\nclassical complex Teichmüller theory [cf. the discussion of [IUTchI], Remark 3.9.3,\n(iii), (iv)].\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 79\nExample 3.3.2: The Jacobi identity for the classical theta function.\n(i) Write z = x + iy for the standard coordinate on the upper half-plane\nH\ndef\n= {z = x+ iy ∈ C | y > 0}. Recall the theta function on H\nΘ(q)\ndef\n=\n+∞∑\nn=−∞\nq\n1\n2n\n2\n.\n— where we write q\ndef\n= e2πiz. Restricting to the imaginary axis [i.e., x = 0] yields\na function\nθ(t)\ndef\n=\n+∞∑\nn=−∞\ne−πn\n2t.\n— where we write t\ndef\n= y.\n(ii) Next, let us observe that\nι\ndef\n=\n(\n0 1\n−1 0\n)\n∈ C× ⊆ GL+(R)\nmaps z \b→ −z−1, hence iy \b→ iy−1, i.e., t \b→ t−1, while, for λ ∈ R≥1,(\nλ 0\n0 1\n)\n∈ C× ⊆ GL+(R)\nmaps z \b→ λ · z, hence iy \b→ iλ · y, i.e., t \b→ λ · t.\n(iii) Next, we observe the following:\n· As t → +∞, the terms in the series for θ(t) are rapidly decreasing,\nand θ(t) → +0. In particular, the series for θ(t) is relatively easy to\ncompute.\n· As t → +0, the terms in the series for θ(t) decrease very slowly,\nand θ(t) → +∞. In particular, the series for θ(t) is very difficult to\ncompute.\nThus, in summary, the “flow\/dilation”\n(\nλ 0\n0 1\n)\nalong the imaginary axis may be re-\ngarded as a sort of “link”, in the context of the theta function θ(t), between small\nvalues [i.e., θ(t) → +0 as t → +∞] and large values [i.e., θ(t) → +∞ as t → +0].\nThat is to say, this flow\/dilation along the imaginary axis behaves in a way that\nis strongly reminiscent of the Θ-link of inter-universal Teichmüller theory [cf. the\ndiscussion of (InfH)].\n(iv) The Jacobi identity for the theta function θ(t)\nθ(t) = t−\n1\n2 · θ(t−1)\nallows one to analyze the behavior of θ(t) as t → +0, which is very difficult to\ncompute [cf. (iii)], in terms of the behavior of θ(t) as t → +∞, which is relatively\neasy to compute [cf. (iii)] — cf. the discussion of the Jacobi identity in [Pano], §3,\n80 SHINICHI MOCHIZUKI\n§4; [Alien], §4.1, (i). Observe that this identity may be understood as a sort of\ninvariance with respect to ι [cf. (ii)], up to a certain easily computed factor [i.e.,\nt−\n1\n2 ]. Note that ι “juggles”, or “rotates\/permutes”, the two dimensions of R2. This\naspect of ι is strongly reminiscent of the log-link of inter-universal Teichmüller\ntheory, which “juggles”, or “rotates\/permutes”, the two underlying dimensions of\nthe ring structures in a vertical column of the log-theta-lattice [cf., e.g., the discus-\nsion of [Alien], Example 2.12.3, (v)]. By contrast, we note that the theta function\nθ(t) does not satisfy any interesting properties of invariance with respect to the\ndilations\n(\nλ 0\n0 1\n)\n.\n(v) Relative to the analogy with the Θ-link and log-link of inter-universal\nTeichmüller theory discussed in (iii), (iv), the ι-invariance interpretation of the\nJacobi identity discussed in (iv) is strongly reminiscent of the central role played\nby log-link invariance in the construction of the multiradial representation\nin inter-universal Teichmüller theory [cf. the discussion surrounding (logORInd),\n(Di\/NDi) in §3.11 below; the discussion of the Jacobi identity in [Pano], §3, §4].\nHere, we note that the factor t−\n1\n2 in the Jacobi identity may be understood as\ncorresponding, relative to the analogy with inter-universal Teichmüller theory, to\nthe indeterminacies (Ind1), (Ind2), (Ind3) acting on the log-shells in the multiradial\nrepresentation. Indeed, both\n· the factor t− 12 in the Jacobi identity — which amounts, in essence,\nto the well-known interpretation of the theta function θ(t) as a modular\nform on H — and\n· the log-shells in the multiradial representation — cf. the discussion\nof the relationship between scheme-theoretic Hodge-Arakelov theory and\ninter-universal Teichmüller theory in [Alien], Example 2.14.3; [Alien], §3.9,\n(i), (ii) —\nare closely related to the notion of “differentials”.\n(vi) At a more technical level, the crucial ι-invariance property of (iv), (v)\nmay be understood as a consequence of the Fourier transform invariance of\nthe Gaussian “e−\u0005\n2\n” on the real line [cf. the discussion of the Jacobi identity\nin [Pano], §3, §4]. This Fourier transform invariance in turn may be understood\nas a consequence of the quadratic form “\u00072” in the exponent of the Gaussian\n“e−\u0005\n2\n”, which may be thought of as the first Chern class of the ample line bundle\nwhose section determines, via the canonical theta trivialization of the line bundle,\nthe theta function under consideration [cf. the classical theory of complex theta\nfunctions as exposed, for instance, in [Mumf1], Chapter I]. When this quadratic\nform “\u00072” is multiplied by a factor t ∈ R>0, application of the Fourier transform\ngives rise, up to suitable factors [involving, in particular, the Gaussian integral!\n— cf. the discussion of (v), as well as [Alien], §3.8], to the transformation\ne−t·\u0005\n2 \u0004 e−t−1·\u00052\nthat underlies the Jacobi identity. In this context, it is of central importance to\nobserve that the transformation “t\u0004 t−1” in the above display [i.e., as opposed to\na transformation of the form “t \u0004 c · t−1”, for some c ∈ R>0] is indicative of —\nand indeed in some sense essentially equivalent to — an absolute notion of “1”\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 81\n[i.e., the unique invariant element of the transformation R>0 \u0016 t \b→ t−1 ∈ R>0]\nin the copy of the real numbers that appears in the exponent “−\u00072” of “e−\u00052”.\nFinally, we observe that the fundamental role played by the quadratic form “\u00072”\nof the above discussion in the proof of the Fourier transform invariance of the\nGaussian that underlies the Jacobi identity is strongly reminiscent of the crucial\nrigidity properties in the theory of the étale theta function [cf. [Alien], §3.4,\n(iii), (iv); the discussion of the Jacobi identity in [Pano], §3, §4] that underlie\nthe multiradial representation of inter-universal Teichmüller theory: Indeed,\nthese rigidity properties may be understood as consequences of the theta group\nsymmetries, which also arise, in essence, from the étale-theoretic version of the\nquadratic form “\u00072” of the above discussion [cf. the discussion at the end of [Alien],\n§3.4, (iv)].\n(vii) Before continuing our discussion, we pause briefly to make the following\nelementary observation:\nLet V be a 1-dimensional R-vector space. Then a [topological] ring\/field\nstructure on V may be understood as a multiplication map V ⊗R V ∼→ V\ngiven by an isomorphism of R-vector spaces. By tensoring with the dual\nvector space to V , one verifies immediately that such a multiplication map\nmay be understood as an isomorphism of R-vector spaces V\n∼→ R, i.e., as\nthe choice of a nonzero element in V given by the image of 1 ∈ R.\nIn particular, it follows immediately from this elementary observation that the\nabsolute notion of “1 ∈ R” discussed in (vi) may be interpreted as a [topological]\nring\/field structure on the copy of the real numbers that appears in the exponent\n“(−)” of the Gaussian “e(−)”. This interpretation is strongly reminiscent of the\ncentral importance, in inter-universal Teichmüller theory, of working with the first\npower of [the reciprocal of the l-th root of] the theta function [cf., e.g., the\ndiscussion of [Alien], §3.4, (iii)], which makes it possible to consider the truncated\n“mod N” Kummer theory of the theta function: indeed, this truncatibility of\nthe Kummer theory of the theta function is closely related to the [topological]\nring\/field structure of the local fields that appear in the context of the log-link\nof inter-universal Teichmüller theory [cf. the analogy between the log-link and “ι”\ndiscussed in (iv); the discussion of Example 3.8.4 below; the discussion of the final\nportion of [Alien], §3.6, (ii)].\n(viii) Another important technical aspect of the Fourier transform discussed\nin (vi) is the factor “eixy” [where x, y ∈ R] that appears in this Fourier transform.\nIndeed,\n· this “exponentiation of a complex unit ∈ S1 ⊆ C×” to a power\ngiven by some indeterminate real number — i.e., the real number\nthat corresponds to the variable of integration in the Fourier ransform —\nis strongly reminiscent of\n· the (Ind2) indeterminacy action on the local units “O×μ” in inter-\nuniversal Teichmüller theory — i.e., which amounts, in essence, to expo-\nnentiation of these local units to a power given by some indeterminate\nelement ∈ Ẑ×\n82 SHINICHI MOCHIZUKI\n[cf. the discussion of the Jacobi identity in [Pano], §3, §4]. Moreover, in this\ncontext, it is of interest to note that the integration that occurs in the Fourier\ntransform may be understood as corresponding to the logical OR\/XOR “∨\/∨̇”\naspect of the indeterminacies that occur in inter-universal Teichmüller theory —\ncf. the correspondence of logical XOR “∨̇” with addition [where we recall that\n“integration” may be understood as a sort of “topological addition” operation], as\ndiscussed in Example 2.4.6, (iii), as well as in (∧(∨̇)-Chn) in §3.10 below.\nExample 3.3.3: Theta functions and multiplicative structures.\n(i) One fundamental reason for the central role played by theta functions in\nthe essential logical structure of inter-universal Teichmüller theory lies in the fact\nthat\n(ThMlt) the main properties of interest of the theta functions that appear in\ninter-universal Teichmüller theory — most notably,\n(ZrPl) the well-known description of the zeroes\/poles of theta func-\ntions at the cusps [cf. [EtTh], Proposition 1.4, (i); [IUTchIII],\nRemark 2.3.3, (vi), (vii); [Alien], §3.4, (iii)], which yields — by\napplying the well-known intersection theory of divisors supported\non the special fiber of the universal topological covering of the\nTate curve [cf. the discussion preceding [EtTh], Proposition 1.1,\nof divisors supported on the special fiber] — a “divisor-theoretic”\ncharacterization of these theta functions [up to translation by a\ndeck transformation of the universal topological covering];\n(SymTh) the well-known symmetries of theta functions [cf. [EtTh],\nProposition 1.4, (ii)];\n(GalEv) the Kummer-compatible Galois evaluation properties of\ntheta functions [cf. [IUTchII], Remark 1.12.4; the discussion at\nthe beginning of [Alien], §3.6; [EtTh], Remark 1.10.4, (i)], which\ngive rise to the canonical splittings of theta monoids, as well as to\nthe construction of the Gaussian monoids [cf. [IUTchII], Corol-\nlaries 2.5, 2.6, 3.5, 3.6; [Alien], §3.4, (iii); [Alien], §3.6, (ii)]\n— may be expressed entirely in terms of the multiplicative structures\nof the various rings that appear, i.e., without invoking the additive struc-\ntures of these rings.\nThis expressibility purely in terms of multiplicative structures plays an essential role\nin establishing the\n· multiradial unit group\/value group splittings\/decouplings [cf.\n[IUTchII], Remark 1.12.2, (vi); [Alien], §3.4, (iii)] and\n· non-interference properties [cf. [Alien], §3.7, (i)]\nthat underlie the definition of the Θ-link [cf. the discussion of the present §3.3 pre-\nceding (InfH)] and log-Kummer-correspondence [cf. [IUTchII], Remark 1.12.2,\n(iv); [IUTchIII], Remark 1.2.3].\n(ii) In the context of (ZrPl), it is important to recall the central importance of\nthe fact that\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 83\n(ZrPlOrd) the signed order [i.e., “+” for zeroes, “−” for poles] of the theta function\nat each of the cusps is precisely one [cf. [IUTchIII], Remark 2.3.3, (vi),\n(vii); [Alien], §3.4, (iii)].\nThis property (ZrPlOrd) of the theta functions that appear in inter-universal Te-\nichmüller theory is closely related to the properties discussed in Example 3.3.2,\n(vi), (vii), in the case of complex theta functions. In inter-universal Teichmüller\ntheory, this property (ZrPlOrd) ensures that the mono-theta-theoretic cyclo-\ntomic rigidity algorithms that arise from the theory of theta functions are\n· compatible with the topology of the tempered fundamental group\nand, moreover, are\n· not subject to {±1}-indeterminacies\n— properties that are not satisfied by the cyclotomic rigidity isomorphisms that\narise from the theory of algebraic rational functions [cf. [IUTchIII], Remark 2.3.3,\n(vi), (vii); [IUTchIII], Remark 3.11.4; [Alien], §3.4, (iii), as well as the discussion of\nExamples 3.3.2, (vi), (vii); 3.8.4, (iv), (v), (vi), below, of the present paper]. Here,\nit is of interest to recall that\n(MltAdd) the anabelian reconstruction algorithms of [AbsTopIII], Theorem 1.9, im-\nply that the additive structure of the function field of an algebraic curve\nmay in fact be reconstructed from the multiplicative structure of the func-\ntion field of algebraic rational functions on the curve, equipped with the\n[multiplicative!] valuation maps and [multiplicative!] evaluation maps at\nthe closed points of the curve [cf. “Uchida’s Lemma”, i.e., [AbsTopIII],\nProposition 1.3].\nThat is to say, (MltAdd) means that once one allows oneself to work with algebraic\nrational functions — i.e., as opposed to theta functions — the issue emphasized in\n(i) of expressibility purely in terms of multiplicative structures in some\nsense ceases to be well-defined\/non-vacuous. By contrast, if one restricts oneself,\nas is indeed the case in the discussion of (i), to considering theta functions — as\nis necessary, in order to apply the essential property (ZrPlOrd) discussed above!\n— then (MltAdd) is no longer applicable, so there are no longer any obstructions\nto the “well-definedness\/non-vacuousness” of the notion of “expressibility purely in\nterms of multiplicative structures”.\nFinally, we recall that, in any vertical line of log-links in the log-theta-lattice,\n· the discrepancy between the [holomorphic] Frobenius-like copies of objects\non either side of a log-link [cf. (RC-log)], as well as\n· the discrepancy between [holomorphic] Frobenius-like copies of objects\nand [holomorphic] étale-like copies of objects [cf. (RC-FrÉt)],\nmay be understood as the extent to which the diagram of arrows that constitutes\nthe log-Kummer-correspondence associated to this vertical line of log-links fails\nto commute.\nThis failure to commute may be estimated by means of the indeterminacy\n(Ind3), i.e., by interpreting this failure to commute as a sort of “upper semi-\ncommutativity”. This indeterminacy (Ind3) is highly nontrivial and, in particu-\nlar, gives rise to the inequality that appears in the final computation of log-volumes\n84 SHINICHI MOCHIZUKI\nin inter-universal Teichmüller theory [cf. [IUTchIV], Corollary 3.12]. In this con-\ntext, it is important to recall that the theory surrounding this indeterminacy (Ind3)\ndepends, in an essential way, on the absolute anabelian geometry of [AbsTopIII], §1,\ni.e., which allows one to reconstruct a hyperbolic curve X over a number field or\nmixed characteristic local field from the abstract profinite group determined by the\nétale fundamental group π1(X) of the curve. That is to say, in summary, this\nabsolute anabelian geometry allows one to show that\nthe discrepancies between the various [holomorphic] Frobenius-like and\n[holomorphic] étale-like copies of objects in a vertical line of log-links [cf.\n(RC-log), (RC-FrÉt)] in the log-theta-lattice are “bounded by” the [rel-\natively mild] indeterminacy (Ind3).\nOn the other hand, this absolute anabelian geometry most certainly does not imply\nthat these discrepancies are trivial\/non-existent, i.e., as asserted in (RC-log), (RC-\nFrÉt) — cf. the discussion of the falsity of (RC-log), (RC-FrÉt) in §3.2 and the\npresent §3.3.\n§3.4. RCS-redundant copies in the domain\/codomain of the Θ-link\nThe Θ-link of inter-universal Teichmüller theory\n• Θ−→ •\nis defined as a gluing between the (Θ±ellNF-)Hodge theater “•” in the domain\nof the arrow and the (Θ±ellNF-)Hodge theater “•” in the codomain of the arrow\nalong F\u0002\u0003×μ-prime-strips “∗” that arise from the Θ-pilot object “Θ-plt” in the\ndomain and the q-pilot object “q-plt” in the codomain. Here, it is important to\nnote that this gluing is obtained by regarding these F\u0002\u0003×μ-prime-strips “∗” as\nbeing known only up to isomorphism. This point of view, i.e., of regarding these\nF\u0002\u0003×μ-prime-strips “∗” as being known only up to isomorphism, is implemented\nformally by taking the gluing to be the full poly-isomorphism — i.e., the set of\nall isomorphisms — between the F\u0002\u0003×μ-prime-strips arising from the domain and\ncodomain of the Θ-link. Here, we recall that\n· q-plt essentially amounts to the arithmetic line bundle determined by [the\nideal generated by] some 2l-th root q\nv\nof the q-parameter at the valuations\nv ∈ Vbad, while\n· Θ-plt essentially amounts to the collection of arithmetic line bundles de-\ntermined by [the ideals generated by] the collection {qj2\nv\n}, as j ranges over\nthe integers 1, . . . , l\u0002\ndef\n= l−1\n2\n[where l is the prime number that appears in\nthe initial Θ-data under consideration], and v ranges over the valuations\n∈ Vbad.\nAlso, we recall that each (Θ±ellNF-)Hodge theater “•” gives rise to an associated\nmodel “Ring” of the ring\/scheme theory surrounding the elliptic curve under con-\nsideration. In the following discussion, we shall write\n· †• for the “•” in the domain of the Θ-link,\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 85\n· ‡• for the “•” in the codomain of the Θ-link,\n· \u0007 for an arbitrary element of the set consisting of “†”, “‡”, and the\n“empty symbol” [i.e., no symbol at all],\n· \u0005Θ-plt ∈ \u0005Ring for the Θ-pilot arising from the collection “{\u0005qj2\nv\n}” that\nappears in the model of ring\/scheme theory associated to \u0005•, and\n· \u0005q-plt ∈ \u0005Ring for the q-pilot arising from the “\u0005q\nv\n” that appears in\nthe model of ring\/scheme theory associated to \u0005•.\nFinally, we recall that since, for j \n= 1, the valuation [at each valuation v ∈ Vbad]\nof qj\n2\nv\ndiffers from that of q\nv\n, the arithmetic degrees of the line bundles constituted\nby q-plt and Θ-plt differ.\nThus, at a more formal level, the above description of the gluing that consti-\ntutes the Θ-link may be summarized as follows:\n†Ring \u0016 †Θ-plt ←: ∗ :→ ‡q-plt ∈ ‡Ring\nRing \u0016 q-plt \n= Θ-plt ∈ Ring\n[where “←:” and “:→” denote the assignments that consitute the gluing discussed\nabove].\nIn this context, we note the following fundamental observation, which un-\nderlies the entire logical structure of inter-universal Teichmüller theory [cf. the\ndiscussion of [IUTchIII], Remark 3.12.2, (citw), (fitw); [Alien], §3.11, (iv)]:\n(AOΘ1) the following condition holds:(\n∗ :→ †Θ-plt ∈ †Ring\n)\n∧\n(\n∗ :→ ‡q-plt ∈ ‡Ring\n)\n.\nBy contrast, if one simply deletes the distinct labels “†”, “‡” [cf. (RC-Θ)!], then\n(AOΘ2) the following condition holds:(\n∗ :→ Θ-plt ∈ Ring\n)\n∨\n(\n∗ :→ q-plt ∈ Ring\n)\n.\nOf course,\n(AOΘ3) the essential mathematical content discussed in this condition (AOΘ2)\nmay be formally described as a condition involving the AND relator “∧”:(\nq-plt ∈ {q-plt,Θ-plt}\n)\n∧\n(\nΘ-plt ∈ {q-plt,Θ-plt}\n)\n.\nOn the other hand, precisely as a consequence of the fact [discussed above] that\nRing \u0016 q-plt \n= Θ-plt ∈ Ring,\n86 SHINICHI MOCHIZUKI\n(AOΘ4) the following condition does not hold:(\n∗ :→ Θ-plt ∈ Ring\n)\n∧\n(\n∗ :→ q-plt ∈ Ring\n)\n.\nThat is to say, the operation of identifying †•, ‡• [hence also †Ring, ‡Ring] — e.g.,\non the grounds of “redundancy” [i.e., as asserted in (RC-Θ)!] — by deleting the\ndistinct labels “†”, “‡” has the effect of passing from a situation in which\nthe AND relator “∧” holds [cf. (AOΘ1)]\nto a situation in which\nthe OR relator “∨” holds [cf. (AOΘ2), (AOΘ3)], but\nthe AND relator “∧” does not hold [cf. (AOΘ4)]!\nIn particular, relative to the correspondences\n†•, †Ring ←→ †I; ∗ ←→ γJ; ‡•, ‡Ring ←→ ‡I\n†Θ-plt ←→ †β; ‡q-plt ←→ ‡α\n[cf. the correspondences (StR1) ∼ (StR6) discussed in §3.2; the correspondences\ndiscussed in Example 2.4.5, (ii); the discussion of [Alien], §3.11, (iv)], one obtains\nvery precise structural resemblances\n(AOΘ1) ←→ (AOL1),\n(AOΘ2) ←→ (AOL2),\n(AOΘ3) ←→ (AOL3),\n(AOΘ4) ←→ (AOL4)\nwith the situation discussed in Example 2.4.1, (i), (ii). Thus, in summary,\nthe falsity of (RC-Θ) may be understood as a consequence of the falsity\n[cf. (AOΘ4)] of the crucial AND relator “∧” in the absence of distinct\nlabels, in stark contrast to the truth [cf. (AOΘ1)] of the crucial AND\nrelator “∧” as an essentially tautological consequence of the use of the\ndistinct labels “†”, “‡”.\nIn the context of the central role played in the logical structure of inter-universal\nTeichmüller theory by the validity of (AOΘ1), it is important to note [cf. the\nproperty discussed in (AOΘ4)!] that\n(NoRng) there does not exist an isomorphism of ring structures †Ring ∼→ ‡Ring\nthat induces, on value groups of corresponding local rings, the desired\nassignment {†qj2\nv\n} \b→ ‡q\nv\n[i.e., that appears in the Θ-link].\nOn the other hand, if, instead of considering the full ring structures of †Ring, ‡Ring,\none considers [cf. the discussion of [Rpt2018], §6]\n· certain suitable subquotients — i.e., in the notation of [Alien], §3.3,\n(vii), (aq), (aΘ), “O×\nk\n” — of the underlying multiplicative monoids\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 87\nof corresponding local fields, as well as\n· the absolute Galois groups — i.e., in the notation of [Alien], §3.3, (vii),\n(aq), (aΘ), “Gk” — associated to corresponding local rings, regarded as\nabstract topological groups [that is to say, not as Galois groups, or\nequivalently\/alternatively, as groups of field automorphisms! — cf. the\ndiscussion of §3.8 below],\nthen one obtains structures — i.e., the structures that constitute the F\u0002\u0003×μ-prime-\nstrips that appear in the Θ-link — that are simultaneously associated [as “un-\nderlying structures”] to both †Ring and ‡Ring via isomorphisms [i.e., of certain\nsuitable multiplicative monoids equipped with actions by certain suitable abstract\ntopological groups] that restrict, on the subquotient monoids that correspond to the\nrespective value groups, to the desired assignment {†qj2\nv\n} \b→ ‡q\nv\n. It is this crucial\nsimultaneity that yields, as a tautological consequence, the validity of the AND\nrelator “∧” in (AOΘ1).\nWorking, as in the discussion above, with multiplicative monoids equipped\nwith actions by abstract topological groups, necessarily gives rise to certain inde-\nterminacies, called (Ind1), (Ind2), that play an important role in inter-universal\nTeichmüller theory. Certain aspects of these indeterminacies (Ind1), (Ind2) will be\ndiscussed in more detail in §3.5 below. In this context, we recall that one central\nassertion of the RCS [cf. the discussion of Example 3.2.2; the discussion of (SSInd),\n(SSId) in [Rpt2018], §7, §10] is to the effect that\n(NeuRng) these indeterminacies (Ind1), (Ind2) may be eliminated, without affecting\nthe essential logical structure of inter-universal Teichmüller theory, by tak-\ning the multiplicative monoids and abstract topological groups that appear\nin the F\u0002\u0003×μ-prime-strips of the above discussion to be equipped with\nrigidifications by regarding them as arising from some fixed “neutral”\nring structure \u0006Ring.\nOn the other hand, as discussed in (NoRng) above, there does not exist any ring\nstructure that is compatible [i.e., in the sense discussed in (NoRng)], with the desired\nassignment {†qj2\nv\n} \b→ ‡q\nv\n. That is to say, in summary,\n(NeuORInd) working with such a fixed “neutral” ring structure \u0006Ring as in (NeuRng)\nmeans either that\n(NeuORInd1) there is no relationship between “∗” and \u0006Ring, or that\n(NeuORInd2) the relationship between “∗” and \u0006Ring is always necessarily\nsubject to an indeterminacy [cf. (AOΘ2), (AOΘ3)!](\n∗ :→ \u0006Θ-plt ∈ \u0006Ring\n)\n∨\n(\n∗ :→ \u0006q-plt ∈ \u0006Ring\n)\n[cf. the situation discussed in Example 3.2.2; the situation discussed in\n[Rpt2018], §10, (SSId)].\nHere, we observe that whichever of these “options”\/“indeterminacies” that appear\nin (NeuORInd) [i.e., (NeuORInd1), (NeuORInd2)] one chooses to adopt, one is\nforced to contend with an indeterminacy that is, in some sense, much more drastic\n88 SHINICHI MOCHIZUKI\nthan the relatively mild indeterminacies (Ind1), (Ind2) whose elimination formed\nthe original motivation for the introduction of \u0006Ring!\nFinally, we observe that this much more drastic indeterminacy (NeuORInd)\nmeans [cf. the discussion of Example 2.4.4!] that throughout any argument, one\nmust always take the position that the only possible relationship between “∗” and\n\u0006Θ-plt, \u0006q-plt is one in which\n(PltRel) “∗” maps either to \u0006Θ-plt or — i.e., “∨”! — to \u0006q-plt, but not both!\nSince ‡Ring may be thought of as a ring structure in which “∗” tautologically maps\nto ‡Θ-plt, while †Ring may be thought of as a ring structure in which “∗” tauto-\nlogically maps to ‡q-plt, one may rephrase the above observation as the observation\nthat one must always take the position that the only possible relationship between\n\u0006Ring, on the one hand, and ‡Ring, †Ring, on the other, is one in which\n(RngRel) the ring structure \u0006Ring is identified either with the ring structure ‡Ring\nor — i.e., “∨”! — with the ring structure †Ring, but not both!\nAt this point, let us recall [cf., e.g., the discussion of §3.5, §3.11, below; [Rpt2018],\n§9, (GIUT), (ΘCR)] that\ninter-universal Teichmüller theory requires, in an essential way, the use of\nthe log-links, hence, in particular, [in order to define the power series of\nthe various p-adic logarithm functions that constitute these log-links!] the\nring structures †Ring, ‡Ring on both sides — i.e., “∧”! — of the Θ-link\n[cf. the discussion surrounding (InfH) of the two vertical lines of log-links in the\n“infinite H” on either side of the Θ-link]. In particular, we conclude formally that\nit is impossible to implement the arguments of inter-universal Teichmüller\ntheory once this sort of much more drastic indeterminacy (NeuORInd)\nhas been imposed.\n§3.5. Gluings, indeterminacies, and pilot discrepancy\nAs discussed in §3.4, the Θ-link involves a gluing\n{†qj2\nv\n} \b→ ‡q\nv\nthat identifies ‡q\nv\n[i.e., 2l-th roots of the q-parameters at primes of multiplicative\nreduction of the [copy belonging to ‡Ring of the] elliptic curve under consideration]\nwith elements, i.e., the †qj\n2\nv\n’s, which, when j \n= 1, have different valuations from\nthe valuation of ‡q\nv\n.\nOn the other hand, in inter-universal Teichmüller theory, by applying the mul-\ntiradial representation of [IUTchIII], Theorem 3.11, which involves various in-\ndeterminacies (Ind1), (Ind2), (Ind3), and then forming [cf. [IUTchIII], Corollary\n3.12, and its proof] the holomorphic hull of the union of possible images of the\nΘ-pilot in this multiradial representation,\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 89\n(ΘGl) one may treat both sides of the Θ-link gluing of the above display\nas belonging to a single ring theory without disturbing [cf. the crucial\nAND relator “∧” property discussed in §3.4!] the gluing.\nAlternative ways to understand the essential mathematical content of (ΘGl) include\nthe following:\n(NonInf) One may think of (ΘGl) as a statement concerning the mutually non-\ninterference or simultaneous executatibility of the Kummer theo-\nries surrounding the q-pilot and Θ-pilot relative to the gluing of abstract\nF\u0002\u0003×μ-prime-strips constituted by the Θ-link, i.e., when the Kummer\ntheory surrounding the q-pilot is held fixed, and one allows the Kummer\ntheory surrounding the Θ-pilot to be subject to various indeterminacies.\n(Cohab) One may think of (ΘGl) as a statement concerning the “cohabitation”,\nor “coexistence”, of the q-pilot and Θ-pilot — relative to the gluing\nof abstract F\u0002\u0003×μ-prime-strips constituted by the Θ-link — within the\ncommon container obtained by applying the multiradial representation\nof the Θ-pilot, forming the holomorphic hull [relative to the holomorphic\nstructure [i.e., (Θ±ellNF-)Hodge theater] that gave rise to the q-pilot under\nconsideration], and finally taking log-volumes.\nIn this context, it is important to recall that this sort of phenomenon — i.e.,\nof computations of global degrees\/heights of elliptic curves in sit-\nuations where a certain “confusion”, up to suitable indeterminacies,\nis allowed between q-parameters of the elliptic curves and certain large\npositive powers of these q-parameters [i.e., as in (ΘGl)]\n— may be seen in various classical examples such as\n· the proof by Faltings of the invariance of heights of abelian varieties\nunder isogeny [cf. the discussion of [Alien], §2.3, §2.4, as well as the\ndiscussion of Example 3.2.1 in the present paper],\n· the classical proof in characteristic zero of the geometric version of the\nSzpiro inequality via the Kodaira-Spencer morphism, phrased in terms of\nthe theory of crystals [cf. the discussion of [Alien], §3.1, (v)], and\n· Bogomolov’s proof over the complex numbers of the geometric version\nof the Szpiro inequality [cf. the discussion of [Alien], §3.10, (vi)]\n— cf. also the discussion of [Rpt2018], §16. Moreover, in the case of crystals, we\nobserve that, relative to the notation introduced in Example 2.4.5, (v), (vi), we\nhave correspondences as follows:\n(CrAND) The logical AND “∧” that appears in themultiradial representation of the\nΘ-pilot in IUT (= AND-IUT)may be understood as being analogous to\nthe fact that crystals, i.e., “∧-crystals”, may be thought of as objects\n[on infinitesimal neighborhoods of the diagonal inside products of two\ncopies of the scheme under consideration] that may be simultaneously\ninterpreted, up to isomorphism, as pull-backs via one projection morphism\nand [cf. “∧”!] as pull-backs via the other projection morphism [cf. the\ndiscussion of (∧(∨̇)-Chn1) in §3.10 below; the discussion of [Alien], §3.11,\n90 SHINICHI MOCHIZUKI\n(iv), (2and), concerning the interpretation of the discussion of crystals in\n[Alien], §3.1, (v), (3KS), in terms of the logical relator “∧”].\n(CrOR) Thus, from the point of view of the analogy discussed in (CrAND), the\nlogical OR “∨” that appears throughout OR-IUT may be understood as\ncorresponding to working with “∨-crystals” [i.e., as opposed to crystals\n(= ∧-crystals)], that is to say, with objects [on infinitesimal neighbor-\nhoods of the diagonal inside products of two copies of the scheme under\nconsideration] that may be interpreted, up to isomorphism, as pull-backs\nvia one projection morphism or [cf. “∨”!] as pull-backs via the other pro-\njection morphism. Here, we observe that this defining “∨” condition of an\n∨-crystal is essentially vacuous since one may obtain ∨-crystals from ar-\nbitrary objects on the scheme under consideration simply by pulling back\nsuch an object to the infinitesimal neighborhood of the diagonal under\nconsideration via one of the two projection morphisms.\n(CrRCS) In a similar vein, from the point of view of the analogies discussed in\n(CrAND) and (CrOR), RCS-IUT may be understood as corresponding\nto the modified version of the usual theory of crystals obtained by re-\nplacing the infinitesimal neighborhoods of the diagonal inside products\nof two copies of the scheme under consideration [i.e., that appear in the\nusual theory of crystals!] by the diagonal itself. Such a replacement\nclearly renders the usual theory of crystals trivial\/meaningless, in a\nfashion that is essentially very similar to the triviality of ∨-crystals dis-\ncussed in (CrOR). Finally, we observe that this similarity between the\nmodified versions of the usual theory of crystals discussed in (CrOR) and\nthe present (CrRCS) is entirely analogous to the equivalence OR-IUT\n⇐⇒ RCS-IUT observed in Example 2.4.5, (v), (XOR\/RCS).\nUnfortunately, however, the situation summarized above in (ΘGl) has resulted\nin certain frequently voiced misunderstandings by some mathematicians. One\nsuch frequently voiced misunderstanding is to the effect that\n(CnfInd1+2) the situation summarized in (ΘGl) may be explained as a consequence of\na “confusion” between q-parameters and large positive powers of these\nq-parameters that results from the indeterminacies (Ind1), (Ind2).\nIn fact, however, as discussed in Example 3.5.1, (iii), below,\nat least in the case of q-parameters of sufficiently small valuation [i.e.,\nsufficiently large positive order, in the sense of loc. cit.], such a “con-\nfusion” [i.e., between q-parameters and large positive powers of these\nq-parameters] can never occur as a consequence of (Ind1), (Ind2), i.e.,\nboth of which amount to automorphisms of the [underlying topological\nmodule of the] log-shells involved\n[cf. also the discussion of (ΘInd) in [Rpt2018], §11]. In this context, we note that\nthis misunderstanding (CnfInd1+2) appears to be caused in many cases, at least\nin part, by a more general misunderstanding concerning the operation of passage\nto underlying structures [cf. Example 3.5.2 below]. A more detailed discussion of\nthe operation of passage to underlying structures may be found in §3.9 below.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 91\nAs discussed in [Rpt2018], §11, the “confusion” summarized in (ΘGl) occurs\nin inter-universal Teichmüller theory as a consequence not only of the local in-\ndeterminacies (Ind1), (Ind2), (Ind3), but also of the constraints imposed by the\nglobal realified Frobenioid portions of the F\u0002\u0003×μ-prime-strips that appear in\nthe Θ-link. In this context, it is of particular importance to observe that\n(CnfInd3) the indeterminacy (Ind3), which constrains one to restrict one’s atten-\ntion to upper bounds [i.e., but not lower bounds!] on the log-volume\nthat is the subject of the computation of [IUTchIII], Corollary 3.12, al-\nready by itself — i.e., without considering (Ind1), (Ind2), or global reali-\nfied Frobenioids! [cf. the discussion of (Ind3>1+2) in §3.11 below] — is\nsufficient to account for the possibility of a “confusion” of the sort\nsummarized in (ΘGl) [i.e., between q-parameters and large positive powers\nof these q-parameters].\nIndeed, the indeterminacy (Ind3) is defined in precisely such a way as to identify\nthe ideals generated by arbitrary positive powers of the q-parameters.\nExample 3.5.1: Bounded nature of log-shell automorphism indetermina-\ncies. Write Zp for the ring of p-adic integers, for some prime number p; Qp for\nthe field of fractions of Zp.\n(i) Let M be a finitely generated free Zp-module, which, in the following dis-\ncussion, we shall think of as being embedded in MQp\ndef\n= M ⊗Zp Qp;\nα : M\n∼→ M\nan automorphism of the Zp-module M . For n ∈ Z, write\nU(M,n) def= {x ∈MQp | x ∈ pn ·M, x \n∈ pn+1 ·M} ⊆ MQp .\nThen observe that α induces a bijection\nU(M,n) ∼→ U(M,n)\nfor every n ∈ Z.\n(ii) In the notation of (i), suppose, for simplicity, that p is odd. Let k be a finite\nfield extension of Qp. Write Ok ⊆ k for the ring of integers of k; O×k ⊆ Ok for the\ngroup of units of Ok; mk ⊆ Ok for the maximal ideal of k; Ik ⊆ k for the log-shell\nassociated to k [cf., e.g., the discussion of [IUTchIII], Remark 1.2.2, (i)], i.e., the\nresult of multiplying by p−1 the image logp(O×k ) of O×k by the p-adic logarithm\nlogp(−). Thus,\nOk ⊆ Ik ⊆ p−c · Ok\nfor some nonnegative integer of c that depends only on the isomorphism class of\nthe field k [cf. [IUTchIV], Proposition 1.2, (i)]. In particular, there exists a positive\ninteger s that depends only on the isomorphism class of the field k such that for\nany automorphism\nφ : Ik ∼→ Ik\n92 SHINICHI MOCHIZUKI\nof the Zp-module Ik and any n ∈ Z, it holds that\nφ(U(Ok, n)) ⊆\ns⋃\ni=−s\nU(Ok, n+ i)\n[where i ranges over the integers between −s and s].\n(iii) In the situation of (ii), we define the order of a nonzero element x ∈ k to be\nthe unique n ∈ Z such that x ∈ mnk , x \n∈ mn+1k . One thus concludes from the final\nportion of the discussion of (ii) that there exists a positive integer t that depends\nonly on the isomorphism class of the field k such that for any automorphism\nφ : Ik ∼→ Ik\nof the Zp-module Ik and any nonzero element q ∈ Ok [i.e., such as the q-parameter\nof a Tate curve over k!], the absolute value of the difference between the orders of\nq and φ(q) is ≤ t, i.e., in words,\nautomorphisms of the Zp-module Ik only give rise to bounded discrep-\nancies in the orders of nonzero elements of Ok.\nExample 3.5.2: Examples of gluings. Distinct auxiliary structures on some\ncommon [i.e., “∧”!] underlying structure may be thought of as gluings of the dis-\ntinct auxiliary structures along the common underlying structure. Here, we observe\nthat, in general, distinct auxiliary structures on a common underlying structure are\nnot necessarily mapped to one another by some automorphism of the common un-\nderlying structure. Concrete examples of these generalities may be found in quite\nsubstantial abundance throughout arithmetic geometry and include, in particular,\nthe examples (i), (ii), (iii), (iv), (v) given below, as well as the elementary Exam-\nples 2.3.2, 2.4.1, 2.4.2, 2.4.3, 2.4.7, 3.3.1 discussed in §2.3, §2.4, §3.3 [cf. also the\ndiscussion of [Rpt2018], §11]. In passing, we observe that\nthese examples may also be understood as interesting examples of the\nsort of gluing\/logical AND “∧” relation that appears in the Θ-\/log-\nlinks of inter-universal Teichmüller theory, i.e., examples of situations that\nare qualitatively similar to the Θ-\/log-links of inter-universal Teichmüller\ntheory in the sense that they involve distinct auxiliary structures that are\nglued together along some common auxiliary structure\n[cf. the discussion of (StR1) ∼ (StR6) in §3.2; the discussion of §3.4; the portion of\nthe present §3.5 preceding Example 3.5.1].\n(i) The group structures of the finite abelian groups Z\/2Z×Z\/2Z and Z\/4Z\nare not mapped to one another by any isomorphism of sets, despite the fact that\nthe underlying sets of these two groups are indeed isomorphic to one another. This\nexample is also of interest in light of the discussion of truncated Witt vectors\nin Example 2.4.6, (iii).\n(ii) The scheme structures of non-isomorphic algebraic curves over a com-\nmon algebraically closed field are not mapped to one another by any isomorphism\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 93\nof topological spaces, despite the fact that the underlying topological spaces of al-\ngebraic curves over a common algebraically closed field are indeed isomorphic to\none another.\n(iii) The holomorphic structures of non-isomorphic compact Riemann sur-\nfaces of the same genus are not mapped to one another by any isomorphism of\ntopological spaces, despite the fact that the underlying topological spaces of such\nRiemann surfaces are indeed isomorphic to one another. [This example is in fact\nessentially similar to the situation discussed in Example 3.3.1, except that in the\nsituation of Example 3.3.1, the “two” Riemann surfaces involved are both isomor-\nphic to the complex plane, hence, in particular, isomorphic to one another.]\n(iv) The field structures of non-isomorphic mixed-characteristic local\nfields [which, by local class field theory, may be regarded as [the formal union with\n“{0}” of] some suitable subquotient of their respective absolute Galois groups] are\nnot, in general, mapped to one another by any isomorphism of profinite groups\nbetween the respective absolute Galois groups [cf., e.g., [Ymgt], §2, Theorem, for\nan example of this phenomenon].\n(v) In the notation of Example 3.5.1, (i), let X be a proper smooth curve of\ngenus ≥ 2 over Fp def= Zp\/pZp. Thus, X may be thought of as an “underlying\nstructure” associated to any lifting of X to Zp, i.e., any flat Zp-scheme Y equipped\nan isomorphism of Fp-schemes Y ×ZpFp ∼→ X. Then observe that non-isomorphic\nliftings ofX to Zp are not, in general, mapped to one another by any automorphism\nof the Fp-scheme X. [Indeed, this is particularly easy to see if one chooses X such\nthat X does not admit any nontrivial automorphisms.] Finally, we note that this\nexample may be regarded as a sort of p-adic analogue of the example of (iii).\n§3.6. Chains of logical AND relations\nFrom the point of view of the simple qualitative model of inter-universal Te-\nichmüller theory given in Example 2.4.5, the discussion of §3.4 concerns the AND\nrelator “∧” in the “Θ-link” portion of Example 2.4.5, (ii). On the other hand,\nstrictly speaking, this portion of inter-universal Teichmüller theory only concerns\nthe initial definition of the Θ-link. That is to say, the bulk of the theory developed\nin [IUTchI-III] concerns, from the point of view of the simple qualitative model of\ninter-universal Teichmüller theory given in Example 2.4.5, (ii), the preservation\nof the AND relator “∧” as one passes from\n· the “Θ-link” portion of Example 2.4.5, (ii), to\n· the “multiradial representation” portion of Example 2.4.5, (ii).\nBy contrast, the passage from the “multiradial representation” portion of Example\n2.4.5, (ii), to the “final numerical estimate” portion of Example 2.4.5, (ii) — i.e.,\nwhich corresponds to the passage from [IUTchIII], Theorem 3.11, to [IUTchIII],\nCorollary 3.12 — is [cf. the discussion of the final portion of Example 2.4.5, (ii)!]\nrelatively straightforward [cf. the discussion of §3.10, §3.11, below].\nAt this point, it is perhaps of interest to consider “typical symptoms” of\nmathematicians who are operating under fundamental misunderstandings con-\ncerning the essential logical structure of inter-universal Teichmüller theory. Such\n94 SHINICHI MOCHIZUKI\ntypical symptoms, which are in fact closely related to one another, include the\nfollowing:\n(Syp1) a sense of unjustified and acutely harsh abruptness in the pas-\nsage from [IUTchIII], Theorem 3.11, to [IUTchIII], Corollary 3.12 [cf. the\ndiscussion of the final portions of Example 2.4.5, (ii), (iii)!];\n(Syp2) a desire to see the “proof” of some sort of commutative diagram or\n“compatibility property” to the effect that taking log-volumes of pilot\nobjects in the domain and codomain of the Θ-link yields the same real\nnumber [a property which, in fact, can never be proved since it is false! —\ncf. the discussion of §3.5];\n(Syp3) a desire to see the inequality of the final numerical estimate obtained\nas the result of concatenating some chain of intermediate inequali-\nties, i.e., as is often done in proofs in real\/complex\/functional analysis or\nanalytic number theory.\nHere, it should be noted that (Syp2) and (Syp3) often occur as approaches to\nmitigating the “harsh abruptness” of (Syp1).\nWith regard to (Syp3), it should be emphasized that it is entirely unrealistic\nto attempt to obtain the inequality of the final numerical estimate as the result of\nconcatenating some chain of intermediate inequalities since this is simply not the\nway in which the logical structure of inter-universal Teichmüller theory is organized.\nThat is to say, in a word, the logical structure of inter-universal Teichmüller theory\ndoes not proceed by concatenating some sort of chain of intermediate inequalities.\nRather,\n(∧-Chn) the logical structure of inter-universal Teichmüller theory proceeds by\nobserving a chain of AND relations “∧”\n[cf. the discussion of [IUTchIII], Remark 3.9.5, (viii), (ix); [IUTchIII], Remark\n3.12.2, (citw), (fitw); [Alien], §3.11, (iv), (v)]. As observed in Example 2.4.5, (ii),\n(iii), once one follows this chain of AND relations “∧” up to and including the\nmultiradial representation of the Θ-pilot [i.e., [IUTchIII], Theorem 3.11], the pas-\nsage to the final numerical estimate [i.e., [IUTchIII], Corollary 3.12] is relatively\nstraightforward [i.e., as one might expect, from the use of the word “corollary”!].\nOne essentially formal consequence of (∧-Chn) is the following: Since the defi-\nnition of the Θ-link, the construction of themultiradial representation of the Θ-pilot,\nand the ultimate passage to the final numerical estimate consist of a finite number\nof steps, one natural and effective way to analyze\/diagnose [cf. the discussion\nof §1.4!] the precise content of misunderstandings of inter-universal Teichmüller\ntheory is to determine\n(∧-Dgns) precisely where in the finite sequence of steps that appear is the first\nstep at which the person feels that the preservation of the crucial AND\nrelator “∧” is no longer clear.\nIn some sense, the starting point of the various AND relations “∧” that appear\nin the multiradial algorithm of [IUTchIII], Theorem 3.11, is the observation that\n(∧-Input) the input data for this multiradial algorithm consists solely of an\nabstract F\u0002\u0003×μ-prime-strip; moreover, this multiradial algorithm is\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 95\nfunctorial with respect to arbitrary isomorphisms between F\u0002\u0003×μ-prime-\nstrips\n[cf. [IUTchIII], Remark 3.11.1, (ii); the final portion of [Alien], §3.7, (i)]. This\nproperty (∧-Input) means that the multiradial algorithm may be applied to any\nF\u0002\u0003×μ-prime-strip that appears, or alternatively\/equivalently, that any F\u0002\u0003×μ-\nprime-strip may serve as the gluing data [cf. the “γJ” in the analogies discussed\nin §3.2, (StR3), (StR4), as well as Example 2.4.5, (ii)!] between a given situation\n[i.e., such as the (Θ±ellNF-)Hodge theater in the codomain of the Θ-link!] and the\ncontent of the multiradial algorithm.\nOn the other hand, in order to conclude that the multiradial algorithm yields\noutput data satisfying suitable AND relations “∧”, it is necessary also to examine\nin detail the content of this output data, i.e., in particular, in the context of the\ncentral IPL and SHE properties discussed in [IUTchIII], Remark 3.11.1, (iii), as\nwell as the chain of (sub)quotients aspect of the SHE property [cf. [IUTchIII],\nRemark 3.11.1, (iii); [IUTchIII], Remark 3.9.5, (viii), (ix)]. In a word, the essential\n“principle” that is applied throughout the various steps of the multiradial algorithm\nin order to derive new AND relations “∧” from old AND relations “∧” is the\nfollowing “principle of extension of indeterminacies”:\n(ExtInd) If A, B, and C are propositions, then it holds [that B =⇒ B ∨ C and\nhence] that\nA ∧B =⇒ A ∧ (B ∨ C).\nOne important tool that is frequently used in inter-universal Teichmüller theory in\na fashion that is closely related to (ExtInd) is the notion of a poly-morphism [cf.\nthe discussion of §3.7 below for more details].\nIn the context of (ExtInd), it is interesting to note that, from the point of view\nof the discussion of §3.4,\nthe “∨” that appears in the conclusion — i.e., A∧ (B ∨C) — of (ExtInd)\nmay be understood as amounting to essentially the same phenomenon as\nthe “∨” that appears in (NeuORInd2) [e.g., by taking “C” to be A].\nThat is to say, instead of generating AND relations “∧” tautologically by means of\nthe introduction of distinct labels [i.e., as in (AOΘ1)] — i.e., say, by introducing a\nnew distinct label for “C” so as to conclude a tautological relation\nA ∧B ∧ C\n— (ExtInd) allows one to generate new AND relations “∧” while avoiding the\nintroduction of new distinct labels. As discussed in §3.4, this point of view [i.e., of\navoiding the introduction of new distinct labels] leads inevitably to OR relations\n“∨”, i.e., as in (NeuORInd2) or as in the conclusion “A ∧ (B ∨ C)” of (ExtInd).\nAs discussed above, the reason that one wishes to avoid the introduction of new\ndistinct labels when applying (ExtInd) is precisely that\n(sQLTL) one wishes to apply (ExtInd) to form “(sub)quotients\/splittings”\nof the log-theta-lattice [cf. the title of [IUTchIII]!], i.e., to project the\nvertical line on the left-hand side of the infinite “H” portion of the log-\ntheta-lattice onto the vertical line on the right-hand side of this infinite\n96 SHINICHI MOCHIZUKI\n“H” by somehow achieving some sort of “crushing together” of distinct\ncoordinates [i.e., “(n,m)”, where n,m ∈ Z] of the log-theta-lattice\n[cf. the discussion of §3.11 below; [IUTchIII], Remark 3.9.5, (viii), (ix); [IUTchIII],\nRemark 3.12.2, (citw), (fitw); [Alien], §3.11, (iv), (v)).\nAt this point, it is of interest to note that there are, in some sense, two ways in\nwhich (ExtInd) is applied during the execution of the various steps of the multiradial\nalgorithm [cf. the discussion of §3.10, §3.11, below, for more details]:\n(ExtInd1) operations that consist of simply adding more possibilites\/indeter-\nminacies [which corresponds to passing from B to B ∨ C] within some\nfixed container;\n(ExtInd2) operations in which one identifies [i.e., “crushes together”, by passing\nfrom B to B ∨ C] objects with distinct labels, at the cost of passing to\na situation in which the object is regarded as being only known up to\nisomorphism.\nTypical examples of (ExtInd1) include the upper semi-continuity of (Ind3), as well\nas the passage to holomorphic hulls. Typically, such applications of (ExtInd1) play\nan important role in establishing various symmetry or invariance properties such\nas multiradiality. This sort of establishment of various symmetry or invariance prop-\nerties by means of (ExtInd1) then allows one to apply label crushing operations\nas in (ExtInd2). Put another way,\n· (ExtInd1) may be understood as a sort of operation whose purpose is to\nprepare suitable descent data, while\n· (ExtInd2) may be thought of as a sort of actual descent operation, i.e.,\nfrom data that depends on the specification of a member of some collection\nof distinct labels to data that is independent of such a label specification.\n[We refer to the discussion of §3.8 below for more details on foundational aspects\nof (ExtInd2) and to the discussion of §3.9 below for more details concerning the\nnotion of “descent”.] Typical examples of (ExtInd2) in inter-universal Teichmüller\ntheory are the following [cf. the notational conventions of [IUTchI], Definition 3.1,\n(e), (f)]:\n· identifying “Πv”’s [where v ∈ V] at different vertical coordinates [i.e.,\n“(n,m)” and “(n,m′)”, for n,m,m′ ∈ Z] of the log-theta-lattice, which\nresults in a “Πv regarded up to isomorphism” that is labeled by a new label\n“(n, ◦)”;\n· identifying “Gv”’s [where v ∈ V] at different horizontal or vertical coor-\ndinates [i.e., “(n,m)” and “(n′,m′)”, for n, n′,m,m′ ∈ Z] of the log-theta-\nlattice, which results in a “Gv regarded up to isomorphism” that is labeled\nby a new label “(◦, ◦)”;\n· identifying the F\u0002\u0003×μ-prime-strips in the Θ-link that arise from the Θ-\nand q-pilot objects in distinct (Θ±ellNF-)Hodge theaters [i.e., the (Θ±ellNF-\n)Hodge theaters in the domain and codomain of the Θ-link] by working\nwith these F\u0002\u0003×μ-prime-strips up to isomorphism.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 97\nIn some sense, the most nontrivial instances of the application of (ExtInd) in\nthe context of the multiradial algorithm occur in relation to the log-Kummer-\ncorrespondence [i.e., in the vertical line on the left-hand side of the infinite “H”]\nand closely related operations of Galois evaluation [cf. the discussion of §3.11\nbelow]. The Kummer theories that appear in this log-Kummer-correspondence —\ni.e., Kummer theories for\n· multiplicative monoids of nonzero elements of rings of integers in mixed-\ncharacteristic local fields,\n· mono-theta environments\/theta monoids, and\n· pseudo-monoids of κ-coric functions\n— involve the construction of various [Kummer] isomorphisms between\n· Frobenius-like data and\n· corresponding data constructed via anabelian algorithms from étale-like\nobjects.\nThe output of such algorithms typically involves constructing the “corresponding\ndata” as one possibility among many. Here, we note that\neither of these Frobenius-like\/étale-like versions of “corresponding data” is\n— unlike, for instance, the data that constitutes an F\u0002\u0003×μ-prime-strip!\n— sufficiently robust that it completely determines [even when only\nregarded up to isomorphism!] the [usual] embedding of the Θ-pilot.\nThat is to say, taken as a whole, the multiradial algorithm — and, especially, the\nportion of the multiradial algorithm that involves the log-Kummer correspondence\nand closely related operations of Galois evaluation — plays the role of\nexhibiting the Frobenius-like Θ-pilot as one possibility within a collec-\ntion of possibilities constructed via anabelian algorithms from étale-like\ndata.\nThus, in this situation, one obtains the crucial preservation of the AND relation\n“∧” by applying (ExtInd) twice, i.e., by applying\n· (ExtInd1) to the enlargement of the collection of possibilities under\nconsideration and\n· (ExtInd2) to the Kummer isomorphisms involved, when one passes\nfrom Frobenius-like object labels “(n,m)” [where n,m ∈ Z] to étale-like\nobject labels “(n, ◦)” [where n ∈ Z].\nThis is precisely what is meant by the chain of (sub)quotients aspect of the SHE\nproperty [cf. [IUTchIII], Remark 3.11.1, (iii); [IUTchIII], Remark 3.9.5, (viii), (ix)]\ndiscussed above [cf. also the discussion of §3.10, §3.11, below].\n§3.7. Poly-morphisms and logical AND relations\nPoly-morphisms— i.e., sets of morphisms between objects — appear through-\nout inter-universal Teichmüller theory as a tool for facilitating\nthe explicit enumeration of a collection of possibilities.\n98 SHINICHI MOCHIZUKI\nComposable ordered pairs of poly-morphisms [i.e., pairs for which the domain of\nthe first member in the pair coincides with the codomain of the second member\nin the pair] may be composed by considering the set of morphisms obtained by\ncomposing the morphisms that belong to the sets of morphisms that constitute the\ngiven pair of poly-morphisms. Such compositions of poly-morphisms allow one to\nkeep track — in a precise and explicit fashion — of collections of possibilities under\nconsideration.\nFrom the point of view of chains of AND relations “∧”, as discussed in\n§3.6,\nthe collections of possibilities enumerated by poly-morphisms are to\nbe understood as being related to one another via OR relations “∨”.\nThat is to say, poly-morphisms may be thought of as a sort of indeterminacy,\nwhich is used in inter-universal Teichmüller theory to produce structures that satisfy\nvarious symmetry or invariance properties, hence yield suitable descent data\n[cf. the discussion of (ExtInd1) in §3.6; the discussion of §3.9 below].\nThus, for instance, in the case of the full poly-isomorphism that constitutes the\nΘ-link, one may understand the fundamental AND relation “∧” of (AOΘ1) —\nwhich, for simplicity, we denote by\nA ∧B\n[where A and B correspond, respectively, in the notation of the discussion of §3.4,\nto “∗ :→ ‡q-plt ∈ ‡Ring” and “∗ :→ †Θ-plt ∈ †Ring”] — may be understood as\na relation “A ∧ (B1 ∨B2 ∨ . . . )”, i.e., a relation to the effect that\nif one fixes the q-pilot ‡q-plt, then this q-pilot is glued, via the Θ-link,\nto the Θ-pilot †Θ-plt by means of one isomorphism [of the full poly-\nisomorphism that constitutes the Θ-link] or another isomorphism, or yet\nanother isomorphism, etc.\n[Here, the various possible gluings that constitute B are denoted by B1, B2, . . . .]\nIn particular, as discussed in (∧-Chn), if one starts with the Θ-link and then con-\nsiders various subsequent logical AND relations “∧” that arise — for instance, by\nconsidering various composites of poly-morphisms! — by applying (ExtInd), then\n(∧(∨)-Chn) the essential logical structure of inter-universal Teichmüller theory, as\ndiscussed in (∧-Chn), may be understood as follows:\nA ∧B = A ∧ (B1 ∨B2 ∨ . . . )\n=⇒ A ∧ (B1 ∨B2 ∨ . . . ∨B′1 ∨B′2 ∨ . . . )\n=⇒ A ∧ (B1 ∨B2 ∨ . . . ∨B′1 ∨B′2 ∨ . . . ∨B′′1 ∨B′′2 ∨ . . . )\n...\nFinally, we recall that various “classical examples” of the notion of a poly-\nmorphism include\n· the collection of maps between topological spaces that constitutes a ho-\nmotopy class, or stable homotopy class, of maps;\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 99\n· the collection of morphisms between complexes that constitutes a mor-\nphism of the associated derived category;\n· the collection of morphisms obtained by considering some sort of orbit by\nsome sort of group action on the domain or codomain of a given morphism\n[cf. the discussion of [Rpt2018], §13, (PMEx1), (PMEx2), (PMQut)]. Also, in this\ncontext, it is useful to recall [cf. the discussion of [Alien], §4.1, (iv)] that\n· gluings via poly-morphisms are closely related to the sorts of gluings\nthat occur in the construction of algebraic stacks [i.e., algebraic stacks\nwhich are not algebraic spaces].\n§3.8. Inter-universality and logical AND relations\nOne fundamental aspect of inter-universal Teichmüller theory lies in the con-\nsideration of distinct universes that arise naturally when one considers Galois\ncategories— i.e., étale fundamental groups— associated to various schemes. Here,\nit is important to note that, when phrased in this way,\nthis fundamental aspect of inter-universal Teichmüller theory is, at least\nfrom the point of view of mathematical foundations, no different from\nthe situation that arises in [SGA1].\nOn the other hand, the fundamental difference between the situation considered\nin [SGA1] and the situations considered in inter-universal Teichmüller theory lies\nin the fact that, whereas in [SGA1], the various distinct schemes that appear are\nrelated to one another by means of morphisms of schemes or rings,\nthe various distinct schemes that appear in inter-universal Teichmüller\ntheory are related to one another, in general, by means of relations —\nsuch as the log- and Θ-links — that are non-ring\/scheme-theoretic in\nnature, i.e., in the sense that they do not arise from morphisms of schemes\nor rings.\nIn general, when considering relations between distinct mathematical objects, it\nis of fundamental importance to specify those mathematical structures that are\ncommon — i.e., in the terminology of inter-universal Teichmüller theory, coric —\nto the various distinct mathematical objects under consideration. Here, we observe\nthat\nthis notion of being “common”\/“coric” to the various distinct mathemat-\nical objects under consideration constitutes, when formulated at a formal,\nsymbolic level, a logical AND relation “∧”.\n— cf. the discussion of §3.2 [cf., especially, Example 3.2.2], §3.4, §3.5, §3.6, §3.7.\nThus, in the situations considered in [SGA1], the ring\/scheme structures of the\nvarious distinct schemes that appear are coric and hence allow one to relate the\nuniverses\/Galois categories\/étale fundamental groups associated to these distinct\nschemes in a way that makes use of the common ring\/scheme structures between\nthese schemes. At a concrete level, this means that\n100 SHINICHI MOCHIZUKI\nin the situations considered in [SGA1], étale fundamental groups may be\nrelated to one in such a way that the only indeterminacies that occur are\ninner automorphism indeterminacies.\nMoreover, these inner automorphism indeterminacies are by no means superfluous\n— cf. the discussion of Examples 3.8.1, 3.8.2, 3.8.3, 3.8.4 below.\nExample 3.8.1: Inevitability of inner automorphism indeterminacies.\nThe unavoidability of inner automorphism indeterminacies may be understood in\nvery elementary terms, as follows.\n(i) Let k be a perfect field; k an algebraic closure of k; N ⊆ Gk def= Gal(k\/k)\na normal closed subgroup of Gk; σ ∈ Gk such that the automorphism ισ : N ∼→ N\nof N given by conjugating by σ is not inner. [One verifies immediately that, for\ninstance, if k is a number field or a mixed-characteristic local field, then such N , σ do\nindeed exist.] Write kN ⊆ k for the subfield of N -invariants of k, GkN def= N ⊆ Gk,\nQN\ndef\n= Gk\/GkN . Then observe that this situation yields an example of a situation\nin which one may verify directly that\nthe functoriality of the étale fundamental group only holds if one\nallows for inner automorphism indeterminacies in the definition of\nthe étale fundamental group.\nIndeed, let us first observe that the “basepoints” of k and kN determined by k\nallows us to regard Gk and GkN , respectively, as the étale fundamental groups of\nk and kN . Thus, if one assumes that the functoriality of the étale fundamental\ngroup holds even in the absence of inner automorphism indeterminacies, then the\ncommutative diagram of schemes\nSpec(kN )\nσ−→ Spec(kN )\n↘ ↙\nSpec(k)\n[where the diagonal morphisms are the natural morphisms] induces a commutative\ndiagram of profinite groups\nGkN\nισ−→ GkN\n↘ ↙\nGk\n— which [since the natural inclusion N = GkN ↪→ Gk is injective!] implies that ισ\nis the identity automorphism, in contradiction to our assumption concerning σ!\n(ii) The phenomenon discussed in (i) may be understood as a consequence of\nthe fact that, whereas Spec(k) is coric in the commutative diagram of schemes that\nappears in (i) [i.e., in the sense that this diagram does indeed commute!], Spec(k)\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 101\nis not coric in the diagram of schemes\nSpec(k)\n↙ ↘\nSpec(kN )\nσ−→ Spec(kN )\n↘ ↙\nSpec(k)\n[where the diagonal morphisms are the natural morphisms], i.e., in the sense that\nthe upper portion of this diagram does not commute!\n(iii) Finally, we consider the natural exact sequence\n1 −→ GkN −→ Gk −→ QN −→ 1\nof profinite groups. Then observe that the inner automorphisms indeterminacies\nof Gk [cf. the discussion of (i), (ii)!] induce outer automorphism indeterminacies\nof GkN that will not, in general, be inner. That is to say,\nif one considers GkN in the context of this natural exact sequence, then\none must in fact consider GkN [not only up to inner automorphism in-\ndeterminacies, i.e., as discussed in (i), (ii), but also] up to certain outer\nautomorphism indeterminacies.\nRelative to the point of view of the discussion of (ii), these outer automorphism\nindeterminaciesmay be understood as a consequence of the fact that, in the context\nof the field extensions k ⊆ kN ⊆ k and the automorphisms of these field extensions\ninduced by elements of Gk,\nthe field k is coric, whereas the field kN is not coric\n— i.e., in the context of these field extensions and automorphisms of field extensions,\nthe relationship of k to the various field extensions that appear is constant and\nfixed, whereas the relationship of kN to the various field extensions that appear is\nvariable, i.e., subject to indeterminacies arising from the action of elements of\nGk.\nExample 3.8.2: Inter-universality and the structure of (Θ±ellNF-)Hodge\ntheaters. In the following discussion of (Θ±ellNF-)Hodge theaters, we fix a col-\nlection of initial Θ-data\n(F\/F, XF , l, CK , V, V\nbad\nmod, \u0002)\nas in [IUTchI], Definition 3.1, and apply the notational conventions of [IUTchI],\nDefinition 3.1. In particular, we recall that E\ndef\n= EF is an elliptic curve over the\nnumber field F ; F is an algebraic closure of F ; l is a prime number; K ⊆ F is\nthe extension field of F determined by the composite of the fields of definition of\nthe closed points of the finite group scheme E[l] ⊆ E of l-torsion points of E. For\nsimplicity, we assume that l > 5.\n102 SHINICHI MOCHIZUKI\n(i) We begin by recalling the following:\n(i-a) The point of view of classical Galois theory with regard to constructing\nfinite Galois extensions of fields may be summarized, in the case of the\nGalois extension K\/F , as follows:\n· one starts with a base field F ;\n· one then constructs a finite field extension K of F that is\nsaturated with respect to Galois conjugation over F .\nThus, relative to this classical point of view, one is constrained to viewing\nthe situation from the point of view of the base field F . This constraint\nobliges one to always take into account the entirety of Galois conjugates\n[over F ] of objects associated to K.\nThe point of view of (i-a) is fundamentally incompatible with the main goal of\nthe construction of (Θ±ellNF-)Hodge theaters in [IUTchI], namely, the simulation\nof a global multiplicative subspace of E[l] [cf. the discussion of global multi-\nplicative subspaces in [IUTchI], §I1; [Alien], §2.3, §2.4; [Alien], §3.3, (iv), as well\nas Example 3.2.1, (vi), of the present paper], together with a global canonical\ngenerator, up to ±1, of the quotient of E[l] by the global multiplicative subspace\n[cf. the discussion of global canonical generators in [IUTchI], §I1; [Alien], §3.3, (iv),\nas well as Example 3.2.1, (vi), of the present paper]. In some sense, the technical\nstarting point of the “simulation of a global multiplicative subspace” implemented\nin [IUTchI] may be summarized as follows:\n(i-b) The “simulation of a global multiplicative subspace” given in [IUTchI] is\nachieved by, in some sense, reversing the flow of the classical construction\nreviewed in (i-a), i.e., by\n· viewing the situation [not from the point of view of the base field\nF , but rather] from the point of view of the hyperbolic orbicurve\nCK\n— which may be thought of as data that amounts to K, together\nwith a fixed choice of a quotient “Q” [cf. [IUTchI], Definition\n3.1, (f)] of E[l], i.e., whose kernel is to serve as the “simulated\nglobal multiplicative subspace” — and\n· regarding the base field Fmod — or, at the level of hyperbolic\norbicurves, CFmod [cf. [IUTchI], Remark 3.1.7, (i)] — as a finite\nétale quotient of K [or, at the level of hyperbolic orbicurves,\nCK ], i.e., which amounts to thinking in terms of [compatible]\nfinite étale quotients\nSpec(K)\u0003 Spec(Fmod), CK → CFmod\n— which are regarded as objects constructed from Spec(K), CK .\nThis approach allows one to concentrate on a fixed [simulated global mul-\ntiplicative] subspace and hence [unlike the situation discussed in (i-a)!]\nto exclude the various nontrivial Galois conjugates over F of this fixed\nsimulated global multiplicative subspace.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 103\nThe approach of (i-b) has numerous important technical consequences [to be dis-\ncussed in (ii), (iii), (iv), below].\n(ii) From the point of view of étale-like objects [i.e., arithmetic fundamental\ngroups], constructing a quotient CK → CFmod as in (i-b) corresponds to constructing\na profinite group “ΠCFmod ” from the profinite group ΠCK that contains ΠCK as an\nopen subgroup. In light of the well-known slimness of ΠCFmod [cf., e.g., [AbsTopI],\nProposition 2.3, (ii)], such a construction of “ΠCFmod ” amounts to the construction\nof a finite group of outer automorphisms of some open subgroup of ΠCK . This finite\ngroup may be thought of as a finite quotient group ΠCFmod \u0003 Γmod of ΠCFmod . If\nwe think of the absolute Galois group GFmod of the number field Fmod as a quotient\nΠCFmod \u0003 GFmod of ΠCFmod , then this finite quotient group Γmod determines a finite\nquotient group GFmod \u0003 ΓGalmod of GFmod . Here, we recall from the construction\nof [IUTchI], Example 4.3, (i), that ΓGalmod has a natural subquotient that may be\nidentified with F\u0002l\ndef\n= F×l \/{±1}, i.e., which corresponds to the multiplicative F\u0002l -\nsymmetry of the (Θ±ellNF-)Hodge theater. In particular, ΓGalmod, hence also Γmod,\nis a finite group of order > 2, which implies, by well-known properties of absolute\nGalois groups of number fields [cf., e.g., [NSW], Theorem 12.1.7] that\n(NoSpl) The surjection GFmod \u0003 ΓGalmod of profinite groups does not admit a\nsplitting.\nHere, we note that [in light of the well-known slimness of GFmod — cf., e.g., [Ab-\nsTopI], Theorem 1.7, (iii)] this non-existence of a splitting may be reformulated\nas the assertion that the natural outer action of ΓGalmod on the kernel Ker(GFmod \u0003\nΓGalmod) does not admit a lifting to an action of Γ\nGal\nmod on Ker(GFmod \u0003 ΓGalmod), i.e.,\nto an action that is free of inner automorphism indeterminacies. In particular, it\nfollows [a fortiori!] that the natural outer action of Γmod on Ker(ΠCFmod \u0003 Γmod)\ndoes not admit a lifting to an action of Γmod on Ker(ΠCFmod \u0003 Γmod), i.e., to an\naction that is free of inner automorphism indeterminacies. That is to say, in sum-\nmary, the inner automorphism indeterminacies in these natural outer actions\nare essential and unavoidable.\n(iii) The existence of the inner automorphism indeterminacies discussed in (ii)\nimplies, in particular, that the permutations of prime-strips in the multiplicative\nsymmetry portion of a (Θ±ellNF-)Hodge theater induced by the F\u0002l -symmetries\nof the (Θ±ellNF-)Hodge theater necessarily give rise to inner automorphism indeter-\nminacies in the isomorphisms between the copies of local absolute Galois groups Gv\n[where v ∈ Vnon] that appear in prime-strips with distinct labels ∈ F\u0002l [cf. [IUTchI],\nRemark 4.5.1, (iii); [IUTchII], Remark 2.5.2, (iii); [IUTchII], Remarks 4.7.2, 4.7.6;\n[Alien], §3.6, (iii)]. Put another way,\n(NoSyn) there is no well-defined synchronization between these copies of Gv\nthat appear in prime-strips at distinct labels ∈ F\u0002l that is free of inner\nautomorphism — i.e., conjugacy — indeterminacies.\nIn this context, we recall that such a conjugate synchronization is of fundamen-\ntal importance in inter-universal Teichmüller theory since it is necessary in order to\nconstruct the data that appears in the unit group portion of the F\u0002\u0003×μ-prime-strip\nthat appears in the domain of the Θ-link, i.e., data that is required to be free of\nany dependence on the distinct labels ∈ F\u0002l . Such a conjugate synchronization is\n104 SHINICHI MOCHIZUKI\nachieved by applying the F\u0003±l -symmetries [where we recall that F\n\u0003±\nl\ndef\n= Fl\u0003{±1},\ni.e., relative to the natural action of {±1} on Fl] in the additive symmetry portion of\nthe (Θ±ellNF-)Hodge theater under consideration [cf. [IUTchII], Corollary 3.5, (i);\n[IUTchII], Remark 3.5.2, (iii); [IUTchII], Remark 4.5.3, (i); [IUTchIII], Theorem\n1.5, (iii); [IUTchIII], Remark 1.5.1, (i); [Alien], §3.6, (ii)]. Here, we observe that in\norder to achieve this conjugate synchronization via the F\u0003±l -symmetry of the various\ncopies of Gv that appear in prime-strips with distinct labels, it is of fundamental\nimportance to keep these copies of Gv isolated from the absolute Galois groups\nof number fields that appear in the discussion of (ii) [i.e., since, as observed in\n(ii), it is precisely the intrinsic structure of these global absolute Galois groups that\ngives rise to the unwanted inner automorphism\/conjugacy indeterminacies!]. This\nlocal-global isolation requirement — i.e., in effect, the requirement that\n(LGIsl) these copies of the local absolute Galois group Gv be regarded not as\nsubgroups of some global absolute Galois group, but rather as coric\nobjects that are treated as being independent of any sort of embedding\ninto a global absolute Galois group\n[cf. [IUTchII], Remark 4.7.6; [Alien], §3.6, (iii)] — will have important consequences,\nas we shall see in the discussion of (iv) below.\n(iv) As discussed in (iii), the issue\n(SymIsl) of isolating the F\u0003±l -symmetry from the F\n\u0002\nl -symmetry in order to\nachieve conjugate synchronization\nis one important reason for imposing the local-global isolation requirement (LGIsl)\nin the context of the construction of (Θ±ellNF-)Hodge theaters. In fact, however,\nthis property (LGIsl) is fundamental to the entire structure of a (Θ±ellNF-)Hodge\ntheater [cf., [IUTchI], Fig. 6.5; [Alien], Fig. 3.8]. That is to say, the issue (SymIsl)\nmay be thought of as being reflected in the gluing along certain collections of prime-\nstrips between the additive and multiplicative symmetry portions of the (Θ±ellNF-\n)Hodge theater [cf., [IUTchI], Fig. 6.5; [Alien], Fig. 3.8]. In fact, however,\n(SctNF) even within the multiplicative symmetry portion of a (Θ±ellNF-)Hodge\ntheater, the goal of simulating a global canonical generator requires\none to treat the various prime-strips that appear in the multiplicative\nsymmetry portion of the (Θ±ellNF-)Hodge theater as “sections”, in some\nsuitable sense, of the finite étale quotient Spec(K)\u0003 Spec(Fmod)\n— a point of view that is fundamentally incompatible with the prime decompo-\nsition trees of the number fields K, Fmod, hence again requires one to impose\n(LGIsl) [cf. [IUTchI], Remarks 4.3.1, 4.3.2; [Alien], §3.3, (iv)]. On the other hand,\nlet us recall that the ring structure of the nonarchimedean local field that gives rise\nto Gv cannot be reconstructed from the abstract topological group Gv [cf. [NSW],\nthe Closing Remark preceding Theorem 12.2.7; [AbsTopIII], §I3; [Alien], Example\n2.12.3, (i)]. In particular, once one imposes (LGIsl), the crucial reconstruction of\nthe ring structures of the nonarchimedean local fields that give rise to the various\ncopies of Gv — where we recall that such ring structures play a fundamental and\nindispensable role in the definition of the log-link! — can only be conducted if one\napplies the absolute anabelian algorithms of [AbsTopIII], §1, locally at each v ∈ Vnon\n[not to Gv, but rather] to Πv, i.e., one must always regard each coric copy of Gv as\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 105\na “certain quotient” of a corresponding coric copy of Πv. Indeed, from a historical\npoint of view [cf. the discussion of [IUTchI], Remark 4.3.2],\nit was precisely these local-global isolation aspects — i.e., surround-\ning (LGIsl), as discussed in (iii) and the present (iv) — of the structure\nof (Θ±ellNF-)Hodge theaters that motivated the author to develop the\nabsolute anabelian algorithms of [AbsTopIII], §1, in the first place!\nExample 3.8.3: Truncated vs. profinite Kummer theory and compatibil-\nity with the p-adic logarithm. In the following discussion, we fix notation as\nfollows: Let k be a finite extension of Qp, for some prime number p; k an algebraic\nclosure of k. Write\n· Gk def= Gal(k\/k);\n· Ok for the ring of integers of k, with maximal ideal mk ⊆ Ok;\n· O\u0004\nk\n⊆ Ok for the multiplicative monoid of nonzero elements of Ok;\n· O×\nk\n⊆ O\u0004\nk\nfor the group of invertible elements of O\u0004\nk\n;\n· O×\nk\n\u0003 O×μ\nk\nfor the quotient of O×\nk\nby the subgroup μ∞ of torsion\nelements [i.e., roots of unity] of O×\nk\n;\n· logk : O×k \u0003 k for the p-adic logarithm on O\n×\nk\n.\nThus, Gk acts naturally on O×μ\nk\n\u0006 O×\nk\n⊆ O\u0004\nk\n⊆ Ok. Let Π\u0003 Gk be a topological\ngroup equipped with a surjection onto Gk, which determines natural actions of Π\non O×μ\nk\n\u0006 O×\nk\n⊆ O\u0004\nk\n⊆ Ok. We shall often think of the pair Gk \u0002 O\u0004k or the pair\nΠ \u0002 O\u0004\nk\n“abstractly” as a pair consisting of an abstract topological monoid [i.e.,\nO\u0004\nk\n] equipped with a continuous action by an abstract topological group [i.e., Gk\nor Π]. Also, we shall write N≥1 for the multiplicative monoid of positive natural\nnumbers.\n(i) The three types of Kummer theory that occur in inter-universal Teichmüller\ntheory [cf. the discussion of Example 3.8.4, (i-a), (i-b), (i-c), below] involve Kum-\nmer towers that consist of N-th power maps, for N ∈ N≥1, on the monoids\ninvolved. In the case of the pair Gk \u0002 O\u0004\nk\n, one observes immediately that such\nN -th power maps satisfy the following properties [cf. the discussion of [IUTchII],\nRemark 3.6.4, (i)]:\n(i-a) the N -th power map O\u0004\nk\n\u0003 O\u0004\nk\nis not a ring homomorphism, i.e., is not\ncompatible with the additive structure underlying the ring structure\non O\u0004\nk\n∪ {0};\n(i-b) the N -th power map O\u0004\nk\n\u0003 O\u0004\nk\nis (Π\u0003) Gk-equivariant, hence may be\nthought of as inducing on étale-like cyclotomes constructed via anabelian\nalgorithms from Gk or Π the isomorphism functorially induced by some\n— at least from an a priori point of view — indeterminate automor-\nphism [cf. (i-a), which implies that Gk or Π must be treated as abstract\n106 SHINICHI MOCHIZUKI\ntopological groups, that is to say, as opposed to groups of ring\/field auto-\nmorphisms, i.e., “Galois groups\/arithmetic fundamental groups”; the dis-\ncussion preceding Example 3.8.1; the discussion following Example 3.8.4\nbelow; the discussion of (vi-c) below] of Gk or Π;\n(i-c) the N -th power map O\u0004\nk\n\u0003 O\u0004\nk\nalters, at least from an a priori point\nof view, cyclotomic rigidity isomorphisms — but not synchroniza-\ntions between collections of cyclotomic rigidity isomorphisms! — between\nétale-like cyclotomes [cf. (i-b)] and [Frobenius-like] cyclotomes arising the\ntorsion subgroup of O\u0004\nk\n— cf., e.g., the [in general] nontrivial action of\nthe N -th power map on the n-th roots of unity in O\u0004\nk\nfor n ∈ N≥1 prime\nto N .\nNote that it follows from (i-a) that, if we think of a “basepoint” as a particular\n“rigid” choice of an algebraic closure that is free of any conjugacy or N -th power\nmap indeterminacies, then whereas\n(i-d) the p-adic logarithm\nk ⊇ O\u0004\nk\n⊇ O×\nk\nlog\nk−→ k ⊇ O\u0004\nk\n⊇ O×\nk\nyields a precise, well-defined — and, in particular, free of any conjugacy or\nN -th power map indeterminacies! — set-theoretic [but not ring-theoretic!]\nrelationship between\n· the basepoint of the [“abstract”] copy ofGk \u0002 O\u0004\nk\nin the domain\nof logk and\n· the basepoint of the [“abstract”] copy of Gk \u0002 O\u0004\nk\nin the\ncodomain of logk\n[where we think of both of these copies of Gk \u0002 O\u0004\nk\nas constituent objects\nin the respective Kummer towers in the domain\/codomain of logk, as\ndiscussed above], i.e.,\na single unified basepoint\nthat is simultaneously valid for both the domain and codomain of logk,\n(i-e) the map Logk induced by the p-adic logarithm logk on inverse limits of\nthe Kummer tower\nlim←−\nN\nk ⊇ lim←−\nN\nO\u0004\nk\n⊇ lim←−\nN\nO×\nk\nLog\nk−→ k ⊇ O\u0004\nk\n⊇ O×\nk\n[where the inverse limits are overN ∈ N≥1, and we recall that raising to the\nN -th power on the “O\u0004\nk\n” in the domain of logk corresponds to multiplying\nby N on the “k” in the codomain of logk] only yields a relationship between\n· the inverse limit basepoint in the domain of Logk and· the basepoint associated to a single constituent Kummer tower\nobject [with a fixed additive structure!] in the codomain of Logk\n[i.e., “O×\nk\n” as opposed to “lim←−\nN\nO×\nk\n”].\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 107\nNote, moreover, that it follows from (i-b), (i-c) that the basepoint shifts that\noccur as one passes between different constituent Kummer tower objects via various\nN -th power maps are indeed — at least from an a priori point of view — substan-\ntive\/nontrivial in the context of cyclotomic rigidity isomorphisms or synchroniza-\ntions between cyclotomes.\n(ii) The situation discussed in (i-e) may be understood as a consequence of the\nfact that\n(ii-a) the Kummer tower inverse limit “lim←−” [where we omit the subscript\nN to simplify notation] is biased toward the multiplicative structure\nof the rings involved — i.e., at the expense of the additive structures of\nthese rings [cf. (i-a)] — hence fundamentally incompatible with the\n“juggling\/rotation\/permutation” of the additive and multiplicative\nstructures that arises from the log-link [cf. the discussion of Example\n3.3.2, (iv)].\nIn the situation of (i-e), we observe, moreover, that the problem of constructing\nsome sort of “hyper-multiplicative tower” of copies of, say, the lim←− O×k in the domain\nof Logk that lifts [i.e., relative to Logk] the [multiplicative] Kummer tower of copies\nof O×\nk\n⊆ O\u0004\nk\n⊆ k in the codomain of Logk appears to be unrealistically intractable:\nIndeed, compatibility, relative to Logk, with the [multiplicative] Kummer tower\nin the codomain of Logk would imply that the transition maps of such a “hyper-\nmultiplicative tower” would, at least at a purely formal computational level (!),\nnecessarily be of the form\nx = exp(log(x)) \b→ exp({log(x)}M )\n= exp({log(x)} · {log(x)}M−1) = x{log(x)}M−1\n— where M ∈ N≥1, and the notation “exp(−)” and “log(−)” is intended in a purely\nformal computational sense (!). On the other hand,\n(ii-b) it seems difficult to conceive of any sort of natural approach to construct-\ning such “hyper-multiplicative transition maps”\nlim←− O×k → lim←− O\n×\nk\nthat realize the purely formal computation “x \b→ x{log(x)}M−1” for\nx ∈ lim←− O×k and, moreover, allow one to relate, in some natural way, the\n[multiplicative] Kummer theory associated to the lim←− O×k in the do-\nmain of the transition map to the corresponding [multiplicative] Kummer\ntheory associated to the lim←− O×k in the codomain of the transition map.\n(iii) Note that the conditions imposed on the “hyper-multiplicative transition\nmaps” in (ii-b) are stated in a somewhat rough and imprecise way. Although it is not\nclear at the time of writing how to make these conditions completely precise, it does,\nhowever, seem natural to consider the possibility of the existence of commutative\ndiagrams as in (iii-a) below, i.e., where one thinks of “Z” as a sort of candidate for\nthe inverse limit of the “hyper-multiplicative transition maps” of (ii-b). That is to\n108 SHINICHI MOCHIZUKI\nsay, the existence of such a commutative diagram may be thought of as a sort of\nnecessary condition for the existence of a suitable system of “hyper-multiplicative\ntransition maps” as in (ii-b). [Here, we note that the surjectivity condition of (iii-\na) below may be understood as a sort of “very weak necessary” version of the\ndomain\/codomain Kummer theory-relatability condition of (ii-b).] In fact, however,\nwe observe that, in the situation of (ii-b) [and the surrounding discussion], such a\ncommutative diagram does not exist:\n(iii-a) Let Z be a set with an action by Gk, ζ : Z → lim←− O×k a Gk-equivariant\nmap of sets that induces a surjection from the J-invariants in the domain\nof ζ to the J-invariants in the codomain of ζ for every closed subgroup\nJ ⊆ Gk that acts trivially on μ∞ ⊆ k. [Thus, ζ itself is necessarily\nsurjective.] Then there does not exist any commutative diagram of the\nform\nZ\nλ−→ lim←− k⏐⏐\nζ ⏐⏐\nψ\nlim←− O×k\nLog\nk−→ k\n—where λ is a Gk-equivariantmap of sets, and ψ is the natural projection.\n(iii-b) The map Logk does not admit any factorization\nlim←−\nN\nO×\nk\nλ−→ lim←−\nN\nk\nψ−→ k\n—where λ is a Gk-equivariantmap of sets, and ψ is the natural projection.\nIndeed, since (iii-b) follows formally from (iii-a), it suffices to verify (iii-a). Suppose\nthat a commutative diagram as in (iii-a) exists. Write\n· Ok def= Ok ∩ k, O×k\ndef\n= O×\nk\n∩ k, mk def= mk ∩ k;\n· F def= k(μ∞) ⊆ k.\nLet x ∈ p2 · Ok be a nonzero element. Write\n· f def= 1 + x ∈ O×k , (0 \n=) y\ndef\n= logk(f) ∈ mk;\n· E ⊆ k for the field extension of F obtained by adjoining all roots of f\nin k.\nThus, f lifts [via the natural projection] to an element of the codomain of ζ that\nis fixed by the action of J\ndef\n= Gal(k\/E) ⊆ Gk, hence also [by our surjectivity\nassumption on ζ] to an element fZ ∈ Z of the domain of ζ that is fixed by the\naction of J . Since λ is Gk-equivariant, we thus conclude that λ(fZ) is fixed by the\naction of J and maps via the natural projection ψ to an element z ∈ E ⊆ k that\ndefines a divisible, hence l-divisible element of E×, for any prime number l \n= p.\nOn the other hand, it follows immediately from the commutativity of the diagram\nthat y = z. Next, observe that since [the unit!] f is already clearly l-divisible in\nk×, hence also in F×, the Galois group Gal(E\/F ) is isomorphic to a quotient of\nZp. But this implies that all l-power roots of [the non-unit!] y = z ∈ k× ⊆ F× are\ncontained in F , in contradiction to the easily verified fact that the value group of\nthe valued field F is isomorphic to Z[p−1]. This completes the proof of (iii-a).\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 109\n(iv) The qualitatively different behavior that occurs in (i-d) and (i-e) may be\nunderstood as a being a consequence of the fact [cf. (i-a)] that whereas\n(iv-a) the ring structure on O\u0004\nk\n∪ {0} [which makes it possible to define\nthe well-known power series for the p-adic logarithm logk] remains intact\nat any particular constituent Kummer tower object [cf. the situation of\n(i-d)],\n(iv-b) the natural multiplicative structure on\nlim←−\nN\nk ⊇ lim←−\nN\nO\u0004\nk\n⊇ lim←−\nN\nO×\nk\ndoes not admit any corresponding additive structure that gives rise\nto a ring structure [i.e., that would make it possible to define the well-\nknown power series for the logarithm, hence a factorization as in (iii-b)]\non any of the three inverse limits in the above display that is compatible\nwith the natural action by Gk and the various natural projections to k [cf.\nthe situation of (i-e)].\nMoreover, we observe that\n(iv-c) the various natural Gk-equivariant projections of multiplicative monoids\nlim←−\nN\nk \u0003 k; lim←−\nN\nO\u0004\nk\n\u0003 O\u0004\nk\n; lim←−\nN\nO×\nk\n\u0003 O×\nk\ndo not admit splittings [as may be seen, for instance, by restricting such\na splitting to the roots of unity, where the existence of such a splitting\nwould amount, in particular, to a splitting of the natural surjection Qp \u0003\nQp\/Zp].\nThat is to say, there is no natural way to relate the finite Kummer theory for a single\nconstituent Kummer tower object “k”, “O\u0004\nk\n”, “O×\nk\n” in the codomain of the map\nLogk of (i-e) to the corresponding profinite Kummer theory obtained by passing to\nthe inverse limit “lim←−\nN\n” of the associated Kummer tower.\n(v) In the context of (iv-b) [and the surrounding discussion], it is also of interest\nto observe that in fact\n(v-a) neither of the inverse limits\nIk\ndef\n= lim←−\nN\nk; IO×\nk\ndef\n= lim←−\nN\n(\nO×\nk\n∪ {0}\n)\nadmits a field structure that is stabilized by the natural action of Gk,\nand whose underlying multiplicative structure is the natural multiplicative\nstructure on the inverse limit.\nIndeed, suppose that I ∈ {Ik, IO×\nk\n} admits such a field structure. Let l be an odd\nprime that does not divide the order of the finite group [cf., e.g., [Tsjm], Lemma\nD, (iii)] of roots of unity of k. Write μl ⊆ k for the group of l-th roots of unity\nof k, (μl ⊆) μl∞ ⊆ k for the group of l-power roots of unity of k, K def= k(μl∞) ⊆\n110 SHINICHI MOCHIZUKI\nk, Gk ⊇ GK def= Gal(k\/K), GK\/k def= Gk\/GK , L def= IGK [i.e., the subfield of\nGK-invariants of I]. Then our assumption on l implies that the natural faithful\naction of GK\/k on μl∞ [which allows us to think of GK\/k as a closed subgroup of\nZ×l ] induces a nontrivial action of GK\/k on μl, hence [in light of the well-known\nstructure of the profinite group Z×l ∼= Zl×F×l for odd primes l] that GK\/k contains\na nontrivial finite closed subgroup H ⊆ GK\/k. Next, observe that the group of\ndivisible elements of the multiplicative module K× is equal to μl∞ [cf. the fact\nthat GK\/k is isomorphic to a closed subgroup of Z\n×\nl ; [Tsjm], Lemma D, (iii), (iv)].\nThis implies that the multiplicative GK\/k-module L\n× is naturally isomorphic to the\nGK\/k-module Ml\ndef\n= Hom(Ql\/Zl, μl∞) ⊗Zl Ql [where we note that as an abstract\nmodule, Ml is isomorphic to Ql], hence, in particular, that the field L is of infinite\ncardinality. On the other hand, it follows from elementary Galois theory that L is\na finite Galois extension of the subfield LH ⊆ L of H-invariants of L. Moreover,\nsince H acts nontrivially on μl, hence also nontrivially on Ml, we thus conclude\n— from the corresponding fact for the action of nontrivial subgroups of the group\nof Teichmüller representatives [F×l ] ⊆ Z×l on Ql — that LH = {0, 1} is a set of\ncardinality two, hence that the infinite field L is a finite field, a contradiction. This\ncompletes the proof of (v-a). Note that\n(v-b) if it was indeed the case that IO×\nk\nadmits a topological field structure as\nin (v-a), then it would be possible to consider the well-known power series\nfor the logarithm [cf. (iv-a), (iv-b)].\nOf course, it follows from (v-a) that such a topological field structure does not exist.\nIn particular, the content of (v-a) may be understood as pointing roughly in the\nsame direction as (iii-a), (iii-b), (iv-a), (iv-b).\n(vi) Thus, in summary,\n(vi-a) the fundamental dichotomy, in the context of the p-adic logarithm,\nbetween\n· [finitely] truncated Kummer theory [as in (i-d)] and\n· profinite Kummer theory [as in (i-e)]\nmay be understood in terms of the existence [cf. the “k” in the domain\nof logk in (i-d)] versus non-existence [cf. the “correspondence” (Z, ζ, λ)\nof (iii-a)] of\na single unified basepoint\nfor the Kummer theories in the domain\/codomain of logk or Logk, i.e., a\nsingle set equipped with an action by Gk that is “sufficiently rich” as to\nadmit subquotients [i.e., where we think of λ as in (iii-a) as being surjective]\nthat correspond to the Kummer theories in the domain\/codomain of logk\nor Logk.\nThat is to say,\n(vi-b) in the case of profinite Kummer theory, the non-existence of such a\nsingle unified basepoint means that one must treat the Kummer theories\nin the domain\/codomain of Logk — i.e., at a more concrete level, the sets\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 111\nIk or IO×\nk\nequipped with their natural multiplicative structures, profinite\ncyclotomes, and Gk-actions — as being independent of one another.\nIn particular, it follows, essentially formally, from (vi-b) that\n(vi-c) one must think of the copies of “Gk” that appear in the Kummer theories\nin the domain\/codomain of the p-adic logarithm — which may in fact arise\nas quotients of copies of some topological group Π in the domain\/codomain\nof the p-adic logarithm — as being related to one another\nnot as groups of automorphisms of the various monoids that\nappear in the Kummer theories in the domain\/codomain of the\np-adic logarithm, but rather as abstract topological groups,\ni.e., which may be related to one another only by means of some\nindeterminate isomorphism of topological groups Π\n∼→ Π\n— cf. the discussion preceding Example 3.8.1, as well as the discus-\nsion following Example 3.8.4, concerning the necessity of working, in the\ncontext of the log- and Θ-links, with abstract topological groups, that is\nto say, as opposed to groups of ring\/field automorphisms, i.e., “Galois\ngroups\/arithmetic fundamental groups”.\nHere, we observe that the isomorphism indeterminacy discussed in (vi-c) includes,\nin particular, inner automorphisms of Π. This chain of observations (vi-a), (vi-b),\n(vi-c) forms the starting point of the discussion of Example 3.8.4 below.\nExample 3.8.4: Symmetrizing isomorphisms, truncatibility, and the log-\nKummer- correspondence. We maintain the notation of Examples 3.8.2, 3.8.3.\nAlso, we shall write “Out(−)” for the group of outer automorphisms [i.e., arbi-\ntrary automorphisms considered up to inner automorphisms] of a topological group\n“(−)”.\n(i) In the following discussion, we consider the issue of compatibility between\n· the various symmetrizing isomorphisms arising from the action of the\nF\u0002l - and F\n\u0003±\nl -symmetries on [“abstract”] copies of the pair Gk \u0002 O\u0004k\n[cf. the discussion of Examples 3.8.2, 3.8.3; the theory of [IUTchI], §4, §5,\n§6; [Alien], §3.3, (v); [Alien], §3.6, (i), (ii), (iii)] and\n· the log-Kummer-correspondence\n. . . → • → • → • → . . .\n. . . ↘ ↓ ↙ . . .\n◦\n[cf. [IUTchIII], Theorem 3.11, (ii)]\nfor the three types of Kummer theory, namely,\n(i-a) the Kummer theory for [“abstract”] copies of the pair Gk \u0002 O\u0004\nk\n, which is\nbased on the classical theory ofBrauer groups\/local class field theory\nfor p-adic local fields [cf. [Alien], §3.4, (v)];\n112 SHINICHI MOCHIZUKI\n(i-b) the Kummer theory surrounding theta functions and theta values\n[cf. [Alien], §3.4, (iii), (iv); [Alien], §3.6, (ii)];\n(i-c) the Kummer theory surrounding κ-coric functions and copies of the\nnumber field Fmod [cf. the discussion of Example 3.8.2; [IUTchI], Defi-\nnition 3.1, (b); [Alien], §3.4, (ii); [Alien], §3.6, (iii)]]\nthat appear in inter-universal Teichmüller theory [cf., e.g., [Alien], Fig. 3.10]. Be-\nfore proceeding, we recall that the symmetrizing isomorphisms arising from the\naction of the F\u0002l - and F\n\u0003±\nl -symmetries on [“abstract”] copies of the pair Gk \u0002 O\u0004k\ndiffer in that\n· whereas the symmetrizing isomorphisms arising from the F\u0003±l -symmetries\nare free of inner automorphism indeterminacies and hence give rise to\nconjugate synchronizations,\n· the symmetrizing isomorphisms arising from the F\u0002l -symmetries neces-\nsarily involve inner automorphism indeterminacies\n[cf. the discussion of Example 3.8.2]. On the other hand, it follows immediately —\ni.e., by considering the symmetrizing isomorphisms induced by arbitrary elements\nof Aut(XK)\n∼→ Out(ΠXK ) [cf. the notation of [IUTchI], Definition 3.1, (d); [Alien],§3.3, (v)] — that\nif one forgets about the issue of conjugate synchronization and just thinks\nin terms of arbitrary [indeterminate] isomorphisms between [“abstract”]\ncopies of the pair Gk \u0002 O\u0004\nk\n, then the symmetrizing isomorphisms arising\nfrom the action of the F\u0002l - and F\n\u0003±\nl -symmetries on [“abstract”] copies of\nthe pair Gk \u0002 O\u0004\nk\nin fact coincide.\n(ii) In the case of the Kummer theory of (i-a),\n(ii-a) compatibility between the F\u0003±l -symmetrizing isomorphisms —\ni.e., without conjugacy indeterminacies! [cf. the final portion of (i)] —\nand the Kummer theories of (i-a) in the domain\/codomain of the log-\nlink then follows formally by applying transport of structure via the F\u0003±l -\nsymmetries to the truncated Kummer theories in the domain\/codomain\nof the log-link, computed relative to the single unified basepoint dis-\ncussed in Example 3.8.3, (i-d), (vi-a), at each evaluation label “t ∈ Fl”\n[cf. the discussion of [IUTchIII], Remark 2.3.3, (viii); [Alien], §3.6, (ii)]. Here, we\nrecall that\n(ii-b) this compatibility is necessary to define the diagonal “0\/ \u001f \/ >” — i.e.,\nwith respect to the evaluation labels “t ∈ Fl” — local data that is used\nto construct the local unit group [i.e., “O×μ”] portion of the gluing data\nthat appears in the Θ-link\n[cf. [IUTchIII], Theorem 1.5, (iii); [IUTchIII], Remark 1.5.1, (i); [Alien], §3.6, (ii)].\nIn this context, it is important to recall that\n(ii-c) this local unit group portion of the Θ-link gluing data satisfies the crucial\nproperty of being independent of the evaluation labels “t ∈ Fl” — which\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 113\nare only well-defined internally within a particular (Θ±ellNF-)Hodge the-\nater! — hence allows one to construct the containers [that is to say, in\nthe form of tensor-packets of log-shells] that appear in the multiradial\nrepresentation of the Θ-pilot, i.e., the containers that make it possible\nto represent the Θ-pilot in the domain (Θ±ellNF-)Hodge theater of the\nΘ-link in terms of “external” data arising from the codomain (Θ±ellNF-\n)Hodge theater of the Θ-link.\nPut another way,\n(ii-d) if it were the case that the containers of (ii-c) could only be constructed\nfrom the domain (Θ±ellNF-)Hodge theater of the Θ-link in a way that\ninvolves independent conjugacy indeterminacies\n· at the distinct evaluation labels “t ∈ Fl” [cf. (ii-b), (ii-c)] or\n· in the domain\/codomain of the log-link [cf. (ii-a)]\n— i.e., all of which are only well-defined internally within the domain\n(Θ±ellNF-)Hodge theater of the Θ-link! — then it would follow that these\ncontainers cannot be constructed in a way that is well-defined externally\nto the domain (Θ±ellNF-)Hodge theater of the Θ-link, e.g., in terms of data\narising from the codomain (Θ±ellNF-)Hodge theater of the Θ-link.\nIn particular, we observe that the truncatibility of the Kummer theory of (i-\na) plays a fundamental role in the logical structure of inter-universal Teichmüller\ntheory.\n(iii) The discussion of (ii-b), (ii-c), (ii-d) centers on the issue of synchro-\nnizing the conjugacy indeterminacies at the diagonal label “\u001f \/ >” in the do-\nmain\/codomain of the log-link. This focus of attention on the diagonal label thus\nprompts the following question:\n(iii-a) If one is only interested in synchronizing the conjugacy indeterminacies at\nthe diagonal label “\u001f \/ >” in the domain\/codomain of the log-link, then\nwhy does it not suffice to relate the profinite [i.e., rather than truncated!]\nversions of the Kummer theory of (i-a) for the diagonal label “\u001f \/ >”\nin the domain\/codomain of the log-link via a single [i.e., corresponding to\nthe “single” diagonal label] indeterminate isomorphism “Π\n∼→ Π” as in\nExample 3.8.3, (vi-c)?\nIn fact, however, the approach described in (iii-a) is not sufficient for the following\nreason:\n(iii-b) Ultimately, in inter-universal Teichmüller theory, one is interested in con-\nstructing the multiradial representation of the Θ-pilot [cf. [IUTchIII],\nTheorem 3.11] — which involves the theta values\n“qj\n2\nv\n”\nat v ∈ Vbad — i.e., whose construction depends, in an essential way, on\nthe use of [global] independent labels “t ∈ Fl” [cf. (ii-d)] at which the\nconjugacy indeterminacies are nevertheless synchronized relative to\na single basepoint arising from the global additive symmetry portion\n114 SHINICHI MOCHIZUKI\n“D\u0007±” of the (Θ±ellNF-)Hodge theater under consideration [cf. [IUTchI],\nDefinition 6.1, (v); [IUTchII], Corollary 4.5, (iii), (iv); [IUTchII], Corollary\n4.6, (iii), (iv)].\nThat is to say,\n(iii-c) the requirements discussed in (iii-b) are satisfied by the approach that is\nactually taken in inter-universal Teichmüller theory [cf. (ii-a); [IUTchIII],\nProposition 1.3, (i); [IUTchIII], Remark 1.3.2], i.e., of synchronizing the\nconjugacy indeterminacies in the domain\/codomain of the log-link locally\n“label by label”, for the various labels “t ∈ Fl” [cf. (ii-d)], so that arbitrary\nconjugacy indeterminacy synchronizations in the domain of the log-link\nare reflected faithfully in the codomain of the log-link.\nOn the other hand,\n(iii-d) the approach discussed in (iii-c) can be implemented precisely because\nof the existence of the canonical single unified basepoints for the do-\nmain\/codomain of the log-link discussed in Example 3.8.3, (vi-a), i.e.,\nwhich are available only in the case of truncated Kummer theory,\nsince the profinite Kummer theory conjugacy indeterminacies of Example\n3.8.3, (vi-c), would give rise [in the situation of the approach discussed\nin (iii-c)] to independent conjugacy indeterminacies at the various labels\n“t ∈ Fl”.\nThus, in summary, the discussion of the present (iii) sheds further light on the\nfundamental role played by the truncatibility of the Kummer theory of (i-a) in\nthe logical structure of inter-universal Teichmüller theory.\n(iv) In the case of the Kummer theory\/Galois evaluation of (i-c), let us first\nrecall from the discussion of [IUTchIII], Remark 2.3.3, (vi), (vii) [cf. also [Alien],\n§3.4, (ii)] that\n(iv-a) the fact that the submonoid\nIord ⊆ N× {±1}\ngenerated by the set of orders of the zeroes\/poles [considered as signed\nelements of N × {±1}] of the rational functions that appear in (i-c)\ncontains — i.e., unlike the case with the theta functions that appear in\n(i-b)! — elements \n∈ {±1}, as well as elements \n∈ N, means that the cyclo-\ntomic rigidity isomorphisms obtained in (i-c) may only be constructed\nin a fashion consistent with the anabelian reconstruction algorithms\nof [AbsTopIII], Theorem 1.9 [cf. also [IUTchI], Remark 3.1.2, (ii), (iii)] if\none constructs these cyclotomic rigidity isomorphisms\n· via profinite Kummer theory [i.e., by applying the fact that\nQ>0 ∩ Ẑ× = {1}] and\n· up to an indeterminacy given by multiplication by elements\nof the image Iord± = {±1} of the projection Iord \u0003 {±1} to the\nsecond factor.\n— i.e., in particular, relative to the constraints discussed in Example\n3.8.3, (vi-b), (vi-c).\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 115\nThat is to say, it follows from the discussion of Example 3.8.3, (vi-b), (vi-c) — cf.\nalso\n· the splitting\/decoupling of the unit group portion from the pseudo-\nmonoid of κ-coric rational functions [as discussed in [IUTchI], Example\n5.1, (v); [Alien], §3.4, (ii)];\n· the non-interference properties satisfied by the Frobenius-like copies of\nF×mod in the domain\/codomain of the log-link [as discussed in [IUTchIII],\nProposition 3.10, (ii); [Alien], §3.7, (i)]\n— that\n(iv-b) the Kummer theories\/Galois evaluation operations of (i-c) in the do-\nmain\/codomain of the log-link\n· involve [pseudo-]monoids that must be treated indepen-\ndently of one another, hence, in particular,\n· may be related to one another only up to indeterminacies that\ninvolve indeterminate isomorphisms between corresponding Ga-\nlois groups\/arithmetic fundamental groups in the domain\/codomain\nof the log-link\n[cf., especially, the discussion of the final portion of Example 3.8.3, (vi)].\nOn the other hand,\n(iv-c) the compatibility between the F\u0002l -symmetrizing isomorphisms —\ni.e., with conjugacy indeterminacies! [cf. the discussion of the final portion\nof (i)] — and the Kummer theories\/Galois evaluation operations of\n(i-c) in the domain\/codomain of the log-link follows formally by applying\ntransport of structure via the F\u0002l -symmetries to the profinite Kummer\ntheories of (i-c) in the domain\/codomain of the log-link, together with the\ncompatibility of the log-link with these F\u0002l -symmetrizing isomorphisms [cf.\n[IUTchIII], Proposition 1.3, (i), (ii); [IUTchIII], Remark 1.3.3, (ii)].\nAs a result of the various indeterminacies of (iv-b), (iv-c),\n(iv-d) in the case of the Kummer theories\/Galois evaluation operations of (i-\nc), the only diagonal “0\/ \u001f \/ >” — i.e., with respect to the evaluation\nlabels “j ∈ F\u0002l ” — number field inside the algebraic closure F that is\nwell-defined externally to the domain (Θ±ellNF-)Hodge theater of the\nΘ-link [cf. the discussion of the multiradial representation in (ii-c), (ii-d)]\nis Fmod [cf. [Alien], §3.6, (iii)].\nHere, we observe that\n(iv-e) the Iord± -indeterminacies of (iv-a) may be synchronized\n· relative to the F\u0002l -symmetry by applying the “linear dis-\njointness” of the theory of κ-coric functions from SL2(Fl) [cf.\nthe second display of [Alien], §3.6, (iii)];\n· relative to the distinct valuations v ∈ V in local-global com-\nparisons of the Kummer theories\/Galois evaluation operations of\n116 SHINICHI MOCHIZUKI\n(i-c) by means of comparison with the cyclotomic rigidity iso-\nmorphisms arising from the Kummer theories of (i-a) [i.e., which\nare not subject to Iord± -indeterminacies]\n[cf. the discussion of the final portion of [IUTchIII], Remark 2.3.3, (vi)],\nbut\n· not relative to the domain\/codomain of the log-link since the\nrelevant [Frobenius-like] monoids in the domain\/codomain of the\nlog-link are not set-theoretically related to one another via the\nlog-link\n[cf. (iv-b)].\nIn particular, we note that one fundamental aspect of the Kummer theory\/Galois\nevaluation of (i-c) that underlies the compatibility and synchronization properties\nof (iv-c), (iv-e) is the F\u0002l -symmetricity of the Kummer theory\/Galois evaluation\nof (i-c) [cf. the discussion of [IUTchII], Remark 1.1.1, (v); the first display of\n[IUTchIII], Remark 2.3.3, (iii); the discussion of [Alien], §3.6, (iii)].\n(v) The Kummer theory\/Galois evaluation of (i-b), which does not satisfy\nthe condition of F\u0003±l -symmetricity, exhibits qualitatively fundamentally different\nbehavior from the F\u0002l -symmetric Kummer theory\/Galois evaluation of (i-c) [cf. the\ndiscussion of [IUTchII], Remark 1.1.1, (v); the first display of [IUTchIII], Remark\n2.3.3, (iii)]. If, for instance,\n(v-a) one attempts to apply the approach via profinite Kummer theory\nof (iv-b), (iv-c) to the task of verifying some sort of compatibility be-\ntween the F\u0003±l -symmetrizing isomorphisms — i.e., with conjugacy\nindeterminacies arising from log-link domain\/codomain comparisons, as\ndiscussed in Example 3.8.3, (vi-b), (vi-c) [cf. also the discussion of (i), (ii),\n(iii), especially (iii-d), of the present Example 3.8.4]! — and the Kummer\ntheories\/Galois evaluation operations of (i-b) in the domain\/codomain of\nthe log-link,\nthen one encounters the following situation:\n(v-b) In order to compare, via the F\u0003±l -symmetry, the Kummer theories\/Galois\nevaluation operations at different t ∈ Fl [i.e., for distinct theta values\n“qj\n2\nv\n”, where v ∈ Vbad], it is necessary to establish a(n) — a priori self-\ncontradictory!— situation in which the F\u0003±l -symmetry permutes the\nlabels “t ∈ Fl” in a nontrivial fashion, but acts trivially on the [non-\nF\u0003±l -symmetric!] étale theta function!\nHere, we recall [cf. the theory of [IUTchII], §2] that\n(v-c) in the context of Galois evaluation, the étale theta function may be\nthought of as a cohomology class\n∈ H1\n(\nsome closed subgroup\nof Πv\n,\nsome cyclotome associated to the\nGalois evaluation label t ∈ Fl\n)\n— that is to say, where [cf. (v-b)!] the F\u0003±l -symmetry is to act nontriv-\nially on the Galois evaluation cyclotome [i.e., the second argument of\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 117\n“H1(−)”], but trivially on the closed subgroup of Πv [i.e., the first\nargument of “H1(−)”]!\nBefore proceeding, it is also of interest to recall that\n(v-d) a profinite Kummer theory situation of the sort discussed in (v-a), (v-b),\n(v-c) occurs, for instance, if one replaces the theta functions that occur in\ninter-universal Teichmüller theory by their N-th powers, where N is an\ninteger ≥ 2 [cf. the discussion of [IUTchIII], Remark 2.3.3, (vi), (vii)].\nNow returning to the discussion of (v-b), (v-c), we observe that\n(v-e) it is essentially a tautology that the only “solution” to the [a priori]\nself-contradictory simultaneous “trivial\/nontrivial action” condi-\ntion “(SimCon)” of (v-b), (v-c) lies in working with objects that are\ninvariant with respect to the F\u0003±l -symmetry, i.e., at a more concrete\nlevel, with a\nsingle unified set-theoretic basepoint\n— which allows one to compute the various F\u0003±l -symmetrizing isomor-\nphisms by projecting to the single copy of “Gv” determined by the base-\npoint.\nOn the other hand,\n(v-f) the existence of the conjugacy indeterminacies inherent in the profi-\nnite Kummer theory F\u0003±l -symmetrizing isomorphisms of (v-a) means that\nany “single unified basepoint” as in (v-e) is well-defined only up to conju-\ngacy indeterminacies — a situation that is unacceptable in inter-universal\nTeichmüller theory since it would mean that the coefficient cyclotomes in\n(v-c) are well-defined only up to certain Ẑ×-multiples, i.e., that the étale\ntheta functions and theta values [i.e., “qj\n2\nv\n”, where v ∈ Vbad] in the\ntheory are well-defined only up to certain Ẑ×-powers [cf. the discussion\nof [IUTchIII], Remark 2.1.1, (v)]!\nThat is to say,\n(v-g) the only way to avoid the pathologies of (v-f) is to replace the profinite\nKummer theory in the discussion of (v-a), (v-b), (v-c), (v-e), (v-f) by the\ntruncated Kummer theory of (i-b) [cf. the discussion of the final por-\ntion of [Alien], §3.6, (ii)], so that we can apply the single unified basepoint\nof Example 3.8.3, (i-d), (vi-a) [cf. also the discussion of (ii) in the present\nExample 3.8.4] to obtain a\na single unified set-theoretic basepoint\nas in (v-e), but which is well-defined up to geometric fundamental\ngroup conjugacy indeterminacies [i.e., indeterminacies arising from\nconjugation by elements of the geometric fundametric fundamental groups\ninvolved], which are, at any rate, inherent in the F\u0003±l -symmetrizing iso-\nmorphisms and, moreover, [unlike the situation discussed in (v-f)!] do\nnot have any effect on the computation of these F\u0003±l -symmetrizing iso-\nmorphisms by projecting to the single copy of “Gv” determined by the\nbasepoint\n118 SHINICHI MOCHIZUKI\n— cf. [IUTchII], Remark 1.1.1, (iv), (v); [IUTchII], Remark 2.6.1, (i), (ii). Before\nproceeding, it is interesting to observe that\n(v-h) the “tautological resolution” of the [a priori] self-contradictory simul-\ntaneity condition (SimCon) discussed in (v-e), (v-g) by considering\n“invariants” — i.e., in the situation of (v-e), (v-g), a single unified set-\ntheoretic basepoint — is formally highly reminiscent of the “tautological\nresolution” of the log-shifts in the left- and right-hand columns of the\n“infinite H” of (InfH) by means of the construction of invariants [cf. the\ndiscussion surrounding (logORInd), (Di\/NDi) in §3.11 below; the discus-\nsion of (Stp7) in §3.10 below].\nThus, in summary, the truncatibility of the Kummer theory\/Galois evaluation\noperations of (i-b) [cf. the discussion of [IUTchIII], Remark 2.3.3, (vi), (vii), (viii);\nthe final portion of [Alien], §3.6, (ii)] plays a fundamental role in the logical structure\nof inter-universal Teichmüller theory [cf. the discussion of Example 3.3.2, (vii)].\n(vi) Recall that the single unified basepoint of (v-g) is completely deter-\nmined by the connected subgraph “Γ\u0003X ⊆ ΓX”, or, equivalently, the connected\nsubgraph “Γ\u0003\nŸ\n⊆ ΓŸ ” [cf. [IUTchII], Remark 2.6.1, (i), (ii); [IUTchII], Remark\n2.6.3, (i)]. Here, we recall further from [IUTchII], Remark 2.6.3, (i), that ΓX or ΓŸ\nmay be thought of as a “copy Γ of the real line R”, in which the integers Z ⊆ R\nare taken to be the vertices, and the line segments joining the integers are taken\nto be the edges. In light of the central role played by the singled unified basepoint\nof (v-g) in the discussion of (v) — and indeed in the entire logical structure\nof inter-universal Teichmüller theory! — it is of interest to recall from the discus-\nsion of [IUTchII], Remark 2.6.3, that this subgraph “Γ\u0003” is [essentially] uniquely\ndetermined by various natural conditions, which must be satisfied in order for\nthe theory to operate in the desired fashion. Indeed, let us first recall from the\ndiscussion of [IUTchII], Remark 2.5.2, (i), (ii), (iii), (iv); [IUTchII], Remark 2.6.3,\n(ii), that\n(vi-a) the various geometric fundamental groups “Δ” that appear act on the\nvarious vertices t ∈ {−l\u0002, . . . ,−1, 0, 1, . . . , l\u0002} ⊆ Γ independently\n— where we recall that l\u0002 = l−1\n2\n, and that\n(vi-b) the actions referred to in (vi-a) of geometric fundamental groups “Δ”\non the “copy Γ of the real line R” amount to the conventional action of\nGΓ\ndef\n= l · Z \u0003 {±1} [i.e., the group generated by translations by elements\nof l · Z and multiplication by −1] on the real line R.\nWrite Γ\u0003Z\ndef\n= Γ\u0003 ∩ Z; ±Γ\u0003Z def= Γ\u0003Z ∪ −Γ\u0003Z ⊆ R. Next, recall that\n(vi-c) since the vertex 0 ∈ Γ is the unique vertex in Γ that may be used to\ndefine the crucial splittings of theta monoids discussed in [IUTchII],\nCorollary 2.6, (ii), it is necessary that 0 ∈ Γ\u0003Z [cf. [IUTchII], Remark 2.6.3,\n(i)].\nOn the other hand, (vi-c), together with the existence of the independent GΓ-\nindeterminacies of (vi-a), (vi-b), imply that\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 119\n(vi-d) if Γ\u0003 contains a connected component Γ\u0003∗ that is not contained in l ·Z (⊆\nΓ), then any theta value\nqj\n2\nv\nobtained via Galois evaluation for j ∈ Γ\u0003∗ ∩ Z ⊆ Γ, is well-defined only\nup to a GΓ-indeterminacy, i.e., up to a possible confusion between j and\nGΓ-translates j\n′ of j.\n(vi-e) any theta value\nqj\n2\nv\nobtained via Galois evaluation for j ∈ Γ\u0003Z , is well-defined only up to\na GΓ-indeterminacy within Γ\n\u0003\nZ , i.e., up to a possible confusion between j\nand GΓ-translates j\n′ of j that lie inside Γ\u0003Z .\nHere, it is important to recall [cf. [IUTchI], Remark 3.5.1, (ii); [IUTchII], Remark\n2.6.3, (iv); [IUTchII], Corollary 4.5, (v); [IUTchII], Corollary 4.6, (v)] that\n(vi-f) any indeterminacies concerning “j”, “j′” of the sort described in (vi-d),\n(vi-e) for nonzero j, j′ ∈ Z with distinct absolute values would result in\nindeterminacies in the “vector of ratios” [cf. [IUTchII], Corollary 4.5, (v)]\nthat determines the structure of the global realified Gaussian Frobenioids,\ni.e., would result in violations of the global product formula relating\nthe value groups at different v ∈ V.\nIn particular, (vi-c), (vi-d), (vi-e), (vi-f) imply that\n(vi-g) the natural map\n±Γ\u0003Z \/{±1} → Z\/GΓ\n— i.e., from the set of {±1}-orbits of ±Γ\u0003Z to the set of GΓ-orbits in Z —\nis injective [cf. [IUTchII], Remark 2.6.3, (ii)];\n(vi-h) the subgraph Γ\u0003 ⊆ Γ is connected [cf. [IUTchII], Remark 2.6.3, (i)].\nNow one verifies immediately that (vi-c), (vi-g), (vi-h) imply formally that\n(vi-i) Γ\u0003 ⊆ Γ is a connected subgraph that contains 0 and is contained in\nthe closed interval [−l\u0002, l\u0002].\nNext, we observe that, in light of (vi-e), (vi-i),\n(vi-j) replacing Γ\u0003Z by ±Γ\u0003Z does not have any effect on the validity of (vi-i) or\non the way in which the subgraph Γ\u0003Z is used in inter-universal Teichmüller\ntheory; in particular, we may assume, without loss of generality, that the\nsymmetry condition Γ\u0003Z = ±Γ\u0003Z holds.\nOn the other hand,\n(vi-k) the estimates that are ultimately obtained in inter-universal Teichmüller\ntheory [cf. [IUTchIII], Corollary 3.12; [IUTchIV], Theorem 1.10] are op-\ntimized precisely when the average of the squares j2 of the nonzero ele-\nments j ∈ Γ\u0003Z = ±Γ\u0003Z is maximized [cf. [IUTchII], Remark 2.6.3, (ii)],\ni.e., when Γ\u0003 = ±Γ\u0003Z is given precisely by the closed interval [−l\u0002, l\u0002]\n[which implies that the natural map of (vi-g) is bijective].\n120 SHINICHI MOCHIZUKI\nThat is to say, in summary, when subject to the symmetry condition Γ\u0003Z = ±Γ\u0003Z\nof (vi-j) and the optimization condition of (vi-k), the subgraph Γ\u0003 ⊆ Γ is in fact\nuniquely determined [cf. [IUTchII], Remark 2.6.3, (v)].\nUnlike the situations considered in [SGA1] [cf. the discussion of Example\n3.8.1], in which the ring\/scheme structures of the various distinct schemes that\nappear are coric, the ring structures of the rings that appear on either side of the\nlog- and Θ-links of inter-universal Teichmüller theory — i.e., such as number fields\nor completions of number fields at various valuations — are not coric with respect\nto the respective links. This leads one naturally to consider weaker structures\n[cf. the discussion of Example 3.2.2, (i), (ii), (iv)] the discussion of Example 3.8.2,\n(iii), (iv); the discussion of Example 3.8.3, (vi)] such as\n· abstract topological groups, in the case of the profinite Kummer theories\nin the domain\/codomain of the log-link, or\n· sets equipped with a topology and a continuous action of a topological\ngroup, in the case of the log-link, or\n· realified Frobenioids or topological monoids equipped with a continuous\naction of a topological group, in the case of the Θ-link,\nwhich are indeed coric with respect to the respective links. Indeed, it is pre-\ncisely this sort of consideration — i.e., of weaker coric structures to relate the uni-\nverses\/Galois categories\/étale fundamental groups associated to ring\/scheme struc-\ntures on opposite sides of the links under consideration [cf. the discussion preceding\nExample 3.8.1] — that gave rise to the term “inter-universal”.\nHere, we note that it is of fundamental importance that these topological groups\n[which typically in fact arise as Galois groups or arithmetic fundamental groups\nof schemes] be treated as abstract topological groups, rather than as Galois\ngroups or arithmetic fundamental groups [cf. the discussion at the beginning of\n§3.2; the discussion of Example 3.8.2, (iii), (iv); the discussion of Example 3.8.3,\n(vi)]. That is to say, to treat these topological groups as Galois groups or arithmetic\nfundamental groups requires the use of the ring\/scheme structures involved, i.e.,\nthe use of structures which are not available since they are not common\/coric to\nthe rings\/schemes that appear on opposite sides of the log-\/Θ-link [cf. the discussion\nof [Alien], §2.10; [IUTchIII], Remarks 1.1.2, 1.2.4, 1.2.5; [IUTchIV], Remarks 3.6.1,\n3.6.2, 3.6.3]. In this context, it is also of fundamental importance to observe that it\nis precisely because these topological groups must be treated as abstract topological\ngroups that anabelian results play a central role in inter-universal Teichmüller\ntheory.\nOne consequence of the constraint [discussed above] that one must typically\nwork, in inter-universal Teichmüller theory, with structures that are substantially\nweaker than ring structures is the necessity, in inter-universal Teichmüller theory,\nof allowing for various indeterminacies, such as (Ind1), (Ind2), (Ind3), that are\nsomewhat more involved than the relatively simple inner automorphism indetermi-\nnacies that occur in [SGA1]. Here, we recall that from the discussion of (∧(∨)-Chn)\nin §3.7 that\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 121\nit is precisely the numerous indeterminacies that arise in inter-universal\nTeichmüller theory that give rise to the numerous logical OR relations\n“∨” in the display of (∧(∨)-Chn).\nOn the other hand, once one takes such indeterminacies into account, i.e.,\nonce one consents to work with various objects “up to certain suitable\nindeterminacies” — e.g., by means of poly-morphisms, as discussed in\n§3.7 — it is natural to identify, by applying (ExtInd2) [as discussed in\n§3.6], objects that are related to one another by means of collections of\nisomorphisms [i.e., poly-isomorphisms] that are uniquely determined up\nto suitable indeterminacies.\nHere, we observe that this sort of (ExtInd2) identification that occurs re-\npeatedly in inter-universal Teichmüller theory [cf. the discussion of §3.6] may at\nfirst glance appear somewhat novel. In fact, however, from the point of view of\nmathematical foundations — i.e., just as in the discussion of inter-universality given\nabove! — this sort of (ExtInd2) identification is qualitatively very similar to nu-\nmerous classical constructions such as the following:\n(AlgCl) the notion of an algebraic closure of a field [cf. the discussion of\nExample 3.8.1], which is not constrained to be a specific set constructed\nfrom the field;\n(DrInv) various categorical constructions such as direct and inverse limits [i.e.,\nsuch as fiber products of schemes] that are defined by means of some sort of\nuniversal property, and which are not constrained to be specific sets even\nwhen the given direct or inverse systems are specified set-theoretically;\n(HomRs) various constructions of (co)homology modules in homological al-\ngebra that depend on the use of resolutions that satisfy certain abstract\nproperties, but which are not constrained in a strict set-theoretic sense\neven if the original objects resolved by such resolutions are specified set-\ntheoretically.\nThat is to say, in each of the classical constructions, the “output object” is, strictly\nspeaking, from the point of view of mathematical foundations, not well-defined as\na particular set, but rather as a collection of sets [where we note that, typically,\nthis “collection” is not a set!] that are related to one another — and hence, in\ncommon practice, identified with one another, in the fashion of (ExtInd2)! —\nvia unique [modulo, say, some sort of well-defined indeterminacy] isomorphisms by\nmeans of some sort of “universal” property.\nIn this context, it is also important to note that, from a foundational point of\nview, the sort of “(sub)quotient” obtained by applying (ExtInd2) [cf. the discussion\nof “(sub)quotients” in (sQLTL) and indeed throughout §3.6] must be regarded, a\npriori, as a formal (sub)quotient, i.e., as some sort of diagram of arrows. That\nis to say, at least from an a priori point of view,\n(NSsQ) any explicit construction of a “naive set-theoretic (sub)quotient”\nnecessarily requires the use of some sort of set-theoretic enumeration\nof each of the individual [set-theoretic] objects that are identified, up to\n122 SHINICHI MOCHIZUKI\nisomorphism, via an application of (ExtInd2). On the other hand, as\nis well-known, typically such set-theoretic enumerations — which often\nreduce, roughly speaking, to consideration of the “set of all sets”! — lead\nimmediately to a contradiction.\nIndeed, it is precisely this aspect of the constructions of inter-universal Teichmüller\ntheory that motivated the author to include the discussion of species in [IUTchIV],\n§3.\nFinally, we recall [cf. also the discussion of §3.10 below] that\n(LVsQ) it is only in the final portion of inter-universal Teichmüller theory, i.e.,\nonce one obtains a formal (sub)quotient that forms a “closed loop”, that\none may pass from this formal (sub)quotient to a “coarse\/set-theoretic\n(sub)quotient” by taking the log-volume\n[cf. the discussion of [Alien], §3.11, (v); [IUTchIII], Remark 3.9.5, (ix); Step (x) of\nthe proof of [IUTchIII], Corollary 3.12].\n§3.9. Passage and descent to underlying structures\nOne fundamental aspect of inter-universal Teichmüller theory lies in the use of\nnumerous functorial algorithms that consist of the construction\ninput data \u0004 output data\nof certain output data associated to given input data. When one applies such func-\ntorial algorithms, there are two ways in which the output data may be treated\n[cf. [Alien], §2.7, (iii); the discussion of “post-anabelian structures” in [IUTchII],\nRemark 1.11.3, (iii), (v); [IUTchIII], Remark 1.2.2, (vii)] :\n(UdOut) One may consider the output data independently of the given input data\nand functorial algorithms used to construct the output data. In this case,\nthe output data may be regarded as a sort of “underlying structure”\nassociated to the input data.\n(InOut) One may consider the output data as data equipped with the additional\nstructure constituted by the input data, together with the functorial algo-\nrithm that gave rise to the output data by applying the algorithm to the\ninput data.\nTypical examples of this phenomenon in inter-universal Teichmüller theory are the\nfollowing [cf. the notational conventions of [IUTchI], Definition 3.1, (e), (f)]:\n(sQGOut) Functorial algorithms that associate to Πv [where v ∈ Vnon] some sub-\nquotient group of Πv, such as, for instance, the quotient Πv \u0003 Gv:\nIn this sort of situation, treatment of the output data [i.e., subquotient\ngroup of Πv] according to (InOut) is indicated by a “(Πv)” following the\nnotation for the particular subquotient under consideration; by contrast,\ntreatment of the output data [i.e., subquotient group of Πv] according to\n(UdOut) is indicated by the omission of this “(Πv)”.\n(MnOut) Functorial algorithms that associate to Πv [where v ∈ Vnon] some sort of\n[abelian] monoid equipped with a continuous action by Πv, such as, for\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 123\ninstance, [data isomorphic to] various subquotient monoids [i.e., “O\u0004”,\n“O×”, “O×μ”, etc.] of the multiplicative monoid F×v : In this sort of\nsituation, treatment of the output data [i.e., monoid equipped with an\naction by Πv] according to (InOut) is indicated by a “(Πv)” following the\nnotation for the particular monoid equipped with an action by Πv under\nconsideration; by contrast, treatment of the output data [i.e., monoid\nequipped with an action by Πv] according to (UdOut) is indicated by the\nomission of this “(Πv)”.\n(PSOut) Functorial algorithms that associate some sort of prime-strip to some\nsort of input data: In this sort of situation, treatment of the output data\n[i.e., some sort of prime-strip] according to (InOut) is indicated by a “(−)”\n[where “−” is the given input data] following the notation for the particular\nprime-strip under consideration; by contrast, treatment of the output data\n[i.e., some sort of prime-strip] according to (UdOut) is indicated by the\nomission of this “(−)”.\nPerhaps the most central example of (PSOut) in inter-universal Teichmüller theory\nis the notion of the “q-\/Θ-intertwinings” on an F\u0002\u0003×μ-prime-strip [cf. the\ndiscussion of [Alien], §3.11, (v); [IUTchIII], Remark 3.9.5, (viii), (ix); [IUTchIII],\nRemark 3.12.2, (ii)]:\n(ItwOut) This terminology refers to the treatment of the F\u0002\u0003×μ-prime-strip ac-\ncording to (InOut), relative to the functorial algorithm for constructing\nthe q-pilot F\u0002\u0003×μ-prime-strip [in the case of the “q-intertwining”] or\nthe Θ-pilot F\u0002\u0003×μ-prime-strip [in the case of the “Θ-intertwining”] from\nsome Θ±ellNF- or D-Θ±ellNF-Hodge theater.\nIn any situation in which one considers a construction from the point of view\nof (UdOut) — that is to say, as a construction that produces “underlying data”\n[i.e., “output data”] from “original data” [i.e., “input data”]\ninput data \u0004 output data\n|| ||\noriginal data underlying data\n— it is natural to consider the issue of descent to [a functorial algorithm in] the\nunderlying data of a functorial algorithm in the original data. Here, we say that\na functorial algorithm Φ in the original data descends to a functorial\nalgorithm Ψ in the underlying data if there exists a functorial isomorphism\nΦ\n∼→ Ψ|original data\nbetween Φ and the restriction of Ψ, i.e., relative to the given construction\noriginal data \u0004 underlying data.\nThat is to say, roughly speaking, to say that the functorial algorithm Φ in the\noriginal data descends to the underlying data means, in essence, that although the\nconstruction constituted by Φ depends, a priori, on the “finer” original data, in\nfact, up to natural isomorphism, it only depends on the “coarser” underlying data.\n124 SHINICHI MOCHIZUKI\nOne elementary example of the phenomenon of descent may be seen in the\nsituation discussed in (HomRs) in §3.8:\n(HmDsc) The various constructions of (co)homology modules in homological\nalgebra are, strictly speaking, constructions that require as input data not\njust some given module [whose (co)homology is computed by the construc-\ntion], but also some sort of resolution of the given module that satisfies\ncertain properties. In fact, however, such constructions of (co)homology\nmodules typically descend, up to unique isomorphism, to constructions\nwhose input data consists solely of the given module.\nAnother illustrative elementary example of the phenomenon of descent is the fol-\nlowing:\nExample 3.9.1: Categories of open subschemes. Let X be a scheme, T a\ntopological space. Write\n· |X| for the underlying topological space of X,\n· Open(X) for the category of open subschemes of X and open immersions\nover X,\n· Open(T ) for the category of open subsets of T and open immersions over\nT .\nThen the functorial algorithm\nX \b→ Open(X)\n— defined, say, on the category of schemes and morphisms of schemes — is eas-\nily verified to descend, relative to the construction X \u0004 |X|, to the functorial\nalgorithm\nT \b→ Open(T )\n— defined, say, on the category of topological spaces and continuous maps of topo-\nlogical spaces. That is to say, one verifies immediately that there is a natural\nfunctorial isomorphism\nOpen(X)\n∼→ Open(|X|)\n[i.e., in this case, following the conventions employed in inter-universal Teichmüller\ntheory, a natural functorial isomorphism class of equivalences of categories — cf.\nthe discussion of “Monoids and Categories” in [IUTchI], §0].\nOn the other hand, perhaps the most fundamental example, in the context of\ninter-universal Teichmüller theory, of this phenomenon of descent is the following\n[cf. the notational conventions of [IUTchI], Definition 3.1, (e), (f)]:\n(MnDsc) The topological multiplicative monoid determined by the topological ring\ngiven by [the union with {0} of] O\u0004(ΠX) [cf. [Alien], Example 2.12.3,\n(iii)] — that is to say, a construction that, a priori, from the point of view\nof [AbsTopIII], Theorem 1.9; [AbsTopIII], Corollary 1.10, is a functorial\nalgorithm in the topological group\nΠX\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 125\n[i.e., “Πv”, from the point of view (sQGOut)] — in fact descends [cf.\nthe discussion at the beginning of [Alien], §2.12; the discussion of [Alien],\nExample 2.12.3, (i)], relative to passage to the underlying quotient group\ndiscussed in (SQGOut), to a functorial algorithm in the topological group\nGk\n[i.e., “Gv”, from the point of view (sQGOut)].\nFinally, we remark that often, in inter-universal Teichmüller theory, the out-\nput data of the functorial algorithm Φ of the above discussion is regarded “stack-\ntheoretically”. That is to say, the output data is not a single “set-theoretic object”,\nbut rather a collection [which is not necessarily a set!] of set-theoretic objects\nlinked by uniquely determined poly-isomorphisms of some sort. Typically, this sort\nof situation arises when one applies (ExtInd2) — cf. the discussion of (NSsQ) in\n§3.8. The most central example of this phenomenon in inter-universal Teichmüller\ntheory is the multiradial algorithm — and, especially, the portion of the multi-\nradial algorithm that involves the log-Kummer-correspondence and closely related\noperations of Galois evaluation — which plays the role of\nexhibiting the Frobenius-like Θ-pilot as one possibility within a collec-\ntion of possibilities constructed via anabelian algorithms from étale-like\ndata\n[cf. the discussion at the end of §3.6, as well as the discussion of §3.10, §3.11, be-\nlow]. That is to say, the log-Kummer-correspondence and closely related operations\nof Galois evaluation exhibit the Frobenius-like Θ-pilot as one possibility within a\ncollection of possibilities constructed via anabelian algorithms from étale-like data\nnot in a set-theoretic sense [i.e., one possibility\/element contained in a set of\npossibilities], but rather in a “stack-theoretic sense”, in accordance with various\napplications of (ExtInd2) [cf. the discussion at the end of §3.6], i.e., as\none possibility, up to isomorphism, within some [not necessarily set-\ntheoretic!] collection of possibilities.\nAs discussed in (LVsQ) in §3.8, one arrives at a set-theoretic situation — i.e., one\npossibility\/element contained in a set of possibilities — only after one obtains a\n“closed loop”, which allows one to pass to a “coarse\/set-theoretic (sub)quotient” by\ntaking the log-volume.\n§3.10. Detailed description of the chain of logical AND relations\nWe begin the present §3.10 with the following well-known and, in some sense,\nessentially tautological observation: Just as every form of data — i.e., ranging from\ntext files and webpages to audiovisual data — that can be processed by a computer\ncan, ultimately, be expressed as a [perhaps very long!] chain of “0’s” and “1’s”, the\nwell-known functional completeness, in the sense of propositional calculus, of the\ncollection of Boolean operators consisting of logical AND “∧”, logical OR “∨”, and\nnegation “¬” motivates the point of view that one can, in principle, express\nthe essential logical structure of any mathematical argument or\ntheory in terms of elementary logical relations, i.e., such as logical\nAND “∧”, logical OR “∨”, and negation “¬”.\n126 SHINICHI MOCHIZUKI\nIndeed, it is precisely this point of view that formed the central motivation and\nconceptual starting point of the exposition given in the present paper.\nFrom the point of view of the correspondence with the terminology and modes\nof expression that actually appear in [IUTchI-III] and [Alien], the representation\ngiven in the present paper of the essential logical structure of inter-universal Te-\nichmüller theory in terms of elementary logical relations, i.e., such as logical AND\n“∧” and logical OR “∨”, may be understood as follows:\n· Logical AND “∧” corresponds to such terms as\n· simultaneous execution and\n· gluing\n[cf. [IUTchIII], Remark 3.11.1, (ii); [IUTchIII], Remark 3.12.2, (ii), (citw),\n(fitw); the final portion of [Alien], §3.7, (i); [Alien], §3.11, (iv)].\n· Logical OR “∨” corresponds to such terms as\n· indeterminacies,\n· poly-morphisms, and\n· projection\/(sub)quotient\/splitting\n[cf. §3.7; the title of [IUTchIII]; [IUTchIII], Remark 3.9.5, (xiii), (ix);\n[Alien], §3.11, (v); [Alien], §4.1, (iv)].\nRecall the essential logical structure of inter-universal Teichmüller theory sum-\nmarized in (∧(∨)-Chn)\nA ∧B = A ∧ (B1 ∨B2 ∨ . . . )\n=⇒ A ∧ (B1 ∨B2 ∨ . . . ∨B′1 ∨B′2 ∨ . . . )\n=⇒ A ∧ (B1 ∨B2 ∨ . . . ∨B′1 ∨B′2 ∨ . . . ∨B′′1 ∨B′′2 ∨ . . . )\n...\n[cf. the discussion of §3.6, §3.7]. Observe that if the description of the various\n“possibilities” related via “∨’s” in the above displays is suitably formulated, i.e.,\nwithout superfluous overlaps, then in fact these logical OR “∨’s”may be understood\nas logical XOR “∨̇’s”, i.e., we conclude the following:\n(∧(∨̇)-Chn) The essential logical structure of inter-universal Teichmüller theory may\nbe summarized as follows:\nA ∧B = A ∧ (B1 ∨̇ B2 ∨̇ . . . )\n=⇒ A ∧ (B1 ∨̇ B2 ∨̇ . . . ∨̇ B′1 ∨̇ B′2 ∨̇ . . . )\n=⇒ A ∧ (B1 ∨̇ B2 ∨̇ . . . ∨̇ B′1 ∨̇ B′2 ∨̇ . . . ∨̇ B′′1 ∨̇ B′′2 ∨̇ . . . )\n...\nHere, we observe the following:\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 127\n(∧(∨̇)-Chn1) The “∧’s” in the above display\n· arise from the Θ-link, which may be thought of as a re-\nlationship between certain portions of the multiplica-\ntive structures of the ring structures arising from the\n(Θ±ellNF-)Hodge theaters in the domain and codomain\nof the Θ-link that are common [cf. “∧”!] to these ring\nstructures.\nThis situation is reminiscent of\n· the fact that from the point of view of Boolean alge-\nbras, “∧” corresponds to the multiplicative structure\nof the field F2, which may be regarded, via the splitting\ndetermined by Teichmüller representatives, as a multi-\nplicative structure that is common [cf. “∧”!] to Z and\nF2 [cf. Example 2.4.6, (iii)], as well as of\n· the discussion of [Alien], §3.11, (iv), (2and), concern-\ning the interpretation of the discussion of crystals in\n[Alien], §3.1, (v), (3KS), in terms of the logical rela-\ntor “∧”, i.e., as objects that may be simultaneously\ninterpreted, up to isomorphism, as pull-backs via one\nprojection morphism and [cf. “∧”!] as pull-backs via\nthe other projection morphism.\n(∧(∨̇)-Chn2) The “∨̇’s” in the above display may be understood as corre-\nsponding to\n· various indeterminacies that arise mainly from the log-\nKummer-correspondence, i.e., from sequences of it-\nerates of the log-link, which may be thought of as a\ndevice for constructing additive log-shells. The addi-\ntive structures of the ring structures arising from the\n(Θ±ellNF-)Hodge theaters in the domain and codomain\nof the Θ-link are structures which, unlike the corre-\nsponding multiplicative structures, are not common\n[cf. “∨̇”!] to these ring structures in the domain and\ncodomain of the Θ-link.\nThis situation is reminiscent of\n· the fact that from the point of view of Boolean al-\ngebras, “∨̇” corresponds to the additive structure of\nthe field F2, which is an additive structure that is not\nshared [cf. “∨̇”!], relative to the splitting determined\nby Teichmüller representatives, by Z and F2 [cf. Exam-\nple 2.4.6, (iii)], as well as of\n· the discussion of [Alien], §3.11, (iv), (2and), concerning\nthe interpretation of the discussion of crystals in [Alien],\n§3.1, (v), (3KS) in terms of the logical relator “∧”, i.e.,\nwhere we recall that the two pull-backs of the rank\n128 SHINICHI MOCHIZUKI\none Hodge subbundle [cf. [Alien], §3.1, (v), (5KS);\nthe discussion of Hodge structures in [IUTchI], §I2] do\nnot, in general, coincide [cf. “∨̇”!], but rather differ\nby an additive “deformation discrepancy”, namely, the\nKodaira-Spencer morphism.\n(∧(∨̇)-Chn3) Taken together, (∧(∨̇)-Chn1) and (∧(∨̇)-Chn2) may be under-\nstood as expressing the fact that the “∨̇’s” and “∧’s” of the above\ndisplay correspond, respectively, to the two underlying com-\nbinatorial dimensions — i.e., addition and multiplication\n— of a ring or, alternatively, to the two-dimensional nature of\nthe log-theta-lattice [cf. the discussion of [IUTchIII], Remark\n3.12.2, (i); the latter portion of [Alien], §3.3, (ii)]. Thus, these\ntwo dimensions may be understood, alternatively, as correspond-\ning to\n· the arithmetically intertwined Θ-link and log-\nlink of inter-universal Teichmüller theory, which give\nrise to the multiradial representation up to suitable\nindeterminacies [cf. “∧(∨̇)”!] of the Θ-pilot;\n· the description given in Example 2.4.6, (iii), of the\ncarry-addition operation on the truncated ring of Witt\nvectors Z\/4Z in terms of “∧” and “∨̇” [cf. “∧(∨̇)”!];\n· the filtered crystal discussed in [Alien], §3.1, (v),\n(5KS), where one may think of the filtration [i.e., rank\none Hodge subbundle] as “being well-defined up to inde-\nterminacies” [cf. “∧(∨̇)”!], i.e., up to a “discrepancy”,\nwhich is given by the Kodaira-Spencer morphism.\n(∧(∨̇)-Chn4) The two dimensions discussed in (∧(∨̇)-Chn3) may be under-\nstood as corresponding to the two dimensions — i.e.,\n· the successive iterates of the Frobenius mor-\nphism in positive characteristic and\n· successive extensions to mixed characteristic\n— of a ring of Witt vectors [cf. the discussion of the latter por-\ntion of [Alien], §3.3, (ii)]. This relationship to the two dimensions\nof a ring of Witt vectors is entirely consistent with the way in\nwhich truncated rings of Witt vectors occur in the discussion of\nExample 2.4.6, (iii), i.e., with the expression\n∨̈ = (∧, ∨̇)\nof mixed characteristic “carry-addition” as a sort of “inter-\ntwining” between addition and multiplication in the field F2\nobtained by “stacking” multiplication “∧” on top of addition\n“∨̇”.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 129\nMoreover, in this context, we note that the various correspondences observed in\n(∧(∨̇)-Chn3) and (∧(∨̇)-Chn4) are particularly fascinating in that they assert that\nthe “arithmetic intertwining” in a ring between addition and multiplication\n— i.e., the mathematical structure which is in some sense the main object of\nstudy in inter-universal Teichmüller theory — may be elucidated by means of a the-\nory [i.e., inter-universal Teichmüller theory!] whose essential logical structure,\nwhen written symbolically in terms of Boolean operators such as “∧” and “∨̇”,\namounts precisely to the description [cf. the discussion of (∨̈= ∧ ∨̇) in Example\n2.4.6, (iii)] of the “Boolean intertwining” that appears in Boolean carry-addition\n“∨̈” between Boolean addition “∨̇” and Boolean multiplication “∧”. Put another\nway, it is as if\n(TrHrc) the arithmetic intertwining which is the main object of study in inter-\nuniversal Teichmüller theory somehow “induces”\/“is reflected in” a sort of\n“structural carry operation”, or “trans-hierarchical similitude”,\nto the Boolean intertwining that constitutes the essential logical struc-\nture of the theory [i.e., inter-universal Teichmüller theory] that is used to\ndescribe it:\narithmetic intertwining \u0004 Boolean intertwining!\nFinally, we observe that it also interesting to note that the essential mechanism\nunderlying the Kummer theory of theta functions — which plays a central\nrole in inter-universal Teichmüller theory, i.e., in inducing the trans-hierarchical\nsimilude discussed in (TrHrc) — namely, the correspondence\nKummer theory of theta functions ←→ one valuation\/cusp\nKummer theory of algebraic rational functions ←→ multiple valuations\/cusps\ndiscussed in [IUTchIII], Remark 2.3.3, (viii), (ix), may itself be thought of as a\nsort of trans-hierarchical similitude between number fields\/local fields and\nfunction fields over number fields\/local fields.\nAt a more technical level, the essential logical structure of inter-universal Te-\nichmüller theory summarized symbolically in (∧(∨̇)-Chn) may be understood as\nconsisting of the following steps:\n(Stp1) log-Kummer-correspondence and Galois evaluation: This step con-\nsists of\nexhibiting the Frobenius-like Θ-pilot at the lattice point (0, 0)\nof the log-theta-lattice — i.e., the data that gives rise to the\nF\u0002\u0003×μ-prime-strip that appears in the domain of the Θ-link\n— as one possibility within a collection of possibilities [cf.\n(ExtInd1)!] constructed via anabelian algorithms from holomor-\nphic [relative to the 0-column] étale-like data.\nIn this context, it is perhaps worth mentioning that it is a logical tautology\nthat the content of the above display may, equivalently, be phrased as\nfollows: this step consists of\n130 SHINICHI MOCHIZUKI\nthe negation “¬” of the assertion of the non-existence of the\nFrobenius-like Θ-pilot at the lattice point (0, 0) of the log-\ntheta-lattice — i.e., the data that gives rise to the F\u0002\u0003×μ-prime-\nstrip that appears in the domain of the Θ-link — within the col-\nlection of possibilities constructed via certain anabelian algo-\nrithms from holomorphic [relative to the 0-column] étale-like\ndata\n[cf. also the discussion of (RcnLb) below]. At the level of labels of lat-\ntice points of the log-theta-lattice, this step corresponds to the descent\noperation\n(0, 0) \u0004 (0, ◦)\n[cf. the discussion at the end of §3.6; the discussion at the end of §3.9;\n[IUTchIII], Remark 3.9.5, (viii), (sQ1), (sQ2); [IUTchIII], Theorem 3.11,\n(ii), (iii)]. Finally, we recall that this step already involves the introduc-\ntion of the (Ind3) indeterminacy, which may be understood as a sort\nof coarse approximation [cf. (ExtInd1), as well as the discussion of\n(logORInd) in §3.11 below] of the complicated apparatus constituted by\nthe log-Kummer-correspondence and Galois evaluation.\n(Stp2) Introduction of (Ind1): This step consists of observing that\nthe anabelian construction algorithms of (Stp1) in fact descend\nto — i.e., are equivalent to algorithms that only require as in-\nput data the weaker data constituted by [cf. the discussion of\n“descent” in §3.9] — the associated mono-analytic étale-like\ndata, i.e., in the notation of (sQGOut), the “Gv’s”.\nAt the level of labels of lattice points of the log-theta-lattice, this step\ncorresponds to the descent operation\n(0, ◦) \u0004 (0, ◦)\b\n[cf. [IUTchIII], Remark 3.9.5, (viii), (sQ1), (sQ2); [IUTchIII], Theorem\n3.11, (i), as well as the references to [IUTchIII], Theorem 3.11, (i), in\n[IUTchIII], Theorem 3.11, (iii)]. Finally, we recall that this step involves\nthe introduction of the (Ind1) indeterminacy, which [very mildly! —\ncf. the discussion of (Ind3>1+2) in §3.11 below] increases the collection\nof possibilities under consideration [cf. (ExtInd1)].\n(Stp3) Introduction of (Ind2): This step consists of observing that\nthe anabelian construction algorithms of (Stp2) in fact descend\nto — i.e., are equivalent to algorithms that only require as in-\nput data the weaker data constituted by [cf. the discussion of\n“descent” in §3.9] — the associated mono-analytic Frobenius-\nlike data, i.e., in the notation of (sQGOut) and (MnOut), the\n“Gv \u0002 O×μF v ’s”.\n[That is to say, one constructs log-shells, for instance, as submonoids of\n“O×μ\nFv\n”, as opposed to subquotients of “Gv”.] At the level of labels of\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 131\nlattice points of the log-theta-lattice, this step corresponds to the descent\noperation\n(0, ◦)\b \u0004 (0, 0)\b\n[cf. [IUTchIII], Remark 3.9.5, (viii), (sQ1), (sQ2); [IUTchIII], Theorem\n3.11, (i), as well as the references to [IUTchIII], Theorem 3.11, (i), in\n[IUTchIII], Theorem 3.11, (iii)]. Since the Θ-link may be thought of as a\nsort of equivalence of labels\n(0, 0)\b ⇐⇒ (1, 0)\b\n— i.e., corresponding to the full poly-isomorphism of F\u0002\u0003×μ-prime-strips\nconstituted by the Θ-link — this descent operation means that the algo-\nrithm under consideration may be regarded as an algorithm whose input\ndata is the mono-analytic Frobenius-like data (1, 0)\b arising from\nthe codomain of the Θ-link. This step involves the introduction of the\n(Ind2) indeterminacy, which [very mildly! — cf. the discussion of\n(Ind3>1+2) in §3.11 below] increases, at least from an a priori point of\nview, the collection of possibilities under consideration [cf. (ExtInd1)].\nFinally, we recall that this step plays the important role of\nisolating the log-link indeterminacies in the domain [i.e., the\n(Ind3) indeterminacy of (Stp1)] and the codomain [i.e., the log-\nshift adjustment discussed in (Stp7) below] of the Θ-link from\none another\n[cf. the discussion of [IUTchIII], Remark 3.9.5, (vii), (Ob7-2); [Alien],\n§3.6, (iv)]. Here, we recall [cf. the discussion of the final portion of [Alien],\n§3.3, (ii)] that these log-link indeterminacies on either side of the Θ-link\nmay be understood, in the context of the discussion of (InfH) in §3.3, as\ncorresponding to the copies “C×” on either side of the double coset space\n“C×\\GL+2 (R)\/C×”.\n(Stp4) Passage to the holomorphic hull: The passage from the collection of\npossible regions that appear in the output data of (Stp3) to the collection\nof regions contained in the holomorphic hull — relative to the 1-column\nof the log-theta-lattice — of the union of possible regions of the output\ndata of (Stp3) [cf. [IUTchIII], Remark 3.9.5, (vi); [IUTchIII], Remark\n3.9.5, (vii), (Ob5); [IUTchIII], Remark 3.9.5, (viii), (sQ3)] is a simple,\nstraighforward application of (ExtInd1), that is to say, of increasing the\nset of possibilities [i.e., of “∨̇’s”]. The purpose of this step, together with\n(Stp5) below, is to pass from arbitrary regions to regions corresponding\nto arithmetic vector bundles [cf. [IUTchIII], Remark 3.9.5, (vii), (Ob1),\n(Ob2)].\n(Stp5) Passage to hull-approximants: This step consists of passing from the\ncollection of arbitrary regions contained the holomorphic hull of (Stp4) to\nhull-approximants, i.e., regions that have the same global log-volume\nas the original “arbitrary regions”, but which correspond to arithmetic\nvector bundles [cf. [IUTchIII], Remark 3.9.5, (vii), (Ob6); [IUTchIII],\nRemark 3.9.5, (viii), (sQ3)]. This operation does not affect the logical\n132 SHINICHI MOCHIZUKI\n“∧\/∨” structure of the algorithm since this operation of passing to hull-\napproximants does\nnot affect the collection of possible value group portions —\ni.e., “F\u0002\u0003-prime-strips” — of F\u0002\u0003×μ-prime-strips determined\nby forming the log-volume of these regions\n[cf. the discussion of [IUTchIII], Remark 2.4.2; the discussion of (IPL) in\n[IUTchIII], Remark 3.11.1, (iii)].\n(Stp6) Passage to a suitable positive rational tensor power of the de-\nterminant: This step consists of passing from the [regions corresponding\nto] arithmetic vector bundles obtained in (Stp4), (Stp5) to a suitable tensor\npower root of a tensor power of the determinant arithmetic line bundle of\nsuch an arithmetic vector bundle [cf. [IUTchIII], Remark 3.9.5, (vii),\n(Ob3), (Ob4); [IUTchIII], Remark 3.9.5, (viii), (sQ3)]. Just as in the case\nof (Stp5), this operation does not affect the logical “∧\/∨” structure of the\nalgorithm since this operation of passing to a suitable positive rational\ntensor power of the determinant does\nnot affect the collection of possible value group portions —\ni.e., “F\u0002\u0003-prime-strips” — of F\u0002\u0003×μ-prime-strips determined\nby forming the log-volume of these regions\n[cf. the discussion of [IUTchIII], Remark 2.4.2; the discussion of (IPL) in\n[IUTchIII], Remark 3.11.1, (iii)].\n(Stp7) Log-shift adjustment: The arithmetic line bundles that appear in\n(Stp6) occur with respect to the arithmetic holomorphic structure — i.e.,\nin effect, ring structure — at the label (1, 1) of the log-theta-lattice, i.e.,\nat a label vertically shifted by +1 relative to the label (1, 0) that forms the\ncodomain of the Θ-link [cf. the discussion of [IUTchIII], Remark 3.9.5,\n(vii), (Ob8); [IUTchIII], Remark 3.9.5, (viii), (sQ4)]. That is to say, by\napplying the algorithm discussed in (Stp1) ∼ (Stp6) at each lattice point\n(1,m) [where m ∈ Z] of the 1-column of the log-theta-lattice, we obtain\nalgorithms with input data at (1,m) and output data at (1,m + 1) — cf.\nthe diagonal arrows of the diagram shown below. Thus, the totality of all\nof these diagonal arrows may be thought of a sort of endomorphism of\nthe 1-column of the log-theta-lattice, i.e., an algorithm whose input data\nis the 1-column of the log-theta-lattice, and whose output data lies in the\nsame 1-column of the log-theta-lattice. In particular, we obtain a closed\nloop [cf. the discussion of Example 3.1.1, (iii); the discussion below of\n(DstMp), (FxGl), (NoCmpIss), (Englf); the discussion of [IUTchIII], Re-\nmark 3.9.5, (ix); [Alien], §3.11, (v)]. Put another way, in the language of\nthe discussion surrounding (DltLb) in §3.11 below, we obtain a situation\nthat simulates — via the introduction of suitable indeterminacies [cf.\nthe discussion of (Stp8) below] — a situation in which the distinct labels\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 133\non the domain and codomain of the Θ-link have been eliminated.\n1-\ncolumn\n1-\ncolumn\n...\n...\n• •\u000b⏐⏐log ↗ \u000b⏐⏐log\n• •\u000b⏐⏐log ↗ \u000b⏐⏐log\n• •\u000b⏐⏐log ↗ \u000b⏐⏐log\n• •\n...\n...\n(Stp8) Passage to log-volumes: The closed loop of (Stp7) [cf. also the\ndiscussion of Example 3.1.1, (iii); the discussion below of (DstMp), (FxGl),\n(NoCmpIss), (Englf)] implies that the crucial logical AND “∧” relation\ncarefully maintained throughout the execution of (Stp1) ∼ (Stp7) yields,\nupon taking the log-volume, a\nlogical AND “∧” relationship between the original q-pilot\ninput F\u0002\u0003-prime-strip and a certain algorithmically constructed\ncollection of possible output F\u0002\u0003-prime-strips within the\nsame container, i.e., some copy of the real numbers “R”\n[cf. [IUTchIII], Remark 3.9.5, (vii), (Ob9); [IUTchIII], Remark 3.9.5,\n(viii), (sQ5); [IUTchIII], Remark 3.9.5, (ix); the discussion of Substeps\n(xi-d), (xi-e) of the proof of [IUTchIII], Corollary 3.12; the discussion\nof [IUTchIII], Remark 3.12.2, (ii); [Alien], §3.11, (v)]. The inequality\nin the statement of [IUTchIII], Corollary 3.12, then follows as a formal\nconsequence [cf. the discussion of Substeps (xi-f), (xi-g) of the proof of\n[IUTchIII], Corollary 3.12]. Here, we observe that the various indetermi-\nnacies [such as (Ind1), (Ind2), (Ind3)] that arise in the course of applying\n(Stp1) ∼ (Stp7) may be thought of as a sort of monodromy associated\nto the closed loop of (Stp7) [cf. also the discussion below of (DstMp),\n(FxGl), (NoCmpIss), (Englf)]. Moreover, in this context, we note that\nthe diagram of arrows “↗” from the 1-column to the 1-column in (Stp7)\nadmits symmetries\n(1,m) \b→ (1,m+ 1)\n[where m ∈ Z] that are compatible with all of the arrows in the diagram,\nas well as with the various arrows of the log-Kummer-correspondence in\nthe 1-column. These symmetries allow one to synchronize the various\n134 SHINICHI MOCHIZUKI\n“monodromy indeterminacies” associated to each “↗” [i.e., to each\napplication of (Stp1) ∼ (Stp6)], so that one may think in terms of a single,\nsynchronized collection of “monodromy indeterminacies” associated to\nthe totality of “↗’s” in (Stp7).\nBefore proceeding, we pause to examine the meaning of the term “closed\nloop” in (Stp7), (Stp8), which is sometimes a source of confusion. The intended\nmeaning of this term is that the sequence of mathematics objects on which the series\nof operations in (Stp1) ∼ (Stp6) [cf. also [IUTchIII], Fig. I.8] are performed forms\na closed loop in the sense that the ultimate output data lies in the same container\n[i.e., up to a log-shift in the 1-column, as discussed in (Stp7)] as the input data.\nOn the other hand, at the level of the actual mathematical objects that\none is working with, the term “closed loop” has the potential to result in certain\nfundamental misunderstandings, since it may be [mistakenly!] interpreted as\nsuggesting that\n(DstMp) one is considering two distinct mappings of abstract prime-strips to\n[Θ, q-]pilot prime-strips.\nOnce one takes this point of view (DstMp), there is inevitably an issue of diagram\ncommutativity, i.e., the issue discussed in §3.6, (Syp2), that one must contend\nwith. As discussed in Example 2.4.5, (iv), (v), (vi), (vii), (viii) [cf. also Exam-\nple 3.10.1 below], this point of view (DstMp) corresponds to EssOR-IUT [i.e.,\n“essentially OR IUT”], which, as the name suggests, is essentially [thought not\nprecisely!] equivalent to OR-IUT, and in particular, constitutes a fundamental\nmisunderstanding of the logical structure of inter-universal Teichmüller theory.\nIndeed, the chain of AND relations “∧” discussed in §3.6, as well as the\npresent §3.10, which lies at the heart of the essential logical structure of inter-\nuniversal Teichmüller theory, consists precisely of\n(FxGl) fixing the Frobenius-like q-pilot at the lattice point (1, 0), as well as the\ngluing [i.e., “∧”!] via the Θ-link of this q-pilot at (1, 0) to the Frobenius-\nlike Θ-pilot at the lattice point (0, 0) [cf. [IUTchIII], Remark 3.12.2, (ii),\n(citw), (eitw), (fitw)].\nOne then proceeds to add to this Θ-pilot at the lattice point (0, 0) more and more\npossibilities\/indeterminacies [i.e., “∨”, or, alternatively, “∨̇”!] in order to obtain\ndata that descends to the same label [i.e., up to a log-shift in the 1-column, as\ndiscussed in (Stp7)] as the q-pilot at (1, 0). That is to say,\n(NoCmpIss) there is never any issue of compatibility between two distinct map-\npings of abstract prime-strips, as in (DstMp).\nFrom a pictorial point of view, at the level of mathematical objects, one is\nworking in (Stp1) ∼ (Stp8) — not with a “closed loop” (!), but rather — a single\nfixed line segment\n• = = ∧= = = •\ncorresponding to the gluing [i.e., “∧”!] of (FxGl) [so the “•’s” on the left and right\ncorrespond, respectively, to the Θ-pilot at (0, 0) and the q-pilot at (1, 0)], which\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 135\nis then subjected to subsequent “fuzzifications” [i.e., “∨”, or alternatively, “∨̇”!]\nof the Θ-pilot at (0, 0) [denoted in the following display by the notation “(. . . )”,\nwhich may be thought of as representing a “fuzzy disc” that contains the “•” on\nthe left]\n• = = ∧= = = •\n\u0004 ( ∨ • =) = ∧= = = •\n\u0004 ( ∨ ∨ • = =) ∧= = = •\n\u0004 ( ∨ ∨ ∨ • = = ∧=) = = •\n\u0004 ( ∨ ∨ ∨ ∨ • = = ∧= =) = •\n\u0004 ( ∨ ∨ ∨ ∨ ∨ • = = ∧= = =) •\n\u0004 ( ∨ ∨ ∨ ∨ ∨ ∨ • = = ∧= = = • )\nthat terminate in a situation [cf. the final line of the above display] in which\n(Englf) the fuzzifications engulf the q-pilot at (1, 0), i.e., a situation in which\nthe distinct labels may be eliminated and nontrivial consequences may be\nobtained [cf. (Stp7), (Stp8), as well as the discussion surrounding (DltLb)\nin §3.11 below].\nThus, in summary, throughout the series of fuzzification operations constituted by\n(Stp1) ∼ (Stp8), the line segment representing the gluing [i.e., “∧”!] of (FxGl)\nremains fixed, so [cf. (NoCmpIss), (Englf)] there is never any issue of compat-\nibility between two distinct mappings of abstract prime-strips, as in (DstMp).\nExample 3.10.1: Symmetries as a fundamental non-formal aspect of glu-\nings. One psychological aspect, and indeed [in many cases] possible cause, of\nthe fundamental misunderstandings discussed above [cf. the discussion above sur-\nrounding (DstMp), as well as the discussion surrounding (RfsDlg), (DngPrc) in\n§1.12] concerning the essential logical structure of inter-universal Teichmüller the-\nory — i.e., the [erroneous!] point of view that this essential logical structure of\ninter-universal Teichmüller theory should be understood as centering around an\nissue of diagram commutativity — is the following. In any sort of gluing sit-\nuation — i.e., from a category-theoretic point of view, any sort of situation [cf.\nthe numerous examples of gluings discussed throughout the present paper, e.g., in\nExamples 2.3.2, 2.4.1, 2.4.2, 2.4.3, 2.4.7, 3.3.1, 3.5.2] in which one considers the\n[possibly formal] inductive limit I of a diagram of the form\nY1 ←− X −→ Y2\n— there is a certain tendency to fall into the “mental trap” of believing that it is\na “tautology” that\n(UnvPrp) any conceivable nontrivial\/interesting property of I should be\nunderstood in terms of the universal property of an inductive limit, i.e.,\n136 SHINICHI MOCHIZUKI\nthe property to the effect that any arrow from I to some object Z should\nbe understood in terms of the issue of commutativity of a diagram of\nthe form\nX −→ Y2⏐⏐\n \u0002 ? ⏐⏐\n\nY1 −→ Z\n— i.e., where the left-hand vertical and upper horizontal arrows are the\narrows in the definition of I.\nHere, we observe that\n(UnvPrpOR) the diagram commutativity issue that arises when one considers the uni-\nversal property discussed in (UnvPrp) is essentially a logical OR “∨”\nsituation: that is to say, one can pass from X to Z in the diagram of\n(UnvPrp) via Y1 OR via Y2, but there is, a priori, no way of passing from\nsuch an “OR situation” to the desired “AND” situation that arises when\ncommutativity holds, i.e., the situation where one knows that there is a\nsingle arrow from X to Z that is simultaneously equal to the composite\nof the two arrows that pass through Y1 and equal to the composite of the\ntwo arrows that pass through Y2\n— cf. the discussion of Example 2.4.5, (iv), (v), (vi), (vii), (viii). Of course, it\nis indeed a tautology that I satisfies a universal property as in (UnvPrp), but the\npoint is that\n(FlsUnv) the fact that I satisfies such a universal property does not by any means\nimply — i.e., as one might falsely conclude from the nuance carried by the\nword “universal” in ordinary, non-technical contexts! — that any con-\nceivable nontrivial\/interesting property of I is best understood\nin terms of this universal property.\nThat is to say,\n(SymPrp) there are numerous examples in mathematics of objects “I” that are\nconstructed as gluings, but that satisfy important nontrivial properties\nsuch as symmetry properties — e.g., of the sort discussed in §3.2 [cf.,\nespecially, Example 3.2.2!] — that do not admit any natural “general\nnonsense” formulation in terms of the universal property of (UnvPrp).\nIn the spirit of (UnvPrpOR), it is of interest to note that\n(SymPrpAND) symmetry properties typically concern some sort of common, or\n“coric”, structure that is shared throughout various “localizations” of\nthe diagram that gives rise to I, where we recall that it is essentially a\ntautology that this sort of notion of a commonly shared\/“coric” property\nis a logical AND “∧” situation\n— cf. the discussion of Example 2.4.5, (iv), (v), (vi), (vii), (viii), as well as the dis-\ncussion of (FxGl), (NoCmpIss), (Englf), above. Well-known elementary examples\nof the phenomenon discussed in (SymPrp) include\n(i) the projective general linear symmetries [i.e., “PGL2”] of the pro-\njective line, which may be constructed as a gluing of two copies of the\naffine line [cf. Example 2.4.7, (iv), (v)];\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 137\n(ii) the group structure of an elliptic curve, which, as is well-known,\nmay be constructed as a gluing of two cubic planar affine curves;\n(iii) the group of general linear oriented symmetries GL+2 (R) of the two-\ndimensional R-vector space R2 — cf. the discussion of the double coset\nspace\nC×\\GL+2 (R)\/C×\nin §3.3, (InfH) — in the situation of the complex Teichmüller defor-\nmations discussed in Example 3.3.1, where we recall that this situation\nmay be regarded [cf. the discussion at the beginning of Example 3.5.2] as\na “gluing” of two distinct copies of the complex plane C along a common\nunderlying two-dimensional R-vector space R2 [cf. also Example 3.5.2,\n(iii)], and we observe that these symmetries GL+2 (R) allow one to re-\ngard each of the two holomorphic structures [i.e., copies of C] that appear\nin this gluing as indeterminate GL+2 (R)-conjugates of the subgroup\nC× ⊆ GL+2 (R) inside the common container GL+2 (R);\n(iv) the common holomorphic structure that appears in the classical\ntheory of analytic continuation of one-variable complex holomorphic\nfunctions — cf., e.g., the discussion of the theory of analytic continuation\nof the complex logarithm in the discussion surrounding (FxEuc) in §3.1,\nas well as in the historical discussion of §1.5 and the discussion of the toral\nrotations “C×” in §3.3, (InfH) — where we observe that such analytic\ncontinuations may be regarded as gluings of copies of the complex open\nunit disc, and that the common holomorphic structure may be regarded as\na sort of common symmetry of such gluings, i.e., if one thinks in terms\nof “almost complex structures”, that is to say, in terms of the symmetry\nof the tangent bundle given by multiplication by i =\n√−1.\nHere, we observe in passing that it is of interest to note, in the context of (ii), that\nthe group structure of an elliptic curve, as well as the existence of invariant [i.e.,\nwith respect to this group structure] differentials on the elliptic curve, play a central\nrole in the argument discussed in Example 3.2.1 — an argument that itself played\na fundamental role in motivating the essential logical structure of inter-universal\nTeichmüller theory [cf. Example 3.2.1, (viii)]. Of course, as discussed extensively\nin §3.3 [cf. (InfH); Examples 3.3.1, 3.3.2], the situations discussed in (iii), (iv) also\nexhibit numerous important structural similarities to various important aspects of\ninter-universal Teichmüller theory.\n§3.11. The central importance of the log-Kummer-correspondence\nIn the context of the discussion of §3.10, it is important to recall that, whereas\n(Stp2) ∼ (Stp8) are technically trivial in the sense that they concern operations that\nare very elementary and only require a few lines to describe, the log-Kummer-\ncorrespondence and Galois evaluation operations that comprise (Stp1) depend\non the highly nontrivial theory of [EtTh] and [AbsTopIII]. Moreover, the technical\ndescription of these operations that comprise (Stp1) occupies the bulk of [IUTchI-\nIII]. The central importance of (Stp1) may also be seen in the subordinate nature\nof (Ind1), (Ind2) [which occur in (Stp2), (Stp3)] relative to (Ind3) [which occurs in\n(Stp1)], i.e., in the sense that\n138 SHINICHI MOCHIZUKI\n(Ind3>1+2) once one constructs the output of the multiradial representation of\nthe Θ-pilot [cf. [IUTchIII], Theorem 3.11, (ii)] via tensor-packets of log-\nshells in such a way that each local portion of this output is stable with\nrespect to the indeterminacy (Ind3), these local portions of the output\nare automatically “essentially stable” [i.e., stable up to discrepancies at\nthe valuations ∈ Vbad that affect the resulting log-volumes only up to\nvery small\/essentially negligible order] with respect to the indeterminacies\n(Ind1), (Ind2) [cf. [IUTchIII], Theorem 3.11, (i)].\nFinally, we observe that this property (Ind3>1+2) is strongly reminiscent of the\ndiscussion of (CnfInd1+2) and (CnfInd3) in §3.5.\nOne way to understand the content of the operations of (Stp1) is as follows.\nThese operations may be regarded as a sort of\n(logORInd) saturation of the Frobenius-like Θ-pilot at the lattice point (0, 0)\nof the log-theta-lattice — i.e., which is linked, via the Θ-link, to the\nFrobenius-like q-pilot at (0, 1) — with respect to all of the possibil-\nities that occur in the 0-column of the log-theta-lattice, i.e., all of the\npossibilities that arise from a possible confusion between the domain and\ncodomain of the log-links in the 0-column [cf. the description of (Stp1)].\nIn this sense, the content of (Stp1) is formally reminiscent of the “(NeuORInd)”\nthat appeared in the discussion of §3.4, i.e., which may be understood as a sort of\n(ΘORInd) saturation of the Frobenius-like Θ-pilot at the lattice point (0, 0) of\nthe log-theta-lattice with respect to all of the possibilities— i.e., “Θ-plt”,\n“q-plt” [cf. (NeuORInd2)] — that arise from a possible confusion between\nthe domain and codomain of the Θ-link joining the lattice points (0, 0)\nand (0, 1).\n[In this context, we note that the logical OR “∨’s” that appear in (logORInd),\n(ΘORInd) may in fact be understood as logical XOR “∨̇’s” — cf. the discussion\nsurrounding (∧(∨̇)-Chn) in §3.10.] On the other hand, whereas, as observed in the\ndiscussion at the end of §3.4, (ΘORInd) yields a meaningless\/useless situation that\ndoes not give rise to any interesting mathematical consequences, (logORInd), by\ncontrast, is a highly potent technical device that forms the technical core of\ninter-universal Teichmüller theory.\nBefore preceding, we observe that, in this context, it is interesting to note that\nboth of these “saturation operations” (logORInd) and (ΘORInd) are\nin some sense qualitatively similar to the label crushing operation\n(ExtInd2).\nIndeed, (ExtInd2) consists, roughly speaking, of regarding mathematical objects of\na certain type up to isomorphism, i.e., of saturating within an isomorphism class\nof mathematical objects of a certain type [cf. the discussion of (ExtInd2) in §3.8,\nas well as the discussion of (DltLb) below].\nThe stark contrast between the potency of (logORInd) and the utterly mean-\ningless nature of (ΘORInd) is highly reminiscent of the central role played, in\nExample 3.3.2, (iv), by invariance with respect to\nι =\n(\n0 1\n−1 0\n)\n∈ C× ⊆ GL+2 (R)\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 139\n[where we recall from (InfH) that C× corresponds to the log-link!], which lies\nin stark contrast to the utterly meaningless nature of considering invariance with\nrespect to dilations\n(\nλ 0\n0 1\n) ∈ GL+2 (R) [where we recall from (InfH) that such dilations\ncorrespond to the Θ-link].\nOne way to witness the potency of (logORInd) is as follows. Recall that the\nΘ-link, by definition [cf. [IUTchIII], Definition 3.8, (ii)], consists of\n· a dilation applied to the local value group portions of the ring structures\nin its domain and codomain, coupled with\n· a full poly-isomorphism — which preserves log-volumes, hence is non-\ndilating! — between the local “O×μ’s”, i.e., the local unit group portions,\nof these ring structures.\nBy contrast, the log-links in the 0-column of the log-theta-lattice have the effect\nof “juggling\/rotating\/permuting” the local value group portions and local unit group\nportions of the ring structures that appear in this 0-column [cf., e.g., the discussion\nof [Alien], Example 2.12.3, (v)]. From this point of view, the tautologically vertically\ncoric — i.e., invariant with respect to the application of the log-link! — nature of\nthe output data of (logORInd) is already somewhat “shocking” in nature. That is\nto say, the tautologically vertically coric nature of this output data of (logORInd)\nsuggests that\n(Di\/NDi) this output data already exhibits some sort of equivalence, up to per-\nhaps some sort of mild discrepancy, between the dilated and non-dilated\nportions of the Θ-link.\nSuch an equivalence already strongly suggests that some sort of bound on heights\nshould follow as a formal consequence, i.e., in the style of the classical argument\nthat implies the isogeny invariance of heights of elliptic curves [cf. the discussion of\n[Alien], §2.3, §2.4, as well as the discussion of Example 3.2.1 in the present paper;\nthe discussion of §3.5 in the present paper].\nFinally, we conclude by emphasizing that, in inter-universal Teichmüller theory,\n(DltLb) ultimately one does want to find some way in which to delete\/eliminate\nthe distinct labels on the Θ- and q-pilot objects [i.e., “Θ-plt” and “q-plt”]\nin the domain and codomain of the Θ-link\n[cf. the discussion of Example 3.1.1, (iii); the discussion of (AOL4), (AOΘ4) in §3.4;\n(Stp7), (Stp8) in §3.10], that is to say, not via the naive, simple-minded approach of\n(ΘORInd) [i.e., (NeuORInd2) in the discussion of §3.4], but rather via the indirect\napproach of applying descent operations\n(0, 0)\n(Stp1)\u0004 (0, ◦) (Stp2)\u0004 (0, ◦)\b (Stp3)\u0004 (0, 0)\b (Stp3)⇐⇒ (1, 0)\b\nas discussed in (Stp1) ∼ (Stp8) [cf., especially, (Stp7), (Stp8)] of §3.10, i.e., an\napproach that centers around (logORInd). This approach is based on the vari-\nous anabelian reconstruction algorithms discussed in (Stp1) ∼ (Stp3), which\nallow one to exhibit the Frobenius-like Θ-pilot object at (0, 0) as one possibil-\nity among some broader collection of possibilities that arise from the introduction\nof various types of indeterminacy. In this context, we observe [cf. the discus-\nsion of (ExtInd2), (NSsQ) at the end of §3.9] that since such anabelian recon-\nstruction algorithms only reconstruct various types of mathematical objects [i.e.,\n140 SHINICHI MOCHIZUKI\nmonoids\/pseudo-monoids\/mono-theta environments, etc.] not “set-theoretically on\nthe nose” [i.e., not in the sense of strict set-theoretic equality], but rather up to [a\ntypically essentially unique, if one allows for suitable indeterminacies] isomorphism,\nit is not immediately clear\n(RcnLb) in what sense such anabelian reconstruction algorithms yield a recon-\nstruction of the crucial labels — i.e., such as “(0, 0)” — that underlie\nthe crucial logical AND “∧” structure discussed in §3.4 [cf., especially,\n(AOL1), (AOΘ1)].\nThe point here is that indeed such anabelian reconstruction algorithms are not\ncapable of reconstructing such labels “set-theoretically on the nose”.\nOn the other hand, in this context, it is important to recall the essential\nsubstantive content of the various labels involved:\n(HolFrLb) (0, 0): The holomorphic Frobenius-like data labeled by (0, 0) con-\nsists of various monoids\/pseudo-monoids\/mono-theta environments, etc.,\nregarded as abstract monoids\/pseudo-monoids\/mono-theta environments,\netc., i.e., as objects that are not equipped with the auxiliary data of how\nthey might have been reconstructed via anabelian algorithms from holo-\nmorphic étale-like data labeled (0, ◦) [cf. the discussion of (UdOut), (In-\nOut), (PSOut), (ItwOut) in §3.9]. In particular, such monoids\/pseudo-\nmonoids\/mono-theta environments, etc., are not invariant with respect\nto the “juggling\/rotating\/permuting” of local value group portions and lo-\ncal unit group portions effected by the log-links in the 0-column of the\nlog-theta-lattice, but rather correspond to a temporary cessation [cf.\nthe label (0, 0) as opposed to the label (0, ◦)!] of this operation of jug-\ngling\/rotation\/permutation.\n(MnAlyLb) (0, 0)\b: The mono-analytic Frobenius-like data labeled by (0, 0)\b\nconsists of the F\u0002\u0003×μ-prime-strip determined by the Frobenius-like Θ-\npilot at (0, 0), regarded as an abstract F\u0002\u0003×μ-prime-strip [cf. the discus-\nsion of (UdOut), (InOut), (PSOut), (ItwOut) in §3.9]. Thus, the transition\nof labels\n(0, 0) \u0004 (0, 0)\b\nconsists of an operation of forgetting some sort of auxiliary structure [cf.\nthe discussion of (UdOut) in §3.9]. Here, we recall that this construction\nof the F\u0002\u0003×μ-prime-strip determined by the Frobenius-like Θ-pilot at\n(0, 0) is technically possible precisely because of the “temporary cessation”\ndiscussed above [cf. the discussion of the definition of the Θ-link in [Alien],\n§3.3, (ii), as well as in §3.3 of the present paper].\nThus, the nontrivial substantive content of the anabelian reconstruction algorithms\nof (Stp1) ∼ (Stp3) — and hence of the descent operations\n(0, 0)\n(Stp1) ∼ (Stp3)\u0004 (0, 0)\b\nthat result from these anabelian reconstruction algorithms — consists of statements\nto the effect that\n(FrgInv) the operation of forgetting discussed in (MnAlyLb) can in fact, if one\nallows for suitable indeterminacies, be inverted.\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 141\nIt is precisely this invertibility (FrgInv), up to suitable indeterminacies, of the\noperation of forgetting discussed in (MnAlyLb), together with the fact that\n(GluDt) the only data appearing in the reconstruction algorithms [i.e., in the\n0-column] that is glued [cf. the discussion of [IUTchIII], Remark 3.11.1,\n(ii); the final portion of [Alien], §3.7, (i), as well as the discussion of the\n(Ind2) indeterminacy in the final portion of §3.11, (Stp3), in the present\npaper] to data in the 1-column is the F\u0002\u0003×μ-prime-strip labeled (0, 0)\b,\nthat ensures that the descent operations discussed above do indeed preserve the\ncrucial logical AND “∧” relations discussed in §3.4, §3.6, §3.7, §3.10, i.e., even\nthough the reconstruction algorithms underlying these descent operations do not\nyield reconstructions of the various labels “(0, 0)”, etc., “set-theoretically on the\nnose”.\nBibliography\n[AnCnCv] Illustration of analytic continuation along a curve, available at the following\nURL [where we note that when “copy-and-pasting” this URL, typically it will\nbe necessary to re-enter the underscore character “ ” manually, as well as to\ndelete stray characters that appear as a result of the carriage return between\nthe first and second lines of the URL]:\nhttps:\/\/commons.m.wikimedia.org\/wiki\/File:Analytic continuation\nalong a curve.png\n[AnCnLg] Illustration of analytic continuation of the complex logarithm, available at\nthe following URL [where we note that when “copy-and-pasting” this URL,\ntypically it will be necessary to re-enter the underscore character “ ” manually,\nas well as to delete stray characters that appear as a result of the carriage return\nbetween the first and second lines of the URL]:\nhttps:\/\/ja.m.wikipedia.org\/wiki\/%E3%83%95%E3%82%A1%E3%82%A4%E3%\n83%AB:Imaginary log analytic continuation.png\n[Btt] U. Bottazzini, “Algebraic truths” vs. “geometric fantasies”: Weierstrass’ re-\nsponse to Riemann, Proceedings of the International Congress of Mathemati-\ncians, Vol. III (Beijing, 2002), Higher Ed. Press (2002), pp. 923-934.\n[EMSCOP] European Mathematical Society, Code of Practice, approved by the EMS Ex-\necutive Committee (October 2012), available at the following URL:\nhttps:\/\/euro-math-soc.eu\/sites\/default\/files\/COP.pdf\n[FC] G. Faltings and C.-L. Chai, Degenerations of Abelian Varieties, Springer-Verlag\n(1990).\n[FsADT] I. Fesenko, Arithmetic deformation theory via arithmetic fundamental groups\nand nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki,\nEur. J. Math. 1 (2015), pp. 405-440.\n[GeoSph] Illustration of the metric geometry of a sphere, available at the following URL\n[where we note that when “copy-and-pasting” this URL, typically it will be\nnecessary to re-enter the underscore character “ ” manually, as well as to delete\nstray characters that appear as a result of the carriage return between the first\n142 SHINICHI MOCHIZUKI\nand second lines of the URL]:\nhttps:\/\/ja.m.wikipedia.org\/wiki\/%E3%83%95%E3%82%A1%E3%82%A4%E3\n%83%AB:Sphere wireframe 10deg 6r.svg\n[FsDss] I. Fesenko, About certain aspects of the study and dissemination of Shinichi\nMochizuki’s IUT theory, available at the following URL:\nhttps:\/\/ivanfesenko.org\/wp-content\/uploads\/2021\/10\/rapg.pdf\n[FsPio] I. Fesenko, On pioneering mathematical research, on the occasion of the an-\nnounced publication of the IUT papers by Shinichi Mochizuki, available at the\nfollowing URL:\nhttps:\/\/ivanfesenko.org\/wp-content\/uploads\/2021\/10\/rpp.pdf\n[AnHst] E. Hairer and G. Wanner, Analysis by its history, Springer-Verlag (2008).\n[FKvid] F. Kato, abc Conjecture and New Mathematics - Prof. Fumiharu Kato, Oct 7,\n2017 (with English subtitles), video lecture in Japanese (with English subtitles)\navailable at the following URL:\nhttps:\/\/www.youtube.com\/watch?v=fNS7N04DLAQ&feature=youtu.be\n[HASurI] S. Mochizuki, A Survey of the Hodge-Arakelov Theory of Elliptic Curves I,\nArithmetic Fundamental Groups and Noncommutative Algebra, Proceedings of\nSymposia in Pure Mathematics 70, American Mathematical Society (2002),\npp. 533-569.\n[EtTh] S. Mochizuki, The Étale Theta Function and its Frobenioid-theoretic Manifes-\ntations, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349.\n[GenEll] S. Mochizuki, Arithmetic Elliptic Curves in General Position,Math. J. Okayama\nUniv. 52 (2010), pp. 1-28.\n[AbsTopIII] S. Mochizuki, Topics in Absolute Anabelian Geometry III: Global Reconstruc-\ntion Algorithms, J. Math. Sci. Univ. Tokyo 22 (2015), pp. 939-1156.\n[IUTchI] S. Mochizuki, Inter-universal Teichmüller Theory I: Construction of Hodge\nTheaters, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 3-207.\n[IUTchII] S. Mochizuki, Inter-universal Teichmüller Theory II: Hodge-Arakelov-theoretic\nEvaluation, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 209-401.\n[IUTchIII] S. Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of\nthe Log-theta-lattice, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 403-626.\n[IUTchIV] S. Mochizuki, Inter-universal Teichmüller Theory IV: Log-volume Computa-\ntions and Set-theoretic Foundations, Publ. Res. Inst. Math. Sci. 57 (2021), pp.\n627-723.\n[ExpEst] S. Mochizuki, I. Fesenko, Y. Hoshi, A. Minamide, W. Porowski, Explicit Es-\ntimates in Inter-universal Teichmüller Theory, Kodai Math. J. 45 (2022), pp.\n175-236.\n[Rpt2013] S. Mochizuki,On the verification of inter-universal Teichmüller theory: a progress\nreport (as of December 2013), available at the following URL:\nhttp:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/IUTeich%20Verification\n%20Report%202013-12.pdf\nLOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 143\n[Rpt2014] S. Mochizuki,On the verification of inter-universal Teichmüller theory: a progress\nreport (as of December 2014), available at the following URL:\nhttp:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/IUTeich%20Verification\n%20Report%202014-12.pdf\n[Pano] S. Mochizuki, A Panoramic Overview of Inter-universal Teichmüller Theory,\nAlgebraic number theory and related topics 2012, RIMS Kōkyūroku Bessatsu\nB51, Res. Inst. Math. Sci. (RIMS), Kyoto (2014), pp. 301-345.\n[Alien] S. Mochizuki, The Mathematics of Mutually Alien Copies: from Gaussian\nIntegrals to Inter-universal Teichmüller Theory, Inter-universal Teichmuller\nTheory Summit 2016, RIMS Kōkyūroku Bessatsu B84, Res. Inst. Math. Sci.\n(RIMS), Kyoto (2021), pp. 23-192.\n[Rpt2018] S. Mochizuki, Report on discussions, held during the period March 15 - 20,\n2018, concerning inter-universal Teichmüller theory (IUTch), available at the\nfollowing URL:\nhttp:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Rpt2018.pdf\n[Dsc2018] Webpage “March 2018 Discussions on IUTeich”, available at the following\nURL:\nhttp:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/IUTch-discussions-2018\n-03.html\n[Mumf1] D. Mumford, Abelian Varieties, Oxford Univ. Press (1974).\n[Mumf2] D. Mumford, An Analytic Construction of Degenerating Abelian Varieties over\nComplete Rings, Appendix to [FC].\n[NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields,Grundlehren\nder Mathematischen Wissenschaften 323, Springer-Verlag (2000).\n[SGA1] A. Grothendieck et al., Revêtement étales et groupe fondamental, Séminaire\nde Géometrie Algébrique du Bois Marie 1960-1961 (SGA1), dirigé par A.\nGrothendieck, augmenté de deux exposés de M. Raynaud, Lecture Notes in\nMathematics 224, Springer-Verlag (1971).\n[Tsjm] S. Tsujimura, Combinatorial Belyi cuspidalization and arithmetic subquotients\nof the Grothendieck-Teichmüller group, Publ. Res. Inst. Math. Sci. 56 (2020),\npp. 779-829.\n[Ymgt] S. Yamagata, A counterexample for the local analogy of a theorem by Iwasawa\nand Uchida, Proc. Japan Acad. 52 (1976), pp. 276-278.\n"}
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{"id":"2MTvp116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/The%20Intrinsic%20Hodge%20Theory%20of%20p-adic%20Hyperbolic%20Curves%20(Nara%201998).pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":131,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"The%20Intrinsic%20Hodge%20Theory%20of%20p-adic%20Hyperbolic%20Curves%20(Nara%201998)","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["MANU.DVI"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 12:25:46 UTC 2011","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":1815396872500918942,"exact_signature_unique_b":true,"fuzzy_signature_l":-3272141865464500992,"fuzzy_signature_unique_b":true,"wordcount_i":2706,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:54 UTC 2022","fresh_date_dt":"Thu Sep 23 19:01:58 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":79983,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086041037012992,"text_t":"The Intrinsic Hodge Theory of\np-adic Hyperbolic Curves\nby Shinichi Mochizuki\nContents:\n§1. Uniformization Theory as a Hodge Theory at Arithmetic Primes\n(A.) Uniformization as a Catalogue of Rational Points\n(B.) “Intrinsic” Hodge Theories\n(C.) Completion at Arithmetic Primes\n§2. The Physical Aspect: Embedding by Automorphic Forms\n(A.) The Complex Case\n(B.) The Arithmetic Fundamental Group\n(C.) The Main Theorem\n(D.) Comparison with the Case of Abelian Varieties\n§3. The Modular Aspect: Canonical Frobenius Actions\n(A.) The Complex Case\n(B.) Teichmüller Theory in Characteristic p\n(C.) Canonical p-adic Liftings\n§1. Uniformization Theory as a Hodge Theory at Arithmetic Primes\n(A.) Uniformization as a Catalogue of Rational Points\nWe begin our discussion by posing the following elementary problem concerning alge-\nbraic varieties over the complex numbers (where, roughly speaking, an “algebraic variety\nover the complex numbers” is a geometric object defined by polynomial equations with\ncoefficients which are complex numbers):\nProblem: Given an algebraic variety Z over C, it is possible to give some sort\nof natural explicit catalogue of the rational points Z(C) of Z?\n1\nTo gain a sense of what is meant by the expression “a natural explicit catalogue,” it is\nuseful to begin by thinking about some basic examples. Perhaps the simplest nontrivial\nexamples of algebraic varieties are plane curves, i.e., subvarieties of A2C (two-dimensional\naffine space over C) defined by a single polynomial equation\nf(X,Y ) = 0\nin two variables. In this case, the set of rational points Z(C) of the corresponding variety\nZ is given by\nZ(C) = {(x, y) ∈ C2| f(x, y) = 0}\nMoreover, we can classify plane curves by the degree of the defining equation f(X,Y ). We\nthen see that the resulting sets Z(C) may be explicitly described as follows:\n(1.) The Linear Case (deg(f) = 1): Up to coordinate transformations, this is the\ncase given by the equation f(X,Y ) = X. In this case, we then obtain an explicit\ncatalogue of the rational points by:\n(0, ?) : C\n∼→ Z(C)\n(i.e., mapping z ∈ C to (0, z) ∈ Z(C)).\n(2.) The Quadratic Case (deg(f) = 2): Up to coordinate transformations (and ruling\nout degenerate cases), we see that this is essentially the case where the equation\nf(X,Y ) = X ·Y −1. In this case, an explicit catalogue is given by the exponential\nmap:\nexp : C → Z(C) = C×\n(In fact, the map may be defined intrinsically, without using coordinate transfor-\nmations to render the defining equation in the “standard form” X · Y = 1.)\n(3.) The Cubic Case (deg(f) = 3): Up to adding the point(s) at infinity, this is\nessentially the case where we are dealing with an elliptic curve E. In this case,\nas well, we have a natural exponential map:\nexpE : TE → E(C)\n(where TE is the one-dimensional complex vector space given by the tangent space\nto some fixed point – “the origin” – of E). This map allows us to think of E as\nbeing of the form “C\/Λ” (where Λ ∼= Z2 is a lattice in C).\n2\n(4.) Higher Degree: If deg(f) ≥ 4, or we wish to consider lower degree cases with lots\nof points removed, then we are led naturally to the following notion:\nA hyperbolic curve Z is a smooth, proper connected algebraic curve of genus\ng with r points removed, where we assume that 2g − 2 + r > 0.\nAccording to the uniformization theorem of Köbe, hyperbolic curves may be uni-\nformized by the upper half-plane H\ndef\n= {z ∈ C | Im(z) > 0}, i.e., we have a\nsurjective (holomorphic) covering map:\nH → Z(C)\nwhich allows us to think of Z(C) as being of the form H\/Γ, where Γ is some\ndiscrete group acting on H.\nKOEBE.EPS\n(B.) “Intrinsic” Hodge Theories\nIn the preceding section, we posed the problem of explicitly cataloguing the rational\npoints of a variety (over C). By looking at various examples, we saw that this problem\nmay also be worded – in perhaps more familiar terms – as the problem of finding natural\nuniformizations of varieties. In the present section, we would like to further refine our\nunderstanding of the problem of finding natural uniformizations\/explicit catalogues of\nrational points by rewording this problem in terms of the language of “Hodge theory.”\nFirst, let us discuss what we mean in general by the notion of a “Hodge theory.” By\na Hodge theory, we shall mean an equivalence of the following form:\n⎛\n⎜⎜⎜⎝\nalgebraic\ngeometry\n(e.g.,\nrational\npoints)\n⎞\n⎟⎟⎟⎠ ⇐⇒\n⎛\n⎜⎜⎜⎜⎜⎝\nOver C: topology +\ndifferential geometry\nOver the p-adics:\npro-p étale\ntopology + Galois action\n⎞\n⎟⎟⎟⎟⎟⎠\nThe most familiar example of such an equivalence is the Hodge theory of cohomolo-\ngies. Over C, this amounts to “classical Hodge theory,” i.e., the well-known isomorphism\nbetween the de Rham cohomology of an algebraic variety (which is well-known to be\nan algebro-geometric invariant of the variety) and the singular cohomology of the variety\n(which is a topological invariant). More recently, the p-adic Hodge theory of cohomolo-\ngies has been developed by Fontaine et al. (cf., e.g., [Falt2,3]). This theory asserts an\n3\nequivalence between the (algebraic) de Rham cohomology of an algebraic variety over a\nfinite extension K of Qp and the p-adic étale cohomology of the variety, equipped with its\nnatural Galois action (i.e., action of Gal(K)).\nThis “Hodge theory of cohomologies” is the most basic example of a “Hodge theory”\nas defined above. In the present manuscript, however, we would like to consider a different\nkind of Hodge theory which we shall call an intrinsic Hodge theory, or IHT, for short.\nBy an intrinsic Hodge theory, we mean a Hodge theory – i.e., an equivalence of the form\ndiscussed above – in which the invariant which appears on the “algebraic geometry” side\nis the “variety itself.”\nThere are several different ways to interpret the phrase “the variety itself.” In the\npresent manuscript, we shall consider the following two interpretations:\n(1.) The Physical Interpretation: In this interpretation, one takes the phrase the\n“variety itself” to mean the “rational points of the variety.”\n(2.) The Modular Interpretation: In this interpretation, one takes the phrase the\n“variety itself” to mean the “moduli of the variety.”\nNote that (it is a tautology of terminology that) a physical IHT essentially amounts to\nsome sort of explicit description of the rational points of the variety in terms of topology\nand geometry. Thus, one may summarize the above discussion as follows:\nphysical IHT = uniformization of the variety\nmodular IHT = uniformization of the moduli space of deformations of the variety\nBefore concluding this section, we make some remarks on the relationship between\nthe “IHT’s” that we wish to discuss here and the more well-known “Hodge theories of\ncohomologies.” First of all, although in general, IHT’s are not the same as Hodge theories\nof cohomologies, typically in proving theorems which realize IHT’s, the technical tools of\nthe corresponding Hodge theory of cohomologies (e.g., in the p-adic case, the techniques\nof so-called “p-adic Hodge theory” as in [Falt2]) play an important role. Secondly, in the\ncase of Gm = V (X · Y − 1) (i.e., Example (2) in §1, (A.)), as well as in the case of elliptic\ncurves (i.e., Example (3) in §1, (A.)), it just so happens that the first cohomology module\nof the curve “is” the curve itself, i.e., in more sophisticated language, in these cases the\ncurve in question is a 1-motive. Thus, in these cases, it turns out that the notions of IHT\nand Hodge theory of H1 coincide. In particular, in these cases, the well-known Hodge\ntheory of H1 already provides a uniformization of the curve. Note that this differs quite\nsubstantially from the case of higher genus (Example (4) in §1, (A.)).\n4\n(C.) Completion at Arithmetic Primes\nSo far in our discussion, we have ignored the important issue of what sort of ground\nfield should be considered in our discussion of uniformizations\/explicit catalogues of ratio-\nnal points\/IHT’s. In the examples of §1, (A.), we considered the situation over the complex\nnumber field, since this is the most elementary and well-known example of a ground field\nover which IHT’s may be realized.\nOf course, ideally, one would like to realize IHT’s over any field, for instance, over a\nnumber field (i.e., a finite extension of Q) – a case for which the problem of determining\nthe set of rational points explicitly is of prime interest. Unfortunately, however, typically,\nin order to realize an IHT (or, indeed, any sort of “Hodge theory”), one must work over a\nbase which is complete with respect to some sort of “arithmetic prime.” The three main\nexamples of this sort of base are the following:\n(i.) the complex number field C (this also covers the case of R by working with\nobjects over C equipped with an action of complex conjugation, i.e., of Gal(C\/R))\n(ii.) a p-adic field K (i.e., a finite extension of Qp)\n(iii.) power series over Z – typically arising as the completion of some sort of mod-\nuli space at a Z-valued point corresponding to a degenerate object parametrized\nby the moduli space.\nIndeed, all completions of number fields fall under cases (i.) and (ii.). Thus, if one is\nultimately interested in rational points of number fields, IHT’s over bases as in (i.) and\n(ii.) are of natural interest. Also, since the coefficients of the powers series appearing in\n(iii.) are integers, case (iii.) is also of substantial arithmetic interest.\nIn the following discussion, the following principle will serve as an important guide:\nGuiding Principle: For every type of arithmetic prime (i.e., cases (i.), (ii.), and\n(iii.) above), one expects that there should be a canonical uniformization theory\nat that prime.\nIn general, however, one does not expect that the canonical uniformization theories at\ndifferent primes should be compatible with one another. We will return to this point in\nmore detail later.\nIn terms of types of varieties, the main cases in which one has well-developed physical\nand modular IHT’s are the following:\n(1.) abelian varieties\n(2.) hyperbolic curves.\nThe physical and modular IHT’s in these cases may be roughly summarized in the following\ncharts:\n5\n(1.) Abelian Varieties\nC p-adic Degenerate Object\nexponential map\nof abelian varieties\/\nSiegel upper\nhalf-plane\nuniformization\nTate’s\ntheorem\/\nSerre-\nTate\ntheory\nSchottky\nuniformizations\nof Tate\/\nMumford\n(2.) Hyperbolic Curves\nC p-adic Degenerate Object\nFuchsian\nuniformization\/\nuniformizations of\nTeichmüller\nand Bers\n§2 (anabelian\nconjecture)\/\n§3 (p-adic\nTeichmüller\ntheory)\nformal\nalgebraic\nSchottky\nuniformization\nof Mumford\nOf these two examples, undoubtedly the example of abelian varieties is the more\nwell-known. Over C, the exponential map of an abelian variety gives a uniformization of\nthe abelian variety by a complex linear space. This generalizes Example (3) of §1, (A.).\nMoreover, by using the periods that one obtains from this uniformization, one obtains\na uniformization of the moduli space of abelian varieties by the Siegel upper half-plane.\nThus, we see both the physical and modular aspects of the IHT of abelian varieties in\nevidence.\nIn the p-adic case, the IHT of an abelian variety essentially amounts to the p-adic\nHodge theory of H1 of the abelian variety. Although the p-adic Hodge theory of H1\nof an abelian variety has many different aspects, most of these may be traced to the\nfundamental paper of Tate ([Tate]) in the late 1960’s. In this paper, the main theorem\n(“Tate’s theorem” in the chart) states that homomorphisms between formal groups (e.g.,\nthe formal groups arising from abelian varieties) over p-adic fields are essentially equivalent\nto homomorphisms between the corresponding Tate modules. In some sense, this result is\nthe analogue for abelian varieties of the main theorem (Theorem 1) of §2 below, and may\nbe regarded as a sort of physical IHT for abelian varieties. On the other hand, the modular\naspect of the IHT of abelian varieties may be seen most easily in Serre-Tate theory, which\ngives rise to p-adic parameters on the moduli space of abelian varieties that are very much\nanalogous to the Siegel upper half-plane uniformization in the complex case.\nFinally, in a neighborhood of a point in the moduli space corresponding to a degen-\nerating abelian variety, one has the theory of Tate curves, generalized by Mumford and\nFaltings\/Chai ([FC]). Moreover, it turns out that in the case of abelian varieties, the com-\nplex, p-adic, and degenerating object theories are all compatible with one another. For\ninstance, if one specializes the uniformizing parameters that one obtains on the moduli\n6\nspace in a neighborhood of a point corresponding to a degenerating abelian variety to a p-\nadic (respectively, complex) base, one obtains parameters compatible with the Serre-Tate\nparameters (respectively, the Siegel upper half-plane uniformization).\nNext, we consider the case of hyperbolic curves. Over C, the physical aspect of the\nIHT of hyperbolic curves (cf. Example (4) of §1, (A.)) essentially amounts to the Fuchsian\nuniformization. Then just as the exponential map uniformization of an abelian variety\n“induces” the Siegel upper half-plane uniformization of the moduli space of abelian vari-\neties, the Fuchsian uniformization of a hyperbolic curve “induces” the Bers uniformization\nof the moduli space of hyperbolic curves (cf. §3, (A.) below, as well as the Introduction of\n[Mzk4]).\nIn a neighborhood of a point in the moduli space corresponding to a totally degenerate\n(proper) hyperbolic curve, one has the theory of [Mumf]. Note, however, that this theory\nis not compatible with the theory of the Fuchsian uniformization in the following sense:\nIf one specializes the (Z-coefficient) power series in the base ring to some C-valued point\nin a neighborhood of a point in the moduli space corresponding to a totally degenerate\ncurve, the resulting uniformization over C that one obtains is the so-called Schottky uni-\nformization of the curve. This uniformization is completely different from the Fuchsian\nuniformization.\nFinally, we come to the p-adic case. It seems that the IHT of p-adic hyperbolic curves\nhas not been studied extensively until relatively recently ([Mzk1-5]). Thus,\nIt is the goal of this manuscript to report on recent developments concerning the\nintrinsic Hodge theory (IHT) of p-adic hyperbolic curves.\nThe physical aspect, which concerns a theorem (Theorem 1) that gives a strong solution to\nGrothendieck’s anabelian conjecture, will be the topic of §2. The modular aspect, which\nconcerns a theory – which we call p-adic Teichmüller theory – which may be regarded as\neither the hyperbolic curve analogue of Serre-Tate theory or the p-adic analogue of the\ntheory of the Fuchsian and Bers uniformizations, will be discussed in §3. We remark here\nthat this p-adic Teichmüller theory is not compatible with the specialization of the theory\nof [Mumf] to the p-adic case. This may disappoint some readers, but is, in fact, natural\nin view of the fact that even over C, the specialization of the theory of [Mumf] to the\ncomplex numbers is not compatible (as remarked above) with the theory of the Fuchsian\nuniformization. Moreover, it is in line with the general Guiding Principle discussed above\nthat to each sort of arithmetic prime there should correspond a natural uniformization\ntheory of hyperbolic curves. Thus, it seems to the author that it is meaningless to argue as\nto whether it is the specialization of the [Mumf] to the p-adic case or the theory of [Mzk1-4]\nwhich is the “true” p-adic analogue of the Fuchsian uniformization. That is to say, it seems\nmore natural to the author to regard the theory of [Mumf] as the “true” analogue of the\nFuchsian uniformization at the “degenerating object prime,” and the theory of [Mzk1-4]\nas the “true” analogue of the Fuchsian uniformization at “the prime p.”\n7\n§2. The Physical Aspect: Embedding by Automorphic Forms\n(A.) The Complex Case\n(B.) The Arithmetic Fundamental Group\n(C.) The Main Theorem\n(D.) Comparison with the Case of Abelian Varieties\n§3. The Modular Aspect: Canonical Frobenius Actions\n(A.) The Complex Case\n(B.) Teichmüller Theory in Characteristic p\n(C.) Canonical p-adic Liftings\n8\nBibliography\n[BK] Bloch, S. and Kato, K., L-Functions and Tamagawa Numbers of Motives in The\nGrothendieck Festschrift, Volume I, Birkhäuser (1990), pp. 333-400.\n[Falt1] Faltings, G., Endlichkeitssätze für Abelschen Varietäten über Zahlkörpern, Inv. Math.\n73 (1983), pp. 349-366.\n[Falt2] Faltings, G., p-adic Hodge Theory, Journal of the Amer. Math. Soc. 1, No. 1 (1988),\npp. 255-299.\n[Falt3] G. Faltings, Crystalline Cohomology and p-adic Galois Representations, JAMI Con-\nference, Johns Hopkins Univ. Press (1990), pp. 25-79.\n[FC] G. Faltings and C.-L. Chai, Degenerations of Abelian Varieties, Springer Verlag (1990).\n[Ihara] Ihara, Y., On the Differentials Associated to Congruence Relations and the Schwarzian\nEquations Defining Uniformizations, Jour. Fac. Sci. Univ. Tokyo, Sect. IA Math. 21\n(1974), pp. 309-332.\n[LS] Lochak, P. and Schneps, L., Geometric Galois Actions: 1. Around Grothendieck’s\nEsquisse d’un Programme, London Math. Soc. Lect. Note Ser. 242, Cambridge Univ.\nPress, 1997.\n[Mumf] D. Mumford, An Analytic Construction of Degenerating Curves over Complete Local\nRings, Compositio Math. 24, pp. 129-174 (1972).\n[Mzk1] Mochizuki, S., A Theory of Ordinary p-adic Curves, RIMS Preprint 1033 (September\n1995); Publ. of RIMS 32, No. 6 (1996), pp. 957-1151.\n[Mzk2] Mochizuki, S., The Generalized Ordinary Moduli of p-adic Hyperbolic Curves, RIMS\nPreprint 1051 (December 1995); 281 pp.\n[Mzk3] Mochizuki, S., Combinatorialization of p-adic Teichmüller Theory, RIMS Preprint\n1076 (April 1996); 32 pp.\n[Mzk4] Mochizuki, S., Foundations of p-adic Teichmüller Theory, to appear (as a book) from\nInternational Press.\n[Mzk5] Mochizuki, S., The Local Pro-p Anabelian Geometry of Curves, RIMS Preprint 1097\n(August 1996); 84 pp.\n[NTM] Nakamura, H., Tamagawa, A., and Mochizuki, S., The Grothendieck Conjecture on\nthe Fundamental Groups of Algebraic Curves, Sugaku 50, No.2 (1998), pp. 1-17 (in\nJapanese).\n[Pop] Pop, F., On Grothendieck’s conjecture of birational anabelian geometry II, Preprint\n(1995).\n[SGA 1] A. Grothendieck et al., Revêtements étales et groupe fondamental, Lecture Notes in\nMathematics 224, Springer, 1971.\n9\n[Tama1] Tamagawa, A., The Grothendieck conjecture for affine curves, Compositio Math. 109,\nNo. 2 (1997), pp. 135-194.\n[Tama2] Tamagawa, A., On the fundamental groups of curves over algebraically closed fields\nof characteristic > 0, RIMS Preprint 1182 (January 1998).\n[Tate] J. Tate, p-Divisible Groups, Proceedings of a Conference on Local Fields, Driebergen,\nSpringer Verlag (1967), pp. 158-183.\n10\n"}
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{"id":"OZ4e_116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/The%20Algebraic%20and%20Anabelian%20Geometry%20of%20Configuration%20Spaces.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":121,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"The%20Algebraic%20and%20Anabelian%20Geometry%20of%20Configuration%20Spaces","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["config-space.dvi"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 12:13:03 UTC 2011","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Jun 03 00:00:00 UTC 2003"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":3220078813043962462,"exact_signature_unique_b":true,"fuzzy_signature_l":-2184408631634615443,"fuzzy_signature_unique_b":true,"wordcount_i":20147,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:55 UTC 2022","fresh_date_dt":"Thu Sep 23 19:08:19 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":385947,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086041997508608,"text_t":"THE ALGEBRAIC AND ANABELIAN\nGEOMETRY OF CONFIGURATION SPACES\nShinichi Mochizuki and Akio Tamagawa\nJuly 2007\n\u0000\u0002\u0003\u0004\u0005\u0006\u0007\u0004\b In this paper, we study the pro-Σ fundamental groups of configuration\nspaces, where Σ is either the set of all prime numbers or a set consisting of a single\nprime number. In particular, we show, via two somewhat distinct approaches, that, in\nmany cases, the “fiber subgroups” of such fundamental groups arising from the various\nnatural projections of a configuration space to lower-dimensional configuration spaces\nmay be characterized group-theoretically.\nContents:\n§0. Notations and Conventions\n§1. Surface Groups\n§2. Configuration Space Groups\n§3. Direct Products of Profinite Groups\n§4. Product-theoretic Quotients\n§5. Divisors and Units on Coverings of Configuration Spaces\n§6. Nearly Abelian Groups\n§7. A Discrete Analogue\nIntroduction\nLet n ≥ 1 be an integer; X a hyperbolic curve of type (g, r) [where 2g−2+r > 0]\nover an algebraically closed field k of characteristic 0. Denote by\nXn ⊆ Pn\nthe n-th configuration space associated to X , i.e., the open subscheme of the direct\nproduct Pn of n copies of X obtained by removing the various diagonals from Pn\n2000 Mathematical Subject Classification. Primary 14H30; Secondary 14H10.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\n[cf. Definition 2.1, (i)]. By omitting the factors corresponding to various subsets of\nthe set of n copies of X , we obtain various natural projection morphisms\nXn → Xm\nfor nonnegative integers m ≤ n [cf. Definition 2.1, (ii)]. Next, let ΣC be either\nthe set of all prime numbers or a set consisting of a single prime number. Write\nC for the class of all finite groups of order a product of primes ∈ ΣC . Then by\nconsidering the maximal pro-C quotient of the étale fundamental group, which we\ndenote by “πC1 (−)”, we obtain various natural surjections\nπC1 (Xn) \u0002 πC1 (Xm)\narising from the natural projection morphisms considered above. We shall refer to\nthe kernel of such a surjection πC1 (Xn) \u0002 πC1 (Xm) as a fiber subgroup of πC1 (Xn)\nof length n − m and co-length m [cf. Definition 2.3, (iii)]. Also, we shall refer to\na closed subgroup of πC1 (Xn) that arises as the inverse image of a closed subgroup\nof πC1 (Pn) via the natural surjection π\nC\n1 (Xn) \u0002 πC1 (Pn) [induced by the inclusion\nXn ↪→ Pn] as product-theoretic [cf. Definition 2.3, (ii)].\nThe present paper is concerned with the issue of the group-theoretic character-\nization of these fiber subgroups. Our main results [cf. Corollaries 4.8, 6.3] may be\nsummarized as follows:\n(i) Suppose that g ≥ 2. Let H ⊆ πC1 (Xn) be a product-theoretic open sub-\ngroup. Then the subgroups H\n⋂\nF of H — where F ranges over the various\nfiber subgroups of πC1 (Xn) — may be characterized group-theoretically [cf.\nCorollary 4.8].\n(ii) Suppose that (g, r) is not equal to (0, 3) or (1, 1). Then the fiber subgroups\nof πC1 (Xn) may be characterized group-theoretically [cf. Corollary 6.3].\nThe proof of (i) is obtained as a consequence of the following result [cf. Theorem\n4.7]:\n(iii) In the notation of (i), every normal closed subgroup J ⊆ H such that the\nquotient group H\/J is abelian and torsion-free is, in fact, product-theoretic.\nThe proof of (iii) is based on a slightly complicated computation involving Chern\nclasses [cf. §4], together with the well-known fact that the action of the Galois\ngroup of a finite Galois covering of a curve of genus ≥ 2 on the Tate module of the\nJacobian of the covering curve contains the regular representation [cf. Proposition\n1.3]. On the other hand, the proof of (ii), due to the second author, makes essential\nuse to the notion of a “nearly abelian group”, i.e., a profinite group G which admits\na normal closed subgroup N ⊆ G which is topologically normally generated by a\nsingle element ∈ G such that G\/N contains an open abelian subgroup [cf. Definition\n6.1]. It is worth noting that at the time of writing, we are unable to prove either\nan analogue of (i) for g < 2 or an analogue of (ii) when (g, r) is equal to (0, 3) or\n(1, 1).\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 3\nThe original proof of (i) [due to the first author] given in §4 may be regarded as\na consequence of various explicit group-theoretic manifestations of certain algebro-\ngeometric properties. This proof of (i) motivated the second author to develop a\nmore direct approach to understanding these essentially purely algebro-geometric\nproperties. This approach, which is exposed in §5, allows one to prove a stronger\nversion [cf. Theorem 5.6], in the case of proper hyperbolic curves, of Theorem\n4.7 and, moreover, implies certain interesting consequences concerning the non-\nexistence of units on finite étale coverings of a sufficiently generic hyperbolic curve\n[cf. Corollary 5.7].\nThe contents of the present paper may be summarized as follows: Basic well-\nknown facts concerning the profinite fundamental groups of hyperbolic curves and\nconfiguration spaces, including a certain mild generalization of a theorem of Lubotzky-\nMelnikov-van den Dries, are reviewed in §1, §2, respectively. In §3, we discuss the\ngroup-theoreticity of direct product decompositions of profinite groups. In §4, §6,\nwe present the proofs, via somewhat different techniques, of the main results (i),\n(ii) discussed above. In §5, we discuss the algebraic geometry of divisors and units\non configuration spaces, a theory which yields an alternate approach to the theory\nof §4, in the case of proper hyperbolic curves. Finally, in §7, we observe that these\nresults (i), (ii) imply a certain discrete analogue [cf. Corollary 7.4] of (i), (ii).\nSection 0: Notations and Conventions\nNumbers:\nThe notation Q will be used to denote the field of rational numbers. The nota-\ntion Z ⊆ Q will be used to denote the set, group, or ring of rational integers. The\nnotation N ⊆ Z will be used to denote the set or [additive] monoid of nonnegative\nintegers. If l is a prime number, then the notation Ql (respectively, Zl) will be used\nto denote the l-adic completion of Q (respectively, Z). The [topological] field of\ncomplex numbers will be denoted C.\nTopological Groups:\nLet G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us\nwrite\nZG(H)\ndef\n= {g ∈ G | g · h = h · g, ∀ h ∈ H}\nfor the centralizer of H in G. Also, we shall write Z(G)\ndef\n= ZG(G) for the center of\nG.\nWe shall say that a profinite group G is slim if for every open subgroup H ⊆ G,\nthe centralizer ZG(H) is trivial. Note that every finite normal closed subgroup\nN ⊆ G of a slim profinite group G is trivial. [Indeed, this follows by observing that\nfor any normal open subgroup H ⊆ G such that N ⋂H = {1}, consideration of the\n4 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\ninclusion N ↪→ G\/H reveals that the conjugation action of H on N is trivial, i.e.,\nthat N ⊆ ZG(H) = {1}.]\nWe shall write Gab for the abelianization of G, i.e., the quotient of G by the\nclosure of the commutator subgroup of G. We shall denote the group of automor-\nphisms of G by Aut(G). Conjugation by elements of G determines a homomorphism\nG → Aut(G) whose image consists of the inner automorphisms of G. We shall de-\nnote by Out(G) the quotient of Aut(G) by the [normal] subgroup consisting of\nthe inner automorphisms. In particular, if G is center-free, then we have an exact\nsequence 1 → G → Aut(G) → Out(G) → 1.\nCurves:\nSuppose that g ≥ 0 is an integer. Then if S is a scheme, a family of curves of\ngenus g\nX → S\nis defined to be a smooth, proper, geometrically connected morphism of schemes\nX → S whose geometric fibers are curves of genus g.\nSuppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denote\nthe moduli stack of r-pointed stable curves of genus g over Z (where we assume\nthe points to be ordered) by Mg,r [cf. [DM], [Knud] for an exposition of the\ntheory of such curves]. The open substack Mg,r ⊆ Mg,r of smooth curves will\nbe referred to as the moduli stack of smooth r-pointed stable curves of genus g or,\nalternatively, as the moduli stack of hyperbolic curves of type (g, r). The divisor\nat infinity Mg,r\\Mg,r of Mg,r is a divisor with normal crossings on the Z-smooth\nalgebraic stack Mg,r, hence determines a log structure on Mg,r; denote the resulting\nlog stack by Mlogg,r. For any integer r′ > r, the operation of “forgetting the last\nr′ − r points” determines a [1-]morphism of log algebraic stacks\nMlogg,r′ → M\nlog\ng,r\nwhich factors as a composite of structure morphisms of various tautological log\nstable curves [cf. [Knud]], hence is log smooth.\nA family of hyperbolic curves of type (g, r)\nX → S\nis defined to be a morphism which factors X ↪→ Y → S as the composite of an\nopen immersion X ↪→ Y onto the complement Y \\D of a relative divisor D ⊆ Y\nwhich is finite étale over S of relative degree r, and a family Y → S of curves of\ngenus g. One checks easily that, if S is normal, then the pair (Y, D) is unique up to\ncanonical isomorphism. We shall refer to Y (respectively, D) as the compactification\n(respectively, divisor of cusps) of X . A family of hyperbolic curves X → S is\ndefined to be a morphism X → S such that the restriction of this morphism to\neach connected component of S is a family of hyperbolic curves of type (g, r) for\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 5\nsome integers g, r as above. A family of hyperbolic curve of type (0, 3) will be\nreferred to as a tripod.\nSection 1: Surface Groups\nIn the present §1, we discuss various well-known preliminary facts concerning\nthe sorts of profinite groups that arise from étale fundamental groups of hyperbolic\ncurves.\nDefinition 1.1. Let C be a family of finite groups containing the trivial group;\nΣ a set of prime numbers.\n(i) We shall refer to a finite group as a Σ-group if every prime dividing its order\nbelongs to Σ. We shall refer to a finite group belonging to C as a C-group and to\na profinite group every finite quotient of which is a C-group as a pro-C group. We\nshall refer to C as a full formation [cf. [FJ], p. 343] if it is closed under taking\nquotients, subgroups, and extensions.\n(ii) Suppose that C is a full formation; write ΣC for the set of primes p such\nthat Z\/pZ is a C-group and Ẑ \u0002 ẐC for the maximal pro-C quotient of Ẑ. Then\nwe shall say that the formation C is nontrivial if there exists a nontrivial C-group\n[or, equivalently, if ΣC is nonempty]. We shall say that the formation C is primary\nif ΣC is of cardinality one. We shall say that the formation C is solvable if every\nC-group is solvable. We shall say that the formation C is total if every finite group\nis a C-group. We shall say that C is a PT-formation if it is either primary or total.\nWe shall say that C is invertible on a scheme S if every prime of ΣC is invertible\non S.\n(iii) Suppose that C is a full formation; let G be a profinite group. If G admits\nan open subgroup H which is abelian, then we shall say that G is almost abelian.\nIf G admits an open subgroup H which is pro-C, then we shall say that G is almost\npro-C. We shall refer to a quotient G \u0002 Q as almost pro-C-maximal if for some\nopen subgroup H ⊆ G with maximal pro-C quotient [cf. [FJ], p. 344] H \u0002 P ,\nwe have Ker(G \u0002 Q) = Ker(H \u0002 P ). [Thus, any almost pro-C-maximal quotient\nof G is almost pro-C.] If G is topologically finitely generated, and, moreover, the\nabelianization Jab of every open subgroup J ⊆ G is torsion-free, then we shall say\nthat G is strongly torsion-free.\nRemark 1.1.1. The notion of a full formation is a special case of the notion of a\nMelnikov formation [cf. [FJ], p. 343]. In the present paper, [partly for the sake of\nsimplicity] we restrict ourselves to full formations.\nRemark 1.1.2. Let C be a full formation. Then [it follows immediately from\nthe definitions that] a solvable finite group is a ΣC-group [cf. Definition 1.1, (ii)]\n6 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nif and only if it is a C-group. In particular, if C is solvable, then it is completely\ndetermined by the set of primes ΣC .\nRemark 1.1.3. Recall that every finite group whose order is a prime power is\nnilpotent, hence, in particular, solvable. Thus, [cf. Remark 1.1.2] a primary full\nformation C is completely determined by the unique prime number ∈ ΣC .\nRemark 1.1.4. One verifies immediately that in the various definitions in Def-\ninition 1.1, (iii), of terms of the form “almost P”, where “P” is some property,\nan equivalent definition is obtained if one requires the open subgroup “H” to be\nnormal.\nRemark 1.1.5. If G is a strongly torsion-free profinite group, then one verifies\nimmediately [by considering abelianizations of open subgroups of G] that G is\ntorsion-free in the usual sense, i.e., that G has no nontrivial elements of finite\norder.\nDefinition 1.2. Let C be a full formation. We shall say that a profinite\ngroup is a [pro-C] surface group (respectively, an almost pro-C-surface group) if\nit is isomorphic to the maximal pro-C quotient (respectively, to some almost pro-\nC-maximal quotient) of the étale fundamental group of a hyperbolic curve [cf. §0]\nover an algebraically closed field of characteristic zero [or, equivalently, the profinite\ncompletion of the topological fundamental group of a hyperbolic Riemann surface\nof finite type]. We shall refer to an almost pro-C-surface group as open (respectively,\nclosed) if it admits (respectively, does not admit) a pro-C free [cf. [FJ], p. 345] open\nsubgroup.\nRemark 1.2.1. Thus, in the notation of Definition 1.2, every pro-C surface group\nis an almost pro-C-surface group. On the other hand, if C is not total, then one\nverifies immediately that there exist almost pro-C-surface groups which are not pro-\nC surface groups. Nevertheless, every almost pro-C-surface group admits a normal\nopen subgroup which is a pro-C surface group.\nRemark 1.2.2. We recall that if Π is a pro-C surface group arising from a\nhyperbolic curve [cf. Definition 1.2] of type (g, r), then Π is topologically generated\nby 2g + r generators subject to a single [well-known!] relation, and Πab [cf. §0] is a\nfree abelian pro-C group of rank 2g − 1 + r (if r > 0), 2g (if r = 0). In particular,\n[since every open subgroup of Π is again a pro-C surface group, it follows that] Π is\nstrongly torsion-free. Moreover, for any l ∈ ΣC , the l-cohomological dimension of Π\nis equal to 1 (if r > 0), 2 (if r = 0); dimQl(H\n2(Π, Ql)) = dimFl(H\n2(Π, Fl)) is equal\nto 0 (if r > 0), 1 (if r = 0). In particular, the quantity\nχ(Π) =\n2∑\ni=0\n(−1)i · dimQl(H i(Π, Ql)) =\n2∑\ni=0\n(−1)i · dimFl(H i(Π, Fl)) = 2 − 2g − r\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 7\nis a group-theoretic invariant of Π which [as is well-known] satisfies the property\nthat\nχ(Π1) = [Π : Π1] · χ(Π)\nfor any open subgroup Π1 ⊆ Π. Finally, we recall that this formula admits a\nrepresentation-theoretic generalization, which will play a crucial role in §4 below,\nin the form of the following elementary consequence:\nProposition 1.3. (Inclusion of the Regular Representation) Let Y → X\nbe a finite [possibly ramified] Galois covering of smooth proper hyperbolic\ncurves over an algebraically closed field k of characteristic prime to the order of\nG\ndef\n= Gal(Y\/X); l a prime number that is invertible in k. Write V for the G-module\ndetermined by the first étale cohomology module H1ét(Y, Ql). Then the G-module V\ncontains the regular representation of G as a direct summand.\nProof. Indeed, this follows immediately from the computation of the Galois module\nV in [Milne], p. 187, Corollary 2.8 [cf. also [Milne], p. 187, Remark 2.9], in light\nof our assumption that X is proper hyperbolic, hence of genus ≥ 2. ©\nProposition 1.4. (Slimness) Let C be a nontrivial full formation. Then every\nalmost pro-C-surface group Π is slim.\nProof. Indeed, this follows immediately by considering the conjugation action of\nΠ\/N on Nab ⊗ Zl, where l ∈ ΣC , for sufficiently small normal open subgroups\nN ⊆ Π [cf. Remark 1.2.1]. That is to say, in light of the interpretation of a certain\nquotient of Nab ⊗ Zl as the Tate module arising from the l-power torsion points\nof the Jacobian of the compactification of the covering determined by N of any\nhyperbolic curve that gives rise to Π [cf. the proof of [Mzk3], Lemma 1.3.1], it\nfollows that this conjugation action is faithful. Another [earlier] approach to the\nslimness of surface groups may be found in [Naka], Corollary 1.3.4. ©\nRemark 1.4.1. The property involving the regular representation discussed in\nProposition 1.3 may be regarded as a stronger version [in the case of coverings of\ncurves of genus ≥ 2] of the faithfulness of the action of Π\/N on [a certain quotient\nof] Nab ⊗ Zl that was applied in the proof of Proposition 1.4, hence, in particular,\nas a stronger version of the slimness of surface groups.\nNext, we give a mild generalization to arbitrary surface groups of a well-known\nresult for free pro-C groups due to Lubotzky-Melnikov-van den Dries. In particular,\nthe argument given below in the proof of Theorem 1.5 may be regarded as a short\nelementary proof of [a certain portion of] the theorem of Lubotzky-Melnikov-van den\nDries, as exposed in [FJ], Proposition 24.10.3; [FJ], Proposition 24.10.4, (a).\nTheorem 1.5. (Normal Closed Subgroups of Surface Groups) Let C be\na full formation; Π an almost pro-C-surface group; N ⊆ Π a topologically\n8 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nfinitely generated normal closed subgroup. Then N is either trivial or of finite\nindex.\nProof. First, we observe that we may assume without loss of generality that C\nis nontrivial. Since Π is slim, hence does not contain any nontrivial finite normal\nclosed subgroups [cf. §0], it follows that we may always replace Π by an open\nsubgroup of Π. In particular, [cf. Remark 1.2.1] we may assume, without loss of\ngenerality, that Π is a pro-C surface group. Now suppose that N is nontrivial and\nof infinite index. Then there exists an l ∈ ΣC such that N contains a nontrivial\nsubgroup A ⊆ N which is a quotient of Zl. In particular, there exists a normal\nopen subgroup Π1 ⊆ Π such that the image of A in Π\/Π1 is nontrivial. Now set\nΠA\ndef\n= Π1 · A ⊆ Π, NA def= N\n⋂\nΠA [so ΠA, NA are open subgroups of Π, N ,\nrespectively]. Then NA is a topologically finitely generated normal closed subgroup\nof infinite index of ΠA such that A ⊆ NA surjects onto the [nontrivial, abelian!]\nimage of ΠA in Π\/Π1. In particular, by replacing N ⊆ Π by NA ⊆ ΠA, we may\nassume without loss of generality that the image of N in Πab is nontrivial.\nSince Π is topologically finitely generated, there exists a descending sequence\nof normal open subgroups\n. . . ⊆ Hn ⊆ . . . ⊆ Π\n[where n ranges over the positive integers] of Π which is, moreover, exhaustive,\ni.e.,\n⋂\nn Hn = {1}. Thus, if we set Nn\ndef\n= Hn · N [for n ≥ 1], then [one verifies\nimmediately that] we obtain a descending sequence of normal open subgroups\n. . . ⊆ Nn ⊆ . . . ⊆ Π\n[where n ranges over the positive integers] of Π such that\n⋂\nn Nn = N [cf. the fact\nthat N is closed!]. Since N is of infinite index in Π, it follows that [Π : Nn] → ∞ as\nn → ∞, hence [cf. Remark 1.2.2] that |χ(Nn)| → ∞ as n → ∞. In particular, there\nexists an n such that the rank [as a free abelian pro-C group] of Nabn is ≥ s + 2,\nwhere we write s for any positive integer such that there exist s elements of N that\ntopologically generate N . Since, moreover, the image of N in Πab, hence a fortiori\nin Nabn is nontrivial, it follows that there exists, for some l ∈ ΣC, a nontrivial\nhomomorphism Zl → Nabn that factors through N . Now write\nNn \u0002 Π∗\nfor the maximal pro-l quotient of Nn [so Π\n∗ is a pro-l surface group], N∗ ⊆ Π∗\nfor the image of N in Π∗. Thus, N∗ ⊆ Π∗ is a topologically finitely generated\nnormal closed subgroup whose image in [the free Zl-module of finite rank] (Π\n∗)ab\nis a nontrivial Zl-submodule M ⊆ (Π∗)ab whose rank is ≤ s, hence ≤ the rank\nof (Π∗)ab minus 2. In particular, there exists an element x ∈ Π∗ such that if we\ndenote by F ∗ ⊆ Π∗ the [necessarily topologically finitely generated!] closed subgroup\ntopologically generated by N∗ and x, then we obtain inclusions of closed subgroups\nN∗ ⊆ F ∗ ⊆ Π∗\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 9\nsuch that N∗ is of infinite index in F ∗, and F ∗ is of infinite index in Π∗ [as may\nbe seen by considering the ranks of the images of these subgroups in Πab].\nNow observe that for any two open subgroups J2 ⊆ J1 ⊆ Π∗, the induced\nmorphism H2(J1, Zl) → H2(J2, Zl) maps a generator of H2(J1, Zl) to [J1 : J2]\ntimes a generator H2(J2, Zl) [where we recall that H\n2(J1, Zl), H\n2(J2, Zl) are either\nboth zero or both isomorphic to Zl, depending on whether Π is an open or a\nclosed surface group]. [Indeed, this follows immediately by thinking about degrees\nof coverings of proper hyperbolic curves! We refer to Remark 4.1.1; Lemma 4.2,\n(i) [and its proof], below, for more details on this well-known circle of ideas.] In\nparticular, since F ∗ is a subgroup of infinite index in Π∗, it follows immediately\n[by considering open subgroups J ⊆ Π∗ containing F ∗] that F ∗ is a pro-l group\nwhose [l-]cohomological dimension is ≤ 1. Thus, by [RZ], Theorem 7.7.4, F ∗ is\na topologically finitely generated free pro-l group [i.e., in particular, a pro-l open\nsurface group], and N∗ ⊆ F ∗ is a nontrivial topologically finitely generated closed\nnormal subgroup of infinite index. Put another way, it suffices to prove Theorem\n1.5 in the case where Π is a pro-l open surface group.\nThus, we return to the original notation of the statement of Theorem 1.5, under\nthe further assumption that Π is a pro-l open surface group [and N is nontrivial\nof infinite index]. Since N is of infinite index in Π, by replacing Π by an open\nsubgroup containing N [cf. the argument above involving the “Hn” and “Nn”], we\nmay assume that the rank of Π is > the rank of N . Thus, [cf., e.g., [RZ], the proof of\nTheorem 7.7.4] dimFl(Π\nab⊗Fl) > dimFl(Nab ⊗Fl). Next, observe that there exists\na normal open subgroup H ⊆ Π such that the natural surjection N \u0002 Nab ⊗ Fl\nfactors through N\/(N\n⋂\nH)\n∼→ (N · H)\/H . But this implies that the inclusion\nN ↪→ N · H induces a homomorphism Nab ⊗ Fl → (N · H)ab ⊗ Fl that admits a\nsplitting (N ·H)ab ⊗Fl \u0002 ((N ·H)\/H)ab ⊗Fl ∼→ (N\/(N\n⋂\nH))ab ⊗Fl ∼→ Nab ⊗Fl.\nIn particular, by replacing Π by the open subgroup N · H [where we note that\ndimFl((N ·H)ab⊗Fl) ≥ dimFl(Πab⊗Fl)], we may assume, without loss of generality,\nthat the natural homomorphism Nab⊗Fl → Πab⊗Fl is injective, but not surjective.\nThus, it follows [cf., e.g., [RZ], the proof of Theorem 7.7.4] that there exists a\ncollection of free generators {γi}i∈I [where I is a finite set] of Π such that for some\nnonempty proper subset I ′ ⊆ I, the {γi}i∈I′ form a collection of free generators of\nN . But, as is well-known [and easily verified, by applying the universal property\nof free pro-l groups, together with the existence of non-normal cyclic subgroups of\nfinite l-groups!], this contradicts the normality of N . ©\nSection 2: Configuration Space Groups\nIn the present §2, we discuss various well-known preliminary facts concerning\nthe sorts of profinite groups that arise from étale fundamental groups of configura-\ntion spaces associated to hyperbolic curves.\nFirst, let us suppose that we have been given a log scheme\nZ log\n10 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nwhich is log regular [cf., [Kato2], Definition 2.1]; write UZ ⊆ Z for the interior of\nZ log [i.e., the open subscheme on which the log structure of Z log is trivial]. By\nabuse of notation, we shall often use the notation for a scheme to denote the log\nscheme with trivial log structure determined by the scheme. If C is a full formation\nthat is invertible on Z, and Z is connected, then we shall write\nπC1 (Z\nlog)\nfor the maximal pro-C quotient of the étale fundamental group [obtained by con-\nsidering Kummer log étale coverings, for some choice of basepoint — cf. [Ill] for\nmore details] of Z log. Thus, by the log purity theorem of Fujiwara-Kato [cf. [Ill];\n[Mzk1], Theorem B], the natural morphism UZ → Z log induces a [continuous outer]\nisomorphism πC1 (UZ)\n∼→ πC1 (Z log).\nNext, suppose that S is a regular scheme, and that\nX → S\nis a family of hyperbolic curves of type (g, r) over S, with compactification X ↪→\nY → S and divisor of cusps D ⊆ Y [cf. §0]. For simplicity, we assume that the\nfinite étale covering D → S is split. Let n ∈ N.\nDefinition 2.1.\n(i) For positive integers i, j ≤ n such that i < j, write\nπi,j : Pn\ndef\n= X ×S . . . ×S X → X ×S X\nfor the projection of the product Pn of n copies of X → S to the i-th and j-th\nfactors. Write E for the set [of cardinality n] of factors of Pn. Then we shall refer\nto as the n-th configuration space associated to X → S the S-scheme\nXn → S\nwhich is the open subscheme determined by the complement in Pn of the union\nof the various inverse images via the πi,j [as (i, j) ranges over the pairs of positive\nintegers ≤ n such that i < j] of the image of the diagonal embedding X ↪→ X×S X .\nWe shall refer to as the n-th log configuration space associated to X → S the [log\nsmooth] log scheme over S\nZ logn → S\nobtained by pulling back the [log smooth] [1-]morphism Mlogg,r+n → M\nlog\ng,r given by\n“forgetting the last n points” [cf. §0] via the classifying [1-]morphism S → Mlogg,r\ndetermined [up to a permutation of the r remaining points] by X → S. We shall\nrefer to E as the index set of the configuration space Xn, or, alternatively, of the\nlog configuration space Z logn .\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 11\n(ii) In the notation of (i), let E′ ⊆ E be a subset of cardinality n′; E′′ def= E\\E′;\nn′′ def= n− n′. Then by “forgetting” the factors of E that belong to E′, we obtain a\nnatural projection morphism\npE′ = p\nE′′ : Xn → Xn′′\n[and similarly in the logarithmic case], which we shall refer to as the projection\nmorphism of profile E′, or, alternatively, the projection morphism of co-profile E′′.\nAlso, in this situation, we shall refer to n′ (respectively, n′′) as the length (respec-\ntively, co-length) of this projection morphism.\nRemark 2.1.1. One verifies immediately that in the notation of Definition 2.1,\n(i), Xn may be naturally identified with the interior of Z\nlog\nn .\nRemark 2.1.2. One verifies immediately that in the notation of Definition\n2.1, (ii), each projection morphism pE′ = p\nE′′ : Xn → Xn′′ is itself the n′-th\nconfiguration space associated to a family of hyperbolic curves of type (g, r + n′′)\nover Xn′′ that embeds as a dense open subscheme of the pull-back via Xn′′ → S of\nthe original family of hyperbolic curves X → S.\nProposition 2.2. (Fundamental Groups of Configuration Spaces) In the\nnotation of the above discussion, suppose further that the following conditions hold:\n(a) S is connected;\n(b) C is a PT-formation which is invertible on S;\n(c) for each l ∈ ΣC, the images of the cyclotomic character π1(S) → F×l\nand the natural Galois action\nπ1(S) → Aut(π1(Ys)ab ⊗ Fl)\narising from the family of curves Y → S are C-groups [a condition which\nis vacuous if C is total].\nLet n ≥ 1 be an integer, s a geometric point of S, and x a geometric point of\nXn−1 [where we write X0\ndef\n= S]; we shall denote the fibers over geometric points by\nmeans of subscripts. Then:\n(i) Any projection morphism Xn → Xn−1 of length one determines a natural\nexact sequence\n1 → πC1 ((Xn)x) → πC1 (Xn) → πC1 (Xn−1) → 1\nof pro-C groups.\n12 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\n(ii) The profinite group πC1 ((Xn)s) is slim and topologically finitely gen-\nerated.\n(iii) The natural sequence\n1 → πC1 ((Xn)s) → πC1 (Xn) → πC1 (S) → 1\nis exact.\n(iv) Suppose that k, k′ are separably closed fields; k ⊆ k′; S = Spec(k);\nS′ def= Spec(k′); s (respectively, s′) is the geometric point of S (respectively, S′)\ndetermined by the identity morphism of S (respectively, S′). Then the natural\nmorphism πC1 ((Xn ×S S′)s′) → πC1 ((Xn)s) is an isomorphism.\n(v) Suppose that S = Spec(R), where R is a complete discrete valuation\nring; that s arises from an algebraic closure of the residue field of R; and that η is\na geometric point of S that arises from an algebraic closure of the quotient field K\nof R. Then the operation of specialization of the normalization of Xn in a covering\nof Xn ×R K determines an isomorphism πC1 ((Xn)η) ∼→ πC1 ((Xn)s).\nProof. First, let us observe that since the kernel of the natural surjection πC1 (Xs) \u0002\nπC1 (Ys) is topologically normally generated by the inertia groups of the cusps [which\nare isomorphic to ẐC(1), where the “(1)” denotes a “Tate twist”, and “ẐC” is as\nin Definition 1.1, (ii)], condition (c) [together with our assumption that the divisor\nof cusps of X → S is split] implies that for each l ∈ ΣC , the image of the natural\nGalois action\nπ1(S) → Aut(π1(Xs)ab ⊗ Fl)\narising from the family of hyperbolic curves X → S is a C-group.\nNow we claim that to complete the proof of Proposition 2.2, it suffices to\nverify assertion (v). Indeed, let us assume that assertion (v) holds and reason by\ninduction on n ≥ 1. [That is to say, if n ≥ 2, then we assume that assertions\n(i), (ii), and (iii) have already been verified for “n − 1”.] Then let us first observe\nthat assertion (iv) follows from assertion (v) by a standard argument in elementary\nalgebraic geometry [cf., e.g., [Mzk4], Proposition 2.3, (ii), in the case where k is\nof characteristic zero, n = 1; since C is invertible on UZ , and we are free to apply\nassertion (v), the case of positive characteristic k and arbitrary n is entirely similar].\nNext, observe that [in light of Remark 2.1.2; the easily verified fact that the family\nXn → Xn−1 also satisfies conditions (a), (b), (c)] assertion (i) is a special case of\nassertion (iii) for “n = 1”; thus, [by applying the induction hypothesis] we may\nassume that assertion (i) holds if n ≥ 2. Since, moreover, the property of being\na slim topologically finitely generated profinite group holds for a profinite group\nwhich is an extension of a profinite group G1 by a profinite group G2 whenever\nit holds for G1 and G2, assertion (ii) [for “n”] follows immediately, by applying\nthe induction hypothesis, from assertion (i) (when n ≥ 2) and Proposition 1.4.\nAs for assertion (iii), let us first observe that by assertions (iv), (v) [and various\nstandard arguments in elementary algebraic geometry], we may assume without\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 13\nloss of generality that s arises from an algebraic closure of the function field K\nof S. Thus, by considering the natural action of GK\ndef\n= π1(Spec(K), s) on s, we\nobtain a natural outer action\nGK → Out(πC1 ((Xn)s))\nwhich is compatible with the natural outer action of GK on π\nC\n1 ((Pn)s) [which may\nbe identified with the product of n copies of πC1 (Xs)], relative to the natural in-\nclusion Xn ↪→ Pn [cf. Definition 2.1, (i)]. In particular, since [by Zariski-Nagata\npurity — i.e., the classical non-logarithmic version of the “log purity theorem”\nquoted above] the kernel of the natural surjection πC1 ((Xn)s) \u0002 πC1 ((Pn)s) is topo-\nlogically normally generated by the inertia groups of the divisors of (Pn)s lying in\nthe complement (Pn\\Xn)s [which are isomorphic to quotients of ẐC(1)], condition\n(c) [together with the observation at the beginning of the present proof] implies\nthat for each l ∈ ΣC , the image of the natural Galois action\nGK → Aut(π1((Xn)s)ab ⊗ Fl)\nis a C-group, hence [cf. Remark 1.1.3 when C is primary] that the homomor-\nphism GK → Out(πC1 ((Xn)s)) factors through the maximal pro-C quotient GCK\nof GK . Note, moreover, that [again] by Zariski-Nagata purity [i.e., the classical\nnon-logarithmic version of the “log purity theorem” quoted above], the kernel of\nthe natural surjection GCK \u0002 πC1 (S) is topologically normally generated by the var-\nious inertia groups determined by the prime divisors of S. On the other hand,\nby assertion (v), the images of these inertia groups in Out(πC1 ((Xn)s)) are trivial.\nThus, we obtain a homomorphism πC1 (S) → Out(πC1 ((Xn)s)), hence — by pulling\nback the natural exact sequence\n1 → πC1 ((Xn)s) → Aut(πC1 ((Xn)s)) → Out(πC1 ((Xn)s)) → 1\n[cf. assertion (ii); §0] via this homomorphism — an exact sequence as in assertion\n(iii). This completes the proof of the claim.\nFinally, we consider assertion (v). First, we remark that assertion (v) is a\nspecial case of the more general result of [Vid], Théorème 2.2; since, however, [Vid]\nhas yet to be published at the time of writing, we give a self-contained [modulo\npublished results] proof of assertion (v), as follows. We begin by observing that by\nthe log purity theorem, we have natural isomorphisms\nπC1 ((Xn)s)\n∼→ πC1 ((Z logn )s); πC1 ((Xn)η) ∼→ πC1 ((Z logn )η)\n[cf. Definition 2.1, (i); Remark 2.1.1]. Now suppose that W\nlog\n0 → (Z logn )s is a\nconnected Kummer log étale covering. Since (Z logn )s is log regular, it thus follows\nthat W\nlog\n0 is also log regular, hence, in particular, normal. By the definition of\n“log étale”, one may deform this covering to a formal Kummer log étale covering\nover the mR-completion [where mR is the maximal ideal of R] of Z\nlog\nn . Moreover,\nthe underlying scheme of this formal covering may be algebrized [cf. [EGA III],\nThéorème 5.4.5; the easily verified fact that Zn is projective], hence determines a\n14 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nfinite morphism W → Zn. Now it follows from the well-known local structure of\nKummer log étale coverings that the formal covering that gave rise to W is S-flat,\nhence that W itself is S-flat, with normal special fiber Ws ∼= W0. Since S is, of\ncourse, normal, we thus conclude [cf. [EGA IV], Corollaire 6.5.4, (ii)] that W is\nnormal and connected, hence irreducible. By considering the formal covering that\ngave rise to W at completions of closed points of Zn lying in the interior Xn ⊆ Zn,\nit follows, moreover, that W → Zn is generically étale. Thus, it makes sense to\nspeak of the ramification divisor in Zn of W → Zn. On the other hand, again by\nconsidering the formal covering that gave rise to W , it follows immediately that\nthis ramification divisor is contained in the complement of Xn in Zn, hence [by\nthe log purity theorem!] that W → Zn determines a Kummer log étale covering\nW log → Z logn whose special fiber W logs → (Z logn )s may be naturally identified with\nthe given covering W\nlog\n0 → (Z logn )s. Thus, by algebrizing morphisms between formal\nKummer log étale coverings [cf. [EGA III], Théorème 5.4.1], we conclude that the\ndeformation and algebrization procedure just described determines an equivalence\nof categories between the categories of Kummer log étale coverings of (Z logn )s, Z\nlog\nn .\nIn particular, we obtain a natural isomorphism πC1 ((Z\nlog\nn )s)\n∼→ πC1 ((Z logn )).\nOn the other hand, again by the log purity theorem, it follows immediately that\nwe obtain an isomorphism\nπC1 ((Z\nlog\nn )η)\n∼→ lim←−\nS′\nπC1 (Z\nlog\nn ×S S′)\n[where S′ ranges over the normalizations of S in the various finite extensions of K\nin the function field of η], hence, by applying the isomorphisms\nπC1 (Z\nlog\nn ×S S′) ∼→ πC1 ((Z logn )s)\n[where we regard s as a geometric point of the various S′] obtained above, we obtain\nan isomorphism πC1 ((Z\nlog\nn )η)\n∼→ πC1 ((Z logn )s), as desired. ©\nRemark 2.2.1. Another proof of Proposition 2.2, (iii), in the case n = 1 may be\nfound in [Stix], Proposition 2.3.\nDefinition 2.3. Let C be a PT-formation.\n(i) We shall say that a profinite group is a [pro-C] configuration space group if\nit is isomorphic to the maximal pro-C quotient of the étale fundamental group\nπC1 (Xn)\nof the n-th configuration space Xn for some n ≥ 1 [cf. Definition 2.1, (i)] of a\nhyperbolic curve X over an algebraically closed field of characteristic \r∈ ΣC [where\nwe note that in this situation, if we take S to be the spectrum of this algebraically\nclosed field, then the conditions (a), (b), (c) of Proposition 2.2 are satisfied].\n(ii) Let X be a hyperbolic curve over an algebraically closed field of charac-\nteristic \r∈ ΣC; Xn the n-th configuration space [for some n ≥ 1] associated to X .\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 15\nThen we shall refer to a closed subgroup H ⊆ πC1 (Xn) as being product-theoretic if\nH arises as the inverse image via the natural surjection\nπC1 (Xn) \u0002 πC1 (Pn)\n[cf. Definition 2.1, (i)] of a closed subgroup of πC1 (Pn).\n(iii) Let X , Xn be as in (ii); write E for the index set of Xn. Let E\n′ ⊆ E be\na subset of cardinality n′; E′′ def= E\\E′; n′′ def= n − n′; pE′ = pE′′ : Xn → Xn′′ the\nprojection morphism of profile E′. Then we shall refer to the kernel\nF ⊆ πC1 (Xn)\nof the induced surjection πC1 (Xn) \u0002 πC1 (Xn′′) [cf. Remark 2.1.2; Proposition 2.2,\n(iii)] as the fiber subgroup of πC1 (Xn) of profile E\n′, or, alternatively, as the fiber\nsubgroup of πC1 (Xn) of co-profile E\n′′. Also, we shall refer to n′ (respectively, n′′) as\nthe length (respectively, co-length) of F .\nProposition 2.4. (Fiber Subgroups of Configuration Spaces) Let C be\na PT-formation; X a hyperbolic curve over an algebraically closed field of\ncharacteristic \r∈ ΣC; Xn the n-th configuration space [for some n ≥ 1] associated\nto X; E the index set of Xn; Π\ndef\n= πC1 (Xn); E\n′\n1, E\n′\n2 ⊆ E subsets whose respective\ncomplements we denote by E′′1 , E\n′′\n2 ⊆ E; F1, F2 ⊆ Π the fiber subgroups with\nrespective profiles E′1, E\n′\n2 ⊆ E. Then:\n(i) The description of Remark 2.1.2 determines on F2 (respectively, Π\/F2) a\nstructure of configuration space group with index set E′2 (respectively, E\n′′\n2 ).\n(ii) F1 ⊆ F2 if and only if E′1 ⊆ E′2. Moreover, in this situation, F1 ⊆ F2 is\nthe fiber subgroup of F2 with profile E\n′\n1 ⊆ E′2 [i.e., relative to the structure of\nF2 as the “π\nC\n1 (−)” of a configuration space that arises from the description given\nin Remark 2.1.2].\n(iii) The image of F1 in Π\/F2 is the fiber subgroup of Π\/F2 with profile\nE′1\n⋂\nE′′2 ⊆ E′′2 [i.e., relative to the structure of Π\/F2 as the “πC1 (−)” of a configu-\nration space that arises from the description given in Remark 2.1.2].\n(iv) The subgroup of Π topologically generated by F1, F2 is the fiber sub-\ngroup F3 with profile E\n′\n3\ndef\n= E′1\n⋃\nE′2. In particular, if E\n′′\n1 , E\n′′\n2 are disjoint and of\ncardinality one, then F1, F2 topologically generate Π.\n(v) In the situation of (iv), suppose that the length of F1, F2 is equal to\n1. Then there exists a normal closed subgroup K ⊆ Π satisfying the following\nproperties: (a) K ⊆ F3; (b) K is topologically normally generated in F3 by a\nsingle element; (c) the images of F1, F2 in F3\/K commute.\n(vi) F2 is topologically generated by the fiber subgroups [of Π] of length 1\nwhose profiles are contained in E′2. In particular, Π is topologically generated\nby its fiber subgroups of length 1.\n16 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nProof. Assertions (i), (ii) are immediate from the definitions [and Remark 2.1.2].\nNext, let us consider assertion (vi). In light of assertions (i), (ii), it suffices to\nverify assertion (vi) in the case where F2 = Π; also, we may assume without loss\nof generality that F1 is of length 1. Then, by induction on n [cf. also assertion (i)],\nΠ\/F1 is topologically generated by its fiber subgroups of length 1. Since the inverse\nimage in Π of any fiber subgroup of length 1 of Π\/F1 is clearly a fiber subgroup\nof length 2, it follows [cf. assertions (i), (ii)] that we may assume without loss\nof generality that n = 2. But then it suffices to observe that if Fα, Fβ ⊆ Π are\nfiber subgroups whose profiles E′α, E\n′\nβ ⊆ E are disjoint subsets of length 1, then\nthe natural morphism Fα ⊆ Π \u0002 Π\/Fβ [which is simply the morphism induced\non “πC1 (−)’s” by an open immersion of hyperbolic curves] is a surjection. This\ncompletes the proof of assertion (vi). Now assertion (iv) follows formally from\nassertion (vi); also, in light of assertion (vi), assertion (iii) follows immediately\nfrom the definitions.\nFinally, we consider assertion (v). First, let us observe that when n = 2, as-\nsertion (v) follows by observing that the kernel of the natural surjection πC1 (X2) \u0002\nπC1 (P2) [cf. Definition 2.3, (ii)] is topologically normally generated by the inertia\ngroup of the diagonal divisor of X2, which is isomorphic to ẐC(1) [hence topolog-\nically generated by a single element]. Now assertion (v) follows immediately for\narbitrary n, by applying assertions (i), (ii), (iv). ©\nRemark 2.4.1. Note that it follows immediately from Proposition 2.2, (ii);\nProposition 2.4, (i) [or, alternatively, (vi)], that the fiber subgroups of πC1 (Xn) are\ntopologically finitely generated normal closed subgroups.\nSection 3: Direct Products of Profinite Groups\nIn the present §3, we study quotients of products of profinite groups. In partic-\nular, we show that, in certain cases, the product decomposition of a direct product\nof profinite groups is “group-theoretic”.\nDefinition 3.1. Let G be a profinite group. Then we shall say that G is\nindecomposable if, for any isomorphism of profinite groups G\n∼→ H ×J , where H, J\nare profinite groups, it follows that either H or J is the trivial group. We shall say\nthat G is strongly indecomposable if every open subgroup of G is indecomposable.\nProposition 3.2. (The Indecomposability of Surface Groups) Let C\nbe a nontrivial full formation. Then every almost pro-C-surface group Π is\nstrongly indecomposable.\nProof. Since every open subgroup of an almost pro-C-surface group is again an\nalmost pro-C-surface group, it suffices to show that Π is indecomposable. Suppose\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 17\nthat we have an isomorphism of profinite groups Π ∼= H × J , where H, J are\nnonabelian [since Π is slim — cf. Proposition 1.4!] infinite [again since Π is slim,\nhence does not contain any nontrivial finite normal closed subgroups — cf. §0]\nprofinite groups. Note that since H, J , are infinite, it follows that for any open\nsubgroup Π1, we may always replace Π by an open subgroup of Π1. In particular,\n[cf. Remark 1.2.1] we may assume, without loss of generality, that Π is a pro-C\nsurface group arising from a curve of genus ≥ 2. Now we claim that for every prime\nnumber l ∈ ΣC, there exist finite quotients H \u0002 QH , J \u0002 QJ such that l divides\nthe order of QH , QJ . Indeed, suppose that l does not divide the order of any finite\nquotient of H. Then there exists a proper normal open subgroup NH ⊆ H such\nthat if we set N\ndef\n= NH × J ⊆ Π, then the conjugation action of Π\/N ∼= H\/NH on\nNab ⊗Zl ∼= (NabH ⊗Zl)× (Jab ⊗Zl) ∼= Jab ⊗Zl is trivial, which, as was seen in the\nproof of Proposition 1.4, leads to a contradiction. This completes the proof of the\nclaim.\nThus, by replacing Π by the maximal pro-l quotient of a suitable open subgroup\nof Π for some l ∈ ΣC [and replacing C by the primary formation determined by l],\nwe may assume without loss of generality that Π, H, J are pro-l groups. Note,\nmoreover, that since H, J are nonabelian pro-l groups, it follows that dimFl(H\nab ⊗\nFl) ≥ 2, dimFl(Jab ⊗ Fl) ≥ 2 [cf., e.g., [RZ], Proposition 7.7.2]. On the other hand,\nobserve that the cup product morphism\nH1(H, Fl) ⊗ H1(J, Fl) → H2(Π, Fl)\nis an injection. [Indeed, this follows immediately by considering the spectral se-\nquences associated to the surjections Π ∼= H × J \u0002 J , H \u0002 {1}, where we note\nthat the latter surjection may be regarded as a quotient of the former surjection.]\nBut this implies that dimFl(H\n2(Π, Fl)) ≥ 2, which [cf. Remark 1.2.2] is absurd.\nThis completes the proof of Proposition 3.2. ©\nRemark 3.2.1. Note that the strong indecomposability of Proposition 3.2 may\nalso be derived as an immediate consequence of Theorem 1.5, in light of the slimness\nof Proposition 1.4.\nProposition 3.3. (Quotients of Direct Products) Let G1, . . . , Gn be profi-\nnite groups, where n ≥ 1 is an integer;\nφ : Π\ndef\n=\nn∏\ni=1\nGi \u0002 Q\na surjection of profinite groups. Then there exist normal closed subgroups Hi ⊆\nGi [for i = 1, . . . , n], N ⊆ Q such that N ⊆ Z(Q) [cf. §0], and the composite\nΠ \u0002 Q\/N of φ with the surjection Q \u0002 Q\/N induces an isomorphism\nΠ\ndef\n=\nn∏\ni=1\nGi\n∼→ Q\/N\n18 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\n— where we write Gi\ndef\n= Gi\/Hi. In particular, if Q is center-free, then we obtain\nan isomorphism Π\n∼→ Q; if Q is center-free and indecomposable, then we obtain\nan isomorphism Gi\n∼→ Q for some i ∈ {1, . . . , n}.\nProof. Indeed, write I\ndef\n= Ker(φ) ⊆ Π; Ii ⊆ Gi for the inverse image of I via the\nnatural injection ιi : Gi ↪→ Π into the i-th factor; Hi ⊆ Gi for the image of I under\nthe natural projection πi : Π \u0002 Gi to the i-th factor [where i ∈ {1, . . . , n}]. Thus,\nwe have inclusions\nΠI\ndef\n=\nn∏\ni=1\nIi ⊆ I ⊆ ΠH def=\nn∏\ni=1\nHi ⊆ Π\ninside Π. Now observe that the commutator of any element\n(1, . . . , 1, gi, 1, . . . , 1) ∈ Π\n[i.e., all of whose components, except possibly the i-th component gi ∈ Gi, are equal\nto 1] with an element h ∈ I yields an element of I [since I is normal in Π] which lies\nin the image of ιi, hence determines an element of Ii ⊆ Gi, which is in fact equal\nto the commutator [gi, πi(h)] ∈ Gi [where we observe that πi(h) ∈ Hi] computed\nin Gi. In particular, since gi ∈ Gi is arbitrary, and any element of Hi arises as\nsuch a “πi(h)”, it follows that the commutator subgroup [Gi, Hi] is contained in\nIi. But this implies that the commutator subgroup [Π, ΠH ] is normally generated\nin Π by elements of ΠI ⊆ I, hence [since I is normal in Π] is contained in I. Put\nanother way, if we set N ⊆ Q equal to the image in Π\/I ∼→ Q of ΠH , then it follows\nthat N ⊆ Z(Q). On the other hand, it is immediate from the definitions that φ\ndetermines an isomorphism\n∏n\ni=1 (Gi\/Hi)\n∼→ Q\/N , as desired. ©\nRemark 3.3.1. Proposition 3.3 may be regarded as being motivated by the\nfollowing elementary fact concerning products of rings: If R1, . . . , Rn [where n ≥ 1\nis an integer] are [not necessarily commutative] rings with unity and\nφ : R\ndef\n=\nn∏\ni=1\nRi \u0002 Q\nis a surjection of rings with unity, then there exist two-sided ideals Ii ⊆ Ri [for\ni = 1, . . . , n] such that φ induces an isomorphism\nR\ndef\n=\nn∏\ni=1\nRi\n∼→ Q\n— where we write Ri\ndef\n= Ri\/Ii. [Indeed, this follows immediately by observing that\nif, for i = 1, . . . , n, we write ei ∈ R for the element whose i-th component is 1 and\nwhose other components are 0, then any element f ∈ Ker(φ) may be written in the\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 19\nform f = f · e1 + . . .+ f · en, where each f · ei ∈ Ker(φ) [since Ker(φ) is a two-sided\nideal!].]\nRemark 3.3.2. Proposition 3.3 is due to the second author. We observe in\npassing that when, in the notation of Proposition 3.3, Q is an almost pro-C-surface\ngroup for some nontrivial full formation C [hence slim and strongly indecomposable\n— cf. Propositions 1.4, 3.2], and the Gi are topologically finitely generated, one may\ngive a different proof of Proposition 3.3 by applying Theorem 1.5 to the images Ji\nof the various composites of φ with the natural inclusions ιi : Gi ↪→ Π — which\nallows one to conclude [in light of the slimness of Q!] that only one of the Ji [as\ni ranges over the integers 1, . . . , n] can be nontrivial. In fact, this argument was\nthe approach originally taken by the first author to proving Proposition 3.3 and,\nmoreover, underlies the proof of the main result of this paper via the approach of\nthe second author given in §6 below. On the other hand, this argument [unlike\nthe very elementary proof of Proposition 3.3 given above!] has the drawback that\nit depends on the generalization of the result of Lubotzky-Melnikov-van den Dries\ngiven in Theorem 1.5. This drawback was pointed out by the second author to the\nfirst author when the first author first informed the second author of this restricted\nversion of Proposition 3.3 and, indeed, served to motivate the second author to\nobtain the more elementary proof of Proposition 3.3 given above.\nCorollary 3.4. (Group-theoreticity of Product Decompositions) Let\nn, m ≥ 1 be integers;\nG1, . . . , Gn; H1, . . . , Hm\nnontrivial profinite groups which are slim and strongly indecomposable [e.g.,\nalmost pro-C-surface groups for some nontrivial full formation C — cf. Propo-\nsitions 1.4, 3.2];\nG ⊆ ΠG def=\nn∏\ni=1\nGi; H ⊆ ΠH def=\nm∏\nj=1\nHj\nopen subgroups;\nα : G\n∼→ H\nan isomorphism of profinite groups. For i = 1, . . . , n; j = 1, . . . , m, write G\u0002i ⊆\nGi, H\n\u0002\nj ⊆ Hj for the respective images of G, H via the natural projections ΠG \u0002\nGi, ΠH \u0002 Hj. Then n = m; there exist a unique permutation σ of the set\n{1, . . . , n} and unique isomorphisms of profinite groups αi : G\u0002i ∼→ H\u0002σ(i) [for\ni = 1, . . . , n] such that the restriction of [the composite with the inclusion into ΠH\nof] the isomorphism\n(α1, . . . , αn) :\n(\nΠG ⊇\n) n∏\ni=1\n(G\u0002i )\n∼→\nn∏\ni=1\n(H\u0002σ(i))\n(\n⊆ ΠH\n)\nto G coincides with [the composite with the inclusion into ΠH of] α.\n20 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nProof. First, we observe that the uniqueness assertions follow immediately from\nthe nontriviality and slimness of the profinite groups G1, . . . , Gn, H1, . . . , Hm.\nThus, it suffices to verify the existence of σ and the αi. For i = 1, . . . , n; j =\n1, . . . , m, write\nG=i ⊆ ΠG; H=j ⊆ ΠH\nfor the respective intersections of G, H with the images of the natural injections\nGi ↪→ ΠG, Hj ↪→ ΠH ;\nG\n\u0006=\ni ⊆ ΠG; H \u0006=j ⊆ ΠH\nfor the respective intersections of G, H with the kernels of the natural projections\nΠG \u0002 Gi, ΠH \u0002 Hj . Now we claim that for each j = 1, . . . , m, the kernel of the\ncomposite\nψj : G → Hj\nof α with the natural projection (H ⊆) ΠH \u0002 Hj contains G\u0006=i , for a unique\ni ∈ {1, . . . , n}. Indeed, since the image of ψj is open, hence slim, it follows [cf.\n§0] that this image has no nontrivial finite normal closed subgroups; since the G\u0006=i\nare normal closed subgroups of G, it thus suffices to prove that the kernel of the\nrestriction of ψj to the open subgroup of G ⊆ ΠG determined by the direct product\nof the G=i′ [for i\n′ = 1, . . . , n] contains the intersection of this open subgroup with\nG\n\u0006=\ni , for a unique i. But [since Hj is slim and strongly indecomposable] this follows\nformally from Proposition 3.3. This completes the proof of the claim.\nNote, moreover, that [in the notation of the claim] the assignment j \u000f→ i\ndetermines a map {1, . . . , m} → {1, . . . , n}, which, in light of the injectivity of\nα, is easily verified to be surjective. But this implies that m ≥ n; thus, by ap-\nplying this argument to α−1, we obtain that m = n. In particular, the map\n{1, . . . , m} → {1, . . . , n} considered above is a bijection, whose inverse we denote\nby σ. By rearranging the indices, we may assume without loss of generality that σ\nis the identity.\nNow it follows from the definition of [the map that gave rise to] σ that we\nobtain a surjection\nαi : G\n\u0002\ni \u0002 H\u0002i\nfor each i = 1, . . . , n, such that the restriction of [the composite with the inclusion\ninto ΠH of] the surjection\n(α1, . . . , αn) :\n(\nΠG ⊇\n) n∏\ni=1\n(G\u0002i ) \u0002\nn∏\ni=1\n(H\u0002i )\n(\n⊆ ΠH\n)\nto G coincides with [the composite with the inclusion into ΠH of] α. In particular,\nsince α is injective, it follows that the kernel of each αi is a finite closed normal\nsubgroup of an open subgroup of Gi. Thus, by the slimness of Gi, we conclude [cf.\n§0] that the αi are injective, as desired. ©\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 21\nSection 4: Product-theoretic Quotients\nIn the present §4, we show that in the case of genus ≥ 2, the [closure of the]\ncommutator subgroup of a product-theoretic open subgroup of a configuration space\ngroup is, up to torsion, again product-theoretic [cf. Theorem 4.7]. This result,\ncombined with the theory of §3, implies a rather strong result, in the case of genus\n≥ 2, concerning the group-theoreticity of the various fiber subgroups associated to a\nconfiguration space group [cf. Corollary 4.8].\nLet Y be a connected smooth variety over an algebraically closed field k which\n[for simplicity] we assume to be of characteristic zero.\nDefinition 4.1. Let j ≥ 1 be an integer. Then we shall refer to Y as j-good if\nfor every positive integer j′ ≤ j and every class\nη ∈ Hj′ét(Y, Z\/NZ)\n[where “H\nj′\nét(−)” denotes étale cohomology, and N ≥ 1 is an integer], there exists\na finite étale covering Y ′ → Y such that η|Y ′ = 0.\nRemark 4.1.1. As is well-known, it follows immediately from the Hochschild-\nSerre spectral sequence in étale cohomology [cf., e.g., [Milne], p. 105, Theorem\n2.20] that one has a natural isomorphism\nHj\n′\n(π1(Y ), Ẑ)\n∼→ Hj′ét(Y, Ẑ)\nfor all nonnegative integers j′ ≤ j whenever Y is j-good. Also, we observe that it\nis immediate from the definitions that the condition “1-good” is vacuous.\nLet\nf : Z → Y\nbe a family of hyperbolic curves over Y ; y ∈ Y (k). We shall denote fibers over y by\nmeans of a subscript “y”. Suppose that we have also been given a section\ns : Y → Z\nof f , whose image we denote by Ds ⊆ Z. Write UZ ⊆ Z for the open subscheme\ngiven by the complement of Ds; L def= OZ(Ds); L× → Z for the complement of the\nzero section of the geometric line bundle determined by L;\nUZ → L×\nfor the morphism determined by the natural inclusion OZ ↪→ OZ(Ds) = L. Thus,\nUZ → Y is also a family of hyperbolic curves. Now if we denote by “π1(−)” the\n22 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nétale fundamental group [for an appropriate choice of basepoint], then we have a\nnatural commutative diagram\n1 −→ π1((UZ)y) −→ π1(UZ) −→ π1(Y ) −→ 1⏐⏐\u000b ⏐⏐\u000b ⏐⏐\u000b\n1 −→ π1(L×y ) −→ π1(L×) −→ π1(Y ) −→ 1⏐⏐\u000b ⏐⏐\u000b ⏐⏐\u000b\n1 −→ π1(Zy) −→ π1(Z) −→ π1(Y ) −→ 1\nin which the first and third horizontal sequences are exact [cf. Proposition 2.2,\n(iii)]. Write Is ⊆ π1(UZ) for the inertia group [well-defined up to conjugation in\nπ1(UZ)] associated to the divisor Ds. Thus, Is ∼= Ẑ(1) [where the “(1)” denotes a\n“Tate twist”].\nLemma 4.2. (The Line Bundle Associated to a Cusp) In the notation\nof the above discussion, suppose further that Y is j-good, for some integer j ≥ 2.\nThen:\n(i) Z is j-good.\n(ii) π1(L\n×) fits into a short exact sequence:\n1 → Ẑ(1) → π1(L×) → π1(Z) → 1\nMoreover, the resulting extension class ∈ H2(π1(Z), Ẑ(1)) ∼= H2ét(Z, Ẑ(1)) [cf. (i);\nRemark 4.1.1] is the first Chern class of the line bundle L.\n(iii) The sequence 1 → π1(L×y ) → π1(L×) → π1(Y ) → 1 of the above commu-\ntative diagram is exact.\n(iv) The morphism of fundmental groups π1(UZ) → π1(L×) induces an iso-\nmorphism Is\n∼→ Ker(π1(L×) → π1(Z)). In particular, the vertical arrows of the\ncommutative diagram of the above discussion are surjections.\n(v) Write π1(UZ\/Z)\ndef\n= Ker(π1(UZ) \u0002 π1(Z)) ⊆ π1((UZ)y). Then the quotient\nof π1(UZ\/Z) by\nπ1(UZ\/L\n×) def= Ker(π1(UZ) → π1(L×)) ⊆ π1(UZ\/Z) (⊆ π1(UZ))\nis the maximal quotient of π1(UZ\/Z) on which the conjugation action by π1((UZ)y)\nis trivial.\nProof. First, we consider assertion (i). In light of the exact sequence 1 → π1(Zy) →\nπ1(Z) → π1(Y ) → 1 [together with the Leray-Serre spectral sequence for Z → Y ],\nit follows immediately that to show that Z is j-good, it suffices to show that Zy is\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 23\nj-good. But this follows immediately from the fact that the cohomological dimen-\nsion of Zy is equal to 1 when Zy is affine [cf., e.g., [Milne], p. 253, Theorem 7.2]\nand from the well-known isomorphism H2ét(Zy, Z\/NZ)\n∼= (Z\/NZ)(−1) determined\nby considering fundamental classes of points [together with the fact that the coho-\nmological dimension of Zy is equal to 2 — cf., e.g., [Milne], p. 276, Theorem 11.1],\nwhen Zy is proper. This completes the proof of assertion (i).\nIn light of assertion (i), assertion (ii) follows from [Mzk2], Lemmas 4.4, 4.5.\nAssertion (iii) follows immediately by considering the natural commutative diagram\n1 −→ Ẑ(1) −→ π1(L×y ) −→ π1(Zy) −→ 1⏐⏐\u000b ⏐⏐\u000b ⏐⏐\u000b\n1 −→ Ẑ(1) −→ π1(L×) −→ π1(Z) −→ 1\n[in which the rows are exact, by assertion (ii); the vertical arrow on the left is an\nisomorphism], together with the exact sequence 1 → π1(Zy) → π1(Z) → π1(Y ) →\n1. Assertion (iv) (respectively, (v)) follows immediately from the argument of the\nproof of [Mzk5], Lemma 4.2, (ii) (respectively, [Mzk5], Lemma 4.2, (iii)). ©\nNow let l be a prime number; suppose that Y is 2-good. Also, let us suppose\nthat, for i = 1, . . . , m [where m ≥ 1 is an integer], we have been given a section\nsi : Y → Z\nof f , whose image we denote by Dsi ⊆ Z. Write Ui ⊆ Z for the open subscheme\ngiven by the complement of Dsi ; WZ\ndef\n=\n⋂m\ni=1 Ui ⊆ Z; Li def= OZ(Dsi); L×i → Z for\nthe complement of the zero section of the geometric line bundle determined by Li;\nWZ → L×i\nfor the morphism determined by the natural inclusion OZ ↪→ OZ(Dsi) = Li. Also,\nlet us suppose that WZ → Y is a family of hyperbolic curves [i.e., that the images\nof the si do not intersect]. By forming the quotient of the exact sequence of Lemma\n4.2, (ii), by the pro-prime-to-l portion of Ẑ(1), we obtain extensions\n1 −→ Zl(1) −→ Ei,y −→ π1(Zy) −→ 1⏐⏐\u000b ⏐⏐\u000b ⏐⏐\u000b\n1 −→ Zl(1) −→ Ei −→ π1(Z) −→ 1\nfor i = 1, . . . , m. Also, let us write\nκi ∈ H2ét(Z, Zl(1))\nfor the fundamental class associated to Dsi [i.e., the first Chern class of the line\nbundle Li — cf. Lemma 4.2, (ii)].\n24 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nLemma 4.3. (Multi-section Splittings) In the notation of the above discus-\nsion:\n(i) The natural homomorphism\nπ1(WZ) →\nm∏\ni=1\nEi\n[where the product is a fiber product over π1(Z)] is surjective.\n(ii) The natural quotient π1(WZ) \u0002 π1(WZ)ab ⊗ Zl factors through the quo-\ntient determined by the surjection of (i).\n(iii) For i = 1, . . . , m, let λi ∈ Zl. Then there exists a surjection π1(WZ) \u0002\nZl(1) — which, by (ii), necessarily factors through the surjection of (i), hence\ndetermines a surjection\nm∏\ni=1\nEi \u0002 Zl(1)\n— that restricts to multiplication by λi on the copy of Zl(1) in Ei if and only if the\nclass\nm∑\ni=1\nλi · κi ∈ H2ét(Z, Zl(1))\nvanishes.\nProof. First, we consider assertion (i). In light of the exact sequences of Proposi-\ntion 2.2, (iii), and Lemma 4.2, (iii), it suffices to show the surjectivity of π1((WZ)y) →∏m\ni=1 Ei,y. But this follows immediately, in light of Lemma 4.2, (iv), by considering\nthe various inertia groups ⊆ π1((WZ)y) of the cusps of (WZ)y. This completes the\nproof of assertion (i). Assertion (ii) follows immediately, in light of Lemma 4.2,\n(iv) [and induction on n], from the fact that the kernel of the natural surjection\nπ1(WZ) \u0002 π1(Z) is topologically normally generated by the inertia groups of cusps.\nFinally, we observe that assertion (iii) follows immediately from the definitions. ©\nLemma 4.4. (The Section Arising from the Graph of a Morphism) In\nthe notation of the above discussion, suppose further that Z → Y is given by the\nprojection to the second factor C ×k C → C, where we write C def= Zy, that C is\nproper, and that s : Y → Z is given by the graph of a k-morphism σ : C → C.\nThen the component of the first Chern class of L in the middle direct summand of\nH2ét(Z, Zl(1))\n∼= H2ét(C, Zl(1)) ⊕ (H1ét(C, Zl) ⊗ H1ét(C, Zl(1))) ⊕ H2ét(C, Zl(1))\n[cf. the Künneth isomorphism in étale cohomology, discussed, e.g., in [Milne], p.\n258, Theorem 8.5] is given by applying the endomorphism σ∗ ⊗ id of the module\nH1ét(C, Zl) ⊗ H1ét(C, Zl(1)) to the element of this module determined by the mor-\nphism HomZl(H\n1\nét(C, Zl(1)), Zl) → H1ét(C, Zl) given by the inverse of the morphism\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 25\nH1ét(C, Zl) → HomZl(H1ét(C, Zl(1)), Zl) arising from the cup product H1ét(C, Zl) ⊗\nH1ét(C, Zl(1)) → H2ét(C, Zl(1)) ∼= Zl in étale cohomology.\nProof. Indeed, this follows immediately from [Milne], p. 287, Lemma 12.2. ©\nLemma 4.5. (Linear Independence for Vector Spaces) Let G be a finite\ngroup, whose order we denote by |G|; K a field; V a finite-dimensional K-vector\nspace equipped with a linear action by G such that the G-module V contains the\nregular representation of G as a direct summand; N ≥ 1 an integer. Write\nW\ndef\n= V ⊕ . . .⊕ V\nfor the direct sum of N copies of V ; ιi ∈ HomK(V, W ) [where i = 1, . . . , N ] for the\ninclusion V ↪→ W into the i-th factor. Then the N · |G| elements\nιi ◦ g\n[where i = 1, . . . , N ; g ∈ G] of HomK(V, W ) are linearly independent.\nProof. Indeed, any nontrivial linear relation between these elements implies — by\napplying the various linear morphisms HomK(V, W ) → HomK(V, V ) obtained by\nprojecting onto the various factors of V in W — a nontrivial linear relation between\nthe endomorphisms ∈ HomK(V, V ) determined by the elements of G, in contradic-\ntion to the assumption that the G-module V contains the regular representation of\nG as a direct summand. ©\nLemma 4.6. (Linear Independence for Configuration Spaces) In the\nnotation of the above discussion, suppose further that:\n(a) there exists a commutative diagram\nZ −→ Y⏐⏐\u000b ⏐⏐\u000b\nX ×k Xn −→ Xn\nwhere the upper horizontal arrow is the given morphism Z → Y ; the lower\nhorizontal arrow is the projection to the second factor; n ≥ 1 is an integer;\nXn is the n-th configuration space associated to some hyperbolic\ncurve X over k; the vertical arrows are finite étale Galois coverings\narising from the coverings of X×kXn, Xn determined by taking the direct\nproduct of copies of a finite étale Galois covering Z0 → X [so Zy may\nbe identified with Z0];\n(b) the genus of the compactification B of X is ≥ 2;\n26 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\n(c) if we write C → B for the normalization of B in Z0, then we have\nm = n · deg(C\/B), and the si : Y → Z are the various liftings of the n\ntautological sections Xn → X ×k Xn arising from the definition of the\nconfiguration space Xn.\n[Thus, the fact that WZ → Y is a family of hyperbolic curves follows imme-\ndiately from Remark 2.1.2; the fact that Xn, hence also Y , is 2-good follows, by\ninduction on n, from Lemma 4.2, (i). Moreover, WZ forms a finite étale covering\nof Xn+1 that arises from a product-theoretic open subgroup of π1(Xn+1).] Then:\n(i) The morphisms of the commutative diagram of (a) determine an isomor-\nphism of k-schemes Z\n∼→ Z0 ×k Y .\n(ii) The isomorphism of (i) determines an isomorphism of first cohomology\ngroups H1ét(Z, Zl)\n∼→ H1ét(Z0, Zl) × H1ét(Y, Zl).\n(iii) The images of the κi in H\n2\nét(Z, Ql(1)) are linearly independent [over\nQl].\nProof. Note that the projection to the first factor X ×k Xn → X determines a\nmorphism Z → Z0 (⊆ C). Thus, we obtain an isomorphism as in assertion (i);\nassertion (ii) follows immediately from assertion (i). Now it remains to verify asser-\ntion (iii). Suppose that the section s : Y → Z arises from a point ∈ Z0(k). Write\nκ ∈ H2ét(Z, Zl(1)) for the fundamental class associated to Ds. For i = 1, . . . , m, set\nκ′i\ndef\n= κi − κ. Next, let us observe that κ and the κi all map [cf. the Leray-Serre\nspectral sequence for Z → Y ] to the same element of H2ét(Zy, Ql(1)) — a Ql-vector\nspace of dimension 0 [cf., e.g., [Milne], p. 253, Theorem 7.2] if Zy is affine and\ndimension 1 [cf., e.g., [Milne], p. 276, Theorem 11.1, (a)] if Zy is proper. In par-\nticular, it follows that κ′i maps to 0 in H\n2\nét(Zy, Ql(1)). Note, moreover, that [since\nκ = 0 whenever Zy is affine] this observation also implies that to verify the linear\nindependence of the images of the κi in H\n2\nét(Z, Ql(1)), it suffices to verify the linear\nindependence of the images of the κ′i in H\n2\nét(Z, Ql(1)). Thus, we conclude that the\nκ′i determine classes\nηi ∈ H1ét(Y, H1ét(Zy, Ql(1))) ∼= H1ét(Y, Ql) ⊗ H1ét(Z0, Ql(1))\n[cf. the Leray-Serre spectral sequence for Z → Y ], and that to verify the linear\nindependence of the images of the κi in H\n2\nét(Z, Ql(1)), it suffices to verify that the\nηi are linearly independent.\nOn the other hand, it follows immediately from the definitions that the κi arise\nas pull-backs via the various projections Y → Z0 ↪→ C, Z → Z0 ↪→ C of the classes\n[cf. Lemma 4.4] determined by the graphs ⊆ C ×k C of the various σ : C → C, for\nσ ∈ Gal(C\/B). In particular, the ηi arise as pull-backs via these various projections\nof the classes in\nH1ét(C, Ql) ⊗ H1ét(C, Ql(1)) (↪→ H1ét(Y, Ql) ⊗ H1ét(Z0, Ql(1)))\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 27\ndetermined [cf. Lemma 4.4] by the graphs of the various σ ∈ G def= Gal(C\/B). On\nthe other hand, by Proposition 1.3 [cf. our assumption that the genus of B is ≥ 2!],\nit follows that the G-module V\ndef\n= H1ét(C, Ql) contains the regular representation of\nG as a direct summand. Note, moreover, that the n inclusions V = H1ét(C, Ql) ↪→\nH1ét(Y, Ql) [determined up to composition with the action of G on V ] arising from\nthe n projections Xn → X determine a map of Ql-vector spaces\n(⊕\nH1ét(C, Ql)\n)\n→ H1ét(Y, Ql)\n[where the direct sum is over n copies of H1ét(C, Ql)] which is injective. Thus, we\nare, in effect, in the situation of Lemma 4.5, so the linear independence of the ηi\nfollows from the linear independence asserted in Lemma 4.5. ©\nTheorem 4.7. (Strongly Torsion-free Pro-solvable Product-theoreticity)\nLet X be a hyperbolic curve of genus ≥ 2 over an algebraically closed field k\nof characteristic zero; n ≥ 1 an integer; Xn the n-th configuration space as-\nsociated to X; H ⊆ π1(Xn) a product-theoretic open subgroup; G a strongly\ntorsion-free pro-solvable profinite group. Then the kernel of any continuous\nhomomorphism\nH → G\nis product-theoretic.\nProof. First, we claim that it suffices to verify Theorem 4.7 in the case where\nG = Zl [for some prime number l]. Indeed, since G is topologically finitely gen-\nerated [cf. Definition 1.1, (iii)], and arbitrary intersections of product-theoretic\nclosed subgroups of π1(Xn) are clearly product-theoretic, Theorem 4.7 for arbi-\ntrary [torsion-free] abelian G follows immediately from the case “G = Zl”. Thus,\nby replacing H, G successively by appropriate open subgroups of H, G, Theorem\n4.7 for arbitrary [strongly torsion-free] pro-solvable G follows immediately from the\n[torsion-free] abelian case. This completes the proof of the claim. Thus, in the\nfollowing, we assume that G = Zl.\nNow observe that Theorem 4.7 is vacuous for n = 1. Thus, by induction on\nn, it suffices to verify Theorem 4.7 for “n + 1” under the assumption that it holds\nfor “n”. Next, let us observe that it follows immediately from the definition of\n“product-theoretic” that any covering of Xn+1 that arises from a product-theoretic\nopen subgroup J ⊆ π1(Xn+1) is dominated by a covering of the form “WZ → Xn+1”\nfor WZ as in Lemma 4.6. Thus, to complete the proof of Theorem 4.7, it suffices\nto show that the kernel of any quotient J \u0002 Jab ⊗Zl \u0002 Zl is product-theoretic, for\nJ corresponding to a covering “WZ → Xn+1” as in Lemma 4.6. In particular, by\napplying Lemma 4.3, (iii), in light of the linear independence asserted in Lemma\n4.6, (iii), and the induction hypothesis [which may be applied to “Y ”, via Lemma\n4.6, (ii)], we conclude that the kernel of such a quotient J \u0002 Zl is product-theoretic,\nas desired. ©\n28 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nRemark 4.7.1. Note that Theorem 4.7 is false if the genus of X is < 2 and n ≥ 2.\nIndeed, to construct a counter-example for arbitrary n ≥ 2, it suffices to construct\na counter-example for n = 2. If, moreover, U is the hyperbolic curve determined\nby an open subscheme of X , then consideration of the natural morphism U2 ↪→ X2\nshows that the existence of a counter-example for X2 implies the existence of a\ncounter-example for U2. Thus, we may assume, without loss of generality, that\nn = 2, and X is either of type (0, 3) or of type (1, 1). But, in either of these cases,\nit is well-known that there exists a dominant map X2 → X that extends to a map\nX ×k X → B [where B is a compactification of X ] that maps the open subscheme\nX2 ⊆ X ×k X into X ⊆ B. Thus, by pulling back an appropriate infinite cyclic\ncovering of some finite étale covering of X , one obtain an infinite cyclic covering of\nsome finite étale covering of X ×k X that is [infinitely] ramified over the diagonal\nof X ×k X .\nCorollary 4.8. (Group-theoreticity of Projections of Configuration\nSpaces I) Let C be a PT-formation. For ξ = α, β, let Xξ be a hyperbolic\ncurve of genus ≥ 2 over an algebraically closed field kξ of characteristic\nzero; nξ ≥ 1 an integer; Xξnξ the nξ-th configuration space associated to Xξ;\nEξ the index set of X\nξ\nnξ\n; Hξ ⊆ Πξ def= πC1 (Xξnξ) a product-theoretic open\nsubgroup. Let\nγ : Hα\n∼→ Hβ\nbe an isomorphism of profinite groups. Then γ induces a bijection σ :\nEα\n∼→ Eβ [so nα = nβ] such that\nγ(Fα\n⋂\nHα) = Fβ\n⋂\nHβ\nfor all fiber subgroups Fα ⊆ Πα, Fβ ⊆ Πβ, whose respective profiles E′α ⊆ Eα,\nE′β ⊆ Eβ correspond via σ.\nProof. First, let us observe that to complete the proof of Corollary 4.8, it suffices\nto construct a bijection σ : Eα\n∼→ Eβ [so nα = nβ] such that\nγ(Fα\n⋂\nHα) = Fβ\n⋂\nHβ\nfor all fiber subgroups Fα ⊆ Πα, Fβ ⊆ Πβ of co-length one whose respective profiles\ncorrespond via σ. Indeed, this follows immediately by applying induction on n\ndef\n=\nnα = nβ [cf. also Proposition 2.4, (i), (ii)].\nNext, for j = 1, . . . , nξ, let us write\nK\nξ\nj ⊆ Hξ\nfor the intersection with Hξ of the fiber subgroup ⊆ Πξ of co-length one with co-\nprofile given by the element of Eξ labeled by j. Thus, [cf. Proposition 2.4, (iv);\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 29\nthe fact that fiber subgroups of co-length one are normal closed subgroups of infi-\nnite index] for distinct j, j′ ∈ {1, . . . , nξ}, Kξj , Kξj′ topologically generate an open\nsubgroup of Πξ; in particular, K\nξ\nj is not contained in K\nξ\nj′ .\nNow we claim that to complete the proof of Corollary 4.8, it suffices to prove\nthat the following statement holds [in general]:\nFor each i ∈ Eα, there exists a j ∈ Eβ such that Kβj ⊆ γ(Kαi ).\nIndeed, by applying this statement to γ, γ−1, we conclude that for each i ∈ Eα,\nthere exist j ∈ Eβ , i′ ∈ Eα such that γ(Kαi′ ) ⊆ Kβj ⊆ γ(Kαi ), hence that Kαi′ ⊆ Kαi .\nBut, as observed above, this implies that i′ = i, hence that Kβj = γ(K\nα\ni ). Moreover,\nthis relation “K\nβ\nj = γ(K\nα\ni )” determines an assignment i \u000f→ j, hence a mapping\nσ : Eα → Eβ, which is a bijection, relative to which intersections with Hα, Hβ of\nfiber subgroups of co-length one with corresponding profiles correspond via γ. This\ncompletes the proof of the claim.\nTo verify the “statement” of the above claim, we reason as follows: Let l ∈ ΣC .\nWrite Hα\/K\nα\ni \u0002 G for the maximal pro-l quotient of Hα\/Kαi ;\nφ : Hβ\n∼→ Hα \u0002 Hα\/Kαi \u0002 G\nfor the surjection determined by γ−1. Then observe that since Hα\/Kαi is a pro-C\nsurface group, it follows that G is a pro-l surface group, hence strongly torsion-\nfree [cf. Remark 1.2.2] and pro-solvable [cf. Remark 1.1.3]. Thus, it follows from\nTheorem 4.7 that φ factors through the quotient Hβ \u0002 Qβ determined by the\nquotient “πC1 (X\nβ\nnβ\n) \u0002 πC1 (P βnβ )” [i.e., the image of Hβ ⊆ πC1 (Xβnβ) in πC1 (P βnβ )]\ncorresponding to the quotient that was denoted “πC1 (Xn) \u0002 πC1 (Pn)” in Definition\n2.3, (ii). In particular, since Qβ admits an open subgroup with a direct product\ndecomposition induced by the natural direct product decomposition of πC1 (P\nβ\nnβ\n), it\nthus follows [since G is slim and strongly indecomposable — cf. Propositions 1.4,\n3.2] from Proposition 3.3 that there exists a j ∈ Eβ such that the image of Kβj in G\nis a finite normal closed subgroup, hence trivial [since G is slim — cf. Proposition\n1.4, §0]. But this implies that the image of Kβj in the pro-C-surface group Hα\/Kαi\nis a topologically finitely generated [cf. Proposition 2.2, (ii)] normal closed subgroup,\nwhich [since G is infinite] is of infinite index, hence — by Theorem 1.5 — trivial.\nThis completes the proof of Corollary 4.8. ©\nRemark 4.8.1. It is interesting to note that in [NT], Theorem 3.1 [cf., especially,\n[NT], Lemmas 3.3 – 3.6], a certain analogue of Theorem 4.8 is shown for graded\nLie algebras. More generally, the idea of studying configuration spaces from an\nanabelian point of view dates back at least to [Naka].\nRemark 4.8.2. The original motivation, for the first author, for developing the\ntheory applied to prove Corollary 4.8 was the idea that by combining Corollary\n30 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\n4.8 with the techniques of [Mzk6], [Mzk7], one could obtain results in the absolute\nanabelian geometry of configuration spaces over p-adic local fields. It is the intention\nof the first author to carry out this application of Corollary 4.8 in a subsequent\npaper.\nSection 5: Divisors and Units on Coverings of Configuration Spaces\nIn the present §5, we discuss a certain generalization [cf. Theorem 5.6; Remark\n5.6.1], in the case of proper hyperbolic curves, of Theorem 4.7 [due to the second\nauthor]. Unlike the proof of Theorem 4.7 given in §4, the proof of this generaliza-\ntion does not rely on the notion of “goodness” or properties involving the “regular\nrepresentation”. In this sense, the approach given in the present §5 is more effi-\ncient and relies on direct algebro-geometric properties — such as the disjointness of\ndivisors — of which the properties involving the “regular representation” applied\nin §4 may be thought of as a sort of “étale-topological translation”. On the other\nhand, the approach of §4 [which was discovered first, by the first author], though\nless efficient, is applicable to both affine and proper hyperbolic curves, and, more-\nover, has the virtue of relying on explicit group-theoretic manifestations of these\nalgebro-geometric properties; it was this explicitness that served to render the ap-\nproach of §4 more readily accessible to the intuition of the first author. Finally,\nwe discuss certain consequences [cf. Corollary 5.7] of the theory of the present §5\nconcerning the “non-existence of units” on finite étale coverings of a sufficiently\ngeneric hyperbolic curve.\nWe begin by reviewing some essentially well-known generalities concerning log\nschemes.\nDefinition 5.1. Let X log be a fine log scheme [cf. [Kato1]].\n(i) Denote by MX the étale sheaf of monoids on X that defines the log structure\non X log. Thus, we have a natural injection O×X ↪→ MX , which we shall use to regard\nO×X as a subsheaf of MX . We shall refer to the quotient sheaf of monoids\nM charX\ndef\n= MX\/O×X\nas the characteristic of X log and to the associated sheaf of groupifications\nM\nchar-gp\nX\nas the group-characteristic of X log. Thus, [since X log is fine] the fibers of M charX\n(respectively, M\nchar-gp\nX ) are finitely generated torsion-free abelian monoids (respec-\ntively, abelian groups). For n ∈ N, we shall denote by\nU\n[n]\nX ⊆ X\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 31\nand refer to as the n-interior of X log the subset [cf. Proposition 5.2, (i), (ii) below]\nof points [of the scheme X lying under geometric points of the scheme X ] at which\nthe fiber of M\nchar-gp\nX is of rank ≤ n. Thus, U [0]X is the interior UX ⊆ X of X log [i.e.,\nthe open subscheme of points at which the log structure of X log is trivial].\n(ii) Let M be a finitely generated [abstract] abelian monoid; N ≥ 1 an integer.\nWe shall say that M is Q-regular [with exponent N ] if for some n ∈ N, the map\nNn → Nn\n[where Nn is the monoid determined by the product of n copies of N] given by\nmultiplication by N factors as a composite of injections of monoids Nn ↪→ M ↪→ Nn.\nWe shall say that X log is weakly Q-regular (respectively, strongly Q-regular) if,\nfor every geometric point x of X , the fiber of M charx at x is a Q-regular monoid\n(respectively, Q-regular monoid with exponent invertible in the residue field of x).\nProposition 5.2. (Generalities on Log Schemes) Let X log be a fine log\nscheme; n ∈ N; l a prime number invertible on X. Then:\n(i) The n-interior U\n[n]\nX ⊆ X is open.\n(ii) Suppose that X log is log regular. Then the complement of the n-interior\nU\n[n]\nX is a closed subset of X of codimension > n; the complement DX\ndef\n= X\\UX\n[equipped with the reduced induced scheme structure] is a divisor on X.\n(iii) Suppose that X log is log regular and weakly Q-regular. Then X is\nlocally Q-factorial [i.e., every Weil divisor on X admits a positive multiple which\nis Cartier]. Moreover, if X is connected, and G is any torsion-free profinite\ngroup, then any homomorphism of profinite groups\nφ : π1(U\n[1]\nX ) → G\nfactors through the natural surjection of étale fundamental groups π1(U\n[1]\nX ) \u0002\nπ1(X).\n(iv) Suppose that X log is log smooth over a field k [equipped with the trivial log\nstructure] and strongly Q-regular; let F ⊆ X be a closed subset of codimension\n≥ n. Then the natural map on étale cohomology\nH\nj\nét(X, Ql) → Hjét(X\\F, Ql)\nis an isomorphism for j ≤ 2n − 2 and an injection for j = 2n − 1.\n(v) Under the assumptions of (iv), suppose further that X is connected; write\nDX =\n⋃\ni∈I\nDX,i\n32 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\n— where I is a finite set; the DX,i ⊆ X are irreducible divisors. [Thus, since\nX log is log regular, hence normal [cf. [Kato2], Theorem 4.1], we have a natural\nsurjection of étale fundamental groups\nπ1(UX) \u0002 π1(X)\nwhose kernel contains the inertia groups of the DX,i; the maximal pro-l quotient\nof each of these inertia groups is naturally isomorphic to some quotient of Zl(1).]\nThen we have a natural exact sequence\n0 → Hom(π1(X), Ql(1)) → Hom(π1(UX), Ql(1))\n→\n⊕\ni∈I\nQl → H2ét(X, Ql(1)) → H2ét(UX , Ql(1))\n— where the “Hom’s” denote the modules of continuous homomorphisms of topo-\nlogical groups; the second arrow is the arrow determined by the natural surjection\nπ1(UX) \u0002 π1(X); the third arrow is the arrow determined by the [copies of Zl(1)\nthat naturally surject onto the] inertia groups of the DX,i [and the natural iden-\ntification of Ql with Hom(Zl(1), Ql(1))]; the fourth arrow is the arrow that sends\nthe 1 ∈ Ql in the direct summand labeled “i” to the fundamental class c(DXi) of\nthe Weil divisor DX,i [which is well-defined, by (iii)].\nProof. Indeed, assertion (i) follows immediately from the definition of a fine\nlog scheme [cf. [Kato1], §2.1-3]. In light of assertion (i), the portion of asser-\ntion (ii) concerning U\n[n]\nX follows immediately from the inequality dim(OX,x) ≥\nrankZ((M\nchar-gp\nX )x) [where x is a geometric point of X ; OX,x is the corresponding\nstrict henselization of a local ring of X ] — cf. the definition of “log regular” in\n[Kato2], Definition 2.1. Since X is normal [cf. [Kato2], Theorem 4.1], the portion\nof assertion (ii) concerning DX follows immediately from the description given in\n[Kato2], Theorem 11.6, of the monoid MX in terms of rational functions on X .\nTo verify assertion (iii) (respectively, (iv)), let us first observe that it follows\nimmediately from our assumptions that X log is log regular (respectively, log smooth\nover k) and weakly Q-regular (respectively, strongly Q-regular) that every point of X\nadmits an étale neighborhood V → X such that there exists a finite (respectively,\n[finite] Kummer log étale) dominant morphism W log → V log [where we equip V\nwith the log structure pulled back from X ] such that the scheme W is regular\n(respectively, smooth over k) and connected, and the log structure of W log arises\nfrom a divisor with normal crossings on W . Now the local Q-factoriality portion\nof assertion (iii) follows immediately by pulling back a given Weil divisor on X\nto the regular scheme W [which yields a Cartier divisor on W ] and then pushing\nforward via W → V [which multiplies the original divisor on V by the degree of\nthe morphism W → V ]. To verify the portion of assertion (iii) concerning étale\nfundamental groups, we may assume without loss of generality that X , V , and W\nare strictly henselian, and that the morphism V → X is an isomorphism. Now by\nZariski-Nagata purity [i.e., the classical non-logarithmic version of the “log purity\ntheorem” quoted in §2], it follows that π1(U [1]W ) ∼→ π1(W ) = {1}. On the other\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 33\nhand, since the morphism W → X is finite, it follows that the natural morphism\n{1} = π1(U [1]W ) → π1(U [1]X ) has open image, hence that π1(U [1]X ) is finite. Thus, our\nassumption that G is torsion-free implies that φ is trivial. This completes the proof\nof assertion (iii).\nTo verify assertion (iv), let us first observe that assertion (iv) holds when X\nis smooth over k. Indeed, in this case, by applying noetherian induction to F\nand possibly base-changing to a finite inseparable extension of k, we may assume\nwithout loss of generality that F is smooth over k; but then the content of assertion\n(iv) is well-known [cf., e.g., [Milne], p. 244, Remark 5.4, (b)]. In the case of arbitrary\nX log, we argue as follows: Write ι : XF\ndef\n= X\\F ↪→ X for the natural inclusion.\nThen [by applying a well-known exact sequence in étale cohomology] it suffices to\nverify that Rjιét,∗(Ql) = 0 for 0 < j ≤ 2n − 1. Since we have already verified\nassertion (iv) for k-smooth X , we may assume that F\n⋂\nUX = ∅. In particular,\nit suffices [cf. [Milne], p. 88, Theorem 1.15] to verify, for an arbitrary strictly\nhenselization V of V at a closed point of V , that Hj(V F , Ql) = 0 for 0 < j ≤ 2n−1\n[where we write V F\ndef\n= XF ×X V ]. On the other hand, let us observe that since\nζ log : W\nlog\nF\ndef\n= V F ×V W log → V logF def= V F ×V V log is Kummer log étale, it follows\nthat one may define a “trace morphism”\nζét,∗((Ql)W F ) → (Ql)V F\n[where we use the subscripts “W F ”, “V F ” to denote the constant sheaf on WF ,\nV F ] that restricts, relative to ζ\n∗\nét, to multiplication by the degree deg(ζ) of ζ on\n(Ql)V F . [Indeed, this is immediate for the restriction ζU : UW F → UV F to the\nrespective interiors, since this restriction is finite étale. On the other hand, since V F\nis normal, we have a natural isomorphism (Ql)V F\n∼→ θ∗θ∗((Ql)V F ), where we write\nθ : UV F ↪→ V F for the natural inclusion of the interior. Thus, we obtain a trace\nmorphism as desired by restricting to the interiors, applying the trace morphism on\nthe interiors, and then applying this natural isomorphism.] Thus, by taking étale\ncohomology, one obtains a trace morphism τ : Hj(W F , Ql) → Hj(V F , Ql) such that\nthe composite τ ◦ ρ with the restriction morphism ρ : Hj(V F , Ql) → Hj(WF , Ql)\nis equal to multiplication by deg(ζ) on Hj(V F , Ql). Since, moreover, we have\nalready verified assertion (iv) for k-smooth X , it follows that Hj(WF , Ql) = 0 for\n0 < j ≤ 2n − 1, hence that Hj(V F , Ql) = 0 for 0 < j ≤ 2n − 1, as desired.\nFinally, we consider assertion (v). When X = U\n[1]\nX [so X , DX are smooth over\nk], assertion (v) follows immediately by applying the well-known Gysin sequence in\nétale cohomology [cf., e.g., [Milne], p. 244, Remark 5.4, (b)]\n0 → H1ét(X, Ql(1)) → H1ét(UX , Ql(1)) →\n⊕\ni∈I\nQl → H2ét(X, Ql(1)) → H2ét(UX , Ql(1))\nand the natural isomorphisms H1ét((−), Ql(1)) ∼= Hom(π1((−)), Ql(1)), for “(−)”\nequal to X , UX . For arbitrary X\nlog, we reduce immediately to the case where\nX = U\n[1]\nX by applying assertions (ii), (iv). ©\n34 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nRemark 5.2.1. We recall in passing that the local Q-factoriality portion of\nProposition 5.2, (iii), is false for arbitrary [not necessarily weakly Q-regular] log\nregular X log. Indeed, such an example appears in the Remark following [Mzk1],\nCorollary 1.8.\nNow we return to our discussion of configuration spaces. Let X be a proper\nhyperbolic curve of genus gX over an algebraically closed field k of characteristic\nzero, n ≥ 1 an integer, l a prime number; write Xn ⊆ Pn for the associated n-th\nconfiguration space, Z logn for the associated n-th log configuration space, and E for\nthe index set of Xn, Z\nlog\nn [cf. Definition 2.1, (i)]. Thus, Xn may be identified with\nthe interior UZn of Z\nlog\nn .\nProposition 5.3. (The Logarithmic Geometry of the Log Configuration\nSpace) In the notation of the above discussion: Write\nV\ndef\n= U\n[1]\nZn\n; V log\ndef\n= Z logn ×Zn V\nfor the 1-interior of Z logn and the log scheme obtained by restricting the log structure\nof Z logn . For j ≥ 1 an integer, let us denote by ∧jE the set of subsets of E of\ncardinality j [so E may be identified with ∧1E] and by\n∧∗E def=\nn⋃\nj=1\n∧jE\nthe [disjoint] union of the subsets of cardinality j ≥ 1. Then:\n(i) We shall refer to a divisor on Zn obtained as the pull-back via a projection\nmorphism Zn → X of co-length 1 [and co-profile e ∈ ∧1E = E] of a point ∈ X(k)\nas a fiber divisor [of co-profile e] [on Zn]. Then all fiber divisors of co-profile\ne ∈ ∧1E = E on Zn determine the same fundamental class\nηe ∈ H2ét(Zn, Ql(1))\n— which we shall refer to as the fiber class of co-profile e [on Zn].\n(ii) The irreducible divisors on Zn contained in the divisor DZn defining the\nlog structure of Z logn are in natural bijective correspondence with the elements\nof (∧∗E)\\E. That is to say, a point of V belongs to the irreducible divisor Dε ⊆ V\ncorresponding to an element ε ∈ (∧∗E)\\E if and only if it corresponds to a stable\ncurve with precisely two irreducible components, one isomorphic to X, the other\nof genus zero, such that the marked points that lie on X are precisely the marked\npoints determined by the factors e ∈ ε′ def= E\\ε. In particular, we obtain a natural\nisomorphism of schemes\nDε ∼= X|ε′|+1 × Q|ε|−2\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 35\n— where |ε|, |ε′| are the cardinalities of of ε, ε′, respectively; the projection Dε →\nX|ε′|+1 is induced by any projection Xn → X|ε′|+1 of co-profile ε+, for ε+ ∈ ∧|ε′|+1E\nan element such that ε′ ⊆ ε+; Q|ε|−2 is the (|ε| − 2)-th configuration space [i.e.,\nSpec(k), when |ε| = 2] of “the” tripod [cf. §0] over k. In particular, the index set\nof the configuration space X|ε′|+1 appearing in this isomorphism may be naturally\nidentified with the set “E\/ε” obtained from E by identifying the elements of ε to a\nsingle element [ε] ∈ E\/ε. We shall refer to the irreducible divisor on Zn contained\nin DZn that corresponds to ε ∈ (∧∗E)\\E as the log-prime divisor of co-profile\nε [on Zn]; we shall refer to the fundamental class\nηε ∈ H2ét(Zn, Ql(1))\nof the log-prime divisor of co-profile ε as the log-prime class of co-profile ε [on\nZn].\n(iii) Let\nW log → V log\nbe a connected [finite] Kummer log étale covering. Then W log is log smooth\nover k and strongly Q-regular, and W , DW are smooth over k. We shall\nalso refer to irreducible divisors on W that lie over fiber divisors on Zn as fiber\ndivisors on W , and to fundamental classes of fiber divisors on W as fiber classes\non W ; in a similar vein, we shall refer to irreducible divisors on W that lie over\nlog-prime divisors on Zn as log-prime divisors on W , and to fundamental classes\nof log-prime divisors on W as log-prime classes on W . Then a fiber class on W\nis completely determined by its co-profile. Also, we shall refer to a class on W or\nZn as a log-characteristic class if it is either a fiber class or a log-prime class.\nProof. Assertion (i) follows, for instance, from [Milne], p. 276, Theorem 11.1,\n(a). Assertion (ii) follows immediately from the definition of Z logn involving the\n[log] moduli stack of stable curves. Assertion (iii) follows immediately from the\ndefinitions and assertion (i). ©\nLemma 5.4. (Line Bundles on Log-prime Divisors) In the notation of\nProposition 5.3, (ii):\n(i) The isomorphism Dε ∼= X|ε′|+1×Q|ε|−2 of Proposition 5.3, (ii), determines\nan isomorphism of Picard groups Pic(Dε)\n∼→ Pic(X|ε′|+1).\n(ii) The co-normal bundle of Dε is isomorphic [cf. (i)] to the line bundle\nobtained by pulling back the canonical bundle ωX of X via the [unique!] projection\nX|ε′|+1 → X that arises from a projection morphism Xn → X of co-length 1 whose\nco-profile is not contained in ε′.\nProof. First, we consider assertion (i). Since Q|ε|−2 is an open subscheme of the\naffine space [of dimension |ε| − 2] over k, it follows that Dε ∼= X|ε′|+1 × Q|ε|−2 is\nisomorphic to an open subscheme of the affine space [of dimension |ε| − 2] over\n36 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nX|ε′|+1. Thus, the assertion concerning Picard groups follows immediately from\nelementary algebraic geometry [cf., e.g., [Fulton], Theorem 3.3, (a)].\nAs for assertion (ii), we observe that the description of the stable curves\nparametrized by Dε given in Proposition 5.3, (ii), implies [in light of the well-\nknown local structure of a node] that the co-normal bundle in question is naturally\nisomorphic to the tensor product of the pull-back of the canonical bundle described\nin the statement of assertion (ii) with some [necessarily trivial — by assertion (i)]\nline bundle on Dε pulled back from the natural projection to Q|ε|−2. This completes\nthe proof of assertion (ii). ©\nLemma 5.5. (Linear Independence of Log-characteristic Classes) In\nthe notation of Proposition 5.3, (iii): Set\nIW\ndef\n= IfiberW\n⋃\nI\nlog-prime\nW\n— where we write IfiberW\ndef\n= E and I\nlog-prime\nW for the set of log-prime divisors on\nW . Thus, if we think of the elements of IfiberW as the co-profiles of fiber classes of\nH2ét(W, Ql(1)), then we obtain a [fiber or log-prime] class\nηi ∈ H2ét(W, Ql(1))\nfor each element i ∈ IW . Then:\n(i) The log-characteristic classes {ηi}i∈IW are linearly independent over\nQl.\n(ii) The restricted fiber classes {ηi|UW }i∈IfiberW are linearly independent [in\nH2ét(UW , Ql(1))] over Ql.\nProof. First, we observe that the description of the kernel of the restriction map\nH2ét(W, Ql(1)) → H2ét(UW , Ql(1)) given in the exact sequence of Proposition 5.2, (v),\nimplies that assertion (ii) follows immediately from assertion (i). For i ∈ I log-primeW ,\nwrite\nDi\nfor the divisor [tautologically!] determined by i. Thus, [by the definition of V log,\nW log]\nDi\n⋂\nDj = ∅\nfor all i, j ∈ I log-primeW such that i \r= j. Also, we observe that it follows immediately\nfrom the definitions that, for i ∈ I log-primeW , the covering W log → V log determines a\nfinite étale morphism\nDi → Dε ∼= X|ε′|+1 × Q|ε|−2\n[cf. Proposition 5.3, (ii)]. Finally, we observe that we may assume without loss of\ngenerality that the Kummer log étale covering W log → V log is Galois, with Galois\ngroup Γ\ndef\n= Gal(W log\/V log).\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 37\nNow let us verify assertion (i) by induction on n. Let\n∑\ni∈IW\nci · ηi = 0\n[where the ci ∈ Ql] be a linear relation among the ηi. The case n = 1 is immediate\nfrom the definitions [cf. also [Milne], p. 276, Theorem 11.1, (a)]. Next, we consider\nthe case n = 2. Let j ∈ I log-primeW . Since n = 2, it follows immediately from the\ndefinitions that Dj is a proper hyperbolic curve such that the covering W\nlog → V log\ninduces a finite étale morphism of Dj onto the diagonal of V = Zn = X × X . In\nparticular, it follows that we may restrict the above linear relation to Dj to obtain\na linear relation\n−cj · ηj |Dj =\n∑\ni∈IfiberW\nci · ηi|Dj\namong classes of H2ét(Dj , Ql(1))\n∼= Ql [cf. [Milne], p. 276, Theorem 11.1, (a)].\nMoreover, since the class ηj |Dj is the pull-back to Dj of some Q×l -multiple of the\nfundamental class of the diagonal of X ×X [cf. Lemma 5.4, (ii)], we thus conclude\nthat ηj |Dj \r= 0. Next, let us observe that, for i ∈ IfiberW , the classes ηi are fixed\nby the natural action of Γ on H2ét(W, Ql(1)). Thus, if we identify H\n2\nét(Dj , Ql(1))\nwith Ql via the natural isomorphism H\n2\nét(Dj , Ql(1))\n∼= Ql, then we conclude that\nthe elements ηi|Dj , ηj |Dj ∈ Ql [where i ∈ IfiberW ] are independent of the choice of j\namong all Γ-conjugates of j, hence that the element cj ∈ Ql is independent of the\nchoice of j among all Γ-conjugates of j. But this implies that the linear relation∑\ni∈IW ci · ηi = 0 arises as the pull-back to W of a similar linear relation on V .\nThat is to say, we may assume without loss of generality that W log = V log. But\nthen the Ql-linear independence of the unique log-prime class ηΔ and the two fiber\nclasses η1, η2 follows, for instance, from the [easily verified] non-singularity of the\nmatrix of intersection numbers among the classes ηΔ, η1, η2 [where we recall that\nηΔ · η1 = ηΔ · η2 = η1 · η2 = 1, η1 · η1 = η2 · η2 = 0, ηΔ · ηΔ = 2 − 2gX ]. This\ncompletes the proof of the case n = 2.\nNow we assume that n ≥ 3. Let Dj [where j ∈ I log-primeW ] be a log-prime\ndivisor of co-length 2 — i.e., which projects to a log-prime divisor of V whose co-\nprofile ε is of cardinality 2. Then the covering W log → V log determines a finite\nétale morphism Dj → Xn−1; that is to say, Dj appears as a “UW ” that arises\nin the case “n − 1”. Moreover, it follows immediately from Lemma 5.4, (ii), that\nηj |Dj is a Ql-multiple of the restriction to Dj of the fiber class of Z logn−1 of co-profile\n[ε] ∈ E\/ε [cf. Proposition 5.3, (ii)]. On the other hand, for i ∈ I log-primeW such that\ni \r= j, ηi|Dj = 0; for i ∈ IfiberW of co-profile e ∈ E, ηi|Dj is a Q×l -multiple of the\nrestriction to Dj of the fiber class of Z\nlog\nn−1 whose co-profile is the image of e in E\/ε.\nIn particular, by restricting the linear relation\n∑\ni∈IW ci · ηi = 0 to Dj , it follows\nby applying assertion (ii) for “n − 1” [i.e., here we apply the induction hypothesis\non n] that ci = 0 for all i ∈ ε′ ⊆ E = IfiberW . Since n ≥ 3, it follows that ε′ \r= ∅.\nThus, by varying j [i.e., varying ε], we conclude that ci = 0 for all i ∈ IfiberW .\nNow to complete the proof of Lemma 5.5, it suffices to show that the coeffi-\ncients of any linear relation\n∑\ni∈Ilog-primeW ci · ηi = 0 vanish. On the other hand,\n38 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nby restricting to Dj , for j ∈ I log-primeW , we obtain relations cj · ηj |Dj = 0 for each\nj ∈ I log-primeW . Thus, to complete the proof of Lemma 5.5, it suffices to show that\nηj |Dj \r= 0\nfor j ∈ I log-primeW . On the other hand, by Lemma 5.4, (ii), it follows that ηj |Dj\nis a Q×l -multiple of the pull-back to Dj of a class on X|ε′|+1 × Q|ε|−2 [for some\nε ∈ (∧∗E)\\E, ε′ def= E\\ε] that arises as the pull-back to X|ε′|+1 × Q|ε|−2 via the\nprojection morphism X|ε′|+1 × Q|ε|−2 → X|ε′|+1 of the restriction to X|ε′|+1 of a\nfiber class of Z|ε′|+1. Since |ε′| + 1 ≤ n − 1, it thus follows from assertion (ii) for\n“|ε′|+1” [i.e., here we apply the induction hypothesis on n] that the class ηj |Dj \r= 0,\nas desired. This completes the proof of Lemma 5.5. ©\nTheorem 5.6. (Extendability of Coverings) Let X be a proper hyperbolic\ncurve over an algebraically closed field k of characteristic zero; n ≥ 1 an\ninteger; Xn ⊆ Pn the n-th configuration space associated to X; (X∗n)log def= Z logn\nthe n-th log configuration space associated to X;\nY → Xn\na finite étale morphism, where Y is connected; Y ∗ → X∗n the normalization\nof X∗n in Y ; G a strongly torsion-free profinite group;\nφ : π1(Y ) → G\na continuous homomorphism. Then:\n(i) The homomorphism φ factors uniquely through the natural surjection\nπ1(Y ) \u0002 π1(Y ∗) induced by the open immersion Y ↪→ Y ∗.\n(ii) Suppose further that the covering Y → Xn arises from a product-theoretic\nopen subgroup of π1(Xn). Then the kernel of φ is product-theoretic.\nProof. First, we consider assertion (i). In the notation of the discussion preceding\nTheorem 5.6: By the log purity theorem [cf. the discussion of §2], we have a natural\nisomorphism π1(UV )\n∼→ π1(V log), where we observe that [it follows immediately\nfrom the definitions that] the interior UV of V\nlog may be identified with Xn. Thus,\nthe finite étale covering Y → Xn may be identified with the interior UW of a\nKummer log étale covering W log → V log. In particular, by the log purity theorem,\nwe conclude that the finite morphism Y ∗ → X∗n determines a Kummer log étale\ncovering (Y ∗)log → (X∗n)log. Thus, in summary, we obtain a commutative diagram\nUW = Y ↪→ W log ↪→ (Y ∗)log⏐⏐\u000b ⏐⏐\u000b ⏐⏐\u000b\nUV = Xn ↪→ V log ↪→ (X∗n)log\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 39\nin which the “hooked horizontal arrows” are open immersions; the complements of\nthe open immersions W ↪→ Y ∗, V ↪→ X∗n are of codimension ≥ 2; the vertical arrows\nare Kummer log étale coverings; by abuse of notation, we use notation for schemes\nto denote the corresponding log schemes with trivial log structure. In particular,\nit follows immediately from the fact that the log structure of (X∗n)log arises from\na divisor with normal crossings that (Y ∗)log is log smooth over k and strongly Q-\nregular. Thus, by Proposition 5.2, (iii) [which is applicable since (Y ∗)log is log\nregular and weakly Q-regular], it follows that, to complete the proof of assertion (i),\nit suffices to show that φ factors through the natural surjection π1(Y ) \u0002 π1(W ).\nNext, we claim that we may assume without loss of generality that G is abelian.\nIndeed, to show that φ factors through the natural surjection π1(Y ) \u0002 π1(W ), it\nsuffices [by Zariski-Nagata purity, since W , DW are smooth over k] to show that that\nφ(ξ) = 1 for each element ξ ∈ π1(Y ) of the inertia group of an irreducible divisor of\nDW = W\\Y . Let Ξ ⊆ π1(Y ) be the closed subgroup topologically generated by ξ;\nsuppose that φ(Ξ) \r= {1}. Then [cf. the argument applied at the beginning of the\nproof of Theorem 1.5] there exists an open subgroup Gξ ⊆ G containing φ(Ξ) such\nthat φ(Ξ) [i.e., Ξ] has nontrivial image in Gabξ . On the other hand, by applying\nassertion (i) to the situation where we replace Y by the finite étale covering Yξ → Y\ndetermined [via φ] by the open subgroup Gξ ⊆ G, replace G by Gabξ , and replace W\nby the normalization Wξ of W in Yξ [where we note that ξ ∈ π1(Yξ) still satisfies the\nproperty of being an element of an inertia group of an irreducible divisor of Wξ\\Yξ],\nwe conclude that Ξ has trivial image in Gabξ , a contradiction. This completes the\nproof of the claim.\nThus, [cf. the proof of Theorem 4.7] it suffices to verify assertion (i) in the case\nwhere G = Zl, for some prime number l. But in this case, the fact that the given\nhomomorphism π1(Y ) = π1(UW ) → G factors through π1(UW ) \u0002 π1(W ) follows\nimmediately from the exact sequence of Proposition 5.2, (v), in light of the Ql-linear\nindependence asserted in Lemma 5.5, (i). This completes the proof of assertion (i).\nNext, we consider assertion (ii). Write\nY † → Pn\nfor the normalization of Pn (⊇ Xn) in Y . Since Y ∗ is normal and maps to X∗n,\nhence to Pn, we thus obtain a birational morphism Y\n∗ → Y †. Since the morphism\nX∗n → Pn is proper, it thus follows that the morphism Y ∗ → Y † is proper. On\nthe other hand, since Y → Xn arises from a product-theoretic open subgroup of\nπ1(Xn), one verifies immediately that Y\n† is smooth over k, and that the kernel\nof the natural surjection π1(Y ) → π1(Y †) is product-theoretic. In particular, by\nZariski-Nagata purity [i.e., the classical non-logarithmic version of the “log purity\ntheorem” quoted above], we conclude that the natural surjection π1(Y\n∗) \u0002 π1(Y †)\nis an isomorphism. But this implies that the given homomorphism π1(Y ) → G\nfactors through the natural surjection π1(Y ) \u0002 π1(Y †), hence, in particular, is\nproduct-theoretic. This completes the proof of assertion (ii). ©\nRemark 5.6.1. Theorem 5.6, (ii), generalizes Theorem 4.7, in the case of proper\nhyperbolic curves, to the case where “G” is not necessarily pro-solvable. This is\n40 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\npossible precisely because, unlike the theory of §4, the theory of the present §5\nis applicable to coverings of configuration spaces that are not necessarily product-\ntheoretic [cf. the reduction in the proof of Theorem 5.6, (i), to the case where G is\nabelian].\nRemark 5.6.2. In an earlier version of the present manuscript, Theorem 5.6 was\nstated in a form that applied to affine hyperbolic curves of genus ≥ 2 as well. In\nfact, however, the proof given for affine curves was found to contain a gap which\nthe authors were unable to repair. Nevertheless, although such a gap exists for the\nanalogue of Theorem 5.6, (i), for affine hyperbolic curves, this gap does not appear\nwhen the covering Y → Xn of Theorem 5.6 is assumed to be product-theoretic\n[i.e., the situation of Theorem 5.6, (ii)], under the further assumption that G be\npro-solvable [cf. Remark 5.6.1]. In particular, it is a routine task to modify the\nargument given in the present §5 so as to yield a proof of Theorem 5.6, (ii) [under\nthe further assumption that G be pro-solvable] — i.e., a new proof of Theorem 4.7\n— for arbitrary hyperbolic curves of genus ≥ 2.\nRemark 5.6.3. It is not clear to the authors at the time of writing whether or not\none should expect the analogue of Theorem 5.6 to hold for affine hyperbolic curves.\nNevertheless, one way to think about the “extendability of coverings” property\ndiscussed in Theorem 5.6 is the following: This property holds [by Theorem 5.6]\nfor the “tautological affine curve” with a given fixed hyperbolic compactification —\ni.e., the tautological affine curve over a configuration space associated to a proper\nhyperbolic curve. That is to say, it may be the case that this sort of special\nproperty only holds for such affine curves with “universal\/tautological cusps” and\nnot for affine curves whose cusps are “non-tautological”.\nCorollary 5.7. (Non-existence of Generic Units) In the notation of Theo-\nrem 5.6, let us fix a projection morphism\nφX : Xn → B def= Xn−1\nof length 1, which allows us to regard Xn as a family of hyperbolic curves over\nB. Denote the [scheme determined by the] generic point of B by η; write k(η) for\nthe residue field of η, X def= Xn ×B η. Let k(η′) be an arbitrary field extension of\nk(η), η′ → η the resulting morphism of schemes, Y a hyperbolic curve over η′,\nand\nY → Xη′ def= X ×η η′ = Xn ×B η′\na finite étale covering over η′. Then every unit u ∈ Γ(Y ,O×Y ) on Y is constant,\ni.e., is contained in the image of k(η′) in OY .\nProof. One reduces immediately by well-known elementary algebraic geometry\narguments [i.e., replacing k(η′) by a finitely generated field extension of k(η), ex-\ntending Y → Xη′ over some variety that admits k(η′) as its function field, and\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 41\nrestricting to a closed point of this variety] to the case where the morphism η′ → η\nis finite étale. Now let us observe that by the exact sequence of Proposition 2.2,\n(i), it follows that, after possibly replacing Y by an appropriate connected finite\nétale covering of Y , we may assume without loss of generality that there exists a\ncommutative diagram\nY −→ Xn⏐⏐\u000b ⏐⏐\u000b\nC −→ B\nin which the horizontal arrows are connected finite étale coverings; the vertical\narrows are families of hyperbolic curves; the divisor of cusps DY ⊆ Y in the [relative,\ni.e., over C] compactification Y → C of Y → C forms a split finite étale covering of\nC; the covering Y → X factors through YηC def= Y ×C ηC [where ηC def= C ×B η] in\nsuch a way that the induced covering\nY → YηC\nis obtained by base-changing the curve YηC over ηC via a morphism η\n′ → ηC . In\nparticular, since the divisor of zeroes and poles of u on the compactification of the\nhyperbolic curve Y descends [by our “splitness” assumption on DY ] to a divisor\non the compactification of the hyperbolic curve YηC , it follows immediately that u\nis constant if and only if its norm relative to this covering Y → YηC [which forms\na unit on YηC ] is constant. Thus, in summary, we may assume without loss of\ngenerality that Y = YηC , and that u is a unit on YηC .\nNow since u may be regarded as a rational function on Y , the divisor of zeroes\nand poles of this rational function on Y may be written in the form\nDcusp + Dbase\n— where Dcusp is a divisor supported on DY ; D\nbase is a divisor on Y that arises as\nthe pull-back to Y of a divisor DbaseC on C. In particular, we obtain a relation\n0 = c(Dcusp) + c(Dbase) ∈ H2ét(Y , Ql(1))\n— where l is a prime number; we write c(−) for the Chern class of the line bundle\ndetermined by a divisor on a k-smooth scheme [such as Y , Y , C]. Next, let us\npull-back this relation via a section s : C → Y whose image Ds is contained in DY .\nThis yields a relation\nλ · s∗(c(Ds)) = s∗(c(Dbase)) = c(DbaseC ) ∈ H2ét(C, Ql(1))\nfor some λ ∈ Ql. On the other hand, let us observe that, relative to the notation\nof the discussion preceding Theorem 5.6, if we take UW → UV to be the covering\nY → Xn, then Y may be regarded as an open subscheme of W , and Ds ⊆ Y as a\nlog-prime divisor of W . From this point of view, it follows from Lemma 5.4, (ii),\nthat s∗(c(Ds)) is a Ql-multiple of the pull-back to C via C → B = Xn−1 of a fiber\n42 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\nclass on B = Xn−1. In particular, we conclude that c(DbaseC ) is a Ql-multiple of the\npull-back to C of a fiber class on B = Xn−1, hence that c(Dbase) ∈ H2ét(Y , Ql(1))\nis a Ql-multiple of the pull-back to Y ⊆ W of a fiber class on W .\nThus, in summary, the relation 0 = c(Dcusp) + c(Dbase) in H2ét(Y , Ql(1))\nconstitutes a Ql-linear relation between certain log-prime classes [i.e., c(D\ncusp)]\nand certain fiber classes [i.e., c(Dbase)] on Y . Since [it follows immediately from\nthe definitions that] the complement of Y in W is a [disjoint!] union of cer-\ntain log-prime divisors on W , the description of the kernel of the restriction map\nH2ét(W, Ql(1)) → H2ét(Y , Ql(1)) given in the exact sequence of Proposition 5.2, (v),\nimplies that this Ql-linear relation 0 = c(D\ncusp) + c(Dbase) in H2ét(Y , Ql(1)) arises\nfrom some Ql-linear relation in H\n2\nét(W, Ql(1)) obtained by appending some Ql-\nlinear combination of the log-prime classes arising from the log-prime divisors in\nthe complement W\\Y . On the other hand, by the Ql-linear independence asserted\nin Lemma 5.5, (i), the coefficients of such a Ql-linear relation necessarily vanish.\nIn particular, since Dcusp is a Z-linear combination of log-prime divisors of W that\nlie in Y , we thus conclude that the coefficients ∈ Z of this Z-linear combination\nvanish, i.e., that Dcusp = 0. But this amounts precisely to the assertion that the\nunit u is constant. This completes the proof of Corollary 5.7. ©\nSection 6: Nearly Abelian Groups\nIn the present §6, we discuss another approach, based on the notion of a “nearly\nabelian” profinite group, to verifying the group-theoreticity of the various fiber sub-\ngroups associated to a configuration space group.\nDefinition 6.1. We shall say that a profinite group G is nearly abelian if it\nadmits a normal closed subgroup N ⊆ G which is topologically normally generated\nby a single element ∈ G such that G\/N is almost abelian.\nProposition 6.2. (Nearly Abelian Surface Groups) Let C be a nontrivial\nfull formation. Then a pro-C surface group Π is nearly abelian if and only\nif it is a free pro-C group on two generators — i.e., it arises from a hyperbolic\ncurve which is either of type (0, 3) or of type (1, 1) [cf. Remark 1.2.2].\nProof. Since the sufficiency of the condition given in the statement of Proposi-\ntion 6.2 is immediate from the definitions, it suffices to verify the necessity of this\ncondition. Thus, we suppose that Π is nearly abelian. Now observe that for any\nl ∈ ΣC , the maximal pro-l quotient of Π is again a nearly abelian [pro-l] surface\ngroup. Thus, we may assume without loss of generality that C is primary. Write\nΣC = {l}; Π def= Π\/[Π, [Π, Π]]; Π1 def= Πab ∼= Πab; Π2 def= [Π, Π] ⊆ Π. Thus, we have\nan exact sequence\n1 → Π2 → Π → Π1 → 1\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 43\nand a surjection ∧2 Π1 \u0002 Π2. Here, we regard Π1, Π2 as finitely generated free\nZl-modules. Note [cf. Remark 1.2.2] that if Π1 is of rank d, then Π2 is of rank\n1\n2\nd(d − 1) − \u0016, where \u0016 = 0 if Π is open, and \u0016 = 1 if Π is closed. Also, we observe\nthat d ≥ 2 if \u0016 = 0, and d ≥ 4 if \u0016 = 1 [cf. Remark 1.2.2]. Thus, to complete the\nproof of Proposition 6.2, it suffices to show that d = 2.\nNext, let us observe that Π is also nearly abelian. Thus, there exists an ele-\nment γ ∈ Π such that if we write N ⊆ Π for the subgroup topologically normally\ngenerated by γ, then Π\/N is almost abelian. In particular, [since the commutator\nsubgroup of any open subgroup of Π forms an open subgroup of Π2] it follows im-\nmediately that N\n⋂\nΠ2 ⊆ Π forms an open subgroup of Π2 — i.e., a Zl-module of\nthe same rank as Π2. If γ ∈ Π2, then N\n⋂\nΠ2 = N is of rank ≤ 1, so we obtain that\n1\n2\nd(d−1)−\u0016 ≤ 1, i.e., d(d−1) ≤ 2(1+\u0016) ≤ 4, so d ≤ 2, i.e., d = 2, as desired. Thus,\nit remains to consider the case where γ \r∈ Π2. In this case, one verifies immediately\nthat N\n⋂\nΠ2 ⊆ Π2 is given by the image of the morphism\n[γ,−] : Π1 → Π2\ngiven by forming the commutator with γ. Since this morphism clearly vanishes on\nγ, its image is of rank ≤ d − 1. Thus, we conclude that\nd − 1 ≥ 1\n2\nd(d − 1) − \u0016\n— i.e., that 2 · \u0016 ≥ (d − 1)(d − 2), which implies that d ≤ 2 if \u0016 = 0, and d ≤ 3 if\n\u0016 = 1. But this is enough to conclude that d = 2 [and \u0016 = 0]. This completes the\nproof of Proposition 6.2. ©\nCorollary 6.3. (Group-theoreticity of Projections of Configuration\nSpaces II) Let C be a PT-formation. For ξ = α, β, let Xξ be a hyperbolic\ncurve whose type is neither (0, 3) nor (1, 1) over an algebraically closed field\nkξ of characteristic zero; nξ ≥ 1 an integer; Xξnξ the nξ-th configuration\nspace associated to Xξ; Πξ\ndef\n= πC1 (X\nξ\nnξ\n); Eξ the index set of X\nξ\nnξ\n. Let\nγ : Πα\n∼→ Πβ\nbe an isomorphism of profinite groups. Then γ induces a bijection σ :\nEα\n∼→ Eβ [so nα = nβ] such that\nγ(Fα) = Fβ\nfor all fiber subgroups Fα ⊆ Πα, Fβ ⊆ Πβ, whose respective profiles E′α ⊆ Eα,\nE′β ⊆ Eβ correspond via σ.\nProof. Just as in the proof of Corollary 4.8, to complete the proof of Corollary 6.3,\nit suffices to verify that the image via γ of any fiber subgroup of Πα of co-length\n44 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\none is contained in a fiber subgroup of Πβ of co-length one. For j = 1, . . . , nξ, let\nus write\nK\nξ\nj ⊆ Πξ\nfor the fiber subgroup ⊆ Πξ of co-length one with co-profile given by the element of\nEξ labeled by j, and\nJ\nξ\nj ⊆ Πξ\nfor the fiber subgroup ⊆ Πξ of length one with profile given by the element of Eξ\nlabeled by j. Thus, [cf. Proposition 2.4, (vi)] to complete the proof of Corollary\n6.3, it suffices to verify the following statement [in general]:\nFor each i ∈ Eα, there exists a j ∈ Eβ such that Jβj′ ⊆ γ(Kαi ) for all\nj′ ∈ Eβ such that j′ \r= j.\nTo verify this statement, we reason as follows: Write\nφ : Πβ\n∼→ Πα \u0002 G def= Πα\/Kαi\nfor the surjection determined by γ−1. Then it suffices to show that there do not exist\ntwo distinct elements j1, j2 ∈ Eβ such that Jβj1 , Jβj2 have nontrivial image under\nφ. Thus, let us suppose that the images J1, J2 of J\nβ\nj1\n, J\nβ\nj2\nunder φ are nontrivial.\nSince J\nβ\nj1\n, J\nβ\nj2\nare topologically finitely generated normal closed subgroups of Πβ, it\nfollows from Theorem 1.5 that J1, J2 are open in G [cf. Remark 3.3.2]. Moreover,\nby Proposition 2.4, (v), it follows that there exists a normal closed subgroup N ⊆ G\nthat is topologically normally generated by a single element such that the images\nof J1, J2 in G\/N commute. Thus, G\/N contains an abelian open subgroup, i.e., is\nalmost abelian. On the other hand, since Kαi ⊆ Πα is a fiber subgroup of co-length\none, it follows that G = Πα\/Kαi is a surface group. Thus, in summary, we conclude\nthat G is a nearly abelian surface group, which, by Proposition 6.2, contradicts\nour hypothesis concerning the type of the hyperbolic curve Xα. This completes the\nproof of Corollary 6.3. ©\nRemark 6.3.1. Unlike the case with Corollary 4.8, it seems unrealistic at the\ntime of writing to extend the technique of the proof of Corollary 6.3 to the case\nof arbitrary product-theoretic open subgroups ⊆ πC1 (Xξnξ) [cf. Corollary 4.8], since\nthis would require an analogue of Proposition 6.2 for surface groups that become\nalmost abelian after forming the quotient by a subgroup topologically normally\ngenerated by a very large number of elements [roughly, on the order of the index of\nthe product-theoretic open subgroups under consideration].\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 45\nSection 7: A Discrete Analogue\nIn the present §7, we discuss various consequences of Theorems 4.7, 5.6 and\nCorollaries 4.8, 6.3 [cf. Corollaries 7.3, 7.4 below] for the topological fundamental\ngroups of configuration spaces over the complex number field C.\nIn the following discussion, if Z is a connected scheme of finite type over C,\nthen we shall use the notation\nπ\ntop\n1 (Z)\nto denote the “topological fundamental group” [i.e., the fundamental group in the\nusual sense of algebraic topology], for some choice of basepoint, of the topological\nspace of C-rational points Z(C) [equipped with the topology determined by the\ntopology of C].\nLet X be a hyperbolic curve over C, n ≥ 1 an integer. Write Xn ⊆ Pn for\nthe n-th configuration space associated to X [cf. the notation of Definition 2.1,\n(i)]. Since the complement in a connected complex manifold of any submanifold of\n[complex] codimension ≥ 1 is clearly connected, it thus follows that the inclusion\nXn ↪→ Pn induces a natural surjection πtop1 (Xn) \u0002 πtop1 (Pn).\nThe following “discrete analogue” of [a certain portion of] Proposition 2.2 is\nwell-known:\nProposition 7.1. (Topological Fundamental Groups of Configuration\nSpaces) In the notation of the above discussion:\n(i) Any projection morphism Xn → Xn−1 of length one determines a natural\nexact sequence\n1 → πtop1 ((Xn)x) → πtop1 (Xn) → πtop1 (Xn−1) → 1\n[where we write X0\ndef\n= Spec(C); x is a C-valued geometric point of Xn−1].\n(ii) The natural morphism\nπ\ntop\n1 (Xn) → π1(Xn)\nto the étale fundamental group π1(Xn) is injective, i.e., π\ntop\n1 (Xn) is residually\nfinite.\nProof. Assertion (i) is discussed, for instance, in [Birm], Theorem 1.4. To verify\nassertion (ii), observe that [by induction on n] it follows from the exact sequences of\nassertion (i) and the analogue of assertion (i) for π1(Xn) [cf. Proposition 2.2, (i)],\nthat we may assume without loss of generality that n = 1. Now let us recall that\nπ\ntop\n1 (X) may be embedded [by considering the well-known uniformization of X(C)\nby the upper half-plane] into SL2(R)\/{±1}. That is to say, πtop1 (X) is a “finitely\n46 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\ngenerated linear group”, so the desired residual-finiteness follows from a well-known\ntheorem of Mal’cev [cf., e.g., [Wehr], Theorem 4.2]. ©\nDefinition 7.2.\n(i) We shall refer to a subgroup H ⊆ πtop1 (Xn) as being product-theoretic if\nH arises as the inverse image via the natural surjection π\ntop\n1 (Xn) \u0002 π\ntop\n1 (Pn) of a\nsubgroup of π\ntop\n1 (Pn).\n(ii) Write E for the index set of Xn. Let E\n′ ⊆ E be a subset of cardinality\nn′; E′′ def= E\\E′; n′′ def= n − n′; pE′ = pE′′ : Xn → Xn′′ the projection morphism of\nprofile E′. Then we shall refer to the kernel\nF ⊆ πtop1 (Xn)\nof the induced surjection π\ntop\n1 (Xn) \u0002 π\ntop\n1 (Xn′′) [cf. Proposition 7.1, (i)] as the\nfiber subgroup of π\ntop\n1 (Xn) of profile E\n′.\nRemark 7.2.1. Note that by the injectivity of Proposition 7.1, (ii), it fol-\nlows immediately that the fiber subgroup of profile E′ of πtop1 (Xn) [cf. the no-\ntation of Definition 7.2, (ii)] is equal to the inverse image via the natural injection\nπ\ntop\n1 (Xn) ↪→ π1(Xn) of Proposition 7.1, (ii), of the fiber subgroup of π1(Xn) of\nprofile E′.\nCorollary 7.3. (Discrete Extendability of Coverings) Let X be a hyper-\nbolic curve of genus ≥ 2 over C; n ≥ 1 an integer; Xn ⊆ Pn the n-th configu-\nration space associated to X; (X∗n)\nlog def= Z logn the n-th log configuration space\nassociated to X;\nY → Xn\na finite étale morphism, where Y is connected; Y ∗ → X∗n the normalization\nof X∗n in Y ; G a strongly torsion-free profinite group;\nφ : π\ntop\n1 (Y ) → G\na homomorphism [of abstract groups!]. Then:\n(i) Suppose that X is proper. Then the homomorphism φ factors uniquely\nthrough the natural morphism π\ntop\n1 (Y ) → πtop1 (Y ∗) induced by the open immersion\nY ↪→ Y ∗.\n(ii) Suppose that one of the following holds: (a) X is proper; (b) G is pro-\nsolvable. Also, let us suppose that the covering Y → Xn arises from a product-\ntheoretic subgroup of finite index of π\ntop\n1 (Xn). Then the kernel of φ is product-\ntheoretic.\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 47\nProof. Indeed, since the homomorphism φ : π\ntop\n1 (Y ) → G necessarily factors\nthrough the profinite completion of π\ntop\n1 (Y ), the conclusion of assertion (i) (respec-\ntively, assertion (ii) when (a) holds; assertion (ii) when (b) holds) follows immedi-\nately from Theorem 5.6, (i) (respectively, Theorem 5.6, (ii); Theorem 4.7). ©\nCorollary 7.4. (Group-theoreticity of Projections of Configuration\nSpaces III) For ξ = α, β, let Xξ be a hyperbolic curve over C whose type is\nneither (0, 3) nor (1, 1); nξ ≥ 1 an integer; Xξnξ the nξ-th configuration space\nassociated to Xξ; Eξ the index set of X\nξ\nnξ\n; Hξ ⊆ Πξ def= πtop1 (Xξnξ) a product-\ntheoretic subgroup of finite index. Let\nγ : Hα\n∼→ Hβ\nbe an isomorphism of groups. Moreover, if either Hα \r= Πα or Hβ \r= Πβ, then\nwe assume that Xξ is of genus ≥ 2, for ξ = α, β. Then γ induces a bijection\nσ : Eα\n∼→ Eβ [so nα = nβ] such that\nγ(Fα\n⋂\nHα) = Fβ\n⋂\nHβ\nfor all fiber subgroups Fα ⊆ Πα, Fβ ⊆ Πβ, whose respective profiles E′α ⊆ Eα,\nE′β ⊆ Eβ correspond via σ.\nProof. Indeed, in light of Remark 7.2.1, Corollary 7.4 follows immediately by\napplying Corollaries 4.8, 6.3 to the isomorphism induced by γ between the profinite\ncompletions of Hα, Hβ. ©\nRemark 7.4.1. There is a partial overlap between the content of Corollary 7.4\nabove and Theorems 1, 2 of [IIM].\nBibliography\n[Birm] J. Birman, Braids, Links, and Mapping Class Groups, Princeton University\nPress (1974).\n[DM] P. Deligne and D. Mumford, The Irreducibility of the Moduli Space of Curves\nof Given Genus, IHES Publ. Math. 36 (1969), pp. 75-109.\n[Fulton] W. Fulton, Intersection Theory, Springer-Verlag (1984).\n[EGA III] A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collabo-\nration de Jean Dieudonné): III. Étude cohomologique des faisceaux cohérents,\nPremière partie, Publ. Math. IHES 11 (1961), pp. 5-167.\n48 SHINICHI MOCHIZUKI AND AKIO TAMAGAWA\n[EGA IV] A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collabora-\ntion de Jean Dieudonné): IV. Étude locale des schémas et des morphismes des\nschémas, Seconde partie, Publ. Math. IHES 24 (1965), pp. 5-231.\n[FJ] M. Fried and M. Jarden, Field Arithmetic (second edition), Springer-Verlag\n(2005).\n[Ill] L. Illusie, An Overview of the Work of K. Fujiwara, K. Kato and C. Nakayama\non Logarithmic Etale Cohomology, in Cohomologies p-adiques et applications\narithmétiques (II), P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, M. Rapoport,\neds, Asterisque 279 (2002), pp. 271-322.\n[IIM] E. Irmak, N. Ivanov, J. D. McCarthy, Automorphisms of surface braid groups,\npreprint (arXiv:math.GT\/0306069v1 3 Jun 2003).\n[Kato1] K. Kato, Logarithmic Structures of Fontaine-Illusie, Proceedings of the First\nJAMI Conference, Johns Hopkins Univ. Press (1990), pp. 191-224.\n[Kato2] K. Kato, Toric Singularities, Amer. J. Math. 116 (1994), pp. 1073-1099.\n[Knud] F. F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, II,\nMath. Scand. 52 (1983), pp. 161-199.\n[Milne] J. S. Milne, Étale Cohomology, Princeton Mathematical Series 33, Princeton\nUniversity Press (1980).\n[Mzk1] S. Mochizuki, Extending Families of Curves over Log Regular Schemes, J. reine\nangew. Math. 511 (1999), pp. 43-71.\n[Mzk2] S. Mochizuki, Topics Surrounding the Anabelian Geometry of Hyperbolic Curves,\nGalois Groups and Fundamental Groups, Mathematical Sciences Research In-\nstitute Publications 41, Cambridge University Press (2003), pp. 119-165.\n[Mzk3] S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois\nTheory and Modular Forms, Kluwer Academic Publishers (2003), pp. 77-122.\n[Mzk4] S. Mochizuki, The Absolute Anabelian Geometry of Canonical Curves, Kazuya\nKato’s fiftieth birthday, Doc. Math. 2003, Extra Vol., pp. 609-640.\n[Mzk5] S. Mochizuki, Galois Sections in Absolute Anabelian Geometry, Nagoya Math.\nJ. 179 (2005), pp. 17-45.\n[Mzk6] S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku\nMath. J. 59 (2007), pp. 455-479.\n[Mzk7] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves,\nJ. Math. Kyoto Univ. 47 (2007), pp. 451-539.\n[Naka] H. Nakamura, Galois rigidity of pure sphere braid groups and profinite calculus,\nJ. Math. Sci. Univ. Tokyo 1 (1994), pp. 71-136.\n[NT] H. Nakamura and N. Takao, Galois Rigidity of pro-l Pure Braid Groups of\nAlgebraic Curves, Trans. Amer. Math. Soc. 350 (1998), pp. 1079-1102.\n[RZ] L. Ribes and P. Zaleskii, Profinite Groups, Ergebnisse der Mathematik und\nihrer Grenzgebiete 3, Springer-Verlag (2000).\nTHE PROFINITE FUNDAMENTAL GROUPS OF CONFIGURATION SPACES 49\n[Stix] J. Stix, A monodromy criterion for extending curves, Int. Math. Res. Not. 29\n(2005), pp. 1787-1802.\n[Vid] I. Vidal, Contributions à la cohomologie étale des schémas et des log schémas,\nthèse (2002).\n[Wehr] B. Wehrfritz, Infinite Linear Groups, Ergebnisse der Mathematik und ihrer\nGrenzgebiete 76, Springer-Verlag (1973).\n"}
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{"id":"tS3y2116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Galois%20Sections.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":64,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Galois%20Sections","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Galois Sections.pdf"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 07:03:15 UTC 2011","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-7443309609369345872,"exact_signature_unique_b":true,"fuzzy_signature_l":4933152672501951646,"fuzzy_signature_unique_b":true,"wordcount_i":8935,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:55 UTC 2022","fresh_date_dt":"Thu Sep 23 21:43:14 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":219399,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086042322567168,"text_t":"Galois Sections in Absolute\nAnabelian Geometry\nShinichi Mochizuki\nIntroduction\nIn this paper, we continue our study of the absolute anabelian geometry of\nhyperbolic curves over p-adic local fields [i.e., finite extensions of the field of p-adic\nnumbers, for some prime number p], begun in [Mzk2], [Mzk3]. In [Mzk3], Theorem\n2.4, it was shown, as a consequence of the main theorem of [Mzk1], that certain\ncategories of finite étale correspondences associated to a hyperbolic curve XK over a\np-adic local field K may be recovered from the profinite group structure of the étale\nfundamental group ΠXK of XK . In the present paper, we generalize this result to\nshow [again as a consequence of the main theorem of [Mzk1]] that certain categories\nof arbitrary dominant [i.e., not necessarily finite étale] correspondences associated\nto XK may be recovered from the profinite group structure of ΠXK [cf. Theorem\n2.3]. We then apply this result to study the extent to which the decomposition\ngroups associated to closed points of XK may be recovered from the profinite group\nstructure of ΠXK [cf. Corollaries 2.5, 2.6, 3.2]. One result that is representative of\nthese techniques is the following special case of Corollary 3.2:\nTheorem A. Let K be a finite extension of Qp; XK a hyperbolic curve of\ngenus zero over K which is, in fact, defined over a number field. Write ΠXK\nfor the étale fundamental group of XK. Then any automorphism of the profinite\ngroup ΠXK preserves the decomposition groups ⊆ ΠXK associated to the\nclosed points of XK.\nThis result may be regarded as a sort of [very] weak version of the “Section Con-\njecture” [cf., e.g., [Mzk1], §19 for more on the “Section Conjecture”]. Finally, in\n§4, we show, in the notation of Theorem A, that various canonical auxiliary struc-\ntures associated to the decomposition groups of cusps of XK are also preserved by\narbitrary automorphisms of ΠXK [cf. Corollary 4.11].\nAcknowledgements:\nI would like to thank Akio Tamagawa for various useful comments, especially\nconcerning the statement of Theorem A; Corollaries 2.8, 3.2 and the proof of Lemma\n4.6.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\nSection 0: Notations and Conventions\nNumbers:\nIf p is a prime number, then we shall denote by Qp the field of p-adic numbers,\ni.e., the completion of the field of rational numbers Q with respect to the p-adic\nvaluation of Q. We shall refer to a field which is isomorphic to a finite extension\nof Qp for some p as a local field. [In particular, in this paper, all “local fields” are\nnonarchimedean.] A number field is defined to be a finite extension of the field of\nrational numbers Q.\nTopological Groups:\nLet G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us\nwrite\nZG(H)\ndef\n= {g ∈ G | g · h = h · g, ∀ h ∈ H}\nfor the centralizer of H in G;\nNG(H)\ndef\n= {g ∈ G | g · H · g−1 = H}\nfor the normalizer of H in G; and\nCG(H)\ndef\n= {g ∈ G | (g · H · g−1)\n⋂\nH has finite index in H, g · H · g−1}\nfor the commensurator of H in G. Note that: (i) ZG(H), NG(H) and CG(H) are\nsubgroups of G; (ii) we have inclusions\nH, ZG(H) ⊆ NG(H) ⊆ CG(H)\nand (iii) H is normal in NG(H). If H = CG(H), then we shall say that H is\ncommensurably terminal in G. Note that ZG(H), NG(H) are always closed in G,\nwhile CG(H) is not necessarily closed in G.\nIf G1, G2 are Hausdorff topological groups, then an outer homomorphism\nG1 → G2 is defined to be an equivalence class of continuous homomorphisms\nG1 → G2, where two such homomorphisms are considered equivalent if they differ\nby composition with an inner automorphism of G2.\nCategories:\nLet C be a category. We shall denote the collections of objects and arrows of C\nby\nOb(C); Arr(C)\nrespectively. If A ∈ Ob(C) is an object of C, then we shall denote by\nCA\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 3\nthe category whose objects are morphisms B → A of C and whose morphisms (from\nan object B1 → A to an object B2 → A) are A-morphisms B1 → B2 in C.\nWe shall refer to a natural transformation between functors [from one category\nto another] all of whose component morphisms are isomorphisms as an isomorphism\nbetween the functors in question. A functor φ : C1 → C2 between categories C1, C2\nwill be called rigid if φ has no nontrivial automorphisms. A category C will be\ncalled slim if the natural functor CA → C is rigid, for every A ∈ Ob(C).\nGiven two arrows fi : Ai → Bi (where i = 1, 2) in a category C, we shall refer\nto a commutative diagram\nA1\n∼→ A2⏐⏐\u0004f1\n⏐⏐\u0004f2\nB1\n∼→ B2\n— where the horizontal arrows are isomorphisms in C — as an abstract equivalence\nfrom f1 to f2. If there exists an abstract equivalence from f1 to f2, then we shall\nsay that f1, f2 are abstractly equivalent and write f1\nabs≈ f2.\nLet G be a profinite group. Then we recall that the category B(G) of finite sets\nwith continuous G-action and morphisms of G-sets is slim if and only if ZG(H) =\n{1} for all open subgroups H ⊆ G.\nCurves:\nSuppose that g ≥ 0 is an integer. Then if S is a scheme, a family of curves of\ngenus g\nX → S\nis defined to be a smooth, proper, geometrically connected morphism of schemes\nX → S whose geometric fibers are curves of genus g.\nSuppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denote\nthe moduli stack of r-pointed stable curves of genus g (where we assume the points\nto be unordered) by Mg,r [cf. [DM], [Knud] for an exposition of the theory of such\ncurves; strictly speaking, [Knud] treats the finite étale covering of Mg,r determined\nby ordering the marked points]. The open substack Mg,r ⊆ Mg,r of smooth curves\nwill be referred to as the moduli stack of smooth r-pointed stable curves of genus g\nor, alternatively, as the moduli stack of hyperbolic curves of type (g, r). The divisor\nat infinity Mg,r\\Mg,r of Mg,r determines a log structure on Mg,r; denote the\nresulting log stack by Mlogg,r.\nA family of hyperbolic curves of type (g, r)\nX → S\nis defined to be a morphism which factors X ↪→ Y → S as the composite of an\nopen immersion X ↪→ Y onto the complement Y \\D of a relative divisor D ⊆ Y\n4 SHINICHI MOCHIZUKI\nwhich is finite étale over S of relative degree r, and a family Y → S of curves of\ngenus g. One checks easily that, if S is normal, then the pair (Y, D) is unique up\nto canonical isomorphism. (Indeed, when S is the spectrum of a field, this fact is\nwell-known from the elementary theory of algebraic curves. Next, we consider an\narbitrary connected normal S on which a prime l is invertible (which, by Zariski\nlocalization, we may assume without loss of generality). Denote by S′ → S the fi-\nnite étale covering parametrizing orderings of the marked points and trivializations\nof the l-torsion points of the Jacobian of Y . Note that S′ → S is independent of\nthe choice of (Y, D), since (by the normality of S), S′ may be constructed as the\nnormalization of S in the function field of S′ (which is independent of the choice\nof (Y, D) since the restriction of (Y, D) to the generic point of S has already been\nshown to be unique). Thus, the uniqueness of (Y, D) follows by considering the\nclassifying morphism (associated to (Y, D)) from S′ to the finite étale covering of\n(Mg,r)Z[ 1l ] parametrizing orderings of the marked points and trivializations of the\nl-torsion points of the Jacobian [since this covering is well-known to be a scheme,\nfor l sufficiently large].) We shall refer to Y (respectively, D; D; D) as the compact-\nification (respectively, divisor at infinity; divisor of cusps; divisor of marked points)\nof X . A family of hyperbolic curves X → S is defined to be a morphism X → S\nsuch that the restriction of this morphism to each connected component of S is a\nfamily of hyperbolic curves of type (g, r) for some integers (g, r) as above. If the\ndivisor of cusps of a family of hyperbolic curves X → S forms a split finite étale\ncovering over S, then we shall say that this family of hyperbolic curves is cuspidally\nsplit. A family of hyperbolic curves X → S of type (0, 3) (respectively, (1, 1)) will\nbe referred to as a tripod (respectively, once-punctured elliptic curve).\nIf XK (respectively, YL) is a hyperbolic curve over a field K (respectively, L),\nthen we shall say that XK is isogenous to YL if there exists a hyperbolic curve ZM\nover a field M together with finite étale morphisms ZM → XK , ZM → YL.\nSection 1: Brief Review of Anabelian Geometry\nLet K, L be local fields [cf. §0]; XK (respectively, YL) a hyperbolic curve [cf. §0]\nover K (respectively, L). Any choice of basepoint for XK determines, up to inner\nautomorphism, the étale fundamental group ΠXK\ndef\n= π1(XK) of XK . Moreover,\nΠXK fits into a natural exact sequence\n1 → ΔX → ΠXK → GK → 1\nwhere GK is the absolute Galois group of K; ΔX , which is often referred to as the\ngeometric fundamental group of XK , is defined so as to make the sequence exact.\nAny choice of basepoint for YL determines a similar exact sequence for YL.\nProposition 1.1. (First Properties)\n(i) ΠXK is slim [cf. §0].\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 5\n(ii) Every isomorphism of profinite groups α : ΠXK\n∼→ ΠYL fits into a unique\ncommutative diagram\nΠXK\nα−→ ΠYL⏐⏐\u0004\n⏐⏐\u0004\nGK −→ GL\nwhere the vertical arrows are the surjections of the natural exact sequence(s) dis-\ncussed above; the horizontal arrows are isomorphisms.\nProof. Assertion (i) (respectively, (ii)) follows from [Mzk2], Lemma 1.3.1 (respec-\ntively, [Mzk2], Lemma 1.3.8). ©\nTheorem 1.2. (Anabelian Theorem for Hyperbolic Curves over Local\nFields) The étale fundamental group functor determines a bijection between the set\nof dominant morphisms of schemes\nXK → YL\nand the set of open outer homomorphisms φ : ΠXK → ΠYL that fit into a\ncommutative diagram\nΠXK\nφ−→ ΠYL⏐⏐\u0004\n⏐⏐\u0004\nGK −→ GL\nfor which the induced morphism GK → GL is an open immersion [i.e., an iso-\nmorphism onto an open subgroup of GL] which arises from an embedding of fields\nL ↪→ K.\nProof. Recall that given a local field M , the topology of M may be always be\nrecovered solely from the field structure of M by observing that the ring of integers\nOM of M is additively generated by O×M , and that O×M ⊆ M is equal to the\nsubgroup of elements of M× that are infinitely divisible by powers of some prime\nnumber. In particular, the Qp-algebra structure of M [for some suitable prime\nnumber p], as well as the prime number p itself [i.e., the unique prime number l\nsuch that OM is not infinitely divisible by powers of l], may be recovered from the\nfield structure of M . In a similar vein, given a function field in one variable M ′\nover M , consideration of the discrete valuations on M ′ with trivial restriction to\nM reveals that the subfield M ⊆ M ′ may be recovered — solely from the field\nstructure of M ′ — as the subfield generated by the elements of (M ′)× that are\ninfinitely divisible by powers of some prime number. In light of these remarks,\nTheorem 1.2 follows formally from [Mzk1], Theorem A. ©\nNext, let us write XK ↪→ XK for the compactification [cf. §0] of XK . Let\nx ∈ XK\n6 SHINICHI MOCHIZUKI\nbe a closed point. Thus, x determines, up to conjugation by an element of ΠXK , a\ndecomposition group:\nDx ⊆ ΠXK\nWe shall refer to a closed subgroup of ΠXK which arises in this way as a decom-\nposition group of ΠXK . If x is a cusp, then we shall refer to the decomposition\ngroup Dx as cuspidal. Note that Dx always surjects onto an open subgroup of GK .\nMoreover, the subgroup\nIx\ndef\n= Dx\n⋂\nΔX\nis isomorphic to Ẑ(1) [i.e., the profinite completion of Z, Tate twisted once] (respec-\ntively, {1}) if x is (respectively, is not) a cusp. We shall refer to a closed subgroup of\nΠXK which is equal to “Ix” for some cusp x as a cuspidal geometric decomposition\ngroup.\nTheorem 1.3. (Decomposition Groups)\n(i) (Determination of the Point) The closed point x is completely deter-\nmined by the conjugacy class of the closed subgroup Dx ⊆ ΠXK . If x is a cusp, then\nx is completely determined by the conjugacy class of the closed subgroup Ix ⊆ ΠXK .\n(ii) (Commensurable Terminality) The subgroup Dx is commensurably ter-\nminal in ΠXK . If x is a cusp, then Dx = CΠXK (H) for any open subgroup H ⊆ Ix.\n(iii) (Absoluteness of Cuspidal Decomposition Groups) Every isomor-\nphism of profinite groups\nα : ΠXK\n∼→ ΠYL\npreserves cuspidal decomposition groups and cuspidal geometric decomposition groups.\n(iv) (Cuspidal and Noncuspidal Decomposition Groups) No noncuspi-\ndal decomposition group of ΠXK is contained in a cuspidal decomposition group of\nΠXK .\nProof. The first half of assertion (i) follows, for instance, formally from [Mzk1],\nTheorem C; the second half of assertion (i) follows from elementary facts about\nfundamental groups of topological surfaces. Assertion (ii) follows formally from\nassertion (i) and the definition of a “decomposition group”. Assertion (iii) follows\nfrom assertion (ii) and [Mzk2], Lemma 1.3.9. As for assertion (iv), we may assume,\nby passing to a finite étale covering of XK , that XK is of genus ≥ 2, so that XK is\nstill hyperbolic. Then assertion (iv) follows from assertion (i). ©\nSection 2: Categories of Dominant Morphisms\nLet XK be a hyperbolic curve over a field K. Write XK ↪→ XK for the\ncompactification of XK .\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 7\nDefinition 2.1.\n(i) We shall refer to an open immersion\nXK ↪→ YK\nas a partial compactification, or PC, for short, of XK if the natural open immersion\nXK ↪→ XK factors as the composite of the given morphism XK ↪→ YK with some\nopen immersion YK ↪→ XK . By abuse of notation, we shall also often speak of\n“YK” as a PC of XK .\n(ii) If XK ↪→ YK is a PC such that YK is a hyperbolic curve, then we shall say\nthat XK ↪→ YK [or YK ] is a hyperbolic partial compactification, or HPC, of XK .\n(iii) If XK ↪→ YK is a PC such that the arrow “↪→” is an isomorphism, then\nwe shall say that XK ↪→ YK [or YK ] is a trivial partial compactification of XK .\nNow we define a “category of dominant localizations”\nDLoc(XK)\nassociated to the hyperbolic curve XK as follows: The objects of this category are\nthe hyperbolic partial compactifications\nY ↪→ Z\nwhere Y is a hyperbolic curve over some field [which is necessarily a finite separable\nextension of K] that arises as a finite étale covering Y → XK . The morphisms of\nthis category from an object Y ↪→ Z to an object Y ′ ↪→ Z ′ are diagrams of the\nform\nY Y ′⏐⏐\u0004\n⏐⏐\u0004\nZ −→ Z ′\nwhere the vertical morphisms are the given morphisms, and the horizontal mor-\nphism is a dominant morphism of schemes. By abuse of notation, we shall often\nsimply refer to the horizontal arrow Z → Z ′ as being a morphism of DLoc(XK).\nSimilarly, by stipulating that all schemes appearing in the definition of the cat-\negory DLoc(XK) given in the preceding paragraph be equipped with K-structures\n[where we take the K-structure on XK to be the given K-structure] and that all\nmorphisms be K-morphisms, we obtain a category\nDLocK(XK)\ntogether with a natural faithful functor DLocK(XK) → DLoc(XK).\n8 SHINICHI MOCHIZUKI\nRemark 2.2.0. Thus, the category DLoc(XK) is reminiscent of the category\n“Loc(XK)” of [Mzk3], §2. Indeed, there is a natural faithful functor\nLoc(XK) → DLoc(XK)\nwhose essential image consists of the objects Y ↪→ Z which are trivial partial\ncompactifications and the dominant morphisms Z → Z ′ which are finite étale. In\nparticular, if we denote by Ét(XK) the category of finite étale coverings of XK and\nmorphisms over XK , then we have natural faithful functors:\nÉt(XK) → Loc(XK) → DLoc(XK)\nSimilarly, we have natural faithful functors: Ét(XK) → LocK(XK) → DLocK(XK).\nProposition 2.2. (Slimness of the Category of Dominant Localizations)\nSuppose that K is a local field. Then the categories DLoc(XK), DLocK(XK) are\nslim.\nProof. Indeed, by using the various copies of “Ét(Z)” [where, say, Y ↪→ Z is an\nobject of DLoc(XK)] lying inside DLoc(XK), DLocK(XK) [cf. Remark 2.2.0], the\nslimness of the categories DLoc(XK), DLocK(XK) follows formally from Proposi-\ntion 1.1, (i) [cf. also the discussion of slimness in §0]. ©\nNext, let us consider the category DLocGK (ΠXK ) defined as follows: An object\nof this category is a surjection of profinite groups\nH \u0002 J\nwhere H ⊆ ΠXK is an open subgroup; J is the quotient of H by the closed normal\nsubgroup generated by some collection of cuspidal geometric decomposition groups;\nand we assume that J is “hyperbolic”, in the sense that the image of ΔX\n⋂\nH in\nJ is nonabelian. Given two objects Hi \u0002 Ji, where i = 1, 2, of this category, a\nmorphism in this category is defined to be a diagram of the form\nH1 H2⏐⏐\u0004\n⏐⏐\u0004\nJ1 −→ J2\nwhere the vertical morphisms are the given morphisms, and the horizontal mor-\nphism is an open outer homomorphism that is compatible with the various natural\n[open] outer homomorphisms from the Hi, Ji to GK .\nNow we have the following analogue of [Mzk3], Theorem 2.4:\nTheorem 2.3. (Group-theoreticity of the Category of Dominant Local-\nizations) Let K, L be local fields; XK (respectively, YL) a hyperbolic curve\nover K (respectively, L). Then:\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 9\n(i) The étale fundamental group functor determines equivalences of cate-\ngories\nDLocK(XK)\n∼→ DLocGK (ΠXK ); DLocL(YL) ∼→ DLocGL(ΠYL)\n(ii) Every isomorphism of profinite groups\nα : ΠXK\n∼→ ΠYL\ninduces an equivalence of categories\nDLocGK (ΠXK )\n∼→ DLocGL(ΠYL)\nhence also [by applying the equivalences of (i)] an equivalence of categories\nDLocK(XK)\n∼→ DLocL(YL)\nin a fashion that is functorial, up to unique isomorphisms of equivalences of cat-\negories, with respect to α.\nProof. Indeed, assertion (i) follows formally from Theorem 1.2, while assertion\n(ii) follows, in light of Proposition 1.1, (ii); Theorem 1.3, (iii), formally from the\ndefinition of the categories “DLocGK (ΠXK )”, “DLocGL(ΠYL)”. [Here, we note that\nthe uniqueness of the isomorphisms of equivalences of categories involved follows\nfrom Proposition 2.2.] ©\nNext, let\nDx ⊆ ΠXK\nbe a decomposition group associated to some closed point x ∈ XK .\nDefinition 2.4. We shall say that x or Dx is of DLoc-type if Dx admits an\nopen subgroup that arises as the image via a morphism Z → XK of DLocK(XK)\nof some cuspidal decomposition group of ΠZ .\nCorollary 2.5. (Group-theoreticity of Decomposition Groups of DLoc-\ntype) In the notation of Theorem 2.3, the isomorphism\nα : ΠXK\n∼→ ΠYL\npreserves the decomposition groups of DLoc-type.\nProof. This follows immediately from the definitions; Theorem 2.3 [and its proof];\nTheorem 1.3, (ii), (iii). ©\n10 SHINICHI MOCHIZUKI\nCorollary 2.6. (The Case of Once-punctured Elliptic Curves) In the\nnotation of Theorem 2.3, let us suppose further that XK, YL are once-punctured\nelliptic curves. Then the isomorphism\nα : ΠXK\n∼→ ΠYL\npreserves the decomposition groups of the “torsion closed points” — i.e., the closed\npoints that arise from torsion points of the underlying elliptic curve. Moreover,\nthe resulting bijection between torsion closed points of XK, YL is compatible with\nthe isomorphism on abelianizations of geometric fundamental groups ΔabX\n∼→ ΔabY\n— i.e., “Tate modules” — induced by α.\nProof. Indeed, if n ≥ 1 is an integer, write\nφ : ZK → XK\nfor the finite étale covering determined by “multiplication by n”. Note that this\ncovering may also be described more group-theoretically as the covering associated\nto the open subgroup H ⊆ ΠXK [which is easily verified to be unique, up to\nconjugation in ΠXK ] such that: (i) H contains a cuspidal decomposition group of\nΠXK ; (ii) H\n⋂\nΔX is equal to the inverse image in ΔX of the subgroup n·ΔabX ⊆ ΔabX .\nObserve that ZK admits XK as an HPC, by “filling in” all of the cusps other\nthat the “origin”. Thus, we obtain an open immersion\nψ : ZK ↪→ XK\n— i.e., an object of DLocK(XK), which exhibits the closed points of XK that arise\nfrom n-torsion points of the underlying elliptic curve as closed points of DLoc-type\ntype. Thus, by transporting φ, ψ via the equivalences of Theorem 2.3, (i), and\napplying Theorem 1.3, (ii), (iii) [as in the proof of Corollary 2.5], we conclude that\nα preserves the decomposition groups of the torsion closed points. Finally, the\ncompatibility with the induced morphism on Tate modules follows by considering\nthe automorphisms of ZK over [i.e., relative to φ] XK , after possibly enlarging K.\n©\nDefinition 2.7. We shall say that a closed point x ∈ XK is algebraic if, for\nsome finite extension L of K, some hyperbolic curve YF over a number field F ⊆ L,\nand some L-isomorphism XL\n∼→ YL [where XL def= XK ×K L, YL def= YF ×F L], x\nlies under a closed point xL ∈ XL which maps to a closed point of Y F under the\ncomposite XL\n∼→ Y L → Y F .\nRemark 2.7.1. One verifies immediately that if a closed point x ∈ XK is\nalgebraic, then given any L′-isomorphism\nXL′\n∼→ Y ′L′\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 11\n[where XL′\ndef\n= XK ×K L′; Y ′L′ def= Y ′F ′ ×F ′ L′; L′ is a finite extension of K; Y ′F ′ is a\nhyperbolic curve over a number field F ′ ⊆ L′], it holds that any point xL′ ∈ XL′\nlying over x maps to a closed point of Y\n′\nF ′ under the composite XL′\n∼→ Y ′L′ → Y\n′\nF ′ .\nCorollary 2.8. (The Case of Genus Zero) In the notation of Theorem 2.3, let\nus suppose further that XK (respectively, YL) is isogenous [cf. §0] to a hyperbolic\ncurve of genus zero. Then the isomorphism\nα : ΠXK\n∼→ ΠYL\npreserves the decomposition groups of the algebraic closed points. In particular,\nXK is defined over a number field [or, equivalently: XK has at least one alge-\nbraic point] if and only if YL is.\nProof. By Theorem 1.3, (ii), and [the “LocK(−) portion” — already contained\nin [Mzk3], Theorem 2.4 — of] Theorem 2.3, (ii), one reduces immediately to the\ncase where both XK and YL are of genus zero. Also, by Theorem 1.3, (ii), we may\nalways enlarge K, L without loss of generality; in particular, we may assume that\nXK , YL are cuspidally split, so that both curves admit a [cuspidally split] tripod as\nan HPC. Then we argue as in the proof of Corollary 2.6: That is to say, given any\nalgebraic x ∈ XK , we observe that [after possibly enlarging K] there exists, by the\ndefinition of “algebraic” and the famous main result of [Belyi], a “Belyi map”\nβ : XK → XK\nthat maps x, as well as all of the cusps of XK , to cusps of XK , and, moreover, is\nunramified over the open subscheme of XK determined by the tripod that forms an\nHPC for XK . In particular, β is unramified over the open subscheme XK ⊆ XK .\nPut another way, there exists an open immersion φ : ZK ↪→ XK [i.e., an HPC]\nsuch that the composite β ◦ φ factors through XK ⊆ XK in such a way that the\nresulting morphism βZ : ZK → XK is finite étale. In particular, βZ exhibits φ as an\nobject of DLocK(XK), and so φ exhibits x as a closed point of DLoc-type. Thus, by\ntransporting φ, βZ via the equivalences of Theorem 2.3, (i), and applying Theorem\n1.3, (ii), (iii) [as in the proof of Corollary 2.5], we conclude that α preserves the\ndecomposition groups of algebraic closed points, as desired. ©\nRemark 2.8.1. In fact, tracing through the proofs of Corollaries 2.6, 2.8 shows\nthat in these proofs, we did not actually need to use the full “Hom” version of\nTheorem 1.2. That is to say, for these proofs, in fact the “isomorphism version”\nof Theorem 1.2 [i.e., the bijection between isomorphisms “XK\n∼→ YL” and certain\nisomorphisms “ΠXK\n∼→ ΠYL”], applied in combination with Theorem 1.3, (iii), is\nsufficient. Indeed, if we use the natural faithful functor discussed in Remark 2.2.0\nto think of LocK(XK) as a [not necessarily full!] subcategory of DLocK(XK), then\nlet us denote by\nArr(LocK(XK)) ⊆ OFLocK(XK) ⊆ Arr(DLocK(XK))\n12 SHINICHI MOCHIZUKI\nthe collection of arrows Z → Z ′ of DLocK(XK) which factor as the composite of an\narrow Z → Z ′′ [of DLocK(XK)] which is an open immersion [i.e., an HPC] with an\narrow Z ′′ → Z ′ [of DLocK(XK)] which is finite étale. We shall refer to the arrows\nof OFLocK(XK) as arrows of OF-type [i.e., “open immersion + finite étale” type].\nSimilarly, we define\nOFLocK(ΠXK ) ⊆ Arr(DLocGK (ΠXK ))\nto be the collection of arrows J1 → J2 of DLocGK (ΠXK ) that factor as the composite\nof a surjection J1 \u0002 J3 [in DLocGK (ΠXK )] whose kernel is normally topologically\ngenerated by some collection of cuspidal geometric decomposition groups, with an\nopen immersion J3 ↪→ J2 [in DLocGK (ΠXK )]. Then [cf. Theorem 2.3, (ii), and its\nproof] we obtain an equivalence of categories\nDLocGK (ΠXK )\n∼→ DLocGL(ΠYL)\nwhose induced map on “Arr(−)’s” maps OFLocK(ΠXK ) into OFLocL(ΠYL) by\napplying Proposition 1.1, (ii); Theorem 1.3, (iii) [i.e., without using Theorem 1.2\nat all!]. Moreover, the isomorphism portion of Theorem 1.2 implies that the étale\nfundamental group functor induces a natural commutative diagram\nOFLocK(XK) ⊆ Arr(DLocK(XK))⏐⏐\u0004\n⏐⏐\u0004\nOFLocK(ΠXK ) ⊆ Arr(DLocGK (ΠXK ))\nsuch that the vertical arrow on the left is “essentially surjective” — i.e., more\nprecisely: induces a bijection on abstract equivalence [cf. §0] classes [defined relative\nto the category structures of DLocK(XK), DLocGK (ΠXK )] lying in OFLocK(XK),\nOFLocK(ΠXK ). Since the proofs of Corollaries 2.6, 2.8 only make use of arrows\nof OF-type, the bijection of abstract equivalence classes just observed, together\nwith the equivalence DLocGK (ΠXK )\n∼→ DLocGL(ΠYL) — all of which involves only\nthe isomorphism portion of Theorem 1.2 — are sufficient for the proofs of these\ncategories, as claimed.\nSection 3: Limits of Galois Sections\nLet XK be a hyperbolic curve over a local field K. As in §1, 2, we have an\nexact sequence:\n1 → ΔX → ΠXK → GK → 1\nSince ΔX is topologically finitely generated, it follows that there exists a sequence\nof characteristic open subgroups\n. . . ⊆ ΔX [j + 1] ⊆ ΔX [j] ⊆ . . . ⊆ ΔX\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 13\n[where j ranges over the positive integers] of ΔX such that\n⋂\nj ΔX [j] = {1}. In\nparticular, given any section\nσ : GK → ΠXK\nwe obtain open subgroups\nΠXK [j,σ]\ndef\n= Im(σ) · ΔX [j] ⊆ ΠXK\n[where Im(σ) denotes the image of σ in ΠXK ] corresponding to a tower of finite\nétale coverings\n. . . → XK [j + 1, σ] → XK [j, σ] → . . . → XK\nof XK by hyperbolic curves over K.\nThe following lemma is reminiscent of the techniques of [Tama], [Mzk1]:\nLemma 3.1. (Criterion for Galois Sections Associated to Rational\nPoints) Suppose that XK is defined over a number field, i.e., there exists a\nhyperbolic curve XK over a number field F ⊆ K such that XK = XF ×F K. Let\nσ : GK → ΠXK be a section such that Im(σ) is not contained in any cuspidal\ndecomposition group of ΠXK . Then the following conditions on σ are equivalent:\n(i) σ arises from a point x ∈ XK(K) [i.e., “Im(σ) = Dx”].\n(ii) For every integer j ≥ 1, XK [j, σ](K) \n= ∅.\n(iii) For every integer j ≥ 1, XK [j, σ](K)alg \n= ∅ [where the superscript “alg”\ndenotes the subset of algebraic [K-rational] closed points].\n(iv) For every integer j ≥ 1, ΠXK [j,σ] contains a decomposition group [i.e.,\nrelative to ΠXK ] of an algebraic closed point of XK that surjects onto GK .\nProof. (i) =⇒ (ii): It follows from the definitions that x ∈ XK(K) lifts to a point\nof ∈ XK [j, σ](K), for all j ≥ 1, which implies (ii).\n(iii) =⇒ (ii), (iv); (iv) =⇒ (iii): Immediate from the definitions.\n(ii) =⇒ (i): For j ≥ 1, choose points xj ∈ XK [j, σ](K). Since the topological\nspace ∏\nj≥1\nXK [j, σ](K)\nis compact, it follows that there exists some infinite set of positive integers J ′ such\nthat for any j ≥ 1, the images of the xj′ , where j′ ≥ j, in\nXK [j, σ](K)\nconverge to a point yj ∈ XK [j, σ](K). Moreover, note that, by the definition of yj ,\nit follows that if j1 > j2, then yj1 maps to yj2 in XK [j2, σ](K). In particular, if we\nwrite y ∈ XK(K) for the image of the yj in XK(K), then it follows formally from\n14 SHINICHI MOCHIZUKI\nthe fact that the yj form a compatible sequence of points of the sets XK [j, σ](K)\nthat Im(σ) is contained in the decomposition group [well-defined up to conjugation]\nDy. On the other hand, by our assumption that Im(σ) is not contained in any\ncuspidal decomposition group of ΠXK , we conclude that y is not a cusp, hence that\n“Im(σ) = Dy”, as desired.\n(ii) =⇒ (iii): Given a point xj ∈ XK [j, σ](K) with image x ∈ XK(K) =\nXF (K), it follows from “Krasner’s lemma” [cf., e.g., [Kobl], p. 69-70] that one may\napproximate x by a point x′ ∈ XF (F ′) ⊆ XF (K) = XK(K), where F ′ ⊆ K is a\nfinite extension of F , which is sufficiently close to x that [just like x] it lifts to a\npoint x′j ∈ XK [j, σ](K), which is necessarily algebraic, as desired. ©\nCorollary 3.2. (Absoluteness of Decomposition Groups for Genus Zero)\nLet K, L be local fields; XK (respectively, YL) a hyperbolic curve over K\n(respectively, L), which is, in fact, defined over a number field. Suppose,\nmoreover, that XK (respectively, YL) is isogenous [cf. §0] to a hyperbolic curve of\ngenus zero. Then every isomorphism of profinite groups\nα : ΠXK\n∼→ ΠYL\npreserves the decomposition groups of the closed points.\nProof. Indeed, Corollary 3.2 follows formally from Corollary 2.8; Theorem 1.3,\n(iii), (iv); and the equivalence (i) ⇐⇒ (iv) of Lemma 3.1. ©\nRemark 3.2.1. Since any once-punctured elliptic curve is isogenous to a hy-\nperbolic curve of genus zero, one might think, at first glance, that Corollary 2.6 is\n[essentially] a “special case” of Corollary 3.2. In fact, however, this is false, since\nCorollary 2.6 applies even to curves which are not necessarily defined over a number\nfield.\nSection 4: Discrete and Integral Structures at Cusps\nLet XK be a hyperbolic curve over a local field K; write XK ↪→ XK for the\ncompactification of XK . Also, if p is the residue characteristic of K, then we shall\nwrite Ẑ′ def= Ẑ\/Zp. Let\nDx ⊆ ΠXK\nbe a decomposition group associated to some cusp x ∈ XK(K). Then we have an\nexact sequence\n1 → Ix (∼= Ẑ(1)) → Dx → GK → 1\nwhose splittings form a torsor over\nH1(GK , Ẑ(1)) ∼= (K×)∧\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 15\n[where the “∧” denotes the profinite completion]. If ωx denotes the cotangent space\nto XK at x, then any choice of a nonzero θ ∈ ωx determines a splitting of this\ntorsor by considering the Ẑ(1)-torsor over the formal completion (XK)x [i.e., of\nXK at x] given by taking N -th roots [as N ranges over the positive integers] of any\nlocal coordinate t ∈ mXK ,x such that dt|x = θ. In particular, if the pointed stable\ncurve associated to XK has stable reduction over OK , then the cotangent module to\nthis stable reduction at the OK-valued point determined by x determines a natural\nintegral structure on ωx [i.e., a rank one free OK -submodule of the one-dimensional\nK-vector space ωx]. In particular, this integral structure determines a reduction of\nthe structure group of the torsor of splittings considered above from (K×)∧ to O×K .\nDefinition 4.1.\n(i) If (K×)∧ → A is a continuous homomorphism of topological groups, then\nthe torsor obtained from the torsor of splittings considered above by changing the\nstructure group via this homomorphism will be referred to as the A-torsor at x. If,\nmoreover, B ⊆ A is a closed subgroup, then any reduction of the structure group\nof the A-torsor at x from A to B will be referred to as a B-torsor structure at x.\n(ii) A O×K - (respectively, K×-) torsor structure on the (K×)∧-torsor at x will\nbe referred to as a(n) integral (respectively, discrete) structure on the cuspidal\ndecomposition group Dx. Let us think of (K\n×)∧ ⊗ Ẑ′ as a quotient of (K×)∧;\nwrite (O×K)′, (K×)′ for the images of O×K , K×, respectively, in (K×)∧ ⊗ Ẑ′. Then\na (O×K)′- (respectively, (K×)′-) torsor structure on the (K×)∧ ⊗ Ẑ′-torsor at x will\nbe referred to as a(n) tame integral (respectively, tame discrete) structure on the\ncuspidal decomposition group Dx.\n(iii) If XK has stable reduction over OK (respectively, XK is arbitrary), then\nthe particular integral (respectively, discrete) structure on Dx arising [as discussed\nabove] from a generator of the rank one free OK -submodule of ωx determined\nby the stable reduction of XK (respectively, any nonzero element of ωx) will be\nreferred to as the canonical integral (respectively, discrete) structure on the cuspidal\ndecomposition group Dx. The canonical integral (respectively, discrete) structure\non Dx induces a tame integral (respectively, tame discrete) structure on Dx which\nwe shall also refer to as canonical.\n(iv) An arbitrary closed point x′ of XK will be referred to as absolute if, for\nevery YL, α as in Theorem 2.3, there exists a closed point y\n′ of Y L such that\nα(Dx′) = Dy′ . A nonconstant unit U ∈ Γ(XK ,O×XK ) on XK will be called coab-\nsolute if XK admits an absolute point at which U is invertible. The hyperbolic\ncurve XK will be called coabsolute if it admits a coabsolute unit. The hyperbolic\ncurve XK will be called quasi-coabsolute if it is isogenous to a coabsolute hyperbolic\ncurve. If XK has stable reduction over OK (respectively, XK is arbitrary), then the\ncusp x will be called integrally absolute (respectively, discretely absolute) if, for ev-\nery YL, α as in Theorem 2.3, the isomorphism Dx\n∼→ Dy [where y is a cusp of YL\n— cf. Theorem 1.3, (iii)] induced by α is compatible with the canonical integral\n16 SHINICHI MOCHIZUKI\n(respectively, discrete) structures on Dx, Dy. Similarly, one has a notion of tamely\nintegrally absolute and tamely discretely absolute cusps.\n(v) The cusp x will be called subprincipal if it is contained in the support of a\ncuspidal principal divisor on [i.e., principal divisor supported in the cusps of] XK .\nThe hyperbolic curve XK will be called subprincipally ample if every cusp of XK\nis subprincipal. The hyperbolic curve XK will be called subprincipally quasi-ample\nif it is isogenous to a subprincipally ample hyperbolic curve.\nRemark 4.1.1. By Theorem 1.3, (iii), cusps are always absolute. By Corollaries\n2.6, 3.2, once-punctured elliptic curves, as well as hyperbolic curves that are isoge-\nnous to a hyperbolic curve of genus zero which is defined over a number field, have\ninfinitely many absolute points.\nNext, let us write\nL def= OXK (x)\nfor the line bundle determined by the cusp x;\nL → XK\nfor the geometric line bundle determined by L; and\n(L ⊇) L× → XK\nfor the complement of the zero section in L. Thus, the natural inclusion OXK ↪→OXK (x) determines a section\nXK → L\nwhose restriction to XK determines a section XK → L×, hence a morphism of\nfundmental groups:\nΠXK → ΠL× def= π1(L×)\nLemma 4.2. (The Line Bundle Associated to a Cusp) Suppose that XK\nis of type (g, r), where g ≥ 2, r = 1. Then:\n(i) ΠL× fits into a short exact sequence:\n1 → Ẑ(1) → ΠL× → ΠXK → 1\nMoreover, the resulting extension class ∈ H2(ΠXK , Ẑ(1)) is the first Chern class of\nthe line bundle L.\n(ii) The morphism of fundmental groups ΠXK → ΠL× induces an isomorphism\nIx\n∼→ Ker(ΠL× → ΠXK ). In particular, the morphism ΠXK → ΠL× is surjective.\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 17\n(iii) Write ΔX\/X\ndef\n= Ker(ΠXK \u0002 ΠXK ). Then the quotient of ΔX\/X by\nKer(ΠXK → ΠL×) ⊆ ΔX\/X\nis the maximal quotient of ΔX\/X on which the conjugation action by ΔX is\ntrivial.\nProof. Assertion (i) follows from [Mzk4], Lemmas 4.3, 4.4, 4.5. Assertion (ii)\nis immediate from the discussion preceding Definition 4.1 involving roots of local\ncoordinates. As for assertion (iii), write Q1\ndef\n= ΔX\/X\/Ker(ΠXK → ΠL×); Q2 for\nthe maximal quotient of ΔX\/X on which the conjugation action by ΔX is trivial.\nThus, we have a natural surjection Q2 \u0002 Q1. Now assertion (iii) follows from\nassetion (ii) and the well-known fact that ΔX\/X is topologically generated by the\nΔX -conjugates of Ix. ©\nNext, let us recall the notation of [Mzk2], §1.2: By local class field theory, we\nhave a natural isomorphism\n(K×)∧ ∼→ GabK\nwhich we may use to think of the group of roots of unity of (K×)∧ as a subgroup:\nμQ\/Z(K) ⊆ GabK\nAlso, we recall [cf. [Mzk2], Proposition 1.2.1, (iv)] that the subgroup K× ⊆\n(K×)∧ ∼→ GK may be recovered group-theoretically from the profinite group struc-\nture of GK . Allowing “K” to vary among the various finite extensions of a given\nK inside an algebraic closure K of K, we obtain groups:\nμQ\/Z(K); μ\u0000Z(K)\ndef\n= Hom(Q\/Z, μQ\/Z(K)); μ\u0000Z′(K)\ndef\n= μ\n\u0000Z\n(K) ⊗ Ẑ′\nIn particular, by considering roots of local coordinates as in the discussion preceding\nDefinition 4.1, we obtain a natural isomorphism μ\n\u0000Z\n(K)\n∼→ Ix.\nTheorem 4.3. (Rigidity of Cuspidal Geometric Decomposition Groups)\nIn the notation of Theorem 2.3, suppose that α induces isomorphisms\nIx\n∼→ Iy; μ\u0000Z(K)\n∼→ μ\n\u0000Z\n(L)\nwhere x ∈ XK(K) (respectively, y ∈ Y L(L)) is a cusp. Then these isomorphisms\nare compatible with the natural isomorphisms μ\n\u0000Z\n(K)\n∼→ Ix; μ\u0000Z(L)\n∼→ Iy.\nProof. Indeed, by replacing XK , YL by finite étale coverings, one reduces immedi-\nately to the case where both curves are of genus ≥ 2. By “filling in” [cf. Theorem\n1.3, (iii)] all of the cusps other than those of interest [i.e., x, y], we may assume,\n18 SHINICHI MOCHIZUKI\nmoreover, that XK , YL satisfy the hypotheses of Lemma 4.2. Thus, by Lemma 4.2,\nwe conclude that the morphism\nH2(ΔX , Ix)\n∼→ H2(ΔY , Iy)\ninduced by α is compatible with the extension classes of Lemma 4.2. On the other\nhand, by [Mzk2], Lemma 2.5, (ii), the morphism\nH2(ΔX , μ\u0000Z(K))\n∼→ H2(ΔY , μ\u0000Z(L))\ninduced by α is compatible with the elements determined by the Chern class of a\npoint on either side. Since all of these “H2’s” are isomorphic to Ẑ, we thus obtain\nthe compatibility asserted in the statement of Theorem 4.3. ©\nProposition 4.4. (Tame Integral Absoluteness) Suppose that XK has\nstable reduction over OK . Then:\n(i) Every cusp of XK is tamely integrally absolute.\n(ii) A cusp of XK is discretely absolute if and only if it is integrally\nabsolute.\nProof. Assertion (ii) follows formally from assertion (i) and the fact that the\nrestriction of the projection Ẑ \u0002 Ẑ′ to Z ⊆ Ẑ is injective. Now we consider\nassertion (i). First, let us observe that it is immediate from the definitions that it\nsuffices to prove assertion (i) after replacing XK by a finite étale covering of XK\nthat extends to an admissible covering of the stable model of XK . In particular,\nwe may assume without loss of generality that every irreducible component of the\nnormalization of the geometric special fiber of this stable model has genus ≥ 1.\nNext, let us recall the “étale quotient”\nΠXK \u0002 ΠetXK\nof [Mzk2], §2. Thus, the finite quotients of ΠetXK correspond to the coverings of\nXK that arise from finite étale coverings of the stable model of XK that are tamely\nramified at the cusps. In particular, the quotient of GK determined by Π\net\nXK\nis\nthe natural quotient GK \u0002 Gk, where k is the residue field of K. If x is a cusp\nof XK , then [in light of our assumption that every irreducible component of the\nnormalization of the geometric special fiber of the stable model has genus ≥ 1] the\nquotient\nDx \u0002 D′x\ndetermined by ΠetXK fits into an exact sequence:\n1 → I ′x → D′x → Gk → 1\n[where I ′x\ndef\n= Ix ⊗ Ẑ′]. In particular, the splittings of this exact sequence form a\ntorsor over H1(Gk, I\n′\nx)\n∼= k×. These splittings may be thought of as elements of\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 19\nH1(D′x, I\n′\nx) whose restriction to I\n′\nx is equal to the identity element of H\n1(I ′x, I\n′\nx) =\nHom(I ′x, I\n′\nx). Thus, unraveling the definitions, one verifies immediately that the\npull-back to Dx of any such element of H\n1(D′x, I ′x) forms an element of H1(Dx, I ′x)\nwhich determines the canonical tame integral structure on Dx. Since the étale\nquotient is compatible with isomorphisms α as in Theorem 2.3 [cf. [Mzk2], Lemma\n2.2, (ii)], we thus conclude that x is tamely integrally absolute, as desired. ©\nProposition 4.5. (Absoluteness and Coverings) Let Z → XK be a finite\nétale covering. Let z be a closed point of the compactification Z of Z that maps\nto a closed point x of XK. Then:\n(i) z is absolute (respectively, a discretely absolute cusp) if and only if x\nis.\n(ii) Suppose that XK , Z have stable reduction [over the rings of integers of\ntheir respective fields of constants]. Then z is an integrally absolute cusp if and\nonly if x is.\nProof. Assertion (i) is immediate from the definitions; [the “LocK(−) portion” —\nalready contained in [Mzk3], Theorem 2.4 — of] Theorem 2.3, (ii) [cf. the proof\nof Corollary 2.8]; Theorem 1.3, (ii); the fact that Ẑ\/Z is divisible. Assertion (ii) is\nimmediate from assertion (i) and Proposition 4.4, (ii). ©\nBefore proceeding, we recall the following well-known result:\nLemma 4.6. (Vanishing of Galois Invariants of the Tate Module) We\nhave: H0(GK , H\n1(ΔX , μ\u0000Z(K))) = 0.\nProof. Since TX\ndef\n= Δab\nX\n, i.e., the Tate module of the Jacobian JX of XK , is\nisomorphic to its Cartier dual, it suffices to show that H0(GK , TX) = 0, i.e., that\nthe torsion subgroup of JX(K) is finite. Since JX is a proper group scheme over\nK, it follows that the p-adic topology on K determines a p-adic topology on JX(K)\nwith respect to which JX(K) forms a compact p-adic Lie group. As is well-known\n[cf., e.g., [Serre], Chapter V, §7], the exponential map for this p-adic Lie group\ndetermines an isomorphism of a certain open neighborhood of the identity of JX(K)\nwith a free Zp-module of finite rank. Thus, the desired finiteness follows formally\nfrom this isomorphism, together with the compactness of JX(K). ©\nRemark 4.6.1. The author wishes to thank A. Tamagawa for informing him of\nthe simple proof of Lemma 4.6 given above.\nNext, let us observe that for any integer N ≥ 1, the Kummer exact sequence\n1 → μN → Gm → Gm → 1\n20 SHINICHI MOCHIZUKI\n[where Gm → Gm denotes the N -th power map on Gm; μN is defined so as to make\nthe sequence exact] on the étale site of XK determines a long exact sequence in\ncohomology, hence, in particular, by letting N vary, an injection\nHX\ndef\n= Γ(XK ,O×XK ) ↪→ H∧X ↪→ H1(ΠXK , μ\u0000Z(K))\n[where we use the easily verified fact that HX is residually finite]. On the other\nhand, the Leray spectral sequence for the quotient ΠXK → GK yields an exact\nsequence:\n0 → (K×)∧ → H1(ΠXK , μ\u0000Z(K)) → DX\ndef\n= H0(GK , H\n1(ΔX , μ\u0000Z(K)))\nMoreover, since, by Lemma 4.6, H0(GK , H\n1(ΔX , μ\u0000Z(K))) = 0, it follows that, if we\nassume, for simplicity, that XK is cuspidally split, then restriction to the various\n“Ix” in ΔX determines [by applying the natural isomorphisms Ix\n∼→ μ\n\u0000Z\n(K)] an\ninjection\nDX ↪→ PX def=\n∏\nx\nẐ\n[where the product ranges over the cusps x of XK ]. In particular, we obtain exact\nsequences:\n0 → K× → HX → PX ; 0 → (K×)∧ → H∧X → PX\nWrite EX\ndef\n= Im(HX) ⊆ PX for the image of HX in PX [so we obtain an induced\ninjection E∧X ↪→ PX ]. Thus, the maps\nHX → PX ; H∧X → PX\nare the maps obtained by associating to a function in HX its divisor of zeroes\nand poles. Put another way, EX ⊆ PX may be characterized as the submodule of\ncuspidal principal divisors.\nProposition 4.7. (Principal Cuspidal Divisors) In the notation of Theorem\n2.3, assume that XK, YL are cuspidally split. Then the isomorphism\nPX\n∼→ PY\ninduced [cf. Theorem 1.3, (iii)] by α maps EX onto EY .\nRemark 4.7.1. In the statement of Proposition 4.7, as well as in the discussion to\nfollow, we shall use similar notation for the objects associated to YL to the notation\nused for the various objects just defined for XK .\nProof. Write JX (respectively, AX) for the Jacobian (respectively, Albanese vari-\nety) of XK . Thus, the natural map XK → AX determines a surjection on funda-\nmental groups ΠXK \u0002 ΠAX\ndef\n= π1(AX) whose kernel is the kernel of ΔX \u0002 ΔabX .\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 21\nIn particular, any pair of sections of ΠXK → GK determines a pair of sections\nof ΠAX → GK whose difference determines an element of H1(GK , ΔabX ). More-\nover, if these sections arise from points ∈ XK(K), then the resulting element of\nH1(GK , Δ\nab\nX\n) completely determines the point of JX(K) given by forming the dif-\nference of these two points [cf., e.g., [Mzk1], the discussion preceding Definition\n6.2; [BK], Example 3.11]. More generally, given any divisor of cusps on XK with\nZ-coefficients of degree 0, the divisor is principal if and only if the resulting element\nof H1(GK , Δ\nab\nX\n) vanishes. Since the sections of ΠXK → GK arising from cusps\nare preserved by α [cf. Theorem 1.3, (iii)], we thus conclude that the isomorphism\nPX → PY induced by α maps EX onto EY , as desired. ©\nDefinition 4.8. We shall say that XK is unitwise absolute if, in the notation of\nTheorem 2.3, the isomorphism\nH1(ΠXK , μ\u0000Z(K))\n∼→ H1(ΠYL , μ\u0000Z(L))\ninduced by α maps the image of Γ(XK ,O×XK ) via the Kummer map onto the image\nof Γ(YL,O×YL) via the Kummer map.\nCorollary 4.9. (Divisor-Theoretic Properties) In the notation of Theorem\n2.3, let x, y (respectively, A ∈ EX , B ∈ EY ) be cusps (respectively, cuspidal\nprincipal divisors) of XK , YL, respectively, that correspond via α [cf. Proposition\n4.7]. Then:\n(i) x is subprincipal if and only if y is.\n(ii) A is the divisor of a coabsolute unit if and only if B is.\n(iii) XK is coabsolute (respectively, quasi-coabsolute) if and only if YL is.\n(iv) XK is subprincipally ample (respectively, subprincipally quasi-ample)\nif and only if YL is.\nProof. In light of Proposition 4.7, all of these statements follow formally from\nthe definitions. Also, we note that for the various “quasi-” properties, one must\napply [the “LocK(−) portion” — already contained in [Mzk3], Theorem 2.4 — of]\nTheorem 2.3, (ii), as in the proof of Corollary 2.8. ©\nTheorem 4.10. (Units and Canonical Integral Structures) Let XK be a\nhyperbolic curve over a local field K. Then:\n(i) If XK is quasi-coabsolute, then it admits a discretely absolute cusp.\n(ii) If XK admits a discretely absolute cusp or an absolute noncusp [i.e.,\nan absolute point which is not a cusp], then XK is unitwise absolute.\n22 SHINICHI MOCHIZUKI\n(iii) If XK is unitwise absolute and subprincipally ample, then every\ncusp of XK is discretely absolute.\n(iv) Suppose that XK has stable reduction over OK . Then if XK is quasi-\ncoabsolute and subprincipally quasi-ample, then every cusp of XK is inte-\ngrally absolute.\nProof. First, we consider assertion (i). In light of Proposition 4.5, (i), we may\nassume that XK is coabsolute. Let U ∈ HX be a coabsolute unit of XK ; let x be a\ncusp of XK at which U fails to be invertible. If U has a zero of order (Z \u0012) n \n= 0\nat x, then the restriction of the class\nηU ∈ H1(ΠXK , μ\u0000Z(K))\ndetermined by U to Dx determines a splitting of the torsor obtained by applying a\nchange of structure group to the (K×)∧-torsor at x via the map (K×)∧ → (K×)∧\ngiven by multiplication by n. Since n \n= 0, and Ẑ\/Z is divisible, it thus follows\nthat this splitting is sufficient to determine the canonical discrete structure on Dx.\nLet us write \fU ∈ EX for the image of ηU in EX . Then \fU determines the set\n(K×)∧ · ηU . On the other hand, since U is coabsolute, it follows that XK admits\nan absolute point x′ at which U is invertible. Thus, the subset\nK× · ηU ⊆ (K×)∧ · ηU\nmay be characterized as the set of elements of (K×)∧ · ηU whose restriction to Dx′\n— which [by the invertibility of U at x′] necessarily lies in\n(K×)∧ ∼= H1(GK , μ\u0000Z(K)) ⊆ H1(Dx′ , μ\u0000Z(K))\n— in fact lies inside K× ⊆ (K×)∧. Thus, Theorem 4.3, Proposition 4.7, together\nwith the absoluteness of x′, imply that x is discretely absolute, as desired.\nNext, we consider assertion (ii). Let x be a discretely absolute cusp or an\nabsolute noncusp of XK . Then, as in the argument applied in the proof of asser-\ntion (i), the image of HX in H\n1(ΠXK , μ\u0000Z(K)) may be characterized as the set of\nelements lying over elements of EX whose restriction to Dx determines a class in\nH1(Dx, μ\u0000Z(K)) that lies in the submodule of this cohomology module generated by\nthe elements that define splittings “compatible with the canonical discrete structure\non Dx” [where in the noncuspidal case, we take this compatibility to mean that\nthe restriction to Dx lies in K\n× ⊆ (K×)∧, as in the proof of assertion (i)]. Thus,\nassertion (ii) follows from Theorem 4.3, Proposition 4.7, together with the discrete\nabsoluteness [in the cuspidal case] or absoluteness [in the noncuspidal case] of x.\nNext, we observe that assertion (iii) follows via the argument applied in the\nproof of assertion (i), since the hypothesis that XK is unitwise absolute and sub-\nprincipally ample implies that for every cusp x of XK , there exists a unit U ∈ HX\nthat is not invertible at x and whose class in H1(ΠXK , μ\u0000Z(K)) is mapped [in the\nnotation of Theorem 2.3] to a class in H1(ΠYL , μ\u0000Z(L)) that lies in the image of HY .\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 23\nFinally, we observe that assertion (iv) is a formal consequence of Proposition\n4.4, (ii); Proposition 4.5, (i); assertions (i), (ii), (iii). ©\nCorollary 4.11. (The Case of Genus Zero) Let XK be a hyperbolic\ncurve over a local field K which is isogenous to a hyperbolic curve of genus\nzero. Then XK is quasi-coabsolute, subprincipally quasi-ample, and unit-\nwise absolute. In particular, if XK has stable reduction over OK , then every\ncusp of XK is integrally absolute.\nProof. In light of Theorem 4.10, it suffices to show that if XK is of genus zero and\ncuspidally split, then XK is coabsolute and subprincipally ample. But since cusps\nare always absolute [cf. Theorem 1.3, (iii)], these properties follow formally from\nthe following two elementary facts: (a) every divisor of degree 0 on XK is principal;\n(b) XK has at least 3 cusps. ©\nIn certain situations, the O×K -torsor determined by the canonical integral struc-\nture on the cuspidal decomposition group Dx admits an even “finer reduction of\nstructure group”, as follows:\nCorollary 4.12. (The Case of Once-punctured Elliptic Curves) Let XK be\na once-punctured elliptic curve over a local field K of residue characteristic\n\n= 2. Suppose that XK has stable reduction over OK . Also, if n ≥ 1 is an\ninteger, we shall write μn(K) ⊆ O×K for the subgroup of n-th roots of unity. Then\nthere exists a μ12(K)-torsor structure at the unique cusp x of XK which is\ncompatible with the canonical integral structure arising from the stable model\nX log and, moreover, is preserved by arbitrary automorphisms of ΠXK .\nProof. We may assume without loss of generality that all of the 2-torsion points\nof the underlying elliptic curve of XK are defined over K. Write YK → XK for\nthe Galois covering of degree 4 determined by the “multiplication by 2” map on\nthe underlying elliptic curve of XK [so YK is hyperbolic of type (1, 4)] and Y log for\nthe stable model over Spec(OK)log [where the log structure on Spec(OK) is that\ndetermined by the closed point] of the smooth log curve Y\nlog\nK determined by YK .\nAlso, let us write e1, e2, e3, e4 for the four cusps of YK .\nLet α : ΠXK\n∼→ ΠXK be an automorphism of ΠXK . Note that, by Theorem 1.3,\n(ii), any μ12(K)-torsor structure at x is preserved by arbitrary inner automorphisms\nof ΠXK . Thus, we may assume [by composing with a suitable inner automorphism\nthat induces a suitable element of Gal(YK\/XK)] that the natural action of α on\nthe cusps of YK [cf. Theorem 1.3, (iii)] preserves e1.\nNext, let us observe that [by the well-known definition of the group law on an\nelliptic curve; the definition of YK → XK ] the divisor D def= 2[e1] − 2[e2] on YK is\nprincipal. Thus, there exists a unique rational function f on YK whose divisor of\nzeroes and poles is D and whose value at e3 is 1. Since D has multiplicity 2 at\n24 SHINICHI MOCHIZUKI\ne1, it follows that f determines a μ2(K)-torsor structure at e1, hence also at x.\nWrite η for the Kummer class [i.e., the image under the Kummer map] of f . In the\nfollowing, we shall write Kummer classes additively.\nNow, observe that, by Proposition 4.7; Theorem 1.3, (iii), if α fixes all four\ncusps of YK , then it follows that α preserves the class η, hence also the μ2(K)-torsor\nstructure at x determined by η.\nNext, let us write Σ for the group of permutations of the three cusps e2, e3, e4\nthat arise from automorphisms β ∈ Aut(ΠXK ) that preserve e1. Thus, the order s\nof Σ divides 6. Let β1, . . . , βs ∈ Aut(ΠXK ) be a collection of automorphisms that\ngive rise to the elements of Σ. Set:\nη′ def= (6\/s) ·\ns∑\nj=1\nηβj\nSince YK is unitwise absolute by Corollary 4.11, it follows that η\n′ arises from a\nrational function f ′ on YK which has a pole of order 12 at e1. In particular, η′\ndetermines a μ12(K)-torsor structure at e1, hence also at x. Moreover, it follows\nformally from the preceding observation concerning automorphisms α that fix all\nfour cusps of YK that arbitrary α [i.e., that are only assumed to fix e1] preserve the\nμ12(K)-torsor structure determined by η\n′. Finally, the fact that this μ12(K)-torsor\nstructure is compatible with the canonical integral structure follows from the easily\nverified fact that the rational function f is generically invertible [in light of our\nassumption that the residue charactertistic of K is \n= 2] on the special fiber of Y .\nThis completes the proof. ©\nRemark 4.12.1. The number “12” appearing in Corollary 4.12 is interesting\nin light of the well-known fact that the line bundle on the moduli stack of elliptic\ncurves determined by the cotangent bundle at the origin of the tautological family\nof elliptic curves has order 12 in the Picard group of this moduli stack.\nRemark 4.12.2. It seems natural to expect that a(n) [perhaps somewhat more\ncomplicated] analogue of Corollary 4.12 should hold for more general hyperbolic\ncurves XK . This topic, however, lies beyond the scope of this paper.\nBibliography\n[Belyi] G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Math. USSR-\nIzv. 14 (1980), pp. 247-256.\n[BK] S. Bloch and K. Kato, L-Functions and Tamagawa Numbers in The Grothen-\ndieck Festschrift, Volume I, Birkhäuser (1990), pp. 333-400.\nGALOIS SECTIONS IN ABSOLUTE ANABELIAN GEOMETRY 25\n[DM] P. Deligne and D. Mumford, The Irreducibility of the Moduli Space of Curves\nof Given Genus, IHES Publ. Math. 36 (1969), pp. 75-109.\n[Knud] F. F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, II,\nMath. Scand. 52 (1983), pp. 161-199.\n[Kobl] N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions, Graduate\nTexts in Mathematics 58, Springer Verlag (1977).\n[Mzk1] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math.\n138 (1999), pp. 319-423.\n[Mzk2] S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois\nTheory and Modular Forms, Kluwer Academic Publishers (2003), pp. 77-122.\n[Mzk3] S. Mochizuki, The Absolute Anabelian Geometry of Canonical Curves, Kazuya\nKato’s fiftieth birthday, Doc. Math. 2003, Extra Vol., pp. 609-640.\n[Mzk4] S. Mochizuki, Topics Surrounding the Anabelian Geometry of Hyperbolic Curves,\nGalois Groups and Fundamental Groups, Mathematical Sciences Research In-\nstitute Publications 41, Cambridge University Press (2003), pp. 119-165.\n[Serre] J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Mathematics 1500,\nSpringer Verlag (1992).\n[Tama] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio\nMath. 109 (1997), pp. 135-194.\n"}
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{"id":"y0wcp116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Combinatorial%20Grothendieck%20Conjecture.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":88,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Combinatorial%20Grothendieck%20Conjecture","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["comb-gc.dvi"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 06:55:25 UTC 2011","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":3848027714947710626,"exact_signature_unique_b":true,"fuzzy_signature_l":2862153407930612053,"fuzzy_signature_unique_b":true,"wordcount_i":11761,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:55 UTC 2022","fresh_date_dt":"Thu Sep 23 21:47:10 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":248286,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086042644480000,"text_t":"A COMBINATORIAL VERSION OF\nTHE GROTHENDIECK CONJECTURE\nShinichi Mochizuki\nApril 2007\n\u0000\u0002\u0003\u0004\u0005\u0006\u0007\u0004\b We study the “combinatorial anabelian geometry” that governs the rela-\ntionship between the dual semi-graph of a pointed stable curve and various associated\nprofinite fundamental groups of the pointed stable curve. Although many results of\nthis type have been obtained previously in various particular situations of interest\nunder unnecessarily strong hypotheses, the goal of the present paper is to step back\nfrom such “typical situations of interest” and instead to consider this topic in the\nabstract — a point of view which allows one to prove results of this type in much\ngreater generality under very weak hypotheses.\nIntroduction\nIn this paper, we apply the language of anabelioids [cf. [Mzk5], [Mzk7]] to study\nthe “profinite combinatorial group theory” arising from the relationship between\nthe semi-graph of anabelioids associated to a pointed stable curve [i.e., a “semi-\ngraph of anabelioids of PSC-type” — cf. Definition 1.1, (i), below for more details]\nand a certain associated profinite fundamental group [cf. Definition 1.1, (ii)]. In\nparticular, we show that:\n(i) The cuspidal portion of the semi-graph may be recovered group-theoretical-\nly from the associated profinite fundamental group, together with certain\nnumerical information [roughly speaking, the number of cusps of the var-\nious finite étale coverings of the given semi-graph of anabelioids] — cf.\nTheorem 1.6, (i).\n(ii) The entire “semi-graph of anabelioids structure” may be recovered group-\ntheoretically from the associated profinite fundamental group, together\nwith a certain filtration [arising from this “semi-graph of anabelioids struc-\nture”] of the abelianizations of the various finite étale coverings of the given\nsemi-graph of anabelioids — cf. Theorem 1.6, (ii).\n2000 Mathematical Subject Classification. Primary 14H30; Secondary 14H25.\nPartly supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education,\nCulture, Sports, Science and Technology, Japan.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\nMoreover, we show that from the point of view of “weights” [i.e., logarithms of\nabsolute values of eigenvalues of the action of the Frobenius element of the Galois\ngroup of a finite field], the data necessary for (i) (respectively, (ii)) above may be\nrecovered from very weak assumptions concerning the “weights” — cf. Corollary\n2.7, (i), (ii). In particular, [unlike the techniques of [Mzk4], Lemmas 1.3.9, 2.3,\nfor example] these very weak assumptions do not even require the existence of a\nparticular Frobenius element. Alternatively, when there are no cusps, the data\nnecessary for (ii) may be recovered from very weak assumptions concerning the l-\nadic inertia action [cf. Corollary 2.7, (iii)] — i.e., one does not even need to consider\nweights. This sort of result may be regarded as a strengthening of various results\nto the effect that a curve has good reduction if and only if the l-adic inertia action\nis trivial [cf., e.g., [Tama1], Theorem 0.8].\nOne consequence of this theory is the result [cf. Corollary 2.7, (iv)] that the\nsubgroup of the group of outer automorphisms of the associated fundamental group\nconsisting of the graphic outer automorphisms [i.e., the automorphisms that are\ncompatible with the “semi-graph of anabelioids structure”] is equal to its own com-\nmensurator within the entire group of outer automorphisms. This result may be\nregarded as a sort of “anabelian analogue” of a well-known “linear algebra fact”\nconcerning the general linear group [cf. Remark 2.7.1].\nThe original motivation for the development of the theory of the present paper\nis as follows: Frequently, in discussions of the anabelian geometry of hyperbolic\ncurves, one finds it necessary to reconstruct the cusps [cf., e.g., [Naka1], Theorem\n3.4; [Mzk4], Lemma 1.3.9; [Tama2], Lemma 2.3, Proposition 2.4] or the entire dual\nsemi-graph associated to a pointed stable curve [cf., e.g., [Mzk2], §1 – 5; [Mzk4],\nLemma 2.3] group-theoretically from some associated profinite fundamental group.\nMoreover, although the techniques for doing this in various diverse situations are\nquite similar and only require much weaker assumptions than the assumptions that\noften hold in particular situations of interest, up till now, there was no unified\npresentation or general results concerning this topic — only a collection of papers\ncovering various “particular situations of interest”. Thus, the goal of the present\npaper is to prove results concerning this topic in maximum possible generality, in\nthe hope that this may prove useful in applications to situations not covered in\nprevious papers [cf., e.g., Corollaries 2.8, 2.9, 2.10; Remarks 2.8.1, 2.8.2].\nThe author would like to thank Akio Tamagawa for various helpful comments\nconcerning the material presented in this paper.\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 3\nSection 0: Notation and Conventions\nNumbers:\nThe notation Q will be used to denote the field of rational numbers. The\nnotation Z ⊆ Q will be used to denote the set, group, or ring of rational integers.\nThe notation N ⊆ Z will be used to denote the submonoid of integers ≥ 0. If l is\na prime number, then the notation Ql (respectively, Zl) will be used to denote the\nl-adic completion of Q (respectively, Z).\nTopological Groups:\nLet G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us\nwrite\nZG(H)\ndef\n= {g ∈ G | g · h = h · g for any h ∈ H}\nfor the centralizer of H in G;\nNG(H)\ndef\n= {g ∈ G | g ·H · g−1 = H}\nfor the normalizer of H in G; and\nCG(H)\ndef\n= {g ∈ G | (g ·H · g−1)\n⋂\nH has finite index in H, g ·H · g−1}\nfor the commensurator of H in G. Note that: (i) ZG(H), NG(H) and CG(H) are\nsubgroups of G; (ii) we have inclusions\nH, ZG(H) ⊆ NG(H) ⊆ CG(H)\nand (iii) H is normal in NG(H). We shall say that H is centrally (respectively,\nnormally; commensurably) terminal in G if ZG(H) = H (respectively, NG(H) = H;\nCG(H) = H).\nWe shall denote the group of automorphisms of G by Aut(G). Conjugation by\nelements of G determines a homomorphism G → Aut(G) whose image consists of\nthe inner automorphisms of G. We shall denote by Out(G) the quotient of Aut(G)\nby the [normal] subgroup consisting of the inner automorphisms.\nCurves:\nSuppose that g ≥ 0 is an integer. Then if S is a scheme, a family of curves of\ngenus g\nX → S\nis defined to be a smooth, proper, geometrically connected morphism of schemes\nX → S whose geometric fibers are curves of genus g.\n4 SHINICHI MOCHIZUKI\nSuppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denote\nthe moduli stack of r-pointed stable curves of genus g (where we assume the points\nto be unordered) by Mg,r [cf. [DM], [Knud] for an exposition of the theory of such\ncurves; strictly speaking, [Knud] treats the finite étale covering of Mg,r determined\nby ordering the marked points]. The open substack Mg,r ⊆ Mg,r of smooth curves\nwill be referred to as the moduli stack of smooth r-pointed stable curves of genus g\nor, alternatively, as the moduli stack of hyperbolic curves of type (g, r). The divisor\nat infinity Mg,r\\Mg,r of Mg,r determines a log structure on Mg,r; denote the\nresulting log stack by Mlogg,r.\nA family of hyperbolic curves of type (g, r)\nX → S\nis defined to be a morphism which factors X ↪→ Y → S as the composite of an\nopen immersion X ↪→ Y onto the complement Y \\D of a relative divisor D ⊆ Y\nwhich is finite étale over S of relative degree r, and a family Y → S of curves of\ngenus g. One checks easily that, if S is normal, then the pair (Y,D) is unique up\nto canonical isomorphism. (Indeed, when S is the spectrum of a field, this fact is\nwell-known from the elementary theory of algebraic curves. Next, we consider an\narbitrary connected normal S on which a prime l is invertible (which, by Zariski\nlocalization, we may assume without loss of generality). Denote by S′ → S the finite\nétale covering parametrizing orderings of the marked points and trivializations of\nthe l-torsion points of the Jacobian of Y . Note that S′ → S is independent of\nthe choice of (Y,D), since (by the normality of S), S′ may be constructed as the\nnormalization of S in the function field of S′ (which is independent of the choice\nof (Y,D), since the restriction of (Y,D) to the generic point of S has already been\nshown to be unique). Thus, the uniqueness of (Y,D) follows by considering the\nclassifying morphism (associated to (Y,D)) from S′ to the finite étale covering\nof (Mg,r)Z[1\/l] parametrizing orderings of the marked points and trivializations\nof the l-torsion points of the Jacobian [since this covering is well-known to be a\nscheme, for l sufficiently large].) We shall refer to Y (respectively, D; D; D) as the\ncompactification (respectively, divisor at infinity; divisor of cusps; divisor of marked\npoints) of X . A family of hyperbolic curves X → S is defined to be a morphism\nX → S such that the restriction of this morphism to each connected component of\nS is a family of hyperbolic curves of type (g, r) for some integers (g, r) as above.\nWrite\nCg,r → Mg,r\nfor the tautological curve over Mg,r; Dg,r ⊆ Mg,r for the corresponding tautological\ndivisor of marked points. The divisor given by the union of Dg,r with the inverse\nimage in Cg,r of the divisor at infinity of Mg,r determines a log structure on Cg,r;\ndenote the resulting log stack by Clogg,r. Thus, we obtain a morphism of log stacks\nClogg,r → M\nlog\ng,r\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 5\nwhich we refer to as the tautological log curve over Mlogg,r. If Slog is any log scheme,\nthen we shall refer to a morphism\nClog → Slog\nwhich is obtained as the pull-back of the tautological log curve via some [necessarily\nuniquely determined — cf., e.g., [Mzk1], §3] classifying morphism Slog → Mlogg,r as a\nstable log curve. If C has no nodes, then we shall refer to Clog → Slog as a smooth\nlog curve.\nIf XK (respectively, YL) is a hyperbolic curve over a field K (respectively, L),\nthen we shall say that XK is isogenous to YL if there exists a hyperbolic curve ZM\nover a field M together with finite étale morphisms ZM → XK , ZM → YL.\n6 SHINICHI MOCHIZUKI\nSection 1: Criterion for Graphicity\nIn the present §1, we state and prove a criterion for an isomorphism between the\nprofinite fundamental groups of pointed stable curves to arise from an isomorphism\nof [semi-]graphs of groups. To do this, we shall find it convenient to use the language\nof anabelioids [cf. [Mzk5]], together with the theory of semi-graphs of anabelioids\nof [Mzk7].\nLet Σ be a nonempty set of prime numbers. Denote by\nẐΣ\nthe pro-Σ completion of Z. Let G be a semi-graph of anabelioids [cf. [Mzk7],\nDefinition 2.1], whose underlying semi-graph we denote by G. Thus, for each vertex\nv (respectively, edge e) of G, we are given a connected anabelioid [i.e., a Galois\ncategory] Gv (respectively, Ge), and for each branch b of an edge e abutting to a\nvertex v, we are given a morphism of anabelioids Ge → Gv.\nDefinition 1.1.\n(i) We shall refer to G as being of pro-Σ PSC-type [i.e., “pointed stable curve\ntype”] if it arises as the pro-Σ completion [cf. [Mzk7], Definition 2.9, (ii)] of the semi-\ngraph of anabelioids determined by the “dual semi-graph of profinite groups with\ncompact structure” [i.e., the object denoted “GcX” in the discussion of pointed stable\ncurves in [Mzk4], Appendix] of a pointed stable curve over an algebraically closed\nfield whose characteristic \u0006∈ Σ. [Thus, the vertices (respectively, closed edges; open\nedges) of G correspond to the irreducible components (respectively, nodes; cusps [i.e.,\nmarked points]) of the pointed stable curve.] We shall refer to G as being of PSC-\ntype if it is of pro-Σ PSC-type for some nonempty set of prime numbers Σ. If G is a\nsemi-graph of anabelioids of PSC-type, then we shall refer to the open (respectively,\nclosed) edges of the underlying semi-graph G of G as the cusps (respectively, nodes)\nof G [or G] and write r(G) (respectively, n(G)) for the cardinality of the set of cusps\n(respectively, nodes) of G; if r(G) = 0 (respectively, n(G) = 0), then we shall say\nthat G is noncuspidal (respectively, nonnodal). Also, we shall write i(G) for the\ncardinality of the set of vertices of G.\n(ii) Suppose that G is of pro-Σ PSC-type. Then we shall denote by\nΠG\nand refer to as the PSC-fundamental group of G the maximal pro-Σ quotient of the\nprofinite fundamental group of G [cf. [Mzk7], the discussion following Definition\n2.2]; we shall refer to a finite étale covering of G that arises from an open subgroup of\nΠG as a [finite étale] ΠG-covering of G. A vertex (respectively, edge) of G determines,\nup to conjugation, a closed subgroup of ΠG ; we shall refer to such subgroups as\nverticial (respectively, edge-like). An edge-like subgroup that arises from a closed\nedge will be referred to as nodal; an edge-like subgroup that arises from an open\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 7\nedge will be referred to as cuspidal. Write MG for the abelianization of ΠG. Then\nthe cuspidal, edge-like, and verticial subgroups of ΠG determine submodules\nM cuspG ⊆M edgeG ⊆MvertG ⊆MG\nof MG , which we shall refer to as cuspidal, edge-like, and verticial, respectively. We\nshall refer to any cyclic finite étale covering of G which arises from a finite quotient\nMG \u0002 Q that factors through MG\/M cuspG and induces a surjection M\nedge\nG \/M\ncusp\nG \u0002\nQ as module-wise nodal. If one forms the quotient of ΠG by the closed normal\nsubgroup generated by the cuspidal [cf. the first “\u0002” in the following display],\nedge-like [cf. the composite of the first two “\u0002’s” in the following display], or\nverticial [cf. the composite of the three “\u0002’s” in the following display] subgroups,\nthen one obtains arrows as follows:\nΠG \u0002 ΠcptG \u0002 ΠunrG \u0002 Π\ngrph\nG\nWe shall refer to Π\ncpt\nG (respectively, Π\nunr\nG ; Π\ngrph\nG ) as the compactified (respectively,\nunramified; graph-theoretic) quotient of ΠG . We shall refer to a ΠG-covering of G\nthat arises from an open subgroup of Π\ncpt\nG (respectively, Π\nunr\nG ; Π\ngrph\nG ) as a Π\ncpt\nG -\n(respectively, ΠunrG -; Π\ngrph\nG -) covering of G. We shall refer to the images of the ver-\nticial (respectively, verticial; edge-like) subgroups of ΠG in Π\ncpt\nG (respectively, Π\nunr\nG ;\nΠ\ncpt\nG ) as the compactified verticial (respectively, unramified verticial; compactified\nedge-like) subgroups. If the abelianization of every unramified verticial subgroup\nof ΠunrG is free of rank ≥ 2 over ẐΣ, then we shall say that G is sturdy.\nRemark 1.1.1. It is immediate from the definitions that any connected finite\nétale covering of a semi-graph of anabelioids of PSC-type is again a semi-graph of\nanabelioids of PSC-type.\nRemark 1.1.2. Note that if G is a semi-graph of anabelioids of pro-Σ PSC-type,\nwith associated PSC-fundamental group ΠG , then Σ may be recovered either from\nΠG or from any verticial or edge-like subgroup of ΠG as the set of prime numbers\nthat occur as factors of orders of finite quotients of ΠG or a verticial or edge-like\nsubgroup of ΠG .\nRemark 1.1.3. It is immediate [cf. the discussion in [Mzk4], Appendix] that ΠG\nis the pro-Σ fundamental group of some hyperbolic curve over an algebraically closed\nfield of characteristic \u0006∈ Σ [or, alternatively, of some hyperbolic Riemann surface of\nfinite type], and that every open subgroup of an edge-like (respectively, verticial)\nsubgroup of ΠG is isomorphic to ẐΣ (respectively, nonabelian). In particular, [by\n[Naka2], Corollary 1.3.4] ΠG is center-free [cf. also [Mzk4], Lemma 1.3.1, for the\ncase where Σ is the set of all prime numbers; the case of arbitrary Σ may be proven\nsimilarly]. Moreover, G has cusps if and only if ΠG is a finitely generated, free pro-Σ\ngroup. On the other hand, Π\ngrph\nG is naturally isomorphic to the pro-Σ fundamental\n8 SHINICHI MOCHIZUKI\ngroup of the underlying semi-graph G. In particular, Π\ngrph\nG is a finitely generated,\nfree pro-Σ group of rank n(G) − i(G) + 1.\nRemark 1.1.4. It is immediate from the well-known structure of fundamen-\ntal groups of Riemann surfaces that, in the notation of Definition 1.1, (ii), the\nẐΣ-modules MG , MG\/M\ncusp\nG [i.e., the abelianization of Π\ncpt\nG ], MG\/M\nvert\nG [i.e., the\nabelianization of Π\ngrph\nG ], M\nvert\nG \/M\nedge\nG [i.e., the direct sum, over the set of vertices\nof G, of the abelianizations of the corresponding unramified verticial subgroups of\nΠunrG ] are all free and finitely generated over Ẑ\nΣ. That is to say, all of the subquo-\ntients of the following filtration are free and finitely generated over ẐΣ:\nM cuspG ⊆M edgeG ⊆MvertG ⊆MG\nRemark 1.1.5. From the point of view of Definition 1.1, (i), the condition that\na semi-graph of anabelioids G of PSC-type be sturdy corresponds to the condition\nthat every irreducible component of the pointed stable curve that gives rise to G be\nof genus ≥ 2. [Indeed, this follows immediately from the well-known structure of\nfundamental groups of Riemann surfaces.] In particular, one verifies immediately\nthat, even if G is not sturdy, there always exists a characteristic open subgroup\nH ⊆ ΠG which satisfies the following property: Every G′ which arises as a ΠG-\ncovering G′ → G such that ΠG′ ⊆ H ⊆ ΠG is sturdy. In fact, [it is a routine exercise\nto show that] one may even bound the index [ΠG : H] explicitly in terms of say, the\nrank [over ẐΣ] of MG.\nRemark 1.1.6. Suppose that G is sturdy. Then observe that the quotient\nΠG \u0002 ΠcptG determines a new semi-graph of anabelioids G′ of PSC-type, which we\nshall refer to as the compactification of G. That is to say, the underlying semi-graph\nG′ of G′ is obtained from the underlying semi-graph G of G by omitting the cusps.\nThe anabelioids at the vertices and edges of G′ are then obtained from G as the\nsubcategories of the corresponding anabelioids of G determined by the quotients of\nthe corresponding verticial and edge-like subgroups of ΠG induced by the quotient\nΠG \u0002 ΠcptG . Thus, it follows immediately that we obtain a natural isomorphism\nΠ\ncpt\nG\n∼→ ΠG′ .\nProposition 1.2. (Commensurable Terminality) Suppose that G is of PSC-\ntype, with associated PSC-fundamental group ΠG. For i = 1, 2, let Ai ⊆ ΠG be a\nverticial (respectively, edge-like) subgroup of ΠG arising from a vertex vi (respec-\ntively, an edge ei) of ΠG; write Bi for the image of Ai in ΠunrG . Then the following\nhold.\n(i) If A1\n⋂\nA2 is open in A1, then v1 = v2 (respectively, e1 = e2). In the non-\nresp’d case, under the further assumption that G is sturdy, if B1\n⋂\nB2 is open in\nB1, then v1 = v2.\n(ii) The Ai are commensurably terminal [cf. §0] in ΠG. In the non-resp’d\ncase, under the further assumption that G is sturdy, the Bi are commensurably\nterminal in ΠunrG .\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 9\nProof. First, we observe that assertion (ii) follows formally from assertion (i) [cf.\nthe derivation of [Mzk7], Corollary 2.7, (i), from [Mzk7], Proposition 2.6]. Now the\nproof of assertion (i) is entirely similar to the proof of [Mzk7], Proposition 2.6: That\nis to say, upon translating the group-theory of ΠG into the language of finite étale\ncoverings of G and possibly replacing G by some finite étale covering of G [which\nallows us, in particular, to replace the words “open in” in assertion (i) by the words\n“equal to”], one sees that to prove assertions (i), (ii), it suffices to prove, under the\nfurther assumption that G is sturdy [cf. Remark 1.1.5], that if v1 \u0006= v2 (respectively,\ne1 \u0006= e2), then there exists a finite étale ΠunrG - (respectively, ΠG-) covering G′ → G\nwhose restriction to the anabelioid Gv2 (respectively, Ge2) is trivial [i.e., isomorphic\nto a disjoint union of copies of Gv2 (respectively, Ge2)], but whose restriction to\nthe anabelioid Gv1 (respectively, Ge1) is nontrivial. But, in light of our assumption\nthat G is sturdy, one verifies immediately that by gluing together appropriate finite\nétale coverings of the anabelioids Gv, Ge, one may construct a finite étale covering\nG′ → G with the desired properties. \u0003\nProposition 1.3. (Duality) Let G be a noncuspidal semi-graph of anabelioids\nof PSC-type. Then the cup product in group cohomology\nH1(ΠG, ẐΣ) ×H1(ΠG, ẐΣ) → H2(ΠG , ẐΣ) ∼= ẐΣ\ndetermines a perfect pairing on MG ∼= Hom(H1(ΠG, ẐΣ), ẐΣ), well-defined up to\nmultiplication by a unit of ẐΣ. Moreover, relative to this perfect pairing, the sub-\nmodules M edgeG , M\nvert\nG of MG are mutual annihilators.\nProof. Since G is noncuspidal, it follows [cf. Remark 1.1.3] that ΠG is the pro-Σ\nfundamental group of some compact Riemann surface, so the existence of a perfect\npairing as asserted follows from the well-known Poincaré duality of such a compact\nRiemann surface. To see that the submodules M edgeG , M\nvert\nG of MG are mutual\nannihilators, we reason as follows: Since the isomorphism class of G is manifestly\ndetermined by purely combinatorial data, we may assume without loss of generality\n[by possibly replacing G by the “pro-Σ′ completion” of G, for some subset Σ′ ⊆ Σ]\nthat G arises from a stable curve over a finite field k whose characteristic \u0006∈ Σ. Write\nq for the cardinality of k; Gk for the absolute Galois group of k. We shall say that an\naction of Gk on a finitely generated, free Ẑ\nΣ-module is of weight w if the eigenvalues\nof the Frobenius element ∈ Gk are algebraic integers all of whose complex absolute\nvalues are equal to qw\/2. Now one has a natural action of Gk on G [cf. Remark\n2.5.1 below for a more detailed description of this action], and hence a natural\naction on MG which preserves M\nedge\nG , M\nvert\nG . By replacing k by a finite extension\nof k, we may assume that Gk acts trivially on the underlying semi-graph G. Thus,\nthe action of Gk on MG\/MvertG (respectively, M\nedge\nG ) is trivial [cf. Remark 1.1.3]\n(respectively, of weight 2). On the other hand, by the “Riemann hypothesis” for\nabelian varieties over finite fields [cf., e.g., [Mumf], p. 206], it follows [cf. Remark\n1.1.4] that the action of Gk on M\nvert\nG \/M\nedge\nG is of weight 1. Note, moreover, that the\naction of Gk on H\n2(ΠG , ẐΣ) is of weight −2. [Indeed, this follows by considering\n10 SHINICHI MOCHIZUKI\nthe first Chern class [cf., e.g., [FK], Chapter II, Definition 1.2] of a line bundle\nof degree one on some irreducible component of the given stable curve over k —\ncf., e.g., [Mzk4], the proof of Lemma 2.6.] Thus, since the subquotients of the\nfiltration M edgeG ⊆MvertG ⊆MG are all free over ẐΣ, the fact that M edgeG and MvertG\nare mutual annihilators follows immediately by consideration of the weights of the\nmodules involved. \u0003\nRemark 1.3.1. By Proposition 1.3 [applied to the semi-graph of anabelioids of\nPSC-type G′ obtained by “compactifying” G — cf. Remark 1.1.6], it follows that\nif G is a [not necessarily noncuspidal!] sturdy semi-graph of anabelioids of PSC-\ntype, then the ranks [over ẐΣ] of M edgeG \/M\ncusp\nG , MG\/M\nvert\nG coincide. This implies\nthat the rank [over ẐΣ] of M cuspG may be computed as the difference between the\nranks [over ẐΣ] of M edgeG , MG\/M\nvert\nG . Moreover, it follows immediately from the\ndefinitions that if G has cusps, then the rank of M cuspG is equal to r(G) − 1. Also,\n[again it follows immediately from the definitions that] G is noncuspidal if and only\nif M cuspG′ = 0 for all finite étale ΠG-coverings G′ → G. Thus, in summary, it follows\nthat one may compute r(G) as soon as one knows the difference between the ranks\n[over ẐΣ] of M edgeG′ , MG′\/M\nvert\nG′ for all finite étale ΠG-coverings G′ → G.\nDefinition 1.4. Suppose that G, H are of PSC-type; denote the respective\nassociated PSC-fundamental groups by ΠG , ΠH and the respective underlying semi-\ngraphs by G, H. Let\nα : ΠG\n∼→ ΠH; β : ΠunrG ∼→ ΠunrH\nbe isomorphisms of profinite groups.\n(i) We shall say that α is graphic if it arises from an isomorphism of semi-graphs\nof anabelioids G ∼→ H.\n(ii) We shall say that α is numerically cuspidal if, for any pair of finite étale\ncoverings G′ → G, H′ → H which correspond via α, we have r(G′) = r(H′).\n(iii) We shall say that α is graphically filtration-preserving (respectively, verti-\ncially filtration-preserving; edge-wise filtration-preserving) if, for any pair of finite\nétale coverings G′ → G, H′ → H which correspond via α, the isomorphism\nMG′\n∼→ MH′\ninduced by α induces an isomorphism between the respective verticial and edge-\nlike (respectively, verticial; edge-like) submodules. We shall say that β is verticially\nfiltration-preserving if, for any pair of finite étale coverings G′ → G, H′ → H which\ncorrespond via β, the isomorphism\nMG′\/M\nedge\nG′\n∼→ MH′\/M edgeH′\ninduced by β induces an isomorphism MvertG′ \/M\nedge\nG′\n∼→ MvertH′ \/M edgeH′ .\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 11\n(iv) We shall say that α is group-theoretically cuspidal (respectively, group-\ntheoretically edge-like; group-theoretically verticial) if and only if it maps each cusp-\nidal (respectively, edge-like; verticial) subgroup of ΠG isomorphically onto a cuspidal\n(respectively, edge-like; verticial) subgroup of ΠH, and, moreover, every cuspidal\n(respectively, edge-like; verticial) subgroup of ΠH arises in this fashion. We shall\nsay that β is group-theoretically verticial if and only if it maps each unramified\nverticial subgroup of ΠunrG isomorphically onto an unramified verticial subgroup of\nΠunrH , and, moreover, every verticial subgroup of Π\nunr\nH arises in this fashion.\n(v) Let G′ → G be a Galois finite étale covering. Then we shall say that G′ → G\nis cuspidally (respectively, nodally; verticially) purely totally ramified if there exists\na cusp e (respectively, node e; vertex v) of G such that G′ → G restricts to a trivial\ncovering over Ge′ (respectively, Ge′ ; Gv′) for all cusps e′ \u0006= e (respectively, nodes\ne′ \u0006= e; vertices v′ \u0006= v) of G and to a connected covering over Ge (respectively,\nGe; Gv). We shall say that G′ → G is cuspidally (respectively, nodally; verticially)\ntotally ramified if there exists a cusp e (respectively, node e; vertex v) of G such\nthat G′ → G restricts to a connected covering over Ge (respectively, Ge; Gv).\n(vi) If A ⊆ ΠG is a closed subgroup, and A′ ⊆ A is an open subgroup of A, then\nwe shall say that the inclusion A′ ⊆ A descends to a finite étale covering G′ → G′′\nif the arrow G′ → G′′ is a morphism of finite étale ΠG-coverings of G such that the\ncorresponding open subgroups ΠG′ ⊆ ΠG′′ ⊆ ΠG satisfy A ⊆ ΠG′′ , A\n⋂\nΠG′ = A′,\n[A : A′] = [ΠG′′ : ΠG′ ]. We shall use similar terminology when, in the preceding\nsentence, “Π” is replaced by “Πunr”.\nRemark 1.4.1. Thus, by Proposition 1.3, it follows that, if, in the notation of\nDefinition 1.4, G, H are noncuspidal, then the following three conditions on α are\nequivalent: (a) α is graphically filtration-preserving; (b) α is verticially filtration-\npreserving; (c) α is edge-wise filtration-preserving.\nRemark 1.4.2. Let G′ → G be a Galois finite étale covering of degree a power\nof l, where G is of pro-Σ PSC-type, Σ = {l}. Then one verifies immediately that\nG′ → G is cuspidally purely totally ramified if and only if the equality\nr(G′) = deg(G′\/G) · (r(G) − 1) + 1\nis satisfied. Similarly, if G′ → G is a finite étale ΠunrG -covering [so n(G′) = n(G) ·\ndeg(G′\/G)], then one verifies immediately that G′ → G is verticially purely totally\nramified if and only if the equality\ni(G′) = deg(G′\/G) · (i(G) − 1) + 1\nis satisfied. Also, we observe that this last equality is equivalent to the following\nequality involving the expression “i(. . . ) − n(. . . )” [cf. Remark 1.1.3]:\ni(G′) − n(G′) = deg(G′\/G) · (i(G) − n(G) − 1) + 1\n12 SHINICHI MOCHIZUKI\nRemark 1.4.3. Let G′ → G be as in Remark 1.4.2; assume further that this\ncovering is a cuspidally (respectively, nodally; verticially) totally ramified ΠG- (re-\nspectively, ΠG-; ΠunrG -) covering, and that G is arbitrary (respectively, arbitrary;\nsturdy). Let e (respectively, e; v) be a cusp (respectively, node; vertex) of G such\nthat G′ → G restricts to a connected covering of Ge (respectively, Ge; Gv). Then\nobserve that:\nThere exists a finite étale ΠG- (respectively, ΠG-; ΠunrG -) covering G′′ → G\nsuch that: (a) G′′ → G is trivial over Ge (respectively, Ge; Gv); (b) the\nsubcovering G′′′ → G′′ of the composite covering G′′′ → G of the coverings\nG′′ → G and G′ → G is cuspidally (respectively, nodally; verticially) purely\ntotally ramified.\nIndeed, the construction of such a covering is immediate [cf. the proof of Proposition\n1.2].\nRemark 1.4.4. Let G′ → G be as in Remark 1.4.2; assume further that this\ncovering is cyclic, and that G is noncuspidal. Then it is immediate that G′ → G\nis module-wise nodal if and only if it is nodally totally ramified. In particular, it\nfollows that:\n(i) Any closed subgroup B ⊆ ΠG is contained in some nodal edge-like sub-\ngroup if and only if, for every open normal subgroup B′ ⊆ B, the inclusion\nB′ ⊆ B descends to a module-wise nodal finite étale covering.\n(ii) A closed subgroup A ⊆ ΠG is a nodal edge-like subgroup of ΠG if and only\nif it satisfies the condition of (i) above [i.e., where one takes “B” to be A],\nand, moreover, is maximal among closed subgroups B ⊆ ΠG satisfying the\ncondition of (i).\nIndeed, the necessity of (i) is immediate. The sufficiency of (i) follows by observing\nthat since the set of nodes of a finite étale covering of G is always finite, an exhaustive\ncollection of open normal subgroups of B thus determines — by considering the\nnodes at which the “total ramification” occurs — [at least one] compatible system\nof nodes of the finite étale ΠG-coverings of G; but this implies that B is contained\nin some nodal edge-like subgroup. In light of (i), the necessity of (ii) is immediate\nfrom the definitions and Proposition 1.2, (i) [which implies maximality], while the\nsufficiency of (ii) follows immediately from the assumption of maximality.\nProposition 1.5. (Incidence Relations) We maintain the notation of Defi-\nnition 1.4. Then the following hold.\n(i) An edge-like subgroup of ΠG is cuspidal (respectively, not cuspidal) if\nand only if it is contained in precisely one (respectively, precisely two) verticial\nsubgroup(s).\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 13\n(ii) α is graphic if and only if it is group-theoretically edge-like and\ngroup-theoretically verticial. Moreover, in this case, α arises from a unique\nisomorphism of semi-graphs of anabelioids G ∼→ H.\nProof. First, we consider assertion (i). Observe that it is immediate from the\ndefinitions that a cuspidal (respectively, noncuspidal) edge-like subgroup of ΠG is\ncontained in at least one (respectively, at least two) verticial subgroup(s). To prove\nthat these lower bounds also serve as upper bounds, it suffices [by possibly replacing\nG by a finite étale covering of G] to show that if e is a cuspidal (respectively, nodal)\nedge of G that does not abut to a vertex v, then there exists a finite étale ΠG-covering\nG′ → G which is trivial over Gv, but nontrivial over Ge. But this is immediate [cf.\nthe proof of Proposition 1.2, (i)].\nNext, we consider assertion (ii). Necessity is immediate. To prove sufficiency,\nwe reason as follows: The assumption that α is group-theoretically edge-like and\ngroup-theoretically verticial implies, by considering conjugacy classes of verticial and\nedge-like subgroups [and applying Proposition 1.2, (i)], that α induces a bijection\nbetween the vertices (respectively, edges) of the underlying semi-graphs G, H. By\nassertion (i), this bijection maps cuspidal (respectively, nodal) edges to cuspidal\n(respectively, nodal) edges and is compatible with the various “incidence relations”\nthat define the semi-graph structure [i.e., the data of which vertices an edge abuts\nto]. Thus, α induces an isomorphism of semi-graphs G\n∼→ H. Finally, by Proposition\n1.2, (ii), one concludes that α arises from a unique isomorphism G ∼→ H, as desired.\n\u0003\nTheorem 1.6. (Criterion for Graphicity) We maintain the notation of\nDefinition 1.4. Then the following hold.\n(i) α is numerically cuspidal if and only if it is group-theoretically cus-\npidal.\n(ii) α is graphic if and only if it is graphically filtration-preserving.\n(iii) Assume that G, H are sturdy. Then β is verticially filtration-preser-\nving if and only if it is group-theoretically verticial.\nProof. First, we consider assertion (i). Sufficiency is immediate [cf. Proposition\n1.2, (i)]. The proof of necessity is entirely similar to the latter half of the proof of\n[Mzk4], Lemma 1.3.9: Let l ∈ Σ [where G, H are of pro-Σ PSC-type]. Since the\ncuspidal edge-like subgroups may be recovered as the stabilizers of cusps of finite\nétale coverings of G, H, it suffices to show that α induces a functorial bijection\nbetween the sets of cusps of G, H. In particular, we may assume, without loss of\ngenerality, that Σ = {l}.\nThen given pairs of finite étale ΠG- or ΠH-coverings that correspond via α\nG′′ → G′ → G; H′′ → H′ → H\n14 SHINICHI MOCHIZUKI\nsuch that G′′ is Galois over G′, and H′′ is Galois over H′, it follows from the\nassumption that α is numerically cuspidal that G′′ → G′ is cuspidally purely totally\nramified if and only if H′′ → H′ is [cf. Remark 1.4.2]. Now observe that the\ncuspidal edge-like subgroups of ΠG (respectively, ΠH) are precisely the maximal\nclosed subgroups A such that, for every open normal subgroup A′ ⊆ A, the inclusion\nA′ ⊆ A descends to a cuspidally purely totally ramified Galois finite étale covering.\nIndeed, in light of Remark 1.4.3 [which implies that, in the preceding sentence,\none may remove the word “purely” without affecting the validity of the assertion\ncontained in this sentence], this follows by a similar argument to the argument\napplied in the case of nodes in Remark 1.4.4. Thus, we thus conclude that α is\ngroup-theoretically cuspidal, as desired.\nNext, we consider assertion (ii). Necessity is immediate. To prove sufficiency,\nlet us first observe that by functoriality; Proposition 1.2, (ii); Proposition 1.5, (ii),\nit follows that we may always replace G, H by finite étale ΠG- or ΠH-coverings that\ncorrespond via α. In particular, by Remark 1.1.5, we may assume without loss\nof generality that G, H are sturdy. Next, let us observe that by Proposition 1.3\n[cf. Remark 1.3.1], the assumption that α is graphically filtration-preserving implies\nthat α is numerically cuspidal, hence [by assertion (i)] that α is group-theoretically\ncuspidal. Thus, by replacing G, H by their respective compactifications [cf. Remark\n1.1.6], and replacing α by the isomorphism induced by α between the respective\nquotients ΠG \u0002 ΠcptG , ΠH \u0002 Π\ncpt\nH , we may assume, without loss of generality, that\nG, H are noncuspidal and sturdy. Also, as in the proof of assertion (i), we may\nassume that Σ = {l}. Now by Proposition 1.5, (ii), it suffices to prove that α is\ngroup-theoretically edge-like and group-theoretically verticial. But by Remark 1.4.4,\nthe assumption that α is edge-wise filtration-preserving implies that α is group-\ntheoretically edge-like. In particular, α induces a verticially filtration-preserving\nisomorphism ΠunrG\n∼→ ΠunrH . Now to prove that α is group-theoretically verticial, it\nsuffices to prove [cf. the proof of assertion (i)] that α induces a functorial bijection\nbetween the sets of vertices of G, H. Thus, to complete the proof of assertion (ii),\nit suffices to prove that β is group-theoretically verticial, that is to say, it suffices to\nverify assertion (iii).\nFinally, we consider assertion (iii). Sufficiency is immediate. On the other\nhand, necessity follows from Remark 1.4.2, by observing that the unramified verti-\ncial subgroups are precisely the maximal closed subgroups A of ΠunrG or Π\nunr\nH such\nthat, for every open normal subgroup A′ ⊆ A, the inclusion A′ ⊆ A descends to\na verticially purely totally ramified Galois finite étale covering. Indeed, in light\nof Remark 1.4.3 [which implies that, in the preceding sentence, one may remove\nthe word “purely” without affecting the validity of the assertion contained in this\nsentence], this follows by a similar argument to the argument applied in the case of\nnodes in Remark 1.4.4. This completes the proof of assertion (ii). \u0003\nRemark 1.6.1. The essential content of Theorem 1.6, (i), is, in many respects,\nsimilar to the essential content of [Tama2], Lemma 2.3 [cf. the use of this lemma\nin [Tama2], Proposition 2.4].\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 15\nSection 2: The Group of Graphic Outer Automorphisms\nIn this §, we study the consequences of the theory of §1 for the group of\nautomorphisms of a semi-graph of anabelioids of PSC-type.\nLet G be a semi-graph of anabelioids of pro-Σ PSC-type [with underlying semi-\ngraph G]. In the following discussion, G, G will remain fixed until further notice to\nthe contrary [in Corollary 2.7].\nDenote by Aut(G) the group of automorphisms of the semi-graph of anabelioids\nG. Here, we recall that an automorphism of a semi-graph of anabelioids consists of\nan automorphism of the underlying semi-graph, together with a compatible system\nof isomorphisms between the various anabelioids at each of the vertices and edges\nof the underlying semi-graph, which are compatible with the various morphisms of\nanabelioids associated to the branches of the underlying semi-graph — cf. [Mzk7],\nDefinition 2.1; [Mzk7], Remark 2.4.2. Then, by Proposition 1.5, (ii), we obtain an\ninjective homomorphism\nAut(G) ↪→ Out(ΠG)\nwhose image we shall denote by\nOutgrph(ΠG) ⊆ Out(ΠG)\nand refer to as the group of graphic outer automorphisms of ΠG . Since ΠG is topo-\nlogically finitely generated [cf. Remark 1.1.3], it follows that Out(ΠG) is equipped\nwith a natural profinite topology, which thus induces a natural topology on the sub-\ngroup Outgrph(ΠG) ⊆ Out(ΠG), which is manifestly closed, by Proposition 1.5, (ii).\nIn particular, Aut(G) ∼= Outgrph(ΠG) is equipped with a natural profinite topology.\nSince ΠG is center-free [cf. Remark 1.1.3], we have a natural exact sequence\n1 → ΠG → Aut(ΠG) → Out(ΠG) → 1, which we may pull-back via Aut(G) ↪→\nOut(ΠG) to obtain an exact sequence as follows:\n1 → ΠG → ΠAutG → Aut(G) → 1\nIf G′ → G is a sturdy [i.e., G′ is sturdy] finite étale ΠG-covering which arises from a\ncharacteristic open subgroup ΠG′ ⊆ ΠG , then there is a natural action of ΠAutG on\nG′. In particular, we obtain, for every l ∈ Σ, a natural action of ΠAutG on the free\nZl-module of rank one [i.e., since G′ is sturdy] H2(ΠcptG′ ,Zl).\nLemma 2.1. (Construction of the Cyclotomic Character) This ac-\ntion of ΠAutG on H\n2(Π\ncpt\nG′ ,Zl) factors through the quotient Π\nAut\nG \u0002 Aut(G), hence\ndetermines a continuous homomorphism Aut(G) → Z×l , whose inverse\nχl : Aut(G) → Z×l\nwe shall refer to as the pro-l cyclotomic character of Aut(G). Moreover, this\ncharacter is independent of the choice of sturdy ΠG-covering G′ → G.\n16 SHINICHI MOCHIZUKI\nProof. To verify the asserted independence of covering, it suffices to observe that\nany two sturdy ΠG-coverings G′ → G, G′′ → G may be dominated by a third sturdy\nΠG-covering G′′′ → G, which induce isomorphisms of free Ql-modules of rank one\nH2(ΠcptG′ ,Zl) ⊗ Q → H2(ΠcptG′′′ ,Zl) ⊗ Q; H2(ΠcptG′′ ,Zl) ⊗ Q → H2(ΠcptG′′′ ,Zl) ⊗ Q\nwhich are compatible with the various actions by ΠAutG .\nTo show that the action of ΠAutG factors through Aut(G), we may assume\nwithout loss of generality that Σ is the set of all primes. On the other hand, by\nthe independence of covering already verified, it follows that we may compute the\nΠAutG -action in question by using a covering G′ → G of degree prime to l(l−1). Since\nthe action in question amounts to a continuous homomorphism ΠAutG → Z×l which\nclearly factors through ΠAutG \/ΠG′ , the desired factorization follows from the fact\nthat [consideration of orders implies that] every homomorphism Gal(G′\/G) → Z×l\nis trivial. \u0003\nProposition 2.2. (The Double of a Semi-Graph of Anabelioids of PSC-\ntype) Suppose that r(G) \u0006= 0. Let H be the semi-graph of anabelioids defined\nas follows: The underlying semi-graph H is obtained by taking the disjoint union\nof two copies of G and, for each cusp e of G abutting to a vertex v of G, replacing\nthe corresponding pairs of cusps lying in these two copies of G by a node [i.e., a\nclosed edge] that joins the pairs of vertices corresponding to v in these two copies.\nWe shall refer to the newly appended nodes as bridges. Away from the bridges,\nwe take the semi-graph of anabelioids structure of H to be the structure induced\nby G, and, at each branch of a bridge of H, we take the semi-graph of anabelioids\nstructure of H to be the structure induced by G at the corresponding cusp e of G, by\ngluing the two copies of Ge in question by means of the inversion automorphism\nGe → Ge [induced by “multiplication by −1” on the abelian fundamental group of\nGe]. We shall refer to H as the double of G. Then the following hold.\n(i) H is a noncuspidal semi-graph of anabelioids of PSC-type.\n(ii) Restriction of finite étale coverings of H to each of the copies of G used to\nconstruct H determines a natural injective continuous outer homomorphism\nΠG ↪→ ΠH.\n(iii) The homomorphism of (ii) maps verticial (respectively, edge-like) sub-\ngroups of ΠG isomorphically onto verticial (respectively, edge-like) subgroups of ΠH.\n(iv) The homomorphism of (ii) induces an injection\nMG ↪→MH\nthat maps M edgeG (respectively, M\nvert\nG ) into M\nedge\nH (respectively, M\nvert\nH ).\nProof. Assertion (i) is immediate from the definitions. Note, relative to Definition\n1.1, (i), that there is a corresponding construction of a “double” of a pointed stable\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 17\ncurve. This explains the need for “gluing by means of the inversion automorphism”\nin the definition of H: Over, say, a complete discrete valuation ring A with alge-\nbraically closed residue field, the completion of a generically smooth pointed stable\ncurve at a node is isomorphic to the formal spectrum of the complete local ring\nA[[x, y]]\/(xy − s), where x, y are indeterminates and s lies in the maximal ideal\nof A. Then the action of the local tame Galois group at each of the branches of\nthe node considered independently is of the form x \t→ ζ · x, y \t→ ζ · y, where ζ\nis some root of unity. On the other hand, since the Galois action on coverings of\nthe entire formal spectrum of A[[x, y]]\/(xy − s) [i.e., where one does not treat the\nbranches of the node independently] necessarily fixes elements of the base ring [i.e.,\nthe normalization of A in some finite extension of its quotient field], it follows that\nthis action must be of the form x \t→ ζ · x, y \t→ ζ−1 · y.\nAs for assertion (ii), it is immediate that we obtain a natural homomorphism\nΠG → ΠH. The asserted injectivity may be verified as follows [cf. also the proof\nof injectivity in [Mzk7], Proposition 2.5, (i)]: Given any finite étale ΠG-covering\nG′ → G, one may construct a finite étale ΠH-covering H′ → H which induces\nG′ → G via the “restriction procedure” of (ii) by gluing together two copies of G′\nover the two copies of G used to construct H. Note that to carry out this gluing,\none must choose a [noncanonical!] isomorphism, at each cusp e of G, between the\nrestriction of G′ → G to Ge and the pull-back via the inversion automorphism of\nthis restriction. [Note that it is immediate that such an isomorphism always exists.]\nAssertion (iii) is immediate from the construction of the double.\nFinally, we consider assertion (iv). To verify that the homomorphism MG →\nMH induced by the homomorphism of (ii) is an injection, it suffices to observe that\nthe gluing procedure discussed in the proof of the injectivity of (ii) determines a\nsplitting of the homomorphism MG → MH. Indeed, if the finite étale ΠG-covering\nG′ → G in question is abelian, with Galois group A, then the resulting H′ → H\nadmits a natural action by A, by letting A act via the identity A → A on one\ncopy of G′ and via “multiplication by −1” A → A on the other copy of G′. [Put\nanother way, if we think of the covering G′ → G as corresponding to the A-set A,\nthen we glue the set A to the set A at the bridges by means of the automorphism\n“multiplication by −1”.] This completes the proof of injectivity. The fact that this\ninjection maps M edgeG (respectively, M\nvert\nG ) into M\nedge\nH (respectively, M\nvert\nH ) follows\nimmediately from assertion (iii). \u0003\nRemark 2.2.1. Certain aspects of Proposition 2.2 are related to the results of\n[Asada].\nRemark 2.2.2. It is quite possible that various aspects of Proposition 2.2 may\nbe generalized from the case of “two copies of G” treated in Proposition 2.2 to the\ncase of gluing together arbitrary finite collections of semi-graphs of anabelioids of\nPSC-type. This topic, however, lies beyond the scope of the present paper.\nDefinition 2.3. Let J be a profinite group which acts continuously on G [i.e.,\n18 SHINICHI MOCHIZUKI\nwe are given a continuous homomorphism J → Aut(G)]. Set:\nΠJG\ndef\n= ΠAutG ×Aut(G) J\nLet M be a continuous Zl[J ]-module [i.e., a topological module equipped with\ncontinuous actions by Zl, J ], where l ∈ Σ.\n(i) We shall refer to a [continuous] character ψ : J → Z×l as quasi-cyclotomic\n(respectively, Q-cyclotomic) if ψ (respectively, some positive power of ψ) coincides\nwith the restriction to J of the character χl (respectively, some integer power of\nthe character χl) of Lemma 2.1 on some open subgroup J\n′ ⊆ J of J . If ψ : J → Z×l\nis a [continuous] character, then we shall denote by\nM(ψ)\nthe ψ-twist of M . That is to say, the underlying topological Zl-modules of M ,\nM(ψ) are identical; if the action of γ ∈ J on M maps m ∈ M to γ ·m ∈ M , then\nthe action of γ ∈ J on M(ψ) maps m \t→ ψ(γ) · (γ ·m) ∈ M = M(ψ). If n ∈ Z,\nthen we shall write M(n)\ndef\n= M((χl|J)n), where χl|J denotes the restriction of the\ncyclotomic character χl of Lemma 2.1 to J . We shall say that M is quasi-trivial if\nsome open subgroup J ′ ⊆ J acts trivially on M . We shall say that M is quasi-toral\nif M(−1) is quasi-trivial. If, for some open subgroup J ′ ⊆ J , there exists a finite\nfiltration of Zl[J\n′]-submodules\nMn ⊆Mn−1 ⊆ . . . ⊆M j ⊆ . . . ⊆M1 ⊆M0 = M\nsuch that each M j\/M j+1 is torsion-free and, moreover, either is quasi-trivial [over\nJ ′] or has no quasi-trivial J ′′-subquotients for any open subgroup J ′′ ⊆ J ′, then we\nshall refer to the [possibly infinite] sum\n∑\nMj\/Mj+1 quasi-trivial\ndimQl (M\nj\/M j+1 ⊗ Ql)\n[which is easily verified to be independent of the choice of a subgroup J ′ ⊆ J and\na filtration {M j} satisfying the above properties] as the quasi-trivial rank of M .\n(ii) We shall say that [the action of] J is l-cyclotomically full if the image of\nthe homomorphism χl|J : J → Z×l is open. Suppose that J is l-cyclotomically\nfull. Then it makes sense to speak of the weight w of a Q-cyclotomic character\nψ : J → Z×l : i.e., w is the unique rational number that may be written in the form\n2a\/b, where a, b are integers such that b \u0006= 0, ψb = (χl|J)a. If w > 0 (respectively,\nw = 0; w < 0), then we shall say that ψ is positive (respectively, null; negative).\nIf w ∈ Q, and ψ : J → Z×l is a Q-cyclotomic character of weight w, then we shall\nrefer to the quasi-trivial rank of M(ψ−1) as the l-weight w rank of M . [One verifies\nimmediately that the l-weight w rank is independent of the choice of ψ.] If the\nl-weight w rank of M is nonzero, then we shall say that w is an associated l-weight\nof M . Write\nwl(M) ⊆ Q\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 19\nfor the set of associated l-weights of M .\n(iii) Suppose that J is l-cyclotomically full. Observe that if ΠG′ ⊆ ΠG is any\ncharacteristic open subgroup, then ΠJG acts naturally on ΠG′ , hence also onMG′⊗Zl.\nSet\nwl(J)\ndef\n=\n⋃\nG′\nwl(MG′ ⊗ Zl)\n[where the union ranges over characteristic open subgroups ΠG′ ⊆ ΠG ]. We shall\nrefer to wl(J) as the set of associated l-weights of [the action of] J . If every w ∈\nwl(J) satisfies 0 ≤ w ≤ 2, then we shall say that [the action of] J is weakly l-\ngraphically full. If, for every characteristic open subgroup ΠG′ ⊆ ΠG , it holds that\nwl((M\nvert\nG′ \/M\nedge\nG′ ) ⊗ Zl) ⊆ (0, 2)Q def= {w ∈ Q | 0 < w < 2}\nthen we shall say that [the action of] J is l-graphically full. [Thus, “J l-graphically\nfull” implies “J weakly l-graphically full” — cf. Proposition 2.4, (i), (ii), below.]\nRemark 2.3.1. The purpose of the introduction of the notion “l-cyclotomically\nfull” is to allow us to describe, in compact form, that situation in which it makes\nsense to speak of “weights” in a fashion similar to the case where the action of\nJ arises from scheme theory. Once it makes sense to speak of “weights”, one\nmay introduce the notion of “l-graphically full” (respectively, “weakly l-graphically\nfull”), which asserts, in essence, that the weights behave as one would expect in the\ncase of precisely one (respectively, at least one, i.e., possibly two nested, as in the\nsituations of Corollaries 2.8, 2.10 below) degeneration(s) of the hypothetical family\nof hyperbolic curves under consideration.\nProposition 2.4. (Quasi-triviality and Quasi-torality) Let J be as in\nDefinition 2.3; l ∈ Σ. Write m(G) for the rank [over ẐΣ] of the finitely generated,\nfree ẐΣ-module MG. Then the following hold.\n(i) (MG\/MvertG ) ⊗ Zl is quasi-trivial.\n(ii) M cuspG ⊗Zl, M edgeG ⊗Zl are quasi-toral. In particular, if J is l-cyclotomi-\ncally full, and 2 \u0006∈ wl(J), then the submodule M edgeG ⊗ Zl ⊆MG ⊗ Zl is zero.\n(iii) Assume that G is sturdy. Then there exists a positive integer m ≤ 2m(G)\nsuch that det(MG ⊗ Zl)⊗2(−m) is quasi-trivial.\n(iv) Assume that G is sturdy. Then a character ψ : J → Z×l is Q-cyclotomic\nif and only if it admits a positive power that coincides with the aψ-th power of\nthe character obtained by the natural action of J on det(MG ⊗ Zl)⊗2, for some\naψ ∈ Z. Suppose further that J is l-cyclotomically full. Then a Q-cyclotomic\nψ is positive (respectively, null; negative) if and only if aψ may be taken to\nbe positive (respectively, zero; negative). Finally, any two Q-cyclotomic characters\nJ → Z×l of the same weight necessarily coincide on some open subgroup J ′ ⊆ J .\n20 SHINICHI MOCHIZUKI\n(v) Assume that the image of J in Aut(G) is open. Then J is l-graphically\nfull.\n(vi) Assume that J is l-cyclotomically full. Then 2 \u0006= w ∈ wl(J) implies\n2 − w ∈ wl(J). If, moreover, G is noncuspidal, then w ∈ wl(J) implies 2 − w ∈\nwl(J).\n(vii) Assume that J is l-cyclotomically full, and that G has cusps [i.e., that\nΠG is free — cf. Remark 1.1.3]. Then for every sufficiently small open subgroup\nΠG′ ⊆ ΠG, the subset\n{0}\n⋃\nwl(MG′ ⊗ Zl) ⊆ Q\nis invariant with respect to the automorphism λ \t→ 2 − λ of Q; in particular, the\nsum of the maximum and minimum elements of this [finite] subset is equal to\n2.\n(viii) Assume that J is l-graphically full. Then M edgeG ⊗ Zl ⊆ MG ⊗ Zl is\nthe maximal quasi-toral Zl[J ]-submodule of MG ⊗ Zl.\n(ix) Assume that J is l-graphically full. Then MG ⊗Zl \u0002 (MG\/MvertG )⊗Zl\nis the maximal torsion-free quasi-trivial Zl[J ]-quotient module of MG⊗Zl.\nProof. Assertion (i) follows immediately from Remarks 1.1.3, 1.1.4. Now when G is\nnoncuspidal, assertion (ii) follows from assertion (i); Proposition 1.3. For arbitrary\nG, assertion (ii) follows from assertion (ii) in the noncuspidal case, together with\nProposition 2.2, (iv). Assertion (iii) follows immediately from assertion (ii) [applied\ntoM cuspG ⊗Zl]; Proposition 1.3 [applied to (MG\/M cuspG )⊗Zl, which is possible in light\nof the sturdiness assumption — cf. Remark 1.1.6]. Assertion (iv) follows formally\nfrom assertion (iii); the definitions; the fact that Z×l contains a torsion-free open\nsubgroup.\nTo verify assertion (v), it suffices to consider the case where G arises from a\npointed stable curve over a finite field k [cf. the proof of Proposition 1.3], and J\nis equal to an open subgroup of Aut(G). Then assertion (v) follows from the fact\nthat [in the notation and terminology of loc. cit.] the action of Gk on M\nvert\nG \/M\nedge\nG\nis of weight 1. Assertion (vi) follows from assertion (ii); Proposition 1.3, applied to\nthe compactification [cf. Remark 1.1.6] of a sturdy finite étale ΠG-covering of G.\nNext, we consider assertion (vii). Suppose that ΠG′ ⊆ ΠG is an open subgroup\nsuch that r(G′) ≥ 2. Thus, M cuspG′ \u0006= 0 [cf. Remark 1.3.1], so [by assertion (ii)]\n0, 2 ∈ EG′ def= {0}\n⋃\nwl(MG′ ⊗ Zl). Thus, if we set E′G′ def= wl(MvertG′ \/M edgeG′ ⊗ Zl),\nthen [by assertions, (i), (ii)], it follows that EG′ = {0, 2}\n⋃\nE′G′ . Moreover, by\nProposition 1.3 [applied to the compactification [cf. Remark 1.1.6] of G], E′G′ is\ninvariant with respect to the automorphism λ \t→ 2 − λ of Q. But this implies the\ndesired invariance of EG′ with respect to this automorphism of Q. This completes\nthe proof of assertion (vii). Finally, assertions (viii), (ix) follow immediately from\nassertions (i), (ii); the definitions. \u0003\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 21\nExample 2.5. Stable Log Curves over a Logarithmic Point. Let Slog\nbe a log scheme, with underlying scheme S\ndef\n= Spec(k), where k is a field, and log\nstructure given by a chart N \u000b 1 \t→ 0 ∈ k [cf. the theory of [Kato]]. Let\nX log → Slog\nbe a stable log curve over Slog [cf. §0]. Let T log → Slog be a “separable closure”\nof Slog, i.e., the underlying scheme T of T log is of the form T = Spec(k), where k\nis a separable closure of k; the log structure of T log is given by a chart M \u000b 1 \t→\n0 ∈ k, where M ⊆ Q is the monoid of positive rational numbers with denominators\ninvertible in k; the morphism T log → Slog arises from the natural maps k ↪→ k,\nN ↪→ M. Thus, if we write Gklog def= Aut(T log\/Slog), then we have a natural exact\nsequence\n1 → Iklog → Gklog → Gk → 1\nwhere Gk\ndef\n= Gal(k\/k); Iklog\ndef\n= Hom(Q\/Z, k\n×\n). Now the admissible coverings of\nX log [with tame ramification at the cusps] determine an admissible fundamental\ngroup ΠXlog which fits into a natural exact sequence:\n1 → ΔXlog → ΠXlog → Gklog → 1\n[The theory of admissible coverings is discussed in detail in [Mzk1], §3; [Mzk2], §2;\n[Mzk4], §2; [Mzk4], Appendix. It follows, in particular, from this theory that, if\none chooses a lifting of X log → Slog to some generically smooth stable log curve\nX loglift → Sloglift\n— where Slift is the spectrum of a complete discrete valuation ring with residue\nfield k; the log structure on Sloglift is the log structure determined by the monoid\nof generically invertible functions — then the coverings arising from ΠXlog may\nbe realized as coverings of the generically smooth curve X loglift that satisfy certain\nproperties.] Moreover, if Σ is a set of primes that does not contain the residue\ncharacteristic of k, and we denote by G the semi-graph of anabelioids of pro-Σ\nPSC-type arising from the pointed stable curve over k determined by X log, then\nthe maximal pro-Σ quotient of ΔXlog may be naturally identified with the PSC-\nfundamental group ΠG . In particular, one obtains a natural outer action of Gklog\non ΠG, the automorphisms of which are easily seen [by the functoriality of the\nvarious fundamental groups involved!] to be graphic. That is to say, we obtain\ncontinuous homomorphisms as follows:\nGklog → Aut(G) ∼= Outgrph(ΠG) ⊆ Out(ΠG)\nNow suppose that H ⊆ Gklog is a closed subgroup such that the restriction to H of\nthe homomorphism Gklog → Aut(G) factors through some quotient H \u0002 J :\nH \u0002 J → Aut(G)\nFor l ∈ Σ, we shall refer to the image in J of the intersection of H with the pro-l\ncomponent of Iklog as the l-inertia subgroup of J ; we shall say that [the action\n22 SHINICHI MOCHIZUKI\non G of] J is l-logarithmically full if the l-inertia subgroup of J is infinite [hence\nisomorphic to Zl(1)]. If H is an open subgroup Gklog , then we shall say that [the\naction on G of] J is arithmetically full and refer to k as the base field.\nRemark 2.5.1. Note that from the point of view of Example 2.5, one may think\nof the action of Gk on G appearing in the proof of Proposition 1.3 as the restriction\nof the action of Gklog on G discussed in Example 2.5 to some section of Gklog \u0002 Gk.\nProposition 2.6. (The Logarithmic Inertia Action) In the notation of\nExample 2.5, Iklog acts quasi-unipotently [i.e., an open subgroup of Iklog acts\nunipotently] on MG ⊗ Zl, and, moreover, the submodule\nMvertG ⊗ Zl ⊆MG ⊗ Zl\nis the maximal quasi-trivial Zl[Iklog ]-submodule of MG ⊗Zl [i.e., the maximal\nsubmodule on which some open subgroup of Iklog acts trivially].\nProof. Let us first observe that if G is noncuspidal, then the asserted quasi-\nunipotency (respectively, quasi-triviality) of the action of Iklog on MG ⊗ Zl (re-\nspectively, MvertG ⊗ Zl) follows immediately from the well-known theory of Galois\nactions on torsion points of degenerating abelian varieties [cf., e.g., [FC], Chapter\nIII, Corollary 7.3; here, we note that, in the terminology of loc. cit., the submodule\nMvertG ⊗Zl corresponds to the submodule determined by the “Raynaud extension”].\nThus, one obtains the asserted quasi-unipotency\/quasi-triviality in the case of not\nnecessarily noncuspidal G by applying the theory of the “double” [cf. Proposition\n2.2, (iv)]. Now it remains to prove the asserted maximality. But this follows again\nfrom [FC], Chapter III, Corollary 7.3 [i.e., the fact that the period matrix of a\ndegenerating abelian variety is always nondegenerate]. \u0003\nNow, by combining Theorem 1.6 with the theory of the present §2 [cf., in\nparticular, Proposition 2.4], we obtain the following result.\nCorollary 2.7. (Graphicity) Let G, H be semi-graphs of anabelioids of\npro-Σ PSC-type; JG → Aut(G), JH → Aut(H) continuous homomorphisms.\nSuppose, moreover, that we have been given isomorphisms of profinite groups\nα : ΠG\n∼→ ΠH; ι : JG ∼→ JH\nwhich are compatible, with respect to the respective outer actions of JG, JH on\nΠG, ΠH. Then, for l ∈ Σ:\n(i) Suppose that the respective actions of JG, JH on G, H are l-cyclotomically\nfull. Then α is group-theoretically cuspidal.\n(ii) Suppose that the respective actions of JG , JH on G, H are l-graphically\nfull [cf., e.g., Proposition 2.4, (v)]. Then α is graphic.\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 23\n(iii) Suppose that the respective actions of JG, JH on G, H arise from data\nas in Example 2.5; that G, H are noncuspidal; and that, in the terminology of\nExample 2.5, these actions are l-logarithmically full, and, moreover, ι maps the\nl-inertia subgroup of JG isomorphically onto that of JH. Then α is graphic.\n(iv) Outgrph(ΠG) is commensurably terminal in Out(ΠG).\nProof. First, we consider assertion (i). By Theorem 1.6, (i), it suffices to prove that\nα is numerically cuspidal, under the further assumption that G, H have cusps [i.e.,\nthat ΠG , ΠH are free — cf. Remark 1.1.3]. By passing to sturdy finite étale coverings\nof G, H that correspond via α [cf. Remark 1.1.5], it follows from Proposition 2.4,\n(iv), that ι preserves positive and null Q-cyclotomic characters to Z×l . Thus, by\nProposition 2.4, (vii), it follows that ι preserves the Q-cyclotomic characters to Z×l\nof weight 2. Now, by applying Proposition 1.3 to the compactifications [cf. Remark\n1.1.6] of sturdy finite étale coverings G′ → G, H′ → H that correspond via α [cf.\nRemark 1.1.5], we conclude that the rank of M cuspG′ (respectively, M\ncusp\nH′ ) may be\ncomputed as the difference between the l-weight 2 and l-weight 0 ranks of MG′\n(respectively, MH′) [cf. Proposition 2.4, (ii); Remark 1.3.1]; moreover, [cf. Remark\n1.3.1] this data allows one to compute r(G′) (respectively, r(H′)). This completes\nthe proof of assertion (i).\nNext, we consider assertion (ii). By assertion (i), it follows that α is group-\ntheoretically cuspidal. Thus, by replacing G, H by the compactifications [cf. Remark\n1.1.6] of sturdy finite étale coverings of G, H that correspond via α [cf. Remark\n1.1.5], we may assume without loss of generality that G, H are noncuspidal. Thus,\nby Theorem 1.6, (ii); Remark 1.4.1, it suffices to prove that α is verticially filtration-\npreserving. But this follows from Proposition 2.4, (ix). This completes the proof of\nassertion (ii). Assertion (iv) follows formally from assertion (ii) [by taking H def= G;\nJG , JH to be open subgroups of Outgrph(ΠG) — cf. Proposition 2.4, (v)].\nFinally, we consider assertion (iii). By Theorem 1.6, (ii); Remark 1.4.1, it\nsuffices to prove that α is verticially filtration-preserving. But this follows from\nProposition 2.6 and the assumptions concerning the l-inertia subgroups. This com-\npletes the proof of assertion (iii). \u0003\nRemark 2.7.1. Corollary 2.7, (iv), may be regarded as a sort of anabelian\nanalogue of the well-known linear algebra fact that, if k is an algebraically closed\nfield, then parabolic subgroups of the general linear group GLn(k), where n ≥ 2 —\ne.g., the subgroups that preserve some filtration of a k-vector space of dimension n\n— are normally terminal in GLn(k) [cf., e.g., [Hum], p. 179].\nRemark 2.7.2. Note that the group-theoretic cuspidality of [Mzk4], Lemma 1.3.9\n(respectively, the graphicity of [Mzk4], Lemma 2.3) may be regarded as a [rather\nweak] special case of Corollary 2.7, (i) (respectively, Corollary 2.7, (ii)) — cf. the\nproof of Proposition 2.4, (v), above.\nCorollary 2.8. (Graphicity over an Arithmetic Logarithmic Point) Let\nG, H be semi-graphs of anabelioids of pro-Σ PSC-type; JG → Aut(G), JH →\n24 SHINICHI MOCHIZUKI\nAut(H) continuous homomorphisms that arise from data as in Example 2.5 such\nthat [in the terminology of Example 2.5] the resulting actions are l-logarithmically\nfull, for some l ∈ Σ, and arithmetically full, with base field isomorphic to\na subfield of a finitely generated extension of Fp or Qp, for some prime p \u0006∈ Σ\n[where we allow p to differ for G, H]. Suppose, moreover, that we have been given\nisomorphisms of profinite groups\nα : ΠG\n∼→ ΠH; ι : JG ∼→ JH\nwhich are compatible, with respect to the respective outer actions of JG, JH on\nΠG, ΠH, and satisfy the property that ι maps the l-inertia subgroup of JG iso-\nmorphically onto that of JH. Then the respective actions of JG, JH on G, H are\nweakly l-graphically full, and α is graphic.\nProof. Indeed, by using the Frobenius elements of the Galois group of a finitely\ngenerated extension of Fp or Qp containing the base field in question [cf. the proof\nof Proposition 2.4, (v)], one obtains that JG , JH are weakly l-graphically full. [Note\nthat, unlike the situation in the proof of Proposition 2.4, (v), the pointed stable\ncurve over a finite field that one uses here to conclude weak l-graphic fullness will,\nin general, be a degeneration of the original pointed stable curve over the base\nfield appearing in Example 2.5. This is the reason why [unlike the situation in\nthe proof of Proposition 2.4, (v)] in the present context, one may only conclude\nweak l-graphic fullness.] By Corollary 2.7, (i), we thus conclude that α is group-\ntheoretically cuspidal. Moreover, this allows us [by passing to compactifications of\nsturdy finite étale coverings] to reduce to the noncuspidal case, hence to conclude\nthat α is graphic by Corollary 2.7, (iii). \u0003\nRemark 2.8.1. In the situation of Corollary 2.8, suppose further that the base\nfield in question is sub-p-adic [i.e., isomorphic to a subfield of a finitely generated\nextension of Qp], and that ι lies over an isomorphism between the absolute Galois\ngroups of the respective base fields that arises from an isomorphism between the\nrespective base fields. Then one may apply the main result of [Mzk3] — just as the\nmain result of [Tama1] was applied in [Mzk2], §7 — to the various verticial subgroups\nto obtain a version of the Grothendieck conjecture for pointed stable curves over a\nsub-p-adic field. Note that in this situation, when Σ is the set of all primes, one may\nalso reconstruct the log structures at the nodes by considering the decomposition\ngroups at the nodes [cf. the theory of [Mzk2], §6]. We leave the routine details to\nthe interested reader.\nRemark 2.8.2. In the situation of Corollary 2.8, suppose further that the base\nfield in question is a finite extension of Qp [which may differ for G, H], and that\nΣ is the set of all primes. Then observe that it follows from [Mzk4], Lemma 1.1.4,\n(ii), that ι lies over an isomorphism between the absolute Galois groups of the re-\nspective base fields [that does not necessarily arise from an isomorphism between\nthe respective base fields!]. Now suppose further that the hyperbolic curve consti-\ntuted by [the complement of the nodes and cusps in] each irreducible component\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 25\nof the pointed stable curves over the respective base fields that give rise to the\ndata in question is isogenous [cf. §0] to a hyperbolic curve of genus zero. Then\nit follows from the theory of [Mzk6], §4 — more precisely, the “rigidity” of the\ncuspidal edge-like subgroups implied by [Mzk6], Theorem 4.3, together with the\nintegral absoluteness of [Mzk6], Corollary 4.11 — that one may reconstruct the log\nstructures at the nodes by considering the decomposition groups at the nodes [cf.\nthe theory of [Mzk2], §6]. We leave the routine details to the interested reader.\nCorollary 2.9. (Unramified Graphicity) Let G, H be sturdy semi-graphs\nof anabelioids of pro-Σ PSC-type; JG → Aut(G), JH → Aut(H) continu-\nous homomorphisms which determine l-graphically full actions, for some l ∈ Σ.\nSuppose, moreover, that we have been given factorizations\nJG \u0002 J ′G → Out(ΠunrG ); JH \u0002 J ′H → Out(ΠunrH )\n[where the composite homomorphisms are the natural homomorphisms; the first ar-\nrow of each factorization is a surjection], together with isomorphisms of profinite\ngroups\nβ : ΠunrG\n∼→ ΠunrH ; ι′ : J ′G ∼→ J ′H\nwhich are compatible, with respect to the respective outer actions of J ′G, J\n′\nH on\nΠunrG , Π\nunr\nH . Then β is group-theoretically verticial.\nProof. By Theorem 1.6, (iii), it suffices to prove that α is verticially filtration-\npreserving. But this follows from Proposition 2.4, (ix). \u0003\nRemark 2.9.1. Note that the group-theoretic verticiality of [Mzk2], Proposition\n1.4 may be regarded as a [rather weak] special case of Corollary 2.9 — cf. the proof\nof Proposition 2.4, (v), above.\nFinally, we observe the following consequence of the theory of the present paper\nconcerning anabelian geometry over finite extensions of the quotient field of the ring\nof Witt vectors of an algebraic closure of a finite field.\nCorollary 2.10. (Inertia Action in the Case of Two Primes) For i = 1, 2,\nlet Ki be a finite extension of the quotient field of the ring of Witt vectors W (Fpi)\nwith coefficients in an algebraic closure Fpi of the finite field of cardinality pi, where\npi is a prime number; Ki an algebraic closure of Ki; Gi\ndef\n= Gal(Ki\/Ki); Xi a hy-\nperbolic curve over Ki whose corresponding stable log curve extends to a stable\nlog curve X logi over the spectrum of the ring of integers OKi of Ki [equipped with\nthe log structure determined by the closed point of Spec(OKi)]; Σ a set of prime\nnumbers such that pi ∈ Σ; Δi the maximal pro-Σ quotient of the étale funda-\nmental group of (Xi) ×Ki Ki [so Δi may be regarded as the profinite fundamental\n26 SHINICHI MOCHIZUKI\ngroup of a semi-graph of anabelioids Gi of pro-Σ PSC-type with precisely one vertex\nand no closed edges];\nαG : G1\n∼→ G2; αΔ : Δ1 ∼→ Δ2\na pair of isomorphisms of profinite groups that are compatible with the nat-\nural outer action of Gi on Δi. Then the following hold.\n(i) We have p1 = p2 [so we shall write p\ndef\n= p1 = p2]; for i = 1, 2, the action of\nGi on Gi is weakly p-graphically full; αΔ is group-theoretically cuspidal.\n(ii) Suppose that the cardinality of Σ is ≥ 2. Then αΔ induces a functorial\n[i.e., with respect to the pair (αG, αΔ)] isomorphism of the “dual semi-graphs\nwith compact structure” [cf. [Mzk4], Appendix] of the special fibers of the X logi .\n(iii) Suppose that the cardinality of Σ is ≥ 2. Write πtemp1 ((Xi) ×Ki Ki)\nfor the tempered fundamental group of [André], §4 [cf. also [Mzk7], Examples\n3.10, 5.6];\nΔ\ntemp\ni\ndef\n= lim←−\nN\nπtemp1 ((Xi) ×Ki Ki)\/N\nfor the “Σ-tempered fundamental group” — i.e., the inverse limit where N\nvaries over the open normal subgroups of πtemp1 ((Xi)×Ki Ki) such that the quotient\nπtemp1 ((Xi)×Ki Ki)\/N is an extension of a finite group whose order is a product\nof primes ∈ Σ by a discrete free group. [Here, we recall that such a discrete\nfree group corresponds to a “combinatorial covering” determined by the graph of the\nspecial fiber of some stable reduction of a covering of Xi — cf. [André], Proposition\n4.3.1; [André], the proof of Lemma 6.1.1.] Thus, we have a natural continuous outer\naction of Gi on Δ\ntemp\ni ; Δi is the pro-Σ completion of Δ\ntemp\ni . Then the operation\nof pro-Σ completion determines a surjection from the set of compatible pairs\nof isomorphisms of topological groups\nβG : G1\n∼→ G2; βΔtemp : Δtemp1 ∼→ Δtemp2\nconsidered up to inner automorphisms of the Δ\ntemp\ni to the set of compatible pairs\nof isomorphisms of topological groups\nγG : G1\n∼→ G2; γΔ : Δ1 ∼→ Δ2\nconsidered up to inner automorphisms of the Δi.\nProof. First, we consider assertion (i). Since [as is well-known] Gi is an extension\nof an abelian profinite group by a nonabelian pro-pi group, it follows that pi may\nbe characterized as the unique prime number p′ such that Gi contains a nonabelian\npro-p′ closed subgroup. Thus, the existence of αG implies that p1 = p2; write\np\ndef\n= p1 = p2. Now since the tensor product with Qp of the abelianization of\nany open subgroup of Δi admits a filtration [cf., e.g., [FC], Chapter III, Corollary\n7.3] each of whose subquotients is Hodge-Tate [relative to the action of some open\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 27\nsubgroup of Gi], with Hodge-Tate decomposition only involving “Tate twists” by\nthe zero-th or first power of the cyclotomic character [cf. [Tate], §4, Corollary 2],\nit follows that Gi is [relative to its outer action on Δi] weakly p-graphically full.\nThus, the remainder of assertion (i) follows from Corollary 2.7, (i).\nNext, we consider assertion (ii). Let l ∈ Σ be distinct from p def= p1 = p2. Write\nΔi \u0002 Δ(l)i for the maximal pro-l quotient of Δi. Since the subgroup of Out(Δ\n(l)\ni )\nthat induces the identity on the tensor product with Fl of the abelianization of Δ\n(l)\ni\nis [easily seen to be] a pro-l group, it follows that by replacing Gi by an open sub-\ngroup of Gi, we may assume that the natural map Gi → Out(Δ(l)i ) factors through\nthe maximal pro-l quotient Gi \u0002 G(l)i of Gi. Thus, the data given by the outer\naction of G\n(l)\ni on Δ\n(l)\ni is l-logarithmically full data of the type considered in Exam-\nple 2.5. In particular, in the noncuspidal case, assertion (ii) follows immediately\nfrom Corollary 2.7, (iii). On the other hand, even if we are not in the noncuspidal\ncase, by passing to compactifications of sturdy finite étale coverings of the Xi and\napplying the fact that a cuspidal edge-like subgroup belongs to a unique verticial\nsubgroup [cf. Proposition 1.5, (i)], we reduce immediately [via assertion (i)] to the\nnoncuspidal case. This completes the proof of assertion (ii).\nFinally, we observe that assertion (iii) follows formally from assertion (ii) via\nthe same argument applied [in the case where Σ is the set of all primes, and the\nbase fields are finite extensions of Qp] in the proof of [Mzk7], Theorem 6.6, to derive\nthe “surjectivity portion” of [Mzk7], Theorem 6.6, from [Mzk4], Lemma 2.3. \u0003\nRemark 2.10.1. Since free discrete groups inject into their pro-Σ completions\n[cf. [RZ], Proposition 3.3.15], the natural map Δ\ntemp\ni → Δi is injective [cf. the\nproof of [Mzk7], Corollary 3.11]. On the other hand, unlike the situation of [Mzk7],\nTheorem 6.6, we are unable to conclude that the surjection of Corollary 2.10, (iii),\nis a bijection since [unlike the profinite case — cf. [André], Corollary 6.2.2] it is not\nclear that the [image in Δi of] Δ\ntemp\ni is equal to its own normalizer in Δi.\n28 SHINICHI MOCHIZUKI\nBibliography\n[André] Y. André, On a geometric description of Gal(Qp\/Qp) and a p-adic avatar of\nĜT , Duke Math. J. 119 (2003), 1–39.\n[Asada] M. Asada, The compatibility of the filtration of mapping class groups of two\nsurfaces pasted along the boundaries, Topology and its Applications 108 (2000)\n217–232.\n[DM] P. Deligne and D. Mumford, The irreducibility of the moduli space of curves\nof given genus, IHES Publ. Math. 36 (1969), 75–109.\n[FC] G. Faltings and C.-L. Chai, Degenerations of abelian varieties, Springer-Verlag\n(1990).\n[FK] E. Freitag and R. Kiehl, Étale cohomology and the Weil conjecture, Springer\nVerlag (1988).\n[Hum] J. E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics 21,\nSpringer-Verlag (1975).\n[Kato] K. Kato, Logarithmic structures of Fontaine-Illusie, Proceedings of the First\nJAMI Conference, Johns Hopkins Univ. Press (1990), 191–224.\n[Knud] F. F. Knudsen, The projectivity of the moduli space of stable curves, II, Math.\nScand. 52 (1983), 161–199.\n[Mzk1] S. Mochizuki, The Geometry of the compactification of the Hurwitz scheme,\nPubl. Res. Inst. Math. Sci. 31 (1995), 355–441.\n[Mzk2] S. Mochizuki, The profinite Grothendieck conjecture for closed hyperbolic\ncurves over number fields, J. Math. Sci., Univ. Tokyo 3 (1996), 571–627.\n[Mzk3] S. Mochizuki, The local pro-p anabelian geometry of curves, Invent. Math. 138\n(1999), 319–423.\n[Mzk4] S. Mochizuki, The absolute anabelian geometry of hyperbolic curves, Galois\ntheory and modular forms, Kluwer Academic Publishers (2003), 77–122.\n[Mzk5] S. Mochizuki, The geometry of anabelioids, Publ. Res. Inst. Math. Sci. 40\n(2004), 819–881.\n[Mzk6] S. Mochizuki, Galois sections in absolute anabelian geometry, Nagoya Math.\nJ. 179 (2005), 17–45.\n[Mzk7] S. Mochizuki, Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006),\n221–322.\n[Mumf] D. Mumford, Abelian varieties, Oxford Univ. Press (1974).\n[Naka1] H. Nakamura, Galois rigidity of the étale fundamental groups of punctured\nprojective lines, J. reine angew. Math. 411 (1990), 205–216.\n[Naka2] H. Nakamura, Galois rigidity of pure sphere braid groups and profinite calculus,\nJ. Math. Sci. Univ. Tokyo 1 (1994), 71–136.\nA COMBINATORIAL VERSION OF THE GROTHENDIECK CONJECTURE 29\n[RZ] Ribes and Zaleskii, Profinite groups, Ergeb. Math. Grenzgeb. 3, Springer-\nVerlag (2000).\n[Tama1] A. Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math.\n109 (1997), 135–194.\n[Tama2] A. Tamagawa, On the fundamental groups of curves over algebraically closed\nFields of characteristic > 0, Internat. Math. Res. Notices 16 (1999), 853–873.\n[Tate] J. Tate, p-divisible groups, Proceedings of a Conference on Local Fields, Drieber-\ngen, Springer-Verlag (1967), 158–183.\nResearch Institute for Mathematical Sciences\nKyoto University\nKyoto 606-8502\nJapan\nmotizuki@kurims.kyoto-u.ac.jp\n"}
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{"id":"gZKz5116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Categorical%20Representation%20of%20Locally%20Noetherian%20Log%20Schemes%20(comments).pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":132,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Categorical%20Representation%20of%20Locally%20Noetherian%20Log%20Schemes%20(comments)","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["logsch-comments-2014-10.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Tue Oct 28 12:15:53 UTC 2014","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-2935620786255893086,"exact_signature_unique_b":true,"fuzzy_signature_l":-9069399207919165257,"fuzzy_signature_unique_b":true,"text_t":"COMMENTS ON “CATEGORICAL REPRESENTATION\nOF LOCALLY NOETHERIAN LOG SCHEMES”\nShinichi Mochizuki\nOctober 2014\n(1.) The theory exposed in §2 contains errors, as follows:\n· the necessity asserted in Proposition 2.3, is false;\n· the isomorphism asserted [unfortunately, without an explicit proof!] in\nLemma 2.6, (ii), is false.\nAt the level of main results of the present paper, these errors in the theory of §2,\ndo not affect the proof of Theorem A given in §1, but they do affect the proof — al-\nthough not the validity! — of Theorem B. This result Theorem B is given a correct\nproof in [MnLgSch], §3 [cf. [MnLgSch], Theorem 3.8, (iii)]. Also, the errors referred\nto above are discussed in more detail in [MnLgSch], Example 0.3; [MnLgSch], Re-\nmark 1.4.1. At the level of main results of papers of the author subsequent to the\npresent paper, the only place where the errors in the theory of §2 have an affect is\nin the portion of the proof of the main result of [ArLgSch] [i.e., [ArLgSch], Theorem\n5.1] that involves the theory of [ArLgSch], §4. The affected portions of [ArLgSch],\n§4, are discussed in more detail in the introduction to [MnLgSch], §4. The main\nresult [ArLgSch], Theorem 5.1, of [ArLgSch] is given a correct proof in [MnLgSch],\n§4 [cf. [MnLgSch], Theorem 4.8, (iv)]. More details on the affected portions of the\npresent paper, as well as of [ArLgSch], may be found in the Appendix to [MnLgSch].\n(2.) In the first paragraph of the proof of Lemma 2.5, the notation “Q ∈ VZ” should\nread “Q ⊆ VZ”.\n(3.) In the second sentence of the statement of Proposition 2.7, the phrase “(nec-\nessarily uniquely!) though” should read “(necessarily uniquely!) through”. In the\nsecond paragraph of the proof of Proposition 2.7, the phrase “(uniquely) though”\nshould read “(uniquely) through”.\n(4.) In the discussion immediately following the display in the statement of Corol-\nlary 2.12, (ii), the notation “Sch(X log)” should read “Schlog(X log)”.\n(5.) In Lemma 2.16, (iii), the phrase “a monomorphism” should read “a scheme-\nlike monomorphism”.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\nBibliography\n[ArLgSch] S. Mochizuki, Categories of log schemes with archimedean structures, J. Math.\nKyoto Univ. 44 (2004), pp. 891-909.\n[MnLgSch] S. Mochizuki, Monomorphisms in Categories of Log Schemes, preprint.\n","wordcount_i":341,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:55 UTC 2022","fresh_date_dt":"Sun Mar 29 19:06:56 UTC 2026","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":42039,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086042659160064}
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{"id":"A2QPd116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/The%20Geometry%20of%20Anabelioids.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":80,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"The%20Geometry%20of%20Anabelioids","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["The Geometry of Anabelioids.pdf"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 12:19:20 UTC 2011","dates_in_content_count_i":3,"dates_in_content_dts":["Fri Jan 01 00:00:00 UTC 2010","Sat Jan 01 00:00:00 UTC 2011","Sun Jan 01 00:00:00 UTC 2012"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":8155987609594160176,"exact_signature_unique_b":true,"fuzzy_signature_l":-3052188145203575590,"fuzzy_signature_unique_b":true,"wordcount_i":18898,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:58 UTC 2022","fresh_date_dt":"Thu Sep 23 19:05:17 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":375972,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086045913939968,"text_t":"The Geometry of Anabelioids\nShinichi MOCHIZUKI*\nContents:\n§0. Notations and Conventions\n§1. Anabelioids\n§1.1. The Notion of an Anabelioid\n§1.2. Finite Étale Morphisms\n§2. Cores and Quasi-Cores\n§2.1. Localizations and Cores\n§2.2. Holomorphic Structures and Commensurable Terminality\n§2.3. Quasi-Cores and Intrinsic Exhaustivity\n§2.4. Canonical Construction of the Fundamental Group\n§3. Anabelioids Arising from Hyperbolic Curves\n§3.1. Anabelioid-Theoretic Interpretation of Scheme-Theoretic Cores\n§3.2. The Logarithmic Special Fiber via Quasi-Cores\nCommunicated by T. Kawai. Received October 20, 2003. Revised January 15, 2004.\n2000 Mathematics Subject Classification: 14H30\n*Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI*\nIntroduction\nThe goal of the present manuscript is to consider the following question:\nTo what extent can the fundamental group of a Galois category be con-\nstructed in a canonical fashion which is independent of a choice of\nbasepoint?\nPut another way, we would like to consider the extent to which the elements (consid-\nered, say, up to conjugation) of the fundamental group may be assigned canonical\nnames, or labels.\nIn §1, §2, we consider this issue from a very general point of view. That is to say,\nwe develop the general theory of “anabelioids” — i.e., “multi-Galois categories”\nin the terminology of [SGA1]— with an eye to giving an answer to this question.\nWe use the new terminology “anabelioid”, partly because it is shorter than “(multi-\n)Galois category”, and partly because we wish to emphasize that we would like to\ntreat such objects from a fundamentally different point of view — a point of view\npartially motivated by Grothendieck’s anabelian philosophy (cf. [Groth]; Remark\n1.1.4.1 below) — from the point of view taken in [SGA1]: Namely, we would like to\nregard anabelioids as the primary geometric objects of interest, which themselves\nform a category [i.e., not as a category containing as objects the primary geometric\nobjects of interest].\nOur main result in §1, §2, is Theorem 2.4.3, which states that:\nWhen an anabelioid possesses a “faithful quasi-core” (cf. Definition 2.3.1),\nthen its fundamental group may be constructed in a canonical fashion as\na profinite group.\nThe notion of a “quasi-core” is motivated partly by the notion of a “hyperbolic core”\n(cf. [Mzk3]) and partly by the “Motivating Example” given below. The condition\nfor a quasi-core states, roughly speaking, that a certain “forgetful functor” from a\ncategory of geometric objects equipped with some special auxiliary structure to the\ncategory of the same geometric objects not equipped with this auxiliary structure\nis, in fact, an equivalence. Indeed, this general pattern of considering such forgetful\nfunctors which are, in fact, equivalences is an important theme in the present man-\nuscript (cf. Definition 2.3.1, as well as Theorem 2.4.2). One elementary example\nof this sort of phenomenon — which was, in fact, one of the main motivations for\nthe introduction of the notion of a “quasi-core” — is the following example from\nelementary complex analysis:\nMotivating Example: Metrics on Hyperbolic Riemann Surfaces. A\nconnected Riemann surface is called hyperbolic if its universal covering is biholomor-\nphic to the upper half plane. An arbitrary Riemann surface will be called hyperbolic\nif every connected component of this Riemann surface is hyperbolic. Let us write\nLochyp\nTHE GEOMETRY OF ANABELIOIDS 3\nfor the category whose objects are hyperbolic Riemann surfaces and whose mor-\nphisms are étale morphisms [i.e., holomorphic maps with everywhere nonvanishing\nderivative]. If X ∈ Ob(Lochyp) is a object of Lochyp, then we shall refer to the\nmetric on its tangent bundle determined by the standard Poincaré metric on the\nupper half plane [which is biholomorphic to the universal covering of every con-\nnected component of X] as the canonical metric on X. If f : X → Y is a morphism\nin Lochyp, then we shall say that this morphism f is integral if the norm of its\nderivative [when measured with respect to the canonical metrics on the tangent\nbundles of X, Y ] is ≤ 1. Let us write\nLoc\nhyp\nint ⊆ Lochyp\nfor the subcategory whose objects are the objects of Lochyp and whose morphisms are\nthe integral morphisms of Lochyp. Then it follows from the “theory of the Kobayashi\nhyperbolic metric” that the natural inclusion functor\nLoc\nhyp\nint ↪→ Lochyp\nis, in fact, an equivalence. At a more concrete level, one verifies easily that the\nessential substantive fact that one needs to show this equivalence is the well-known\nSchwarz lemma of elementary complex analysis [to the effect that any holomorphic\nfunction φ : D → C on the open unit disc D in the complex plane satisfying\nφ(0) = 0, |φ(z)| ≤ 1 (for all z ∈ D), necessarily satisfies |φ′(0)| ≤ 1]. This lemma\nof Schwarz in turn may be regarded as a formal consequence of the well-known\n“maximum modulus principle” of elementary complex analysis.\nFrom a category-theoretic point of view, the point of this example is that,\nalthough the object of Lochyp determined by the upper half plane is by no means\na terminal object in Lochyp — i.e., a “core” (cf. the theory of [Mzk3]) — the fact\nthat the inclusion functor Loc\nhyp\nint ↪→ Lochyp is an equivalence serves to express,\nin a rigorous mathematical fashion, the sentiment that, in some sort of “metric”\nsense, the upper half plane “almost serves as a core” for Lochyp. It is this set of\ncircumstances that led the author to the introduction of the notion of a “quasi-\ncore”.\nThis example also suggests an interesting relationship between the notions\nof uniformization and of canonical labels for elements of the fundamental group:\nNamely, the Koebe uniformization theorem for hyperbolic Riemann surfaces gives\nrise to “canonical labels” (up to an ambiguity arising from some sort of conjuga-\ntion action) via 2 by 2 matrices since it induces an embedding of the topological\nfundamental group of the Riemann surface into PSL2(R).\nThis leads us to the content of §3: In §3, we discuss the theory of §1, §2, in\nthe case of hyperbolic curves over p-adic and number fields. In this case, our main\nresult — Theorem 3.1.6 — states that:\nIf a non-proper hyperbolic curve over such a field is a “geometric core”\n(i.e., a core as in [Mzk3]), then its associated anabelioid admits a faithful\nquasi-core.\n4 SHINICHI MOCHIZUKI*\nThis allows us to assign “canonical names” to the elements of its arithmetic\nfundamental group in a fashion reminiscent of the way in which the Koebe uni-\nformization theorem allows one to assign “canonical names” to the elements of the\ntopological fundamental group of a hyperbolic Riemann surface. This main result\nis, in essence, a formal consequence of Theorem A of [Mzk6], and may be regarded\nas an interpretation of the main result of [Mzk9], §2, via the geometry of anabelioids.\nAcknowledgements: I would like to thank A. Tamagawa for countless stimulating\ndiscussions concerning the theory presented in this paper and especially for his com-\nments concerning the theory of quasi-cores and intrinsic exhaustivity (cf. Remarks\n2.3.6.2, 2.3.7.1 in the text). Also, I would like to thank T. Tsuji for pointing out\nto me (cf. [Mzk2], §1) an error in [Mzk1], Chapter IX — an event which opened\nmy eyes to the importance (in the context of the arithmetic Kodaira-Spencer mor-\nphism) of keeping track of basepoints. Finally, I would like to thank H. Nakamura for\norganizing a conference on arithmetic fundamental groups at RIMS, Kyoto Univer-\nsity, in October 2001, at which the author was afforded the opportunity to present\na “preliminary version” (cf. [Mzk7]) of the ideas exposed in the present paper.\nTHE GEOMETRY OF ANABELIOIDS 5\nSection 0: Notations and Conventions\nNumbers:\nWe will denote by N the set of natural numbers, by which we mean the set of\nintegers n ≥ 0. A number field is defined to be a finite extension of the field of\nrational numbers Q.\nTopological Groups:\nLet G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us\nwrite\nZG(H)\ndef\n= {g ∈ G | g · h = h · g, ∀ h ∈ H}\nfor the centralizer of H in G;\nNG(H)\ndef\n= {g ∈ G | g · H · g−1 = H}\nfor the normalizer of H in G; and\nCG(H)\ndef\n= {g ∈ G | (g · H · g−1)\n⋂\nH has finite index in H, g · H · g−1}\nfor the commensurator of H in G. Note that: (i) ZG(H), NG(H) and CG(H) are\nsubgroups of G; (ii) we have inclusions\nH, ZG(H) ⊆ NG(H) ⊆ CG(H)\nand (iii) H is normal in NG(H).\nNote that ZG(H), NG(H) are always closed in G, while CG(H) is not neces-\nsarily closed in G.\nIndeed, one may construct such an example as follows: Let\nM\ndef\n=\n∏\nN\nZp\nendowed with the product topology (of the various copies of Zp equipped with their\nusual topology). Thus, M is a Hausdorff topological group. For n ∈ N, write\nFn(M) ⊆ M for the sub-topological group given by the product of the copies\nof Zp indexed by m ≥ n. Write AutF (M) for the set of automorphisms of the\ntopological group M that preserve the filtration F ∗(M) on M . If α ∈ AutF (M),\nthen for every n ∈ N, α induces a continuous homomorphism αn : M\/Fn(M) →\nM\/Fn(M) which is clearly surjective, hence an isomorphism (since M\/Fn(M) is\nprofinite and topologically finitely generated — cf. [FJ], Proposition 15.3). It\nthus follows that α induces an isomorphism Fn(M)\n∼→ Fn(M), hence that the\ninverse of α also lies in AutF (M). In particular, we conclude that AutF (M) is\n6 SHINICHI MOCHIZUKI*\na group. Equip AutF (M) with the coarsest topology for which all of the homo-\nmorphisms AutF (M) → Aut(M\/Fn(M)) (where Aut(M\/Fn(M)) ∼= GLn(Zp) is\nequipped with its usual topology) are continuous. Note that relative to this topol-\nogy, AutF (M) forms a Hausdorff topological group. Now define G to be the semi-\ndirect product of M with AutF (M) (so G is a Hausdorff topological group), and H\nto be ∏\nn∈N\npn · Zp ⊆\n∏\nN\nZp = M\n(so H ⊆ G is a closed subgroup). Then CG(H) is not closed in G. For instance,\nif one denotes by en ∈\n∏\nN Zp the vector with a 1 in the n-th place and zeroes\nelsewhere, then the limit A∞ (where\nA∞(en)\ndef\n= en + en+1\nfor all n ∈ N) of the automorphisms Am ∈ CG(H) (where Am(en) def= en + en+1 if\nn ≤ m, Am(en) def= en if n > m) is not contained in CG(H).\nCurves:\nSuppose that g ≥ 0 is an integer. Then a family of curves of genus g\nX → S\nis defined to be a smooth, proper, geometrically connected morphism X → S whose\ngeometric fibers are curves of genus g.\nSuppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denote\nthe moduli stack of r-pointed stable curves of genus g (where we assume the points\nto be unordered) by Mg,r (cf. [DM], [Knud] for an exposition of the theory of such\ncurves; strictly speaking, [Knud] treats the finite étale covering of Mg,r determined\nby ordering the marked points). The open substack Mg,r ⊆ Mg,r of smooth curves\nwill be referred to as the moduli stack of smooth r-pointed stable curves of genus g\nor, alternatively, as the moduli stack of hyperbolic curves of type (g, r).\nA family of hyperbolic curves of type (g, r)\nX → S\nis defined to be a morphism which factors X ↪→ Y → S as the composite of an\nopen immersion X ↪→ Y onto the complement Y \\D of a relative divisor D ⊆ Y\nwhich is finite étale over S of relative degree r, and a family Y → S of curves of\ngenus g. One checks easily that, if S is normal, then the pair (Y,D) is unique up\nto canonical isomorphism. (Indeed, when S is the spectrum of a field, this fact\nis well-known from the elementary theory of algebraic curves. Thus, the asserted\nuniqueness follows formally from the normality of S and the fact that Mg,r is a\nseparated algebraic stack (cf. [DM], [Knud]).) We shall refer to Y (respectively,\nD; D; D) as the compactification (respectively, divisor at infinity; divisor of cusps;\nTHE GEOMETRY OF ANABELIOIDS 7\ndivisor of marked points) of X. A family of hyperbolic curves X → S is defined to\nbe a morphism X → S such that the restriction of this morphism to each connected\ncomponent of S is a family of hyperbolic curves of type (g, r) for some integers (g, r)\nas above.\nNext, we would like to consider “orbicurves”. We shall say that an algebraic\nstack is generically scheme-like if it admits an open dense algebraic substack which\nis isomorphic to a scheme. Let X be a smooth, geometrically connected, generically\nscheme-like algebraic stack of finite type over a field k of characteristic zero. Then\nwe shall say that X is an orbicurve if it is of dimension 1. We shall say that X is a\nhyperbolic orbicurve if it is an orbicurve which admits a compactification X ↪→ X\n(necessarily unique!) by a proper orbicurve X over k such that if we denote the\nreduced divisor X\\X by D ⊆ X, then X is scheme-like near D, and, moreover,\nthe line bundle ωX\/k(D) on X has positive degree. An example of a hyperbolic\norbicurve over k is given by the quotient — in the sense of stacks — of a hyperbolic\ncurve over k by the action of a finite group which acts freely on all but a finite\nnumber of points of the curve.\nNow suppose that\nX\nis a hyperbolic orbicurve over a field k (of characteristic zero), with compactification\nX ↪→ X. Let k be an algebraic closure of k. Write\nX → X ′\nfor the “coarse moduli space” (cf. [FC], Chapter I, Theorem 4.10) associated to X.\nThus, X\n′\nis a smooth, proper, geometrically connected curve over k. Denote the\nopen subscheme of X\n′\nwhich is the image of X by X ′. Write:\nN∞\ndef\n= (N\\{0, 1})\n⋃\n{∞}\nThen we shall say that the hyperbolic curve X is of type\n(g, \u0006r)\nif X\n′\nis of genus g and \u0006r : N∞ → N is the function with finite support [i.e., which is\n0 away from some finite subset of N∞] defined as follows: \u0006r(∞) is the cardinality\nof (X\n′\\X ′)(k). For every positive integer e ∈ N∞, \u0006r(e) is the cardinality of the\nset of k-valued points of X ′ over which X is (necessarily tamely) ramified with\nramification index e.\nWhen k = k, it is well-known (and easily verified) that the isomorphism class of\nthe algebraic fundamental group π1(X) is completely determined by the type (g, \u0006r).\n8 SHINICHI MOCHIZUKI*\nCategories:\nWe shall say that two arrows fi : Ai → Bi (where i = 1, 2) in a category C\nare abstractly equivalent — and write f1\nabs≈ f2 — if there exists a commutative\ndiagram:\nA1\n∼→ A2⏐⏐\u0007f1 ⏐⏐\u0007f2\nB1\n∼→ B2\n(where the horizontal arrows are isomorphisms in C).\nWe shall refer to a natural transformation between functors all of whose com-\nponent morphisms are isomorphisms as an isomorphism between the functors in\nquestion. A functor φ : C1 → C2 between categories C1, C2 will be called rigid if φ\nhas no nontrivial automorphisms.\nA diagram of functors between categories will be called 1-commutative if the\nvarious composite functors in question are rigid and isomorphic. When such a\ndiagram “commutes in the literal sense” we shall say that it 0-commutes. Note\nthat when a diagram “1-commutes”, it follows from the rigidity hypothesis that\nany isomorphism between the composite functors in question is necessarily unique.\nThus, to state that the diagram 1-commutes does not result in any “loss of infor-\nmation” by comparison to the datum of a specific isomorphism between the various\ncomposites in question.\nWe shall say that two rigid functors φi : Ci → C′i (where i = 1, 2; the Ci, C′i\nare categories) are abstractly equivalent — and write φ1\nabs≈ φ2 — if there exists a\n1-commutative diagram\nC1 ∼→ C2⏐⏐\u0007φ1 ⏐⏐\u0007φ2\nC′1 ∼→ C′2\n(in which the horizontal arrows are equivalences of categories).\nTHE GEOMETRY OF ANABELIOIDS 9\nSection 1: Anabelioids\n§1.1. The Notion of an Anabelioid\nWe begin by fixing a (Grothendieck) universe V , in the sense of set-theory (cf.,\ne.g., [McLn1]; [McLr], §12.1), in which we shall work. Also, let us assume that we\nare given a V -small category Ensf of finite sets.\nLet G be a (V -small) profinite group — that is to say, the underlying profinite\nset of G is an inverse limit of V -sets indexed by a V -set. Then to G, we may\nassociate the (V -small) category\nB(G)\nof (V -small) finite sets ∈ Ob(Ensf) with continuous G-action. This category is a(n)\n(elementary) topos (in the sense of topos theory). In fact, it forms a rather special\nkind of topos called a Galois category (cf. [John1] for an exposition of the general\ntheory of topoi and, in particular, of Galois categories; cf. also [SGA1], Exposé V).\nDefinition 1.1.1. We shall refer to as a connected anabelioid any category X\nwhich is equivalent to a category of the form B(G) for some profinite group G.\nRemark 1.1.1.1. Thus, a “connected anabelioid” is the same as a Galois category\n(as defined, for instance, in [John1], p. 285) — i.e., a “Boolean topos” that admits\nan “exact, isomorphism reflecting functor” to the category of finite sets.\nLet X be a connected anabelioid. Then recall (cf. [SGA1], Exposé V, §5) the\nnotion of a fundamental functor\nβ∗ : X → Ensf\n— i.e., an exact functor. Here, we recall that an exact functor is a functor that\npreserves finite limits and finite colimits. Note that (since X is assumed to be a\nconnected anabelioid) an exact functor β∗ : X → Ensf is necessarily isomorphism\nreflecting (i.e., a morphism α of X is an isomorphism if and only if β∗(α) is).\nRecall, moreover, that if X def= B(G), and β∗ : B(G) → Ensf is the functor defined\nby forgetting the G-action, then G may be recovered, up to inner automorphism,\nfrom X , β as the group:\nAut(β∗)\nAlso, let us recall that any two fundamental functors are isomorphic. Note that\nEnsf itself is a connected anabelioid (i.e., the result of applying B(−) to the trivial\ngroup), so we may think of fundamental functors as “basepoints” in the following\nway:\n10 SHINICHI MOCHIZUKI*\nDefinition 1.1.2.\n(i) If X and Y are connected anabelioids, then we define a morphism φ : X → Y\nto be an exact functor φ∗ : Y → X (cf. [SGA1], Exposé V, Proposition 6.1). An\nisomorphism between connected anabelioids is a morphism whose corresponding\nfunctor in the opposite direction is an equivalence of categories.\n(ii) We define a basepoint of a connected anabelioid X to be a morphism β :\nEnsf → X . If β is a basepoint of X , then we refer to the group Aut(β) as the\nfundamental group π1(X , β) of (X , β).\nRemark 1.1.2.1. Thus, the “category of (V -small) connected anabelioids” is a 2-\ncategory (cf., e.g., [John1], §0.1; [McLr], Chapter 12; [McLn2], XII), hence requires\nspecial care, for instance, when considering composites, etc. Also, we remark,\nrelative to the standard terminology of category theory, that if φ : X → Y is an\nisomorphism (of connected anabelioids), it will not, in general, be the case that φ∗\nis an isomorphism of categories (i.e., an equivalence for which the correspondence\nbetween classes of objects in the two categories is a bijection — cf. [McLn2], IV,\n§4).\nRemark 1.1.2.2. Since the isomorphism class of the fundamental group π1(X , β)\nis independent of the choice of basepoint β, we will also speak of the “fundamental\ngroup π1(X ) of X” when the choice of basepoint is irrelevant to the issue under\ndiscussion.\nRemark 1.1.2.3. Note that a functor φ∗ : Y → X which is an equivalence is\nalways necessarily exact. Thus, an isomorphism of anabelioids φ : X → Y is simply\nan equivalence φ∗ : Y → X in the opposite direction.\nExample 1.1.3. Anabelioids Associated to Schemes. Let X be a (V -small)\nconnected locally noetherian scheme. Then we shall denote by\nÉt(X)\nthe category whose objects are (V -small) finite étale coverings of X and whose\nmorphisms are morphisms of schemes over X. Then it is well-known (cf. [SGA1],\nExposé V, §7) that Ét(X) is a connected anabelioid.\nIf G is a profinite group, then we shall use the notation\nAut(G); Inn(G); Out(G)\ndef\n= Aut(G)\/Inn(G)\nto denote the group of (continuous) automorphisms (respectively, inner automor-\nphisms; (continuous) outer automorphisms) of G. If H is another profinite group,\nTHE GEOMETRY OF ANABELIOIDS 11\nthen we shall write Hom(G,H) for the set of continuous homomorphisms G → H,\nand\nHomOut(G,H)\nfor the set of continuous outer homomorphisms G → H, i.e., the quotient of\nHom(G,H) by the natural action of H from the right. Also, we shall write\nHomOut(G,H)\nfor the (V -small) category whose objects are the elements of the set Hom(G,H)\nand for which the morphisms\nMorHomOut(ψ1, ψ2)\nfrom an object ψ1 : G → H to an object ψ2 : G → H are the elements h ∈ H such\nthat ψ2(g) = h · ψ1(g) · h−1, ∀g ∈ G. Thus, HomOut(G,H) may be thought of as\nthe set of isomorphism classes of the category HomOut(G,H).\nProposition 1.1.4. (The “Grothendieck Conjecture” for Connected\nAnabelioids) Let X def= B(G), Y def= B(H) (where G, H are profinite groups), and\nβ : Ensf → X , γ : Ensf → Y be the tautological basepoints of X , Y, respectively,\ndetermined by the definition of the notation “B(−)”. Then:\n(i) There is a natural equivalence of categories:\nHomOut(G,H)\n∼→ Mor(X ,Y)\nwhich induces a natural bijection:\nHomOut(G,H)\n∼→ Mor(X ,Y)\nHere, Mor(X ,Y) (respectively, Mor(X ,Y)) denotes the category (respectively, set\nof isomorphism classes) of morphisms X → Y.\n(ii) There is a natural bijection:\nHom(G,H)\n∼→ Mor{(X , β); (Y, γ)}\nHere, Mor{(X , β); (Y, γ)} denotes the set of (isomorphism classes of) morphisms\nφ : X → Y such that φ ◦ β = γ.\nProof. Let us first consider the situation of (2). Given a homomorphism ψ : G →\nH, composition with ψ induces a continuous action of G on any finite set with\ncontinuous H-action. Moreover, this operation does not affect the underlying finite\nset, so we get an element ψMor ∈ Mor{(X , β); (Y, γ)}. This defines the morphism of\n(2). On the other hand, given an element φ ∈ Mor{(X , β); (Y, γ)}, it follows from\nthe definitions that φ induces a morphism Aut(β) → Aut(γ). One checks easily\n12 SHINICHI MOCHIZUKI*\nthat this correspondence defines a two-sided inverse (well-defined up to composition\nwith an inner automorphism of Aut(γ) ∼= H) to the correspondence ψ \f→ ψMor.\nNext, we consider the situation of (1). By the above paragraph, we get a\nmorphism\nHom(G,H) → Mor(X ,Y)\nLet us first verify that this morphism is a surjection. Denote by S the pro-object\nof Y whose underlying profinite set Sset = H and whose H-action is given by the\nusual action on Sset = H from the left. Note that the group of automorphisms\nAutY(S) of S (as a pro-object of Y) may be identified with H via the action of H\non Sset = H from the right. In fact, this action of H on S (from the right) endows\nS with a structure of “H-torsor object” of Y. Thus, if φ : X → Y is a morphism,\nthen T\ndef\n= φ∗(S) is an H-torsor object of X . If we think of T as a profinite set\nTset equipped with a G-action from the left and an H-action from the right, then\nlet us observe that, by fixing some element t ∈ Tset, we may identify the group of\nautomorphisms AutH(Tset) of the profinite set Tset that commute with the H-action\nfrom the right with H via its action from the left. Here, we observe that such an\nidentification\nAutH(Tset) ∼= H\nis determined by the choice of a “basepoint” t ∈ Tset, hence is well-defined, up\nto composition with an inner automorphism of H. It thus follows that the action\nof G on Tset from the left determines a continuous outer homomorphism G →\nAutH(Tset) = H which (cf. the preceding paragraph) gives rise to a morphism\nX → Y isomorphic to φ. This completes our verification of surjectivity.\nThus, to complete the proof of (1), it suffices to verify that there is a natural\nbijection between the set of isomorphisms between the morphisms φ1, φ2 : X → Y\narising from two continuous homomorphisms\nψ1, ψ2 : G → H\nand the subset MorHomOut(ψ1, ψ2) ⊆ H. To verify this, let us observe that if we\npull-back the H-torsor object S of Y (cf. the preceding paragraph) via φ1, φ2 to\nobtain H-torsor objects T1\ndef\n= φ∗1(S), T2\ndef\n= φ∗2(S) of X , then it is a tautology that\nisomorphisms φ1\n∼→ φ2 are in natural bijective correspondence with isomorphisms\nT1\n∼→ T2 of H-torsor objects of X . Thus, the desired bijection is a consequence of\nLemma 1.1.5 below. ©\nLemma 1.1.5. (Two-Sided Group Actions) Let\nψ1, ψ2 : G → H\nbe continuous homomorphisms. For i = 1, 2, denote by Yi a copy of H equipped with\nthe usual action of H from the right and the action of G determined by composing\nthe usual action of H from the left with ψi; write ti for the copy of “1” in Yi. Then\nξ \f→ h ∈ H\nTHE GEOMETRY OF ANABELIOIDS 13\n— where h satisfies ξ(t1) = t2 ·h — determines a bijection from the set of (G,H)-\nequivariant bijections ξ : Y1\n∼→ Y2 to the subset MorHomOut(ψ1, ψ2) ⊆ H.\nProof. Indeed,\nt2 · h · ψ1(g) = ξ(t1) · ψ1(g) = ξ(t1 · ψ1(g)) = ξ(g · t1) = g · ξ(t1) = t2 · ψ2(g) · h\ni.e., h ·ψ1(g) ·h−1 = ψ2(g), ∀ g ∈ G. Thus, h ∈ MorHomOut(ψ1, ψ2) ⊆ H. Similarly,\nif ψ1 and ψ2 differ by composition with an inner automorphism of H defined by\nan element h ∈ MorHomOut(ψ1 , ψ2), then t1 \f→ t2 · h defines a (G,H)-equivariant\nbijection ξ, as desired. ©\nRemark 1.1.4.1. Many readers may feel that Proposition 1.1.4 is “trivial” and\n“well-known”. The reason that we nevertheless chose to give a detailed exposition\nof this fact here is that it represents the essential spirit that we wish to convey\nin the term “anabelioid”. That is to say, we wish to think of anabelioids X as\ngeneralized spaces (which is natural since they are, after all, topoi — cf. [John2])\nwhose geometry just happens to be “completely determined by their fundamental\ngroups” (albeit somewhat tautologically!). This is meant to recall the notion of an\nanabelian variety (cf. [Groth]), i.e., a variety whose geometry is determined by its\nfundamental group. The point here (which will become clear as the manuscript\nprogresses) is that:\nThe introduction of anabelioids allows us to work with both “algebro-\ngeometric anabelioids” (i.e., anabelioids arising from (anabelian) varieties\n— cf. Example 1.1.3) and “abstract anabelioids” (i.e., those which do not\nnecessarily arise from an (anabelian) variety) as geometric objects on\nan equal footing.\nThe reason that it is important to deal with “geometric objects” as opposed to\ngroups, is that:\nWe wish to study what happens as one varies the basepoint of one of\nthese geometric objects.\nThat is to say, groups are determined only once one fixes a basepoint. Thus, it\nis difficult to describe what happens when one varies the basepoint solely in the\nlanguage of groups.\nNext, let\nφ : X → Y\nbe a morphism between connected anabelioids. Write\nIφ ⊆ X\n14 SHINICHI MOCHIZUKI*\nfor the smallest subcategory of X that contains all subquotients of objects in the\nessential image of the pull-back functor φ∗. One verifies immediately that Iφ is a\nconnected anabelioid. Note that the morphism φ : X → Y factors naturally as a\ncomposite\nX → Iφ → Y\nwith the property that if we choose a basepoint βX of X and denote the result-\ning basepoints of Iφ, Y, by βIφ , βY , respectively, then the induced morphisms of\nfundamental groups\nπ1(X , βX ) \u0002 π1(Iφ, βIφ) ↪→ π1(Y, βY )\nare a surjection followed by an injection. Moreover, we note the following conse-\nquence of Proposition 1.1.4, (i):\nCorollary 1.1.6. (Automorphism of an Arrow Between Connected\nAnabelioids) The set of automorphisms Aut(X → Y) of a 1-arrow X → Y between\nconnected anabelioids is in natural bijective correspondence with the centralizer in\nthe fundamental group of Y of the image of the fundamental group of X .\nDefinition 1.1.7.\n(i) We shall refer to Iφ as the image of X in Y.\n(ii) We shall refer to a morphism φ : X → Y between connected anabelioids\nas a π1-epimorphism (respectively, π1-monomorphism) if the morphism Iφ → Y\n(respectively, X → Iφ) is an equivalence.\nNow let I be a finite set. Assume that for each i ∈ I, we are given a connected\nanabelioid Xi. Write\nXI def=\n∏\ni∈I\nXi\nfor the product of the categories Xi. In the terminology of [SGA1], Exposé V, §9,\nthis XI is a “multi-Galois category”. In particular, XI is a topos.\nDefinition 1.1.8. Let X be a topos, and S ∈ Ob(X ) an object of X . Write\n0X (respectively, 1X ) for the initial (respectively, terminal) object of X . Then any\ncollection of data\nS ∼=\n∐\na∈A\nSa\n(where 0X \u000e∼= Sa ∈ Ob(X ); the index set A is finite) will be called a decomposition\nof S. The object S will be called connected if the index set of any decomposition\nof S has cardinality one. The topos X will be called connected if 1X is connected.\nNext, let us observe that:\nTHE GEOMETRY OF ANABELIOIDS 15\nThe set I and the categories Xi (for i ∈ I), as well as the equivalence of\ncategories between XI with the product of the Xi may be recovered entirely\nfrom the abstract category XI.\nIndeed, let us denote (for i ∈ I) the object of XI obtained by taking the product\nof 1Xi with the 0Xj (for j \u000e= i) by \fi. Thus, we obtain a decomposition\n1XI =\n∐\ni∈I\n\fi\nof the object 1XI . Moreover, this decomposition is clearly maximal with respect\nto the partial ordering on decompositions of 1XI determined by the (obviously\ndefined) notion of refinements of decompositions of 1XI . Thus, we see that this\ndecomposition may be recovered solely from internal structure of the category XI .\nIn particular, the finite set I may be recovered category-theoretically from the\ncategory XI . Moreover, the category Xi may be recovered category-theoretically\nfrom the category XI as the subcategory of objects over \fi. Finally, it is clear that\nthese subcategories determine the equivalence of categories between XI with the\nproduct of the Xi.\nDefinition 1.1.9. We shall refer to the Xi as the connected components of XI\nand to π0(XI) def= I as the (finite) index set of connected components.\nDefinition 1.1.10.\n(i) We shall refer to a category equivalent to a category of the form XI as an\nanabelioid. We shall denote the 2-category of V -small anabelioids by\nAnabV\n(or simply Anab, when there is no danger of confusion).\n(ii) A morphism between anabelioids is defined to be an exact functor in the\nopposite direction. An isomorphism between anabelioids is a morphism whose\ncorresponding functor in the opposite direction is an equivalence of categories.\nNext, let us observe that if we are given a finite set J , together with connected\nanabelioids Yj for each j ∈ J , and morphisms\nζ : I → J ; φi : Xi → Yζ(j)\nwe get an exact functor φ∗I : YJ → XI (by forming the product of the φi), which\nwe would like to regard as a morphism φI : XI → YJ .\nProposition 1.1.11. (Morphisms of Not Necessarily Connected Anabe-\nlioids) The association\n{ζ, φi} \f→ φI\n16 SHINICHI MOCHIZUKI*\ndefines an equivalence between the category of data on the left and the category of\narrows ψ : XI → YJ .\nProof. Indeed, this follows immediately by considering the pull-back of 1YJ , as\nwell as of its maximal decomposition, via the exact functor ψ∗, in light of the fact\n(observed above) that\n1XI =\n∐\ni∈I\n\fi\nis the maximal (relative to refinement) decomposition of 1XI . ©\nDefinition 1.1.12. We shall refer to a morphism between anabelioids as a π1-\nepimorphism (respectively, π1-monomorphism) if each of the component morphisms\n(cf. Proposition 1.1.11) between connected anabelioids is a π1-epimorphism (respec-\ntively, π1-monomorphism).\n§1.2. Finite Étale Morphisms\nIn this §, we consider the notion of a “finite étale morphism” in the context of\nanabelioids.\nLet X be an anabelioid. Let S ∈ Ob(X ). We will denote the category of objects\nover S by\nXS\n(i.e., the objects of XS are arrows T → S in X ; the arrows of XS between T → S\nand T ′ → S are S-morphisms T → T ′). Let us write\njS : XS → X\nfor the forgetful functor (i.e., the functor that maps T → S to T ) and\ni∗S : X → XS\nfor the functor given by taking the product with S.\nProposition 1.2.1. (The Extension Functor)\n(i) The category XS is an anabelioid whose connected components are in\nnatural bijective correspondence with the connected components of S.\n(ii) The functor jS is left adjoint to the functor i\n∗\nS.\n(iii) The functor i∗S is exact, hence defines a morphism of anabelioids iS :\nXS → X .\n(iv) Suppose that S is the coproduct of a finite number of copies of 1X (indexed,\nsay, by a (V -)set A). Then each connected component of XS may be identified with\nTHE GEOMETRY OF ANABELIOIDS 17\nX ; the set of connected components of XS may be identified with A. Moreover, jS\nmaps a collection of objects {Sa}a∈A of X indexed by A to the coproduct object∐\na∈A\nSa\nin X .\n(v) Suppose that X = B(G) (where G is a (V -small) profinite group) and\nthat S is given by the G-set G\/H, where H ⊆ G is an open subgroup. Then\niS : XS → X may be identified with [i.e., is “abstractly equivalent” (cf. §0) — in a\nnatural fashion — to] the morphism\nB(H) → B(G)\ninduced by the inclusion H ↪→ G. Moreover, if T ∈ Ob(XS) is represented by an\nH-set Tset, then jS(T ) is isomorphic to the G-set given by\n(G × Tset)\/H\nwhere H \u000f h acts on G × Tset \u000f (g, t) via (g, t) \f→ (hg, ht), and the G-action is the\naction induced on (G× Tset)\/H by letting G \u000f g act on G by multiplication by g−1\nfrom the right.\nProof. These assertions all follow immediately from the definitions. ©\nThus, Proposition 1.2.1, (ii), shows that if φ : Y → X factors as the com-\nposite of an isomorphism α : Y ∼→ XS with the morphism iS : XS → X for some\nS ∈ Ob(X ), then there is a natural choice for the isomorphism α, namely, the\nisomorphism induced by the left adjoint φ! : Y → X to the functor φ∗. Indeed,\nit follows from Proposition 1.2.1, (ii), that such a left adjoint φ! always exists and\nthat φ! induces an isomorphism Y ∼→ XSφ , where Sφ def= φ!(1Y).\nDefinition 1.2.2.\n(i) A morphism of anabelioids φ : Y → X will be called finite étale if it factors\nas the composite of an isomorphism α : Y ∼→ XS with the morphism iS : XS → X\nfor some S ∈ Ob(X ).\n(ii) Suppose that φ : Y → X is a finite étale morphism. Then we shall refer\nto the left adjoint functor φ! to the pull-back functor φ\n∗ as the extension functor\nassociated to φ.\nRemark 1.2.2.1. Thus, the morphism B(H) → B(G) induced by a continuous\nhomomorphism φ : H → G is finite étale if and only if φ is an injection onto an\nopen subgroup of G. Moreover, any finite étale morphism of connected anabelioids\n18 SHINICHI MOCHIZUKI*\nmay be written in this form (by choosing appropriate basepoints for the domain\nand range). The characterization of Definition 1.2.2, (i), however, has the virtue of\nbeing independent of choices of basepoints.\nDefinition 1.2.3. Let φ : Y → X be a finite étale morphism of anabelioids.\nThen we shall say that φ is a covering (respectively, relatively connected) if the\ninduced morphism π0(Y) → π0(X ) on connected components (cf. Definition 1.1.9)\nis surjective (respectively, an bijective).\nDefinition 1.2.4.\n(i) Let G be a profinite group. Then we shall say that G is slim if the centralizer\nZG(H) of any open subgroup H ⊆ G in G is trivial.\n(ii) Let X be an anabelioid. Then we shall say that X is slim if the fundamental\ngroup π1(Xi) of every connected component i ∈ π0(X ) of X is slim.\n(iii) A morphism of anabelioids whose corresponding pull-back functor is rigid\nwill be called rigid. A 2-category of anabelioids will be called slim if every 1-\nmorphism in the 2-category is rigid.\n(iv) If C is a 2-category, we shall write\n|C|\nfor the associated 1-category whose objects are objects of C and whose morphisms\nare isomorphism classes of morphisms of C. We shall also refer to |C| as the coar-\nsification of C.\nRemark 1.2.4.1. The name “coarsification” is motivated by the theory of “coarse\nmoduli spaces” associated to (say) “fine moduli stacks”.\nRemark 1.2.4.2. Thus, a diagram of rigid morphisms of anabelioids “1-commutes”\n(cf. §0) if and only if it commutes in the coarsification.\nIn a word, the theory of coverings of anabelioids is easiest to understand when\nthe anabelioid in question is slim. For instance:\nProposition 1.2.5. (Slim Anabelioids) Let X be a slim anabelioid. Then:\n(i) The pull-back and extension functors associated to a finite étale morphism\nbetween slim anabelioids are rigid (cf. §0). In particular, if we write\nEt(X ) ⊆ Anab\nTHE GEOMETRY OF ANABELIOIDS 19\nfor the 2-category whose (0-)objects are finite étale morphisms Y → X and whose\n(1-)morphisms are finite étale arrows Y1 → Y2 “over” X [i.e., in the sense of\n“1-commutativity” — cf. §0], then Et(X ) is slim. Write: Ét(X ) def= |Et(X )|.\n(ii) The functor\nFX : X → Ét(X )\nS \f→ (XS → X )\n(where S ∈ Ob(X )) is an equivalence (i.e., fully faithful and essentially surjec-\ntive).\nProof. Indeed, (i) follows formally from Corollary 1.1.6 and Definition 1.2.4, (i),\n(ii), (iii). As for (ii), essential surjectivity follows formally from Definition 1.2.2, (i).\nTo prove fully faithfulness, it suffices to compute, when X = B(G), Y1 = B(H1),\nY2 = B(H2), and H1, H2 are open subgroups of G, the subset\nMorX (Y1,Y2) ⊆ Mor(Y1,Y2)\n[i.e., of isomorphism classes of morphisms “over” X ] via Proposition 1.1.4, (i).\nThis computation yields that the set in question is equal to the quotient, via the\nconjugation action by H2, of the set of morphisms H1 → H2 induced by conjugation\nby an element g ∈ G such that H1 ⊆ g ·H2 ·g−1. But this quotient may be identified\nwith the subset of elements g · H2 ∈ G\/H2 such that H1 ⊆ g · H2 · g−1. Note that\nhere we must apply the assumption of slimness, to conclude that it is not necessary\nto quotient G\/H2 any further by various centralizers in G of conjugates of H1. On\nthe other hand, this quotient is simply another description of the set\nHomG(G\/H1, G\/H2)\nas desired. ©\nRemark 1.2.5.1. By Proposition 1.2.5, (i), it follows that, at least when we\nrestrict our attention to finite étale morphisms of slim anabelioids, we do not “lose\nany essential information” by working in the coarsification (of Anab). Thus,\nin the following discussion, we shall often do this, since this simplifies things sub-\nstantially. For instance, if φ : Y → X and ψ : Z → X are arbitrary finite étale\nmorphism of slim anabelioids, then [if we work in the coarsification] it makes sense\nto speak of the pull-back (of φ via ψ), or fiber product (of Y, Z over X ):\nY ×X Z\nIndeed, such an object may be defined by the formula:\nZψ∗(φ!(1Y)) → Z\n20 SHINICHI MOCHIZUKI*\nBy thinking of φ : Y → X as some “XS → X ” as in the above discussion, one\nverifies easily that this definition satisfies all the expected properties. One verifies\neasily that all conceivable compatibilities are satisfied [e.g., when one interchanges\nthe roles of φ and ψ].\nRemark 1.2.5.2. In fact, essentially all of the anabelioids that we shall actually\ndeal with in this paper will be slim. Thus, in some sense, it might have been more\nnatural to take the notion of a “slim anabelioid” as our definition of the term\n“anabelioid”. There are two reasons why we chose not to do this: First, this would\nrequire us to prove slimness every time that we wish to use term “anabelioid”, which\nwould, in some sense, be rather unnatural, just as having to prove separatedness\nevery time one uses the term “scheme” (if, as in the earlier terminology, one defines\na scheme to be a “separated scheme” (in the current terminology)). Second, just as\nwith the separatedness of schemes, which is not a Zariski local notion, the notion\nof slimness of an anabelioid is not (finite) étale local. (That is to say, a non-\nslim anabelioid may admit a finite étale covering which is slim.) Thus, requiring\nanabelioids to be slim would mean that the notion of an anabelioid is not “finite\nétale local”, which would again be unnatural.\nRemark 1.2.5.3. Note that although FX is fully faithful and essentially surjective,\nsubstantial care should be exercised when speaking of FX as an “equivalence”. The\nreason for this is that:\nThe collection of objects of Ét(X ) or Et(X ) necessarily belongs to a larger\nGrothendieck universe — that is to say, unlike X , the category Ét(X )\nis no longer V -small — than the collection of objects of X .\nPut another way, FX , i.e., the passage from X to Ét(X ), may be thought of as a sort\nof “change of Grothendieck universe, while keeping the internal category structure\nintact”.\nJust as in the theory of schemes, one often wishes to work not just with finite\nétale coverings, but also with “profinite étale coverings” (i.e., projective systems of\nétale coverings). In the case of anabelioids, we make the following\nDefinition 1.2.6. We shall refer to as a pro-anabelioid any “pro-object” (indexed\nby a set)\nX = lim←−\nα\nXα\nrelative to the coarsified category\nAnab\ndef\n= |Anab|\nin which all of the transition morphisms Xα → Xβ are finite étale coverings of\nslim anabelioids. Here, by “pro-object”, we mean an equivalence class of projective\nsystems (relative to the evident notion of equivalence).\nTHE GEOMETRY OF ANABELIOIDS 21\nRemark 1.2.6.1. Thus, (for us) pro-anabelioids only exist at the “coarsified\nlevel” (unlike anabelioids, which may be treated either in Anab or in Anab).\nRemark 1.2.6.2. Given a pro-anabelioid\nX = lim←−\nα\nXα\nit is natural to define the set of connected components of X by:\nπ0(X ) def= lim←−\nα\nπ0(Xα)\nIn general, π0(X ) will be a profinite set. Moreover, for each i ∈ I, one obtains a\nconnected pro-anabelioid\nXi\nby forming\nlim←−\nα\nof the compatible system of connected components of the Xα indexed by i.\nRemark 1.2.6.3. Given two pro-anabelioids X = lim←−α Xα; Y = lim←−β Yβ , by the\ndefinition of a “pro-object”, it follows that:\nMor(X ,Y) = lim←−\nβ\nlim−→\nα\nMor(Xα,Yβ)\nNote that this formula also applies in the case when one or both of X , Y is an\nanabelioid, by thinking of anabelioids as pro-anabelioids indexed by the set with\none element.\nSuppose that we are given a connected anabelioid X def= B(G) (where G is\na profinite group). Let us write β : Ensf → X for the tautological basepoint of\nB(G). Then one important example of a pro-anabelioid which forms a profinite\nétale covering of X is the “universal covering” X̃β, defined as follows: For each\nopen subgroup H ⊆ G, let us write XH def= B(H). (In other words, XH is the\ncategory XS associated to the object S ∈ Ob(X ) determined by the G-set G\/H.)\nThus, if H ′ ⊆ H, then we have a natural morphism XH ′ → XH . Moreover, these\nmorphisms form a projective system whose transition morphisms are clearly finite\nétale coverings. Hence, we obtain a pro-anabelioid\nX̃β def= lim←−\nH\nXH\n(where H ranges over the open subgroups of G), together with a “profinite étale\ncovering”\nX̃β → X\n22 SHINICHI MOCHIZUKI*\nwhich (just as in the case of schemes) has the property that the pull-back via\nthis covering of any finite étale covering Y → X splits (i.e., is isomorphic to the\ncoproduct of a finite number of copies of the base).\nDefinition 1.2.7. Let X be an anabelioid, and Y a pro-anabelioid. Then a\nprofinite étale covering Y → X will be referred to as a universal covering of X if\nit is relatively connected [i.e., given by a projective system of relatively connected\nfinite étale morphisms] and satisfies the property that the pull-back to Y of any\nfinite étale covering of X splits.\nNote that by Proposition 1.2.5, (ii), it follows that when X = B(G) is slim, the\nset\nMorX (X̃β ,XH)\nmay be identified with G\/H. In particular, we obtain the result that the base-\npoint β is naturally equivalent to the restriction to the image of the functor FX of\nProposition 1.2.5, (ii), of the basepoint of Ét(X ) defined by the formula:\nMorX (X̃β ,Y)\n(where Y → X is an object of Ét(X )).\nProposition 1.2.8. (Basic Properties of Universal Coverings) Let X be\na slim anabelioid. Then:\n(i) There exists a universal covering Y → X .\n(ii) Any two universal coverings Y → X , Y ′ → X are isomorphic over X .\n(iii) Suppose that X is connected. Then the formula\nβ\n\u0000X (S)\ndef\n= MorX (X̃ ,XS)\n(where S ∈ Ob(X )) defines an equivalence of categories between the category\nof universal coverings X̃ → X (whose morphisms are isomorphisms X̃ ∼→ X̃ ′\nover X ) and the category of basepoints β : Ensf → X (whose morphisms β ∼→ β′\nare isomorphisms of functors (β′)∗ ∼→ β∗). In particular, if X̃ → X determines the\nbasepoint β\n\u0000X , then\nAutX (X̃ ) = Aut(β \u0000X ) = π1(X , β \u0000X )\n(where AutX (X̃ ) is the set of automorphisms relative to the category of universal\ncoverings just defined).\n(iv) Suppose that X def= B(G), X ′ def= B(G′) are slim connected anabelioids. Let\nX̃ → X , X̃ ′ → X ′ be the universal coverings determined by the tautological base-\npoints β, β′. Then, if we denote by Isom(X̃ , X̃ ′) the set of isomorphisms X̃ ∼→ X̃ ′\nTHE GEOMETRY OF ANABELIOIDS 23\nwhich do not necessarily lie over some isomorphism X ∼→ X ′, we have a natural\nisomorphism\nIsog((X , β); (X ′, β′)) def= Isom(X̃ , X̃ ′) ∼→ lim−→\nH\n{open injections H ↪→ G′}\n(where H ranges over the open subgroups of G).\nProof. Assertions (i), (ii) follow formally from the definitions and the above discus-\nsion. Now let us consider assertion (iv). Suppose that we are given an isomorphism\nφ : X̃ ∼→ X̃ ′. By Proposition 1.1.4, (i), such a morphism arises from some homo-\nmorphism H → H ′, determined up to conjugation with an inner automorphism\narising from H ′. Here, we take H ⊆ G, H ′ ⊆ G′ to be normal open subgroups.\nIf K ′ ⊆ G is another normal open subgroup contained in H ′, then there exists a\nnormal open subgroup K ⊆ G contained in H, together with a homomorphism\nK → K ′ (determined by φ, up to conjugation with an inner automorphism arising\nfrom K ′) such that the outer homomorphism H → H ′ is compatible with the outer\nhomomorphism K → K ′. Note, moreover, that since X ′ is slim, a unique homo-\nmorphism H → H ′ up to conjugation with an inner automorphism arising from K ′\nis determined by the homomorphism K → K ′. (Indeed, this follows by considering\nthe faithful actions (by conjugation) of H, H ′ on K, K ′, respectively.) Thus, by\ntaking K ′ to be arbitrarily small, we see that φ determines a unique homomorphism\nH → H ′ ⊆ G′. Consideration of the inverse to φ shows that this homomorphism\nH → G′ is necessarily an open injection. On the other hand, any open injection\nH ↪→ G′ clearly determines an isomorphism φ. This completes the proof of (iv).\nFinally, we consider property (iii). Since it is clear that any isomorphism be-\ntween universal coverings induces an isomorphism of the corresponding basepoints,\nit suffices to prove property (iii) in the “automorphism” case. For simplicity, we\nshall write X = B(G), and assume that the basepoint β in question is the tautolog-\nical basepoint. By property (iv), any isomorphism φ : X̃ ∼→ X̃ arises from an open\ninjection H ↪→ G. The fact that the composite of φ with X̃ → X is isomorphic\nto X̃ → X implies (cf. Proposition 1.1.4, (i)) that this open injection H ↪→ G is\ninduced by conjugation by a unique (by slimness) element of G. On the other hand,\nconjugation by an element of G clearly determines an element of AutX (X̃ ). Thus,\nAutX (X̃ ) = G, as desired. ©\nRemark 1.2.8.1. When (cf. Proposition 1.2.8, (iv)) β, β′ are fixed throughout\nthe discussion, we shall write\nIsog(X ,X ′)\nfor Isog((X , β); (X ′, β′)). When (X , β) = (X ′, β′), we shall write Isog(X ) for\nIsog(X ,X ′).\nFinally, before proceeding, we present the following:\n24 SHINICHI MOCHIZUKI*\nDefinition 1.2.9.\n(i) We shall say that a continuous homomorphism of Hausdorff topological\ngroups G → H is relatively slim if the centralizer in H of the image of every open\nsubgroup of G is trivial.\n(ii) We shall say that a morphism of anabelioids U → V is relatively slim if\nthe induced morphism between fundamental groups of corresponding connected\ncomponents of U , V is relatively slim.\nRemark 1.2.9.1. Thus, X is slim if and only if the identity morphism X → X is\nrelatively slim. Also, if U → V is relatively slim, then the arrow U → V is rigid [i.e.,\nhas no nontrivial automorphisms — cf. Corollary 1.1.6]. If U → V is a relatively\nslim morphism between connected anabelioids, then it follows that V is slim; if,\nmoreover, U → V is a π1-monomorphism, then it follows that U is also slim.\nRemark 1.2.9.2. The construction of a pull-back, or fiber product, discussed in\nRemark 1.2.5.1 generalizes immediately to the case where φ : Y → X is a finite\nétale morphism of slim anabelioids, and ψ : Z → X is an arbitrary relatively slim\nmorphism of slim anabelioids, via the formula of loc. cit.:\nZψ∗(φ!(1Y)) → Z\nOne verifies immediately that all conceivable compatibilities are satisfied.\nTHE GEOMETRY OF ANABELIOIDS 25\nSection 2: Cores and Quasi-Cores\n§2.1. Localizations and Cores\nIn this §, we discuss the notion of a core in the context of slim anabelioids.\nThis notion will play a central role in the theory of the present paper.\nLet X be a slim anabelioid. Let us write\nLoc(X ) (⊆ Anab)\nfor the 2-category whose (0-)objects are (necessarily slim) anabelioids Y that admit\na finite étale morphism to X , and whose (1-)morphisms are finite étale morphisms\nY1 → Y2 (that do not necessarily lie over X !). Note that given an object of Loc(X ),\nthe set of connected components of this object may be recovered entirely category-\ntheoretically from the coarsification\nLoc(X ) def= |Loc(X )|\nof the 2-category Loc(X ) (cf. Proposition 1.1.11).\nProposition 2.1.1. (Categories of Localizations) Let X be a slim anabe-\nlioid. Then:\n(i) Loc(X ) is slim.\n(ii) Denote by\nLoc(X)\nthe 2-category whose (0-)objects Z are slim anabelioids which arise as finite\nétale quotients of objects in Loc(X ) [i.e., there exists a finite étale morphism Y →\nZ, where Y ∈ Ob(Loc(X ))] and whose (1-)morphisms are finite étale morphisms.\nThen the 2-category Loc(X ) is slim. Write: Loc(X ) def= |Loc(X )|.\n(iii) The 2-category Loc(X ) (respectively, category Loc(X )) may be recon-\nstructed entirely category-theoretically from Loc(X ) (respectively, Loc(X )) by\nconsidering the “2-category (respectively, category) of objects of Loc(X ) (respec-\ntively, Loc(X )) equipped with a finite étale equivalence relation”.\n(iv) Suppose that we arbitrarily choose finite étale structure morphisms to\nX for all of the objects of Loc(X ). Then every morphism Y1 → Y2 of Loc(X )\nmay be written as the composite of an isomorphism Y1 ∼→ Y3 with a finite étale\nmorphism Y3 → Y2 over X .\nProof. Assertions (i) and (ii) are formal consequences of Corollary 1.1.6. Asser-\ntions (iii) and (iv) follow formally from the definitions. ©\nLet X be a slim anabelioid. Then:\n26 SHINICHI MOCHIZUKI*\nDefinition 2.1.2.\n(i) We shall say that X is a(n) (absolute) core if X is a terminal object in\nLoc(X ).\n(ii) We shall say that X admits a(n) (absolute) core if there exists a terminal\nobject Z in Loc(X ). In this case, Loc(X ) = Loc(Z) = Loc(Z), so we shall say that\nZ is a core.\nRemark 2.1.2.1. Note that in Proposition 2.1.1, (ii), it is important to assume\nthat the quotients Z that one considers are slim. Indeed, if one did not impose\nthis condition, then by “forming quotients of slim anabelioids by the trivial actions\nof finite groups”, one verifies easily that the 1-category associated to the resulting\n2-category never admits a terminal object — i.e., “no slim anabelioid would admit\na core”. From the point of view of anabelian varieties — e.g., hyperbolic orbicurves\n— this condition of slimness amounts to the condition that the algebraic stacks\nthat one works with are generically schemes (cf. [Mzk9], §2).\nRemark 2.1.2.2. Note that the definability of Loc(X ), Loc(X ) is one of the most\nfundamental differences between the theory of finite étale coverings of anabelioids\nas discussed in §1.2 and the theory of finite étale coverings from the point of view\nof “Galois categories”, as given in [SGA1]. Indeed, from the point of view of the\ntheory of [SGA1], it is only possible to consider “Ét(X )” — i.e., finite étale coverings\nand morphisms that always lie over X . That is to say, in the context of the theory\nof [SGA1], it is not possible to consider diagrams such as:\nZ\n↙ ↘\nX Y\n(where the arrows are finite étale) that do not necessarily lie over any specific\ngeometric object. We shall refer to such a diagram as a correspondence or isogeny\nbetween X and Y. When there exists an isogeny between X and Y, we shall say\nthat X and Y are isogenous.\nNext, we would like to consider universal coverings. Let β, γ be basepoints of\na connected slim anabelioid X . Write\nπβ : X̃β → X ; πγ : X̃γ → X\nfor the associated universal coverings (cf. the discussion of §1.2). In the following\ndiscussion, we would also like to consider an isomorphism\nξ : X̃β ∼→ X̃γ\n(cf. Proposition 1.2.8, (iv)).\nTHE GEOMETRY OF ANABELIOIDS 27\nDefinition 2.1.3. We shall refer to an isomorphism ξ : X̃β ∼→ X̃γ as above as an\nouter path from β to γ. If ξ arises from a commutative [i.e., at the coarsified level]\ndiagram of anabelioids\nX̃β ξ−→ X̃γ⏐⏐\u0007 ⏐⏐\u0007\nX idX−→ X\nthen we shall refer to ξ as an inner path from β to γ. An outer (respectively, inner)\npath from β to itself will be referred to as an (X̃β-valued) open (respectively, closed)\npath.\nRemark 2.1.3.1. Thus, inner paths are precisely the paths of [SGA1], Exposé V,\n§7. Note that the difference between an “inner” path and an “outer” path depends\nessentially on the “identity” of β, γ — i.e., what appears to be an outer path if one\nthinks of β and γ as in fact being “equal” may appear to be an inner path if one\nthinks of β and γ as “distinct”. Put another way:\nThe distinction between inner and outer paths depends essentially on the\n“model of set theory” under consideration — i.e., on the labels that\none uses to describe the various sets involved in the discussion.\nIt is the hope of the author to pursue this point of view in more detail in a future\npaper.\nRemark 2.1.3.2. Note that an inner path is a special case of an outer path. The\ndifference between an inner path and an arbitrary outer path is easiest to analyze\nwhen β = γ (but cf. Remark 2.1.3.1!). In this case, an (X̃β-valued) closed path is\nsimply an element of the fundamental group π1(X , β).\nOn the other hand, the motivation for the terminology “open path” is the\nfollowing. Let K be a perfect field; L a finite Galois extension of K; and K an\nalgebraic closure of K. Then to give a K-valued basepoint β of L is to give an\nembedding ιβ : L ↪→ K. If we are then given a K-linear isomorphism σ : K ∼→ K\n(i.e., an element σ ∈ Gal(K\/K)), then the composite of σ with ιβ determines\nanother embedding ιγ : L ↪→ K. Of course, the two basepoints β, γ of Spec(L)\ndefined by ιβ , ιγ map to the same basepoint of Spec(K) — i.e., “if one applies the\nprojection Spec(L) → Spec(K), then σ becomes a closed path in Spec(K)”. This is\nintended to be reminiscent of the analogy between Galois groups in field theory and\nfundamental groups in algebraic topology (where we recall that the theory of the\nlatter may be formulated not just in terms of covering groups, but also in terms of\nliteral closed paths, i.e., topological images of the circle S1, in the space in question).\nThus, it is natural to regard σ — when working with σ as an object associated to\nSpec(L) — as an open path (valued in K), i.e., the analogue of a topological image\nof the interval [0, 2π] as opposed to the circle S1, on Spec(L).\n28 SHINICHI MOCHIZUKI*\nIncidentally, this example also shows the reason for the choice of terminology\n“inner\/outer path”. That is to say, inner\/closed paths induce (via “parallel trans-\nport”) inner automorphisms of the fundamental group, while outer\/open paths\narise from arbitrary (outer) automorphisms, or even isogenies, of the fundamental\ngroup.\nProposition 2.1.4. (The Totality of Basepoints) Let X be a connected\nslim anabelioid. Let X̃ → X be a universal covering of X , that determines\nsome basepoint β of X . Then:\n(i) The subgroup\nΠX\ndef\n= π1(X , β) = AutX (X̃ ) ⊆ Aut(X̃ ) = Isog(X )\nis commensurable with all of its conjugates in Isog(X ). Moreover, the open\nsubgroups of ΠX define a basis for a topology on Isog(X ) with respect to which\nIsog(X ) forms a Hausdorff topological group. Finally, the subgroup ΠX ⊆\nIsog(X ) is both open and closed with respect to this topology.\n(ii) Isog(X ) acts transitively on the set of X̃ -valued basepoints — i.e.,\n(isomorphism classes of) profinite étale morphisms X̃ → X — of X . Moreover,\nthis action determines a bijection between the set of X̃ -valued basepoints and the\ncoset space:\nIsog(X )\/ΠX\n(iii) Suppose that X is a core. Then ΠX = Isog(X ). That is to say, X admits\nprecisely one X̃ -valued basepoint. In particular, all open paths on X are, in fact,\nclosed. Moreover, the natural functors\nEt(X ) → Loc(X ) → Loc(X ); Ét(X ) → Loc(X ) → Loc(X )\nare equivalences.\nProof. These assertions are all formal consequences of the definitions (cf. also\nProposition 1.2.8, (iv)). ©\nRemark 2.1.4.1. Note, however, that the subgroup of Isog(X ) generated by ΠX\nand some conjugate of ΠX does not necessarily contain either of these two groups\nas a finite index subgroup. Perhaps the most famous example of this phenomenon\nis the theorem of Ihara (cf., e.g., [Serre1], II, §1.4, Corollary 1; [Ihara]) expressing\nSL2(Qp) as an amalgam of two copies of SL2(Zp), amalgamated along a subgroup\nwhich is open in both copies of SL2(Zp). In the notation of the present discussion,\nthis example corresponds to the case\nX def= B(SL±2 (Zp))\nTHE GEOMETRY OF ANABELIOIDS 29\n(where, instead of SL2(Zp), we use its quotient SL\n±\n2 (Zp) by ±1 to ensure that X\nis slim). Note that this example shows that Isog(X ) does not necessarily admit a\nnatural structure of profinite group. Indeed, in the case of SL±2 (Zp), one checks\neasily (by applying the theory of p-adic Lie groups — cf., e.g., [Serre2], Chapter V,\n§7) that Isog(X ) = PGL2(Qp) (which is not profinite).\nRemark 2.1.4.2. The above example of SL2(Zp) highlights one of the major\nthemes of the present paper, i.e., that:\nopen paths ⇐⇒ Isog(X ) ⇐⇒ correspondences\n— that is to say, just as (in the “classical theory” of the étale fundamental group\ngiven in [SGA1]) closed paths (i.e., elements of π1) correspond to elements of ΠX ,\nopen paths corresponds to elements of Isog(X ), i.e., “correspondences”.\nRemark 2.1.4.3. It is interesting to note relative to Proposition 2.1.4, (ii) (cf.\nalso Proposition 1.2.8, (iii); Remark 2.1.3.1) that the cardinality of the collection\nof basepoints Ensf → X is the same as that of the collection of profinite étale\nmorphisms X̃ → X . Indeed, both collections have the same cardinality as the\ncollection of morphisms Ensf → Ensf.\n§2.2. Holomorphic Structures and Commensurable Terminality\nIn this §, we wish to discuss a relative version of the theory of §2.1. Let X , Q\nbe slim anabelioids.\nDefinition 2.2.1.\n(i) A Q-holomorphic structure on X is the datum of a relatively slim morphism\n(cf. Definition 1.2.9, (ii)) X → Q, which we shall refer to as the structure morphism.\n(ii) A slim anabelioid equipped with a Q-holomorphic structure will be referred\nto as a Q-anabelioid.\n(iii) A Q-holomorphic morphism (or “Q-morphism” for short) between Q-\nanabelioids is a morphism of anabelioids compatible with the Q-holomorphic struc-\ntures.\n(iv) A Q-holomorphic structure\/Q-anabelioid will be called faithful if its struc-\nture morphism is a π1-monomorphism.\nRemark 2.2.1.1. Here, we note that the term “compatible” in Definition 2.2.1,\n(iii), makes sense, precisely because of the assumption of relative slimness in Defi-\nnition 2.2.1, (i) (cf. Corollary 1.1.6).\n30 SHINICHI MOCHIZUKI*\nLet us write\nLocQ(X )\nfor the 2-category whose (0-)objects Y → Q are Q-anabelioids that admit a Q-\nholomorphic finite étale morphism Y → X to X , and whose (1-)morphisms are\narbitrary finite étale Q-morphisms (that do not necessarily lie over X !). Now we\nhave the “Q-holomorphic analogue” of Proposition 2.1.1:\nProposition 2.2.2. (Categories of Holomorphic Localizations) Let Q be\na slim, connected anabelioid; X a Q-anabelioid. Then:\n(i) LocQ(X ) is slim. Write: LocQ(X ) def= |LocQ(X )|.\n(ii) Denote by\nLocQ(X)\nthe 2-category whose (0-)objects Z → Q are Q-anabelioids which arise as finite\nétale quotients of objects in LocQ(X ) [i.e., there exists a finite étale Q-morphism\nY → Z, where Y ∈ Ob(LocQ(X ))] and whose (1-)morphisms are finite étale Q-\nmorphisms. Then the 2-category LocQ(X ) is slim. Write: LocQ(X ) def= |LocQ(X )|.\n(iii) The 2-category LocQ(X ) (respectively, category LocQ(X )) may be recon-\nstructed entirely category-theoretically from LocQ(X ) (respectively, LocQ(X ))\nby considering the “2-category (respectively, category) of objects of LocQ(X ) (re-\nspectively, LocQ(X )) equipped with a finite étale equivalence relation”.\n(iv) Suppose that we arbitrarily choose finite étale structure morphisms to\nX for all of the objects of LocQ(X ). Then every morphism Y1 → Y2 of LocQ(X )\nmay be written as the composite of an isomorphism Y1 ∼→ Y3 (over Q) with a\nfinite étale morphism Y3 → Y2 over X .\nLet X be a Q-anabelioid. Then:\nDefinition 2.2.3.\n(i) We shall say that X is a Q-core if X [i.e., X → Q] is a terminal object in\nLocQ(X ).\n(ii) We shall say that X admits a Q-core if there exists a terminal object Z in\nLocQ(X ). In this case, LocQ(X ) = LocQ(Z) = LocQ(Z), so we shall say that Z is\na Q-core.\nDefinition 2.2.4.\n(i) We shall say that a closed subgroup H ⊆ G of a profinite group G is\ncommensurably (respectively, normally) terminal if the commensurator CG(H) (re-\nspectively, normalizer NG(H)) is equal to H.\nTHE GEOMETRY OF ANABELIOIDS 31\n(ii) We shall say that a π1-monomorphism of anabelioids U → V is commen-\nsurably (respectively, normally) terminal if the image of the induced morphism\nbetween fundamental groups of corresponding connected components of U , V is\ncommensurably (respectively, normally) terminal.\nRemark 2.2.4.1. Thus, it is a formal consequence of the definitions that:\ncommensurably terminal =⇒ normally terminal\nand that\ncommensurably terminal with slim domain =⇒ relatively slim\n(where the “domain” is the group H (respectively, anabelioid U) in Definition 2.2.4,\n(i) (respectively, (ii))).\nProposition 2.2.5. (Commensurable Terminality and Holomorphic\nCores) Let X be a connected faithful Q-anabelioid; assume that Q is also\nconnected. Then X is a Q-core if and only if its structure morphism is commen-\nsurably terminal.\nProof. Without loss of generality, we may write X = B(H), Q = B(G), where H ⊆\nG is a closed subgroup. First, we verify sufficiency. By Proposition 1.1.4, it suffices\nto prove that, if H ′ ⊆ H is an open subgroup, then any continuous homomorphism\nφ : H ′ → G whose image lies in H and which factors as the composite of the natural\ninclusion H ′ ↪→ G with conjugation by an element g ∈ G is, in fact, equal to the\nto composite of the natural inclusion H ′ ↪→ G with conjugation by an element\nh ∈ H. But this follows immediately from Definition 2.2.4, (i), which implies that\ng ∈ H. Finally, necessity follows by reversing the preceding argument in the evident\nfashion. ©\nLet X be a connected Q-anabelioid. For simplicity, we also assume that Q\nis connected. Suppose that we are given a universal covering Q̃ → Q of Q and\nconsider the resulting cartesian diagram:\nQ̃|X −→ Q̃⏐⏐\u0007 ⏐⏐\u0007\nX −→ Q\nNote that ΠQ\ndef\n= AutQ(Q̃) acts (compatibly) on Q̃ over Q, as well as on Q̃|X over\nX . On the other hand, if we consider a connected component X̃ of Q̃|X as an\nindependent geometric object, even if the Q-holomorphic structure on X̃ remains\nfixed, in general X̃ will admit distinct (profinite) étale morphisms to X . Put another\n32 SHINICHI MOCHIZUKI*\nway, in general, X admits distinct X̃ -valued Q-holomorphic basepoints. That is to\nsay, we have the Q-holomorphic analogue of Proposition 2.1.4:\nProposition 2.2.6. (The Totality of Q-Holomorphic Basepoints) Let X\nbe a connected faithful Q-anabelioid, where Q is also connected. Let Q̃ → Q be\na universal covering of Q; X̃ → X a connected component of Q̃|X → X . Write\nΠQ\ndef\n= AutQ(Q̃), ΠX def= AutX (X̃ ). Thus, we have a natural inclusion ΠX ⊆ ΠQ.\nThen:\n(i) The subgroup\nΠX = AutX (X̃ ) ⊆ IsogQ(X ) def= AutQ(X̃ ) = CΠQ(ΠX )\nis commensurable with all of its conjugates in IsogQ(X ). Moreover, the open\nsubgroups of ΠX define a basis for a topology on IsogQ(X ) with respect to which\nIsogQ(X ) forms a Hausdorff topological group. Finally, the subgroup ΠX ⊆\nIsogQ(X ) (respectively, IsogQ(X ) ⊆ Isog(X )) is both open and closed (respectively,\nopen) with respect to this topology.\n(ii) IsogQ(X ) acts transitively on the set of X̃ -valued Q-holomorphic\nbasepoints — i.e., (isomorphism classes of) profinite étale Q-morphisms X̃ → X\n— of X . Moreover, this action determines a bijection between the set of X̃ -valued\nQ-holomorphic basepoints and the coset space:\nIsogQ(X )\/ΠX\n(iii) Suppose that X is a Q-core. Then ΠX = IsogQ(X ). That is to say, X\nadmits precisely one X̃ -valued Q-holomorphic basepoint. In particular, all “Q-\nholomorphic” open paths on X are, in fact, closed. Moreover, the natural functors\nEt(X ) → LocQ(X ) → LocQ(X ); Ét(X ) → LocQ(X ) → LocQ(X )\nare equivalences.\nRemark 2.2.6.1. Thus, at a more intuitive level, just as “(absolute) cores\nhave essentially only one basepoint”, if X is a Q-core, then every basepoint of Q\ndetermines an essentially unique (up to renaming) Q-holomorphic basepoint of X .\nRemark 2.2.6.2. The topology of Proposition 2.2.6, (i), is not to be confused\nwith the topology on CΠQ(ΠX ) induced by the topology of ΠQ. For instance, if ΠX\nis the profinite free group on 2 generators (which is easily seen to be slim — cf.,\ne.g., [Mzk8], Lemma 1.3.1) and ΠQ = Aut(ΠX ) (which also has a natural structure\nof profinite group), then ΠQ = CΠQ(ΠX ), but ΠX is not open [i.e., relative to the\nprofinite topology of ΠQ] in ΠQ. Here, we note that Out(ΠX ) = Aut(ΠX )\/ΠX ,\nhence also ΠQ, is infinite and slim. [Indeed, the slimness of ΠQ may be shown, for\nTHE GEOMETRY OF ANABELIOIDS 33\ninstance, as follows: By [Tama], Theorem 0.4, applied to the projective line minus\nthree points over the field of rational numbers, it follows that the centralizer of\nany open subgroup of Out(ΠX ) is contained in the subgroup of Out(ΠX ) obtained\nby considering the permutation group of the three points. On the other hand, by\nprojecting to Out(ΠabX ) ∼= GL2(Ẑ), one sees that any element of this permutation\ngroup that centralizes an open subgroup of GL2(Ẑ) must be trivial.]\n§2.3. Quasi-Cores and Intrinsic Exhaustivity\nIn order to define the fundamental group of a (connected slim) anabelioid X ,\nit is necessary to choose a basepoint for X . As we saw in Proposition 1.2.8, this\nis equivalent to choosing a universal cover X̃ → X of X . On the other hand, in\ngeneral, there is nothing special that distinguishes a given profinite étale X̃ → X\nfrom another X̃ → X obtained from the first by composition with some element of\nAut(X̃ ) = Isog(X ). That is to say, the difference between these two X̃ → X is a\n“matter of arbitrary choices of labels”. Thus, the question naturally arises:\nTo what extent is it possible to construct the fundamental group of a (con-\nnected slim) anabelioid in a canonical fashion that does not depend on\nsuch arbitrary choices?\nIn this § and the next, we would like to analyze this issue in more detail. Our main\nresult (cf. Theorem 2.4.3 below) states that when the anabelioid in question admits\na “faithful quasi-core” (cf. Definition 2.3.1), then its fundamental group can indeed\nbe constructed in a rather canonical fashion. In addition to quasi-cores, we also\nconsider the notion of intrinsic exhaustivity, which provides a convenient, intrinsic\nnecessary condition for an anabelioid to admit a faithful quasi-core.\nIn the following, we shall always consider morphisms between anabelioids in\nthe coarsification Anab of Anab.\nDefinition 2.3.1. Let X be a Q-anabelioid (so X , Q are slim). For simplicity,\nwe also assume that the fundamental group of every irreducible component of Q is\ncountably (topologically) generated.\n(i) We shall say that X admits (Q as) a quasi-core if the natural functor\nLocQ(X ) → Loc(X )\n(given by forgetting the Q-holomorphic structure) is an equivalence.\n(ii) We shall say that X admits (Q as) a faithful quasi-core if X admits Q as\na quasi-core, and, moreover, the Q-structure on X is faithful.\nNext, let us recall that if G is a slim profinite group, then it admits a natural\ninjection\nG ↪→ Isog(G) def= Isog(B(G))\n34 SHINICHI MOCHIZUKI*\n(cf. Propositions 1.2.8, (iv); 2.1.4, (i)). Thus, in the following discussion, we shall\nregard G as a subgroup of Isog(G).\nDefinition 2.3.2. We shall refer to as a profinite subgroup K ⊆ Isog(G) a sub-\ngroup K of the abstract group Isog(G) which is equipped with a structure of profinite\ngroup such that the intersection K\n⋂\nG is a closed subgroup of both G and K\nwhose induced topologies from G and K coincide.\nRemark 2.3.2.1. If K ⊆ Isog(G) is a profinite subgroup which is, moreover,\ncommensurable to a closed subgroup F ⊆ G (i.e., K ⋂ F is open in F , K), then\none verifies easily that the topology on K is the unique topology with respect to\nwhich K ⊆ Isog(G) is a profinite subgroup.\nRemark 2.3.2.2. One verifies immediately that if G′ ⊆ Isog(G) is a profinite\nsubgroup commensurable to G — so that one has a natural identification Isog(G) =\nIsog(G′) — then the profinite subgroups of Isog(G) are the same (relative to this\nidentification) as the profinite subgroups of Isog(G′).\nWe will also make use of the following definitions:\nDefinition 2.3.3.\n(i) A profinite group G will be called weakly intrinsically exhaustive if for every\nopen subgroup H ⊆ G and every open embedding ι : H ↪→ G, we have:\n[G : H] = [G : ι(H)]\n(ii) A slim profinite group G will be called intrinsically exhaustive if there exists\na filtration\n. . . ⊆ Gn+1 ⊆ Gn ⊆ . . . ⊆ G\n(where n ranges over the positive integers) of open normal subgroups Gn of G such\nthat ⋂\nn\nGn = {1}\nand, moreover, for any profinite subgroup K ⊆ Isog(G) commensurable to G, there\nexists an integer nK — depending only on the profinite subgroup K — such that\nGn ⊆ K for n ≥ nK , and, for any open subgroup H ⊆ Gn (where n ≥ nK) and\nany open embedding ι : H ↪→ K, we have ι(H) ⊆ Gn (⊆ K).\n(ii) An anabelioid will be called intrinsically exhaustive (respectively, weakly\nintrinsically exhaustive) if the fundamental group of every connected component of\nthe anabelioid is intrinsically exhaustive (respectively, weakly intrinsically exhaus-\ntive).\nDefinition 2.3.4. Let X be a Q-anabelioid. Then we shall refer to a finite étale\n(necessarily Galois) covering Y → X obtained as the direct summand of the pull-\nback via the structure morphism X → Q of a finite étale Galois covering R → Q\nas Q-Galois.\nTHE GEOMETRY OF ANABELIOIDS 35\nProposition 2.3.5. (Basic Properties of Quasi-Cores and Intrinsic Ex-\nhaustivity)\n(i) Suppose that a slim anabelioid X admits a quasi-core X → Q. Then the\nnatural functor\nLocQ(X ) → Loc(X )\n(given by forgetting the Q-holomorphic structure) is an equivalence. Moreover, any\nrelatively slim composite X → Q′ of X → Q with a morphism Q → Q′ of slim\nanabelioids is also a quasi-core for X .\n(ii) If a slim anabelioid X admits a core X → Q, then X → Q is a faithful\nquasi-core for X .\n(iii) Suppose that X and Y are slim, connected anabelioids which are isoge-\nnous. Then X admits a quasi-core (respectively, admits a faithful quasi-core) if\nand only if Y does.\n(iv) Suppose that X and Y are slim, connected anabelioids which are isoge-\nnous. Then X is intrinsically exhaustive (respectively, weakly intrinsically exhaus-\ntive) if and only if Y is.\n(v) If X is intrinsically exhaustive, then it is weakly intrinsically exhaustive.\n(vi) Suppose that X is weakly intrinsically exhaustive. Then there is a\nunique map\ndegX : Ob(Loc(X )) → Q>0\nsuch that\ndegX (X ) = 1; deg(Y1\/Y2) = degX (Y1)\/degX (Y2)\nfor all morphisms Y1 → Y2 of Loc(X ). In particular, if Y → X is a finite étale\nmorphism of connected anabelioids of degree > 1, then Y is not isomorphic to X .\n(vii) Let X be a slim, connected, weakly intrinsically exhaustive anabelioid\nthat admits a quasi-core X → Q. Let\nφ : Y → X\nbe a connected Q-Galois covering. Then any finite étale (not necessarily Galois!)\nmorphism ψ : Y → X is abstractly equivalent (cf. §0) to φ.\n(viii) If X admits a faithful quasi-core, then X is intrinsically exhaustive.\nIn particular, if X admits a core, then X is intrinsically exhaustive.\nProof. Assertions (i), (ii), (iv), and (vi) are immediate from the definitions. As-\nsertion (iii) follows from the definitions and assertion (i). Next, we verify assertion\n(v). Let H ⊆ G be an open subgroup, and ι : H ↪→ G be an open embedding.\n36 SHINICHI MOCHIZUKI*\nSuppose that (for some large n) Gn (as in Definition 2.3.3) is contained in H, so\nι(Gn) ⊆ Gn. Then:\n∞ > [G : ι(H)] · [H : Gn] = [G : ι(H)] · [ι(H) : ι(Gn)]\n= [G : ι(Gn)] = [G : Gn] · [Gn : ι(Gn)]\n≥ [G : Gn] = [G : H] · [H : Gn]\nThus, [G : ι(H)] ≥ [G : H]. On the other hand, if we apply this inequality to\nι−1 : ι(H) ↪→ G, then we obtain the reverse inequality. This implies equality, as\ndesired.\nNext, we turn to assertion (vii). Suppose that X → Q is a quasi-core for X .\nWithout loss of generality, we may assume that X = B(G), Q = B(A), and that\nX → Q is induced by a continuous homomorphism G → A which factors:\nG \u0002 GA ⊆ A\nIf B ⊆ A is an open normal subgroup of A, and HA def= GA\n⋂\nB, H\ndef\n= G ×A B,\nthen for any open embedding ι : H ↪→ G, it follows from Definition 2.3.1, (i),\nthat the image of the composite of ι with the homomorphism G → A is equal to\na · HA · a−1 (for some element a ∈ A). Thus, since B is normal in A, we conclude\nthat a · HA · a−1 ⊆ GA\n⋂\nB = HA (for some a ∈ A). On the other hand, this\nimplies that ι factors through H, hence — by assertion (vi) — that ι(H) = H, as\ndesired.\nFinally, we turn to assertion (viii). Suppose that X → Q is a faithful quasi-\ncore for X . Without loss of generality, we may assume that X = B(G), Q = B(A),\nwhere G ⊆ A is a closed subgroup of a profinite group A. Let\n. . . ⊆ An+1 ⊆ An ⊆ . . . ⊆ A\n(where n ranges over the positive integers) be a descending sequence of open normal\nsubgroups of A (which exists since A is assumed to be countably (topologically)\ngenerated — cf. Definition 2.3.1) such that:⋂\nn\nAn = {1}\nLet Gn\ndef\n= G\n⋂\nAn. Then for any profinite subgroup K ⊆ Isog(G) commensurable\nto G, it follows from assertion (i) that K\n⋂\nG ⊆ G ⊆ A extends uniquely to\nan inclusion K ⊆ A. Now take nK to be sufficiently large that Gn = Kn def=\nK\n⋂\nAn (⊆ K), for all n ≥ nK . Then for any open subgroup H ⊆ Gn (where\nn ≥ nK) and any open embedding ι : H ↪→ K, it follows from Definition 2.3.1, (i),\nthat the composite of ι with the inclusion K ⊆ A is induced by conjugation by an\nelement a ∈ A. Thus, (since An is normal in A) we obtain the desired inclusion:\nι(H) = a · H · a−1 ⊆ K\n⋂\nAn = Kn = Gn\nTHE GEOMETRY OF ANABELIOIDS 37\n©\nRemark 2.3.5.1. Thus, in words (cf. Definition 2.3.3; Proposition 2.3.5, (vi)),\nweak intrinsic exhaustivity means, with respect to finite étale localization on B(G),\nthat:\nThe property of “being sufficiently local as to be finite étale over B(G) of\ndegree N” is intrinsic.\nOn the other hand, intrinsic exhaustivity means that:\nThe property of “being sufficiently local as to be finite étale over B(Gn)”\nis intrinsic.\nMoreover, we have implications (cf. Proposition 2.3.5, (v), (viii)):\nexistence of a faithful quasi-core =⇒ intrinsic exhaustivity\n=⇒ weak intrinsic exhaustivity\nHere, the second implication is strict (cf. Example 2.3.7, (ii), (iii), below), but it is\nnot clear to the author at the time of writing to what extent the first implication\nis strict (but cf. Theorem 3.1.3, (iii); Corollary 3.1.7).\nProposition 2.3.6. (Quasi-Cores and the Group of Isogenies) Let G be a\nslim profinite group.\n(i) Suppose that Isog(G) is profinite (i.e., “Isog(G) ⊆ Isog(G) is a profinite\nsubgroup” — cf. Definition 2.3.2). Then B(G) → B(Isog(G)) is a quasi-core.\n(ii) Suppose that G is intrinsically exhaustive; let {Gn} be as in Definition\n2.3.3, (ii). Then the natural inclusions . . . ⊆ Aut(Gn) ⊆ Aut(Gn+1) ⊆ . . . ⊆\nIsog(G) (where n ≥ nG) induce an isomorphism of abstract groups:\nlim−→\nn\nAut(Gn)\n∼→ Isog(G)\n(iii) Suppose that G is a closed subgroup of a slim profinite group A such\nthat the inclusion G ↪→ A is relatively slim. Then the following are equivalent:\n(a) B(G) → B(A) is a faithful quasi-core.\n(b) The natural inclusion CA(G) ↪→ Isog(G) is surjective.\n(c) The homomorphism of abstract groups G ↪→ A factors through G ↪→\nIsog(G).\n38 SHINICHI MOCHIZUKI*\nProof. These assertions are all formal consequences of the definitions. ©\nRemark 2.3.6.1. Relative to Proposition 2.3.6, (ii), we note that Aut(Gn) is also\nequal to the normalizer of Gn in Isog(G). When G (hence also the Gn) is topo-\nlogically finitely generated, then it follows that G admits an exhaustive descending\nsequence of characteristic open subgroups . . . ⊆ Hm ⊆ . . . ⊆ G, hence that Aut(G)\n(hence also the Gn) admits a natural structure of profinite group (by considering\nthe inverse limit of the images of Aut(G) in the various Aut(G\/Hm)). On the other\nhand, this profinite topology on Aut(Gn) does not, in general, coincide with the\ntopology induced by the topology of Isog(G) discussed in Proposition 2.1.4, (i) —\ncf. Remark 2.2.6.2. Moreover, (relative to Proposition 2.3.6, (ii)) the work of [TSH]\n— involving inductive limits of topological groups whose inductive limit topology\n(in the category of topology spaces) is not necessarily compatible with the group\nstructure of the inductive limit — shows that the topology of inductive limits of\ntopological groups can, in general, be a rather subtle issue.\nRemark 2.3.6.2. The observations given in Proposition 2.3.6, (i), (iii); Remark\n2.3.6.1 were related to the author by A. Tamagawa.\nExample 2.3.7. Non-Intrinsically Exhaustive Profinite Groups. Let p\nbe a prime number.\n(i) Take A\ndef\n= Zp\n×, B def= Zp. Let A act on B in the usual fashion. Take\nG\ndef\n= B \u0002 A. Note that G is slim. Then the open subgroup\nH\ndef\n= (p · B) \u0002 A ⊆ G\nis clearly isomorphic to G, hence violates Proposition 2.3.5, (vi). Thus, G fails to\nbe weakly intrinsically exhaustive.\n(ii) Let G\ndef\n= PGL2(Zp). Note that G is slim. For m a positive integer, write\nCm ⊆ G for the subgroup determined by the matrices congruent to the identity\nmatrix modulo pm. Then G fails to be intrinsically exhaustive. Indeed, if {Gn} is\nas in Definition 2.3.3, then there exist positive integers m ≥ n ≥ nG such that:\nCm ⊆ Gn ⊆ C1\nThus, for all open embeddings ι : Cm ↪→ G, we should have: ι(Cm) ⊆ C1. But this\ninclusion fails to hold if we take ι to be the embedding given by conjugation by\nthe matrix\n(\npm 0\n0 1\n)\n. On the other hand, (it is an easy exercise to show that) in\nthis case, the unimodularity of the action by conjugation of GL2(Qp) on M2(Qp)\nimplies that G is weakly intrinsically exhaustive.\n(iii) For n ≥ 2, let G def= F̂n, the free profinite group on n generators. Then G\nis slim (cf., e.g., [Naka], Corollary 1.3.4; [Mzk8], Lemma 1.3.1). Moreover, since,\nTHE GEOMETRY OF ANABELIOIDS 39\nfor any n,m ≥ 2, F̂n, F̂m admit isomorphic open subgroups, in order to prove\nthat G is not intrinsically exhaustive for all n, it suffices to prove that G fails to\nbe intrinsically exhaustive for some n (cf. Proposition 2.3.5, (iv)). On the other\nhand, there exists an n such that G is isomorphic to an open subgroup of the\nprofinite completion SL2(Z)\n∧ of SL2(Z). Thus, one may show that to assume the\nintrinsic exhaustivity of any open subgroup of such a G leads to a contradiction\nby conjugating by “Hecke operator-type matrices” — an operation which preserves\nthe quotient SL2(Z)\n∧ \u0002 SL2(Zp) — as in (ii), above. Note, however, that in this\ncase, the Nielsen-Schreier formula (cf., e.g., [FJ], Proposition 15.25) implies that\nG is weakly intrinsically exhaustive.\n(iv) The anabelioid Ét(A1Fp) (notation as in Example 1.1.3) associated to the\naffine line over Fp fails to be weakly intrinsically exhaustive. Indeed, the existence\nof the finite étale morphism A1Fp → A1Fp defined by\nT \f→ T p + T\n(where T is the standard coordinate on A1Fp) contradicts Proposition 2.3.5, (vi).\n(v) If K is a finite extension of Qp, then the associated anabelioid Ét(K) is\nweakly intrinsically exhaustive (cf., e.g., [Mzk5], Proposition 1.2), but fails to be\nintrinsically exhaustive, at least when p > 2. Indeed, to see that GK (the absolute\nGalois group of K) fails to be intrinsically exhaustive, let us first recall the following\ntheorem of [JR]:\nLet K1, K2 be finite extensions of Qp (where p > 2) which contain the\nroots of unity of order p. Then GK1\n∼→ GK2 if and only if [K1 : Qp] =\n[K2 : Qp] and K1\n⋂\n(Qp\nab) = K2\n⋂\n(Qp\nab) (where Qp\nab is the maximal\nabelian extension of Qp).\nNow suppose that {Gn} is a sequence of open normal subgroups of GK as in Def-\ninition 2.3.3, (ii). Without loss of generality (cf. Proposition 2.3.5, (iv)), we may\nassume that K contains the roots of unity of order p, and that [K : Qp] ≥ 3. Let\nL be the finite Galois extension of K corresponding to some Gn. Write M ⊆ L\nfor the maximal tamely ramified subextension of L over K. By taking n to be suf-\nficiently large, we may assume that the extension L of M is not cyclotomic, i.e.,\nthat L \u000e= L ⋂ (M · Qpab). Since [M : Qp] ≥ [K : Qp] ≥ 3, it thus follows from\nlocal class field theory (cf., e.g., [Serre3]) that the wild inertia subgroup of GabM has\nrank ≥ 3 over Zp, hence that there exists a wildly ramified abelian extension L′ of\nM such that:\n[L′ : M ] = [L : M ]; L′ \u000e= L; L′\n⋂\n(M · Qpab) = L\n⋂\n(M · Qpab)\nThus, (by the theorem of [JR] quoted above) we conclude that GL′\n∼→ GL = Gn\ndespite the fact that GL′ \u000e= GL. But this contradicts Definition 2.3.3, (ii).\nRemark 2.3.7.1. Examples (iv) and (v) were related to the author by A.\nTamagawa.\n40 SHINICHI MOCHIZUKI*\n§2.4. Canonical Construction of the Fundamental Group\nLet X be a slim, connected anabelioid. In this §, we would like to examine the\nextent to which the fundamental group of X may be constructed in a canonical\nfashion, independent of a choice of basepoint.\nWe begin by introducing some notation. Let us write\nLocbp(X )\nfor the category each of whose objects is an arrow U → Y, where Y is a connected\nobject of Loc(X ), and U → Y is a universal covering of Y (cf. Definition 1.2.7),\nand whose morphisms from an arrow U1 → Y1 to an arrow U2 → Y2 are pairs of\narrows αU : U1 ∼→ U2, αY : Y1 → Y2 such that the diagram\nU1 αU−→ U2⏐⏐\u0007 ⏐⏐\u0007\nY1 αY−→ Y2\ncommutes; αU is an isomorphism; and αY is finite étale. Thus, in particular, by\nmapping U → Y to Y, we obtain a functor\nΦX : Locbp(X ) → Loc(X )0\n— where the superscript “0” is to denote the full subcategory consisting of con-\nnected objects — which (by definition) is surjective on objects.\nOn the other hand, if we define\nSGp\nto be the category whose objects are pairs (G,H), where G is a group, and H is\na subgroup of G, and whose morphisms from (G1,H1) to (G2,H2) are homomor-\nphisms φ : G1 → G2 such that φ(H1) ⊆ H2, then we obtain a natural functor\nΨX : Locbp(X ) → SGp\nby mapping an arrow U → Y to the pair\n(Aut(U),AutY(U) ⊆ Aut(U))\nand a morphism from U1 → Y1 to U2 → Y2 to the isomorphism Aut(U1) ∼→ Aut(U2).\nThus:\nLocbp(X ) may be thought of as the “category of objects of Loc(X ) equipped\nwith a basepoint” and ΨX may be thought of as the standard construction\nof the fundamental group (in the presence of a basepoint).\nTHE GEOMETRY OF ANABELIOIDS 41\nWhen it is necessary to specify the universe V relative to which we are working\n— i.e., relative to which we take all of our (pro-)anabelioids (respectively, groups)\nto be V -small (respectively, V -sets) — we shall write LocVbp(X ), LocV (X ) (respec-\ntively, SGpV ). [Similarly, we shall write ΦVX , Ψ\nV\nX .] Thus, we observe, in particular,\nthat the categories LocVbp(X ), LocV (X ) are not V -small.\nProposition 2.4.1. (Dependence of the Fundamental Group on the\nChoice of Universal Covering) Let V be a universe [which is, therefore, in\nparticular, a “set” in some ambient model of set theory]. Let X be a V -small slim,\nconnected anabelioid such that the subgroup ΠX ⊆ Isog(X ) (cf. Proposition 2.1.4,\n(i)) is not normal. Then there exist distinct objects of LocVbp(X ) that map via\nΦVX to the same object of Loc\nV (X )0, but via ΨVX to distinct objects of SGpV . In\nparticular, the functor ΨVX does not factor through Φ\nV\nX .\nProof. Indeed, let π : X̃ → X be a universal covering; let α ∈ Aut(X̃ ) be an\nelement that does not normalize ΠX\ndef\n= AutX (X̃ ). Then π′ def= π ◦ α−1 is also a\nuniversal covering of X . Moreover, we have\nΨVX (π) = (Aut(X̃ ),ΠX ) \u000e= ΨVX (π′) = (Aut(X̃ ), α · ΠX · α−1)\nbut ΦVX (π) = Φ\nV\nX (π\n′) = X . ©\nRemark 2.4.1.1. Thus, the proof of Proposition 2.4.1 suggests, in particular,\nthat, in order to obtain a factorization of ΨVX through Φ\nV\nX — i.e., to obtain a\n“canonical construction” of the fundamental group that does not depend on the\nchoice of basepoint — it is necessary to modify ΨVX so that it takes values in\nsome sort of “quotient” in which subgroups of Aut(X̃ ) are identified with their\nconjugates. This motivates the following discussion.\nLet V , X be as in Proposition 2.4.1. Then let us denote by\nSGpVX\nthe category each of whose objects is an assignment A\nU \f→ AU\n— where U ranges over all V -small universal coverings of X [i.e., all domains of\narrows in Locbp(X )], and AU is a collection of subgroups of Aut(U) — such that\nfor every isomorphism U1 ∼→ U2, the induced isomorphism Aut(U1) ∼→ Aut(U2)\nmaps AU1 onto AU2 ; and whose morphisms Hom(A,A\n′) are defined as follows: The\ncardinality of Hom(A,A′) is always ≤ 1; we take the cardinality of Hom(A,A′) to\nbe 1 if and only if the following condition is satisfied: for every U , every H ∈ AU ,\nthere exists an H ′ ∈ A′U such that H ⊆ H ′. Note that this category SGpVX is not\nV -small.\n42 SHINICHI MOCHIZUKI*\nThus, we obtain a natural functor\nΞ̃VX : Loc\nV\nbp(X ) → SGpVX\nby mapping an arrow U → Y to the assignment that maps a universal covering V to\nthe conjugacy class of subgroups of Aut(V) determined by the subgroup AutY(U) ⊆\nAut(U) and an isomorphism Aut(U) ∼→ Aut(V) which is induced by an isomorphism\nU ∼→ V . [Note that this conjugacy class is independent of the choice of isomorphism\nU ∼→ V .] Moreover, it is evident from the definition of Ξ̃VX that:\nTheorem 2.4.2. (Canonical Fundamental Groups up to Isogeny) Let V ,\nX be as in Proposition 2.4.1. Then there exists a functor\nΞVX : Loc\nV (X )0 → SGpVX\nsuch that ΞVX = Ξ̃\nV\nX ◦ ΦVX .\nRemark 2.4.2.1. Thus, the functor of Theorem 2.4.2 yields a functorial [i.e., with\nrespect to finite étale coverings] construction of the fundamental group as a group\nof transformations of some geometric object [i.e., the universal covering], albeit up\nto a certain indeterminacy, given by the action of Isog(X ). On the other hand,\nthis functor has the drawback that it only constructs the fundamental group as an\n“abstract group”, i.e., not as a profinite group, as one might ideally wish.\nNow let us assume that X is a connected Q-anabelioid. For simplicity, we\nassume that Q is also connected. In the following discussion, we would like to show\nthat (certain quotients) of the fundamental group of X may be constructed in a\nvery canonical fashion complete with their profinite structure, under the assumption\nthat X → Q is a quasi-core for X .\nFirst, let us choose an explicit system of finite étale Galois coverings\n. . . → Qn+1 → Qn → . . . → Q\nof Q which, when regarded as a pro-anabelioid Q∞, forms a universal covering of\nQ. For each n, choose a coherent system of connected components\nXn ↪→ Qn|X\nof Qn|X (cf. the proof of Proposition 2.3.5, (viii)). This system thus defines a\npro-anabelioid X∞, together with a morphism X∞ → Q∞.\nNow observe that, since X → Q is a quasi-core, it follows that any automor-\nphism α : Xn ∼→ Xn necessarily lies over Q, hence that the natural morphism\nXn → Qn\nTHE GEOMETRY OF ANABELIOIDS 43\n[obtained by composing the inclusion Xn ↪→ Qn|X with the projection Qn|X → Qn]\nis preserved by composition on the the left with arbitrary automorphisms of Xn,\nup to the action of a (unique) element of Gal(Qn\/Q) def= AutQ(Qn). That is to\nsay, there is a unique element αQn ∈ Gal(Qn\/Q) for which the following diagram\ncommutes: Xn → Qn⏐⏐\u0007α ⏐⏐\u0007αQn\nXn → Qn\nMoreover, the uniqueness of this element implies that the assignment α \f→ αQn is a\nhomomorphism. Thus, in summary, we see that we obtain an outer homomorphism\nρAutn : Aut(Xn) → Gal(Qn\/Q)\nwhich is entirely determined (as an outer homomorphism) by the isomorphism class\nof Xn. In particular, restricting to Gal(Xn\/X ) ⊆ Aut(Xn), we obtain an outer\nhomomorphism\nρGaln : Gal(Xn\/X ) → Gal(Qn\/Q)\nwhich is entirely determined (as an outer homomorphism) by the abstract equiv-\nalence class of the morphism Xn → X , hence, in particular, by the isomorphism\nclass of X plus the covering Qn → Q (since X → Q is a quasi-core).\nSince the above construction is clearly “functorial in n”, by passing to the limit\nover n, we thus obtain an outer homomorphism\nρGal∞ : Gal(X∞\/X ) → ΠQ def= Gal(Q∞\/Q)\nwhose image is entirely determined (up to conjugacy) by the isomorphism class of\nX . Let us denote this image (well-defined up to conjugacy) by:\nΠX\/Q ⊆ ΠQ\nMoreover, since the above construction is determined entirely by the isomorphism\nclass of X , it follows (cf. Proposition 2.1.1, (iv)) that the assignment X \f→ ΠX\/Q\nis functorial with respect to finite étale coverings X1 → X2 of connected objects of\nLoc(X ) in the sense that such a covering induces an inclusion\nΠX1\/Q ⊆ ΠX2\/Q (⊆ ΠQ)\nwhich is well-defined up to conjugation by elements of ΠQ. (That is to say, one\nallows an indeterminacy with respect to distinguishing between, say, a given inclu-\nsion ΠX1\/Q ⊆ ΠX2\/Q and some other inclusion ΠX1\/Q ⊆ π · ΠX2\/Q · π−1, where\nπ ∈ ΠQ.)\nIf G is a Hausdorff topological group, then let us write\nSub(G)\n44 SHINICHI MOCHIZUKI*\nfor the category whose objects are conjugacy classes of closed subgroups H ⊆ G,\nand whose morphisms H → H ′ are inclusions of H into a (conjugate of) H ′. That\nis to say, the cardinality of the set of morphisms between two objects of Sub(G) is\neither 0 or 1.\nThen the above discussion may be summarized as follows:\nTheorem 2.4.3. (Canonically Constructed Fundamental Groups via\nQuasi-Cores) Let Q be a slim, connected anabelioid. Suppose that X is a connected\nQ-anabelioid for which X → Q is a quasi-core. Then there is a functor\nLoc(X )0 → Sub(ΠQ)\nY \f→ {ΠY\/Q ⊆ ΠQ}\nsuch that B(ΠY\/Q) is isomorphic to the image of Y in Q (cf. Definition 1.1.7). In\nparticular, if X → Q is a faithful quasi-core, then Y ∼= B(ΠY\/Q).\nRemark 2.4.3.1. Thus, Theorem 2.4.3 yields a canonical construction of\nthe fundamental group of a slim, connected X which admits a faithful quasi-core\nQ. Moreover, this construction has the virtue that it is compatible [cf. the above\ndiscussion!] with the profinite structure of the fundamental group of X . That is\nto say, more concretely:\nThe functor of Theorem 2.4.3 may be written as an inverse limit of a\ncompatible system of functors to the categories\nSub(ΠQ\/Hn)\nwhere . . . ⊆ Hn ⊆ . . . ⊆ ΠQ is an exhaustive descending sequence of open\nnormal subgroups of ΠQ.\nThis compatibility with the profinite structure is closely related to the the intrin-\nsicity of “knowing how local one is” (cf. Remark 2.3.5.1).\nOn the other hand, one drawback of the construction of Theorem 2.4.3 is that\nit depends on the arbitrary choice of a universal covering Q∞ → Q for Q as an\n“input datum”. This motivates the following definition:\nDefinition 2.4.4. Let X be a slim, connected anabelioid. Then we shall refer to\na closed subgroup Δ ⊆ ΠX , considered as a subgroup of Isog(X ), as an intrinsic\nprofinite subgroup if it is topologically finitely generated, normal in Isog(X ), and,\nmoreover, the continuous inclusion of Hausdorff topological groups Δ ↪→ Isog(X )\nis relatively slim.\nRemark 2.4.4.1. Note that since Δ is topologically finitely generated, it follows\n(cf. Remark 2.3.6.1) that Aut(Δ) has a natural structure of profinite group.\nTHE GEOMETRY OF ANABELIOIDS 45\nProposition 2.4.5. (The Faithful Quasi-Core Associated to an Intrinsic\nProfinite Subgroup) Let Δ ⊆ Isog(X ) be an intrinsic profinite subgroup.\nThen the action by conjugation of ΠX on Δ yields a morphism\nX ∼= B(ΠX ) → B(Aut(Δ))\nwhich is a faithful quasi-core for X .\nProof. Indeed, this is a formal consequence of Proposition 2.3.6, (iii), (a) ⇐⇒ (c).\n©\nRemark 2.4.5.1. Thus, when the quasi-core of Theorem 2.4.3 is obtained as in\nProposition 2.4.5, one can replace the functor of Theorem 2.4.3 by a functor in the\nstyle of Proposition 2.4.2: That is to say, instead of considering a conjugacy class\nof subgroups of a particular profinite group ΠQ, we observe that for any universal\ncovering U , we obtain a natural profinite subgroup\nΔU ⊆ Aut(U)\n(determined by conjugating Δ by some isomorphism U ∼→ X̃ of U to the universal\ncovering X̃ used to define Isog(X )) such that any isomorphism U1 ∼→ U2 maps ΔU1\nto ΔU2. In particular, we obtain an assignment\nU \f→ AU def= Aut(ΔU )\nwhich is functorial in isomorphisms U1 ∼→ U2. Then instead of obtaining a con-\njugacy class of subgroups [as in Theorem 2.4.3] in a particular ΠQ, we obtain a\nconjugacy class of subgroups of AU , for each U , which is compatible with all iso-\nmorphisms U1 ∼→ U2. [We leave the routine details to the reader.] At any rate, this\nyields a construction of the canonical fundamental groups of Theorem 2.4.3 which\nis independent of the choice of any universal covering of Q.\n46 SHINICHI MOCHIZUKI*\nSection 3: Anabelioids Arising from Hyperbolic Curves\n§3.1. Anabelioid-Theoretic Interpretation of Scheme-Theoretic Cores\nIn the following discussion, we wish to translate the scheme-theoretic theory\nof cores in the context of hyperbolic curves (cf. [Mzk9], §2) into the language\nof anabelioids (cf. the profinite group-theoretic approach to such a translation\ngiven in [Mzk9], §2). The main technical tool that will enable us to do this is the\n“Grothendieck Conjecture” — i.e., Theorem A of [Mzk6].\nFor i = 1, 2, let Fi be either Q or Qpi (for some prime number pi). Let Ki be a\nfinite extension of Fi. Let (Xi)Ki be a hyperbolic orbicurve over Ki. Assume that\nwe have chosen basepoints of the (Xi)Ki , which thus induce basepoints\/algebraic\nclosures Ki of the Ki and determine fundamental groups Π(Xi)Ki\ndef\n= π1((Xi)Ki) and\nGalois groups GKi\ndef\n= Gal(Ki\/Ki). Thus, for i = 1, 2, we have an exact sequence:\n1 → ΔXi → Π(Xi)Ki → GKi → 1\n(where ΔXi ⊆ Π(Xi)Ki is defined so as to make the sequence exact). Here, we\nshall think of GKi as a quotient of Π(Xi)Ki (i.e., not as an independent group to\nwhich Π(Xi)Ki happens to surject). One knows (cf. [Mzk8], Lemma 1.3.8) that this\nquotient Π(Xi)Ki → GKi is an intrinsic invariant of the profinite group Π(Xi)Ki .\nNext, we would like to introduce anabelioids into our discussion. Write:\nXi def= Ét((Xi)Ki); Si def= Ét(Ki)\nNote that Xi, Si are slim (cf. [Mzk8], Theorem 1.1.1, (ii); [Mzk8], Lemma 1.3.1),\nand that the structure morphisms Xi → Si are relatively slim (cf. [Mzk8], Theorem\n1.1.1, (ii)). Thus, we may think of Xi as an Si-anabelioid (cf. §2.2). In particular,\nwe may consider the categories\nLocSi(Xi); LocSi(Xi)\nof §2.2. In the following discussion, we shall work with anabelioids “at the coarsified\nlevel” [i.e., in Anab].\nCorollary 3.1.1. (Anabelioid-Theoretic Preservation of Arithmetic\nQuotients) Any finite étale morphism\nα : X1 → X2\ninduces a commutative diagram\nX1 α−→ X2⏐⏐\u0007 ⏐⏐\u0007\nS1 αS−→ S2\nTHE GEOMETRY OF ANABELIOIDS 47\n(where the horizontal morphisms are finite étale), hence pull-back\nLocS2 (X2) → LocS1 (X2 ×S2 S1); LocS2(X2) → LocS1(X2 ×S2 S1)\nand extension functors\nLocS1 (X1) ↪→ LocS1(X2 ×S2 S1); LocS1(X1) ∼→ LocS1(X2 ×S2 S1)\nwhich are equivalences whenever α is an isomorphism. Here, the extension functor\non “Loc(−)’s” (respectively, “Loc(−)’s”) is a full embedding (respectively, equiva-\nlence).\nProof. Indeed, this is a formal consequence of [Mzk8], Lemma 1.3.8 (and Propo-\nsition 2.2.2, (iv)). ©\nTheorem 3.1.2. (Anabelioid-Theoreticity of Correspondences) Let K\nbe a finite extension of Qp or Q; XK a hyperbolic orbicurve over K; write\nX def= Ét(XK), S def= Ét(K). Then the natural functor\nLocK(XK) −→ LocS(X )\nZ \f→ Ét(Z)\n(defined by applying “Ét(−)”) is an equivalence of categories. A similar asser-\ntion holds for “Loc(−)” replaced by “Loc(−)”. In particular, XK is (respectively,\nadmits) a K-core if and only if X is (respectively, admits) an S-core.\nProof. Since “Loc(−)” may be categorically reconstructed from “Loc(−)” via\nthe same recipe for both schemes and anabelioids, it suffices to prove the asserted\nequivalence in the case of “Loc(−)”.\nIn this case, it is immediate from the definitions that the functor in question is\nessentially surjective. It follows from the injectivity of [Mzk6], Theorem A (cf. also\nProposition 1.1.4) that this functor is faithful. Thus, it suffices to prove that this\nfunctor is full. Since “fullness” follows from Proposition 1.2.5, (ii), for morphisms\nover X , it suffices (by Proposition 2.2.2, (iv)) to prove that every S-isomorphism\nY ∼→ Z\n(where Y, Z are anabelioids representing objects of LocS(X )) arises from a mor-\nphism of schemes in LocK(XK). But this is a formal consequence of [Mzk6], The-\norem A (cf. also Proposition 1.1.4). ©\nTheorem 3.1.3. (Absolute Cores over Number Fields) Let K be a num-\nber field; XK a hyperbolic orbicurve over K; write X def= Ét(XK). Then:\n48 SHINICHI MOCHIZUKI*\n(i) XK is an [absolute] core if and only if XK is a K-core, and, moreover, K\nis a minimal field of definition for XK.\n(ii) Applying “Ét(−)” induces an equivalence of categories:\nLoc(XK)\n∼→ Loc(X )\nIn particular, X is (respectively, admits) an [absolute] core if and only if XK is\n(respectively, admits) an [absolute] core.\n(iii) Suppose that XK is non-proper. Then X admits a core if and only if\nit is intrinsically exhaustive.\nProof. Assertion (i) follows formally from [Mzk9], Definition 2.1 and [Mzk9],\nRemark 2.1.1. Assertion (ii) follows, in light of [Mzk8], Theorem 1.1.3, by the same\nargument as that used to prove Theorem 3.1.2. To prove assertion (iii), let us recall\nfrom the theory of [Mzk3] (cf. [Mzk9], Remark 2.1.2) that XK [or, equivalently, by\nassertion (ii), X ] fails to admit a core if and only if XK is isogenous to a Shimura\ncurve. Since XK is assumed to be non-proper, this Shimura curve may be taken to\nbe the moduli stack of hemi-elliptic curves (cf. [Take], p. 396, second paragraph).\nThus, if XK fails to admit a core, one may show that X fails to be intrinsically\nexhaustive by using Hecke correspondences on the moduli stack of hemi-elliptic\ncurves, as in Example 2.3.7, (ii), (iii) (cf. Proposition 2.3.5, (iv)). On the other\nhand, if X admits a core, then it follows from Proposition 2.3.5, (ii), (viii), that X\nis intrinsically exhaustive. ©\nRemark 3.1.3.1. One expects that the assumption that XK be non-proper in\nTheorem 3.1.3, (iii), is inessential. We made this assumption only to technically\nsimplify the proof that X fails to be intrinsically exhaustive (when it is assumed\nto fail to admit a core). The point of Theorem 3.1.3, (iii), was to give an example\nwhere the existence of a core is equivalent to intrinsic exhaustivity (cf. Remark\n2.3.5.1), since this contrasts with the situation that occurs in the p-adic case (cf.\nRemark 3.1.6.1, Corollary 3.1.7 below).\nCorollary 3.1.4. (Anabelioid-Theoreticity of Cores) Let\nα : X1 → X2\nbe a finite étale morphism. Then:\n(i) α induces — in a fashion functorial with respect to α — a pull-back functor\nLocK2((X2)K2 ) → LocK1((X1)K1)\nwhich is an equivalence whenever S1 → S2 is an isomorphism, and is equal to\nthe usual scheme-theoretic pull-back functor whenever α arises from a finite étale\nmorphism of schemes (X1)K1 → (X2)K2 .\nTHE GEOMETRY OF ANABELIOIDS 49\n(ii) (X1)K1 is K1-arithmetic if and only if (X2)K2 is K2-arithmetic. Simi-\nlarly, if X1 → X2 ×S2 S1 is an isomorphism, then (X1)K1 is a K1-core if and only\nif (X2)K2 is a K2-core.\n(iii) If a finite étale morphism (X2)K2 → (Z2)K2 to a K2-core (Z2)K2 maps\n(via the functor of (i)) to a finite étale morphism (X1)K1 → (Z1)K1 , then (Z1)K1\nis a K1-core, and, moreover, the morphism X1 → X2 ×S2 S1 extends uniquely\nto a commutative diagram:\nX1 → X2 ×S2 S1 → X2⏐⏐\u0007 ⏐⏐\u0007 ⏐⏐\u0007\nZ1 ∼→ Z2 ×S2 S1 → Z2\n(where Zi def= Ét((Zi)Ki), and the lower horizontal arrow on the left is an isomor-\nphism).\nProof. The functor of (i) is obtained by composing the pull-back functor on\n“Loc(−)’s” of Corollary 3.1.1 with an inverse to the extension functor on “Loc(−)’s”\nof Corollary 3.1.1 (which is an equivalence), and then applying the equivalences of\nTheorem 3.1.2 to the domain and codomain of this composite. Assertion (ii) is a\nformal consequence of assertion (i); [Mzk9], Definition 2.1; [Mzk9], Remark 2.1.1;\nand [Mzk9], Proposition 2.3, (i). To prove assertion (iii), we may assume, for sim-\nplicity, (cf. Proposition 2.1.1, (iv)) that S1 → S2 is an isomorphism. Then it follows\nthat the pull-back functor on “Loc(−)’s” of (i) is an equivalence:\nLocK2 ((X2)K2 )\n∼→ LocK1((X1)K1 )\nThus, the existence of an extension as in assertion (iii) follows formally by think-\ning of Xi, Zi as subcategories of LocKi((Xi)Ki) (cf. Proposition 1.2.5, (ii)). The\nuniqueness of such an extension is a formal consequence of the slimness of Zi. ©\nProposition 3.1.5. (Absolute Degrees) For i = 1, 2, set:\ndegarith(Xi) def= [Ki : Fi]\nand deggeo(Xi) equal to the Euler characteristic of (Xi)Ki . [That is to say,\nif (Xi)Ki is a hyperbolic curve of type (gi, ri), then we set deggeo(Xi) equal to\n2gi−2+ri; more generally, if (Xi)Ki is only an orbicurve, then we take its deggeo(−)\nto be the deggeo(−) of some degree d finite étale covering of (Xi)Ki which is a curve,\ndivided by d.] Then for any finite étale morphism α : X1 → X2 (which thus induces\na commutative diagram as in Corollary 3.1.1), we have:\ndeggeo(X1) = deggeo(X2)·(deg(α)\/deg(αS)); degarith(X1) = degarith(X2)·deg(αS)\nIn particular, Xi is weakly intrinsically exhaustive. We shall refer to deggeo(Xi)\n(respectively, degarith(Xi)) as the absolute geometric (respectively, absolute\narithmetic) degree of Xi.\n50 SHINICHI MOCHIZUKI*\nProof. Indeed, this follows from [Mzk8], Lemma 1.3.9, (for the absolute geometric\ndegree) and [Mzk8], Proposition 1.2.1, (i), (v) (for the absolute arithmetic degree).\n©\nRemark 3.1.5.1. Proposition 3.1.5 already suggests the possibility that, under\nthe further assumption that (Xi)Ki admits a Ki-core, Xi should admit a faithful\nquasi-core. In the remainder of the present §, we shall show that this is, in fact, the\ncase (at least when (Xi)Ki is non-proper) — cf. Theorem 3.1.6 below. In light of\nProposition 2.3.5, (ii); Theorem 3.1.3, (ii), this fact is primarily of interest in the\ncase where Ki is a p-adic local field (although we shall not assume this to be the\ncase in the following discussion).\nIn the following discussion, we would like to assume that:\n(a) The hyperbolic orbicurve (Xi)Ki admits a Ki-core (Zi)Ki (where i =\n1, 2).\n(b) The anabelioids X1, X2 are isogenous.\nChoose basepoints for (Zi)Ki , so that we obtain, for i = 1, 2, exact sequences:\n1 → ΔZi → Π(Zi)Ki → GKi → 1\nWrite Zi def= Ét((Zi)Ki). Then assumptions (a), (b); Corollary 3.1.4, (iii); and\n[Mzk8], Lemma 1.3.9, imply that (Z1)K1 , (Z2)K2 are hyperbolic orbicurves of the\nsame type (g, \u0006r). Let us choose once and for all a model\nΠ̂g,\tr\nof the geometric fundamental group of a hyperbolic orbicurve of type (g, \u0006r) (in char-\nacteristic 0). To simplify notation, in the following discussion, we shall simply write\nΠ̂ for Π̂g,\tr .\nThus, we have (noncanonical) isomorphisms Π̂ ∼= ΔZi . Such isomorphisms\ninduce an outer homomorphism Π(Zi)Ki → Aut(Π̂) which is independent (as an\nouter homomorphism) of the choice of such isomorphism and, moreover, fits into a\ncommutative diagram:\n1 −→ ΔXi −→ Π(Xi)Ki −→ GKi −→ 1⏐⏐\u0007 ⏐⏐\u0007 ⏐⏐\u0007\n1 −→ ΔZi −→ Π(Zi)Ki −→ GKi −→ 1⏐⏐\u0007 ⏐⏐\u0007 ⏐⏐\u0007\n1 −→ Π̂ −→ Aut(Π̂) −→ Out(Π̂) −→ 1\nTHE GEOMETRY OF ANABELIOIDS 51\nHere, we observe that the vertical arrows between the first and second lines are al-\nways injective. If, moreover, (Xi)Ki is non-proper, then the vertical arrows between\nthe second and third lines are also injective (by the theory of [Mtmo] — cf. [Mzk8],\nTheorem 1.3.6). If we then set\nZcom def= B(Aut(Π̂)) → Mcom def= B(Out(Π̂))\n— i.e., we wish to think of Zcom → Mcom as a sort of “universal combinatorial\nmodel” of Zi → Si — then we obtain a commutative diagram of connected slim\nanabelioids Xi −→ Zi −→ Zcom⏐⏐\u0007 ⏐⏐\u0007 ⏐⏐\u0007\nSi id−→ Si −→ Mcom\nin which the horizontal arrows are all relatively slim (cf. [Mzk8], Theorem 1.1.1, (ii);\n[Mzk8], Lemma 1.3.1; [Mzk6], Theorem A). Next, let us observe that the intrinsic\nnature of the anabelioid associated to a geometric core (cf. Corollary 3.1.4, (iii))\nimplies that the morphism Xi → Zcom is functorial with respect to arbitrary finite\nétale morphisms X1 → X2.\nFinally, let us observe that Aut(Π̂) (hence also Out(Π̂)) is countably (topo-\nlogically) generated. Indeed, to show this, it suffices to show the existence of a\ndescending sequence of open subgroups\n. . . ⊆ An+1 ⊆ An ⊆ . . . ⊆ Aut(Π̂)\nsuch that\n⋂\nn An = {1}. To this end, let us note that Π̂ admits a descending\nsequence of open characteristic subgroups\n. . . ⊆ Π̂[n + 1] ⊆ Π̂[n] ⊆ . . . ⊆ Aut(Π̂)\nsuch that\n⋂\nn Π̂[n] = {1}. Thus, if we set\nAn\ndef\n= Ker(Aut(Π̂) → Aut(Π̂\/Π̂[n]))\nwe obtain a sequence {An} with the desired properties.\nThus, in summary, we see that we have proven (most of) the following:\nTheorem 3.1.6. (The Quasi-Core Associated to a Geometric Core) Let\nK be a finite extension of Qp or Q; XK a hyperbolic orbicurve over K which\nadmits a K-core ZK of type (g, \u0006r). Write:\nX def= Ét(XK); Z def= Ét(ZK); Zcom def= B(Aut(Π̂g,\tr)); Mcom def= B(Out(Π̂g,\tr))\nThen:\n52 SHINICHI MOCHIZUKI*\n(i) ZK determines a Zcom-holomorphic structure X → Zcom on X which is a\nquasi-core for X . In particular, the theory of §2.3, 2.4 may be applied to X .\n(ii) If XK is non-proper, then this quasi-core is faithful and, moreover,\nobtained as the quasi-core associated to an intrinsic profinite subgroup (cf.\nProposition 2.4.5). Finally, if K is a number field which is a minimal field of\ndefinition for ZK , then the morphism Z → Zcom is commensurably terminal.\nProof. It remains only to observe that the final part of (ii) is a formal consequence\nof Theorem 3.1.3, (i), (ii); Proposition 2.2.5. ©\nRemark 3.1.6.1. In the case of p-adic local fields, one does not expect Z def=\nÉt(ZK) to be a core (even if K is a minimal extension of Qp over which ZK is\ndefined). Nevertheless, Theorem 3.1.6 shows that Z has the interesting property\nof being “closer to being a core” than, for instance, PGL2(Qp) (cf. Example 2.3.7,\n(ii), (iii); Theorem 3.1.3, (iii); Corollary 3.1.7 below).\nRemark 3.1.6.2. Our use of [Mzk8], Theorem 1.3.6 [i.e., the main result of\n[Mtmo]] in the above construction of a faithful quasi-core — which (by the the-\nory of §2.4) allows us to construct “canonical fundamental groups”, i.e., to assign\ncanonical names, or labels (up to conjugacy) to the elements of the funda-\nmental group — is reminiscent of the essential idea lying behind the theory of\nthe Grothendieck-Teichmüller group, which applies this same injectivity to assign\ncanonical names (up to conjugacy) to the elements of GQ. It is interesting to\nnote, however, that although this theory of the Grothendieck-Teichmüller group\nis typically applied to analyzing GQ, in fact, (by the “Neukirch-Uchida Theorem”\n— cf., e.g., [Mzk8], Theorem 1.1.3) the elements of GQ already possess intrinsic,\ncanonically determined names (up to conjugacy). Thus, the ability to assign\ncanonically determined names has much greater significance in the case of p-adic\nlocal fields.\nRemark 3.1.6.3. Relative to Remark 3.1.6.2, it is also interesting to note that,\njust as the theory of §2.4 only applies in the case where the curve in question\nadmits a geometric core, the theory of the Grothendieck-Teichmüller group centers\naround considering not just the projective line minus three points — a curve which\nfails to admit a geometric core — but instead a certain system of moduli stacks\nof hyperbolic curves, which includes, for instance, the moduli stack of hyperbolic\ncurves of type (0, 5) which (by [Mzk3], Theorem C) does admit a geometric core.\nFinally, we have the following analogue of Theorem 3.1.3, (iii), which is valid\nin the local p-adic case as well:\nCorollary 3.1.7. (Criteria for the Existence of a Geometric Core) Let\nK be a finite extension of Qp or Q; XK a non-proper hyperbolic orbicurve\nover K; write X def= Ét(XK). Then the following assertions are equivalent:\nTHE GEOMETRY OF ANABELIOIDS 53\n(i) XK admits a K-core.\n(ii) X admits a faithful quasi-core.\n(iii) X is intrinsically exhaustive.\nProof. This is a formal consequence of Theorem 3.1.6, (i), (ii); Proposition 2.3.5,\n(viii); and the existence of Hecke correspondences (cf. the proof of Theorem 3.1.3,\n(iii)) when XK does not admit a K-core. ©\nRemark 3.1.7.1. The implication (i) =⇒ (iii) of Corollary 3.1.7 (in the p-adic\ncase) is somewhat surprising in light of Example 2.3.7, (v). That is to say, Corollary\n3.1.7 implies that (when XK admits a K-core) the rigidity of Ét(XK) is sufficiently\nstrong to eliminate the non-intrinsic exhaustivity of Ét(K). In particular, we con-\nclude in this case that the natural inclusion\nCOut(ΔZ)(GK) ↪→ Isog(GK)\nfails to be surjective (cf. Propositions 2.3.5, (viii); 2.3.6, (iii)).\n§3.2. The Logarithmic Special Fiber via Quasi-Cores\nIn this §, we interpret the results of [Mzk8], §2, from the point of view of the\ntheory of quasi-cores — cf. §2.3, 2.4.\nLet XK be a hyperbolic curve over a finite extension K of Qp. Denote the ring\nof integers (respectively, residue field) of K by OK (respectively, k); also we shall\nuse notation such as “klog”, “(klog)∼”, as in [Mzk8], §2.\nAssume that XK admits a stable model over OK (cf. [Mzk8], §2), as well as\na K-core ZK , and that XK is Galois over ZK . Then we define the stable model of\nZK to be the quotient — in the sense of [log] stacks — of the stable model of XK\nby Gal(XK\/ZK). Let us denote the logarithmic special fibers of the stable models\nof XK , ZK by X\nlog\nk , Z\nlog\nk , respectively. Write:\nX def= Ét(XK); Z def= Ét(ZK)\nAlso, let us write Zcom for the quasi-core (for X , Z) of Theorem 3.1.6.\nNow recall from [Mzk8], the discussion following Remark 2.5.3, the “universal\nadmissible covering”\nX̃\nlog\nk → X logk\nof X\nlog\nk determined by the admissible quotient ΠXK \u0002 ΠadmXK . Put another way, this\ncovering is the composite of all admissible coverings (cf. [Mzk4], §3) of X logk ×klog\n54 SHINICHI MOCHIZUKI*\n(klog)∼. In the following discussion, let us denote the category of (disjoint unions\nof coverings isomorphic to) subcoverings of this universal admissible covering (re-\nspectively, subcoverings of the “geometric universal admissible covering” X̃\nlog\nk →\nX\nlog\nk ×klog (klog)∼) by:\nÉtadm(X\nlog\nk ) (respectively, Ét\nadm(X\nlog\nk ×klog (klog)∼))\nTo keep the notation simple, we set:\nX0 def= Étadm(X logk ); X 0 def= Étadm(X logk ×klog (klog)∼)\n[so the fundamental group of X0 (respectively, X 0) may be identified with ΠadmXK\n(respectively, the geometric portion ΔadmXK ⊆ ΠadmXK of ΠadmXK — cf. [Mzk8], §2)].\nSimilarly, we may construct\nZ0 def= Étadm(Z logk )\n[for instance, as the quotient of X0 by the faithful action on X0 of the finite group\nGal(XK\/ZK)].\nNext, let us write\nQ0\nfor the “anabelioid quotient” of X 0 by the natural action on X 0 by the profinite\ngroup\nAut(X\nlog\nk ×klog (klog)∼)\n[i.e., the group of automorphisms of the abstract log scheme which do not neces-\nsarily lie over klog or (klog)∼!]. That is to say, at the level of profinite groups,\nthe fundamental group of the anabelioid Q0 is the extension of the profinite group\nAut(X\nlog\nk ×klog (klog)∼) by the [slim! — cf. [Mzk8], Lemma 2.2, (i)] fundamental\ngroup of X 0 determined by the natural outer action of the former profinite group\non the latter. Note that by the definition of “Aut”, the slimness of X 0, and the\nslimness of Gal((klog)∼\/klog) [cf. [Mzk8], Proposition 1.2.3, (iii)], it follows that Q0\nis also slim.\nThus, we have a commutative diagram of natural relatively slim morphisms of\nslim, connected anabelioids\nX → Z⏐⏐\u0007 ⏐⏐\u0007\nX0 → Z0 → Q0\nin which the horizontal morphisms are all finite étale.\nTHE GEOMETRY OF ANABELIOIDS 55\nTheorem 3.2.1. (The Admissible Quotient as Quasi-Core) Assume that\nX → Z is Zcom-Galois. Then the morphism\nX → Q0\nis a quasi-core. In particular, the theory of §2.3, 2.4 may be applied to this\nmorphism.\nProof. By the functoriality of the anabelioid associated to a geometric core [(cf.\nCorollary 3.1.4, (iii)] and our hypothesis that X → Z is Zcom-Galois [cf. Proposi-\ntion 2.3.5, (vii)], it follows that it suffices to consider, for K ′ a finite extension of\nK, the behavior of automorphisms of the quotient\n(ΠX ⊇) ΠXK′ \u0002 ΠadmXK′ (⊆ ΠX0)\ninduced by arbitrary automorphisms of ΠXK′ . By [Mzk8], Theorem 2.7, it follows\nthat such automorphisms of ΠadmXK′\nnecessarily arise from automorphisms of the\nlogarithmic special fiber of XK′ . Thus, we conclude by the definition of Q0 and the\neasily verified fact that base-change to totally wildly ramified extensions K ′′ of K ′\ndoes not affect the automorphism group of the logarithmic special fiber. ©\nRemark 3.2.1.1. Note that the anabelian nature of the logarithmic special fiber\n(i.e., [Mzk8], Theorem 2.7) is applied in Theorem 3.2.1 in a fashion similar to the\nway in which the anabelian nature of hyperbolic curves over number fields is applied\nin Theorem 3.1.3, (ii).\n56 SHINICHI MOCHIZUKI*\nBibliography\nUnpublished RIMS preprints are available as .ps files at:\nhttp:\/\/www.kurims.kyoto-u.ac.jp\/∼kenkyubu\/paper\/all.html\n[DM] P. Deligne and D. Mumford, The Irreducibility of the Moduli Space of Curves\nof Given Genus, IHES Publ. Math. 36 (1969), pp. 75-109.\n[FC] G. Faltings and C.-L. Chai, Degenerations of Abelian Varieties, Springer-Verlag\n(1990).\n[FJ] M. Fried and M. Jarden, Field Arithmetic, Springer-Verlag (1986).\n[Groth] A. Grothendieck, Letter to G. Faltings (June 1983) in Lochak, L. Schneps, Geo-\nmetric Galois Actions; 1. Around Grothendieck’s Esquisse d’un Programme,\nLondon Math. Soc. Lect. Note Ser. 242, Cambridge Univ. Press (1997).\n[Ihara] Y. Ihara, On discrete subgroups of the two by two projective linear group over\np-adic fields, J. Math. Soc. Japan 18 (1966), pp. 219-235.\n[JR] M. Jarden and J. Ritter, On the characterization of local fields by their absolute\nGalois groups, J. Number Theory 11 (1979), pp. 1-13.\n[John1] P. T. Johnstone, P. T, Topos theory, London Mathematical Society Monographs\n10, Academic Press (1977).\n[John2] P. T. Johnstone, How general is a generalized space?, Aspects of topology,\nLondon Math. Soc. Lecture Note Ser. 93, Cambridge Univ. Press (1985), pp.\n77-111.\n[Knud] F. F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, II,\nMath. Scand. 52 (1983), 161-199.\n[McLn1] S. MacLane, One Universe as a Foundation for Category Theory, Reports of the\nMidwest Category Seminar III, Lecture Notes in Mathematics 106, Springer-\nVerlag (1969).\n[McLn2] S. MacLane, Categories for the working mathematician (Second edition), Grad-\nuate Texts in Mathematics 5, Springer-Verlag (1998).\n[McLr] C. McLarty, Elementary Categories, Elementary Toposes, Oxford Logic Guides\n21, Oxford University Press (1992).\n[Mtmo] M. Matsumoto, Galois representations on profinite braid groups on curves, J.\nReine Angew. Math. 474 (1996), pp. 169-219.\n[Mzk1] S. Mochizuki, The Hodge-Arakelov Theory of Elliptic Curves: Global Discretiza-\ntion of Local Hodge Theories, RIMS Preprint Nos. 1255, 1256 (October 1999).\n[Mzk2] S. Mochizuki, The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic\nCurve, RIMS Preprint No. 1287 (July 2000).\nTHE GEOMETRY OF ANABELIOIDS 57\n[Mzk3] S. Mochizuki, Correspondences on Hyperbolic Curves, Journ. Pure Appl. Al-\ngebra 131 (1998), pp. 227-244.\n[Mzk4] S. Mochizuki, The Geometry of the Compactification of the Hurwitz Scheme,\nPubl. of RIMS 31 (1995), pp. 355-441.\n[Mzk5] S. Mochizuki, A Version of the Grothendieck Conjecture for p-adic Local Fields,\nThe International Journal of Math. 8 (1997), pp. 499-506.\n[Mzk6] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math.\n138 (1999), pp. 319-423.\n[Mzk7] S. Mochizuki, Anabelian geometry in the Hodge-Arakelov theory of elliptic\ncurves, Communications in arithmetic fundamental groups (Kyoto, 1999\/2001),\nSūrikaisekikenkyūsho Kōkyūroku 1267 (2002), pp. 96-111.\n[Mzk8] S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois\nTheory and Modular Forms, Kluwer Academic Publishers (2003), pp. 77-122.\n[Mzk9] S. Mochizuki, The Absolute Anabelian Geometry of Canonical Curves, RIMS\nPreprint 1379, October 2002.\n[Naka] H. Nakamura, Galois rigidity of pure sphere braid groups and profinite calculus,\nJ. Math. Sci. Univ. Tokyo 1 (1994), pp. 71-136.\n[Serre1] J.-P. Serre, Trees, Springer-Verlag (1980).\n[Serre2] J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Mathematics 1500,\nSpringer-Verlag (1992).\n[Serre3] J.-P. Serre, Local Class Field Theory in Algebraic Number Theory, ed. J.W.S.\nCassels and A. Fröhlich, Academic Press (1967).\n[SGA1] Revêtement étales et groupe fondamental, Séminaire de Géometrie Algébrique\ndu Bois Marie 1960-1961 (SGA1), dirigé par A. Grothendieck, augmenté de\ndeux exposés de M. Raynaud, Lecture Notes in Mathematics 224, Springer-\nVerlag (1971).\n[Take] K. Takeuchi, Arithmetic Fuchsian Groups with Signature (1; e), Journ. Math.\nSoc. Japan 35 (1983), pp. 381-407.\n[Tama] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio\nMath. 109 (1997), pp. 135-194.\n[TSH] N. Tatsuuma, H. Shimomura, T. Hirai, On group topologies and unitary rep-\nresentations of inductive limits of topological groups and the case of the group\nof diffeomorphisms, J. Math. Kyoto Univ. 38 (1998), pp. 551-578.\n"}
{"index":{}}
{"id":"4Z0Tg116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Galois%20Sections%20(comments).pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":77,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Galois%20Sections%20(comments)","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["abssect-rzp-comments-2014-08-09.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Sat Aug 02 16:31:32 UTC 2014","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-7924704812718663522,"exact_signature_unique_b":true,"fuzzy_signature_l":6549230937687108129,"fuzzy_signature_unique_b":true,"text_t":"COMMENTS ON “GALOIS SECTIONS IN\nABSOLUTE ANABELIAN GEOMETRY”\nShinichi Mochizuki\nAugust 2014\n(1.) In the statement of Lemma 3.1, the second “XK” should read “XF ”.\n(2.) In the statement of Corollary 4.12, one should add the hypothesis that “K\ncontain a primitive 12-th root of unity”.\nTypeset by AMS-TEX\n1\n","wordcount_i":43,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:58 UTC 2022","fresh_date_dt":"Tue May 12 04:59:12 UTC 2026","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":24480,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086045955883008}
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{"id":"xLt4d116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Topics%20in%20Absolute%20Anabelian%20Geometry%20III%20(comments).pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":111,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Topics%20in%20Absolute%20Anabelian%20Geometry%20III%20(comments)","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["abs-anab-topics-3-comments-2019-06-01.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Sun May 26 08:00:14 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":8524889849037060735,"exact_signature_unique_b":true,"fuzzy_signature_l":6438227518639608035,"fuzzy_signature_unique_b":true,"text_t":"COMMENTS ON “TOPICS IN\nABSOLUTE ANABELIAN GEOMETRY III:\nGLOBAL RECONSTRUCTION ALGORITHMS”\nShinichi Mochizuki\nJune 2019\n(1.) In Remark 5.10.2, (iv), “On the other,” should read “On the other hand,”.\n(2.) In Theorem 1.11, (b), the text “totally ramified at precisely one closed point\nbut unramified elsewhere” should read “totally ramified at some closed point”.\n(3.) In the final sentences of Proposition 5.8, (iii), (vi), the phrase “despite the fact\none does not” should read “despite the fact that one does not”.\n(4.) In Remark 3.7.5, (ii), (2), “muliplication” should read “multiplication”.\n(5.) In the statement of Proposition 3.2, (iv), the phrase “if T = TM, or” should\nread as follows:\n“if T = TM, and (Π \u0002 MT), (Π\n∗\n\u0002 M∗\nT\n) are of hyperbolic orbicurve\ntype, or”\nIn the proof of Proposition 3.2, (iv), the phrase “[cf. also the first displayed iso-\nmorphism of Corollary 1.10, (b)]” should read as follows:\n“[cf. also the first displayed isomorphism of Corollary 1.10, (b); [Mzk20],\nTheorem 2.6, (v)]”\nIn the statement of Proposition 3.3, (ii), the phrase “if T = TCG. If T = TLG,”\nshould read as follows:\n“if T = TCG, and (Π \u0002 M), (Π∗ \u0002 M∗) are of hyperbolic orbicurve\ntype. If T = TLG, and (Π \u0002 M), (Π∗ \u0002 M∗) are of hyperbolic\norbicurve type,”\nThese changes have no effect either on the subsequent discussions and results of\nthe paper under consideration or on subsequent papers.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\n(6.) In the statement of Theorem 1.9, (b), the phrase “is as in (i)” should read “is\nas in (a)”.\n(7.) In the second sentence of the statement of Proposition 5.7, (i), the phrase\n“compact open subsets” should read “nonempty compact open subsets”.\nSimilarly, in the second sentence of the statement of Proposition 5.7, (i), (c), the\nphrase “an open subset” should read “a nonempty open subset”.\n","wordcount_i":278,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:58 UTC 2022","fresh_date_dt":"Fri Dec 15 09:14:50 UTC 2023","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":38692,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086045964271616}
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{"id":"Bih_t116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Combinatorial%20Anabelian%20Topics%20I.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":85,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Combinatorial%20Anabelian%20Topics%20I","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["comb-topics-1-2012-08-28.pdf"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Fri Nov 09 04:49:00 UTC 2012","dates_in_content_count_i":2,"dates_in_content_dts":["Thu Mar 31 00:00:00 UTC 2011","Wed Dec 28 00:00:00 UTC 2011"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":4467440739908461741,"exact_signature_unique_b":true,"fuzzy_signature_l":7387490896200874382,"fuzzy_signature_unique_b":true,"wordcount_i":45403,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:50:07 UTC 2022","fresh_date_dt":"Tue Mar 23 22:50:39 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.3 (Windows)","language_s":"en","size_i":928246,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086055498973184,"text_t":"Topics Surrounding the Combinatorial Anabelian\nGeometry of Hyperbolic Curves I: Inertia Groups\nand Profinite Dehn Twists\nYuichiro Hoshi and Shinichi Mochizuki\nReceived March 31, 2011.\nRevised December 28, 2011.\n2010 Mathematics Subject Classification. Primary 14H30; Secondary\n14H10.\nKey words and phrases. anabelian geometry, combinatorial anabelian geom-\netry, profinite Dehn twist, semi-graph of anabelioids, inertia group, hyperbolic\ncurve, configurationa space.\nThe first author was supported by Grant-in-Aid for Young Scientists (B),\nNo. 22740012, Japan Society for the Promotion of Science.\n2\nAbstract.\nLet Σ be a nonempty set of prime numbers. In the present pa-\nper, we continue our study of the pro-Σ fundamental groups of hyper-\nbolic curves and their associated configuration spaces over algebraically\nclosed fields of characteristic zero. Our first main result asserts, roughly\nspeaking, that if an F-admissible automorphism [i.e., an automorphism\nthat preserves the fiber subgroups that arise as kernels associated to the\nvarious natural projections of the configuration space under consider-\nation to configuration spaces of lower dimension] of a configuration\nspace group arises from an F-admissible automorphism of a configura-\ntion space group [arising from a configuration space] of strictly higher\ndimension, then it is necessarily FC-admissible, i.e., preserves the cus-\npidal inertia subgroups of the various subquotients corresponding to\nsurface groups. After discussing various abstract profinite combinato-\nrial technical tools involving semi-graphs of anabelioids of PSC-type\nthat are motivated by the well-known classical theory of topological\nsurfaces, we proceed to develop a theory of profinite Dehn twists, i.e.,\nan abstract profinite combinatorial analogue of classical Dehn twists\nassociated to cycles on topological surfaces. This theory of profinite\nDehn twists leads naturally to comparison results between the abstract\ncombinatorial machinery developed in the present paper and more clas-\nsical scheme-theoretic constructions. In particular, we obtain a purely\ncombinatorial description of the Galois action associated to a [scheme-\ntheoretic!] degenerating family of hyperbolic curves over a complete\nequicharacteristic discrete valuation ring of characteristic zero. Finally,\nwe apply the theory of profinite Dehn twists to prove a “geometric ver-\nsion of the Grothendieck Conjecture” for — i.e., put another way, we\ncompute the centralizer of the geometric monodromy associated to —\nthe tautological curve over the moduli stack of pointed smooth curves.\nContents\nIntroduction 3\n0. Notations and Conventions 11\n1. F-admissibility and FC-admissibility 14\n2. Various operations on semi-graphs of anabelioids of PSC-type 30\n3. Synchronization of cyclotomes 49\n4. Profinite Dehn multi-twists 74\n5. Comparison with scheme theory 92\n6. Centralizers of geometric monodromy 122\nCombinatorial anabelian topics I 3\n§ Introduction\nLet Σ ⊆ Primes be a nonempty subset of the set of prime numbers\nPrimes. In the present paper, we continue our study [cf. [SemiAn],\n[CmbGC], [CmbCsp], [MT], [NodNon]] of the anabelian geometry of\nsemi-graphs of anabelioids of [pro-Σ] PSC-type, i.e., semi-graphs of an-\nabelioids that arise from a pointed stable curve over an algebraically\nclosed field of characteristic zero. Roughly speaking, such a “semi-\ngraph of anabelioids” may be thought of as a slightly modified, Galois\ncategory-theoretic formulation of the “graph of profinite groups” asso-\nciated to such a pointed stable curve that takes into account the cusps\n[i.e., marked points] of the pointed stable curve, and in which the profi-\nnite groups that appear are regarded as being defined only up to inner\nautomorphism. At a more conceptual level, the notion of a semi-graph of\nanabelioids of PSC-type may be thought of as a sort of abstract profi-\nnite combinatorial analogue of the notion of a hyperbolic topolog-\nical surface of finite type, i.e., the underlying topological surface of\na hyperbolic Riemann surface of finite type. One central object of study\nin this context is the notion of an outer representation of IPSC-type [cf.\n[NodNon], Definition 2.4, (i)], which may be thought of as an abstract\nprofinite combinatorial analogue of the scheme-theoretic notion of a de-\ngenerating family of hyperbolic curves over a complete discrete valuation\nring. In [NodNon], we studied a purely combinatorial generalization of\nthis notion, namely, the notion of an outer representation of NN-type [cf.\n[NodNon], Definition 2.4, (iii)], which may be thought of as an abstract\nprofinite combinatorial analogue of the topological notion of a family\nof hyperbolic topological surfaces of finite type over a circle.\nHere, we recall that such families are a central object of study in the\ntheory of hyperbolic threefolds.\nAnother central object of study in the combinatorial anabelian ge-\nometry of hyperbolic curves [cf. [CmbCsp], [MT], [NodNon]] is the no-\ntion of a configuration space group [cf. [MT], Definition 2.3, (i)], i.e.,\nthe pro-Σ fundamental group of the configuration space associated to a\nhyperbolic curve over an algebraically closed field of characteristic zero,\nwhere Σ is either equal to Primes or of cardinality one. In [MT], it\nwas shown [cf. [MT], Corollary 6.3] that, if one excludes the case of\nhyperbolic curves of type (g, r) ∈ {(0, 3), (1, 1)}, then, up to a permu-\ntation of the factors of the configuration space under consideration, any\nautomorphism of a configuration space group is necessarily F-admissible\n[cf. [CmbCsp], Definition 1.1, (ii)], i.e., preserves the fiber subgroups\nthat arise as kernels associated to the various natural projections of the\n4 Yuichiro Hoshi and Shinichi Mochizuki\nconfiguration space under consideration to configuration spaces of lower\ndimension.\nIn §1, we prove our first main result [cf. Corollary 1.9], by means of\ntechniques that extend the techniques of [MT], §4, i.e., techniques that\ncenter around applying the fact that the first Chern class associated\nto the diagonal divisor in a product of two copies of a proper hyper-\nbolic curve consists, in essence, of the identity matrix [cf. Lemma 1.3,\n(iii)]. This result asserts, roughly speaking, that if an F-admissible auto-\nmorphism of a configuration space group arises from an F-admissible\nautomorphism of a configuration space group [arising from a configu-\nration space] of strictly higher dimension, then it is necessarily\nFC-admissible [cf. [CmbCsp], Definition 1.1, (ii)], i.e., preserves the\ncuspidal inertia subgroups of the various subquotients corresponding to\nsurface groups.\nTheorem A (F-admissibility and FC-admissibility). Let Σ be\na set of prime numbers which is either of cardinality one or equal\nto the set of all prime numbers; n a positive integer; (g, r) a pair\nof nonnegative integers such that 2g − 2 + r > 0; X a hyperbolic curve\nof type (g, r) over an algebraically closed field k of characteristic \u0004∈ Σ;\nXn the n-th configuration space of X; Πn the maximal pro-Σ quotient\nof the fundamental group of Xn; “Out\nFC(−)”, “OutF(−)” ⊆ “Out(−)”\nthe subgroups of FC- and F-admissible [cf. [CmbCsp], Definition 1.1,\n(ii)] outomorphisms [cf. the discussion entitled “Topological groups” in\n§0] of “(−)”. Then the following hold:\n(i) Let α ∈ OutF(Πn+1). Then α induces the same outomor-\nphism of Πn relative to the various quotients Πn+1 \u0002 Πn by\nfiber subgroups of length 1 [cf. [MT], Definition 2.3, (iii)]. In\nparticular, we obtain a natural homomorphism\nOutF(Πn+1) −→ OutF(Πn) .\n(ii) The image of the homomorphism\nOutF(Πn+1) −→ OutF(Πn)\nof (i) is contained in\nOutFC(Πn) ⊆ OutF(Πn) .\nFor the convenience of the reader, we remark that our treatment of\nTheorem A in §1 does not require any knowledge of the theory of semi-\ngraphs of anabelioids. On the other hand, in a sequel to the present pa-\nper, we intend to prove a substantial stengthening of Theorem A, whose\nCombinatorial anabelian topics I 5\nproof makes quite essential use of the theory of [CmbGC], [CmbCsp], and\n[NodNon] [i.e., in particular, of the theory of semi-graphs of anabelioids\nof PSC-type].\nIn §2 and §3, we develop various technical tools that will play a cru-\ncial role in the subsequent development of the theory of the present pa-\nper. In §2, we study various fundamental operations on semi-graphs\nof anabelioids of PSC-type. A more detailed description of these opera-\ntions may be found in the discussion at the beginning of §2, as well as in\nthe various illustrations referred to in this discussion. Roughly speaking,\nthese operations may be thought of as abstract profinite combinatorial\nanalogues of various well-known operations that occur in the theory of\n“surgery” on topological surfaces — i.e.,\n• restriction to a subsurface arising from a decomposition, such\nas a “pants decomposition”, of the surface or to a [suitably\npositioned] cycle;\n• partially compactifying the surface by adding “missing\npoints”;\n• cutting a surface along a [suitably positioned] cycle;\n• gluing together two surfaces along [suitably positioned] cy-\ncles.\nMost of §2 is devoted to the abstract combinatorial formulation of these\noperations, as well as to the verification of various basic properties in-\nvolving these operations.\nIn §3, we develop the local theory of the second cohomology group\nwith compact supports associated to various sub-semi-graphs and com-\nponents of a semi-graph of anabelioids of PSC-type. Roughly speaking,\nthis theory may be thought of as a sort of abstract profinite combina-\ntorial analogue of the local theory of orientations on a topological\nsurface S, i.e., the theory of the locally defined cohomology modules\n(U, x) \u0006→ H2(U,U \\ {x};Z) (∼= Z)\n— where U ⊆ S is an open subset, x ∈ U . In the abstract profinite\ncombinatorial context of the present paper, the various locally defined\nsecond cohomology groups with compact supports give rise to cyclo-\ntomes, i.e., copies of quotients of the once-Tate-twisted Galois module\nẐ(1). The main result that we obtain in §3 concerns various canoni-\ncal synchronizations of cyclotomes [cf. Corollary 3.9], i.e., canon-\nical isomorphisms between these cyclotomes associated to various local\n6 Yuichiro Hoshi and Shinichi Mochizuki\nportions of the given semi-graph of anabelioids of PSC-type which are\ncompatible with the various fundamental operations studied in §2.\nIn §4, we apply the technical tools developed in §2, §3 to define\nand study the notion of a profinite Dehn [multi-]twist [cf. Defini-\ntion 4.4; Theorem 4.8, (iv)]. This notion is, needless to say, a natural\nabstract profinite combinatorial analogue of the usual notion of a Dehn\ntwist in the theory of topological surfaces. On the other hand, it is de-\nfined, in keeping with the spirit of the present paper, in a fashion that\nis purely combinatorial, i.e., without resorting to the “crutch” of consid-\nering, for instance, profinite closures of Dehn twists associated to cycles\non topological surfaces. Our main results in §4 [cf. Theorem 4.8, (i),\n(iv); Proposition 4.10, (ii)] assert, roughly speaking, that profinite Dehn\ntwists satisfy a structure theory of the sort that one would expect from\nthe analogy with the topological case, and that this structure theory is\ncompatible, in a suitable sense, with the various fundamental operations\nstudied in §2.\nTheorem B (Properties of profinite Dehn multi-twists). Let\nΣ be a nonempty set of prime numbers and G a semi-graph of anabelioids\nof pro-Σ PSC-type. Write\nAut|grph|(G) ⊆ Aut(G)\nfor the group of automorphisms of G which induce the identity automor-\nphism on the underlying semi-graph of G and\nDehn(G) def= {α ∈ Aut|grph|(G) |αG|v = idG|v for any v ∈ Vert(G) }\n— where we write αG|v for the restriction of α to the semi-graph of\nanabelioids G|v of pro-Σ PSC-type determined by v ∈ Vert(G) [cf. Def-\ninitions 2.1, (iii); 2.14, (ii); Remark 2.5.1, (ii)]; we shall refer to an\nelement of Dehn(G) as a profinite Dehn multi-twist of G. Then the\nfollowing hold:\n(i) (Normality) Dehn(G) is normal in Aut(G).\n(ii) (Structure of the group of profinite Dehn multi-twists)\nWrite\nΛG\ndef\n= Hom\nẐΣ\n(H2c (G, ẐΣ), ẐΣ)\nfor the cyclotome associated to G [cf. Definitions 3.1, (ii),\n(iv); 3.8, (i)]. Then there exists a natural isomorphism\nDG : Dehn(G) ∼−→\n⊕\nNode(G)\nΛG\nCombinatorial anabelian topics I 7\nthat is functorial, in G, with respect to isomorphisms of semi-\ngraphs of anabelioids. In particular, Dehn(G) is a finitely\ngenerated free ẐΣ-module of rank Node(G)\u0002. We shall\nrefer to a nontrivial profinite Dehn multi-twist whose image\n∈ ⊕Node(G) ΛG lies in a direct summand [i.e., in a single “ΛG”]\nas a profinite Dehn twist.\n(iii) (Exact sequence relating profinite Dehn multi-twists\nand glueable outomorphisms) Write\nGlu(G) ⊆\n∏\nv∈Vert(G)\nAut|grph|(G|v)\nfor the [closed] subgroup of “glueable” collections of outomor-\nphisms of the direct product\n∏\nv∈Vert(G) Aut\n|grph|(G|v) consist-\ning of elements (αv)v∈Vert(G) such that χv(αv) = χw(αw) for\nany v, w ∈ Vert(G) — where we write G|v for the semi-graph of\nanabelioids of pro-Σ PSC-type determined by v ∈ Vert(G) [cf.\nDefinition 2.1, (iii)] and χv : Aut(G|v) → (ẐΣ)∗ for the pro-\nΣ cyclotomic character of v ∈ Vert(G) [cf. Definition 3.8,\n(ii)]. Then the natural homomorphism\nAut|grph|(G) −→ ∏v∈Vert(G) Aut|grph|(G|v)\nα \u0006→ (αG|v )v∈Vert(G)\nfactors through Glu(G) ⊆ ∏v∈Vert(G) Aut|grph|(G|v), and, more-\nover, the resulting homomorphism ρVertG : Aut\n|grph|(G) →\nGlu(G) [cf. (i)] fits into an exact sequence of profinite groups\n1 −→ Dehn(G) −→ Aut|grph|(G) ρ\nVert\nG−→ Glu(G) −→ 1 .\nThe approach of §2, §3, §4 is purely combinatorial in nature. On the\nother hand, in §5, we return briefly to the world of [log] schemes in or-\nder to compare the purely combinatorial constructions of §2, §3, §4\nto analogous constructions from scheme theory. The main techinical\nresult [cf. Theorem 5.7] of §5 asserts that the purely combinatorial\nsynchronizations of cyclotomes constructed in §3, §4 for the profinite\nDehn twists associated to the various nodes of the semi-graph of anabe-\nlioids of PSC-type under consideration coincide with certain natural\nscheme-theoretic synchronizations of cyclotomes. This technical\nresult is obtained, roughly speaking, by applying the various fundamen-\ntal operations of §2 so as to reduce to the case where the semi-graph of\n8 Yuichiro Hoshi and Shinichi Mochizuki\nanabelioids of PSC-type under consideration admits a symmetry that\npermutes the nodes [cf. Fig. 6]; the desired coincidence of synchro-\nnizations is then obtained by observing that both the combinatorial and\nthe scheme-theoretic synchronizations are compatible with this symme-\ntry. One way to understand this fundamental coincidence of synchro-\nnizations is as a sort of abstract combinatorial analogue of the cyclotomic\nsynchronization given in [GalSct], Theorem 4.3; [AbsHyp], Lemma 2.5,\n(ii) [cf. Remark 5.9.1, (i)]. Another way to understand this fundamental\ncoincidence of synchronizations is as a statement to the effect that\nthe Galois action associated to a [scheme-theoretic!]\ndegenerating family of hyperbolic curves over a com-\nplete equicharacteristic discrete valuation ring of\ncharacteristic zero — i.e., “an outer representation\nof IPSC-type” — admits a purely combinatorial\ndescription [cf. Corollary 5.9, (iii)].\nThat is to say, one central problem in the theory of outer Galois repre-\nsentations associated to hyperbolic curves over arithmetic fields is pre-\ncisely the problem of giving such a “purely combinatorial description”\nof the outer Galois representation. Indeed, this point of view plays a\ncentral role in the theory of the Grothendieck-Teichmüller group. Thus,\nalthough an explicit solution to this problem is well out of reach at\nthe present time in the case of number fields or mixed-characteristic lo-\ncal fields, the theory of §5 yields a solution to this problem in the case\nof complete equicharacteristic discrete valuation fields of characteristic\nzero. One consequence of this solution is the following criterion for an\nouter representation to be of IPSC-type [cf. Corollary 5.10].\nTheorem C (Combinatorial\/group-theoretic nature of\nscheme-theoreticity). Let (g, r) be a pair of nonnegative integers such\nthat 2g − 2 + r > 0; Σ a nonempty set of prime numbers; R a complete\ndiscrete valuation ring whose residue field k is separably closed of char-\nacteristic \u0004∈ Σ; Slog the log scheme obtained by equipping S def= SpecR\nwith the log structure determined by the maximal ideal of R; (Mg,r)S\nthe moduli stack of r-pointed stable curves of genus g over S whose\nr marked points are equipped with an ordering; (Mg,r)S ⊆ (Mg,r)S\nthe open substack of (Mg,r)S parametrizing smooth curves; (M\nlog\ng,r)S the\nlog stack obtained by equipping (Mg,r)S with the log structure associ-\nated to the divisor with normal crossings (Mg,r)S \\ (Mg,r)S ⊆ (Mg,r)S;\nx ∈ (Mg,r)S(k) a k-valued point of (Mg,r)S; Ô the completion of the\nlocal ring of (Mg,r)S at the image of x; T log the log scheme obtained by\nCombinatorial anabelian topics I 9\nequipping T\ndef\n= Spec Ô with the log structure induced by the log struc-\nture of (Mlogg,r)S; tlog the log scheme obtained by equipping the closed\npoint of T with the log structure induced by the log structure of T log;\nX\nlog\nt the stable log curve over t\nlog corresponding to the natural strict (1-\n)morphism tlog → (Mlogg,r)S; IT log the maximal pro-Σ quotient of the log\nfundamental group π1(T\nlog) of T log; ISlog the maximal pro-Σ quotient\nof the log fundamental group π1(S\nlog) of Slog; G\nXlog\nthe semi-graph of\nanabelioids of pro-Σ PSC-type determined by the stable log curve X\nlog\nt\n[cf. [CmbGC], Example 2.5]; ρuniv\nX\nlog\nt\n: IT log → Aut(GXlog) the natural\nouter representation associated to X\nlog\nt [cf. Definition 5.5]; I a profinite\ngroup; ρ : I → Aut(G\nXlog\n) an outer representation of pro-Σ PSC-type\n[cf. [NodNon], Definition 2.1, (i)]. Then the following conditions are\nequivalent:\n(i) ρ is of IPSC-type [cf. [NodNon], Definition 2.4, (i)].\n(ii) There exist a morphism of log schemes φlog : Slog → T log over\nS and an isomorphism of outer representations of pro-Σ\nPSC-type ρ\n∼→ ρuniv\nX\nlog\nt\n◦ Iφlog [cf. [NodNon], Definition 2.1, (i)]\n— where we write Iφlog : ISlog → IT log for the homomorphism\ninduced by φlog — i.e., there exist an automorphism β of\nG\nXlog\nand an isomorphism α : I\n∼→ I logS such that the diagram\nI\nρ−−−−→ Aut(G\nXlog\n)\nα\n⏐⏐\b\u0005 ⏐⏐\b\u0005\nISlog\nρ\nX\nlog\nt\n◦I\nφlog\n−−−−−−−→ Aut(G\nXlog\n)\n— where the right-hand vertical arrow is the automorphism of\nAut(G\nXlog\n) induced by β — commutes.\n(iii) There exist a morphism of log schemes φlog : Slog → T log over\nS and an isomorphism α : I\n∼→ I logS such that ρ = ρunivXlogt ◦Iφlog◦\nα — where we write Iφlog : ISlog → IT log for the homomorphism\ninduced by φlog — i.e., the automorphism “β” of (ii) may be\ntaken to be the identity.\nBefore proceeding, in this context we observe that one fundamen-\ntal intrinsic difference between outer representations of IPSC-type and\nmore general outer representations of NN-type is that, unlike the case\nwith outer representations of IPSC-type, the period matrices associated\n10 Yuichiro Hoshi and Shinichi Mochizuki\nto outer representations of NN-type may, in general, fail to be nonde-\ngenerate — cf. the discussion of Remark 5.9.2.\nHere, we remark in passing that in a sequel to the present paper,\nthe theory of §5 will play an important role in the proofs of certain\napplications to the theory of tempered fundamental groups developed in\n[André].\nFinally, in §6, we apply the theory of profinite Dehn twists developed\nin §4 to prove a “geometric version of the Grothendieck Conjec-\nture” for — i.e., put another way, we compute the centralizer of the\ngeometric monodromy associated to — the tautological curve over the\nmoduli stack of pointed smooth curves [cf. Theorems 6.13; 6.14].\nTheorem D (Centralizers of geometric monodromy groups\narising from moduli stacks of pointed curves). Let (g, r) be a\npair of nonnegative integers such that 2g − 2 + r > 0; Σ a nonempty\nset of prime numbers; k an algebraically closed field of characteristic\nzero. Write (Mg,r)k for the moduli stack of r-pointed smooth curves\nof genus g over k whose r marked points are equipped with an order-\ning; Cg,r → Mg,r for the tautological curve over Mg,r [cf. the dis-\ncussion entitled “Curves” in §0]; ΠMg,r\ndef\n= π1((Mg,r)k) for the étale\nfundamental group of the moduli stack (Mg,r)k; Πg,r for the maximal\npro-Σ quotient of the kernel Ng,r of the natural surjection π1((Cg,r)k) \u0002\nπ1((Mg,r)k) = ΠMg,r ; ΠCg,r for the quotient of the étale fundamen-\ntal group π1((Cg,r)k) of (Cg,r)k by the kernel of the natural surjection\nNg,r \u0002 Πg,r; OutC(Πg,r) for the group of outomorphisms [cf. the dis-\ncussion entitled “Topological groups” in §0] of Πg,r which induce bijec-\ntions on the set of cuspidal inertia subgroups of Πg,r. Thus, we have a\nnatural exact sequence of profinite groups\n1 −→ Πg,r −→ ΠCg,r −→ ΠMg,r −→ 1 ,\nwhich determines an outer representation\nρg,r : ΠMg,r −→ Out(Πg,r) .\nThen the following hold:\n(i) Let H ⊆ ΠMg,r be an open subgroup of ΠMg,r . Suppose that\none of the following two conditions is satisfied:\n(a) 2g − 2 + r > 1, i.e., (g, r) \u0004∈ {(0, 3), (1, 1)}.\n(b) (g, r) = (1, 1), 2 ∈ Σ, and H = ΠMg,r .\nCombinatorial anabelian topics I 11\nThen the composite of natural homomorphisms\nAut(Mg,r)k((Cg,r)k) −→ AutΠMg,r (ΠCg,r )\/Inn(Πg,r)\n∼−→ ZOut(Πg,r)(Im(ρg,r)) ⊆ ZOut(Πg,r)(ρg,r(H))\n[cf. the discussion entitled “Topological groups” in §0] deter-\nmines an isomorphism\nAut(Mg,r)k((Cg,r)k)\n∼−→ ZOutC(Πg,r)(ρg,r(H)) .\nHere, we recall that Aut(Mg,r)k((Cg,r)k) is isomorphic to⎧⎨\n⎩\nZ\/2Z× Z\/2Z if (g, r) = (0, 4);\nZ\/2Z if (g, r) ∈ {(1, 1), (1, 2), (2, 0)};\n{1} if (g, r) \u0004∈ {(0, 4), (1, 1), (1, 2), (2, 0)} .\n(ii) Let H ⊆ OutC(Πg,r) be a closed subgroup of OutC(Πg,r) that\ncontains an open subgroup of Im(ρg,r) ⊆ Out(Πg,r). Suppose\nthat\n2g − 2 + r > 1, i.e., (g, r) \u0004∈ {(0, 3), (1, 1)}.\nThen H is almost slim [cf. the discussion entitled “Topolog-\nical groups” in §0]. If, moreover,\n2g − 2 + r > 2, i.e., (g, r) \u0004∈ {(0, 3), (0, 4), (1, 1), (1, 2), (2, 0)},\nthen H is slim [cf. the discussion entitled “Topological groups”\nin §0].\n§0. Notations and Conventions\nSets: If S is a set, then we shall denote by 2S the power set of S and\nby S\u0005 the cardinality of S.\nNumbers: The notation Primes will be used to denote the set of all\nprime numbers. The notation N will be used to denote the set or [ad-\nditive] monoid of nonnegative rational integers. The notation Z will be\nused to denote the set, group, or ring of rational integers. The notation\nQ will be used to denote the set, group, or field of rational numbers.\nThe notation Ẑ will be used to denote the profinite completion of Z.\nIf p ∈ Primes, then the notation Zp (respectively, Qp) will be used to\ndenote the p-adic completion of Z (respectively, Q). If Σ ⊆ Primes, then\nthe notation ẐΣ will be used to denote the pro-Σ completion of Z.\n12 Yuichiro Hoshi and Shinichi Mochizuki\nMonoids: We shall write Mgp for the groupification of a monoid M .\nTopological groups: Let G be a topological group and P a property\nof topological groups [e.g., “abelian” or “pro-Σ” for some Σ ⊆ Primes].\nThen we shall say that G is almost P if there exists an open subgroup\nof G that is P.\nLet G be a topological group and H ⊆ G a closed subgroup of\nG. Then we shall denote by ZG(H) (respectively, NG(H); CG(H)) the\ncentralizer (respectively, normalizer; commensurator) of H in G, i.e.,\nZG(H)\ndef\n= { g ∈ G | ghg−1 = h for any h ∈ H } ,\n(respectively, NG(H)\ndef\n= { g ∈ G | g ·H · g−1 = H } ;\nCG(H)\ndef\n= { g ∈ G | H ∩ g·H·g−1 is of finite index inH and g·H·g−1 } );\nwe shall refer to Z(G)\ndef\n= ZG(G) as the center of G. It is immediate\nfrom the definitions that\nZG(H) ⊆ NG(H) ⊆ CG(H) ; H ⊆ NG(H) .\nWe shall say that the closed subgroup H is centrally terminal (respec-\ntively, normally terminal; commensurably terminal) in G if H = ZG(H)\n(respectively, H = NG(H); H = CG(H)). We shall say that G is slim if\nZG(U) = {1} for any open subgroup U of G.\nLet G be a topological group. Then we shall write Gab for the\nabelianization of G, i.e., the quotient of G by the closure of the commu-\ntator subgroup of G.\nLet G be a topological group. Then we shall write Aut(G) for the\ngroup of [continuous] automorphisms of G, Inn(G) ⊆ Aut(G) for the\ngroup of inner automorphisms of G, and Out(G)\ndef\n= Aut(G)\/Inn(G).\nWe shall refer to an element of Out(G) as an outomorphism of G. Now\nsuppose that G is center-free [i.e., Z(G) = {1}]. Then we have an exact\nsequence of groups\n1 −→ G ( ∼→ Inn(G)) −→ Aut(G) −→ Out(G) −→ 1 .\nIf J is a group and ρ : J → Out(G) is a homomorphism, then we shall\ndenote by\nG\nout\n\u0002 J\nthe group obtained by pulling back the above exact sequence of profinite\ngroups via ρ. Thus, we have a natural exact sequence of groups\n1 −→ G −→ G\nout\n\u0002 J −→ J −→ 1 .\nCombinatorial anabelian topics I 13\nSuppose further that G is profinite and topologically finitely generated.\nThen one verifies easily that the topology of G admits a basis of char-\nacteristic open subgroups, which thus induces a profinite topology on the\ngroups Aut(G) and Out(G) with respect to which the above exact se-\nquence relating Aut(G) and Out(G) determines an exact sequence of\nprofinite groups. In particular, one verifies easily that if, moreover, J\nis profinite and ρ : J → Out(G) is continuous, then the above exact\nsequence involving G\nout\n\u0002 J determines an exact sequence of profinite\ngroups.\nLet G, J be profinite groups. Suppose that G is center-free and\ntopologically finitely generated. Let ρ : J → Out(G) be a continuous\nhomomorphism. Write AutJ(G\nout\n\u0002 J) for the group of [continuous]\nautomorphisms of G\nout\n\u0002 J that preserve and induce the identity auto-\nmorphism on the quotient J . Then one verifies easily that the operation\nof restricting to G determines an isomorphism of profinite groups\nAutJ(G\nout\n\u0002 J)\/Inn(G)\n∼−→ ZOut(G)(Im(ρ)) .\nLet G and H be topological groups. Then we shall refer to a homo-\nmorphism of topological groups φ : G → H as a split injection (respec-\ntively, split surjection) if there exists a homomorphism of topological\ngroups ψ : H → G such that ψ ◦ φ (respectively, φ ◦ ψ) is the identity\nautomorphism of G (respectively, H).\nLog schemes: When a scheme appears in a diagram of log schemes,\nthe scheme is to be understood as the log scheme obtained by equipping\nthe scheme with the trivial log structure. If X log is a log scheme, then\nwe shall refer to the largest open subscheme of the underlying scheme of\nX log over which the log structure is trivial as the interior of X log. Fiber\nproducts of fs log schemes are to be understood as fiber products taken\nin the category of fs log schemes.\nCurves: We shall use the terms “hyperbolic curve”, “cusp”, “stable log\ncurve”, “smooth log curve”, and “tripod” as they are defined in [CmbGC],\n§0; [Hsh], §0. If (g, r) is a pair of nonnegative integers such that 2g −\n2 + r > 0, then we shall denote by Mg,r the moduli stack of r-pointed\nstable curves of genus g over Z whose r marked points are equipped with\nan ordering, by Mg,r ⊆ Mg,r the open substack of Mg,r parametrizing\nsmooth curves, by Mlogg,r the log stack obtained by equipping Mg,r with\nthe log structure associated to the divisor with normal crossings Mg,r \\\n14 Yuichiro Hoshi and Shinichi Mochizuki\nMg,r ⊆ Mg,r, by Cg,r → Mg,r the tautological curve over Mg,r, and\nby Dg,r ⊆ Cg,r the corresponding tautological divisor of marked points\nof Cg,r → Mg,r. Then the divisor given by the union of Dg,r with the\ninverse image in Cg,r of the divisor Mg,r \\ Mg,r ⊆ Mg,r determines\na log structure on Cg,r; denote the resulting log stack by C\nlog\ng,r. Thus,\nwe obtain a (1-)morphism of log stacks Clogg,r → M\nlog\ng,r. We shall denote\nby Cg,r ⊆ Cg,r the interior of C\nlog\ng,r. Thus, we obtain a (1-)morphism of\nstacks Cg,r → Mg,r. Let S be a scheme. Then we shall write (Mg,r)S def=\nMg,r ×SpecZ S, (Mg,r)S def= Mg,r ×SpecZ S, (M\nlog\ng,r)S\ndef\n= Mlogg,r ×SpecZ\nS, (Cg,r)S def= Cg,r ×SpecZ S, (Cg,r)S def= Cg,r ×SpecZ S, and (C\nlog\ng,r)S\ndef\n=\nClogg,r ×SpecZ S.\nLet n be a positive integer and X log a stable log curve of type (g, r)\nover a log scheme Slog. Then we shall refer to the log scheme obtained\nby pulling back the (1-)morphism Mlogg,r+n → M\nlog\ng,r given by forgetting\nthe last n points via the classifying (1-)morphism Slog → Mlogg,r of X log\nas the n-th log configuration space of X log.\n§1. F-admissibility and FC-admissibility\nIn the present §, we consider the FC-admissibility [cf. [CmbCsp],\nDefinition 1.1, (ii)] of F-admissible automorphisms [cf. [CmbCsp], Def-\ninition 1.1, (ii)] of configuration space groups [cf. [MT], Definition 2.3,\n(i)]. Roughly speaking, we prove that if an F-admissible automorphism\nof a configuration space group arises from an F-admissible automor-\nphism of a configuration space group [arising from a configuration space]\nof strictly higher dimension, then it is necessarily FC-admissible, i.e.,\npreserves the cuspidal inertia subgroups of the various subquotients cor-\nresponding to surface groups [cf. Theorem 1.8, Corollary 1.9 below].\nLemma 1.1 (Representations arising from certain families\nof hyperbolic curves). Let (g, r) be a pair of nonnegative integers such\nthat 2g − 2 + r > 0; l a prime number; k an algebraically closed field\nof characteristic \u0004= l; B and C hyperbolic curves over k of type (g, r);\nn a positive integer. Suppose that (r, n) \u0004= (0, 1). For i = 1, · · · , n, let\nfi : B\n∼→ C be an isomorphism over k; si the section of B ×k C\npr1→\nB determined by the isomorphism fi. Suppose that, for any i \u0004= j,\nCombinatorial anabelian topics I 15\nIm(si) ∩ Im(sj) = ∅. Write\nZ\ndef\n= B ×k C \\\n⋃\ni=1,··· ,n\nIm(si) ⊆ B ×k C\nfor the complement of the images of the si’s, where i ranges over the\nintegers such that 1 ≤ i ≤ n; pr for the composite Z ↪→ B ×k C\npr1→ B\n[thus, pr: Z → B is a family of hyperbolic curves of type (g, r +\nn)]; ΠB (respectively, ΠC ; ΠZ) the maximal pro-l quotient of the étale\nfundamental group π1(B) (respectively, π1(C); π1(Z)) of B (respectively,\nC; Z); pr: ΠZ \u0002 ΠB for the surjection induced by pr; ΠZ\/B for the\nkernel of pr; ρZ\/B : ΠB → Out(ΠZ\/B) for the outer representation of\nΠB on ΠZ\/B determined by the exact sequence\n1 −→ ΠZ\/B −→ ΠZ\npr−→ ΠB −→ 1 .\nLet b be a geometric point of B and Zb the geometric fiber of pr: Z →\nB at b. For i = 1, · · · , n, fix an inertia subgroup [among its various\nconjugates] of the étale fundamental group π1(Zb) of Zb associated to\nthe cusp of Zb determined by the section si and denote by\nIsi ⊆ ΠZ\/B\nthe image in ΠZ\/B of this inertia subgroup of π1(Zb). Then the following\nhold:\n(i) (Fundamental groups of fibers) The quotient ΠZ\/B of the\nétale fundamental group π1(Zb) of the geometric fiber Zb co-\nincides with the maximal pro-l quotient of π1(Zb).\n(ii) (Abelianizations of the fundamental groups of fibers)\nFor i = 1, · · · , n, write Jsi ⊆ ΠabZ\/B for the image of Isi ⊆ ΠZ\/B\nin Πab\nZ\/B\n. Then the composite Isi ↪→ ΠZ\/B \u0002 ΠabZ\/B determines\nan isomorphism Isi\n∼→ Jsi ; moreover, the inclusions Jsi ↪→\nΠab\nZ\/B\ndetermine an exact sequence\n1 −→ (\nn⊕\ni=1\nJsi)\/Jr −→ ΠabZ\/B −→ ΠabC −→ 1\n— where\nJr ⊆\nn⊕\ni=1\nJsi\n16 Yuichiro Hoshi and Shinichi Mochizuki\nis a Zl-submodule such that\nJr \f\n{\nZl if r = 0,\n0 if r \u0004= 0,\nand, moreover, if r = 0 and i = 1, · · · , n, then the composite\nJr ↪→\nn⊕\ni=1\nJsi\nprsi\u0002 Jsi\nis an isomorphism.\n(iii) (Unipotency of a certain natural representation) The\naction of ΠB on Π\nab\nZ\/B\ndetermined by ρZ\/B preserves the exact\nsequence\n1 −→ (\nn⊕\ni=1\nJsi)\/Jr −→ ΠabZ\/B −→ ΠabC −→ 1\n[cf. (ii)] and induces the identity automorphisms on the\nsubquotients (\n⊕n\ni=1 Jsi)\/Jr and Π\nab\nC ; in particular, the nat-\nural homomorphism ΠB → AutZl(ΠabZ\/B) factors through a\nuniquely determined homomorphism\nΠB −→ HomZl\n(\nΠabC , (\nn⊕\ni=1\nJsi)\/Jr\n)\n.\nProof. Assertion (i) follows immediately from the [easily verified]\nfact that the natural action of π1(B) on π1(Zb)\nab ⊗\nẐ\nZl is unipotent\n— cf., e.g., [Hsh], Proposition 1.4, (i), for more details. [Note that\nalthough [Hsh], Proposition 1.4, (i), is only stated in the case where\nthe hyperbolic curves corresponding to B and C are proper, the same\nproof may be applied to the case where these hyperbolic curves are\naffine.] Assertion (ii) follows immediately, in light of our assumption\nthat (r, n) \u0004= (0, 1), from assertion (i), together with the well-known\nstructure of the maximal pro-l quotient of the fundamental group of\na smooth curve over an algebraically closed field of characteristic \u0004= l.\nFinally, we verify assertion (iii). The fact that the action of ΠB on ΠZ\/B\npreserves the exact sequence appearing in the statement of assertion (iii)\nfollows immediately from the fact that the surjection Πab\nZ\/B\n\u0002 ΠabC is\ninduced by the open immersion Z ↪→ B ×k C over B. The fact that the\naction in question induces the identity automorphism on (\n⊕n\ni=1 Jsi)\/Jr\nCombinatorial anabelian topics I 17\n(respectively, ΠabC ) follows immediately from the fact that the fi’s are\nisomorphisms (respectively, the fact that the surjection Πab\nZ\/B\n\u0002 ΠabC is\ninduced by the open immersion Z ↪→ B ×k C over B). Q.E.D.\nLemma 1.2 (Maximal cuspidally central quotients of certain\nfundamental groups). In the notation of Lemma 1.1, for i = 1, · · · , n,\nwrite\nΠZ\/B \u0002 Π(Z\/B)[i] (\u0002 ΠC)\nfor the quotient of ΠZ\/B by the normal closed subgroup topologically\nnormally generated by the Isj ’s, where j ranges over the integers such\nthat 1 ≤ j ≤ n and j \u0004= i;\nΠ(Z\/B)[i] \u0002 E(Z\/B)[i]\nfor the maximal cuspidally central quotient [cf. [AbsCsp], Defini-\ntion 1.1, (i)] relative to the surjection Π(Z\/B)[i] \u0002 ΠC determined by the\nnatural open immersion Z ↪→ B ×k C;\nIEsi ⊆ E(Z\/B)[i]\nfor the kernel of the natural surjection E(Z\/B)[i] \u0002 ΠC ; and\nEZ\/B\ndef\n= E(Z\/B)[1] ×ΠC · · · ×ΠC E(Z\/B)[n] .\nThen the following hold:\n(i) (Cuspidal inertia subgroups) Let 1 ≤ i, j ≤ n be integers.\nThen the homomorphism Isi → IEsj determined by the compos-\nite Isi ↪→ ΠZ\/B \u0002 E(Z\/B)[j] is an isomorphism (respectively,\ntrivial) if i = j (respectively, i \u0004= j).\n(ii) (Surjectivity) The homomorphism ΠZ\/B → EZ\/B determined\nby the natural surjections ΠZ\/B \u0002 E(Z\/B)[i] — where i ranges\nover the integers such that 1 ≤ i ≤ n — is surjective.\n(iii) (Maximal cuspidally central quotients and abelianiza-\ntions) The quotient ΠZ\/B \u0002 EZ\/B of ΠZ\/B [cf. (ii)] coin-\ncides with the maximal cuspidally central quotient [cf.\n[AbsCsp], Definition 1.1, (i)] relative to the surjection ΠZ\/B \u0002\nΠC determined by the natural open immersion Z ↪→ B ×k C.\nIn particular, the natural surjection ΠZ\/B \u0002 ΠabZ\/B factors\n18 Yuichiro Hoshi and Shinichi Mochizuki\nthrough the surjection ΠZ\/B \u0002 EZ\/B, and the resulting sur-\njection EZ\/B \u0002 ΠabZ\/B fits into a commutative diagram\n1 −−−−→ ⊕ni=1 IEsi −−−−→ EZ\/B −−−−→ ΠC −−−−→ 1⏐⏐\b ⏐⏐\b ⏐⏐\b\n1 −−−−→ (⊕ni=1 Jsi)\/Jr −−−−→ ΠabZ\/B −−−−→ ΠabC −−−−→ 1\n— where the horizontal sequences are exact, and the vertical\narrows are surjective. Moreover, the left-hand vertical arrow\ncoincides with the surjection induced by the natural isomor-\nphisms Isi\n∼→ Jsi [cf. Lemma 1.1, (ii)] and Isi\n∼→ IEsi [cf. (i)].\nFinally, if r \u0004= 0, then the right-hand square is cartesian.\nProof. Assertion (i) follows immediately from the definition of the\nquotient E(Z\/B)[j] of ΠZ\/B , together with the well-known structure of the\nmaximal pro-l quotient of the fundamental group of a smooth curve over\nan algebraically closed field of characteristic \u0004= l [cf., e.g., [MT], Lemma\n4.2, (iv), (v)]. Assertion (ii) follows immediately from assertion (i).\nAssertion (iii) follows immediately from assertions (i), (ii) [cf. [AbsCsp],\nProposition 1.6, (iii)]. Q.E.D.\nLemma 1.3 (The kernels of representations arising from cer-\ntain families of hyperbolic curves). In the notation of Lemmas 1.1,\n1.2, suppose that r \u0004= 0. Then the following hold:\n(i) (Unipotency of a certain natural outer representation)\nConsider the action of ΠB on EZ\/B determined by the natural\nisomorphism\nEZ\/B\n∼−→ ΠabZ\/B ×ΠabC ΠC\n[cf. Lemma 1.2, (iii)], together with the natural action of ΠB\non Πab\nZ\/B\ninduced by ρZ\/B and the trivial action of ΠB on ΠC .\nThen the outer action of ΠB on EZ\/B induced by this ac-\ntion coincides with the natural outer action of ΠB on EZ\/B\ninduced by ρZ\/B. In particular, relative to the natural identifi-\ncation Isi\n∼→ IEsi [cf. Lemma 1.2, (i)], the above action of ΠB\non EZ\/B factors through the homomorphism\nΠB −→ HomZl\n(\nΠC ,\nn⊕\ni=1\nIsi\n) ∼−→ HomZl(ΠabC , n⊕\ni=1\nIsi\n)\nCombinatorial anabelian topics I 19\nobtained in Lemma 1.1, (iii).\n(ii) (Homomorphisms arising from a certain extension) For\ni = 1, · · · , n, write φi for the composite\nΠB −→ HomZl\n(\nΠabC ,\nn⊕\nj=1\nIsj\n)\n−→ HomZl\n(\nΠabC , Isi\n)\n— where the first arrow is the homomorphism of (i), and the\nsecond arrow is the homomorphism determined by the projec-\ntion pri :\n⊕n\nj=1 Isj \u0002 Isi . Then the homomorphism φi co-\nincides with the image of the element of H2(ΠB × ΠC , Isi)\ndetermined by the extension\n1 −→ Isi −→ ΠEZ[i] −→ ΠB ×ΠC −→ 1\n— where we write ΠE\nZ[i]\ndef\n= ΠZ\/Ker(ΠZ\/B \u0002 EZ\/B[i]) — of\nΠB ×ΠC by Isi\n∼→ IEsi [cf. Lemma 1.2, (i)] via the composite\nH2(ΠB ×ΠC , Isi)\n∼→ H1(ΠB , H1(ΠC , Isi))\n∼→ Hom\n(\nΠB ,Hom(ΠC , Isi)\n)\n— where the first arrow is the isomorphism determined by\nthe Hochschild-Serre spectral sequence relative to the surjection\nΠB ×ΠC\npr1\u0002 ΠB.\n(iii) (Factorization) Write B (respectively, C) for the compactifi-\ncation of C (respectively, B) and ΠB (respectively, ΠC) for the\nmaximal pro-l quotient of the étale fundamental group π1(B)\n(respectively, π1(C)) of B (respectively, C). Then the homo-\nmorphism φi of (ii) factors as the composite\nΠB \u0002 ΠabB\n∼→ Πab\nC\n∼→ HomZl\n(\nΠab\nC\n, Isi\n)\n↪→ HomZl\n(\nΠabC , Isi\n)\n— where the first (respectively, second; fourth) arrow is the\nhomomorphism induced by B ↪→ B (respectively, fi : B ∼→ C;\nC ↪→ C), and the third arrow is the isomorphism determined\nby the Poincaré duality isomorphism in étale cohomology, rel-\native to the natural isomorphism Isi\n∼→ Zl(1). [Here, the “(1)”\ndenotes a “Tate twist”.]\n(iv) (Kernel of a certain natural representation) The kernel\nof the homomorphism ΠB → AutZl(ΠabZ\/B) determined by ρZ\/B\ncoincides with the kernel of the natural surjection ΠB \u0002 ΠabB .\n20 Yuichiro Hoshi and Shinichi Mochizuki\nProof. Assertions (i), (ii) follow immediately from the various defi-\nnitions involved. Next, we verify assertion (iii). It follows from assertion\n(ii), together with [MT], Lemma 4.2, (ii), (v) [cf. also the discussion\nsurrounding [MT], Lemma 4.2], that, relative to the natural isomor-\nphism Isi\n∼→ Zl(1), the image of φi ∈ Hom(ΠB ,HomZl(ΠabC , Isi)) via the\nisomorphisms\nHom(ΠB ,HomZl(Π\nab\nC , Isi))\n∼→ Hom(ΠB ,HomZl(ΠabC ,Zl(1)))\n∼← H2(ΠB ×ΠC ,Zl(1)) ∼→ H2(B ×k C,Zl(1))\n— where the first (respectively, second) isomorphism is the isomor-\nphism induced by the above isomorphism Isi\n∼→ Zl(1) (respectively, the\nHochschild-Serre spectral sequence relative to the surjection ΠB×ΠC\npr1\u0002\nΠB) — is the first Chern class of the invertible sheaf associated to the\ndivisor determined by the scheme-theoretic image of si : Bi ↪→ B ×k C.\nThus, since the section si extends uniquely to a section si : B ↪→ B×kC,\nwhose scheme-theoretic image we denote by Im(si), it follows that the\nhomomorphism φi ∈ Hom(ΠB ,HomZl(ΠabC , Isi)) coincides with the im-\nage of the first Chern class of the invertible sheaf on B ×k C associated\nto the divisor Im(si) via the composite\nH2(B ×k C,Zl(1)) ∼← H2(B ×k C, Isi) → H2(B ×k C, Isi)\n∼← H2(ΠB ×ΠC , Isi)\n∼→ Hom\n(\nΠB ,HomZl(ΠC , Isi)\n)\n—where the first arrow is the isomorphism induced by the above isomor-\nphism Isi\n∼→ Zl(1), and the second arrow is the homomorphism induced\nby the natural open immersion B ×k C ↪→ B ×k C. In particular, as-\nsertion (iii) follows immediately from [Mln], Chapter VI, Lemma 12.2\n[cf. also the argument used in the proof of [MT], Lemma 4.4]. Finally,\nwe verify assertion (iv). To this end, we recall that by Lemma 1.1, (iii),\nthe homomorphism ΠB → AutZl(ΠabZ\/B) factors through the homomor-\nphism ΠB → HomZl\n(\nΠabC ,\n⊕n\ni=1 Jsi\n)\nof assertion (i). Thus, assertion\n(iv) follows immediately from assertion (iii). This completes the proof\nof assertion (iv). Q.E.D.\nDefinition 1.4. For \u0003 ∈ {◦, •}, let Σ\u0002 be a set of prime numbers\nwhich is either of cardinality one or equal to the set of all prime numbers;\n(g\u0002, r\u0002) a pair of nonnegative integers such that 2g\u0002 − 2 + r\u0002 > 0; X\u0002\na hyperbolic curve of type (g\u0002, r\u0002) over an algebraically closed field of\nCombinatorial anabelian topics I 21\ncharacteristic \u0004∈ Σ\u0002; d\u0002 a positive integer; X\u0002\nd\u0002 the d\n\u0002-th configuration\nspace of X\u0002 [cf. [MT], Definition 2.1, (i)]; Π\u0002\nd\u0002 the pro-Σ\n\u0002 configura-\ntion space group [cf. [MT], Definition 2.3, (i)] obtained by forming the\nmaximal pro-Σ\u0002 quotient of the étale fundamental group π1(X\u0002d\u0002) of\nX\u0002\nd\u0002 .\n(i) We shall say that an isomorphism of profinite groups α : Π◦d◦\n∼→\nΠ•d• is PF-admissible [i.e., “permutation-fiber-admissible”] if α\ninduces a bijection between the set of fiber subgroups [cf. [MT],\nDefinition 2.3, (iii)] of Π◦d◦ and the set of fiber subgroups of Π\n•\nd• .\nWe shall say that an outer isomorphism Π◦d◦\n∼→ Π•d• is PF-\nadmissible if it is determined by a PF-admissible isomorphism.\n(ii) We shall say that an isomorphism of profinite groups α : Π◦d◦\n∼→\nΠ•d• is PC-admissible [i.e., “permutation-cusp-admissible”] if\nthe following condition is satisfied: Let\n{1} = Kd◦ ⊆ Kd◦−1 ⊆ · · · ⊆ Km ⊆ · · · ⊆ K2 ⊆ K1 ⊆ K0 = Π◦d◦\nbe the standard fiber filtration of Π◦d◦ [cf. [CmbCsp], Definition\n1.1, (i)]; then for any integer 1 ≤ a ≤ d◦, the image α(Ka) ⊆\nΠ•d• is a fiber subgroup of Π\n•\nd• of length d\n◦ − a [cf. [MT], Defi-\nnition 2.3, (iii)], and, moreover, the isomorphism Ka−1\/Ka\n∼→\nα(Ka−1)\/α(Ka) determined by α induces a bijection between\nthe set of cuspidal inertia subgroups of Ka−1\/Ka and the set\nof cuspidal inertia subgroups of α(Ka−1)\/α(Ka). [Note that it\nfollows immediately from the various definitions involved that\nthe profinite group Ka−1\/Ka (respectively, α(Ka−1)\/α(Ka))\nis equipped with a natural structure of pro-Σ◦ (respectively,\npro-Σ•) surface group [cf. [MT], Definition 1.2].] We shall say\nthat an outer isomorphism Π◦d◦\n∼→ Π•d• is PC-admissible if it is\ndetermined by a PC-admissible isomorphism.\n(iii) We shall say that an isomorphism of profinite groups α : Π◦d◦\n∼→\nΠ•d• is PFC-admissible [i.e., “permutation-fiber-cusp-admissi-\nble”] if α is PF-admissible and PC-admissible. We shall say\nthat an outer isomorphism Π◦d◦\n∼→ Π•d• is PFC-admissible if it\nis determined by a PFC-admissible isomorphism.\n22 Yuichiro Hoshi and Shinichi Mochizuki\n(iv) We shall say that an isomorphism of profinite groups α : Π◦d◦\n∼→\nΠ•d• is PF-cuspidalizable if there exists a commutative diagram\nΠ◦d◦+1\n∼−−−−→ Π•d•+1⏐⏐\b ⏐⏐\b\nΠ◦d◦\n∼−−−−→\nα\nΠ•d•\n— where the upper horizontal arrow is a PF-admissible iso-\nmorphism, and the left-hand (respectively, right-hand) vertical\narrow is the surjection obtained by forming the quotient by a\nfiber subgroup of length 1 [cf. [MT], Definition 2.3, (iii)] of\nΠ◦d◦+1 (respectively, Π\n•\nd•+1). We shall say that an outer iso-\nmorphism Π◦d◦\n∼→ Π•d• is PF-cuspidalizable if it is determined\nby a PF-cuspidalizable isomorphism.\nRemark 1.4.1. It follows immediately from the various definitions\ninvolved that, in the notation of Definition 1.4, an automorphism α of\nΠ◦d◦ is PF-admissible (respectively, PC-admissible; PFC-admissible) if\nand only if there exists an automorphism σ of Π◦d◦ that lifts the outo-\nmorphism [cf. the discussion entitled “Topological groups” in §0] of Π◦d◦\nnaturally determined by a permutation of the d◦ factors of the config-\nuration space involved such that the composite α ◦ σ is F-admissible\n(respectively, C-admissible; FC-admissible) [cf. [CmbCsp], Definition\n1.1, (ii)]. In particular, a(n) F-admissible (respectively, C-admissible;\nFC-admissible) automorphism of Π◦d◦ is PF-admissible (respectively, PC-\nadmissible; PFC-admissible):\nF-admissible ⇐= FC-admissible =⇒ C-admissible\n⇓ ⇓ ⇓\nPF-admissible ⇐= PFC-admissible =⇒ PC-admissible .\nProposition 1.5 (Properties of PF-admissible isomor-\nphisms). In the notation of Definition 1.4, let α : Π◦d◦\n∼→ Π•d• be an\nisomorphism. Then the following hold:\n(i) Σ◦ = Σ•.\nCombinatorial anabelian topics I 23\n(ii) Suppose that the isomorphism α is PF-admissible. Let 1 ≤\nn ≤ d◦ be an integer and H ⊆ Π◦d◦ a fiber subgroup of length\nn of Π◦d◦ . Then the subgroup α(H) ⊆ Π•d• is a fiber subgroup\nof length n of Π•d• . In particular, it holds that d\n◦ = d•.\n(iii) Write Ξ◦ ⊆ Π◦d◦ (respectively, Ξ• ⊆ Π•d•) for the normal closed\nsubgroup of Π◦d◦ (respectively, Π\n•\nd•) obtained by taking the in-\ntersection of the various fiber subgroups of length d◦ − 1 (re-\nspectively, d•−1). Then the isomorphism α is PF-admissible\nif and only if α induces an isomorphism Ξ◦ ∼→ Ξ•.\nProof. Assertion (i) follows immediately from the [easily verified]\nfact that Σ\u0002 may be characterized as the smallest set of primes Σ∗ for\nwhich Π\u0002\nd\u0002 is pro-Σ\n∗. Assertion (ii) follows immediately from the various\ndefinitions involved. Finally, we verify assertion (iii). The necessity of\nthe condition follows immediately from assertion (ii). The sufficiency\nof the condition follows immediately from a similar argument to the\nargument used in the proof of [CmbCsp], Proposition 1.2, (i). This\ncompletes the proof of assertion (iii). Q.E.D.\nLemma 1.6 (C-admissibility of certain isomorphisms). In the\nnotation of Definition 1.4, let α2 : Π\n◦\n2\n∼→ Π•2, α11 : Π◦1\n∼→ Π•1, α21 : Π◦1\n∼→\nΠ•1 be isomorphisms of profinite groups which, for i = 1, 2, fit into a\ncommutative diagram\nΠ◦2\nα2−−−−→ Π•2\npr◦{i}\n⏐⏐\b ⏐⏐\bpr•{i}\nΠ◦1\nαi1−−−−→ Π•1\n— where the vertical arrow “pr\u0002{i}” is the surjection induced by the pro-\njection “X\u00022 → X\u00021 ” obtained by projecting to the i-th factor. Then the\nisomorphism α11 is C-admissible. In particular, (g\n◦, r◦) = (g•, r•).\nProof. Write Σ\ndef\n= Σ◦ = Σ• [cf. Proposition 1.5, (i)]. Now it fol-\nlows from the well-known structure of the maximal pro-Σ quotient of\nthe fundamental group of a smooth curve over an algebraically closed\nfield of characteristic \u0004∈ Σ that Π\u00021 is a free pro-Σ group if and only if\nr\u0002 \u0004= 0 [cf. [CmbGC], Remark 1.1.3]. Thus, if r◦ = r• = 0, then it is\nimmediate that α11 is C-admissible; moreover, it follows, by considering\nthe rank of the abelianization of Π\u00021 [cf. [CmbGC], Remark 1.1.3], that\ng◦ = g•. In particular, to verify Lemma 1.6, we may assume without loss\n24 Yuichiro Hoshi and Shinichi Mochizuki\nof generality that r◦, r• \u0004= 0. Then it follows from [CmbGC], Theorem\n1.6, (i), that, to verify Lemma 1.6, it suffices to show that α11 is numer-\nically cuspidal [cf. [CmbGC], Definition 1.4, (ii)], i.e., to show that the\nfollowing assertion holds:\nLet Π◦Y ⊆ Π◦1 be an open subgroup of Π◦1. Write\nΠ•Y\ndef\n= α11(Π\n◦\nY ) ⊆ Π•1, Y ◦ → X◦ (respectively, Y • →\nX•) for the connected finite étale covering of X◦ (re-\nspectively, X•) corresponding to the open subgroup\nΠ◦Y ⊆ Π◦1 (respectively, Π•Y ⊆ Π•1), and (g◦Y , r◦Y ) (re-\nspectively, (g•Y , r\n•\nY )) for the type of Y\n◦ (respectively,\nY •). Then it holds that r◦Y = r\n•\nY .\nOn the other hand, in the notation of the above assertion, one verifies\neasily that for any l ∈ Σ and \u0003 ∈ {◦, •}, if Π\u0002Y ′ ⊆ Π\u00021 is an open\nsubgroup of Π\u00021 contained in Π\n\u0002\nY , then the natural inclusion Π\n\u0002\nY ′ ↪→ Π\u0002Y\ninduces a surjection\nKer((Π\u0002Y ′)\nab \u0002 (Π\u0002\nY\n′)ab)⊗\nẐΣ\nQl \u0002 Ker((Π\u0002Y )ab \u0002 (Π\u0002Y )\nab)⊗\nẐΣ\nQl\n— where we write (Π\u0002\nY\n), (Π\u0002\nY\n′) for the maximal pro-Σ quotients of the\nétale fundamental groups of the compactifications Y , Y\n′\nof Y , Y ′, re-\nspectively. Thus, since any open subgroup of Π◦1 contains a character-\nistic open subgroup of Π◦1, it follows immediately from the well-known\nfact that for \u0003 ∈ {◦, •}, (Π\u0002Y )ab (respectively, (Π\u0002Y )\nab) is a free ẐΣ-\nmodule of rank 2g\u0002Y + r\n\u0002\nY − 1 (respectively, 2g\u0002Y ) [cf., e.g., [CmbGC],\nRemark 1.1.3] that to verify the above assertion, it suffices to ver-\nify that if Π◦Y ⊆ Π◦1 in the above assertion is characteristic, then the\nisomorphism Π◦Y\n∼→ Π•Y determined by α11 induces an isomorphism of\nKer((Π◦Y )\nab \u0002 (Π◦\nY\n)ab)⊗\nẐΣ\nQl with Ker((Π\n•\nY )\nab \u0002 (Π•\nY\n)ab)⊗\nẐΣ\nQl for\nsome l ∈ Σ.\nTo this end, for \u0003 ∈ {◦, •}, write Π\u0002Z ⊆ Π\u00022 for the normal open\nsubgroup of Π\u00022 obtained by forming the inverse image via the surjection\n(pr\u0002{1}, pr\n\u0002\n{2}) : Π\n\u0002\n2 \u0002 Π\u00021 ×Π\u00021\nof the image of the natural inclusion Π\u0002Y × Π\u0002Y ↪→ Π\u00021 × Π\u00021 ; Z\u0002 →\nX\u00022 for the connected finite étale covering corresponding to this normal\nopen subgroup Π\u0002Z ⊆ Π\u00022 ; Π\u0002Z\/Y for the kernel of the natural surjection\nΠ\u0002Z \u0002 Π\u0002Y induced by the composite Z\u0002 → X\u00022 ↪→ X\u0002 ×k X\u0002\npr1→ X\u0002.\nThen the natural surjection Π\u0002Z \u0002 Π\u0002Y determines a representation\nΠ\u0002Y −→ Aut((Π\u0002Z\/Y )ab) ;\nCombinatorial anabelian topics I 25\nmoreover, the isomorphisms α2, α\n1\n1, and α\n2\n1 determine a commutative\ndiagram\nΠ◦Y −−−−→ Aut((Π◦Z\/Y )ab)⏐⏐\b ⏐⏐\b\nΠ•Y −−−−→ Aut((Π•Z\/Y )ab)\n— where the vertical arrows are isomorphisms. [Here, we note that since\nΠ•Y is a characteristic subgroup of Π\n•\n1, and the composite α\n2\n1 ◦ (α11)−1\nis an automorphism of Π•1, it follows that Π\n•\nY = α\n2\n1(Π\n◦\nY ), hence that\nα2 induces an isomorphism Π\n◦\nZ\n∼→ Π•Z .] On the other hand, it follows\nfrom the definition of Z\u0002 that Z\u0002 is isomorphic to the open subscheme\nof Y \u0002 ×k Y \u0002 obtained by forming the complement of the graphs of the\nvarious elements of Aut(Y \u0002\/X\u0002). Thus, it follows from Lemma 1.3, (iv)\n— by replacing the various profinite groups involved by their maximal\npro-l quotients for some l ∈ Σ — that the isomorphism Π◦Y\n∼→ Π•Y\ndetermined by α11 induces an isomorphism of Ker((Π\n◦\nY )\nab \u0002 (Π◦\nY\n)ab)⊗\nẐΣ\nQl with Ker((Π\n•\nY )\nab \u0002 (Π•\nY\n)ab)⊗\nẐΣ\nQl for some l ∈ Σ. This completes\nthe proof of Lemma 1.6. Q.E.D.\nLemma 1.7 (PFC-admissibility of certain PF-admissible iso-\nmorphisms). In the notation of Definition 1.4, let α : Π◦d◦\n∼→ Π•d• be\na PF-admissible isomorphism. Then the following condition implies\nthat the isomorphism α is PFC-admissible:\nLet H◦ ⊆ Π◦d◦ be a fiber subgroup of length 1 [cf.\n[MT], Definition 2.3, (iii)]. Write H• def= α(H◦) ⊆\nΠ•d• for the fiber subgroup of length 1 obtained as\nthe image of H◦ via α [cf. Proposition 1.5, (ii)].\n[Thus, it follows immediately from the various defini-\ntions involved that H◦ (respectively, H•) is equipped\nwith a natural structure of pro-Σ◦ (respectively, pro-\nΣ•) surface group.] Then the isomorphism H◦ ∼→\nH• induced by α is C-admissible.\nProof. Let \u0003 ∈ {◦, •}. Then one may verify easily that the follow-\ning fact holds:\nLet 1 ≤ a ≤ d\u0002 be an integer and F ′ ⊆ F ⊆ Π\u0002\nd\u0002\nfiber subgroups of Π\u0002\nd\u0002 such that F is of length a,\nand F ′ is of length a − 1. Then there exists a fiber\n26 Yuichiro Hoshi and Shinichi Mochizuki\nsubgroup H ⊆ F ⊆ Π\u0002\nd\u0002 of Π\n\u0002\nd\u0002 of length 1 such that\nthe composite\nH ↪→ F \u0002 F\/F ′\narises from a natural open immersion of a hyperbolic\ncurve of type (g\u0002, r\u0002+d\u0002−1) into a hyperbolic curve\nof type (g\u0002, r\u0002 + d\u0002 − a). [Note that it follows im-\nmediately from the various definitions involved that\nH (respectively, F\/F ′) is equipped with a natural\nstructure of pro-Σ\u0002 surface group.] In particular, the\ncomposite is a surjection whose kernel is topologically\nnormally generated by suitable cuspidal inertia sub-\ngroups of H; moreover, any cuspidal inertia subgroup\nof F\/F ′ may be obtained as the image of a cuspidal\ninertia subgroup of H.\nOn the other hand, one may verify easily that Lemma 1.7 follows im-\nmediately from the above fact. This completes the proof of Lemma 1.7.\nQ.E.D.\nTheorem 1.8 (PFC-admissibility of certain isomorphisms).\nFor \u0003 ∈ {◦, •}, let Σ\u0002 be a set of prime numbers which is either of\ncardinality one or equal to the set of all prime numbers; (g\u0002, r\u0002)\na pair of nonnegative integers such that 2g\u0002−2+r\u0002 > 0; X\u0002 a hyperbolic\ncurve of type (g\u0002, r\u0002) over an algebraically closed field of characteristic\n\u0004∈ Σ\u0002; d\u0002 a positive integer; Π\u0002\nd\u0002 the pro-Σ\n\u0002 configuration space group\n[cf. [MT], Definition 2.3, (i)] obtained by forming the maximal pro-Σ\u0002\nquotient of the étale fundamental group of the d\u0002-th configuration space\nof X\u0002;\nα : Π◦d◦\n∼−→ Π•d•\nan isomorphism of [abstract] groups. If\n{(g◦, r◦), (g•, r•)} ∩ {(0, 3), (1, 1)} \u0004= ∅ ,\nthen we suppose further that the isomorphism α is PF-admissible [cf.\nDefinition 1.4, (i)]. Then the following hold:\n(i) Σ◦ = Σ•.\n(ii) The isomorphism α is an isomorphism of profinite groups.\n(iii) The isomorphism α is PF-admissible. In particular, d◦ = d•.\nCombinatorial anabelian topics I 27\n(iv) If α is PF-cuspidalizable [cf. Definition 1.4, (iv)], then α\nis PFC-admissible [cf. Definition 1.4, (iii)]. In particular,\n(g◦, r◦) = (g•, r•).\nProof. Assertion (ii) follows from [NS], Theorem 1.1. In light of\nassertion (ii), assertion (i) follows from Proposition 1.5, (i). Assertion\n(iii) follows from Proposition 1.5, (ii); [MT], Corollary 6.3, together with\nthe assumption appearing in the statement of Theorem 1.8. Assertion\n(iv) follows immediately from Lemmas 1.6, 1.7. Q.E.D.\nCorollary 1.9 (F-admissibility and FC-admissibility). Let Σ\nbe a set of prime numbers which is either of cardinality one or equal\nto the set of all prime numbers; n a positive integer; (g, r) a pair\nof nonnegative integers such that 2g − 2 + r > 0; X a hyperbolic curve\nof type (g, r) over an algebraically closed field k of characteristic \u0004∈ Σ;\nXn the n-th configuration space of X; Πn the maximal pro-Σ quotient\nof the fundamental group of Xn; “Out\nFC(−)”, “OutF(−)” ⊆ “Out(−)”\nthe subgroups of FC- and F-admissible [cf. [CmbCsp], Definition 1.1,\n(ii)] outomorphisms [cf. the discussion entitled “Topological groups” in\n§0] of “(−)”. Then the following hold:\n(i) Let α ∈ OutF(Πn+1). Then α induces the same outomor-\nphism of Πn relative to the various quotients Πn+1 \u0002 Πn by\nfiber subgroups of length 1 [cf. [MT], Definition 2.3, (iii)]. In\nparticular, we obtain a natural homomorphism\nOutF(Πn+1) −→ OutF(Πn) .\n(ii) The image of the homomorphism\nOutF(Πn+1) −→ OutF(Πn)\nof (i) is contained in\nOutFC(Πn) ⊆ OutF(Πn) .\nProof. First, we verify assertion (i). Let H1, H2 ⊆ Πn+1 be two\ndistinct fiber subgroups of Πn+1 of length 1. Observe that the normal\nclosed subgroup H ⊆ Πn+1 of Πn+1 topologically generated by H1 and\nH2 is a fiber subgroup of Πn+1 of length 2 [cf. [MT], Proposition 2.4,\n(iv)], hence is equipped with a natural structure of pro-Σ configuration\nspace group, with respect to which Hi ⊆ H may be regarded as a fiber\n28 Yuichiro Hoshi and Shinichi Mochizuki\nsubgroup of length 1 [cf. [MT], Proposition 2.4, (ii)]. Moreover, it follows\nimmediately from the scheme-theoretic definition of the various config-\nuration space groups involved that one has natural outer isomorphisms\nΠn+1\/H\ni ∼→ Πn and H\/H1 ∼→ H\/H2. Thus, since for i ∈ {1, 2}, we\nhave natural outer isomorphisms\nΠn\n∼← Πn+1\/Hi ∼→ (H\/Hi)\nout\n\u0002 Πn+1\/H\n[cf. the discussion entitled “Topological groups” in §0] which are com-\npatible with the various natural outer isomorphisms discussed above,\none verifies easily [cf. the argument given in the first paragraph of the\nproof of [CmbCsp], Theorem 4.1] that to complete the proof of assertion\n(i), by replacing Πn+1 by H, it suffices to verify assertion (i) in the case\nwhere n = 1. The rest of the proof of assertion (i) is devoted to verifying\nassertion (i) in the case where n = 1.\nLet α̃ ∈ AutF(Π2) be an F-admissible automorphism of Π2; α̃1,\nα̃2 ∈ Aut(Π1) the automorphisms of Π1 induced by α̃ relative to the\nquotients Π2 \u0002 Π2\/H1 ∼→ Π1, Π2 \u0002 Π2\/H2 ∼→ Π1, respectively. Now\nit is immediate that to complete the proof of assertion (i), it suffices to\nverify that the difference α̃1 ◦ (α̃2)−1 ∈ Aut(Π1) is Π1-inner. Therefore,\nit follows immediately from [JR], Theorem B, that to complete the proof\nof assertion (i), it suffices to verify that\n(∗1): for any normal open subgroup N ⊆ Π1 of Π1,\nit holds that α̃1(N) = α̃2(N).\nTo this end, let N ⊆ Π1 be a normal open subgroup of Π1. Write\nΠN\ndef\n= Π2 ×Π1 N for the fiber product of Π2 \u0002 Π2\/H1\n∼→ Π1 and\nN ↪→ Π1 and FN for the kernel of the composite ΠN = Π2 ×Π1 N\npr1\n↪→\nΠ2 \u0002 Π2\/H2 ∼→ Π1. Then the surjection ΠN \u0002 N ×Π1 determined by\nthe natural surjection ΠN \u0002 ΠN\/FN ∼→ Π1 and the second projection\nΠN = Π2 ×Π1 N\npr2\u0002 N fits into a commutative diagram of profinite\ngroups\n1 −−−−→ FN −−−−→ ΠN −−−−→ Π1 −−−−→ 1⏐⏐\b ⏐⏐\b ∥∥∥\n1 −−−−→ N −−−−→ N ×Π1\npr2−−−−→ Π1 −−−−→ 1\n— where the horizontal sequences are exact, and the vertical arrows are\nsurjective. Write ρN : Π1 → Aut(F abN ) for the natural action determined\nby the upper horizontal sequence and VN ⊆ F abN for the kernel of the\nnatural surjection F abN \u0002 Nab induced by the left-hand vertical arrow.\nNow we claim that\nCombinatorial anabelian topics I 29\n(∗2): the action ρN of Π1 on F abN preserves VN ⊆\nF abN , and, moreover, the resulting action ρ\nV\nN : Π1 →\nAut(VN ) factors as the composite\nΠ1 \u0002 Π1\/N ↪→ Aut(VN )\n— where the second arrow is injective.\nIndeed, the fact that the action ρN of Π1 on F\nab\nN preserves VN ⊆ F abN\nfollows immediately from the definition of ρN [cf. also the above com-\nmutative diagram]. Next, let us observe that it follows immediately from\nthe various definitions involved that if we write f : Y → X for the con-\nnected finite étale Galois covering of X corresponding to N ⊆ Π1, then\nthe right-hand square of the above diagram arises from a commutative\ndiagram of schemes\n(Y ×k X) \\ Γf\npr2−−−−→ X⏐⏐\b ∥∥∥\nY ×k X\npr2−−−−→ X\n— where we write Γf ⊆ Y ×k X for the graph of f , and the left-hand\nvertical arrow is the natural open immersion. Thus, it follows imme-\ndiately from a similar argument to the argument used in the proof of\nLemma 1.1, (i) [cf. also [Hsh], Proposition 1.4, (i)], that FN , N are\nnaturally isomorphic to the maximal pro-Σ quotients of the étale fun-\ndamental groups of geometric fibers of the families of hyperbolic curves\nY ×k X \\ Γf , Y ×k X\npr2→ X over X, respectively. Therefore, by the\nwell-known structure of the maximal pro-Σ quotient of the fundamental\ngroup of a smooth curve over an algebraically closed field of character-\nistic \u0004∈ Σ, we conclude — by considering the natural action of Π1 on the\nset of cusps of the family of hyperbolic curves Y ×k X \\ Γf\npr2→ X —\nthat the resulting action ρVN : Π1 → Aut(VN ) factors as the composite\nΠ1 \u0002 Π1\/N → Aut(VN ), and that if X is affine (respectively, proper),\nthen for any l ∈ Σ, the resulting representation Π1\/N → Aut(VN⊗ẐΣQl)\nis isomorphic to\nthe regular representation of Π1\/N over Ql (respec-\ntively, the quotient of the regular representation of\nΠ1\/N over Ql by the trivial subrepresentation [of di-\nmension 1]).\nIn particular, as is well-known, the homomorphism Π1\/N → Aut(VN⊗ẐΣ\nQl), hence also the homomorphism Π1\/N → Aut(VN ), is injective. This\ncompletes the proof of the claim (∗2).\n30 Yuichiro Hoshi and Shinichi Mochizuki\nNext, let us observe that since α̃ is F-admissible, it follows imme-\ndiately from the definition of “ρVN” that the automorphism α̃ induces a\ncommutative diagram\nΠ1\nρVN−−−−→ Aut(VN )\nα̃2\n⏐⏐\b\u0005 ⏐⏐\b\u0005\nΠ1\nρV\nα̃1(N)−−−−→ Aut(Vα̃1(N))\n— where the vertical arrows are isomorphisms that are induced by α̃.\nThus, by considering the kernels of ρVN , ρ\nV\nα̃1(N)\n, one concludes from the\nclaim (∗2) that α̃1(N) = α̃2(N). This completes the proof of (∗1), hence\nalso of assertion (i).\nAssertion (ii) follows immediately from Theorem 1.8, (iv) [cf. also\nRemark 1.4.1]. This completes the proof of Corollary 1.9. Q.E.D.\nRemark 1.9.1. The discrete versions of Theorem 1.8, Corollary 1.9\nwill be discussed in a sequel to the present paper.\n§2. Various operations on semi-graphs of anabelioids of PSC-\ntype\nIn the present §, we study various operations on semi-graphs of\nanabelioids of PSC-type. These operations include the following:\n(Op1) the operation of restriction to a sub-semi-graph [satisfying cer-\ntain conditions] of the underlying semi-graph [cf. Definition 2.2,\n(ii); Fig. 2 below],\n(Op2) the operation of partial compactification [cf. Definition 2.4, (ii);\nFig. 3 below],\n(Op3) the operation of resolution of a given set [satisfying certain\nconditions] of nodes [cf. Definition 2.5, (ii); Fig. 4 below], and\n(Op4) the operation of generization [cf. Definition 2.8; Fig. 5 below].\nA basic reference for the theory of semi-graphs of anabelioids of PSC-\ntype is [CmbGC]. We shall use the terms “semi-graph of anabelioids of\nPSC-type”, “PSC-fundamental group of a semi-graph of anabelioids of\nCombinatorial anabelian topics I 31\nPSC-type”, “finite étale covering of semi-graphs of anabelioids of PSC-\ntype”, “vertex”, “edge”, “cusp”, “node”, “verticial subgroup”, “edge-like\nsubgroup”, “nodal subgroup”, “cuspidal subgroup”, and “sturdy” as they\nare defined in [CmbGC], Definition 1.1. Also, we shall apply the vari-\nous notational conventions established in [NodNon], Definition 1.1, and\nrefer to the “PSC-fundamental group of a semi-graph of anabelioids of\nPSC-type” simply as the “fundamental group” [of the semi-graph of an-\nabelioids of PSC-type]. That is to say, we shall refer to the maximal\npro-Σ quotient of the fundamental group of a semi-graph of anabelioids\nof pro-Σ PSC-type [as a semi-graph of anabelioids!] as the “fundamental\ngroup of the semi-graph of anabelioids of PSC-type”.\nLet Σ be a nonempty set of prime numbers and G a semi-graph of\nanabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph\nof G, ΠG for the [pro-Σ] fundamental group of G, and G̃ → G for the\nuniversal covering of G corresponding to ΠG . Then since the fundamental\ngroup ΠG of G is topologically finitely generated, the profinite topology\nof ΠG induces [profinite] topologies on Aut(ΠG) and Out(ΠG) [cf. the\ndiscussion entitled “Topological groups” in §0]. If, moreover, we write\nAut(G)\nfor the automorphism group of G, then by the discussion preceding\n[CmbGC], Lemma 2.1, the natural homomorphism\nAut(G) −→ Out(ΠG)\nis an injection with closed image. [Here, we recall that an automorphism\nof a semi-graph of anabelioids consists of an automorphism of the un-\nderlying semi-graph, together with a compatible system of isomorphisms\nbetween the various anabelioids at each of the vertices and edges of the\nunderlying semi-graph which are compatible with the various morphisms\nof anabelioids associated to the branches of the underlying semi-graph\n— cf. [SemiAn], Definition 2.1; [SemiAn], Remark 2.4.2.] Thus, by\nequipping Aut(G) with the topology induced via this homomorphism by\nthe topology of Out(ΠG), we may regard Aut(G) as being equipped with\nthe structure of a profinite group.\nDefinition 2.1.\n(i) For z ∈ VCN(G) such that z ∈ Vert(G) (respectively, z ∈\nEdge(G)), we shall say that a closed subgroup of ΠG is a VCN-\nsubgroup of ΠG associated to z ∈ VCN(G) if the closed sub-\ngroup is a verticial (respectively, an edge-like) subgroup of\n32 Yuichiro Hoshi and Shinichi Mochizuki\nΠG associated to z ∈ VCN(G). For z̃ ∈ VCN(G̃) such that\nz̃ ∈ Vert(G̃) (respectively, z̃ ∈ Edge(G̃)), we shall say that a\nclosed subgroup of ΠG is the VCN-subgroup of ΠG associated\nto z̃ ∈ VCN(G̃) if the closed subgroup is the verticial (respec-\ntively, edge-like) subgroup of ΠG associated to z̃ ∈ VCN(G̃) [cf.\n[NodNon], Definition 1.1, (vi)].\n(ii) For z ∈ VCN(G), we shall write\nGz\nfor the anabelioid corresponding to z ∈ VCN(G).\n(iii) For v ∈ Vert(G), we shall write\nG|v\nfor the semi-graph of anabelioids of pro-Σ PSC-type defined as\nfollows [cf. Fig. 1 below]: We take Vert(G|v) to consist of the\nsingle element “v”, Cusp(G|v) to be the set of branches of G\nwhich abut to v, and Node(G|v) to be the empty set. We take\nthe anabelioid of G|v corresponding to the unique vertex “v”\nto be Gv [cf. (ii)]. For each edge e ∈ E(v) of G and each branch\nb of e that abuts to the vertex v, we take the anabelioid of G|v\ncorresponding to the branch b to be a copy of the anabelioid\nGe [cf. (ii)]. For each edge e ∈ E(v) of G and each branch b of e\nthat abuts, relative to G, to the vertex v, we take the morphism\nof anabelioids (G|v)eb → (G|v)v of G|v — where we write eb for\nthe cusp of G|v corresponding to b — to be the morphism of\nanabelioids Ge → Gv associated, relative to G, to the branch b.\nThus, one has a natural morphism\nG|v −→ G\nof semi-graphs of anabelioids.\nRemark 2.1.1. Let v ∈ Vert(G) be a vertex of G and Πv ⊆ ΠG\na verticial subgroup of ΠG associated to v ∈ Vert(G). Then it follows\nimmediately from the various definitions involved that the fundamental\ngroup of G|v is naturally isomorphic to Πv, and that we have a natural\nidentification\nAut(Gv) \f Out(Πv)\nCombinatorial anabelian topics I 33\nand a natural injection\nAut(G|v) ↪→ Aut(Gv) .\nFigure 1: G|v\nG\nv\n⇒•\n◦\n• G|v\n•\n◦\n◦\n◦\n◦\nDefinition 2.2 (cf. the operation (Op1) discussed at the beginning\nthe present §2).\n(i) Let K be a [not necessarily finite] semi-graph and H a sub-semi-\ngraph of K [cf. [SemiAn], the discussion following the figure\nentitled “A Typical Semi-graph”]. Then we shall say that H is\nof PSC-type if the following three conditions are satisfied:\n(1) H is finite [i.e., the set consisting of vertices and edges of\nH is finite] and connected.\n(2) H has at least one vertex.\n(3) If v is a vertex of H, and e is an edge of K that abuts\nto v, then e is an edge of H. [Thus, if e abuts both to a\nvertex lying in H and to a vertex not lying in H, then the\nresulting edge of H is a “cusp”, i.e., an open edge.]\nThus, a sub-semi-graph of PSC-type H is completely deter-\nmined by the set of vertices that lie in H.\n(ii) Let H be a sub-semi-graph of PSC-type [cf. (i)] of G. Then one\nmay verify easily that the semi-graph of anabelioids obtained\nby restricting G to H [cf. the discussion preceding [SemiAn],\nDefinition 2.2] is of pro-Σ PSC-type. Here, we recall that the\nsemi-graph of anabelioids obtained by restricting G to H is the\nsemi-graph of anabelioids such that the underlying semi-graph\nis H; for each vertex v (respectively, edge e) of H, the anabe-\nlioid corresponding to v (respectively, e) is Gv (respectively,\n34 Yuichiro Hoshi and Shinichi Mochizuki\nGe) [cf. Definition 2.1, (ii)]; for each branch b of an edge e\nof H that abuts to a vertex v of H, the morphism associated\nto b is the morphism Ge → Gv associated to the branch of G\ncorresponding to b. We shall write\nG|H\nfor this semi-graph of anabelioids of pro-Σ PSC-type and refer\nto G|H as the semi-graph of anabelioids of pro-Σ PSC-type ob-\ntained by restricting G to H [cf. Fig. 2 below]. Thus, one has\na natural morphism\nG|H −→ G\nof semi-graphs of anabelioids.\nFigure 2: Restriction\nH: the sub-semi-graph of PSC-type whose set of vertices = {v}\nG\nv\n⇓\nG|H\n×\n×\n×\n×\nCombinatorial anabelian topics I 35\nDefinition 2.3. Let (g, r) be a pair of nonnegative integers such\nthat 2g − 2 + r > 0.\n(i) We shall say that G is of type (g, r) if G arises from a stable log\ncurve of type (g, r) over an algebraically closed field of charac-\nteristic \u0004∈ Σ, i.e., Cusp(G) is of cardinality r, and, moreover,\nrank\nẐΣ\n(ΠabG ) = 2g +Cusp(G)\u0005 − cG\n— where\ncG\ndef\n=\n{\n0 if Cusp(G) = ∅,\n1 if Cusp(G) \u0004= ∅.\n[Here, we recall that it follows from the discussion of [CmbGC],\nRemark 1.1.3, that ΠabG is a free Ẑ\nΣ-module of finite rank.]\n(ii) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)].\nThen we shall say that H is of type (g, r) if the semi-graph of\nanabelioids G|H, which is of pro-Σ PSC-type [cf. Definition 2.2,\n(ii)], is of type (g, r) [cf. (i)].\n(iii) Let v ∈ Vert(G) be a vertex. Then we shall say that v is\nof type (g, r) if the semi-graph of anabelioids G|v, which is of\npro-Σ PSC-type [cf. Definition 2.1, (iii)], is of type (g, r) [cf.\n(i)].\n(iv) We shall say that G is totally degenerate if each vertex of G is\nof type (0, 3) [cf. (iii)].\n(v) One may verify easily that there exists a unique, up to isomor-\nphism, semi-graph of anabelioids of pro-Σ PSC-type that is of\ntype (g, r) [cf. (i)] and has no node. We shall write\nGmodelg,r\nfor this semi-graph of anabelioids of pro-Σ PSC-type.\nRemark 2.3.1. It follows immediately from the various definitions\ninvolved that there exists a unique pair (g, r) of nonnegative integers\nsuch that G is of type (g, r) [cf. Definition 2.3, (i)].\nDefinition 2.4 (cf. the operation (Op2) discussed at the beginning\nthe present §2).\n36 Yuichiro Hoshi and Shinichi Mochizuki\n(i) We shall say that a subset S ⊆ Cusp(G) of Cusp(G) is omit-\ntable if the following condition is satisfied: For each vertex\nv ∈ Vert(G) of G, if v is of type (g, r) [cf. Definition 2.3, (iii);\nRemark 2.3.1], then it holds that 2g − 2 + r− (E(v)∩ S)\u0005 > 0.\n(ii) Let S ⊆ Cusp(G) be a subset of Cusp(G) which is omittable\n[cf. (i)]. Then by eliminating the cusps [i.e., the open edges]\ncontained in S, and, for each vertex v of G, replacing the an-\nabelioid Gv corresponding to v by the anabelioid of finite étale\ncoverings of Gv that restrict to a trivial covering over the cusps\ncontained in S that abut to v, we obtain a semi-graph of an-\nabelioids\nG•S\nof pro-Σ PSC-type. We shall refer to G•S as the partial compact-\nification of G with respect to S [cf. Fig. 3 below]. Thus, for each\nv ∈ Vert(G) = Vert(G•S), the pro-Σ fundamental group of the\nanabelioid (G•S)v corresponding to v ∈ Vert(G) = Vert(G•S)\nmay be naturally identified, up to inner automorphism, with\nthe quotient of a verticial subgroup Πv ⊆ ΠG of ΠG associated\nto v ∈ Vert(G) = Vert(G•S) by the subgroup of Πv topolog-\nically normally generated by the Πe ⊆ Πv for e ∈ E(v) ∩ S.\nIf, moreover, we write ΠG•S for the [pro-Σ] fundamental group\nof G•S and NS ⊆ ΠG for the normal closed subgroup of ΠG\ntopologically normally generated by the cuspidal subgroups of\nΠG associated to elements of S, then we have a natural outer\nisomorphism\nΠG\/NS\n∼−→ ΠG•S .\nRemark 2.4.1.\n(i) Let S1 ⊆ S2 ⊆ Cusp(G) be subsets of Cusp(G). Then it fol-\nlows immediately from the various definitions involved that the\nomittability of S2 [cf. Definition 2.4, (i)] implies the omittability\nof S1.\n(ii) If G is sturdy, then it follows from the various definitions in-\nvolved that Cusp(G), hence also any subset of Cusp(G) [cf. (i)],\nis omittable. Moreover, the partial compactification of G with\nCombinatorial anabelian topics I 37\nrespect to Cusp(G) coincides with the compactification of G [cf.\n[CmbGC], Remark 1.1.6; [NodNon], Definition 1.11].\nFigure 3: Partial compactification\nG\n× × ×\nc1 c2\n⇓\nG•{c1,c2}\n×\nDefinition 2.5 (cf. the operation (Op3) discussed at the beginning\nthe present §2). Let S ⊆ Node(G) be a subset of Node(G).\n(i) We shall say that S is of separating type if the semi-graph\nobtained by removing the closed edges corresponding to the\nelements of S from G is disconnected. Moreover, for each node\ne ∈ Node(G), we shall say that e is of separating type if {e} ⊆\nNode(G) is of separating type.\n(ii) Suppose that S is not of separating type [cf. (i)]. Then one\nmay define a semi-graph of anabelioids of pro-Σ PSC-type as\nfollows: We take the underlying semi-graph G\bS to be the\nsemi-graph obtained by replacing each node e of G contained\nin S such that V(e) = {v1, v2} ⊆ Vert(G) — where v1, v2\nare not necessarily distinct — by two cusps that abut to v1,\nv2 ∈ Vert(G), respectively. We take the anabelioid correspond-\ning to a vertex v (respectively, node e) of G\bS to be Gv (respec-\ntively, Ge). [Note that the set of vertices (respectively, nodes)\n38 Yuichiro Hoshi and Shinichi Mochizuki\nof G\bS may be naturally identified with Vert(G) (respectively,\nNode(G)\\S).] We take the anabelioid corresponding to a cusp\nof G\bS arising from a cusp e of G to be Ge. We take the an-\nabelioid corresponding to a cusp of G\bS arising from a node\ne of G to be Ge. For each branch b of G\bS that abuts to a\nvertex v of a node e (respectively, of a cusp e that does not\narise from a node of G), we take the morphism associated to\nb to be the morphism Ge → Gv associated to the branch of\nG corresponding to b. For each branch b of G\bS that abuts\nto a vertex v of a cusp of G\bS that arises from a node e of\nG, we take the morphism associated to b to be the morphism\nGe → Gv associated to the branch of G corresponding to b. We\nshall denote the resulting semi-graph of anabelioids of pro-Σ\nPSC-type by\nG\bS\nand refer to G\bS as the semi-graph of anabelioids of pro-Σ\nPSC-type obtained from G by resolving S [cf. Fig. 4 below].\nThus, one has a natural morphism\nG\bS −→ G\nof semi-graphs of anabelioids.\nRemark 2.5.1.\n(i) Let S1 ⊆ S2 ⊆ Node(G) be subsets of Node(G). Then it follows\nimmediately from the various definitions involved that if S2 is\nnot of separating type [cf. Definition 2.5, (i)], then S1 is not of\nseparating type.\n(ii) Let v ∈ Vert(G) be a vertex of G. Then one may verify eas-\nily that there exists a unique sub-semi-graph of PSC-type [cf.\nDefinition 2.2, (i)] Gv of G such that the set of vertices of\nGv is equal to {v}. Moreover, one may also verify easily that\nNode(G|Gv ) [cf. Definition 2.2, (ii)] is not of separating type [cf.\nDefinition 2.5, (i)], relative to G|Gv , and that the semi-graph\nof anabelioids of pro-Σ PSC-type\n(G|Gv )\bNode(G|Gv )\n[cf. Definition 2.5, (ii)] is naturally isomorphic to G|v [cf. Def-\ninition 2.1, (iii)].\nCombinatorial anabelian topics I 39\nFigure 4: Resolution\nG e\n×\n×\n×\n×\n⇓\nG\u0013{e}\n×\n×\n40 Yuichiro Hoshi and Shinichi Mochizuki\nDefinition 2.6.\n(i) Let S ⊆ VCN(G) be a subset of VCN(G). Then we shall denote\nby\nAutS(G) ⊆ Aut(G)\nthe [closed] subgroup of Aut(G) consisting of automorphisms α\nof G such that the automorphism of the underlying semi-graph\nG of G induced by α preserves S and by\nAut|S|(G) ⊆ AutS(G)\nthe [closed] subgroup of Aut(G) consisting of automorphisms\nα of G such that the automorphism of the underlying semi-\ngraph G of G induced by α preserves and induces the identity\nautomorphism of S. Moreover, we shall write\nAut|grph|(G) def= Aut|VCN(G)|(G) .\n(ii) Let H ⊆ ΠG be a closed subgroup of ΠG . Then we shall denote\nby\nOutH(ΠG) ⊆ Out(ΠG)\nthe [closed] subgroup of Out(ΠG) consisting of outomorphisms\n[cf. the discussion entitled “Topological groups” in §0] of ΠG\nwhich preserve the ΠG-conjugacy class of H ⊆ ΠG . Moreover,\nwe shall denote by\nAutH(G) def= Aut(G) ∩OutH(ΠG) .\n(iii) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)]\nof G. Then VCN(G|H) [cf. Definition 2.2, (ii)] may be regarded\nas a subset of VCN(G). We shall write\nAut|H|(G) def= Aut|VCN(G|H)|(G)\n⊆ AutH(G) def= AutVCN(G|H)(G)\n= AutVert(G|H)(G) .\nProposition 2.7 (Subgroups determined by sets of compo-\nnents). Let S ⊆ VCN(G) be a nonempty subset of VCN(G). Then:\nCombinatorial anabelian topics I 41\n(i) It holds that\nAut|S|(G) =\n⋂\nz∈S\nAutΠz (G)\n— where we use the notation Πz to denote a VCN-subgroup\n[cf. Definition 2.1, (i)] of ΠG associated to z ∈ VCN(G).\n(ii) It holds that\nAut|grph|(G) =\n⋂\nz∈VCN(G)\nOutΠz (ΠG)\n— where we use the notation Πz to denote a VCN-subgroup of\nΠG associated to z ∈ VCN(G).\n(iii) The closed subgroups Aut|S|(G), AutS(G) ⊆ Aut(G) are open\nin Aut(G). Moreover, the closed subgroup Aut|S|(G) ⊆ AutS(G)\nis normal in AutS(G). In particular, Aut|grph|(G) ⊆ Aut(G)\nis normal in Aut(G).\nProof. Assertion (i) follows immediately from [CmbGC], Proposi-\ntion 1.2, (i). Next, we verify assertion (ii). It follows immediately from\n[CmbGC], Proposition 1.5, (ii), that the right-hand side of the equality\nin the statement of assertion (ii) is contained in Aut(G). Thus, asser-\ntion (ii) follows immediately from assertion (i). Assertion (iii) follows\nimmediately from the finiteness of the semi-graph G, together with the\nvarious definitions involved. Q.E.D.\nDefinition 2.8 (cf. the operation (Op4) discussed at the beginning\nthe present §2). Let S ⊆ Node(G) be a subset of Node(G). Then we\ndefine the semi-graph of anabelioids of pro-Σ PSC-type\nG\u0003S\nas follows:\n(i) We take Cusp(G\u0003S) def= Cusp(G).\n(ii) We take Node(G\u0003S) def= Node(G) \\ S.\n(iii) We take Vert(G\u0003S) to be the set of connected components of\nthe semi-graph obtained from G by omitting the edges e ∈\nEdge(G) \\ S. Alternatively, one may take Vert(G\u0003S) to be\n42 Yuichiro Hoshi and Shinichi Mochizuki\nthe set of equivalence classes of elements of Vert(G) with re-\nspect to the equivalence relation “∼” defined as follows: for v,\nw ∈ Vert(G), v ∼ w if either v = w or there exist n elements\ne1, · · · , en ∈ S of S and n+ 1 vertices v0, v1, · · · , vn ∈ Vert(G)\nof G such that v0 def= v, vn def= w, and, for 1 ≤ i ≤ n, it holds\nthat V(ei) = {vi−1, vi}.\n(iv) For each branch b of an edge e ∈ Edge(G\u0003S) (= Edge(G) \\ S\n— cf. (i), (ii)) and each vertex v ∈ Vert(G\u0003S) of G\u0003S , b abuts,\nrelative to G\u0003S, to v if b abuts, relative to G, to an element of\nthe equivalence class v [cf. (iii)].\n(v) For each edge e ∈ Edge(G\u0003S) (= Edge(G) \\ S — cf. (i), (ii))\nof G\u0003S , we take the anabelioid of G\u0003S corresponding to e ∈\nEdge(G\u0003S) to be Ge [cf. Definition 2.1, (ii)].\n(vi) Let v ∈ Vert(G\u0003S) be a vertex of G\u0003S . Then one verifies\neasily that there exists a unique sub-semi-graph of PSC-type\n[cf. Definition 2.2, (i)] Hv of G such that the set of vertices of\nHv consists of the elements of the equivalence class v [cf. (iii)].\nWrite\nTv\ndef\n= Node(G|Hv ) \\ (S ∩Node(G|Hv ))\n[cf. Definition 2.2, (ii)]. Then we take the anabelioid of G\u0003S\ncorresponding to v ∈ Vert(G\u0003S) to be the anabelioid deter-\nmined by the finite étale coverings of\n(G|Hv )\bTv\n[cf. Definition 2.5, (ii)] of degree a product of primes ∈ Σ.\n(vii) Let b be a branch of an edge e ∈ Edge(G\u0003S) (= Edge(G)\\S —\ncf. (i), (ii)) that abuts to a vertex v ∈ Vert(G\u0003S). Then since b\nabuts to v, one verifies easily that there exists a unique vertex w\nof G which belongs to the equivalent class v [cf. (iii)] such that b\nabuts to w relative to G. We take the morphism of anabelioids\nassociated to b, relative to G\u0003S , to be the morphism naturally\ndetermined by the morphism of anabelioids\nGe → Gw\ncorresponding to the branch b relative to G and the morphism\nof semi-graphs of anabelioids of pro-Σ PSC-type\nG|w → (G|Hv )\bTv\nCombinatorial anabelian topics I 43\n[cf. (vi); Definition 2.1, (iii)]. Here, we recall that the anabe-\nlioid obtained by considering the connected finite étale cov-\nerings of G|w may be naturally identified with Gw [cf. Re-\nmark 2.1.1].\nWe shall refer to this semi-graph of anabelioids of pro-Σ PSC-type G\u0003S\nas the generization of G with respect to S [cf. Fig. 5 below].\nFigure 5: Generization\nG\ne\n⇓\n×\n×\n×\n×\nG\u0004{e}\nRemark 2.8.1. It follows immediately from the various definitions\ninvolved that if G is of type (g, r) [cf. Definition 2.3, (i)], then the\ngenerization G\u0003Node(G) of G with respect to Node(G) is isomorphic to\nGmodelg,r [cf. Definition 2.3, (v)].\nProposition 2.9 (Specialization outer isomorphisms). Let S\n⊆ Node(G) be a subset of Node(G). Write ΠG\u0003S for the [pro-Σ] funda-\nmental group of the generization G\u0003S of G with respect to S [cf. Defi-\nnition 2.8]. Then the following hold:\n44 Yuichiro Hoshi and Shinichi Mochizuki\n(i) There exists a natural outer isomorphism of profinite\ngroups\nΦG\u0003S : ΠG\u0003S\n∼−→ ΠG\nwhich satisfies the following three conditions:\n(1) ΦG\u0003S induces a bijection between the set of cuspidal sub-\ngroups of ΠG\u0003S and the set of cuspidal subgroups of ΠG.\n(2) ΦG\u0003S induces a bijection between the set of nodal sub-\ngroups of ΠG\u0003S and the set of nodal subgroups of ΠG as-\nsociated to the elements of Node(G) \\ S.\n(3) Let v ∈ Vert(G\u0003S) be a vertex of G\u0003S; Hv, Tv as in\nDefinition 2.8, (vi). Then ΦG\u0003S induces a bijection be-\ntween the ΠG\u0003S -conjugacy class of any verticial subgroup\nΠv ⊆ ΠG\u0003S of ΠG\u0003S associated to v ∈ Vert(G\u0003S) and the\nΠG-conjugacy class of subgroups determined by the image\nof the outer homomorphism\nΠ(G|Hv )\u0004Tv −→ ΠG\ninduced by the natural morphism (G|Hv )\bTv → G [cf. Def-\ninitions 2.2, (ii); 2.5, (ii)] of semi-graphs of anabelioids\nof pro-Σ PSC-type.\nMoreover, any two outer isomorphisms ΠG\u0003S\n∼→ ΠG that sat-\nisfy the above three conditions differ by composition with a\ngraphic [cf. [CmbGC], Definition 1.4, (i)] outomorphism [cf.\nthe discussion entitled “Topological groups” in §0] of ΠG\u0003S .\n(ii) The isomorphism\nOut(ΠG)\n∼−→ Out(ΠG\u0003S )\ninduced by the natural outer isomorphism of (i) determines an\ninjection\nAutS(G) ↪→ Aut(G\u0003S)\n[cf. Definition 2.6, (i)].\nProof. First, we verify assertion (i). An outer isomorphism that\nsatisfies the three conditions of assertion (i) may be obtained by ob-\nserving that, after sorting through the various definitions involved, a\nfinite étale covering of G\u0003S amounts to the same data as a finite étale\ncovering of G. The final portion of assertion (i) follows immediately,\nin light of the three conditions in the statement of assertion (i), from\nCombinatorial anabelian topics I 45\n[CmbGC], Proposition 1.5, (ii). This completes the proof of assertion\n(i). Assertion (ii) follows immediately from [CmbGC], Proposition 1.5,\n(ii), together with the three conditions in the statement of assertion (i).\nThis completes the proof of Proposition 2.9. Q.E.D.\nDefinition 2.10. Let S ⊆ Node(G) be a subset of Node(G). Write\nΠG\u0003S for the [pro-Σ] fundamental group of the generization G\u0003S of G\nwith respect to S [cf. Definition 2.8]. Then we shall refer to the natural\nouter isomorphism\nΦG\u0003S : ΠG\u0003S\n∼−→ ΠG\nobtained in Proposition 2.9, (i), as the specialization outer isomorphism\nwith respect to S.\nProposition 2.11 (Commensurable terminality of closed\nsubgroups determined by certain semi-graphs). Let H be a\nsub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G and S ⊆\nNode(G|H) [cf. Definition 2.2, (ii)] a subset of Node(G|H) that is not of\nseparating type [cf. Definition 2.5, (i)]. Then the natural morphism\n(G|H)\bS → G [cf. Definitions 2.2, (ii); 2.5, (ii)] of semi-graphs of an-\nabelioids of pro-Σ PSC-type determines an outer injection of profinite\ngroups\nΠ(G|H)\u0004S ↪→ ΠG .\nMoreover, the image of this outer injection is commensurably termi-\nnal in ΠG [cf. the discussion entitled “Topological groups” in §0].\nProof. Write H def= (G|H)\bS and T def= Node(G|H) \\ S. Note that it\nfollows from the definition of G|H that T may be regarded as the subset\nof Node(G) determined by Node(H); for simplicity, we shall identify T\nwith Node(H). Now it follows immediately from the definition of “G\u0003T ”\nthat the composite\nΠH\nΦH,S−→ ΠG\nΦ\n−1\nG\u0003T∼−→ ΠG\u0003T\nfactors through a verticial subgroup Πv ⊆ ΠG\u0003T of ΠG\u0003T associated to\na vertex v ∈ Vert(G\u0003T ), and that the composite\nΠH −→ Π(G\u0003T )|v\nof the resulting outer homomorphism ΠH → Πv [which is well-defined in\nlight of the commensurable terminality of Πv in ΠG\u0003S — cf. [CmbGC],\n46 Yuichiro Hoshi and Shinichi Mochizuki\nProposition 1.2, (ii)] and the natural outer isomorphism Πv \f Π(G\u0003T )|v\n[cf. Remark 2.1.1] may be identified with “Φ−1H\u0003T ” [cf. Definition 2.10].\nThus, Proposition 2.11 follows immediately from the fact that ΦH\u0003T\nis an outer isomorphism, together with the fact that Πv ⊆ ΠG\u0003S is\ncommensurably terminal in ΠG\u0003S [cf. [CmbGC], Proposition 1.2, (ii)].\nThis completes the proof of Proposition 2.11. Q.E.D.\nLemma 2.12 (Restrictions of outomorphisms). Let H ⊆ ΠG\nbe a closed subgroup of ΠG which is normally terminal [cf. the discus-\nsion entitled “Topological groups” in §0] and α ∈ OutH(ΠG) [cf. Defini-\ntion 2.6, (ii)]. Then the following hold:\n(i) There exists a lifting α̃ ∈ Aut(ΠG) of α such that α̃ preserves\nthe closed subgroup H ⊆ ΠG. Moreover, such a lifting α̃ is\nuniquely determined up to composition with an H-inner\nautomorphism of ΠG.\n(ii) Write αH for the outomorphism [cf. the discussion entitled\n“Topological groups” in §0] of H determined by the restriction\nof a lifting α̃ as obtained in (i) to the closed subgroup H ⊆ ΠG.\nThen the map\nOutH(ΠG) −→ Out(H)\ngiven by assigning α \u0006→ αH is a homomorphism.\n(iii) The homomorphism\nOutH(ΠG) −→ Out(H)\nobtained in (ii) depends only on the conjugacy class of the\nclosed subgroup H ⊆ ΠG, i.e., if we write Hγ def= γ ·H · γ−1 for\nγ ∈ ΠG, then the diagram\nOutH(ΠG) −−−−→ Out(H)∥∥∥ ⏐⏐\b\nOutH\nγ\n(ΠG) −−−−→ Out(Hγ)\n— where the upper (respectively, lower) horizontal arrow is the\nhomomorphism given by mapping α \u0006→ αH (respectively, α \u0006→\nαHγ ), and the right-hand vertical arrow is the isomorphism\nCombinatorial anabelian topics I 47\nobtained by mapping φ ∈ Out(H) to\nHγ\nInn(γ−1)\n∼−→ H\nφ\n∼−→ H\nInn(γ)\n∼−→ Hγ\n— commutes.\nProof. Assertion (i) follows immediately from the normal termi-\nnality of H in ΠG . Assertion (ii) follows immediately from assertion\n(i). Assertion (iii) follows immediately from the various definitions in-\nvolved. Q.E.D.\nDefinition 2.13. Let H ⊆ ΠG be a [closed] subgroup of ΠG which\nis normally terminal [cf. the discussion entitled “Topological groups” in\n§0]. Then we shall write\nOut|H|(ΠG) ⊆ OutH(ΠG)\nfor the closed subgroup of OutH(ΠG) consisting of outomorphisms [cf.\nthe discussion entitled “Topological groups” in §0] α of ΠG such that the\nimage αH of α via the homomorphism Out\nH(G) → Out(H) obtained in\nLemma 2.12, (ii), is trivial. Also, we shall write\nAut|H|(G) def= Out|H|(ΠG) ∩Aut(G) .\nDefinition 2.14.\n(i) Let T ⊆ Cusp(G) be an omittable [cf. Definition 2.4, (i)] sub-\nset of Cusp(G). Write ΠG•T for the [pro-Σ] fundamental group\nof G•T [cf. Definition 2.4, (ii)] and NT ⊆ ΠG for the nor-\nmal closed subgroup of ΠG topologically normally generated\nby the cuspidal subgroups of ΠG associated to elements of T .\nThen one verifies easily that the natural outer isomorphism\nΠG\/NT\n∼→ ΠG•T [cf. Definition 2.4, (ii)] induces a homomor-\nphism OutNT (ΠG) → Out(ΠG•T ) that fits into a commutative\ndiagram\nAutT (G) −−−−→ Aut(G•T )⏐⏐\b ⏐⏐\b\nOutNT (ΠG) −−−−→ Out(ΠG•T )\n48 Yuichiro Hoshi and Shinichi Mochizuki\n— where the vertical arrows are the natural injections. For\nα ∈ OutNT (ΠG), we shall write\nαG•T ∈ Out(ΠG•T )\nfor the image of α via the lower horizontal arrow in the above\ncommutative diagram. If, moreover, α ∈ AutT (G), then, in\nlight of the injectivity of the right-hand vertical arrow in the\nabove diagram, we shall write [by abuse of notation]\nαG•T ∈ Aut(G•T )\nfor the image of α via the upper horizontal arrow in the above\ncommutative diagram.\n(ii) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)]\nof G and S ⊆ Node(G|H) [cf. Definition 2.2, (ii)] a subset of\nNode(G|H) that is not of separating type [cf. Definition 2.5, (i)].\nWrite Π(G|H)\u0004S for the [pro-Σ] fundamental group of (G|H)\bS\n[cf. Definition 2.5, (ii)]. Then the natural outer homomor-\nphism Π(G|H)\u0004S → ΠG is an outer injection whose image is\ncommensurably terminal [cf. Proposition 2.11]. Thus, it fol-\nlows from Lemma 2.12, (iii), that we have a homomorphism\nOut\nΠ(G|H)\u0004S (ΠG) → Out(Π(G|H)\u0004S ) that fits into a commuta-\ntive diagram\nAutH\bS(G) def= AutH(G) ∩AutS(G) −−−−→ Aut((G|H)\bS)⏐⏐\b ⏐⏐\b\nOut\nΠ(G|H)\u0004S (ΠG) −−−−→ Out(Π(G|H)\u0004S )\n— where the vertical arrows are the natural injections. For\nα ∈ OutΠ(G|H)\u0004S (ΠG), we shall write\nα(G|H)\u0004S ∈ Out(Π(G|H)\u0004S )\nfor the image of α via the lower horizontal arrow in the above\ncommutative diagram. If, moreover, α ∈ AutH\bS(G), then, in\nlight of the injectivity of the right-hand vertical arrow in the\nabove diagram, we shall write [by abuse of notation]\nα(G|H)\u0004S ∈ Aut((G|H)\bS)\nfor the image of α via the upper horizontal arrow in the above\ncommutative diagram. Finally, if T ⊆ Cusp((G|H)\bS) is an\nCombinatorial anabelian topics I 49\nomittable subset of Cusp((G|H)\bS), then we shall write\nAutH\bS•T (G) ⊆ AutH\bS(G)\nfor the inverse image of the closed subgroup AutT ((G|H)\bS) ⊆\nAut((G|H)\bS) of Aut((G|H)\bS) in AutH\bS(G) via the upper\nhorizontal arrow AutH\bS(G) → Aut((G|H)\bS) of the above\ncommutative diagram; thus, we have a natural homomorphism\n[cf. (i)]\nAutH\bS•T (G) −→ Aut(((G|H)\bS)•T )\nα \u0006→ α((G|H)\u0004S)•T .\n(iii) Let z ∈ VCN(G) be an element of VCN(G) and Πz ⊆ ΠG\na VCN-subgroup of ΠG associated to z ∈ VCN(G). Then it\nfollows from [CmbGC], Proposition 1.2, (ii), that the closed\nsubgroup Πz ⊆ ΠG is commensurably terminal. Thus, it fol-\nlows from Lemma 2.12, (iii), that we obtain a homomorphism\nOutΠz (ΠG) → Out(Πz) that fits into a commutative diagram\nAut{z}(G) −−−−→ Aut(Gz)⏐⏐\b ⏐⏐\b\u0005\nOutΠz (ΠG) −−−−→ Out(Πz)\n— where the left-hand vertical arrow is injective, and the right-\nhand vertical arrow is an isomorphism. For α ∈ OutΠz (ΠG),\nwe shall write\nαz ∈ Out(Πz)\nfor the image of α via the lower horizontal arrow in the above\ncommutative diagram.\n§3. Synchronization of cyclotomes\nIn the present §, we introduce and study the notion of the second\ncohomology group with compact supports of a semi-graph of anabelioids\nof PSC-type [cf. Definition 3.1, (ii), (iii) below]. In particular, we show\nthat such cohomology groups are compatible with graph-theoretic lo-\ncalization [cf. Definition 3.4, Lemma 3.5 below]. This leads naturally\n50 Yuichiro Hoshi and Shinichi Mochizuki\nto a discussion of the phenomenon of synchronization among the vari-\nous cyclotomes [cf. Definition 3.8 below] arising from a semi-graph of\nanabelioids of PSC-type [cf. Corollary 3.9 below].\nLet Σ be a nonempty set of prime numbers and G a semi-graph of\nanabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph\nof G, ΠG for the [pro-Σ] fundamental group of G, and G̃ → G for the\nuniversal covering of G corresponding to ΠG .\nDefinition 3.1. Let M be a finitely generated ẐΣ-module and v ∈\nVert(G) a vertex of G.\n(i) We shall write\nH2(G,M) def= H2(ΠG ,M)\n— where we regardM as being equipped with the trivial action\nof ΠG — and refer toH2(G,M) as the second cohomology group\nof G.\n(ii) Let s be a section of the natural surjection Cusp(G̃) \u0002\nCusp(G). Given a central extension of profinite groups\n1 −→M −→ E −→ ΠG −→ 1 ,\nand a cusp e ∈ Cusp(G), we shall refer to a section of this\nextension over the edge-like subgroup Πs(e) ⊆ ΠG of ΠG deter-\nmined by s(e) ∈ Cusp(G̃) as a trivialization of this extension\nat the cusp e. We shall write\nH2c (G,M)\nfor the set of equivalence classes\n[E, (ιe : Πs(e) → E)e∈Cusp(G)]\nof collections of data (E, (ιe : Πs(e) → E)e∈Cusp(G)) as follows:\n(a) E is a central extension of profinite groups\n1 −→M −→ E −→ ΠG −→ 1 ;\n(b) for each e ∈ Cusp(G), ιe is a trivialization of this extension\nat the cusp e. The equivalence relation “∼” is then defined as\nfollows: for two collections of data (E, (ιe)) and (E\n′, (ι′e)), we\nshall write (E, (ιe)) ∼ (E′, (ι′e)) if there exists an isomorphism\nCombinatorial anabelian topics I 51\nof profinite groups α : E\n∼→ E′ over ΠG which induces the iden-\ntity automorphism ofM , and, moreover, for each e ∈ Cusp(G),\nmaps ιe to ι\n′\ne. We shall refer to H\n2\nc (G,M) as the second coho-\nmology group with compact supports of G.\n(iii) We shall write\nH2c (v,M)\ndef\n= H2c (G|v,M)\n[cf. (ii); Definition 2.1, (iii)] and refer to H2c (v,M) as the\nsecond cohomology group with compact supports of v.\n(iv) The set H2c (G,M) is equipped with a natural structure of ẐΣ-\nmodule defined as follows:\n• Let [E, (ιe)], [E′, (ι′e)] ∈ H2c (G,M). Then the fiber prod-\nuct E ×ΠG E′ of the surjections E \u0002 ΠG , E′ \u0002 ΠG is\nan extension of ΠG by M ×M . Thus, the quotient S of\nE ×ΠG E′ by the image of the composite\nM ↪→ M ×M ↪→ E ×ΠG E′\nm \u0006→ (m,−m)\nis an extension of ΠG byM . On the other hand, it follows\nfrom the definition of S that for each e ∈ Cusp(G), the sec-\ntions ιe and ι\n′\ne naturally determine a section ι\nS\ne : Πs(e) → S\nover Πs(e). Thus, we define\n[E, (ιe)] + [E\n′, (ι′e)]\ndef\n= [S, (ιSe )] .\nHere, one may verify easily that the equivalence class\n[S, (ιSe )] depends only on the equivalence classes [E, (ιe)],\n[E′, (ι′e)], and that this definition of “+” determines a\nmodule structure on H2c (G,M).\n• Let [E, (ιe)] ∈ H2c (G,M) be an element of H2c (G,M) and\na ∈ ẐΣ. Now the composite E ×M\npr1\u0002 E \u0002 ΠG deter-\nmines an extension of ΠG by M ×M . Thus, the quotient\nP of E ×M by the image of the composite\nM ↪→ M ×M ↪→ E ×M\nm \u0006→ (m,−am)\nis an extension of ΠG byM . On the other hand, it follows\nfrom the definition of P that for each e ∈ Cusp(G), the\n52 Yuichiro Hoshi and Shinichi Mochizuki\nsection ιe and the zero homomorphism Πs(e) → M natu-\nrally determine a section ιPe : Πs(e) → P over Πs(e). Thus,\nwe define\na · [E, (ιe)] def= [P, (ιPe )] .\nHere, one may verify easily that the equivalence class\n[P, (ιPe )] depends only on the equivalence class [E, (ιe)]\nand a ∈ ẐΣ, and that this definition of “·” determines a\nẐΣ-module structure on H2c (G,M).\nFinally, we note that it follows from Lemma 3.2 below that\nthe ẐΣ-module “H2c (G,M)” does not depend on the choice of\nthe section s. More precisely, the ẐΣ-module “H2c (G,M)” is\nuniquely determined by G andM up to the natural isomorphism\nobtained in Lemma 3.2.\nLemma 3.2 (Independence of the choice of section). Let M\nbe a finitely generated ẐΣ-module and s, s′ sections of the natural sur-\njection Cusp(G̃) \u0002 Cusp(G). Write H2c (G,M, s), H2c (G,M, s′) for the\nẐΣ-modules “H2c (G,M)” defined in Definition 3.1 by means of the sec-\ntions s, s′, respectively. Then there exists a natural isomorphism of\nẐΣ-modules\nH2c (G,M, s)\n∼−→ H2c (G,M, s′) .\nProof. Let [E, (ιe)] ∈ H2c (G,M, s) be an element of H2c (G,M, s).\nNow it follows from the various definitions involved that, for each e ∈\nCusp(G), there exists an element γe ∈ ΠG such that Πs′(e) = γe · Πs(e) ·\nγ−1e . For each e ∈ Cusp(G), fix a lifting γ̃e ∈ E of γe ∈ ΠG and write\nι′e : Πs′(e) → E for the section given by\nΠs′(e) = γe ·Πs(e) · γ−1e −→ E\nγeaγ\n−1\ne \u0006→ γ̃eιe(a)γ̃−1e .\nThen it follows immediately from the fact that M ⊆ E is contained in\nthe center Z(E) of E that this section ι′e does not depend on the choice of\nthe lifting γ̃e ∈ E of γe ∈ ΠG . Moreover, it follows immediately from the\nvarious definitions involved that the assignment “[E, (ιe)] \u0006→ [E, (ι′e)]”\ndetermines an isomorphism of ẐΣ-modules\nH2c (G,M, s)\n∼−→ H2c (G,M, s′) .\nThis completes the proof of Lemma 3.2. Q.E.D.\nCombinatorial anabelian topics I 53\nLemma 3.3 (Exactness of certain sequences). Let M be a\nfinitely generated ẐΣ-module. Suppose that Cusp(G) \u0004= ∅. Then the\nnatural inclusions Πe ↪→ ΠG — where e ranges over the cusps of G, and,\nfor each cusp e ∈ Cusp(G), we use the notation Πe to denote an edge-like\nsubgroup of ΠG associated to the cusp e — determine an exact sequence\nof ẐΣ-modules\nHom\nẐΣ\n(ΠabG ,M) −→\n⊕\ne∈Cusp(G)\nHom\nẐΣ\n(Πe,M) −→ H2c (G,M) −→ 0 .\nProof. Let s be a section of the natural surjection Cusp(G̃) \u0002\nCusp(G). Then given an element\n(φe : Πe →M)e∈Node(G) ∈\n⊕\ne∈Cusp(G)\nHom\nẐΣ\n(Πe,M) ,\none may construct an element\n[M ×ΠG (\npr2\u0002 ΠG), (ιe : Πs(e) →M ×ΠG)e∈Node(G)]\n— where we write ιe : Πs(e) → M × ΠG for the section determined by\nφe : Πs(e) → M and the natural inclusion Πs(e) ↪→ ΠG — of H2c (G,M).\nIn particular, we obtain a map\n⊕\ne∈Cusp(G) HomẐΣ(Πe,M) → H2c (G,M),\nwhich, as is easily verified, is a homomorphism of ẐΣ-modules. Now the\nexactness of the sequence in question follows immediately from the fact\nthat ΠG is free pro-Σ [cf. [CmbGC], Remark 1.1.3]. This completes the\nproof of Lemma 3.3. Q.E.D.\nDefinition 3.4. Let M be a finitely generated ẐΣ-module.\n(i) Let E be a semi-graph of anabelioids. Denote by VCN(E) the\nset of components of E [i.e., the set of vertices and edges of E ]\nand, for each z ∈ VCN(E), by ΠEz the fundamental group of\nthe anabelioid Ez of E corresponding to z ∈ VCN(E). Then we\ndefine a central extension of G by M to be a collection of data\n(E , α = (αz : M ↪→ ΠEz )z∈VCN(E), β : E\/α\n∼→ G)\nas follows:\n(a) For each z ∈ VCN(E), αz : M ↪→ ΠEz is an injective ho-\nmomorphism of profinite groups whose image is contained\n54 Yuichiro Hoshi and Shinichi Mochizuki\nin the center Z(ΠEz ) of ΠEz . [Thus, the image of αz is a\nnormal closed subgroup of ΠEz .]\n(b) For each branch b of an edge e that abuts to a vertex v of\nE , we assume that the outer homomorphism ΠEe → ΠEv\nassociated to b is injective and fits into a commutative\ndiagram of [outer] homomorphisms of profinite groups\nM M\nαe\n⏐⏐\b ⏐⏐\bαv\nΠEe −−−−→ ΠEv\n— i.e., where the lower horizontal arrow is the outer in-\njection associated to b.\n(c) Write E\/α for the semi-graph of anabelioids defined as\nfollows: We take the underlying semi-graph of E\/α to be\nthe underlying semi-graph of E ; for each z ∈ VCN(E), we\ntake the anabelioid (E\/α)z of E\/α corresponding to z ∈\nVCN(E) to be the anabelioid determined by the profinite\ngroup ΠEz\/Im(αz) [cf. condition (a)]; for each branch b\nof an edge e that abuts to a vertex v of E , we take the\nassociated morphism of anabelioids (E\/α)e → (E\/α)v to\nbe the morphism of anabelioids naturally determined by\nthe morphism Ee → Ev associated, relative to E , to b [cf.\ncondition (b)].\n(d) β : E\/α ∼→ G is an isomorphism of semi-graphs of anabe-\nlioids.\nThere is an evident notion of isomorphisms of central exten-\nsions of G by M . Also, given a central extension of G by M ,\nand a section s of the natural surjection Cusp(G̃) \u0002 Cusp(G),\nthere is an evident notion of trivialization of the given cen-\ntral extension of G by M at a cusp of G [cf. the discussion of\nDefinition 3.1, (ii), (iv)].\n(ii) Let\n1 −→M −→ E −→ ΠG −→ 1\nbe a central extension of ΠG by M . Then we shall define a\nsemi-graph of anabelioids\nGE\nCombinatorial anabelian topics I 55\n— which we shall refer to as the semi-graph of anabelioids\nassociated to the central extension E — as follows: We take the\nunderlying semi-graph of GE to be the underlying semi-graph of\nG. We take the anabelioid of GE corresponding to z ∈ VCN(G)\nto be the anabelioid determined by the fiber product E×ΠG Πz\nof the surjection E → ΠG and a natural inclusion Πz ↪→ ΠG —\nwhere we use the notation Πz ⊆ ΠG to denote a VCN-subgroup\n[cf. Definition 2.1, (i)] of ΠG associated to z ∈ VCN(G); for\neach branch b of an edge e that abuts to a vertex v of G, if\nwe write (GE)v, (GE)e for the anabelioids of GE corresponding\nto v, e, respectively, then we take the morphism of anabelioids\n(GE)e → (GE)v associated to the branch b to be the morphism\nnaturally determined by the morphism of anabelioids Ge → Gv\nassociated, relative to G, to b.\n(iii) In the notation of (ii), one may verify easily that the semi-\ngraph of anabelioids GE associated to the central extension E\nis equipped with a natural structure of central extension of G\nby M . More precisely, for each z ∈ VCN(G), if we denote\nby αz : M ↪→ Π(GE)z = E ×ΠG Πz the homomorphism deter-\nmined by the natural inclusion M ↪→ E and the trivial homo-\nmorphism M → Πz, then there exists a natural isomorphism\nβ : GE\/(αz)z∈VCN(G) ∼→ G such that the collection of data\n(GE , (αz)z∈VCN(G), β)\nforms a central extension of G by M , which we shall refer to\nas the central extension of G by M associated to the central\nextension E.\nLemma 3.5 (Graph-theoretic localizability of central exten-\nsions of fundamental groups). Let M be a finitely generated ẐΣ-\nmodule. Then the following hold:\n(i) (Exactness and centrality) Let\n(E , α = (αz : M ↪→ ΠEz )z∈VCN(E), β : E\/α\n∼→ G) (‡1)\n56 Yuichiro Hoshi and Shinichi Mochizuki\nbe a central extension of G by M [cf. Definition 3.4, (i)].\nWrite ΠE for the pro-Σ fundamental group of E, i.e., the max-\nimal pro-Σ quotient of the fundamental group of E [cf. the dis-\ncussion preceding [SemiAn], Definition 2.2]. Then the compos-\nite E → E\/α\nβ\n∼→ G determines an exact sequence of profinite\ngroups\n1 −→M −→ ΠE −→ ΠG −→ 1 (‡2)\nwhich is central.\n(ii) (Natural isomorphism I) In the notation of (i), the central\nextension of G byM associated to the central extension (‡2) [cf.\nDefinition 3.4, (iii)] is naturally isomorphic, as a central\nextension of G by M , to (‡1).\n(iii) (Natural isomorphism II) Let\n1 −→M −→ E −→ ΠG −→ 1\nbe a central extension of ΠG by M . Then the pro-Σ fun-\ndamental group of the semi-graph of anabelioids GE associated\nto the central extension E [cf. Definition 3.4, (ii)] — i.e., the\nmaximal pro-Σ quotient of the fundamental group of GE — is\nnaturally isomorphic, over ΠG, to E.\n(iv) (Equivalence of categories) The correspondences of (i), (ii),\n(iii) determine a natural equivalence of categories between\nthe category of central extensions of G byM and the category of\ncentral extensions of ΠG by M . [Here, we take the morphisms\nin both categories to be the isomorphisms of central exten-\nsions of the sort under consideration.] Moreover, this equiva-\nlence extends to a similar natural equivalence of categories\nbetween categories of central extensions equipped with trivial-\nizations at the cusps of G [cf. Definitions 3.1, (ii); 3.4,\n(i)].\nProof. First, we verify assertion (i). If Node(G) = ∅, then assertion\n(i) is immediate; thus, suppose that Node(G) \u0004= ∅. For each connected\nfinite étale covering E ′ → E of E , denote by ΠE′ the pro-Σ fundamental\ngroup of E ′, by VCN(E ′) the set of components of E ′ [i.e., the set of\nvertices and edges of E ′], and by Vert(E ′) the set of vertices of E ′; for\neach z ∈ VCN(E ′), denote by E ′z the anabelioid of E ′ corresponding to\nz ∈ VCN(E ′) and by ΠE′z the fundamental group of E ′z. Now we claim\nthat\nCombinatorial anabelian topics I 57\n(∗1): the composite in question E → E\/α ∼→ G in-\nduces an isomorphism between the underlying semi-\ngraphs, as well as an outer surjection ΠE \u0002 ΠG .\nIndeed, the fact that the composite in question determines an isomor-\nphism between the underlying semi-graphs follows from conditions (c),\n(d) of Definition 3.4, (i). In particular, we obtain a bijection VCN(E) ∼→\nVCN(G). Now for each z ∈ VCN(E) ∼→ VCN(G), again by conditions\n(c), (d) of Definition 3.4, (i), the composite E → E\/α ∼→ G induces\nan outer surjection ΠEz \u0002 Πz, where we use the notation Πz ⊆ ΠG\nto denote a VCN-subgroup [cf. Definition 2.1, (i)] of ΠG associated to\nz ∈ VCN(G). Therefore, in light of the isomorphism verified above be-\ntween the semi-graphs of E and G, one may verify easily that the natural\nouter homomorphism ΠE → ΠG is surjective. This completes the proof\nof the claim (∗1).\nFor each vertex v ∈ Vert(E) ∼→ Vert(G) [cf. claim (∗1)], it follows\nfrom the assumption that Node(G) \u0004= ∅ that any verticial subgroup\nΠv ⊆ ΠG of ΠG associated to a vertex v ∈ Vert(G) is a free pro-Σ\ngroup [cf. [CmbGC], Remark 1.1.3]; thus, there exists a section of the\nnatural surjection ΠEv \u0002 Πv. Now for each vertex v ∈ Vert(G), let\nus fix such a section of the natural surjection ΠEv \u0002 Πv, hence also\n— since the extension ΠEv of Πv by M is central [cf. condition (a) of\nDefinition 3.4, (i)] — an isomorphism tv : M × Πv ∼→ ΠEv . Let G1 → G\nbe a connected finite étale Galois covering of G and write E1 def= E ×G\nG1. Then it follows from the claim (∗1) that E1 is connected; moreover,\none may verify easily that the structure of central extension of G by\nM on E naturally determines a structure of central extension of G1 by\nM on E1, and that for each vertex v ∈ Vert(E) ∼→ Vert(G) and each\nvertex w ∈ Vert(E1) ∼→ Vert(G1) that lies over v, the normal closed\nsubgroup Π(E1)w ⊆ ΠEv corresponds to M × Πw ⊆ M × Πv relative\nto the isomorphism tv : M × Πv ∼→ ΠEv fixed above, i.e., we obtain an\nisomorphism tw : M ×Πw ∼→ Π(E1)w .\nNow for a finite quotient M \u0002 Q of M and a connected finite étale\nGalois covering G1 → G of G, we shall say that a connected finite étale\ncovering E2 → E of E satisfies the condition (†Q,G1) if the following two\nconditions are satisfied:\n(†1Q,G1) E2 → E factors through E1\ndef\n= E ×G G1 → E , the resulting\ncovering E2 → E1 is Galois, and for each vertex v ∈ VCN(E1),\nthe composite\nM ↪→ Π(E1)v → ΠE1 \u0002 ΠE1\/ΠE2\n58 Yuichiro Hoshi and Shinichi Mochizuki\nis surjective, with kernel equal to the kernel of M \u0002 Q.\n(†2Q,G1) E2 → E is Galois.\nThen we claim that\n(∗2): for any finite quotient M \u0002 Q of M and any\nconnected finite étale Galois covering G1 → G, there\nexists — after possibly replacing G1 → G by a con-\nnected finite étale Galois covering of G that factors\nthrough G1 → G — a connected finite étale covering\nof E which satisfies the condition (†Q,G1).\nIndeed, let M \u0002 Q be a finite quotient of M , G1 → G a connected finite\nétale Galois covering of G, and E1 def= E ×G G1. For each vertex v ∈\nVert(E1) ∼→ Vert(G1) [cf. the above discussion], denote by Π(E1)v \u0002 Qv\nthe quotient of Π(E1)v obtained by forming the composite\nΠ(E1)v\ntv∼←M ×Πv\npr1\u0002 M \u0002 Q .\nThus, we have a natural isomorphism Q\n∼→ Qv. Next, let e be a node of\nE1; b, b′ the two distinct branches of e; v, v′ the [not necessarily distinct]\nvertices of E1 to which b, b′ abut. Then since the quotient Q [\f Qv \f\nQv′ ] is finite, one may verify easily that — after possibly replacing G1 →\nG by a connected finite étale Galois covering of G that factors through\nG1 → G — the kernels of the two composites Π(E1)e ↪→ Π(E1)v \u0002 Qv,\nΠ(E1)e ↪→ Π(E1)v′ \u0002 Qv′ — where Π(E1)e ↪→ Π(E1)v , Π(E1)e ↪→ Π(E1)v′\nare the natural outer injections corresponding to b, b′, respectively —\ncoincide. Moreover, if we write Ne ⊆ Π(E1)e for this kernel, then it\nfollows immediately from condition (b) of Definition 3.4, (i), that the\nactions ofQ induced by the natural isomorphismsQ\n∼→ Qv ∼← Π(E1)e\/Ne,\nQ\n∼→ Qv′ ∼← Π(E1)e\/Ne on the connected finite étale Galois covering\nof (E1)e corresponding to Ne ⊆ Π(E1)e coincide. Therefore, since the\nunderlying semi-graph of E1 is finite, by applying this argument to the\nvarious nodes of E1 and then gluing the connected finite étale Galois\ncoverings of the various (E1)v’s corresponding to the quotients Π(E1)v \u0002\nQv to one another by means of Q-equivariant isomorphisms, we obtain\na connected finite étale Galois covering E2 → E1 which satisfies the\ncondition (†1Q,G1).\nWrite E02 → E for the Galois closure of the connected finite étale\ncovering E2 → E ; thus, since E1 is Galois over E , we have connected\nfinite étale Galois coverings E02 → E2 → E1 of E1. Now it follows\nCombinatorial anabelian topics I 59\nimmediately from the condition (†1Q,G1) that E2 → E1 induces an iso-\nmorphism between the underlying semi-graphs. In particular, it fol-\nlows from Lemma 3.6 below, in light of the claim (∗1), that the natural\nouter homomorphisms ΠE2 ↪→ ΠE1 \u0002 ΠG1 induce outer isomorphisms\nΠE2\/Π\nvert\nE2\n∼→ ΠE1\/ΠvertE1\n∼→ ΠG1\/ΠvertG1 \f π\ntop\n1 (G1)\nΣ, where we write\n“Πvert\n(−) ⊆ Π(−)” for the normal closed subgroup of “Π(−)” topologically\nnormally generated by the verticial subgroups and π\ntop\n1 (G1)\nΣ for the\npro-Σ completion of the [discrete] topological fundamental group of the\nunderlying semi-graph G1 of G1. On the other hand, since for each ver-\ntex v ∈ Vert(E) ∼→ Vert(G) and each vertex w ∈ Vert(E1) ∼→ Vert(G1)\nthat lies over v, the isomorphism tw : M ×Πw ∼→ Π(E1)w arises from the\nisomorphism tv : M × Πv ∼→ ΠEv , one may verify easily that the closed\nsubgroup Π(E2)w ⊆ ΠEv is normal. [Here, we regard w ∈ Vert(E1) as\nan element of Vert(E2) by the bijection Vert(E2) ∼→ Vert(E1) induced\nby E2 → E1.] In particular, it follows immediately that the connected\nfinite étale Galois covering E02 → E2 arises from a normal open sub-\ngroup of the quotient ΠE2 \u0002 ΠE2\/ΠvertE2\n∼→ π1(G1)Σ. Therefore, there\nexists a connected finite étale Galois covering G′1 → G that factors\nthrough G1 → G [and arises from a normal open subgroup of the quo-\ntient ΠG1 \u0002 πtop1 (G1)Σ] such that the connected finite étale covering\nE2 ×G1 G′1 of E is Galois. Now it follows immediately from the fact that\nE2 → E satisfies the condition (†1Q,G1) that E2 ×G1 G′1 → E satisfies both\nconditions (†1Q,G′1) and (†\n2\nQ,G′1), as desired. This completes the proof of\nthe claim (∗2).\nNext, we claim that\n(∗3): the composite E → E\/α ∼→ G, together with the\ncomposites\nM ↪→ ΠEv → ΠE\nfor v ∈ Vert(E), determine an exact sequence of profi-\nnite groups\n1 −→M −→ ΠE −→ ΠG −→ 1 .\nIndeed, it follows immediately from the claim (∗2) — by arguing as in\nthe final portion of the proof of (∗2) — that any connected finite étale\nGalois covering of E is a subcovering of a covering of E which satisfies\nthe condition (†Q,G1) for some finite quotient M \u0002 Q of M and some\nconnected finite étale Galois covering G1 of G. Therefore, the exactness of\nthe sequence in question follows immediately from the various definitions\ninvolved, together with the claim (∗1). This completes the proof of the\nclaim (∗3).\n60 Yuichiro Hoshi and Shinichi Mochizuki\nFinally, we claim that\n(∗4): the exact sequence of profinite groups\n1 −→M −→ ΠE −→ ΠG −→ 1\nof (∗3) is central, i.e., if we write ρ : ΠG → Aut(M)\nfor the representation of ΠG onM determined by this\nextension ΠE , then ρ is trivial.\nIndeed, it follows immediately from condition (a) of Definition 3.4, (i),\nthat ΠvertG ⊆ Ker(ρ), where we write ΠvertG ⊆ ΠG for the normal closed\nsubgroup of ΠG topologically normally generated by the verticial sub-\ngroups of ΠG . On the other hand, it follows immediately from con-\ndition (b) of Definition 3.4, (i), by “parallel transporting” along loops\non G, that the restriction to π\ntop\n1 (G) ⊆ πtop1 (G)Σ of the representa-\ntion [ΠG \u0002 ΠG\/ΠvertG\n∼→] πtop1 (G)Σ → Aut(M) [cf. Lemma 3.6 below]\ninduced by ρ — where we write π\ntop\n1 (G) for the [discrete] topological\nfundamental group of the semi-graph G and π\ntop\n1 (G)\nΣ for the pro-Σ\ncompletion of π\ntop\n1 (G) — is trivial. In particular, since the subgroup\nπ\ntop\n1 (G) ⊆ πtop1 (G)Σ is dense, the representation ρ is trivial, as desired.\nThis completes the proof of the claim (∗4), hence also the proof of as-\nsertion (i).\nAssertion (ii) follows immediately from the various definitions in-\nvolved. Next, we verify assertion (iii). It follows immediately from\nassertion (i), together with Definition 3.4, (iii), that if we write ΠGE\nfor the pro-Σ fundamental group of GE , then we have a natural exact\nsequence of profinite groups\n1 −→M −→ ΠGE −→ ΠG −→ 1 .\nOn the other hand, it follows immediately from the definition of GE\nthat one may construct a tautological profinite covering of GE [i.e., a\npro-object of the category B(GE) that appears in the discussion fol-\nlowing [SemiAn], Definition 2.1] equipped with a tautological action by\nE. In particular, one obtains an outer surjection ΠGE \u0002 E that is\ncompatible with the respective outer surjections to ΠG . Thus, one con-\ncludes from the “Five Lemma” that this outer surjection ΠGE \u0002 E is\nan outer isomorphism, as desired. This completes the proof of assertion\n(iii). Assertion (iv) follows immediately, in light of assertions (i), (ii),\n(iii), from the various definitions involved. This completes the proof of\nLemma 3.5. Q.E.D.\nCombinatorial anabelian topics I 61\nLemma 3.6 (Quotients by verticial subgroups). Let H be a\nsemi-graph of anabelioids. Write ΠH for the pro-Σ fundamental group of\nH [i.e., the pro-Σ quotient of the fundamental group of H] and ΠvertH ⊆\nΠH for the normal closed subgroup of ΠH topologically normally gen-\nerated by the verticial subgroups of ΠH. Then the natural injection\nΠvertH ↪→ ΠH determines an exact sequence of profinite groups\n1 −→ ΠvertH −→ ΠH −→ πtop1 (H)Σ −→ 1\n— where we write π\ntop\n1 (H)\nΣ for the pro-Σ completion of the [discrete]\ntopological fundamental group π\ntop\n1 (H) of the underlying semi-graph H\nof H.\nProof. This follows immediately from the various definitions in-\nvolved. Q.E.D.\nTheorem 3.7 (Properties of the second cohomology group\nwith compact supports). Let Σ be a nonempty set of prime num-\nbers, G a semi-graph of anabelioids of pro-Σ PSC-type, and M a finitely\ngenerated ẐΣ-module. Then the following hold:\n(i) (Change of coefficients) There exists a natural isomor-\nphism of ẐΣ-modules\nH2c (G,M)\n∼−→ H2c (G, ẐΣ)⊗ẐΣ M\nthat is functorial with respect to isomorphisms of the pair\n(G,M). If, moreover, Cusp(G) = ∅, then there exists a natu-\nral isomorphism of ẐΣ-modules\nH2c (G,M)\n∼−→ H2(G,M)\nthat is functorial with respect to isomorphisms of the pair\n(G,M).\n(ii) (Structure as an abstract profinite group) The second co-\nhomology group with compact supports H2c (G,M) of G is [non-\ncanonically] isomorphic to M .\n(iii) (Synchronization with respect to generization) Let S ⊆\nNode(G) be a subset of Node(G). Then the specialization\nouter isomorphism ΦG\u0003S : ΠG\u0003S\n∼→ ΠG with respect to S [cf.\nDefinition 2.10] determines a natural isomorphism\nH2c (G,M)\n∼−→ H2c (G\u0003S ,M)\n62 Yuichiro Hoshi and Shinichi Mochizuki\nthat is functorial with respect to isomorphisms of the triple\n(G, S,M).\n(iv) (Synchronization with respect to “surgery”) Let H be\na sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G,\nS ⊆ Node(G|H) [cf. Definition 2.2, (ii)] a subset of Node(G|H)\nthat is not of separating type [cf. Definition 2.5, (i)], and\nT ⊆ Cusp((G|H)\bS) [cf. Definition 2.5, (ii)] an omittable [cf.\nDefinition 2.4, (i)] subset of Cusp((G|H)\bS). Then there exists\na natural isomorphism — given by “extension by zero”\n—\nH2c (((G|H)\bS)•T ,M)\n∼−→ H2c (G,M)\n[cf. Definition 2.4, (ii)] that is functorial with respect to iso-\nmorphisms of the quintuple (G,H, S, T,M). In particular, for\neach vertex v ∈ Vert(G) of G, there exists a natural isomor-\nphism of ẐΣ-modules\nH2c (v,M)\n∼−→ H2c (G,M)\n[cf. Remark 2.5.1, (ii)] that is functorial with respect to iso-\nmorphisms of the triple (G, v,M).\n(v) (Homomorphisms induced by finite étale coverings) Let\nH → G be a connected finite étale covering of G. Then the\nimage of the natural homomorphism\nH2c (G,M) −→ H2c (H,M)\nis given by\n[ΠG : ΠH] ·H2c (H,M) .\nProof. Assertion (iii) follows immediately from condition (1) of\nProposition 2.9, (i).\nNext, we verify assertions (i), (ii) in the case where Cusp(G) \u0004= ∅.\nThe existence of a natural isomorphism H2c (G,M)\n∼→ H2c (G, ẐΣ)⊗ẐΣ M\nfollows immediately from Lemma 3.3. On the other hand, the fact that\nH2c (G,M) is [noncanonically] isomorphic toM follows immediately from\nLemma 3.3, together with the following well-known facts [cf. [CmbGC],\nRemark 1.1.3]:\n(A) ΠG is a free pro-Σ group.\nCombinatorial anabelian topics I 63\n(B) For any cusp e0 ∈ Cusp(G) of G, the natural homomorphism\nof ẐΣ-modules ⊕\ne∈Cusp(G)\\{e0}\nΠe −→ ΠabG\nis a split injection of free ẐΣ-modules [cf. the discussion entitled\n“Topological groups” in §0], and its image contains the image\nof Πe0 in Π\nab\nG .\nThis completes the proof of assertions (i), (ii) in the case where\nCusp(G) \u0004= ∅.\nNext, we verify assertions (i), (ii) in the case where Cusp(G) = ∅.\nThe existence of a natural isomorphism H2c (G,M)\n∼→ H2(G,M) is well-\nknown [cf., e.g., [NSW], Theorem 2.7.7]. Now it follows from assertion\n(iii) that to verify assertions (i), (ii) in the case where Cusp(G) = ∅, we\nmay assume without loss of generality — by replacing G by G\u0003Node(G)\n— that Node(G) = ∅. Then the existence of a natural isomorphism\nH2c (G,M)\n∼→ H2c (G, ẐΣ) ⊗ẐΣ M and the fact that H2c (G,M) is [non-\ncanonically] isomorphic to M follow immediately from the existence of\na natural isomorphism H2c (G,M)\n∼→ H2(G,M) and the fact that any\ncompact Riemann surface of genus \u0004= 0 is a “K(π, 1)” space [i.e., its uni-\nversal covering is contractible], together with the well-known structure\nof the second cohomology group of a compact Riemann surface. This\ncompletes the proof of assertions (i), (ii) in the case where Cusp(G) = ∅.\nNext, we verify assertion (iv) in the case where H = G and S = ∅,\ni.e., ((G|H)\bS)•T = G•T . Thus, suppose that H = G and S = ∅. Now\ndefine a homomorphism of ẐΣ-modules\nH2c (G•T ,M) −→ H2c (G,M)\nas follows: Let G̃•T → G•T be a universal covering of G•T which is\ncompatible [in the evident sense] with the universal covering G̃ → G\nof G, s• a section of the natural surjection Cusp(G̃•T ) \u0002 Cusp(G•T ),\nand [E•, (ι•e : Πs•(e) → E•)e∈Cusp(G•T )] ∈ H2c (G•T ,M) an element of\nH2c (G•T ,M). Write E for the fiber product of the surjection E• \u0002 ΠG•T\nand the natural surjection ΠG \u0002 ΠG•T [arising from the compatibility\nof the respective universal coverings]. Next, we introduce notation as\nfollows:\n• for e ∈ Cusp(G•T ) (= Cusp(G) \\ T ⊆ Cusp(G)), denote by\nιe : Πe → E — where we use the notation Πe ⊆ ΠG to de-\nnote an edge-like subgroup of ΠG associated to e such that the\n64 Yuichiro Hoshi and Shinichi Mochizuki\ncomposite Πe ↪→ ΠG \u0002 ΠG•T determines an isomorphism of\nΠe with Πs•(e) ⊆ ΠG•T — the section over Πe naturally deter-\nmined by the composite\nΠe\n∼−→ Πs•(e)\nι•e−→ E• ,\nand\n• for e ∈ Cusp(G) \\ Cusp(G•T ) (= T ⊆ Cusp(G)), denote by\nιe : Πe → E—where we use the notation Πe ⊆ ΠG to denote an\nedge-like subgroup of ΠG associated to e— the section over Πe\nnaturally determined by the trivial homomorphism Πe → E•.\nThen it follows immediately from the various definitions involved that\nthe assignment “[E•, (ι•e)e∈Cusp(G•T )] \u0006→ (E, (ιe)e∈Cusp(G))” determines a\nhomomorphism of ẐΣ-modules\nH2c (G•T ,M) −→ H2c (G,M) ,\nas desired.\nNext, we verify that this homomorphism H2c (G•T ,M) → H2c (G,M)\nis an isomorphism. First, let us observe that it follows from assertion (ii)\nthat, to verify that the homomorphism in question is an isomorphism, it\nsuffices to verify that it is surjective. The rest of the proof of assertion\n(iv) in the case where H = G and S = ∅ is devoted to verifying this\nsurjectivity. To verify the desired surjectivity, by induction on the car-\ndinality T \u0005 of the finite set T , we may assume without loss of generality\nthat T \u0005 = 1, i.e., T = {e0} for some e0 ∈ Cusp(G).\nTo verify the desired surjectivity, let [E, (ιe)e∈Cusp(G)] ∈ H2c (G,M)\nbe an element of H2c (G,M). Then since ΠG is a free pro-Σ group, there\nexists a continuous section ΠG → E of the surjection E \u0002 ΠG , hence also\n— since the extension E of ΠG is central— an isomorphismM×ΠG ∼→ E.\nWrite ΠG \u0002 Π for the maximal cuspidally central quotient [cf. [AbsCsp],\nDefinition 1.1, (i)] relative to the surjection ΠG \u0002 ΠG•T , EΠ for the\nquotient of E by the normal closed subgroup of E corresponding to\n{1} × Ker(ΠG \u0002 Π) ⊆ M × ΠG [thus, EΠ ∼← M × Π], and N ⊆ EΠ for\nthe image of the composite\nΠs(e0)\nιe0\n↪→ E \u0002 EΠ .\nNow we claim that N ⊆ EΠ is contained in the center Z(EΠ) of EΠ,\nhence also normal in EΠ. Indeed, since the composite\nΠs(e0) ↪→ ΠG \u0002 Π\nCombinatorial anabelian topics I 65\nis injective, and its image coincides with the kernel of the natural sur-\njection Π \u0002 ΠG•T , it holds that the image of the composite\nΠs(e0)\nιe0\n↪→ E \u0002 EΠ ∼←M ×Π\nis contained in M × Ker(Π → ΠG•T ). On the other hand, since the\nextension E of ΠG is central, it follows from the definition of the quotient\nΠ of ΠG that the image of M ×Ker(Π \u0002 ΠG•T ) in EΠ via M ×Π\n∼→ EΠ\nis contained in the center Z(EΠ) of EΠ. This completes the proof of the\nabove claim.\nNow it follows from the definition of N ⊆ EΠ, together with the\nabove claim, that we obtain a commutative diagram of profinite groups\n1 −−−−→ M −−−−→ E −−−−→ ΠG −−−−→ 1∥∥∥ ⏐⏐\b ⏐⏐\b\n1 −−−−→ M −−−−→ EΠ\/N −−−−→ ΠG•T −−−−→ 1\n— where the horizontal sequences are exact, and the vertical arrows are\nsurjective. In particular, we obtain an extension EΠ\/N of ΠG•T by M ,\nwhich is central since the extension E is central. For e ∈ Cusp(G•T ) =\nCusp(G) \\ {e0}, write Π•e ⊆ ΠG•T for the edge-like subgroup of ΠG•T\n[associated to e ∈ Cusp(G•T )] determined by the image of Πs(e) ⊆ ΠG\nand ι•e for the section Π\n•\ne → EΠ\/N over Π•e determined by ιe : Πs(e) → E.\nThen it follows immediately from the various definitions involved that\nthe image of\n[EΠ\/N, (ι\n•\ne)e′∈Cusp(G•T )] ∈ H2c (G•T ,M)\nin H2c (G,M) is [E, (ιe)e∈Cusp(G)] ∈ H2c (G,M). This completes the proof\nof the desired surjectivity and hence of assertion (iv) in the case where\nH = G and S = ∅.\nNext, to complete the proof of assertion (iv) in the general case,\none verifies immediately that it suffices to verify assertion (iv) in the\ncase where T = ∅, i.e., ((G|H)\bS)•T = (G|H)\bS . Thus, suppose that\nT = ∅. Write H def= (G|H)\bS . To define a natural homomorphism of ẐΣ-\nmodules H2c (H,M) → H2c (G,M), let H̃ → H be a universal covering of\nH which is compatible [in the evident sense] with the universal covering\nG̃ → G of G, sH a section of the natural surjection Cusp(H̃) \u0002 Cusp(H),\nand [EH, (ιHe : ΠsH(e) → EH)e∈Cusp(H)] ∈ H2c (H,M) an element of\nH2c (H,M). Since the extension EH of ΠH by M is central, the sec-\ntion ιHe : ΠsH(e) → EH naturally determines an isomorphism\nM ×ΠsH(e)\n∼−→ EH ×ΠH ΠsH(e)\n66 Yuichiro Hoshi and Shinichi Mochizuki\nof the direct product M ×ΠsH(e) with the fiber product EH×ΠH ΠsH(e)\nof the surjection EH \u0002 ΠH and the natural inclusion ΠsH(e) ↪→ ΠH.\nWrite GEH for the semi-graph of anabelioids associated to the central\nextension EH [cf. Definition 3.4, (ii)]. Then one may define a central\nextension of G by M\n(E , α, β : E\/α ∼→ G)\n[cf. Definition 3.4, (i)] whose restriction to H, relative to the isomor-\nphism β : E\/α ∼→ G, is isomorphic to the semi-graph of anabelioids\nGEH as follows: We take the underlying semi-graph of E to be the\nunderlying semi-graph of G; for each vertex v ∈ Vert(G|H), we take\nthe anabelioid Ev of E corresponding to the vertex v ∈ Vert(G|H) to\nbe the anabelioid (GEH)v of GEH corresponding to the vertex v; for\neach vertex v ∈ Vert(G) \\ Vert(G|H), we take the anabelioid Ev of E\ncorresponding to v ∈ Vert(G) \\ Vert(G|H) to be the anabelioid associ-\nated to the profinite group M × Πv. Then the above isomorphisms\nM × ΠsH(e)\n∼→ EH ×ΠH ΠsH(e) induced by the various ιHe ’s naturally\ndetermine the remaining data [i.e., consisting of anabelioids associated\nto edges and morphisms of anabelioids associated to branches] necessary\nto define a semi-graph of anabelioids E which is naturally equipped with\na structure of central extension of G by M whose restriction to H is\nnaturally isomorphic to the semi-graph of anabelioids GEH , as desired.\nNow it follows from Lemma 3.5, (i), that if we denote by ΠE the\npro-Σ fundamental group of E — i.e., the maximal pro-Σ quotient of the\nfundamental group of E — then ΠE is a central extension of ΠG by M .\nThus, it follows from the equivalences of categories of Lemma 3.5, (iv),\nthat the sections ιHe — where e ranges over the cusps of G that abut to a\nvertex of G|H — and the tautological sections Πe′ ↪→M ×Πe′ = ΠEe′ —\nwhere e′ ranges over the cusps of G that do not abut to a vertex of G|H —\nnaturally determine an equivalence class [ΠE , (ιe)e∈Cusp(G)] ∈ H2c (G,M).\nIn particular, we obtain a map\nH2c (H,M) −→ H2c (G,M)\nby assigning [EH, (ιHe )e∈Cusp(H)] \u0006→ [ΠE , (ιe)e∈Cusp(G)]. Moreover, it fol-\nlows immediately from the various definitions involved that this map is\na homomorphism of ẐΣ-modules, as desired.\nNext, we verify that this homomorphism H2c (H,M) → H2c (G,M)\nis an isomorphism. Since, for any vertex v ∈ Vert(G|H), the natural\nmorphism G|v → G factors through (G|H)\bS = H → G, by replacing H\nby G|v [cf. Remark 2.5.1, (ii)], we may assume without loss of generality\nthat H = G|v. Moreover, if Node(G) = ∅, then assertion (iv) in the\ncase where T = ∅ is immediate; thus, we may assume without loss\nCombinatorial anabelian topics I 67\nof generality that Node(G) \u0004= ∅. On the other hand, it follows from\nassertion (ii) that to verify that the homomorphism in question is an\nisomorphism, it suffices to verify that it is surjective. The rest of the\nproof of assertion (iv) in the case where T = ∅ is devoted to verifying\nthe surjectivity of the homomorphism H2c (v,M) → H2c (G,M).\nLet J be a semi-graph of anabelioids of pro-Σ PSC-type such that\nthere exist a vertex w ∈ Vert(J ) and an “omittable” cusp e ∈ C(w)\n[i.e., a cusp that abuts to w such that {e} is omittable] such that J•{e}\nis isomorphic to G, and, moreover, the isomorphism J•{e} ∼→ G induces\nan isomorphism of (J |w)•{e} ∼→ G|v. [Note that one may verify easily\nthat such a semi-graph of anabelioids of pro-Σ PSC type always exists.]\nThen it follows immediately from assertion (iv) in the case where H = G\nand S = ∅, together with the various definitions involved, that we have\na commutative diagram\nH2c (v,M)\n∼−−−−→ H2c ((J |w)•{e},M)\n∼−−−−→ H2c (w,M)⏐⏐\b ⏐⏐\b\nH2c (G,M)\n∼−−−−→ H2c (J•{e},M)\n∼−−−−→ H2c (J ,M)\n— where the left-hand horizontal arrows are isomorphisms induced by\nthe isomorphisms (J |w)•{e} ∼→ G|v, J•{e} ∼→ G, respectively, and the\nright-hand horizontal arrows are isomorphisms obtained by applying as-\nsertion (iv) in the case where H = G and S = ∅. In particular, to verify\nthe desired surjectivity of the homomorphism H2c (v,M) → H2c (G,M),\nby replacing G (respectively, v) by J (respectively, w), we may assume\nwithout loss of generality that C(v) \u0004= ∅.\nTo verify the desired surjectivity of the homomorphism H2c (v,M) →\nH2c (G,M) in the case where C(v) \u0004= ∅, let [E, (ιe)e∈Cusp(G)] ∈ H2c (G,M)\nbe an element of H2c (G,M). Now it follows from Lemma 3.3, together\nwith the assumption that C(v) \u0004= ∅, that we have two exact sequences\nof ẐΣ-modules\nHom\nẐΣ\n(ΠabG ,M) −→\n⊕\ne∈Cusp(G)\nHom\nẐΣ\n(Πe,M) −→ H2c (G,M) −→ 0 ;\nHom\nẐΣ\n(ΠabG|v ,M) −→\n⊕\ne∈Cusp(G|v)\nHom\nẐΣ\n(Πe,M) −→ H2c (v,M) −→ 0 .\nLet e0 ∈ C(v) be a cusp of G that abuts to v. Here, note that it fol-\nlows immediately from the definition of G|v that e0 may be regarded\nas a cusp of G|v. Then it follows immediately from the facts (A), (B)\nused in the proof of assertions (i), (ii) in the case where Cusp(G) \u0004= ∅\n68 Yuichiro Hoshi and Shinichi Mochizuki\nthat there exists a lifting (φe)e∈Cusp(G) ∈\n⊕\ne∈Cusp(G) HomẐΣ(Πe,M)\nof [E, (ιe)e∈Cusp(G)] ∈ H2c (G,M) [with respect to the first exact se-\nquence of the above display] such that if e \u0004= e0, then φe = 0. Write\n(ψe)e∈Cusp(G|v) ∈\n⊕\ne∈Cusp(G|v) HomẐΣ(Πe,M) for the element such that\nψe0 = φe0 , ψe = 0 for e \u0004= e0. Then it follows immediately from\nthe definitions of the above exact sequences and the homomorphism\nH2c (v,M) → H2c (G,M) in question that the image of (ψe)e∈Cusp(G|v) ∈⊕\ne∈Cusp(G|v) HomẐΣ(Πe,M) inH\n2\nc (v,M) is mapped to [E, (ιe)e∈Cusp(G)]\n∈ H2c (G,M) via the homomorphism H2c (v,M) → H2c (G,M) in question.\nThis completes the proof of assertion (iv) in the case where T = ∅, hence\nalso of assertion (iv) in the general case.\nFinally, we verify assertion (v). If Cusp(G) = ∅, then it follows im-\nmediately from a similar argument to the argument used in the proof\nof assertions (i), (ii) in the case where Cusp(G) = ∅, together with the\nwell-known structure of the second cohomology group of a compact Rie-\nmann surface, that assertion (v) holds. Next, suppose that Cusp(G) \u0004= ∅.\nWrite G\u0004 for the double of G [cf. [CmbGC], Proposition 2.2, (i)] — i.e.,\nthe analogue in the theory of semi-graphs of anabelioids of pro-Σ PSC-\ntype to the well-known “double” of a Riemann surface with boundary.\nWrite H\u0004 for the double of H. Then it follows from the various defini-\ntions involved that the connected finite étale covering H → G determines\na connected finite étale covering H\u0004 → G\u0004 of degree [ΠG : ΠH]. Next,\nlet us observe G (respectively, H) may be naturally identified with the\nrestriction [cf. Definition 2.2, (ii)] of G\u0004 (respectively, H\u0004) to a suit-\nable sub-semi-graph of PSC-type of the underlying semi-graph of G\u0004\n(respectively, H\u0004). Thus, it follows from assertion (iv) that we have a\ncommutative diagram of ẐΣ-modules\nH2c (G,M)\n∼−−−−→ H2c (G\u0004,M)⏐⏐\b ⏐⏐\b\nH2c (H,M)\n∼−−−−→ H2c (H\u0004,M)\n— where the horizontal arrows are the isomorphisms of assertion (iv),\nand the vertical arrows are the homomorphisms induced by the con-\nnected finite étale coverings H → G, H\u0004 → G\u0004, respectively — and\nhence that assertion (v) in the case where Cusp(G) \u0004= ∅ follows immedi-\nately from assertion (v) in the case where Cusp(G) = ∅. This completes\nthe proof of assertion (v). Q.E.D.\nCombinatorial anabelian topics I 69\nDefinition 3.8.\n(i) We shall write\nΛG\ndef\n= Hom\nẐΣ\n(H2c (G, ẐΣ), ẐΣ)\nand refer to ΛG as the cyclotome associated to G. For a vertex\nv ∈ Vert(G) of G, we shall write\nΛv\ndef\n= Hom\nẐΣ\n(H2c (v, Ẑ\nΣ), ẐΣ)\nand refer to Λv as the cyclotome associated to v ∈ Vert(G).\nNote that it follows from Theorem 3.7, (ii), that the cyclotomes\nΛG and Λv are free ẐΣ-modules of rank 1.\n(ii) We shall write\nχG : Aut(G) −→ Aut(ΛG) \f (ẐΣ)∗\nfor the natural homomorphism induced by the natural action of\nAut(G) on H2c (G, ẐΣ) and refer to χG as the pro-Σ cyclotomic\ncharacter of G. For a vertex v ∈ Vert(G) of G, we shall write\nχv\ndef\n= χG|v : Aut(G|v) −→ Aut(Λv) \f (ẐΣ)∗\nand refer to χv as the pro-Σ cyclotomic character of v.\nRemark 3.8.1. One verifies easily that if l ∈ Σ, then the composite\nAut(G) χG→ (ẐΣ)∗ \u0002 Z∗l\ncoincides with the pro-l cyclotomic character of Aut(G) defined in the\nstatement of [CmbGC], Lemma 2.1.\nCorollary 3.9 (Synchronization of cyclotomes). Let Σ be a\nnonempty set of prime numbers and G a semi-graph of anabelioids of\npro-Σ PSC-type. Then the following hold:\n(i) (Synchronization with respect to generization) Let S ⊆\nNode(G) be a subset of Node(G). Then the specialization outer\n70 Yuichiro Hoshi and Shinichi Mochizuki\nisomorphism ΦG\u0003S : ΠG\u0003S\n∼→ ΠG with respect to S [cf. Defini-\ntion 2.10] determines a natural isomorphism\nΛG\u0003S\n∼−→ ΛG\nthat is functorial with respect to isomorphisms of the pair\n(G, S).\n(ii) (Synchronization with respect to “surgery”) Let H be\na sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G,\nS ⊆ Node(G|H) [cf. Definition 2.2, (ii)] a subset of Node(G|H)\nthat is not of separating type [cf. Definition 2.5, (i)], and\nT ⊆ Cusp((G|H)\bS) [cf. Definition 2.5, (ii)] an omittable [cf.\nDefinition 2.4, (i)] subset of Cusp((G|H)\bS). Then there exists\na natural isomorphism — given by “extension by zero”\n—\nΛG\n∼−→ Λ((G|H)\u0004S)•T\n[cf. Definition 2.4, (ii)] that is functorial with respect to iso-\nmorphisms of the quadruple (G,H, S, T ). In particular, [by tak-\ning the inverse of this isomorphism] we obtain, for each vertex\nv ∈ Vert(G) of G, a natural isomorphism of ẐΣ-modules\nsyn v : Λv\n∼−→ ΛG\nthat is functorial with respect to isomorphisms of the pair\n(G, v).\n(iii) (Synchronization with respect to finite étale coverings)\nLet H → G be a connected finite étale covering of G. Then there\nexists a natural isomorphism\nΛH\n∼−→ ΛG\nthat is functorial with respect to isomorphisms of the pair\n(G,H).\n(iv) (Synchronization of cyclotomic characters) Let v ∈\nVert(G) be a vertex of G and α ∈ Aut{v}(G) [cf. Definition 2.6,\n(i)]. Then it holds that\nχG(α) = χv(αG|v )\n[cf. Definitions 2.14, (ii); 3.8, (ii); Remark 2.5.1, (ii)].\nCombinatorial anabelian topics I 71\n(v) (Synchronization associated to branches) Let e ∈\nEdge(G) be an edge of G, b a branch of e that abuts to a vertex\nv ∈ V(e), and Πe ⊆ ΠG an edge-like subgroup of ΠG associated\nto e ∈ Edge(G). Then there exists a natural isomorphism\nsyn b : Πe\n∼−→ Λv\nthat is functorial with respect to isomorphisms of the quadru-\nple (G, b, e, v).\n(vi) (Difference between two synchronizations associated\nto the two branches of a node) Let e ∈ Node(G) be a\nnode of G with branches b1 \u0004= b2 that abut to vertices v1, v2 ∈\nVert(G), respectively. Then the two composites\nΠe\nsyn b1∼−→ Λv1\nsyn v1∼−→ ΛG ; Πe\nsyn b2∼−→ Λv2\nsyn v2∼−→ ΛG\ndiffer by the automorphism of ΛG given by multiplication by\n−1 ∈ ẐΣ.\nProof. Assertion (i) (respectively, (ii)) follows immediately from\nTheorem 3.7, (iii) (respectively, Theorem 3.7, (iv)). Assertion (iv) fol-\nlows immediately from assertion (ii).\nNext, we verify assertion (iii). It follows immediately from Theo-\nrem 3.7, (v), that the homomorphism of ẐΣ-modules ΛH → ΛG obtained\nby applying the functor “Hom\nẐΣ\n(−, ẐΣ)” to the induced homomorphism\nH2c (G, ẐΣ) → H2c (H, ẐΣ) and dividing by the index [ΠG : ΠH] is an iso-\nmorphism. This completes the proof of assertion (iii).\nNext, we verify assertion (v). First, we observe that to verify as-\nsertion (v), by replacing G by G|v and e ∈ Edge(G) by the cusp of\nG|v corresponding to b, we may assume without loss of generality that\ne ∈ Cusp(G). Then we have homomorphisms of ẐΣ-modules\nHom\nẐΣ\n(Πe,Πe) ↪→\n⊕\ne′∈Cusp(G) HomẐΣ(Πe′ ,Πe)\n\u0002 H2c (G,Πe)\n∼→ Hom\nẐΣ\n(ΛG ,Πe)\n— where the first arrow is the natural inclusion into the component\nindexed by e, and the second arrow is the surjection appearing in the\nexact sequence of Lemma 3.3 in the case where M = Πe. Here, we note\nthat it follows immediately from the facts (A), (B) used in the proof of\nTheorem 3.7, (i), (ii), that the composite of these homomorphisms is an\n72 Yuichiro Hoshi and Shinichi Mochizuki\nisomorphism. Therefore, we obtain a natural isomorphism\nsyn b : Πe\n∼−→ ΛG\nby forming the inverse of the image of the identity automorphism of\nΠe via the composite of the homomorphisms of the above display. This\ncompletes the proof of assertion (v).\nFinally, we verify assertion (vi). First, we observe that one may\nverify easily that there exist\n• a semi-graph of anabelioids of pro-Σ PSC-type H†,\n• a sub-semi-graph of PSC-type K† of the underlying semi-graph\nof H†,\n• an omittable subset S† ⊆ Cusp((H†)|K†), and\n• an isomorphism\n((H†)|K†)•S†\n∼−→ G\nsuch that the node eH† ∈ Node(H†) of H† corresponding, relative to\nthe isomorphism ((H†)|K†)•S†\n∼→ G, to the node e ∈ Node(G) is not of\nseparating type. [Note that it follows immediately from the various def-\ninitions involved that Node(G) ∼← Node(((H†)|K†)•S†) may be regarded\nas a subset of Node(H†).] Thus, it follows immediately from assertions\n(i), (ii) — by replacing G (respectively, e) by (H†)\u0003Node(H†)\\{eH†} (re-\nspectively, eH†) — that to verify assertion (vi), we may assume without\nloss of generality that Node(G) = {e}, and that e is not of separating\ntype.\nNext, we observe that one may verify easily that there exists a semi-\ngraph of anabelioids of pro-Σ PSC-type H‡ such that\n• Node(H‡) consists of precisely two elements eH‡ , e′H‡ ;\n• V(eH‡) consists of precisely one element vH‡ of type (0, 3) [cf.\nDefinition 2.3, (iii)].\n• e′H‡ is of separating type;\n• (H‡)\u0003{e′H‡} is isomorphic to G.\nThus, if we write K‡ for the unique sub-semi-graph of PSC-type of the\nunderlying semi-graph of H‡ whose set of vertices = {vH‡}, then it fol-\nlows immediately from assertions (i), (ii) — by replacing G (respectively,\ne) by H‡|K‡ (respectively, eH‡) — that to verify assertion (vi), we may\nassume without loss of generality that Node(G) = {e}, that e is not of\nCombinatorial anabelian topics I 73\nseparating type [so Vert(G) consists of precisely one element], and that\nG is of type (1, 1).\nWrite v ∈ Vert(G) for the unique vertex of G. Note that it follows\nimmediately from the various assumptions on G that G|v is of type (0, 3).\nWrite e1, e2 ∈ Cusp(G|v) for the cusps of G|v corresponding, respectively,\nto the two branches b1, b2 of the node e; write e3 ∈ Cusp(G|v) for the\nunique element of Cusp(G|v) \\ {e1, e2}. Then since G|v is of type (0, 3),\nthere exists a graphic isomorphism of G|v with the semi-graph of an-\nabelioids of pro-Σ PSC-type [without nodes] determined by the tripod\n[cf. the discussion entitled “Curves” in §0] P1k \\ {0, 1,∞} over an alge-\nbraically closed field k of characteristic \u0004∈ Σ such that the cusps e1, e2 of\nG|v correspond to the cusps 0, ∞ of P1k \\ {0, 1,∞}, respectively, relative\nto the graphic isomorphism. Thus, by considering the automorphism of\nP1k \\ {0, 1,∞} over k given by “t \u0006→ 1\/t”, we obtain an automorphism\nτv ∈ Aut(G|v) of G|v that maps e1 \u0006→ e2, e2 \u0006→ e1. Moreover, since\nthis automorphism of P1k \\ {0, 1,∞} induces an automorphism of the\nstable log curve of type (1, 1) obtained by identifying the cusps 0 and\n∞ of P1k \\ {0, 1,∞}, we also obtain an automorphism τG ∈ Aut(G) of\nG. Note that it follows immediately from the definition of τv, together\nwith the well-known structure of the étale fundamental group of the\ntripod P1k \\ {0, 1,∞}, that the automorphism τv induces the identity\nautomorphism of the anabeloid (G|v)e3 corresponding to e3.\nNext, let us observe that it follows immediately from the definition\nof G|v, together with the proof of assertion (v), that for i = 1, 2, there\nexists a natural isomorphism Πe\n∼→ Πei — where we use the notations\nΠe, Πei to denote edge-like subgroups of ΠG , ΠG|v associated to e, ei,\nrespectively — such that the composite\nΠe\n∼−→ Πei\nsyn b′i∼−→ Λv [= ΛG|v ]\nsyn v∼−→ ΛG\n— where we write b′i for the [unique] branch of ei — coincides with the\ncomposite in question\nΠe\nsyn bi∼−→ Λv\nsyn v∼−→ ΛG .\nNext, let us observe that it follows immediately from the functori-\nality portion of assertion (v) that the automorphisms τv, τG induce a\n74 Yuichiro Hoshi and Shinichi Mochizuki\ncommutative diagram of ẐΣ-modules\nΠe\n∼−−−−→ Πe1\nsyn b′i−−−−→ Λv [= ΛG|v ]\nsyn v−−−−→ ΛG\n\u0005\n⏐⏐\b \u0005⏐⏐\b ⏐⏐\b\u0005 ⏐⏐\b\u0005\nΠe\n∼−−−−→ Πe2\nsyn b′\n2−−−−→ Λv [= ΛG|v ]\nsyn v−−−−→ ΛG\n— where the vertical arrows are the isomorphisms induced by the au-\ntomorphisms τv, τG . Now by considering the well-known local structure\nof a stable log curve in a neighborhood of a node, one may verify easily\nthat the left-hand vertical arrow in the above diagram is the automor-\nphism of Πe given by multiplication by −1 ∈ ẐΣ. Thus, to complete the\nverification of assertion (vi), it suffices, in light of the commutativity\nof the above diagram, to verify that τv ∈ Aut(G|v) induces the identity\nautomorphism of ΛG|v = Λv. On the other hand, this follows immedi-\nately from assertion (v), applied to the cusp e3, together with the fact\nthat the automorphism τv induces the identity automorphism of (G|v)e3 .\nThis completes the proof of assertion (vi). Q.E.D.\n§4. Profinite Dehn multi-twists\nIn the present §, we introduce and discuss the notion of a profinite\nDehn multi-twist. Although our definition of this notion [cf. Defini-\ntion 4.4 below] is entirely group-theoretic in nature, our main result\nconcerning this notion [cf. Theorem 4.8 below] asserts, in effect, that\nthis group-theoretic notion coincides with the usual geometric notion of\na “Dehn multi-twist”.\nLet Σ be a nonempty set of prime numbers and G a semi-graph of\nanabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph\nof G, ΠG for the [pro-Σ] fundamental group of G, and G̃ → G for the\nuniversal covering of G corresponding to ΠG .\nDefinition 4.1. We shall say that G is cyclically primitive (respec-\ntively, noncyclically primitive) if Node(G)\u0005 = 1, and the unique node\nof G is not of separating type (respectively, is of separating type) [cf.\nDefinition 2.5, (i)].\nCombinatorial anabelian topics I 75\nRemark 4.1.1. If G is cyclically primitive (respectively, noncycli-\ncally primitive), then Vert(G)\u0005 = 1 (respectively, 2), and the [discrete]\ntopological fundamental group π\ntop\n1 (G) of the underlying semi-graph G\nof G is noncanonically isomorphic to Z (respectively, is trivial).\nLemma 4.2 (Structure of the fundamental group of a non-\ncyclically primitive semi-graph of anabelioids of PSC-type).\nSuppose that G is noncyclically primitive [cf. Definition 4.1]. Let\nv, w ∈ Vert(G) be the two distinct vertices of G [cf. Remark 4.1.1];\nẽ, ṽ, w̃ ∈ VCN(G̃) elements of VCN(G̃) such that ṽ(G) = v, w̃(G) = w,\nand, moreover, ẽ ∈ N (ṽ) ∩ N (w̃). Then the natural inclusions Πẽ, Πṽ,\nΠw̃ ↪→ ΠG determine an isomorphism of pro-Σ groups\nlim−→(Πṽ ←↩ Πẽ ↪→ Πw̃)\n∼−→ ΠG\n— where the inductive limit is taken in the category of pro-Σ groups.\nProof. This may be thought of as a consequence of the “van Kam-\npen Theorem” in elementary algebraic topology. At a more combinato-\nrial level, one may reason as follows: It follows immediately from the\nsimple structure of the underlying semi-graph G that there is a natural\nequivalence of categories between\n• the category of finite sets with continuous ΠG-action [and ΠG-\nequivariant morphisms] and\n• the category of finite sets with continuous actions of Πṽ, Πw̃\nwhich restrict to the same action on Πẽ [and Πṽ-,\nΠw̃-equivariant morphisms].\nThe isomorphism between ΠG and the inductive limit appearing in the\nstatement of Lemma 4.2 now follows formally from this equivalence of\ncategories. Q.E.D.\nLemma 4.3 (Infinite cyclic tempered covering of a cyclically\nprimitive semi-graph of anabelioids of PSC-type). Suppose that\nG is cyclically primitive [cf. Definition 4.1]. Denote by πtemp1 (G) the\ntempered fundamental group of G [cf. the discussion preceding [SemiAn],\nProposition 3.6], by π\ntop\n1 (G) [\f Z — cf. Remark 4.1.1] the [discrete]\ntopological fundamental group of the underlying semi-graph G of G, and\nby G∞ → G the connected tempered covering of G corresponding to the\nnatural surjection π\ntemp\n1 (G) \u0002 πtop1 (G) [where we refer to [SemiAn], §3,\n76 Yuichiro Hoshi and Shinichi Mochizuki\nconcerning tempered coverings of a semi-graph of anabelioids]. Then the\nfollowing hold:\n(i) (Exact sequence) The natural morphism G∞ → G induces\nan exact sequence\n1 −→ πtemp1 (G∞) −→ πtemp1 (G) −→ πtop1 (G) −→ 1 .\nMoreover, the subgroup π\ntemp\n1 (G∞) ⊆ πtemp1 (G) of πtemp1 (G) is\ncharacteristic.\n(ii) (Automorphism groups) There exist natural injective ho-\nmomorphisms\nAut|grph|(G) ↪→ Aut|grph|(G∞) , πtop1 (G) ↪→ Aut(G∞)\n— where we write Aut|grph|(G∞) for the group of automor-\nphisms of G∞ that induce the identity automorphism of the\nunderlying semi-graph of G∞. Moreover, the centralizer of\nπ\ntop\n1 (G) in Aut\n|grph|(G∞) satisfies the equality\nZAut|grph|(G∞)(π\ntop\n1 (G)) = Aut\n|grph|(G) .\n(iii) (Action of the fundamental group of the underlying\nsemi-graph) Let γ∞ ∈ πtop1 (G) ⊆ Aut(G∞) [cf. (ii)] be a\ngenerator of π\ntop\n1 (G) \f Z. Write Vert(G∞), Node(G∞), and\nCusp(G∞) for the sets of vertices, nodes [i.e., closed edges],\nand cusps [i.e., open edges] of G∞, respectively. Then there\nexist bijections\nV : Z\n∼−→ Vert(G∞) , N : Z ∼−→ Node(G∞) ,\nC : Z× Cusp(G) ∼−→ Cusp(G∞)\nsuch that, for each a ∈ Z,\n• the set of edges that abut to the vertex V (a) is equal to\nthe disjoint union of {N(a), N(a+ 1)} and {C(a, z) | z ∈\nCusp(G)};\n• the automorphism of Vert(G∞) (respectively, Node(G∞);\nCusp(G∞)) induced by γ∞ ∈ Aut(G∞) maps V (a) (respec-\ntively, N(a); C(a, z)) to V (a+1) (respectively, N(a+1);\nC(a+ 1, z)).\nCombinatorial anabelian topics I 77\n(iv) (Restriction to a finite sub-semi-graph) Let a ≤ b ∈ Z be\nintegers. Denote by G[a,b] the [uniquely determined] sub-\nsemi-graph of PSC-type [cf. Definition 2.2, (i)] of the under-\nlying semi-graph of G∞ such that the set of vertices of G[a,b]\nis equal to {V (a), V (a + 1), · · · , V (b)} [cf. (iii)]; denote by\nG[a,b] the semi-graph of anabelioids obtained by restricting G∞\nto G[a,b] [cf. the discussion preceding [SemiAn], Definition 2.2].\nThen G[a,b] is a semi-graph of anabelioids of pro-Σ PSC-\ntype. Moreover, G[a,a+1] is noncyclically primitive.\n(v) (Restriction to a sub-semi-graph having precisely one\nvertex) Let a ≤ c ≤ b ∈ Z be integers. Then the natural mor-\nphism of semi-graphs of anabelioids G[c,c] → G[a,b] [cf. (iv)] de-\ntermines an isomorphism G[c,c] ∼→ G[a,b]|V (c) — where we regard\nV (c) ∈ Vert(G∞) as a vertex of G[a,b]. Moreover, if we write\nv ∈ Vert(G) for the unique vertex of G [cf. Remark 4.1.1],\nthen the composite of natural morphisms of semi-graphs of an-\nabelioids G[c,c] → G∞ → G determines an isomorphism of G[c,c]\nwith G|v.\n(vi) (Natural isomorphisms between restrictions to finite\nsub-semi-graphs) Let a ≤ b ∈ Z be integers and γ∞ ∈\nπ\ntop\n1 (G) ⊆ Aut(G∞) the automorphism of G∞ appearing in\n(iii). Then γ∞ determines an isomorphism G[a,b] ∼→ G[a+1,b+1].\nProof. First, we verify assertion (i). To show that the natural\nmorphism G∞ → G induces an exact sequence\n1 −→ πtemp1 (G∞) −→ πtemp1 (G) −→ πtop1 (G) −→ 1 ,\nit suffices to verify that every tempered covering of G∞ determines, via\nthe morphism G∞ → G, a tempered covering of G. But this follows\nimmediately, in light of the definition of a tempered covering, from the\nfiniteness of the underlying semi-graph G and the topologically finitely\ngenerated nature of the verticial subgroups of the tempered fundamental\ngroup π\ntemp\n1 (G∞) of G∞. On the other hand, the fact that the subgroup\nπ\ntemp\n1 (G∞) ⊆ πtemp1 (G) is characteristic follows immediately from the\nobservation that the quotient π\ntemp\n1 (G) \u0002 πtemp1 (G)\/πtemp1 (G∞) may be\ncharacterized as the maximal discrete free quotient of π\ntemp\n1 (G) [cf. the\nargument of [André], Lemma 6.1.1]. This completes the proof of asser-\ntion (i).\nNext, we verify assertion (ii). The existence of a natural injec-\ntion π\ntop\n1 (G) ↪→ Aut(G∞) follows immediately from the definition of\n78 Yuichiro Hoshi and Shinichi Mochizuki\nthe connected tempered covering G∞ → G, together with the fact that\nπ\ntop\n1 (G) is abelian. On the other hand, it follows immediately from\nassertion (i), together with the various definitions involved, that any\nelement of Aut|grph|(G) determines — up to composition with an el-\nement of π\ntop\n1 (G) ⊆ Aut(G∞) — an automorphism of G∞. There-\nfore, by composing with a suitable element of π\ntop\n1 (G) ⊆ Aut(G∞), one\nobtains a uniquely determined element of Aut|grph|(G∞), hence also a\nnatural injective homomorphism Aut|grph|(G) ↪→ Aut|grph|(G∞). Next,\nto verify the equality ZAut|grph|(G∞)(π\ntop\n1 (G)) = Aut\n|grph|(G), observe\nthat π\ntemp\n1 (G∞) is center-free [cf. [SemiAn], Example 2.10; [SemiAn],\nProposition 3.6, (iv)]; this implies that we have a natural isomorphism\nπ\ntemp\n1 (G) \f πtemp1 (G∞)\nout\n\u0002 π\ntop\n1 (G) [cf. the discussion entitled “Topo-\nlogical groups” in §0]. Thus, in light of the [easily verified] inclusion\nAut|grph|(G) ⊆ ZAut|grph|(G∞)(π\ntop\n1 (G)), the desired equality follows im-\nmediately from [CmbGC], Proposition 1.5, (ii). This completes the proof\nof assertion (ii).\nAssertions (iii), (iv), (v), and (vi) follow immediately from the defi-\nnition of the connected tempered covering G∞ → G. Q.E.D.\nDefinition 4.4. We shall write\nDehn(G) def= {α ∈ Aut|grph|(G) |αG|v = idG|v for any v ∈ Vert(G) }\n— where we refer to Definitions 2.1, (iii); 2.14, (ii); Remark 2.5.1, (ii),\nconcerning “αG|v”. We shall refer to an element of Dehn(G) as a profinite\nDehn multi-twist of G.\nProposition 4.5 (Equalities concerning the group of profi-\nnite Dehn multi-twists). It holds that\nDehn(G) = ⋂v∈Vert(G) Aut|Πv|(G) = ⋂z∈VCN(G) Aut|Πz|(G)\n=\n⋂\nz∈VCN(G) Out\n|Πz|(ΠG) ⊆ Aut|grph|(G)\n[cf. Definitions 2.13; 2.6, (i); [CmbGC], Proposition 1.2, (ii)] — where\nwe use the notation “Π(−)” to denote a VCN-subgroup [cf. Defini-\ntion 2.1, (i)] of ΠG associated to “(−)” ∈ VCN(G).\nProof. The first equality follows immediately from the various def-\ninitions involved [cf. also [CmbGC], Proposition 1.2, (i)]. The second\nCombinatorial anabelian topics I 79\nequality follows immediately from the fact that any edge-like subgroup\nis contained in a verticial subgroup. The third equality follows imme-\ndiately from Proposition 2.7, (ii). This completes the proof of Proposi-\ntion 4.5. Q.E.D.\nLemma 4.6 (Construction of certain homomorphisms). Let\nẽ ∈ Node(G̃), e def= ẽ(G) ∈ Node(G). Then the following hold:\n(i) Let α ∈ Dehn(G) be a profinite Dehn multi-twist of G and\nṽ ∈ V(ẽ) ⊆ Vert(G̃). Write w̃ for the unique element of the\ncomplement V(ẽ) \\ {ṽ} [cf. [NodNon], Remark 1.2.1, (iii)].\nThen there exists a unique lifting α[ṽ] ∈ Aut(ΠG) of α which\npreserves the verticial subgroup Πṽ ⊆ ΠG of ΠG associated to\nṽ ∈ Vert(G̃) and induces the identity automorphism of Πṽ.\nMoreover, this lifting α[ṽ] preserves the verticial subgroup\nΠw̃ ⊆ ΠG of ΠG associated to w̃ ∈ Vert(G̃), and there exists\na unique element δẽ,ṽ ∈ Πẽ of the edge-like subgroup Πẽ ⊆ ΠG\nof ΠG associated to ẽ ∈ Node(G̃) such that the restriction of\nα[ṽ] to Πw̃ is the inner automorphism determined by δẽ,ṽ ∈ Πẽ\n(⊆ Πw̃).\n(ii) For ṽ ∈ V(ẽ), denote by Dẽ,ṽ : Dehn(G) → ΛG the composite of\nthe map\nDehn(G) −→ Πẽ\ngiven by assigning α \u0006→ δẽ,ṽ ∈ Πẽ [cf. (i)] and the isomorphism\nΠe\nsyn b∼−→ Λv\nsyn v∼−→ ΛG\n[cf. Corollary 3.9, (ii), (v)] — where we write v\ndef\n= ṽ(G) and\nb for the branch of e determined by the unique branch of ẽ\nwhich abuts to ṽ. Then the map Dẽ,ṽ : Dehn(G) → ΛG is a\nhomomorphism of profinite groups which does not depend\non the choice of the element ṽ ∈ V(ẽ), i.e., if w̃ ∈ V(ẽ) \\\n{ṽ}, then Dẽ,ṽ = Dẽ,w̃. Moreover, the homomorphism Dẽ,ṽ\n(= Dẽ,w̃) depends only on e ∈ Node(G), i.e., it does not\ndepend on the choice of the element ẽ ∈ Node(G̃) such that\nẽ(G) = e.\nProof. First, we verify assertion (i). The fact that there exists a\nunique lifting α[ṽ] ∈ Aut(ΠG) of α which preserves Πṽ and induces the\n80 Yuichiro Hoshi and Shinichi Mochizuki\nidentity automorphism of Πṽ follows immediately, in light of the slimness\nof Πṽ [cf. [CmbGC], Remark 1.1.3] and the commensurable terminality\nof Πṽ in ΠG [cf. [CmbGC], Proposition 1.2, (ii)], from the fact that\nα ∈ Out|Πṽ|(ΠG) [cf. Proposition 4.5]. The fact that α[ṽ] preserves\nΠw̃ follows immediately, in light of the graphicity of α[ṽ], from the fact\nthat Πw̃ is the unique verticial subgroup H of ΠG such that H \u0004= Πṽ\nand Πẽ ⊆ H [cf. [NodNon], Remark 1.2.1, (iii); [NodNon], Lemma 1.7],\ntogether with the fact that α[ṽ] preserves Πṽ, Πẽ ⊆ ΠG . The fact that\nthere exists a unique element δẽ,ṽ ∈ Πẽ of Πẽ such that the restriction\nof α[ṽ] to Πw̃ is the inner automorphism determined by δẽ,ṽ follows\nimmediately, in light of the slimness of Πw̃ [cf. [CmbGC], Remark 1.1.3]\nand the commensurable terminality of Πẽ [cf. [CmbGC], Proposition\n1.2, (ii)], from the fact that α ∈ Out|Πw̃|(ΠG) [cf. Proposition 4.5].\nThis completes the proof of assertion (i). Next, we verify assertion (ii).\nThe fact that the map Dẽ,ṽ is a homomorphism follows immediately\nfrom the various uniqueness properties discussed in assertion (i). The\nfact that the map Dẽ,ṽ does not depend on the choice of the element\nṽ ∈ V(ẽ) follows immediately from Corollary 3.9, (vi). The fact that\nthe homomorphism Dẽ,ṽ does not depend on the choice of the element\nẽ ∈ Node(G̃) such that ẽ(G) = e follows immediately from the definition\nof the map Dẽ,ṽ. This completes the proof of assertion (ii). Q.E.D.\nDefinition 4.7. For each node e ∈ Node(G) of G, we shall write\nDe\ndef\n= Dẽ,ṽ : Dehn(G) −→ ΛG\nfor the homomorphism obtained in Lemma 4.6, (ii). [Note that it follows\nfrom Lemma 4.6, (ii), that this homomorphism depends only on e ∈\nNode(G).] We shall write\nDG\ndef\n=\n⊕\ne∈Node(G)\nDe : Dehn(G) −→\n⊕\nNode(G)\nΛG .\nTheorem 4.8 (Properties of profinite Dehn multi-twists).\nLet Σ be a nonempty set of prime numbers and G a semi-graph of an-\nabelioids of pro-Σ PSC-type. Then the following hold:\n(i) (Normality) Dehn(G) is normal in Aut(G).\nCombinatorial anabelian topics I 81\n(ii) (Compatibility with generization) Let S ⊆ Node(G).\nThen — relative to the inclusion AutS(G) ⊆ Aut(G\u0003S) [cf.\nDefinition 2.8] induced by the specialization outer isomorphism\nΠG\n∼→ ΠG\u0003S with respect to S [cf. Proposition 2.9, (ii)] — we\nhave a diagram of inclusions\nDehn(G) ←↩ Dehn(G\u0003S)\n∩ ∩\nAutS(G) ↪→ Aut(G\u0003S) .\nMoreover, if we regard Node(G\u0003S) as a subset of Node(G),\nthen the above inclusion Dehn(G\u0003S) ↪→ Dehn(G) fits into a\ncommutative diagram of profinite groups\nDehn(G\u0003S) −−−−→ Dehn(G)\nDG\u0003S\n⏐⏐\b ⏐⏐\bDG⊕\nNode(G\u0003S) ΛG −−−−→\n⊕\nNode(G) ΛG\n— where the lower horizontal arrow is the natural inclusion\ndetermined by the inclusion Node(G\u0003S) ↪→ Node(G) and the\nnatural isomorphism ΛG\u0003S\n∼→ ΛG [cf. Corollary 3.9, (i)].\n(iii) (Compatibility with “surgery”) Let H be a sub-semi-graph\nof PSC-type [cf. Definition 2.2, (i)] of G, S ⊆ Node(G|H) [cf.\nDefinition 2.2, (ii)] a subset of Node(G|H) that is not of sepa-\nrating type [cf. Definition 2.5, (i)], and T ⊆ Cusp((G|H)\bS)\n[cf. Definition 2.5, (ii)] an omittable [cf. Definition 2.4, (i)]\nsubset of Cusp((G|H)\bS). Then the natural homomorphism\nAutH\bS•T (G) −→ Aut(((G|H)\bS)•T )\nα \u0006→ α((G|H)\u0004S)•T\n[cf. Definitions 2.4, (ii); 2.14, (ii)] induces a homomorphism\nDehn(G) −→ Dehn(((G|H)\bS)•T ) .\nMoreover, if we regard Node(((G|H)\bS)•T ) as a sub-\nset of Node(G), then the above homomorphism Dehn(G) →\n82 Yuichiro Hoshi and Shinichi Mochizuki\nDehn(((G|H)\bS)•T ) fits into a commutative diagram of profi-\nnite groups\nDehn(G) −−−−→ Dehn(((G|H)\bS)•T )\nDG\n⏐⏐\b ⏐⏐\bD((G|H)\u0004S)•T⊕\nNode(G) ΛG −−−−→\n⊕\nNode(((G|H)\u0004S)•T ) ΛG\n— where the lower horizontal arrow is the natural projection,\nand we apply the natural isomorphism ΛG\n∼→ Λ((G|H)\u0004S)•T [cf.\nCorollary 3.9, (ii)].\n(iv) (Structure of the group of profinite Dehn multi-twists)\nThe homomorphism defined in Definition 4.7\nDG : Dehn(G) −→\n⊕\nNode(G)\nΛG\nis an isomorphism of profinite groups that is functorial,\nin G, with respect to isomorphisms of semi-graphs of anabe-\nlioids of pro-Σ PSC-type. In particular, Dehn(G) is a finitely\ngenerated free ẐΣ-module of rank Node(G)\u0002. We shall\nrefer to a nontrivial profinite Dehn multi-twist whose image\n∈ ⊕Node(G) ΛG lies in a direct summand [i.e., in a single “ΛG”]\nas a profinite Dehn twist.\n(v) (Conjugation action on the group of profinite Dehn\nmulti-twists) The action of Aut(G) on ⊕Node(G) ΛG\nAut(G) −→ Aut(Dehn(G)) ∼−→ Aut(\n⊕\nNode(G)\nΛG)\ndetermined by conjugation by elements of Aut(G) [cf. (i)] and\nthe isomorphism of (iv) coincides with the action of Aut(G) on⊕\nNode(G) ΛG determined by the action χG of Aut(G) on ΛG\nand the natural action of Aut(G) on the finite set Node(G).\nProof. Assertions (i), (ii), and (iii) follow immediately from the\nvarious definitions involved. Next, we verify assertion (iv). The functo-\nriality of the homomorphism DG follows immediately from the various\ndefinitions involved. The rest of the proof of assertion (iv) is devoted\nto verifying that the homomorphism DG is an isomorphism. First, we\nclaim that\nCombinatorial anabelian topics I 83\n(∗1): if G is noncyclically primitive [cf. Definition 4.1],\nthen the homomorphism DG is injective.\nIndeed, this follows immediately from Lemma 4.2, together with the\ndefinition of the homomorphism DG . This completes the proof of the\nabove claim (∗1).\nNext, we claim that\n(∗2): if G is cyclically primitive [cf. Definition 4.1],\nthen the homomorphism DG is injective.\nIndeed, let α ∈ Ker(DG) ⊆ Out(ΠG) be an element of Ker(DG). Since\nwe are in the situation of Lemma 4.3, we shall apply the notational con-\nventions established in Lemma 4.3. Denote by α∞ ∈ Aut|grph|(G∞) the\nautomorphism of G∞ determined by α [cf. Lemma 4.3, (ii)]; for integers\na ≤ b ∈ Z, denote by α[a,b] ∈ Aut|grph|(G[a,b]) the automorphism of G[a,b]\nobtained by restricting α∞ ∈ Aut|grph|(G∞). Then since α is a profinite\nDehn multi-twist, one may verify easily that α[a,b] is a profinite Dehn\nmulti-twist of G[a,b]. Thus, since G[a,a+1] is noncyclically primitive [cf.\nLemma 4.3, (iv)], it follows immediately from the fact that α ∈ Ker(DG),\ntogether with the claim (∗1), that α[a,a+1] is trivial. Moreover, for any\na < b ∈ Z, it follows — by applying induction on b− a and considering,\nin light of the claim (∗1), the various generizations [cf. assertion (ii)] of\nG[a,b] with respect to sets of the form “Node(G[a.b]) \\ {e}” — that the\nprofinite Dehn multi-twist α[a,b], hence also the automorphism α∞, is\ntrivial. In particular, it holds that α is trivial [cf. Lemma 4.3, (ii)], as\ndesired. This completes the proof of the above claim (∗2).\nNext, we claim that\n(∗3): for arbitrary G, the homomorphism DG is injec-\ntive.\nWe verify this claim (∗3) by induction on Node(G)\u0005. If Node(G)\u0005 ≤ 1,\nthen the claim (∗3) follows formally from the claims (∗1) and (∗2). Now\nsuppose that Node(G)\u0005 > 1, and that the induction hypothesis is in force.\nLet e ∈ Node(G) be a node of G. Write H for the unique sub-semi-graph\nof PSC-type of G whose set of vertices is V(e). Then one may verify\neasily that S\ndef\n= Node(G|H) \\ {e} is not of separating type as a subset\nof Node(G|H). Thus, since (G|H)\bS has precisely one node, it follows\nimmediately from assertion (iii), together with the claims (∗1) and (∗2),\nthat the profinite Dehn multi-twist α(G|H)\u0004S of (G|H)\bS determined by\nα ∈ Dehn(G) is trivial. In particular, it follows immediately from the def-\ninition of a generization [cf., especially, the definition of the anabelioids\ncorresponding to the vertices of a generization given in Definition 2.8,\n(vi)], together with the definition of a profinite Dehn multi-twist, that\n84 Yuichiro Hoshi and Shinichi Mochizuki\nthe automorphism αG\u0003{e} of the generization G\u0003{e} determined by α\n[cf. Proposition 2.9, (ii)] is a profinite Dehn multi-twist. Therefore,\nsince Node(G\u0003{e})\u0005 < Node(G)\u0005, it follows immediately from assertion\n(ii), together with the induction hypothesis, that αG\u0003{e} ∈ Ker(DG\u0003{e}),\nhence also α ∈ Ker(DG), is trivial. This completes the proof of the claim\n(∗3).\nNext, we claim that\n(∗4): if G is noncyclically primitive [cf. Definition 4.1],\nthen the homomorphism DG is surjective.\nIndeed, this follows immediately from Lemma 4.2, together with the\nvarious definitions involved. This completes the proof of the claim (∗4).\nNext, we claim that\n(∗5): if G is cyclically primitive [cf. Definition 4.1],\nthen the homomorphism DG is surjective.\nIndeed, let λ ∈ ΛG be an element of ΛG . Since we are in the situation\nof Lemma 4.3, we shall apply the notational conventions established\nin Lemma 4.3. Then it follows immediately from Corollary 3.9, (ii),\ntogether with Lemma 4.3, (v), that for any integers a ≤ 0 < b ∈ Z,\nthe natural morphisms G[0,0] → G[a,b] and G[0,0] → G∞ → G induce\nisomorphisms ΛG[a,b]\n∼← ΛG[0,0]\n∼→ ΛG . By abuse of notation, write λ ∈\nΛG[a,b] for the element of ΛG[a,b] corresponding to λ ∈ ΛG . Now since\nG[0,1] is noncyclically primitive [cf. Lemma 4.3, (iv)], it follows from the\nclaims (∗1), (∗4) that there exists a unique profinite Dehn multi-twist\nλ[0,1] ∈ Dehn(G[0,1]) such that DG[0,1](λ[0,1]) = λ.\nNext, we claim that\n(†) : for any a ≤ 0 < b ∈ Z, there exists a [necessarily\nunique — cf. claim (∗3)] profinite Dehn multi-twist\nλ[a,b] ∈ Dehn(G[a,b]) such thatDe(λ[a,b]) = λ for every\nnode e ∈ Node(G[a,b]).\nWe verify this claim (†) by induction on b−a. If b−a = 1, or equivalently,\n[a, b] = [0, 1], then we have already shown the existence of a profinite\nDehn multi-twist λ[0,1] ∈ Dehn(G[0,1]) of the desired type. Now suppose\nthat 1 < b−a, and that for I ∈ {[a, b−1], [a+1, b]}, there exists a profi-\nnite Dehn multi-twist λI ∈ Dehn(GI) such that De(λI) = λ for every\nnode e ∈ Node(GI). Then one may verify easily that Node(GI) may be\nregarded as a subset of Node(G[a,b]), that H[a,b] def= (G[a,b])\u0003Node(GI) is\nnoncyclically primitive, and that, if one allows v to range over the [two]\nvertices of H[a,b], then the resulting semi-graphs of anabelioids (H[a,b])|v\nare naturally isomorphic to HI def= (GI)\u0003Node(GI) and G[cI ,cI ], where we\nwrite cI for b (respectively, a) if I = [a, b−1] (respectively, I = [a+1, b]).\nCombinatorial anabelian topics I 85\nLet ΠeI ⊆ ΠHI be a cuspidal subgroup of ΠHI corresponding to the\ncusp eI determined by the unique node of H[a,b]; Πe[cI ,cI ] ⊆ ΠG[cI ,cI ]\na cuspidal subgroup of ΠG[cI ,cI ] corresponding to the cusp e[cI ,cI ] de-\ntermined by the unique node of H[a,b]; λ̃I ∈ Aut(ΠHI ) a lifting of the\noutomorphism of ΠHI determined by λI ∈ Dehn(GI) ↪→ Aut(HI) [cf.\nProposition 2.9, (ii)] which preserves ΠeI and induces the identity au-\ntomorphism of ΠeI . [Note that since λI ∈ Dehn(GI), one may verify\neasily that such a lifting λ̃I ∈ Aut(ΠHI ) exists.] Then for any element\nδ ∈ Πe[cI ,cI ] of Πe[cI ,cI ] , it follows immediately from Lemma 4.2 that by\ngluing — by means of the natural isomorphism ΠeI\n∼→ Πe[cI ,cI ] — the\nautomorphism λ̃I ∈ Aut(ΠHI ) to the inner automorphism of ΠG[cI ,cI ] by\nδ ∈ Πe[cI ,cI ] , we obtain an outomorphism λ[a,b][δ] of ΠH[a,b] , which —\nin light of [CmbGC], Proposition 1.5, (ii), together with the fact that\nλI ∈ Dehn(GI) — is contained in\nDehn(G[a,b]) ⊆ Aut|grph|(G[a,b]) ↪→ Aut|grph|(H[a,b]) ⊆ Out(ΠH[a,b])\n[cf. Proposition 2.9, (ii)]. Now it follows immediately from the definition\nof the homomorphism “De” that the assignment δ \u0006→ DeG[a,b] (λa,b[δ]) —\nwhere we write eG[a,b] for the node of G[a,b] corresponding to the unique\nnode of H[a,b] — determines a bijection Πe[cI ,cI ]\n∼→ ΛG . Thus, since\nDe(λI) = λ for every node e ∈ Node(GI), we conclude that there exists\na unique element δ ∈ Πe[cI ,cI ] of Πe[cI ,cI ] such that De(λ[a,b][δ]) = λ for\nevery node e ∈ Node(G[a,b]). This completes the proof of the claim (†).\nWrite λ∞ ∈ Aut|grph|(G∞) for the automorphism of G∞ determined\nby the λ[a,b]’s of the claim (†). Now since De(λ[a,b]) = λ for arbitrary\na < b ∈ Z and e ∈ Node(G[a,b]), one may verify easily, by applying\nthe claim (∗3), that the automorphism λ∞ commutes with the natural\naction of π\ntop\n1 (G) \f Z on G∞. Thus, the automorphism λ∞ determines\nan automorphism λG ∈ Aut|grph|(G) of G [cf. Lemma 4.3, (ii)]. Moreover,\nit follows immediately from the definition of λG , together with the fact\nthat De(λ[a,b]) = λ for arbitrary a < b ∈ Z and e ∈ Node(G[a,b]), that\nλG is a profinite Dehn multi-twist such that DG(λG) = λ ∈ ΛG . This\ncompletes the proof of the claim (∗5).\nFinally, we claim that\n(∗6): for arbitrary G, the homomorphism DG is sur-\njective.\n86 Yuichiro Hoshi and Shinichi Mochizuki\nFor each node e ∈ Node(G) of G, it follows from assertion (ii) that we\nhave a commutative diagram of profinite groups\nDehn(G\u0003Node(G)\\{e}) −−−−→ Dehn(G)\nDG\u0003Node(G)\\{e}\n⏐⏐\b ⏐⏐\bDG\nΛG −−−−→\n⊕\ne′∈Node(G) ΛG\n— where the lower horizontal arrow is the natural inclusion into the\ncomponent indexed by e. Now since Node(G\u0003Node(G)\\{e})\u0005 = 1, it follows\nfrom the claims (∗4), (∗5) that the left-hand vertical arrowDG\u0003Node(G)\\{e}\nin the above commutative diagram is surjective. Therefore, by allowing\n“e” to vary among the elements of Node(G), we conclude that DG is\nsurjective. This completes the proof of the claim (∗6) — hence also, in\nlight of the claim (∗3) — of assertion (iv).\nFinally, assertion (v) follows immediately from the various defini-\ntions involved, together with assertion (iv). This completes the proof of\nTheorem 4.8. Q.E.D.\nRemark 4.8.1. In the notation of Theorem 4.8, denote by\nπ\ntemp\n1 (G) the tempered fundamental group of G [cf. the discussion pre-\nceding [SemiAn], Proposition 3.6], by π\ntop\n1 (G) the [discrete] topological\nfundamental group of the underlying semi-graph G of G, by G∞ → G\nthe connected tempered covering of G corresponding to the natural sur-\njection π\ntemp\n1 (G) \u0002 πtop1 (G) [where we refer to [SemiAn], §3, concerning\ntempered coverings of a semi-graph of anabelioids], by Aut|grph|(G∞) the\ngroup of automorphisms of G∞ that induce the identity automorphism\nof the underlying semi-graph of G∞, and by Dehn(G∞) ⊆ Aut|grph|(G∞)\nthe group of “profinite Dehn multi-twists” of G∞ — i.e., automorphisms\nof G∞ which induce the identity automorphism on the underlying semi-\ngraph of G∞, as well as on the anabelioids of G∞ corresponding to the\nvertices of G∞. Then the following hold:\n(i) The natural morphism G∞ → G induces an exact sequence\n1 −→ πtemp1 (G∞) −→ πtemp1 (G) −→ πtop1 (G) −→ 1 .\nMoreover, the subgroup π\ntemp\n1 (G∞) ⊆ πtemp1 (G) of πtemp1 (G) is\ncharacteristic.\n(ii) There exist natural injections\nAut|grph|(G) ↪→ Aut|grph|(G∞) , Dehn(G) ↪→ Dehn(G∞) ,\nCombinatorial anabelian topics I 87\nπ\ntop\n1 (G) ↪→ Aut(G∞)\n— where the third injection is determined up to composition\nwith a π\ntop\n1 (G)-inner automorphism — which satisfy the equal-\nities\nZAut|grph|(G∞)(π\ntop\n1 (G)) = Aut\n|grph|(G) ;\nDehn(G) = Aut|grph|(G) ∩Dehn(G∞) .\n(iii) There exists a natural isomorphism\nDehn(G∞) ∼→\n∏\nNode(G∞)\nΛG .\nIndeed, assertion (i) (respectively, (ii)) follows immediately from a simi-\nlar argument to the argument used in the proof of Lemma 4.3, (i) (respec-\ntively, Lemma 4.3, (ii)), together with the various definitions involved.\nOn the other hand, the existence of the natural isomorphism asserted\nin assertion (iii) follows immediately from the fact that the various ho-\nmomorphisms D(G∞)|H — where H ranges over the sub-semi-graphs of\nPSC-type [cf. Definition 2.2, (i)] of the underlying semi-graph of G∞,\nand we write (G∞)|H for the semi-graph of anabelioids obtained by re-\nstricting G∞ to H [cf. the discussion preceding [SemiAn], Definition 2.2],\nwhich [as is easily verified] is of pro-Σ PSC-type — are isomorphisms.\n[Note that since (G∞)|H is of pro-Σ PSC-type, the fact that D(G∞)|H is an\nisomorphism is a consequence of Theorem 4.8, (iv). However, since H is\na tree, it follows from the simple structure of H that one may verify that\nD(G∞)|H is an isomorphism in a fairly direct fashion, by arguing as in the\nproofs of the claims (∗1), (∗4) that appear in the proof of Theorem 4.8,\n(iv).]\nIn particular, it follows immediately from assertions (ii), (iii) that\none may recover the natural isomorphism\nDehn(G) ∼→ Z∏\nNode(G∞) ΛG\n(π\ntop\n1 (G))\n∼→\n∏\nNode(G)\nΛG\nof Theorem 4.8, (iv).\nDefinition 4.9. We shall write\nGlu(G) ⊆\n∏\nv∈Vert(G)\nAut|grph|(G|v)\n88 Yuichiro Hoshi and Shinichi Mochizuki\nfor the [closed] subgroup of “glueable” collections of outomorphisms\nof the direct product\n∏\nv∈Vert(G) Aut\n|grph|(G|v) consisting of elements\n(αv)v∈Vert(G) such that χv(αv) = χw(αw) [cf. Definition 3.8, (ii)] for any\nv, w ∈ Vert(G).\nProposition 4.10 (Properties of automorphisms that fix the\nunderlying semi-graph).\n(i) (Factorization) The natural homomorphism\nAut|grph|(G) −→ ∏v∈Vert(G) Aut|grph|(G|v)\nα \u0006→ (αG|v )v∈Vert(G)\n[cf. Definition 2.14, (ii); Remark 2.5.1, (ii)] factors through\nthe closed subgroup Glu(G) ⊆ ∏v∈Vert(G) Aut|grph|(G|v).\n(ii) (Exact sequence relating profinite Dehn multi-twists\nand glueable outomorphisms) The resulting homomor-\nphism ρVertG : Aut\n|grph|(G) → Glu(G) [cf. (i)] fits into an exact\nsequence of profinite groups\n1 −→ Dehn(G) −→ Aut|grph|(G) ρ\nVert\nG−→ Glu(G) −→ 1 .\n(iii) (Surjectivity of cyclotomic characters) The restriction of\nthe pro-Σ cyclotomic character χG of G [cf. Definition 3.8,\n(ii)] to Aut|grph|(G) ⊆ Aut(G)\nχG |Aut|grph|(G) : Aut|grph|(G) −→ (ẐΣ)∗\n— hence also χG — is surjective.\n(iv) (Liftability of automorphisms) Let H be a sub-semi-graph\nof PSC-type [cf. Definition 2.2, (i)] of G and S ⊆ Node(G|H)\n[cf. Definition 2.2, (ii)] a subset of Node(G|H) that is not of\nseparating type [cf. Definition 2.5, (i)]. Then the homo-\nmorphism\nAut|grph|(G) −→ Aut|grph|((G|H)\bS)\nα \u0006→ α(G|H)\u0004S\n[cf. Definitions 2.5, (ii); 2.14, (ii)] is surjective.\nCombinatorial anabelian topics I 89\nProof. Assertion (i) follows immediately from Corollary 3.9, (iv).\nNext, we verify assertion (ii). It follows immediately from the various\ndefinitions involved that Ker(ρVertG ) = Dehn(G) ⊆ Aut|grph|(G). Thus,\nto complete the proof of assertion (ii), it suffices to verify that the ho-\nmomorphism ρVertG is surjective.\nNow we claim that\n(∗1): if G is noncyclically primitive [cf. Definition 4.1],\nthen the homomorphism ρVertG is surjective.\nIndeed, this follows immediately from Corollary 3.9, (v); Lemma 4.2,\ntogether with the various definitions involved. This completes the proof\nof the claim (∗1).\nNext, we claim that\n(∗2): if G is cyclically primitive [cf. Definition 4.1],\nthen the homomorphism ρVertG is surjective.\nIndeed, since we are in the situation of Lemma 4.3, we shall apply the\nnotational conventions established in Lemma 4.3. Then it follows im-\nmediately from the fact that Vert(G)\u0005 = 1 [cf. Remark 4.1.1], together\nwith Lemma 4.3, (v), that the composite of natural morphisms G[0,0] →\nG∞ → G determines a natural identification Glu(G) ∼→ Aut|grph|(G[0,0]).\nLet α = α[0,0] ∈ Glu(G) ∼→ Aut|grph|(G[0,0]) be an element of Glu(G) ∼→\nAut|grph|(G[0,0]). For each a ∈ Z, denote by α[a,a] ∈ Aut(G[a,a]) the au-\ntomorphism of G[a,a] determined by conjugating the automorphism α of\nG[0,0] by the isomorphism γa∞ : G[0,0]\n∼→ G[a,a] [cf. Lemma 4.3, (iii), (vi)].\nThen for any c < b ∈ Z, it follows from the various definitions involved\nthat the various α[a,a]’s satisfy the gluing condition necessary to apply\nthe claim (∗1), hence that we may glue them together [cf. the proof of\nthe claim (∗3) below for more details concerning this sort of gluing argu-\nment] to obtain a(n) [not necessarily unique] element of Aut|grph|(G[c,b]).\nThus, by allowing c < b ∈ Z to vary, we obtain a(n) [not necessarily\nunique] element α∞ ∈ Aut|grph|(G∞) of Aut|grph|(G∞). Now it follows\nimmediately from the definition of α∞ that for any γ ∈ πtop1 (G), the\nautomorphism [α∞, γ]\ndef\n= α∞ · γ · α−1∞ · γ−1 of G∞ is a “profinite Dehn\nmulti-twist” of G∞, i.e., [α∞, γ] ∈ Dehn(G∞) [cf. Remark 4.8.1]. More-\nover, one may verify easily that the assignment γ \u0006→ [α∞, γ] determines\na 1-cocycle π\ntop\n1 (G) → Dehn(G∞). Thus, by Remark 4.8.1, (iii), together\nwith the [easily verified] fact that\nH1(Z,\n∏\nZ\nẐΣ) = {0}\n90 Yuichiro Hoshi and Shinichi Mochizuki\n— where we take the action of Z on\n∏\nZ Ẑ\nΣ to be the action determined\nby the trivial action of Z on ẐΣ and the action of Z on the index set\nZ given by addition — we conclude that there exists an element β ∈\nDehn(G∞) such that the automorphism β ◦α∞ commutes with the nat-\nural action of π\ntop\n1 (G) on G∞. In particular, it follows from Lemma 4.3,\n(ii), that β◦α∞ determines an element αG ∈ Aut|grph|(G) of Aut|grph|(G).\nNow since β ∈ Dehn(G∞), it follows immediately from the various def-\ninitions involved that ρVertG (αG) = α ∈ Glu(G)\n∼→ Aut|grph|(G[0,0]). This\ncompletes the proof of the claim (∗2).\nFinally, we claim that\n(∗3): for arbitrary G, the homomorphism ρVertG is sur-\njective.\nWe verify this claim (∗3) by induction on Node(G)\u0005. If Node(G)\u0005 ≤ 1,\nthen this follows immediately from the claims (∗1), (∗2). Now sup-\npose that Node(G)\u0005 > 1, and that the induction hypothesis is in force.\nLet e ∈ Node(G) be a node of G. Write H for the unique sub-semi-\ngraph of PSC-type of G whose set of vertices is V(e). Then one may\nverify easily that S\ndef\n= Node(G|H) \\ {e} is not of separating type as\na subset of Node(G|H). Thus, since (G|H)\bS has precisely one node,\nand (αv)v∈V(e) may be regarded as an element of Glu((G|H)\bS), it fol-\nlows from the claims (∗1), (∗2) that there exists an automorphism β ∈\nAut|grph|((G|H)\bS) of (G|H)\bS such that ρVert(G|H)\u0004S (β) = (αv)v∈V(e) ∈\nGlu((G|H)\bS). Write β\u0003{e} ∈ Aut|grph|(((G|H)\bS)\u0003{e}) for the auto-\nmorphism of ((G|H)\bS)\u0003{e} determined by β ∈ Aut|grph|((G|H)\bS) [cf.\nProposition 2.9, (ii)]. Then it follows immediately from Corollary 3.9,\n(i), together with the definition of a generization [cf., especially, the def-\ninition of the anabelioids corresponding to the vertices of a generization\ngiven in Definition 2.8, (vi)], that the element\nγ\ndef\n= (β\u0003{e}, (αv)v \n∈V(e))\n∈ Aut|grph|(((G|H)\bS)\u0003{e}) ×\n∏\nv \n∈V(e) Aut\n|grph|(G|v)\nmay be regarded as an element of Glu(G\u0003{e}). Now since Node(G\u0003{e})\u0005\n< Node(G)\u0005, it follows from the induction hypothesis that there ex-\nists an automorphism α\u0003{e} ∈ Aut|grph|(G\u0003{e}) of G\u0003{e} such that\nρVertG\u0003{e}(α\u0003{e}) = γ ∈ Glu(G\u0003{e}). On the other hand, since β\u0003{e}\narises from an element β of Aut|grph|((G|H)\bS), it follows immediately\nfrom [CmbGC], Proposition 1.5, (ii), that α\u0003{e} ∈ Aut|grph|(G\u0003{e}) is\ncontained in the image of Aut|grph|(G) ↪→ Aut|grph|(G\u0003{e}) [cf. Proposi-\ntion 2.9, (ii)]. Moreover, since ρVert\n(G|H)\u0004S (β) = (αv)v∈V(e) ∈ Glu((G|H)\bS),\nCombinatorial anabelian topics I 91\nit follows immediately from our original characterization of α\u0003{e} that\nρVertG (α\u0003{e}) = (αv)v∈Vert(G) ∈ Glu(G). Thus, we conclude that ρVertG is\nsurjective, as desired. This completes the proof of the claim (∗3), hence\nalso of assertion (ii).\nNext, we verify assertion (iii). First, let us observe that one may\nverify easily that there exist a semi-graph of anabelioids of pro-Σ PSC-\ntype H that is totally degenerate [cf. Definition 2.3, (iv)], a subset S ⊆\nNode(H), and an isomorphism of semi-graphs of anabelioids H\u0003S ∼→\nG. Now since we have a natural injection Aut|grph|(H) ↪→\nAut|grph|(H\u0003S) ∼→ Aut|grph|(G) [cf. Proposition 2.9, (ii)], it follows\nimmediately from Corollary 3.9, (i), that to verify assertion (iii), by\nreplacing G by H, we may assume without loss of generality that G is\ntotally degenerate. On the other hand, it follows immediately from as-\nsertion (ii), together with Corollary 3.9, (ii), that to verify assertion (iii),\nit suffices to verify the surjectivity of χG|v for each v ∈ Vert(G). Thus, to\nverify assertion (iii), by replacing G by G|v, we may assume without loss\nof generality that G is of type (0, 3) [cf. Definition 2.3, (i)]. But assertion\n(iii) in the case where G is of type (0, 3) follows immediately by consid-\nering the natural outer action of the absolute Galois group Gal(Q\/Q)\nof the field of rational numbers Q — where we use the notation Q to\ndenote an algebraic closure of Q — on the semi-graph of anabelioids of\npro-Σ PSC-type associated to the tripod P1\nQ\n\\ {0, 1,∞} over Q. This\ncompletes the proof of assertion (iii).\nFinally, we verify assertion (iv). Write H def= (G|H)\bS . Then it\nfollows immediately from assertion (ii), together with Theorem 4.8, (iii),\nthat the homomorphism Aut|grph|(G) → Aut|grph|(H) in question fits\ninto a commutative diagram of profinite groups\n1 −−−−→ Dehn(G) −−−−→ Aut|grph|(G) ρ\nVert\nG−−−−→ Glu(G) −−−−→ 1⏐⏐\b ⏐⏐\b ⏐⏐\b\n1 −−−−→ Dehn(H) −−−−→ Aut|grph|(H) ρ\nVert\nH−−−−→ Glu(H) −−−−→ 1\n— where the horizontal sequences are exact. Now since the left-hand ver-\ntical arrow is surjective [cf. Theorem 4.8, (iii), (iv)], to verify assertion\n(iv), it suffices to verify the surjectivity of the right-hand vertical arrow.\nBut this follows immediately from assertion (iii), together with the defi-\nnition of “Glu(−)”. This completes the proof of assertion (iv). Q.E.D.\n92 Yuichiro Hoshi and Shinichi Mochizuki\n§5. Comparison with scheme theory\nIn the present §, we discuss [cf. Proposition 5.6; Theorem 5.7; Corol-\nlaries 5.9, 5.10 below] the relationship between intrinsic, group-theoretic\nproperties of profinite Dehn multi-twists [such as length, nondegener-\nacy, and positive definiteness — cf. Definitions 5.1; 5.8, (ii), (iii) below]\nand scheme-theoretic characterizations of properties of outer representa-\ntions of pro-Σ PSC-type [such as length, strict nodal nondegeneracy, and\nIPSC-ness — cf. Definition 5.3, (ii) below; [NodNon], Definition 2.4, (i),\n(iii)]. The resulting theory leads naturally to a proof of the graphicity\nof C-admissible outomorphisms contained in the commensurator of the\ngroup of profinite Dehn multi-twists [cf. Theorem 5.14 below].\nLet Σ be a nonempty set of prime numbers and G a semi-graph of\nanabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph\nof G, ΠG for the [pro-Σ] fundamental group of G, and G̃ → G for the\nuniversal covering of G corresponding to ΠG .\nDefinition 5.1. Let ρ : I → Aut(G) (⊆ Out(ΠG)) be an outer rep-\nresentation of pro-Σ PSC-type [cf. [NodNon], Definition 2.1, (i)] which\nis of NN-type [cf. [NodNon], Definition 2.4, (iii)] and ẽ ∈ Node(G̃) an\nelement of Node(G̃). Write ΠI def= ΠG\nout\n\u0002 I [cf. the discussion entitled\n“Topological groups” in §0]; ṽ, w̃ ∈ Vert(G) for the two distinct elements\nof Vert(G̃) such that V(ẽ) = {ṽ, w̃} [cf. [NodNon], Remark 1.2.1, (iii)];\nIẽ, Iṽ, Iw̃ ⊆ ΠI for the inertia subgroups of ΠI associated to ẽ, ṽ, w̃,\nrespectively, i.e., the centralizers of Πẽ, Πṽ, Πw̃ ⊆ ΠI in ΠI , respectively\n[cf. [NodNon], Definition 2.2]. Then it follows from condition (3) of\n[NodNon], Definition 2.4, that the natural homomorphism Iṽ × Iw̃ → Iẽ\nis an open injection. Write\nlngΣG (ẽ, ρ)\ndef\n= [Iẽ : Iṽ × Iw̃]\nfor the index of Iṽ × Iw̃ in Iẽ; we shall refer to lngΣG (ẽ, ρ) as the Σ-length\nof ẽ with respect to ρ. Note that it follows immediately from the various\ndefinitions involved that the Σ-length of ẽ with respect to ρ depends only\non e\ndef\n= ẽ(G) ∈ Node(G) and ρ. Write\nlngΣG (e, ρ)\ndef\n= lngΣG (ẽ, ρ) ;\nwe shall refer to lngΣG (e, ρ) as the Σ-length of e ∈ Node(G) with respect\nto ρ.\nCombinatorial anabelian topics I 93\nLemma 5.2 (Outer representations of SVA-type and profi-\nnite Dehn multi-twists). Let ρ : I → Aut(G) (⊆ Out(ΠG)) be an outer\nrepresentation of pro-Σ PSC-type which is of SVA-type [cf. [NodNon],\nDefinition 2.4, (ii)] and ẽ ∈ Node(G̃) an element of Node(G̃). Write\nΠI\ndef\n= ΠG\nout\n\u0002 I [cf. the discussion entitled “Topological groups” in\n§0]; ṽ, w̃ ∈ Vert(G̃) for the two distinct elements of Vert(G̃) such that\nV(ẽ) = {ṽ, w̃} [cf. [NodNon], Remark 1.2.1, (iii)]; Iẽ, Iṽ, Iw̃ ⊆ ΠI for\nthe inertia subgroups of ΠI associated to ẽ, ṽ, w̃, respectively; e\ndef\n= ẽ(G);\nv\ndef\n= ṽ(G). Then the following hold:\n(i) (Outer representations of SVA-type and profinite Dehn\nmulti-twists) The outer representation ρ factors through the\nclosed subgroup Dehn(G) ⊆ Aut(G). By abuse of notation,\nwrite ρ for the resulting homomorphism I → Dehn(G).\n(ii) (Outer representations of SVA-type and homomor-\nphisms of Dehn coordinates) The natural inclusions Iṽ,\nIw̃ ↪→ Iẽ and the composite Iẽ ↪→ ΠI \u0002 I determine a diagram\nof profinite groups\nIṽ × Iw̃⏐⏐\b\n1 −−−−→ Πẽ −−−−→ Iẽ −−−−→ I −−−−→ 1\n— where the lower horizontal sequence is exact, and the closed\nsubgroups Iṽ, Iw̃ ⊆ Iẽ determine sections of the surjection\nIẽ \u0002 I, respectively — hence also homomorphisms\nI\n∼← Iṽ → Iẽ\/Iw̃ ∼← Πẽ = Πe\nsyn bṽ∼→ Λv\nsyn v∼→ ΛG\n— where the first “\n∼←” denotes the isomorphism given by the\ncomposite Iṽ ↪→ ΠI \u0002 I, and bṽ denotes the branch of e deter-\nmined by the [unique] branch of ẽ that abuts to ṽ. Moreover,\nthe composite of these homomorphisms\nI → ΛG\ncoincides with the composite\nI\nρ−→ Dehn(G) De−→ ΛG\n[cf. (i); Definition 4.7]. In particular, if ρ is of SNN-type\n[cf. [NodNon], Definition 2.4, (iii)], then the image of the\n94 Yuichiro Hoshi and Shinichi Mochizuki\ncomposite I\nρ→ Dehn(G) De→ ΛG coincides with lngΣG (e, ρ) ·ΛG ⊆\nΛG.\n(iii) (Centralizers and cyclotomic characters) Suppose that ρ\nis of SNN-type [cf. [NodNon], Definition 2.4, (iii)]. Let\ne ∈ Node(G) be a node of G. Then χG(α) = 1 [cf. Defini-\ntion 3.8, (ii)] for any α ∈ ZAut{e}(G)(Im(ρ)) ⊆ Aut{e}(G) [cf.\nDefinition 2.6, (i)].\nProof. Assertion (i) follows immediately from condition (2′) of\n[NodNon], Definition 2.4. Next, we verify assertion (ii). The fact that\nthe natural inclusions Iṽ, Iw̃ ↪→ Iẽ and the composite Iẽ ↪→ ΠI \u0002 I give\nrise to the diagram and homomorphisms of the first and second displays\nin the statement of assertion (ii) follows immediately from [NodNon],\nLemma 2.5, (iv); condition (2′) of [NodNon], Definition 2.4. On the\nother hand, it follows immediately from the various definitions involved\nthat the image of each β ∈ I via the composite of I ∼← Iṽ with the\naction Iṽ → Aut(ΠG) given by conjugation coincides with the “α[ṽ]” of\nLemma 4.6, (i), in the case where one takes “α” to be ρ(β). Thus, it\nfollows immediately from the definition of Iw̃ that the image of β ∈ I\nvia the composite I\n∼← Iṽ → Iẽ\/Iw̃ ∼→ Πẽ coincides with the “δẽ,ṽ” of\nLemma 4.6, (i), in the case where one takes “α” to be ρ(β). Therefore, it\nfollows immediately from the definition of De that the homomorphisms\nof the final two displays of assertion (ii) coincide. Thus, the final portion\nof assertion (ii) concerning ρ of SNN-type follows immediately from the\ndefinition of Σ-length. This completes the proof of assertion (ii). To ver-\nify assertion (iii), let us first observe that, by Theorem 4.8, (v), the con-\njugation action of α ∈ Aut{e}(G) on the ΛG ⊆\n⊕\nNode(G) ΛG\n∼← Dehn(G)\nindexed by e ∈ Node(G) is given by multiplication by χG(α). On the\nother hand, since N \u0015 lngΣG (e, ρ) \u0004= 0, it follows from the final portion of\nassertion (ii) that the projection of Im(ρ) to the coordinate indexed by\ne is open. Thus, the fact that α lies in the centralizer ZAut{e}(G)(Im(ρ))\nimplies that χG(α) = 1, as desired. This completes the proof of assertion\n(iii). Q.E.D.\nDefinition 5.3. Let R be a complete discrete valuation ring whose\nresidue field is separably closed of characteristic \u0004∈ Σ; π ∈ R a prime\nelement of R; vR the discrete valuation of R such that vR(π) = 1;\nSlog the log scheme obtained by equipping S\ndef\n= SpecR with the log\nstructure defined by the maximal ideal (π) ⊆ R of R; slog the log scheme\nCombinatorial anabelian topics I 95\nobtained by equipping the spectrum s of the residue field of R with\nthe log structure induced by the log structure of Slog via the natural\nclosed immersion s ↪→ S; X log a stable log curve over Slog; G\nXlog\nthe\nsemi-graph of anabelioids of pro-Σ PSC-type determined by the special\nfiber X logs\ndef\n= X log ×Slog slogof the stable log curve X log [cf. [CmbGC],\nExample 2.5]; ISlog (\f ẐΣ) the maximal pro-Σ completion of the log\nfundamental group π1(S\nlog) of Slog.\n(i) One may verify easily that the natural outer representation\nISlog → Aut(GXlog) associated to the stable log curve X log\nover Slog factors through Dehn(G\nXlog\n) ⊆ Aut(G\nXlog\n). We shall\nwrite\nρ\nX\nlog\ns\n: ISlog −→ Dehn(GXlog)\nfor the resulting homomorphism.\n(ii) It follows from the well-known local structure of a stable log\ncurve in a neighborhood of a node that for each node e of the\nspecial fiber of X log, there exists a nonzero element ae \u0004= 0 of\nthe maximal ideal (π) ⊆ R such that the completion ÔX,e of\nthe local ring OX,e at e is isomorphic to R[[s1, s2]]\/(s1s2 − ae)\n— where s1, s2 denote indeterminates. Write\nlngXlog(e)\ndef\n= vR(ae); lng\nΣ\nXlog(e)\ndef\n= [ẐΣ : lngXlog(e) · ẐΣ].\nWe shall refer to lngXlog(e) as the length of e and to lng\nΣ\nXlog(e)\nas the Σ-length of e. One verifies easily that lngXlog(e), hence\nalso lngΣXlog(e), depends only on e, i.e., is independent of the\nchoice of the isomorphism ÔX,e \f R[[s1, s2]]\/(s1s2 − ae).\nLemma 5.4 (Local geometric universal outer representa-\ntions). In the notation of Definition 5.3, suppose that G\nXlog\nis of type\n(g, r) [cf. Definition 2.3, (i); Remark 2.3.1]. Write N\ndef\n= Node(G\nXlog\n)\u0005\nand σlog : Slog → (Mlogg,r)S [cf. the discussion entitled “Curves” in §0]\nfor the classifying morphism of the stable log curve X log over Slog. Then\nthe following hold:\n(i) (Local structure of the moduli stack of pointed sta-\nble curves) Write Ô for the completion of the local ring of\n96 Yuichiro Hoshi and Shinichi Mochizuki\n(Mg,r)S at the image of the closed point of S via the underly-\ning (1-)morphism of stacks σ of σlog and T log for the\n[fs] log scheme obtained by equipping T\ndef\n= Spec Ô with the log\nstructure induced by the log structure of (Mlogg,r)S. [Thus, we\nhave a tautological strict [cf. [Illu], 1.2] (1-)morphism T log →\n(Mlogg,r)S.] Then there exists an isomorphism of R-algebras\nR[[t1, · · · , t3g−3+r]] ∼→ Ô such that the following hold:\n• The log structure of the log scheme T log is given by the\nfollowing chart:⊕\ne∈Node(G\nXlog\n) Ne −→ R[[t1, · · · , t3g−3+r]]\n∼→ Ô\n(ne1 , · · · , neN ) \u0006→ t\nne1\n1 · · · t\nneN\nN\n— where we write Ne for the copy of N indexed by e ∈\nNode(G\nXlog\n).\n• For 1 ≤ i ≤ N , the homomorphism of R-algebras Ô → R\ninduced by the morphism σ maps ti to aei [cf. Defini-\ntion 5.3, (ii)].\n(ii) (Log-scheme-theoretic description of log fundamental\ngroups) Write IT log for the maximal pro-Σ quotient of the\nlog fundamental group π1(T\nlog) of T log. Then we have natural\nisomorphisms\nISlog\n∼→ Hom\n(\nNgp, ẐΣ(1)\n)\n;\nIT log\n∼→ Hom\n(⊕\ne∈Node(G\nXlog\n) N\ngp\ne , Ẑ\nΣ(1)\n)\n∼→ ⊕e∈Node(G\nXlog\n) Hom\n(\nNgpe , Ẑ\nΣ(1)\n)\n,\nand the homomorphism ISlog → IT log induced by the classify-\ning morphism σlog is the homomorphism obtained by applying\nthe functor “Hom\nẐΣ\n((−)gp, ẐΣ(1))” to the homomorphism of\nmonoids⊕\ne∈Node(G\nXlog\n) Ne −→ N\n(ne1 , · · · , neN ) \u0006→\n∑N\ni=1 nei lngXlog(ei)\n.\n(iii) (Local geometric universal outer representations) The\nnatural outer representation IT log → Aut(GXlog) associ-\nated to the stable log curve over T log determined\nCombinatorial anabelian topics I 97\nby the tautological strict morphism T log → (Mlogg,r)S factors\nthrough Dehn(G\nXlog\n) ⊆ Aut(G\nXlog\n); thus, we have a homomor-\nphism IT log → Dehn(GXlog). Moreover, the homomorphism\nρ\nX\nlog\ns\n: ISlog → Dehn(GXlog) factors as the composite of the ho-\nmomorphism ISlog → IT log induced by σlog and this homomor-\nphism IT log → Dehn(GXlog).\nProof. Assertion (i) follows immediately from the well-known local\nstructure of the log stack (Mlogg,r)S [cf. [Knud], Theorem 2.7]. Assertion\n(ii) follows immediately from assertion (i), together with the well-known\nstructure of the log fundamental groups of Slog and T log. Assertion (iii)\nfollows immediately from the various definitions involved. Q.E.D.\nDefinition 5.5. In the notation of Definition 5.3, Lemma 5.4, we\nshall write tlog for the log scheme obtained by equipping the closed point\nt of T with the log structure naturally induced by the log structure of\nT log; X\nlog\nt for the stable log curve over t\nlog corresponding to the natural\nstrict morphism tlog (↪→ T log) → (Mlogg,r)S ;\nρuniv\nX\nlog\nt\n: IT log −→ Dehn(GXlog)\nfor the homomorphism obtained in Lemma 5.4, (iii).\nProposition 5.6 (Outer representations arising from stable\nlog curves). In the notation of Definition 5.3, Lemma 5.4, the following\nhold:\n(i) (Compatibility of Σ-lengths) For each node e ∈\nNode(G\nXlog\n) of G\nXlog\n, it holds that\nlngΣG\nXlog\n(e, ρ\nX\nlog\ns\n) = lngΣXlog(e)\n[cf. Definitions 5.1; 5.3, (ii)].\n(ii) (Isomorphicity of local geometric universal outer rep-\nresentations) The homomorphism\nρuniv\nX\nlog\nt\n: IT log −→ Dehn(GXlog)\n[cf. Definition 5.5] is an isomorphism of profinite groups.\n98 Yuichiro Hoshi and Shinichi Mochizuki\n(iii) (Compatibility with generization) Let Q ⊆ Node(G\nXlog\n)\nbe a subset of Node(G\nXlog\n). Then there exist a stable log curve\nY log over Slog and an isomorphism of semi-graphs of anabe-\nlioids (G\nXlog\n)\u0003Q\n∼→ G\nY log\nthat fit into a commutative diagram\nof profinite groups\nI\nT\nlog\nY\nρuniv\nY\nlog\nt−−−−→ Dehn(G\nY log\n)⏐⏐\b ⏐⏐\b\nI\nT\nlog\nX\nρuniv\nX\nlog\nt−−−−→ Dehn(G\nXlog\n)\n— where we write I\nT\nlog\nX\n, I\nT\nlog\nY\nfor the “IT log” associated to X\nlog,\nY log, respectively; the right-hand vertical arrow is the natural\ninclusion induced, via the isomorphism (G\nXlog\n)\u0003Q\n∼→ G\nY log\n, by\nthe natural inclusion of Theorem 4.8, (ii); the left-hand vertical\narrow is the injection induced, via the [relevant] isomorphism\nof Lemma 5.4, (ii), by the natural projection of monoids⊕\ne∈Node(G\nXlog\n)\nNe \u0002\n⊕\ne∈Node(G\nY log\n)\nNe .\n[Note that it follows immediately from the various definitions\ninvolved that Node(G\nY log\n)\n∼← Node((G\nXlog\n)\u0003Q) may be regarded\nas a subset of Node(G\nXlog\n).]\n(iv) (Compatibility with specialization) Let H be a semi-graph\nof anabelioids of pro-Σ PSC-type, Q ⊆ Node(H), and H\u0003Q ∼→\nG\nXlog\nan isomorphism of semi-graphs of anabelioids. Then\nthere exist a stable log curve Y log over Slog and an isomor-\nphism of semi-graphs of anabelioids H ∼→ G\nY log\nthat fit into a\ncommutative diagram of profinite groups\nI\nT\nlog\nX\nρuniv\nX\nlog\nt−−−−→ Dehn(G\nXlog\n)⏐⏐\b ⏐⏐\b\nI\nT\nlog\nY\nρuniv\nY\nlog\nt−−−−→ Dehn(G\nY log\n)\n— where we write I\nT\nlog\nX\n, I\nT\nlog\nY\nfor the “IT log” associated to X\nlog,\nY log, respectively; the right-hand vertical arrow is the natural\nCombinatorial anabelian topics I 99\ninclusion induced, via the isomorphisms H\u0003Q ∼→ GXlog and\nH ∼→ G\nY log\n, by the natural inclusion of Theorem 4.8, (ii); the\nleft-hand vertical arrow is the injection induced, via the [rele-\nvant] isomorphism of Lemma 5.4, (ii), by the natural projection\nof monoids ⊕\ne∈Node(G\nY log\n)\nNe \u0002\n⊕\ne∈Node(G\nXlog\n)\nNe .\n[Note that it follows immediately from the various definitions\ninvolved that Node(G\nXlog\n)\n∼← Node(H\u0003Q) may be regarded as\na subset of Node(G\nY log\n)\n∼← Node(H).]\n(v) (Input compatibility with “surgery”) Let H be a sub-\nsemi-graph of PSC-type [cf. Definition 2.2, (i)] of the un-\nderlying semi-graph of G\nXlog\n, Q ⊆ Node((G\nXlog\n)|H) [cf. Def-\ninition 2.2, (ii)] a subset of Node((G\nXlog\n)|H) that is not of\nseparating type [cf. Definition 2.5, (i)], and U ⊆\nCusp(((G\nXlog\n)|H)\bQ) [cf. Definition 2.5, (ii)] an omittable\n[cf. Definition 2.4, (i)] subset of Cusp(((G\nXlog\n)|H)\bQ). Then\nthere exist a stable log curve Y log over Slog and an isomor-\nphism (((G\nXlog\n)|H)\bQ)•U ∼→ GY log [cf. Definition 2.4, (ii)] that\nfit into a commutative diagram of profinite groups\nISlog −−−−→ IT logX\nρuniv\nX\nlog\nt−−−−→ Dehn(G\nXlog\n)∥∥∥ ⏐⏐\b ⏐⏐\b\nISlog −−−−→ IT logY\nρuniv\nY\nlog\nt−−−−→ Dehn(G\nY log\n)\n— where we write I\nT\nlog\nX\n, I\nT\nlog\nY\nfor the “IT log” associated to\nX log, Y log, respectively; the left-hand horizontal arrows are the\nhomomorphisms induced by the classifying morphisms associ-\nated to X log, Y log, respectively; the right-hand vertical arrow\nis the natural surjection induced, via the isomorphism\n(((G\nXlog\n)|H)\bQ)•U ∼→ GY log , by the natural surjection of The-\norem 4.8, (iii); the middle vertical arrow is the surjection in-\nduced, via the [relevant] isomorphism of Lemma 5.4, (ii), by\nthe natural inclusion of monoids⊕\ne∈Node(G\nY log\n)\nNe ↪→\n⊕\ne∈Node(G\nXlog\n)\nNe .\n100 Yuichiro Hoshi and Shinichi Mochizuki\n[Note that it follows immediately from the various definitions\ninvolved that Node(G\nY log\n)\n∼← Node((((G\nXlog\n)|H)\bQ)•U ) may be\nregarded as a subset of Node(G\nXlog\n).]\n(vi) (Output compatibility with “surgery”) Let H be a semi-\ngraph of anabelioids of pro-Σ PSC-type, K a sub-semi-graph\nof PSC-type [cf. Definition 2.2, (i)] of the underlying semi-\ngraph of H, Q ⊆ Node(H|K) [cf. Definition 2.2, (ii)] a subset\nof Node(H|K) that is not of separating type [cf. Defini-\ntion 2.5, (i)], U ⊆ Cusp((H|K)\bQ) [cf. Definition 2.5, (ii)] an\nomittable [cf. Definition 2.4, (i)] subset of Cusp((H|K)\bQ),\nand ((H|K)\bQ)•U ∼→ GXlog [cf. Definition 2.4, (ii)] an isomor-\nphism of semi-graphs of anabelioids. Then there exist a stable\nlog curve Y log over Slog and an isomorphism of semi-graphs of\nanabelioids H ∼→ G\nY log\nthat fit into a commutative diagram of\nprofinite groups\nISlog −−−−→ IT logY\nρuniv\nY\nlog\nt−−−−→ Dehn(G\nY log\n)∥∥∥ ⏐⏐\b ⏐⏐\b\nISlog −−−−→ IT logX\nρuniv\nX\nlog\nt−−−−→ Dehn(G\nXlog\n)\n— where we write I\nT\nlog\nX\n, I\nT\nlog\nY\nfor the “IT log” associated to\nX log, Y log, respectively; the left-hand horizontal arrows are\nthe homomorphisms induced by the classifying morphisms as-\nsociated to Y log, X log, respectively; the right-hand vertical ar-\nrow is the natural surjection induced, via the isomorphisms\n((H|K)\bQ)•U ∼→ GXlog and H\n∼→ G\nY log\n, by the natural surjec-\ntion of Theorem 4.8, (iii); the middle vertical arrow is the sur-\njection induced, via the [relevant] isomorphism of Lemma 5.4,\n(ii), by the natural inclusion of monoids⊕\ne∈Node(G\nXlog\n)\nNe ↪→\n⊕\ne∈Node(G\nY log\n)\nNe .\n[Note that it follows immediately from the various definitions\ninvolved that Node(G\nXlog\n)\n∼← Node(((H|K)\bQ)•U ) may be re-\ngarded as a subset of Node(G\nY log\n)\n∼← Node(H).]\nProof. Assertion (i) follows immediately from the well-known lo-\ncal structure of a stable log curve in a neighborhood of a node. Next,\nCombinatorial anabelian topics I 101\nwe verify assertion (ii). By allowing “ρ\nX\nlog\ns\n” to vary among the natu-\nral outer representations ISlog → Dehn(GXlog) associated to stable log\ncurves “X log” over Slog whose classifying morphisms “σ” coincide with\nthe given σ on the closed point s of S, one concludes that the surjec-\ntivity of ρuniv\nX\nlog\nt\nfollows immediately from the final portion of Lemma 5.2,\n(ii), concerning ρ of SNN-type [cf. also assertion (i); Theorem 4.8, (iv)].\n[Here, we recall that ρ\nX\nlog\ns\nis of IPSC-type [cf. [NodNon], Definition\n2.4, (i)], hence also of SNN-type [cf. [NodNon], Remark 2.4.2].] On the\nother hand, since both Dehn(G\nXlog\n) and IT log are free Ẑ\nΣ-modules of\nrank Node(G\nXlog\n)\u0005 [cf. Theorem 4.8, (iv); Lemma 5.4, (ii)], assertion (ii)\nfollows immediately from this surjectivity of ρuniv\nX\nlog\nt\n. This completes the\nproof of assertion (ii).\nAssertion (iii) (respectively, (iv)) follows immediately, in light of\nthe well-known structure of (Mlogg,r)S [cf. also the discussion entitled\n“The Étale Fundamental Group of a Log Scheme” in [CmbCsp], §0,\nconcerning the specialization isomorphism on fundamental groups, as\nwell as Remark 5.6.1 below], by considering a lifting to Slog of a sta-\nble log curve over slog obtained by deforming the nodes of the special\nfiber X logs\ndef\n= X log ×Slog slog corresponding to the nodes contained in\nQ (respectively, degenerating the moduli of X logs so as to obtain nodes\ncorresponding to the nodes contained in Q) [cf. also Proposition 4.10,\n(iii)].\nNext, we verify assertion (v). First, we observe that one may verify\neasily that if H is the underlying semi-graph of G\nXlog\n, and Q = ∅, then\nthe stable log curve Y log over Slog obtained by omitting the cusps ofX log\ncontained in U and the resulting natural isomorphism (G\nXlog\n)•U\n∼→ G\nY log\nsatisfy the conditions given in the statement of assertion (v). Thus, one\nverifies immediately that to verify assertion (v), we may assume without\nloss of generality that U = ∅.\nWrite H def= ((G\nXlog\n)|H)\bQ and V def= Vert(GXlog) \\Vert((GXlog)|H) ⊆\nVert(G\nXlog\n). Denote by (gH, rH) the type of H, and, for each v ∈ V , by\n(gv, rv) the type of v [cf. Definition 2.3, (i), (iii); Remark 2.3.1]. Then\nit follows immediately from the general theory of stable log curves that\nthere exists a “clutching (1-)morphism” corresponding to the operations\n“(−)|H” and “(−)\bQ” [i.e., obtained by forming appropriate composites\nof the clutching morphisms discussed in [Knud], Definition 3.6]\nN def= (MgH,rH)s ×s\n( ∏\nv∈V\n(Mgv,rv )s\n)\n−→ (Mg,r)s\n102 Yuichiro Hoshi and Shinichi Mochizuki\n— where the fiber product “\n∏\nv∈V ” is taken over s — that satisfies\nthe following condition: write N log for the log stack obtained by equip-\nping the stack N with the log structure induced by the log structure\nof (Mlogg,r)s via the above clutching morphism; then there exists an slog-\nvalued point σ\nlog\nN ∈ N log(slog) of N log such that the image of σ\nlog\nN via\nthe natural strict (1-)morphism N log → (Mlogg,r)s coincides with the slog-\nvalued point of (Mlogg,r)s obtained by restricting the classifying morphism\nσlog ∈ (Mlogg,r)S(Slog) of X log to slog. If, moreover, we write Y logs for the\nstable log curve over slog corresponding to the image of σ\nlog\nN ∈ N log(slog)\nvia the composite of (1-)morphisms\nN log −→ N log def= (MloggH,rH)s ×s\n( ∏\nv∈V\n(Mgv,rv )s\n)\npr\nlog\n1−→ (MloggH,rH)s\n— where the first arrow is the (1-)morphism of log stacks obtained by\n“forgetting” the portion of the log structure of N log that arises from [the\nportion of the log structure of (Mlogg,r)s determined by] the irreducible\ncomponents of the divisor (Mg,r)s \\ (Mg,r)s which contain the image of\nN → (Mg,r)s — then one verifies immediately that, for any stable log\ncurve Y log over Slog that lifts Y logs , there exists a natural identification\nisomorphism H = ((G\nXlog\n)|H)\bQ ∼→ GY log .\nNext, let us observe that by applying the various definitions in-\nvolved, together with the fact that the (1-)morphism N log → (Mlogg,r)S\nis strict, one may verify easily that the restrictions of the natural (1-\n)morphisms of log stacks\n(MloggH,rH)s\npr\nlog\n1←− N log ←− N log −→ (Mlogg,r)s\nto a suitable étale neighborhood of the underlying morphism of stacks\nof σ\nlog\nN ∈ N log(slog) induce the following morphisms between the charts\nof (MloggH,rH)s, N log, N\nlog, and (Mlogg,r)s determined by the chart of\n“(Mlogg•,r•)s” given in Lemma 5.4, (i):⊕\ne∈Node(H) Ne\n∼→\n(⊕\ne∈Node(H) Ne\n)\n⊕ {0}\n↪→ ⊕e∈Node(G\nXlog\n) Ne\n∼← ⊕e∈Node(G\nXlog\n) Ne\n— where we use the notation Ne to denote a copy of the monoid N\nindexed by e, and the “↪→” is the natural inclusion determined by the\nCombinatorial anabelian topics I 103\nnatural inclusion Node(H) ↪→ Node(G). Thus, by applying the func-\ntor “Hom\nẐΣ\n((−)gp, ẐΣ(1))” to the homomorphism ⊕e∈Node(H) Ne ↪→⊕\ne∈Node(G\nXlog\n) Ne obtained by composing the morphisms of the above\ndisplay and considering the [relevant] isomorphism of Lemma 5.4, (ii),\nwe obtain a homomorphism I\nT\nlog\nX\n→ I\nT\nlog\nY\n, which makes the left-hand\nsquare of the diagram in the statement of assertion (v) commute.\nOn the other hand, to verify the commutativity of the right-hand\nsquare of the diagram in the statement of assertion (v), let us observe\nthat by Theorem 4.8, (iv), it suffices to verify that for any node e ∈\nNode(G\nY log\n) of G\nY log\n, the two composites\nI\nT\nlog\nX\nρuniv\nX\nlog\nt−→ Dehn(GXlog)\nDeX−→ ΛG\nXlog\n∼−→ ΛG\nY log\n;\nI\nT\nlog\nX\n−→ I\nT\nlog\nY\nρuniv\nY\nlog\nt−→ Dehn(GY log)\nDe−→ ΛG\nY log\n— where we write eX for the node of GXlog corresponding to the node\ne ∈ Node(G\nY log\n) via the natural inclusion Node(G\nY log\n) ↪→ Node(G\nXlog\n)\n— coincide. But this follows immediately by comparing the natural\naction of I\nT\nlog\nX\non the portion of G\nXlog\ncorresponding to {eX} ∪ V(eX)\nwith the natural action of I\nT\nlog\nY\non the portion of G\nY log\ncorresponding to\n{e} ∪ V(e). This completes the proof of assertion (v).\nFinally, we verify assertion (vi). First, we observe that one may\nverify easily that if K is the underlying semi-graph of H, and Q = ∅,\nthen the stable log curve Y log over Slog obtained by equipping X log with\nsuitable cusps satisfies, for a suitable choice of isomorphism H ∼→ G\nY log\n,\nthe conditions given in the statement of assertion (vi). Thus, one verifies\nimmediately that to verify assertion (vi), we may assume without loss\nof generality that U = ∅.\nWrite V\ndef\n= Vert(H) \\ Vert(H|K) ⊆ Vert(H). Denote by (gH, rH)\nthe type of H, and, for each v ∈ V , by (gv, rv) the type of v. Then\nit follows immediately from the general theory of stable log curves that\nthere exists a clutching “(1-)morphism” corresponding to the operations\n“(−)|K” and “(−)\bQ” [i.e., obtained by forming appropriate composites\nof the clutching morphisms discussed in [Knud], Definition 3.6]\nN def= (Mg,r)s ×s\n( ∏\nv∈V\n(Mgv,rv )s\n)\n−→ (MgH,rH)s\n— where the fiber product “\n∏\nv∈V ” is taken over s — that satisfies\nthe following condition: write N log for the log stack obtained by equip-\nping the stack N with the log structure induced by the log structure\n104 Yuichiro Hoshi and Shinichi Mochizuki\nof (MloggH,rH)s via the above clutching morphism; then there exists an\nslog-valued point σ\nlog\nN ∈ N log(slog) of N log such that the image of σ\nlog\nN ∈\nN log(slog) via the composite of (1-)morphisms\nN log −→ N log def= (Mlogg,r)s ×s\n( ∏\nv∈V\n(Mgv,rv )s\n)\npr\nlog\n1−→ (Mlogg,r)s\n— where the first arrow is the (1-)morphism of log stacks obtained by\n“forgetting” the portion of the log structure of N log that arises from\n[the portion of the log structure of (MloggH,rH)s determined by] the irre-\nducible components of the divisor (MgH,rH)s \\ (MgH,rH)s which con-\ntain the image of N → (MgH,rH)s — coincides with the slog-valued\npoint of (Mlogg,r)s obtained by restricting the classifying morphism σlog ∈\n(Mlogg,r)S(Slog) of X log to slog. If, moreover, we write Y logs for the stable\nlog curve over slog corresponding to the image of σ\nlog\nN ∈ N log(slog) via\nthe natural strict (1-)morphism N log → (MloggH,rH)s, then one verifies\nimmediately that, for any stable log curve Y log over Slog that lifts Y logs ,\nthere exist a sub-semi-graph of PSC-type K′ of the underlying semi-graph\nof G\nY log\n, a subset Q′ ⊆ Node((G\nY log\n)|K′), and an isomorphism of semi-\ngraphs of anabelioids H ∼→ G\nY log\nthat satisfy the following conditions:\n(a) ((G\nY log\n)|K′)\bQ′ may be naturally identified with GXlog .\n(b) The isomorphism H ∼→ G\nY log\ninduces an isomorphism K\n∼→ K′\nand a bijectionQ\n∼→ Q′, hence also an isomorphism (H|K)\bQ ∼→\n((G\nY log\n)|K′)\bQ′ .\n(c) The automorphism of G\nXlog\ndetermined by the composite\nGXlog\n∼←− (H|K)\bQ ∼−→ ((GY log)|K′)\bQ′\n∼−→ GXlog\n— where the first arrow is the isomorphism given in the state-\nment of assertion (vi); the second arrow is the isomorphism of\n(b); the third arrow is the natural isomorphism arising from the\nnatural identification of (a) — is contained in Aut|grph|(G\nXlog\n),\nand, moreover, the automorphism of ΛG\nXlog\ninduced by this au-\ntomorphism of G\nXlog\nis the identity automorphism [cf. Propo-\nsition 4.10, (iii)].\nThus, by applying a similar argument to the argument used in the proof\nof assertion (v), one verifies easily that the stable log curve Y log and the\nCombinatorial anabelian topics I 105\nisomorphism H ∼→ G\nY log\nsatisfy the conditions given in the statement of\nassertion (vi). This completes the proof of assertion (vi). Q.E.D.\nRemark 5.6.1. Here, we take the opportunity to correct a minor\nmisprint in the discussion entitled “The Étale Fundamental Group of a\nLog Scheme” in [CmbCsp], §0. In the third paragraph of this discussion,\nthe field K should be defined as a maximal algebraic extension of K◦\namong those extensions which are unramified over US◦ [i.e., but not\nnecessarily over R◦].\nTheorem 5.7 (Compatibility of scheme-theoretic and ab-\nstract combinatorial cyclotomic synchronizations). Let (g, r) be a\npair of nonnegative integers such that 2g−2+r > 0; Σ a nonempty set of\nprime numbers; R a complete discrete valuation ring whose residue field\nis separably closed of characteristic \u0004∈ Σ; Slog the log scheme obtained by\nequipping S\ndef\n= SpecR with the log structure defined by its closed point;\nX log a stable log curve of type (g, r) over Slog; G\nXlog\nthe semi-graph\nof anabelioids of pro-Σ PSC-type determined by the special fiber of the\nstable log curve X log [cf. [CmbGC], Example 2.5]; Ô the completion of\nthe local ring of (Mg,r)S [cf. the discussion entitled “Curves” in §0] at\nthe image of the closed point of S via the underlying (1-)morphism of\nstacks σ : S → (Mg,r)S of the classifying morphism of X log; T log for\nthe log scheme obtained by equipping T\ndef\n= Spec Ô with the log struc-\nture induced by the log structure of (Mlogg,r)S [cf. the discussion entitled\n“Curves” in §0]; IT log the maximal pro-Σ quotient of the log fundamental\ngroup π1(T\nlog) of T log. Then there exists an isomorphism\nsynXlog : Λ\nΣ def= Hom(Ngp, ẐΣ(1))\n∼−→ ΛG\nXlog\n[cf. Definition 3.8, (i)] such that the composite\nIT log\n∼→ ⊕e∈Node(G\nXlog\n) Λ\nΣ[e]⊕\nsyn\nXlog∼→ ⊕e∈Node(G\nXlog\n) ΛGXlog\nDG\nXlog∼← Dehn(G\nXlog\n)\n[cf. Definitions 4.4; 4.7] — where we use the notation ΛΣ[e] to denote\na copy of ΛΣ indexed by e ∈ Node(G\nXlog\n), and the first arrow is the\n[relevant] isomorphism of Lemma 5.4, (ii) — coincides with the outer\nrepresentation ρuniv\nX\nlog\nt\n: IT log → Dehn(GXlog) [cf. Definition 5.5] associated\n106 Yuichiro Hoshi and Shinichi Mochizuki\nto the stable log curve over T log corresponding to the tautological strict\n(1-)morphism T log → (Mlogg,r)S.\nProof. In light of Theorem 4.8, (ii), (iv); Proposition 5.6, (ii), by\napplying Proposition 5.6, (iii), to the various generizations of the form\n“(G\nXlog\n)\u0003Node(G\nXlog\n)\\{e}”, it follows immediately that for each node e ∈\nNode(G\nXlog\n), there exists a(n) [necessarily unique] isomorphism\nsynXlog [e] : Λ\nΣ[e]\n∼−→ ΛG\nXlog\n— where ΛΣ[e] is a copy of ΛΣ indexed by e ∈ Node(G\nXlog\n) — such that\nthe composite\nIT log\n∼→ ⊕e∈Node(G\nXlog\n) Λ\nΣ[e]⊕\ne synXlog [e]∼→ ⊕e∈Node(G\nXlog\n) ΛGXlog\nDG\nXlog∼← Dehn(G\nXlog\n)\n— where the first “\n∼→” is the [relevant] isomorphism of Lemma 5.4, (ii)\n— coincides with ρuniv\nX\nlog\nt\n.\nThus, to complete the proof of Theorem 5.7, it suffices to verify that\nthis isomorphism synXlog [e] is independent of the choice of e. Now if\nNode(G\nXlog\n)\u0005 ≤ 1, then this independence is immediate. Thus, suppose\nthat Node(G\nXlog\n)\u0005 > 1 and fix two distinct nodes e1, e2 ∈ Node(GXlog)\nof G\nXlog\n. The rest of the proof of Theorem 5.7 is devoted to verifying\nthat\n(‡): the two isomorphisms\nΛΣ[e1]\nsyn\nXlog\n[e1]\n∼−→ ΛG\nXlog\n, ΛΣ[e2]\nsyn\nXlog\n[e2]\n∼−→ ΛG\nXlog\ncoincide.\nNext, let us observe that one may verify easily that there exist\n• a semi-graph of anabelioids of pro-Σ PSC-type H∗,\n• a sub-semi-graph of PSC-type K∗ of the underlying semi-graph\nof H∗,\n• an omittable subset Q∗ ⊆ Cusp((H∗)|K∗), and\n• an isomorphism\n((H∗)|K∗)•Q∗ ∼−→ GXlog\nCombinatorial anabelian topics I 107\nsuch that the subset U∗ ⊆ Node(H∗) corresponding, relative to the iso-\nmorphism ((H∗)|K∗)•Q∗ ∼→ GXlog , to the subset {e1, e2} ⊆ Node(GXlog)\nis not of separating type. Thus, it follows immediately from Propo-\nsition 5.6, (vi) — i.e., by replacing X log (respectively, e1, e2) by the\nstable log curve “Y log” obtained by applying Proposition 5.6, (vi), to\nthe isomorphism ((H∗)|K∗)•Q∗ ∼→ GXlog (respectively, by the two nodes\n∈ Node(G\nY log\n) corresponding to the two nodes ∈ U∗) — that to verify\nthe above (‡), we may assume without loss of generality that the subset\n{e1, e2} ⊆ Node(GXlog) is not of separating type.\nThus, it follows immediately from Proposition 5.6, (iii) — i.e., by\nreplacing X log (respectively, e1, e2) by the stable log curve “Y\nlog” ob-\ntained by applying Proposition 5.6, (iii), to (G\nXlog\n)\u0003Node(G\nXlog\n)\\{e1,e2}\n(respectively, by the two nodes ∈ Node(G\nY log\n) corresponding to e1, e2)\n— that to verify the above (‡), we may assume without loss of general-\nity that Node(G\nXlog\n) = {e1, e2}, and that Node(GXlog) = {e1, e2} is not\nof separating type. One verifies easily that these hypotheses imply that\nVert(G\nXlog\n)\u0005 = 1.\nNext, let us observe that one may verify easily that there exist [cf.\nFig. 6 below]\n• a semi-graph of anabelioids of pro-Σ PSC-type H†,\n• two distinct cusps c†1, c†2 ∈ Cusp(H†) of H†,\n• three distinct nodes f†1 , f†2 , f†3 ∈ Node(H†) of H†, and\n• an isomorphism\n(H†\u0003{f†1 ,f†2 ,f†3})•{c†1,c†2}\n∼−→ GXlog\nsuch that\n• Vert(H†) = {v†1, v†2, v†3, v†4};\n• for i ∈ {1, 2}, if we write e†i ∈ Node(H†) for the node corre-\nsponding, relative to the isomorphism (H†\u0003{f†1 ,f†2 ,f†3})•{c†1,c†2}\n∼→\nG\nXlog\n, to ei ∈ Node(GXlog), then it holds that V(e\n†\ni ) = {v\n†\ni };\n• V(f†1 ) = {v†1, v†3}, V(f†2 ) = {v†2, v†3}, V(f†3 ) = {v†3, v†4};\n• V(c†1) = V(c†2) = {v†4};\n• for i ∈ {1, 2, 3}, v†i is of type (0, 3) [cf. Definition 2.3, (iii)].\n108 Yuichiro Hoshi and Shinichi Mochizuki\nFigure 6: The underlying semi-graph of H†\nv†1 v\n†\n2\nv†3\nv†4\ne†1 e\n†\n2\nf †1 f\n†\n2\nf †3\n•\n•\n• •\n◦◦ · · · · · ·\nOne verifies easily that these hypotheses imply that (N (v†1) ∩\nN (v†3))\u0005 = (N (v†2) ∩ N (v†3))\u0005 = 1. Thus, it follows immediately from\nProposition 5.6, (iv), (vi) — i.e., by replacing X log (respectively, e1, e2)\nby the stable log curve “Y log” obtained by applying Proposition 5.6, (iv),\n(vi), to the isomorphism (H†\u0003{f†1 ,f†2 ,f†3})•{c†1,c†2}\n∼→ G\nXlog\n(respectively, by\nthe two nodes ∈ Node(G\nY log\n) corresponding to the two nodes e\n†\n1, e\n†\n2) —\nthat to verify the above (‡), we may assume without loss of generality\nthat there exist vertices v1, v2, v3 of GXlog such that\n• for i ∈ {1, 2}, V(ei) = {vi};\n• for i ∈ {1, 2, 3}, vi is of type (0, 3);\n• (N (v1) ∩N (v3))\u0005 = (N (v2) ∩N (v3))\u0005 = 1.\nWrite H for the sub-semi-graph of PSC-type of the underlying semi-\ngraph of G\nXlog\nwhose set of vertices = {v1, v2, v3}. Then one verifies eas-\nily that these hypotheses imply that Node((G\nXlog\n)|H) = {e1, e2, f1, f2},\nwhere we write {f1} = N (v1) ∩N (v3), {f2} = N (v2) ∩N (v3).\nThus, it follows immediately from Proposition 5.6, (v) — i.e., by\nreplacing X log (respectively, e1, e2) by the stable log curve “Y\nlog” ob-\ntained by applying Proposition 5.6, (v), to (G\nXlog\n)|H (respectively, by\nthe two nodes ∈ Node(G\nY log\n) corresponding to e1, e2) — that to verify\nthe above (‡), we may assume without loss of generality that there exist\nthree distinct vertices v1, v2, v3 of GXlog such that\n• for i ∈ {1, 2}, V(ei) = {vi};\nCombinatorial anabelian topics I 109\n• for i ∈ {1, 2, 3}, vi is of type (0, 3);\n• Node(G\nXlog\n) = {e1, e2, f1, f2}, where we write {f1} = (N (v1)∩\nN (v3)), {f2} = (N (v2) ∩N (v3)).\nOne verifies easily that these hypotheses imply that there exists a cusp\nc of G\nXlog\nsuch that Cusp(G\nXlog\n) = {c} = C(v3).\nThen it follows immediately from the explicit structure of G\nXlog\nthat\nthere exists an automorphism τ of X\nlog\nt [cf. Definition 5.5] such that the\nautomorphism of Node(G\nXlog\n) = {e1, e2, f1, f2} (respectively, IT log\n∼→\nHom\nẐΣ\n((Ne1⊕Ne2⊕Nf1⊕Nf2)gp, ẐΣ(1)) [cf. Lemma 5.4, (ii)]) induced by\nτ is given by mapping e1 \u0006→ e2, e2 \u0006→ e1, f1 \u0006→ f2, f2 \u0006→ f1, (respectively,\nby the corresponding permutation of factors of Ne1 ⊕Ne2 ⊕Nf1 ⊕Nf2),\nand, moreover, τ preserves the cusp corresponding to c. Now it follows\nimmediately from Corollary 3.9, (v), together with the fact that the\nautomorphism of the anabelioid (G\nXlog\n)c corresponding to the cusp c\ninduced by τ is the identity automorphism [cf. the argument used in the\nfinal portion of the proof of Corollary 3.9, (vi)], that the automorphism\nof ΛG\nXlog\ninduced by τ is the identity automorphism. Thus, by applying\nthe evident functoriality of the homomorphism ρuniv\nX\nlog\nt\nwith respect to the\nautomorphism of G\nXlog\ninduced by τ , one concludes immediately from\nthe above description of τ , together with Theorem 4.8, (v), that the\nassertion (‡) holds. This completes the proof of Theorem 5.7. Q.E.D.\nDefinition 5.8. Let α ∈ Dehn(G) be a profinite Dehn multi-twist\nof G and u ∈ ΛG a topological generator of ΛG .\n(i) Let e ∈ Node(G) be a node of G. Then since ΛG is a free ẐΣ-\nmodule of rank 1 [cf. Definition 3.8, (i)], there exists a unique\nelement ae ∈ ẐΣ of ẐΣ such that De(α) = aeu. We shall refer\nto ae ∈ ẐΣ as the Dehn coordinate of α indexed by e with\nrespect to u.\n(ii) We shall say that a profinite Dehn multi-twist α ∈ Dehn(G) is\nnondegenerate if, for each node e ∈ Node(G) of G, the Dehn\ncoordinate of α indexed by e with respect to u [cf. (i)] topo-\nlogically generates an open subgroup of ẐΣ. Note that it is\nimmediate that if α is nondegenerate, then the Dehn coordi-\nnate (∈ ẐΣ ∼→ ∏l∈Σ Zl ⊆ ∏l∈Σ Ql) of α indexed by e with\nrespect to u is contained in\n∏\nl∈Σ Q\n∗\nl .\n110 Yuichiro Hoshi and Shinichi Mochizuki\n(iii) We shall say that a profinite Dehn multi-twist α ∈ Dehn(G) is\npositive definite if α is nondegenerate [cf. (ii)], and, moreover,\nthe following condition is satisfied: For each node e ∈ Node(G)\nof G, denote by ae ∈ ẐΣ the Dehn coordinate of α indexed by\ne with respect to u [cf. (i)]. [Thus, ae ∈\n∏\nl∈Σ Q\n∗\nl — cf. (ii).]\nThen for any e, e′ ∈ Node(G), ae\/ae′ is contained in the image\nof the diagonal map Q>0\ndef\n= { a ∈ Q | a > 0 } ↪→ ∏l∈Σ Q∗l .\nRemark 5.8.1. One may verify easily that the notions defined in\nDefinition 5.8, (ii), (iii), are independent of the choice of the topological\ngenerator u of ΛG .\nCorollary 5.9 (Properties of outer representations of PSC–\ntype and profinite Dehn multi-twists). Let Σ be a nonempty set\nof prime numbers and ρ : I → Aut(G) an outer representation of pro-Σ\nPSC-type [cf. [NodNon], Definition 2.1, (i)]. Suppose that I is isomor-\nphic to ẐΣ. Then the following hold:\n(i) (Outer representations of SVA-type and profinite Dehn\nmulti-twists) The following three conditions are equivalent:\n(i-1) ρ is of SVA-type [cf. [NodNon], Definition 2.4, (ii)].\n(i-2) The image of any topological generator of I is a profinite\nDehn multi-twist [cf. Definition 4.4].\n(i-3) There exists a topological generator of I whose image via\nρ is a profinite Dehn multi-twist.\n(ii) (Outer representations of SNN-type and nondegener-\nate profinite Dehn multi-twists) The following three con-\nditions are equivalent [cf. the related discussion of [NodNon],\nRemark 2.14.1]:\n(ii-1) ρ is of SNN-type [cf. [NodNon], Definition 2.4, (iii)].\n(ii-2) The image of any topological generator of I is a non-\ndegenerate [cf. Definition 5.8, (ii)] profinite Dehn\nmulti-twist.\n(ii-3) There exists a topological generator of I whose image via\nρ is a nondegenerate profinite Dehn multi-twist.\nCombinatorial anabelian topics I 111\n(iii) (Outer representations of IPSC-type and positive defi-\nnite profinite Dehn multi-twists) The following three con-\nditions are equivalent [cf. Remark 5.10.1 below; the related\ndiscussion of [NodNon], Remark 2.14.1]:\n(iii-1) ρ is of IPSC-type [cf. [NodNon], Definition 2.4, (i)].\n(iii-2) The image of any topological generator of I is a posi-\ntive definite [cf. Definition 5.8, (iii)] profinite Dehn\nmulti-twist.\n(iii-3) There exists a topological generator of I whose image via\nρ is a positive definite profinite Dehn multi-twist.\n(iv) (Synchronization associated to outer representations\nof IPSC-type) Suppose that ρ is of IPSC-type. Write\n(ẐΣ)+ ⊆ (ẐΣ)∗\nfor the intersection of the images of the diagonal map Q>0\ndef\n=\n{ a ∈ Q | a > 0 } ↪→ ∏l∈Σ Ql and the composite of natural\nmorphisms (ẐΣ)∗ ↪→ ẐΣ ∼→ ∏l∈Σ Zl ⊆ ∏l∈Σ Ql. [Thus, when\nΣ = Primes, it holds that (ẐΣ)+ = {1}.] Then there exists a\nnatural (ẐΣ)+-orbit of isomorphisms of ẐΣ-modules\nsyn ρ : I\n∼−→ ΛG\nthat is functorial, in ρ, with respect to isomorphisms of outer\nrepresentations of PSC-type [cf. [NodNon], Definition 2.1,\n(ii)].\n(v) (Compatibility of synchronizations with finite étale\ncoverings) In the situation of (iv), let Π ⊆ ΠI def= ΠG\nout\n\u0002 I [cf.\nthe discussion entitled “Topological groups” in §0] be an open\nsubgroup of ΠI such that if we write H → G for the connected\nfinite étale covering of G corresponding to Π ∩ ΠG [so ΠH =\nΠ ∩ ΠG], then the outer representation ρΠ : J def= Π\/ΠH →\nOut(ΠH) is of IPSC-type. Then the diagram of ẐΣ-modules\nJ\nsyn ρΠ−−−−→ ΛH⏐⏐\b ⏐⏐\b\u0005\nI\nsyn ρ−−−−→ ΛG\n112 Yuichiro Hoshi and Shinichi Mochizuki\n— where the left-hand vertical arrow is the natural inclusion;\nthe right-hand vertical arrow is the isomorphism of Corol-\nlary 3.9, (iii) — commutes up to multiplication by an el-\nement ∈ Q>0.\nProof. Assertion (i) follows immediately from condition (2′) of\n[NodNon], Definition 2.4. Next, we verify assertions (ii) and (iii). The\nimplication\n(ii-1) =⇒ (ii-2) , (respectively, (iii-1) =⇒ (iii-2))\nfollows immediately from the final portion of Lemma 5.2, (ii), concern-\ning ρ of SNN-type (respectively, Lemma 5.4, (ii); Theorem 5.7). The\nimplications\n(ii-2) =⇒ (ii-3) , (iii-2) =⇒ (iii-3)\nare immediate.\nNext, we verify the implication\n(ii-3) =⇒ (ii-1) .\nIt follows from the implication (i-3) ⇒ (i-1) that ρ is of SVA-type. Thus,\nto show the implication in question, it suffices to verify that ρ satisfies\ncondition (3) of [NodNon], Definition 2.4. Let ẽ ∈ Node(G̃) be an ele-\nment of Node(G̃); ΠI def= ΠG\nout\n\u0002 I [cf. the discussion entitled “Topological\ngroups” in §0]; ṽ, w̃ ∈ Vert(G̃) the two distinct elements of Vert(G̃) such\nthat V(ẽ) = {ṽ, w̃} [cf. [NodNon], Remark 1.2.1, (iii)]; Iẽ, Iṽ, Iw̃ ⊆ ΠI\nthe inertia subgroups of ΠI associated to ẽ, ṽ, w̃, respectively. Then\nsince the homomorphisms of the final two displays of Lemma 5.2, (ii),\ncoincide, and ΛG\nXlog\nand Iṽ are free Ẑ\nΣ-modules of rank 1 [cf. Defi-\nnition 3.8, (i); [NodNon], Lemma 2.5, (i)], it follows immediately from\nthe definition of nondegeneracy that the composite of the second display\nof Lemma 5.2, (ii), is an open injection. Thus, it follows immediately\nthat the natural homomorphism Iṽ × Iw̃ → Iẽ has open image, and that\nIṽ ∩ Iw̃ = {1}, i.e., that Iṽ × Iw̃ → Iẽ is injective. That is to say, ρ\nsatisfies condition (3) of [NodNon], Definition 2.4. This completes the\nproof of the implication in question.\nNext, we verify the implication\n(iii-3) =⇒ (iii-1) .\nLet u ∈ ΛG be a topological generator of ΛG . Then it follows immediately\nfrom Lemma 5.4, (i), (ii), and Theorem 5.7 — by considering the stable\nCombinatorial anabelian topics I 113\nlog curve over Slog corresponding to a suitable homomorphism of R-\nalgebras Ô \f R[[t1, · · · , t3g−g+r]] → R [cf. Lemma 5.4, (i)] — that to\ncomplete the proof of the implication in question, it suffices to verify\nthat there exists a topological generator α ∈ I of I which satisfies the\nfollowing condition (∗):\n(∗): The Dehn coordinates of ρ(α) with respect to u\n[cf. Definition 5.8, (i)] ∈ N\n=0 def= N \\ {0}.\nTo this end, let α ∈ I be a topological generator of I such that ρ(α) is\na positive definite profinite Dehn multi-twist of G [cf. condition (iii-3)].\nFor each node e ∈ Node(G) of G, denote by ae ∈ ẐΣ the Dehn coordinate\nof ρ(α) indexed by e with respect to u. Now since ρ(α) is nondegenerate,\nit follows immediately from the definition of nondegeneracy that for each\nnode e ∈ Node(G) of G, it holds that ae ∈ N\n=0 · (ẐΣ)∗. Thus, it follows\nimmediately that for a given node f ∈ Node(G) of G, by replacing α\nby a suitable topological generator of I, we may assume without loss of\ngenerality that af ∈ N \n=0. In particular, it follows immediately from the\ndefinition of positive definiteness that there exists an element a ∈ N \n=0\nsuch that for each node e ∈ Node(G) of G, it holds that a · ae ∈ N\n=0.\nMoreover, again by replacing α by a suitable topological generator of\nI, we may assume that every prime number dividing a belongs to Σ.\nBut then it follows from the fact that ae ∈ ẐΣ ∩ ( 1a · N \n=0) that ae is\na positive rational number that is integral at every element of Primes,\ni.e., that ae ∈ N \n=0, as desired. In particular, the topological generator\nα ∈ I of I satisfies the above condition (∗). This completes the proof of\nthe implication in question, hence also of assertions (ii) and (iii).\nNext, we verify assertion (iv). It follows immediately from the fi-\nnal portion of Lemma 5.2, (ii), concerning ρ of SNN-type that for each\ne ∈ Node(G), the homomorphism syn ρ : I → ΛG obtained by dividing\nthe composite I\nρ→ Dehn(G) De→ ΛG by lngΣG (e, ρ) is an isomorphism.\nMoreover, by “translating into group theory” the scheme-theoretic con-\ntent of Lemma 5.4, (ii), by means of the correspondence between group-\ntheoretic and scheme-theoretic notions given in Proposition 5.6, (i); The-\norem 5.7, one concludes that syn ρ is independent — up to multiplication\nby an element of (ẐΣ)+ — of the choice of the node e ∈ Node(G). Now\nthe functoriality of syn ρ follows immediately from the functoriality of\nthe homomorphism De [cf. Theorem 4.8, (iv)], together with the group-\ntheoreticity of lngΣG (e, ρ). This completes the proof of assertion (iv).\nFinally, assertion (v) follows immediately, in light of the group-\ntheoretic construction of “syn ρ” given in the proof of assertion (iv),\nfrom the various definitions involved. Q.E.D.\n114 Yuichiro Hoshi and Shinichi Mochizuki\nRemark 5.9.1.\n(i) Corollary 5.9, (iv), may be regarded as a sort of abstract com-\nbinatorial analogue of the cyclotomic synchronization given in\n[GalSct], Theorem 4.3 [cf. also [AbsHyp], Lemma 2.5, (ii)].\n(ii) It follows from Theorem 5.7 that one may think of the isomor-\nphisms of Corollary 5.9, (iv), as a sort of abstract combinato-\nrial construction of the various identification isomorphisms be-\ntween the various copies of “ẐΣ(1)” that appear in Lemma 5.4,\n(ii). Such identification isomorphisms are typically “taken for\ngranted” in conventional discussions of scheme theory.\nRemark 5.9.2.\n(i) Consider the exact sequence of free ẐΣ-modules\n0 −→MvertG −→MG\ndef\n= ΠabG −→M combG\ndef\n= MG\/MvertG −→ 0\n— where we write MvertG ⊆ MG for the ẐΣ-submodule of MG\ntopologically generated by the images of the verticial subgroups\nof ΠG [cf. [CmbGC], Remark 1.1.4]. Then one verifies easily\nthat any profinite Dehn multi-twist α ∈ Dehn(G) preserves and\ninduces the identity automorphism on MvertG , M\ncomb\nG . In par-\nticular, the homomorphismMG →MG obtained by considering\nthe difference of the automorphism of MG induced by α and\nthe identity automorphism on MG naturally determines [and\nis determined by!] a homomorphism\nαcomb,vert : M combG −→MvertG .\nWrite M\nedge\nG ⊆MvertG for the ẐΣ-submodule topologically gen-\nerated by the image of the edge-like subgroups of ΠG . Then\nthe following two facts are well-known:\n• If Cusp(G) = ∅, then Poincaré duality MG ∼→\nHom\nẐΣ\n(MG ,ΛG) determines an isomorphism M\nedge\nG\n∼→\nHom\nẐΣ\n(M combG ,ΛG) [cf. [CmbGC], Proposition 1.3].\n• The natural homomorphism\nDehn(G) −→ Hom\nẐΣ\n(M combG ,M\nvert\nG )\ngiven by mapping α \u0006→ αcomb,vert factors through the sub-\nmodule Hom\nẐΣ\n(M combG ,M\nedge\nG ) ⊆ HomẐΣ(M combG ,MvertG ).\nCombinatorial anabelian topics I 115\n[Indeed, this may be verified, for instance, by applying\na similar argument to the argument used in the proof of\n[CmbGC], Proposition 1.3, involving weights.]\nThus, if Cusp(G) = ∅, then we obtain a homomorphism\nΩG : Dehn(G) −→M edgeG ⊗ẐΣ M\nedge\nG ⊗ẐΣ HomẐΣ(ΛG , ẐΣ)\nthat is manifestly functorial, in G, with respect to isomor-\nphisms of semi-graphs of anabelioids of pro-Σ PSC-type. The\nmatrices that appear in the image of this homomorphism ΩG\nare often referred to as period matrices.\n(ii) Now let us recall that [CmbGC], Proposition 2.6, plays a key\nrole in the proof of the combinatorial version of the\nGrothendieck conjecture given in [CmbGC], Corollary 2.7, (iii).\nMoreover, the proof of [CmbGC], Proposition 2.6, is essentially\na formal consequence of the nondegeneracy of the period matrix\nassociated to a positive definite profinite Dehn multi-twist —\ni.e., of the injectivity of the homomorphism\nαcomb,vert : M combG −→MvertG\nof (i) in the case where α ∈ Dehn(G) is positive definite [cf.\nCorollary 5.9, (iii)].\n(iii) In general, the period matrix associated to a profinite Dehn\nmulti-twist may fail to be nondegenerate even if the profinite\nDehn multi-twist is nondegenerate. Indeed, suppose that Σ\u0005 =\n1, that G is the double [cf. [CmbGC], Proposition 2.2, (i)] of a\nsemi-graph of anabelioids of pro-Σ PSC-type H such that\n(Vert(H)\u0005,Node(H)\u0005,Cusp(H)\u0005) = (1, 0, 2) .\nSuppose, moreover, that H admits an automorphism which\npermutes the two cusps of H and extends to an automorphism\nφ of G. [One verifies easily that such data exist.] Then one may\nverify easily that Node(G)\u0005 = 2, that Cusp(G)\u0005 = 0, and that\nthe free ẐΣ-module M combG , hence also M\nedge\nG ⊗ẐΣ M\nedge\nG ⊗ẐΣ\nHom\nẐΣ\n(ΛG , ẐΣ) [cf. (i)], is of rank 1. Now let us recall that the\nperiod matrix associated to a positive definite profinite Dehn\nmulti-twist is necessarily nondegenerate [cf. Corollary 5.9, (iii);\nthe proof of [CmbGC], Proposition 2.6]. Thus, since Σ\u0005 = 1, it\nfollows immediately from the functoriality of ΩG [cf. (i)] and\n116 Yuichiro Hoshi and Shinichi Mochizuki\nDG [cf. Theorem 4.8, (iv)] with respect to φ that the kernel of\nthe composite of natural homomorphisms\n⊕\nNode(G)\nΛG\nDG∼←− Dehn(G) ΩG−→M edgeG ⊗ẐΣM\nedge\nG ⊗ẐΣHomẐΣ(ΛG , ẐΣ)\nis a free ẐΣ-submodule of\n⊕\nNode(G) ΛG of rank 1 that is stabi-\nlized by φ. On the other hand, since profinite Dehn multi-twists\nof the form (u, u) ∈ ⊕Node(G) ΛG , where u ∈ ΛG , are [mani-\nfestly!] positive definite, we thus conclude that the kernel in\nquestion is equal to\n{ (u,−u) ∈\n⊕\nNode(G)\nΛG |u ∈ ΛG } .\nIn particular, any nonzero element of this kernel yields an ex-\nample of a nondegenerate profinite Dehn multi-twist whose as-\nsociated period matrix fails to be nondegenerate.\nCorollary 5.10 (Combinatorial\/group-theoretic nature of\nscheme-theoreticity Let (g, r) be a pair of nonnegative integers such\nthat 2g − 2 + r > 0; Σ a nonempty set of prime numbers; R a complete\ndiscrete valuation ring whose residue field k is separably closed of charac-\nteristic \u0004∈ Σ; Slog the log scheme obtained by equipping S def= SpecR with\nthe log structure determined by the maximal ideal of R; x ∈ (Mg,r)S(k)\na k-valued point of the moduli stack of curves (Mg,r)S of type (g, r)\nover S [cf. the discussion entitled “Curves” in §0). ; Ô the completion of\nthe local ring of (Mg,r)S at the image of x; T log the log scheme obtained\nby equipping T\ndef\n= Spec Ô with the log structure induced by the log struc-\nture of (Mlogg,r)S [cf. the discussion entitled “Curves” in §0]; tlog the log\nscheme obtained by equipping the closed point of T with the log struc-\nture induced by the log structure of T log; X\nlog\nt the stable log curve over\ntlog corresponding to the natural strict (1-)morphism tlog → (Mlogg,r)S;\nIT log the maximal pro-Σ quotient of the log fundamental group π1(T\nlog)\nof T log; ISlog the maximal pro-Σ quotient of the log fundamental group\nπ1(S\nlog) of Slog; G\nXlog\nthe semi-graph of anabelioids of pro-Σ PSC-type\ndetermined by the stable log curve X\nlog\nt [cf. [CmbGC], Example 2.5];\nρuniv\nX\nlog\nt\n: IT log → Aut(GXlog) the natural outer representation associated to\nX\nlog\nt [cf. Definition 5.5]; I a profinite group; ρ : I → Aut(GXlog) an\nCombinatorial anabelian topics I 117\nouter representation of pro-Σ PSC-type [cf. [NodNon], Definition 2.1,\n(i)]. Then the following conditions are equivalent:\n(i) ρ is of IPSC-type.\n(ii) There exist a morphism of log schemes φlog : Slog → T log over\nS and an isomorphism of outer representations of pro-Σ\nPSC-type ρ\n∼→ ρuniv\nX\nlog\nt\n◦ Iφlog [cf. [NodNon], Definition 2.1, (i)]\n— where we write Iφlog : ISlog → IT log for the homomorphism\ninduced by φlog — i.e., there exist an automorphism β of\nG\nXlog\nand an isomorphism α : I\n∼→ I logS such that the diagram\nI\nρ−−−−→ Aut(G\nXlog\n)\nα\n⏐⏐\b\u0005 ⏐⏐\b\u0005\nISlog\nρ\nX\nlog\nt\n◦I\nφlog\n−−−−−−−→ Aut(G\nXlog\n)\n— where the right-hand vertical arrow is the automorphism of\nAut(G\nXlog\n) induced by β — commutes.\n(iii) There exist a morphism of log schemes φlog : Slog → T log over\nS and an isomorphism α : I\n∼→ I logS such that ρ = ρunivXlogt ◦Iφlog◦\nα — where we write Iφlog : ISlog → IT log for the homomorphism\ninduced by φlog — i.e., the automorphism “β” of (ii) may be\ntaken to be the identity.\nProof. The equivalence (i) ⇔ (ii) follows from the definition of the\nterm “IPSC-type” [cf. [NodNon], Definition 2.4, (i)]. The implication\n(iii) ⇒ (ii) is immediate. The implication (ii) ⇒ (iii) follows immedi-\nately, in light of the functoriality asserted in Theorem 4.8, (iv), from\nLemma 5.4, (i), (ii), and Theorem 5.7. Q.E.D.\nRemark 5.10.1.\n(i) The equivalence of Corollary 5.10 essentially amounts to the\nequivalence\n“IPSC-type ⇐⇒ positive definite”\nwhich was discussed in [HM], Remark 2.14.1, without proof.\n118 Yuichiro Hoshi and Shinichi Mochizuki\n(ii) One way to understand the equivalence of Corollary 5.10 is as\nthe statement that the property that an outer representation\nof PSC-type be of scheme-theoretic origin may be formulated\npurely in terms of combinatorics\/group theory.\nIn the final portion of the present §5, we apply the theory devel-\noped so far [i.e., in particular, the equivalences of Corollary 5.9, (ii),\n(iii)] to derive results [cf. Theorem 5.14] concerning normalizers and\ncommensurators of groups of profinite Dehn multi-twists.\nDefinition 5.11. Let M ⊆ H ⊆ Out(ΠG) be closed subgroups of\nOut(ΠG). Suppose further that M is an abelian pro-Σ group [such as\nDehn(G) — cf. Theorem 4.8, (iv)].\n(i) We shall write\nN scalH (M) ⊆ NH(M) ⊆ H\nfor the [closed] subgroup ofH consisting of α ∈ H satisfying the\nfollowing condition: α ∈ NH(M), and, moreover, the action of\nα onM by conjugation coincides with the automorphism ofM\ngiven by multiplication by an element of (ẐΣ)∗. We shall refer\nto N scalH (M) as the scalar-normalizer of M in H.\n(ii) We shall write\nCscalH (M) ⊆ CH(M) ⊆ H\nfor the subgroup of H consisting of α ∈ H satisfying the fol-\nlowing condition: there exists an open ẐΣ-submoduleM ′α ⊆M\nof M [possibly depending on α] such that the action of α on\nH by conjugation determines an automorphism of M ′α given\nby multiplication by an element of (ẐΣ)∗. We shall refer to\nCscalH (M) as the scalar-commensurator of M in H.\nLemma 5.12 (Scalar-normalizers and scalar-commensura-\ntors). Let M ⊆ H ⊆ Out(ΠG) be closed subgroups of Out(ΠG). Suppose\nfurther that M is an abelian pro-Σ group. Then:\nCombinatorial anabelian topics I 119\n(i) It holds that\nM ⊆ ZH(M) ⊆ N scalH (M) ⊆ CscalH (M) .\n(ii) If M ′ ⊆M is a ẐΣ-submodule of M , then\nN scalH (M) ⊆ N scalH (M ′) ; CscalH (M) ⊆ CscalH (M ′) .\nIf, moreover, M ′ ⊆M is open in M , then\nCscalH (M) = C\nscal\nH (M\n′) .\nProof. These assertions follow immediately from the various defi-\nnitions involved. Q.E.D.\nDefinition 5.13. Let H ⊆ Out(ΠG) be a closed subgroup of\nOut(ΠG). Then we shall say that H is IPSC-ample (respectively, NN-\nample) if H contains a positive definite (respectively, nondegenerate) [cf.\nDefinition 5.8] profinite Dehn multi-twist ∈ Dehn(G).\nRemark 5.13.1. It follows immediately from Theorem 4.8, (iv),\nthat any open subgroup of Dehn(G) is IPSC-ample, hence also NN-ample\n[cf. Definition 5.13].\nTheorem 5.14 (Normalizers and commensurators of groups\nof profinite Dehn multi-twists). Let Σ be a nonempty set of prime\nnumbers, G a semi-graph of anabelioids of pro-Σ PSC-type, OutC(ΠG)\nthe group of group-theoretically cuspidal [cf. [CmbGC], Definition\n1.4, (iv)] outomorphisms of ΠG, and M ⊆ OutC(ΠG) a closed subgroup\nof OutC(ΠG) which is abelian pro-Σ. Then the following hold:\n(i) Suppose that one of the following two conditions is satisfied:\n(1) M is IPSC-ample [cf. Definition 5.13].\n(2) M is NN-ample [cf. Definition 5.13], and Cusp(G) \u0004= ∅.\n120 Yuichiro Hoshi and Shinichi Mochizuki\nThen it holds that\nN scal\nOutC(ΠG)(M) ⊆ C\nscal\nOutC(ΠG)(M) ⊆ Aut(G)\n[cf. Definition 5.11]. If, moreover, M ⊆ Dehn(G) [cf. Defini-\ntion 4.4], then\nAut|Node(G)|(G) ⊆ N scal\nOutC(ΠG)(M) ⊆ C\nscal\nOutC(ΠG)(M) ⊆ Aut(G)\n[cf. Definition 2.6, (i)]. In particular,\nN scal\nOutC(ΠG)(M) , C\nscal\nOutC(ΠG)(M) ⊆ Aut(G)\nare open subgroups of Aut(G).\n(ii) If M is an open subgroup of Dehn(G), then it holds that\nAut(G) = COutC(ΠG)(M) .\nIf, moreover, Node(G) \u0004= ∅, then\nAut|Node(G)|(G) ∩ Ker(χG) = ZOutC(ΠG)(M)\n[cf. Definition 3.8, (ii)].\n(iii) It holds that\nAut(G) = NOutC(ΠG)(Dehn(G)) = COutC(ΠG)(Dehn(G)) .\nProof. First, we verify the inclusion Cscal\nOutC(ΠG)\n(M) ⊆ Aut(G) as-\nserted in assertion (i). Suppose that condition (1) (respectively, (2)) is\nsatisfied. Let α ∈ Cscal\nOutC(ΠG)\n(M). Then since α ∈ Cscal\nOutC(ΠG)\n(M), and\nM is IPSC-ample (respectively, NN-ample), it follows immediately that\nthere exists an element β ∈M of M such that both β and αβα−1 = βλ,\nwhere λ ∈ (ẐΣ)∗, are positive definite (respectively, nondegenerate) profi-\nnite Dehn multi-twists. Thus, the graphicity of α follows immediately\nfrom [NodNon], Remark 4.2.1, together with Corollary 5.9, (iii) (respec-\ntively, from [NodNon], Theorem A, together with Corollary 5.9, (ii)).\nThis completes the proof of the inclusion Cscal\nOutC(ΠG)\n(M) ⊆ Aut(G), hence\nalso, by Lemma 5.12, (i), of the two inclusions in the first display of as-\nsertion (i).\nIf, moreover, M ⊆ Dehn(G), then the inclusion Aut|Node(G)|(G) ⊆\nN scal\nOutC(ΠG)\n(M) follows immediately from Theorem 4.8, (v). Thus, since\nCombinatorial anabelian topics I 121\nAut|Node(G)|(G) is an open subgroup of Aut(G) [cf. Proposition 2.7, (iii)],\nit follows immediately that N scal\nOutC(ΠG)\n(M), hence also Cscal\nOutC(ΠG)\n(M), is\nan open subgroup of Aut(G). This completes the proof of assertion (i).\nNext, we verify the equality Aut(G) = COutC(ΠG)(M) in the first\ndisplay of assertion (ii). It follows immediately from Theorem 4.8, (i),\nthat Aut(G) ⊆ NOutC(ΠG)(Dehn(G)) ⊆ COutC(ΠG)(M). Thus, to verify\nthe equality Aut(G) = COutC(ΠG)(M), it suffices to verify the inclusion\nCOutC(ΠG)(M) ⊆ Aut(G). To this end, let α ∈ COutC(ΠG)(M). Then it\nfollows from Lemma 5.12, (ii), that\nCscal\nOutC(ΠG)(M) = C\nscal\nOutC(ΠG)(α ·M · α\n−1) = α · Cscal\nOutC(ΠG)(M) · α\n−1 ,\ni.e., α ∈ NOutC(ΠG)(CscalOutC(ΠG)(M)). Thus, since C\nscal\nOutC(ΠG)\n(M) is an\nopen subgroup of Aut(G) [cf. assertion (i); Remark 5.13.1], we conclude\nthat α ∈ COutC(ΠG)(Aut(G)). Thus, the fact that α ∈ Aut(G) follows\nfrom the commensurable terminality of Aut(G) in Out(ΠG), i.e., the\nequality Aut(G) = COut(ΠG)(Aut(G)) [cf. [CmbGC], Corollary 2.7, (iv)].\nThis completes the proof of the equality Aut(G) = COutC(ΠG)(M).\nNext, we verify the equality\nAut|Node(G)|(G) ∩ Ker(χG) = ZOutC(ΠG)(M)\nin the second display of assertion (ii). Now it follows immediately from\nTheorem 4.8, (v), that Aut|Node(G)|(G) ∩ Ker(χG) ⊆ ZOutC(ΠG)(M).\nThus, to show the equality in question, it suffices to verify the inclu-\nsion ZOutC(ΠG)(M) ⊆ Aut\n|Node(G)|(G) ∩ Ker(χG). To this end, let us\nobserve that since Aut(G) = COutC(ΠG)(M) [cf. the preceding para-\ngraph], it holds that ZOutC(ΠG)(M) ⊆ Aut(G). Thus, since the action\nof ZOutC(ΠG)(M) on M by conjugation preserves and induces the iden-\ntity automorphism on the intersection of M with each direct summand\nof\n⊕\ne∈Node(G) ΛG\nDG∼← Dehn(G) [i.e., each “ΛG”], it follows immediately\nfrom Theorem 4.8, (v), in light of our assumption that Node(G) \u0004= ∅,\nthat ZOutC(ΠG)(M) ⊆ Aut\n|Node(G)|(G) ∩ Ker(χG). This completes the\nproof of assertion (ii).\nAssertion (iii) follows immediately from assertion (ii), together with\nTheorem 4.8, (i). This completes the proof of Theorem 5.14. Q.E.D.\nRemark 5.14.1. In the notation of Theorem 5.14, (i) (respectively,\nTheorem 5.14, (ii)), in general, the inclusion\nCscal\nOutC(ΠG)(M) ⊆ Aut(G)\n122 Yuichiro Hoshi and Shinichi Mochizuki\n[hence, a fortiori, by the inclusions of the first display of Theorem 5.14,\n(i), the inclusion N scal\nOutC(ΠG)\n(M) ⊆ Aut(G)] (respectively, in general, the\ninclusion\nNOutC(ΠG)(M) ⊆ Aut(G) )\nis strict. Indeed, suppose that there exist a node e ∈ Node(G) and an\nautomorphism α ∈ Aut(G) of G such that α does not stabilize e, and\nχG(α) = 1. [For example, in the notation of the final paragraph of the\nproof of Theorem 5.7, the node e1 and the automorphism induced by τ\nof G\nXlog\nsatisfy these conditions.] Now fix a prime number l ∈ Σ; write\nM\ndef\n= l · (ΛG)e ⊕\n(⊕\nf \n=e\nΛG\n)\n⊆\n⊕\nf∈Node(G)\nΛG\nDG∼← Dehn(G)\n— where we use the notation (ΛG)e to denote a copy of ΛG indexed by\ne ∈ Node(G). ThenM is an open subgroup of Dehn(G), hence also IPSC-\nample [cf. Remark 5.13.1], but it follows immediately from Theorem 4.8,\n(v), that α \u0004∈ Cscal\nOutC(ΠG)\n(M) (respectively, α \u0004∈ NOutC(ΠG)(M)).\n§6. Centralizers of geometric monodromy\nIn the present §, we study the centralizer of the image of certain\ngeometric monodromy groups. As an application, we prove a “geometric\nversion of the Grothendieck conjecture” for the universal curve over the\nmoduli stack of pointed smooth curves [cf. Theorem 6.13 below].\nDefinition 6.1. Let Σ be a nonempty set of prime numbers and Π\na pro-Σ surface group [cf. [MT], Definition 1.2]. Then we shall write\nOutC(Π) = OutFC(Π) = OutPFC(Π)\nfor the group of outomorphisms of Π which induce bijections on the\nset of cuspidal inertia subgroups of Π. We shall refer to an element of\nOutC(Π) = OutFC(Π) = OutPFC(Π) as a C-, FC-, or PFC-admissible\noutomorphism of Π.\nRemark 6.1.1. In the notation of Definition 6.1, suppose that ei-\nther Σ\u0005 = 1 or Σ = Primes. Then it follows from the various def-\ninitions involved that Π is equipped with a natural structure of pro-\nΣ configuration space group [cf. [MT], Definition 2.3, (i)]. Thus, the\nCombinatorial anabelian topics I 123\nterms “C-\/FC-\/PFC-admissible outomorphism of Π” and the notation\n“OutC(Π) = OutFC(Π)” have already been defined in [CmbCsp], Defini-\ntion 1.1, (ii), and Definition 1.4, (iii), of the present paper. In this case,\nhowever, one may verify easily that these definitions coincide.\nLemma 6.2 (Extensions arising from log configuration\nspaces). Let (g, r) be a pair of nonnegative integers such that 2g−2+r >\n0; 0 < m < n positive integers; ΣF ⊆ ΣB nonempty sets of prime\nnumbers; k an algebraically closed field of characteristic zero; (Spec k)log\nthe log scheme obtained by equipping Spec k with the log structure given\nby the fs chart N → k that maps 1 \u0006→ 0; X log = X log1 a stable log\ncurve of type (g, r) over (Spec k)log. Suppose that ΣF ⊆ ΣB satisfy one\nof the following two conditions:\n(1) ΣF and ΣB determine PT-formations [i.e., are either of car-\ndinality one or equal to Primes — cf. [MT], Definition 1.1,\n(ii)].\n(2) n−m = 1 and ΣB = Primes.\nWrite\nX logn , X\nlog\nm\nfor the n-th, m-th log configuration spaces of the stable log curve X log\n[cf. the discussion entitled “Curves” in §0], respectively; Πn, ΠB def= Πm\nfor the respective maximal pro-ΣB quotients of the kernels of the natu-\nral surjections π1(X\nlog\nn ) \u0002 π1((Spec k)log), π1(X logm ) \u0002 π1((Spec k)log);\nΠn\/m ⊆ Πn for the kernel of the surjection Πn \u0002 ΠB = Πm induced\nby the projection X logn → X logm obtained by forgetting the last (n − m)\nfactors; ΠF for the maximal pro-ΣF quotient of Πn\/m; ΠT for the quo-\ntient of Πn by the kernel of the natural surjection Πn\/m \u0002 ΠF. Thus,\nwe have a natural exact sequence of profinite groups\n1 −→ ΠF −→ ΠT −→ ΠB −→ 1 ,\nwhich determines an outer representation\nρn\/m : ΠB −→ Out(ΠF) .\nThen the following hold:\n(i) The isomorphism class of the exact sequence of profinite groups\n1 −→ ΠF −→ ΠT −→ ΠB −→ 1\n124 Yuichiro Hoshi and Shinichi Mochizuki\ndepends only on (g, r) and the pair (ΣF,ΣB), i.e., if 1 →\nΠ•F → Π•T → Π•B → 1 is the exact sequence “1 → ΠF →\nΠT → ΠB → 1” associated, with respect to the same (ΣF,ΣB),\nto another stable log curve of type (g, r) over (Spec k)log, then\nthere exists a commutative diagram of profinite groups\n1 −−−−→ ΠF −−−−→ ΠT −−−−→ ΠB −−−−→ 1\n\u0005\n⏐⏐\b \u0005⏐⏐\b \u0005⏐⏐\b\n1 −−−−→ Π•F −−−−→ Π•T −−−−→ Π•B −−−−→ 1\n— where the vertical arrows are isomorphisms which may be\nchosen to arise scheme-theoretically.\n(ii) The profinite group ΠB is equipped with a natural structure\nof pro-ΣB configuration space group [cf. [MT], Definition\n2.3, (i)]. If, moreover, ΣF ⊆ ΣB satisfies condition (1) (re-\nspectively, (2)), then the profinite group ΠF is equipped with a\nnatural structure of pro-ΣF configuration space group (re-\nspectively, surface group [cf. [MT], Definition 1.2]).\n(iii) The outer representation ρn\/m : ΠB → Out(ΠF) factors through\nthe closed subgroup OutC(ΠF) ⊆ Out(ΠF) [cf. Definition 6.1;\n[CmbCsp], Definition 1.1, (ii)].\nProof. Assertion (i) follows immediately by considering a suitable\nspecialization isomorphism [cf. the discussion preceding [CmbCsp], Def-\ninition 2.1, as well as Remark 5.6.1 of the present paper]. Assertion (ii)\nfollows immediately from assertion (i), together with the various def-\ninitions involved. Assertion (iii) follows immediately from the various\ndefinitions involved. This completes the proof of Lemma 6.2. Q.E.D.\nDefinition 6.3. In the notation of Lemma 6.2 in the case where\n(m,n,ΣB) = (1, 2,Primes),\nlet x ∈ X(k) be a k-valued point of the underlying scheme X of X log.\n(i) We shall denote by\nG\nthe semi-graph of anabelioids of pro-Primes PSC-type deter-\nmined by the stable log curve X log; by\nGx\nCombinatorial anabelian topics I 125\nthe semi-graph of anabelioids of pro-ΣF PSC-type determined\nby the geometric fiber of X\nlog\n2 → X log over xlog\ndef\n= x×X X log;\nby ΠG , ΠGx the [pro-Primes, pro-ΣF] fundamental groups of G,\nGx, respectively. Thus, we have a natural outer isomorphism\nΠB\n∼−→ ΠG\nand a natural Im(ρ2\/1) (⊆ Out(ΠF))-torsor of outer isomor-\nphisms\nΠF\n∼−→ ΠGx .\nLet us fix isomorphisms ΠB\n∼→ ΠG , ΠF ∼→ ΠGx that belong to\nthese collections of isomorphisms.\n(ii) Denote by\ncFdiag,x ∈ Cusp(Gx)\nthe cusp of Gx [i.e., the cusp of the geometric fiber of X log2 →\nX log over xlog] determined by the diagonal divisor of X\nlog\n2 .\nFor v ∈ Vert(G) (respectively, c ∈ Cusp(G)) [i.e., which cor-\nresponds to an irreducible component (respectively, a cusp) of\nX log], denote by\nvFx ∈ Vert(Gx) (respectively, cFx ∈ Cusp(Gx))\nthe vertex (respectively, cusp) of Gx that corresponds naturally\nto v ∈ Vert(G) (respectively, c ∈ Cusp(G)).\n(iii) Let e ∈ Edge(G), v ∈ Vert(G), S ⊆ VCN(G), and z ∈ VCN(G).\nThen we shall say that x lies on e if the image of x is the cusp\nor node corresponding to e ∈ Edge(G). We shall say that x lies\non v if x does not lie on any edge of G, and, moreover, the image\nof x is contained in the irreducible component corresponding\nto v ∈ Vert(G). We shall write x \u0003 S if x lies on some s ∈ S.\nWe shall write x\u0003 z if x\u0003 {z}.\nLemma 6.4 (Cusps and vertices of fibers). In the notation\nof Definition 6.3, let x, x′ ∈ X(k) be k-valued points of X. Then the\nfollowing hold:\n(i) The isomorphism ΠGx\n∼→ ΠGx′ obtained by forming the compos-\nite of the isomorphisms ΠGx\n∼← ΠF ∼→ ΠGx′ [cf. Definition 6.3,\n126 Yuichiro Hoshi and Shinichi Mochizuki\n(i)] is group-theoretically cuspidal [cf. [CmbGC], Defini-\ntion 1.4, (iv)].\n(ii) The injection Cusp(G) ↪→ Cusp(Gx) given by mapping c \u0006→ cFx\ndetermines a bijection\nCusp(G) ∼−→ Cusp(Gx) \\ {cFdiag,x}\n[cf. Definition 6.3, (ii)]. Moreover, if we regard Cusp(G) as a\nsubset of each of the sets Cusp(Gx), Cusp(Gx′) by means of the\nabove injections, then the bijection Cusp(Gx) ∼→ Cusp(Gx′) de-\ntermined by the group-theoretically cuspidal isomorphism\nΠGx\n∼→ ΠGx′ of (i) maps cFdiag,x \u0006→ cFdiag,x′ and induces the\nidentity automorphism on Cusp(G). Thus, in the remain-\nder of the present §, we shall omit the subscript “x” from the\nnotation “cFx” and “c\nF\ndiag,x”.\n(iii) The injection Vert(G) ↪→ Vert(Gx) given by mapping v \u0006→ vFx\n[cf. Definition 6.3, (ii)] is bijective if and only if x\u0003 Vert(G)\n[cf. Definition 6.3, (iii)]. If x \u0003 Edge(G), then the comple-\nment of the image of Vert(G) in Vert(Gx) is of cardinality one;\nin this case, we shall write\nvFnew,x ∈ Vert(Gx) \\Vert(G)\nfor the unique element of Vert(Gx) \\Vert(G).\n(iv) Suppose that x \u0003 Cusp(G) (respectively, Node(G)). Then\nit holds that cFdiag ∈ C(vFnew,x) [cf. (iii)], and (C(vFnew,x)\u0005,\nN (vFnew,x)\u0005) = (2, 1) (respectively, = (1, 2)). Moreover, for any\nelement eF ∈ N (vFnew,x), it holds that V(eF)\u0005 = 2.\nProof. These assertions follow immediately from the various defi-\nnitions involved. Q.E.D.\nDefinition 6.5. In the notation of Definition 6.3:\n(i) Write\nCuspF(G) def= Cusp(G) \u0019 {cFdiag}\n[cf. Definition 6.3, (ii); Lemma 6.4, (ii)].\nCombinatorial anabelian topics I 127\n(ii) Let α ∈ OutC(ΠF) be an C-admissible outomorphism of ΠF [cf.\nDefinition 6.1; Lemma 6.2, (ii)]. Then it follows immediately\nfrom Lemma 6.4, (ii), that for any k-valued point x ∈ X(k)\nof X, the automorphism of CuspF(G) [cf. (i)] obtained by\nconjugating the natural action of α on Cusp(Gx) by the natural\nbijection CuspF(G) ∼→ Cusp(Gx) implicit in Lemma 6.4, (ii),\ndoes not depend on the choice of x. We shall refer to this\nautomorphism of CuspF(G) as the automorphism of CuspF(G)\ndetermined by α. Thus, we have a natural homomorphism\nOutC(ΠF) → Aut(CuspF(G)).\n(iii) For c ∈ CuspF(G) [cf. (i)], we shall refer to a closed subgroup\nof ΠF obtained as the image — via the isomorphism ΠGx\n∼← ΠF\n[cf. Definition 6.3, (i)] for some k-valued point x ∈ X(k) —\nof a cuspidal subgroup of ΠGx associated to the cusp of Gx\ncorresponding to c ∈ CuspF(G) as a cuspidal subgroup of ΠF\nassociated to c ∈ CuspF(G). Note that it follows immediately\nfrom Lemma 6.4, (ii), that the ΠF-conjugacy class of a cuspidal\nsubgroup of ΠF associated to c ∈ CuspF(G) depends only on\nc ∈ CuspF(G), i.e., does not depend on the choice of x or on\nthe choices of isomorphisms made in Definition 6.3, (i).\nLemma 6.6 (Images of VCN-subgroups of fibers). In the no-\ntation of Definition 6.3, let ΠcFdiag\n⊆ ΠF be a cuspidal subgroup of ΠF\nassociated to cFdiag ∈ CuspF(G) [cf. Definition 6.5, (i), (iii)], x ∈ X(k)\na k-valued point of X, zF ∈ VCN(Gx) \\ {cFdiag}, and ΠzF ⊆ ΠGx a VCN-\nsubgroup of ΠGx associated to z\nF. Write Ndiag ⊆ ΠF for the normal\nclosed subgroup of ΠF topologically normally generated by ΠcFdiag\n. [Note\nthat it follows immediately from Lemma 6.4, (i), that Ndiag is normal\nin ΠT.] Then the following hold:\n(i) Write GΣF for the semi-graph of anabelioids of pro-ΣF PSC-\ntype obtained by forming the pro-ΣF completion of G [cf.\n[SemiAn], Definition 2.9, (ii)]. Then there exists a natural\nouter isomorphism ΠF\/Ndiag\n∼→ ΠGΣF that satisfies the fol-\nlowing conditions:\n• Suppose that x\u0003 Vert(G) [cf. Definition 6.3, (iii)]. Then\nthe ΠGΣF -conjugacy class of the image of the composite\nΠzF ↪→ ΠGx\n∼← ΠF \u0002 ΠF\/Ndiag ∼→ ΠGΣF\n128 Yuichiro Hoshi and Shinichi Mochizuki\ncoincides with the ΠGΣF -conjugacy class of any VCN-sub-\ngroup of ΠGΣF associated to the element of VCN(GΣF) =\nVCN(G) naturally determined by zF.\n• Suppose that x \u0003 e ∈ Edge(G), and that zF \u0004∈ {vFnew,x} ∪\nE(vFnew,x) (respectively, zF ∈ {vFnew,x} ∪ E(vFnew,x)) [cf.\nLemma 6.4, (iii)]. Then the ΠGΣF -conjugacy class of the\nimage of the composite\nΠzF ↪→ ΠGx\n∼← ΠF \u0002 ΠF\/Ndiag ∼→ ΠGΣF\ncoincides with the ΠGΣF -conjugacy class of any VCN-sub-\ngroup of ΠGΣF associated to the element of VCN(GΣF) =\nVCN(G) natural determined by zF (respectively, associated\nto e ∈ Edge(GΣF) = Edge(G)).\n(ii) The image of the composite\nΠzF ↪→ ΠGx\n∼← ΠF \u0002 ΠF\/Ndiag\nis commensurably terminal.\n(iii) Suppose that either\n• zF ∈ Edge(Gx),\nor\n• zF = vFx for v ∈ Vert(G) such that x does not lie on v.\nThen the composite\nΠzF ↪→ ΠGx\n∼← ΠF \u0002 ΠF\/Ndiag\nis injective.\n(iv) Let Π(z′)F ⊆ ΠGx be a VCN-subgroup of ΠGx associated to an\nelement (z′)F ∈ VCN(Gx) \\ {cFdiag}. Suppose that either\n• x\u0003 Vert(G),\nor\n• x\u0003 Edge(G), and zF, (z′)F \u0004∈ {vFnew,x} ∪ E(vFnew,x).\nThen if the ΠF\/Ndiag-conjugacy classes of the images of ΠzF ,\nΠ(z′)F ⊆ ΠGx via the composite\nΠGx\n∼← ΠF \u0002 ΠF\/Ndiag\nCombinatorial anabelian topics I 129\ncoincide, then zF = (z′)F.\nProof. Assertion (i) follows immediately from the various defini-\ntions involved. Assertion (ii) follows immediately from [CmbGC], Propo-\nsition 1.2, (ii), and assertion (i), together with the various definitions\ninvolved. Assertion (iii) follows immediately from assertion (i), together\nwith the various definitions involved. Assertion (iv) follows immediately\nfrom [CmbGC], Proposition 1.2, (i), and assertion (i), together with the\nvarious definitions involved. Q.E.D.\nLemma 6.7 (Outomorphisms preserving the diagonal). In\nthe notation of Definition 6.3, let H ⊆ ΠB be an open subgroup of ΠB,\nα̃ an automorphism of ΠT|H def= ΠT ×ΠB H over H, αF ∈ Out(ΠF) the\noutomorphism of ΠF determined by the restriction α̃|ΠF of α̃ to ΠF ⊆\nΠT|H , and ΠcFdiag ⊆ ΠF a cuspidal subgroup of ΠF associated to c\nF\ndiag ∈\nCuspF(G) [cf. Definition 6.5, (i), (iii)]. Then the following hold:\n(i) Suppose that α̃ preserves ΠcFdiag\n⊆ ΠF. Then the automorphism\nof ΠF\/Ndiag [where we refer to the statement of Lemma 6.6\nconcerning Ndiag] induced by α̃ is the identity automor-\nphism. If, moreover, αF is C-admissible [cf. Definition 6.1;\nLemma 6.2, (ii)], then the automorphism of CuspF(G) induced\nby αF [cf. Definition 6.5, (ii)] is the identity automor-\nphism.\n(ii) Let e ∈ Edge(G), x ∈ X(k) be such that x \u0003 e. Suppose\nthat αF is C-admissible, and that Edge(G) = {e} ∪ Cusp(G).\nThen it holds that αF ∈ Aut(Gx) (⊆ Out(ΠGx)\n∼← Out(ΠF)).\nIf, moreover, α̃ preserves ΠcFdiag\n⊆ ΠF, then αF ∈ Aut|grph|\n(Gx) (⊆ Aut(Gx)).\nProof. First, we verify assertion (i). Now let us observe that it\nfollows immediately from a similar argument to the argument used in\nthe proof of [CmbCsp], Proposition 1.2, (iii) — i.e., by considering the\naction of α̃ on the decomposition subgroup D ⊆ ΠT|H of ΠT|H associated\nto the diagonal divisor of X\nlog\n2 such that ΠcFdiag\n⊆ D, and applying the\nfact that D = NΠT|H (ΠcFdiag) ⊆ ΠT|H — that α̃ induces the identity au-\ntomorphism on some normal open subgroup J ⊆ ΠF\/Ndiag of ΠF\/Ndiag.\nThus, it follows immediately from the slimness [cf. [CmbGC], Remark\n130 Yuichiro Hoshi and Shinichi Mochizuki\n1.1.3] of ΠGΣF\n∼← ΠF\/Ndiag ↪→ Aut(J) that α̃ induces the identity au-\ntomorphism on ΠF\/Ndiag. This completes the proof of the fact that α̃\ninduces the identity automorphism of ΠF\/Ndiag. On the other hand, if,\nmoreover, αF is C-admissible, then since α̃ induces the identity automor-\nphism of ΠF\/Ndiag, it follows immediately from [CmbGC], Proposition\n1.2, (i), applied to the cuspidal inertia subgroups of ΠF\/Ndiag\n∼→ ΠGΣF\n[cf. Lemma 6.6, (i)] that the automorphism of CuspF(G) induced by αF\nis the identity automorphism. This completes the proof of assertion (i).\nNext, we verify assertion (ii). Let Πe ⊆ ΠG ∼← ΠB be an edge-\nlike subgroup associated to the edge e ∈ Edge(G). By abuse of nota-\ntion, we shall write H ∩ Πe ⊆ ΠB for the intersection of H with the\nimage of Πe in ΠB. Now since αF is C-admissible, and α̃ is an au-\ntomorphism of ΠT|H over H, it holds that αF ∈ ZOutC(ΠF)(ρ2\/1(H))\n[cf. the discussion entitled “Topological groups” in §0], hence also that\nαF ∈ ZOutC(ΠF)(ρ2\/1(H ∩ Πe)). On the other hand, in light of the\nwell-known structure of X log in a neighborhood of the cusp or node cor-\nresponding to e, one verifies easily — by applying [NodNon], Proposition\n2.14, together with our assumption that Edge(G) = {e} ∪ Cusp(G) —\nthat the image of the composite\nΠe ↪→ ΠG ∼← ΠB\nρ2\/1→ Out(ΠF) ∼→ Out(ΠGx) ,\nhence also the image ρ2\/1(H∩Πe) ⊆ Out(ΠF) ∼→ Out(ΠGx), is NN-ample\n[cf. Definition 5.13; Theorem 5.9, (ii)]. Thus, since cFdiag ∈ Cusp(Gx) \u0004=\n∅, it follows immediately from Theorem 5.14, (i), that αF ∈ Aut(Gx).\nThis completes the proof of the fact that αF ∈ Aut(Gx). Now suppose,\nmoreover, that α̃ preserves ΠcFdiag\n⊆ ΠF. Then it follows from assertion\n(i) that αF fixes the cusps of Gx, hence that it fixes vFnew,x. On the\nother hand, since α̃ induces the identity automorphism of ΠF\/Ndiag [cf.\nassertion (i)], it follows from Lemma 6.6, (iii), (iv), that αF fixes the\nvertices of Gx that are \u0004= vFnew,x, as well as [cf. [CmbGC], Proposition\n1.2, (i)] the branches of nodes of Gx that abut to such vertices. Thus,\nαF ∈ Aut|grph|(Gx), as desired. This completes the proof of assertion\n(ii). Q.E.D.\nLemma 6.8 (Triviality of certain outomorphisms). In the no-\ntation of Definition 6.3, let ΠcFdiag\n⊆ ΠF be a cuspidal subgroup of ΠF\nassociated to cFdiag ∈ CuspF(G) [cf. Definition 6.5, (i), (iii)], H ⊆ ΠB an\nopen subgroup of ΠB, and α ∈ ZOutC(ΠF)(ρ2\/1(H)) [cf. Definition 6.1;\nCombinatorial anabelian topics I 131\nLemma 6.2, (ii)]. Suppose that α preserves the ΠF-conjugacy class of\nΠcFdiag\n⊆ ΠF. Then α is the identity outomorphism.\nProof. The following argument is essentially the same as the argu-\nment applied in [CmbCsp], [NodNon] to prove [CmbCsp], Corollary 2.3,\n(ii); [NodNon], Corollary 5.3.\nLet ΠT|H def= ΠT ×ΠB H and α̃ ∈ AutH(ΠT|H) a lifting of α ∈\nZOutC(ΠF)(ρ2\/1(H)) ⊆ ZOut(ΠF)(ρ2\/1(H))\n∼← AutH(ΠT|H)\/Inn(ΠF) [cf.\nthe discussion entitled “Topological groups” in §0]. Since we have as-\nsumed that α preserves the ΠF-conjugacy class of ΠcFdiag\n⊆ ΠF, it fol-\nlows from Lemma 6.7, (i), (ii), that by replacing α̃ by a suitable ΠF-\nconjugate of α̃, we may assume without loss of generality that α̃ pre-\nserves ΠcFdiag\n⊆ ΠF, and, moreover, that\n(a) the automorphism of ΠF\/Ndiag induced by α̃ is the identity\nautomorphism;\n(b) for e ∈ Edge(G), x ∈ X(k) such that x \u0003 e, if Edge(G) = {e}\n∪ Cusp(G), then α ∈ Aut|grph|(Gx) (⊆ Out(ΠGx)\n∼← Out(ΠF)).\nNext, we claim that\n(∗1): if (g, r) = (0, 3), then α is the identity outomor-\nphism.\nIndeed, write c1, c2, c3 ∈ Cusp(G) for the three distinct cusps of G;\nv ∈ Vert(G) for the unique vertex of G. For i ∈ {1, 2, 3}, let xi ∈ X(k)\nbe such that xi \u0003 ci. Next, let us observe that since our assumption\nthat (g, r) = (0, 3) implies that Node(G) = ∅, it follows immediately\nfrom (b) that for i ∈ {1, 2, 3}, the outomorphism α of ΠGxi\n∼← ΠF is\n∈ Aut|grph|(Gxi) (⊆ Out(ΠGxi )\n∼← Out(ΠF)). Next, let us fix a verticial\nsubgroup ΠvFx2\n⊆ ΠGx2\n∼← ΠF associated to vFx2 ∈ Vert(Gx2) [cf. Defini-\ntion 6.3, (ii)]. Then since α ∈ Aut|grph|(Gx2), it follows immediately from\nthe commensurable terminality of the image of the composite ΠvFx2\n↪→\nΠGx2\n∼← ΠF \u0002 ΠF\/Ndiag [cf. Lemma 6.6, (ii)], together with (a), that\nthere exists an Ndiag-conjugate β̃ of α̃ such that β̃(ΠvFx2\n) = ΠvFx2\n. Thus,\nsince the composite ΠvFx2\n↪→ ΠGx2\n∼← ΠF \u0002 ΠF\/Ndiag is injective [cf.\nLemma 6.6, (iii)], it follows immediately from (a) that β̃ induces the\nidentity automorphism on ΠvFx2\n⊆ ΠGx2\n∼← ΠF. Next, let ΠcF1 ⊆ ΠF be a\ncuspidal subgroup of ΠF associated to c1 ∈ CuspF(G) [cf. Definition 6.5,\n(iii)] which is contained in ΠvFx2\n⊆ ΠGx2\n∼← ΠF; ΠvFx3 ⊆ ΠGx3\n∼← ΠF a\n132 Yuichiro Hoshi and Shinichi Mochizuki\nverticial subgroup associated to vFx3 ∈ Vert(Gx3) that contains ΠcF1 ⊆ ΠF.\nThen since β̃ induces the identity automorphism on ΠvFx2\n⊆ ΠGx2\n∼← ΠF,\nit follows from the inclusion ΠcF1 ⊆ ΠvFx2 that β̃(ΠcF1 ) = ΠcF1 . Thus, since\nthe verticial subgroup ΠvFx3\n⊆ ΠGx3\n∼← ΠF is the unique verticial sub-\ngroup of ΠGx3\n∼← ΠF associated to vFx3 ∈ Vert(Gx3) which contains ΠcF1\n[cf. [CmbGC], Proposition 1.5, (i)], it follows immediately from the fact\nthat α ∈ Aut|grph|(Gx3) that β̃(ΠvFx3 ) = ΠvFx3 . In particular, since the\ncomposite ΠvFx3\n↪→ ΠF \u0002 ΠF\/Ndiag is injective [cf. Lemma 6.6, (iii)], it\nfollows immediately from (a) that β̃ induces the identity automorphism\non ΠvFx3\n⊆ ΠGx3\n∼← ΠF. On the other hand, since ΠF is topologically\ngenerated by ΠvFx2\n⊆ ΠGx2\n∼← ΠF and ΠvFx3 ⊆ ΠGx3\n∼← ΠF [cf. [CmbCsp],\nLemma 1.13], this implies that β̃ induces the identity automorphism on\nΠF. This completes the proof of the claim (∗1).\nNext, we claim that\n(∗2): for arbitrary (g, r), α is the identity outomor-\nphism.\nIndeed, we verify the claim (∗2) by induction on 3g−3+r. If 3g−3+r =\n0, i.e., (g, r) = (0, 3), then the claim (∗2) amounts to the claim (∗1). Now\nsuppose that 3g−3+r > 1, and that the induction hypothesis is in force.\nSince 3g − 3 + r > 1, one verifies easily that there exists a stable log\ncurve Y log of type (g, r) over (Spec k)log such that Y log has precisely one\nnode. Thus, it follows immediately from Lemma 6.2, (i), that to verify\nthe claim (∗2), by replacing X log by Y log, we may assume without loss\nof generality that Node(G)\u0005 = 1. Let e be the unique node of G and\nx ∈ X(k) such that x \u0003 e. Now let us observe that since Node(G)\u0005 =\n1, and e ∈ Node(G), it follows from (b) that α ∈ Aut|grph|(Gx) (⊆\nOut(ΠGx)\n∼← Out(ΠF)). Write {eF1 , eF2 } = N (vFnew,x) [cf. Lemma 6.4,\n(iv)]. Also, for i ∈ {1, 2}, denote by vi ∈ Vert(G) the vertex of G\nsuch that (vi)\nF\nx ∈ Vert(Gx) is the unique element of V(eFi ) \\ {vFnew,x}\n[cf. Lemma 6.4, (iv)]; by Hi the sub-semi-graph of PSC-type of the\nunderlying semi-graph Gx of Gx whose set of vertices = {vFnew,x, (vi)Fx};\nand by Si\ndef\n= Node((Gx)|Hi) \\ {eFi } ⊆ Node((Gx)|Hi) the complement\nof {eFi }. [Thus, if G is noncyclically primitive (respectively, cyclically\nprimitive) [cf. Definition 4.1], then Hi \u0004= Gx and Si = ∅ (respectively,\nHi = Gx and Si = {eF3−i}). In particular, Si ⊆ Node((Gx)|Hi) is not of\nseparating type.]\nNext, let us observe that to complete the proof of the above claim\n(∗2), it suffices to verify that\nCombinatorial anabelian topics I 133\n(†): α ∈ Dehn(Gx), and, moreover, for i ∈ {1, 2}, α\nis contained in the kernel of the natural surjection\nDehn(Gx) \u0002 Dehn(((Gx)|Hi)\bSi) [cf. Theorem 4.8,\n(iii), (iv)].\nIndeed, since [as is easily verified] Node(Gx) = N (vFnew,x) = {eF1 , eF2 }, it\nfollows immediately from Theorem 4.8, (iii), (iv), that\n2⋂\ni=1\nKer\n(\nDehn(Gx) \u0002 Dehn(((Gx)|Hi)\bSi)\n)\n= {1} .\nIn particular, the implication (†) ⇒ (∗2) holds. The remainder of the\nproof of the claim (∗2) is devoted to verifying the above (†).\nFor i ∈ {1, 2}, let Π(vi)Fx ⊆ ΠGx\n∼← ΠF be a verticial subgroup of\nΠGx\n∼← ΠF associated to the vertex (vi)Fx ∈ V(eFi ) \\ {vFnew,x}. Then\nsince α ∈ Aut|grph|(Gx), it follows that α̃ preserves the ΠF-conjugacy\nclass of Π(vi)Fx ⊆ ΠGx\n∼← ΠF. Thus, since the image of the composite\nΠ(vi)Fx ↪→ ΠF \u0002 ΠF\/Ndiag is commensurably terminal [cf. Lemma 6.6,\n(ii)], it follows immediately from (a) that there exists an Ndiag-conjugate\nβ̃i [which may depend on i ∈ {1, 2}!] of α̃ such that β̃i(Π(vi)Fx ) = Π(vi)Fx .\nTherefore, since the composite Π(vi)Fx ↪→ ΠF \u0002 ΠF\/Ndiag is injective\n[cf. Lemma 6.6, (iii)], it follows from (a) that β̃i induces the identity\nautomorphism of Π(vi)Fx .\nNext, let ΠeFi ⊆ Π(vi)Fx be a nodal subgroup of ΠGx\n∼← ΠF associated\nto eFi ∈ Node(Gx) that is contained in Π(vi)Fx ; ΠvFnew,x;i ⊆ ΠGx\n∼← ΠF\na verticial subgroup [which may depend on i ∈ {1, 2}!] associated to\nvFnew,x ∈ Vert(Gx) which contains ΠeFi :\nΠvFnew,x;i ⊇ ΠeFi ⊆ Π(vi)Fx ⊆ ΠGx\n∼← ΠF .\nThen since β̃i preserves and induces the identity automorphism on Π(vi)Fx ,\nit follows from the inclusion ΠeFi ⊆ Π(vi)Fx that β̃i(ΠeFi ) = ΠeFi . More-\nover, since ΠvFnew,x;i is the unique verticial subgroup of ΠGx\n∼← ΠF\nassociated to vFnew,x which contains ΠeFi [cf. [CmbGC], Proposition\n1.5, (i)], it follows immediately from the fact that α ∈ Aut|grph|(Gx)\nthat β̃i(ΠvFnew,x;i) = ΠvFnew,x;i. Thus, β̃i preserves the closed subgroup\nΠFi ⊆ ΠF of ΠF obtained by forming the image of the natural homo-\nmorphism\nlim−→\n(\nΠvFnew,x;i ←↩ ΠeFi ↪→ Π(vi)Fx\n)\n−→ ΠF\n134 Yuichiro Hoshi and Shinichi Mochizuki\n— where the inductive limit is taken in the category of pro-ΣF groups.\nNow one may verify easily that the ΠF-conjugacy class of ΠFi ⊆ ΠF\ncoincides with the ΠF-conjugacy class of the image of the natural outer\ninjection Π((Gx)|Hi )\u0004Si ↪→ ΠGx\n∼← ΠF discussed in Proposition 2.11; in\nparticular, ΠFi is commensurably terminal in ΠF [cf. Proposition 2.11].\nMoreover, by applying a similar argument to the argument used in\n[CmbCsp], Definition 2.1, (iii), (vi), or [NodNon], Definition 5.1, (ix),\n(x) [i.e., by considering the portion of the underlying scheme X2 of X\nlog\n2\ncorresponding to the underlying scheme (Xvi)2 of the 2-nd log configu-\nration space (Xvi)\nlog\n2 of the stable log curve X\nlog\nvi\ndetermined by G|vi ],\none concludes that there exists a verticial subgroup Πvi ⊆ ΠG\n∼← ΠB\nassociated to vi ∈ Vert(G) such that the outer representation of Πvi on\nΠF determined by the composite Πvi ↪→ ΠB\nρ2\/1→ Out(ΠF) preserves the\nΠF-conjugacy class of ΠFi ⊆ ΠF [so we obtain a natural outer repre-\nsentation Πvi → Out(ΠFi) — cf. Lemma 2.12, (iii)], and, moreover,\nthat if we write ΠTi\ndef\n= ΠFi\nout\n\u0002 Πvi (⊆ ΠT) [cf. the discussion enti-\ntled “Topological groups” in §0], then ΠTi is naturally isomorphic to the\n“ΠT” obtained by taking “G” to be G|vi .\nNow since β̃i(ΠFi) = ΠFi , and α ∈ ZOutC(ΠF)(ρ2\/1(H)), one may\nverify easily that the outomorphism of ΠFi determined by β̃i|ΠFi [cf.\nLemma 2.12, (iii)] is ∈ ZOutC(ΠFi )(ρ2\/1(H ∩ Πvi)) — where, by abuse\nof notation, we write H ∩ Πvi ⊆ ΠB for the intersection of H with\nthe image of Πvi in ΠB. Therefore, since the quantity “3g − 3 + r”\nassociated to G|vi is < 3g − 3 + r, by considering a similar diagram to\nthe diagram in [CmbCsp], Definition 2.1, (vi), or [NodNon], Definition\n5.1, (x), and applying the induction hypothesis, we conclude that β̃i|ΠFi\nis a ΠFi-inner automorphism. In particular, it follows immediately [by\nallowing i ∈ {1, 2} to vary] that the outomorphism α is ∈ Dehn(Gx), and,\nmoreover — by considering the natural identification outer isomorphism\nΠFi\n∼→ Π((Gx)|Hi ))\u0004Si — that α is contained in the kernel of the natural\nsurjection Dehn(Gx) \u0002 Dehn(((Gx)|Hi))\bSi), as desired. This completes\nthe proof of (†), hence also of Lemma 6.8. Q.E.D.\nDefinition 6.9. In the notation of Definition 6.3:\n(i) Suppose that 2g − 2 + r > 1, i.e., (g, r) \u0004∈ {(0, 3), (1, 1)}. Then\nwe shall write\nAg,r\ndef\n= {1} ⊆ Aut(CuspF(G))\nCombinatorial anabelian topics I 135\n[cf. Definition 6.5, (i)].\n(ii) Suppose that (g, r) = (1, 1). Then we shall write\n(Z\/2Z \f) Ag,r def= Aut(CuspF(G)) .\n(iii) Suppose that (g, r) = (0, 3). Then we shall write\n(Z\/2Z× Z\/2Z \f) Ag,r ⊆ Aut(CuspF(G))\nfor the subgroup of Aut(CuspF(G)) obtained as the image of\nthe subgroup of the symmetric group on 4 letters\n{id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} ⊆ S4\nvia the isomorphismS4\n∼→ Aut(CuspF(G)) arising from a bijec-\ntion {1, 2, 3, 4} ∼→ CuspF(G). [Note that since the above sub-\ngroup of S4 is normal, the subgroup Ag,r ⊆ Aut(CuspF(G))\ndoes not depend on the choice of the bijection {1, 2, 3, 4} ∼→\nCuspF(G).]\nLemma 6.10 (Permutations of cusps arising from certain\nC-admissible outomorphisms). In the notation of Definition 6.3,\nlet H ⊆ ΠB be an open subgroup of ΠB. Then the following hold:\n(i) The composite\nZOutC(ΠF)(ρ2\/1(H)) ↪→ Out\nC(ΠF) → Aut(CuspF(G))\n[cf. Definition 6.5, (ii)] factors through the subgroup Ag,r ⊆\nAut(CuspF(G)) [cf. Definition 6.9], hence determines a homo-\nmorphism\nZOutC(ΠF)(Im(ρ2\/1)) −→ Ag,r .\n(ii) The composite\nAutXlog(X\nlog\n2 ) −→ ZOutC(ΠF)(Im(ρ2\/1)) −→ Ag,r\nof the natural homomorphism\nAutXlog(X\nlog\n2 ) −→ ZOutC(ΠF)(Im(ρ2\/1))\n136 Yuichiro Hoshi and Shinichi Mochizuki\nwith the homomorphism of (i) is an isomorphism. In par-\nticular, the homomorphism ZOutC(ΠF)(Im(ρ2\/1)) → Ag,r of (i)\nis a split surjection [cf. the discussion entitled “Topological\ngroups” in §0].\nProof. First, we verify assertion (i). If (g, r) = (1, 1), then since\nAg,r = Aut(Cusp\nF(G)), assertion (i) is immediate. On the other hand, if\nr = 0, then since CuspF(G)\u0005 = 1, assertion (i) is immediate. Thus, in the\nremainder of the proof of assertion (i), we suppose that (g, r) \u0004= (1, 1),\nr ≥ 1.\nNow we verify assertion (i) in the case where r = 1. Let us observe\nthat it follows immediately from Lemma 6.2, (i), that by replacing X log\nby a suitable stable log curve of type (g, r) over (Spec k)log, we may as-\nsume without loss of generality [cf. our assumption that r = 1, which im-\nplies that (g, r) \u0004= (0, 3)] that G is cyclically primitive [cf. Definition 4.1].\nLet c ∈ Cusp(G) be the unique cusp of G, e ∈ Node(G) the unique node\nof G, x ∈ X(k) such that x \u0003 e, and α ∈ ZOutC(ΠF)(ρ2\/1(H)). Then\nlet us observe that it follows immediately from our assumption that G is\ncyclically primitive of type (g, r) \u0004= (1, 1) (respectively, the various defi-\nnitions involved) that the vertex of Gx to which cF (respectively, cFdiag)\nabuts is not of type (0, 3) (respectively, is of type (0, 3)). Moreover, it\nfollows immediately from Lemma 6.7, (ii), that the outomorphism α of\nΠGx\n∼← ΠF is ∈ Aut(Gx). Thus, we conclude that the automorphism of\nCuspF(G) induced by α is the identity automorphism. This completes\nthe proof of assertion (i) in the case where r = 1.\nNext, we verify assertion (i) in the case where r > 1. Let us observe\nthat it follows immediately from Lemma 6.2, (i), that by replacing X log\nby a suitable stable log curve of type (g, r) over (Spec k)log, we may\nassume without loss of generality that Node(G) = ∅. Let v ∈ Vert(G)\nbe the unique vertex of G [cf. our assumption that Node(G) = ∅] and\nα ∈ ZOutC(ΠF)(ρ2\/1(H)). Now let us observe that for any c ∈ Cusp(G),\nx ∈ X(k) such that x \u0003 c, it follows immediately from the various def-\ninitions involved that Vert(Gx) = {vFx , vFnew,x}; C(vFnew,x) = {cF, cFdiag};\nC(vFx ) = Cusp(Gx)\\{cF, cFdiag}; vFx is of type (g, r); vFnew,x is of type (0, 3).\nMoreover, it follows immediately from Lemma 6.7, (ii), that the outo-\nmorphism α of ΠGx\n∼← ΠF is ∈ Aut(Gx). Thus, if (g, r) \u0004= (0, 3), then\nsince vFx is of type (g, r), and v\nF\nnew,x is of type (0, 3), it follows imme-\ndiately that α induces the identity automorphism on Vert(Gx), hence\nthat α preserves the subset {c, cFdiag} ⊆ CuspF(G) corresponding to\nC(vFnew,x) = {cF, cFdiag}. In particular, if (g, r) \u0004= (0, 3), (respectively,\n(g, r) = (0, 3)), then — by allowing “c” to vary among the elements of\nCombinatorial anabelian topics I 137\nCusp(G) — one may verify easily that the automorphism of CuspF(G)\ninduced by α is the identity automorphism (respectively, satisfies the\ncondition that\nfor any subset S ∈ CuspF(G) of cardinality 2, the au-\ntomorphism of CuspF(G) induced by α determines\nan automorphism of the set {S,CuspF(G) \\ S} ⊆\n2Cusp\nF(G),\nhence, by Lemma 6.11 below, is contained in Ag,r ⊆ Aut(CuspF(G))).\nThis completes the proof of assertion (i) in the case where r > 1, hence\nalso of assertion (i).\nNext, we verify assertion (ii). One verifies easily that the composite\nof natural homomorphisms\nAutXlog(X\nlog\n2 ) → AutΠB(ΠT)\/Inn(ΠF)\n∼→ ZOut(ΠF)(Im(ρ2\/1))\n[cf. the discussion entitled “Topological groups” in §0] factors through\nZOutC(ΠF)(Im(ρ2\/1)) ⊆ ZOut(ΠF)(Im(ρ2\/1)). In particular, we obtain a\nnatural homomorphism AutXlog(X\nlog\n2 ) → ZOutC(ΠF)(Im(ρ2\/1)). Now the\nfact that the composite\nAutXlog(X\nlog\n2 ) → ZOutC(ΠF)(Im(ρ2\/1)) ↪→ Out\nC(ΠF) → Aut(CuspF(G))\ndetermines a surjection AutXlog(X\nlog\n2 ) \u0002 Ag,r is well-known and easily\nverified. To verify that this surjection is injective, observe that an el-\nement of the kernel of this surjection determines an automorphism of\nthe trivial family X log ×(Spec k)log X log → X log over X log that preserves\nthe image of the diagonal. On the other hand, since the relative tangent\nbundle of this trivial family has no nonzero global sections, one con-\ncludes immediately that such an automorphism is constant, i.e., arises\nfrom a single automorphism of the fiber X log over (Spec k)log that is\ncompatible with the diagonal, hence [as is easily verified] equal to the\nidentity automorphism, as desired. This completes the proof of asser-\ntion (ii). Q.E.D.\nLemma 6.11 (A subgroup of the symmetric group on 4 let-\nters). Write G ⊆ S4 for the subgroup of the symmetric group on 4\nletters S4 consisting of g ∈ S4 such that\n(∗): for any subset S ⊆ {1, 2, 3, 4} of cardinality 2,\nthe automorphism g of {1, 2, 3, 4} determines an au-\ntomorphism of the set {S, {1, 2, 3, 4} \\S} ⊆ 2{1,2,3,4}.\n138 Yuichiro Hoshi and Shinichi Mochizuki\nThen\nG = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} .\nProof. First, let us observe that one may verify easily that\n{id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} ⊆ G .\nThus, to verify Lemma 6.11, it suffices to verify that G\u0005 = 4. Next, let\nus observe that it follows immediately from the condition (∗) that for\nany element g ∈ G, it holds that g4 = id; in particular, by the Sylow\nTheorem, together with the fact that S\u00054 = 2\n3 · 3, we conclude that G is\na 2-group. Thus, to verify Lemma 6.11, it suffices to verify that G\u0005 \u0004= 8.\nNext, let us observe that it follows immediately from the condition (∗)\nthat G ⊆ S4 is normal. Thus, if G\u0005 = 8, then since S\u00054 = 23 · 3, and\n(1 2) ∈ S4 is of order 2, again by the Sylow Theorem, we conclude that\n(1 2) ∈ G, in contradiction to the fact that (1 2) does not satisfy the\ncondition (∗). This completes the proof of Lemma 6.11. Q.E.D.\nTheorem 6.12 (Centralizers of geometric monodromy\ngroups arising from configuration spaces). Let (g, r) be a pair of\nnonnegative integers such that 2g − 2 + r > 0; 0 < m < n positive\nintegers; ΣF ⊆ ΣB nonempty sets of prime numbers; k an algebraically\nclosed field of characteristic zero; (Spec k)log the log scheme obtained\nby equipping Spec k with the log structure given by the fs chart N → k\nthat maps 1 \u0006→ 0; X log = X log1 a stable log curve of type (g, r) over\n(Spec k)log. Suppose that ΣF ⊆ ΣB satisfy one of the following two\nconditions:\n(1) ΣF and ΣB determine PT-formations [i.e., are either of car-\ndinality one or equal to Primes — cf. [MT], Definition 1.1,\n(ii)].\n(2) n−m = 1 and ΣB = Primes.\nWrite\nX logn , X\nlog\nm\nfor the n-th, m-th log configuration spaces of the stable log curve X log\n[cf. the discussion entitled “Curves” in §0], respectively; Πn, ΠB def= Πm\nfor the respective maximal pro-ΣB quotients of the kernels of the natu-\nral surjections π1(X\nlog\nn ) \u0002 π1((Spec k)log), π1(X logm ) \u0002 π1((Spec k)log);\nΠn\/m ⊆ Πn for the kernel of the surjection Πn \u0002 ΠB = Πm induced\nby the projection X logn → X logm obtained by forgetting the last (n − m)\nCombinatorial anabelian topics I 139\nfactors; ΠF for the maximal pro-ΣF quotient of Πn\/m; ΠT for the quo-\ntient of Πn by the kernel of the natural surjection Πn\/m \u0002 ΠF. Thus,\nwe have a natural exact sequence of profinite groups\n1 −→ ΠF −→ ΠT −→ ΠB −→ 1 ,\nwhich determines an outer representation\nρn\/m : ΠB −→ Out(ΠF) .\nThen the following hold:\n(i) Let H ⊆ ΠB be an open subgroup of ΠB. Recall that X logn →\nX logm may be regarded as the (n−m)-th log configuration space\nof the family of stable log curves X\nlog\nm+1 → X logm over X logm .\nThen the composite of natural homomorphisms\nAut\nX\nlog\nm\n(X\nlog\nm+1) −→ AutXlogm (X\nlog\nn ) −→ AutΠB(ΠT)\/Inn(ΠF)\n∼−→ ZOut(ΠF)(Im(ρn\/m)) ⊆ ZOut(ΠF)(ρn\/m(H))\n— where the first arrow is the homomorphism arising from the\nfunctoriality of the construction of the log configuration space;\nthe third arrow is the isomorphism appearing in the discussion\nentitled “Topological groups” in §0 — determines an isomor-\nphism\nAut\nX\nlog\nm\n(X\nlog\nm+1)\n∼−→ ZOutFC(ΠF)(ρn\/m(H))\n— where we write OutFC(ΠF) for the group of FC-admissible\n[cf. Definition 6.1; [CmbCsp], Definition 1.1, (ii)] outomor-\nphisms of ΠF [cf. Lemma 6.2, (ii)]. Here, we recall that the\nautomorphism group Aut\nX\nlog\nm\n(X\nlog\nm+1) is isomorphic to⎧⎨\n⎩\nZ\/2Z× Z\/2Z if (g, r,m) = (0, 3, 1);\nZ\/2Z if (g, r,m) = (1, 1, 1);\n{1} if (g, r,m) \u0004∈ {(0, 3, 1), (1, 1, 1)}.\n(ii) The isomorphism of (i) and the natural inclusion Sn−m ↪→\nZOutPFC(ΠF)(ρn\/m(H)) — where we write Out\nPFC(ΠF) for the\ngroup of PFC-admissible [cf. Definitions 1.4, (iii); 6.1] out-\nomorphisms of ΠF [cf. Lemma 6.2, (ii)] — determine an iso-\nmorphism\nAut\nX\nlog\nm\n(X\nlog\nm+1)×Sn−m\n∼−→ ZOutPFC(ΠF)(ρn\/m(H)) .\n140 Yuichiro Hoshi and Shinichi Mochizuki\n(iii) Let H be a closed subgroup of OutPFC(ΠF) that contains an\nopen subgroup of Im(ρn\/m) ⊆ Out(ΠF). Then H is almost\nslim [cf. the discussion entitled “Topological groups” in §0]. If,\nmoreover, H ⊆ OutFC(ΠF), and (g, r,m) \u0004∈ {(0, 3, 1), (1, 1, 1)},\nthen H is slim [cf. the discussion entitled “Topological groups”\nin §0].\nProof. First, we verify assertion (i). We begin by observing that\nthe description of the automorphism group Aut\nX\nlog\nm\n(X\nlog\nm+1) given in the\nstatement of assertion (i) follows immediately from Lemma 6.10, (ii).\nNext, let us observe that\n(∗1): to verify assertion (i), it suffices to verify asser-\ntion (i) in the case where ΣB = Primes.\nIndeed, this follows immediately from the various definitions involved.\nThus, in the remainder of the proof of assertion (i), we suppose that\nΣB = Primes.\nNext, we claim that\n(∗2): the composite homomorphism of assertion (i)\ndetermines an injection\nAut\nX\nlog\nm\n(X\nlog\nm+1) ↪→ ZOutFC(ΠF)(ρn\/m(H)) .\nIndeed, one verifies easily that the composite as in assertion (i) factors\nthrough ZOutFC(ΠF)(ρn\/m(H)). On the other hand, by considering the\naction of Aut\nX\nlog\nm\n(X\nlog\nm+1) on the set of conjugacy classes of cuspidal\ninertia subgroups of suitable subquotients [arising from fiber subgroups]\nof ΠF, it follows immediately that the composite as in assertion (i) is\ninjective [cf. Lemma 6.10, (ii)]. This completes the proof of the claim\n(∗2).\nNext, we claim that\n(∗3): the injection of (∗2) is an isomorphism.\nIndeed, it follows immediately from the various definitions involved that\nif NB ⊆ ΠB is a fiber subgroup of ΠB of length 1 [cf. Lemma 6.2,\n(ii); [MT], Definition 2.3, (iii)], then the natural surjection ΠT ×ΠB\nNB \u0002 NB may be regarded as the “ΠT \u0002 ΠB” obtained by taking\n“(g, r,m, n)” to be (g, r + m − 1, 1, n − m + 1). Thus, by applying\nthe inclusion ZOutFC(ΠF)(ρn\/m(H)) ⊆ ZOutFC(ΠF)(ρn\/m(H ∩ NB)) and\nreplacing ΠT \u0002 ΠB by ΠT ×ΠB NB \u0002 NB, we may assume without\nloss of generality that m = 1. On the other hand, it follows immediately\nfrom the various definitions involved that if NF ⊆ ΠF is a fiber subgroup\nof ΠF of length n − 2, then the natural surjection ΠT\/NF \u0002 ΠB may\nCombinatorial anabelian topics I 141\nbe regarded as the “ΠT \u0002 ΠB” obtained by taking “(g, r,m, n)” to\nbe (g, r, 1, 2). Thus, since the natural homomorphism OutFC(ΠF) →\nOutFC(ΠF\/NF) is injective [cf. [NodNon], Theorem B], by replacing\nΠT \u0002 ΠB by ΠT\/NF \u0002 ΠB, we may assume without loss of generality\nthat (m,n) = (1, 2). In particular — in light of our assumption that\nΣB = Primes [cf. (∗1)] — we are in the situation of Definition 6.3.\nLet α ∈ ZOutFC(ΠF)(ρn\/m(H)). Then it follows immediately from\nLemma 6.10, (ii), that there exists an element β of the image of the\ninjection of (∗2) such that α ◦ β ∈ ZOutFC(ΠF)(ρn\/m(H)) induces the\nidentity automorphism of CuspF(G) [cf. Definition 6.5, (i), (ii)]. In par-\nticular, α ◦ β preserves the ΠF-conjugacy class of a cuspidal subgroup\nΠcFdiag\n⊆ ΠF of ΠF associated to cFdiag ∈ CuspF(G) [cf. Definition 6.5,\n(iii)]. Thus, it follows from Lemma 6.8 that α ◦ β is the identity outo-\nmorphism of ΠF. In particular, we conclude that the injection of (∗2)\nis surjective. This completes the proof of the claim (∗3), hence also of\nassertion (i).\nNext, we verify assertion (ii). First, let us observe that by consider-\ning the action of ZOutPFC(ΠF)(ρn\/m(H)) on the set of fiber subgroups of\nΠF of length 1, we obtain an exact sequence of profinite groups\n1 −→ ZOutFC(ΠF)(ρn\/m(H)) −→ ZOutPFC(ΠF)(ρn\/m(H)) −→ Sn−m .\nNow by considering the action of Sn−m on X logn over X\nlog\nm obtained by\npermuting the first n−m factors of X logn , we obtain a section Sn−m ↪→\nZOutPFC(ΠF)(ρn\/m(H)) of the third arrow in the above exact sequence;\nin particular, the third arrow is surjective. On the other hand, it follows\nfrom [NodNon], Theorem B, that the image of the section Sn−m ↪→\nZOutPFC(ΠF)(ρn\/m(H)) commutes with ZOutFC(ΠF)(ρn\/m(H)). Thus, the\ncomposite of natural homomorphisms\nAut\nX\nlog\nm\n(X\nlog\nm+1)\n∼→ ZOutFC(ΠF)(ρn\/m(H)) ↪→ ZOutFPC(ΠF)(ρn\/m(H))\n[cf. assertion (i)] and the section Sn−m ↪→ ZOutPFC(ΠF)(ρn\/m(H)) de-\ntermine an isomorphism as in the statement of assertion (ii). This com-\npletes the proof of assertion (ii). Assertion (iii) follows immediately\nfrom assertions, (i), (ii). This completes the proof of Theorem 6.12.\nQ.E.D.\nRemark 6.12.1. By considering a suitable specialization isomor-\nphism, one may replace the expression “k an algebraically closed field of\ncharacteristic zero” in the statement of Theorem 6.12 by the expression\n“k an algebraically closed field of characteristic \u0004∈ ΣB”.\n142 Yuichiro Hoshi and Shinichi Mochizuki\nTheorem 6.13 (Centralizers of geometric monodromy\ngroups arising from moduli stacks of pointed curves). Let (g, r)\nbe a pair of nonnegative integers such that 2g−2+ r > 0; Σ a nonempty\nset of prime numbers; k an algebraically closed field of characteristic\nzero. Write ΠMg,r\ndef\n= π1((Mg,r)k) for the étale fundamental group of\nthe moduli stack (Mg,r)k [cf. the discussion entitled “Curves” in §0];\nΠg,r for the maximal pro-Σ quotient of the kernel Ng,r of the natural sur-\njection π1((Cg,r)k) \u0002 π1((Mg,r)k) = ΠMg,r [cf. the discussion entitled\n“Curves” in §0]; ΠCg,r for the quotient of the étale fundamental group\nπ1((Cg,r)k) of (Cg,r)k by the kernel of the natural surjection Ng,r \u0002 Πg,r.\nThus, we have a natural exact sequence of profinite groups\n1 −→ Πg,r −→ ΠCg,r −→ ΠMg,r −→ 1 ,\nwhich determines an outer representation\nρg,r : ΠMg,r −→ Out(Πg,r) .\nThen the following hold:\n(i) The profinite group Πg,r is equipped with a natural structure of\npro-Σ surface group [cf. [MT], Definition 1.2].\n(ii) Let H ⊆ ΠMg,r be an open subgroup of ΠMg,r . Suppose that\n2g − 2 + r > 1, i.e., (g, r) \u0004∈ {(0, 3), (1, 1)}.\nThen the composite of natural homomorphisms\nAut(Mg,r)k((Cg,r)k) −→ AutΠMg,r (ΠCg,r )\/Inn(Πg,r)\n∼−→ ZOut(Πg,r)(Im(ρg,r)) ⊆ ZOut(Πg,r)(ρg,r(H))\n[cf. the discussion entitled “Topological groups” in §0] deter-\nmines an isomorphism\nAut(Mg,r)k((Cg,r)k)\n∼−→ ZOutC(Πg,r)(ρg,r(H))\n[cf. (i); Definition 6.1]. Here, we recall that the automorphism\ngroup Aut(Mg,r)k((Cg,r)k) is isomorphic to⎧⎨\n⎩\nZ\/2Z× Z\/2Z if (g, r) = (0, 4);\nZ\/2Z if (g, r) ∈ {(1, 2), (2, 0)};\n{1} if (g, r) \u0004∈ {(0, 4), (1, 2), (2, 0)} .\nCombinatorial anabelian topics I 143\n(iii) Let H ⊆ OutC(Πg,r) be a closed subgroup of OutC(Πg,r) that\ncontains an open subgroup of Im(ρg,r) ⊆ Out(Πg,r). Suppose\nthat\n2g − 2 + r > 1, i.e., (g, r) \u0004∈ {(0, 3), (1, 1)}.\nThen H is almost slim [cf. the discussion entitled “Topolog-\nical groups” in §0]. If, moreover,\n2g − 2 + r > 2, i.e., (g, r) \u0004∈ {(0, 3), (0, 4), (1, 1), (1, 2), (2, 0)},\nthen H is slim [cf. the discussion entitled “Topological groups”\nin §0].\nProof. Assertion (i) follows immediately from the various defini-\ntions involved. Next, we verify assertion (ii). First, we recall that the\ndescription of the automorphism group Aut(Mg,r)k((Cg,r)k) given in the\nstatement of assertion (ii) is well-known [cf., e.g., [CorHyp], Theorem B,\nif 2g− 2 + r > 2, i.e., (g, r) \u0004∈ {(0, 4), (1, 2), (2, 0)}]. Next, we claim that\n(∗1): the composite homomorphism of assertion (ii)\ndetermines an injection\nAut(Mg,r)k((Cg,r)k) ↪→ ZOutC(Πg,r)(ρg,r(H)) .\nIndeed, one verifies easily that the composite as in assertion (ii) factors\nthrough ZOutC(Πg,r)(ρg,r(H)). Thus, the claim (∗1) follows immediately\nfrom the well-known fact that any nontrivial automorphism of a hyper-\nbolic curve over an algebraically closed field of characteristic \u0004∈ Σ induces\na nontrivial outomorphism of the maximal pro-Σ quotient of the étale\nfundamental group of the hyperbolic curve [cf., e.g., [LocAn], the proof\nof Theorem 14.1]. This completes the proof of the claim (∗1).\nNext, we claim that\n(∗2): if r > 0, then the injection of (∗1) is an isomor-\nphism.\nIndeed, write N ⊆ ΠMg,r for the kernel of the surjection ΠMg,r \u0002\nπ1((Mg,r−1)k) determined by the (1-)morphism (Mg,r)k → (Mg,r−1)k\nobtained by forgetting the last section. Then it follows immediately from\nthe various definitions involved that there exists a commutative diagram\nof profinite groups\n1 −−−−→ Πg,r −−−−→ E −−−−→ N −−−−→ 1\n\u0005\n⏐⏐\b \u0005⏐⏐\b \u0005⏐⏐\b\n1 −−−−→ ΠF −−−−→ ΠT −−−−→ ΠB −−−−→ 1\n144 Yuichiro Hoshi and Shinichi Mochizuki\n— where the upper sequence is the exact sequence obtained by pulling\nback the exact sequence 1 → Πg,r → ΠCg,r → ΠMg,r → 1 by the\nnatural inclusion N ↪→ ΠMg,r ; the lower sequence is the exact sequence\n“1 → ΠF → ΠT → ΠB → 1” obtained by applying the procedure given\nin the statement of Theorem 6.12 in the case where (m,n,ΣF,ΣB) =\n(1, 2,Σ,Primes) to a stable log curve of type (g, r − 1) over (Spec k)log;\nthe vertical arrows are isomorphisms. Thus, it follows immediately from\nTheorem 6.12, (i), that ZOutC(Πg,r)(ρg,r(H ∩ N)) is isomorphic to the\nautomorphism group AutXlog(X\nlog\n2 ) for the stable log curve X\nlog over\n(Spec k)log of type (g, r− 1). In particular, by the claim (∗1), we obtain\nthat\nAut(Mg,r)k((Cg,r)k) ↪→ ZOutC(Πg,r)(ρg,r(H))\n⊆ ZOutC(Πg,r)(ρg,r(H ∩N))\n∼← AutXlog(X log2 ) .\nThus, by comparing (Aut(Mg,r)k((Cg,r)k))\u0005 with AutXlog(X\nlog\n2 )\n\u0005 [cf. The-\norem 6.12, (i)], we conclude that the injection of the claim (∗1) is an\nisomorphism. This completes the proof of the claim (∗2). Moreover, it\nfollows immediately from the proof of the claim (∗2) that\n(∗3): if α ∈ ZOutC(Π0,4)(ρ0,4(H)) induces the identity\nautomorphism on the set of conjugacy classes of cus-\npidal inertia subgroups of Π0,4, then α is the identity\noutomorphism of Π0,4.\nIn light of the claim (∗2), in the remainder of the proof of assertion\n(ii), we assume that\nr = 0, hence also that g ≥ 2.\nFor x ∈ (Mg,0)k(k), write\nGx\nfor the semi-graph of anabelioids of pro-Σ PSC-type associated to the\ngeometric fiber of (Clogg,0)k → (M\nlog\ng,0)k over x\nlog def= x ×(Mg,0)k (M\nlog\ng,0)k;\nthus, we have a natural Im(ρg,0) (⊆ Out(Πg,0))-torsor of outer isomor-\nphisms Πg,0\n∼→ ΠGx . Let us fix an isomorphism Πg,0\n∼→ ΠGx that belongs\nto this collection of isomorphisms. Moreover, for x ∈ (Mg,0)k(k), we\nshall say that x satisfies the condition (†) if\n(†1) Vert(Gx) = {v1, v2}; Node(Gx) = {e1, e2, · · · , eg+1};\n(†2) N (v1) = N (v2) = Node(Gx);\n(†3) v1 and v2 are of type (0, g + 1);\nCombinatorial anabelian topics I 145\nwe shall say that x satisfies the condition (‡) if\n(‡1) Vert(Gx) = {v∗1 , v∗2 , w∗}; Node(Gx) = {e∗1, e∗2, · · · , e∗g+1, f∗};\n(‡2) N (v∗1) = {e∗1, e∗2, · · · , e∗g+1}; N (v∗2) = {e∗1, e∗2, · · · , e∗g−1, f∗};\nN (w∗) = {e∗g, e∗g+1, f∗};\n(‡3) v∗1 is of type (0, g + 1), v∗2 is of type (0, g), and w∗ is of type\n(0, 3).\nLet us observe that one may verify easily that there exists a k-valued\npoint x ∈ (Mg,0)k(k) that satisfies (†); if, moreover, g > 2, then there\nexists a k-valued point x ∈ (Mg,0)k(k) that satisfies (‡).\nLet x ∈ (Mg,0)k(k) be a k-valued point. Then we claim that\n(∗4): if x satisfies (†), and, relative to the isomor-\nphism Πg,0\n∼→ ΠGx fixed above,\nα ∈ ZOutC(Πg,0)(ρg,0(H))\ndetermines an element of Aut|grph|(Gx) (⊆ Out(ΠGx)∼← Out(Πg,0)), then for any e ∈ Node(Gx), the image\nαe of α via the natural inclusion Aut\n|grph|(Gx) ↪→\nAut|grph|((Gx)\u0003{e}) [cf. Proposition 2.9, (ii)] satisfies\nαe ∈ Dehn((Gx)\u0003{e}) .\nIndeed, let e ∈ Node(Gx) and y ∈ (Mg,0)k(k) a k-valued point such that\nGy corresponds to (Gx)\u0003{e} [cf. the special fibers of the stable log curves\nover “Slog” that appear in Proposition 5.6, (iii)]. Write v ∈ Vert(Gy)\nfor the unique vertex of Gy. [Note that it follows from the definition\nof the condition (†) that Vert(Gy)\u0005 = 1.] Then it follows immediately\nfrom the general theory of stable log curves that there exist a “clutching\n(1-)morphism” corresponding to the operation of resolving the nodes of\nGy [i.e., obtained by forming appropriate composites of the clutching\nmorphisms discussed in [Knud], Definition 3.6]\n(M0,2g)k −→ (Mg,0)k\nand a k-valued point ỹ ∈ (M0,2g)k(k) such that the image of ỹ via the\nabove clutching morphism coincides with y, and, moreover, Gỹ is natu-\nrally isomorphic to (Gy)|v. Write (Mlog0,2g)k for the log stack obtained by\nequipping (M0,2g)k with the log structure induced by the log structure\nof (Mlogg,0)k via the above clutching morphism. Then one verifies easily\n146 Yuichiro Hoshi and Shinichi Mochizuki\nthat the composite\nΠM0,2g\ndef\n= π1((Mlog0,2g)k) −→ π1((M\nlog\ng,0)k)\n∼←− ΠMg,0\nρg,0−→ Out(Πg,0)\n— where the first arrow is the outer homomorphism induced by the\nabove clutching morphism, and the second arrow is the outer isomor-\nphism obtained by applying the “log purity theorem” to the natural\n(1-)morphism (Mg,0)k ↪→ (M\nlog\ng,0)k [cf. [ExtFam], Theorem B] — fac-\ntors through Aut|grph|(Gy) ⊆ Out(ΠGy )\n∼← Out(Πg,0). Moreover, the\nresulting homomorphism ΠM0,2g → Aut\n|grph|(Gy) fits into a commuta-\ntive diagram of profinite groups\nΠM0,2g −−−−→ ΠM0,2g⏐⏐\b ⏐⏐\b\nAut|grph|(Gy)\nρVertGy−−−−→ Glu(Gy) = Aut|grph|((Gy)|v)\n[cf. Definition 4.9; Proposition 4.10, (ii)] — where the upper hori-\nzontal arrow is the outer homomorphism induced by the (1-)morphism\n(Mlog0,2g)k → (M0,2g)k obtained by forgetting the log structure. More-\nover, one verifies easily that there exists a natural outer isomorphism\nΠ(Gy)|v\n∼→ Π0,2g such that the homomorphism ΠM0,2g → Out(Π0,2g) ob-\ntained by conjugating the outer action implicit in the right-hand vertical\narrow of the above diagram ΠM0,2g → Aut|grph|((Gy)|v) ⊆ Out(Π(Gy)|v )\nby the outer isomorphism Π(Gy)|v\n∼→ Π0,2g coincides with ρ0,2g. Thus,\nby considering the image in ΠM0,2g of the inverse image of H ⊆ ΠMg,0\nin ΠM0,2g [cf. the diagrams of the above displays], it follows immedi-\nately from the claims (∗2) [in the case where “(g, r)”= (0, 2g)] and (∗3)\n[in the case where g = 2], together with the various definitions involved,\nthat if α ∈ ZOutC(Πg,0)(ρg,0(H)) determines an element of Aut\n|grph|(Gx)\n(⊆ Out(ΠGx)\n∼← Out(Πg,0)), then the image of α via\nAut|grph|(Gx) ↪→ Aut|grph|((Gx)\u0003{e}) ∼→ Aut|grph|(Gy)\nρVertGy\u0002 Glu(Gy) = Aut|grph|((Gy)|v)\n[cf. Proposition 2.9, (ii)] is trivial. In particular, it follows from Propo-\nsition 4.10, (ii), that the image αe of α via Aut\n|grph|(Gx) ↪→\nAut|grph|((Gx)\u0003{e}) satisfies αe ∈ Dehn((Gx)\u0003{e}). This completes the\nproof of the claim (∗4).\nNext, we claim that\nCombinatorial anabelian topics I 147\n(∗5): if x satisfies (†), and α ∈ ZOutC(Πg,0)(ρg,0(H))\ndetermines an element of Aut|grph|(Gx) (⊆ Out(ΠGx)∼← Out(Πg,0)), then α is the identity outomorphism\nof Πg,0.\nIndeed, it follows from the claim (∗4) that\nα ∈\n⋂\ne∈Node(Gx)\nIm\n(\nDehn((Gx)\u0003{e}) → Dehn(Gx)\n)\n[cf. Theorem 4.8, (ii)]. On the other hand, it follows immediately from\nTheorem 4.8, (ii), (iv), that the right-hand intersection is = {1}. This\ncompletes the proof of the claim (∗5).\nNext, we claim that\n(∗6): we have\nZOutC(Πg,0)(ρg,0(H)) ⊆ Aut\n|Node(Gx)|(Gx) (⊆ Out(ΠGx)\n∼← Out(Πg,0)) ;\nif, moreover, x satisfies (‡), then\nZOutC(Πg,0)(ρg,0(H)) ⊆ Aut\n|grph|(Gx) .\nIndeed, it follows immediately from Proposition 5.6, (ii), together with\nthe definition of xlog = x×(Mg,0)k (M\nlog\ng,0)k, that the composite\nπ1(x\nlog) −→ π1((M\nlog\ng,0)k)\n∼←− ΠMg,0\nρg,0−→ Out(Πg,0)\n— where the second arrow is the outer isomorphism obtained by apply-\ning the “log purity theorem” to the natural (1-)morphism (Mg,0)k ↪→\n(Mlogg,0)k [cf. [ExtFam], Theorem B] — determines a surjection π1(xlog)\n\u0002 Dehn(Gx) (⊆ Out(ΠGx)\n∼← Out(Πg,0)) [i.e., which induces an iso-\nmorphism between the respective maximal pro-Σ quotients]. Thus, it\nfollows immediately from the various definitions involved that there ex-\nists an open subgroup M ⊆ Dehn(Gx) such that ZOutC(Πg,0)(ρg,0(H)) ⊆\nZOutC(ΠGx )(M) relative to the identification Out\nC(Πg,0)\n∼→ OutC(ΠGx)\narising from our choice of an isomorphism Πg,0\n∼→ ΠGx . Therefore,\nthe inclusion ZOutC(Πg,0)(ρg,0(H)) ⊆ Aut\n|Node(Gx)|(Gx) follows immedi-\nately from Theorem 5.14, (ii). This completes the proof of the inclusion\nZOutC(Πg,0)(ρg,0(H)) ⊆ Aut\n|Node(Gx)|(Gx). On the other hand, if, more-\nover, x satisfies (‡), then it follows immediately from the definition of\nthe condition (‡) that Aut|grph|(Gx) = Aut|Node(Gx)|(Gx). In particular,\n148 Yuichiro Hoshi and Shinichi Mochizuki\nwe obtain that ZOutC(Πg,0)(ρg,0(H)) ⊆ Aut\n|grph|(Gx). This completes\nthe proof of the claim (∗6).\nNext, we claim that\n(∗7): if x satisfies (†), then for any α ∈\nZOutC(Πg,0)(ρg,0(H)), there exists an element β of the\nimage of the injection of (∗1) such that the outo-\nmorphism α ◦ β of Πg,0 ∼→ ΠGx is ∈ Aut|grph|(Gx)\n(⊆ Out(ΠGx)\n∼← Out(Πg,0)).\nIndeed, suppose that g > 2. Then by the definitions of (†), (‡), one\nmay verify easily that there exist y ∈ (Mg,0)k(k) and f ∈ Node(Gy)\nsuch that y satisfies (‡), and, moreover, Gx corresponds to (Gy)\u0003{f} [cf.\nProposition 5.6, (iv)]. Thus, it follows immediately from the claim (∗6)\nthat ZOutC(Πg,0)(ρg,0(H)) ⊆ Aut\n|grph|(Gy) ↪→ Aut|grph|(Gx) [cf. Proposi-\ntion 2.9, (ii)], i.e., so we may take β to be the identity outomorphism.\nThis completes the proof of the claim (∗7) in the case where g > 2.\nNext, suppose that g = 2. Write Gx for the underlying semi-graph\nof Gx and Aut|Node|(Gx) for the group of automorphisms of Gx which\ninduce the identity automorphism of the set of nodes of Gx. Then one\nmay verify easily from the explicit structure of Gx [cf. the definition of\nthe condition (†)] that Aut|Node|(Gx) is isomorphic to Z\/2Z. Thus, since\nthe automorphism group Aut(M2,0)k((C2,0)k) is isomorphic to Z\/2Z, it\nfollows immediately from the claim (∗6), together with the various defi-\nnitions involved, that — to complete the proof of the claim (∗7) in the\ncase where g = 2 — it suffices to verify that the composite of natural\nhomomorphisms\nAut(M2,0)k((C2,0)k) −→ Aut(Gx) −→ Aut(Gx)\nfactors through Aut|Node|(Gx) ⊆ Aut(Gx) and is injective. Now the\nfact that the composite in question factors through Aut|Node|(Gx) ⊆\nAut(Gx) follows immediately from the claim (∗6), applied to elements\nof the image of the injection of (∗1). On the other hand, the injectivity\nof the composite in question follows immediately from the injectivity\nof the natural homomorphism Aut(M2,0)k((C2,0)k) → Aut(Gx) [cf. the\nproof of the claim (∗1)] and the claim (∗5). This completes the proof\nof the claim (∗7) in the case where g = 2, hence also — in light of the\nabove proof of the claim (∗7) in the case where g > 2 — of the claim\n(∗7). Thus, the surjectivity of the injection of (∗1) follows immediately\nfrom the claims (∗5) and (∗7). This completes the proof of assertion (ii).\nAssertion (iii) follows immediately from assertion (ii). This completes\nthe proof of Theorem 6.13. Q.E.D.\nCombinatorial anabelian topics I 149\nRemark 6.13.1. In the notation of Theorem 6.13, since ΠM0,3 =\n{1}, it is immediate that a similar result to the results stated in Theo-\nrem 6.13, (ii), (iii), does not hold in the case where (g, r) = (0, 3). On\nthe other hand, it is not clear to the authors at the time of writing\nwhether or not a similar result to the results stated in Theorem 6.13,\n(ii), (iii), holds in the case where (g, r) = (1, 1). Nevertheless, we are able\nto obtain a conditional result concerning the centralizer of the geometric\nmonodromy group in the case where (g, r) = (1, 1) [cf. Theorem 6.14,\n(iii), (iv) below].\nTheorem 6.14 (Centralizers of geometric monodromy\ngroups arising from moduli stacks of punctured semi-ellptic\ncurves). In the notation of Theorem 6.13, write (C±1,1)k for the stack-\ntheoretic quotient of (C1,1)k by the natural action of Aut(M1,1)k((C1,1)k)\nover the moduli stack (M1,1)k; Π±1,1 for the maximal pro-Σ quotient\nof the kernel N±1,1\ndef\n= Ker(π1((C±1,1)k) \u0002 π1((M1,1)k) = ΠM1,1) of the\nnatural surjection π1((C±1,1)k) \u0002 π1((M1,1)k) = ΠM1,1 ; ΠC±1,1 for the\nquotient of the étale fundamental group π1((C±1,1)k) of the stack (C±1,1)k\nby the kernel of the natural surjection N±1,1 \u0002 Π±1,1. Thus, we have a\nnatural exact sequence of profinite groups\n1 −→ Π±1,1 −→ ΠC±1,1 −→ ΠM1,1 −→ 1 ,\nwhich determines an outer representation\nρ±1,1 : ΠM1,1 −→ Out(Π±1,1) .\nWrite OutC(Π±1,1) for the group of outomorphisms of Π\n±\n1,1 which induce\nbijections on the set of cuspidal inertia subgroups of Π±1,1. Suppose that\n2 ∈ Σ.\nThen the following hold:\n(i) The profinite group Π±1,1 is slim [cf. the discussion entitled\n“Topological groups” in §0].\n(ii) Let H ⊆ ΠM1,1 be an open subgroup of ΠM1,1 . Then the com-\nposite of natural homomorphisms\nAut(M1,1)k((C±1,1)k) −→ AutΠM1,1 (ΠC±1,1)\/Inn(Π\n±\n1,1)\n∼−→ ZOut(Π±1,1)(Im(ρ\n±\n1,1)) ⊆ ZOut(Π±1,1)(ρ\n±\n1,1(H))\n150 Yuichiro Hoshi and Shinichi Mochizuki\n[cf. (i); the discussion entitled “Topological groups” in §0] de-\ntermines an isomorphism\nAut(M1,1)k((C±1,1)k)\n∼−→ ZOutC(Π±1,1)(ρ\n±\n1,1(H)) .\nHere, we recall that Aut(M1,1)k((C±1,1)k) = {1}.\n(iii) Let H ⊆ ΠM1,1 be an open subgroup of ΠM1,1 . Then the com-\nposite of natural homomorphisms\nAut(M1,1)k((C1,1)k) −→ AutΠM1,1 (ΠC1,1)\/Inn(Π1,1)\n∼−→ ZOut(Π1,1)(Im(ρ1,1)) ⊆ ZOut(Π1,1)(ρ1,1(H))\n[cf. Theorem 6.13, (i); the discussion entitled “Topological\ngroups” in §0] determines an injection\nAut(M1,1)k((C1,1)k) ↪→ ZOutC(Π1,1)(ρ1,1(H)) .\nMoreover, the image of this injection is centrally terminal\nin ZOutC(Π1,1)(ρ1,1(H)) [cf. the discussion entitled “Topologi-\ncal groups” in §0]. Here, we recall that Aut(M1,1)k((C1,1)k) \f\nZ\/2Z.\n(iv) The composite of natural homomorphisms\nAut(M1,1)k((C1,1)k) −→ AutΠM1,1 (ΠC1,1)\/Inn(Π1,1)\n∼−→ ZOut(Π1,1)(Im(ρ1,1))\n[cf. Theorem 6.13, (i); the discussion entitled “Topological\ngroups” in §0] determines an isomorphism\nAut(M1,1)k((C1,1)k)\n∼−→ ZOutC(Π1,1)(Im(ρ1,1)) .\nProof. Assertion (i) follows immediately from a similar argument to\nthe argument used in the proof of [MT], Proposition 1.4. This completes\nthe proof of assertion (i).\nNext, we verify assertion (ii). First, let us recall that the description\nof the automorphism group Aut(M1,1)k((C±1,1)k) given in the statement\nof assertion (ii) is well-known and easily verified. Write E → (M1,1)k\nfor the family of elliptic curves determined by the family of hyperbolic\ncurves (C1,1)k → (M1,1)k of type (1, 1); U → (C1,1)k for the restriction\nof the finite étale covering E → E over (M1,1)k given by multiplica-\ntion by 2 to (C1,1)k ⊆ E . Then one verifies easily that the action of\nCombinatorial anabelian topics I 151\nAut(M1,1)k((C1,1)k) on (C1,1)k lifts naturally to an action [i.e., given by\n“multiplication by ±1”] on U over (M1,1)k. Write P for the stack-\ntheoretic quotient of U by the action of Aut(M1,1)k((C1,1)k) on U ; ΠP\/M\nfor the maximal pro-Σ quotient of the kernel of the natural surjection\nπ1(P) \u0002 π1((M1,1)k);\nρP\/M : ΠM1,1 −→ Out(ΠP\/M)\nfor the natural pro-Σ outer representation arising from the family of\nhyperbolic curves P → (M1,1)k. Thus, since 2 ∈ Σ, one verifies easily\nthat ΠP\/M may be regarded as a normal open subgroup of Π\n±\n1,1. Now\nlet us observe that one verifies easily that\n(∗1): P → (M1,1)k is a family of hyperbolic curves of\ntype (0, 4). If, moreover, we denote by T → (M1,1)k\nthe connected finite étale covering that trivializes the\nfinite étale covering determined by the four cusps\nof P → (M1,1)k, then the classifying (1-)morphism\nT → (M0,4)k of P ×(M1,1)k T → T [which is well-\ndefined up to the natural action of S4 on (M0,4)k] is\ndominant.\nNow we claim that\n(∗2): every element of OutC(Π±1,1) preserves the nor-\nmal open subgroup ΠP\/M ⊆ Π±1,1.\nIndeed, let us observe that one verifies easily that the natural surjections\nΠ±1,1 \u0002 Π±1,1\/Π1,1, Π±1,1\/ΠP\/M determine an isomorphism\n(Π±1,1)\nab ⊗\nẐΣ\nZ\/2Z\n∼−→ (Π±1,1\/Π1,1)× (Π±1,1\/ΠP\/M) .\nMoreover, it follows immediately from the various definitions involved\nthat the natural action of (Π±1,1)\nab ⊗\nẐΣ\nZ\/2Z on the set of conjugacy\nclasses of cuspidal inertia subgroups of the kernel of the natural surjec-\ntion Π±1,1 \u0002 (Π±1,1)ab ⊗ẐΣ Z\/2Z [which is equipped with a natural struc-\nture of pro-Σ surface group of type (1, 4)] factors through (Π±1,1)\nab ⊗\nẐΣ\nZ\/2Z\n∼→ (Π±1,1\/Π1,1)× (Π±1,1\/ΠP\/M)\npr2\u0002 (Π±1,1\/ΠP\/M), and that the re-\nsulting action of (Π±1,1\/ΠP\/M) is faithful. Thus, we conclude that every\nelement of OutC(Π±1,1) preserves the normal open subgroup ΠP\/M ⊆\nΠ±1,1. This completes the proof of the claim (∗2).\nTo verify assertion (ii), take an element α± ∈ ZOutC(Π±1,1)(ρ\n±\n1,1(H)).\nThen it follows from the claim (∗2) that α± naturally determines an ele-\nment αP ∈ Aut(ΠP\/M)\/Inn(Π±1,1). Let us fix a lifting β ∈ OutC(ΠP\/M)\n152 Yuichiro Hoshi and Shinichi Mochizuki\nof αP . Next, let us observe that since Π±1,1\/ΠP\/M is finite, to ver-\nify assertion (ii), by replacing H by an open subgroup of ΠM1,1 con-\ntained in H, we may assume without loss of generality that β com-\nmutes with ρP\/M(H) ⊆ Out(ΠP\/M), i.e., β ∈ ZOutC(ΠP\/M)(ρP\/M(H)).\nThen it follows immediately from Theorem 6.13, (ii), in the case where\n(g, r) = (0, 4), together with (∗1), that β is contained in the image of the\nnatural injection Π±1,1\/ΠP\/M ↪→ Out(ΠP\/M) obtained by conjugation.\nThus, αP , hence also — by the manifest injectivity [cf. assertion (i)]\nof the homomorphism OutC(Π±1,1) → Aut(ΠP\/M)\/Inn(Π±1,1) implicit in\nthe content of the claim (∗2) — α±, is trivial. This completes the proof\nof assertion (ii).\nNext, we verify assertion (iii). First, recall that the description of\nAut(M1,1)k((C1,1)k) given in the statement of assertion (iii) is well-known\nand easily verified. Next, let us observe that the fact that the composite\nin the statement of assertion (iii) determines an injection\nAut(M1,1)k((C1,1)k) ↪→ ZOutC(Π1,1)(ρ1,1(H))\nfollows immediately from a similar argument to the argument used in the\nproof of the claim (∗1) in the proof of Theorem 6.13, (ii), together with\nthe various definitions involved. Next, let us observe that by applying\nthe natural outer isomorphism Π±1,1\n∼→ Π1,1\nout\n\u0002 Aut(M1,1)k((C1,1)k), we\nobtain an exact sequence of profinite groups\n1 −→ Aut(M1,1)k((C1,1)k) −→ ZOut(Π1,1)(Aut(M1,1)k((C1,1)k))\n−→ Out(Π±1,1)\n— where we regard Aut(M1,1)k((C1,1)k) as a closed subgroup of Out(Π1,1)\nby means of the injection “↪→” of the above display. Thus, the central\nterminality asserted in the statement of assertion (iii) follows immedi-\nately, in light of the above exact sequence, from assertion (ii). This\ncompletes the proof of assertion (iii).\nFinally, we verify assertion (iv). It follows immediately from asser-\ntion (iii) that the image of the homomorphism Aut(M1,1)k((C1,1)k) ↪→\nZOutC(Π1,1)(Im(ρ1,1)) determined by the composite in the statement of\nassertion (iv) is centrally terminal. On the other hand, as is well-\nknown, this image of Aut(M1,1)k((C1,1)k) in Out(Π1,1) is contained in\nIm(ρ1,1) ⊆ Out(Π1,1). [Indeed, recall that there exists a natural outer\nisomorphism SL2(Z)\n∧ ∼→ ΠM1,1 , where we write SL2(Z)∧ for the profi-\nnite completion of SL2(Z), such that the image of\n(\n−1 0\n0 −1\n)\n∈ SL2(Z)∧\nCombinatorial anabelian topics I 153\nin Out(Π1,1) coincides with the image of the unique nontrivial element\nof Aut(M1,1)k((C1,1)k) \f Z\/2Z in Out(Π1,1).] Now assertion (iv) follows\nimmediately. This completes the proof of assertion (iv). Q.E.D.\nRemark 6.14.1. The authors hope to be able to address the issue\nof whether or not a similar result to the results stated in Theorem 6.13,\n(ii), (iii), holds for other families of pointed curves [e.g., the universal\ncurves over moduli stacks of hyperelliptic curves or more general Hurwitz\nstacks] in a sequel to the present paper.\nReferences\n[André] Y. André, On a geometric description of Gal(Qp\/Qp) and a p-adic\navatar of ĜT, Duke Math. J. 119 (2003), no. 1, 1-39.\n[Hsh] Y. Hoshi, Absolute anabelian cuspidalizations of configuration spaces of\nproper hyperbolic curves over finite fields, Publ. Res. Inst. Math. Sci. 45\n(2009), no. 3, 661-744.\n[NodNon] Y. Hoshi and S. Mochizuki, On the combinatorial anabelian geometry\nof nodally nondegenerate outer representations, Hiroshima Math. J. 41\n(2011), no. 3, 275-342.\n[Illu] L. Illusie, An overview of the work of K. Fujiwara, K. Kato, and C.\nNakayama on logarithmic étale cohomology, Astérisque 279 (2002), 271-\n322.\n[JR] M. Jarden and J. Ritter, Normal automorphisms of absolute Galois groups\nof p-adic fields. Duke Math. J. 47 (1980), no. 1, 47-56.\n[Knud] F. Knudsen, The projectivity of the moduli space of stable curves. II.\nThe stacks Mg,n. Math. Scand. 52 (1983), no. 2, 161-199.\n[Mln] J. S. Milne, Étale cohomology, Princeton Mathematical Series, 33, Prince-\nton University Press, Princeton, N.J, 1980.\n[CorHyp] S. Mochizuki, Correspondences on Hyperbolic Curves, J. Pure Appl.\nAlgebra 131 (1998), no. 3, 227-244.\n[LocAn] S. Mochizuki, The local pro-p anabelian geometry of curves, Invent.\nMath. 138 (1999), no. 2, 319-423.\n[ExtFam] S. Mochizuki, Extending families of curves over log regular schemes,\nJ. Reine Angew. Math. 511 (1999), 43-71.\n[AbsHyp] S. Mochizuki, The absolute anabelian geometry of hyperbolic curves,\nGalois theory and modular forms, 77-122, Dev. Math, 11, Kluwer Acad.\nPubl, Boston, MA, 2004.\n[GalSct] S. Mochizuki, Galois sections in absolute anabelian geometry, Nagoya\nMath. J. 179 (2005), 17-45.\n154 Yuichiro Hoshi and Shinichi Mochizuki\n[SemiAn] S. Mochizuki, Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci.\n42 (2006), no. 1, 221-322.\n[CmbGC] S. Mochizuki, A combinatorial version of the Grothendieck conjecture,\nTohoku Math J. 59 (2007), 455-479.\n[AbsCsp] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyper-\nbolic curves, J. Math. Kyoto Univ. 47 (2007), no. 3, 451-539.\n[CmbCsp] S. Mochizuki, On the Combinatorial Cuspidalization of Hyperbolic\nCurves, Osaka J. Math. 47 (2010), 651-715.\n[MT] S. Mochizuki and A. Tamagawa, The Algebraic and Anabelian Geometry\nof Configuration Spaces, Hokkaido Math. J. 37 (2008), no. 1, 75-131.\n[NSW] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields,\nSecond edition, Grundlehren der Mathematischen Wissenschaften, 323,\nSpringer-Verlag, Berlin, 2008.\n[NS] N. Nikolov and D. Segal, Finite index subgroups in profinite groups, C. R.\nMath. Acad. Sci. Paris 337 (2003), no. 5, 303-308.\n(Yuichiro Hoshi) Research Institute for Mathematical Sciences, Kyoto University,\nKyoto 606-8502, JAPAN\nE-mail address: yuichiro@kurims.kyoto-u.ac.jp\n(Shinichi Mochizuki) Research Institute for Mathematical Sciences, Kyoto Univer-\nsity, Kyoto 606-8502, JAPAN\nE-mail address : motizuki@kurims.kyoto-u.ac.jp\n"}
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{"id":"-R7HS116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/On%20Semi-positivity%20and%20Filtered%20Frobenius%20Crystals.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":107,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"On%20Semi-positivity%20and%20Filtered%20Frobenius%20Crystals","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Semipos.dvi"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 07:25:09 UTC 2011","dates_in_content_count_i":2,"dates_in_content_dts":["Thu Feb 24 00:00:00 UTC 1994","Wed Jul 20 00:00:00 UTC 1994"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-6307430338007434056,"exact_signature_unique_b":true,"fuzzy_signature_l":7270790819211236086,"fuzzy_signature_unique_b":true,"wordcount_i":5986,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:50:08 UTC 2022","fresh_date_dt":"Thu Sep 23 21:32:37 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":147758,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086056025358336,"text_t":"SEMI-POSITIVITY AND FROBENIUS CRYSTALS\nOn Semi-Positivity and Filtered Frobenius Crystals\nby Shinichi MOCHIZUKI*\n§0. Introduction\nThe purpose of this paper is to prove that certain subquotients of the MF∇-objects\nof Faltings ([1]) are semi-positive. In particular, we obtain an algebraic proof of a result\n(generalizing those of [5], [9]) on the semi-positivity of the higher direct images of certain\nkinds of sheaves for semistable families of algebraic varieties. Our Main Theorem, proven\nin §3, is as follows:\nTheorem 3.4: Let f : (X,E) → (S,D) be a semistable family of varieties of relative\ndimension d, with S a smooth, proper scheme over a field L of characteristic zero. Let\n(A,∇A, F i(A)) be a globally crystalline filtered vector bundle with connection on (X,E).\nThen for any nonnegative integer α, the coherent sheaf of OS-modules\nRαf∗(ωX\/S ⊗OX (A\/F 1(A))∨)\nis a semi-positive vector bundle, as are all of its tensor powers.\n(The term “semistable” (respectively, “globally crystalline”) is defined in §1 (respectively,\n§3).) In particular, this Theorem implies the following:\nCorollary 3.5: Let f : (X,E) → (S,D) be as above. Then for any nonnegative\ninteger α, the coherent sheaf of OS-modules\nf∗((Ω\nlog\nX\/S\n)α)\nis a semi-positive vector bundle, as are all of its tensor powers.\nThus, Fujita and Kawamata proved the above Corollary in the case α = d.\nReceived February 24, 1994. Revised July 20, 1994.\n1991 Mathematics Subject Classification: 14F30\n* Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan\n1\nRoughly speaking, the idea of the proof is as follows. We restrict to a curve, and\nconsider the possibility of quotients of the bundle in question that have negative degree. If\nsuch a quotient did exist, it would mean that we could construct a large number of sections\nof a space of bounded dimension. This concludes the proof.\nSince the proof is substantially simpler and less high-powered in the case when the\nfibres of f , as well as the base S, are one-dimensional and the filtered vector bundle\nis trivial, we present the proof in that special case in an Appendix (which is logically\nindependent of the rest of the paper, except for basic facts and definitions given in §1). We\nbelieve that this special case sheds light on the general case. In essence, what makes the\nproof easier in this case is the fact that we have a very physical and concrete realization\nof 1-motives and Hodge theory (for trivial coefficients), namely, the theory of semiabelian\nschemes and the Picard group. It is the lack of such explicit tools in the higher-dimensional\ncase with nontrivial coefficients that requires one, if one is to carry out an analogous proof,\nto employ more abstract tools.\nIn [7], Illusie has proven similar results using essentially equivalent techniques to our\nown, although from a somewhat different point of view. Namely, Illusie works modulo p,\nand obtains the following result:\nTheorem : (given as Remark 4.18 in [7]) If f : (X,E) → (S,D) is as in Theorem\n3.4, and L is an ample line bundle on S, then for all i ≥ 1, j ≥ 0, we have:\nHi(S,L ⊗Rjf∗(ωX\/L)) = 0.\n(In fact, as was pointed out to the author by the referee, such vanishing results can also be\nproven by the same technique as that used in [2] to prove the Kodaira Vanishing Theorem.)\nWhen the base S is projective, then Kollar (in [10]) shows how to pass from a vanishing\nresult such as the result of Illusie just quoted to a semi-positivity result such as the one\ngiven in Theorem 3.4. On the other hand, if a vector bundle E on S is found to be semi-\npositive, and L is an ample line bundle on S, then F = E ⊗OS L forms an ample vector\nbundle on S. It is not difficult to see that if one applies Kodaira vanishing to the bundle\nO(1) on the projective bundle P → S associated to F , one then gets a vanishing result\nsimilar to the above quoted result of Illusie, but somewhat weaker. Moreover, when S is a\ncurve, it follows immediately from Serre duality that Theorem 3.4 (for the case of trivial\ncoefficients) is equivalent to the vanishing result of Illusie. However, in the case of higher\ndimensional S, especially, when S is not projective, it is not clear to the author what the\nprecise relationship is between Illusie’s result quoted above and Theorem 3.4.\nThe author would like to express his gratitude to Prof. B. Mazur for reading over a\npreliminary draft of this paper and informing him of the above mentioned paper of Illusie;\nand to Profs. J. Harris, D. Abramovich, L. Illusie, and the referee for their comments and\nfor pointing out some errors in early versions of this paper. Finally, we thank Prof. A.\nOgus for suggesting a significant simplification of the original argument.\n2\n§1. Basic Algebraic Lemmas\nLet A be a regular noetherian ring. Let X, S be smooth, proper schemes over A, and\nlet D ⊆ S, E ⊆ X be relative divisors with normal crossings over A. Let f : X → S\nbe a morphism which is smooth over S − D, and such that (schematically) E = f−1(D).\nNote that these assumptions together imply that f is, in fact, smooth in codimension one.\nTo simplify things, we also assume that f has geometrically connected fibres and constant\nrelative dimension. Then, using the terminology of log schemes as given in [8], we make\nthe following\nDefinition 1.1: We say that (f : X → S;E;D) (or just f : X → S, for short) is a\nsemistable family of varieties (or just semistable for short) if, when we give X (respectively,\nS) the log structure arising from the divisor with normal crossings E (respectively, D), the\nmorphism of log schemes induced by f is log-smooth.\nIf f : X → S is semistable, then we denote the relative sheaf of logarithmic differentials by\nΩ\nlog\nX\/S\n. If the relative dimension of f is d, then we denote the dth exterior power of Ω\nlog\nX\/S\nby ωX\/S . Note that ωX\/S is the relative canonical sheaf of X over S in the sense of duality\ntheory.\nProposition 1.2: Suppose that A is an algebraically closed field, that S is one-\ndimensional (i.e., a curve), and that L is an ample line bundle on S. Then for any N ∈ N,\nthere exists a finite, flat morphism π : T → S which has the following properties:\n(1) T is a smooth, proper, connected A-scheme;\n(2) the ramification divisor R ⊆ S of π avoids D\n(3) dimAH\n0(T,LT ) ≥ N .\nProof: This is the standard technique of passing to a ramified cyclic covering, used, for\ninstance, in algebraic proofs of the Kodaira Vanishing Theorem (see [2]). Namely, one\ntakes a very high tensor power of L, say M = L⊗M (where we take M to be very large,\nand invertible in A), such that M is very ample. Then there exists a section σ of M\nover S whose zero divisor is étale over A and avoids D. Then taking the cyclic covering\nof S′ given by extracting an M th root of σ gives us a scheme T as above such that the\ndimension of the space of sections over T of L is one greater than it was before. Repeating\nthis process gives the inequality desired. ©\nLet us go back to assuming that A is arbitrary regular noetherian ring. We now review\nsome basic well-known facts about blow-ups. Suppose that we are given a A-flat subscheme\n3\nF ⊆ S. Let us say that F is flatly positioned (respectively, transversely positioned) with\nrespect to the divisor with normal crossings D if for any étale morphism U → S such that\nD|U splits into a union of smooth components, the schematic intersection of F |U with\nany irreducible component of D|U is flat over A (respectively, and the restriction to any\nconnected component I ⊆ F |U of F |U of the union of irreducible components of D|U that\ndo not contain I is a relative divisor with normal crossings on I).\nLemma 1.3 : Let us suppose that A is a domain, and that we are given a section\nσ : Spec(A) → S whose image is flatly positioned with respect to D. Let π : S̃ → S be\nthe blow-up of S at the image of σ. Then S̃ is smooth over A, and D̃ = π−1(D)red ⊆ S̃ is\na relative divisor with normal crossings over A.\nProof: Indeed, it suffices to take local coordinates x0, x1, . . . , xn for S over A at some\npoint in the image of σ such that the image of σ is defined by the ideal generated by the\nx’s, and D is defined by some product of the x’s. The result follows by direct computation.\n©\nLemma 1.4 : Suppose that A is an algebraically closed field, and that we are\ngiven an A-smooth curve (i.e., an A-smooth, closed subscheme of dimension one) C ⊆ S.\nThen, after applying a series of blow-ups of S at closed points, we can obtain a birational\nmorphism π : S̃ → S such that if C̃ is the strict transform of C in S̃, and D̃ = π−1(D)red,\nthen C̃ is transversely positioned with respect to D̃.\nProof: Clearly we can assume (by using étale localization) that D splits into a union of\nsmooth divisors: D =\n⋃\nDi. Now we claim that after a series of blow-ups, we may assume\nthat each Di either contains C or intersects C transversely. Indeed, if Di intersects C\nat a point with multiplicity r, then a simple calculation reveals that after blowing up at\nthat point, the strict transform of Di intersects the strict transform of C with multiplicity\n≤ r−1, while the exceptional divisor intersects C transversely. Thus, repeating this process\nproves the claim.\nThus, we assume that each Di either contains C or intersects C transversely. For\nsimplicity, by localizing, we can also assume that C fails to be transversely positioned with\nrespect to D at only one point. With these assumptions, the only thing that could keep C\nfrom being transversely positioned is if two or more distinct Di’s intersect C transversely\nat the same point. If this happens, then let us blow up the point. It is then easy to see that\nthe strict transforms of those Di’s do not meet the strict transform of C. This completes\nthe proof. ©\nNow let us go back to assuming that A is a regular noetherian ring. Let us suppose that\nC is a smooth, geometrically connected curve over A which is embedded as a transversely\npositioned closed subscheme C ⊆ S with respect to the divisor D. Let us denote by Slog\nthe log scheme (see [8] for details) obtained by putting on S the log structure associated\n4\nto the divisor with normal crossings D. Let Cvir be the log scheme obtained by placing on\nC the log structure obtained by pulling back the log structure of Slog to C. (Here “vir”\nstands for virtual, since the log structure involves “virtual divisors” that don’t really exist\non C.) Finally, let Cphs be the log scheme obtained by putting on C the log structure\nobtained étale locally on C by restricting to C only the components of D that do not\ncontain C. (Here “phs” stands for “physical” since the log structure arises from actual\n“physical” divisors on C.) Thus, we have natural morphisms of log schemes Cvir → Slog,\nand ν : Cvir → Cphs. Note that ν induces, via pull-back, a locally split morphism of\nvector bundles ν∗ : ΩCphs\/A → ΩCvir\/A on C; let us denote the cokernel vector bundle by\nResCvir\/A. Thus, intuitively speaking, ResCvir\/A corresponds to the residues with respect\nto the components of D that contain C. It is easy to see that ResCvir\/A becomes trivial\nafter pull-back to some finite étale morphism G → C. Indeed, one need merely take a\ncovering G → C on which the étale local system of “smooth irreducible components of D\nthat contain C” splits. On such a G, one then obtains a global (i.e., over all of G) basis\nfor ResCvir\/A by considering the logarithmic differentials of functions defining the various\nsmooth irreducible components. Thus, in the future, often it will be convenient to assume,\nin this situation, that ResCvir\/A already is trivial, and it is always possible to assume this\nby replacing C by the finite étale cover G.\n§2. Faltings’ Extension of Fontaine-Laffaille Theory\nIn this Section, we review the definition of Faltings’ category MF∇[a,b](S◦) of “loga-\nrithmic crystalline filtered vector bundles with Frobenius action,” and study certain func-\ntoriality properties of this category which are not mentioned in [1]. In this Section and\nthroughout the rest of the paper (except the Appendix), we adopt the following conven-\ntions: k is a finite field with q = pn elements, A = W (k) (the ring of Witt vectors with\nresidue field k), and 0 < dimA(S) ≤ dimA(X) ≤ p − 2 (where dimA(S) is the relative\ndimension of S over A).\nLet us choose an affine open U ⊆ S, with U = Spec(R). Let us suppose that we are\ngiven an A-semilinear (i.e., with respect to the natural Frobenius on A) endomorphism\nΦ : R̂ → R̂ of the p-adic completion R̂ of R that lifts the Frobenius on R\/pR and fixes\nthe divisor D ⊆ S. Let a, b ∈ N with a ≤ b and b − a ≤ p − 1. Then by definition ([1],\nII, c)) an object of MF∇[a,b](R◦) consists of the data of a collection of finitely generated\np-torsion R-modules M , F i(M) (for all i ∈ Z), together with an integrable logarithmic\nconnection ∇ : M → M⊗RΩlogR\/A, plus locally split R-linear injections F i+1(M) → F i(M),\nF i(M) → M , and R-linear morphisms φi : F i(M) ⊗\nR̂,Φ R̂ → M that satisfy the following\naxioms:\n(1) the composite of F i+1(M) → F i(M) with F i(M) → M is the morphism\nF i+1(M) → M ;\n(2) the morphism F i(M) → M is an isomorphism if i ≤ a; F i(M) = 0 if\ni > b;\n5\n(3) if we let M̃ be the inductive limit of the diagram:\n. . . → F i+1(M) ← F i+1(M) → F i(M) ← F i(M) → F i−1(M) ← . . .\nwhere the right arrows are the morphisms in the definition and the\nleft arrows are multiplication by p, then we assume that the φi in-\nduce an isomorphism φ : M̃ ⊗\nR̂,Φ R̂ → M ; (thus, here we require that\nφi−1|F i(M) = pφi;)\n(4) the connection ∇ satisfies Griffiths transversality, i.e., ∇(F i(M)) ⊆\nF i−1(M) ⊗R ΩlogR\/A;\n(5) the isomorphism φ : M̃ ⊗R̂,Φ R̂ → M are parallel with respect to the\nconnection ∇ on M , and the connection naturally induced by ∇ on\nM̃ ⊗R̂,Φ R̂ (as in [1], §2).\nAlthough this local definition depends on the choice of our Frobenius lifting Φ, it is possible\nto prove ([1], Theorem 2.3 – which holds since we are assuming that p > 2) that for any two\nchoices of Φ, there is a canonical isomorphism between the two resulting MF∇[a,b](R◦)’s.\nThus, we can glue these local categories together and thus obtain a global category, which\nis denoted MF∇[a,b](S◦).\nOne natural way to obtain objects in MF∇[a,b](S◦) is by taking the higher direct image\nsheaves with respect to f◦cr of logarithmic crystals on X. Indeed, we may rephrase this\nstatement in greater detail as follows. Let E be an object of MF∇[0,a](X◦). Now by\nregarding X and S as being endowed with logarithmic structures arising from the divisors\nE and D, we may consider (following [8], §4) the relative crystalline sites of X and S\n(with their respective log structures) over A. Since E is endowed, in particular, with a\nconnection, it defines a crystal on the relative crystalline site of X◦ over A. Thus, we can\ntake the higher direct image sheaves Ri(f◦cr)∗E with respect to the morphism of topoi f◦cr\nfrom the logarithmic crystalline site of X to that of S. It is then a result of [1], Theorem\n6.2, that these higher direct images Ri(f◦cr)∗E come with a natural filtration and Frobenius\naction in such a way that we may regard Ri(f◦cr)∗E as being an object of MF∇[0,a+i](S◦),\nas long as max(a + d, a + i) ≤ p− 2. A typical example is the case where E is taken to be\nOXn = OX ⊗Zp Z\/pnZ (with the obvious “trivial” filtration, connection, and Frobenius\naction), and a = 0.\nLater, following a suggestion of Ogus (to simplify the original proof) we will make use\nof the following:\nLemma 2.1 : Let (M,F i(M),∇, φi) be an object of MF∇[0,a](S◦), where S is\nsmooth and proper, and a ≤ p − 2. Then every element of H0(S,M\/F 1(M)) lifts to a\nhorizontal element of H0(S,M).\n6\nProof: This follows from the degeneration of the Hodge spectral sequence ([1], Theorem\n4.1). ©\nThis completes our review of the definition of MF∇[a,b](S◦). We now wish to discuss\ncertain functoriality properties of the category MF∇[a,b]. Let us assume that we are given\nan A-smooth closed, one-dimensional subscheme C ⊆ S which is transversely positioned\n(see §1 for a definition) with respect to D. Let F ⊆ C be the divisor (necessarily étale\nover A) obtained by restricting to C the irreducible components of D that do not contain\nC. (In fact, these irreducible components may only exist after étale localization, so one\nmay need to apply étale descent to construct F , but this is no problem.) With the choice\nof divisor F ⊆ C, we may consider the category MF∇[a,b](C◦). We wish to construct a\nrestriction functor\nρ : MF∇[a,b](S◦) → MF∇[a,b](C◦)\nas follows. Clearly, we can use étale descent, and thus assume that S = Spec(R), C =\nSpec(T ) and that D splits as a union of smooth divisors. Also, we use the notation of “vir”\nand “phs” to denote the various log structures on C defined at the end of §1. We start\nwith R-modules M , F i(M), together with various split injections among these modules\nand an integrable logarithmic connection ∇ : M → M ⊗R ΩlogR\/A. These restrict to modules\nMC , F\ni(M)C , along with various split injections and an integrable logarithmic connection\n∇virC : MC → MC ⊗T ΩlogCvir\/A. Of course, ideally, we would like an object with connection\non the log scheme Cphs, not Cvir. To obtain such an object we reason as follows. First, by\n(finite) étale descent, we assume that ResCvir\/A is a trivial vector bundle, and we choose\na basis of generating sections of ResCvir\/A over C each of which is locally obtained as the\nlogarithmic differential of a function defining a smooth irreducible component of D, and we\nlet s1, . . . , sr be the corresponding basis of the dual vector bundle Res\n∨\nCvir\/A\n. By applying\n∇virC to each of the si, we obtain a collection of commuting (because of the integrability\nof ∇) OC-linear endomorphisms Ni : MC → MC , for i = 1, . . . , r. Moreover, since they\nrepresent the monodromy of an object of MF∇[a,b], the Ni are necessarily nilpotent. Indeed,\nto see this, we reason as follows: Let N be one of the Ni’s. Note that there exists (by\ncondition (3) above) a “Verschiebung” morphism V : M → M⊗R̂,Φ R̂ such that φa◦V = pj\nand V ◦ φa = pj , where j def= b − a. Thus, (by condition (5) above) N j = φa ◦ N j ◦ V ,\nso N2j = pj · φa ◦ N2j ◦ V . Thus, by recursion, N2j = 0. This proves that the Ni are\nnilpotent.\nThus, the Ni define a filtration with r indices, which we call the monodromy filtration\non MC , and which we denote by:\nμi1,...,iα,...,ir (MC) = Ker(N\ni1\n1 . . . N\niα\nα . . . N\nir\nr )\nNow let Grμ(MC) denote the graded object associated to the monodromy filtration. It is\nclear that the filtration F i(M)C defines a natural filtration F\ni(Grμ(MC)) on Grμ(MC),\n7\nand that ∇virC induces a connection ∇gr : Grμ(MC) → Grμ(MC) ⊗T ΩlogCphs\/A, as desired.\nIt is easy to see that (Grμ(MC), F\ni(Grμ(MC)),∇gr) satisfy all the necessary axioms for\nan object of MF∇[a,b](C◦), and all that remains to do in defining ρ is the construction of\nthe Frobenius action on Grμ(MC).\nFirst of all, because of our assumption that C is transversely positioned, one sees\neasily that (after possible further étale localization) it is possible to choose a Frobenius\nlifting Φ : R̂ → R̂ that not only fixes D, but also fixes C (and F ). Thus, Φ restricts to a\nFrobenius lifting ΦC : T̂ → T̂ , so that tensoring φ over R with T (via the natural projection\nR → T ) gives an isomorphism φC : M̃C ⊗T,ΦC T ∼= (M̃ ⊗R,Φ R) ⊗R T → MC . Since this\nFrobenius action on MC clearly commutes with the Ni, it follows that the Frobenius action\non MC induces a Frobenius action on Grμ(MC). One checks easily that the Frobenius that\nwe obtain on Grμ(MC) is independent of the choice of Φ (as long as we deal with Φ that\nfix C and D): indeed, the morphism α constructed in the proof of [1], Theorem 2.3, is\neasily seen to commute with restriction to C. Finally, it is easy to check that this Frobenius\naction satisfies all the remaining properties in the definition of the category MF∇: Indeed,\nthis follows from using the fact that MF (i.e., the category of [1], §2, that is similar to\nMF∇, but lacks the datum of a connection) is abelian ([1], Theorem 2.1, Corollary). This\ncompletes the definition of the functor ρ.\nNow we wish to note that functoriality also holds for blow-ups π : S̃ → S at a\nsection σ : Spec(A) → S which is flatly positioned. The procedure is the same as in the\nprevious paragraph: one pulls back the necessary modules, morphisms, and connection in\nthe obvious fashion; the only trick is pulling back the Frobenius action. The key is to note\nthat since σ is flatly positioned, one can choose the Frobenius lifting Φ : R̂ → R̂ in such a\nway that not only D, but also σ is fixed by Φ. It is then easy to see that the Frobenius\naction on R̂ extends to a Frobenius action on the p-adic completion of S̃ in such a way\nas to be functorial with respect to π. It remains to check that π∗ is independent of the\nchoice of Frobenius lifting Φ. But the only thing which is unclear is whether or not the\nFrobenius action depends on Φ, and this follows from restricting to a dense open subset of\nS over which π is an isomorphism. Thus, we see that we get a natural functor\nπ∗ : MF∇[a,b](S◦) → MF∇[a,b]((S̃)◦)\nas desired.\nNow let C → Spec(A) be a proper, smooth curve, with geometrically connected\nfibres, and let F ⊆ C be an étale divisor over A. Let us suppose that we are given a finite\nmorphism ψ : C → S. Then we shall say that ψ is admissibly factorizable if it factors into\nthe composite of the following form:\n(1) we have a morphism ψ1 : S̃ → S, which is the composite of a sequence\nof blow-ups at A-sections that are flatly positioned over A; let D̃ =\nψ−11 (D)red;\n8\n(2) for some C′ → Spec(A) (also a smooth curve with geometrically con-\nnected fibres) and some F ′ ⊆ C′ (an étale divisor over A), we have a\nclosed immersion ψ2 : C\n′ ↪→ S̃ such that C′ is transversely positioned\nwith respect to D̃, and F ′ is the restriction to C′ of the components of\nD̃ that do not contain C′; we also assume that C′ is not contained in\nany of the exceptional divisors of the morphism ψ1;\n(3) we have a morphism ψ3 : C → C′ such that ψ−13 (F ′)red ⊆ F and ψ3 is\nétale outside of F with tame ramification at F ;\nso that ψ = ψ1 ◦ ψ2 ◦ ψ3.\nLet us suppose that ψ is admissibly factorizable. Then by the functoriality consider-\nations reviewed above, it is clear that by composition we obtain a functor\nψ∗ : MF∇[a,b](S◦) → MF∇[a,b](C◦)\n(Indeed, only for morphisms like ψ3 did we not discuss functoriality; but functoriality for\nmorphisms of this type is the most obvious.) It remains to see that ψ∗ is independent of the\nfactorization ψ = ψ1 ◦ ψ2 ◦ ψ3. This is clear for all the data involved except (perhaps) for\nthe Frobenius action and the formation of Grμ(−); but this follows, as usual, by restricting\nto the open dense subset C−F , where the fact that the Frobenius action and the formation\nof Grμ(−) are independent of the factorization is clear. We thus summarize our findings\nin the following\nProposition 2.2: For all admissibly factorizable morphisms ψ : (C,F ) → (S,D),\nwe have a canonical functor\nψ∗ : MF∇[a,b](S◦) → MF∇[a,b](C◦)\nfor all b − a ≤ p − 1.\n§3. The Main Theorem\nBefore stating and proving the Main Theorem of this paper, we need to introduce\nsome new definitions. Unfortunately, these definitions are somewhat ad hoc, but at least\nthey allow us to prove a theorem with a somewhat broader range of applicability than\nprevious statements of such results (as given in [5], [9]).\nLet L be a field of characteristic zero. Let X be a proper, smooth L-scheme; E ⊆ X\na divisor with normal crossings. Fix a ∈ N. Let us suppose that we are given a vector\nbundle A on X together with a filtration F a(A) ⊆ F a−1(A) ⊆ . . . F 1(A) ⊆ F 0(A) = A\n9\nby subbundles such that all the inclusions are locally split, and a connection ∇A with\nlogarithmic poles at E. Then we make the following\nDefinition 3.1: (i) Suppose that L is a number field (i.e., a finite extension of Q).\nLet OL be its ring of integers. Then for some positive rational integer N , the objects X,\nE, A, and ∇A are defined over R def= OL[N−1] and satisfy the same hypotheses as they\ndid over L (i.e. being proper and smooth, being a relative divisor with normal crossings,\nbeing a vector bundle, being a locally split inclusion, being a connection with logarithmic\npoles at E, etc.). Let us fix such models XR, ER, AR, and (∇A)R over R, and denote by\nmeans of a subscript the result of base-changing these objects from R to some R-algebra.\nThen we shall say that (A,∇A, F i(A)) is globally crystalline if for almost all primes ℘ of\nL, when we base-change from R to the completion A℘ of OL (the ring of integers) at ℘, the\nresulting filtered vector bundle with connection arises from an inverse system of objects\n{A℘,n}n∈N (where A℘,n is flat over Z\/pnZ) of MF∇[0,a](X◦A℘).\n(ii) If L is any field (of characteristic zero), then we say that (A,∇A, F i(A)) is globally\ncrystalline if there is a finitely generated Q-subalgebra R ⊆ L over which X, E, A, ∇A,\nand the F i(A) are defined and satisfy the same hypotheses as they did over L (i.e. being\nproper and smooth, being a relative divisor with normal crossings, being a vector bundle,\nbeing a locally split inclusion, being a connection with logarithmic poles at E, etc.) such\nthat at all closed points of Spec(R) (whose residue fields are necessarily number fields),\nthe restricted filtered vector bundle is globally crystalline (as in (i)).\nNote that if (A,∇A, F i(A)) is globally crystalline, then the OX -dual of A, which we\ndenote by A∨, endowed with the dual filtration (shifted over a places) and connection, gives\nrise to a filtered vector bundle with connection (A∨,∇A∨ , F i(A∨)) which is again globally\ncrystalline. Continuing with our field L of characteristic zero, suppose that X is proper\nand geometrically connected, and that we are also given a smooth, proper, geometrically\nconnected L-scheme S and a divisor with normal crossings D ⊆ S, plus a morphism\nf : (X,E) → (S,D) which is semistable in the sense of Definition 1.1.\nLemma 3.2 : We denote by d the relative dimension of X over S. Suppose that\n(A,∇A, F i(A)) is globally crystalline. Then for any i ≥ 0, the de Rham cohomology\nRd−i(f◦DR)∗(A,∇A), together with the Gauss-Manin connection, and the Hodge filtration\n(induced by F i(A)) constitute a globally crystalline object on (S,D). Let (B,∇B, F i(B))\nbe the dual object to this crystalline object on (S,D). Then,\n(B\/F 1(B))∨ = Rif∗(ωX\/S ⊗OX (A\/F 1(A))∨)\nProof: This follows from [1], Theorem 6.2, and Grothendieck-Serre duality applied to\nthe morphism f (for which ωX\/S is the dualizing sheaf). ©\n10\nLemma 3.3 : Assume that L is algebraically closed. Suppose that we have\na vector bundle with connection (A,∇A) on a X with logarithmic poles at E. Then\ndimL H\n0\nDR(X,A) ≤ rank(A), where by the subscript “DR,” we mean the horizontal\nglobal sections.\nProof: By formal integration using the connection, and the fact that in characteristic\nzero, the exterior derivative vanishes only on constant functions, it follows that a horizontal\nglobal section is determined by its value at a single point. ©\nWe now review some basic definitions about semi-positivity from [5]. Let V be a\nprojective variety over a field L. Then a line bundle L on V is semi-positive if for any\nfinite morphism g : C → V , where C is a smooth, projective curve, the degree of g∗L is\nnonnegative. If E is a vector bundle on V , then E is semi-positive if the O(1)-bundle on\nPV (E) is semi-positive. It follows immediately from the definitions that if E is semi-positive\nand L is an ample line bundle, then E ⊗OV L is an ample vector bundle on V .\nWe are now ready to state the Main Theorem of this paper. Various forms of this\ntheorem were proven previously by Fujita ([5]) and Kawamata ([9]). Moreover, as Viehweg\nremarks in [11], §1.21, the techniques of J. Kollar, when taken with the results of Deligne on\nthe degeneration of the Hodge-Deligne spectral sequence, constitute an algebraic proof of\nthis sort of result. Strictly speaking, the result given here does not quite imply Kawamata’s\nresult because our definition of semistability (involving a geometric condition on the divi-\nsors at infinity, as opposed to a monodromy-type condition) is stronger than Kawamata’s.\n(Kollar uses a similar definition to our own.) Finally, as mentioned in the Introduction,\nthere is the paper [7]. The proof given here is a simplification of the author’s original\nproof, following a suggestion of Prof A. Ogus.\nTheorem 3.4: Let f : (X,E) → (S,D) be a semistable family of varieties of relative\ndimension d, with S a smooth, proper scheme over a field L of characteristic zero. Let\n(A,∇A, F i(A)) be a globally crystalline filtered vector bundle with connection on (X,E).\nThen for any nonnegative integer α, the coherent sheaf of OS-modules\nRαf∗(ωX\/S ⊗OX (A\/F 1(A))∨)\nis a semi-positive vector bundle, as are all of its tensor powers.\nProof: First of all, by Lemma 3.2, since taking de Rham cohomology and tensor products\ndoes not take us out of the category MF∇, we may assume that f is the identity, and\nwe may prove the result for the “first tensor power,” i.e., the original globally crystalline\nobject. Also, without loss of generality, let us assume that L is algebraically closed. Let S\nbe the vector bundle (A\/F 1(A))∨. Let r = rank(A).\nLet us assume that S is not semi-positive. Then there exists some smooth, proper,\nconnected curve C over L, together with a finite morphism ψ : C → S and a nonzero\n11\nmorphism SC → Q, where Q is a line bundle of negative degree. By Proposition 1.2, after\nreplacing C by a finite covering of C, we may assume that h0(C,Q−1) ≥ 1 + r. Thus, by\ndualizing the morphism SC → Q, we see that h0(C,S∨C) ≥ 1 + r. On the other hand, by\nLemma 1.4, there exists a birational map π : S̃ → S obtained as a sequence of blow-ups at\nclosed points such that the morphism C → S̃ (obtained from ψ by the valuative criterion of\nproperness) factors through a smooth, proper, connected curve C′ embedded in S̃: that is,\nwe have a finite morphism ψ1 : C → C′ and an embedding ψ2 : C′ ↪→ S̃ whose composite\nis the morphism C → S̃. Moreover, by Lemma 1.4, we can assume that the image of C′ is\ntransversely positioned with respect to D̃ = π−1(D)red. It thus follows that by spreading\nout, specializing, completing, we obtain objects CA, SA, DA, etc. over a ring A satisfying\nthe hypotheses given at the beginning of §2, together with a divisor FA ⊆ CA such that\nψA : (CA, FA) → (SA,DA) is admissibly factorizable (in the sense of the definition given\ndirectly before Proposition 2.2). Thus, for the rest of the proof, we shall work over A, and\nomit the subscripts for all the objects that have been base changed into objects over A.\nThus, in summary, we have an admissibly factorizable morphism ψ : (C,F ) → (S,D),\nsuch that h0(CK ,S∨CK ) ≥ 1 + r, where K, as usual, is the quotient field of A. Now\nrecall that S∨ = A\/F 1(A). Let us consider the inverse system (M,∇M , F i(M), φi) of\nobjects of MF∇[0,a](S◦) associated to (A,∇A, F j(A)) whose existence is guaranteed by the\nfact that (A,∇A, F j(A)) is globally crystalline. (We shall regard this inverse system as\na single object, which is flat over Zp.) Let us apply the pull-back functor of Proposi-\ntion 2.2 to (M,∇M , F i(M), φi) to obtain a Zp-flat object (actually, an inverse system)\n(N,∇N , F i(N), ψi) of MF∇[0,a](C◦). Now since h0(CK ,S∨CK ) ≥ 1 + r, and N\/F 1(N) is a\ndirect sum of subquotients of the monodromy filtration on S∨C = ψ∗(M\/F 1(M)), we thus\nobtain that\nh0(CK , (N\/F\n1(N))K) ≥ 1 + r\nBy Lemma 2.1, this implies that dimK H\n0\nDR(CK , NK) ≥ 1 + r. Since r is the rank of the\nvector bundle M , and hence also of the vector bundle N , we thus conclude from Lemma\n3.3 that we have a contradiction. This completes the proof. ©\nIn particular, let us consider (under the hypotheses of Theorem 3.4) the globally\ncrystalline triple (B, F i(B),∇B) (on (S,D)) given by R2d−i(f◦DR)∗(OX), together with\nits Hodge filtration and the Gauss-Manin connection. Let (A, F i(A),∇A) be the triple\nobtained from (B, F i(B),∇B) by shifting the filtration down by d − i places. Just as in\nLemma 3.2, we see that\n(A\/F 1(A))∨ = (Rdf∗(ΩlogX\/S)d−i)∨ = f∗((ΩlogX\/S)i)\nThus, if we apply Theorem 3.4 to (A, F i(A),∇A) and the identity morphism on S, we see\nobtain the following:\n12\nCorollary 3.5: Let f : (X,E) → (S,D) be as above. Then for any nonnegative\ninteger α, the coherent sheaf of OS-modules\nf∗((Ω\nlog\nX\/S\n)α)\nis a semi-positive vector bundle, as are all of its tensor powers.\nAppendix: The Case of One-Dimensional Fibres\nLet f : X → S, D, and E be as in §1, with f semistable and of relative dimension\none, and suppose that S is a curve. Let us assume that the ring A in §1 is an algebraically\nclosed field k of characteristic zero. Then we have the following\nTheorem : The vector bundle f∗ωX\/S on S is semi-positive in the sense that there\ndo not exist nonzero morphisms φ : f∗ωX\/S → Q, where Q is a line bundle on S of negative\ndegree.\nProof: Let us assume that such a φ exists. By Proposition 1.2, we may assume that\ndimkH\n0(S,Q−1) ≥ g + 1, where g is the genus of the generic fibre of f . Thus, by con-\nsidering φ∨ : Q−1 → R1f∗OX , we see that dimkH0(S,R1f∗OX) ≥ g + 1. By the Leray-\nSerre spectral sequence for f applied to the sheaf OX , we see that we have, in fact,\ndimkH\n1(X,OX) ≥ q + g + 1, where q is the genus of S. On the other hand, since we are\nin characteristic zero, dimkH\n1(X,OX) is exactly the dimension of Pic0(X) over k. Thus,\nwe see that dimk(Pic\n0(X)) ≥ q + g + 1.\nNow we recompute the dimension of Pic0(X) by means of l-adic cohomology to see\nthat this is absurd. Indeed, by applying the Leray-Serre spectral sequence for fet to\nthe étale sheaf Ql(1), we see that dimQlH\n1(Xet,Ql(1)) ≤ 2q + 2g. On the other hand,\nKummer theory tells us that dimQlH\n1(Xet,Ql(1)) is equal to twice the dimension (over\nk) of Pic0(X) (since Pic0(X) is an abelian variety over k). Putting this together with the\ninequality derived in the first paragraph, we obtain a contradiction. ©\nBibliography\n[1] G. Faltings, Crystalline Cohomology and p-adic Galois Representations, Proceedings\nof the First JAMI Conference, Johns Hopkins University Press (1990), 25-79.\n[2] G. Faltings, p-adic Hodge Theory, Journal of the AMS 1 (1988), 255-299.\n13\n[3] J.M. Fontaine and G. Laffaille, Construction de représentations p-adiques, Ann. Sci.\nENS. 15 (1982), 547-608.\n[4] J.M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois\nd’un corps local: construction d’un anneau de Barsotti-Tate, Ann. of Math. 115 (1982),\n529-577.\n[5] T. Fujita, On Kähler Fibre Spaces over Curves, J. Math. Soc. Japan 30 (1978), 779-\n794.\n[6] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-\nVerlag, 1977.\n[7] L. Illusie, Réduction semi-stable et décomposition de complexes de de Rham à coeffi-\ncients, Duke Math. Journal, Vol. 60, No. 1 (1990), 139-185.\n[8] K. Kato, Logarithmic Structures of Fontaine-Illusie, Proceedings of the First JAMI\nConference, Johns Hopkins University Press (1990), 191-224.\n[9] Y. Kawamata, Characterization of Abelian Varieties, Compositio Math. 43 (1981),\n253-276.\n[10] J. Kollar, Higher Direct Image Sheaves I, Ann. of Math. 123 (1986), pp. 11-42.\n[11] E. Viehweg, Weak Positivity and the Stability of Certain Hilbert Points, Invent. Math.\n96 (1989), 639-667.\n14\n"}
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{"id":"gyfGg116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Monomorphisms%20in%20Categories%20of%20Log%20Schemes.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":99,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Monomorphisms%20in%20Categories%20of%20Log%20Schemes","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["mono-logsch.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Sun Nov 23 13:14:14 UTC 2014","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":254848530146327641,"exact_signature_unique_b":true,"fuzzy_signature_l":-1363626959734908990,"fuzzy_signature_unique_b":true,"wordcount_i":28295,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:50:10 UTC 2022","fresh_date_dt":"Mon Mar 16 18:38:07 UTC 2026","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":487083,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086057841491968,"text_t":"Monomorphisms in\nCategories of Log Schemes\nShinichi Mochizuki\nNovember 2014\nAbstract. In the present paper, we study category-theoretic properties ofmonomor-\nphisms in categories of log schemes. This study allows one to give a purely category-\ntheoretic reconstruction of the log scheme that gave rise to the category under con-\nsideration. We also obtain analogous results for categories of schemes of locally finite\ntype over the ring of rational integers that are equipped with “archimedean struc-\ntures”. Such reconstructions were discussed in two previous papers by the author,\nbut these reconstructions contained some errors, which were pointed out to the au-\nthor by C. Nakayama and Y. Hoshi. These errors revolve around certain elementary\ncombinatorial aspects of fan decompositions of two-dimensional rational polyhedral\ncones — i.e., of the sort that occur in the classical theory of toric varieties — and\nmay be repaired by applying the theory developed in the present paper.\nContents:\n§0. Notations and Conventions\n§1. Generalities on Monomorphisms and Minimal Points\n§2. The Scheme Structure of Submonic Log Schemes\n§3. Seamless Partitions of Orientable Log Schemes\n§4. Category-theoretic Representation of Archimedean Structures\nAppendix\nIntroduction\nThe purpose of the present paper is to study, in some detail, various aspects\nof the structure of categories of log schemes that revolve around the behavior\nof monomorphisms in such categories. This study leads naturally to a purely\ncategory-theoretic reconstruction of the log scheme that gave rise to the category\nunder consideration. Our main result is the following [cf. Theorem 3.8, (iii)].\nTheorem A. (Category-theoretic reconstruction of log schemes) For\ni = 1, 2, let X\nlog\ni be a locally noetherian fs log scheme [cf. the discussion\nentitled “Log schemes” in §0]. For i = 1, 2, we shall write Schlog(X logi ) for the\n2010 Mathematics Subject Classification. Primary 14A15; Secondary 14G40.\nKey words and phrases. Log scheme, scheme, category, anabelian geometry, archimedean\nstructure.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\ncategory of noetherian fs log schemes of finite type over X\nlog\ni and morphisms of\nfinite type [cf. the discussion at the beginning of §1 for more details]. Let\nΦ : Schlog(X\nlog\n1 )\n∼→ Schlog(X log2 )\nbe an [arbitrary!] equivalence of categories. Then there exists a unique iso-\nmorphism of log schemes\nX\nlog\n1\n∼→ X log2\nsuch that Φ is isomorphic to the equivalence of categories induced by this isomor-\nphism of log schemes X\nlog\n1\n∼→ X log2 .\nWe also obtain analogous results for categories of locally noetherian fs log\nschemes\n“SCHlog(−)”\n[cf. Theorem 4.6, (iv)], as well as for versions\n“Sch\nlog\n(−)”, “SCHlog(−)”\nof the categories “Schlog(−)”, “SCHlog(−)” for schemes of locally finite type over\nZ that are equipped with “archimedean structures” [cf. Theorem 4.8, (iv)].\nThe theory exposed in the present paper arose as an attempt to correct errors,\npointed out to the author by Chikara Nakayama and Yuichiro Hoshi in June 2013,\nin the theory of [LgSch], §2. These errors concern the category-theoretic properties\nof monomorphisms in categories of log schemes and are discussed in more detail in\nExample 0.3 and Remark 1.4.1 of the present paper.\nAt the level of main results of the paper [LgSch], these errors in the theory of\n[LgSch], §2, do not affect the proof of [LgSch], Theorem A, given in [LgSch], §1,\nbut they do affect the proof — although not the validity! — of [LgSch], Theorem B.\nThis result [LgSch], Theorem B, is given a correct proof in §3 of the present paper\nand corresponds precisely to Theorem A [stated above].\nAt the level of main results of papers of the author subsequent to [LgSch], the\nonly place where the errors in the theory of [LgSch], §2, have an effect is in the\nportion of the proof of the main result of [ArLgSch] [i.e., [ArLgSch], Theorem 5.1]\nthat involves the theory of [ArLgSch], §4. The affected portions of [ArLgSch], §4,\nare discussed in more detail in the introduction to §4 of the present paper. The\nmain result [ArLgSch], Theorem 5.1, of [ArLgSch] is given a correct proof in §4 of\nthe present paper and corresponds precisely to Theorem 4.8, (iv) [quoted above].\nAt the level of individual propositions, lemmas, corollaries, theorems, and ex-\namples [i.e., which do not necessarily qualify as “main results” of the paper under\nconsideration], a detailed discussion of the affected portions of [LgSch] and [ArL-\ngSch] may be found in the Appendix to the present paper.\nOne important invariant of the structure of an fs log scheme is the rank of the\ngroupification of the fiber of the characteristic sheaf associated to the log structure\nat a geometric point of the underlying scheme of the log scheme [cf. Definition\n1.2, (i)]. For instance, when this rank is equal to 0 at all geometric points, the log\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 3\nstructure of the fs log scheme under consideration is trivial. One central theme of\nthe theory of the present paper consists of the phenomenon that\nthe theory of category-theoretic properties of monomorphisms exhibits\nquite substantive qualitative differences, depending upon whether or\nnot it holds that the ranks just referred to are ≤ 1.\nWhen it holds that these rank are ≤ 1, the fs log scheme under consideration will\nbe referred to in the present paper as submonic [cf. Definition 1.2, (i)].\nThus, in some sense, the simplest “borderline case” between submonic and\nnon-submonic fs log schemes is the case of a log scheme whose underlying scheme\nis the spectrum of a field whose absolute Galois group acts trivially on geometric\nfibers of the characteristic sheaf associated to the log structure, and for which the\nrank of the groupification of each such geometric fiber of the characteristic sheaf is\nequal to 2. In this case, the log scheme under consideration will be referred to as\nlog-nodal [cf. Definition 1.2, (i)].\nOne important feature of the category-theoretic properties of monomorphisms\nin categories of log schemes lies in the observation that\nthese category-theoretic properties of monomorphisms take on a partic-\nularly straightforward and intuitive form whenever it holds that the\nvarious fs log schemes under consideration are all submonic.\nThis observation is one of the main themes of the theory discussed in §1 of the\npresent paper. Roughly speaking, the errors pointed out by Nakayama and Hoshi\nin the theory of [LgSch], §2, may be summarized as follows:\nthe author wrote [LgSch], §2, under the misunderstanding that this\n“straightforward” and “intuitive” approach to category-theoretic proper-\nties of monomorphisms holds even if the various fs log schemes under\nconsideration are not necessarily submonic.\nOn the other hand, it turns out [cf. the theory of §2 of the present paper] that the\nvarious complications that occur in the study of the category-theoretic properties\nof monomorphisms of arbitrary non-submonic fs log schemes already appear in the\ncase of log-nodal fs log schemes. Moreover, it turns out that\nthese complications essentially revolve around various combinatorial as-\npects of fan decompositions of two-dimensional rational polyhedral\ncones, i.e., of the sort that occur in the classical theory of toric varieties.\nThese elementary combinatorial aspects are reviewed in §0 of the present paper.\nThe theory developed in the present paper may be summarized as follows.\nIn §1, we introduce basic terminology and discuss various generalities concern-\ning monomorphisms in categories of log schemes. In particular, we discuss [cf.,\nespecially, Lemma 1.5] how the elementary combinatorics of two-dimensional fan\ndecompositions reviewed in §0 may be interpreted in the context of categories of log\nschemes. In §2, we apply these elementary combinatorics of two-dimensional fan\ndecompositions [cf. Proposition 2.3] to show, in effect, that certain connectedness\nproperties of such fan decompositions allow one to give a category-theoretic char-\nacterization of submonic fs log schemes. We then proceed to give, in Theorem\n4 SHINICHI MOCHIZUKI\n2.6, a category-theoretic reconstruction of the scheme structure of a submonic\nfs log scheme. This reconstruction is quite “straightforward” and “intuitive” and\namounts, in essence, to an application of the techniques of [LgSch], §2. In the\nremainder of §2, we show [cf. Corollary 2.12] that the various complications that\narise in the case of arbitrary non-submonic fs log schemes amount, in essence, to\nthe issue of giving a category-theoretic algorithm that allows one\nto distinguish a log-nodal fs log scheme from a nontrivial log étale\nlocalization of such a log-nodal fs log scheme [i.e., of the sort that arises\nfrom a nontrivial two-dimensional fan decomposition].\nSuch a category-theoretic algorithm is furnished, in effect, by the theory of seam-\nless partitions of orientable log schemes developed in §3 [cf. Theorem 3.6]. This\ntheory may be regarded as a translation into category theory of the elementary\nobservation that\na nontrivial two-dimensional fan decomposition may be distinguished from\na trivial two-dimensional fan decomposition by considering the “seamless\npartition” constituted by the various constituent cones of the fan decom-\nposition.\nFinally, in §4, we observe that the theory developed in §1, §2, §3 may be generalized,\nwithout any essential complications, to the case of fs log schemes of locally finite\ntype over Z that are equipped with archimedean structures [cf. Theorems 4.3, 4.8].\nSuch generalizations allow one to avoid the difficulties that arise from applying the\nerroneous portions of [LgSch], §2, in the theory of [ArLgSch], §4, i.e., by, in essence,\nisolating the [easily resolved] submonic aspects of these difficulties from the [more\nsubtle!] non-submonic aspects of these difficulties.\nAcknowledgements:\nThis paper owes its existence to the discovery by Chikara Nakayama and\nYuichiro Hoshi of various errors [cf. Example 0.3; Remark 1.4.1] in the arguments\nof [LgSch], §2. The author wishes to express his gratitude to Nakayama and Hoshi\nfor their careful reading of [LgSch].\nSection 0: Notations and Conventions\nNumbers:\nWe will denote by N the set of natural numbers, by which we mean the set of\nintegers n ≥ 0, and by Z the ring of rational integers. By a slight abuse of notation,\nwe shall also use the notation N, Z to denote the corresponding monoids. We shall\ndenote by Q≥0 the additive monoid of nonnegative rational numbers.\nGeneralities on monoids:\nWe shall refer to a finitely generated, saturated [cf. [Kato2], §1.1] monoid that\nhas no nonzero invertible elements as an fs monoid. Thus, if P is an fs monoid,\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 5\nthen the natural homomorphism of monoids P → P gp from P to its groupification\nP gp is injective, and P gp is a finitely generated free abelian group. We shall refer\nto the rank of P gp as the rank rk(P ) of the fs monoid P .\nA homomorphism of monoids φ : P → Q between monoids P , Q will be\ncalled positive if φ maps every nonzero element of P to a nonzero element of Q.\nA nonzero element a ∈ P of a monoid P will be called a sum-dominator if there\nexists a positive integer n such that n · a may be written as the sum of a finite\ncollection of generators of P . Thus, if φ : P → Q is a nonzero homomorphism\n[i.e., a homomorphism that maps any collection of generators of P to a subset of\nQ that contains at least one nonzero element!] from an arbitrary monoid P to an\nfs monoid Q, and a ∈ P is a sum-dominator, then φ(a) \u0006= 0. We shall say that a\nhomomorphism of monoids φ : P → Q is sum-dominating if it maps every nonzero\nelement of P to a sum-dominator of Q. Thus, a sum-dominating homomorphism is\nnecessarily positive.\nLet P be an fs monoid. Thus, in the terminology of the discussion entitled\n“Monoids” of [FrdI], §0, P is sharp, integral, and saturated. In particular, it makes\nsense to speak of the perfection P pf of P , as well as of the set of primes Prime(P )\nof P — cf. the discussion entitled “Monoids” of [FrdI], §0, for more details.\nRank two fs monoids:\nNow let us suppose that P is an fs monoid of rank two. Then we recall that\nthere exists an isomorphism of monoids\nP pf\n∼→ Q≥0 ⊕ Q≥0\n[cf. [ExtFam], Proposition 1.7]. In particular, one verifies immediately that the\nset of primes Prime(P ) = Prime(P pf) is of cardinality two. Write Prime(P ) =\nPrime(P pf) = {p1, p2}. Thus, for each i = 1, 2, pi may be regarded as a collection\nof elements of P pf, which generates a submonoid P\npf\npi ⊆ P pf. For simplicity, let us\nwrite Pi\ndef\n= P\npf\npi . Then one verifies immediately that the two direct summands of\nthe codomain of the isomorphism of the above display correspond precisely to P1,\nP2, i.e., we have a natural isomorphism\nP1 ⊕ P2 ∼→ P pf\nand noncanonical isomorphisms of abstract monoids P1 ∼= P2 ∼= Q≥0. In particular,\nthese two direct summands are preserved, up to possible permutation, by any auto-\nmorphism of the monoid P pf. Note that [since the monoid Q≥0 has no nontrivial\nautomorphisms of finite order] these observations imply that\nany finite subgroup of Aut(P pf) — or, indeed, of Aut(P ) (↪→ Aut(P pf))\n— is of order ≤ 2.\nNext, let\nφ0 : P → J0 def= N\n6 SHINICHI MOCHIZUKI\nbe a positive homomorphism that induces a surjection on groupifications φ\ngp\n0 :\nP gp \u0002 Jgp0 = Z. Thus, Ker(φ\ngp\n0 )\n∼= Z. Fix a nonzero element a ∈ Ker(φgp0 ) ⊆ P gp.\nFor i = 1, 2, write\n(P ⊆) Ji ⊆ P gp\nfor the saturation [cf. [LgSch], Lemma 2.5, (ii)] of the submonoid of P gp generated\nby P and a if i = 1 (respectively, −a if i = 2) and\nφi : P ↪→ Ji\nfor the natural inclusion. Thus, P gp = J\ngp\ni for i = 1, 2. One verifies immediately\nthat, up to a possible permutation of the indices “1” and “2”, the submonoids J1\nand J2 of P\ngp are independent of the choice of a. Moreover, we observe that it\nfollows immediately from the definition of J1 and J2 that\nif i = 0 (respectively, i = 1, i = 2), then a positive homomorphism φ : P →\nN factors, via φi : P → Ji, through a positive homomorphism Ji → N if\nand only if the homomorphism induced on groupifications φgp : P gp → Z\nsatisfies the condition φgp(a) = 0 (respectively, φgp(a) > 0; φgp(a) < 0).\nIn this situation, we shall refer to J1 and J2 as bisecting monoids of P at φ0.\nBefore proceeding, we observe the following “continuity property” of bisecting\nmonoids:\nSuppose that P ∗ ⊆ P gp is a rank two fs monoid that arises as a submonoid\nof P gp that contains P . For i = 1, 2, suppose that there exists a homo-\nmorphism ψi : P\n∗ → N whose restriction to P factors, via φi : P → Ji,\nthrough a positive homomorphism Ji → N. Then φ0 : P → N extends to\na positive homomorphism ψ0 : P\n∗ → N.\nIndeed, if φ0 does not admit such an extension ψ0, then it follows that there exist\nnonzero elements b ∈ P , c ∈ P ∗ such that a + b + c = 0 for some element a ∈\nKer(φ\ngp\n0 ) ⊆ P gp. Then it follows from the above discussion of bisecting monoids\nthat, for some i ∈ {1, 2}, ψgpi (a) ≥ 0. Since the restriction of ψi to P is a positive\nhomomorphism, we thus conclude that 0 = ψ\ngp\ni (a) + ψ\ngp\ni (b) + ψ\ngp\ni (c) > 0 ∈ N, a\ncontradiction. This completes the proof of this “continuity property”.\nBisecting monoids may be understood more explicitly if one passes to perfec-\ntions. Indeed, by restricting our attention to perfections, one verifies immediately\nthat we may assume without loss of generality that\nP pf = Q≥0 ⊕Q≥0, P1 = Q≥0 ⊕ 0, P2 = 0⊕Q≥0,\nand that φ\npf\n0 : P\npf → Q≥0 is the homomorphism determined by sending (1, 0) and\n(0, 1) to 1. Then one computes easily that, if one takes a\ndef\n= (1,−1), then Jpf1 is\nequal to the perfection of the submonoid of (P pf)gp = Q⊕Q generated by (0, 1) and\n(1,−1), while Jpf2 is equal to the perfection of the submonoid of (P pf)gp = Q ⊕ Q\ngenerated by (1, 0) and (−1, 1). Thus, if φpf maps\n(1, 0) \n→ α; (0, 1) \n→ β\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 7\nfor α, β ∈ Q≥0, then one verifies immediately that φpf : P pf → Q≥0 factors, via\nφ\npf\ni : P\npf → Jpfi , through a positive homomorphism Jpfi → Q≥0\nfor i = 0 (respectively, i = 1; i = 2) ⇐⇒ α = β (respectively, α > β; α < β).\nIn the present paper, we shall often consider certain sequences of submonoids\nsatisfying certain special properties, as in the following examples.\nExample 0.1. Submonoids converging from one side. Let P be an fs\nmonoid of rank two, ∞P ⊆ P gp a bisecting monoid of P at some positive homomor-\nphism ∞φ : P → N. Then there exists an infinite descending sequence\nP ⊆ ∞P ⊆ . . . ⊆ nP ⊆ . . . ⊆ 1P ⊆ 0P\n— where n ∈ N — of submonoids of P gp such that every positive homomorphism\nφ : ∞P → N factors through a positive homomorphism nP → N for some n [which\nmay depend on φ], and, moreover, for each m ∈ N, mP is a bisecting monoid\nof P [hence, in particular, an fs monoid of rank two] whose image ∞φgp(mP ) via\n∞φgp : P gp → Z contains both positive and negative elements. Indeed, by reasoning\nas in the above discussion, one reduces immediately to the verification — say, in the\ncase where P pf = Q≥0⊕Q≥0, ∞φpf is the homomorphism P pf = Q≥0⊕Q≥0 → Q≥0\ngiven by (α, β) \n→ α + β, and ∞P pf is the perfection of the submonoid of Q ⊕ Q\ngenerated by (−1, 1) and (1, 0) — of the existence of an infinite descending sequence\nP pf ⊆ ∞P pf ⊆ . . . ⊆ nP pf ⊆ . . . ⊆ 1P pf ⊆ 0P pf\n—where n ∈ N— of submonoids of (P pf)gp such that every positive homomorphism\nψ : ∞P pf → Q≥0 factors through a positive homomorphism nP pf → Q≥0 for some\nn [which may depend on ψ], and, moreover, for each m ∈ N, mP pf is the perfection\nof a finitely generated submonoid of Q⊕Q such that mP def= mP pf ∩ P gp [so mP pf\nmay be identified with the perfection of mP , as the notation suggests!] is a bisecting\nmonoid of P whose image ∞φgp(mP ) contains both positive and negative elements.\nSuch an infinite descending sequence may be obtained, for instance, by taking nP pf\nto be the perfection of the submonoid of Q ⊕ Q generated by (−1, 1 − 1n+2 ) and\n(1, 0).\nExample 0.2. Submonoids converging from the center. Let P be an fs\nmonoid of rank two. Then there exists an infinite descending sequence\nP ⊆ . . . ⊆ nP ⊆ . . . ⊆ 1P ⊆ 0P\n— where n ∈ N — of submonoids of P gp such that every positive homomorphism\nφ : P → N factors through a positive homomorphism nP → N for some n [which\nmay depend on φ], and, moreover, for each m ∈ N, the inclusion P ↪→ mP is\na sum-dominating homomorphism of fs monoids. Indeed, by reasoning as in the\nabove discussion, one reduces immediately to the verification, in the case where\nP pf = Q≥0 ⊕Q≥0, of the existence of an infinite descending sequence\nP pf ⊆ . . . ⊆ nP pf ⊆ . . . ⊆ 1P pf ⊆ 0P pf\n8 SHINICHI MOCHIZUKI\n— where n ∈ N — of perfections of finitely generated submonoids of (P pf)gp such\nthat every positive homomorphism ψ : P pf → Q≥0 factors through a positive ho-\nmomorphism nP pf → Q≥0 for some n [which may depend on ψ], and, moreover,\nfor each m ∈ N, the inclusion P ↪→ mP def= mP pf ∩ P gp [so mP pf may be identi-\nfied with the perfection of mP , as the notation suggests!] induced by the inclusion\nP pf ↪→ mP pf is a sum-dominating homomorphism of fs monoids. Such an infinite\ndescending sequence may be obtained, for instance, by taking nP pf to be the per-\nfection of the submonoid of Q⊕Q generated by (1,− 1n+2 ) and (− 1n+2 , 1). Finally,\nwe observe that this explicit construction shows that the nP may be chosen so as\nto be preserved by any finite group of automorphisms of P .\nLog schemes:\nIf X is a scheme, then we shall write\nXred ⊆ X\nfor the closed subscheme determined by equipping the underlying topological space\nof the scheme X with the reduced induced scheme structure. If X is the underlying\nscheme of a log scheme X log [cf. [Kato1], §1.2], then we shall write X logred for the log\nscheme determined by restricting the log structure of X log to Xred ⊆ X.\nWe shall use the terms log étale (respectively, log smooth) to refer to morphisms\nbetween log schemes which are “étale” (respectively, “smooth”) in the sense of\n[Kato1], §3.3 (respectively, [Kato1], §3.3; [Kato2], §8.1).\nWe use the term “fs log scheme” to refer to a log scheme which is fine [cf.\n[Kato1], §2.3] and saturated [cf. [the evident étale generalization of] [Kato2], §1.5].\nWe shall refer to a log scheme as noetherian (respectively, locally noetherian) if\nits underlying scheme is noetherian (respectively, locally noetherian). We shall\nsay that a morphism of log schemes is of finite type if its underlying morphism\nof schemes is of finite type. We shall say that a morphism of log schemes is an\nopen immersion if its underlying morphism of schemes is an open immersion, and,\nmoreover, the log structure on its domain is obtained as the pull-back of the log\nstructure on its codomain. We shall say that a morphism of log schemes is dominant\nif its underlying morphism of schemes is dominant.\nWe recall from [LgSch], Lemma 2.6, (i), (ii), (iii), that the natural morphism\nfrom the underlying scheme of any fiber product in the category of locally noetherian\nfs log schemes to the corresponding fiber product of underlying schemes is finite.\nOn the other hand, this natural morphism is not necessarily surjective! That is to\nsay, the isomorphism asserted [unfortunately, without an explicit proof!] in [LgSch],\nLemma 2.6, (ii), is false. Indeed, the following example constitutes a counterexample\nto this isomorphism.\nExample 0.3. Empty fiber products of log schemes. Consider the fiber\nproduct determined by the diagram of log schemes\nX log → Z log ← Y log\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 9\nobtained by equipping the diagram of schemes\nX\ndef\n= Spec(k) → Z def= Spec(k) ← Y def= Spec(k)\n— where k is a field, and the arrows are the identity morphisms — with the log\nstructures determined by the diagram of monoids\nPX\ndef\n= 〈(1, 0); (−1, 1)〉 ⊇ PZ def= N⊕ N ⊆ PY def= 〈(1,−1); (0, 1)〉\n— where the notation “〈−〉” denotes the submonoid of P gpZ = Ngp ⊕ Ngp = Z ⊕ Z\ngenerated by the element(s) in brackets — and the morphisms of monoids PX → k,\nPY → k, PZ → k that map 0 \n→ 1 ∈ k and all nonzero elements of the domain\nto 0 ∈ k. Then one verifies immediately that this fiber product is, in fact, empty,\ndespite the fact that X ×Y Z = Spec(k) \u0006= ∅.\nSection 1: Generalities on Monomorphisms and Minimal Points\nIn the present §1, we discuss various definitions and generalities related to\nmonomorphisms and “minimal points” in categories of log schemes.\nWe suppose that we are in the situation of [LgSch], §2. That is to say, let X log\nbe a locally noetherian fs log scheme [cf. the discussion entitled “Log schemes” in\n§0]. Then we denote by\nSchlog(X log)\nthe category whose objects are morphisms of log schemes of finite type Y log → X log,\nwhere Y log is a noetherian fs log scheme, and whose morphisms [from an object\nY\nlog\n1 → X log to an object Y log2 → X log] are morphisms of finite type Y log1 → Y log2\nlying over X log. To simplify the exposition, we shall often refer to the domain\nY log of an arrow Y log → X log which is an object of Schlog(X log) as an “object of\nSchlog(X log)”.\nRecall the category Sch(X) of [LgSch], §1, i.e., the category whose objects are\nmorphisms of finite type Y → X, where Y is a noetherian scheme, and whose\nmorphisms [from an object Y1 → X to an object Y2 → X] are morphisms of finite\ntype Y1 → Y2 lying over X. Note that by associating to an object Y → X of\nSch(X) the object Y log → X log of Schlog(X log) obtained by equipping Y with the\nlog structure obtained by pulling back the log structure on X log via Y → X, we\nobtain a natural embedding\nSch(X) ↪→ Schlog(X log)\n— which thus allows us to regard Sch(X) as a full subcategory of Schlog(X log).\nLet Y log be an fs log scheme. Then we shall denote its underlying scheme\n(respectively, the morphism of monoids that constitutes its log structure) by Y\n(respectively, expY : MY → OY ). Thus, we have an exact sequence of étale sheaves\nof monoids on Y\n0 → O×Y → MY → PY → 0\n10 SHINICHI MOCHIZUKI\n— where the “characteristic sheaf” PY is defined so as to make the sequence exact.\nIt follows immediately from the fact that Y log is an fs log scheme that the fibers\nof PY (respectively, the groupification P\ngp\nY of PY ) at geometric points of Y are fs\nmonoids [cf. the discussion entitled “Generalities on monoids” in §0] (respectively,\nare finitely generated free abelian groups). In particular, we have natural injections\nPY ↪→ P gpY ; MY ↪→ MgpY\n— where the superscript “gp” denotes the groupification associated to a sheaf of\nmonoids. In the following, we shall use similar notation for objects associated to\narbitrary fs log schemes “(−)log”.\nIn this situation, we shall apply the terminology introduced in [LgSch], §2:\nDefinition 1.1. In the notation of the above discussion:\n(i) If Y is reduced (respectively, one-pointed — cf. [LgSch], Proposition 1.1),\nthen we shall say that Y log is reduced (respectively, one-pointed). If Y log is reduced\nand one-pointed, i.e., Y is equal to the spectrum of a field k, then one may think\nof PY as consisting of a [discrete] monoid equipped with a continuous action of the\nabsolute Galois group Gk of k; when this action is trivial, we shall say that Y\nlog is\nsplit and, by a slight abuse of notation, denote Γ(Y, PY ) by PY .\n(ii) An object Y log → X log of Schlog(X log) will be called minimal if it is\nnon-initial and satisfies the property that any monomorphism Z log \u0003 Y log in\nSchlog(X log), where Z log is non-initial, is necessarily an isomorphism [cf. [LgSch],\nProposition 2.4].\n(iii) Suppose that Y log is a one-pointed object of the category Schlog(X log).\nThen a monomorphism H log \u0003 Y log in Schlog(X log) will be called a hull for Y log\nif every morphism Slog → Y log in Schlog(X log) from a minimal object Slog to\nY log factors [necessarily uniquely!] through the given monomorphism H log \u0003 Y log\n[cf. [LgSch], Proposition 2.7]. A hull H log \u0003 Y log will be called a minimal hull if\nevery monomorphism H\nlog\n1 \u0003 H log in Schlog(X log) for which the composite H\nlog\n1 \u0003\nH log \u0003 Y log is a hull is necessarily an isomorphism [cf. [LgSch], Proposition 2.7]. A\none-pointed object H log of Schlog(X log) will be called a minimal hull if the identity\nmorphism H log → H log is a minimal hull for H log. [The notions of “hull”\/“minimal\nhull” will not be used in the present paper, but are reviewed here for the sake of\ncomparison with the notions of “point-hull”\/“minimal point-hull”, which do play\nan important role in the present paper — cf. Definition 2.9, (iii).]\n(iv) Suppose that f log : Z log → Y log is a morphism of Schlog(X log). Then\n[cf. [LgSch], Definition 2.11, (i), (ii)]: f log will be called log-like if the underlying\nmorphism of schemes f : Z → Y is an isomorphism; f log will be called scheme-\nlike if the log structure on Z log is the pull-back of the log structure on Y log via\nthe underlying morphism of schemes f : Z → Y [i.e., in the terminology of many\nauthors, if f log is strict]. Write\nSchlog(X log)|sch-lk ⊆ Schlog(X log)\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 11\nfor the full subcategory of objects of Schlog(X log) determined by scheme-like mor-\nphisms Y log → X log. Thus, one verifies immediately that the natural embedding\nSch(X) ↪→ Schlog(X log) discussed above admits a natural factorization as the com-\nposite of a natural equivalence of categories\nSch(X)\n∼→ Schlog(X log)|sch-lk\nwith the natural inclusion Schlog(X log)|sch-lk ↪→ Schlog(X log).\nAlso, we introduce some new terminology as follows:\nDefinition 1.2. In the notation of the above discussion:\n(i) Let n ∈ N. Then we shall say that Y log is of rank ≤ n (respectively, of rank\nn) and write\nrk(Y log) ≤ n (respectively, rk(Y log) = n)\nif every fiber of PY at a geometric point of Y is of rank ≤ n (respectively, rank\nn) [cf. the discussion entitled “Generalities on monoids” in §0]. We shall say that\nY log is submonic if it is of rank ≤ 1. If Y log is locally noetherian, then we define\nthe submonic dimension of Y log to be the supremum\ndimsm(Y log)\ndef\n= sup\nZlog\u0002Y log\ndim(Z) ∈ N ∪ {−∞,+∞}\n— where Z log \u0003 Y log ranges over the monomorphisms of Schlog(Y log) such that\nZ log is submonic, and “dim(Z)” denotes the scheme-theoretic dimension of the\nunderlying locally noetherian scheme Z of Z log. Thus, the submonic dimension is\nequal to −∞ if and only if it holds that the underlying scheme of every “Z log” that\nappears in the supremum of the above display is the empty scheme. We shall say\nthat Y log is log-nodal if it is reduced, one-pointed, split, and of rank two.\n(ii) Suppose that Y log arises from an object Y log → X log of Schlog(X log). Then\na minimal point Z log \u0003 Y log of Y log is defined to be a monomorphism Z log \u0003 Y log\nof Schlog(X log) such that Z log is a minimal object of Schlog(X log). Thus, a minimal\npoint of Y log may be thought of as an object of Schlog(Y log). We shall write\nMinPt(Y log)\nfor the set of isomorphism classes [i.e., as objects of Schlog(Y log)] of minimal points\nof Y log.\nProposition 1.3. (Empty and connected underlying schemes) Suppose\nthat Y log is an object of Schlog(X log). Then:\n(i) The underlying scheme Y of Y log is empty if and only if Y log is an initial\nobject in the category Schlog(X log).\n(ii) The underlying scheme Y of Y log is connected if and only if the object\nY log of Schlog(X log) is non-initial and, moreover, does not admit a representation\nas a coproduct of two non-initial objects of Schlog(X log).\n12 SHINICHI MOCHIZUKI\nProof. Assertions (i) and (ii) follow immediately from the definitions. ©\nProposition 1.4. (First properties of monomorphisms) Suppose that\nf log : Z log → Y log is a morphism of Schlog(X log). Thus, the underlying morphism\nf : Z → Y of f log may be regarded as a morphism of Sch(X). Then:\n(i) The property of being a monomorphism in the category of fs log schemes\n(respectively, in the category Schlog(X log)) is stable under base-change in the\ncategory of fs log schemes (respectively, in the category Schlog(X log)).\n(ii) Let M → N be a morphism of finitely generated, saturated monoids\nsuch that the induced morphism Mgp → Ngp is surjective. Then the induced\nmorphism of fs log schemes\nSpec(Z[N ])log → Spec(Z[M ])log\n— where we use the superscript “log” to denote the log structures determined by\nthe tautological charts M ↪→ Z[M ], N ↪→ Z[N ] — is a monomorphism in the\ncategory of fs log schemes.\n(iii) If f log is a monomorphism in Schlog(X log), then the induced morphism\nof sheaves of abelian groups P\ngp\nY |Z → P gpZ is surjective.\n(iv) Suppose that Y log is submonic, and that the morphism P\ngp\nY |Z → P gpZ\ninduced by f log is surjective. Then Z log is submonic, and f log is scheme-like.\n(v) Suppose that f log is scheme-like. Then f log is a monomorphism in\nSchlog(X log) if and only if f is a monomorphism in Sch(X).\n(vi) Suppose that Y log is submonic, and that f log is a monomorphism in\nSchlog(X log). Then the morphism P\ngp\nY |Z → P gpZ induced by f log is surjective; Z log\nis submonic; f log is scheme-like [which, in fact, implies the surjectivity of the\nmorphism P\ngp\nY |Z → P gpZ ]; and f is a monomorphism in Sch(X).\n(vii) Suppose that f is a monomorphism in Sch(X), and that the morphism\nP\ngp\nY |Z → P gpZ induced by f log is surjective. Then f log is a monomorphism in\nSchlog(X log).\nProof. Assertions (i) and (v) follow immediately from the definitions. Next, before\nproceeding, let us recall that, for instance in the case of the log scheme Y log,\n(∗sys) the sheaf of monoids that defines the log structure of Y log may be thought\nof as the restriction to PY ⊆ P gpY of a certain system of line bundles [i.e.,\na system of Gm-torsors] parametrized by the sheaf of abelian groups P\ngp\nY .\nNow assertion (ii) follows immediately from (∗sys). Assertion (iii) follows from the\nargument given in the proof of [LgSch], Proposition 2.3 [but cf. Remark 1.4.1\nbelow!]: That is to say, one reduces immediately to the case where Z and Y are\nequal to Spec(k) for some field k; then, under the assumption that the asserted\nsurjectivity fails to hold, one constructs scheme-like morphisms W log → Z log, where\nW log is an fs log scheme whose underlying scheme is an artinian k-algebra, whose\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 13\nexistence contradicts the assumption that f log is a monomorphism in Schlog(X log).\nAssertion (iv) follows immediately from the simple and well-understood structure\nof the monoid N. Assertion (vi) follows formally from assertions (iii), (iv), and (v).\nFinally, assertion (vii) follows from the definitions, together with the observation\n(∗sys) discussed above. ©\nRemark 1.4.1. Suppose that we are in the situation of Proposition 1.4. Then in\ngeneral,\nit is not necessarily the case that the assumption that f log is a monomor-\nphism in Schlog(X log) implies that f is a monomorphism in Sch(X).\nThat is to say, the corresponding portion of the necessity asserted in [LgSch], Propo-\nsition 2.3, is false as stated. Such an example may be obtained by considering the\nmonomorphism constructed in Proposition 1.4, (ii), in the case where the morphism\nof monoids M → N is taken to be the morphism\nM\ndef\n= N⊕ N → N def= N⊕ N\nthat maps M \u0017 (1, 0) \n→ (1, 1) ∈ N and M \u0017 (0, 1) \n→ (0, 1) ∈ N , i.e., in which\ncase the resulting morphism of schemes is a “blow-up morphism” that has fibers of\ndimension one.\nLemma 1.5. (Well-known generalities concerning fs monoids and\nassociated log schemes) Let k be a field; ksep a separable closure of k;\nGk\ndef\n= Gal(ksep\/k); P an fs monoid [cf. the discussion entitled “Generalities\non monoids” in §0] equipped with a continuous action by Gk [i.e., relative to the\ndiscrete topology on P ]; Mgp an extension, in the category of topological abelian\ngroups equipped with continuous Gk-actions, of P\ngp by (ksep)× [i.e., the multi-\nplicative group of nonzero elements of ksep, equipped with the discrete topology];\nM\ndef\n= Mgp ×P gp P . Write T log for the reduced, one-pointed fs log scheme whose\nunderlying scheme is equal to T = Spec(ksep), and whose log structure is given by\nthe homomorphism of monoids M → ksep that restricts to the natural inclusion\n(ksep)× ↪→ ksep on (ksep)× ⊆ M and maps non-invertible elements of M to 0 ∈ k.\nThus, the associated characteristic sheaf PT is the constant sheaf on T determined\nby P ; the log scheme T log admits a natural Gk-action, which may be regarded as a\ncollection of [pro-]finite étale descent data that gives rise to a reduced, one-pointed\nfs log scheme Slog whose underlying scheme is S = Spec(k). Then:\n(i) Suppose that the action of Gk on P is trivial. Then the extension of\nGk-modules 1 → (ksep)× → Mgp → P gp → 1 splits.\n(ii) Suppose that rk(P ) ≥ 1. Then there exists a positive [cf. the discussion\nentitled “Generalities on monoids” in §0], Gk-equivariant [i.e., with respect to the\ntrivial action of Gk on N] homomorphism φ : P → N that induces a surjection on\ngroupifications φgp : P gp \u0002 Ngp. Now fix such a homomorphism φ : P → N, and\nassume, moreover, that rk(P ) ≥ 2. Then there exists a positive homomorphism\nψ : P → N that induces a surjection on groupifications ψgp : P gp \u0002 Ngp such that\nKer(φgp) \u0006= Ker(ψgp).\n14 SHINICHI MOCHIZUKI\n(iii) Suppose that rk(P ) ≥ 2. Then there exist an fs monoid Q of rank\ntwo and a positive homomorphism ξ : P → Q that induces a surjection on\ngroupifications ξgp : P gp \u0002 Qgp, and, moreover, satisfies the following property:\nLet ζ : Q → R be a positive homomorphism of fs monoids of rank ≥ 1 and\nσ ∈ Gk such that the composite homomorphism ζ ◦ ξ ◦ σ : P → R factors\nas the composite ζσ ◦ ξ of ξ : P → Q with some positive homomorphism\nζσ : Q → R. Then σ stabilizes the subquotient P gp \u0002 Qgp ⊇ Q and\ninduces the identity on Q.\nIn particular, if τ ∈ Gk stabilizes the subquotient P gp \u0002 Qgp ⊇ Q, then τ induces\nthe identity on Q.\n(iv) Let ξ : P → Q be a positive homomorphism of fs monoids that induces\na surjection on groupifications ξgp : P gp \u0002 Qgp. Write Ξsep for the subfunc-\ntor of the contravariant functor determined by the terminal object [i.e., T log] of\nSchlog(T log) that consists of objects Z log → T log of Schlog(T log) such that the com-\nposite homomorphism P gp → Γ(T, P gpT ) → Γ(Z,P gpZ ) induces, via ξ, a homomor-\nphism Q → Γ(Z,PZ); write Ξsep+ ⊆ Ξsep for the subfunctor corresponding to the\ncondition that, for each fiber PZ,z of PZ at a geometric point z of Z, the resulting\nhomomorphism Q → PZ,z is positive. Then Ξsep may be represented by the object\nof Schlog(T log) determined by a log étale monomorphism\nT log[ξ] \u0003 T log\nof Schlog(T log). If, moreover, Q coincides with the saturation of the image of\nξ in Qgp, then the following properties hold: Ξ\nsep\n+ = Ξ\nsep; the closed subscheme\nT [ξ]red ⊆ T [ξ] [cf. the discussion entitled “Log schemes” in §0] of the underlying\nscheme T [ξ] of T log[ξ] is a torus over ksep of dimension rk(P ) − rk(Q); the\ncharacteristic sheaf PT log[ξ] is isomorphic to the constant sheaf on T [ξ] determined\nby Q; if we write M [ξ]\ndef\n= Mgp×P gp Ker(ξgp), then the group of invertible functions\non the torus T [ξ]red may be naturally identified with M [ξ].\n(v) Suppose that we are in the situation of (iv). Write H ⊆ Gk for the open\nsubgroup of elements that stabilize the subquotient P gp \u0002 Qgp ⊇ Q determined\nby ξ; S\nlog\nH for the reduced, one-pointed fs log scheme obtained by descending T\nlog\nvia H ⊆ Gk; Ξ for the subfunctor of the contravariant functor determined by the\nterminal object [i.e., S\nlog\nH ] of Sch\nlog(S\nlog\nH ) that consists of objects Z\nlog → SlogH of\nSchlog(S\nlog\nH ) such that the object Z\nlog ×SlogH T\nlog → T log of Schlog(T log) determined\nby base-changing from S\nlog\nH to T\nlog determines an element of Ξsep(Z log ×SlogH T\nlog);\nΞ+ for the subfunctor of Ξ determined by the subfunctor Ξ\nsep\n+ of Ξ\nsep. Then Ξ may\nbe represented by the object of Schlog(S\nlog\nH ) determined by a log étale monomor-\nphism\nSlog[ξ] \u0003 SlogH\nof Schlog(S\nlog\nH ) which may be obtained, via [pro-]finite étale descent, from the nat-\nural H-action on the monomorphism T log[ξ] \u0003 T log of (iv).\n(vi) Suppose that we are in the situation of (v). Let S\nlog\n+ [ξ] \u0003 Slog[ξ] be some\nmonomorphism of Schlog(S\nlog\nH ) that determines an element of Ξ+(−) ⊆ Ξ(−). [That\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 15\nis say, we do not make any assumption to the effect that S\nlog\n+ [ξ] admits some sort\nof “special functorial interpretation”!] Then if either rk(Q) = 1 or ξ is as in (iii),\nthen the composite\nS\nlog\n+ [ξ] \u0003 Slog[ξ] \u0003 SlogH → Slog\n— where the second arrow is the monomorphism of the final display of (v); the\nthird arrow is the natural morphism S\nlog\nH → Slog — is a monomorphism in\nSchlog(Slog).\n(vii) Suppose that rk(P ) = 2, and that we have been given a positive ho-\nmomorphism φ0 : P → J0 def= N that induces a surjection on groupifications\nφ\ngp\n0 : P\ngp \u0002 Jgp0 = Z. Then, in the notation of the discussion entitled “Rank\ntwo fs monoids” in §0, for i = 0, 1, 2, let us write φi : P → Ji for the associ-\nated positive homomorphism of fs monoids [which is well-defined, up to possible\npermutation of the indices “1” and “2”]. For i = 0, 1, 2, write Φ\nsep\ni for the sub-\nfunctor of the contravariant functor determined by the terminal object [i.e., T log]\nof Schlog(T log) that consists of objects Z log → T log of Schlog(T log) such that, for\neach fiber PZ,z of PZ at a geometric point z of Z, the composite homomorphism\nP gp → Γ(T, P gpT ) → Γ(Z,P gpZ ) → P gpZ,z induces, via φi : P → Ji, a positive\nhomomorphism Ji → PZ,z. If E ⊆ {0, 1, 2} is a subset, then write ΦsepE for the sub-\nfunctor of the contravariant functor determined by the terminal object [i.e., T log]\nof Schlog(T log) that consists of the [disjoint!] union of the Φ\nsep\ni , for i ∈ E. Then,\nfor any E ⊆ {0, 1, 2} such that 0 ∈ E, ΦsepE may be represented by the object of\nSchlog(T log) determined by a log étale monomorphism\nT log[φE ] \u0003 T log\nof Schlog(T log) which satisfies the following properties: T log[φE ] is connected\n[hence nonempty]. If E = {0}, then T log[φE ] \u0003 T log may be identified with\nthe morphism T log[φ0] \u0003 T log of (iv); in particular, in this case, the closed sub-\nscheme T [φE ]red ⊆ T [φE ] of the underlying scheme T [φE ] of T log[φE ] is a one-\ndimensional torus over ksep. Finally, if 0 ∈ E ⊆ E∗ ⊆ {0, 1, 2}, then the\nresulting morphism of log schemes T log[φE ] → T log[φE∗ ] is a dominant open\nimmersion [cf. the discussion entitled “Log schemes” in §0].\n(viii) Suppose that we are in the situation of (vii). Write H ⊆ Gk for the open\nsubgroup of elements that stabilize the subquotient P gp \u0002 Jgp0 ⊇ J0 determined\nby φ0; S\nlog\nH for the reduced, one-pointed fs log scheme obtained by descending T\nlog\nvia H ⊆ Gk. Thus, H acts naturally on Prime(P ), hence also on the set of indices\n{0, 1, 2} [where we regard the index “0” as being stabilized by the action of H]. Let\nE ⊆ {0, 1, 2} be a subset that is stabilized by this natural action of H. Write ΦE\nfor the subfunctor of the contravariant functor determined by the terminal object\n[i.e., S\nlog\nH ] of Sch\nlog(S\nlog\nH ) that consists of objects Z\nlog → SlogH of Schlog(SlogH ) such\nthat the object Z log ×SlogH T\nlog → T log of Schlog(T log) determined by base-changing\nfrom S\nlog\nH to T\nlog determines an element of Φ\nsep\nE (Z\nlog ×SlogH T\nlog). Suppose that\n0 ∈ E. Then ΦE may be represented by the object of Schlog(SlogH ) determined by a\nlog étale monomorphism\nSlog[φE ] \u0003 SlogH\n16 SHINICHI MOCHIZUKI\nof Schlog(S\nlog\nH ) which may be obtained, via [pro-]finite étale descent, from the nat-\nural H-action on the monomorphism T log[φE ] \u0003 T log of (vii).\n(ix) Suppose that we are in the situation of (viii). Suppose further that φ0 :\nP → J0 satisfies the following property:\nLet ζ : P → N be a positive homomorphism of fs monoids; σ ∈ Gk;\ni0, i1 ∈ {0, 1}. Suppose that, for m ∈ {0, 1}, ζ ◦ σm : P → N factors, via\nφim : P → Jim , through a positive homomorphism Jim → N. Then σ\nacts trivially on P .\nThen [one verifies immediately, by taking “ζ” to be φ0 that] H fixes the index “1”.\nMoreover, the composite Slog[φ{0,1}] \u0003 SlogH → Slog of the monomorphism of the\nfinal display of (viii) with the natural morphism S\nlog\nH → Slog is a monomorphism\nin Schlog(Slog).\nProof. Since P gp is a finitely generated free abelian group, assertion (i) follows\nimmediately from the assumption that the action of Gk on P is trivial, together\nwith the well-known fact from elementary Galois theory [i.e., Hilbert’s “Theorem\n90”] that H1(Gk, (k\nsep)×) = 0. Next, we consider assertion (ii). The existence of φ\nfollows immediately from [LgSch], Lemma 2.5, (iii), i.e., by considering the [finite!]\nsum of the Gk-conjugates of a positive homomorphism P → N of the sort discussed\nin [LgSch], Lemma 2.5, (iii); the existence of ψ then follows by applying [LgSch],\nLemma 2.5, (iii), to two distinct elements of P that map, via φ, to the same nonzero\nelement of N. This completes the proof of assertion (ii).\nNext, we consider assertion (iii). First, we observe that the final portion of\nassertion (iii) concerning τ ∈ Gk follows immediately from the property in the\ndisplay of assertion (iii) by taking ζ : Q → R to be the identity automorphism of\nQ. Next, we observe that the homomorphisms φ and ψ of assertion (ii) determine\na positive homomorphism (φ, ψ) : P → N⊕ N whose image I ⊆ N⊕ N generates a\nrank two subgroup Igp of Ngp ⊕Ngp = Z⊕ Z. Thus, for some positive integer n, it\nholds that n · Ngp ⊕ n · Ngp ⊆ Igp. In particular, we have n · N ⊕ n · N ⊆ Q def=\nIgp ∩ (N ⊕ N) ⊆ N ⊕ N; I ⊆ Q; Qgp = Igp [since Igp ⊆ Qgp ⊆ Igp]. One verifies\nimmediately that this implies that this monoid Q ⊆ N⊕N is an fs monoid of rank\ntwo. Write ξ : P → Q for the resulting positive homomorphism of monoids. Note\nthat ξ induces a surjection on groupifications ξgp : P gp \u0002 Qgp (= Igp).\nNow suppose that ζ : Q → R is a positive homomorphism of fs monoids of\nrank ≥ 1 and σ ∈ Gk such that the composite homomorphism ζ ◦ ξ ◦ σ : P → R\nfactors as the composite ζσ ◦ ξ of ξ : P → Q with some positive homomorphism\nζσ : Q → R; in the following, we shall show that σ stabilizes the subquotient\nP gp \u0002 Qgp ⊇ Q and induces the identity on Q. Here, we note that, by applying\nassertion (ii) in the case where we take “P”, “k”, and “Mgp” to be R, ksep, and\nRgp × (ksep)×, respectively, we may assume without loss of generality that R = N.\nAlso, by replacing R by a suitable submonoid of R, we may assume without loss of\ngenerality that ζ, ζσ induce surjections ζ\ngp, ζgpσ : Q\ngp \u0002 Rgp = Ngp = Z. Next, let\nus observe that, by restricting the first projection N ⊕ N \u0002 N to Q ⊆ N ⊕ N, one\nmay regard φ : P → N as the composite η ◦ ξ of ξ : P → Q with a homomorphism\nof monoids η : Q → N. Since η vanishes on 0 ⊕ n · N ⊆ Q, it follows that η is not\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 17\npositive, and hence that Ker(ηgp) \u0006= Ker(ζgp), Ker(ηgp) \u0006= Ker(ζgpσ ). Since ξgp is\nsurjective, we thus conclude that, if we write θ\ndef\n= ζ◦ξ, θσ def= ζσ◦ξ, then Ker(φgp) \u0006=\nKer(θgp), Ker(φgp) \u0006= Ker(θgpσ ), and hence that both Ker(φgp) ∩ Ker(θgp) ⊆ P gp\nand Ker(φgp) ∩ Ker(θgpσ ) ⊆ P gp are submodules of rank rk(P gp)− 2 that contain\nKer(ξgp). Since Ker(ξgp) is also a submodule of P gp of rank rk(P gp) − 2, we thus\nconclude [since P gp\/Ker(ξgp)\n∼→ Qgp is torsion-free] that Ker(φgp) ∩ Ker(θgp) =\nKer(φgp) ∩ Ker(θgpσ ) = Ker(ξgp). But, since φ is Gk-equivariant, this implies that\nKer(ξgp) is stabilized by σ, i.e., that σ induces an automorphism of the quotient\nξgp : P gp \u0002 Qgp, as well as of the quotient ηgp : Qgp \u0002 Ngp = Z, and maps the\nquotient ζgp : Qgp \u0002 Ngp = Z to the quotient ζgpσ : Qgp \u0002 Ngp = Z.\nNow to complete the proof of assertion (iii), it suffices to verify that σ induces\nthe identity on Qgp. Thus, we suppose that σ does not induce the identity on\nQgp. Then since σ clearly stabilizes the fs monoid of rank two obtained by forming\nthe saturation of the image of ξ : P → Q in Q [cf. [LgSch], Lemma 2.5, (ii)],\nit follows [cf. the discussion entitled “Rank two fs monoids” in §0] that σ acts\non Qgp as an automorphism of order 2, and hence that σ permutes the quotients\ndetermined by ζgp, ζgpσ . In particular, σ stabilizes the kernel of the homomorphism\non groupifications ζ\ngp\n+ : Q\ngp → Z determined by the positive homomorphism ζ+ :\nQ → N obtained by forming the sum of ζ, ζσ. Since σ acts nontrivially on Qgp,\nthis implies that Ker(ζ\ngp\n+ ) = Ker(η\ngp). Thus, the positivity of ζ+ contradicts the\nnon-positivity of η. This completes the proof of assertion (iii).\nNext, we observe that assertions (iv), (v), (vii), and (viii) are immediate con-\nsequences of the well-known correspondence between the theory of log schemes and\nthe classical theory of toric varieties. Next, we consider assertion (vi). First of all,\ngiven an object Z log of Schlog(Slog) and two Slog-morphisms α : Z log → Slog+ [ξ],\nβ : Z log → Slog+ [ξ], to verify that α = β, it suffices to verify that α and β coincide\nafter base-change from k to ksep. Moreover, since the morphism SH → S is finite\nétale, and the morphism S\nlog\n+ [ξ] \u0003 SlogH is already known to be a monomorphism,\none verifies immediately that we may assume without loss of generality that Z log\nis reduced and one-pointed — an assumption which reduces the assertion under\nconsideration to an assertion concerning fs monoids, i.e., the assertion that if, for\nsome σ ∈ Gk, there exist positive homomorphisms of fs monoids ζ : Q → R and\nζσ : Q → R such that ζ ◦ ξ ◦ σ = ζσ ◦ ξ : P → R, then σ stabilizes the subquotient\nP gp \u0002 Qgp ⊇ Q determined by ξ. But this assertion concerning fs monoids follows\nimmediately, i.e., if one assumes either that rk(Q) = 1 or that ξ satisfies the prop-\nerties stated in (iii). This completes the proof of assertion (vi). Finally, we observe\nthat assertion (ix) may be verified by a similar argument to the argument applied\nin the proof of assertion (vi). ©\nProposition 1.6. (Minimal objects) Suppose that Y log is an object of\nSchlog(X log). Then:\n(i) Suppose that Y log is a nonempty object of Schlog(X log). Then there exists\na minimal point Z log → Y log such that Z log is submonic. Now fix such a\nminimal point Z log → Y log, and assume, moreover, that Y log is not submonic.\nThen there exists a minimal point W log → Y log, where W log is submonic, that\nis not isomorphic to Z log → Y log.\n18 SHINICHI MOCHIZUKI\n(ii) Y log is a minimal object of Schlog(X log) if and only if Y log is reduced,\none-pointed, and submonic. Put another way, Y log is a minimal object of\nSchlog(X log) if and only if, for some field k, Y log is either equal to Spec(k) equipped\nwith the trivial log structure or equal to Spec(k) equipped with the log structure\nN \u0017 1 \n→ 0 ∈ k.\n(iii) Suppose that Y log and Z log are minimal objects of Schlog(X log). If f log :\nZ log → Y log is a morphism in Schlog(X log), then let us write\nMinLg(f log) ∈ N ∪ {+∞}\nfor the “minimal length” of f log: that is to say, we set MinLg(f log)\ndef\n= 0 if f log\nis an isomorphism; if f log is not an isomorphism, then we take MinLg(f log) to be\nthe supremum of the set of positive integers n such that f log admits a factorization\nZ logn\ndef\n= Z log → Z logn−1 → . . . → Z log1 → Z log0 def= Y log\nas a composite of morphisms of Schlog(X log) which are not isomorphisms such\nthat, for each i = 1, . . . , n, Z\nlog\ni is a minimal object of Sch\nlog(X log). Then Y log\nis of rank one if and only if MinLg(f log) is finite for every morphism f log in\nSchlog(X log) with codomain equal to Y log [and domain given by some minimal ob-\nject].\nProof. First, we consider assertion (i). Observe that we may assume without loss\nof generality that Y log is reduced and one-pointed [cf. [LgSch], Proposition 1.1,\n(i)], and hence that the underlying scheme Y of Y log may be written in the form\nSpec(kY ), for a suitable field kY . Next, let us consider the situation discussed in\nLemma 1.5, (v), in the case where\n· one takes the data that gives rise to “Slog” to be the data that arises\nfrom Y log [so “k” corresponds to kY ];\n· if rk(Y log) = 0, then one takes the positive homomorphism “ξ” to be the\nidentity morphism;\n· if rk(Y log) ≥ 1, then one takes the positive homomorphism “ξ” to be the\nhomomorphism “φ : P → N” of Lemma 1.5, (ii).\nThen one verifies immediately from the description of the torus “T [ξ]red” in Lemma\n1.5, (iv), that any splitting as in Lemma 1.5, (i), over a suitable finite separable\nextension of k — which, in the terminology of [ExtFam], Definition 1.3, may be\nregarded as a “Galois-equivariant clean chart” — determines a closed point of S[ξ].\nIn particular, by restricting the log structure of the submonic log scheme Slog[ξ] to\nthis closed point, we obtain, by Proposition 1.4, (vii); Lemma 1.5, (vi), a monomor-\nphism f log : Z log → Y log in Schlog(X log), for some submonic Z log. Since, by [LgSch],\nProposition 2.4, (ii), (iii), Z log is necessarily minimal, we thus conclude that the\nmorphism f log determines a minimal point of Y log, as desired. In a similar vein, if\nY log is not submonic [i.e., is of rank n ≥ 2], then we consider the situation discussed\nin Lemma 1.5, (v), in the case where one takes the data that gives rise to “Slog”\nto be the data that arises from Y log [so “k” corresponds to kY ], and one takes the\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 19\npositive homomorphism “ξ” to be the homomorphism “ψ : P → N” of Lemma 1.5,\n(ii). Then a splitting as in Lemma 1.5, (i), over a suitable finite separable extension\nof k determines a closed point of S[ξ] whose residue field kW is a finite separable\nextension field of kY . Now, by restricting the log structure of the submonic log\nscheme Slog[ξ] to this closed point, we obtain, by Proposition 1.4, (vii); Lemma 1.5,\n(vi), a monomorphism W log → Y log in Schlog(X log), for some submonic W log whose\nunderlying scheme W is equal to Spec(kW ), which determines, by [LgSch], Proposi-\ntion 2.4, (iii), a minimal point of Y log that is not isomorphic to f log : Z log → Y log.\nThis completes the proof of assertion (i).\nNext, we consider assertion (ii). The sufficiency portion of assertion (ii) follows\nimmediately from [LgSch], Proposition 2.4, (ii), (iii). Thus, to complete the proof\nof assertion (ii), it suffices to verify the necessity portion of assertion (ii). To this\nend, suppose that Y log is minimal. Then it follows from [LgSch], Proposition 2.4,\n(i), that Y log is reduced and one-pointed, i.e., that Y = Spec(kY ), for some field kY ,\nand hence, from assertion (i), that there exists a minimal point f log : Z log → Y log\nin Schlog(X log), for some submonic Z log. If Y log is not submonic, then it follows\nthat f log is not an isomorphism, i.e., in contradiction to the assumed minimality of\nY log. This completes the proof of assertion (ii).\nFinally, we consider assertion (iii). First, let us observe that it follows from\nassertion (ii) that the underlying scheme Y (respectively, Z) of Y log (respectively,\nZ log) may be written in the form Spec(kY ) (respectively, Spec(kZ)), for a suit-\nable field kY (respectively, kZ). Then if Y\nlog is of rank one, then the finiteness\nof MinLg(f log) follows immediately by considering the finiteness of the extension\ndegree [kZ : kY ], together with the simple, well-understood structure of the monoid\nN. On the other hand, if Y log is of rank zero, but Z log is of rank one, then the fact\nthat MinLg(f log) = +∞ follows by considering the infinite descending sequence of\nsubmonoids N ⊇ 2 ·N ⊇ . . . ⊇ 2n ·N ⊇ . . . , for 1 ≤ n ∈ N. This completes the proof\nof assertion (iii). ©\nProposition 1.7. (Monomorphisms from log-nodal objects into non-\nsubmonic objects) Suppose that Y log is a non-submonic object of Schlog(X log).\nThen there exists a log-nodal object Z log of Schlog(X log) that admits a monomor-\nphism Z log \u0003 Y log.\nProof. As in the proof of Proposition 1.6, (i), one verifies immediately that we\nmay assume without loss of generality that Y log is reduced and one-pointed, i.e.,\nthat Y = Spec(kY ), for some field kY . Now we consider the situation discussed in\nLemma 1.5, (v), in the case where one takes the data that gives rise to “Slog” to\nbe the data that arises from Y log [so “k” corresponds to kY ], and one takes the\npositive homomorphism “ξ” to be the homomorphism “ξ : P → Q” of Lemma 1.5,\n(iii). Then one verifies immediately that any splitting as in Lemma 1.5, (i), over\na suitable finite separable extension of k determines a closed point of S[ξ] whose\nresidue field kZ is a finite separable extension field of kY such that the log scheme\nZ log obtained by restricting the log structure of the log scheme Slog[ξ] to this closed\npoint determines an element of Ξ+(−) ⊆ Ξ(−). Thus, we obtain, by Proposition\n1.4, (vii); Lemma 1.5, (vi), a monomorphism Z log → Y log in Schlog(X log), for some\nreduced, one-pointed, split [cf. the final portion of Lemma 1.5, (iii)] Z log of rank two\n20 SHINICHI MOCHIZUKI\n[cf. Lemma 1.5, (iii)] whose underlying scheme Z is equal to Spec(kZ), as desired.\n©\nProposition 1.8. (Submonic one-pointed log schemes) Suppose that Y log\nis an object of Schlog(X log). Then Y log is submonic and one-pointed if and only\nif MinPt(Y log) is of cardinality one.\nProof. First, we verify necessity. Suppose that Y log is submonic and one-pointed.\nThen it follows that Y\nlog\nred [cf. the discussion entitled “Log schemes” in §0] is reduced,\none-pointed, and submonic, hence, by Proposition 1.6, (ii), that Y\nlog\nred is minimal.\nSince any morphism from a [necessarily reduced, by Proposition 1.6, (ii)!] mini-\nmal object of Schlog(X log) to Y log clearly factors uniquely through Y\nlog\nred , we thus\nconclude that MinPt(Y log) is of cardinality one, and that the unique element of\nMinPt(Y log) arises from the natural inclusion Y\nlog\nred ↪→ Y log. This completes the\nproof of necessity. Next, we verify sufficiency. Suppose that MinPt(Y log) is of\ncardinality one. Then by applying the initial portion of Proposition 1.6, (i), to the\nobjects “Z log” of Schlog(X log) obtained by considering scheme-like monomorphisms\nZ log \u0003 Y log that arise from monomorphisms Z \u0003 Y in Sch(X) for reduced, one-\npointed Z [cf. Proposition 1.4, (vii); [LgSch], Proposition 1.1, (iii)], we conclude\nthat Y log is one-pointed. Thus, by applying the final portion of Proposition 1.6, (i),\nto Y log, we conclude that Y log is submonic. This completes the proof of sufficiency.\n©\nBefore proceeding, we review a well-known consequence of the general theory\nof fs log schemes.\nLemma 1.9. (Specialization morphisms associated to characteristic\nsheaves) Suppose that the underlying scheme Y of Y log is the spectrum of a strict\nhenselian domain A. Write s for the tautological geometric point of Y associated\nto the unique closed point of Y . Let η be a geometric point of Y whose image in\nY is the unique generic point of Y . In the following, we shall use subscripted “s’s”\nand “η’s” to denote the respective fibers at s, η of sheaves on the étale site of Y .\nThen the natural “specialization morphism”\nPY,s → PY,η\nis surjective. In particular, this specialization morphism is an isomorphism if\nand only if rk(PY,s) = rk(PY,η). Finally, if rk(PY,η) ≥ 1, and a ∈ PY,s is a sum-\ndominator [cf. the discussion entitled “Generalities on monoids” in §0] such that,\nfor elements a∗ ∈ MY,s and f ∈ A, it holds that a∗ \n→ a, a∗ \n→ f , then f = 0.\nProof. The asserted surjectivity follows immediately from the existence, étale\nlocally, of charts that give rise to the log structure of Y log. If rk(PY,s) = rk(PY,η),\nthen we thus obtain a surjection P\ngp\nY,s \u0002 P\ngp\nY,η between free abelian groups of the\nsame rank; since such a surjection is necessarily an isomorphism, we thus conclude\nfrom the inclusion PY ↪→ P gpY , that the specialization morphism PY,s → PY,η is an\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 21\nisomorphism, as desired. Finally, we observe that if rk(PY,η) ≥ 1, and MY,s \u0017 a∗ \n→\nf ∈ A, where a∗ lifts a sum-dominator a ∈ PY,s, then, in light of the surjectivity of\nthe specialization morphism PY,s → PY,η, it follows immediately from the discussion\nof sum-dominators in §0 that a maps to a nonzero element b ∈ PY,η. On the other\nhand, if we write K for the quotient field of A, then it follows immediately from\nthe definition of the notion of a log structure that the image f ∈ A ⊆ K of any\nlifting b∗ ∈ MY,η of the element b ∈ PY,η in K is noninvertible, hence 0, as desired.\nThis completes the proof of Lemma 1.9. ©\nProposition 1.10. (Lower bounds on the submonic dimension) Suppose\nthat Y log is an object of Schlog(X log), and that Z log \u0003 Y log is a monomorphism of\nSchlog(X log) such that, for suitable n, d ∈ N, the log scheme Z log is of rank n, and\nthe underlying scheme Z of Z log is of dimension d. Then if n ≥ 1 (respectively,\nn = 0), then the submonic dimensions dimsm(Y log), dimsm(Z log) of Y log, Z log\nsatisfy the inequality\ndimsm(Y log) ≥ dimsm(Z log) = d+ n− 1\n(respectively, dimsm(Y log) ≥ dimsm(Z log) = d).\nProof. First, let us observe that it follows immediately from the definition of\nsubmonic dimension [cf. Definition 1.2, (i)] that dimsm(Y log) ≥ dimsm(Z log). In\nparticular, we may assume without loss of generality that Z log = Y log. Thus, it\nfollows immediately from Lemma 1.9 that the characteristic sheaf PY is locally\nconstant. Next, by replacing Y log by the log scheme determined by a suitable sub-\nscheme of Y , one verifies immediately we may assume without loss of generality\nthat the scheme Y is integral. Now the case where n = 0 is immediate [cf. Propo-\nsition 1.4, (vi)], so we may assume without loss of generality that n ≥ 1. Thus, we\nmay apply the theory reviewed in Lemma 1.5 to the generic point of Y . Moreover,\none verifies immediately from the fact that PY is locally constant that the objects\n[and properties of these objects] discussed in this theory extend to objects [and\nproperties of these objects] over the entire scheme Y [i.e., not just the generic point\nof Y ]. In particular, by applying Lemma 1.5, (iv), (v), (vi), where we take the fs\nmonoid “Q” to be N, we conclude that given any monomorphism W log \u0003 Y log,\nwhere W log is a submonic object of Schlog(X log) whose underlying scheme W is\nintegral, there exists a monomorphism V log \u0003 Y log, where V log is a submonic ob-\nject of Schlog(X log) whose underlying scheme V is a family of (n− 1)-dimensional\ntori [cf. Lemma 1.5, (iv)] over Y , such that the monomorphism W log \u0003 Y log\nfactors as a composite of monomorphisms W log \u0003 V log \u0003 Y log. In particular,\ndim(W ) ≤ dim(V ) = d + n − 1 [cf. Proposition 1.4, (vi)], so we conclude that\ndimsm(Y log) = d+ n− 1, as desired. ©\nThe following generalities on log-like and scheme-like morphisms will be of use\nin the remainder of the present paper.\nProposition 1.11. (Generalities on log-like and scheme-like morphisms)\nLet f log : Z log → Y log be a morphism of Schlog(X log). Then:\n(i) Write U log for the log scheme whose underlying scheme is equal to the\nunderlying scheme Z of Z log and whose log structure is the pull-back of the log\n22 SHINICHI MOCHIZUKI\nstructure of Y log via the underlying morphism of schemes f : Z = U → Y associated\nto f log. Then U log may be regarded, in a natural way, as an object of Schlog(X log),\nand there exists a natural factorization\nZ log\nf log1−→ U log f\nlog\n2−→ Y log\nof f log in Schlog(X log), where f\nlog\n1 is log-like, and f\nlog\n2 is scheme-like.\n(ii) The factorization Z log\nf log1−→ U log f\nlog\n2−→ Y log of (i) may be characterized, up\nto a unique isomorphism, via the following universal property: The morphism\nf\nlog\n2 is scheme-like, and, moreover, if\nZ log\nhlog1−→ V log h\nlog\n2−→ Y log\nis a factorization of f log in Schlog(X log) such that h\nlog\n2 is scheme-like, then there\nexists a unique scheme-like morphism glog : U log → V log such that hlog1 = glog ◦\nf\nlog\n1 , h\nlog\n2 ◦ glog = f log2 .\n(iii) Base-change via the morphism f\nlog\n1 : Z\nlog → U log of (i) determines an\nequivalence of categories\nSchlog(U log)|sch-lk ∼→ Schlog(Z log)|sch-lk\n[cf. the notational conventions of Definition 1.1, (iv)]. The morphism f\nlog\n2 : U\nlog →\nY log of (i) — which may be regarded as an object of Schlog(Y log)|sch-lk — deter-\nmines an equivalence of categories\nSchlog(U log)|sch-lk ∼→\n{\nSchlog(Y log)|sch-lk\n}\nf log2\nof Schlog(U log) with the category\n{\nSchlog(Y log)|sch-lk\n}\nf log2\nof objects of the cat-\negory Schlog(Y log)|sch-lk equipped with a structure morphism to the object f log2 of\nSchlog(Y log)|sch-lk and morphisms of the category Schlog(Y log)|sch-lk that are com-\npatible with the structure morphisms to the object f\nlog\n2 .\nProof. Assertions (i), (ii), and (iii) follow immediately from the various definitions\ninvolved. ©\nSection 2: The Scheme Structure of Submonic Log Schemes\nIn the present §2, we give a category-theoretic reconstruction of the underlying\nscheme structure of submonic objects of the categories of log schemes defined in §1.\nWe maintain the notation of §1.\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 23\nDefinition 2.1. Let f log : Z log → Y log be a morphism of Schlog(X log). Then we\nshall say that the morphism f log is SLEM [i.e., a “submonically log étale monomor-\nphism”] if f log is a monomorphism in Schlog(X log), and, moreover, for any commu-\ntative diagram\nV log −→ Z log⏐⏐\u0005 ⏐⏐\u0005f log\nW log −→ Y log\n— where V log and W log are one-pointed and submonic, and the left-hand ver-\ntical arrow is a monomorphism in Schlog(X log) — of objects and morphisms in\nSchlog(X log), there exists a unique [“lifting”] morphism W log → Z log that renders\nthe two resulting triangles in the above diagram commutative.\nProposition 2.2. (SLEM morphisms and open immersions) Let f log :\nZ log → Y log be a morphism of Schlog(X log). Thus, the underlying morphism f :\nZ → Y of f log may be regarded as a morphism of Sch(X). Then:\n(i) If f log is an open immersion [cf. the discussion entitled “Log schemes”\nin §0], then f log is SLEM.\n(ii) If Y log is submonic, and f log is SLEM, then f log is an open immer-\nsion.\nProof. First, let us observe that any monomorphism between one-pointed objects\nin Sch(X) is necessarily a closed immersion between spectra of artinian rings [cf.,\ne.g., the proof of [LgSch], Corollary 1.2]. In particular, it follows from Proposition\n1.4, (vi), that any monomorphism V log → W log as in Definition 2.1 is necessarily\nscheme-like, and, moreover, that the underlying morphism of schemes associated\nto any monomorphism V log → W log as in Definition 2.1 is necessarily a closed\nimmersion between spectra of artinian rings. Thus, it is immediate that if f log is an\nopen immersion, then f log is SLEM. This completes the proof of assertion (i). Now\nsuppose that Y log is submonic, and f log is SLEM. Thus, it follows from Proposition\n1.4, (vi), that f log is scheme-like, and, moreover, that f is a monomorphism in\nSch(X). In particular, the existence of unique liftings as stipulated in Definition\n2.1 implies that f is an étale monomorphism in Sch(X), hence [cf., e.g., [LgSch],\nCorollary 1.3] an open immersion. This completes the proof of assertion (ii). ©\nProposition 2.3. (Connectedness with respect to SLEM localizations)\n(i) Let Slog be a connected [hence nonempty] object of Schlog(X log) [cf.\nProposition 1.3]; U log, {V logi }i∈N nonempty objects of Schlog(X log); U log \u0003 Slog,\n{V logi \u0003 Slog}i∈N SLEM morphisms of Schlog(X log) such that, for each i ∈ N,\nthe morphism V\nlog\ni \u0003 Slog admits a [necessarily unique] factorization V\nlog\ni \u0003\nV\nlog\ni+1 \u0003 Slog through the morphism V\nlog\ni+1 \u0003 Slog, and, moreover, the fiber product\nU log ×Slog V logi [in Schlog(X log)] is empty. Then the natural map\nMinPt(U log)\n∐ { ⋃\ni∈N\nMinPt(V\nlog\ni )\n}\n→ MinPt(Slog)\n24 SHINICHI MOCHIZUKI\nis injective.\n(ii) In the situation of (i), suppose further that Slog is submonic. Then the\nnatural map of (i) is never surjective.\n(iii) Suppose that Slog is a log-nodal object of Schlog(X log). Then, for suitable\nchoices of U log \u0003 Slog and {V logi \u0003 Slog}i∈N as in (i), the natural map of (i) is\nsurjective.\n(iv) Let T log be an object of Schlog(X log). Then T log is non-submonic if and\nonly if there exist morphisms U log \u0003 Slog and {V logi \u0003 Slog}i∈N as in (i), together\nwith a monomorphism Slog \u0003 T log in Schlog(X log), such that the natural map of\n(i) is surjective.\nProof. First, we consider assertion (i). Let i ∈ N. Then the injectivity of each\nof the natural maps MinPt(U log) → MinPt(Slog), MinPt(V logi ) → MinPt(Slog)\nfollows immediately from the definition of “MinPt(−)”. The fact that the images of\nthese two maps are disjoint follows immediately from the definition of “MinPt(−)”,\ntogether with the assumption that the fiber product U log ×Slog V logi is empty. This\ncompletes the proof of assertion (i).\nNext, we consider assertion (ii). Since Slog is submonic, it follows from Propo-\nsition 2.2, (ii), that the morphisms U log \u0003 Slog, {V logi \u0003 Slog}i∈N are open im-\nmersions. Since [the underlying scheme of] Slog is connected, it thus follows from\nthe assumption that the objects U log, {V logi }i∈N are nonempty, whereas the fiber\nproducts {U log ×Slog V logi }i∈N are empty, that the open subscheme of Slog deter-\nmined by the union of the images of the morphisms U log \u0003 Slog, {V logi \u0003 Slog}i∈N\ndoes not coincide with Slog, and hence [cf. Proposition 1.6, (i)] that the natural\nmap of (i) is not surjective. This completes the proof of assertion (ii).\nNext, we consider assertion (iii). First, let us observe that, in light of the\nvarious assumptions imposed on Slog, one verifies immediately that Slog may be\nregarded as the “Slog” that appears in Lemma 1.5, (viii). Here, the positive homo-\nmorphism φ0 : P → J0 = N of Lemma 1.5, (viii), may be taken to be the positive\nhomomorphism “φ” of Lemma 1.5, (ii). In particular, we also obtain homomor-\nphisms φ1 : P → J1 and φ2 : P → J2. Now we apply Example 0.1, where we take\n“P” to be P and “∞P” to be J2. This yields an infinite descending sequence\nP ⊆ J2 ⊆ . . . ⊆ iP ⊆ . . . ⊆ 1P ⊆ 0P\n— where i ∈ N — of submonoids of P gp satisfying various properties as described\nin Example 0.1. Suppose that, for i ∈ N, iP is obtained as the bisecting monoid of\nP at a positive homomorphism iψ0 : P → N that is assigned the index “2”.\nThus, for i ∈ N, the log étale monomorphism\nSlog[iψ{0,2}] \u0003 Slog\nof Lemma 1.5, (vii), (viii) [where we take “φ0” to be\niψ0] factors through the log\nétale monomorphism\nSlog[i+1ψ{0,2}] \u0003 Slog\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 25\nof Lemma 1.5, (vii), (viii) [where we take “φ0” to be\ni+1ψ0], as well as through the\nlog étale monomorphism\nSlog[φ{0,2}] \u0003 Slog\nof Lemma 1.5, (vii), (viii) [where we take “φ0” to be φ0]. In particular, it fol-\nlows from the fact that Slog[φ{0,1}] ×Slog Slog[φ{0,2}] = Slog[φ{0}] [cf. Lemma 1.5,\n(vii), (viii)], together with the discussion of Example 0.1, that the fiber product\nSlog[φ{0,1}]×Slog Slog[iψ{0,2}] is empty.\nThus, in summary, if we take U log \u0003 Slog to be the morphism\nSlog[φ{0,1}] \u0003 Slog\nand, for i ∈ N, V logi \u0003 Slog to be the morphism\nSlog[iψ{0,2}] \u0003 Slog\ndiscussed above, then we obtain data as in assertion (i). Note, moreover, that it\nfollows immediately from the discussion of Example 0.1 that the natural map of\nassertion (i) is surjective, as desired. This completes the proof of assertion (iii).\nFinally, we observe that the sufficiency (respectively, necessity) portion of as-\nsertion (iv) follows formally from assertion (ii) (respectively, (iii)), together with\nProposition 1.4, (vi) (respectively, together with Proposition 1.7, applied to T log).\nThis completes the proof of assertion (iv). ©\nProposition 2.4. (Characterization of scheme-like morphisms between\nminimal objects) Let hlog : T log → Slog be a morphism between minimal objects\nof Schlog(X log). Set r\ndef\n= rk(Slog) ∈ {0, 1} [cf. Proposition 1.6, (ii)]. Then hlog\nis scheme-like if and only if there exists a connected, submonic object Z log of\nSchlog(X log) such that the domain of every minimal point of Z log is of rank r,\nand, moreover, hlog admits a factorization\nT log → Z log → Slog\nas the composite of a minimal point T log \u0003 Z log of Z log with a morphism Z log →\nSlog that admits a section Slog → Z log [i.e., such that the composite Slog → Z log →\nSlog is the identity morphism].\nProof. First, we observe that since the underlying morphism of schemes T → S\nnecessarily arises from [i.e., by applying “Spec(−)” to] a finite extension of fields,\nthe asserted necessity follows immediately by taking Z log\ndef\n= AN\nZ\n×Z Slog [i.e., N -\ndimensional affine space over Slog, for a suitable positive integer N ]. Here, we note\nthat the fact that “the domain of every minimal point of this Z log is of rank r” fol-\nlows immediately from Proposition 1.4, (vi). Thus, it remains to verify sufficiency.\nFirst, let us observe that it follows from the manifestly constructible nature of the\ncharacteristic sheaf PZ [cf. also Propositions 1.4, (vi); 1.6, (i), (ii)] that the assump-\ntion that “the domain of every minimal point of Z log is of rank r” implies that Z log\nitself is of rank r, and hence [cf. Lemma 1.9] that the characteristic sheaf PZ is\n26 SHINICHI MOCHIZUKI\nlocally constant. Since the monoids 0 and N have no nontrivial automorphisms, we\nthus conclude that the characteristic sheaf PZ is constant, with fibers isomorphic\nto the monoid 0 (respectively, N) if r = 0 (respectively, r = 1). The existence of\nthe section Slog → Z log thus implies that the morphism Z log → Slog is scheme-like.\nSince the monomorphism T log \u0003 Z log is also scheme-like [cf. Proposition 1.4, (vi)],\nwe thus conclude that hlog is scheme-like, as desired. This completes the proof of\nsufficiency and hence of Proposition 2.4. ©\nProposition 2.5. (Characterization of scheme-like morphisms between\nsubmonic objects) Let f log : Z log → Y log be a morphism between submonic\nobjects of Schlog(X log). Then f log is scheme-like if and only if, for every minimal\npoint T log \u0003 Z log of Z log, there exists a minimal point Slog \u0003 Y log of Y log and\na scheme-like morphism T log → Slog of Schlog(X log) that fit into a commutative\ndiagram\nT log \u0003 Z log⏐⏐\u0005 ⏐⏐\u0005f log\nSlog \u0003 Y log\nof objects of Schlog(X log).\nProof. The asserted necessity is immediate from the definitions and Propositions\n1.4, (vi), (vii); 1.6, (ii). The asserted sufficiency follows immediately, in light of\nthe manifestly constructible nature of the characteristic sheaves PZ , PY , from the\ndefinitions and Propositions 1.4, (vi); 1.6, (i), (ii). ©\nTheorem 2.6. (Reconstruction of the scheme structure of submonic\nobjects) For i = 1, 2, let X\nlog\ni be a locally noetherian fs log scheme [cf. the\ndiscussion entitled “Log schemes” in §0]. For i = 1, 2, we shall write Schlog(X logi )\nfor the category defined at the beginning of §1. Let\nΦ : Schlog(X\nlog\n1 )\n∼→ Schlog(X log2 )\nbe an [arbitrary!] equivalence of categories. Then:\n(i) Φ preserves the following:\n(i-a) monomorphisms;\n(i-b) empty objects;\n(i-c) connected objects;\n(i-d) minimal objects;\n(i-e) minimal points;\n(i-f) submonic one-pointed objects;\n(i-g) ranks of minimal objects;\n(i-h) SLEM morphisms;\n(i-i) submonic objects;\n(i-j) scheme-like morphisms between minimal objects;\n(i-k) scheme-like morphisms between submonic objects;\n(i-l) the submonic dimension of objects.\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 27\n(ii) For i = 1, 2, let Y\nlog\ni be an object of Sch\nlog(X\nlog\ni ); write Yi for the under-\nlying scheme of Y\nlog\ni . Suppose further that Φ(Y\nlog\n1 ) = Y\nlog\n2 . Thus, [cf. the portion\nof (i) concerning (i-i)] Y\nlog\n1 is submonic if and only if Y\nlog\n2 is. Suppose that Y\nlog\ni\nis submonic for i = 1, 2. Then Φ induces an equivalence of categories\n(\nSch(Y1)\n∼→\n)\nSchlog(Y\nlog\n1 )|sch-lk ∼→ Schlog(Y log2 )|sch-lk\n( ∼→ Sch(Y2)\n)\n— where the equivalences in parentheses are the natural equivalences of Definition\n1.1, (iv) — that is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nFinally, the composite of the equivalences of categories in the above display induces,\nby applying [LgSch], Theorem 1.7, (ii), an isomorphism of schemes\nY1\n∼→ Y2\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nProof. First, we consider assertion (i). The preservation of (i-a) is a matter of\ngeneral nonsense. The preservation of (i-b) follows from Proposition 1.3, (i). The\npreservation of (i-c) follows from Proposition 1.3, (ii). The preservation of (i-d)\nand (i-e) follows immediately from the preservation of (i-a). The preservation of\n(i-f) follows immediately, in light of Proposition 1.8, from the preservation of (i-e).\nThe preservation of (i-g) follows immediately, in light of Proposition 1.6, (iii), from\nthe preservation of (i-d). The preservation of (i-h) follows immediately from the\npreservation of (i-a) and (i-f). The preservation of (i-i) follows immediately, in light\nof Proposition 2.3, (iv), from the preservation of (i-a), (i-b), (i-c), (i-e), and (i-h).\nThe preservation of (i-j) follows immediately, in light of Proposition 2.4, from the\npreservation of (i-c), (i-d), (i-e), (i-g), and (i-i). The preservation of (i-k) follows\nimmediately, in light of Proposition 2.5, from the preservation of (i-e), (i-i), and\n(i-j). This completes the proof of assertion (i), except for the verification of the\npreservation of (i-l). Assertion (ii) follows immediately [i.e., in the spirit of [LgSch],\nCorollary 2.15] from the portion of assertion (i) concerning the preservation of (i-k).\nHere, we note that the functoriality of the isomorphism of schemes in the final dis-\nplay in the statement of assertion (ii) follows immediately from the characterization\ngiven in Proposition 1.11, (ii), of the factorization discussed in Proposition 1.11, (i),\ntogether with the natural equivalences of categories discussed in Proposition 1.11,\n(iii). Finally, the portion of assertion (i) concerning the preservation of (i-l) follows\nfrom the portion of assertion (i) concerning the preservation of (i-a), (i-i), together\nwith the isomorphisms of schemes obtained in assertion (ii). ©\nLemma 2.7. (Characterization of isomorphisms among positive homo-\nmorphisms) Let ξ : P → Q be a positive homomorphism between fs monoids\nsuch that rk(P ) ≥ rk(Q), and, moreover, the following condition is satisfied:\nEvery positive homomorphism φ : P → N admits a factorization P →\nQ → N as a composite of ξ with a positive homomorphism ψ : Q → N.\nThen ξ is an isomorphism.\n28 SHINICHI MOCHIZUKI\nProof. First, let us observe that, by Lemma 1.5, (ii), there exists a positive ho-\nmomorphism φ† : P → N. Next, let us observe that if p is a prime number, then\ngiven a surjective homomorphism ζ : P gp \u0002 Fp, there exists a homomorphism\nζ : P gp → Z whose composite with the natural surjection Z \u0002 Fp is equal to ζ.\n[Indeed, this follows immediately from the fact that P gp is a finitely generated free\nabelian group — cf. the discussion entitled “Generalities on monoids” in §0.] In\nparticular, it follows from the fact that P is a finitely generated monoid that, for\nsufficiently large n ∈ N, the homomorphism (ζ + pn · (φ†)gp) : P gp → Z coincides\nwith ζ when composed with the natural surjection Z \u0002 Fp and, moreover, deter-\nmines a positive homomorphism φ : P → N. In particular, it follows from the\nhypotheses imposed on ξ that φ admits a factorization P → Q → N as a composite\nof ξ with a positive homomorphism ψ : Q → N. Since the resulting composite\nP gp → Qgp → Z \u0002 Fp coincides with ζ, we thus conclude, by allowing p and ζ\nto vary, that the reduction of the homomorphism of finitely generated free abelian\ngroups ξgp : P gp → Qgp modulo any prime number is injective, and, hence, since\nrk(P ) ≥ rk(Q), that ξgp : P gp → Qgp is an isomorphism. That is to say, P and Q\nmay be regarded as finitely generated saturated monoids within a single Z-module\nP gp\n∼→ Qgp. In particular, it follows from well-known properties of fs monoids [cf.,\ne.g., [LgSch], Lemma 2.5, (iv)] that the hypotheses imposed on ξ imply that ξ is an\nisomorphism, as desired. ©\nProposition 2.8. (Characterization of scheme-like morphisms between\nreduced, one-pointed, non-minimal objects) Let f log : Z log → Y log be a\nmorphism between reduced, one-pointed, non-minimal objects of Schlog(X log).\nThen f log is scheme-like if and only if dimsm(Z log) ≤ dimsm(Y log), and, more-\nover, the following condition is satisfied:\nLet Slog be a minimal object of Schlog(X log), hlog : Slog → Y log a mor-\nphism of Schlog(X log). Then there exists a commutative diagram of mor-\nphisms of Schlog(X log)\nT log → Z log⏐⏐\u0005 ⏐⏐\u0005f log\nSlog\nhlog−→ Y log\nin which the left-hand vertical arrow T log → Slog is a scheme-like mor-\nphism between minimal objects of Schlog(X log).\nProof. First of all, we observe that the asserted necessity follows immediately\nfrom Proposition 1.10, together with the definition of the term “scheme-like”.\nThus, it suffices to verify the sufficiency of the condition that appears in the\nstatement of Proposition 2.8. To this end, let us first observe that it follows [cf.\nProposition 1.6, (ii)] from the assumption that Z log and Y log are non-minimal\nthat rk(Z log) ≥ 2, rk(Y log) ≥ 2. Thus, it follows from Proposition 1.10 that\nrk(Z log) = dimsm(Z log) + 1 ≤ dimsm(Y log) + 1 = rk(Y log). Next, let us observe —\ni.e., by applying Lemma 1.5, (v), as in the proof of Proposition 1.6, (i) — that the\ncondition under consideration implies that the restriction to a geometric point of\nZ log of the morphism of characteristic sheaves PY |Z → PZ induced by f log satisfies\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 29\nthe condition discussed in Lemma 2.7. In particular, we conclude from Lemma 2.7\nthat this morphism PY |Z → PZ is, in fact, an isomorphism, and hence that f log is\nscheme-like, as desired. ©\nDefinition 2.9.\n(i) Let Z be a scheme. Then we shall refer to a point z of the underlying\ntopological space of Z as a locally closed point if z determines a closed point of\nsome open subscheme of Z. Write\nLCPt(Z)\nfor the set of locally closed points of Z.\n(ii) Let Z log be an object of Schlog(X log). For i = 1, 2, let U\nlog\ni be a minimal\nobject of Schlog(X log) and f\nlog\ni : U\nlog\ni → Z log an arrow of Schlog(X log). Then we\nshall say that f\nlog\n1 and f\nlog\n2 are point-equivalent if there exist a morphism f\nlog\nW :\nW log → Z log and, for each i = 1, 2, a morphism hlogi : V logi → U logi between\nminimal objects of Schlog(X log) such that W log is log-nodal, and, moreover, for each\ni = 1, 2, the composite morphism f\nlog\ni ◦ hlogi : V logi → Z log admits a factorization\nV\nlog\ni → W log → Z log through f logW : W log → Z log.\n(iii) Let Z log be an object of Schlog(X log) whose underlying scheme we de-\nnote by Z, z ∈ LCPt(Z log) def= LCPt(Z). Then a monomorphism H log \u0003 Z log in\nSchlog(X log) will be called a point-hull at z if H log is one-pointed, and, moreover,\nevery morphism Slog → Z log in Schlog(X log) from a minimal object Slog to Z log\nthat maps the unique point of the underlying scheme S of Slog to z factors [nec-\nessarily uniquely!] through the given monomorphism H log \u0003 Z log. A point-hull\nH log \u0003 Z log at z will be called a minimal point-hull at z if every monomorphism\nH\nlog\n1 \u0003 H log in Schlog(X log) for which the composite H\nlog\n1 \u0003 H log \u0003 Z log is a\npoint-hull at z is necessarily an isomorphism. An arrow of Schlog(X log) which is a\nminimal point-hull at some element of LCPt(−) of the codomain of the arrow will\nbe referred to as a minimal point-hull. Thus, if Z log is one-pointed, and one restricts\none’s attention to monomorphisms with one-pointed domains, then the notion of a\npoint-hull (respectively, minimal point-hull) at z is identical to the notion of a hull\n(respectively, minimal hull) [cf. Definition 1.1, (iii)].\nProposition 2.10. (Point-classes and minimal point-hulls) Let Z log be an\nobject of Schlog(X log). For i = 1, 2, let U\nlog\ni be a minimal object of Sch\nlog(X log)\nand f\nlog\ni : U\nlog\ni → Z log an arrow of Schlog(X log). For i = 1, 2, write Z, Ui for the\nunderlying schemes of Z log, U\nlog\ni , respectively. Then:\n(i) Z log is one-pointed if and only if the set LCPt(Z log) = LCPt(Z) is of\ncardinality one.\n(ii) Let z be a point of the underlying topological space of Z. Then the following\nconditions are equivalent: (ii-a) z is locally closed; (ii-b) z appears as the image of\na morphism U → Z of Sch(X) for some minimal object U [cf. [LgSch], Proposition\n30 SHINICHI MOCHIZUKI\n1.1, (ii)] of Sch(X); (ii-c) z appears as the image of a morphism U log → Z log of\nSchlog(X log) for some minimal object U log of Schlog(X log).\n(iii) Write zi for the image in Z via [the underlying morphism of schemes\nassociated to] f\nlog\ni of the unique point of Ui. Then the arrows f\nlog\n1 and f\nlog\n2 are\npoint-equivalent if and only if z1 = z2. In particular, the notion of point-\nequivalence determines an equivalence relation on the collection [i.e., which, strictly\nspeaking, is not necessarily a set!] of arrows in Schlog(X log) from minimal objects\nof Schlog(X log) to Z log. Write\nPtCl(Z log)\nfor the set of equivalence classes of such arrows. We shall refer to an element of\nPtCl(Z log) as a point-class of Z log.\n(iv) If f log : U log → Z log is an arrow that determines a point-class of Z log, then\nlet us write Im(f log) for the image in Z via [the underlying morphism of schemes\nassociated to] f log of the unique point of the underlying scheme U of U log. Then\nthe assignment f log \n→ Im(f log) determines a bijection of sets\nPtCl(Z log)\n∼→ LCPt(Z log) = LCPt(Z)\nthat is functorial [in the evident sense] with respect to Z log.\n(v) Let z ∈ LCPt(Z). Write zlog for the reduced, one-pointed object of\nSchlog(X log) obtained by restricting the log structure of Z log to z. Then a monomor-\nphism hlog : H log \u0003 Z log in Schlog(X log) is a minimal point-hull at z if and\nonly if hlog induces an isomorphism H log\n∼→ zlog.\nProof. First, we observe that assertion (i) follows immediately from the various\ndefinitions involved [cf. also [LgSch], Proposition 1.1, (i)]. Next, we consider as-\nsertion (ii). First, we recall from [LgSch], Proposition 1.1, (ii), that an object of\nSch(X) is minimal if and only if it is reduced and one-pointed. Next, we recall from\nProposition 1.6, (ii), that aminimal object of Schlog(X log) is necessarily reduced and\none-pointed. Now the implication (ii-a) =⇒ (ii-b) follows immediately. In a similar\nvein, the implication (ii-a) =⇒ (ii-c) follows immediately, by applying Proposition\n1.6, (i). To verify the implications (ii-b) =⇒ (ii-a), (ii-c) =⇒ (ii-a), it suffices to\nverify that if U is a one-pointed object of Sch(X), then the image via any morphism\nU → Z of Sch(X) of the unique point of U is a locally closed point of Z. Note that,\nby considering the schematic closure of such a morphism in a suitable affine open of\nZ, we may assume without loss of generality that U and Z are affine, and that the\nmorphism [of finite type!] U → Z has dense image. Since this image [which consists\nof a single point!] is necessarily constructible, hence contains a dense open subset of\nthe underlying topological space of Z, we thus conclude that we may assume, after\nreplacing Z by a suitable affine open of Z, that the morphism U → Z is surjective,\ni.e., that Z is one-pointed. This completes the proof of assertion (ii).\nNext, we consider assertion (iii). Since minimal objects of Schlog(X log) are nec-\nessarily one-pointed [cf. Proposition 1.6, (ii)], the necessity portion of the asserted\nequivalence follows immediately from the various definitions involved. Thus, it suf-\nfices to verify the sufficiency portion of the asserted equivalence. To this end, let us\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 31\nfirst observe that, since minimal objects of Schlog(X log) are necessarily reduced [cf.\nProposition 1.6, (ii)], we may assume without loss of generality that Z log is reduced\nand one-pointed. Also, by base-changing to a suitable finite extension of the field\nwhose spectrum is Z, we conclude that we may assume without loss of generality\nthat f\nlog\n1 and f\nlog\n2 are log-like, and that Z\nlog is split. Thus, by considering a suitable\nsplitting as in Lemma 1.5, (i), one verifies immediately that, to complete the proof\nof sufficiency, it suffices to verify the following assertion concerning fs monoids:\nLet P be an fs monoid. For i = 1, 2, let φi : P → N be a positive\nhomomorphism of fs monoids. Then there exist an fs monoid Q of rank\ntwo and a positive homomorphism ψ : P → Q of fs monoids such that, for\ni = 1, 2, the homomorphism 2 · φi : P → N [i.e., the composite of φi with\nthe positive homomorphism N → N given by multiplication by 2] admits\na factorization P → Q → N as the composite of ψ with some positive\nhomomorphism ψi : Q → N.\nThis assertion concerning fs monoids may be verified as follows. For i = 1, 2,\nwrite N\ngp\ni ⊆ P gp for the kernel of the morphism φgpi : P gp → Z. If Ngp1 = Ngp2 ,\nthen one verifies immediately that one obtains data as desired by considering the\nfactorization N → N ⊕ N → N [i.e., determined by the assignments N \u0017 1 \n→\n(1, 1) ∈ N ⊕ N and N ⊕ N \u0017 (a, b) \n→ a + b ∈ N] of the homomorphism N → N\ngiven by multiplication by 2. Thus, we may assume without loss of generality that\nN\ngp\n1 \u0006= Ngp2 . Write Q for the saturation [cf. [LgSch], Lemma 2.5, (ii)] of the image\nof P in (P gp\/N\ngp\n1 ) ⊕ (P gp\/Ngp2 ) (∼= Z ⊕ Z). Thus, we obtain a natural positive\nhomomorphism of monoids ψ : P → Q such that, for i = 1, 2, φi : P → N admits a\nfactorization P → Q → N as the composite of ψ with some positive homomorphism\nψi : Q → N. Here, we note that the positivity of ψi follows immediately from the\npositivity of φi. Also, we observe that the positivity of ψi implies that the monoid\nQ has no nonzero invertible elements. We thus conclude that Q is an fs monoid\nof rank two, as desired. This completes the proof of assertion (iii). Assertion (iv)\nfollows immediately from assertion (iii), together with the equivalence (ii-a) ⇐⇒\n(ii-c) of assertion (ii).\nFinally, we consider assertion (v). First, we consider the sufficiency portion\nof the asserted equivalence. To verify this sufficiency, it suffices to verify that the\nnatural monomorphism hlogz : z\nlog \u0003 Z log [cf. Proposition 1.4, (vii)] is a minimal\npoint-hull at z. The fact that hlogz is a point-hull at z follows immediately from the\nvarious definitions involved. Now suppose that h\nlog\n1 : H\nlog\n1 \u0003 zlog is a monomor-\nphism such that the composite hlogz ◦hlog1 : H log1 \u0003 Z log is a point-hull at z [so both\nzlog and H\nlog\n1 are one-pointed]. Then one verifies immediately that, by applying\nLemma 1.5, (v), as in the proof of Proposition 1.6, (i), it follows from Proposition\n1.4, (iii), and Lemma 2.7 that h\nlog\n1 is scheme-like, and hence, by Proposition 1.4,\n(v); [LgSch], Proposition 1.1, (ii), that h\nlog\n1 is an isomorphism, as desired. Thus, to\ncomplete the proof of assertion (v), it suffices to verify the necessity portion of the\nasserted equivalence. First, let us observe that it follows from the existence of the\nnatural monomorphism H\nlog\nred \u0003 H log, together with the definition of the notion of\na minimal point-hull, that H log is reduced and one-pointed. Thus, it follows imme-\ndiately from Proposition 1.6, (i), that hlog induces a monomorphism H log \u0003 zlog.\nSince we have already verified that hlogz is a minimal point-hull at z, we thus con-\nclude that this monomorphism H log \u0003 zlog is an isomorphism, as desired. This\n32 SHINICHI MOCHIZUKI\ncompletes the proof of assertion (v). ©\nProposition 2.11. (Characterization of scheme-like morphisms between\narbitrary objects) Let f log : Z log → Y log be a morphism between arbitrary\nobjects of Schlog(X log). Then f log is scheme-like if and only if, for every minimal\npoint-hull hlog : T log → Z log, there exists a commutative diagram of morphisms\nof Schlog(X log)\nT log\nhlog−→ Z log⏐⏐\u0005 ⏐⏐\u0005f log\nSlog → Y log\nin which the lower horizontal arrow Slog → Y log is a minimal point-hull, and the\nleft-hand vertical arrow T log → Slog is a scheme-like morphism between reduced,\none-pointed objects of Schlog(X log).\nProof. The asserted equivalence follows immediately, in light of the manifestly\nconstructible nature of the characteristic sheaves PZ , PY , from Proposition 2.10,\n(v), together with the definition of the term “scheme-like”. ©\nCorollary 2.12. (Conditional reconstruction of the scheme structure of\narbitrary objects) Suppose that we are in the situation of Theorem 2.6, and that\nΦ satisfies the following condition:\n(∗nod) an object of Schlog(X log1 ) is log-nodal if and only if its image via Φ is\na log-nodal object of Schlog(X\nlog\n2 ).\nThen:\n(i) Φ preserves the following:\n(i-a) point-equivalent pairs of arrows;\n(i-b) the set-valued functor LCPt(−) [up to natural equivalence];\n(i-c) arrows which are minimal point-hulls;\n(i-d) scheme-like morphisms between arbitrary objects.\n(ii) For i = 1, 2, let Y\nlog\ni be an object of Sch\nlog(X\nlog\ni ); write Yi for the under-\nlying scheme of Y\nlog\ni . Suppose further that Φ(Y\nlog\n1 ) = Y\nlog\n2 . Then Φ induces an\nequivalence of categories\n(\nSch(Y1)\n∼→\n)\nSchlog(Y\nlog\n1 )|sch-lk ∼→ Schlog(Y log2 )|sch-lk\n( ∼→ Sch(Y2)\n)\n— where the equivalences in parentheses are the natural equivalences of Definition\n1.1, (iv) — that is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nFinally, the composite of the equivalences of categories in the above display induces,\nby applying [LgSch], Theorem 1.7, (ii), an isomorphism of schemes\nY1\n∼→ Y2\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 33\nProof. First, we consider assertion (i). The preservation of (i-a) follows imme-\ndiately, in light of the preservation of (i-d) asserted in Theorem 2.6, (i), from the\ncondition (∗nod), together with the definition of the term “point-equivalent”. The\nprservation of (i-b) now follows from the preservation of (i-a), together with the\nbijection of Proposition 2.10, (iv). The preservation of (i-c) then follows from the\npreservation of (i-b) [cf. also the preservation of (i-a), (i-d) asserted in Theorem\n2.6, (i)], together with the equivalence of Proposition 2.10, (i). The preservation of\n(i-d) follows, in light of the preservation of (i-c), from Propositions 2.8; 2.10, (v);\n2.11 [cf. also the preservation of (i-d), (i-j), (i-l) asserted in Theorem 2.6, (i)]. This\ncompletes the proof of assertion (i). Now assertion (ii) follows immediately [i.e., in\nthe spirit of Theorem 2.6, (ii); [LgSch], Corollary 2.15] from the portion of assertion\n(i) concerning the preservation of (i-d). Here, we note that the functoriality of the\nisomorphism of schemes in the final display in the statement of assertion (ii) follows\nimmediately from the characterization given in Proposition 1.11, (ii), of the factor-\nization discussed in Proposition 1.11, (i), together with the natural equivalences of\ncategories discussed in Proposition 1.11, (iii). ©\nSection 3: Seamless Partitions of Orientable Log Schemes\nIn the present §3, we discuss the notion of a seamless partition of an orientable\nlog scheme. This notion leads naturally to a category-theoretic characterization of\nlog-nodal objects, which we apply to eliminate the dependence on the condition\n“(∗nod)” in Corollary 2.12.\nWe maintain the notation of §2.\nDefinition 3.1.\n(i) Suppose that Y log is an object of Schlog(X log). Then we shall say that Y log\nis log-Dedekind if it satisfies the following conditions:\n(i-a) dimsm(Y log) ≤ 1;\n(i-b) if Z log is a minimal object of Schlog(X log) such that there exists a mor-\nphism Z log → Y log in Schlog(X log), then Z log is of rank one;\n(i-c) if Z log is a nonempty submonic object of Schlog(X log), with underly-\ning scheme Z, such that there exists a SLEM morphism Z log \u0003 Y log in\nSchlog(X log), then the closed subscheme Zred ⊆ Z is regular and of positive\ndimension.\nIf y is a point of the underlying scheme Y of a log-Dedekind object Y log, and the\nfiber of PY at some geometric point of Y that maps to y is of rank two, then we\nshall say that y is a nodal point of Y log.\n(ii) Suppose that Y log is a log-Dedekind object of Schlog(X log). For i = 1, 2,\nlet Z\nlog\ni be a connected [hence nonempty], submonic object of Sch\nlog(X log) and\nf\nlog\ni : Z\nlog\ni \u0003 Y log\n34 SHINICHI MOCHIZUKI\na SLEM morphism. We shall say that f\nlog\n1 and f\nlog\n2 are submonically equivalent if\nthe fiber product Z\nlog\n12\ndef\n= Z\nlog\n1 ×Y log Z log2 determined by f log1 and f log2 is nonempty.\n[Here, we note that, for i = 1, 2, the projection Z\nlog\n12 → Z logi , is SLEM, hence, by\nProposition 2.2, (ii), an open immersion, whose image is, by condition (i-c), dense\nwhenever it is nonempty.] One verifies immediately that the notion of submonic\nequivalence determines an equivalence relation on the collection [i.e., which, strictly\nspeaking, is not necessarily a set!] of arrows of Schlog(X log) which are SLEM\nmorphisms from connected, submonic objects of Schlog(X log) to Y log. Write\nSmCp(Y log)\nfor the set of equivalence classes of such arrows. We shall refer to an element of\nSmCp(Y log) as a submonic component of Y log.\n(iii) Suppose that Y log is a log-Dedekind object of Schlog(X log). If hlog : H log \u0003\nY log is a monomorphism of Schlog(X log), then we shall write\nChn(hlog) ⊆ SmCp(Y log)\nfor the subset of submonic components for which there exists a representative arrow\nZ log \u0003 Y log that admits a factorization Z log \u0003 H log \u0003 Y log through hlog :\nH log \u0003 Y log. If C ⊆ SmCp(Y log) is a nonempty subset, then we shall refer to C as\na chain if there exists a SLEM morphism hlog : H log \u0003 Y log of Schlog(X log) such\nthat H log is connected [hence nonempty!], and C = Chn(hlog). If C ⊆ SmCp(Y log)\nis a subset, then we shall refer to C as an N-chain if there exists a collection {Ci}i∈N\nof chains Ci ⊆ SmCp(Y log) such that C = ∪i∈N Ci, and Ci ⊆ Ci+1 for all i ∈ N.\nProposition 3.2. (First properties of log-Dedekind objects) Suppose that\nY log is a log-Dedekind object of Schlog(X log). Then:\n(i) Y log is of rank ≤ 2.\n(ii) The non-nodal points of the underlying scheme Y of Y log form an open\nsubset of the underlying topological space of Y . Write Ysm ⊆ Y for the corre-\nsponding open subscheme and Y logsm for the log scheme obtained by restricting the\nlog structure of Y log to Ysm. Then the complement of Ysm in Y is a closed sub-\nscheme of Y of dimension zero, and Y logsm is submonic. We shall refer to Y\nlog\nsm\nas the submonic locus of Y log.\n(iii) Let Z log be a nonempty submonic object of Schlog(X log) and Z log \u0003\nY log a SLEM morphism. Then the closed subscheme Zred ⊆ Z of the underlying\nscheme Z of Z log is regular and of dimension one, and Z log is of rank one. In\nparticular, [cf. Proposition 2.2, (i)] (Ysm)red is regular and of dimension one, and\nY logsm is of rank one.\n(iv) Let f log : Z log \u0003 Y log be a SLEM morphism from a connected, sub-\nmonic object Z log of Schlog(X log) to Y log. Then f log either admits a factorization\nZ log \u0003 Y logsm \u0003 Y log as the composite of an open immersion Z log \u0003 Y logsm with the\nnatural monomorphism Y logsm \u0003 Y log or maps the entire underlying scheme Z of\nZ log to some nodal point y of Y log. In the former case, we shall say that f log\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 35\nis non-nodal; in the latter case, we shall say that f log is nodal and lies over\ny. We shall also apply this terminology “non-nodal”\/“nodal” to the element of\nSmCp(Y log) determined by f log.\n(v) Let y be a nodal point of Y log. Then the subset\nSmCp(Y log)y ⊆ SmCp(Y log)\nof nodal elements that lie over y forms an N-chain. Moreover, every morphism\nH log → Y log in Schlog(X log) from a minimal object H log to Y log that maps the\nunique point of the underlying scheme of H log to y factors through some represen-\ntative of an element of SmCp(Y log)y.\n(vi) Every element γ ∈ SmCp(Y log) admits a “maximal” representative arrow\nf log : Z log \u0003 Y log, i.e., a representative arrow such that every arrow U log \u0003 Y log\nof Schlog(X log) that is submonically equivalent to f log admits a factorization\nU log \u0003 Z log \u0003 Y log\nas the composite of some open immersion U log \u0003 Z log with f log. If, moreover, γ\nis non-nodal, then such a maximal representative f log : Z log \u0003 Y log arises from\nan isomorphism of Z log onto some connected component of Y logsm .\nProof. First, let us observe that the inequality dimsm(Y log) ≤ 1 of Definition\n3.1, (i-a), together with the restriction imposed by Definition 3.1, (i-b) [cf. also\nPropositions 1.4, (iv); 1.6, (i)], imply that the integers “d” and “n” in Proposition\n1.10 satisfy the following conditions:\n(∗1) n ∈ {1, 2};\n(∗2) n = 2 =⇒ d = 0;\n(∗3) n = 1 =⇒ d ≤ 1.\nAssertion (i) thus follows from (∗1) [cf. also Lemma 1.9]. Assertion (ii) follows from\n(∗1), (∗2) [cf. also Lemma 1.9]. Assertion (iii) follows from (∗1), (∗3), together with\nDefinition 3.1, (i-c) [cf. also Proposition 2.2, (i)].\nNext, we consider assertion (iv). If y is a nodal point of Y log, then write ylog\nfor the log scheme obtained by restricting the log structure of Y log to the closed\nsubscheme, equipped with the reduced induced scheme structure, of Y determined\nby y. Write Z logy\ndef\n= Z log ×Y log ylog. Thus, the underlying scheme Zy of Z logy may\nbe identified with the scheme-theoretic fiber of Z over y. Note that if Zy = ∅ for\nevery nodal point y of Y log, then f log admits a factorization Z log \u0003 Y logsm \u0003 Y log as\nthe composite of a monomorphism Z log \u0003 Y logsm with the natural monomorphism\nY logsm \u0003 Y log; moreover, since f log is SLEM, it follows immediately that the mor-\nphism Z log \u0003 Y logsm is SLEM and hence, by assertion (ii) and Proposition 2.2, (ii),\nan open immersion. Thus, since, by assertion (iii), Zred is regular and of dimension\none, it follows immediately — i.e., by possibly replacing Z log by the log scheme\ndetermined by a suitable dense open subscheme of Z — that, to complete the proof\nof assertion (iv), it suffices to verify, under the additional assumption that Z logy is\nconnected [hence nonempty] for some fixed nodal point y of Y log, that dim(Zy) = 1.\n36 SHINICHI MOCHIZUKI\nTo this end, let us first observe that the natural morphism Z logy → ylog is SLEM.\nSince Z logy is connected and [by assertion (iii)] of rank one, it follows from Lemma\n1.5, (v) [where we take “Slog” to be ylog], that the monomorphism Z logy → ylog\nadmits a factorization as a composite of monomorphisms\nZ logy \u0003 ylogZ \u0003 ylog\n— where y\nlog\nZ is, in the notation of Lemma 1.5, (v), a log scheme of rank one of\nthe form “Slog[ξ]”; y\nlog\nZ \u0003 ylog is the composite monomorphism of Lemma 1.5, (vi)\n[where we take “S\nlog\n+ [ξ] \u0003 Slog[ξ]” to be the identity morphism]. Since Z logy \u0003\nylog is SLEM, it follows immediately that Z logy \u0003 ylogZ is SLEM and hence, by\nProposition 2.2, (ii), an open immersion. Since the underlying scheme of y\nlog\nZ is of\ndimension one [cf. Lemma 1.5, (iv), (v)], we thus conclude that dim(Zy) = 1, as\ndesired. This completes the proof of assertion (iv).\nNext, we consider assertion (v). Write k for the residue field of Y at y, Slog\ndef\n=\nY log ×Y Spec(k) [where the morphism implicit in the right-hand factor of the fiber\nproduct is the tautological morphism Spec(k) → Y associated to y]. Thus, Slog is\na log scheme of the sort that appears in Lemma 1.5, so, in the following discussion,\nwe shall apply the notational conventions introduced at the beginning of Lemma\n1.5. Write O\n̂Y for the complete noetherian local ring obtained by completing Y\nalong y, Ŷ\ndef\n= Spec(O\n̂Y ), Ŷ\nlog def= Y log ×Y Ŷ , ŷ for the unique closed point of\nŶ . Write O\n̂Y sep for the completion of the strict henselization of ÔY determined\nby ksep, Ŷ sep\ndef\n= Spec(O\n̂Y sep) [so Ŷ\nsep is equipped with a natural action by Gk],\n(Ŷ sep)log\ndef\n= Y log ×Y Ŷ sep, ŷsep for the unique closed point of Ŷ sep.\nNext, let us fix a chart P → O\n̂Y sep of (Ŷ\nsep)log that determines a “clean chart”\nin the sense of [ExtFam], Definition 1.3. This chart thus determines a natural\nisomorphism of the fiber at ŷsep of the monoid M\n̂Y sep that defines the log structure\nof (Ŷ sep)log with the product P ×O×\n̂Y sep\n. In particular, the natural action of Gk on\nthis fiber determines an action of Gk on P ×O×\n̂Y sep\n[i.e., which is compatible with the\nfactor {0} × O×\n̂Y sep\n, but not necessarily compatible with the factor P × {1}, of this\nproduct decomposition!], hence also on the groupification P gp×O×\n̂Y sep\nof P ×O×\n̂Y sep\n.\nNote that since Y log is a log-Dedekind object of Schlog(X log), it follows immediately\nfrom assertion (ii) that the support of the closed subscheme Ŷ\nsep\n∗ ⊆ Ŷ sep determined\nby the ideal generated by the image via the chart under consideration of P \\ {0} is\nequal to {ŷsep}.\nNext, let\nQ ⊆ P gp\nbe a finitely generated, saturated submonoid such that P ⊆ Q \u0006= P gp. Write\nGQ ⊆ Gk for the open subgroup of elements that preserve Q [i.e., relative to\nthe natural action of Gk on the quotient (P\ngp × O×\n̂Y sep\n)\/O×\n̂Y sep\n∼→ P gp deter-\nmined, as discussed above, by the chart under consideration!]. Thus, the ac-\ntion of Gk on P\ngp × O×\n̂Y sep\ndetermines an action of GQ ⊆ Gk on the submonoid\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 37\nQ ×O×\n̂Y sep\n⊆ P gp ×O×\n̂Y sep\n. Moreover, we assume further that one of the following\n[mutually exclusive!] conditions holds:\n(v-a) GQ = Gk, and, moreover, the natural inclusion P ⊆ Q is a sum-\ndominating homomorphism of fs monoids [cf. the discussion entitled “Gen-\neralities on monoids” in §0].\n(v-b) There exists a positive homomorphism ξ : P → N which induces a\nsurjection on groupifications ξgp : P gp \u0002 Z such that Q coincides with\nthe saturation [cf. [LgSch], Lemma 2.5, (ii)] of the submonoid of P gp\ngenerated by P and Ker(ξgp).\nThus, even when GQ \u0006= Gk [which implies that condition (v-b) holds], one verifies\nimmediately that the natural inclusion P ⊆ Q is a sum-dominating homomorphism.\nThat is to say, the natural inclusion P ⊆ Q is a sum-dominating homomorphism,\nno matter which of the two conditions (v-a), (v-b) one assumes.\nNext, let us observe that the inclusion P ↪→ Q determines a log étale monomor-\nphism\nZ log[Q]\ndef\n= Spec(O\n̂Y sep [Q])\nlog → Z log[P ] def= Spec(O\n̂Y sep [P ])\nlog\n[cf. the construction discussed in Proposition 1.4, (ii), as well as [Kato1], Proposi-\ntion 3.4]. Thus, one verifies immediately that the actions [determined, as discussed\nabove, by the chart under consideration!] ofGk on P×O×\n̂Y sep\nand ofGQ onQ×O×\n̂Y sep\ndetermine, respectively, actions of Gk on Z\nlog[P ] and GQ on Z\nlog[Q]. Moreover,\nthe chart P → O\n̂Y sep under consideration determines a tautological Gk-equivariant\nmorphism (Ŷ sep)log → Z log[P ] and hence a fiber product [of fs log schemes]\n(Ẑsep)log\ndef\n= (Ŷ sep)log ×Zlog[P ] Z log[Q]\nequipped with a natural action by GQ. This natural GQ-action in turn determines\ndescent data for the projection morphism (Ẑsep)log → (Ŷ sep)log, which may be used\nto descend this projection morphism to a log étale monomorphism Ẑ log → Ŷ logQ ,\nwhere we write ŶQ → Ŷ for the finite étale covering corresponding to the open\nsubgroup GQ ⊆ Gk, Ŷ logQ def= Ŷ log ×̂Y ŶQ.\nNext, let us observe that since Y log is a log-Dedekind object of Schlog(X log) [or,\nequivalently, of Schlog(Y log)], it follows immediately from assertions (ii) and (iii)\nthat any minimal object of Schlog(Ẑ log) is of rank one. Thus, since the inclusion\nP ⊆ Q is sum-dominating, it follows from the final portion of Lemma 1.9 that\nany regular function on the underlying scheme Ẑsep of (Ẑsep)log that arises [i.e.,\nvia the various charts implicit in the above discussion] from an element ∈ P \\\n{0} necessarily vanishes at every point of Ẑsep, hence [since Ẑsep is noetherian] is\nnecessarily nilpotent. Since, as observed above, the support of the closed subscheme\nŶ\nsep\n∗ ⊆ Ŷ sep is equal to {ŷsep}, we thus conclude that the natural morphism Ẑsep →\nŶ sep factors through a closed subscheme of Ŷ sep whose support is equal to {ŷsep}.\nThis in turn implies that, if we write Ẑ for the underlying scheme of Ẑ log, then the\ncomposite morphism Ẑ → ŶQ → Ŷ factors through a closed subscheme of Ŷ whose\nsupport is equal to {ŷ}.\n38 SHINICHI MOCHIZUKI\nNext, I claim that the composite morphism\nẐ log → Ŷ logQ → Ŷ log\nis a log étale monomorphism. Indeed, in light of what has already been verified, it\nsuffices to prove, in the case where GQ \u0006= Gk [which implies that condition (v-b)\nholds], that this composite morphism is a monomorphism. Since the morphism\nẐ log → Ŷ logQ is already known to be a monomorphism, and the morphism Ŷ logQ →\nŶ log is a scheme-like morphism whose underlying morphism of schemes is finite\nétale, one verifies immediately that to complete the proof of the claim, it suffices\nto verify [cf. the argument applied in the proof of Lemma 1.5, (vi); the fact that\nthe composite morphism Ẑ → ŶQ → Ŷ factors through a closed subscheme of Ŷ\nwhose support is equal to {ŷ}] that the base-change of the morphism Ẑ log → Ŷ log\nvia the natural morphism Slog → Ŷ log is a monomorphism. On the other hand, one\nverifies immediately that this base-changed morphism Ẑ log ×\n̂Y log S\nlog → Slog may\nbe identified with the morphism “Slog[ξ] → Slog” of Lemma 1.5, (vi) [where the\nobjects “ξ”, “H” of Lemma 1.5, (vi), correspond, respectively, to ξ and GQ in the\npresent discussion; we observe that it follows immediately from condition (v-b) that\n“Ξ+ = Ξ”]. Thus, the fact that this base-changed morphism Ẑ\nlog×\n̂Y log S\nlog → Slog\nis a monomorphism follows from Lemma 1.5, (vi). This completes the proof of the\nclaim.\nThus, in summary, the composite morphism Ẑ log → Ŷ log → Y log may be\nregarded as a log étale monomorphism of Schlog(Y log), or, indeed, of Schlog(X log).\nIn the following, we shall use the notation\nf log : Z log → Y log\nto denote this composite morphism. Moreover, one computes easily that, if we write\nZ for the underlying scheme of Z log, then Zred×̂YQ Ŷ sep may be identified with the\nreduced closed subscheme of Spec(ksep[Q]) determined by forming the zero locus of\nthe set of functions P \\ {0} ⊆ Q. Thus, if condition (v-a) holds, then one verifies\nimmediately, by applying an isomorphism Qpf\n∼→ Q≥0 ⊕ Q≥0 as in the discussion\nentitled “Rank two fs monoids” in §0 [cf. also Lemma 1.5, (iv)], that Zred×̂YQ Ŷ sep\nmay be regarded as the codomain of a finite surjective morphism whose domain\nconsists of two copies of the affine line over ksep glued together at a single point,\nhence, in particular, is connected. On the other hand, if condition (v-b) holds, then\none verifies immediately that Zred ×̂YQ Ŷ sep is a one-dimensional torus [cf. the\nsituation discussed in Lemma 1.5, (iv)], hence, in particular, is connected.\nThus, in summary, the morphism f log : Z log → Y log is a log étale monomor-\nphism with connected domain such that the resulting chain\nChn(f log) ⊆ SmCp(Y log)\nis contained in SmCp(Y log)y. Now we consider the monoids constructed in Ex-\nample 0.2, where we allow n ∈ N to vary. Then it follows immediately from the\ndiscussion of Example 0.2 that given any element γ ∈ SmCp(Y log)y, it holds that\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 39\nγ ∈ Chn(f log), if, in the notation of Example 0.2, we take Q def= nP — a sub-\nmonoid which, as discussed in Example 0.2, may be constructed in such a way that\ncondition (v-a) holds — for n sufficiently large.\nFinally, let H log → Y log be a morphism in Schlog(X log) from a minimal object\nH log to Y log that maps the unique point of the underlying scheme H of H log to\ny. Thus, if we regard H as the spectrum of a finite subextension of k in the\nperfection of ksep, then the morphism H log → Y log determines, by considering the\ninduced morphism on log structures, a positive homomorphism ξ : P → N and\nsubmonoid Q ⊆ P gp that satisfy condition (v-b). Moreover, it follows immediately\nfrom the construction of f log that Z log is submonic [so f log may be regarded as a\nrepresentative of an element of SmCp(Y log)y], and that the morphism H\nlog → Y log\nfactors through f log. This completes the proof of assertion (v).\nFinally, we consider assertion (vi). If γ is non-nodal, then assertion (vi) follows\nimmediately from assertions (iii) and (iv). Thus, we may assume without loss of\ngenerality that γ is nodal. Then assertion (vi) follows immediately by gluing, in the\nnotation of Definition 3.1, (ii), the various Z\nlog\ni \u0003 Y log that constitute an element\nof SmCp(Y log) along the open immersions Z\nlog\n12 \u0003 Z\nlog\ni . Here, we note that it\nfollows immediately from the fact that the log scheme y\nlog\nZ that appeared in the\nproof of assertion (iv) is noetherian that this gluing process terminates after a finite\nnumber of steps. This completes the proof of assertion (vi). ©\nDefinition 3.3. Suppose that Y log is a connected, non-submonic, log-Dedekind\nobject of Schlog(X log). Let γ ∈ SmCp(Y log). Write\nMono(Y log)\nfor the full subcategory of Schlog(Y log) determined by the arrows H log → Y log of\nSchlog(X log) which are monomorphisms in Schlog(X log).\n(i) Let C1, C2 ⊆ SmCp(Y log) be chains. Then we shall say that the pair of\nchains {C1, C2} forms a partition at γ if the chains C1, C2 satisfy the following\nconditions:\n(i-a) C1 ∪ C2 = SmCp(Y log), C1 ∩ C2 = {γ};\n(i-b) for i = 1, 2, the subset Ci \\ {γ} ⊆ SmCp(Y log) is an N-chain [hence\nnonempty];\n(i-c) the N-chains of (i-b) are “maximal” in the sense that every N-chain\nC ⊆ SmCp(Y log) such that γ \u0006∈ C is contained in Ci for some i ∈ {1, 2};\n(i-d) if, for i = 1, 2, we write Ψi for the subfunctor of the contravariant\nfunctor determined by the terminal object [i.e., Y log] of Mono(Y log) that\nconsists of objects hlog : H log \u0003 Y log of Mono(Y log) such that every\ncomposite morphism H\nlog\n∗ \u0003 H log \u0003 Y log, where H log∗ \u0003 H log is a\nminimal point of H log, factors through some representative of an element\n∈ Ci (⊆ SmCp(Y log)), then Ψi is representable by an object hlogi : Y logi \u0003\nY log of Mono(Y log).\n40 SHINICHI MOCHIZUKI\nWe shall say that Y log is orientable if Y log admits a partition at every element of\nSmCp(Y log).\n(ii) Let {C1, C2} be a partition at γ. Suppose that hlog1 , hlog2 are as in (i-d).\nThen we shall say that the partition {C1, C2} is seamless if the following condition\nis satisfied:\na monomorphism hlog : H log \u0003 Y log in Schlog(X log) is an isomorphism if\nand only if, for i = 1, 2, the projection H log ×Y log Y logi → Y logi associated\nto the fiber product determined by hlog and h\nlog\ni is an isomorphism.\nWe shall say that Y log is homogeneous if Y log is orientable, and, moreover, no\npartition at an element ∈ SmCp(Y log) is seamless.\nRemark 3.3.1. In the situation of Definition 3.3, (i-d), we observe that it\nfollows immediately from Proposition 3.2, (v), (vi), that [the underlying morphism\nof schemes of] the morphism\nh\nlog\n1\n∐\nh\nlog\n2 : Y\nlog\n1\n∐\nY\nlog\n2 → Y log\nis surjective.\nProposition 3.4. (First properties of partitions)\n(i) Suppose that Y log is an orientable object of Schlog(X log). Let {C1, C2} be\na partition at an element γ ∈ SmCp(Y log). Then, up to a possible permutation of\nthe indices “1”, “2”, every partition at γ coincides with {C1, C2}.\n(ii) Suppose that Y log is an orientable object of Schlog(X log). Let {C1, C2}\nbe a partition at a non-nodal element γ ∈ SmCp(Y log); hlog1 : Y log1 \u0003 Y log,\nh\nlog\n2 : Y\nlog\n2 \u0003 Y log monomorphisms as in Definition 3.3, (i-d). Then, for i = 1, 2,\nh\nlog\ni : Y\nlog\ni \u0003 Y log is an open immersion, and the fiber product Y\nlog\n1 ×Y log Y log2\ndetermined by h\nlog\n1 and h\nlog\n2 is a maximal representative for γ, i.e., in the sense\nof Proposition 3.2, (vi). In particular [cf. Remark 3.3.1], the partition {C1, C2} is\nseamless.\n(iii) Suppose that Y log is a homogeneous object of Schlog(X log). Then Y log\nis one-pointed, and Y logsm is empty.\n(iv) Suppose that Y log is a log-nodal object of Schlog(X log). Then Y log is\nhomogeneous, hence, in particular, orientable. Moreover, relative to the nota-\ntional conventions introduced in Definition 1.1, (i), SmCp(Y log) may be naturally\nidentified with the set of positive homomorphisms ξ : PY → N such that ξ induces\na surjection on groupifications ξgp : P\ngp\nY \u0002 Z.\n(v) Suppose that Y log is a reduced, one-pointed, non-split object of rank\ntwo of Schlog(X log). Then Y log is log-Dedekind, but not orientable. In particu-\nlar, Y log is not homogeneous. If, moreover, Y = Spec(kY ) for some field kY , and\nkZ is a finite Galois extension of kY such that Z\nlog def= Y log ×kY kZ is log-nodal,\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 41\nthen SmCp(Y log) may be naturally identified with the set of Gal(kZ\/kY )-orbits of\nthe set SmCp(Z log) [i.e., which was described explicitly in (iv)].\nProof. Assertion (i) follows, by applying entirely formal set-theoretic considera-\ntions, from Definition 3.3, (i-a), (i-b), (i-c). Next, we consider assertion (ii). If\none restricts the morphisms h\nlog\ni : Y\nlog\ni \u0003 Y log to the open subscheme Ysm ⊆ Y [cf.\nProposition 3.2, (ii)], then one verifies immediately that the corresponding “restric-\ntions” [in the evident sense] to Ysm of the properties asserted in assertion (ii) follow\nimmediately from Proposition 3.2, (vi). Next, let y be a nodal point of Y log. Then,\nsince γ is non-nodal, it follows immediately from Propositions 1.6, (i); 3.2, (iv), (v);\nDefinition 3.3, (i-a), (i-c), (i-d), that there exists a j ∈ {1, 2} such that, if i = j (re-\nspectively, i \u0006= j), then SmCp(Y log)y ⊆ Ci (respectively, SmCp(Y log)y ∩ Ci = ∅),\nand, moreover, the restriction of h\nlog\ni to the formal scheme obtained by completing\nY along y is an isomorphism (respectively, has empty domain). Thus, it follows\nimmediately [cf. Proposition 3.2, (ii)] that there exists a Zariski open neighborhood\nU of y in Y such that, for i = 1, 2, the restriction h\nlog\ni |U of hlogi to U is scheme-like,\nand, moreover, the underlying morphism of schemes associated to h\nlog\ni |U is an étale\nmonomorphism [cf. Proposition 1.4, (v)], hence, by elementary scheme theory, an\nopen immersion, whose image contains y if i = j. The seamlessness of the partition\n{C1, C2} thus follows from elementary scheme theory [i.e., an easy case of “Zariski\ndescent”]. This completes the proof of assertion (ii).\nNext, we consider assertion (iii). First, let us observe that it follows formally\nfrom assertion (ii) that every submonic component of a homogeneous object of\nSchlog(X log) is necessarily nodal. It thus follows formally [cf. Proposition 3.2,\n(vi)] that Y logsm is empty and hence, by Proposition 3.2, (ii), that Y is of dimension\nzero. Since homogeneous objects of Schlog(X log) are, by definition, connected [hence\nnonempty], we thus conclude that Y log is one-pointed. This completes the proof of\nassertion (iii).\nNext, we consider assertion (iv). First, let us observe that Y log satisfies the\nhypotheses imposed on the log scheme “Slog” of Lemma 1.5. Thus, Lemma 1.5, (iv),\n(v), which we apply in the case where, in the notation of loc. cit., “Q” is of rank one,\nyields a log étale monomorphism “Slog[ξ] \u0003 Slog”, whose domain is connected and\nsubmonic. In particular, it follows immediately from the existence and functorial\ninterpretation [cf. Lemma 1.5, (iv), (v)] of such monomorphisms “Slog[ξ] \u0003 Slog”\nthat Y log is log-Dedekind [cf. Propositions 1.4, (vi); 2.2, (ii)]. Next, for simplicity,\nlet us write P\ndef\n= PY . Then observe that, since Y\nlog is split, it follows immediately\nfrom the various definitions involved that any element γ ∈ SmCp(Y log) determines\n— i.e., by considering the morphism induced on log structures by a representative\nof γ [cf. Proposition 1.4, (iii)] — a positive homomorphism ξγ : P → N such that\nξγ induces a surjection on groupifications ξ\ngp\nγ : P\ngp \u0002 Z. Moreover, it follows\nimmediately from Proposition 3.2, (vi), together with the various properties of the\nmonomorphisms “Slog[ξ] \u0003 Slog” discussed in Lemma 1.5, (v), that the assignment\nγ \n→ ξγ\njust discussed determines a natural bijection between SmCp(Y log) and the set of\npositive homomorphisms ξ : P → N such that ξ induces a surjection on groupifi-\ncations ξgp : P gp \u0002 Z. In the following, we shall apply this natural bijection to\nidentify these two sets.\n42 SHINICHI MOCHIZUKI\nNext, let γ ∈ SmCp(Y log). Write φ0 : P → J0 def= N for the element ξγ\ndiscussed above. In the notation of the discussion entitled “Rank two fs monoids”\nin §0, for i = 1, 2, let us write φi : P → Ji for the associated positive homomorphism\nof fs monoids [which is well-defined, up to possible permutation of the indices “1”\nand “2”] and Ci ⊆ SmCp(Y log) for the subset of elements δ ∈ SmCp(Y log) such\nthat ξδ : P → N factors through either φ0 or φi. Then I claim that\n{C1, C2} is a partition at γ which is not seamless.\nIndeed, let us first observe that condition (i-a) of Definition 3.3 follows immediately\nfrom the discussion of bisecting monoids in §0. Next, let us observe that, if we\ntake the log scheme “Slog” in Lemma 1.5 to be Y log, then it follows, by applying\nLemma 1.5, (vii), (viii), to φ0, that, for i = 1, 2, the log étale monomorphism\n“Slog[φ{0,i}] \u0003 Slog” yields an object hlogi : Y\nlog\ni \u0003 Y log as in condition (i-d)\nof Definition 3.3. Next, we verify condition (i-c) of Definition 3.3. To this end,\nsuppose that C ⊆ SmCp(Y log) \\ {γ} is a chain that intersects both C1 \\ {γ} and\nC2 \\ {γ}. Then it follows immediately from the connectedness assumption in the\ndefinition of a chain [cf. Definition 3.1, (iii)], together with Proposition 1.4, (iii);\nLemma 1.9, that there exists a rank two fs monoid P ∗ that arises as a submonoid\nof P gp that contains P and, moreover, for i = 1, 2, admits a homomorphism ψi :\nP ∗ → N whose restriction to P determines an element of Ci \\ {γ}. Moreover,\nit follows immediately from the description given above of SmCp(Y log) [i.e., by\nconsidering suitable minimal points — cf. also Proposition 2.2, (ii)] that P ∗ may\nbe chosen so that any positive homomorphism P ∗ → N that induces a surjection\non groupifications determines an element of C. On the other hand, it follows\nimmediately from the “continuity property” of bisecting monoids discussed in §0\nthat φ0 extends to a positive homomorphism P\n∗ → N and hence that γ ∈ C, a\ncontradiction. This completes the verification of condition (i-c) of Definition 3.3.\nNext, we observe that condition (i-b) of Definition 3.3 — i.e., the fact that, for\ni = 1, 2, Ci \\ {γ} is an N-chain — follows immediately by considering the log\nétale monomorphisms “Slog[φ{0,i}] \u0003 Slog” that arise by applying Lemma 1.5,\n(vii), (viii) [for an appropriate choice of the indices “1” and “2”], to a sequence of\nbisecting monoids as in Example 0.1, where we take “P ⊆ ∞P” to be the inclusion\nof monoids P ⊆ Ji that appears in the present discussion. This completes the\nproof of the fact that {C1, C2} is a partition at γ. The fact that this partition is\nnot seamless follows immediately from the existence of the log étale monomorphism\n“Slog[φ{0,1,2}] \u0003 Slog” that arises by applying Lemma 1.5, (vii), (viii), to φ0. This\ncompletes the proof of the claim. Now it follows formally that Y log is homogeneous.\nThis completes the proof of assertion (iv).\nFinally, we consider assertion (v). First, we observe that the fact that Y log\nis log-Dedekind follows immediately from assertion (iv), via a routine étale de-\nscent argument; the description given in the statement of assertion (v) of the\nset SmCp(Y log) also follows immediately, in light of the various definitions in-\nvolved, via a routine étale descent argument [cf. also Proposition 3.2, (vi)]. Now\nlet δ ∈ SmCp(Y log) be an element that arises from a Gal(kZ\/kY )-invariant ele-\nment γ ∈ SmCp(Z log). Here, we note that the existence of such an element of\nSmCp(Z log) follows immediately from the description of SmCp(Z log) given in as-\nsertion (iv), together with Lemma 1.5, (ii), which implies the existence of a suitable\npositive homomorphism ξγ : P\ndef\n= PZ → N. Then to complete the proof that Y log\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 43\nis not orientable, it suffices to verify that Y log does not admit a partition at δ.\nMoreover, to verify that Y log does not admit a partition at δ, it suffices, in light of\nconditions (i-b), (i-c) of Definition 3.3, to show that SmCp(Y log)\\{δ} is an N-chain.\nTo this end, we consider the sequence of bisecting monoids {nP}n∈N of Example\n0.1, where we take “P ⊆ ∞P” to be one of the two bisecting monoids of P at ξγ .\nThus, the homomorphism “∞φ” of Example 0.1 corresponds to ξγ in the present\ndiscussion. Now let us consider the log étale monomorphisms\n“Slog[φ{0,1}] \u0003 Slog”\nthat arise by applying Lemma 1.5, (vii), (viii), (ix), where we take the log scheme\n“Slog” of loc. cit. to be Y log, and we take “φ1 : P → J1” to be the inclusion\nP ⊆ nP , for n ∈ N. Here, we observe that if ζ : P → N and σ are as in the\ncondition of the display of Lemma 1.5, (ix), and σ acts nontrivially on P , then it\nfollows immediately from the Gal(kZ\/kY )-invariance of ξγ [i.e., “\n∞φ”] that σ acts\nnontrivially on Ker(ξgpγ ) (\n∼= Z), and hence [since both ζ and ζ ◦ σ are assumed to\nfactor through J1 and hence through “\n∞P”] that ζgp vanishes on Ker(ξgpγ ); but\nthis implies that we may assume without loss of generality that ζ = ξγ , which in\nturn implies [cf. Example 0.1] that ζgp(J1) = ξ\ngp\nγ (J1) ⊆ Z contains both positive\nand negative elements, in contradiction to the assumptions imposed on ζ. That is\nto say, the condition of the display of Lemma 1.5, (ix), is satisfied.\nThus, in summary, we obtain a collection\n{Z logn \u0003 Y log}n∈N\nof log étale monomorphisms with connected domains [cf. Lemma 1.5, (vii), (viii)]\nsuch that [cf. the discussion of Example 0.1] δ \u0006∈ Chn(Z logn \u0003 Y log) ⊆ SmCp(Y log),\nand, moreover, ∪n∈N Chn(Z logn \u0003 Y log) = SmCp(Y log) \\ {δ}. This completes the\nproof of the fact that SmCp(Y log) \\ {δ} is an N-chain and hence of assertion (v).\n©\nProposition 3.5. (Characterization of log-nodal objects)\n(i) Suppose that Y log is nonempty object of Schlog(X log). Then Y log is one-\npointed if and only if the following condition is satisfied:\nFor i = 1, 2, let U\nlog\ni be a minimal object of Sch\nlog(X log) and f\nlog\ni :\nU\nlog\ni → Y log an arrow of Schlog(X log). Then there exist a morphism\nf\nlog\nW : W\nlog → Y log and, for each i = 1, 2, a morphism hlogi : V logi →\nU\nlog\ni between minimal objects of Sch\nlog(X log) such that W log is homo-\ngeneous, and, moreover, for each i = 1, 2, the composite morphism\nf\nlog\ni ◦ hlogi : V logi → Y log admits a factorization V logi → W log → Y log\nthrough f\nlog\nW : W\nlog → Y log.\n(ii) Suppose that Y log is an object of Schlog(X log). Then Y log is log-nodal if\nand only if Y log is homogeneous, and the identity morphism Y log → Y log is a\nminimal point-hull in Schlog(X log).\n44 SHINICHI MOCHIZUKI\nProof. First, we consider assertion (i). Since, by Proposition 3.4, (iii), homo-\ngeneous objects are one-pointed, one verifies immediately from the sufficency por-\ntion of Proposition 2.10, (iii), that the condition under consideration implies that\nPtCl(Y log) is of cardinality one, and hence, by Proposition 2.10, (i), (iv), that\nY log is one-pointed, as desired. Now suppose that Y log is one-pointed. Then, by\nProposition 2.10, (i), (iv), it follows that PtCl(Y log) is of cardinality one. Since,\nby Proposition 3.4, (iv), log-nodal objects are homogeneous, we thus conclude from\nthe definition of the notion of “point-equivalence” that the condition under consid-\neration is satisfied. This completes the proof of assertion (i).\nNext, we consider assertion (ii). The necessity portion of assertion (ii) follows\nimmediately from Propositions 2.10, (v); 3.4, (iv). The sufficiency portion of asser-\ntion (ii) follows immediately, in light of the definition of the term “homogeneous”,\nfrom Propositions 2.10, (v); 3.2, (i); 3.4, (v). This completes the proof of assertion\n(ii). ©\nTheorem 3.6. (Reconstruction of the scheme structure of arbitrary\nobjects) For i = 1, 2, let X\nlog\ni be a locally noetherian fs log scheme [cf. the\ndiscussion entitled “Log schemes” in §0]. For i = 1, 2, we shall write Schlog(X logi )\nfor the category defined at the beginning of §1. Let\nΦ : Schlog(X\nlog\n1 )\n∼→ Schlog(X log2 )\nbe an [arbitrary!] equivalence of categories. Then:\n(i) Φ preserves the following:\n(i-a) log-Dedekind objects;\n(i-b) the set SmCp(−) associated to a log-Dedekind object;\n(i-c) the subsets of the set SmCp(−) of (i-b) which are [N-]chains;\n(i-d) partitions at elements of the set SmCp(−) of (i-b);\n(i-e) orientable objects;\n(i-f) homogeneous objects;\n(i-g) one-pointed objects;\n(i-h) point-hulls with one-pointed codomains;\n(i-i) minimal point-hulls with one-pointed codomains;\n(i-j) log-nodal objects.\n(ii) For i = 1, 2, let Y\nlog\ni be an object of Sch\nlog(X\nlog\ni ); write Yi for the under-\nlying scheme of Y\nlog\ni . Suppose further that Φ(Y\nlog\n1 ) = Y\nlog\n2 . Then Φ induces an\nequivalence of categories\n(\nSch(Y1)\n∼→\n)\nSchlog(Y\nlog\n1 )|sch-lk ∼→ Schlog(Y log2 )|sch-lk\n( ∼→ Sch(Y2)\n)\n— where the equivalences in parentheses are the natural equivalences of Definition\n1.1, (iv) — that is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nFinally, the composite of the equivalences of categories in the above display induces,\nby applying [LgSch], Theorem 1.7, (ii), an isomorphism of schemes\nY1\n∼→ Y2\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 45\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nProof. First, we consider assertion (i). The preservation of (i-a) follows immedi-\nately from the preservation of (i-b), (i-d), (i-g), (i-h), (i-i), (i-l) asserted in Theorem\n2.6, (i), together with the isomorphisms of schemes obtained in Theorem 2.6, (ii).\nThe preservation of (i-b) follows immediately from the preservation of (i-b), (i-c),\n(i-h), (i-i) asserted in Theorem 2.6, (i). The preservation of (i-c) follows imme-\ndiately, in light of the preservation of (i-b), from the preservation of (i-c), (i-h)\nasserted in Theorem 2.6, (i). The preservation of (i-d) follows immediately, in light\nof the preservation of (i-a), (i-b), (i-c), from the preservation of (i-a), (i-c), (i-e),\n(i-i) asserted in Theorem 2.6, (i). The preservation of (i-e) follows formally from\nthe preservation of (i-b), (i-d). The preservation of (i-f) follows formally from the\npreservation of (i-b), (i-d), (i-e), together with the preservation of (i-a) asserted\nin Theorem 2.6, (i). The preservation of (i-g) follows immediately, in light of the\npreservation of (i-f) and the characterization given in Proposition 3.5, (i), from the\npreservation of (i-b), (i-d) asserted in Theorem 2.6, (i). The preservation of (i-h),\n(i-i) follows immediately, in light of the preservation of (i-g), from the preservation\nof (i-a), (i-d) asserted in Theorem 2.6, (i). The preservation of (i-j) follows immedi-\nately from the preservation of (i-f), (i-i), together with the characterization given in\nProposition 3.5, (ii). Finally, assertion (ii) follows formally, in light of the portion\nof assertion (i) concerning the preservation of (i-j), from Corollary 2.12, (ii). ©\nIt remains to reconstruct, in a category-theoretic fashion, the log structures of\nthe various log schemes under consideration. The approach taken in the present\npaper is essentially similar to the approach taken in [LgSch], but is formulated in\na slightly different way. We begin by introducing notation as in the discussion pre-\nceding [LgSch], Lemma 2.16: Write A1\nZ\n= Spec(Z[t]) [where t is an indeterminate]\nfor the affine line over Z; A\nlog\nZ\nfor the affine line A1\nZ\nover Z equipped with the log\nstructure determined by the divisor V (t) [i.e., “the origin”]; expA : A\nlog\nZ\n→ AZ for\nthe natural morphism determined by “forgetting the log structure”;\nexpY log : A\nlog\nY log\n→ AY log\nfor the “exponentiation morphism” obtained by base-changing expA via the natural\nmorphism Y log → Spec(Z);\nA×\nY log\n↪→ AY log\nfor the open immersion determined by the complement of the origin of AY log ; A\n×\nY ,\nAY for the underlying schemes of A\n×\nY log\n, AY log ;\n0Y : Y → AY , 1Y : Y → AY\nfor the sections determined by the assignments t \n→ 0, t \n→ 1. Thus, the map induced\nby expY log on Y\nlog-valued points may be naturally identified with expY : MY → OY .\nMoreover, one verifies easily that the morphism AZ ×Z AZ → AZ that defines the\nmultiplication operation on the ring scheme AZ → Spec(Z) determines a morphism\nof log schemes over Y log\nA\nlog\nY log\n×Y log AlogY log → AlogY log\n46 SHINICHI MOCHIZUKI\nthat induces, i.e., on Y log-valued points, the monoid operation on MY . In the\nfollowing,\nwe shall always regard AY log as being equipped with the “ring log scheme”\nstructure — i.e., the ring object structure in the category of log schemes\n— determined by the ring scheme structure of AZ → Spec(Z).\nOne verifies immediately that any automorphism of the log scheme AY log that lies\nover the identity automorphism of Y log and is compatible with the ring log scheme\nstructure of AY log is necessarily equal to the identity automorphism. Finally, if Y\nlog\nis an object of Schlog(X log), then we observe that expY log : A\nlog\nY log\n→ AY log may be\nregarded, in a natural way, as an arrow between objects of Schlog(X log).\nProposition 3.7. (Categories of quasi-exponentiation morphisms) We\nmaintain the notation of the above discussion. Suppose that Y log is an object\nof Schlog(X log). Thus, AY log may be regarded, in a natural way, as an object of\nSchlog(X log). Write\nQExp(Y log) ⊆ Schlog(AY log)\nfor the full subcategory of Schlog(AY log) consisting of objects f\nlog : Z log → AY log\n[i.e., “quasi-exponentiation morphisms”] that satisfy the following conditions:\n(a) the morphism Z log → Y log determined by f log is log smooth;\n(b) f log is log-like, i.e., induces an isomorphism f : Z\n∼→ AY between the\nunderlying schemes of Z log, AY log ;\n(c) the base-change of f log via the open immersion A×\nY log\n↪→ AY log is an\nisomorphism;\n(d) if\nT log −→ Z log f\nlog\n−→ AY log⏐⏐\u0005glog ⏐⏐\u0005\nSlog → Y log\nis a commutative diagram of morphisms of Schlog(X log) in which the hor-\nizontal arrows of the square are minimal point-hulls, and the resulting\nfiber product T log ×A\nY log\nA×\nY log\nis the empty object of Schlog(X log), then\nglog is not an isomorphism, and, moreover, if Slog is not a minimal\nobject of rank zero, then, for some reduced, one-pointed object W log of\nSchlog(X log), there exist two distinct morphisms h\nlog\n1 , h\nlog\n2 : W\nlog → T log\nsuch that the two resulting composite morphisms glog ◦ hlog1 , glog ◦ hlog2 :\nW log → T log → Slog coincide and are scheme-like;\n(e) there exists a Y log-morphism Z log ×Y log Z log → Z log in Schlog(X log)\nfor which the induced morphism on underlying schemes coincides, rela-\ntive to the isomorphism f : Z\n∼→ AY of condition (b), with the morphism\nAY ×Y AY → AY determined by the multiplication operation arising\nfrom the ring log scheme structure of AY log .\n[Thus, expY log : A\nlog\nY log\n→ AY log may be regarded as an object of QExp(Y log).]\nThen every object f log : Z log → AY log of QExp(Y log) is isomorphic to the object\nof QExp(Y log) determined by expY log : A\nlog\nY log\n→ AY log . Finally, the morphism\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 47\nZ log ×Y log Z log → Z log of condition (e) is, in fact, uniquely determined by the\nhypotheses imposed in condition (e).\nProof. Proposition 3.7 follows formally from [LgSch], Lemma 2.16. Indeed, one\nverifies immediately that property (i) (respectively, (ii); (iii) [cf. Remark 3.7.1 be-\nlow]; (iv)) of [LgSch], Lemma 2.16, follows, in light of condition (b) in the statement\nof Proposition 3.7, from condition (c) (respectively, (d); (a); (e)) in the statement\nof Proposition 3.7. Here, we note in passing that the argument applied in the final\nparagraph of the proof of [LgSch], Lemma 2.16, may be simplified considerably:\nthat is to say, in the notation of loc. cit., the fact that “the morphism of monoids\nQ → P may be identified with the natural inclusion Q ↪→ Q×N” may be concluded\ndirectly from the isomorphism of rings “k[[Q]][[T ]]\n∼→ k[[P ]]” obtained in the second\nto last paragraph of the proof of [LgSch], Lemma 2.16, by considering an element\nξ ∈ P such that, if we apply this isomorphism to identify the rings k[[Q]][[T ]] and\nk[[P ]], then the set Q∪ {ξ} generates the maximal ideal of the local ring k[[P ]]. ©\nRemark 3.7.1. In the context of Proposition 3.7, we take the opportunity to cor-\nrect a misprint in the statement of [LgSch], Lemma 2.16: In [LgSch], Lemma 2.16,\n(iii), the phrase “a monomorphism” should read “a scheme-like monomorphism”.\nThe following result may be regarded as the culmination of the theory de-\nveloped in the present paper and corresponds to Theorem B [or, more precisely,\nTheorem 2.19, (ii)] of [LgSch], the proof of which [i.e., as given in [LgSch]] is,\nunfortunately, incomplete.\nTheorem 3.8. (Reconstruction of the log scheme structure of arbitrary\nobjects) For i = 1, 2, let X\nlog\ni be a locally noetherian fs log scheme [cf. the\ndiscussion entitled “Log schemes” in §0]. For i = 1, 2, we shall write Schlog(X logi )\nfor the category defined at the beginning of §1. Let\nΦ : Schlog(X\nlog\n1 )\n∼→ Schlog(X log2 )\nbe an [arbitrary!] equivalence of categories. Then:\n(i) Φ preserves the following constructions [i.e., up to, in the case of (i-a),\n(i-c), a unique isomorphism] associated to an object “(−)”:\n(i-a) the ring object A(−);\n(i-b) the full subcategory QExp( (−) ) ⊆ Schlog(A(−));\n(i-c) the exponentiation morphism exp(−) : A\nlog\n(−) → A(−);\n(i-d) the monoid object structure on the object A\nlog\n(−) of (i-c).\n(ii) For i = 1, 2, let Y\nlog\ni be an object of Sch\nlog(X\nlog\ni ); write Yi for the under-\nlying scheme of Y\nlog\ni . Suppose further that Φ(Y\nlog\n1 ) = Y\nlog\n2 . Then Φ induces an\nisomorphism of log schemes\nY\nlog\n1\n∼→ Y log2\n48 SHINICHI MOCHIZUKI\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 and compatible\nwith the isomorphism of schemes of Theorem 3.6, (ii).\n(iii) There exists a unique isomorphism of log schemes\nX\nlog\n1\n∼→ X log2\nsuch that Φ is isomorphic to the equivalence of categories induced by this isomor-\nphism of log schemes X\nlog\n1\n∼→ X log2 .\nProof. First, we consider assertion (i). The preservation of (i-a) follows immedi-\nately from Theorem 3.6, (ii); [LgSch], Proposition 1.6, (iii). To verify the preserva-\ntion of (i-b), it suffices to verify the preservation of the conditions (a), (b), (c), (d),\n(e) in the statement of Proposition 3.7. The preservation of condition (a) follows\nimmediately, in light of the functorial definition of log smoothness [i.e., in terms\nof scheme-like closed immersions, as in [Kato2], §8.1, (i)], from Theorem 3.6, (ii).\nThe preservation of condition (b) follows formally from Theorem 3.6, (ii). The\npreservation of conditions (c) and (e) follows immediately from the preservation of\n(i-a) [i.e., which has already been verified], together with Theorem 3.6, (ii). The\npreservation of condition (d) follows immediately from the preservation of (i-c) [cf.\nalso Proposition 2.10, (v)], (i-d) asserted in Corollary 2.12, (i) [which is applicable\nin light of the preservation of (i-j) asserted in Theorem 3.6, (i)], together with the\npreservation of (i-b), (i-d), (i-g) asserted in Theorem 2.6, (i). This completes the\nproof of the preservation of (i-b). The preservation of (i-c) and (i-d) follows for-\nmally from Proposition 3.7, together with the preservation of (i-b). This completes\nthe proof of assertion (i).\nSince the map induced by the exponentiation morphism exp(−) on (−)-valued\npoints may be naturally identified with the morphism between sheaves of monoids\nthat defines the log structure of “(−)” [cf. the discussion preceding Proposition\n3.7], assertion (ii) follows immediately from assertion (i); Theorem 3.6, (ii). Finally,\nassertion (iii) follows immediately from the existence of the functorial isomorphisms\nof log schemes discussed in assertion (ii), by considering, for i = 1, 2, a suitable ind-\nobject of Schlog(X\nlog\ni )\n{αiY logi }αi∈Ai\n— where the transition morphisms [notation for which was omitted for the sake\nof simplicity!] are assumed to be open immersions — that “represents X\nlog\ni ” in\nSchlog(X\nlog\ni ). [Here, we recall that if X\nlog\ni fails to be quasi-compact, then X\nlog\ni does\nnot determine an object of Schlog(X\nlog\ni ) in the usual sense.] ©\nSection 4: Category-theoretic Representation of Archimedean Structures\nIn the present §4, we explain the relatively minor modifications to the the-\nory developed in the present paper for log schemes that are necessary in order to\naccommodate categories of log schemes equipped with archimedean structures as\ndiscussed in [ArLgSch]. At a more concrete level, we observe that\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 49\n· Theorem 3.1;\n· Proposition 4.3;\n· Proposition 4.4\nof [ArLgSch] depend on the portions of the theory of [LgSch] that [cf. Example 0.3;\nRemark 1.4.1] are in error. Thus, in the present §4, we explain how these results,\nas well as the main theorem of [ArLgSch] [i.e., [ArLgSch], Theorem 5.1], may be\nrepaired by applying the theory developed thus far in the present paper.\nWe begin by reviewing [and slightly modifying] the notation introduced at the\nbeginning of [ArLgSch], §4. Write\nSCH\nfor the category of arithmetic schemes,\nSCH\nlog\nfor the category of arithmetic log schemes [cf. [ArLgSch], Definition 4.2, and the\nfollowing discussion], and\nSCH ⊆ SCH; SCHlog ⊆ SCHlog\nfor the full subcategories determined by the purely nonarchimedean objects [cf.\n[ArLgSch], Definition 4.3, (i)]. Let X\nlog\nbe an object of SCH\nlog\n. Thus, X\nlog\ndeter-\nmines underlying objects X log, X, and X of the categories SCHlog, SCH, and SCH,\nrespectively. Write\nSCH\nlog\n(X\nlog\n)\ndef\n= (SCH\nlog\n)\nX\nlog ; SCHlog(X log)\ndef\n= (SCHlog)Xlog ;\nSCH(X)\ndef\n= SCHX ; SCH(X)\ndef\n= SCHX\nfor the respective categories of “objects over the subscripted objects” [cf. the nota-\ntional conventions introduced in the discussion entitled “Categories” in [ArLgSch],\n§2] and\nSch\nlog\n(X\nlog\n) ⊆ SCHlog(X log); Schlog(X log) ⊆ SCHlog(X log);\nSch(X) ⊆ SCH(X); Sch(X) ⊆ SCH(X)\nfor the full subcategories determined by the noetherian objects. To simplify the\nexposition, we shall often refer to the domain of an arrow which is an object of any\nof the categories of the preceding display as an “object” of the category.\nNote that the notation just introduced is consistent with the notational con-\nventions introduced at the beginning of §1 of the present paper for “Schlog(X log)”\nand “Sch(X)”. Indeed, if X log is any locally noetherian fs log scheme, then one\nmay define [in a fashion consistent with the notation introduced above!]\nSCHlog(X log)\nto be the category whose objects are morphisms of log schemes of locally finite type\nY log → X log, where Y log is a locally noetherian fs log scheme, and whose morphisms\n50 SHINICHI MOCHIZUKI\n[from an object Y\nlog\n1 → X log to an object Y log2 → X log] are morphisms of locally\nfinite type Y\nlog\n1 → Y log2 lying over X log. In a similar vein, if X is any locally\nnoetherian scheme, then one may define [in a fashion consistent with the notation\nintroduced above!]\nSCH(X)\nto be the category whose objects are morphisms of schemes of locally finite type\nY → X, where Y is a locally noetherian scheme, and whose morphisms [from an\nobject Y1 → X to an object Y2 → X] are morphisms of locally finite type Y1 → Y2\nlying over X.\nDefinition 4.1.\n(i) We shall apply similar terminology to data [i.e., such as collections of objects\nand collections of morphisms] associated to any of the categories\nSch\nlog\n(X\nlog\n), SCH\nlog\n(X\nlog\n), Schlog(X log), SCHlog(X log),\nSch(X), SCH(X), Sch(X), SCH(X)\nto the terminology that has already been established earlier in the present paper\nfor “Schlog(X log)” or in [LgSch], §1, for “Sch(X)” whenever this terminology may\nbe defined in an evidently analogous fashion for the category of the above display\nunder consideration. When it is necessary, in order to avoid confusion, to specify\nthe category of the above display with respect to which the terminology is to be\nunderstood, we shall append an appropriate prefix such as\nSch\nlog\n-, SCH\nlog\n-, Schlog-, SCHlog- Sch-, SCH-, Sch-, SCH-\nto the terminology in question. This convention concerning prefixes will be applied,\nin particular, when the terminology is to be understood as being applied to the\nunderlying object in one of the categories of the first display that is determined by\nanother of the categories of the first display.\n(ii) Let Clog ∈ {Schlog, SCHlog}, X log an arithmetic log scheme, Y log an object\nof Clog(X log). Then we shall say that Y log is submonically nonarchimedean if it holds\nthat every submonic one-pointed object Z\nlog\nof Clog(X log) that admits a morphism\nto Y\nlog\nis purely nonarchimedean.\nTheorem 4.2. (Equivalences of categories of schemes) Let C ∈ {Sch, SCH}.\nFor i = 1, 2, let Xi be a locally noetherian scheme. Then, relative to the nota-\ntion introduced at the beginning of the present §4, any equivalence of categories\nΦ : C(X1) ∼→ C(X2)\narises from a unique isomorphism of schemes X1\n∼→ X2.\nProof. When C = Sch, Theorem 4.2 is precisely the content of [LgSch], Theorem\n1.7, (ii). When C = SCH, Theorem 4.2 follows from an entirely similar argument.\n©\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 51\nTheorem 4.3. (Equivalences of categories of arithmetic schemes) Let\nC ∈ {Sch, SCH}. For i = 1, 2, let Xi be an arithmetic scheme [cf. [ArLgSch],\nDefinition 4.2, (i)]. Then, relative to the notation introduced at the beginning of\nthe present §4, any equivalence of categories\nΦ : C(X1) ∼→ C(X2)\narises from a unique isomorphism of arithmetic schemes X1\n∼→ X2.\nProof. If C = Sch, then set C def= Sch; if C = SCH, then set C def= SCH. Then\nTheorem 4.3 follows, in effect, by combining the theory of [LgSch], §1, with the\nnon-logarithmic portion of the theory developed in [ArLgSch], §4, §5. [That is to\nsay, the errors in [ArLgSch] discussed at the beginning of the present §4 concern\nsubtleties that arise from the log structures of the log schemes involved and hence\nhave no effect on the non-logarithmic portion of the theory.] Indeed, let i ∈ {1, 2};\nwrite Xi for the underlying scheme of Xi. Then one verifies immediately that\nthe C-minimal objects of C(Xi) are the purely nonarchimedean objects that arise\nfrom the C-minimal objects of C(Xi). Thus, the one-pointed objects of C(Xi) are\nprecisely the objects Y such that MinPt(Y ) = MinPt(Y ) [where we write Y for\nthe object of C(Xi) determined by the underlying scheme of Y ] is of cardinality\none. This characterization of one-pointed objects of C(Xi) allows one to circum-\nvent the application of [ArLgSch], Proposition 4.3, in the theory of [ArLgSch], §4.\nIn particular, we obtain a category-theoretic characterization of C-minimal point-\nhulls as in [ArLgSch], Proposition 4.4, (iii). One thus obtains — i.e., by considering\nepimorphisms as in [ArLgSch], Proposition 4.5 — a category-theoretic characteri-\nzation of the purely nonarchimedean one-pointed objects of C(Xi) as in [ArLgSch],\nCorollary 4.1, (i), and of the purely archimedean morphisms [cf. [ArLgSch], Def-\ninition 4.3, (ii)] of C(Xi) as in [ArLgSch], Corollary 4.1, (ii). In particular, we\nobtain a category-theoretic characterization, as in [ArLgSch], Corollary 4.2, of the\npurely nonarchimedean objects of C(Xi) and hence, by applying Theorem 4.2, a\ncategory-theoretic reconstruction of the underlying scheme of an object of C(Xi), as\nin [ArLgSch], Corollary 4.3. Now, to complete the proof of Theorem 4.3 [cf. the\nproof of [ArLgSch], Theorem 5.1], it suffices to apply the “non-logarithmic global\ncompatibility” established in [ArLgSch], Lemma 5.1. ©\nNext, we consider analogues of Theorem 2.6 for SCHlog, Sch\nlog\n, and SCH\nlog\n.\nTheorem 4.4. (Reconstruction of the scheme structure of submonic\nobjects for SCHlog) For i = 1, 2, let X\nlog\ni be a locally noetherian fs log scheme\n[cf. the discussion entitled “Log schemes” in §0]. We shall apply the notation\nintroduced at the beginning of the present §4. Let\nΦ : SCHlog(X\nlog\n1 )\n∼→ SCHlog(X log2 )\nbe an [arbitrary!] equivalence of categories. Then:\n(i) Φ preserves the following:\n(i-a) monomorphisms;\n52 SHINICHI MOCHIZUKI\n(i-b) empty objects;\n(i-c) connected objects;\n(i-d) minimal objects;\n(i-e) minimal points;\n(i-f) submonic one-pointed objects;\n(i-g) ranks of minimal objects;\n(i-h) SLEM morphisms;\n(i-i) submonic objects;\n(i-j) scheme-like morphisms between minimal objects;\n(i-k) scheme-like morphisms between submonic objects;\n(i-l) the submonic dimension of objects.\n(ii) For i = 1, 2, let Y\nlog\ni be an object of SCH\nlog(X\nlog\ni ); write Yi for the under-\nlying scheme of Y\nlog\ni . Suppose further that Φ(Y\nlog\n1 ) = Y\nlog\n2 . Thus, [cf. the portion\nof (i) concerning (i-i)] Y\nlog\n1 is submonic if and only if Y\nlog\n2 is. Suppose that Y\nlog\ni\nis submonic for i = 1, 2. Then Φ induces an equivalence of categories\n(\nSCH(Y1)\n∼→\n)\nSCHlog(Y\nlog\n1 )|sch-lk ∼→ SCHlog(Y log2 )|sch-lk\n( ∼→ SCH(Y2)\n)\n— where the equivalences in parentheses are the evident analogues for SCH, SCHlog\nof the natural equivalences of Definition 1.1, (iv) — that is functorial [in the evi-\ndent sense!] with respect to Y\nlog\n1 , Y\nlog\n2 . Finally, the composite of the equivalences of\ncategories in the above display induces, by applying Theorem 4.2, an isomorphism\nof schemes\nY1\n∼→ Y2\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nProof. The proof is entirely similar to the proof of Theorem 2.6. ©\nTheorem 4.5. (Reconstruction of the scheme structure of submonic\nobjects for Sch\nlog\n, SCH\nlog\n) Let Clog ∈ {Schlog, SCHlog}. If Clog = Schlog, then set\nClog def= Schlog, C def= Sch; if Clog = SCHlog, then set Clog def= SCHlog, C def= SCH. For\ni = 1, 2, let X\nlog\ni be an arithmetic log scheme [cf. [ArLgSch], Definition 4.2,\n(ii)]. We shall apply the notation introduced at the beginning of the present §4. Let\nΦ : Clog(X log1 ) ∼→ C\nlog\n(X\nlog\n2 )\nbe an [arbitrary!] equivalence of categories. Then:\n(i) Φ preserves the following:\n(i-a) monomorphisms;\n(i-b) empty objects;\n(i-c) connected objects;\n(i-d) minimal objects;\n(i-e) minimal points;\n(i-f) submonic one-pointed objects;\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 53\n(i-fnon) purely nonarchimedean submonic one-pointed objects;\n(i-g) ranks of minimal objects;\n(i-h) Clog-SLEM morphisms;\n(i-i) submonic objects;\n(i-inon) purely nonarchimedean submonic objects;\n(i-j) Clog-scheme-like morphisms between minimal objects;\n(i-k) Clog-scheme-like morphisms between submonic objects;\n(i-l) the submonic dimension of objects.\n(ii) For i = 1, 2, let Y\nlog\ni be an object of C\nlog\n(X\nlog\ni ); write Y\nlog\ni for the under-\nlying log scheme of Y\nlog\ni , Yi for the underlying scheme of Y\nlog\ni . Suppose further\nthat Φ(Y\nlog\n1 ) = Y\nlog\n2 . Thus, [cf. the portion of (i) concerning (i-i), (i-i\nnon)] Y\nlog\n1 is\nsubmonic if and only if Y\nlog\n2 is; Y\nlog\n1 is purely nonarchimedean submonic if\nand only if Y\nlog\n2 is. Suppose that Y\nlog\ni is submonic for i = 1, 2. Then Φ induces\nan equivalence of categories\n(\nC(Y1) ∼→\n)\nClog(Y log1 )|sch-lk ∼→ Clog(Y log2 )|sch-lk\n( ∼→ C(Y2)\n)\n— where the equivalences in parentheses are the evident analogues for C, Clog of\nthe natural equivalences of Definition 1.1, (iv) — that is functorial [in the evident\nsense!] with respect to Y\nlog\n1 , Y\nlog\n2 . Finally, the composite of the equivalences of\ncategories in the above display induces, by applying Theorem 4.2, an isomorphism\nof schemes\nY1\n∼→ Y2\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nProof. First, we consider assertion (i). The preservation of (i-a), (i-b), (i-c), (i-d),\n(i-e), (i-f), and (i-g) follows from an entirely similar argument to the argument\napplied in the proof of the preservation of the corresponding properties in Theorem\n2.6, (i). Here, we observe that one verifies immediately, by arguing as in [ArLgSch],\nProposition 4.2, that\nthe minimal objects of Clog(X logi ) are precisely the purely nonarchimedean\nobjects of Clog(X logi ) that arise from the minimal objects of Clog(X logi ),\nwhere we write X\nlog\ni for the underlying log scheme of X\nlog\ni .\nThe preservation of (i-fnon) now follows, in light of the preservation of (i-f), from\nan entirely similar argument — i.e., by considering epimorphisms as in [ArLgSch],\nProposition 4.5 — to the argument applied to verify the category-theoretic charac-\nterization of purely nonarchimedean one-pointed objects given in [ArLgSch], Corol-\nlary 4.1, (i). In light of the preservation of (i-fnon), the preservation of (i-h) follows\nfrom an entirely similar argument to the argument applied in the proof of the\npreservation of (i-h) in Theorem 2.6, (i). In light of the preservation of (i-h), the\npreservation of (i-i) follows from an entirely similar argument to the argument ap-\nplied in the proof of the preservation of (i-i) in Theorem 2.6, (i). The preservation\nof (i-inon) now follows from the preservation of (i-f), (i-fnon), (i-i), since [one verifies\n54 SHINICHI MOCHIZUKI\nimmediately that] the purely nonarchimedean submonic objects Y\nlog\nof Clog(X logi )\nmay be characterized as the submonically nonarchimedean submonic objects Y\nlog\nof Clog(X logi ). In light of the preservation of (i-i), the preservation of (i-j), (i-k)\nfollows from an entirely similar argument to the argument applied in the proof of\nthe preservation of (i-j), (i-k) in Theorem 2.6, (i). This completes the proof of\nassertion (i), except for the verification of the preservation of (i-l).\nNext, we consider assertion (ii). Suppose that Y\nlog\ni is submonic for i = 1, 2.\nLet Z\nlog\ni → Y\nlog\ni be a purely archimedean morphism of C\nlog\n(X\nlog\ni ) such that Z\nlog\ni\nis purely nonarchimedean submonic. Here, one verifies immediately that such a\nmorphism Z\nlog\ni → Y\nlog\ni exists, and, moreover, that Z\nlog\ni may be characterized up to\nisomorphism as an object over Y\nlog\ni by the property that any arrow T\nlog → Y logi\nin Clog(X logi ) such that T\nlog\nis purely nonarchimedean submonic admits a unique\nfactorization T\nlog → Z logi → Y\nlog\ni . Thus, it follows from the portion of assertion (i)\nconcerning the preservation of (i-inon) that we may assume without loss of generality\nthat Φ(Z\nlog\n1 ) = Z\nlog\n2 . Moreover, since Z\nlog\ni is purely nonarchimedean, one verifies\nimmediately from the various definitions involved that the full subcategory\nClog(Z logi ) ⊆ C\nlog\n(Y\nlog\ni )\nadmits a natural equivalence of categories Clog(Y logi ) ∼→ C\nlog\n(Z\nlog\ni ) [cf. the state-\nment of [ArLgSch], Corollary 4.3]. Thus, by applying the portion of assertion (i)\nconcerning the preservation of (i-k), one verifies immediately that assertion (ii)\nfollows immediately follows from an entirely similar argument to the argument ap-\nplied to verify Theorem 2.6, (ii). Finally, the portion of assertion (i) concerning\nthe preservation of (i-l) follows from an entirely similar argument to the argument\napplied in the proof of the preservation of (i-l) in Theorem 2.6, (i). ©\nNext, we consider the analogue of Corollary 2.12 and Theorems 3.6, 3.8 for\nSCHlog.\nTheorem 4.6. (Reconstruction of the log scheme structure of arbitrary\nobjects for SCHlog) For i = 1, 2, let X\nlog\ni be a locally noetherian fs log scheme\n[cf. the discussion entitled “Log schemes” in §0]. We shall apply the notation\nintroduced at the beginning of the present §4. Let\nΦ : SCHlog(X\nlog\n1 )\n∼→ SCHlog(X log2 )\nbe an [arbitrary!] equivalence of categories. Then:\n(i) Φ preserves the following:\n(i-a) log-Dedekind objects;\n(i-b) the set SmCp(−) associated to a log-Dedekind object;\n(i-c) the subsets of the set SmCp(−) of (i-b) which are [N-]chains;\n(i-d) partitions at elements of the set SmCp(−) of (i-b);\n(i-e) orientable objects;\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 55\n(i-f) homogeneous objects;\n(i-g) one-pointed objects;\n(i-h) point-hulls with one-pointed codomains;\n(i-i) minimal point-hulls with one-pointed codomains;\n(i-j) log-nodal objects.\n(ii) Φ preserves the following:\n(ii-a) point-equivalent pairs of arrows;\n(ii-b) the set-valued functor LCPt(−) [up to natural equivalence];\n(ii-c) arrows which are minimal point-hulls;\n(ii-d) scheme-like morphisms between arbitrary objects.\n(iii) For i = 1, 2, let Y\nlog\ni be an object of SCH\nlog(X\nlog\ni ); write Yi for the un-\nderlying scheme of Y\nlog\ni . Suppose further that Φ(Y\nlog\n1 ) = Y\nlog\n2 . Then Φ induces an\nequivalence of categories\n(\nSCH(Y1)\n∼→\n)\nSCHlog(Y\nlog\n1 )|sch-lk ∼→ SCHlog(Y log2 )|sch-lk\n( ∼→ SCH(Y2)\n)\n— where the equivalences in parentheses are the evident analogues for SCH, SCHlog\nof the natural equivalences of Definition 1.1, (iv) — that is functorial [in the evi-\ndent sense!] with respect to Y\nlog\n1 , Y\nlog\n2 . Finally, the composite of the equivalences of\ncategories in the above display induces, by applying Theorem 4.2, an isomorphism\nof schemes\nY1\n∼→ Y2\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\n(iv) There exists a unique isomorphism of log schemes\nX\nlog\n1\n∼→ X log2\nsuch that Φ is isomorphic to the equivalence of categories induced by this isomor-\nphism of log schemes X\nlog\n1\n∼→ X log2 .\nProof. In light of Theorem 4.4, the proof of assertion (i) (respectively, assertion\n(ii)) is entirely similar to the proof of Theorem 3.6, (i) (respectively, Corollary 2.12,\n(i)). Now assertion (iii) follows from the portion of assertion (ii) concerning the\npreservation of (ii-d) by applying an entirely similar argument to the argument\napplied to verify Corollary 2.12, (ii). Finally, it follows immediately from assertion\n(iii) that Φ preserves objects whose underlying scheme is noetherian [i.e., quasi-\ncompact], and hence that Φ induces an equivalence of categories\nSchlog(X\nlog\n1 )\n∼→ Schlog(X log2 )\n[i.e., as in Theorem 3.8]. Thus, assertion (iv) follows immediately from Theorem\n3.8, (iii). ©\nFinally, we consider analogues of Theorems 3.6, 3.8 for Sch\nlog\n, SCH\nlog\n. In\norder to formulate and prove these analogues, it will be necessary to introduce\n56 SHINICHI MOCHIZUKI\nsome new terminology [patterned after the terminology introduced in Definition\n3.3], as follows.\nDefinition 4.7. Let Clog ∈ {Schlog, SCHlog}. If Clog = Schlog, then set\nClog def= Schlog; if Clog = SCHlog, then set Clog def= SCHlog. Let X log be an arith-\nmetic log scheme. We shall apply the notation introduced at the beginning of the\npresent §4. Let Y log be a connected, non-submonic, Clog-log-Dedekind, submonically\nnonarchimedean [cf. Remark 4.7.1 below] object of Clog(X log); write Y log for the\nunderlying log scheme of Y\nlog\n. Let γ ∈ Clog-SmCp(Y log) def= SmCp(Y log). Write\nMono(Y\nlog\n)\nfor the full subcategory of Clog(Y log) determined by the arrows H log → Y log of\nClog(X log) which are monomorphisms in Clog(X log).\n(i) Let C1, C2 ⊆ Clog-SmCp(Y log) be Clog-chains. Then we shall say that the\npair of Clog-chains {C1, C2} forms a C̈log-partition at γ if the Clog-chains C1, C2\nsatisfy the following conditions:\n(i-a) C1 ∪ C2 = Clog-SmCp(Y log), C1 ∩ C2 = {γ};\n(i-b) for i = 1, 2, the subset Ci \\ {γ} ⊆ Clog-SmCp(Y log) is a Clog-N-chain\n[hence nonempty];\n(i-c) the Clog-N-chains of (i-b) are “maximal” in the sense that every Clog-N-\nchain C ⊆ Clog-SmCp(Y log) such that γ \u0006∈ C is contained in Ci for some\ni ∈ {1, 2};\n(i-d) if, for i = 1, 2, we write Ψi for the subfunctor of the contravariant functor\ndetermined by the terminal object [i.e., Y\nlog\n] of Mono(Y\nlog\n) that consists\nof objects h\nlog\n: H\nlog \u0003 Y log of Mono(Y log) such that every composite\nmorphism H\nlog\n∗ \u0003 H\nlog \u0003 Y log, where H log∗ \u0003 H\nlog\nis a minimal point\nof H\nlog\n, determines an underlying morphism in Clog(Y log) that factors\nthrough some representative of an element ∈ Ci (⊆ Clog-SmCp(Y log)),\nthen Ψi is representable by an object h\nlog\ni : Y\nlog\ni \u0003 Y\nlog\nof Mono(Y\nlog\n).\nWe shall say that Y\nlog\nis C̈log-orientable if Y log admits a C̈log-partition at every\nelement of Clog-SmCp(Y log).\n(ii) Let {C1, C2} be a C̈log-partition at γ. Suppose that hlog1 , h\nlog\n2 are as in (i-d).\nThen we shall say that the C̈log-partition {C1, C2} is C̈log-seamless if the following\ncondition is satisfied:\na monomorphism h\nlog\n: H\nlog \u0003 Y log in Clog(X log) is an isomorphism if\nand only if, for i = 1, 2, the projection H\nlog ×\nY\nlog Y\nlog\ni → Y\nlog\ni associated\nto the fiber product determined by h\nlog\nand h\nlog\ni is an isomorphism.\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 57\nWe shall say that Y\nlog\nis C̈log-homogeneous if Y log is C̈log-orientable, and, moreover,\nno C̈log-partition at an element ∈ Clog-SmCp(Y log) is C̈log-seamless.\nRemark 4.7.1. Let Y\nlog\nbe as in Definition 4.7, i.e., a connected, non-submonic,\nClog-log-Dedekind, submonically nonarchimedean object of Clog(X log). Write Y log\nfor the underlying log scheme of Y\nlog\n; Y for the underlying arithmetic scheme of\nY\nlog\n; Y for the underlying scheme of Y ; K log for the compact set that determines\nthe archimedean structure of Y\nlog\n[i.e., the set “H” of [ArLgSch], Definition 4.2,\n(ii)]; K for the compact set that determines the archimedean structure of Y [i.e.,\nthe set “H” of [ArLgSch], Definition 4.2, (i)]. Thus, it follows immediately from\nthe various definitions involved that we have a natural surjection K log \u0002 K whose\nfibers are compact [cf. the discussion of such compact subsets in the proof of [ArL-\ngSch], Lemma 4.1] . Now observe that the assumption that Y\nlog\nis submonically\nnonarchimedean implies [cf. Proposition 3.2, (i), (ii)] that\nK is a finite compact set which is supported over the nodal points of Y log.\nSince the finiteness of K implies that any [e.g., open!] subset of K is compact [i.e.,\nrelative to the topology induced by K], we thus conclude that\nany subset of K log that arises as the inverse image via the natural surjec-\ntion K log \u0002 K of a subset of K is compact [i.e., relative to the topology\ninduced by K log].\nIn particular, it follows that any open subscheme Z ⊆ Y determines, in a natural\nway, not only a log scheme Z log\ndef\n= Y log×Y Z, but also an arithmetic scheme Z and\nan arithmetic log scheme Z\nlog\n[i.e., by considering the subsets of K, K log consisting\nof points that map to points of Z (⊆ Y )]. Moreover, one verifies immediately that\nZ\nlog\n(respectively, Z) represents the covariant subfunctor of the functor\nrepresented by Y\nlog\n(respectively, Y ) on the category of arithmetic log\nschemes (respectively, arithmetic schemes) determined by the condition\non a morphism to Y\nlog\n(respectively, Y ) that the associated underlying\nmorphism of schemes maps into Z ⊆ Y .\nThese observations may be applied, for instance, to open subschemes of Y that\narise as images of open immersions of the sort discussed in Proposition 3.4, (ii).\nTheorem 4.8. (Reconstruction of the arithmetic log scheme structure\nof arbitrary objects for Sch\nlog\n, SCH\nlog\n) Let Clog ∈ {Schlog, SCHlog}. If Clog =\nSch\nlog\n, then set Clog def= Schlog; if Clog = SCHlog, then set Clog def= SCHlog. For\ni = 1, 2, let X\nlog\ni be an arithmetic log scheme [cf. [ArLgSch], Definition 4.2,\n(ii)]. We shall apply the notation introduced at the beginning of the present §4. Let\nΦ : Clog(X log1 ) ∼→ C\nlog\n(X\nlog\n2 )\nbe an [arbitrary!] equivalence of categories. Then:\n58 SHINICHI MOCHIZUKI\n(i) Let Y\nlog\n, Z\nlog\nbe objects of Clog(X logi ), for some i ∈ {1, 2}, that are con-\nnected, non-submonic, Clog-log-Dedekind, and submonically nonarchime-\ndean; write Y log, Z log for the underlying log schemes of Y\nlog\n, Z\nlog\n, respectively.\nSuppose further that Z\nlog\nis purely nonarchimedean. Then the following prop-\nerties hold:\n(i-aY ) every C̈log-partition at an element γ ∈ Clog-SmCp(Y log)\ndetermines a Clog-partition at γ;\n(i-aZ) there is a natural bijective correspondence between\nC̈log-partitions at elements ∈ Clog-SmCp(Z log) and Clog-\npartitions at elements ∈ Clog-SmCp(Z log);\n(i-bY ) if Y\nlog\nis C̈log-orientable, then Y log is Clog-orienta-\nble;\n(i-bZ) Z\nlog\nis C̈log-orientable if and only if Z log is Clog-\norientable;\n(i-cZ) a C̈log-partition at an element ∈ Clog-SmCp(Z log) is\nC̈log-seamless if and only if it corresponds to a Clog-\npartition [cf. (i-aZ)] that is Clog-seamless;\n(i-dY ) if Y\nlog\nis C̈log-homogeneous, then it is one-pointed,\nand Y logsm is empty;\n(i-dZ) Z\nlog\nis C̈log-homogeneous if and only if Z log is Clog-\nhomogeneous.\n(ii) Φ preserves the following:\n(ii-a) Clog-log-Dedekind objects;\n(ii-b) the set Clog-SmCp(−) associated to a Clog-log-Dedekind object;\n(ii-c) the subsets of the set Clog-SmCp(−) of (ii-b) which are Clog-[N-]chains;\n(ii-d) C̈log-partitions at elements of the set Clog-SmCp(−) of (ii-b);\n(ii-e) C̈log-orientable objects;\n(ii-f) C̈log-homogeneous objects;\n(ii-g) one-pointed objects;\n(ii-h) point-hulls with one-pointed codomains;\n(ii-i) minimal point-hulls with one-pointed codomains.\n(iii) For i = 1, 2, let Y\nlog\ni be an object of C\nlog\n(X\nlog\ni ); write Y\nlog\ni for the underly-\ning log scheme of Y\nlog\ni . Suppose further that Φ(Y\nlog\n1 ) = Y\nlog\n2 . Then Y\nlog\n1 is purely\nnonarchimedean if and only if Y\nlog\n2 is. In particular, Φ induces an equivalence\nof categories\nClog(Y log1 ) ∼→ Clog(Y log2 )\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 . Finally, the\nequivalence of categories in the above display induces, by applying Theorems 3.8,\n(iii); 4.6, (iv), an isomorphism of log schemes\nY\nlog\n1\n∼→ Y log2\nthat is functorial [in the evident sense!] with respect to Y\nlog\n1 , Y\nlog\n2 .\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 59\n(iv) There exists a unique isomorphism of arithmetic log schemes\nX\nlog\n1\n∼→ X log2\nsuch that Φ is isomorphic to the equivalence of categories induced by this isomor-\nphism of arithmetic log schemes X\nlog\n1\n∼→ X log2 .\nProof. First, we consider assertion (i). Properties (i-aY ), (i-aZ), (i-bY ), (i-bZ),\n(i-cZ), and (i-dZ) follow formally from the definitions [cf. also the first display in\nthe proof of Theorem 4.5]. Property (i-dY ) then follows, in light of properties (i-\naY ) and (i-bY ) [cf. also the first display in the proof of Theorem 4.5], by applying\na similar argument to the argument [i.e., involving Proposition 3.4, (ii)] applied\nin the proof of Proposition 3.4, (iii). Here, we note that one must apply the as-\nsumption [cf. the beginning of Definition 4.7] that any C̈log-homogeneous object\nis submonically nonarchimedean in order to conclude that any C̈log-partition that\ndetermines a Clog-seamless Clog-partition as in [the evident analogue for Clog of]\nProposition 3.4, (ii), is necessarily C̈log-seamless. That is to say, this assumption\nthat any C̈log-homogeneous object is submonically nonarchimedean implies [cf. the\ndiscussion of Remark 4.7.1] that the discrepancy between C̈log-\/Clog-seamless C̈log-\n\/Clog-partitions may — at least in the case of Clog-seamless Clog-partitions as in [the\nevident analogue for Clog of] Proposition 3.4, (ii) — be ignored. This completes the\nproof of assertion (i).\nNext, we observe that, in light of Theorem 4.5, (i), (ii), assertion (ii) follows\nby applying a similar argument to the argument applied to verify Theorem 3.6, (i).\nHere, we observe that the preservation of the crucial property of being submonically\nnonarchimedean [cf. the beginning of Definition 4.7] follows formally from the\nportion of Theorem 4.5, (i), concerning the preservation of (i-f), (i-fnon). Also, we\nobserve, with regard to the preservation of (ii-g), that, by applying\n· the property (i-dY ) of assertion (i) in place of Proposition 3.4, (iii), and\n· the property (i-dZ) of assertion (i), together with the evident analogue\nfor Clog of Proposition 3.4, (iv), in place of Proposition 3.4, (iv),\none obtains a suitable analogue for Clog — i.e., by considering C̈log-homogeneous\nobjects — of the characterization of one-pointed objects given in Proposition 3.5,\n(i). This completes the proof of assertion (ii).\nNext, we consider assertion (iii). First, let us observe that the portion of as-\nsertion (ii) concerning the preservation of (ii-g), (ii-i) allows one to circumvent the\napplication of [ArLgSch], Propositions 4.3, 4.4, in the theory of [ArLgSch], §4. One\nthus obtains — i.e., by considering epimorphisms as in [ArLgSch], Proposition 4.5 —\ncategory-theoretic characterizations of the purely nonarchimedean one-pointed ob-\njects of Clog(X logi ) as in [ArLgSch], Corollary 4.1, (i), and of the purely archimedean\nmorphisms of Clog(X logi ) as in [ArLgSch], Corollary 4.1, (ii) [cf. also Proposition 1.4,\n(iii), (v), of the present paper]. In particular, we obtain a category-theoretic charac-\nterization, as in [ArLgSch], Corollary 4.2, of the purely nonarchimedean objects of\nClog(X logi ) and hence, by applying Theorems 3.8, (iii); 4.6, (iv), a category-theoretic\nreconstruction of the underlying log scheme of an object of Clog(X logi ), as in [ArL-\ngSch], Corollary 4.3. This completes the proof of assertion (iii). Finally, assertion\n60 SHINICHI MOCHIZUKI\n(iv) follows from assertion (iii) [cf. the proof of [ArLgSch], Theorem 5.1], by ap-\nplying the “logarithmic global compatibility” established in [ArLgSch], Lemma 5.2.\n©\nAppendix\nIn the present Appendix, we discuss in more detail, at the level of individual\npropositions, lemmas, corollaries, theorems, and examples, the validity of the theory\ndeveloped in [LgSch] and [ArLgSch].\nFirst, we recall that the errors discussed in the Introduction of the present\npaper have no effect on [LgSch], §1. The effect of these errors on the validity of the\nstatements [second column], as well as on the validity of the proofs [third column],\nof the individual propositions, lemmas, corollaries, and theorems of [LgSch], §2,\nis summarized in Fig. 1 below. Here, the symbol “©” indicates no effect on\nthe validity in question; the symbol “×” indicates some effect on the validity in\nquestion. Certain results that concern equivalences are divided into sufficiency and\nnecessity portions. Moreover, the necessity portion of [LgSch], Proposition 2.3, is\ndivided into\n· a portion concerning whether or not the underlying morphism of schemes\nis a monomorphism in the case where the given morphism of log schemes\nis scheme-like,\n· a portion concerning whether or not the induced morphism of groupifi-\ncations of characteristics is surjective, and\n· a portion concerning whether or not the underlying morphism of schemes\nis a monomorphism in the case where the given morphism of log schemes\nis not scheme-like.\nThe statement of [LgSch], Lemma 2.6, (ii), is also divided into a closed immersion\nportion, an isomorphism portion, and a [final] surjectivity portion. We also indicate,\nin the fourth column of Fig. 1, whether [“©”] or not [“×”] the result in question is\napplied, in an explicit [i.e., via a direct reference] logical sense, in the present paper.\nThe data of this fourth column does not include references for the statements of\ndefinitions\/conditions or references made for the sake of pointing out content that\nis related in an expository sense [i.e., but not in a logical sense!]. The unique “\u0019” in\nthis fourth column in the case of [LgSch], Proposition 2.4, indicates that although\nwe apply this result in an explicit logical sense in the present paper [i.e., despite the\nfact that the proof given in [LgSch] is in error!], this does not result in any logical\ngaps, since the proof of [LgSch], Proposition 2.4, given in [LgSch] may be repaired\nif, instead of applying [LgSch], Proposition 2.3, one applies Proposition 1.4, (vi),\nof the present paper [i.e., which corresponds to the necessity portion of [LgSch],\nProposition 2.3, in the case of submonic log schemes].\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 61\nExplicit\nVali- Vali- logical\nIndividual dity dity appli-\npropositions\/lemmas\/ of of cation\ncorollaries\/theorems state- proof in the\nment present\npaper\n2.3 (surjectivity portion of\nnecessity); 2.5; 2.6, (i), (iii); 2.6, (ii) © © ©\n(closed immersion, [final] surjectivity\nportions); 2.16 [cf. Remark 3.7.1]\n2.1; 2.2; 2.3 (sufficiency);\n2.3 (monomorphism portion\nof necessity: scheme-like case); © © ×\n2.7, (i), (ii); 2.8;\n2.12, (i) (necessity); 2.12, (ii);\n2.17; 2.18; 2.19, (i); 2.20\n2.4 © × \u0019\n2.14; 2.15; 2.19, (ii); 2.13 © × ×\n2.3 (monomorphism portion\nof necessity: non-scheme-like case);\n2.6, (ii) (isomorphism portion); × × ×\n2.7, (iii); 2.9; 2.10;\n2.12, (i) (sufficiency)\nFig. 1: Validity of individual propositions\/lemmas\/corollaries\/theorems\nof [LgSch]\nNext, we consider the effect of the errors discussed in the Introduction of the\npresent paper on [ArLgSch]. Here, we recall that [ArLgSch], §1, consists of an ex-\npository introduction to the theory of [ArLgSch], while [ArLgSch], §2, is devoted to\na discussion of the notations and conventions applied in [ArLgSch]. Thus, it suffices\nto consider the effect of the errors discussed in the Introduction of the present paper\non [ArLgSch], §3, §4, §5. The effect of these errors on the validity of the statements\n[second column], as well as on the validity of the proofs [third column], of the in-\ndividual propositions [“P”], lemmas [“L”], corollaries [“C”], theorems [“T”], and\nexamples [“E”] of [ArLgSch], §3, §4, §5, is summarized in Fig. 2 below. Here, the\nsymbol “©” indicates no effect on the validity in question; the symbol “×” indicates\nsome effect on the validity in question; the numbers in parentheses indicate, for ease\nof reference, the corresponding result in the preprint version [available on the home-\npage http:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/papers-english.html] of\n[ArLgSch]. We also indicate, in the fourth column of Fig. 2, whether [“©”] or\nnot [“×”] the result in question is applied, in an explicit [i.e., via a direct reference]\nlogical sense, in the present paper. The data of this fourth column does not include\nreferences for the statements of definitions\/conditions or references made for the\n62 SHINICHI MOCHIZUKI\nsake of pointing out content that is related in an expository sense [i.e., but not in a\nlogical sense!]. The three “\u0019’s” in this fourth column indicate the following state\nof affairs:\n· The results marked with a “\u0019” are indeed applied in an explicit logical\nsense in the present paper in the proofs of Theorems 4.3, 4.5, 4.8 – i.e.,\ndespite the fact that the proofs given in [ArLgSch] of these results are in\nerror!\n· The explicit logical application of these results in the present paper does\nnot, however, result in any logical gaps for the following reason: In the\ncase of the first and third (respectively, case of the second) “\u0019”, the only\nproblem with the proofs given in [ArLgSch] is that they rely on the recon-\nstruction of one-pointed objects given in [ArLgSch], Proposition 4.3 [i.e.,\nwhich is in error!] (respectively, on [ArLgSch], Corollaries 4.2, 4.3 [i.e.,\nwhose proofs are in error!]). On the other hand, these results are only\napplied in the present paper in situations in which the one-pointed objects\nhave already been reconstructed (respectively, in which results correspond-\ning to [ArLgSch], Corollaries 4.2, 4.3, have already been proven).\n· Put another way, one may think of the application in the present paper of\nthe results marked with a “\u0019” as consisting of a “similar argument” to the\nargument given in [ArLgSch] — i.e., a “similar argument” which does not\nsuffer from the logical gaps of [ArLgSch], since, in the case of the first and\nthird (respectively, case of the second) “\u0019”, this “similar argument” is only\napplied in situations in which the one-pointed objects have already been\nreconstructed (respectively, in which results corresponding to [ArLgSch],\nCorollaries 4.2, 4.3, have already been proven).\nExplicit\nVali- Vali- logical\nIndividual dity dity appli-\npropositions\/lemmas\/ of of cation\ncorollaries\/theorems\/ state- proof in the\nexamples ment present\npaper\nL4.1 (2.5); P4.2 (2.6); C4.1 (2.10), (ii) © © ©\nP4.1 (2.4); P4.4 (2.8), (i), (ii); © © ×\nE5.1 (3.5)\nP4.4 (2.8), (iii); C4.2 (2.11) © × \u0019\nL5.1 (3.2); L5.2 (3.3) © × \u0019\nT3.1 (1.1); T5.1 (3.4); C4.3 (2.12) © × ×\nP4.5 (2.9); C4.1 (2.10), (i) × × \u0019\nP4.3 (2.7) × × ×\nFig. 2: Validity of individual propositions\/lemmas\/corollaries\ntheorems\/examples of [ArLgSch]\nMONOMORPHISMS IN CATEGORIES OF LOG SCHEMES 63\nBibliography\n[Kato1] K. Kato, Logarithmic Structures of Fontaine-Illusie, Algebraic analysis, geom-\netry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press\n(1989), pp. 191-224.\n[Kato2] K. Kato, Toric Singularities, Amer. J. Math. 116 (1994), pp. 1073-1099.\n[ExtFam] S. Mochizuki, Extending Families of Curves over Log Regular Schemes, J. reine\nangew. Math. 511 (1999), pp. 43-71.\n[LgSch] S. Mochizuki, Categorical representation of locally noetherian log schemes,\nAdv. Math. 188 (2004), pp. 222-246.\n[ArLgSch] S. Mochizuki, Categories of log schemes with archimedean structures, J. Math.\nKyoto Univ. 44 (2004), pp. 891-909.\n[FrdI] S. Mochizuki, The Geometry of Frobenioids I: The General Theory, Kyushu J.\nMath. 62 (2008), pp. 293-400.\nResearch Institute for Mathematical Sciences,\nKyoto University,\nKyoto 606-8502,\nJAPAN\nE-mail address: motizuki@kurims.kyoto-u.ac.jp\n"}
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{"id":"SpWJh116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/The%20Geometry%20of%20the%20Compactification%20of%20the%20Hurwitz%20Scheme.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":121,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"The%20Geometry%20of%20the%20Compactification%20of%20the%20Hurwitz%20Scheme","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Text.dvi"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 12:20:53 UTC 2011","dates_in_content_count_i":1,"dates_in_content_dts":["Fri Jun 11 00:00:00 UTC 1993"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-8974615320603905661,"exact_signature_unique_b":true,"fuzzy_signature_l":4667351795835252618,"fuzzy_signature_unique_b":true,"wordcount_i":33656,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:50:11 UTC 2022","fresh_date_dt":"Thu Sep 23 19:04:48 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":541026,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086058830299136,"text_t":"GEOMETRY OF THE HURWITZ SCHEME\nThe Geometry of the Compactification\nof the Hurwitz Scheme\nby Shinichi MOCHIZUKI*\nTable of Contents\nTable of Contents\nIntroduction\n§1. Different Types of Hurwitz Schemes\n§2. Irreducibility\n§3. Log Admissible Coverings\n§3A. Basic Definitions\n§3B. First Properties\n§3C. Global Moduli\n§3D. Admissible Hurwitz Coverings\n§4. Construction of the Boundary Components\nAppendix to §4\nPictorial Appendix\n§5. Cohomology Calculations\n§6. The Main Fibration\n§6A. The Excess Divisors in the Main Fibration\n§6B. Intersection Theory Calculations\n§6C. Ramification Indices\n§7. The Coefficient Matrix\n§8. Arithmetic Applications\nBibliography\nIntroduction\nThe purpose of this paper is to study the geometry of the Harris-Mumford compactifi-\ncation of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings\nReceived June 11, 1993.\n1991 Mathematics Subject Classification: 14H10\n* Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan\n1\nf : C → P1 of the projective line by smooth curves. Thus, from the very outset, one sees\nthat there are essentially two ways to approach the Hurwitz scheme:\n(1) We start with P1 and regard the objects of interest as coverings of P1 ;\n(2) We start with C and regard the objects of interest as morphisms from C to P1.\nOne finds that one can obtain the most information about the Hurwitz scheme and its\ncompactification by exploiting interchangeably these two points of view.\nOur first main result is the following\nTheorem: Let b,d, and g be integers such that b = 2d + 2g − 2, g ≥ 5 and d > 2g + 4.\nLet H be the Hurwitz scheme over Z[ 1\nb! ] parametrizing coverings of the projective line of\ndegree d with b points of ramification. Then Pic(H) is finite.\nRemark: The number g in the statement of the Theorem is the genus of the “curve C\nupstairs” of the coverings in question. Note, however, that the Hurwitz scheme H, and\nhence also the genus g, are completely determined by b and d.\nThis Theorem is stated in §6.7, of the text. Note that although in the statement of the\nTheorem here in the introduction, we spoke of “the” Hurwitz “scheme,” there are in fact\nseveral different Hurwitz schemes used in the literature, some of which are, in fact, not\nschemes, but stacks. For details about the particular type of Hurwitz scheme for which\nthe main theorem is proved, we refer the reader to the exact statement in §6.7, as well as\nto §1 which explains the notation. Finally, we should address the issue of what happens if\nd ≤ 2g + 4. Although our proof uses the somewhat leisurely lower bound of 2g + 4, it may\nbe possible to prove the same result for smaller d using essentially the same techniques,\nbut being just a bit more careful. Since at the time this paper was written, the author\nwas not particularly interested in this issue, he has allowed himself the luxury of taking d\nto be greater than 2g + 4.\nThe main idea of the proof is that by combinatorially analyzing the boundary of\nthe compactification of the Hurwitz scheme, one realizes that there are essentially three\nkinds of divisors in the boundary, which we call excess divisors (§4.7), which are “more\nimportant” than the other divisors in the boundary in the sense that the other divisors\nmap to sets of codimension ≥ 2 under various natural morphisms. On the other hand, we\ncan also consider the moduli stack G (§6.1) of pairs consisting of a smooth curve of genus\ng, together with a linear system of degree d and dimension 1. The subset of G consisting of\nthose pairs that arise from Hurwitz coverings is open in G, and its complement consists of\nthree divisors, which correspond precisely to the excess divisors. Using results of Harer on\nthe Picard group of Mg, we show that these three divisors on G form a basis of Pic(G)⊗ZQ,\nand in fact, we even compute explicitly (§7) the matrix relating these three divisors on G\nto a certain standard basis of Pic(G) ⊗Z Q. The above Theorem then follows formally.\n2\nCrucial to our study of the Hurwitz scheme is its compactification by means of ad-\nmissible coverings, a notion introduced in [19]. In [19], the existence of a coarse moduli\nscheme of such coverings is stated. However, we could not follow a certain key step in the\nconstruction of this coarse moduli scheme, and so we decided that it would be best to give\na treatment of such coverings independent of [19]. In fact, we study a more general sort\nof admissible covering than [19], between stable curves of arbitrary genus, and we prove a\nrather general theorem (§3.22) concerning the existence of a canonical logarithmic (in the\nsense of [21]) algebraic stack (A,MA) parametrizing such coverings:\nTheorem: Fix nonnegative integers g, r, q, s, d such that 2g − 2 + r = d(2q − 2 + s) ≥ 1.\nLet A be the stack over Z defined as follows: For a scheme S, the objects of A(S) are\nadmissible coverings π : C → D of degree d from a symmetrically r-pointed stable curve\n(f : C → S;μf ⊆ C) of genus g to a symmetrically s-pointed stable curve (h : D → S;μh ⊆\nD) of genus q; and the morphisms of A(S) are pairs of S-isomorphisms α : C → C and\nβ : D → D that stabilize the divisors of marked points such that π ◦α = β ◦π. Then A is a\nseparated algebraic stack of finite type over Z. Moreover, A is equipped with a canonical log\nstructure MA → OA, together with a logarithmic morphism (A,MA) → MS\nlog\nq,s (obtained\nby mapping (C;D;π) \b→ D) which is log étale (always) and proper over Z[ 1\nd! ].\nNow we summarize what we do section by section. In §1, we define various Hurwitz\nschemes as well as certain auxiliary objects to be used later. All of these objects have\nboth “combinatorial” (corresponding to the first point of view) and “algebro- geometric”\n(corresponding to the second point of view) definitions. In §2, we use the combinatorial\npoint of view to prove the irreducibility of many of the objects of §1. In §3, we discuss\nadmissible coverings from the point of view of log schemes, and prove the Theorem just\nstated above. In §4, the culmination of our exploitation of the combinatorial point of view,\nwe explicitly enumerate and construct the divisors at infinity of the compactification, and\nbegin the determination of the the divisor class group of the Hurwitz scheme.\nFrom then on, we switch gears to the second, or more algebro-geometric, point of\nview. In §5, we prepare for this by reviewing certain relevant cohomological results. In\n§6, we carry out a detailed study of the “excess divisors” at infinity and thereby complete\nthe proof of the finiteness of the divisor class group of the Hurwitz scheme, modulo a\ntechnical result from the next Section. Finally, in §7, to make our understanding of the\nexcess divisors more explicit, we carry out certain calculations relating the excess divisors\nto other, better known line bundles, which also serve to complete the proof of the technical\nresult needed earlier.\n§8 is purely conjectural and proposes possible applications of the results discussed\npreviously to prove arithmetic results. In particular, the explicit calculations of §7 suggest\na possible application to an effective form of the Mordell conjecture.\nThe reader who is interested in the circle of ideas dealt with in this paper may also\nconsult the related work of Arbarello, Harris, and Diaz on Severi varieties; see, for instance,\n[18]. Although the results of [18] are not literally the same as ours, they are certainly\n3\nclosely philosophically related. Also, D. Edidin has been preparing a paper that gives\nsimilar results, although at the present time, I have not yet seen this paper. Finally,\nalthough we treat here the case when the dimension of the Hurwitz scheme is rather large,\nwhen the dimension is very small (i.e., 1 or 2), one has the results of [6] and [7].\nAn earlier version of this paper was submitted as my doctoral dissertation at Princeton\nUniversity in the spring of 1992. I would like to thank my advisor, Prof. G. Faltings, both\nfor suggesting the topic and for his advice and support during my years as a graduate\nstudent. Also, although none of our conversations contributed directly to the material\nin this paper, my general understanding of algebraic geometry profited greatly from my\nnumerous conversations with Prof. N. Katz; I would, therefore, like to express here my\nthanks to him, as well. Next, I would like to thank Prof. K. Kato for explaining the notion\nof a log structure to me during the summer of 1991. Finally, I would like to express my\ndeep gratitude to Profs. K. Saito and Y. Ihara for encouraging me to publish this paper,\ndespite substantial opposition in certain parts of the algebraic geometry community to the\nuse of stacks.\n§1. Different Types of Hurwitz Schemes\n§1.1. The original purpose of the Hurwitz scheme is to parametrize coverings of the\nprojective line that have at most simple branch points. However, since there are several\ndifferent versions of the Hurwitz schemes running around in the literature, we take the\nopportunity here to standardize and make explicit which version we are using at any\nparticular time. Also, it is necessary to construct Hurwitz-type schemes which parametrize\ncoverings with worse ramification than “simple branch points”. These generalized Hurwitz\nschemes will aid in our elucidation of the geometry at infinity of the ordinary Hurwitz\nscheme. We shall omit some details here since we are essentially reviewing well-known\nmaterial. For more details, see [13].\n§1.2. Remark on characteristic p: In general, in this part of the paper, we will work\nover the ring Z[ 1\nN\n], where N is divisible by all numbers characteristic to the problem.\nThus there will be no essential difference between what we do and what one would do if\none were to work over characteristic zero. Indeed, we shall often prove some results by\nreducing to the case of characteristic zero.\nRemark on Stacks: We will need to employ the notion of an “algebraic stack” (which we\nwill henceforth call simply a “stack” for short) in the sense of Deligne-Mumford [8]; the\nreader who is not satisfied with the treatment given in [8] may also refer to Chapter 1, §4,\nof [11] for basic facts about stacks. In fact, (see [11], Chapter 1, §4.10) étale locally, every\nstack can be formed by taking quotients (in the sense of stacks) of schemes by finite group\nactions. We explain what this means as follows: Let S be a noetherian scheme; let X be an\nS-scheme of finite type; let G be a finite group acting on X by means of S-automorphisms.\n4\nThen we shall denote by [X\/G] the algebraic stack defined by “taking the quotient of X\nby G in the sense of stacks.”\nConcretely, relative to the “Working Definition of Algebraic Stacks” given in [11],\nChapter 1, §4.9, [X\/G] is defined as follows: In the notation of loc. cit., it suffices to\nspecify schemes R and U , together with morphisms s, t : R → U , and μ : R ×U,t,s R → R.\nFor U , we take X; for R, we take G × X (i.e., a disjoint union of copies of X indexed by\nthe elements of G). For t : R → U , we take the morphism G × X → X that defines the\ngroup action. For s : R → U , we take the projection G × X on the second factor. For\nμ : R ×U,t,s R = (G × X) ×X (G × X) → R = G × X, we take the morphism that sends\n(g1, x) × (g2, g1x) to (g2g1, x). Once checks easily that all the necessary hypotheses are\nsatisfied.\nNow let us suppose that X = Spec(A) is affine. In [11], Theorem 4.10, a general\nmethod is described for passing from a noetherian algebraic stack to an associated coarse\nmoduli space (which is general is just an algebraic space, not necessarily a scheme). If one\napplies this Theorem to the algebraic stack [X\/G] just constructed, one sees easily that\none obtains the scheme Y = Spec(AG) (where AG ⊆ A is the subring of functions invariant\nunder the action of G) as the coarse moduli space associated to [X\/G].\n§1.3. We start with the ordinary Hurwitz scheme. Let b,d, and g be natural numbers\nsubject to the relation 2(g − 1) = −2d + b, with b ≥ 4. Psychologically, g is the the\ngenus of the curve upstairs, b is the number of branch point of the covering, and d is the\ndegree of the covering. Suppose we wish to parametrize sets of b distinct points of P1.\nLet R = Z[ 1\nb! ]. (The reason for inverting these primes is so that all of our coverings of\ndegree b or d will have tame ramification, as well as Galois closures which are generically\nseparable.) Then there are (at least) four ways of doing this: we consider the category C\nof R-schemes and the stack (in fact, a covariant functor in the first three cases) on C, that\nassigns to S (an object of C) one of the four following categories (or sets in the first three\ncases):\n(1) “UOb” (=“unrigidified ordered”): ordered sets of sections σ1, . . . , σb :\nS → S ×P1 (i.e., such that composing further with the first projection\nS × P1 → S is the identity) such that the images of σi and σj do not\nintersect when i \t= j;\n(2) “USb” (=“unrigified symmetrized”): divisors D ⊆ S ×P1 étale over S\nof degree b;\n(3) “ROb” (=“rigidified ordered”): isomorphism classes of ordered sets\nof sections as in (1), where by “isomorphism” we mean that we re-\ngard {σ1, . . . , σb} as isomorphic to {σ′1, . . . , σ′b} if there exists an S-\nautomorphism of S ×P1 that carries {σ1, . . . , σb} to {σ′1, . . . , σ′b};\n(4) “RSb” (=“rigidified symmetrized”): the stack whose objects are étale\ndivisors in P1-bundles (in the étale topology) over S, and whose mor-\nphisms are isomorphisms of P1-bundles that preserve the designated\n5\ndivisors. Put another way, this stack is the stack obtained by taking the\nquotient (in the sense of stacks) of the functor defined in (3) by the nat-\nural action of Sb (the symmetric group on b letters) given by permuting\nthe marked sections.\nAlthough the use of the terminology “rigidified” here may at first appear counterintuitive\nin the sense that often one adds a “rigidifying structure,” then forms the quotient by\nthe action of some algebraic group, here we chose to use this terminology in the sense\nthat “rigidification” consists in fixing the first three points at 0,1, and ∞, whereas in\nthe “unrigidified situation” all the points are floating around freely, hence not fixed or\nrigidified.\nIt is elementary that each of the above four functors is representable by a smooth,\nquasi-compact stack (in fact, a scheme for the first three functors) over R. Namely:\n(1) UOb: Here we simply take (P1)b − {diagonals}.\n(2) USb: We form the quotient by the action of Sb, the symmetric group\non b letters acting on UOb by permuting the b factors of P1, to obtain\nthe appropriate scheme, which may be naturally regarded as an open\nsubset of Pb, namely, the complement of the discriminant locus.\n(3) ROb: For 1 ≤ i, j ≤ b − 3 with i \t= j, let Δij ⊆ (P1)b−3 be the (i, j)-\ndiagonal. Let pi : (P\n1)b−3 → P1 for 1 ≤ i ≤ b− 3 be the ith projection.\nThen\n(P1)b−3 −\n{(⋃\ni \u0003=j\nΔij\n)\n∪\n(⋃\ni\np−1i ({0, 1,∞})\n)}\ndoes the job.\n(4) RSb: Clearly Sb acts on ROb in such a way that the action is generi-\ncally free. If we form the quotient by this action in the sense of stacks\n(as reviewed in §1.2), we obtain the desired algebraic stack, which is\ngenerically a scheme.\nWe will refer to any one of these four schemes (resp. functors, stacks as “b-point schemes\n(resp. functors, stacks)”, prefixing this term with the appropriate descriptives “rigidified”,\n“ordered”, etc. when necessary, and omitting the “b” when speaking generally. These four\nb-point stacks fit into the following commutative diagram:\nUOb −→ USb⏐⏐\b ⏐⏐\b\nROb −→ RSb\n(Diagram 1.1A)\n6\nNote that the vertical arrows are PGL(2)-torsors, while the horizontal arrows are Sb-\ntorsors. Thus, when we tensor with C and take the topological fundamental group, we\nget:\nπ\ntop\n1 (UOb,C) −→ πtop1 (USb,C)⏐⏐\b ⏐⏐\b\nπ\ntop\n1 (ROb,C) −→ πtop1 (RSb,C)\n(Diagram 1.1B)\nHere the horizontal arrows are injections and the vertical arrows are surjections; also, by\n“topological fundamental group,” we mean in the sense of stacks, i.e., the fundamental\ngroup formed by considering étale coverings of the stack in question by analytic stacks.\nFinally, let us note that USb is used in [13], while ROb is used in [19].\nLet T be any one of the four b-point stacks (resp. schemes). Then to T, we may\nassociate a Hurwitz stack H in such a way that H is étale over T, hence representable by\nan algebraic stack. Indeed, consider the following stacks (which are schemes in the first\nthree cases, so long as d ≥ 2) over C:\n(1) HUOb,d: data of the following form: an arrow α : C → P1 in the\ncategory of S-schemes such that the induced arrow C → S is a smooth,\ngeometrically connected, proper curve of genus g, and where α is flat of\ndegree d with simple ramification (i.e., the discriminant divisor is étale\nover the base – see [13], §5) exactly at given sections σ1, . . . , σb : S → P1,\nwhere the σi’s are mutually disjoint S-sections;\n(2) HUSb,d: same sort of data as above except that instead of the σi’s, we\nare given a divisor D ⊆ S ×P1 which is finite étale over S of degree d,\nat which the simple ramification is to take place;\n(3) HROb,d: isomorphism classes of the data in (1), where isomorphisms\ninvolve automorphisms over S of P1 that carry one set of sections to the\nother, and over which there is an isomorphism of the respective curves\nC;\n(4) HRSb,d: stack whose objects are collections of data as in (2), except\nthat we replace S×P1 by an arbitrary P1-bundle in the étale topology,\nand whose morphisms are isomorphisms of P1-bundles that preserve the\ndesignated divisors and over which lies an isomorphism of the respective\ncurves C.\nWe see easily by Grothendieck’s representability theorem (as applied in [13], §6) that all\nfour of these Hurwitz stacks are relatively representable by étale morphisms over their\nrespective b-point schemes. The key fact in the proof of loc. cit. (and proven there on p.\n566) is the following result, whose proof we repeat here for the convenience of the reader\n(in a slightly more general form):\n7\nLemma : Let k be an algebraically closed field, f : C → P1k a covering of degree d ≥ 3,\nwhere C is a smooth, connected, proper curve of genus ≥ 2 over k. We assume that C is\nsimply ramified over P1k, except possibly at one point p ∈ P1k, where we allow arbitrary\nramification. If ϕ : C → C is an automorphism such that f ◦ ϕ = f , then ϕ = id.\nProof: (cf. [13], p. 548 — the problem with Fulton’s proof is that it apparently makes\nuse of the characteristic zero assumption, so we trivially generalize his proof here for the\nsake of completeness). Let μ = H1ét(ϕ) denote the induced map on l-adic cohomology,\nwhere l is different from the characteristic of k. Since, as in well-known the automorphism\ngroup of C is finite, we see that μ must have eigenvalues that are roots of unity; hence\n|Tr(μ)| ≤ 2g (under any embedding of an algebraic closure of Q\u0002 into C). Thus, by the\nLefshetz fixed point theorem, the number of fixed points of ϕ is ≤ 2g + 2. But since ϕ\nmust fix the ≥ (2g − 2) + 2d + 1 − (d − 1) = 2g + d ≥ 2g + 3 branch points, we have a\ncontradiction unless ϕ = id. ©\nLet us note that if T ′ is another b-point stack with corresponding Hurwitz stack H ′, and\nif T ′ → T is one of the arrows in Diagram 1.1A, then we have H ′ = H ×T T ′.\n§1.4. We are now going to define generalizations of the Hurwitz stacks, namely, “de-\ngenerate Hurwitz stacks”, which we shall use to make explicit the compactification of the\noriginal Hurwitz stacks. We start by defining “degenerate b-point stacks”. The reason for\nthe use of the descriptor “degenerate” will become clear once these Hurwitz-type stacks\nare defined. As usual, we have four types, of which the first three are schemes:\n(1) DUOb: We take UOb.\n(2) DUSb: Let Sb−1 act on the first b − 1 sections of UOb. Then take the\nquotient of UOb by this action. Note that (1) and (2) can be defined\nalso for b = 3.\n(3) DROb: We take ROb, when b ≥ 4. When b = 3, we take RO3 =\nSpec R.\n(4) DRSb: As before Sb−1 acts on ROb. Take the quotient by this action\nin the sense of stacks if b ≥ 4. When b = 3, take RS3 = Spec R.\nAs before, there is a functorial interpretation of these degenerate b-point stacks. We leave\nit to the reader to work out this interpretation in terms of various sorts of collections of\nsections of divisors in P1. In this interpretation, we shall call the first b− 1 sections (resp.\nthe étale divisor induced by the first b − 1 sections) the marking sections (resp. marking\ndivisor) and the last (i.e., bth) section the clutching section.\nNow fix σ ∈ Sd. Let T = one of the degenerate b-point stacks. Then we denote by\nM ⊆ P1 × T (resp. C ⊆ P1 × T ) the marking (resp. clutching) divisor. Let D = M ∪ C.\nThen M , C, and D are relative T -divisors with normal crossings. Moreover, T is smooth\n8\nover our original ring R. Now by [29], Exposé XIII, Appendice I, we can form F = R1étf∗Sd,\nwhere U = (P1 × T ) − D, f : U → T is the restriction of the projection to the second\nfactor π2 : P\n1 × T → T , and (by abuse of notation) Sd is the constant étale sheaf in\nnoncommutative groups on U with fibre Sd:\nU ⊆ P1 × T ⊇ D ⊇ M,C\nf ↘\n⏐⏐\bπ2\nT\nThus F is a finite étale covering of T. Write g : F → T . For each geometric point t of\nT , g−1(t) can be identified with Hom′(πalg1 (P\n1\nt − Dt), Sd), where the “prime” after the\nHom means “up to inner automorphism” (thus exempting us from the need to choose a\nbase point for our π1). Let us consider the closed and open substack K ⊆ F , which is also\nfinite étale over T and which is such that if k = g|K, then k−1(t) can be identified with\nthose homomorphisms ϕ : π\nalg\n1 (P\n1\nt − Dt) → Sd such that:\n(1) ϕ is surjective.\n(2) ϕ takes the generators of the monodromy groups around the marked\npoints to transpositions.\n(3) ϕ takes the generator of the monodromy group around the clutching\npoint to a conjugate of σ.\nThen K will be our degenerate Hurwitz stack, which we shall denote, depending on the\nsituation by DHUOσb,d; DHUSσb,d; DHROσb,d; or DHRSσb,d (of which all but the last are\nschemes). Note that the degenerate Hurwitz stacks corresponding to conjugate σ may be\nnaturally identified.\n§1.5. Remark: Unlike the original Hurwitz stacks, it is not clear (at least to the\nauthor) that these degenerate Hurwitz stacks should have any modular interpretation at\nall. Indeed, the logic of [13], §1.4, 1.5, does not follow through here because of the more\ncomplicated nature of the ramification involved. The problem is that if we denote by\nF what one might think is the appropriate modular stack (in terms of coverings of the\nprojective line), and by DH one of the degenerate Hurwitz stacks, we get morphisms α :\nF → DH (clear) and β : DH → F (by the “pseudo-universal” covering to be constructed\nin the following paragraph) and one can show that α ◦ β = 1DH, hence that α is smooth.\nBut in order to show that it is étale, one needs, in the case of simple coverings, the fact\nthat the discriminant divisor is étale; since one does not have this fact here, one cannot\nfollow through as before.\nHowever, if DH is one of the degenerate Hurwitz stacks, we can construct what one\nmight call a “pseudo-universal” covering as follows. (Here we use the term “pseudo-\nuniversal” loosely in the sense that although DH does not actually represent the moduli\n9\nstack F of coverings of a certain type (in which case the tautological covering over DH\nwould be called simply “universal”), at least over each closed point of DH the “pseudo-\nuniversal” covering “is” the covering corresponding to that closed point). Indeed, it is\ntautological that there exists an étale covering ψ : V → (P1 ×DH) − DDH such that the\nnormalization V → P1 × DH of P1 × DH in V is what we want. The only thing that is\nnonobvious is that V is a smooth, proper curve over DH. But smoothness follows from\nthe fact that DH is regular (since T is) and by applying Abhyankar’s lemma. Thus, to\nsummarize we have the following diagram:\nC η−→ DH× P1 ⊇ DDH ⊇ MDH, CDH\nζ ↘\n⏐⏐\b\nDH\nwhere C = V , ζ is smooth, proper, with geometrically connected fibres, η is flat, étale\noutside of DDH, has simple ramification outside of CDH, and has ramification of the type\nprescribed by σ over CDH. In particular, by the Lemma of §1.3, it follows that if d ≥ 3,\nthen C has no automorphisms that fix η.\n§1.6. Finally, let us note as before that we have diagrams:\nDUOb −→ DUSb⏐⏐\b ⏐⏐\b\nDROb −→ DRSb\n(Diagram 1.2A)\nwhere the vertical arrows are PGL(2)-torsors, and the horizontal arrows are Sb−1-torsors,\nand\nπ\ntop\n1 (DUOb,C) −→ πtop1 (DUSb,C)⏐⏐\b ⏐⏐\b\nπ\ntop\n1 (DROb,C) −→ πtop1 (DRSb,C)\n(Diagram 1.2B)\nwhere the vertical arrows are surjections, and the horizontal arrows are injections. Note\nthat if D′ → D is an arrow in the first diagram, then with respect to degenerate Hurwitz\nstacks and “pseudo-universal” curves, we have DH′ = DH×D D′ and C′ = C ×DH DH′.\nBefore continuing, we remark that although it is not absolutely necessary for what\nfollows to use all four versions of the Hurwitz scheme, we presented them here in this\nintroductory section in some detail so as to clarify which was which, since to the author’s\nknowledge, there does not yet seem to exist either standard notation or standard termi-\nnology in the literature that allows one to specify precisely which Hurwitz scheme one is\ndealing with at any particular time.\n10\n§2. Irreducibility\n§2.1. The first topological invariant that one wishes to compute about a newly con-\nstructed object is its connectedness, or irreducibility. Here we show that HUSb,d and\nDHUSσb,d – which are schemes, so long as d ≥ 3 – are geometrically irreducible over R.\nIt then follows from Diagrams 1.1A and 1.2A that the stacks HRSb,d and DHRSσb,d are\nalso geometrically irreducible over R. To prove irreducibility, we note that as in [13], pp.\n546–7, we must show that the fundamental group of the appropriate b-point stack acts\ntransitively on the set corresponding to the étale covering which is the Hurwitz scheme; as\nin loc. cit., we see that the fundamental group of the b-point stack has certain canonical\nelements that correspond to “braiding” various pairs of points; hence we are reduced to\nthe following combinatorial Proposition.\n§2.2. Proposition : Fix b, d ∈ N, σ ∈ Sd. Let Aσb,d be the set of ordered b-tuples of\ntranspositions {t1, . . . , tb} of Sd such that t1t2 . . . tb = σ and such that the group generated\nby the ti’s acts transitively on {1, . . . , d}. Define an “elementary move” on Aσb,d as an\nautomorphism of Aσb,d that takes {t1, . . . , ti, ti+1, . . . , tb} to {t1, . . . , ti+1, ti+1titi+1, . . . , tb}\nfor some i. Then the free group F generated by the elementary moves acts transitively on\nAσb,d.\n(Psychological Remark: In the case of degenerate Hurwitz stacks, i.e. σ \t= id, what we\nhave called b here really corresponds to b − 1 in the previous section.)\nNotational Remark: We shall denote chains of transpositions by <?, ?, . . . , ? >, where each\nof the question marks is a transposition, and we shall make elementary moves on such\nchains, where we regard these chains as subchains of the original b-tuple of transpositions.\nIt is clear that an elementary move on such a subchain corresponds to an elementary move\non the original b-tuple. We call two elements t, t′ of Aσb,d equivalent if they lie in the same\ndomain of transitivity of F. We write t ∼ t′.\nBibliographical Remark: It has been brought to the author’s attention that similar results\nwere obtained using similar techniques in the classical papers of [4], [20], and [25], as well\nas the more recent paper of [2].\n§2.3. Lemma : We have the following relations:\n(1) < (12), (23) > ∼ < (13), (12) >\n(2) < (12), (23), (23) > ∼ < (12), (13), (13) >\n(3) < (12), (13), (13) > ∼ < (13), (13), (12) >\n(4) And lastly:\n< (12), (12), (23), (23),(23), (23) >\n∼ < (12), (12), (12), (12), (23), (23) >\n11\nProof: For (1) to (3), we have < (12), (23) > ∼ < (23), (13) > ∼ < (13), (12) >.\nApplying this twice, we get\n< (12), (23), (23) > ∼ < (13), (13), (12) >\nOn the other hand,\n< (12), (13), (13) > ∼ < (13), (23), (13) > ∼ < (13), (13), (12) >\nPutting this together, we obtain (1) through (3). As for (4), one sees by (3) that, up to\nequivalence, both sides of (4) are of the same form in the sense that they differ only in\nthat the digits 1 and 3 have opposite roles on the two sides; moreover, a chain of this form\nmay be uniquely specified by stating which digit (i.e., of 1, 2, or 3) appears twice, which\nfour times, and which six times. For the left hand side, these are, respectively, (1, 3, 2); for\nthe right hand side, they are (3, 1, 2). Thus, if we think of the symmetric group on three\nletters acting on chains of this form by permuting the digits 1, 2, and 3, we see that it\nsuffices to show that the left hand side is equivalent both to “(2, 3, 1)” and to “(1, 2, 3)”.\nBy the part of the lemma already proven, we have that the left hand side\n< (12), (12), (23), (23), (23), (23) > ∼ < (12), (13), (13), (13), (13), (12) > =\nwhat we have referred to as “(2, 3, 1)”. Since < (12), (23) > ∼ < (23), (13) >, we have\nthat the left hand side\n< (12), (12), (23), (23),(23), (23) >\n∼ < (23), (13), (13), (23), (23), (23) >\n∼ < (13), (13), (23), (23), (23), (23) >\nby (3). But this last chain is just “(1, 2, 3)”. ©\n§2.4. Lemma : Let α, β ∈ {1, . . . , d} with α \t= β. Let t ∈ Aσb,d. Then t ∼ t′ such\nthat t′1 = (α β).\nProof: Let us note first that by the transitivity assumption on elements of Aσb,d, we know\nthat α and β must appear somewhere as digits in the transpositions of t. By elementary\nmoves, we can assume that t1 contains α; by subsequent elementary moves toward the\nfront, and the transitivity assumption on elements of Aσb,d, we may assume that t is of the\nfollowing form:\nt1 = (α x2), t2 = (x2x3), t3 = (x3, x4), . . . , ti = (xixi+1), . . . , tr = (xr β), with r ≤ b\nHere the xi’s are distinct from each other and from α and β. Now we have\n< (α x3), (α x2) > ∼ < (α x2), (x2x3) > = < t1, t2 >\n12\nso by substituting < (α x3), (α x2) > for < t1, t2 >, moving the ti’s for 3 ≤ i ≤ r down one\nposition to the left, and sending (what would originally have been) (α x2) out to the right\nbeyond what was tr, we see that we obtain the same situation as what we started with,\nexcept that r is one less than it was before. Now apply induction. ©\n§2.5. Let us prove the Proposition first in the case of the nondegenerate Hurwitz stack,\ni.e., when σ = id. Let t ∈ A(id)b,d . We shall show that t ∼ a chain which consists of a nonzero,\neven number of (12)’s, followed by a nonzero, even number of (23)’s, and so on through\n(d − 1, d). By Lemma 3, (4), all such chains are equivalent, so we are done.\nBy transitivity, Lemma 4 allows us to assume that t1 = (12), t2 = (12) on account of\nthe fact that σ = id. By using inverse elementary moves to bring things to the front, we\nmay assume that t3, . . . , tr contain a “1”, but that ti (∀i > r) do not. We claim that we\nmay move to a situation where (with possibly a new r) t3, . . . , tr are all equal to (12). For,\nif there is an i such that 3 ≤ i ≤ r − 1 and ti = (1a), ti+1 = (1b) with 1 \t= a \t= b, then\n< (1a), (1b) > ∼ < (1b), (ab) >, so we can reduce to the same situation with lower r.\nThus we may assume that t3 = t4 = . . . = tr. If t3 = (12), then we have proven the claim.\nIf t3 = (1a) with a \t= 2, then it is easy to see that r must be even (since ti (∀i > r)\ndoesn’t contain a “1”); hence, applying Lemma 3, (2), proves the claim. Now repeating\nthe above procedure finishes the proof for σ = id.\n§2.6. Proof of Proposition 2:\nStep A: First we introduce some notation. Write σ = γ1 . . . γr where each\nγi = (a\n(i)\n1 , a\n(i)\n2 , . . . , a\n(i)\nei\n)\nis a nontrivial cycle with domain of transitivity Δi such that Δi\n⋂\nΔj = ∅ for i \t= j. Let\nq = r + d − (∑ri=1 ei), N = (∑ri=1 ei) − r. Then let the Δi for r + 1 ≤ i ≤ q be the\nremaining trivial domains of transitivity of σ. For simplicity, let’s write 1i, 2i, etc. for\na\n(i)\n1 , a\n(i)\n2 , . . .. Now define\nZi =< (1i2i), (1i3i), (1i4i), . . . , (1i(ei)i) >∈ Aγi(ei−1),ei\nLet N denote the concatenation of chains Z1,Z2, . . . ,Zr.\nNow let t ∈ Aσb,d. By Lemma 4, we see that we may assume that t begins with N .\n(Note that this same logic already proves the Proposition for A\nγi\n(ei−1),ei .) Thus we have\nnormalized t1, . . . , tN . It remains to deal with ti for i > N .\nStep B: Let D1, . . . ,Ds (⊆ {1, . . . , d}) be the domains of transitivity of the group gen-\nerated by the ti, for i > N , (so\n⋃s\ni=1 Di = {1, . . . , d} and Di\n⋂Dj = ∅ ∀i \t= j). For\n1 ≤ j ≤ s, let\nPj =\n{\ni ∈ {1, . . . , b}| i > N and ti acts nontrivially on Dj\n}\n13\nNote that if we take the subchain τ j of transpositions of t indexed by Pj , then τ j ∈\nA\n(id)\n|Pj |,|Dj |. By §2.5, we may assume that τ\nj consists of a sequence of pairs of identical\ntranspositions. Now we claim:\nClaim : If q \t= 1, then we may assume that every ti (∀i > N) is of the form (xy) with\nx ∈ Δk, y ∈ Δl, k \t= l.\nProof of Claim : Suppose we have a pair ti = ti+1 = (ab), where i ≡ N +1 (mod 2)\nand a, b ∈ Δk with a \t= b. Since all the ti’s together generate a group transitive on\n{1, . . . , d}, there exists a j > N with j ≡ N + 1 (mod 2) such that tj = tj+1 = (cd)\nwith c ∈ Δk and d ∈ Δl where l \t= k. We may move ti, ti+1 so that they are adjacent to\ntj , tj+1 (by Lemma 3, (3)), so we obtain:\n< (cd), (cd), (ab), (ab) >\nBy Lemma 7A below, we may assume that b = c. Then applying Lemma 3, (2), with 1, 2, 3\nreplaced by d, c, a, respectively, we have\n< (cd), (cd), (ac), (ac) > ∼ < (cd), (cd), (ad), (ad) >\nRepeating this procedure proves the claim. ©\n§2.7. Lemma A : Suppose in the above situation (except that q might be one) that\nwe have a pair ti = ti+1 = (ab) with i ≡ N + 1 (mod 2), and a, b ∈ Δk, a \t= b. Then we\nmay move to the same situation, except with a = 1k, b = 2k.\nProof: We may assume that with respect to the ordering 1k < 2k < 3k < . . ., we have\na < b. By moving around chunks of t that commute with each other, we may assume that\ni = N + 1 and we may move Zk right up to the left of tN+1:\nt = Z1Z2 . . .Zk . . .ZrtN+1tN+2 . . . tb⏐⏐\bgoes to\nt = Z1Z2 . . .Zk−1Zk+1 . . .ZrZktN+1tN+2 . . . tb\nSuppose a \t= 1k. Since we have shown the Proposition for Aγk(ek−1),ek , we may assume that\nZk ends with the transposition\n(\n(a − 1), a\n)\n. By Lemma 3, (2),\n<\n(\n(a − 1), a\n)\n, (ab), (ab) > ∼ <\n(\n(a − 1), a\n)\n,\n(\n(a − 1), b\n)\n,\n(\n(a − 1), b\n)\n>\nBy repeating this procedure many times, we can assume that a = 1k. Applying the same\nprocess to b proves the Lemma. ©\n14\nLemma B : In the above situation, suppose that we have a pair ti = ti+1 = (ab) (where\ni ≡ N + 1 (mod 2)) with a ∈ Δk and b ∈ Δl, where k \t= l. Then we may assume that\na = 1k, b = 1l.\nProof: Same as for Lemma A. ©\n§2.8. By Lemma 7B and the claim at the end of §2.6, if q \t= 1, then we may assume\nthat every ti (for i > N) is of the form (1k1l) for k \t= l. If q = 1, then Δ1 = {1, . . . , d}\nand we may assume that every ti (for i > N) is of the form (1121). Thus if q = 1, we have\nfound a standard normalizing form for t, so the Proposition has been proved.\nIf q \t= 1, then we note that if we regard (1k1l) as an element of the permutation group\nof the symbols Δ1, . . . ,Δq, then the ti’s (for i > N) act transitively on the Δj ’s (since all\nthe ti’s together act transitively on {1, . . . , d}); on the other hand, tN+1tN+2 . . . tb = (id).\nThus we are in the situation of dealing with an element of A\n(id)\n(b−N),q. But the Proposition\nhas already been proven in this case. Thus we are done. ©\n§2.9. For ease of reference, we restate the main result of this section as a Theorem:\nTheorem : For all b, d ∈ N such that b ≥ 3, and all σ ∈ Sd, we have that HUSb,d;\nDHUSσb,d; HRSb,d; and DHRSσb,d are all geometrically irreducible over R.\n§2.10. We now turn to the proof of another similar irreducibility result which will be\nuseful later. Let us recall the situation of of §1.5:\nC β−→ D × P1 ⊇ CD\nα ↘\n⏐⏐\bγ\nD⏐⏐\b\nT\n(Here T is the relevant degenerate b-pointed stack.) By §4.3, we see that the reduced\ninverse image E of CD via β is étale over CD ∼= D. (We placed this Lemma in §4 in which\nit is more in context, since this Section is concerned primarily with combinatorics.) Thus\nE → T is an étale covering. It is clear that the portions of E that correspond to different\nramification indices form different connected components, even over R. In fact, we have\nthe following result:\nProposition : The geometric (i.e., after tensoring from R up to C) connected com-\nponents of E are in one-to-one correspondence with the set of ramification indices over\nCD, which is the same as the set of cardinalities of domains of transitivity of σ, where\nD = DHUSσb,d or DHRSσb,d.\n15\nProof: As before, we consider the action of braiding elements of the fundamental group of\nT on the set B corresponding to the étale covering E → T . Simple reflection reveals that\nB may be given the following description: B consists of the set of pairs {(t1, . . . , tb−1),Δ}\nwhere (t1, . . . , tb−1) ∈ Aτb−1,d, with τ conjugate to σ and Δ a domain of transitivity of τ .\nThe action of braiding on the first b−1 factors clearly doesn’t affect τ , and acts via an\n“elementary move” on the Aτb−1,d part and trivially on Δ. The action of double braiding\nbetween the last two factors clearly takes the domain of transitivity Δ of τ to the domain\nof transitivity tb−1(Δ) of tb−1τtb−1. By Proposition 2, we can take any given element of B\nto an element with tb−1 arbitrary via braiding in the first b− 1 factors. By doing this and\nthen double braiding in the last two factors, it is thus clear that we can take any domain\nof transitivity Δ of τ to any other domain of transitivity of τ with the same cardinality.\nThis completes the proof.©\n§3. Log Admissible Coverings\n§3.1. In this paper, we will be interested in studying not only the various Hurwitz\nstacks introduced in §1, but also their compactifications. In [19], a definition of an “ad-\nmissible covering” is proposed as a means of compactifying the Hurwitz scheme HROb,d.\nUnfortunately, since Harris-Mumford are interested only in the coarse moduli scheme of\nsuch admissible coverings, their definition of an admissible covering is not well suited to\nthe study of families and the construction of an algebraic stack of admissible coverings.\nThe main reason for this is that the natural context in which to study such admissible\ncoverings is not the category of schemes, but rather that of log schemes, as defined by\nKato in [21]. Thus, in this Section, we propose to give a foundational treatment of log\nadmissible coverings. In particular, we prove a rather general existence theorem for a log\nalgebraic stack of such coverings. If we forget the log structure of this log algebraic stack,\nand consider the associated coarse moduli space (as in [11], Theorem 4.10), we recover the\ncoarse moduli scheme of Harris-Mumford.\nRemark: In an earlier version of this paper, we referred to the proof of [19] in our construc-\ntion of a moduli stack of admissible coverings. Indeed, the existence of such an algebraic\nmoduli stack is essentially a formal consequence of what [19] claim to have proven. How-\never, various people who read that earlier version complained bitterly that they could not\nfollow the proof in [19]. Upon closer inspection, the author discovered a counterexample\nto one of the key claims in the proof of the existence of a coarse moduli scheme in [19]\n(see §3D below for more details). Thus, it seems that at this point, one must consider the\nproof of this result in [19] to be incomplete. In light of this situation, the author decided\nthat it would be best to give a detailed treatment of the existence of an algebraic moduli\nstack of admissible coverings that is independent of what is done in [19].\n16\n§3A. Basic Definitions\n§3.2. We start by reviewing Knudsen’s notion of an r-pointed stable curve. Let S be\na scheme.\nDefinition: A flat, proper morphism f : C → S of schemes together with r sections\nσ1, . . . , σr : S → C is called an r-pointed stable curve of genus g if\n(1) the geometric fibres Cs of f are connected, reduced, of dimension 1, have\ndimk(s) H\n1(Cs,OCs) = g, and have at most ordinary double points as\nsingularities;\n(2) the images of the sections σ1, . . . , σr are disjoint and land in the smooth\nlocus of f ;\n(3) the relative dualizing sheaf ωC\/S is relatively ample with respect to f .\nWe shall refer to the sections σ1, . . . , σr as the marked points of f : C → S, and to the\ndivisor in C defined by the union of the σi as the divisor of marked points.\nNote that the ampleness condition (3) implies that 2g − 2 + r ≥ 1. [22] shows that there\nexists a smooth, proper (over Z) moduli stack for such data, which we shall denote by\nMg,r. Those r-pointed stable curves for which C → S is smooth define an open substack\nMg,r ⊆ Mg,r. The complement of Mg,r in Mg,r is a divisor with normal crossings. We\nshall call this divisor the divisor at infinity of Mg,r.\nWe shall often find it convenient to use the language of log structures and log schemes\nof [21]. A log structure on S consists of an étale monoid M on S, together with a morphism\nM → OS of monoids that satisfies certain properties. See [21] for more details. We shall\nwrite the monoid operation on sections of M additively, and refer to the given morphism\nM → OS as the exponential map of the log structure.\nThe divisor at infinity of Mg,r thus defines (as in [21], §1.5, (1)) a log structure on\nMg,r which we shall call the canonical log structure on Mg,r. We shall denote the resulting\nlog stack by Mlogg,r .\nLet MMg,r denote the étale monoid that defines the canonical log structure on Mg,r.\nThus MMg,r is precisely the subsheaf of OMg,r consisting of functions invertible on Mg,r.\nLet\nGMg,r ⊆ MMg,r\nbe the subsheaf of functions whose zero locus is smooth (over Z). Thus, O×Mg,r ⊆ GMg,r ,\nand O×Mg,r acts freely on GMg,r . Let NMg,r be the quotient MMg,r\/O\n×\nMg,r\n. Then\n17\nΓMg,r\ndef\n= GMg,r\/O\n×\nMg,r\n⊆ NMg,r\nis a constructible sheaf of sets (in the étale topology). Finally, note that GMg,r generates\nthe sheaf of monoids MMg,r .\nLet (C → Mg,r;σ1, . . . , σr : Mg,r → C) denote the universal r-pointed stable curve\nof genus g. It is shown in [22] that C may be naturally identified with Mg,r+1. Let\nus denote by Clog the log stack obtained by pulling back the canonical log structure on\nMg,r+1 relative to this identification. We shall call this log structure on C the canonical\nlog structure on C. Note that the canonical log structure on C can also be obtained as\nthe log structure associated to the divisor with normal crossings which is the union of the\nimages of the σi, together with the pull-back via C → Mg,r of the divisor at infinity of\nMg,r. Thus, just as for Mg,r, we can define GC ⊆ MC and ΓC def= GC\/O×C ⊆ NC .\n§3.3. When working in the logarithmic category, the following notion is more conve-\nnient than the classical notion of Knudsen reviewed above. Let Slog be a fine log scheme\n(in the sense of [21], §2.3).\nDefinition: A morphism of log stacks κlog : Slog → Mlogg,r will be referred to as the\ndata for an r-pointed stable log-curve of genus g. For such a κlog, let Clog → Slog be the\npull-back (in the logarithmic category) of the universal object Clog → Mlogg,r by means of\nκlog. Such a Clog → Slog will be referred to as an r-pointed stable log-curve of genus g.\nNote that for smooth curves, once one is given a classifying morphism κ : S → Mg,r,\nthe resulting log morphism κlog : Slog → Mlogg,r is uniquely defined, so smooth curves and\nsmooth log-curves are equivalent notions. However, if κ : S → Mg,r does not map into\nMg,r, then it is easy to construct examples of distinct log morphisms Slog → M\nlog\ng,r with\nthe same underlying stack morphism κ : S → Mg,r. Nonetheless, given a morphism\nκ : S → Mg,r, there always exists a natural choice of log structure on S for which κ\nextends to a logarithmic morphism: namely, the pull-back of the log structure on Mlogg,r .\nNow let Clog → Slog be an r-pointed stable log-curve of genus g. Let MC (respectively,\nMS) denote the étale monoid defining the log structure of C\nlog (respectively, Slog). Let\nNC (respectively, NS) be the quotient MC\/O×C (respectively, MS\/O×S ). Then we obtain a\nnatural morphism from ΓC|C (respectively, ΓMg,r |S), to NC (respectively, NS). We denote\nthe image of this morphism by ΓC (respectively, ΓS). Thus, ΓC (respectively, ΓS) forms\na subsheaf (of sets) of NC (respectively, NS). Let GC (respectively, GS) be the inverse\nimage of ΓC (respectively, ΓS) in MC (respectively, MS) via the projection MC → NC\n(respectively, MS → NS). We shall call ΓC (respectively, ΓS ; GC ; GS) the sheaf of\ngenerators of NC (respectively, NS; MC ; MS). Since ΓC is a constructible sheaf of sets on\nC, one can regard (see [28], p. 160) ΓC as a quasi-finite algebraic space over S. Then GC\ndefines an O×ΓC -torsor, hence a line bundle LC on this algebraic space ΓC .\n18\nFinally, by forgetting log structures, we obtain an underlying r-pointed stable curve of\ngenus g: (f : C → S;σ1, . . . , σr : S → C).\n§3.4. Sometimes it is convenient to forget the ordering on the marked points of a\npointed stable curve. That is to say, we have a natural action of Sr (the symmetric group\non r letters) on Mlogg,r given by permuting the marked points. If we form the quotient of\nMlogg,r by Sr (in the sense of stacks), we thus obtain a log stack MS\nlog\ng,r . Note that the\nuniversal log-curve Clog → Mlogg,r descends to a log-curve CSlog → MS\nlog\ng,r . Let S\nlog be a\nfine log scheme.\nDefinition: A morphism of log stacks κlog : Slog → MSlogg,r will be referred to as the data\nfor a symmetrically r-pointed stable log-curve of genus g. For such a κlog, let Clog → Slog\nbe the pull-back (in the logarithmic category) of the universal object CSlog → MSlogg,r by\nmeans of κlog. Such a Clog → Slog will be referred to as a symmetrically r-pointed stable\nlog-curve of genus g. If we forget the log structures of such a log-curve, the resulting\n(f : C → S;μf ⊆ C) (where μf ⊆ C is the divisor of marked points) will be referred to as\na symmetrically r-pointed stable curve of genus g.\nFor instance, when g = 0, the open substack MSg,r ⊆ MSg,r of smooth curves may be\nidentified with the stack RSb of §1.3.\n§3.5. We are now ready to state the main definition of this Section. Let f log : Clog →\nSlog (respectively, hlog : Dlog → Slog) be a symmetrized r- (respectively, s-) pointed stable\nlog-curve of genus g (respectively, q). Let μf ⊆ C (respectively, μh ⊆ D) be the divisor of\nmarked points. Let πlog : Clog → Dlog be a morphism of log schemes.\nDefinition: We shall say that πlog : Clog → Dlog is a log admissible covering of degree\nd if the following conditions are satisfied:\n(1) πlog is log étale (in the sense of [21], §3.3); moreover, the morphism\nπ−1ND → NC induced by the morphism of log structures maps ΓD into\nΓdC\ndef\n=\nd⋃\ni=1\ni · ΓC\n(2) π is finite;\n(3) π is of degree d on a dense open set of every fiber of h : D → S;\n(4) μf ⊆ π−1(μh); π−1(μh) ⊆ d · μf ;\n(5) f is smooth at c ∈ C if and only if h is smooth at π(c).\n19\nWe now make some trivial observations concerning log admissible coverings:\n(1) Of the above conditions, only the first depends on the log structures;\nthe rest depend only on the morphism of schemes π.\n(2) Note that the second inclusion in condition (4) actually follows from\nconditions (1) and (5).\n(3) Let x ∈ C − μf be a point at which f is smooth. Let y = π(x). Then\nthe log structure on Clog (respectively, Dlog) at x (respectively, y) is\njust the pull-back to C (respectively, D) of the log structure on Slog. It\nthus follows that π is étale at x.\n(4) If hlog has the property that every irreducible component of every ge-\nometric fiber Ds of h is smooth, then f\nlog also has this property.\n§3.6. Suppose that now that (f : C → S, μf ⊆ C) and (h : D → S, μh ⊆ D) are\nsymmetrically pointed stable curves. Then we make the following definition\nDefinition: A morphism π : C → D will be called an admissible covering if there exist,\nétale locally on S:\n(1) a fine log scheme Slog with underlying scheme S;\n(2) symmetrically pointed stable log-curves Clog → Slog and Dlog → Slog\nwhose underlying symmetrically pointed stable curves are (f : C →\nS, μf ⊆ C) and (h : D → S, μh ⊆ D);\n(3) a log admissible covering πlog : Clog → Dlog whose underlying scheme\nmorphism is π\nRemark: We shall see below that when D → S is of genus zero, the definition agrees with\nthat of [19], p. 57. The reason that we prefer this definition is that when considering\nmoduli of morphisms of curves, it is very difficult to incorporate conditions that involve\nthe existence of certain local functions on the curves (as do the conditions given in [19],\np. 57). Since a log structure essentially involves some sort of complicated monoid-worth\nof global (on the curve) line bundles, a morphism of log structures amounts to some sort\nof morphism of line bundles, hence (as we shall see below) is relatively easy to consider\nthe moduli of. On the other hand, one draw-back of the above definition of an admissible\ncovering is that the log structure on S is not uniquely or canonically defined. Nonetheless,\nwe shall see below a posteriori that, in fact, there does always exist a canonical log structure\non S with respect to which an admissible covering π extends to a log admissible covering.\n20\n§3B. First Properties\n§3.7. Let A be a local noetherian henselian ring. Let s ∈ mA (the maximal ideal of\nA). Let R be the henselization of A[X,Y ]\/(XY − s) (where X and Y are indeterminates)\nat the ideal generated by mA, X, and Y . Thus, R is a local noetherian henselian ring,\nwith maximal ideal mR. Let x, y ∈ R be the images of X and Y , respectively. Then we\nhave the following\nLemma: Suppose that we are given x′, y′ ∈ R; s′ ∈ A such that x′y′ = s′, and\n(x, y,mA) = (x\n′, y′,mA) (equality of ideals in R). Then there exist units ux, uy ∈ R×\nwith uxuy ∈ A× such that either uxx = x′ and uyy = y′, or uxx = y′ and uyy = x′.\nProof: Let us first consider the case when s = 0. Then the Lemma follows immediately\nfrom the simple structure of the ring A[[X,Y ]]\/(XY ), and faithfully flat descent from\nA[[X,Y ]]\/(XY ) down to R. Now let us return to the case of arbitrary s. By considering\nthe ring R\/sR = A\/sA[[X,Y ]]\/(XY ) (and applying the Lemma to this ring), it thus follows\nthat s′ ∈ (s). Moreover, without loss of generality, we may assume that x′ = x + s · φ and\ny′ = y + s ·ψ, for some φ, ψ ∈ R. By multiplying x′ and y′, we thus obtain that s′ is equal\nto s times a unit in R. Since R is faithfully flat over A, it follows that s′ is equal to s\ntimes a unit in A. Thus, without loss of generality, we may assume that s′ = s. Let R′\n(respectively, A′) be the completion of R of (respectively, A). If we restrict ourselves to\nfinding ux and uy such that uxuy = 1, then it follows from the definition of “henselization,”\nthat it suffices to prove the Lemma with R and A replaced by R′ and A′, respectively.\nLet us review our situation: We have x′, y′ ∈ R′ such that x′y′ = s and (x′, y′,mA′) =\n(x, y,mA′). Suppose we know that x\n′ = x + sif ; y′ = y + sig, for some f, g ∈ R′, and\nsome positive integer i. By multiplying x′ and y′, we obtain that x(sig)+y(sif) ∈ si+1R′.\nThus, x + y annihilates the image of x(sig) ≡ −y(sif) in siR′\/si+1R′. Since x + y is a\nregular element of R′, we thus obtain that xsig ≡ ysif ≡ 0. On the other hand, from\nthe structure of the module siR′\/si+1R′ = siA′\/si+1A′[[X,Y ]]\/(XY ), one then obtains\nthat sif − sixα, sig − siyβ ∈ (si+1), for some α, β ∈ R′ such that α (respectively, β) is a\npower series in x (respectively, y). Now let us note that we can always make substitutions\nof the following sort: replace x′ by x′ · u−1, and y′ by y′ · u (for some unit u ∈ (R′)×),\nwithout affecting the validity of the Lemma. Moreover, if we make such a substitution,\nwith u = 1+ siα− siβ, we find ourselves in the situation reviewed at the beginning of this\nparagraph, except with i replaced by i+1. By induction (and the fact that R′ is complete\nand separated in the s-adic topology), this completes the proof of the Lemma. ©\n§3.8. We continue with the notation of the previous §.\nLemma: Suppose that there exist units ux, uy ∈ R× such that xux = x, yuy = y, and\nuxuy ∈ A×. Then it follows that ux = uy = 1.\nProof: As in the previous §, we prove the Lemma with R and A replaced by their\ncompletions R′ and A′. Suppose that we know that ux ≡ 1, uy ≡ 1 modulo siR′,\n21\nfor some nonnegative integer i. Let dx = 1 − ux; dy = 1 − uy. Since xdx = 0,\nydy = 0, it follows that we can write dx(mod s\ni+1R′) (respectively, dy(mod si+1R′))\nas a power series in positive powers of y (respectively, x) with coefficients in siA′\/si+1A′.\nMoreover, dxdy = 1 − ux − uy + uxuy = dx + dy + (uxuy − 1). It thus follows that\n(uxuy − 1) ∈ A′\n⋂\nsiR′ = siA′, hence that dx +dy ∈ siA′ + si+1R′. But from the represen-\ntation of dx(mod s\ni+1R′) and dy(mod si+1R′) as power series of the form just described,\nit thus follows that dx, dy ∈ si+1R′. By induction (and the fact that R′ is complete and\nseparated in the s-adic topology), this completes the proof of the Lemma. ©\n§3.9. We now consider a slightly different definition of an “admissible covering,” which\nis essentially the same as that of [19]. Let S be a scheme. Let (f : C → S, μf ⊆ C) and\n(h : D → S, μh ⊆ D) be symmetrically pointed stable curves. Then we make the following\ndefinition\nDefinition: A morphism π : C → D will be called an HM-admissible covering if it\nsatisfies conditions (2) through (5) of the definition of a log admissible covering, as well as\nthe following condition, consisting of two parts, which we refer to together as “condition\n(1′)”: π is étale, except:\n(1′A) over μh, where it exhibits tame ramification\n(1′B) over the singular points: if s ∈ S, λ is a node of Cs, and ν = π(λ), then\nthere exist a ∈ mshS,s, x, y ∈ mshC,λ, and u, v ∈ mshD,ν , such that xy = a,\nuv = ae, u = xe, v = ye (for some natural number e (necessarily\nless than d) such that e ∈ O×S,s) and such that x, y (respectively, u, v)\ngenerate msh\nf−1(s),λ (respectively, m\nsh\nh−1(s),ν). Here, by “m\nsh,” we mean\nthe maximal ideal of “Osh,” the strict henselization of the local ring in\nquestion.\nRemark 1: Suppose that condition (1′B) is satisfied. Then let us assume that we are\ngiven elements u′, v′ ∈ msh\nD,π(c) and b\n′ ∈ mshS,s that satisfy u′v′ = b′ and that generate\nmsh\nh−1(s),d. Then we claim that there exist x\n′, y′ ∈ mshC,c, together with a′ ∈ mshS,s that\nsatisfy (x′)e = u′; (y′)e = v′; x′y′ = a′. Indeed, this follows from the fact that the\nunordered pair (u′, v′) differs from the unordered pair (u, v) by multiplication by units\n(Lemma 7). Taking eth roots of these units (which is possible precisely because we are\nworking in the strict henselizations), then multiplying by the original x, y, and a proves\nthe claim.\nRemark 2: Note that unlike in [19], p. 57, in condition (1′B) above, we use strict henseliza-\ntions rather than completions. The main reason for this is that strict henselizations are\noften technically easier to work with. Moreover, whether one uses strict henselizations or\ncompletions of strict henselizations (i.e., one wants the residue field to be separably closed)\n22\nin condition (1′B) makes no difference. Indeed, this follows immediately from the argu-\nment of the preceding Remark: i.e., whether or not condition (1′B) holds is independent\nof which u, v, and b one chooses downstairs; thus, if one knows that a “good” set of u, v,\nand b exist once one passes to the completion of the strict henselization, then it follows\nthat a “good” set already exists in the strict henselization.\n§3.10. We retain the notations of the preceding §. Let π : C → D be an HM-\nadmissible covering. We would now like to define a canonical log structure on S (determined\ncompletely by π), over which π will extend to a log admissible covering πlog. First let us\ndenote by M\nup\nS (respectively, M\ndn\nS ) the monoid which forms the log structure on S obtained\nby pulling back that of MSlogg,r (respectively, MS\nlog\nq,s) via the classifying morphism for\n(f : C → S, μf ) (respectively, (h : D → S, μh) ). Let MupC (respectively, MdnD ) denote the\nmonoid which forms the log structure on C (respectively, D) obtained by pulling back that\nof the universal symmetric stable log-curve. Next let us focus attention at a node λ ∈ Cs.\nLet ν = π(λ). Let us suppose that we have x, y, u, v, a, e as in Condition (1′B) above. Let\nb = ae, so uv = b. Let us choose sections (in an étale neighborhood of λ) log(x), log(y) of\nM\nup\nC , and a section log(a) of M\nup\nS such that log(x) + log(y) = log(a), and such that under\nthe exponential map, log(x), log(y), and log(a) map to x, y, and a, respectively. Note that\nby Lemma 8, these properties determine log(x), log(y), and log(a) (as a triple) uniquely.\nSimilarly, we choose sections log(u), log(v), and log(b).\nWe would like to define a new log structure, whose monoid we denote by MS , on S.\nTo do this, we note that, étale locally on S near s, the log structure M\nup\nS (respectively,\nMdnS ) has a chart (in the sense of [21], §2.9) whose generators are indexed by the nodes\non Cs (respectively, Ds). More concretely, we may take, for the generator indexed by the\nnode λ (respectively, ν), the element log(a) (respectively, log(b)) just defined. We then\ndefine MS to be the log structure associated to the following quotient monoid:\n(M\nup\nS × MdnS )\/ ∼\nwhere the equivalence relation “∼” is that generated by requiring that (0, log(b)) ∼ (e ·\nlog(a), 0). By Lemmas 7 and 8, everything that we are doing is canonical, so even though we\ndefined MS étale locally on S, (the log structure defined by) MS descends to our original\nS. This completes the construction of MS . Note that we have canonical morphisms\nM\nup\nS → MS and MdnS → MS .\nNow let us denote by (C,MC) (respectively, (D,MD)) the log scheme obtained as the\nfiber product (C,M\nup\nC ) ×(S,MupS ) (S,MS) (respectively, (D,M\ndn\nD ) ×(S,MdnS ) (S,MS)). We\nwould like to extend the morphism π to a morphism of log schemes\nπlog : (C,MC) → (D,MD)\nover (S,MS) which is a log admissible covering. We do this as follows. First note that away\nfrom the nodes, π extends naturally to πlog since the only substantive part (i.e., not pulled\n23\nback from S) of the log structure is constituted by the divisors of marked points, which are\ncarried to each other (by the definition of an HM-admissible covering). Moreover, away\nfrom the nodes, the log morphism πlog that we obtain is log étale, by condition (1′A) of\nthe definition of an HM-admissible covering. Thus, it remains to deal with the nodes. We\ncan work étale locally, as long as we do things canonically. Thus, we focus attention at a\nnode λ of Cs, as above. Then the portion of the log structure of MD that does not come\nfrom MS is generated (in a neighborhood of ν) by log(u) and log(v). Thus, it suffices\nto specify what happens to log(u) and log(v) under (πlog)−1, and to check that what we\nspecify satisfies the relevant relations. We do this as follows: we send log(u) to e · log(x)\nand log(v) to e · log(y). The only relation to be checked is log(u) + log(v) = log(b), but\nthis is clearly preserved (by the definition of MS !). By Lemmas 7 and 8, everything is\ncanonical, so this morphism MD → MC descends to C. Moreover, by [21], Theorem 3.5,\nit is clear that the morphism MD → MC thus defined is log étale. Finally, since e ≤ d,\nit follows that the log morphism πlog : (C,MC) → (D,MD) satisfies condition (1) of the\ndefinition of a log admissible covering.\n§3.11. We summarize the above discussion as follows.\nDefinition: We shall call the log structure MS on S the canonical log structure associated\nto the (HM-)admissible covering π.\nProposition: A morphism π : C → D between symmetrically pointed stable curves\n(f : C → S, μf ⊆ C) and (h : D → S, μh ⊆ D) is an HM-admissible covering if and only if\nit is an admissible covering in the sense of Definition 6.\nProof: We have just shown the (nontrivial) “only if” part. The “if” part is trivial:\nit follows immediately by writing out condition (1) of the definition of a log admissible\ncovering, i.e., what it means to have a log étale morphism. ©\n§3.12. Before going on to the next subsection, we would like to consider briefly how\nadmissible coverings behave over discrete valuation rings. Let S = Spec(A), where A is a\nstrict henselian discrete valuation ring.\nLemma: Let A be a strict henselian discrete valuation ring such that n ∈ A×. Let\ns ∈ A be a uniformizer. For each natural number i ≥ 1, let Ri be the strict henselization\nof A[X,Y ]\/(XY − si) at the maximal ideal m = (x, y, s) (where x, y are the images of\nX, Y in Rn). (Here X and Y are indeterminates.) Let Wi = Spec(Ri) − {m}. Then\nπ1(Wn) ∼= Z\/nZ, and the universal cover W∼n of Wn is W1 with the map W∼n → Wn\ninduced by mapping x to xn and y to yn.\nProof: The map W1 → Wn is clearly finite étale with W1 connected. On the other hand,\nR1 is a regular local ring, so π1(W1) is trivial. The lemma follows. ©\n§3.13. We continue with the notations of the preceding §. Let (f : C → S, μf ⊆ C)\nand (h : D → S, μh ⊆ D) be symmetrically pointed stable curves over S that are smooth\nover the generic point η of S. Let πη : Cη → Dη be an admissible covering over η. Then\n24\nLemma: There is at most one extension of πη to an admissible covering π : C → D\nover S. If d! ∈ A×, then (after possibly replacing A by a tamely ramified extension) there\nexists an admissible covering π : C → D that extends πη.\nProof: The uniqueness statement follows immediately from the fact that Cη is schemat-\nically dense in C. Now we consider existence: Let C′ → S be the normalization of D\nin Cη. By Abhyankar’s lemma, we may, after possibly replacing A by a tamely ramified\nextension, assume that C′ → D is not ramified over the generic points of the special fibre\nof D. By purity of the branch locus, the only places where C′ → D might not be étale are\nover μh and over the singular points of Ds (where s, as usual, is the special point of S).\nBy Abhyankar’s lemma (and the fact that d! ∈ A×), we have tame ramification over μh,\nso we need only deal with the singular points of Ds. But at the singular points, it follows\nfrom the Lemma of the preceding § that C′ → D extends to an admissible covering, and\nC′ = C. ©\n§3C. Global Moduli\n§3.14. Let S be a noetherian scheme. Let f : X → S be a flat, proper morphism. Let\nL be a relatively ample line bundle on X. Then we have the following\nLemma: Let φ : F → G be a morphism of coherent sheaves on X. Suppose that G is flat\nover S. Then there exists a unique closed subscheme Z ⊆ S such that for every morphism\nT → S, we have φT = 0 (where the subscript denotes the result of base-changing from S\nto T ) if and only if the morphism T → S factors through Z.\nProof: By uniqueness, it suffices to prove the Lemma Zariski locally on S. Then there\nexists an n such that F(n) (where F(n) = F ⊗OX L⊗n) is generated by global sections.\nThus, it suffices to prove the Lemma in the case F = OX . Moreover, if n is sufficiently\nlarge, then f∗OX(n) and f∗G(n) are vector bundles on S. Let α : f∗OX(n) → f∗G(n) be\nthe morphism of vector bundles induced by twisting, then pushing forward φ. Let Z be\nthe zero locus of α. Then Z has the property asserted in the statement of the Lemma. ©\nCorollary: Let F and G be closed subschemes of X, and suppose that F is flat over\nS. Then there exists a closed subscheme Z ⊆ S such that for every morphism T → S, we\nhave FT ⊆ GT (where the subscripts denote the result of base-changing from S to T ) if\nand only if the morphism T → S factors through Z.\nProof: Apply the above Lemma to the morphism J → OF , where G = V (J ). ©\n§3.15. Suppose now that (f : C → S, μf ⊆ C) and (h : D → S, μh ⊆ D) are\nsymmetrically pointed stable curves. Suppose that we are given an S-morphism π : C → D.\n25\nLemma: There exists a unique subscheme Z ⊆ S such that for any morphism T → S,\nπT : CT → DT satisfies conditions (2), (3), and (4) of the definition of a log admissible\nmorphism if and only if T → S factors through Z.\nProof: For finiteness (condition (2)), one notes that there exists a closed set H ⊆ C such\nthat π restricted to C −H is quasi-finite, while for x ∈ H, the dimension of π−1π(x) at x\nis nonzero. Then taking Z2 = S − f(H) (which is open in S) takes care of condition (2).\nThus, we may assume that π is finite. Note that it then follows (by the local criterion for\nflatness) that π is flat over the smooth locus of h : D → S. Then let us take a collection\nof nonintersecting sections ρ1, . . . , ρN of h : D → S at whose images h is smooth and such\nthat every irreducible component of every fiber of h has a ρi passing through it (note that\none can always find such ρi étale locally on S). Then Ki\ndef\n= C ×D,π,ρi S is finite and\nflat over S. Moreover, condition (3) is equivalent to the condition that all the Ki be of\ndegree d over S. Thus, taking the open subscheme Z3 ⊆ S over which all the Ki have\ndegree d takes care of condition (3). Finally, we can take care of Condition (4) by means\nof Corollary 14 (in the previous §). ©\n§3.16. Suppose now that (f : C → S, μf ⊆ C) and (h : D → S, μh ⊆ D) are\nsymmetrically pointed stable curves, and that we are given an S-morphism π : C → D\nsatisfying conditions (2), (3), and (4), of the definition of a log admissible covering. Now\nlet us note that the fact that condition (4) is satisfied means that π automatically extends\nuniquely to a log morphism πlog over the smooth locus of h : D → S, for any choice of\nlog structure on S. Thus, we can consider the morphism dπ : π∗ΩD\/S(μh) → ΩC\/S(μf ) of\ndifferentials with logarithmic poles at the marked points. Let us denote by ωC\/S the relative\ndualizing sheaf of C over S. Note that ωC\/S(μf ) is naturally isomorphic to ΩClog\/Slog .\nIndeed, this follows immediately from computing dualizing sheaves in the universal case\nC → Mg,r. Thus, we have a natural morphism\nΩC\/S(μf ) → ΩClog\/Slog = ωC\/S(μf )\nwhich is an isomorphism outside the singular locus of f : C → S. In other words, if\nwe denote the zero set of this natural morphism by Csing, then Csing ⊆ C is a closed\nsubscheme, and C − Csing ⊆ C is precisely the open set of points at which f : C → S is\nsmooth.\nSince the smooth locus of f : C → S is schematically dense in C, it follows that if\nthere exists a morphism ψ : π∗ωD\/S(μh) → ωC\/S(μf ) that extends dπ then it is unique.\nWe shall refer to an isomorphism ψ : π∗ωD\/S(μh) → ωC\/S(μf ) that extends dπ as a log\nextension of dπ.\nLemma: There exists a unique subscheme Z ⊆ S such that for any T → S, dπT admits\na log extension if and only if T → S passes through Z.\n26\nProof: Let Z1 ⊆ S be the subscheme such that for any T → S, (π∗ωD\/S(μh))|T ∼=\n(ωC\/S(μf ))|T ⊗OT N (for some line bundle N on T ) if and only if T → S factors through\nZ1. (See [3], Theorem 7.3, for the existence of such a Z1.) By restricting to Z1, we may\nassume that π∗ωD\/S(μh) and ωC\/S(μf ) are already isomorphic. Let ρ : S → C be a section\nsuch that f is smooth at ρ, and h is smooth at π ◦ ρ. By passing to an open subset of S,\nwe may assume that ρ∗(dπ) is an isomorphism. Let ψ be that isomorphism of π∗ωD\/S(μh)\nand ωC\/S(μf ) whose restriction to ρ is equal to ρ\n∗(dπ). The fact that there exists a Z\nas asserted in the statement of the Lemma then follows from applying Lemma 14 to the\ndifference of the two morphisms π∗ΩD\/S(μh) → ωC\/S(μf ) (one obtained from dπ, the\nother from ψ). ©\nSuppose that dπ admits a log extension. Then it follows immediately from direct\ncalculation of ωC\/S at a smooth point and at a node that condition (5) of the definition\nof a log admissible morphism is satisfied.\n§3.17. Before we can continue, we need to make certain observations concerning the\nlog structure of a symmetrically pointed stable log-curve f log : Clog → Slog. As before, we\ndenote by MC the étale monoid on C defining the log structure of C\nlog, and by NC the\nquotient MC\/O×C . Let ν ∈ Csing be a node. Then we make the following\nDefinition: We shall call an (unordered) pair of elements (α, β) of Γ(C,NC) a separating\npair at ν if α · β ∈ Γ(S,NS), and, moreover, (étale locally at ν), there exist sections α̃, β̃\nof MC such that α̃ (respectively, β̃) projects to α (respectively, β) in NC , and moreover,\nthe images of α̃ and β̃ under the morphism MC → OC (that forms the log structure) map\ninto and generate the maximal ideal of OCf(ν),ν .\nNote that if (α, β) forms a separating pair, then it follows from the definitions that both\nα and β necessarily belong to ΓC ⊆ NC , at least in a neighborhood of ν.\nIt is not necessarily the case that a separating pair at ν exists, even étale locally on\nthe base. However, we have the following result:\nProposition: Suppose that removing ν from Cf(ν) breaks Cf(ν) up into two separate\nconnected components. Then, étale locally on S, there exists a separating pair α, β ∈\nΓ(C,ΓC) at ν.\nProof: It suffices to prove this in the universal case. Therefore, we may assume that S\nis étale over Mg,r. In fact, by basic properties of étale sheaves, we may even assume that\nS is the completion of Mg,r at the point defined by Cf(ν), and that the log structure on\nS is that log structure pulled back from the classifying morphism S → Mg,r. Thus, MC\ninjects into OC . In an étale neighborhood of ν, there exist elements x, y of mC,ν (i.e., the\nmaximal ideal of OC,ν) and t of mS,f(ν) such that xy = t, and such that x, y generate\nmCf(ν),ν . We then define α as follows: In a neighborhood of ν where x is defined and there\nare no nodes other than ν, we let α be the element of NC defined by x ∈ MC ⊆ OC . Away\nfrom ν, we define α as follows: Over the connected component where x is not identically\n27\nzero, we take α to be the trivial section of NC . Over the connected component where x is\nidentically zero, we take α to be the section of NC defined by t ∈ MC ⊆ OC . It is easy to\nsee that α, defined in this way is well-defined and forms a section of ΓC . We define β by\nreversing the roles of x and y. It is clear that (α, β) then forms a separating pair at ν. ©\n§3.18. We continue with the notation of the preceding §. So what happens when the\nnode ν does not satisfy the hypotheses of the preceding Proposition? This happens when\nthe dual graph of Cf(ν) has a loop. In this case, however, it is easy to see that there exists\na finite étale covering of degree two: φf(ν) : C̃f(ν) → Cf(ν) such that\n(1) φ−1(ν) consists of two points ν1 and ν2\n(2) over Cf(ν) − ν, C̃f(ν) consists of two disjoint copies of Cf(ν) − ν.\nMoreover, étale locally on S, there exists a finite étale covering of degree two φ : C̃ → C\nwhose restriction to f(ν) is φf(ν). Let us equip C̃ with the log structure pulled back from\nC, and denote the resulting log scheme by C̃log. Moreover, we let Γ\nC̃\n= φ∗ΓC . Then we\nhave the following result:\nProposition: Étale locally on S, there exist elements α, β ∈ Γ(C̃,Γ\nC̃\n) that form a\nseparating pair at both ν1 and ν2.\nProof: The proof is completely analogous to the proof of the Proposition in the preceding\n§. ©\n§3.19. Suppose now that (f : C → S, μf ⊆ C) and (h : D → S, μh ⊆ D) are\nsymmetrically pointed stable curves, and that we are given an S-morphism π : C → D\nsatisfying conditions (2), (3), (4), and (5) of the definition of a log admissible covering. Let\nM\nup\nS and M\ndn\nS be as in §3.10. Let NupS = M\nup\nS \/O×S ; and NdnS = MdnS \/O×S . Also, we have\nsubsheaves of generators Γ\nup\nS ⊆ N\nup\nS , and Γ\ndn\nS ⊆ NdnS , as defined in §3.3. Since Γ\nup\nS and\nΓdnS are constructible sheaves of sets on S, it follows that they can be represented ([28], p.\n160) by quasi-finite étale algebraic spaces over S; in the following we shall identify these\nsheaves with the spaces that represent them. Let Δ\nup\nS be the subsheaf of N\nup\nS obtained by\nconsidering sums of d sections of Γ\nup\nS . Let Z1 → S be the quasi-finite étale algebraic space\nover S parametrizing morphisms\nΓdnS → ΔupS\nthat preserve the zero section. Note that Z1 need not be a separated S-scheme.\nLet us focus attention on a geometric point s of S. Let us assume that we are given a\ncollection γ1, . . . , γN ∈ Γ(S,ΓdnS ) such that the morphism of sheaves of sets {1, . . . , N} →\n28\nΓdnS given by mapping i to γi is surjective, and N is equal to the number of nodes of\nDs. (One can always find such a collection of γi’s by étale localization at s.) Thus, over\nZ1, we have a tautological morphism Σ : Γ\ndn\nS |Z1 → ΔupS |Z1 , together with sections γi ∈\nΓ(Z1,Γ\ndn\nS ) ⊆ Γ(Z1, NdnS ), and sections γ′i = Σ(γi) ∈ Γ(Z1, NupS ). Let Lγi (respectively,\nLγ′\ni\n) be the line bundle on Z1 corresponding to the O×Z1-torsor obtained by considering\nliftings of γ (respectively, γ′) to a section of MdnS (respectively, M\nup\nS ). Let Z2 → Z1 be the\nspace of N -tuples of isomorphisms Lγi ∼= Lγ′i over Z1. Thus, Z2 is a Gm-torsor over Z1.\nNote that if we know that π : C → D is admissible, then we obtained a natural section\nof Z2 in the course of constructing the canonical log structure on S. More concretely, in\nthe notation of §3.10, the morphisms ΓdnS → Δ\nup\nS and Lγi → Lγ′i are those induced by the\nmorphism MdnS → M\nup\nS given by log(b) \b→\n∑\ne · log(a), where the sum is over all nodes\nλ of Cs that map to the node in question ν (i.e., the node for which b is a deformation\nparameter) of Ds.\nLet Z3 ⊆ Z2 be the closed sub-algebraic space over which the tautological isomor-\nphisms Lγi → Lγ′i are compatible with the exponential morphisms Lγi → OZ2 and\nLγ′\ni\n→ OZ2 of the log structures MdnS and M\nup\nS . Let M\nad → OZ3 be the log struc-\nture on Z3 associated to the largest quotient monoid Q of (M\nup\nS × MdnS )|Z3 such that the\nfollowing diagram commutes:\nLγi |Z3 −→ MdnS |Z3\n(0,id)−→ (MupS × MdnS )|Z3 −→ Q⏐⏐\b ⏐⏐\bid\nLγ′\ni\n|Z3 −→ MupS |Z3\n(id,0)−→ (MupS × MdnS )|Z3 −→ Q\nThus, we obtain a log algebraic space (Z3,M\nad), together with logarithmic morphisms\n(Z3,M\nad) → (Z3,MupS |Z3) → MS\nlog\ng,r , and (Z3,M\nad) → (Z3,MdnS |Z3) → MS\nlog\nq,s , which\nthus determine log-curves C\nlog\nZ3\n→ (Z3,Mad) and DlogZ3 → (Z3,M\nad) (whose underlying\ncurves are C → S and D → S, pulled back to Z3). Let us denote the monoids defining the\nlog structures on C\nlog\nZ3\nand D\nlog\nZ3\nby MC and MD, respectively. Thus, as usual, we obtain\nsheaves NC , ΓC , ND, and ΓD.\n§3.20. We maintain the notation of the preceding §. Let us focus attention on a node\nν of Ds. Let us assume that we have a separating pair (α, β) at ν (for the log-curve\nD\nlog\nZ3\n→ (Z3,Mad)). Also, let us assume that γ def= α ·β is equal to the image of γ1 in Mad.\nLet Zν4 → Z3 be the quasi-finite étale algebraic space of pairs of sections α̃, β̃ of ΓdC |Z3 such\nthat α̃ · β̃ = α ·β. (Here we use the fact that the push-forward of a constructible sheaf by a\nproper morphism is again constructible – see, e.g., [28], p. 223, Theorem 2.1.) Let Lα and\nLβ be the O×DZ3 -torsors of liftings to sections of GD of α and β, respectively. Similarly,\nwe have L\nα̃\nand L\nβ̃\non OCZ3 . Let Z\nν\n5 → Zν4 be the relative scheme parametrizing pairs of\nisomorphisms φ : π∗Lα ∼= Lα̃ and ψ : π∗Lβ ∼= Lβ̃ that are compatible with the exponential\nmorphisms of the log structures of Clog and Dlog, and such that the composite morphism\n29\nf∗Lγ ∼= π∗(Lα ⊗ Lβ)\nφ⊗ψ−→ L\nα̃\n⊗ L\nβ̃\n∼= f∗Lγ\n(where the two isomorphisms at either end are those induced by the equalities α · β =\nγ = α̃ · β̃) is the identity morphism. Note that if we know that π : C → D is admissible,\nthen we automatically obtain canonical α̃, β̃, φ, and ψ by considering the morphism of log\nstructures (that we get automatically from Proposition 11).\nOf course, in general, a separating pair (α, β) might not exist. But just as in Proposi-\ntion 18, we can always find such a pair by passing to an appropriate double covering of D.\nThen it is easy to see that we can still define a Zν5 as above (by working with the double\ncoverings of C and D). Thus, we obtain Zν5 ’s for every node ν on Ds. Let Z6 → Z3 be\nthe product over Z3 of the various Z\nν\n5 , as ν varies over the nodes of Ds. Then we have the\nfollowing\nLemma: The morphism π|Z6 : C|Z6 → D|Z6 is an admissible covering.\nProof: Indeed, it suffices to check condition (1′B) of the definition of an HM-admissible\ncovering (§3.9). But this follows immediately from considering local generators (near a\nnode) of the line bundles Lα, Lβ , Lα̃, and Lβ̃ above, and writing out what it means to give\nmorphisms φ and ψ (as in the definition of Zν5 ) that are compatible with the exponential\nmorphisms of the log structures, and whose tensor product φ ⊗ ψ satisfies the condition\nspecified in the definition of Zν5 . ©\n§3.21. We maintain the notations of the preceding §. Now that we know that π|Z6 is\nadmissible, we can consider the canonical morphisms ΓdnS → Δ\nup\nS ; Lγi → Lγ′i ; Lα → Lα̃;\nLβ → Lβ̃ arising from the logarithmic morphisms obtained over the canonical log structures\nof §3.10, 3.11. Let Z7 → Z6 be the open sub-algebraic space on which the canonical\nΓdnS → Δ\nup\nS agrees with the tautological Γ\ndn\nS → Δ\nup\nS given by the definition of Z1. Let\nZ8 ⊆ Z7 be the closed sub-algebraic space on which the canonical Lγi → Lγ′i agrees with\nthe tautological Lγi → Lγ′i given by the definition of Z2. Similarly, we let Z9 ⊆ Z8 be\nthe sub-algebraic space on which the canonical α̃; β̃; Lα → Lα̃; Lβ → Lβ̃ agree with the\ntautological α̃; β̃; Lα → Lα̃; Lβ → Lβ̃ given by the definitions of Z\nν\n4 and Z\nν\n5 . Thus, in\nsummary, we have proven the following:\nLemma: Let T → S be a morphism of schemes. Then πT : CT → DT is admissible if\nand only if T → S factors through Z9. Moreover, this factorization of T → S through Z9\nis unique. In particular, the algebraic space Z9 is, in fact, separated over S. Finally, if d!\nis invertible on S, then Z9 is even a closed subscheme of S.\nProof: Indeed, it suffices to prove the final statement. This follows by noting that Lemma\n13 implies that the morphism Z9 → S is proper. Since it is also a monomorphism, it\n30\nfollows that it is quasi-finite, hence (by Zariski’s main theorem) finite. Thus, Z9 is a closed\nsubscheme of S. ©\nNote finally, that Z9 comes equipped with a canonical log structure, i.e., the log structure\nassociated to the admissible covering π|Z9 .\nRemark: In fact, it is not difficult to show (although we will not use this fact) that the\nmorphism Z9 → S is open (i.e., the image of an open set is open). However, even when d!\nis invertible on S, it is not necessarily the case either that Z9 and S coincide as sets, or\nthat Z9 ↪→ S is an open immersion. Indeed, below (in §3.24), we shall construct examples\n(where d! is invertible on S) of points of S which are not in Z9, as well as of cases where\nthe nilpotent ideal defining the closed subscheme Z9 ⊆ S is nonzero.\n§3.22. We are now ready to state the main Theorem of this Section:\nTheorem: Fix nonnegative integers g, r, q, s, d such that 2g − 2 + r = d(2q − 2 + s) ≥ 1.\nLet A be the stack over Z defined as follows: for a scheme S, the objects of A(S) are\nadmissible coverings π : C → D of degree d from a symmetrically r-pointed stable curve\n(f : C → S;μf ⊆ C) of genus g to a symmetrically s-pointed stable curve (h : D → S;μh ⊆\nD) of genus q; and the morphisms of A(S) are pairs of S-isomorphisms α : C → C and\nβ : D → D that stabilize the divisors of marked points such that π ◦α = β ◦π. Then A is a\nseparated algebraic stack of finite type over Z. Moreover, A is equipped with a canonical log\nstructure MA → OA, together with a logarithmic morphism (A,MA) → MS\nlog\nq,s (obtained\nby mapping (C;D;π) \b→ D) which is log étale (always) and proper over Z[ 1\nd! ].\nProof: Indeed, we start with W log\ndef\n= MSlogg,r ×Z MS\nlog\nq,s . We consider the Hilbert stack\nH → W (i.e., H → W is a relative scheme) of morphisms from the tautological r-pointed\ncurve of genus g to the tautological s-pointed curve of genus q. Applying Lemmas 15 and\n16, we obtain a sub-algebraic stack S ⊆ H, to which we can apply the constructions of\n§3.19. Finally, we have the stack Z9 → S (which was constructed étale locally on S, but\nwhich clearly descends back down to the original S by Lemma 21). This gives us our\nA. The fact that the morphism (A,MA) → MS\nlog\nq,s is log étale is an exercise in tracing\nthrough the definitions of the various log structures involved. The assertion on properness\nthen follows from Lemma 13. ©\nRemark: Applying the construction of [11], Chapter 1, §4.10, to A, one obtains a coarse\nmoduli space Ac. Moreover, over Z[ 1\nd! ], Ac is a projective scheme: Indeed, let Mc be\nthe coarse moduli space associated to MSq,s. It is well-known (at least in characteristic\nzero – see [23]) that Mc is a normal projective scheme. Then by the above Theorem,\nAc → Mc is proper. Also, it is clear (by considering the number of possibilities for an\nadmissible morphism between two given stable curves) that Ac → Mc is quasi-finite. Thus,\nby Zariski’s main theorem, it follows that Ac → Mc is finite. Thus, Ac is a projective\nscheme.\n31\n§3.23. Finally, just as in [19], p. 62, we can make explicit the completion of a local ring\nof A. Let k be an algebraically closed field. Let A = W (k), the ring of Witt vectors with\ncoefficients in k. Let π : C → D be an admissible covering over k. Let us denote by [π] ∈\nA(k); [C] ∈ MSg,r(k); [D] ∈ MSq,s(k) the corresponding k-valued points of the respective\nmoduli stacks. Let O[π] (respectively, O[D]) be the completion of the strict henselization\nof A (respectively, MSq,s) at [π] (respectively, [D]). Suppose that D has exactly N nodes,\nwhich we number from 1 to N . Thus, we have a noncanonical isomorphism\nO[D] ∼= A[[t1, . . . , tN , s1, . . . , s3q−3+s−N ]]\nwhere the t’s and s’s are indeterminates; and ti defines a universal deformation parameter\nfor the ith node on D. Suppose that C has ni nodes over the i\nth node on D, which we\nlabel (i, 1) through (i, ni). Suppose that the ramification index of (either branch of) C\nover D at the node (i, j) is ei,j . Then\nO[π] ∼= O[D][{ti,j}]\/(tei,ji,j − ti)\nwhere we allow i to run from 1 to N , and j to run from 1 to ni. The canonical log\nstructure at [π] is the log structure associated to the quotient monoid of the free monoid on\nthe symbols {log(ti,j); log(ti)} (where i and j vary as before) by the equivalence relation\ngenerated by log(ti) = ei,j · log(ti,j). The exponential morphism of the canonical log\nstructure maps the symbol log(ti,j) to (the image in O[π] of) ti,j and the symbol log(ti) to\n(the image in O[π] of) ti.\nThis sort of explicit structure result gives us the following:\nCorollary: Let As ⊆ A be the open substack where the curves C and D of the tau-\ntological admissible covering π : C → D are smooth. Then As is smooth over Z and\nschematically dense in A. Moreover, A is flat over MSq,s; reduced; and its normalization\nN is smooth over Z and flat over MSq,s. Thus, we have a natural inclusion As ↪→ N .\nFinally, the complement of As in N is a divisor with normal crossings. We shall refer to\nthese divisors as the divisors at infinity of N .\n§3.24. We maintain the notations of the preceding §. In this §, we would like to\nconstruct examples showing the extent to which (in the notation of Lemma 21) Z9 \t= S:\nExample 1: First, we show how to construct k-valued points of Z9 that are not in S.\nNamely, we consider admissible π : C → D such that D has a node ν over which two\ndistinct indices of ramification e and e′ occur (i.e., e \t= e′). Let λ (respectively, λ′) be\nthe point of C where the index of ramification e (respectively, e′) occurs. Let us assume\n(for simplicity) that ν sits on two distinct irreducible components ID and JD of D. Let\nIC (respectively, JC) be the irreducible component of C containing λ that lies over ID\n(respectively, JD). Similarly, we define I\n′\nC and J\n′\nC for λ\n′. Let C̃ be the curve obtained\nfrom C as follows: C̃ is exactly the same as C except at λ and λ′; there we glue IC (at λ)\nto J ′C (at λ\n′) and I ′C (at λ\n′) to JC (at λ). Then we have a map π̃ : C̃ → D (induced by\n32\nπ) which satisfies conditions (2) through (5) of the definition of a log admissible covering,\nbut is obviously not admissible. (In fact, if e ≡ e′ (mod p) (where p = char(k)), then dπ′\neven admits a log extension in the sense of §3.16. Of course, e ≡ e′ (mod p) is impossible\nif d! ∈ k×.)\nExample 2: Let B = k[\u0013]\/(\u00132). Let S = Spec(B); let s ∈ S(k) be the unique point. Let\nD\u000f = D×k S; (μh)\u000f = (μh)×k S ⊆ D\u000f, where μh ⊆ D is the divisor of marked points in D.\nWe would like to consider morphisms π\u000f : C\u000f → D\u000f from a symmetrically pointed stable\ncurve (f\u000f : C\u000f; (μf )\u000f ⊆ C\u000f) to (D\u000f; (μh)\u000f) such that π\u000f|s = π; and π\u000f satisfies conditions\n(2) through (5) of the definition of a log admissible morphism. By deformation theory, it\nfollows that the isomorphism classes of such π\u000f are in one-to-one correspondence with the\nelements of some finite-dimensional k-vector space V , which can be computed by means of\nlocal computations at the nodes. In fact, it is not difficult to calculate V explicitly. One\nthus obtains that\ndimk V =\n∑\ni,j\n2(ei,j − 1) + sgn(ei,j − 1)\nwhere i, j run over all i and j as allowed in §3.23, and sgn(0) = 0; sgn of a positive number\nis 1. On the other hand, the subspace of V corresponding to admissible coverings π\u000f is (by\nwhat we did in §3.23) of dimension ∑i,j sgn(ei,j − 1), i.e., in general, it is much smaller\nthan all of V . Thus, we see that even at a point of S contained in Z9, the ideal defining\nthe closed subscheme Z9 is, in general, nonzero.\n§3D. Admissible Hurwitz Coverings\n§3.25. Let π : C → D be an admissible covering.\nDefinition: We shall call π an admissible Hurwitz covering if the genus of D is zero and\nπ is simply ramified over the marked points of D.\nIn this Section, we have been concerned with proving the very general Theorem 22 above.\nIn the rest of this paper, however, we will be concerned only with Hurwitz coverings (i.e.,\nthe sorts of coverings parametrized by the Hurwitz stacks of §1) and their degenerations,\nso we shall refer to admissible Hurwitz coverings simply as admissible coverings.\n§3.26. Remark: In this §, we explain why the proof of Theorem 4, p. 58 of [19] has\na fundamental gap. On p. 60 of [19], it is claimed that (in the notation of [19]) H4 will\nrepresent the functor HUk,b. In the notation of our Lemma 21, this essentially amounts to\nthe assertion that Z9 = S. Indeed, the only difference between our S and [19]’s H4 is that\nH4 includes projective embedding data for C (as described on p. 59 of [19]). On the other\nhand, we saw (Examples 1 and 2 of §3.24) that in general, Z9 \t= S.\n33\nTo put the issue another way, [19] completely neglect the nontrivial issue of showing\nthat condition (1′B) of the definition of an HM-admissible covering is “scheme-like,” i.e.,\nthat given an S-morphism π : C → D satisfying all the conditions for an admissible\ncovering except (1′B), the set of s ∈ S over which πs : Cs → Ds also satisfies (1′B) has\nthe natural structure of a subscheme Z of S, and, moreover, that over Z, πZ : CZ → DZ\nis an admissible covering. These issues occupy the bulk of our proof of Theorem 22, and\nare the main reason for the introduction of log structures. Indeed, the condition that a\nfunction defined only locally in a neighborhood of a node on C have an eth root (say, étale\nlocally) is in general not a very “scheme-like” condition.\nFor instance, let S = Spec(k[[t]]); X = Spec(k[[x, t]]), where k is an algebraically\nclosed field of characteristic \t= 2. We regard X as an S-scheme. Consider the condition\nthat the function x2 − t2 have a square root étale locally in a neighborhood of the point\nx = 0. This condition is satisfied at both the special and the generic points of S. But it is\ncertainly not satisfied over S.\nThus, in order to obtain a nice moduli problem it is necessary to translate this sort\nof local condition (i.e., condition (1′B)) into a condition which is global on the curve C.\nThis is precisely what is achieved by the introduction of log structures.\n§3.27. Notation: We close by introducing some notation that will be of use in the\nrest of this paper. Let b ≥ 4 be an integer. Let R = Z[ 1\nb! ]. Relative to the notation\nof the present section, we are interested in the case q = 0, s = b. We denote by ROb\n(respectively, RSb) the scheme Mq,s ⊗Z R (respectively, stack MSq,s ⊗Z R), which thus\nforms a compactification of the scheme ROb (respectively, stack RSb) of §1.3. By simple\ncombinatorial analysis, one sees that in the genus zero case, in fact, every irreducible\ncomponent of a stable curve D of genus zero over an algebraically closed field is smooth\nand rational and that the graph obtained by taking the irreducible components of D as\nvertices and the singular points (which necessarily join two distinct components) as edges\nmust have H1 = 0 (in the sense of singular homology). We shall call this graph the “graph\nassociated to D.”\nIt follows from [22] that the complement of ROb (resp. RSb) in ROb (resp. RSb),\ni.e. “the divisor at infinity” of ROb (resp. RSb), is made up of irreducible components\nI, where the generic point of I represents a stable curve that consists of two copies of P1\njoined at one point with i of the marking points on one component and b− i points on the\nother component. In the symmetrized case, the divisors at infinity are thus in one-to-one\ncorrespondence with unordered pairs of natural numbers (i, j) such that i + j = b, i ≥\n2, j ≥ 2. Thus, in the future, we shall speak of the “(i,j)-component” or “(i,j)-divisor”\n(at infinity) and so on.\nLet d ≥ 2 be an integer. Since we have already fixed q = 0 and s = b, Theorem 22\ngives us an algebraic stack A. Let AR = A ⊗Z R. Note that by Abhyankar’s Lemma,\nthere is an open and closed substack CSb,d of AR parametrizing the admissible Hurwitz\ncoverings. Thus, CSb,d contains HRSb,d as a dense open substack (Corollary 23). By §2.9,\nit thus follows that CSb,d is geometrically irreducible over R. Let COb,d → CSb,d be the\nfinite étale covering of degree b! obtained by ordering the marked points of the curve of\n34\ngenus 0 downstairs (i.e., the “D” of π : C → D). Let NSb,d (respectively, NOb,d) be the\nnormalization of CSb,d (respectively, COb,d). Finally, note that we have a commutative\ndiagram\nNOb,d −→ NSb,d −→ Mg⏐⏐\b (∗) ⏐⏐\b ⏐⏐\bid\nCOb,d −→ CSb,d −→ Mg\nwhere the horizontal morphisms on the right assign to an admissible covering π : C → D\nthe curve C, and the square (∗) is Cartesian.\n§3.28. Remark : The compactification of the Hurwitz scheme has many interesting\napplications to the geometry of Mg. For instance, as in [27] (the Appendix to [19]), one\nsees that it allows a direct algebraic proof of the connectedness of Mg. One may also note\nthat the proof of the properness of the compactification COb,d never uses the Semi-Stable\nReduction Theorem. Thus it follows that (at least for high characteristic) we obtain a\nnew, direct, elementary proof of this Theorem.\n§4. Construction of the Boundary Components\n§4.1. We shall refer to the irreducible components of the complement of HRSb,d in\nCSb,d (respectively, NSb,d) as the “divisors at infinity” of CSb,d (respectively, NSb,d). In\nthis Section, we shall give an explicit construction as well as a complete combinatorial\ndescription of the “divisors at infinity” of CSb,d. (One can then classify the divisors at\ninfinity of COb,d according to which divisor at infinity of CSb,d they are mapped to; un-\nfortunately, however, it is not clear that the irreducibility results for divisors at infinity of\nCSb,d extend to the divisors at infinity of COb,d.)\n§4.2. Let S be an R-scheme. Let Xi → S, i = 1, 2 be prestable curves in the sense of\n[22], i.e., they are flat and proper and have geometric fibres that are reduced, of dimenion\none, and have at most ordinary double points as singularities. Let τi : S → Xi be a section\nthat lands in the smooth locus. By loc. cit, Theorem 3.4, p. 181, we can glue together\nthe Xi’s along the τi’s to obtain a new prestable curve X → S. It is clear that this\nconstruction is functorial, so if (Yi, ρi) satisfy the same hypotheses as the (Xi, τi), then\nwe obtain, by clutching, a new prestable curve Y → S as well as a morphism X → Y for\nevery pair of morphisms Xi → Yi, (i = 1, 2) that carry τi to ρi. It thus follows that if we\nalso give sections σ\n(i)\n1 , . . . , σ\n(i)\nbi\n(i = 1, 2) : S → Yi that land in the smooth locus and are\ndisjoint from each other and from ρi, in such a way that (Yi, σ\n(i)\n1 , . . . , σ\n(i)\nbi\n, ρi), (i = 1, 2)\nis a (bi + 1)-pointed stable curve, and if the morphisms Xi → Yi satisfy all the conditions\nfor an admissible covering of degree d over (Yi, σ\n(i)\n1 , . . . , σ\n(i)\nbi\n, ρi) except that:\n35\n(1) the ramification of Xi → Yi over ρi might not be simple; however, it is\ntame, and, moreover, we give sections τ\n(i)\n1 , . . . , τ\n(i)\nl , (i = 1, 2) : S → Xi\nthat land in the smooth locus, are disjoint from each other, map via\nXi → Yi to ρi (i = 1, 2), and are such that over each geometric point s\nof S, τ\n(i)\n1,s; . . . ; τ\n(i)\nl,s are exactly the points of Xi,s which are in the inverse\nimage of ρi,s under Xi,s → Yi,s; moreover, the ramification index of\nXi,s → Yi,s at τ (i)j,s (for all j) is independent of i (for i = 1, 2);\n(2) the fibres of Xi → S might not be geometrically connected, but if we\nglue together X1 and X2 along τ\n(1)\nj and τ\n(2)\nj (for all j), the resulting\nX → S has geometrically connected fibres;\nthen we may construct an admissible covering by taking for X the gluing together of X1\nand X2 along τ\n(1)\nj and τ\n(2)\nj (for all j) and for Y , the gluing together of Y1 and Y2 along\nρ1 and ρ2, and for the map X → Y , the functorially associated map for the pair of maps\nX1 → Y1 and X2 → Y2. In particular, note that Y is necessarily stable.\nNote that instead of gluing together two “almost admissible” coverings to obtain an\nadmissible covering, we could just as well have glued together more than two “almost\nadmissible” coverings (with the appropriate conditions analogous to those above on the\ndata) to obtain an admissible covering. However, we shall have no need for this (only\nclerically) more difficult version of the above construction in this paper.\n§4.3. Lemma : Let S be an excellent Noetherian R-scheme. Let V → S be smooth\nof dimension 1 with section ρ : S → V . Let U → S be smooth of dimension 1 with a finite,\nflat S-morphism ϕ : U → V which is étale except over ρ where it has tame ramification.\nLet W = the scheme ϕ−1S = U ×V S, where S is a V -scheme via ρ. Then if S is regular,\nWred is finite étale over S.\nProof: Using the excellence hypothesis (to ensure that taking “red” commutes with pass-\ning to completion – see EGA IV for a definition and basic facts about excellent rings), we\nreduce immediately to the case where S = Spec A, where A is a complete local ring with\nseparably closed residue field. Let s be the closed point of S. Then let V ∧ = the completion\nof V at the prime ρ(s). Let U∧ = U ×V V ∧. Now V ∧ = Spec A[[T ]], where T is an inde-\nterminate such that the image subscheme of ρ has ideal generated by T . By Abhyankar’s\nlemma, U∧ is the spectrum of a direct product of rings of the form (A[[T ]])[X]\/(Xn − T )\nwhere n is prime to char k(s). The result follows. ©\n§4.4. Definitions : As in §1, we fix b, d, g. We define a layout L to be the following\nset of data:\n(1) An ordered pair of integers (iL, iR) with iL, iR ≥ 2, iL + iR = b (here\nL stands for “left”, R for “right”);\n(2) Two ordered chains CL and CR of iL (resp. iR) transpositions in Sd\nsuch that if we denote by\n∏\nCL (resp.\n∏\nCR) the ordered product of the\n36\nrespective transpositions, then we have\n∏\nCL =\n∏\nCR; and such that\nthe subgroup of Sd generated by all the transpositions of CL and CR\nacts transitively on {1, . . . , d}.\nWhen we need to deal with several layouts at once, we shall use notation like iLL, C\nL\nL ,\netc. to denote the various objects associated to the layout L. Let D1L, . . . ,DnLL (resp.\nD1R, . . . ,D\nnR\nR ) be the domains of transitivity (including those with only one element) of\nthe group of permutations generated by the transpositions of CL (resp. CR). We refer\nto these as the left (resp. right) domains of transitivity. The permutation\n∏\nCL =\n∏\nCR\nwill be called the total clutching permutation σ. The ordering of CL gives us a natural\nbijection of {1, . . . , iL} (resp. {1, . . . , iR}) with CL (resp. CR). Relative to this bijection,\nlet BiL (resp. B\ni\nR), where i = 1, . . . , nL (resp. i = 1, . . . , nR) be the subset of {1, . . . , iL}\n(resp. {1, . . . , iR}) corresponding to those transpositions that act nontrivially on DiL. Let\nbiL (resp. b\ni\nR) = the cardinality of B\ni\nL (resp. B\ni\nR). We call b\ni\nL (resp. b\ni\nR) the i\nth left (resp.\nright) partition number of the layout. The permutation which is the ordered product of\nall the transpositions of CL (resp. CR) that correspond to elements of B\ni\nL (resp. B\ni\nR)\nwill be called the ith left (resp. right) clutching permutation σiL (resp. σ\ni\nR) and will also\nbe written\n∏\nDi\nL\nCL (resp.\n∏\nDi\nR\nCR). Note that the total clutching permutation σ is the\nproduct of the σiL (resp. σ\ni\nR) over i = 1, . . . , nL (resp. i = 1, . . . nR). Two layouts will be\nsaid to be isomorphic if one can be obtained from the other by switching left and right\nand\/ or by conjugating all the data by a some element of Sd. It is clear that isomorphism\nis an equivalence relation. We denote the (finite) set of layouts by Lb,d (or simply L when\nb, d are fixed and clear) and the set of isomorphism classes of layouts by CLb,d (or simply\nCL).\nGiven a layout L, we define its template T to be the following set of data:\n(1) The ordered pair (iL, iR);\n(2) The (unordered) set of “left” (resp. “right”) triples of the following sort:\nEach “left” (resp. “right”) triple consists of a left (resp. right) domain\nof transitivity, its associated clutching permutation, and its associated\npartition number.\nLet Tb,d (or simply T) be the set of templates T corresponding to all possible layouts\nL. Note that we have an analogous notion of isomorphism of templates, corresponding\nto switching left and right and\/ or conjugating by a permutation. Let CTb,d (or simply\nCT) be the set of isomorphism classes of templates. Then we have the following natural\nsurjections:\nΦ : L −→ T\nΨ : CL −→ CT\n§4.5. Theorem : The divisors at infinity of CSb,d are in natural one-to-one corre-\nspondence with the elements of CT.\n37\nProof: The proof will consist of several steps:\nStep A: Construction of divisors at infinity:\nLet T ∈ Tb,d be a template. We start by constructing the “left iL-point stack”. (Of\ncourse, the construction of the right iR-point stack is entirely analogous.) Let LROT =\nRO(iL+1), where L stands for “left”, RO for “rigidified, ordered” (as usual), and T denotes\nthe dependence on the template. Let b1, . . . , bnL be the left partition numbers (in any\norder!) of the template T . Let us partition the set {1, . . . , iL} into disjoint subsets Ai\n(where i = 1, . . . , nL) of bi elements. The sections of P\n1 × LROT corresponding to the\nset Ai will be called sections of the i\nth partition. The last section will be called the\nclutching section. Let πi : LROT → DRObi+1 denote the map corresponding to sending\nthe sections of the ith left partition (in some order) to the marking sections of DRObi+1\nand the clutching section to the clutching section (recall the terminology from §1). Now\nG = Sb1 × Sb2 × . . . × SbnL generically acts freely on LROT by allowing the ith factor to\nact naturally on the sections of the ith partition. We define the left iL-point stack, LRST ,\nto be the quotient (in the sense of stacks) of LROT by G, which is generically a scheme\n(by the generic freeness of the section of G). Note that the πi’s induce natural projections\nϕi : LRST → DRSbi+1.\nNow let σi ∈ Sd be the ith left clutching permutation. Let Di be the ith left do-\nmain of transitivity, and let di = the cardinality of Di. Then we constructed (§1.4, 1.5)\nDHRSσibi+1,di and a “pseudo-universal” covering:\nCi\nβi−→ P1Di ⊇ DDi ⊇ MDi , CDi\nαi ↘\n⏐⏐\b\nDi\nwhere CDi ∼= Di, and we write Di for DHRSσibi+1,di . Recall that CDi is the clutching\nsection, while MDi (which is finite étale of rank bi over Di) is the marking divisor. Since\nDi is regular, it follows by Lemma 3 that the reduced inverse image via βi of CDi is étale\nover CDi ∼= Di; let us denote this reduced inverse image by C∼Di . Now the question arises:\nis C∼Di “geometric” over R in the sense that the étale covering C\n∼\nDi → CDi does not factor\nthrough a covering obtained by tensoring up by an étale cover R∼ of R? I claim that the\nanswer is “yes”. Indeed, recall from Proposition 10 of §2 that the connected components\nover R correspond naturally to the various ramification indices, and these components are,\nin fact, geometric over R.\nWe are now in a position to construct the left degenerate Hurwitz stack associated\nto the template T , which we we shall call LDHT . We define LDHT to be the fibred\nproduct over LRST of the pull-backs via the ϕi of the arrow Di → DRSbi+1. It is clear\nby the results of §2 that LDHT is smooth and geometrically irreducible over R. We can\nalso pull-back the coverings C∼Di → Di to coverings C∧Di → LDH\nT . These coverings are\nétale and “geometric” in the sense used above. Therefore, by basic facts of Galois the-\nory, there exists a (noncanonical) finite étale Galois covering Λ → LDHT that dominates\n38\nall the connected components of all the C∼Di (for all i) and which is geometrically irre-\nducible over R. We denote the pull-backs of the objects Ci, αi, βi,MDi , CDi , C∼Di ,DDi by\nCΛi , γi, δi,MΛi , CΛi , C∼,Λi ,DΛi . We thus obtain a commutative diagram:\nC\n∼,Λ\ni ⊆ CΛi\nδi−→ P1Λ ⊇ DΛi ⊇ MΛi , CΛi ∼= Λ\nγi ↘\n⏐⏐\b\nΛ\n(Diagram 4.1)\nwhere the relative dimension of Λ is iL − 2. Note that C∼,Λi → Λ is split (i.e., every\nconnected component of C\n∼,Λ\ni maps isomorphically to Λ).\nAnalogously, we construct the right degenerate Hurwitz stack RDHT and, in place of\nΛ, we have P (the capital Greek letter “rho”). We thus obtain a diagram:\nC\n∼,P\ni ⊆ CPi\n\u000fi−→ P1P ⊇ DPi ⊇ MPi , CPi ∼= P\nζi ↘\n⏐⏐\b\nP\n(Diagram 4.2)\nwhere (again) the relative dimension of P over R is iR − 2. Thus Kdef=Λ×R P has relative\ndimension over R equal to b − 4. This is just the right dimension for a divisor of CSb,d.\nLet us pull the Diagrams 4.1 and 4.2 back to K via the projections to Λ and P , and\ndenote the objects so obtained by C\n∼,Λ\ni,K ; CΛi,K ; etc. Now we first glue the left and right\ncopies of P1 together along CΛi and C\nP\ni ; using the marking divisors M\nΛ\ni and M\nP\ni we thus\nobtain a b-pointed stable curve B → K. Let us next note that (referring to §4.4) the\nirreducible components of CΛi (resp. C\nP\ni ) correspond naturally to the left (respectively,\nright) domains of transitivity, which is part of the information of the layout that is pre-\nserved in the template. Thus, there is a natural isomorphism CΛi\n∼= CPi . Let us glue\ntogether\n∐\ni=1,...,nL\nCΛi,K and\n∐\ni=1,...,nR\nCPi,K (both prestable curves) along CΛi and CPi via\nthis isomorphism and call the resulting curve X → K. It follows from the conditions in the\ndefinition of a layout and by what we did in §4.2, that we then obtain an admissible cov-\nering f : X → B over K. By the modular definition of CSb,d, we then obtain a morphism\nK → CSb,d which we call MT . It is clear by composition with the morphism CSb,d → RSb\nthat MT is quasi-finite. Thus, for dimensional reasons, the closure of its image must be\nan irreducible divisor of CSb,d, which is flat over R. We shall refer to this divisor as the\ndivisor DT associated to T . We shall also refer to any divisor of NSb,d that lies above\nDT as a “divisor associated to T ”. Finally, it is clear that isomorphic templates define the\nsame divisor. Thus we obtain a map:\nΩ : CT −→ {divisors at infinity of CSb,d}\n39\nStep B: Surjectivity of Ω:\nSince CSb,d → RSb is flat (Corollary 23 of §3), every divisor at infinity of CSb,d lies\nabove a divisor at infinity of RSb,d, whose generic element is necessarily of the form given\nin Diagram 3.1. Thus it suffices to see that every geometric point of CSb,d such that the\nstable curve downstairs is of the form given in Diagram 3.1 (in the future, we shall call\nsuch a covering a basic admissible covering of type (i, j), where (i, j) is as in loc. cit.) is in\nthe image of a geometric point of one the MT ’s. But this is obvious from the construction.\nStep C: Injectivity of Ω:\nIt suffices to see that given any basic admissible covering, we can recover the informa-\ntion contained in a template from the purely geometric data of the covering. But this is\nclear, for left (resp. right) domains of transitivity correspond to the irreducible components\non the left (resp. right) hand side of the curve upstairs, associated clutching permutation\ncorrespond to the ramification data at clutching points, and partition numbers correspond\nto the number simple ramification points of each irreducible component upstairs. ©\n§4.6. Any particular divisor at infinity of CSb,d will be called a Species. If (iL, iR)\nis the first piece of data of a template T (see §4.4), then we call T a “template of type\n(iL, iR)”. We also say that the corresponding Species is of type (iL, iR). Before going\non, it is worthwhile to enumerate the various Species of type (b − 2, 2). By the previous\nTheorem, this amounts to classifying explicitly the various templates that appear for basic\nadmissible coverings of type (b − 2, 2).\n(b-2)   points\nx      x\n2     points\nx  x  x  x  x  .........  x  x  x  x\nLet C be a component on the right hand side of the curve upstairs for such a covering.\nThen either:\nI. C has no “x”’s (i.e., simple ramification points); then C maps isomor-\nphically to the copy of P1 downstairs;\nII. C has exactly one “x”; then it is clear that C ∼= P1 and the map\ndownstairs must be the “squaring map”; there is one clutching point on\nC, where the ramification index is two; or\nIII. C has exactly two “x”’s; then each of those corresponds to a transpo-\nsition; up to renaming elements, these two transpositions are either:\n(A) (12), (23): then C ∼= P1; there is one clutching point\non C with ramification index three; the map down-\nstairs is some uniquely determined map;\n40\n(B) (12), (12): then C ∼= P1; there is no ramification at\nthe clutching points, of which there are two on C; the\nmap downstairs is the “squaring map”;\n(C) (12), (34): the degree of the map from C to the copy\nof P1 downstairs would have to be four, and the de-\ngree of the total ramification divisor on C would also\nhave to be four; thus this case is impossible by the\nRiemann-Hurwitz formula.\nLet us now consider the totality of components on the right. Exactly one of the following\ntwo possibilities must hold:\n(1) There are two components on the right with one “x” each and all the other components\nhave no “x”’s. In this case, let us consider the associated graph of the curve upstairs.\nOn the right, we have d − 2 vertices, each with exactly one edge protruding from it. It\nfollows that if the graph is to be connected (which it must be), then there can only be one\nirreducible component on the left. This component is a smooth component of genus g (the\noriginal genus) and has d − 2 clutching points, one going to each irreducible component\non the right. Of these, there is ramification of degree 2 at two clutching points (which\nwe shall call points of degenerate ramification), and no ramification at any of the d − 4\nothers. It is thus clear that in this case, the template is uniquely determined. We call\nthe corresponding divisor at infinity the divisor of Species 2. A typical basic admissible\ncovering of this Species is shown in the Pictorial Appendix.\n(2) There is exactly one component on the right with two “x”’s, and all the other components\nhave no “x”’s. Then the component with two “x”’s is as in exactly one of the following,\ncorresponding, respectively, to the Cases (A) and (B) of the preceding III:\n(A) It is clear that the graph of the curve upstairs is the same in in (1) above.\nThus there is only one irreducible component on the left; it is smooth\nof genus g and has d − 2 clutching points, one going to each irreducible\ncomponent on the right. Of these, there is ramification of degree 3 at one\nclutching point (which we shall call the point of degenerate ramification)\nand no ramification at any of the d−3 others. It is thus clear that in this\ncase, the template is uniquely determined. We call the corresponding\ndivisor at infinity the divisor of Species 1. A typical basic admissible\ncovering of this Species is shown in the Pictorial Appendix.\n(B) Let us consider the graph associated the curve upstairs. On the right,\nwe have d − 1 vertices; from d − 2 of them, there protrudes exactly one\nedge; from the other one of them there protrude exactly two edges. Now\neither those two edges end up at the same vertex on the left, or they\nend up at different vertices. We thus examine each of these two cases:\n41\nα ) The Same Vertex: Then there is only one vertex\non the left. This corresponds to a smooth, irreducible\ncomponent of genus g−1, since the loop on the graph\ncontributes “one” to the arithmetic genus of the curve\nupstairs. There are exactly d clutching points on this\ncomponent on the left, all of which are unramified.\nOf these d − 2 connect to components on the right\nof type (I) in our classification above of components\nof the curve upstairs, and the other 2 (which we shall\npoints of degenerate ramification) connect to the com-\nponent of type (III) (B) in that classification. Clearly\nthe template is thus uniquely determined. We call the\ncorresponding divisor at infinity the divisor of Species\n3A. A typical basic admissible covering of this Species\nis shown in the Pictorial Appendix.\nβ ) Different Vertices: Then it is clear that these are\nthe only two vertices on the left. They correspond\nto two smooth, irreducible components C1 and C2.\nLet gi be the genus of Ci (where i = 1, 2). If bi =\nthe number of “x”’s on Ci, and di is the degree of Ci\nover the left hand copy of P1, then we have, by the\nHurwitz formula, 2 (gi − 1) + 2di = bi, g1 + g2 = g,\nb1 + b2 = b − 2, d1 + d2 = d. The map downstairs\nCi → P1 thus has only simple ramification. The\nclutching points are all unramified with one clutch-\ning point on each Ci (which we shall refer to as a\npoint of degenerate ramification)going to the compo-\nnent of type (III) (B); the other di−1 clutching points\non each Ci go to components of type (I). Given pairs\nof integers (b1, b2), (d1, d2) such that b1 + b2 = b − 2,\nd1+d2 = d, with bi ≥ 0, di ≥ 1 and such that the gi’s\ncomputed via the Hurwitz formula are nonnegative in-\ntegers (for i = 1, 2) will always appear, and once the\nbi’s and di’s are given, the template is clearly uniquely\ndetermined. The corresponding divisor at infinity will\nbe call the divisor of Species 3B (b1, b2); (d1, d2). It\nwill turn out that the case when b1 = b − 2, b2 =\n0, d1 = d − 1, d2 = 1 (⇒ g1 = g, g2 = 0) will be es-\npecially important; we shall call this case the divisor\nof Species 3B∗. As usual, illustrations are shown in\nthe Pictorial Appendix.\n§4.7. Let us note that among the templates of type (b− 2, 2), the Species that do not\n42\nmap via the map CSb,d → Mg of §3.27, into the boundary at infinity of Mg are exactly\nSpecies 1, 2, and 3B (b1, b2); (d1, d2) when g1 = g, g2 = 0. Among these, it turns out that\nthe divisors of Species 1, 2, and 3B∗ (which constitutes a set of precisely three divisors)\nwill play an important role in the future. We shall refer to these three divisors as the\nexcess divisors.\n§4.8. If F is a ring and E is a set, we shall denote by F [E] the free F -module generated\nby elements of E. By Theorem 5, to every element of the set CT, one can naturally\nassociate a divisor, hence a divisor class of CSb,d, i.e., an element of Cl(CSb,d). We thus\nobtain a map:\nΣ : Z[CT] −→ Cl(CSb,d)\nLater we shall show the following\nTheorem: If d > 2g + 4, g ≥ 5, then the group Cl(CSb,d) is of finite type over Z, and\nΣ is a surjection when tensored with Q.\nClearly it suffices to show that Pic(HRSb,d) is finite. We shall complete the proof in §6.7.\nAppendix to §4\n§4.9. In SGA 7, one finds an exposition of the theory of vanishing cycles for ordinary\n(i.e., 0-pointed) stable curves. Although the theory of vanishing cycles for arbitrary b-\npointed stable curves is also implicit in SGA, we review it here explicitly for the sake of\nclarity. In the case of b-pointed stable curves of genus zero, which is the principal case of\ninterest here, one can often describe the monodromy action much more explicitly than in\nthe 0-pointed higher genus case since the fundamental group involved is a free group.\n§4.10. Let (A,m) be a strictly henselian discrete valuation ring, S = Spec A. Let\n{C f−→S;σ1, . . . , σb : S → C} be a b-pointed stable curve over S. Let U = C −\n⋃b\ni=1 σi(S).\nWe use the notation Uη, Uη, Us, as usual. Then we have the following\nProposition : There is a natural, surjective morphism of specialization:\nπt1(Uη, ρη)\n′ −→ πt1(Us, ρs)′\nwhere ρ : S → U is a section of f , and “t” means the tame fundamental group, and the\n“prime” next to the fundamental group indicates that we are only concerned with the part\nof the fundamental dealing with coverings of degree prime to the residue characteristic of\nA.\n43\nRemark: It has been pointed out by the referee that this result has already been proven\nby Grothendieck in [5].\nProof: Write j : U → C for the natural inclusion. Let G = a finite group of order prime\nto the residue characteristic of A. By [29], Exposé XIII, Theorem 2.4, p. 380, we have\nR1étjs ∗GUs\n∼= (R1étj∗GU )|Cs . By the Leray spectral sequence and the proper base change\ntheorem, we thus see that πt1(Us, ρs)\n∼= πt1(U, ρs) ∼= πt1(U, ρη). The rest of the Proposition\nthen follows trivially. ©\n§4.11. Let us now assume that C is obtained in the following fashion. We are given\na collection of generically distinct sections σ′1, . . . σ\n′\nb : S → P1S such that σ′1(S) intersects\nσ′2(S), but all the other intersections among pairs of σ\n′\ni are empty. Then C is obtained by\nblowing up P1S at the schematic intersection of σ\n′\n1(S) and σ\n′\n2(S); we thus obtain a stable\ncurve {C f−→S;σ1, . . . , σb} where the σi are induced by the σ′i. Now if we fix a base section ρ\n(that doesn’t intersect any of the σi), and assume (for simplicity) that we are working over\nC, then πan1 (Uη, ρη) may be described as being generated by paths γ1, . . . γb around the\nvarious σi,η with the sole relation that γ1γ2 . . . γb = 1, i.e., a free group on b−1 generators.\nOn the other hand, πan1 (Us) may be described as being generated by paths δ1, . . . , δb around\nthe various σi,s with the sole relations δ1δ2 = 1, δ3δ4 . . . δb = 1. An illustration of this\ndegeneration situation is given in the Pictorial Appendix. The morphism of specialization\nis given by γi \b→ δi, so the “vanishing cycle” is clearly γ1γ2 = (γ3γ4 . . . γb)−1.\nPictorial Appendix\nSpecies 1:\n   P1\n   P1\n(b-2)   points\nx      x\n2     points\ngenus\ng\nRamification\n        index:\n1\n1\n3\npoint of\ndegenerate\nramification\none   copy\n         of\n(d-3)  copies\n            of\nLeft Right\nx     x\nx  x  x  x  x  .........  x  x  x  x\nx  x  ......  x  x\n(b-2)  points\n44\nSpecies 2:\n   P1\n   P1\n(b-2)   points\nx      x\n2     points\ngenus\ng\nRamification\n        index:\n1\n1\n         of\n            of\nLeft Right\n (d-4)   copies\ntwo    copies\n2\n2\nx\nx\nx  x  x  x  x  .........  x  x  x  x\nx  x  .......  x  x\n(b-2)  points\nSpecies 3A:\n   P1\n   P1\n(b-2)   points\nx      x\n2     points\ngenus\nRamification\n        index:\n1\n1\n         of\n            of\nLeft Right\n1\n1\n (d-2)   copies\none     copy\ng-1\nx       x\nx  x  x  x  x  .........  x  x  x  x\nx  x  .....  x  x\n(b-2)  points\n45\nSpecies 3B (b1, b2); (d1, d2):\n(b-2)   points\nx      x\n2     points\nLeft Right\nb    points1\nb    points2\ngenus   g\ngenus   g\n1\n2\n (d  -1)   copies\n              of\nP 1\n (d  -1)   copies\n              of\nP\n1\n1\n2\nRamification\n      index:\n1\n1\n1\n1\n1\n1\nx  x  .......  x\nx  x  .......  x\nx  x  x  x  x  .........  x  x  x  x\nx            x\nSpecies 3B∗:\nP 1\n(b-2)   points\nx      x\n2     points\nLeft Right\ngenus   g\nRamification\n      index:\n1\n1\n1\n1\n              of\n  (d-2)   copies\nP 1\n        one  copy\n              of\nP 1\n(b-2)  points\npoints of\ndegenerate\nramification\n1\nx  x  .......  x\nx  x  x  x  x  .........  x  x  x  x\nx              x\n46\nDegeneration Situation with Vanishing Cycle:\nxγ\nγ\n 1\n 2\nγ   γ1 2\nvanishing   cycle\n x\nx\nγ\n b\n 3\nγ\n x\nx\n b\n 3\nδ\nδ\n x\nδ\nδ\n 2\n 1\nx\ndegeneration\nx\nP 1\nP 1\nP 1\n§5. Cohomology Calculations\n§5.1. In this Section, we compute the various cohomology groups of a number of\nuniversal objects, in preparation for the construction of the “main fibration” in the next\nSection. At first, we shall work over C, so that we can use a number of analytic results;\nat the end of the Section, we shall extend the results to the arithmetic situation over Z.\nThus, unless we specify otherwise, all objects are taken to be over the complex numbers\nC.\nOften in this Section, we shall use the language of torsors over a group; for the reader\nunfamiliar with this language, we suggest [15] as a place to start.\n§5.2. Let Mg = the moduli stack of smooth curves of genus g with g ≥ 5. Let\nf : C → Mg be the universal curve of genus g. Let P = Pic (C\/Mg) be the Picard\nstack, a smooth algebraic group stack over Mg, which doesn’t necessarily represent the\nPicard functor Pic (since f doesn’t admit a section), but is, instead, its étale sheafification\n(see, e.g., [12] for a detailed discussion of the Picard functor); thus we have a morphism\nPic → P. Let J = Picτ (C\/Mg)\nϕ−→Mg be the open subgroup which is the universal\n47\nJacobian. Let E = P\/J be the quotient étale group stack, with geometric fibres over Mg\nisomorphic Z. Thus we have the following situation:\nPic⏐⏐\b\nJ −→ P −→ E\nϕ ↘\n⏐⏐\b ↙\nC f−→ Mg\nNow from the existence of the global bundle ω = Ω1C\/Mg on C, or by the existence of the\ndegree map, one sees that E ∼= Mg × Z. Thus P splits into connected components J d,\nwith d ∈ Z, according to the degree. Each J d is thus J -torsor in the étale topology over\nMg, i.e., an element ηd ∈ H1ét(Mg,J ). From the fact that P is a group stack, we see that\nηd = d η1. From the existence of ω, we see that 2(g − 1) η1 = 0. Let χ = 2(g − 1). Then if\nwe “change groups” for the torsor J d via the homomorphism [χ] : J → J multiplication\nby χ, we obtain a new J -torsor J∼,d, which is clearly trivial. Moreover, by the Leray\nspectral sequence, we see that Hian(J d, A) ∼= Hian(J∼,d, A) ∼= Hian(J , A) (where “an”\ndenotes the analytic topology) for any ring A in which χ is invertible.\n§5.3. We recall the following results of [16] and [1]:\n(A) Pic Mg ∼= H2an(Mg,Z) ∼= Z, where the first isomorphism is the first\nChern class map, and Pic Mg is generated by the Hodge bundle λ.\n(B) Pic C ∼= H2an(C,Z) ∼= Z2, where the first isomorphism is the first Chern\nclass map, and Pic C is generated by λ and ω.\n(C) Let F = R1anf∗Z ∼= R1anϕ∗Z. Then H1an(Mg,F) vanishes. (In fact, in\n[16], Harer makes the stronger assertion than in fact the first homology\ngroup of F vanishes; later in a correction (whose existence is mentioned\nin [17], Chapter 7) he retracts this and asserts only that the first ho-\nmology group of F is torsion; but this implies that the first cohomology\ngroup must vanish.)\n(D) Pic (C ×Mg C × . . .× C) is finitely generated for any number of factors\nof C. (In fact, the result in [1] is stated only for the moduli stack of\nb-pointed stable curves of genus g; but one deduces (D) easily from this\nfact using elementary properties of the Picard group.)\n48\nRecall that F is a locally constant sheaf of free Z-modules of rank 2g on Mg. Moreover,\nF is endowed with a natural symplectic form E (“the intersection pairing”).\nLemma : H0an(Mg,R2anϕ∗Z) ∼= Z and is generated by the element corresponding to E\nunder the isomorphism R2anϕ∗Z ∼= ∧2F .\nProof: Since E is not a multiple of any other form, it suffices to prove the result after\ntensoring with Q. If M is the Γ-module (where Γ = πan1 (Mg)) corresponding to F ⊗Z Q,\nthen since ∧2Q ⊆ M ⊗Q M ∼= HomQ(M,M) (where the last isomorphism of Γ-modules\nfollows from the fact that via E, M is a self-dual Γ-module), it suffices to show that M\nis an irreducible Γ-module. But this action factors through the surjection [8], p. 107,\nΓ → Sp(2g,Z), and one sees from looking at well-known sets of generators that Sp(2g,Z)\nis Zariski dense in Sp(2g,Q). Moreover, it is obvious that M is irreducible under the\naction of Sp(2g,Q). The Lemma follows. ©\n§5.4. Proposition : There exists an isomorphism A : J ∼→J ∧ (i.e., the dual abelian\nstack to J ) whose corresponding alternating form is E.\nProof: This, in fact, holds over Z, so for the remainder of §5.4, we work over Z, not C.\nThe Proposition is well-known when there exists a section of C. Thus, let K = the quotient\nfield of Mg; let L be a finite Galois extension of K such that CL admits a section. We\nthus obtain an isomorphism JL ∼→J ∧L with the desired property. Since E is defined over\nK, this isomorphism descends to JK ∼→J ∧K . By [11], Remark 1.10 a), p.7, the isomorphism\nextends over Mg. ©\nCorollary : There exists a line bundle on J whose relative first Chern class (with\nrespect to ϕ) is 2E.\nProof: Pull-back the Poincaré bundle on J ×Mg J ∧ via (1,A) : J → J ×Mg J ∧. ©\n§5.5. Proposition : The first Chern class induces a surjection c1 : Pic (J d) →\nH2an(J ,Z) and we have an exact sequence:\n0 −→ Pic (Mg) ⊗Z Q −→ c1(Pic(J d)) ⊗Z Q −→ QE −→ 0\nwhere the injection is the pull-back via J d → Mg and the surjection is the relative first\nChern class with respect ϕ.\nProof: Clear from the preceding paragraphs. At first it may appear that c1 is only\nsurjective up to torsion, but it follows from the Kummer and exponential exact sequences\nthat the cokernel of c1 is always torsion-free. ©\n§5.6. Let C∼,d = C ×Mg J d. Write ψd : C∼,d → J d. Denote by κα ∈ H2ét(J d,Gm) the\nobstruction to the existence of a universal bundle of degree d on C∼,d. Write id : J d → P\nfor the natural inclusion. From the Leray spectral sequence of ψd for the étale sheaf Gm,\nwe get an exact sequence:\n49\nPic(J d)⏐⏐\bnatural identification\nH1ét(C∼,d,Gm) −→ H0ét(J d,R1étψd,∗Gm)\nγ−→ H2ét(J d,Gm)\nBy general nonsense, κd = γ(id). From the existence of ω on C, one sees that under the\nidentification C∼,d ∼= C∼,d+χ (via ω), we have κd = κd+χ. On the other hand, if d > χ,\nthen the functor Divd of effective divisors of degree d is a Brauer-Severi stack over J d (by\nwhich we mean that it is a projective bundle in the étale topology), whose associated class\nin H2ét(J d,Gm) is clearly κd. Thus we see that for all d, κd is a torsion element.\n§5.7. Proposition : c1 : Pic (J d) → H2an(J d,Z) is injective.\nProof: If suffices to prove that for d large, Pic (J d) is finitely generated (since the kernel\nof c1 must be infinitely divisible). Now if X = C ×Mg × . . . ×Mg C (the d-fold product),\nthen we know that Pic (X ) is finitely generated. Now we have a natural finite flat map\nX → Divd; hence, via the associated norm map Pic (X ) → Pic (Divd), we see that\nPic (Divd) is finitely generated. We also have a natural map π : Divd → J d (which\nis a Brauer-Severi morphism), so if L is a line bundle on J d, then L ∼= π∗π∗L. Thus\nπ∗ : Pic (J d) → Pic (Divd) is an injection, and we are done. ©\n§5.8. We now go back to working over an arbitrary base, which we denote by a sub-\nscript. Let X = C or J d. Then we have the following\nProposition : The natural map Pic (XZ) → Pic (XC) is injective with finite cokernel.\nProof: Injectivity follows from the fact that H0(XR,OXR) = R for any ring R. The rest\nfollows from the preceding paragraph. ©\n§5.9. Given any relative Brauer-Severi stack f : X → S, we obtain a bundle of\nGrassmannians f∼ : X∼ → S (in the étale topology) by considering the lines in the\nprojective spaces of the fibres of f . Applying this construction to Divd → J d (for d > χ),\nwe obtain a map ψ : Gd → J d. Denote by ζ the canonical bundle on Gd for ψ. Write\nPicQ( ? ) for Pic( ? )⊗Z Q. We call elements of PicQ( ? ) “virtual line bundles”. Denote\nby γ ∈ PicQ (Gd) the Q-multiple ζ such that restricted to the fibres of ψ, it is the line\nbundle that defines the Plücker embedding.\nDefinition : We shall refer to the following diagram as the main fibration :\nGd ψ−→ J d ϕ−→ Mg\nWe now wish to define a canonical element Θd ∈ PicQ(J d) which has relative first Chern\nclass with respect to ϕ equal to E. We do this as follows. Since J 0 → Mg has the canonical\nidentity section, we can define Θ0 by requiring that the pull-back via the identity section\nbe trivial. For arbitrary d, we consider μd : J d → J 0 given by composing the map\n50\n“multiplication by χ” from J d to J dχ with the map “subtract off ω⊗d” from J dχ to J 0.\nThen we define Θd to be ( 1\nχ2\n)μ∗dΘ\n0. Note that if χ divides d, with χ = Nd, then Θd = the\npull-back of Θ0 via the map “subtract off ω⊗N” from J d to J 0. When d is fixed, we shall\noften omit it in the symbol Θd.\n§5.10. Theorem : Pic (XR) is finitely generated, for X = Mg; J d; or Gd and R\n= Z or C. Moreover, for such R,\nPicQ Mg ∼= Q, generated by λ;\nPicQ J d ∼= Q2, generated by ϕ∗λ and Θ ;\nPicQ Gd ∼= Q3, generated by γ, (ϕψ)∗λ, and ψ∗Θ.\n(For convenience, we shall write λ instead of ϕ∗λ, etc.) Thus via pull-back, we obtain a\nfiltration, which we shall refer to as the main filtration, on PicQ Gd which has subquotients\nof dimension 1: 0 = F 0 ⊆ F 1 ⊆ F 2 ⊆ F 3 = PicQ Gd, where F 1 ∼= PicQ Mg, and\nF 2 ∼= PicQ J d.\nProof: Clear from the previous results in this Section. ©\n§6. The Main Fibration\n§6.1. In this Section, we study the “the main fibration” defined in §5. We construct\nvarious maps from Hurwitz-type schemes into G, and use these to prove some facts about\nthe Picard group of the Hurwitz scheme and of G. (Here d is fixed, and so we write G for\nGd.)\n§6A. The Excess Divisors in the Main Fibration\n§6.2. We return to working over the ring R of §1. We fix b,d, and g, and assume\nfurther that g ≥ 5, and that d > 2g + 4. For simplicity we shall denote HRSb,d by H and\nNSb,d by H. Let us denote by H∼ the open substack of H that parametrizes admissible\ncoverings with the following two properties:\n(1) the nodal curve upstairs has an irreducible component which is smooth\nof genus g;\n(2) the genus zero curve downstairs has at most two irreducible components.\n51\nLet us note that under the natural map H → Mg (§3.27), H∼ maps into Mg. On the\nother hand, if we denote NOb,d by H\n+\n, then we have an étale covering f : H+ → H. Thus\nf is a Galois covering with Galois group equal to Γ\ndef\n= Sb. We denote by H+, H∼,+ the\npull-backs of H, H∼ via f . Thus we have a commutative diagram:\nH+ −→ H∼,+ −→ H+⏐⏐\b ⏐⏐\b ⏐⏐\bf\nH −→ H∼ −→ H⏐⏐\b ⏐⏐\b\nMg −→ Mg\nwhere all the horizontal arrows are open immersions.\n§6.3. Let us note that the relative dimension of G over R is: 2((d + 1 − g) − 2) + g +\n3(g − 1) = 2d + 2g − 2 − 3 = b − 3 (by the Riemann-Hurwitz formula). But this is the\ndimension (over R) of H. In fact, we have the following\nProposition : There is a natural open immersion κ : H → G.\nProof: Let S be an R-scheme. Then elements of H+(S) correspond to diagrams:\nC\nα−→ P1S\nβ ↘\n⏐⏐\bγ\nS\nwhere C → S is a smooth curve of genus g, and C → P1S is a degree d and has only simple\nramification. By assigning to this element the triple {(C → S) ∈ Mg(S), α∗(OP1\nS\n(1)) ∈\nJ d(S), (γ∗(OP1\nS\n(1)) → β∗(α∗(OP1\nS\n(1)))) ∈ G(S)}, we obtain a morphism H+ → G which\nis clearly invariant under the action of Γ. Hence we get a morphism κ : H → G. It is clear\nfrom the definitions and by looking at the situation when S is the spectrum of an artinian\nring that κ is étale. It is also clear from the definitions that κ is injective on geometric\npoints. Thus κ must be an open immersion. ©\n§6.4. The next order of business is to understand what the complement G−κ(H) looks\nlike. To do this, we work over an algebraically closed field k (that is an R-algebra). Let C\nbe a smooth, proper, connected curve over k, and let L be a line bundle of degree d on C.\nLet V = Γ(C,L), P = Pk(V ∧), where “∧” for us will always mean the dual space. Thus\ndimk V = d + 1 − g, dimk P = d − g, and P may be thought of as the space of effective\ndivisors that give rise to L. A map α : C → P1k (considered up to automorphisms of P1k)\nsuch that α∗(OP1\nS\n(1)) = L may be thought of as a line L in P such that:\n52\n(1) There doesn’t exist any point p ∈ C(k) that appears in every divisor in\nL.\nOn the other hand, what does it mean to say that α has at most simple ramification? It\nmeans that every divisor in L that is not made up of d distinct points has d − 1 distinct\npoints, so only one point has coefficient not equal to 1, and that coefficient is 2. In other\nwords, we must require that:\n(2) L doesn’t include a divisor that contains 3p for some p ∈ C(k);\n(3) L doesn’t include a divisor that contains 2p + 2q for some p, q ∈ C(k).\nNow let G = the Grassmannian of lines in P . Then dimk G = 2(d−g−1). Let δG denote\nthe ample generator of the Picard group of G. If we think of G as parametrizing rank 2\nquotients V ∧ → Q, then δG corresponds to ∧2Q. Let us consider the diagram:\nΔ ⊆ C × C\nπ1 ↙ ↘ π2\nC C\n(Diagram 6.1)\nLet i be such that d − i > 2g − 2. We consider the line bundles M(i) = (π∗1L)(−iΔ).\nLet E i = π2,∗(M(i)); it has rank d − g + 1 − i. Let us denote by G(a vector bundle) the\nGrassmannian of 2-dimensional quotients of the dual of the vector bundle. Now clearly we\nhave a natural locally split injection E1 → E0 which gives rise to a morphism ρ3 : G(E1) →\nG(E0). Now E0 is simply the pull-back to C of the k-vector space V , so G(E0) = C × G.\nIf we compose ρ3 with the projection to G, we obtain a morphism σ3 : G(E1) → G which\nis clearly generically injective and whose image consists precisely of the lines L that do no\nsatisfy (1). Since dimk G(E1) = dimk (G) − 1, this image is some irreducible effective\ndivisor D3, whose degree we will compute in §6B.\nWe also have a natural locally split injection E3 → E0 which gives rise to a morphism\nP(E3,∧) → P(E0,∧) = C × P . If we then compose with the projection to P , we get a\nmorphism ρ1 : P(E3,∧) → P which is clearly generically injective and whose image consists\nexactly of the bad divisors that appear in (2). The image in P is an irreducible variety\nV1 of codimension 2. Let us now consider the incidence variety I ⊆ P × G of points\n(p,L) with p ∈ P (k), L ∈ G(k) such that the point p lies on the line L. Then it is well-\nknown that I is a P1-bundle over G and a P(d−g−1)-bundle over P . Denote by IV1 the\npull-back of I (regarded as an object over P ) via V1 → P . Let σ1 : IV1 → G denote the\ncomposition of the natural inclusions IV1 ⊆ I ⊆ P ×G with the projection to G. It is clear\n(by elementary algebraic geometry) that σ1 is generically injective. On the other hand,\ndimk IV1 = dimk (G) − 1, so the image of IV1 in G is some irreducible divisor D1 whose\npoints are exactly the bad lines that appear in condition (2). We shall compute the degree\nof D1 in §6B.\n53\nIV1 ⊆ I ⊆ P × G⏐⏐\b ⏐⏐\b ↙ ↘\nV1 ⊆ P G\nNow let us consider the following diagram:\nΔ12,Δ13 ⊆ C × C × C\n↙ π12\n⏐⏐\bπ23 π13 ↘\nC × C C × C C × C\n(Diagram 6.2)\nwhere the πij ’s are the respective projections and Δij = π\n−1\nij (Δ). Let us suppose that\ni, j are such that d − i − j > 2g − 2. Let F (i,j) = (π∗1L)(−iΔ12 − jΔ13). Let K(i,j) =\nπ23,∗F (i,j). Then K(i,j) is a vector bundle on C × C of rank d + 1 − g − i − j. Note that\nwe have a natural locally split injection K(2,2) → K(0,0) which gives rise to a morphism\nP(K(2,2),∧) → P(K(0,0),∧) = C × C × P . Composing with the projection P , we get a\nmorphism P(K(2,2),∧) → P which is clearly generically injective after taking the quotient\nby the action of Z\/2Z permuting the two factors of C, and whose image consists exactly of\nthe bad divisors that appear in (3). The image is an irreducible variety V2 of codimension\n2 in P . Forming IV2 and σ2 : IV2 → G as before, we see that σ2 is generically injective,\nand that its image in G is some irreducible divisor D2 whose points are exactly the bad\nlines that appear in condition (3). We shall compute the degree of D2 in §6B.\nIn particular, since G \t= D1\n⋃\nD2\n⋃\nD3, we see that we can always construct a map\nα : C → P1 with at most simple ramification. Thus ψ◦κ is surjective. Note that instead of\nworking over k, then, we could have worked over J d. In summary, we have the following:\nProposition : The complement of κ(H) in G consists of three relative divisors D1,\nD2, and D3 over J d, which are also geometrically irreducible over J d. They correspond,\nrespectively, to the linear pencils that fail to meet one of the conditions (2),(3), or (1)\nlisted at the beginning of this §.\n§6.5. We now wish to relate these divisors of G to the divisors of H constructed in §4.\nTo do this, we shall need to construct some sort of map from H (or at least part of it) into\nG. This is a somewhat delicate task, which we now undertake.\nWe shall call divisors at infinity of H∼,+ or H+ that lie over divisors of species X in\nH∼ or H “divisors of species X”, as well. We propose to extend the map μ′ : H+ → H κ→G\nto a map μ : H∼,+ → G. For any two distinct integers i and j between 1 and b, let us\ndenote by Uij the open substack of H∼,+ parametrizing admissible covers for which the\nmarking sections numbered i and j of the curve downstairs lie in the irreducible component\nof the curve downstairs which is the image of the curve of genus g upstairs. Then it is\neasy to see that, as i, j ranges over all admissible values, the union of the Uij is all of\n54\nH∼,+, while the intersection of the Uij is H+. Thus, to define our extension μ, it suffices\nto define, on each Uij , a morphism Uij → G that restricts to μ′ on H+. Let f : C → P\ndenote the universal admissible covering over H∼,+. Let σ1, . . . , σb : H∼,+ → P denote the\nmarking sections. Let Ei denote the schematic inverse image via f of σi. Let C′ → H∼,+\ndenote the smooth curve of genus g obtained by blowing down C as in the diagram of §3.27;\nthus we have a blow-down map π : C → C′. Let E′i denote the push-forward of the divisor\nEi via π. Since C′ is a regular scheme, E′i defines a line bundle Li on C′, along with a\nglobal section si of Li corresponding to E′i. Now Li defines a morphism νi : H∼,+ → J d\n(the universal Jacobian). Since it is clear that νi and νj coincide when restricted to H+,\nit follows (since H∼,+ is separated and integral) that νi = νj . Now it follows from the\ndefinition of Uij that over Uij , the restrictions of E′i and E′j to any geometric fibre of\nC′ → H∼,+ are distinct divisors. Thus, over Uij , the pair E′i and E′j define a pencil, that\nis, a morphism μij : Uij → G whose composite with the projection G → J d is νi = νj .\nClearly this morphism restricts to μ′ on H+. Thus, gluing together the μij , we obtain our\nμ. The behavior of μ on divisors at infinity is given by the following\nProposition : Suppose that d > 2g + 4. Then μ maps divisors of Species 1 (resp.\n2, 3B∗) into D1 (resp. D2, D3); moreover, the restriction of μ to the excess divisors is a\nquasi-finite morphism. On the other hand, μ maps the nonexcess divisors to subsets of\ncodimension at least two that lie outside the image of κ. In particular, the inverse image\ndivisor via μ of D1 (resp. D2, D3) is a positive linear combination of all the divisors of\nH∼,+ of Species 1 (resp. 2, 3B∗). Finally, the image via μ of any excess divisor is dense\nin the restriction of the corresponding Di to any geometric fibre of G → Mg.\nProof: First, we note that it follows immediately from the definition of μ and the explicit\ndescriptions of the excess divisors given in §4 that μ maps a divisor of Species 1 (resp. 2,\n3B∗) into D1 (resp. D2, D3); the assertion on quasi-finiteness is also clear from the explicit\ndescriptions in §4.\nNow let us prove the final statement. We fix a geometric point c : Spec(k) → Mg,\nwhere k is an algebraically closed field (of characteristic greater than b). Then it suffices\nto show that the generic point of Di|c (for i = 1, 2, 3) corresponds to a morphism C → P1\nwhich appears in an admissible covering of Species 1, 2, or 3B∗. Let us first note that the\nthree irreducible divisors Di|c in G (i = 1, 2, 3) are distinct: there are many ways to see\nthis; one way is simply to note that if any two were equal, then their degrees, as computed\nin Theorem 21 of §6B would coincide; but an elementary calculation reveals that this is\nabsurd. We now proceed case by case.\nFor i = 1, a simple combinatorial check reveals that it suffices to show that the generic\npencil in D1|c does not contain two divisors each of which has a point of multiplicity three.\nBut in the notation of §6.4, this amounts to checking that the set of lines in P that pass\nthrough two points of V1 has codimension at least two in G, but this is clear.\nFor i = 2, there are two points to check: first, that the generic pencil does not contain\ntwo divisors each of which has two points of multiplicity two; but this follows by the same\nargument as in the previous paragraph. Second, we must check that the generic pencil\ndoes not contain a divisor which has three points of multiplicity two; but the subvariety\n55\nV3 ⊆ P consisting of divisors with three points of multiplicity two has codimension three\nin P (since we assume that d > 2g + 4). Thus, the subvariety of G consisting of lines in P\nthat pass through V3 has codimension two, as desired.\nFor i = 3, here we need to show that the generic pencil does not contain two base\npoints. But one sees immediately that the subvariety of G corresponding to pencils with\ntwo base points has dimension 2(d − 3) + 2 = 2d − 4, i.e., has codimension two in G, as\ndesired.\nWe now turn to considering the nonexcess divisors. Let us suppose that x : Spec(k) →\nH∼,+ is a geometric point that lands in a nonexcess divisor. One piece of information of\nan admissible covering that is clearly preserved by μ is the unique nonconstant morphism,\nwhich we shall henceforth refer as the central morphism of the admissible covering, from\nthe curve of genus g upstairs to a copy of P1 downstairs. Thus, let us suppose that μ(x)\nlands inside the image of κ. Then the central morphism of x must be ramified over b\ndistinct points downstairs; but since the curve downstairs in the admissible covering x\nis stable, this could only happen if the curve downstairs in x is, in fact, smooth, which\ncontradicts the assumption that x lies in a nonexcess divisor. Thus, μ(x) must land inside\none of the Di. Now let us suppose that μ(x) lands in a sufficiently generic point of Di.\nThen by our proof of the final statement of the Proposition, it follows that the central\nmorphism of x is ramified over exactly b − 1 distinct points downstairs; thus, since the\ncurve downstairs in the admissible covering x is stable, it follows by the same reasoning as\nbefore, that the curve downstairs in x must be of type (b − 2, 2). On the other hand, we\ngave explicit descriptions in §4 of all the divisors of type (b − 2, 2); going down the list,\none sees that the only possiblity now for the nonexcess divisor in which x sits is Species\n3B with g1 = g, g2 = 0. But if d2 > 1, then the central morphism of x must have at least\ntwo base points, which contradicts what we just saw in the previous paragraph. ©\n§6.6. At this point, we need to employ a result which will not be proven until the end\nof §7. We state it here for ease of reference:\nTheorem : Under the assumption that d > 2g +4, the line bundles defined by D1, D2,\nand D3 are linearly independent in the Picard group of G.\n§6.7. Now by elementary algebraic geometry, we have an exact sequence:\nZD1 ⊕ ZD2 ⊕ ZD3 −→ Pic G −→ Pic H −→ 0\nIf we tensor with Q, and use the fact (§5) that PicQ G = Q3, then we obtain the following\n(again under the assumption that d > 2g + 4)\nCorollary : D1, D2, and D3 form a basis of PicQ G.\nCorollary : Pic (H) is a finite group. The homomorphism Σ of §4.8, is a surjection\nwhen tensored with Q.\n56\nRemark : Suppose we knew a priori or through some other technique that Pic (H) is\nfinite. Then what we have done here would prove that PicQ G has dimension 3. This, in\nturn, would allow us to rederive in a more algebraic way many of the cohomological results\nof Harer (up to torsion) which were recalled in §5.3. Given all the explicit calculations\ndone in the paper, it does not seem unreasonable to expect that one could, by closer\nexamination, prove that Pic (H) is finite in an a priori fashion. At the present time,\nhowever, the author has not succeeded in doing this.\n§6.8. Remark : It is now of interest to compute the coefficient matrix relating the\nbasis {D1,D2,D3} and the basis {λ,Θ, γ}. We shall do this in the next Section.\n§6B. Intersection Theory Calculations\n§6.9. Returning to the situation over an algebraically closed field k as in the beginning\nof §6.4, the purpose of this Subsection is to compute the “degree” of the divisors D1, D2,\nand D3 in G, where by “degree”, we really mean the number N such that a divisor is\nlinearly equivalent to δ⊗NG . We begin by reviewing some basic facts about the geometry of\nGrassmannians.\n§6.10. Lemma : Let X ⊆ P be an irreducible variety of codimension 2. Then if\nwe construct IX and the associated generically injective morphism IX → G as in §6.4, the\ndegree of the image of this morphism in G is equal to the degree of X in P .\nProof: Let e = the degree of X in P . Then one knows (e.g. from [14]) that if one selects\na linear subspace Q ⊆ P of dimension 2 and a point q ∈ Q, the subvariety of G formed of\nlines in Q that pass through q gives rise to a cycle which is dual to δG. Thus it suffices\nto intersect X with a generic linear subspace Q ⊆ P of dimension 2 such that X ⋂Q has\ne points {x1, . . . , xe}, and to choose a generic q ∈ Q that doesn’t sit on any line passing\nthrough two of the xi’s. Then clearly there are exactly e lines in Q that pass through q\nand intersect X, and moreover, the intersection between the image of IX in G and the\ncycle associated to {Q, q} is clearly transverse. Hence the Lemma. ©\n§6.11. Let S be a smooth, proper, connected curve over k; let E be a vector bundle\nof rank r + 1 on S. We use the Chow rings Ai(−). Write \u0013 = c1(E) ∈ A1(S). Let\nP1 = P(E), L1 = OP1(1). Then we have an exact sequence 0 → F → EP1 → L1 →\n0, where the subscript denotes pull-back and F is defined by this exact sequence. Let\nP2 = P(F), L2 = OP2(1). On the other hand, let G′ = G(E), and let Q be the universal\nquotient of E on G′. Let P3 = P(Q), L3 = OP3(1). Then clearly P2 ∼= P3, with L1 and\nL3 corresponding under this isomorphism. We thus have a diagram:\n57\nP2 ∼= P3⏐⏐\b ⏐⏐\b\nP1 G\n′\n↘\n⏐⏐\b\nS\nLet us denote ∧2Q by δ′. Let κi = c1(Li) ∈ A1(Pi) for i = 1, 2, 3. Then we have\nA∗(P1) = A∗(S)[κ1]\/(κ\nr+1\n1 − κr1\u0013), so κN1 = 0 for N ≥ r + 2.\nA∗(P3) = A∗(G′)[κ3]\/(κ23 − c1(Q)κ3 + c2(Q)).\nNote that if we identify A∗(P2) and A∗(P3), we have κ1 = κ3, c1(Q) = κ1 + κ2, c2(Q) =\nκ1κ2. Let us denote the fundamental class of S by η, so \u0013 = deg(E)η.\n§6.12. Lemma : We have the following identities:\n(1) κr+1i = \u0013κ\nr\ni ; hence κ\nr+2\ni = 0 for i = 1, 2.\n(2) κr−11 κ\nr\n2 + κ\nr\n1κ\nr−1\n2 = \u0013κ\nr−1\n1 κ\nr−1\n2 ;\n(3) If \u0013 = 0, then κr−21 κ\nr\n2 + κ\nr\n1κ\nr−2\n2 = −κr−11 κr−12 .\nProof: We give the proof assertion by assertion:\n(1) This is elementary and well-known.\n(2) ct(F) = (1 − \u0013t)(1 − κ1t)−1. If we substitute formally by t = κ−12\nand multiply by κr2, the result will be zero. So we do this and multiply\nfurther by κr−11 : 0 = κ\nr\n2κ\nr−1\n1 (1 − \u000fκ2 )(1 −\nκ1\nκ2\n)−1 = κr2κ\nr−1\n1 + κ\nr−1\n2 κ\nr\n1 +\nκr−22 \u0013κ\nr\n1 − \u0013κr−12 κr−11 − \u0013κr−22 κr1. Rearranging terms gives the result.\n(3) Same technique as in (2), but with \u0013 = 0, and we multiply by κr−21\ninstead of κr−11 : 0 = κ\nr\n2κ\nr−2\n1 (1 − κ1κ2 )\n−1 = κr2κ\nr−2\n1 + κ\nr−1\n2 κ\nr−1\n1 + κ\nr−2\n2 κ\nr\n1.\n©\n§6.13. Lemma : c1(δ′)2r−1 = ( 2r+1 )\n(\n2r−1\nr\n)\ndeg (E).\nProof: Using the previous Lemma:\n58\nc1(δ\n′)2r−1 = (κ1 + κ2)\n2r−1\n= \u0013κr−21 κ\nr\n2\n(\n2r − 1\nr + 1\n)\n+ κr−11 κ\nr\n2\n(\n2r − 1\nr\n)\n+ κr1κ\nr−1\n2\n(\n2r − 1\nr\n)\n+ \u0013κr1κ\nr−2\n2\n(\n2r − 1\nr + 1\n)\n= \u0013κr−11 κ\nr−1\n2\n(\n2r − 1\nr\n)\n− \u0013κr−11 κr−12\n(\n2r − 1\nr + 1\n)\nNow the result follows from the fact that since κr1κ\nr−1\n2 η is clearly the fundamental class\nof P3, κ\nr−1\n1 κ\nr−1\n2 η must be the fundamental class of G\n′, and by the elementary calculation:(\n2r−1\nr\n)\n−\n(\n2r−1\nr+1\n)\n=\n(2r−1)!\nr!(r−2)! (\n1\nr−1 − 1r+1 ) = 2\n(2r−1)!\n(r+1)!(r−1)! = (\n2\nr+1 )\n(\n2r−1\nr\n)\n. ©\n§6.14. Lemma : If \u0013 = 0, then c1(δ′)2(r−1) = 1r−1\n(\n2r−2\nr\n)\n.\nProof: Using §6.12:\nc1(δ\n′)2(r−1) = (κ1 + κ2)\n2r−2\n= κr−21 κ\nr\n2\n(\n2r − 2\nr\n)\n+ κr−11 κ\nr−1\n2\n(\n2r − 2\nr − 1\n)\n+ κr1κ\nr−2\n2\n(\n2r − 2\nr\n)\n= κr−11 κ\nr−1\n2\n{(\n2r − 2\nr − 1\n)\n−\n(\n2r − 2\nr\n)}\nBut\n(\n2r−2\nr−1\n)\n−\n(\n2r−2\nr\n)\n=\n(2r−2)!\n(r−1)!(r−2)! (\n1\nr−1 − 1r ) =\n(2r−2)!\nr!(r−1)! =\n1\nr−1\n(\n2r−2\nr\n)\n. ©\n§6.15. Proposition : [D3] = d δG, where the brackets denote the linear equivalence\nclass.\nProof: In the calculations of §6.11 ∼ 14, set G′ = G(E1,∧). Recall our map σ3 : G′ → G.\nClearly σ∗3(δG) = δ\n′ (i.e., the analogous bundle for G′ to δG on G). Suppose [D3] = N δG.\nSince here r + 1 = d − g, we see from §6.14, that c1(δ′)2r−1 = N(c1(δG))2r = N(1r )\n(\n2r\nr+1\n)\n.\nOn the other hand, by §6.13, c1(δ′)2r−1 = ( 2r+1 )\n(\n2r−1\nr\n)\ndeg (E1,∧). Thus N = −deg (E1).\nThe result thus is a consequence of the following:\n§6.16. Lemma : If d − i > 2g − 2, then deg (E i) = −di − (g − 1)i(i − 1).\nProof: Referring to Diagram 6.1 of §6.4, we see that we have exact sequences:\n0 −→ (π∗1L)(−(i + 1)Δ) −→ (π∗1L)(−iΔ) −→ (ω⊗i ⊗ L)Δ −→ 0\nPushing forward by π2, we get:\n59\n0 −→ E i+1 −→ E i −→ ω⊗i ⊗ L −→ 0\nHence we obtain deg (E i+1) = deg (E i) − d − 2i(g − 1). Since deg (E0) = 0, the result\nfollows. ©\n§6.17. Proposition : [D1] = {3d + 6(g − 1)}δG = 3(b − d)δG.\nProof: By §6.10, it suffices to compute the degree of V1 in P (notation as in the relevant\npart of §6.4). Since ρ∗1(OP (1)) = OP(E3,∧)(1). Thus if α = c1(OP (1)), α′ = c1(OP(E3,∧)(1)),\nand V1 is of degree N , then (α\n′)d−g−2 = Nαd−g. On the other hand, from basic facts about\nprojective bundles, we know that (α′)d−g−2 = deg (E3,∧) = −deg (E3). Now apply §6.16.\n©\n§6.18. Lemma : Assuming, as usual, that d > 2g + 2, we have the following Chern\nclasses on C × C:\nc1(K(2,2)) = −(b − 4)(η1 + η2) + 4δ\nc2(K(2,2)) = (b − 4)(b − 6) − 24g,\nwhere on C ×C, ηi is the pull-back of the fundamental class of the ith factor, for i = 1, 2,\nand δ = c1(Δ) − η1 − η2, where Δ is the diagonal.\nProof: First of all, K(0,2) = π∗2E2. Thus c(K(0,2)) = 1 − {2d + 2(g − 1)}η2. On the other\nhand, we have exact sequences on C × C × C:\n0 −→ F (2,2) −→ F (1,2) −→ π∗12{(ω)Δ} ⊗ π∗13M(2) −→ 0\n0 −→ F (1,2) −→ F (0,2) −→ π∗12(OΔ) ⊗ π∗13M(2) −→ 0\nApplying π23,∗, we obtain exact sequences on C × C:\n0 −→ K(2,2) −→ K(1,2) −→ π∗1(ω) ⊗M(2) −→ 0\n0 −→ K(1,2) −→ K(0,2) −→ π∗1(OC) ⊗M(2) −→ 0\nNow c(M(2)) = 1+(d−2)η1−2η2−2δ, where δ2 = −2g, δ η1 = δ η2 = 0. Thus we obtain:\nc(K(1,2)) = c(K(2,2)){1 + (d + 2g − 4)η1 − 2η2 − 2δ}\nc(K(0,2)) = c(K(1,2)){1 + (d − 2)η1 − 2η2 − 2δ}\n60\nThus we have:\nc(K(2,2)) = {1 − bη2}{1 + (d − 2)η1 − 2η2 − 2δ}−1{1 + (b − d − 2)η1 − 2η2 − 2δ}−1\n= {1 − bη2}{1 − (d − 2)η2 + 2η2 + 2δ − 4(d − 2)η1η2 + 4δ2}\n{1 − (b − d − 2)η1 + 2η2 + 2δ − 4(b − d − 2)η1η2 + 4δ2}\n= {1 − bη2}{1 − (b − 4)η1 + 4η2 + 4δ − 6(b − 4)η1η2 + 12δ2}\n= {1 − (b − 4)(η1 + η2) + 4δ + (b − 4)(b − 6)η1η2 + 12δ2}\n©\n§6.19. Let S be a proper, smooth, connected surface over k. Let E be a vector bundle\non S of rank r + 1. Let α = c1(OP(E∧)(1)). Then we have the following:\nLemma : αr+2 = c21(E) − c2(E).\nProof: We have αr+1 = −c1(E)αr − c2(E)αr−1. Then αr+2 = −c1(E)αr+1 − c2(E)αr =\n{c21(E) − c2(E)}αr, as desired. ©\n§6.20. Proposition : [D2] = 12{b(b − 10) + 8d}δG.\nProof: As in §6.17, we apply §6.10, and we see that if we take S = C ×C and E = K(2,2),\nso that r + 1 = d − g − 3, we have 12αr+2 = [D2]. Thus by §6.19, §6.18, we have 2[D2] =\n2(b − 4)2 − (b − 4)(b − 6) + (−32g) + 24g = (b − 4)(b − 2) − 8g = b(b − 10) + 8d. ©\n§6.21. To summarize, we have the following\nTheorem : The degrees of the excess divisors in G are as follows:\n[D1] = 3(b − d) δG\n[D2] =\n1\n2{b(b − 10) + 8d} δG\n[D3] = d δG\n§6C. Ramification Indices\n§6.22. Let us consider the morphism μ : H∼,+ → G constructed in §6.5. Now H+ ⊆\nH∼,+ as an open subscheme, and the complement of this open subscheme consists of certain\ndivisors at infinity. In this Subsection, we wish to compute the indices of ramification of μ\nat the excess divisors. These computations will be important for §7. Note that we continue\nto assume here that g ≥ 5 and d > 2g + 4.\n61\nNote that all this is taking place over Z[ 1\nN\n], where N = b!. On the other hand, G\nis defined over Z, which prompts the question: What is the nature of the ramification of\nμ′ : H+ → H κ−→G, over the primes p dividing N? In fact, we shall see that there is no\nramification at the odd primes.\n§6.23. We begin with the ramification of μ at the excess divisor of Species 2. Note that\nin this case, the different (though not the discriminant) is étale over the base; thus, the\nprocess of “ordering the sections” gives rise to a covering of G which is étale over D2. Let\nf : C → P denote the universal admissible covering over H∼,+. Let σ1, . . . , σb : H∼,+ → P\nbe the b marking sections; suppose that σi lifts to ρi : H∼,+ → C, at which f is simply\nramified. Let ℘ be a prime divisor of height one of H∼,+ which “is” a divisor at infinity\nof Species 2. Let U be an open substack of H∼,+ that contains ℘ and is so small that the\nnormal sheaves of the σb (which are line bundles on H∼,+) are trivial when restricted to U .\nChoose trivializations τ1, . . . , τb for these normal sheaves over U . Let U\n′ → U be the étale\ncovering that parametrizes trivializations of the normal sheaves of the ρi whose squares are\nthe trivializations τi. Let ℘\n′ be a prime of U ′ that lies over ℘. Note that with this extra\nrigidification, a generic admissible covering of Species 2 loses all its automorphisms, so that\nU ′ is a scheme in a neighborhood of ℘′. Let A be the completion of U ′ at ℘′. It is thus clear\nfrom the general theory of algebraic stacks and their associated coarse moduli spaces (see,\ne.g., [11], p. 23) that to determine the desired index of ramification, we must simply do the\nfollowing: consider the group Γ = Gal(U ′\/U) ∼= (Z\/2Z)b that acts on the added structure\nof choosing “specific square roots of uniformizers over points of simple ramification”. The\ninertia subgroup I at ℘′ is isomorphic to (Z\/2Z) and sits inside the (Z\/2Z)2 ⊆ Γ that\ncorresponds to the two points of simple ramification on the P1’s upstairs. Then from the\nequivalent definition of A as the solution to a certain infinitesimal moduli problem, we see\nthat I acts nontrivially on A. Let B = AI , the ring of invariants. Then B is also a discrete\nvaluation ring, and the desired index of ramification is the the same as that of A over B.\nLooking at the explicit description given in [19],§4, p. 62, however, we see that I takes\nsome uniformizer of A to minus itself; hence the index of ramification must be 2.\nProposition : μ has ramification index 2 at excess divisors of Species 2.\n§6.24. Let us now note that H∼,+ is already a scheme at the generic points of divisors\nof Species 1 and 3B∗; indeed, for those Species, once we order the points, there are no\nautomorphisms.\nNow since generically, in the case of either of these excess divisors, at most two of the\nb sections come together as one degenerates from an ordinary Hurwitz covering, it is clear\nthat the ramification index is either 1 or 2. Thus, to show that in both of these cases the\nramification index is, in fact, 2, it suffices to construct one example where S = Spec C[[x]],\nC → S a proper, smooth curve of genus g, with a diagram:\nCη\nfη−→ P1η\n↘\n⏐⏐\b\nη ⊆ S\n62\nsuch that fη is an ordinary Hurwitz covering which generates to a covering of Species 1 or\n3B∗ and such that if T = S ×G H∼,+, then T → S is ramified.\n§6.25. We construct such examples using the theory of vanishing cycles reviewed in the\nAppendix to §4. Let S = Spec R, with R = C[[x]], Ri = C[[xi]] ⊆ R for i a natural number,\nSi = Spec Ri. Choose distinct nonzero complex numbers λ2, . . . , λb−1 and denote the\ncorresponding sections of P1 by σ2, . . . , σb−1. We will later subject the complex numbers\nλi to a further condition which the generic choice of λi’s will fulfill (assuming d to be large).\nBy abuse of notation, we also denote by σ2, . . . , σb−1 the sections pulled back to S1, S2,\nand S3. Write C[[T ]] for the completed local ring at the point 0 of P\n1\nC. Let (for i = 0, 1) σi\nbe the section of P1S3 which is the graph of the map S3 → P1C given by T \b→ (−1)iX3. Let\nτ be the unique nontrivial element of the Galois group of S3 over S6. Then τ(σi) = σ1−i\nfor i = 0, 1, which τ(σi) = σi for i ≥ 2. Denote by ηi the generic point of Si. Now if\nGi = Gal (k(η)\/k(ηi)), where k(η) = k(ηi) for all i, then we have G1 ⊆ G3 ⊆ G6, and\nG1 ⊆ G2 ⊆ G6. Choosing a root of unity, we get isomorphisms Gi ∼= Z∧ for all i. Let ζi\nbe the canonical generator of Gi under this isomorphism. Then ζ\n2\n6 = ζ3, ζ\n3\n6 = ζ2, ζ\n6\n6 = ζ1.\nMoreover ζ6 acts like τ on the σi’s.\n“Drawing” paths around the σi’s, we see that the fundamental group Γ of P\n1\nη −\n{σ0, σ1, . . . , σb−1} = Uη may be described as the (profinite completion of the) free group\non generators g0, g1, . . . , gb−1 modulo the sole relation g0g1g2 . . . gb−1 = 1. A simple calcu-\nlation reveals the action of ζ6 on Γ is given by g0 \b→ g0, g1 \b→ g−11 g0g1, gi \b→ gi for i ≥ 2.\nA connected étale covering of Uη of degree d is defined by a transitive action of Γ on the\nset {1, . . . , d} of d elements. This covering is rational over ηi if and only if the action of\nΓ obtained by composing the original action with the automorphism of Γ defined by ζi is\nthe same as the original action. Finally, note that the subscheme σ0\n⋃\nσ1 of P\n1\nS3\ndescends\nto a closed, irreducible subscheme of P1S6 , which, as an S6-scheme, is isomorphic to S3,\nhence ramified over S6.\nThe Case of Species 1 : Choose a transitive sequence of b transpositions in Sd whose\nproduct is 1 to assign consecutively to g0, . . . , gb−1, such that the first two transpositions\nare (12). We assume that the λi’s are chosen such that the curve of genus g in the special\nfibre over S in the covering defined by this sequence of transpositions has no nontrivial\nautomorphisms. (This will always be possible for d > 2g + 4 by Proposition 5.) Clearly ζ6\nacts trivially on this homomorphism from Γ to Sd, so we get the example desired over S6.\nThe Case of Species 3B∗ : Choose a transitive sequence of transpositions as above that\ngives rise this time to Species 3B∗, with the first transposition being (12) and the second\n(23). Again we assume the λi’s and sequence of transpositions chosen so that the curve of\ngenus g in the special fibre over S has no nontrivial automorphisms. A simple calculation\nreveals that ζ2 acts trivially on the induced homomorphism from Γ to Sd, so we obtain\nthe desired example over S2.\nWe have thus proven the following\nProposition : μ has ramification index 2 at divisors of Species 1 and 3B∗.\n63\n§6.26. Corollary : Assuming still that g ≥ 5 and d > 2g +4, HROb,d and HUOb,d\nare geometrically connected.\nProof: We treat the rigidified case; the unrigidified case is analogous. If not, there\nwould be a nontrivial maximal unramified extension between HROb,d and HRSb,d, but\nour explicit computation of the ramification shows that this is impossible. The result thus\nfollows from the corresponding result for HRSb,d (§2). ©\nRemark : The referee has informed me that this last result was obtained by Clebsch\ndirectly via purely combinatorial methods similar to those used in §2.\n§6.27. In closing, we shall prove the following:\nProposition : μ is unramified over all odd primes p of Z.\nProof: As G is regular, it suffices to show, by the purity theorem, that there exists a\ngeometric point of H of any odd characteristic such that the covering of this geometric\npoint induced by H+ → H is étale. (Here we note that H can be defined over Z as the\ncomplement in G of the three divisors D1, D2, and D3, which are clearly defined and\nproper over Z.) Translated, this means that given an algebraically closed field k of odd\ncharacteristic p, we must find a smooth, proper, connected curve C over k and a morphism\nf : C → P1k of degree d such that the discriminant of f is étale over k. But this clearly\nfollows from the analysis of §4, which works so long as k has odd characteristic. ©\nRemark : It was the goal of this Subsection to determine the indices of ramification\nof μ at all the primes of height one of G (over Z). We have thus completed this task at\nall primes except 2, where there is wild ramification, thus making our goal at this prime\nbeyond the scope of this paper.\n§7. The Coefficient Matrix\n§7.1. In this Section we shall complete the proof of Theorem 8 of §4 by showing that\nthe excess divisors in G are linearly independent in Pic(G), and, in the process, we shall\nexplicitly compute the coefficients of λ, Θ, and γ in terms of D1, D2, and D3. This\nalso gives us explicitly the “main filtration” (§5.10) in terms of the Di’s. We can thus\ncheck the coefficients of λ and Θ by making sure that they are in the kernel of the map\nF 3 → F 3\/F 2 ∼= Q δG, which was computed explicitly in §6B. As usual, it suffices to work\nover the complex field C, so we will do that in this Section.\n§7.2. Let us note, first of all, that there exists a proper, smooth, connected curve S\n(over C) with the following property: there exist enough morphisms φi : S → H\n+\nthat\navoid nonexcess divisors so that if we write ψi : S → G for the morphism induced by\ncomposing φi with κ, then the matrix {deg(ψ∗i Fj)}, where j = 1, 2, 3; F1 = λ; F2 = Θ;\nF3 = γ, is of rank three. Indeed, we can prove the existence of such an S as follows. First,\n64\ntake any nonconstant morphism f ′ : S′ → Mg, where S′ is a proper, smooth, connect curve\n(over C). (To see the existence of such an f ′, consider the Satake embedding Mg ↪→ PN\nof the coarse moduli space Mg; since the divisors at infinity of Mg map to subsets of\ncodimension ≥ 2 in the closure Mg of Mg in PN , one can construct a proper curve inside\nMg by cutting by hyperplane sections.) Let G\n′ = G ×Mg S′. Now it follows from our\nexplicit analysis of PicQ(G) that the natural morphism PicQ(G) → PicQ(G′) is injective.\nMoreover, it follows from Proposition 5 of §4 that the inverse images via G′ → G of\nthe images via μ of the nonexcess divisors are of codimension at least two in G′. Thus,\nit suffices to prove the original statement for G′ instead of G; but since G′ is clearly a\nprojective scheme, the existence of S and the ψi follows by reducing to the case when G\n′\nis a smooth, proper surface (by cutting with hyperplane sections) and then taking a basis\nof the Q-Néron-Severi group of the surface which consists of smooth very ample divisors.\nThe only remaining technical point is that it may at first seem that we need to use different\ncurves Si with ψi : Si → G′ (or G, or H\n+\n), but in fact, it is elementary that given any\nfinite number of curves Si, there exists a curve S and finite morphisms αi : S → Si. This\ncompletes the proof.\nLet us fix one such morphism φ : S → H+. Now let us note that if we can prove\na linear relation between the degrees of the pull-backs via ψ of certain line bundles and\ndivisors from G for all φ as above, then the relation must, in fact, hold on G.\nNow φ corresponds to some admissible covering f∼ : X∼ → P∼ over S. Let us denote\nby σ0, . . . , σb−1 the sections S → P∼ that make up the discriminant of f∼. The stable\ncurve associated to p∼ : X∼ → S, which we shall call p : X → S (obtained as in §3.27),\nis, in fact, smooth, and the only kinds of degeneracies that occur are those of Species 1,\n2, or 3B∗. Let us denote by U ⊆ X the complement of the finite set of points that are\nthe points of degenerate ramification in the fibres of Species 3B∗ (so there is exactly one\nsuch point in each fibre of Species 3B∗). Then one sees easily that f∼ contracts to a map\nf : U → P , where P is a P1-bundle over S. Thus we have a diagram:\nX ⊇ U f−→ P⏐⏐\bp ↙ π\nS\nBy abuse of notation, we also denote by σi the push-forwards of the original σi to sections\nof π. We denote by τi the unique section of X → S that lies over σi and passes through\nthe points of ramification. We denote by OP (1) some line bundle on P (which we fix for\nthe entire discussion, but the choice of which is noncanonical) whose negative square is the\nrelative canonical bundle for P → S. (Note that we may need to replace S by some finite\nétale cover of S in order for OP (1) to exist.) Observe that the square of the first Chern\nclass of OP (1) is zero. Let L be the unique line bundle on X such that, restricted to U ,\nit is equal to f∗ OP (1). By abuse of notation, we will similarly write f∗ F for other line\nbundles F on P to denote the unique extension to X of the actual bundle f∗ F on U . Also,\n65\nlet R =\n∑b−1\ni=0 τi, in the sense of divisors. Then we see that ωX = (p\n∗ ωS) ⊗ (f∗ ωP\/S)(R)\nand ωX\/S = f\n∗ωP\/S(R), where the various ω’s denote the respective canonical bundles.\nLet hi denote the height of the section σi of π relative to OP (1). Thus, on P , we\nhave σi · σj = hi + hj . Let l =\n∑\nhi. Let us choose two distinct, constant, but sufficiently\ngeneric divisors F1 and F2 on P which are of the form “the sum of a divisor which is a\n(sufficiently generic) section of P → S and various (sufficiently generic) vertical divisors”\nand which is such that OP (1) ∼= OP (F1) ∼= OP (F2). Let Gi be the divisor f−1 Fi on X,\nfor i = 1, 2, so that L ∼= OX(G1) ∼= OX(G2).\n§7.3. We wish to compute the following intersection numbers of divisors on X:\n(A) G21 = G\n2\n2 = G1 · G2\n(B) τi · G1 = τi · G2\n(C) τi · τj (for i \t= j)\n(D) τ2i\nIt is clear that it suffices to work locally over S, i.e., using local intersection theory. Thus\nwe let C = Spec A with A the completion of the local ring at some closed point x of S and\nobtain a diagram\nX ⊇ U −→ P ∼= P1C⏐⏐\bp ↙ π\nC\nwhich is the result of applying the base change C → S to the diagram in the previous §.\nBy abuse of notation, we continue to use the symbols σi, τi for the base-changed sections.\nAlso note that it suffices to compute the result in the universal situation as a function of\nthe σi · σj ’s and that formula will hold in general.\nFinally, note that (in the global situation) if q = the genus of S, then by the adjunction\nformula, we have:\n2(q − 1) = τ2i + τi · ωX\n= τ2i + τ\n2\ni + τi · (R − τi) + τi · p∗ωS + τi · f∗ωP\/S\n= 2τ2i + τi · (R − τi) + 2(q − 1) − 2τi · G1\nThus, τ2i = − 12τi · (R − τi) + τi · G1. Hence it suffices to compute (A), (B), and (C)\nlocally.\n§7.4. We begin with the case of a closed point x ∈ S where the covering does not\ndegenerate. Then it is clear that:\n66\n(A) G1 · G2 = 0\n(B) τi ·G1 = 0 for most points x, except for a finite number of special points;\nthe sum of the local intersections over these points is hi.\n(C) τi · τj = 0 for i \t= j.\n§7.5. Equally simple is the case of Species 2. Here we see that:\n(A) G1 · G2 = 0\n(B) τi · G1 = 0 (since G1 is sufficiently generic)\n(C) τi · τj = 0 for i \t= j.\n§7.6. We now consider the case of Species 1. By reduction to the univeral case, we may\nassume that σi ·σj (where i \t= j) is 0 except when {i, j} = {0, 1} in which case σ0 ·σ1 = 3.\nNow we have the following formulas for divisors:\nf−1 (σ0) = 2τ0 + E0\nf−1 (σ1) = 2τ1 + E1\nwhere E0, E1 are effective horizontal divisors of X of degree d − 2 over C. Thus we get:\n3 = σ0 · σ1 = f∗ (τ0) · σ1 = τ0 · f−1 (σ1) = 2τ0 · τ1 + τ0 · E1. But it is clear that τ0 · τ1\nand τ0 · E1 are both positive integers. Thus τ0 · τ1 = 1. Returning to the general local\ncase of Species 1, we see that if for some m, k, σm · σk \t= 0 while σi · σj = 0 for all other\ncombinations, then:\n(A) G1 · G2 = 0\n(B) τi · G1 = 0 (since G1 is sufficiently generic)\n(C) τm · τk = 13σm · σk, τi · τj = 0 for all other combinations {i, j}.\n§7.7. The case of Species 3B∗ is somewhat more involved. Let us first note that by\nreduction to the universal case, we may assume that σ0 ·σ1 = 1, and σi · σj = 0 (for i \t= j)\nfor all other combinations. Then if we define X∼ to be the scheme obtained from C as a\nresult of two ordinary blow-ups at points taken in a certain fashion, then f will extend to\na morphism f∼ : X∼ → P∼ which is everywhere defined and is the pull-back via C → S of\nthe f∼ of §7.2. Let X1 be the blow-up of X at the point of degenerate simple ramification.\nDenote by D the exceptional curve of this blow-up m : X1 → X . Then X∼ is the blow-up\n67\nk : X∼ → X1 at a point of D. Let C1 be the strict transform of D in X∼, and C2 the\nexceptional curve of k. Let h = m ◦ k. Then we have a diagram:\nX∼1\nk←− X∼ f\n∼\n−→ P∼\nm ↘\n⏐⏐\bh ⏐⏐\bg\nX ⊇ U f−→ P⏐⏐\bp ⏐⏐\bπ\nC id−→ C\nMoreover we have the sections σi of π, and their unique liftings σ\n∼\ni to sections of π ◦ g; on\nthe other hand, we also have the sections τi of p lifting the σi, the sections τ\n(1)\ni of p ◦ m\nlifting the τi, and the sections τ\n∼\ni of p◦h again lifting the τi. Let (for i = 1, 2) G\n(1)\ni and G\n∼\ni\nbe the respective strict transforms on X1 and X∼. Let L(1) = m∗ L, L∼ = h∗ L = k∗ L(1),\nM = g∗ OP(1), M∼ = f∼,∗ M. Now OP(1)·OP(1) = 0, so M·M = 0, and M∼ ·M∼ = 0.\nBut clearly, M∼ = OX∼(G∼1 ) = OX∼(G∼2 ), so (G∼1 )2 = (G∼2 )2 = G∼1 · G∼2 = 0. On\nthe other hand, k∗ D = C1 + C2 and D2 = −1, C22 = −1, C1 · C2 = 1, so we get\n−1 = C21 + 2C1 · C2 + C22 , hence C21 = −2. Also, it is clear that for i = 1, 2, we have\nG∼i ·C1 = 0 and G∼i ·C2 = 1. Now L(1) = O(G\n(1)\ni + D), so that L∼ = O(G∼i + C1 + 2C2),\nwhere i = 1, 2. Hence,\nG1 · G2 = L · L = L∼ · L∼\n= (G∼1 + C1 + 2C2) · (G∼2 + C1 + 2C2)\n= 4 + (C1 + 2C2)\n2 = 4 − 2 − 4 + 4 = 2\nLet us now note the following easy relations: τ∼0 · τ∼1 = 0, τ∼i · C1 = 1, τ∼i · C2 = 0,\nτ∼i · G∼j = 0 for all i ∈ {0, 1}, j ∈ {1, 2}. Also, we see that h∗ τi = τ∼i + C1 + C2, for\ni ∈ {0, 1}. Thus,\nτ0 · τ1 = h∗τ0 · h∗τ1\n= (τ∼0 + C1 + C2) · (τ∼1 + C1 + C2)\n= 2 + (C1 + C2)\n2 = 2 − 1 = 1\nFinally, for any i ∈ {0, 1}, j ∈ {1, 2},\nτi · Gj = h∗(τi) · h∗(Gj)\n= (τ∼i + C1 + C2) · (G∼j + C1 + 2C2) = 1 + 2 − 2 = 1\nReturning to the general local case of Species 3B∗, we see that if σm ·σk \t= 0 for some\nm, k and σi · σj = 0 for all other combinations of i, j with i \t= j, then:\n68\n(A) G1 · G2 = 2σm · σk\n(B) τi · G1 = σm · σk for i ∈ {m, k}, 0 otherwise.\n(C) τm · τk = σm · σk, τi · τj = 0 for all other combinations such that i \t= j.\n§7.8. We are now ready to return to the global situation of §7.2 and compute global\nintersection numbers. Let ν1 (resp. ν2, ν3) denote the number of points of S (with multi-\nplicity) at which a degeneracy of Species 1 (resp. 2, 3B∗) occurs. Thus ν1+ν2+ν3 = l(b−1).\nFirst, we see that L · L = G1 · G2 = 2ν3; hence (f∗ωP\/S)2 = 8ν3. Next, we have\nR · L = ∑b−1i=0 τi · G1 = l + 2ν3; hence R · (f∗ωP\/S) = −2(l + 2ν3). Finally, we have:\nR2 =\nb−1∑\ni=0\nτ2i + 2\n∑\ni<j\nτi · τj\n=\nb−1∑\ni=0\nτ2i +\n1\n2\n(\n∑\ni \u0003=j\nτi · τj) + (\n∑\ni<j\nτi · τj)\n=\nb−1∑\ni=0\n{τ2i +\n1\n2\nτi · (R − τi)} +\n∑\ni<j\nτi · τj\n= R · G1 +\n1\n3\nν1 + ν3\n= l +\n1\n3\nν1 + 3ν3\nWe have, after possibly enlarging S, a morphism α : S → G corresponding to the\ndiagram in §7.2. Let δi = deg (α∗Di). Then by §6C, we see that ν1 = 32δ1, ν2 = δ2,\nν3 =\n1\n2δ3. Thus we summarize as follows:\n(1) 3δ1 + 2δ2 + δ3 = 2l(b − 1)\n(2) (f∗ ωP\/S)\n2 = 4δ3\n(3) R · (f∗ ωP\/S) = −2(l + δ3)\n(4) R2 = l + 12δ1 +\n3\n2δ3.\n§7.9. Let us compute λS = 112 (ωX\/S)2 purely in terms of δ1, δ2, and δ3 (and not l).\nWe have:\n69\nω2X\/S = (f\n∗ ωP\/S)\n2 + R2 + 2R · (f∗ ωP\/S)\n= −3l + 1\n2\nδ1 +\n3\n2\nδ3\n=\n1\n2(b − 1){−3(3δ1 + 2δ2 + δ3) + (b − 1)(δ1 + 3δ3)}\n=\n1\n2(b − 1){(b − 10)δ1 − 6δ2 + 3(b − 2)δ3}\nNow by the reasoning of §7.2, by choosing various φ, we obtain enough possibilities so that\nany relation among line bundles on G that holds for all these possibilities must, in fact,\nhold on G. Thus we obtain the following:\nProposition : λ = 124(b−1){(b − 10)D1 − 6D2 + 3(b − 2)D3}\n§7.10. It follows from the definition and from [26], Corollary 2.7.6 of Exposé II, that\nΘS = − 12 (L − d2(g−1)ωX\/S)2. Let χ = 2(g − 1). Then we see that (−8χ2) ΘS = (χR −\nbωX\/S)\n2. Note that χ = b − 2d. Then\n(−8χ) ΘS = (b − 2d)R2 − 2b R · (ωX\/S) + (\nb2\nχ\n)(ωX\/S)\n2\n= −(b + 2d)(l + 1\n2\nδ1 +\n3\n2\nδ3) + 4b(l + δ3)\n+\nb2\n2(b − 2d)(b − 1){(b − 10)δ1 − 6δ2 + 3(b − 2)δ3}\nThus,\n−16χ(b − 1)ΘS = (3b − 2d)(3δ1 + 2δ2 + δ3) − (b + 2d)(b − 1)δ1 + (b − 1)(5b − 6d)δ3\n+\nb2\n(b − 2d){(b − 10)δ1 − 6δ2 + 3(b − 2)δ3}\n= −{b(b − 10) + 2d(b + 2)}δ1 + 2(3b − 2d)δ2 + {b(5b − 2) − 2d(3b − 2)}δ3\n+\nb2\n(b − 2d){(b − 10)δ1 − 6δ2 + 3(b − 2)δ3}\nThus,\n−16(b − 1)(b − 2d)2ΘS = {b2(b − 10) − b(b − 2d)(b − 10) − 2d(b − 2d)(b + 2)}δ1\n+ {2(3b − 2d)(b − 2d) − 6b2}δ2\n+ {b(5b − 2)(b − 2d) − 2d(3b − 2)(b − 2d) + 3b2(b − 2)}δ3\n= −4d{6b − d(b + 2)}δ1 + 8d(d − 2b)δ2 + 4{2b2(b − 1)\n− 2bd(2b − 1) + d2(3b − 2)}δ3\n70\nBy the same logic as above,\nProposition : Θ = 14(b−1)(b−2d)2\n(\nd{6b − d(b + 2)}D1 − 2d(d − 2b)D2 − {2b2(b − 1) −\n2bd(2b − 1) + d2(3b − 2)}D3\n)\n§7.11. Let E = p∗L. Then E is a vector bundle of rank d + 1 − g on S. Noting by\nRiemann-Roch that π∗OP (1) has degree zero, we see that it follows from the definitions\nthat (d + 1 − g)γS = 2N , where N = deg (E). Let κ = c1(ωX\/S), μ = c1(L). By\nRiemann-Roch,\nN = Td(−κ) ch(μ) = (1 − κ\n2\n+\nκ2\n12\n)(1 + μ +\n1\n2\nμ2)\n=\nκ2\n12\n+\nμ2\n2\n− κμ\n2\nIn our language, then,\n24N = 2(ωX\/S)\n2 + 3(f∗ ωP\/S)\n2 + 6ωX\/S · (f∗ ωP\/S)\n= 2(ωX\/S)\n2 + 9(f∗ ωP\/S)\n2 + 6R · (f∗ ωP\/S)\nHence,\n24(b − 1)N = (b − 10)δ1 − 6δ2 + 3(b − 2)δ3 + {36δ3 − 12δ3 − 12l}(b − 1)\n= (b − 10)δ1 − 6δ2 + 3{(b − 2) + 8(b − 1)}δ3 − 6(3δ1 + 2δ2 + δ3)\n= (b − 28)δ1 − 18δ2 + 9(3b − 4)δ3\nThus we have the following:\nProposition : γ = 16(b−1)(4d−b){(b − 28)D1 − 18D2 + 9(3b − 4)D3}\n§7.12. Remark : This completes our calculation. As remarked previously, we can\ncheck our calculation by making sure that the expressions for λ and Θ go to zero, while\nthat for γ goes to one under the map: D1 \b→ 3(b − d), D2 \b→ b2 (b − 10) + 4d, D3 \b→ d.\n§7.13. We are now in a position to complete the proof of Theorem 6 of §6, and hence of\nits consequences, first mentioned in §4, and restated in §6.7. We wish to show that D1, D2,\nand D3 are linearly independent in the Picard group of G. But from the relations obtained\nalready in this Section, we see that the Q-linear subspace generated by the excess divisors\nalready contains a basis for the whole space. The result thus follows immediately.\n71\n§8. Arithmetic Applications\n§8.1. In this Section, we make suggestions about possible applications of the material\ndiscussed up till now to giving\n(1) An alternate proof of the Shafarevich conjecture\n(2) An effective version of the Mordell conjecture in the style of [30].\nIn particular, we expose a certain kind of argument discovered by the author that allows\none to conclude the above arithmetic results by using the geometry of the compactification\nof the Hurwitz scheme. The argument to be exposed, however, has a couple of gaps that\nthe author has not been able to fill in. We state the facts necessary to fill in these gaps\nas “Wishes,” in the hope that perhaps some reader will see how to prove them, thus\ncompleting the proposed proofs of the above arithmetic results.\nLet B = Spec OF , where OF is the ring of integers of a number field F . Let f : C → B\nbe a smooth curve of genus g ≥ 3. (We assume good reduction to simplify the discussion,\nsince this case already illustrates the essential problem involved.)\n§8.2. If f is hyperelliptic (i.e., the geometric fibres of f are hyperelliptic), then we can\ncanonically write f as a double covering of P1B , so that to every such f we can canonically\nassign (up to automorphisms of P1B) a certain number of points of P\n1\nB which are distinct\nfrom each other outside primes over 2. We can then conclude the “Shafarevich conjecture”\nfor such hyperelliptic curves, i.e., that only a finite number of such curves exist over B, by\napplying Siegel’s theorem on integral points.\nIf f is not hyperelliptic, then no such canonical presentation as a double covering\nexists. Generically, that is, over F , we can write f as a Hurwitz covering of P1F , but\nthen the problem is that we can’t apply Siegel’s theorem unless we know that the points\nof ramification remain distinct, not only generically, but at all primes outside some fixed\nfinite set. Thus it is natural to study in detail degenerations of Hurwitz coverings, i.e., the\ncompactification of the Hurwitz scheme, to see if one can get around this problem. This\nwas the original motivation for this paper.\n§8.3. We can add some canonicality to the situation if we insist that we only consider\ncoverings f : CF → P1F such that f∗ O(1) = ω⊗NCF \/F , where N is some fixed integer. If we\nlet σ : Mg → J Nχ be the section corresponding to ω⊗NCF \/F , and let K be such that the\nfollowing diagram is Cartesian:\nK τ−→ G⏐⏐\bρ ⏐⏐\b\nMg σ−→ J Nχ\n72\nthen ideally, we would like a section ζ of ρ that avoids the Ei = τ∗ Di, for i = 1, 2, 3. This\nis unrealistic, however. In fact, even constructing any section ζ at all is rather difficult.\nAt the other extreme, if we assume that the height (e.g.[10]) of our curve is bounded,\nthen everything becomes trivial. Indeed, if we fix an appropriate metric on K so as to\ngive it an “integral structure” over the “compact curve” Spec OF\n⋃\nS∞ (where S∞ = the\narchimedean places), then we see that if c : B → Mg is the classifying map of our curve,\nthen the number of possibilities for the Grassmann bundle c∗K is bounded as soon as\ndegAr (c\n∗λ) (where “degAr ” means the degree of the arithmetic line bundle) is bounded,\nby the following result of Faltings:\nLemma : The number of closed subvarieties V ⊆ PMB (where M is any positive integer),\nwith π : V → B flat, of degree d (∈ N) as a subvariety over F of PMF , of relative dimension\nn over B, and of height degAr (π∗[O(1)]n+1) ≤ M ′ (for some fixed M ′ ∈ R) is finite.\nProof: (Sketch) The Lemma is obvious for hypersurfaces. But then one can easily reduce\nto the case of hypersurfaces via the method of the Chow variety. ©\n(Here we apply the Lemma by using the Plücker embedding for c∗K and note that\ndegAr c\n∗(ρ∗[γ]\nN+1) ≤ K degAr (c∗λ)\nfor some fixed K (independent of c) and for N = relative dimension of ρ). Similarly, since\nthe total number of possibilities for the entire situation is finite, we can certainly find some\nfinite set S of primes of B such that we can always write curves C with bounded height\nas Hurwitz coverings over B − S. (In fact, the Lemma also shows that the number of\npossibilities for C is, in this case, also finite!).\n§8.4. The problem, then, is to find an intermediate course. First, we must bound the\nnumber of possibilities for the Grassmann bundle c∗K without bounding degAr (c∗λ). If the\nGrassmann bundle is associated to a vector bundle E which is “sufficiently indecomposable”\nin the sense that it is parametrized by some sort of finite “cohomology group” on the\ncompactified curve Spec OF\n⋃\nS∞ (cf. the well-known situation for indecomposable vector\nbundles of rank 2 on curves over a finite field), then the number of possibilities for c∗K will\nbe finite. At present, however, the author does not know how to prove this sort of result,\nso we merely state it as a “wish:”\nWish 1: The number of possibilities for c∗K (as a Grassmann bundle) is finite.\nMoreover, in order to be able to prove the Shafarevich conjecture as in the hyperelliptic\ncase, we also need the following result:\nWish 2 : Let F be a number field, with ring of integers OF . Let G → Spec OF be a\nGrassmann bundle with integral structure at infinity. Let r ∈ N, d1, . . . , dr ∈ N. Fix a\nfinite set of closed primes S ⊆ Spec OF . Then there exists a finite set of closed primes\n(S ⊆)T ⊆ Spec OF with the following property:\n(∗) Let D1, . . . ,Dr ⊆ G be flat divisors over Spec OF with Di of degree di\nfor all i = 1, . . . , r that are geometrically integral and distinct outside\n73\nS. Then there exists a section σ : Spec OF → G that avoids D1, . . . ,Dr\noutside T .\nThe result is known to be true in the following cases:\n(1) If G is a P1-bundle, then it follows from Siegel’s theorem on integral\npoints.\n(2) If G is a P2-bundle, then if we assume that the Di are not only ge-\nometrically integral, but smooth, then it follows from the Shafarevich\nconjecture applied to smooth plane curves.\nIn both these cases, however, one should note that Siegel’s theorem and the Shafarevich\nconjecture are somewhat stronger than Wish 2, and so using them to prove Wish 2 is sort\nof overkill. At this point, the author knows of no way to prove Wish 2 in any generality,\nbut if a proof could be given for Wish 1 and Wish 2, then one could prove the Shafarevich\nconjecture using the technique of Hurwitz coverings.\nIn closing we remark that since the Shafarevich conjecture is, in fact, known [10], the\nnumber of possibilities for f : C → B is, in fact, finite, so we can, indeed, find a fixed,\nfinite set of primes T which is independent of F and such that we can write any f : C → B\nas a Hurwitz covering outside S. Thus, we see that something like Wish 1 and Wish 2\nmust be true, but to approach things in this way is to defeat the purpose of this venture!\n§8.5. We now move on to motivating the possible application to an effective form of\nthe Mordell conjecture. In §7, we computed the coefficients of λ in terms of the Di and\nfound that the coefficients for D1 and D3 were positive, while that for D2 was negative.\nSuppose, however, just for the moment that all three coefficients were positive. Then, since\nthe Di are all effective (i.e., in the sense of divisors, not in the sense of an “effective” form\nof the Mordell conjecture), there would exist a constant K ∈ N such that the inequality\n(∗1) Θ ≤ Kλ\nwould hold on G. Now suppose that our curve f : C → B is fixed and that we have a fixed\nline bundle L on C of relative degree d − 1 for some large d. Then we have a diagram:\nC\nα−→ J d⏐⏐\bf ⏐⏐\b\nB\nc−→ Mg\nwhere α is defined on an S-section τ : S → CS by associating to τ the line bundle L(τ)\n(where τ is regarded as a relative divisor) of relative degree d on CS . Now α\n∗Θ is relatively\n74\nample on C and hence defines a height hΘ(−) on rational points. Now if we pull back the\nGrassmann bundle α∗(G → J ) to a bundle over C, this Grassmann bundle, which a priori\nis locally trivial only in the étale topology, will, in fact, be locally trivial for the Zariski\ntopology; thus, if we are given a section σ of f , and pull-back this Grassmann bundle\nfurther via σ to a Grassmann bundle over B, this bundle over B will always have a section\nover B. Thus we can pull back the relation (∗1) via α ◦ σ to obtain:\n(∗2) hΘ(σ) ≤ KdegAr (c∗λ)\nMoreover, this holds (with the same K) over any number field F and thus gives us a\nrelation of the same type (only stronger) as that conjectured by Vojta [24], Appendix, §5,\np. 176. In particular, it implies the Mordell conjecture.\n§8.6. In fact, however, (∗2) is too strong and it is easy to see that it cannot hold. This\nis not surprising considering that after all, we derived it using the incorrect assumption\nthat the coefficient of D2 for λ is positive. To form a reasonable conjectural inequality like\n(∗2), one must somehow involve the discriminant of the base field. Thus, Vojta conjectures:\n(∗3) hΘ(σ) ≤ K1 discF\/Q + K2\nwhere K1 and K2 are constants independent of F , and discF\/Q is the “logarithmic dis-\ncriminant” of loc. cit..\nOn the other hand, althought the derivation of §8.5 is incorrect, it is very tempting,\nsince it “almost” works in the following metamathematical sense: only the coefficient of D2\nis negative, and, moreover, since d, hence b, is taken to be very large, the absolute value of\nthe coefficient of D2 is much smaller than those of the other, positive coefficients. One is,\nthus, tempted to conjecture that if we write Gσ for the Grassmann bundle σ∗ α∗(G → J d)\nover B, and Di,σ for the various pull-backs of the Di’s to divisors of Gσ, then we have the\nfollowing:\nWish 3: For fixed F , we can choose a section τ : B → Gσ such that degAr (τ∗D2,σ) is\nbounded independent of σ.\nas well as:\nWish 4 : For variable F , there exist K1,K2 ∈ N such that for any σ over any F , there\nexists a section τ : B → Gσ such that degAr (τ∗D2,σ) ≤ K1 discF\/Q + K2 (so K1 and K2\nare independent of σ and F ).\nThus, combined with the reasoning of §8.5, Wish 3 would imply the ordinary Mordell\nconjecture, while Wish 4 would imply Vojta’s generalization (∗3) of that conjecture. It\nseems that Wish 3 and Wish 4 might not necessarily be easier to prove than the more\nnatural statements that they are alleged to imply, but it is hoped that they might provide\nan interesting new perspective on the problem.\n75\nBibliography\n[1] E. Arbarello and M. Cornalba, The Picard Groups of the Moduli Space of Curves,\nTopology, 26 (1987), pp. 153-171.\n[2] E. Arbarello, Weierstrass Points and the Moduli of Curves, Compositio Math., 29\n(1974), pp. 325-342.\n[3] M. Artin, Algebrization of Formal Moduli, I, Global Analysis (Papers in Honor of K.\nKodaira), University of Tokyo Press, 1969, pp. 21-71.\n[4] A. Clebsch, Zür Theorie der Riemann’schen Fläche, Math. Ann., 6 (1872), pp. 216-230.\n[5] A. Grothendieck, Géometrie Formelle et Géometrie Algébrique, Séminaire Bourbaki\n182.\n[6] S. Diaz and R. Donagi, Hurwitz Surfaces with Nontrivial Divisors, Algebraic Geometry,\nSundance 1988, Contemp. Math. 116, Amer. Math. Soc., 1991, pp. 1-8.\n[7] S. Diaz, R. Donagi, and D. Harbater, Every Curve is a Hurwitz Space, Duke J. Math.,\n59 (1989), pp. 737-746.\n[8] P. Deligne and D. Mumford, The Irreducibility of the Space of Curves of Given Genus,\nInstitute des Hautes Etudes Scientifiques Publications Mathématiques, 36 (1969), pp.\n75-109.\n[9] A. Grothendieck and J. Dieudonné, Etude Locale des schémas et des morphismes de\nschémas, Publ. Math. IHES 20 (1964), 24 (1965), 28 (1966), 32 (1967).\n[10] G. Faltings, Endlichkeitssätze für Abelschen Varietäten über Zahlkörpern, Inv. Math.,\n73 (1983), pp. 349-366.\n[11] G. Faltings and C. L. Chai, Degenerations of Abelian Varieties, Springer-Verlag, 1990,\nChapter 1.\n[12] A. Grothendieck, Fondements de la Géometrie Algébrique, Séminaire Bourbaki 1957-\n62, Secrétariat Math., Paris, 1962.\n[13] W. Fulton, Hurwitz Schemes and the Irreducibility of Moduli of Algebraic Curves,\nAnnals of Math., 90 (1969), pp. 542-575.\n[14] P. Griffiths and J. Harris, Algebraic Geometry, Wiley-Interscience, 1978, pp. 193-211.\n[15] J. Girard, Cohomologie non abélienne, Springer-Verlag, 1971.\n[16] J. Harer, The Second Homology Group of the Mapping Class Group of an Orientable\nSurface, Inv. Math., 72 (1982), pp. 221-239.\n[17] J. Harer, The Cohomology of the Moduli Space of Curves, Theory of Moduli (Monte-\ncatini Terme, 1985), E. Sernesi (Ed.), Lecture Notes in Mathematics 1337, Springer-\nVerlag, 1988.\n76\n[18] J. Harris and S. Diaz, The Geometry of the Severi Variety II: Independence of Divisor\nClasses and Examples, Algebraic Geometry (Sundance, UT, 1986), Lecture Notes in\nMathematics 1311, Springer-Verlag, 1988, pp. 23-50.\n[19] J. Harris and D. Mumford, On the Kodaira Dimension of the Moduli Space of Curves,\nInv. Math., 67 (1982), pp. 23-86.\n[20] A. Hurwitz, Über Riemann’schen Fläche mit gegebenen Verzweigungs-punkten, Math.\nAnn., 39 (1891), pp. 1-61.\n[21] K. Kato, Logarithmic Structures of Fontaine-Illusie, Proceedings of the First JAMI\nConference, Johns-Hopkins University Press, 1990, pp. 191-224.\n[22] F. F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, II: The Stacks\nMg,n, Math. Scand., 52 (1983), pp. 161-199.\n[23] F. F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, III: The Line\nBundles on Mg,n, and a Proof of the Projectivity of Mg,n in Characteristic 0, Math.\nScand., 52 (1983), pp. 200-212.\n[24] S. Lang, Introduction to Arakelov Theory, Springer-Verlag, 1988, Appendix by Vojta.\n[25] J. Lüroth, Über Verzweigungsschnitte und Querschnitte in einer Riemannschen Flä-\nche, Math. Ann, 4 (1871), pp. 181-184.\n[26] L. Moret-Bailly, Exposé II in Séminaire sur Les Pinceaux Arithmétiques: La Conjec-\nture de Mordell, edited by L. Szpiro, Astérisque 127, 1985.\n[27] W. Fulton, On the Irreducibility of the Moduli Space of Curves, Inv. Math., 67 (1982),\npp.87-88.\n[28] J. S. Milne, Étale Cohomology, Princeton University Press, 1980.\n[29] A. Grothendieck et al., Revêtements étales et Groupe Fondamental, Lecture Notes in\nMathematics 224, Springer-Verlag, 1971.\n[30] P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes\nin Mathematics 1239, Springer-Verlag, 1987.\n77\n"}
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{"id":"xunqC116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Topics%20in%20Absolute%20Anabelian%20Geometry%20I.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":96,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Topics%20in%20Absolute%20Anabelian%20Geometry%20I","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["abs-anab-topics-1.pdf"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Tue Mar 13 09:32:15 UTC 2012","dates_in_content_count_i":1,"dates_in_content_dts":["Thu Jun 10 00:00:00 UTC 2010"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-1198857030663906133,"exact_signature_unique_b":true,"fuzzy_signature_l":-3675566922606048137,"fuzzy_signature_unique_b":true,"wordcount_i":36776,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:45 UTC 2022","fresh_date_dt":"Thu Jul 22 08:28:28 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.2 (Windows)","language_s":"en","size_i":619112,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086032284549120,"text_t":"TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I:\nGENERALITIES\nShinichi Mochizuki\nMarch 2012\nAbstract. This paper forms the first part of a three-part series in which we treat\nvarious topics in absolute anabelian geometry from the point of view of develop-\ning abstract algorithms, or “software”, that may be applied to abstract profinite\ngroups that “just happen” to arise as [quotients of] étale fundamental groups from\nalgebraic geometry. One central theme of the present paper is the issue of under-\nstanding the gap between relative, “semi-absolute”, and absolute anabelian\ngeometry. We begin by studying various abstract combinatorial properties of profi-\nnite groups that typically arise as absolute Galois groups or arithmetic\/geometric\nfundamental groups in the anabelian geometry of quite general varieties in arbitrary\ndimension over number fields, mixed-characteristic local fields, or finite fields. These\nconsiderations, combined with the classical theory of Albanese varieties, allow us\nto derive an absolute anabelian algorithm for constructing the quotient of an\narithmetic fundamental group determined by the absolute Galois group of the\nbase field in the case of quite general varieties of arbitrary dimension. Next, we\ntake a more detailed look at certain p-adic Hodge-theoretic aspects of the absolute\nGalois groups of mixed-characteristic local fields. This allows us, for instance, to\nderive, from a certain result communicated orally to the author by A. Tamagawa, a\n“semi-absolute” Hom-version — whose absolute analogue is false! — of the an-\nabelian conjecture for hyperbolic curves over mixed-characteristic local fields.\nFinally, we generalize to the case of varieties of arbitrary dimension over arbitrary\nsub-p-adic fields certain techniques developed by the author in previous papers over\nmixed-characteristic local fields for applying relative anabelian results to obtain\n“semi-absolute” group-theoretic contructions of the étale fundamental group\nof one hyperbolic curve from the étale fundamental group of another closely related\nhyperbolic curve.\nContents:\n§0. Notations and Conventions\n§1. Some Profinite Group Theory\n§2. Semi-absolute Anabelian Geometry\n§3. Absolute Open Homomorphisms of Local Galois Groups\n§4. Chains of Elementary Operations\nAppendix: The Theory of Albanese Varieties\n2000 Mathematical Subject Classification. Primary 14H30; Secondary 14H25.\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\nIntroduction\nThe present paper is the first in a series of three papers, in which we con-\ntinue our study of absolute anabelian geometry in the style of the following papers:\n[Mzk6], [Mzk7], [Mzk8], [Mzk9], [Mzk10], [Mzk11], [Mzk13]. If X is a [geometrically\nintegral] variety over a field k, and ΠX\ndef\n= π1(X) is the étale fundamental group of\nX [for some choice of basepoint], then roughly speaking, “anabelian geometry”\nmay be summarized as the study of the extent to which properties of X — such\nas, for instance, the isomorphism class of X — may be “recovered” from [various\nquotients of] the profinite group ΠX . One form of anabelian geometry is “relative\nanabelian geometry” [cf., e.g., [Mzk3]], in which instead of starting from [vari-\nous quotients of] the profinite group ΠX , one starts from the profinite group ΠX\nequipped with the natural augmentation ΠX \u0002 Gk to the absolute Galois group of\nk. By contrast, “absolute anabelian geometry” refers to the study of proper-\nties of X as reflected solely in the profinite group ΠX . Moreover, one may consider\nvarious “intermediate variants” between relative and absolute anabelian geometry\nsuch as, for instance, “semi-absolute anabelian geometry”, which refers to the\nsituation in which one starts from the profinite group ΠX equipped with the kernel\nof the natural augmentation ΠX \u0002 Gk.\nThe new point of view that underlies the various “topics in absolute anabelian\ngeometry” treated in the present three-part series may be summarized as follows.\nIn the past, research in anabelian geometry typically centered around the establish-\nment of “fully faithfulness” results — i.e., “Grothendieck Conjecture-type” results\n— concerning some sort of “fundamental group functor X \u0002→ ΠX” from varieties\nto profinite groups. In particular, the term “group-theoretic” was typically used\nto refer to properties preserved, for instance, by some isomorphism of profinite\ngroups ΠX\n∼→ ΠY [i.e., between the étale fundamental groups of varieties X, Y ].\nBy contrast:\nIn the present series, the focus of our attention is on the development of\n“algorithms” — i.e., “software” — which are “group-theoretic” in the\nsense that they are phrased in language that only depends on the structure\nof the input data as [for instance] a profinite group.\nHere, the “input data” is a profinite group that “just happens to arise” from scheme\ntheory as an étale fundamental group, but which is only of concern to us in its\ncapacity as an abstract profinite group. That is to say,\nthe algorithms in question allow one to construct various objects reminis-\ncent of objects that arise in scheme theory, but the issue of “eventually\nreturning to scheme theory” — e.g., of showing that some isomor-\nphism of profinite groups arises from an isomorphism of schemes — is no\nlonger an issue of primary interest.\nOne aspect of this new point of view is that the main results obtained are no longer\nnecessarily of the form\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 3\n(∗) “some scheme is anabelian” — i.e., some sort of “fundamental group\nfunctor X \u0002→ ΠX” from varieties to profinite groups is fully faithful —\nbut rather of the form\n(†) “some a priori scheme-theoretic property\/construction\/operation\nmay be formulated as a group-theoretic algorithm”, i.e., an algorithm\nthat depends only on the topological group structure of the arithmetic fun-\ndamental groups involved\n— cf., e.g., (2), (4) below. A sort of intermediate variant between (∗) and (†) is\nconstituted by results of the form\n(∗′) “homomorphisms between arithmetic fundamental groups that satisfy\nsome sort of relatively mild condition arise from scheme theory”\n— cf., e.g., (3) below.\nHere, we note that typically results in absolute or semi-absolute anabelian\ngeometry are much more difficult to obtain than corresponding results in relative\nanabelian geometry [cf., e.g., the discussion of (i) below]. This is one reason why\none is frequently obliged to content oneself with results of the form (†) or (∗′), as\nopposed to (∗).\nOn the other hand, another aspect of this new point of view is that, by abol-\nishing the restriction that one must have as one’s ultimate goal the complete recon-\nstruction of the original schemes involved, one gains a greater degree of freedom\nin the geometries that one considers. This greater degree of freedom often results in\nthe discovery of new results that might have eluded one’s attention if one restricts\noneself to obtaining results of the form (∗). Indeed, this phenomenon may already\nbe seen in previous work of the author:\n(i) In [Mzk6], Proposition 1.2.1 [and its proof], various group-theoretic al-\ngorithms are given for constructing various objects associated to the ab-\nsolute Galois group of a mixed-characteristic local field. In this case, we\nrecall that it is well-known [cf., e.g., [NSW], the Closing Remark preceding\nTheorem 12.2.7] that in general, there exist isomorphisms between such\nabsolute Galois groups that do not arise from scheme theory.\n(ii) In the theory of pro-l cuspidalizations given in [Mzk13], §3, “cuspi-\ndalized geometrically pro-l fundamental groups” are “group-theoretically\nconstructed” from geometrically pro-l fundamental groups of proper hy-\nperbolic curves without ever addressing the issue of whether or not the\noriginal curve [i.e., scheme] may be reconstructed from the given geomet-\nrically pro-l fundamental group [of a proper hyperbolic curve].\n(iii) In some sense, the abstract, algorithmic point of view discussed above\nis taken even further in [Mzk12], where one works with certain types of\npurely combinatorial objects — i.e., “semi-graphs of anabelioids” — whose\ndefinition “just happens to be” motivated by stable curves in algebraic ge-\nometry. On the other hand, the results obtained in [Mzk12] are results\n4 SHINICHI MOCHIZUKI\nconcerning the abstract combinatorial geometry of these abstract combina-\ntorial objects — i.e., one is never concerned with the issue of “eventually\nreturning” to, for instance, scheme-theoretic morphisms.\nThemain results of the present paper are, to a substantial extent, “generalities”\nthat will be of use to us in the further development of the theory in the latter\ntwo papers of the present three-part series. These main results center around the\ntheme of understanding the gap between relative, semi-absolute, and absolute\nanabelian geometry and may be summarized as follows:\n(1) In §1, we study various notions associated to abstract profinite groups\nsuch as RTF-quotients [i.e., quotients obtained by successive formation\nof torsion-free abelianizations — cf. Definition 1.1, (i)], slimness [i.e.,\nthe property that all open subgroups are center-free], and elasticity [i.e.,\nthe property that every nontrivial topologically finitely generated closed\nnormal subgroup of an open subgroup is itself open — cf. Definition 1.1,\n(ii)] in the context of the absolute Galois groups that typically appear\nin anabelian geometry [cf. Proposition 1.5, Theorem 1.7].\n(2) In §2, we begin by formulating the terminology that we shall use in our\ndiscussion of the anabelian geometry of quite general varieties of arbitrary\ndimension [cf. Definition 2.1]. We then apply the theory of slimness and\nelasticity developed in §1 to study various variants of the notion of “semi-\nabsoluteness” [cf. Proposition 2.5]. Moreover, in the case of quite general\nvarieties of arbitrary dimension over number fields, mixed-characteristic\nlocal fields, or finite fields, we combine the various group-theoretic consid-\nerations of (1) with the classical theory of Albanese varieties [reviewed in\nthe Appendix] to give various\n“group-theoretic algorithms” for constructing the quotient of an\narithmetic fundamental group determined by the absolute\nGalois group of the base field [cf. Theorem 2.6, Corollary 2.8].\nFinally, in the case of hyperbolic orbicurves, we apply the theory of max-\nimal pro-RTF-quotients developed in §1 to give quite explicit “group-\ntheoretic algorithms” for constructing these quotients [cf. Theorem 2.11].\nSuch maximal pro-RTF-quotients may be thought of as a sort of analogue,\nin the case of mixed-characteristic local fields, of the reconstruction, in the\ncase of finite fields, of the quotient of an arithmetic fundamental group\ndetermined by the absolute Galois group of the base field via the opera-\ntion of “passing to the maximal torsion-free abelian quotient” [cf. Remark\n2.11.1].\n(3) In §3, we develop a generalization of the main result of [Mzk1] concerning\nthe geometricity of arbitrary isomorphisms of absolute Ga-\nlois groups of mixed-characteristic local fields that pre-\nserve the ramification filtration [cf. Theorem 3.5].\nThis generalization allows one to replace the condition of “preserving\nthe ramification filtration” by various more general conditions, certain\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 5\nof which were motivated by a result orally communicated to the author\nby A. Tamagawa [cf. Remark 3.8.1]. Moreover, unlike the main result of\n[Mzk1], this generalization may be applied, in certain cases, to\narbitrary open homomorphisms — i.e., not just isomor-\nphisms! —\nbetween absolute Galois groups of mixed-characteristic local fields, hence\nimplies certain semi-absolute Hom-versions [cf. Corollary 3.8, 3.9] of the\nrelative Hom-versions of the Grothendieck Conjecture given in [Mzk3],\nTheorems A, B. Also, we observe, in Example 2.13, that the corresponding\nabsolute Hom-version of these results is false in general. Indeed, it was\nprecisely the discovery of this counterexample to the “absolute Hom-\nversion” that led the author to the detailed investigation of the “gap be-\ntween absolute and semi-absolute” that forms the content of §2.\n(4) In §4, we study various “fundamental operations” for passing from one\nalgebraic stack to another. In the case of arbitrary dimension, these op-\nerations are the operations of “passing to a finite étale covering”\nand “passing to a finite étale quotient”; in the case of hyperbolic\norbicurves, we also consider the operations of “forgetting a cusp” and\n“coarsifying a non-scheme-like point”. Our main result asserts that\nif one assumes certain relative anabelian results concerning\nthe varieties under consideration, then there exist group-theoretic\nalgorithms for describing the corresponding semi-absolute an-\nabelian operations on arithmetic fundamental groups [cf. The-\norem 4.7].\nThis theory, which generalizes the theory of [Mzk9], §2, and [Mzk13], §2,\nmay be applied not only to hyperbolic orbicurves over sub-p-adic fields [cf.\nExample 4.8], but also to “iso-poly-hyperbolic orbisurfaces” over sub-p-\nadic fields [cf. Example 4.9]. In [Mzk15], this theory will be applied, in an\nessential way, in our development of the theory of Belyi and elliptic cusp-\nidalizations. We also give a tempered version of this theory [cf. Theorem\n4.12].\nFinally, in an Appendix, we review, for lack of an appropriate reference, various well-\nknown facts concerning the theory of Albanese varieties that will play an important\nrole in the portion of the theory of §2 concerning varieties of arbitrary dimension.\nMuch of this theory of Albanese varieties is contained in such classical references\nas [NS], [Serre1], [Chev], which are written from a somewhat classical point of\nview. Thus, in the Appendix, we give a modern scheme-theoretic treatment of this\nclassical theory, but without resorting to the introduction of motives and derived\ncategories, as in [BS], [SS]. In fact, strictly speaking, in the proofs that appear\nin the body of the text [i.e., §2], we shall only make essential use of the portion\nof the Appendix concerning abelian Albanese varieties [i.e., as opposed to semi-\nabelian Albanese varieties]. Nevertheless, we decided to give a full treatment of\nthe theory of Albanese varieties as given in the Appendix, since it seemed to the\nauthor that the theory is not much more difficult and, moreover, assumes a much\n6 SHINICHI MOCHIZUKI\nmore natural form when formulated for “open” [i.e., not necessarily proper] varieties\n[which, roughly speaking, correspond to the case of semi-abelian Albanese varieties]\nthan when formulated only for proper varieties [which, roughly speaking, correspond\nto the case of abelian Albanese varieties].\nAcknowledgements:\nI would like to thank Akio Tamagawa for many helpful discussions concerning\nthe material presented in this paper. Also, I would like to thank Brian Conrad for\ninforming me of the references in the Appendix to [FGA], and Noboru Nakayama\nfor advice concerning non-smooth normal algebraic varieties.\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 7\nSection 0: Notations and Conventions\nNumbers:\nThe notation Q will be used to denote the field of rational numbers. The\nnotation Z ⊆ Q will be used to denote the set, group, or ring of rational integers.\nThe notation N ⊆ Z will be used to denote the set or monoid of nonnegative rational\nintegers. The profinite completion of the group Z will be denoted Ẑ. Write\nPrimes\nfor the set of prime numbers. If p ∈ Primes, then the notation Qp (respectively,\nZp) will be used to denote the p-adic completion of Q (respectively, Z). Also, we\nshall write\nZ(×)p ⊆ Z×p\nfor the subgroup 1 + pZp ⊆ Z×p if p > 2, 1 + p2Zp ⊆ Z×p if p = 2. Thus, we have\nisomorphisms of topological groups\nZ(×)p × (Z×p \/Z(×)p ) ∼→ Z×p ; Z(×)p ∼→ Zp\n— where the second isomorphism is the isomorphism determined by dividing the\np-adic logarithm by p if p > 2, or by p2 if p = 2; Z×p \/Z\n(×)\np\n∼→ F×p if p > 2,\nZ×p \/Z\n(×)\np\n∼→ Z\/pZ if p = 2.\nA finite field extension of Q will be referred to as a number field, or NF, for\nshort. A finite field extension of Qp for some p ∈ Primes will be referred to as a\nmixed-characteristic nonarchimedean local field, or MLF, for short. A field of finite\ncardinality will be referred to as a finite field, or FF, for short. We shall regard the\nset of symbols {NF,MLF,FF} as being equipped with a linear ordering\nNF > MLF > FF\nand refer to an element of this set of symbols as a field type.\nTopological Groups:\nLet G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us\nwrite\nZG(H)\ndef\n= {g ∈ G | g · h = h · g, ∀ h ∈ H}\nfor the centralizer of H in G;\nNG(H)\ndef\n= {g ∈ G | g ·H · g−1 = H}\nfor the normalizer of H in G; and\nCG(H)\ndef\n= {g ∈ G | (g ·H · g−1)⋂H has finite index in H, g ·H · g−1}\nfor the commensurator of H in G. Note that: (i) ZG(H), NG(H) and CG(H) are\nsubgroups of G; (ii) we have inclusions H, ZG(H) ⊆ NG(H) ⊆ CG(H); (iii) H is\n8 SHINICHI MOCHIZUKI\nnormal in NG(H). If H = NG(H) (respectively, H = CG(H)), then we shall say\nthat H is normally terminal (respectively, commensurably terminal) in G. Note\nthat ZG(H), NG(H) are always closed in G, while CG(H) is not necessarily closed\nin G. Also, we shall write Z(G)\ndef\n= ZG(G) for the center of G.\nLet G be a topological group. Then [cf. [Mzk14], §0] we shall refer to a normal\nopen subgroup H ⊆ G such that the quotient group G\/H is a free discrete group\nas co-free. We shall refer to a co-free subgroup H ⊆ G as minimal if every co-free\nsubgroup of G contains H. Thus, any minimal co-free subgroup of G is necessarily\nunique and characteristic.\nWe shall refer to a continuous homomorphism between topological groups as\ndense (respectively, of DOF-type [cf. [Mzk10], Definition 6.2, (iii)]; of OF-type) if\nits image is dense (respectively, dense in some open subgroup of finite index; an\nopen subgroup of finite index). Let Π be a topological group; Δ a normal closed\nsubgroup such that every characteristic open subgroup of finite index H ⊆ Δ admits\na minimal co-free subgroup Hco-fr ⊆ H. Write Π̂ for the profinite completion of Π.\nLet\nΠ̂ \u0002 Q\nbe a quotient of profinite groups. Then we shall refer to as the (Q,Δ)-co-free\ncompletion of Π, or co-free completion of Π with respect to [the quotient Π̂ \u0002] Q\nand [the subgroup] Δ ⊆ Π — where we shall often omit mention of Δ when it is\nfixed throughout the discussion — the inverse limit\nΠQ\/co-fr\ndef\n= lim←−\nH\nImQ(Π\/H\nco-fr)\n— where H ⊆ Δ ranges over the characteristic open subgroups of Δ of finite index;\nĤco-fr ⊆ Π̂ is the closure of the image of Hco-fr in Π̂; Ĥco-frQ ⊆ Q is the image of\nĤco-fr in Q; “ImQ(−)” denotes the image in Q\/Ĥco-frQ of the group in parentheses.\nThus, we have a natural dense homomorphism Π → ΠQ\/co-fr.\nWe shall say that a profinite group G is slim if for every open subgroup H ⊆ G,\nthe centralizer ZG(H) is trivial. Note that every finite normal closed subgroup\nN ⊆ G of a slim profinite group G is trivial. [Indeed, this follows by observing that\nfor any normal open subgroup H ⊆ G such that N ⋂H = {1}, consideration of the\ninclusion N ↪→ G\/H reveals that the conjugation action of H on N is trivial, i.e.,\nthat N ⊆ ZG(H) = {1}.]\nWe shall say that a profinite group G is decomposable if there exists an iso-\nmorphism of profinite groups H1 ×H2 ∼→ G, where H1, H2 are nontrivial profinite\ngroups. If a profinite group G is not decomposable, then we shall say that it is\nindecomposable.\nWe shall write Gab for the abelianization of a profinite group G, i.e., the quo-\ntient of G by the closure of the commutator subgroup of G, and\nGab-t\nfor the torsion-free abelianization of G, i.e., the quotient of Gab by the closure of\nthe torsion subgroup of Gab. Note that the formation of Gab, Gab-t is functorial\nwith respect to arbitrary continuous homomorphisms of profinite groups.\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 9\nWe shall denote the group of automorphisms of a profinite group G by Aut(G).\nConjugation by elements of G determines a homomorphism G → Aut(G) whose\nimage consists of the inner automorphisms of G. We shall denote by Out(G)\nthe quotient of Aut(G) by the [normal] subgroup consisting of the inner auto-\nmorphisms. In particular, if G is center-free, then we have an exact sequence\n1 → G → Aut(G) → Out(G) → 1. If, moreover, G is topologically finitely generated,\nthen it follows immediately that the topology of G admits a basis of characteristic\nopen subgroups, which thus determine a topology on Aut(G), Out(G) with respect\nto which the exact sequence 1 → G → Aut(G) → Out(G) → 1 may be regarded as\nan exact sequence of profinite groups.\nAlgebraic Stacks and Log Schemes:\nWe refer to [FC], Chapter I, §4.10, for a discussion of the coarse space associated\nto an algebraic stack. We shall say that an algebraic stack is scheme-like if it is, in\nfact, a scheme. We shall say that an algebraic stack is generically scheme-like if it\nadmits an open dense substack which is a scheme.\nWe refer to [Kato] and the references given in [Kato] for basic facts and def-\ninitions concerning log schemes. Here, we recall that the interior of a log scheme\nrefers to the largest open subscheme over which the log structure is trivial.\nCurves:\nWe shall use the following terms, as they are defined in [Mzk13], §0: hyperbolic\ncurve, family of hyperbolic curves, cusp, tripod. Also, we refer to [Mzk6], the proof\nof Lemma 2.1; [Mzk6], the discussion following Lemma 2.1, for an explanation of\nthe terms “stable reduction” and “stable model” applied to a hyperbolic curve over\nan MLF.\nIf X is a generically scheme-like algebraic stack over a field k that admits a\nfinite étale Galois covering Y → X, where Y is a hyperbolic curve over a finite\nextension of k, then we shall refer to X as a hyperbolic orbicurve over k. [Thus,\nwhen k is of characteristic zero, this definition coincides with the definition of a\n“hyperbolic orbicurve” in [Mzk13], §0, and differs from, but is equivalent to, the\ndefinition of a “hyperbolic orbicurve” given in [Mzk7], Definition 2.2, (ii). We refer\nto [Mzk13], §0, for more on this equivalence.] Note that the notion of a “cusp of\na hyperbolic curve” given in [Mzk13], §0, generalizes immediately to the notion of\n“cusp of a hyperbolic orbicurve”. If X → Y is a dominant morphism of hyperbolic\norbicurves, then we shall refer to X → Y as a partial coarsification morphism if the\nmorphism induced by X → Y on associated coarse spaces is an isomorphism.\nLet X be a hyperbolic orbicurve over an algebraically closed field; denote its\nétale fundamental group by ΔX . We shall refer to the order of the [manifestly finite!]\ndecomposition group in ΔX of a closed point x of X as the order of x. We shall\nrefer to the [manifestly finite!] least common multiple of the orders of the closed\npoints of X as the order of X. Thus, it follows immediately from the definitions\nthat X is a hyperbolic curve if and only if the order of X is equal to 1.\n10 SHINICHI MOCHIZUKI\nSection 1: Some Profinite Group Theory\nWe begin by discussing certain aspects of abstract profinite groups, as they\nrelate to the Galois groups of finite fields, mixed-characteristic nonarchimedean\nlocal fields, and number fields. In the following, let G be a profinite group.\nDefinition 1.1.\n(i) In the following, “RTF” is to be understood as an abbreviation for “recur-\nsively torsion-free”. If H ⊆ G is a normal open subgroup that arises as the kernel\nof a continuous surjection G \u0002 Q, where Q is a finite abelian group, that factors\nthrough the torsion-free abelianization G \u0002 Gab-t of G [cf. §0], then we shall refer\nto (G,H) as an RTF-pair. If for some integer n ≥ 1, a sequence of open subgroups\nGn ⊆ Gn−1 ⊆ . . . G1 ⊆ G0 = G\nof G satisfies the condition that, for each nonnegative integer j ≤ n− 1, (Gj , Gj+1)\nis an RTF-pair, then we shall refer to this sequence of open subgroups as an RTF-\nchain [from Gn to G]. If H ⊆ G is an open subgroup such that there exists an\nRTF-chain from H to G, then we shall refer to H ⊆ G as an RTF-subgroup [of G].\nIf the kernel of a continuous surjection φ : G \u0002 Q, where Q is a finite group, is an\nRTF-subgroup of G, then we shall say that φ : G \u0002 Q is an RTF-quotient of G.\nIf φ : G \u0002 Q is a continuous surjection of profinite groups such that the topology\nof Q admits a basis of normal open subgroups {Nα}α∈A satisfying the property\nthat each composite G \u0002 Q \u0002 Q\/Nα [for α ∈ A] is an RTF-quotient, then we\nshall say that φ : G \u0002 Q is a pro-RTF-quotient. If G is a profinite group such\nthat the identity map of G forms a pro-RTF-quotient, then we shall say that G is\na pro-RTF-group. [Thus, every pro-RTF-group is pro-solvable.]\n(ii) We shall say that G is elastic if it holds that every topologically finitely\ngenerated closed normal subgroup N ⊆ H of an open subgroup H ⊆ G of G is\neither trivial or of finite index in G. If G is elastic, but not topologically finitely\ngenerated, then we shall say that G is very elastic.\n(iii) Let Σ ⊆ Primes [cf. §0] be a set of prime numbers. If G admits an\nopen subgroup which is pro-Σ, then we shall say that G is almost pro-Σ. We\nshall refer to a quotient G \u0002 Q as almost pro-Σ-maximal if for some normal open\nsubgroup N ⊆ G with maximal pro-Σ quotient N \u0002 P , we have Ker(G \u0002 Q) =\nKer(N \u0002 P ). [Thus, any almost pro-Σ-maximal quotient of G is almost pro-Σ.]\nIf Σ\ndef\n= Primes \\ {p} for some p ∈ Primes, then we shall write “pro-( \u000b= p)” for\n“pro-Σ”. Write\nẐ( \u0005=p)\nfor the maximal pro-( \u000b= p) quotient of Ẑ. We shall say that G is pro-omissive (re-\nspectively, almost pro-omissive) if it is pro-( \u000b= p) for some p ∈ Primes (respectively,\nif it admits a pro-omissive open subgroup). We shall say that G is augmented pro-p\nif there exists an exact sequence of profinite groups 1 → N → G → Ẑ( \u0005=p) → 1,\nwhereN is pro-p; in this case, the image ofN in G is uniquely determined [i.e., as the\nmaximal pro-p subgroup of G]; the quotient G \u0002 Ẑ( \u0005=p) [which is well-defined up to\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 11\nautomorphisms of Ẑ( \u0005=p)] will be referred to as the augmentation of the augmented\npro-p group G. We shall say that G is augmented pro-prime if it is augmented pro-\np for some [not necessarily unique!] p ∈ Primes. If Σ = {p} for some unspecified\np ∈ Primes, we shall write “pro-prime” for “pro-Σ”. If C is the “full formation”\n[cf., e.g., [FJ], p. 343] of finite solvable Σ-groups, then we shall refer to a pro-C\ngroup as a pro-Σ-solvable group.\nProposition 1.2. (Basic Properties of Pro-RTF-quotients) Let\nφ : G1 → G2\nbe a continuous homomorphism of profinite groups. Then:\n(i) If H ⊆ G2 is an RTF-subgroup of G2, then φ−1(H) ⊆ G1 is an RTF-\nsubgroup of G1.\n(ii) If H, J ⊆ G are RTF-subgroups of G, then so is H⋂ J .\n(iii) If H ⊆ G is an RTF-subgroup of G, then there exists a normal [open]\nRTF-subgroup J ⊆ G of G such that J ⊆ H.\n(iv) Every RTF-quotient G \u0002 Q of G factors through the quotient\nG \u0002 GRTF def= lim←−\nN\nG\/N\n— where N ranges over the normal [open] RTF-subgroups of G. We shall refer\nto this quotient G \u0002 GRTF as the maximal pro-RTF-quotient. Finally, the\nprofinite group GRTF is a pro-RTF-group.\n(v) There exists a commutative diagram\nG1\nφ−→ G2⏐⏐\u0005\n⏐⏐\u0005\nGRTF1\nφRTF−→ GRTF2\n— where the vertical arrows are the natural morphisms, and the continuous homo-\nmorphism φRTF is uniquely determined by the condition that the diagram commute.\nProof. Assertion (i) follows immediately from the definitions, together with the\nfunctoriality of the torsion-free abelianization [cf. §0]. To verify assertion (ii), one\nobserves that an RTF-chain from H\n⋂\nJ to G may be obtained by concatenating\nan RTF-chain from H\n⋂\nJ to J [whose existence follows from assertion (i) applied\nto the natural inclusion homomorphism J ↪→ G] with an RTF-chain from J to\nG. Assertion (iii) follows by applying assertion (ii) to some finite intersection of\nconjugates of H. Assertion (iv) follows immediately from assertions (ii), (iii), and\nthe definitions involved. Assertion (v) follows immediately from assertions (i), (iv).\n©\n12 SHINICHI MOCHIZUKI\nProposition 1.3. (Basic Properties of Elasticity)\n(i) Let H ⊆ G be an open subgroup of the profinite group G. Then the elas-\nticity of G implies that of H. If G is slim, then the elasticity of H implies that of\nG.\n(ii) Suppose that G is nontrivial. Then G is very elastic if and only if it\nholds that every topologically finitely generated closed normal subgroup N ⊆ H of\nan open subgroup H ⊆ G of G is trivial.\nProof. Assertion (i) follows immediately from the definitions, together with the\nfact that a slim profinite group has no normal closed finite subgroups [cf. §0]. The\nnecessity portion of assertion (ii) follows from the fact that the existence of a topo-\nlogically finitely generated open subgroup of G implies that G itself is topologically\nfinitely generated; the sufficiency portion of assertion (ii) follows immediately by\ntaking N\ndef\n= G \u000b= {1}. ©\nNext, we consider Galois groups.\nDefinition 1.4. We shall refer to a field k as solvably closed if, for every finite\nabelian field extension k′ of k, it holds that k′ = k.\nRemark 1.4.1. Note that if k̃ is a solvably closed Galois extension of a field k of\ntype MLF or FF [cf. §0], then k̃ is an algebraic closure of k. Indeed, this follows\nfrom the well-known fact that the absolute Galois group of a field of type MLF or\nFF is pro-solvable [cf., e.g., [NSW], Chapter VII, §5].\nProposition 1.5. (Pro-RTF-quotients of MLF Galois Groups) Let k be an\nalgebraic closure of an MLF [cf. §0] k of residue characteristic p; Gk def= Gal(k\/k);\nGk \u0002 GRTFk the maximal pro-RTF-quotient [cf. Proposition 1.2, (iv)] of Gk.\nThen:\n(i) GRTFk is slim.\n(ii) There exists an exact sequence 1 → P → GRTFk → Ẑ → 1, where P is a\npro-p group whose image in GRTFk is equal to the image of the inertia subgroup\nof Gk in G\nRTF\nk . In particular, G\nRTF\nk is augmented pro-p.\nProof. Recall from local class field theory [cf., e.g., [Serre2]] that for any open sub-\ngroup H ⊆ Gk, corresponding to a subfield kH ⊆ k, we have a natural isomorphism\n(k×H)\n∧ ∼→ Hab\n[where the “∧” denotes the profinite completion of an abelian group; “×” denotes\nthe group of units of a ring]; moreover, Hab fits into an exact sequence\n1 → O×kH → Hab → Ẑ → 1\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 13\n[where OkH ⊆ kH is the ring of integers] in which the image of O×kH in Hab coincides\nwith the image of the inertia subgroup of H. Observe, moreover, that the quotient\nof the abelian profinite group O×kH by its torsion subgroup is a pro-p group. Thus,\nassertion (ii) follows immediately from this observation, together with the definition\nof the maximal pro-RTF-quotient. Next, let us observe that by applying the natural\nisomorphism (O×kH ) ⊗ Qp\n∼→ kH , it follows that whenever H is normal in Gk, the\naction of Gk\/H on H\nab-t is faithful. Thus, assertion (i) follows immediately. ©\nThe following result is well-known.\nProposition 1.6. (Maximal Pro-p Quotients of MLF Galois Groups) Let\nk be an algebraic closure of an MLF k of residue characteristic p; Gk\ndef\n= Gal(k\/k);\nGk \u0002 G(p)k the maximal pro-p-quotient of Gk. Then:\n(i) Any almost pro-p-maximal quotient Gk \u0002 Q of Gk is slim.\n(ii) Suppose further that k contains a primitive p-th root of unity. Then\nfor any finite module M annihilated by p equipped with a continuous action by G\n(p)\nk\n[which thus determines a continuous action by Gk], the natural homomorphism\nGk \u0002 G(p)k induces an isomorphism\nHj(G\n(p)\nk ,M)\n∼→ Hj(Gk,M)\non Galois cohomology modules for all integers j ≥ 0.\n(iii) If k contains (respectively, does not contain) a primitive p-th root\nof unity, then any closed subgroup of infinite index (respectively, any closed\nsubgroup of arbitrary index) H ⊆ G(p)k is a free pro-p group.\nProof. Assertion (i) follows from the argument applied to verify Proposition 1.5,\n(i). To verify assertion (ii), it suffices to show that the cohomology module\nHj(J,M) ∼= lim−→\nk′\nHj(Gk′ ,M)\n[where J\ndef\n= Ker(Gk \u0002 G(p)k ); k′ ranges over the finite Galois extensions of k such\nthat [k′ : k] is a power of p; Gk′ ⊆ Gk is the open subgroup determined by k′]\nvanishes for j ≥ 1. By “dévissage”, we may assume that M ∼= Fp with the trivial\nGk-action. Since the cohomological dimension of Gk′ is equal to 2 [cf. [NSW],\nTheorem 7.1.8, (i)], it suffices to consider the cases j = 1, 2. For j = 2, since\nH2(Gk′ ,Fp) ∼= Fp [cf. [NSW], Theorem 7.1.8, (ii); our hypothesis that k contains\na primitive p-th root of unity], it suffices, by the well-known functorial behavior of\nH2(Gk′ ,Fp) [cf. [NSW], Corollary 7.1.4], to observe that k\n′ always admits a cyclic\nGalois extension of degree p [arising, for instance, from an extension of the residue\nfield of k′]. On the other hand, for j = 1, the desired vanishing is a tautology,\nin light of the definition of the quotient Gk \u0002 G(p)k . This completes the proof of\nassertion (ii).\n14 SHINICHI MOCHIZUKI\nFinally, we consider assertion (iii). If k does not contain a primitive p-th root\nof unity, then G\n(p)\nk itself is a free pro-p group [cf. [NSW], Theorem 7.5.8, (i)], so\nany closed subgroup H ⊆ G(p)k is also free pro-p [cf., e.g., [RZ], Corollary 7.7.5].\nThus, let us assume that k contains a primitive p-th root of unity, so we may\napply the isomorphism of assertion (ii). In particular, if J ⊆ G(p)k is an open\nsubgroup such that H ⊆ J , and kJ ⊆ k is the subfield determined by J , then one\nverifies immediately that the quotient GkJ \u0002 J may be identified with the quotient\nGkJ \u0002 G\n(p)\nkJ\n, so we obtain an isomorphism H2(J,Fp)\n∼→ H2(GkJ ,Fp) [where Fp is\nequipped with the trivial Galois action]. Thus, to complete the proof that H is free\npro-p, it suffices [by a well-known cohomological criterion for free pro-p groups —\ncf., e.g., [RZ], Theorem 7.7.4] to show that the cohomology module\nH2(H,Fp) ∼= lim−→\nkJ\nH2(GkJ ,Fp)\n[where Fp is equipped with the trivial Galois action; kJ ranges over the finite\nextensions of k arising from open subgroups J ⊆ G(p)k such that H ⊆ J ] vanishes.\nAs in the proof of assertion (ii), this vanishing follows from the well-known functorial\nbehavior ofH2(GkJ ,Fp), together with the observation that, by our assumption that\nH is of infinite index in G\n(p)\nk , kJ always admits an extension of degree p arising\nfrom an open subgroup of J [where J ⊆ G(p)k corresponds to kJ ] containing H. ©\nTheorem 1.7. (Slimness and Elasticity of Arithmetic Galois Groups)\nLet k̃ be a solvably closed Galois extension of a field k; write Gk\ndef\n= Gal(k̃\/k).\nThen:\n(i) If k is an FF, then Gk ∼= Ẑ is neither elastic nor slim.\n(ii) If k is an MLF of residue characteristic p, then Gk, as well as any\nalmost pro-p-maximal quotient Gk \u0002 Q of Gk, is elastic and slim.\n(iii) If k is an NF, then Gk is very elastic and slim.\nProof. Assertion (i) is immediate from the definitions; assertion (iii) is the content\nof [Mzk11], Corollary 2.2; [Mzk11], Theorem 2.4. The slimness portion of assertion\n(ii) for Gk is shown, for instance, in [Mzk6], Theorem 1.1.1, (ii) [via the same\nargument as the argument applied to prove Proposition 1.5, (i); Proposition 1.6,\n(i)]; the slimness portion of assertion (ii) forQ is precisely the content of Proposition\n1.6, (i). Write p for the residue characteristic of k.\nTo show the elasticity portion of assertion (ii) for Q, let N ⊆ H be a closed\nnormal subgroup of infinite index of an open subgroup H ⊆ Q such that N is\ntopologically generated by r elements, where r ≥ 1 is an integer. Then it suffices\nto show that N is trivial. Since Q has already been shown to be slim [hence has\nno nontrivial finite normal closed subgroups — cf. §0], we may always replace k\nby a finite extension of k. In particular, we may assume that H = Q, and that\nQ is maximal pro-p. Since [Q : N ] is infinite, it follows that there exists an open\nsubgroup J ⊆ Q, corresponding to a subfield kJ ⊆ k, such that N ⊆ J , and\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 15\n[kJ : Qp] ≥ r + 1. Here, we recall from our discussion of local class field theory in\nthe proof of Proposition 1.5 that dimQp(J\nab ⊗ Qp) = [kJ : Qp] + 1 (≥ r + 2). In\nparticular, we conclude that N is necessarily a subgroup of infinite index of some\ntopologically finitely generated closed subgroup P ⊆ J such that [J : P ] is infinite.\n[For instance, one may take P to be the subgroup of J topologically generated by\nN , together with an element of J that maps to a non-torsion element of the quotient\nof Jab by the image of Nab.] Thus, we conclude from Proposition 1.6, (iii), that P\nis a free pro-p group which contains a topologically finitely generated closed normal\nsubgroup N ⊆ P of infinite index. On the other hand, by [a rather easy special\ncase of] the theorem of Lubotzky-Melnikov-van den Dries [cf., e.g., [FJ], Proposition\n24.10.3; [MT], Theorem 1.5], this implies that N is trivial. This completes the proof\nof the elasticity portion of assertion (ii) for Q.\nTo show the elasticity portion of assertion (ii) for Gk, let N ⊆ H be a closed\nnormal subgroup of infinite index of an open subgroup H ⊆ Gk such that N is\ntopologically generated by r elements, where r ≥ 1 is an integer. Then it suffices\nto show that N is trivial. As in the proof of the elasticity of “Q”, we may assume\nthat H = Gk; also, since [Gk : N ] is infinite, by passing to a finite extension of\nk corresponding to an open subgroup of Gk containing N , we may assume that\n[k : Qp] ≥ r. But this implies that the image of N in Gabk ⊗ Zp [which is of rank\n[k : Qp] + 1 ≥ r + 1] is of infinite index, hence that the image of N in any almost\npro-p-maximal quotient Gk \u0002 Q is of infinite index. Thus, by the elasticity of “Q”,\nwe conclude that such images are trivial. Since, moreover, the natural surjection\nGk \u0002 lim←−\nQ\nQ\n[where Q ranges over the almost pro-p-maximal quotients of Gk] is [by the definition\nof the term “almost pro-p-maximal quotient”] an isomorphism, this is enough to\nconclude that N is trivial, as desired. ©\n16 SHINICHI MOCHIZUKI\nSection 2: Semi-absolute Anabelian Geometry\nIn the present §2, we consider the problem of characterizing “group-theoretically”\nthe quotient morphism to the Galois group of the base field of the arithmetic fun-\ndamental group of a variety. In particular, the theory of the present §2 refines the\ntheory of [Mzk6], Lemma 1.1.4, in two respects: We extend this theory to the case of\nquite general varieties of arbitrary dimension [cf. Corollary 2.8], and, in the case of\nhyperbolic orbicurves, we give a “group-theoretic version” of the numerical criterion\nof [Mzk6], Lemma 1.1.4, via the theory of maximal pro-RTF-quotients developed in\n§1 [cf. Corollary 2.12]. The theory of the present §2 depends on the general theory\nof Albanese varieties, which we review in the Appendix, for the convenience of the\nreader.\nSuppose that:\n(1) k is a perfect field, k an algebraic closure of k, k̃ ⊆ k a solvably closed\nGalois extension of k, and Gk\ndef\n= Gal(k̃\/k).\n(2) X → Spec(k) is a geometrically connected, smooth, separated algebraic\nstack of finite type over k.\n(3) Y → X is a connected finite étale Galois covering which is a [necessarily\nseparated, smooth, and of finite type over k] k-scheme such that Gal(Y\/X)\nacts freely on some nonempty open subscheme of Y [so X is generically\nscheme-like — cf. §0].\n(4) Y ↪→ Y is an open immersion into a connected proper k-scheme Y such\nthat Y is the underlying scheme of a log scheme Y\nlog\nthat is log smooth over\nk [where we regard Spec(k) as equipped with the trivial log structure], and\nthe image of Y in Y coincides with the interior [cf. §0] of the log scheme\nY\nlog\n.\nThus, it follows from the log purity theorem [which is exposed, for instance, in\n[Mzk4] as “Theorem B”] that the condition that a finite étale covering Z → Y\nbe tamely ramified over the height one primes of Y is equivalent to the condition\nthat the normalization Z of Y in Z determine a log étale morphism Z\nlog → Y log\n[whose underlying morphism of schemes is Z → Y ]; in particular, one concludes\nimmediately that the condition that Z → Y be tamely ramified over the height\none primes of Y is independent of the choice of “log smooth log compactification”\nY\nlog\nfor Y . Thus, one verifies immediately [by considering the various Gal(Y\/X)-\nconjugates of the “log compactification” Y\nlog\n] that the finite étale coverings of X\nwhose pull-backs to Y are tamely ramified over [the height one primes of] Y form a\nGalois category, whose associated profinite group [relative to an appropriate choice\nof basepoint for X] we denote by πtame1 (X,Y ), or simply\nπtame1 (X)\nwhen Y → X is fixed. In particular, if we use the subscript “k” to denote base-\nchange from k to k, then by choosing a connected component of Y k, we obtain\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 17\na subgroup πtame1 (Xk) ⊆ πtame1 (X) which fits into a natural exact sequence 1 →\nπtame1 (Xk) → πtame1 (X) → Gal(k\/k) → 1.\nNext, let Σ ⊆ Primes be a set of prime numbers; πtame1 (Xk) \u0002 ΔX an al-\nmost pro-Σ-maximal quotient of πtame1 (Xk) whose kernel is normal in π\ntame\n1 (X),\nhence determines a quotient πtame1 (X) \u0002 ΠX ; we also assume that the quotient\nπtame1 (X) \u0002 Gal(Y\/X) admits a factorization πtame1 (X) \u0002 ΠX \u0002 Gal(Y\/X),\nand that the kernel of the resulting homomorphism ΔX → Gal(Y\/X) is pro-Σ.\nThus, Ker(ΔX → Gal(Y\/X)) may be identified with the maximal pro-Σ quotient\nof Ker(πtame1 (Xk) → Gal(Y\/X)); we obtain a natural exact sequence\n1 → ΔX → ΠX → Gal(k\/k) → 1\n— which may be thought of as an extension of the profinite group Gal(k\/k).\nDefinition 2.1.\n(i) We shall refer to any profinite group Δ which is isomorphic to the profinite\ngroup ΔX constructed in the above discussion for some choice of data (k,X, Y ↪→\nY ,Σ) as a profinite group of [almost pro-Σ] GFG-type [where “GFG” is to be under-\nstood as an abbreviation for “geometric fundamental group”]. In this situation, we\nshall refer to any surjection πtame1 (Xk) \u0002 Δ obtained by composing the surjection\nπtame1 (Xk) \u0002 ΔX with an isomorphism ΔX\n∼→ Δ as a scheme-theoretic envelope\nfor Δ; we shall refer to (k,X, Y ↪→ Y ,Σ) as a collection of construction data for\nΔ. [Thus, given a profinite group of GFG-type, there are, in general, many possible\nchoices of construction data for the profinite group.]\n(ii) We shall refer to any extension 1 → Δ → Π → G → 1 of profinite groups\nwhich is isomorphic to the extension 1 → ΔX → ΠX → Gal(k\/k) → 1 constructed\nin the above discussion for some choice of data (k,X, Y ↪→ Y ,Σ) as an extension\nof [geometrically almost pro-Σ] AFG-type [where “AFG” is to be understood as an\nabbreviation for “arithmetic fundamental group”]. In this situation, we shall refer\nto any surjection πtame1 (X) \u0002 Π (respectively, any surjection πtame1 (Xk) \u0002 Δ; any\nisomorphism Gal(k\/k)\n∼→ G) obtained by composing the surjection πtame1 (X) \u0002 ΠX\n(respectively, the surjection πtame1 (Xk) \u0002 ΔX ; the identity Gal(k\/k) = Gal(k\/k))\nwith an isomorphism ΠX\n∼→ Π (respectively, ΔX ∼→ Δ; Gal(k\/k) ∼→ G) arising\nfrom an isomorphism of the extensions 1 → Δ → Π → G → 1, 1 → ΔX → ΠX →\nGal(k\/k) → 1 as a scheme-theoretic envelope for Π (respectively, Δ; G); we shall\nrefer to (k,X, Y ↪→ Y ,Σ) as a collection of construction data for this extension.\n[Thus, given an extension of AFG-type, there are, in general, many possible choices\nof construction data for the extension.]\n(iii) Let 1 → Δ∗ → Π∗ → G∗ → 1 be an extension of AFG-type; N ⊆ G∗\nthe inverse image of the kernel of the quotient Gal(k\/k) \u0002 Gk relative to some\nscheme-theoretic envelope Gal(k\/k)\n∼→ G∗. Suppose further that Δ∗ is slim, and\nthat the outer action of N on Δ∗ [arising from the extension structure] is trivial.\nThus, every element of N ⊆ G∗ lifts to a unique element of Π∗ that commutes with\nΔ∗. In particular, N lifts to a closed normal subgroup NΠ ⊆ Π∗. We shall refer\nto any extension 1 → Δ → Π → G → 1 of profinite groups which is isomorphic\n18 SHINICHI MOCHIZUKI\nto an extension of the form 1 → Δ∗ → Π∗\/NΠ → G∗\/N → 1 just constructed as\nan extension of [geometrically almost pro-Σ] GSAFG-type [where “GSAFG” is to\nbe understood as an abbreviation for “geometrically slim arithmetic fundamental\ngroup”]. In this situation, we shall refer to any surjection πtame1 (X) \u0002 Π (respec-\ntively, πtame1 (Xk) \u0002 Δ; Gal(k\/k) \u0002 G) obtained by composing a scheme-theoretic\nenvelope πtame1 (X) \u0002 Π∗ (respectively, πtame1 (Xk) \u0002 Δ∗; Gal(k\/k)\n∼→ G∗) with\nthe surjection Π∗ \u0002 Π (respectively, Δ∗ \u0002 Δ; G∗ \u0002 G) in the above discus-\nsion as a scheme-theoretic envelope for Π (respectively, Δ; G); we shall refer to\n(k, k̃,X, Y ↪→ Y ,Σ) as a collection of construction data for this extension. [Thus,\ngiven an extension of GSAFG-type, there are, in general, many possible choices of\nconstruction data for the extension.]\n(iv) Given construction data “(k,X, Y ↪→ Y ,Σ)” or “(k, k̃,X, Y ↪→ Y ,Σ)” as\nin (i), (ii), (iii), we shall refer to “k” as the construction data field, to “X” as the\nconstruction data base-stack [or base-scheme, if X is a scheme], to “Y ” as the con-\nstruction data covering, to “Y ” as the construction data covering compactification,\nand to “Σ” as the construction data prime set. Also, we shall refer to a portion\nof the construction data “(k,X, Y ↪→ Y ,Σ)” or “(k, k̃,X, Y ↪→ Y ,Σ)” as in (i),\n(ii), (iii), as partial construction data. If every prime dividing the order of a finite\nquotient group of Δ is invertible in k, then we shall refer to the construction data\nin question as base-prime.\nThe following result is well-known, but we give the proof below for lack of an\nappropriate reference in the case where [in the notation of the above discussion] X\nis not necessarily proper.\nProposition 2.2. (Topological Finite Generation) Any profinite group Δ\nof GFG-type is topologically finitely generated.\nProof. Write (k,X, Y ↪→ Y ,Σ) for a choice of construction data for Δ. Since\na profinite fundamental group is topologically finitely generated if and only if it\nadmits an open subgroup that is topologically finitely generated, we may assume\nthat X = Y ; moreover, by applying de Jong’s theory of alterations [as reviewed,\nfor instance, in Lemma A.10 of the Appendix], we may assume that Y is projective\nand k-smooth, and that D\ndef\n= Y \\ Y is a divisor with normal crossings on Y . Since\nwe are only concerned with Δ, we may assume that k is algebraically closed, hence,\nin particular, infinite. Now suppose that dim(Y ) ≥ 2. Then since Y is smooth\nand projective [over k], it follows that there exists a connected, k-smooth closed\nsubscheme C ⊆ Y obtained by intersecting Y with a sufficiently general hyperplane\nsection H such that D\n⋂\nH forms a divisor with normal crossings on C. Write\nC\ndef\n= C\n⋂\nY ( \u000b= ∅). Now if Z → Y is any connected finite étale covering that is\ntamely ramified over D, then write Z → Y for the normalization of Y in Z. Thus,\nsince Z is tamely ramfied over D — so, by the log purity theorem reviewed above,\none may apply the well-known theory of log étale morphisms to describe the local\nstructure of Z → Y — and D intersects C transversely, it follows immediately\nthat ZC\ndef\n= Z ×Y C is normal. On the other hand, since the closed subscheme\nZC ⊆ Z arises as the zero locus of a nonzero section of an ample line bundle\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 19\non the normal scheme Z, it thus follows [cf., [SGA2], XI, 3.11; [SGA2], XII, 2.4]\nthat ZC is connected, hence [since ZC is normal] irreducible. But this implies\nthat ZC\ndef\n= Z ×Y C = ZC\n⋂\nY is connected. Moreover, this connectedness of ZC\nfor arbitrary choices of the covering Z → Y implies that the natural morphism\nπtame1 (C) → πtame1 (Y ) is surjective. Thus, by induction on dim(Y ), it suffices to\nprove Proposition 2.2 in the case where Y is a curve. But in this case, [as is well-\nknown] Proposition 2.2 follows by deforming Y ↪→ Y to a curve in characteristic\nzero, in which case the desired topological finite generation follows from the well-\nknown structure of the topological fundamental group of a Riemann surface of finite\ntype. ©\nProposition 2.3. (Slimness and Elasticity for Hyperbolic Orbicurves)\n(i) Let Δ be a profinite group of GFG-type that admits partial construction\ndata (k,X,Σ) [consisting of the construction data field, construction data base-\nstack, and construction data prime set] such that X is a hyperbolic orbicurve\n[cf. §0], and Σ contains a prime invertible in k. Then Δ is slim and elastic.\n(ii) Let 1 → Δ → Π → G → 1 be an extension of GSAFG-type that\nadmits partial construction data (k,X,Σ) [consisting of the construction data field,\nconstruction data base-stack, and construction data prime set] such that X is a\nhyperbolic orbicurve, Σ \u000b= ∅, and k is either an MLF or an NF. Then Π is\nslim, but not elastic.\nProof. Assertion (i) is the easily verified “generalization to orbicurves over fields\nof arbitrary characteristic” of [MT], Proposition 1.4; [MT], Theorem 1.5 [cf. also\nProposition 1.3, (i)]. The slimness portion of assertion (ii) follows immediately\nfrom the slimness portion of assertion (i), together with the slimness portion of\nTheorem 1.7, (ii), (iii); the fact that Π is not elastic follows from the existence of\nthe nontrivial, topologically finitely generated [cf. Proposition 2.2], closed, normal,\ninfinite index subgroup Δ ⊆ Π. ©\nDefinition 2.4. For i = 1, 2, let\n1 → Δi → Πi → Gi → 1\nbe an extension which is either of AFG-type or of GSAFG-type. Suppose that\nφ : Π1 → Π2\nis a continuous homomorphism of profinite groups. Then:\n(i) We shall say that φ is absolute if φ is open [i.e., has open image].\n(ii) We shall say that φ is semi-absolute (respectively, pre-semi-absolute) if φ\nis absolute, and, moreover, the image of φ(Δ1) in G2 is trivial (respectively, either\ntrivial or of infinite index in G2).\n20 SHINICHI MOCHIZUKI\n(iii) We shall say that φ is strictly semi-absolute (respectively, pre-strictly semi-\nabsolute) if φ is semi-absolute, and, moreover, the subgroup φ(Δ1) ⊆ Δ2 is open\n(respectively, either open or nontrivial).\nProposition 2.5. (First Properties of Absolute Homomorphisms) For\ni = 1, 2, let\n1 → Δi → Πi → Gi → 1\nbe an extension which is either of AFG-type or of GSAFG-type; (ki, Xi,Σi)\npartial construction data for Πi \u0002 Gi [consisting of the construction data field,\nconstruction data base-stack, and construction data prime set]. Suppose that\nφ : Π1 → Π2\nis a continuous homomorphism of profinite groups. Then:\n(i) The following implications hold:\nφ strictly semi-absolute =⇒ φ pre-strictly semi-absolute =⇒ φ semi-absolute\n=⇒ φ pre-semi-absolute =⇒ φ absolute.\n(ii) Suppose that k2 is an NF. Then “φ semi-absolute” ⇐⇒ “φ pre-semi-\nabsolute” ⇐⇒ “φ absolute”.\n(iii) Suppose that k2 is an MLF. Then “φ semi-absolute” ⇐⇒ “φ pre-semi-\nabsolute”.\n(iv) Suppose that k1 either an FF or an MLF; that X2 is a hyperbolic\norbicurve; and that Σ2 is of cardinality > 1. Then “φ pre-strictly semi-absolute”\n⇐⇒ “φ semi-absolute”.\n(v) Suppose that X2 is a hyperbolic orbicurve, and that Σ2 contains a\nprime invertible in k2. Then “φ strictly semi-absolute” ⇐⇒ “φ pre-strictly semi-\nabsolute”.\nProof. Assertion (i) follows immediately from the definitions. Since Δ1 is topo-\nlogically finitely generated [cf. Proposition 2.2], assertion (ii) (respectively, (iii))\nfollows immediately, in light of assertion (i), from the fact that G2 is very elastic\n[cf. Theorem 1.7, (iii)] (respectively, elastic [cf. Theorem 1.7, (ii)]). To verify\nassertion (iv), it suffices, in light of assertion (i), to consider the case where φ is\nsemi-absolute, but not pre-strictly semi-absolute. Then since Δ2 is elastic [cf. the\nhypothesis on Σ2; Proposition 2.3, (i)], and Δ1 is topologically finitely generated [cf.\nProposition 2.2], it follows that the subgroup φ(Δ1) ⊆ Δ2 is either open or trivial.\nSince φ is not pre-strictly semi-absolute, we thus conclude that φ(Δ1) = {1}, so φ\ninduces an open homomorphism G1 → Π2. That is to say, every sufficiently small\nopen subgroup Δ∗2 ⊆ Δ2 admits a surjection H1 \u0002 Δ∗2 for some closed subgroup\nH1 ⊆ G1. On the other hand, since X2 is a hyperbolic orbicurve, and Σ2 is of\ncardinality > 1, it follows [e.g., from the well-known structure of topological fun-\ndamental groups of hyperbolic Riemann surfaces of finite type] that we may take\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 21\nΔ∗2 such that Δ\n∗\n2 admits quotients Δ\n∗\n2 \u0002 F ′, Δ∗2 \u0002 F ′′, where F ′ (respectively,\nF ′′) is a nonabelian free pro-p′ (respectively, pro-p′′) group, for distinct p′, p′′ ∈ Σ2.\nBut this contradicts the well-known structure of G1, when k1 is either an FF or\nan MLF — i.e., the fact that G1, hence also H1, may be written as an extension\nof a meta-abelian group by a pro-p subgroup, for some prime p. [Here, we recall\nthat this fact is immediate if k1 is an FF, in which case G1 is abelian, and follows,\nfor instance, from [NSW], Theorem 7.5.2; [NSW], Corollary 7.5.6, (i), when k1 is a\nMLF.] Assertion (v) follows immediately from the elasticity of Δ2 [cf. Proposition\n2.3, (i)], together with the topological finite generation of Δ1 [cf. Proposition 2.2].\n©\nTheorem 2.6. (Field Types and Group-theoreticity) Let\n1 → Δ → Π → G → 1\nbe an extension which is either of AFG-type or of GSAFG-type; (k,X,Σ)\npartial construction data [consisting of the construction data field, construction\ndata base-stack, and construction data prime set] for Π \u0002 G. Suppose further that\nk is either an FF, an MLF, or an NF, and that every prime ∈ Σ is invertible\nin k. If H is a profinite group, j ∈ {1, 2}, and l ∈ Primes, write\nδ\nj\nl (H)\ndef\n= dimQl(H\nj(H,Ql)) ∈ N\n⋃ {∞}\n\u0007\nj\nl (Π)\ndef\n= supJ⊆Π {δjl (J)} ∈ N\n⋃ {∞}\nθj(Π)\ndef\n= {l | \u0007jl (Π) ≥ 3− j} ⊆ Primes\n[where J ranges over the open subgroups of Π]; also, we set\nζ(H)\ndef\n= sup\np,p′∈Primes\n{δ1p(H)− δ1p′(H)} ∈ Z\n⋃\n{∞}\nwhenever δ1l (H) < ∞, ∀l ∈ Primes. Then:\n(i) Suppose that k is an FF. Then Π is topologically finitely generated;\nthe natural surjections\nΠab-t \u0002 Gab-t; G \u0002 Gab-t\nare isomorphisms. In particular, the kernel of the quotient Π \u0002 G may be char-\nacterized [“group-theoretically”] as the kernel of the quotient Π \u0002 Πab-t [cf.\n[Tama1], Proposition 3.3, (ii), in the case of curves]. Moreover, for every open\nsubgroup H ⊆ Π, and every prime number l, δ1l (H) = 1.\n(ii) Suppose that k is an MLF of residue characteristic p. Then Π is topologi-\ncally finitely generated; in particular, for every open subgroup H ⊆ Π, and every\nprime number l, δ1l (H) is finite. Moreover, δ\n1\nl (G) = 1 if l \u000b= p, δ1p(G) = [k : Qp]+1;\nthe quantity\nδ1l (Π)− δ1l (G)\nis = 0 if l \/∈ Σ, and is independent of l if l ∈ Σ. Finally, \u00071p(Π) = ∞; in\nparticular, the cardinality of θ1(Π) is always ≥ 1.\n22 SHINICHI MOCHIZUKI\n(iii) Let k be as in (ii). Then θ2(Π) ⊆ Σ. If, moreover, the cardinality of θ1(Π)\nis ≥ 2, then θ2(Π) = Σ.\n(iv) Let k be as in (ii). Then every almost pro-omissive topologically finitely\ngenerated closed normal subgroup of Π is contained in Δ. If, moreover, Σ \u000b=\nPrimes, then the kernel of the quotient Π \u0002 G may be characterized [“group-\ntheoretically”] as the maximal almost pro-omissive topologically finitely gen-\nerated closed normal subgroup of Π.\n(v) Let k be as in (ii). If θ2(Π) \u000b= Primes, then write\nΘ ⊆ Π\nfor the maximal almost pro-omissive topologically finitely generated closed nor-\nmal subgroup of Π, whenever a unique such maximal subgroup exists; if θ2(Π) =\nPrimes, or there does not exist a unique such maximal subgroup, set Θ\ndef\n= {1} ⊆ Π.\nThen\nζ(Π)\ndef\n= ζ(Π\/Θ) = [k : Qp]\n[cf. the finiteness portion of (ii)]. In particular, the kernel of the quotient Π \u0002 G\nmay be characterized [“group-theoretically” — since “θ2(−)”, “ζ(−)”, “ζ(−)”\nare “group-theoretic”] as the intersection of the open subgroups H ⊆ Π such that\nζ(H)\/ζ(Π) = [Π : H].\n(vi) Suppose that k is an NF. Then the natural surjection Πab-t \u0002 Gab-t is an\nisomorphism. The kernel of the quotient Π \u0002 G may be characterized [“group-\ntheoretically”] as the maximal topologically finitely generated closed normal sub-\ngroup of Π. In particular, Π is not topologically finitely generated.\nProof. Write X → A for the Albanese morphism associated to X. [We refer to\nthe Appendix for a review of the theory of Albanese varieties — cf., especially,\nCorollary A.11, Remark A.11.2.] Thus, A is a torsor over a semi-abelian variety\nover k such that the morphism X → A induces an isomorphism\nΔab-t ⊗ Zl ∼→ Tl(A)\nonto the l-adic Tate module Tl(A) of A for all l ∈ Σ. Note, moreover, that for l ∈ Σ,\nthe quotient of Δ determined by the image of Δ in the pro-l completion of Πab-t\nfactors through the quotient\nΔ \u0002 Δab-t ⊗ Zl ∼→ Tl(A) \u0002 Tl(A)\/G\n— where we use the notation “\/G” to denote the maximal torsion-free quotient on\nwhich G acts trivially.\nNext, whenever k is an MLF, let us write, for l ∈ Σ,\nΔab-t \u0002 Δab-t ⊗ Zl ∼→ Tl(A) \u0002 Rl def= R⊗ Zl \u0002 Ql def= Q⊗ Zl\nfor the pro-l portion of the quotients T (A) \u0002 R \u0002 Q of Lemma 2.7, (i), (ii), below\n[in which we take “k” to be k and “B” to be the semi-abelian variety over which\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 23\nA is a torsor]. Here, we observe that Ql is simply the quotient Tl(A)\/G considered\nabove. Thus, the Zl-ranks of Rl, Ql are independent of l ∈ Σ.\nThe topological finite generation portion of assertion (i) follows immediately\nfrom the fact that G ∼= Ẑ, together with the topological finite generation of Δ [cf.\nProposition 2.2]. The remainder of assertion (i) follows immediately from the fact\nthat Tl(A)\/G = 0 [a consequence of the “Riemann hypothesis for abelian varieties\nover finite fields” — cf., e.g., [Mumf], p. 206]. In a similar vein, assertion (vi)\nfollows immediately from the fact that Tl(A)\/G = 0 [again a consequence of the\n“Riemann hypothesis for abelian varieties over finite fields”], together with the fact\nthat G is very elastic [cf. Theorem 1.7, (iii)].\nTo verify assertion (ii), let us first observe that the topological finite generation\nof Π follows from that of Δ [cf. Proposition 2.2], together with that of G [cf. [NSW],\nTheorem 7.5.10]. Next, let us recall the well-known fact that\nδ1l (G) = 1 if l \u000b= p, δ1p(G) = [k : Qp] + 1\n[cf. our discussion of local class field theory in the proofs of Proposition 1.5; Theorem\n1.7, (ii)]; in particular, ζ(G) = [k : Qp]. Moreover, the existence of a rational point\nof A over some finite extension of k [which determines a Galois section of the étale\nfundamental group of A over some open subgroup of G] implies that\nδ1l (Π) = δ\n1\nl (G) + dimQl(Ql ⊗Ql)\n[where we recall that dimQl(Ql ⊗Ql) is independent of l] for l ∈ Σ, δ1l (Π) = δ1l (G)\nfor l \/∈ Σ. Thus, by considering extensions of k of arbitrarily large degree, we obtain\nthat \u00071p(Π) = ∞. This completes the proof of assertion (ii).\nNext, we consider assertion (iii). First, let us consider the “E2-term” of the\nLeray spectral sequence of the group extension 1 → Δ → Π → G → 1. Since G is\nof cohomological dimension 2 [cf., e.g., [NSW], Theorem 7.1.8, (i)], and δ2l (G) = 0\nfor all l ∈ Primes [cf., e.g., [NSW], Theorem 7.2.6], the spectral sequence yields an\nequality δ2l (Π) = 0 if l \/∈ Σ, and a pair of injections\nH1(G,Hom(Rl,Ql)) ↪→ H1(G,Hom(Δab-t,Ql)) ↪→ H2(Π,Ql)\nif l ∈ Σ [cf. Lemma 2.7, (iii), below]. By applying the analogue of this conclusion\nfor an arbitrary open subgroup H ⊆ Π, we thus obtain that δ2l (H) = 0 if l \/∈ Σ,\ni.e., that \u00072l (Π) = 0 if l \/∈ Σ; this already implies that if l \/∈ Σ, then l \/∈ θ2(Π),\ni.e., that θ2(Π) ⊆ Σ. If the cardinality of θ1(Π) is ≥ 2, then there exists some\nopen subgroup H ⊆ Π and some l ∈ Primes such that δ1l (H) ≥ 2, l \u000b= p. Now\nwe may assume without loss of generality that H acts trivially on the quotient\nR; also to simplify notation, we may replace Π by H and assume that H = Π.\nThen [since δ1l (G) = 1, by assertion (ii)] the fact that δ\n1\nl (Π) ≥ 2 implies that\nl ∈ Σ, and dimQl(Rl ⊗Ql) ≥ 1 [cf. our computation in the proof of assertion (ii)].\nBut this implies that for any l′ ∈ Σ, we have dimQl′ (Rl′ ⊗ Ql′) ≥ 1, hence that\nH1(G,Hom(Rl′ ,Ql′)) = H\n1(G,Ql′) ⊗ Hom(Rl′ ,Ql′) \u000b= 0. Thus, by the injections\ndiscussed above, we conclude that \u00072l′(Π) ≥ δ2l′(Π) ≥ 1, so l′ ∈ θ2(Π). This completes\nthe proof of assertion (iii).\n24 SHINICHI MOCHIZUKI\nAssertion (iv) follows immediately from the existence of a surjection G \u0002 Ẑ\n[cf., e.g., Proposition 1.5, (ii)], together with the elasticity of G [cf. Theorem 1.7,\n(ii)], and the topological finite generation of Δ [cf. Proposition 2.2].\nNext, we consider assertion (v). First, let us observe that whenever Σ =\nPrimes, it follows from assertion (ii) that ζ(Π) = ζ(G) = [k : Qp].\nNow we consider the case θ2(Π) = Primes. In this case, Θ = {1} [by def-\ninition], and θ2(Π) = Σ = Primes [by assertion (iii)]. Thus, we obtain that\nζ(Π) = ζ(Π\/Θ) = [k : Qp], as desired [cf. [Mzk6], Lemma 1.1.4, (ii)]. Next,\nwe consider the case θ1(Π) \u000b= {p} [i.e., θ1(Π) is of cardinality ≥ 2 — cf. as-\nsertion (ii)], θ2(Π) \u000b= Primes. In this case, by assertion (iii), we conclude that\nΣ = θ2(Π) \u000b= Primes. Thus, by assertion (iv), Θ = Δ, so ζ(Π\/Θ) = ζ(G) = [k : Qp],\nas desired.\nFinally, we consider the case θ1(Π) = {p} [i.e., θ1(Π) is of cardinality one],\nθ2(Π) \u000b= Primes. If Σ \u000b= Primes, then it follows from the definition of Θ, together\nwith assertion (iv), that Θ = Δ, hence that ζ(Π\/Θ) = ζ(G) = [k : Qp], as desired.\nIf, on the other hand, Σ = Primes, then since θ1(Π) = {p}, it follows [cf. the\ncomputation in the proof of assertion (ii)] that dimQl(Ql ⊗ Ql) = 0 for all primes\nl \u000b= p, hence that dimQp(Qp ⊗ Qp) = 0; but this implies that δ1l (Π) = δ1l (G) for\nall l ∈ Primes. Now since Θ ⊆ Δ [by assertion (iv)], it follows that δ1l (Π) ≥\nδ1l (Π\/Θ) ≥ δ1l (G) for all l ∈ Primes, so we obtain that δ1l (Π) = δ1l (Π\/Θ) = δ1l (G)\nfor all l ∈ Primes. But this implies that ζ(Π) = ζ(Π\/Θ) = ζ(G) = [k : Qp], as\ndesired. This completes the proof of assertion (v). ©\nRemark 2.6.1. When [in the notation of Theorem 2.6] X is a smooth proper\nvariety, the portion of Theorem 2.6, (ii), concerning “δ1l (Π)− δ1l (G)” is essentially\nequivalent to the main result of [Yoshi].\nLemma 2.7. (Combinatorial Quotients of Tate Modules) Suppose that\nk is an MLF [so k = k̃]. Let B be a semi-abelian variety over k. Write\nT (B)\ndef\n= Hom(Q\/Z, B(k))\nfor the Tate module of B. Then:\n(i) The maximal torsion-free quotient module T (B) \u0002 Q of T (B) on which\nGk acts trivially is a finitely generated free Ẑ-module.\n(ii) There exists a quotient Gk-module T (B) \u0002 R such that the following\nproperties hold: (a) R is a finitely generated free Ẑ-module; (b) the action of Gk\non R factors through a finite quotient; (c) no nonzero torsion-free subquotient S of\nthe Gk-module N\ndef\n= Ker(T (B) \u0002 R) satisfies the property that the resulting action\nof Gk on S factors through a finite quotient.\n(iii) If R is as in (ii), then the natural map\nH1(Gk,Hom(R, Ẑ)) → H1(Gk,Hom(T (B), Ẑ))\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 25\nis injective.\nProof. Assertion (i) is literally the content of [Mzk6], Lemma 1.1.5. Assertion\n(ii) follows immediately from the proof of [Mzk6], Lemma 1.1.5 [more precisely, the\n“combinatorial” quotient “Tcom” of loc. cit.]. Assertion (iii) follows by considering\nthe long exact cohomology sequence associated to the short exact sequence 0 →\nHom(R, Ẑ) → Hom(T (B), Ẑ) → Hom(N, Ẑ) → 0, since the fact that N has no\nnonzero torsion-free subquotients on which Gk acts through a finite quotient implies\nthat H0(Gk,Hom(N, Ẑ)) = 0. ©\nCorollary 2.8. (Field Types and Absolute Homomorphisms) For i = 1, 2,\nlet 1 → Δi → Πi → Gi → 1, ki, Xi, Σi, φ : Π1 → Π2 be as in Proposition 2.5.\nSuppose further that ki is either an FF, an MLF, or an NF, and that every prime\n∈ Σi is invertible in ki. Then:\n(i) Suppose further that φ is absolute. Then the field type of k1 is ≥ [cf.\n§0] the field type of k2. If, moreover, it holds either that both k1 and k2 are FF’s\nor that both k1 and k2 are NF’s, then φ is semi-absolute, i.e., φ(Δ1) ⊆ Δ2.\n(ii) Suppose further that φ is an isomorphism. Then the field types of k1,\nk2 coincide, and φ is strictly semi-absolute, i.e., φ(Δ1) = Δ2. If, moreover,\nfor i = 1, 2, ki is an MLF of residue characteristic pi, then p1 = p2.\nProof. Assertion (i) concerning the inequality “≥” follows immediately from the\ntopological finite generation portions of Theorem 2.6, (i), (ii), (vi), together with\nthe estimates of “δ1l (−)”, “\u00071l (−)” in Theorem 2.6, (i), (ii). The final portion of\nassertion follows, in the case of FF’s, from Theorem 2.6, (i), and, in the case of\nNF’s, from Proposition 2.5, (ii). Next, we consider assertion (ii). The fact that\nthe field types of k1, k2 coincide follows from assertion (i) applied to φ, φ\n−1. To\nverify that φ is strictly semi-absolute, let us first observe that every semi-absolute\nisomorphism whose inverse is also semi-absolute is necessarily strictly semi-absolute.\nThus, since the inverse to φ satisfies the same hypotheses as φ, to complete the\nproof of Corollary 2.8, it suffices to verify that φ is semi-absolute. If k1, k2 are\nFF’s (respectively, MLF’s; NF’s), then this follows immediately from the “group-\ntheoretic” characterizations of Πi \u0002 Gi in Theorem 2.6, (i) (respectively, Theorem\n2.6, (v); Theorem 2.6, (vi)). Finally, if, for i = 1, 2, ki is an MLF of residue\ncharacteristic pi, then since φ induces an isomorphism G1\n∼→ G2, the fact that\np1 = p2 follows, for instance, from [Mzk6], Proposition 1.2.1, (i). ©\nRemark 2.8.1. In the situation of Corollary 2.8, suppose further that k2 is an\nMLF of residue characteristic p2; that X2 is a hyperbolic orbicurve; that Σ2 ⊆ {p2}\n[cf. Proposition 2.5, (iv)]; and that if Σ2 = ∅, then k1 is an NF. Then it is not clear\nto the author at the time of writing [but of interest in the context of the theory of\nthe present §2!] whether or not there exists a continuous surjective homomorphism\nG1 \u0002 Π2\n[in which case, by Corollary 2.8, (i), k1 is either an NF or an MLF].\n26 SHINICHI MOCHIZUKI\nThe general theory discussed so far for arbitrary X becomes substantially sim-\npler and more explicit, when X is a hyperbolic orbicurve.\nDefinition 2.9. Let G be a profinite group. Then we shall refer to as an aug-free\ndecomposition of G any pair of closed subgroups H1, H2 ⊆ G that determine an\nisomorphism of profinite groups\nH1 ×H2 ∼→ G\nsuch that H1 is a slim, topologically finitely generated, augmented pro-prime [cf.\nDefinition 1.1, (iii)] profinite group, and H2 is either trivial or a nonabelian pro-Σ-\nsolvable free group for some set Σ ⊆ Primes of cardinality ≥ 2. In this situation,\nwe shall refer to H1 as the augmented subgroup of this aug-free decomposition and\nto H2 as the free subgroup of this aug-free decomposition. If G admits an aug-free\ndecomposition, then we shall say that G is of aug-free type. If G is of aug-free type,\nwith nontrivial free subgroup, then we shall say that G is of strictly aug-free type.\nProposition 2.10. (First Properties of Aug-free Decompositions) Let\nH1 ×H2 ∼→ G\nbe an aug-free decomposition of a profinite group G, in which H1 is the aug-\nmented subgroup, and H2 is the free subgroup. Then:\n(i) Let J be a topologically finitely generated, augmented pro-prime\ngroup; φ : J → G a continuous homomorphism of profinite groups such that φ(J)\nis normal in some open subgroup of G. Then φ(J) ⊆ H1.\n(ii) Aug-free decompositions are unique — i.e., if J1 × J2 ∼→ G is any aug-\nfree decomposition of G, in which J1 is the augmented subgroup, and J2 is the free\nsubgroup, then J1 = H1, J2 = H2.\nProof. First, we consider assertion (i). Suppose that φ(J) is not contained in H1.\nThen the image I ⊆ H2 of φ(J) via the projection to H2 is a nontrivial, topologically\nfinitely generated closed subgroup which is normal in an open subgroup of H2. Since\nH2 is elastic [cf. [MT], Theorem 1.5], it follows that I is open in H2, hence that\nI is a nonabelian pro-Σ-solvable free group for some set Σ ⊆ Primes of cardinality\n≥ 2. On the other hand, since I is a quotient of the augmented pro-prime group J ,\nit follows that there exists a p ∈ Primes such that the maximal pro-( \u000b= p) quotient\nof I is abelian. But this implies that Σ ⊆ {p}, a contradiction. Next, we consider\nassertion (ii). By assertion (i), J1 ⊆ H1, H1 ⊆ J1. Thus, H1 = J1. Now since\nH1 = J1 is slim, it follows that the centralizer ZH1(G) (respectively, ZJ1(G)) is\nequal to H2 (respectively, J2), so H2 = J2, as desired. ©\nTheorem 2.11. (Maximal Pro-RTF-quotients for Hyperbolic Orbi-\ncurves) Let\n1 → Δ → Π → G → 1\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 27\nbe an extension of AFG-type; (k,X,Σ) partial construction data [consisting\nof the construction data field, construction data base-stack, and construction data\nprime set] for Π \u0002 G. Suppose that k is an MLF of residue characteristic p; X is\na hyperbolic orbicurve; Σ \u000b= ∅. For l ∈ Primes, write\nΠ[l] ⊆ Π\nfor the maximal almost pro-l topologically finitely generated closed normal sub-\ngroup of Π, whenever a unique such maximal subgroup exists; if there does not exist\na unique such maximal subgroup, then set Π[l]\ndef\n= {1}.\nIn the following, we shall use a subscript “G” to denote the quotient of a\nclosed subgroup of Π determined by the quotient Π \u0002 G; we shall use the super-\nscript “RTF” to denote the maximal pro-RTF-quotient and the superscripts\n“RTF-aug”, “RTF-free” to denote the augmented and free subgroups of the max-\nimal pro-RTF-quotient whenever this maximal pro-RTF-quotient is of aug-free\ntype. Then:\n(i) Suppose that Π[l] \u000b= {1} for some l ∈ Primes. Then Π[l] = Δ, Σ = {l};\nΠ[l′] = {1} for all l′ ∈ Primes such that l′ \u000b= l.\n(ii) Suppose that Π[l] = {1} for all l ∈ Primes. Then Σ is of cardinality ≥ 2.\nMoreover, for every open subgroup J ⊆ Π, there exists an open subgroup H ⊆ J\nwhich is characteristic as a subgroup of Π such that HRTF is of aug-free type.\nIn particular, [cf. Proposition 2.10, (ii)] the subquotients HRTF-aug, HRTF-free of Π\nare characteristic.\n(iii) Suppose that Π[l] = {1} for all l ∈ Primes. Suppose, moreover, that\nH ⊆ Π is an open subgroup that corresponds to a finite étale covering Z → X,\nwhere Z is a hyperbolic curve, defined over a finite extension kZ of k such\nthat Z has stable reduction [cf. §0] over the ring of integers OkZ of kZ ; that\nZ(kZ) \u000b= ∅; that the dual graph ΓZ of the geometric special fiber of the resulting\nmodel [cf. §0] over OkZ has either trivial or nonabelian topological fundamental\ngroup; and that the Galois action of G on ΓZ is trivial. Thus, the finite Galois\ncoverings of the graph ΓZ of degree a product of primes ∈ Σ determine a pro-Σ\n“combinatorial” quotient H \u0002 ΔcomH ; write ΔcomH \u0002 Δcom-solH for the maximal\npro-solvable quotient of ΔcomH . Then the quotient\nH \u0002 HRTFG ×Δcom-solH\nmay be identified with the maximal pro-RTF-quotient H \u0002 HRTF of H; more-\nover, this product decomposition determines an aug-free decomposition of HRTF.\nFinally, for any open subgroup J ⊆ Π, there exists an open subgroup H ⊆ J which\nis characteristic as a subgroup of Π and, moreover, satisfies the above hypotheses\non “H”.\n(iv) Suppose that Π[l] = {1} for all l ∈ Primes. Let H ⊆ J ⊆ Π be open\nsubgroups of Π such that HRTF, JRTF are of aug-free type. Then we have iso-\nmorphisms\nJRTF-aug\n∼→ JRTFG ; JRTF-free ∼→ Ker(JRTF \u0002 JRTFG )\n28 SHINICHI MOCHIZUKI\n[arising from the natural morphisms involved]; the open homomorphism HRTF →\nJRTF induced by φ maps HRTF-aug (respectively, HRTF-free) onto an open subgroup\nof JRTF-aug (respectively, JRTF-free).\nProof. Since Δ is elastic [cf. Proposition 2.3, (i)], every nontrivial topologically\nfinitely generated closed normal subgroup of Δ is open, hence almost pro-Σ′ for\nΣ′ ⊆ Primes if and only if Σ′ ⊇ Σ. Also, let us observe that by Theorem 2.6, (iv),\nΠ[l] ⊆ Δ for all l ∈ Primes. Thus, if Π[l] \u000b= {1} for some l ∈ Primes, then it follows\nthat Σ = {l}, Π[l] = Δ, and that Π[l′] is finite, hence trivial [since Δ is slim — cf.\nProposition 2.3, (i)] for primes l′ \u000b= l. Also, we observe that if Σ is of cardinality\none, i.e., Σ = {l} for some l ∈ Primes, then Δ = Π[l] \u000b= {1} [cf. Theorem 2.6,\n(iv)]. This completes the proof of assertion (i), as well as of the portion of assertion\n(ii) concerning Σ. Also, we observe that the remainder of assertion (ii) follows\nimmediately from assertion (iii).\nNext, we consider assertion (iii). Suppose that H ⊆ Π satisfies the hypotheses\ngiven in the statement of assertion (iii); write ΔH\ndef\n= Δ\n⋂\nH. Thus, one has the\nquotient H \u0002 ΔcomH , where ΔcomH is either trivial or a nonabelian pro-Σ free group,\nand Σ is of cardinality ≥ 2 [cf. the portion of assertion (ii) concerning Σ]. Write\nΔabH = Δ\nab-t\nH \u0002 R for the maximal pro-Σ quotient of the quotient “R” of Lemma\n2.7, (ii), associated to the Albanese variety of Z.\nNow I claim that the quotient ΔH \u0002 R coincides with the quotient ΔH \u0002\n(ΔcomH )\nab. First, let us observe that by the definition of R [cf. Lemma 2.7, (ii)], it\nfollows that the quotient ΔH \u0002 (ΔcomH )ab factors through the quotient ΔH \u0002 R.\nIn particular, since, for l ∈ Σ, the modules R⊗Zl, (ΔcomH )ab⊗Zl are Zl-free modules\nof rank independent of l ∈ Σ [cf. Lemma 2.7, (ii); the fact that ΔcomH is pro-Σ free],\nit suffices to show that these two ranks are equal, for some l ∈ Σ. Moreover, let\nus observe that for the purpose of verifying this claim, we may enlarge Σ. Thus,\nit suffices to show that the two ranks are equal for some l ∈ Σ such that l \u000b= p.\nBut then the claim follows immediately from the [well-known] fact that by the\n“Riemann hypothesis for abelian varieties over finite fields” [cf., e.g., [Mumf], p.\n206], all powers of the Frobenius element in the absolute Galois group of the residue\nfield of k act with eigenvalues \u000b= 1 on the pro-l abelianizations of the fundamental\ngroups of the geometric irreducible components of the smooth locus of the special\nfiber of the stable model of Z over OkZ . This completes the proof of the claim.\nNow let us write H \u0002 Hcom for the quotient of H by Ker(ΔH \u0002 ΔcomH ). Then\nby applying the above claim to various open subgroups of H, we conclude that\nthe quotient H \u0002 HRTF factors through the quotient H \u0002 Hcom [i.e., we have a\nnatural isomorphism HRTF\n∼→ (Hcom)RTF]. On the other hand, since Z(kZ) \u000b= ∅,\nit follows that H \u0002 HG, hence also Hcom \u0002 HG admits a section s : HG → Hcom\nwhose image lies in the kernel of the quotient Hcom \u0002 ΔcomH [cf. the proof of\n[Mzk3], Lemma 1.4]. In particular, we conclude that the conjugation action of HG\non ΔcomH\n∼= Ker(Hcom \u0002 HG) ⊆ Hcom arising from s is trivial. Thus, s determines\na direct product decomposition\nHcom\n∼→ HG ×ΔcomH\n—hence a direct product decompositionHRTF\n∼→ (Hcom)RTF ∼→HRTFG ×(ΔcomH )RTF.\nMoreover, since ΔcomH is either trivial or nonabelian pro-Σ free, it follows immedi-\nately that the quotient ΔcomH \u0002 (ΔcomH )RTF may be identified with the quotient\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 29\nΔcomH \u0002 Δcom-solH , where Δcom-solH is either trivial or nonabelian pro-Σ-solvable free.\nSince HRTFG is slim, augmented pro-prime, and topologically finitely generated [cf.\nProposition 1.5, (i), (ii); Theorem 2.6, (ii)], we thus conclude that we have obtained\nan aug-free decomposition of HRTF, as asserted in the statement of assertion (iii).\nFinally, given an open subgroup J ⊆ Π, the existence of an open subgroup\nH ⊆ J which satisfies the hypotheses on “H” in the statement of assertion (iii)\nfollows immediately from well-known facts concerning stable curves over discretely\nvalued fields [cf., e.g., the “stable reduction theorem” of [DM]; the fact that Σ \u000b= ∅,\nso that one may assume that ΓZ is as large as one wishes by passing to admissible\ncoverings]. The fact that one can choose H to be characteristic follows immediately\nfrom the characteristic nature of Δ [cf., e.g., Corollary 2.8, (ii)], together with the\nfact that Δ, Π are topologically finitely generated [cf., e.g. Proposition 2.2; Theorem\n2.6, (ii)]. This completes the proof of assertion (iii).\nFinally, we consider assertion (iv). First, we observe that since the augmented\nand free subgroups of any aug-free decomposition are slim [cf. Definition 2.9; [MT],\nProposition 1.4], hence, in particular, do not contain any nontrivial closed normal fi-\nnite subgroups, we may always replaceH by an open subgroup ofH that satisfies the\nsame hypotheses as H. In particular, we may assume that H is an open subgroup\n“H” as in assertion (iii) [which exists, by assertion (iii)]. Then by Proposition 2.10,\n(i), the image of HRTF-aug in JRTF is contained in JRTF-aug, so we obtain a mor-\nphism HRTF-aug → JRTF-aug. By assertion (iii), HRTF-free = Ker(HRTF \u0002 HRTFG ),\nand the natural morphism HRTF-aug → HRTFG is an isomorphism. Since HG → JG,\nhence also HRTFG → JRTFG , is clearly an open homomorphism, we thus conclude\nthat the natural morphism HRTF-aug → JRTFG , hence also the natural morphism\nJRTF-aug → JRTFG , is open. Thus, the image of JRTF-free in JRTFG commutes with an\nopen subgroup of JRTFG [i.e., the image of J\nRTF-aug in JRTFG ], so by the slimness of\nJRTFG [cf. Proposition 1.5, (i)], we conclude that J\nRTF-free ⊆ Ker(JRTF \u0002 JRTFG ).\nIn particular, we obtain a surjection JRTF-aug \u0002 JRTFG , hence an exact sequence\n1 → N → JRTF-aug → JRTFG → 1\n— where we write N\ndef\n= Ker(JRTF-aug \u0002 JRTFG ) ⊆ JRTF-aug ⊆ JRTF. Note,\nmoreover, that since JRTFG is an augmented pro-p group [cf. Proposition 1.5, (ii)]\nwhich admits a surjection JRTFG \u0002 Zp × Zp [cf. the computation of “δ1p(−)” in\nTheorem 2.6, (ii)], it follows immediately that [the augmented pro-prime group]\nJRTF-aug is an augmented pro-p group whose augmentation factors through JRTFG ;\nin particular, we conclude thatN is pro-p. Also, we observe that since the composite\nHRTF-free → HRTFG → JRTFG is trivial, it follows that the projection under the\nquotient JRTF \u0002 JRTF-aug of the image of HRTF-free in JRTF is contained in N .\nNow I claim that to complete the proof of assertion (iv), it suffices to verify\nthat N = {1} [or, equivalently, since JRTF-aug is slim, that N is finite]. Indeed,\nif N = {1}, then we obtain immediately the isomorphisms JRTF-aug ∼→ JRTFG ,\nJRTF-free\n∼→ Ker(JRTF \u0002 JRTFG ). Moreover, by the above discussion, if N = {1},\nthen it follows that the image of HRTF-free in JRTF is contained in JRTF-free. Since\nthe homomorphism HRTF → JRTF is open, this implies that the open homomor-\nphism HRTF \u0002 JRTF induced by φ maps HRTF-aug (respectively, HRTF-free) onto\nan open subgroup of JRTF-aug (respectively, JRTF-free), as desired. This completes\nthe proof of the claim.\n30 SHINICHI MOCHIZUKI\nNext, let J ⊆ J be an open subgroup that arises as the inverse image in J\nof an [open] RTF-subgroup JG ⊆ JG [so the notation “JG” does not lead to any\ncontradictions]. Then one verifies immediately from the definitions that any RTF-\nsubgroup of JG (respectively, J) determines an RTF-subgroup of JG (respectively,\nJ). Thus, the natural morphisms\nJRTFG → JRTFG ; JRTF → JRTF\nare injective. Moreover, the subgroups JRTF-aug\n⋂\nJRTF, JRTF-free of JRTF clearly\ndetermine an aug-free decomposition of JRTF. Thus, from the point of view of\nverifying the finiteness of N , we may replace J by J [and H by an appropriate\nsmaller open subgroup contained in J and satisfying the hypotheses of the “H” of\n(iii)]. In particular, since — by the definition of “RTF” and of the subgroup N ! —\nthere exists a J such that N ⊆ JRTF-aug has nontrivial image in (JRTF-aug)ab-t, we\nmay assume without loss of generality thatN has nontrivial image in (JRTF-aug)ab-t.\nThus, we have\n(δ1p(J) ≥) δ1p(JRTF-aug) > δ1p(JRTFG ) = δ1p(JG)\n[cf. the notation of Theorem 2.6], i.e., sJ\ndef\n= δ1p(J\nRTF-aug) − δ1p(JRTFG ) > 0. By\nTheorem 2.6, (ii), this already implies that p ∈ Σ.\nIn a similar vein, let J ⊆ J be an open subgroup that arises as the inverse\nimage in J of an [open] RTF-subgroup JRTF-free ⊆ JRTF-free. Then one verifies\nimmediately from the definitions that any RTF-subgroup of J determines an RTF-\nsubgroup of J . Thus, the natural morphism JRTF → JRTF is injective, with image\nequal to JRTF-aug×JRTF-free. Moreover, the subgroups JRTF-aug, JRTF-free of JRTF\nclearly determine an aug-free decomposition of JRTF [so the notation “JRTF-free”\ndoes not lead to any contradictions]. Since [by the above discussion applied to\nJ instead of J ] JRTF-free maps to the identity in JRTFG , we thus obtain a quotient\nJRTF \u0002 JRTF-aug = JRTF-aug \u0002 JRTFG , hence a quotient JRTF \u0002 JRTF-aug \u0002 JRTFG\nin which the image of J\n⋂\nΔ is a finite normal closed subgroup, hence trivial [since\nJRTFG is slim — cf. Proposition 1.5, (i)]. That is to say, the surjection J \u0002\nJRTF-aug \u0002 JRTFG to the pro-RTF-group JRTFG factors through JG, hence through\nJRTFG . Thus, we obtain a surjection J\nRTF\nG \u0002 JRTFG whose composite JRTFG \u0002\nJRTFG → JRTFG with the natural morphism induced by the inclusion J ↪→ J is the\nidentity [since JRTFG is slim [cf. Proposition 1.5, (i)], and all of these maps “lie\nunder a fixed J”]. But this implies that the natural morphism JRTFG → JRTFG is an\nisomorphism. In particular, we have an isomorphism of kernels Ker(JRTF-aug \u0002\nJRTFG )\n∼→ Ker(JRTF-aug \u0002 JRTFG ). Thus, from the point of view of verifying the\nfiniteness of N , we may replace J by J [and H by an appropriate smaller open\nsubgroup contained in J and satisfying the hypotheses of the “H” of (iii)]. In\nparticular, since JRTF-aug\n∼→ JRTF-aug, we may assume without loss of generality\nthat the rank rJ of the pro-ΣJ -solvable free group J\nRTF-free [for some subset ΣJ ⊆\nPrimes of cardinality ≥ 2] is either 0 or > δ1p(JRTF-aug). In particular, if l ∈ ΣJ ,\nthen either rJ = 0 or rJ = δ\n1\nl (J\nRTF-free) > δ1p(J\nRTF-aug) ≥ sJ .\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 31\nNow we compute: Since Σ is of cardinality ≥ 2, let l ∈ Σ be a prime \u000b= p. Then:\nδ1l (J\nRTF-free) = δ1l (J\nRTF-free) + δ1l (J\nRTF-aug)− δ1l (JRTFG )\n= δ1l (J\nRTF)− δ1l (JRTFG ) = δ1l (J)− δ1l (JG)\n= δ1p(J)− δ1p(JG) = δ1p(JRTF)− δ1p(JRTFG )\n= δ1p(J\nRTF-free) + δ1p(J\nRTF-aug)− δ1p(JRTFG ) = δ1p(JRTF-free) + sJ\n— where we apply the “independence of l” of Theorem 2.6, (ii). Thus, we conclude\nthat sJ = δ\n1\nl (J\nRTF-free) − δ1p(JRTF-free) — where δ1l (JRTF-free), δ1p(JRTF-free) ∈\n{0, rJ} [depending on whether or not l, p belong to ΣJ ] — is a positive integer. But\nthis implies that 0 < sJ ∈ {0, rJ ,−rJ}, hence that sJ = rJ > 0 — in contradiction\nto the inequality sJ < rJ [which holds if rJ > 0]. This completes the proof of\nassertion (iv). ©\nRemark 2.11.1. One way of thinking about the content of Theorem 2.11,\n(iv), is that it asserts that “aug-free decompositions of maximal pro-RTF-\nquotients play an analogous [though somewhat more complicated] role for absolute\nGalois groups of MLF’s to the role played by torsion-free abelianizations for\nabsolute Galois groups of FF’s” [cf. Theorem 2.6, (i)].\nCorollary 2.12. (Group-theoretic Semi-absoluteness via Maximal Pro-\nRTF-quotients) For i = 1, 2, let 1 → Δi → Πi → Gi → 1, ki, Xi, Σi, φ :\nΠ1 → Π2 be as in Proposition 2.5. Suppose further that ki is an MLF; Xi is a\nhyperbolic orbicurve; Σi \u000b= ∅. Also, for i = 1, 2, let us write\nΘi ⊆ Πi\nfor the maximal almost pro-prime topologically finitely generated closed normal\nsubgroup of Πi, whenever a unique such maximal subgroup exists; if there does not\nexist a unique such maximal subgroup, then we set Θi\ndef\n= {1}. Suppose that φ is\nabsolute. Then:\n(i) For i = 1, 2, Θi ⊆ Δi; Θi \u000b= {1} if and only if Σi is of cardinality\none; if Θi \u000b= {1}, then Θi = Δi. Finally, φ(Θ1) ⊆ Θ2 [so φ induces a morphism\nΠ1\/Θ1 → Π2\/Θ2].\n(ii) In the notation of Theorem 2.11, φ is semi-absolute [or, equivalently,\npre-semi-absolute — cf. Proposition 2.5, (iii)] if and only if the following [“group-\ntheoretic”] condition holds:\n(∗s-ab) For i = 1, 2, let Hi ⊆ Πi\/Θi be an open subgroup such that HRTFi is\nof aug-free type, and [the morphism induced by] φ maps H1 into H2.\nThen the open homomorphism\nHRTF1 → HRTF2\ninduced by φ maps HRTF-free1 into H\nRTF-free\n2 .\n32 SHINICHI MOCHIZUKI\n(iii) If, moreover, Σ2 is of cardinality ≥ 2, then φ is semi-absolute if and\nonly if it is strictly semi-absolute [or, equivalently, pre-strictly semi-absolute\n— cf. Proposition 2.5, (v)].\nProof. First, we consider assertion (i). By Theorem 2.6, (iv), any almost pro-\nprime topologically finitely generated closed normal subgroup of Πi — hence, in\nparticular, Θi — is contained in Δi. Thus, by Theorem 2.11, (i), (ii), Θi \u000b= {1}\nif and only if Σi is of cardinality one; if Θi \u000b= {1}, then Θi = Δi. Now to show\nthat φ(Θ1) ⊆ Θ2, it suffices to consider the case where φ(Θ1) \u000b= {1} [so Σ1 is of\ncardinality one]. Since φ is absolute, it follows that φ(Θ1) is normal in some open\nsubgroup of Π2. Thus, by Theorem 2.6, (iv), we have φ(Θ1) ⊆ Δ2, so we may\nassume that Θ2 = {1} [which implies that Σ2 is of cardinality ≥ 2]. But then the\nelasticity of Δ2 [cf. Proposition 2.3, (i)] implies that φ(Θ1) is an open subgroup\nof Δ2, hence that φ(Θ1) is almost pro-Σ2 [for some Σ2 of cardinality ≥ 2], which\ncontradicts the fact that φ(Θ1) is almost pro-Σ1 [for some Σ1 of cardinality one].\nThis completes the proof of assertion (i).\nNext, we consider assertion (ii). By Proposition 2.5, (iii), one may replace\nthe term “semi-absolute” in assertion (ii) by the term “pre-semi-absolute”. By\nassertion (i), for i = 1, 2, either Θi = {1} or Θi = Δi; in either case, it follows from\nTheorem 2.11, (iv) [cf. also Proposition 1.5, (i), (ii)], that [in the notation of (∗s-ab)]\nthe projection HRTFi \u0002 H\nRTF-aug\ni may be identified with the projection H\nRTF\ni \u0002\n(Hi)\nRTF\nGi\n[which is an isomorphism whenever Θi = Δi]. Thus, the condition (∗s-ab)\nmay be thought of as the condition that the morphismHRTF1 → HRTF2 be compatible\nwith the projection morphisms HRTFi \u0002 (Hi)RTFGi . From this point of view, it\nfollows immediately that the semi-absoluteness of φ implies (∗s-ab), and that (∗s-ab)\nimplies [in light of the existence of H1, H2 — cf. Theorem 2.11, (ii)] the pre-semi-\nabsoluteness of φ. Assertion (iii) follows from Proposition 2.5, (iv), (v). ©\nRemark 2.12.1. The criterion of Corollary 2.12, (ii), may be thought of as a\n“group-theoretic Hom-version”, in the case of hyperbolic orbicurves, of the numerical\ncriterion “ζ(H)\/ζ(Π) = [Π : H]” of Theorem 2.6, (v). Alternatively [cf. the point\nof view of Remark 2.11.1], this criterion of Corollary 2.12, (ii), may be thought of\nas a [necessarily — cf. Example 2.13 below!] somewhat more complicated version\nfor MLF’s of the latter portion of Corollary 2.8, (i), in the case of FF’s or NF’s.\nExample 2.13. A Non-pre-semi-absolute Absolute Homomorphism.\n(i) In the situation of Theorem 2.11, suppose that Σ = Primes. Fix a natural\nnumber N [which one wants to think of as being “large”]. By replacing Π by an open\nsubgroup of Π, we may assume that Π satisfies the hypotheses of the subgroup “H”\nof Theorem 2.11, (iii), and that the dual graph of the special fiber of X is not a tree\n[cf. the discussion preceding [Mzk6], Lemma 2.4]. Thus, we have a “combinatorial”\nquotient Π \u0002 Δcom, where Δcom is a nonabelian profinite free group. In particular,\nthere exists an open subgroup of Δcom which is a profinite free group on > N\ngenerators. Thus, by replacing Π by an open subgroup of Π arising from an open\nsubgroup of Δcom, we may assume from the start that Δcom is a profinite free group\non > N generators.\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 33\n(ii) Now let\n1 → Δ∗ → Π∗ → G∗ → 1\nbe an extension of AFG-type that admits a construction data field which is an MLF.\nThus, Π∗ is topologically finitely generated [cf. Theorem 2.6, (ii)], so it follows that\nthere exists a Π as in (i), together with a surjection of profinite groups\nψ : Π \u0002 Π∗\nthat factors through the quotient Π \u0002 Δcom. In particular, ψ is an absolute\nhomomorphism which is not pre-semi-absolute [hence, a fortiori, not semi-absolute].\nIn light of the appearance of the “combinatorial quotient” in Theorem 2.11,\n(iii), we pause to recall the following result [cf. [Mzk6], Lemma 2.3, in the profinite\ncase].\nTheorem 2.14. (Graph-theoreticity for Hyperbolic Curves) For i = 1, 2,\nlet 1 → Δi → Πi → Gi → 1, ki, Xi, Σi, φ : Π1 → Π2 be as in Proposition 2.5.\nSuppose further that ki is an MLF of residue characteristic pi; that Σi contains a\nprime \u000b= pi; that φ is an isomorphism; and that Xi is a hyperbolic curve with\nstable reduction over the ring of integers Oki of ki. Write Γi for the dual semi-\ngraph with compact structure [i.e., the dual graph, together with additional\nopen edges corresponding to the cusps — cf. [Mzk6], Appendix] of the geometric\nspecial fiber of the stable model of Xi over Oki . Then:\n(i) We have p1 = p2, Σ1 = Σ2; φ induces isomorphisms Δ1\n∼→ Δ2, G1 ∼→ G2;\nφ induces an isomorphism of semi-graphs φΓ : Γ1\n∼→ Γ2 which is functorial\nin φ. In particular, the natural Galois action of G1 on Γ1 is compatible, relative to\nφΓ, with the natural Galois action of G2 on Γ2.\n(ii) For i = 1, 2, suppose that the action of Gi on Γi is trivial. Write Πi \u0002\nΔcomi for the pro-Σi “combinatorial” quotient determined by the finite Galois\ncoverings of the semi-graph Γi of degree a product of primes ∈ Σi. Then φ is\ncompatible with the quotients Πi \u0002 Δcomi .\nProof. First, we consider assertion (i). By Corollary 2.8, (ii), p1 = p2, and φ\ninduces isomorphisms Δ1\n∼→ Δ2, G1 ∼→ G2. Since [by the well-known structure\nof geometric fundamental groups of hyperbolic curves] Σi is the unique minimal\nΣ ⊆ Primes such that Δi is almost pro-Σ, we thus conclude that Σ1 = Σ2. Write\np\ndef\n= p1 = p2, Σ\ndef\n= Σ1 = Σ2; let l ∈ Σ be such that l \u000b= p. Then it follows\nimmediately from the “Riemann hypothesis for abelian varieties over finite fields”\n[cf., e.g., [Mumf], p. 206] that the action of Gi on the maximal pro-l quotient Δi \u0002\nΔ\n(l)\ni is — in the terminology of [Mzk12] — “l-graphically full”. Thus, by [Mzk12],\nCorollary 2.7, (ii), the isomorphism Δ\n(l)\n1\n∼→ Δ(l)2 is — again in the terminology\nof [Mzk12] — “graphic”, hence induces a functorial isomorphism of semi-graphs\nΓ1\n∼→ Γ2, as desired.\nNext, we consider assertion (ii). First, we observe that, by assertion (i), the\ncondition that the action of Gi on Γi be trivial is compatible with φ. Also, let us\n34 SHINICHI MOCHIZUKI\nobserve that if Hi ⊆ Πi is an open subgroup corresponding to a finite étale covering\nZi → Xi of Xi, then the condition that Zi have stable reduction is compatible with\nφ [cf. [Mzk6], the proof of Lemma 2.1; our assumption that there exists an l ∈ Σi\nsuch that l \u000b= pi]. Next, I claim that:\nA finite étale Galois covering Zi → Xi of Xi arises from Δcomi if and only\nif Zi has stable reduction, and the action of Gal(Zi\/Xi) on the dual semi-\ngraph with compact structure of the geometric special fiber of the stable\nmodel of Zi is free.\nIndeed, the necessity of this criterion is clear. To verify the sufficiency of this\ncriterion, observe that, by considering the non-free actions of inertia subgroups of\nthe Galois covering Zi → Xi, it follows immediately that this criterion implies\nthat all of the inertia groups arising from irreducible components and cusps of\nthe geometric special fiber of a stable model of Xi are trivial, hence [cf., e.g.,\n[SGA2], X, 3.4, (i); [Tama2], Lemma 2.1, (iii)] that the covering Zi → Xi extends\nto an admissible covering of the respective stable models. On the other hand, once\none knows that the covering Zi → Xi admits such an admissible extension, the\nsufficiency of this criterion is immediate. This completes the proof of the claim.\nNow assertion (ii) follows immediately, by applying the functorial isomorphisms of\nsemi-graphs of assertion (i). ©\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 35\nSection 3: Absolute Open Homomorphisms of Local Galois Groups\nIn the present §, we give various generalizations of the main result of [Mzk1]\nconcerning isomorphisms between Galois groups of MLF’s. One aspect of these\ngeneralizations is the substitution of the condition given in [Mzk1] for such an\nisomorphism to arise geometrically — a condition that involves the higher ram-\nification filtration — by various other conditions [cf. Theorem 3.5]. Certain of\nthese conditions were motivated by a recent result of A. Tamagawa [cf. Remark\n3.8.1] concerning Lubin-Tate groups and abelian varieties with complex multiplica-\ntion; other conditions [cf. Corollary 3.7] were motivated by a certain application of\nthe theory of the present §3 to be discussed in [Mzk15]. Another aspect of these\ngeneralizations is that certain of the conditions studied below allow one to prove\na “Hom-version” [i.e., involving open homomorphisms, as opposed to just isomor-\nphisms — cf. Theorem 3.5] of the main result of [Mzk1]. Finally, this Hom-version\nof the main result of [Mzk1] implies certain semi-absolute Hom-versions [cf. Corol-\nlary 3.8, 3.9 below] of the absolute Isom-version of the Grothendieck Conjecture\ngiven in [Mzk13], §2, and the relative Hom-version of the Grothendieck Conjecture\nfor function fields given in [Mzk3], Theorem B.\nLet k be an MLF of residue characteristic p; k an algebraic closure of k; Gk\ndef\n=\nGal(k\/k); k̂ the p-adic completion of k; E an MLF of residue characteristic p all of\nwhose Qp-conjugates are contained in k. Write Ik ⊆ Gk (respectively, Iwildk ⊆ Ik) for\nthe inertia subgroup (respectively, wild inertia subgroup) of Gk; G\ntame\nk\ndef\n= Gk\/I\nwild\nk ;\nGunrk\ndef\n= Gk\/Ik (∼= Ẑ).\nDefinition 3.1.\n(i) Let A be an abelian topological group; ρ, ρ′ : Gk → A characters [i.e.,\ncontinuous homomorphisms]. Then we shall write ρ ≡ ρ′ and say that ρ, ρ′ are\ninertially equivalent if, for some open subgroup H ⊆ Ik, the restricted characters\nρ|H , ρ′|H coincide [cf. [Serre3], III, §A.5].\n(ii) Write Emb(E, k) for the set of field embeddings σ : E ↪→ k. Let σ ∈\nEmb(E, k). Then if π is a uniformizer of k, then we shall denote by χσ,π : Gk → E×\nthe composite homomorphism\nGk \u0002 Gabk\n∼→ (k×)∧ ∼→ O×k × Ẑ \u0002 O×k → O×E ⊆ E×\n— where the “∧” denotes the profinite completion; the first “ ∼→ ” is the isomor-\nphism arising from local class field theory [cf., e.g., [Serre2]]; the second “\n∼→ ”\nis the splitting determined by π; the second “\u0002” is the projection to the factor\nO×k , composed with the inverse automorphism on O×k [cf. Remark 3.1.1 below];\nthe homomorphism O×k → O×E is the norm map associated to the field embed-\nding σ. Since [as is well-known, from local class field theory] Ik ⊆ Gk surjects to\nO×k × {1} ⊆ O×k × Ẑ, it follows immediately that the inertial equivalence class of\nχσ,π is independent of the choice of π. Thus, we shall often write χσ to denote χσ,π\nfor some unspecified choice of π.\n36 SHINICHI MOCHIZUKI\n(iii) Let ρ : Gk → E× be a character. Then we shall say that ρ is of qLT-\ntype [i.e., “quasi-Lubin-Tate” type] if there exists an open subgroup H ⊆ Gk,\ncorresponding to a field extension kH of k, and a field embedding σ : E ↪→ kH such\nthat ρ|H ≡ χσ; in this situation, we shall refer to [E : Qp] as the dimension of ρ.\nWe shall say that ρ is of 01-type if it is Hodge-Tate, and, moreover, every weight\nappearing in its Hodge-Tate decomposition ∈ {0, 1}. Write\nχ\ncyclo\nk : Gk → Q×p\nfor the cyclotomic character associated to Gk. We shall say that ρ is of ICD-type\n[i.e., “inertially cyclotomic determinant” type] if its determinant det(ρ) : Gk → Q×p\n[i.e., the composite of ρ with the norm map E× → Q×p ] is inertially equivalent to\nχ\ncyclo\nk .\n(iv) For i = 1, 2, let ki be an MLF of residue characteristic pi; ki an algebraic\nclosure of ki; k̂i the pi-adic completion of ki. We shall use similar notation for the\nvarious subquotients of the absolute Galois group Gki\ndef\n= Gal(ki\/ki) of ki to the\nnotation already introduced for Gk. Let\nφ : Gk1 → Gk2\nbe an open homomorphism. Then we shall say that φ is of qLT-type (respectively, of\n01-qLT-type) if p1 = p2, and, moreover, for every pair of open subgroups H1 ⊆ Gk1 ,\nH2 ⊆ Gk2 such that φ(H1) ⊆ H2, and every character ρ : H2 → F× of qLT-type\n[where F is an MLF of residue characteristic p1 = p2 all of whose conjugates are\ncontained in the fields determined by H1, H2], the restricted character ρ|H1 : H1 →\nF× [obtained by restricting via φ] is of qLT-type (respectively, of 01-type). We\nshall say that φ is of HT-type [i.e., “Hodge-Tate” type] if p1 = p2, and, moreover,\nthe topological Gk1 -module [but not necessarily the topological field!] obtained by\ncomposing φ with the natural action of Gk2 on k̂2 is isomorphic [as a topological\nGk1-module] to k̂1. We shall say that φ is of CHT-type [i.e., “cyclotomic Hodge-\nTate” type] if φ is of HT-type, and, moreover, the cyclotomic characters of Gk1 ,\nGk2 satisfy χ\ncyclo\nk1\n= χ\ncyclo\nk2\n◦ φ. We shall say that φ is geometric if it arises from an\nisomorphism of fields k2\n∼→ k1 that maps k2 into k1 [which implies, by considering\nthe divisibility of the k×i , that p1 = p2, and that the isomorphism k2\n∼→ k1 is\ncompatible with the respective topologies].\n(v) Let 1 → Δ → Π → Gk → 1 be an extension of AFG-type. Then we shall\nsay that this extension Π \u0002 Gk [or, when there is no danger of confusion, that Π] is\nof A-qLT-type [i.e., “Albanese-quasi-Lubin-Tate” type] if for every open subgroup\nH ⊆ Gk, and every character ρ : H → F× of qLT-type [where F is an MLF of\nresidue characteristic p all of whose conjugates are contained in the field determined\nby H], there exists an open subgroup J ⊆ Π ×Gk (Ik\n⋂\nH) [so one has an outer\naction of the image JG of J in Gk on JΔ\ndef\n= J\n⋂\nΔ] such that the JG-module Vρ\nobtained by letting JG act on F via ρ|JG is isomorphic to some subquotient S of\nthe JG-module J\nab\nΔ ⊗Qp.\nRemark 3.1.1. As is well-known, the ρ that arises from a Lubin-Tate group is\nof qLT-type — cf., e.g., [Serre3], III, §A.4, Proposition 4. This is the reason for the\nterminology “quasi-Lubin-Tate”.\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 37\nWe begin by reviewing some well-known facts.\nProposition 3.2. (Characterization of Hodge-Tate Characters) Let ρ :\nGk → E× be a character; write Vρ for the Gk-module obtained by letting Gk act\non E via ρ. Then ρ is Hodge-Tate if and only if\nρ ≡\n∏\nσ∈Emb(E,k)\nχnσσ\nfor some nσ ∈ Z. Moreover, in this case, we have an isomorphism of k̂[Gk]-modules:\nVρ ⊗Qp k̂ ∼=\n⊕\nσ∈Emb(E,k)\nk̂(nσ)\n[where the “(−)” denotes a Tate twist].\nProof. Indeed, this criterion for the character ρ to be Hodge-Tate is precisely the\ncontent of [Serre3], III, §A.5, Corollary. The Hodge-Tate decomposition of Vρ then\nfollows immediately the Hodge-Tate decomposition of “Vρ” in the case where one\ntakes “ρ” to be χσ [cf. [Serre3], III, §A.5, proof of Lemma 2]. ©\nProposition 3.3. (Characterization of Quasi-Lubin-Tate Characters)\nLet ρ, Vρ be as in Proposition 3.2. Then the following conditions on ρ are equiv-\nalent:\n(i) ρ is of qLT-type.\n(ii) We have an isomorphism of k̂[Gk]-modules: Vρ ⊗Qp k̂ ∼= k̂(1)⊕ k̂⊕ . . .⊕ k̂.\n(iii) ρ is of ICD-type and Hodge-Tate; the resulting nσ’s of Proposition 3.2\nare ∈ {0, 1}.\n(iv) ρ is of ICD-type and of 01-type.\nProof. The fact that (i) implies (ii) follows immediately from the description of\nthe Hodge-Tate decomposition of “Vρ” in the case where one takes “ρ” to be χσ\n[cf. [Serre3], III, §A.5, proof of Lemma 2]. Next, let us assume that (ii), (iii), or\n(iv) holds. In either of these cases, it follows that ρ, hence also the determinant\ndet(ρ) : Gk → Q×p of ρ, is Hodge-Tate. Then by applying Proposition 3.2 to ρ, we\nobtain that the associated nσ’s are ∈ {0, 1}; by applying Proposition 3.2 to det(ρ)\n[in which case one takes “E” to be Qp], we obtain that det(ρ) is inertially equivalent\nto the (\n∑\nσ nσ)-th power of χ\ncyclo\nk . But this allows one to conclude [either from\nthe explicit Hodge-Tate decomposition of (ii), or from the assumption that ρ is\nof ICD-type in (iii), (iv)] that\n∑\nσ nσ = 1, hence that there exists precisely one\nσ ∈ Emb(E, k) such that nσ = 1, nσ′ = 0 for σ′ \u000b= σ. Thus, [sorting through the\ndefinitions] we conclude that (i), (ii), (iii), and (iv) hold. This completes the proof\nof Proposition 3.3. ©\n38 SHINICHI MOCHIZUKI\nProposition 3.4. (Preservation of Tame Quotients) In the notation of\nDefinition 3.1, (iv), let φ : Gk1 → Gk2 be an open homomorphism. Then\np1 = p2, and there exists a commutative diagram\nGk1\nφ−→ Gk2⏐⏐\u0005\n⏐⏐\u0005\nGtamek1\nφtame−→ Gtamek2\n— where the vertical arrows are the natural surjections; φtame is an injective\nhomomorphism.\nProof. We may assume without loss of generality that φ is surjective. Next, let\nH2 ⊆ Gk2 be an open subgroup, H1 def= φ−1(H2) ⊆ Gk1 . Then if p1 \u000b= p2, then\n[since we have a surjection H2 \u0002 H1] 1 = δ1l (H2) ≥ δ1l (H1) ≥ 2 for l = p1 [cf.\nTheorem 2.6, (ii)]; thus, we conclude that p1 = p2. Write p\ndef\n= p1 = p2. Since\nGtamek2\n∼= Ẑ(\u0005=p)(1)\u0002 Ẑ [for some faithful action of Ẑ on Ẑ(\u0005=p)(1) — cf., e.g., [NSW],\nTheorem 7.5.2], it follows immediately that every closed normal pro-p subgroup of\nGtamek2 is trivial. Thus, the image of φ(I\nwild\nk1\n) in Gtamek2 is trivial, so we conclude that\nφ induces a surjection φtame : Gtamek1 \u0002 G\ntame\nk2\n. Since, for i = 1, 2, the quotient\nGtameki \u0002 G\nunr\nki\n∼= Ẑ may be characterized as the quotient Gtameki \u0002 (Gtameki )ab-t, it\nthus follows immediately that φtame induces continuous homomorphisms\nẐ ∼= Gunrk1 \u0002 Gunrk2 ∼= Ẑ; Ẑ( \u0005=p)(1) ∼= Ik1\/Iwildk1 → Ik2\/Iwildk2 ∼= Ẑ(\u0005=p)(1)\n— the first of which is surjective, hence an isomorphism [since, as is well-known,\nevery surjective endomorphism of a topologically finitely generated profinite group\nis an isomorphism]. But this implies that the second displayed homomorphism is\nalso surjective, hence an isomorphism. This completes the proof of Proposition 3.4.\n©\nTheorem 3.5. (Criteria for Geometricity) For i = 1, 2, let ki be an MLF\nof residue characteristic pi; ki an algebraic closure of ki; k̂i the pi-adic completion\nof ki. We shall use similar notation for the various subquotients of the absolute\nGalois group Gki\ndef\n= Gal(ki\/ki) of ki to the notation introduced at the beginning of\nthe present §3 for Gk. Let\nφ : Gk1 → Gk2\nbe an open homomorphism. Then:\n(i) The following conditions on φ are equivalent: (a) φ is of CHT-type; (b)\nφ is of 01-qLT-type; (c) φ is of qLT-type; (d) φ is geometric.\n(ii) Suppose that φ is an isomorphism. Then φ is geometric if and only if\nit is of HT-type.\n(iii) For i = 1, 2, let 1 → Δi → Πi → Gki → 1 be an extension of AFG-type;\nψ : Π1 → Π2\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 39\na semi-absolute [or, equivalently, pre-semi-absolute — cf. Proposition 2.5,\n(iii)] homomorphism that lifts φ. Suppose that Π2 is of A-qLT-type. Then φ is\ngeometric.\nProof. First, we observe that by Proposition 3.4, it follows that p1 = p2; write\np\ndef\n= p1 = p2. Also, we may always assume without loss of generality that φ is\nsurjective. Thus, by Proposition 3.4, it follows that φ(Ik1) = Ik2 . In the following,\nwe will use a superscript “Gki” [where i = 1, 2] to denote the submodule of Gki-\ninvariants of a Gki -module.\nNext, we consider assertion (i). First, we observe that it is immediate that\ncondition (d) implies condition (a). Next, let us suppose that condition (a) holds.\nSince ki = k̂\nGki\ni is finite-dimensional over Qp, it follows that, for i = 1, 2, any Gki-\nmodule M which is finite-dimensional over Qp is Hodge-Tate with weights ∈ {0, 1}\nif and only if\ndimQp((M ⊗ k̂i)Gki ) + dimQp((M(−1)⊗ k̂i)Gki ) = dimQp(M) · dimQp(k̂\nGki\ni )\n[where the tensor products are over Qp]. Now suppose that M is a Gk2 -module\nthat arises as a “Vρ” for some character ρ : Gk2 → E× of qLT-type [so M is\nHodge-Tate with weights ∈ {0, 1} — cf. Proposition 3.3, (i) =⇒ (iv)]; write Mφ for\nthe Gk1-module Mφ obtained by composing the Gk2-action on M with φ. Thus,\nit follows immediately from our assumption that φ is of CHT-type that the above\ncondition concerning Qp-dimensions for M implies the above condition concerning\nQp-dimensions for Mφ. Applying this argument to corresponding open subgroups\nof Gk1 , Gk2 thus shows that φ is of 01-qLT-type, i.e., that condition (b) holds.\nNext, let us assume that condition (b) holds. First, I claim that χ\ncyclo\nk1\n≡ χcyclok2 ◦\nφ. Indeed, by condition (b), it follows that the character χ\ncyclo\nk2\n◦ φ : Gk1 → Q×p is\nof 01-type. Thus, by Proposition 3.2, we conclude that χ\ncyclo\nk2\n◦ φ ≡ (χcyclok1 )n, for\nsome n ∈ {0, 1}. On the other hand, the restriction of χcyclok2 to Ik2 clearly has open\nimage; since φ(Ik1) = Ik2 , it thus follows that the restriction of χ\ncyclo\nk2\n◦φ to Ik1 has\nopen image. This rules out the possibility that n = 0, hence completes the proof\nof the claim. Now, by applying this claim, together with Proposition 3.3, (i) ⇐⇒\n(iv), we conclude that φ is of qLT-type, i.e., that condition (c) holds.\nNext, let us assume that condition (c) holds. First, I claim that this already\nimplies that φ is injective [i.e., an isomorphism]. Indeed, let γ ∈ Ker(φ) ⊆ Gk1 be\nsuch that γ \u000b= 1. Then there exists an open subgroup J1 ⊆ Gk1 ⊆ GQp satisfying\nthe following conditions: (1) γ \/∈ J1; (2) J1 is characteristic as a subgroup GQp ; (3)\nthe extension E of Qp determined by J1 contains all Qp-conjugates of k2. Fix an\nembedding σ0 : k2 ↪→ E; write H2 ⊆ Gk2 for the corresponding open subgroup. Let\nH1 ⊆ J1 ⊆ Gk1 be an open subgroup which is normal in Gk1 such that φ(H1) ⊆\nH2; for i = 1, 2, write kHi for the extension of ki determined by Hi. Thus, the\nembedding σ2 : E ↪→ kH2 given by the identity E = kH2 (respectively, σ1 : E ↪→ kH1\ndetermined by the inclusion H1 ⊆ J1) determines a character ρ2 : H2 → E×\n(respectively, ρ1 : H1 → E×) of qLT-type [i.e., the character “χσ2” (respectively,\n“χσ1”)]. Moreover, by condition (c), the character ρ2 ◦(φ|H1) : H1 → E× is of qLT-\ntype, hence is inertially equivalent to τ ◦ρ1 : H1 → E× for some τ ∈ Gal(E\/Qp). In\n40 SHINICHI MOCHIZUKI\nparticular, by replacing σ2 by σ2 ◦ τ−1, we may assume that τ is the identity, hence\nthat ρ2 ◦ (φ|H1) ≡ ρ1. On the other hand, since γ \/∈ J1, hence acts nontrivially on\nthe subfield E ⊆ kH1 [relative to the embedding σ1], it follows that ρ1 ◦κγ ≡ δ ◦ρ1,\nwhere we write κγ for the automorphism of H1 given by conjugating by γ, and\nδ ∈ Gal(E\/Qp) is not equal to the identity. But since φ(γ) = 1 ∈ Gk2 , we thus\nconclude that δ ◦ ρ1 ≡ ρ1 ◦ κγ ≡ ρ2 ◦ (φ|H1) ◦ κγ ≡ ρ2 ◦ (φ|H1) ≡ ρ1, which [since\nρ1 has open image] contradicts the fact that δ ∈ Gal(E\/Qp) is not equal to the\nidentity. This completes the proof of the claim. Thus, we may assume that φ is\nan isomorphism of qLT-type, i.e., we are, in effect, in the situation of [Mzk1], §4.\nIn particular, the fact that φ is geometric, i.e., that condition (d) holds, follows\nimmediately via the argument of [Mzk1], §4. This completes the proof of assertion\n(i).\nNext, we consider assertion (ii). Since φ is an isomorphism, it follows [cf.\n[Mzk1], Proposition 1.1; [Mzk6], Proposition 1.2.1, (vi)] that χ\ncyclo\nk1\n= χ\ncyclo\nk2\n◦ φ. In\nparticular, φ is of HT-type if and only if φ is of CHT-type. Thus, assertion (ii)\nfollows from the equivalence of (a), (d) in assertion (i).\nFinally, we consider assertion (iii). First, let us recall that by a well-known\nresult of Tate [cf. [Tate], §4, Corollary 2], if J ⊆ Π1 is an open subgroup with image\nJG ⊆ Gk1 and intersection JΔ def= J\n⋂\nΔ1, then the JG-module J\nab\nΔ ⊗ Qp is always\nHodge-Tate with weights ∈ {0, 1}. Thus, the condition that Π2 is of A-qLT-type\nimplies that φ is of 01-qLT-type, hence, by assertion (i), geometric. This completes\nthe proof of assertion (iii). ©\nDefinition 3.6.\n(i) If H ⊆ Gk is an open subgroup corresponding to an extension field kH of\nk, then by local class field theory [cf., e.g., [Serre2]], we have a natural isomorphism\nO×kH\n∼→ Tor(H)\n— where we write Tor(H) [i.e., the “toral portion of H”] for the image of Ik\n⋂\nH\nin Hab. Thus, by applying the p-adic logarithm O×kH → kH , we obtain a natural\nisomorphism λH : Tor(H)⊗Qp ∼→ kH .\n(ii) We shall refer to a collection {NH}H , where H ranges over a collection of\nopen subgroups of Gk that form a basis of the topology of Gk, as a uniformly toral\nneighborhood of Gk if there exist nonnegative integers a, b [which are independent\nof H!] such that [in the notation of (i)] NH ⊆ Tor(H) ⊗ Qp is an open subgroup\nsuch that pa · OkH ⊆ λH(NH) ⊆ p−b · OkH ⊆ kH .\n(iii) Let φ : Gk1\n∼→ Gk2 be an isomorphism of profinite groups. Then we shall\nsay that φ is uniformly toral if Gk1 admits a uniformly toral neighborhood {NH}H\nsuch that {Nφ(H) def= φ(NH)}φ(H) forms a uniformly toral neighborhood of Gk2 . We\nshall say that φ is RF-preserving [i.e., “ramification filtration preserving”] if φ is\ncompatible with the filtrations on Gk1 , Gk2 given by the [positively indexed] higher\nramification groups in the upper numbering [cf., [Mzk1], Theorem].\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 41\nCorollary 3.7. (Uniform Torality and Geometricity) In the situation\nof Theorem 3.5, suppose further that φ is an isomorphism. Then the following\nconditions on φ are equivalent: (a) φ is RF-preserving; (b) φ is uniformly\ntoral; (c) φ is geometric.\nProof. First, we observe that by Proposition 3.4, it follows that p1 = p2; write\np\ndef\n= p1 = p2. Also, we observe that it is immediate that condition (c) implies\ncondition (a). Next, we recall that the fact that condition (a) implies condition\n(b) is precisely the content of the discussion preceding [Mzk1], Proposition 2.2.\nThat is to say, for i = 1, 2, the images of appropriate higher ramification groups\nin Tor(H) ⊗ Qp [for open subgroups H ⊆ Gki ] multiplied by appropriate integral\npowers of p yield a uniformly toral neighborhood of Gki that is compatible with φ\nwhenever φ is RF-preserving.\nNext, let us assume that condition (b) holds. For i = 1, 2, let {N iH}H be\na uniformly toral neighborhood of Gki . Again, we take the point of view of the\ndiscussion preceding [Mzk1], Proposition 2.2. That is to say, we think of ki as the\ninductive limit\nIi\ndef\n= lim−→\nH\nTor(H)⊗Qp\n— where H ranges over the open subgroups ⊆ Gki involved in {N iH}H ; the mor-\nphisms in the inductive system are those induced by the Verlagerung, or transfer,\nmap. Write Ni ⊆ Ii for the subgroup generated by the N iH ⊆ Tor(H)⊗Qp. Then\nrelative to the isomorphism [of abstract modules!] λi : Ii\n∼→ ki determined by the\nλH ’s, we have\npa · Oki ⊆ λi(Ni) ⊆ p−b · Oki ⊆ ki\nfor some nonnegative integers a, b [cf. Definition 3.6, (ii)]. In particular, it follows\nthat the topology on Ii determined by the submodules p\nc · Ni, where c ≥ 0 is an\ninteger, coincides, relative to λi, with the p-adic topology on ki [i.e., the topology\ndetermined by the pc · Oki , where c ≥ 0 is an integer]. Write Îi for the completion\nof Ii relative to the topology determined by the p\nc · Ni. Thus, λi determines an\nisomorphism of topological Gki-modules Îi\n∼→ k̂i. In particular, the assumption that\nφ is uniformly toral implies that φ is of HT-type. Thus, by Theorem 3.5, (ii), we\nconclude that φ is geometric, i.e., that condition (c) holds. This completes the proof\nof Corollary 3.7. ©\nRemark 3.7.1. In fact, one verifies immediately that the argument applied in the\nproof of Corollary 3.7 implies that the equivalences of Corollary 3.7 [as well as the\ndefinitions of Definition 3.6] continue to hold when φ is replaced by an isomorphism\nof profinite groups between the maximal pro-p quotients of the Gki . We leave the\nroutine details to the reader.\nCorollary 3.8. (Geometricity of Semi-absolute Homomorphisms for\nHyperbolic Orbicurves) For i = 1, 2, let ki, ki, k̂i, pi, Gki [and its subquotients]\nbe as in Theorem 3.5; 1 → Δi → Πi → Gki → 1 an extension of AFG-type;\n(ki, Xi,Σi) partial construction data [consisting of the construction data field,\n42 SHINICHI MOCHIZUKI\nconstruction data base-stack, and construction data prime set] for Πi \u0002 Gki ; αi :\nπ1(Xi) = π\ntame\n1 (Xi) \u0002 Πi a scheme-theoretic envelope compatible with the\nnatural projections π1(Xi) \u0002 Gki , Πi \u0002 Gki ;\nψ : Π1 → Π2\na semi-absolute [or, equivalently, pre-semi-absolute — cf. Proposition 2.5,\n(iii)] homomorphism that lifts a homomorphism φ : G1 → G2. Suppose further\nthat X2 is a hyperbolic orbicurve, that p2 ∈ Σ2, and that one of the following\nconditions holds:\n(a) φ is of CHT-type;\n(b) φ is of 01-qLT-type;\n(c) φ is of qLT-type;\n(d) φ is an isomorphism of HT-type;\n(e) φ is a uniformly toral isomorphism;\n(f) φ is an RF-preserving isomorphism;\n(g) Π2 is of A-qLT-type.\n(h) φ is geometric;\nThen ψ is geometric, i.e., arises [relative to the αi] from a unique dominant\nmorphism of schemes X1 → X2 lying over a morphism Spec(k1) → Spec(k2).\nProof. Indeed, by Theorem 3.5, (i), (ii), (iii); Corollary 3.7, it follows that any of\nthe conditions (a), (b), (c), (d), (e), (f), (g), (h) implies condition (h). Thus, since\nX2 is a hyperbolic orbicurve, and p2 ∈ Σ2, the fact that ψ is geometric follows from\n[Mzk3], Theorem A. ©\nRemark 3.8.1. One important motivation for the theory of the present §3 is the\nfollowing result, orally communicated to the author by A. Tamagawa:\n(∗A-qLT) Let X be a hyperbolic orbicurve over k that admits a finite étale covering\nY → X by a hyperbolic curve Y such that Y admits a dominant k-\nmorphism Y → P , where P is the projective line minus three points\nover k [i.e., a tripod — cf. §0]. Then the arithmetic fundamental group\nπ1(X) \u0002 Gk of X is of A-qLT-type.\nIn particular, it follows that:\nCorollary 3.8 may be applied [in the sense that condition (g) is satisfied]\nwhenever X2 satisfies the conditions placed on the hyperbolic orbicurve\n“X” of (∗A-qLT).\nIndeed, Tamagawa’s original motivation for considering (∗A-qLT) was precisely the\ngoal of applying the methods of [Mzk1] to obtain an “isomorphism version” of\nCorollary 3.8, (g). Upon learning of these ideas of Tamagawa, the author proceeded\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 43\nto re-examine the theory of [Mzk1]. This led the author to the discovery of the\nvarious generalizations of [Mzk1] — and, in particular, the Hom-version of Corollary\n3.8, (g) — given in the present §3. Tamagawa derives (∗A-qLT) from the following\nresult:\n(∗CM) Given a character ρ : Gk → E× of qLT-type, there exists an abelian\nvariety with complex multiplication A over some finite extension kA of k\nsuch that ρ|GkA is inertially equivalent to some character whose associated\nGkA -module appears as a subquotient of the GkA -module given by the p-\nadic Tate module of A.\nIndeed, to derive (∗A-qLT) from (∗CM), one reasons as follows: Every abelian variety\nwith complex multiplication A is defined over a number field, hence arises as a\nquotient of a Jacobian of a smooth proper curve Z over a number field. Moreover,\nby considering Belyi maps, it follows that some open subscheme UZ ⊆ Z arises as\na finite étale covering of the projective line minus three points. Thus, any Galois\nmodule that appears as a subquotient of the p-adic Tate module of A also appears\nas a subquotient of the p-adic Tate module of the Jacobian of some finite étale\ncovering of the curve P of (∗A-qLT), hence, a fortiori, as a subquotient of the p-adic\nTate module of the Jacobian of some finite étale covering of the curves Y , X of\n(∗A-qLT). Thus, we conclude that π1(X) is of A-qLT-type, as desired.\nCorollary 3.9. (Geometricity of Strictly Semi-absolute Homomor-\nphisms for Function Fields) Assume that either of the results (∗A-qLT), (∗CM) of\nRemark 3.8.1 holds. For i = 1, 2, let ki be an MLF, Ki a function field of tran-\nscendence degree ≥ 1 over ki [so ki is algebraically closed in Ki], Ki an algebraic\nclosure of Ki, ki the algebraic closure of ki determined by Ki, Πi\ndef\n= Gal(Ki\/Ki),\nGi\ndef\n= Gal(ki\/ki), Δi\ndef\n= Ker(Πi \u0002 Gi). Then every open homomorphism\nψ : Π1 → Π2\nthat induces an open homomorphism ψΔ : Δ1 → Δ2 [hence also an open homomor-\nphism φ : G1 → G2] is geometric, i.e., arises from a unique embedding of fields\nK2 ↪→ K1 that induces an embedding of fields k2 ↪→ k1 of finite degree.\nProof. Since every function field of transcendence degree ≥ 1 over k2 contains the\nfunction field of a tripod over k2, it follows from (∗A-qLT), hence also from (∗CM)\n[cf. Remark 3.8.1], that there exists a hyperbolic curve X over k2 whose function\nfield is contained in K2 such that if we write Π2 \u0002 Π3 def= π1(X) for the resulting\nsurjection, then Π3 is of A-qLT-type. Now we wish to apply a “birational analogue”\nof Corollary 3.8, (g), to the composite homomorphism Π1 → Π2 \u0002 Π3 [where the\nfirst arrow is ψ].\nTo verify that such an analogue holds, it suffices to verify that φ is of 01-qLT-\ntype [cf. Theorem 3.5, (i), (b) =⇒ (d)]. To this end, set k3 def= k2, G3 def= G2,\nΔ3\ndef\n= Ker(Π3 \u0002 G3); let us suppose, for i = 1, 3, that Hi ⊆ Δi, Ji ⊆ Gi\nare characteristic open subgroups such that ψΔ(H1) ⊆ H3, φ(J1) ⊆ J3. Thus, if\n44 SHINICHI MOCHIZUKI\nwe write p for the common residue characteristic of k1, k3 [cf. Proposition 3.4],\nthen we obtain a surjection Hab1 ⊗ Qp \u0002 Hab3 ⊗ Qp that is compatible with φ.\nMoreover, it follows immediately from Corollary A.11 [cf. also Proposition A.3,\n(v)] of the Appendix that the J1-module H\nab-t\n1 ⊗ Zp admits a quotient J1-module\nHab-t1 ⊗ Zp \u0002 Q1 such that Q1 is the p-adic Tate module of some abelian variety\nover a finite extension of k1, and, moreover, the kernel Ker(H\nab-t\n1 ⊗ Zp \u0002 Q1) is\ntopologically generated by topologically cyclic subgroups [i.e., “copies of Zp”] on\nwhich some open subgroup of J1 [which may depend on the cyclic subgroup] acts via\nthe cyclotomic character. Next, let us observe that if V3 is any J3-module associated\nto a character of qLT-type of dimension ≥ 2, then V3 does not contain any sub-J3-\nmodules of dimension 1 over Qp. From this observation, it follows immediately that\nany subquotient [cf. Definition 3.1, (v)] of the J3-moduleH\nab\n3 ⊗Qp that is isomorphic\nto the J3-module associated to a character of qLT-type of dimension ≥ 2 determines\na subquotient [not only of the J1-module H\nab-t\n1 ⊗ Qp, but also] of the J1-module\nQ1 ⊗Qp. Thus, we conclude that any such subquotient of the J1-module Q1 ⊗Qp\nis Hodge-Tate with weights ∈ {0, 1}. Moreover, by considering determinants of such\nsubquotients, one concludes that the pull-back of the cyclotomic character J3 → Z×p\nis a character J1 → Z×p which is Hodge-Tate, and whose unique weight w is ≥ 0. If\nw ≥ 2, then the fact that the J3-module determined by the cyclotomic character of\nJ3 occurs as a subquotient of H\nab\n3 ⊗Qp [for sufficiently small H3], hence determines\na J1-module that occurs as a subquotient [not only of the J1-module H\nab-t\n1 ⊗ Qp,\nbut also, in light of our assumption that w ≥ 2!] of the J1-module Q1⊗Qp leads to\na contradiction [since the J1-module Q1 ⊗Qp is Hodge-Tate with weights ∈ {0, 1}].\nThus, we conclude that φ : G1 → G2 = G3 is of 01-qLT-type, hence geometric,\ni.e., arises from a unique embedding of fields k2 ↪→ k1 of finite degree. Finally,\nthe geometricity of φ implies that the geometricity of ψ may be derived from the\n“relative” result given in [Mzk3], Theorem B. ©\nRemark 3.9.1. The proof given above of Corollary 3.9 shows that the “Π2” of\nCorollary 3.9 may, in fact, be taken to be a “Π2” as in Corollary 3.8, (g).\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 45\nSection 4: Chains of Elementary Operations\nIn the present §4, we generalize [cf. Theorems 4.7, 4.12; Remarks 4.7.1, 4.12.1\nbelow] the theory of “categories of dominant localizations” discussed in [Mzk9],\n§2 [cf. also the tempered versions of these categories, discussed in [Mzk10], §6],\nto include “localizations” obtained by more general “chains of elementary opera-\ntions” — i.e., the operations of passing to a finite étale covering, passing to a finite\nétale quotient, “de-cuspidalization”, and “de-orbification” [cf. Definition 4.2 below;\n[Mzk13], §2] — which are applied to some given algebraic stack over a field. The\nfield and algebraic stack under consideration are quite general in nature [by com-\nparison, e.g., to the theory of [Mzk9], §2; [Mzk13], §2], but are subject to various\nassumptions. One key assumption asserts that the algebraic stack satisfies a certain\nrelative version of the “Grothendieck Conjecture”.\nBefore proceeding, we recall the following immediate consequence of [Mzk13],\nLemma 2.1; [Mzk12], Proposition 1.2, (ii).\nLemma 4.1. (Decomposition Groups of Hyperbolic Orbicurves) Let Σ\nbe a nonempty set of prime numbers, Δ a pro-Σ group of GFG-type that admits\nbase-prime [cf. Definition 2.1, (iv)] partial construction data (k,X,Σ) [consisting\nof the construction data field, construction data base-stack, and construction data\nprime set] such that X is a hyperbolic orbicurve [cf. §0], and k is algebraically\nclosed. Let xA (respectively, xB \u000b= xA) be either a closed point or a cusp [cf.\n§0] of X; A ⊆ Δ (respectively, B ⊆ Δ) the decomposition group [well-defined\nup to conjugation in Δ] of xA (respectively, xB). Then:\n(i) A, B are pro-cyclic groups; A\n⋂\nB = {1}. If xA is a closed point of\nX, and A \u000b= {1}, then A is a finite, normally terminal [cf. §0] subgroup of Δ.\nIf xA is a cusp, then A is a torsion-free, commensurably terminal [cf. §0]\ninfinite subgroup of Δ.\n(ii) The order of every finite cyclic closed subgroup C ⊆ Δ divides the\norder of X [cf. §0].\n(iii) Every finite nontrivial closed subgroup C ⊆ Δ is contained in a\ndecomposition group of a unique closed point of X. In particular, the non-\ntrivial decomposition groups of closed points of X may be characterized [“group-\ntheoretically”] as the maximal finite nontrivial closed subgroups of Δ.\n(iv) X is a hyperbolic curve if and only if Δ is torsion-free.\n(v) Suppose that the quotient ψA : Δ \u0002 ΔA of Δ by the closed normal sub-\ngroup of Δ topologically generated by A is slim and nontrivial. If xA is a closed\npoint of X (respectively, a cusp), then we suppose further that Σ = Primes [which\nforces the characteristic of k to be zero] (respectively, that A ⊆ J for some normal\nopen torsion-free subgroup J of Δ). Then ΔA is a profinite group of GFG-type\nthat admits base-prime partial construction data (k,XA,Σ) [consisting of the con-\nstruction data field, construction data base-stack, and construction data prime set]\nsuch that XA is a hyperbolic orbicurve equipped with a dominant k-morphism\nφA : X → XA that is uniquely determined [up to a unique isomorphism] by the\n46 SHINICHI MOCHIZUKI\nproperty that it induces [up to composition with an inner automorphism] ψA. More-\nover, if xA is a closed point of X (respectively, a cusp), then φA is a partial\ncoarsification morphism [cf. §0] which is an isomorphism either over XA or\nover the complement in XA of the point of XA determined by xA (respectively, is an\nopen immersion whose image is the complement of the point of XA determined\nby xA).\n(vi) In the notation of (v), if B \u000b= {1}, then ψA(B) \u000b= {1}.\nProof. First, we recall that by the definition of a profinite group of GFG-type\n[cf. the discussion at the beginning of §2], it follows that there exists a normal\nopen subgroup H ⊆ Δ such that if we write XH → X for the corresponding\nGalois covering, then XH is a hyperbolic curve. Next, let us observe that, in light\nof our assumption that the partial construction data is base-prime, we may lift\nthe entire situation to characteristic zero, hence assume, at least for the proof of\nassertions (i), (ii), (iii), (iv), that k is of characteristic zero. Thus, assertions (i),\n(ii), (iii) when xA, xB are closed points (respectively, cusps) ofX follow immediately\nfrom [Mzk13], Lemma 2.1 (respectively, [Mzk12], Proposition 1.2, (ii)). Next, we\nconsider assertion (iv). First, we observe that the necessity portion of assertion\n(iv) follows immediately from assertion (iii). To verify sufficiency, let us suppose\nthat Δ is torsion-free. Let πtame1 (X) \u0002 Δ be a scheme-theoretic envelope of Δ.\nThen since XH is a scheme, it follows that the nontrivial [finite closed] subgroups\nof πtame1 (X) that arise as decomposition groups of closed points map injectively,\nvia the composite surjection πtame1 (X) \u0002 Δ \u0002 Δ\/H, into Δ\/H, hence, a fortiori,\ninjectively via the surjection πtame1 (X) \u0002 Δ, into Δ [which is torsion-free]. Thus,\nthe decomposition groups in π1(X) = π\ntame\n1 (X) [cf. our assumption that k is\nalgebraically closed of characteristic zero] of closed points of X are trivial. But this\nimplies [by considering, for instance, the Galois covering XH → X] that X is a\nscheme, as desired. This completes the proof of assertion (iv).\nNext, we consider assertion (v). First, let us observe that XA admits a finite\nétale covering YA → XA arising from a normal open subgroup of ΔA such that YA is\na curve, which will necessarily be hyperbolic, in light of the slimness and nontriviality\nof ΔA. Indeed, when xA is a closed point of X [so Σ = Primes; k is of characteristic\nzero], this follows immediately from the equivalence of definitions of a “hyperbolic\norbicurve” discussed in §0; when xA is a cusp, this follows from assertion (iv) and\nour assumption of the existence of the subgroup J ⊆ Δ. Now the remainder of\nassertion (v) follows immediately from the definitions. This completes the proof of\nassertion (v). Finally, we consider assertion (vi). Assertion (vi) is immediate if xB\nis a cusp [cf. assertion (i)]; thus, we may assume that xB is a closed point of X.\nIf ψA(B) = {1}, then it follows that the decomposition group ⊆ ΔA of the image\nof xB in XA is trivial. Since [by assertion (v)] XA admits a finite étale covering\nYA → XA arising from an open subgroup of ΔA such that YA is a hyperbolic curve,\nwe thus conclude that XA is scheme-like in a neighborhood of the image of xB in\nXA, hence [in light of the explicit description of the morphism φA in the statement\nof assertion (v)] that X is scheme-like in a neighborhood of xB . But this implies\nthat B = {1}. This completes the proof of assertion (vi). ©\nRemark 4.1.1. Note that Lemma 4.1, (iv), is false if we only assume that Δ is\nalmost pro-Σ. Indeed, such an example may be constructed by taking X to be a\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 47\nhyperbolic curve over an algebraically closed field k of characteristic zero, Y → X a\nfinite étale Galois covering of degree prime to Σ, and Δ to be the quotient of π1(X)\nby the kernel of the surjection (π1(X) ⊇) π1(Y ) \u0002 π1(Y )(Σ) to the maximal pro-Σ\nquotient π1(Y )\n(Σ) of π1(Y ). Then for any prime p dividing the order of Gal(Y\/X)\n[so p \u000b∈ Σ], it follows by considering Sylow p-subgroups that Δ contains an element\nof order p, despite the fact that X is a curve.\nDefinition 4.2. Let G be a slim profinite group;\n1 → Δ → Π → G → 1\nan extension of GSAFG-type that admits base-prime partial construction data\n(k,X,Σ), where Σ \u000b= ∅; α : πtame1 (X) \u0002 Π is a scheme-theoretic envelope. Thus, if\nwe write πtame1 (X) \u0002 Gk for the quotient given by the absolute Galois group Gk\nof k, then α determines a scheme-theoretic envelope β : Gk \u0002 G. Write X̃ → X\nfor the pro-finite étale covering of X determined by the surjection α; k̃ for the\nresulting field extension of k. In a similar vein, we shall write Π̃ for the projective\nsystem of profinite groups determined by the open subgroups of Π. [Thus, one\nmay consider homomorphisms between Π̃ and a profinite group by thinking of the\nprofinite group as a trivial projective system of profinite groups — cf. the theory\nof “pro-anabelioids”, as in [Mzk8], Definition 1.2.6.] Then:\n(i) We shall refer to as an [X̃\/X-]chain [of length n] [where n ≥ 0 is an integer]\nany finite sequence\nX0 \u0003 X1 \u0003 . . . \u0003 Xn−1 \u0003 Xn\nof generically scheme-like algebraic stacks Xj [for j = 0, . . . , n], each equipped with\na dominant “rigidifying morphism” ρj : X̃ → Xj satisfying the following conditions:\n(0X) X0 = X [equipped with its natural rigidifying morphism X̃ → X].\n(1X) There exists a [uniquely determined] morphism Xj → Spec(kj) com-\npatible with ρj , where kj ⊆ k̃ is a finite extension of k such that Xj is\ngeometrically connected over kj .\n(2X) Each ρj determines a maximal pro-finite étale covering X̃j → Xj such\nthat X̃ → Xj admits a factorization X̃ → X̃j → Xj . The kernel Δj of\nthe resulting natural surjection\nΠj\ndef\n= Gal(X̃j\/Xj) \u0002 Gj def= Gal(k̃\/kj)\nis slim and nontrivial; every prime dividing the order of a finite quotient\ngroup of Δj is invertible in k.\n(3X) Suppose that X is a hyperbolic orbicurve [over k]. Then each Xj is also\na hyperbolic orbicurve [over kj ]. Moreover, each Δj is a pro-Σ group.\n(4X) Each “Xj \u0003 Xj+1” [for j = 0, . . . , n− 1] is an “elementary operation”,\nas defined below.\n48 SHINICHI MOCHIZUKI\nHere, an elementary operation “Xj \u0003 Xj+1” is defined to consist of the datum of\na dominant “operation morphism” φ either from Xj to Xj+1 or from Xj+1 to Xj\nwhich is compatible with ρj , ρj+1, and, moreover, is of one of the following four\ntypes:\n(a) Type \u0004: In this case, the elementary operation Xj \u0003 Xj+1 consists of\na finite étale covering φ : Xj+1 → Xj . Thus, φ determines an open\nimmersion of profinite groups Πj+1 ↪→ Πj .\n(b) Type \u0005: In this case, the elementary operation Xj \u0003 Xj+1 consists of a\nfinite étale morphism φ : Xj → Xj+1 — i.e., a “finite étale quotient”.\nThus, φ determines an open immersion of profinite groups Πj ↪→ Πj+1.\n(c) Type •: This type of elementary operation is only defined if X is a\nhyperbolic orbicurve. In this case, the elementary operation Xj \u0003 Xj+1\nconsists of an open immersion φ : Xj ↪→ Xj+1 [so kj = kj+1] — i.e., a\n“de-cuspidalization” — such that the image of φ is the complement\nof a single kj+1-valued point of Xj+1 whose decomposition group in Δj\nis contained in some normal open torsion-free subgroup of Δj . Thus, φ\ndetermines a surjection of profinite groups Πj \u0002 Πj+1.\n(d) Type \u0006: This type of elementary operation is only defined if X is a\nhyperbolic orbicurve and Σ = Primes [which forces the characteristic of k\nto be zero]. In this case, the elementary operation Xj \u0003 Xj+1 consists\nof a partial coarsification morphism [cf. §0] φ : Xj → Xj+1 [so kj =\nkj+1] — i.e., a “de-orbification” — such that φ is an isomorphism over\nthe complement in Xj+1 of some kj+1-valued point of Xj+1. Thus, φ\ndetermines a surjection of profinite groups Πj \u0002 Πj+1.\nThus, any X̃\/X-chain determines a sequence of symbols ∈ {\u0004,\u0005, •,\u0006} [correspond-\ning to the types of elementary operations in the X̃\/X-chain], which we shall refer\nto as the type-chain associated to the X̃\/X-chain.\n(ii) An isomorphism between two X̃\/X-chains with identical type-chains [hence\nof the same length]\n(X0 \u0003 . . . \u0003 Xn) ∼→ (Y0 \u0003 . . . \u0003 Yn)\nis defined to be a collection of isomorphisms of generically scheme-like algebraic\nstacks Xj\n∼→ Yj [for j = 0, . . . , n] that are compatible with the rigidifying mor-\nphisms. [Here, we note that the condition of compatibility with the rigidifying\nmorphisms implies that every automorphism of an X̃\/X-chain is given by the iden-\ntity, and that every isomorphism of X̃\/X-chains is compatible with the respective\noperation morphisms.] Thus, one obtains a category\nChain(X̃\/X)\nwhose objects are the X̃\/X-chains [with arbitrary associated type-chain], and whose\nmorphisms are the isomorphisms between X̃\/X-chains [with identical type-chains].\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 49\nA terminal morphism between two X̃\/X-chains [with arbitrary associated type-\nchains]\n(X0 \u0003 . . . \u0003 Xn) → (Y0 \u0003 . . . \u0003 Ym)\nis defined to be a dominant k-morphism Xn → Ym. Thus, one obtains a category\nChaintrm(X̃\/X)\nwhose objects are the X̃\/X-chains [with arbitrary associated type-chain], and whose\nmorphisms are the terminal morphisms between X̃\/X-chains; write\nChainiso-trm(X̃\/X) ⊆ Chaintrm(X̃\/X)\nfor the subcategory determined by the terminal isomorphisms [i.e., the isomor-\nphisms of Chaintrm(X̃\/X)]. Thus, it follows immediately from the definitions that\nwe obtain natural functors Chain(X̃\/X) → Chainiso-trm(X̃\/X) → Chaintrm(X̃\/X).\n(iii) We shall refer to as a [Π-]chain [of length n] [where n ≥ 0 is an integer]\nany finite sequence\nΠ0 \u0003 Π1 \u0003 . . . \u0003 Πn−1 \u0003 Πn\nof slim profinite groups Πj [for j = 0, . . . , n], each equipped with an open “rigid-\nifying homomorphism” ρj : Π̃ → Πj [i.e., since we are working with slim profinite\ngroups, an open homomorphism from some open subgroup of Π to Πj ] satisfying\nthe following conditions:\n(0Π) Π0 = Π [equipped with its natural rigidifying homomorphism Π̃ → Π].\n(1Π) There exists a [uniquely determined] surjection Πj \u0002 Gj , where Gj ⊆ G\nis an open subgroup, that is compatible with ρj and the natural composite\nmorphism Π̃ → Π \u0002 G.\n(2Π) Each kernel\nΔj\ndef\n= Ker(Πj \u0002 Gj ↪→ G)\nis slim and nontrivial; every prime dividing the order of a finite quotient\ngroup of Δj is invertible in k.\n(3Π) Suppose that X is a hyperbolic orbicurve [over k]. Then each Δj is a\npro-Σ group. Also, we shall refer to as a cuspidal decomposition group in\nΔj any commensurator in Δj of a nontrivial image via ρj of the inverse\nimage in Π̃ of the decomposition group in Δ [determined by α] of a cusp\nof X.\n(4Π) Each “Πj \u0003 Πj+1” [for j = 0, . . . , n − 1] is an “elementary operation”,\nas defined below.\nHere, an elementary operation “Πj \u0003 Πj+1” is defined to consist of the datum of\nan open “operation homomorphism” φ either from Πj to Πj+1 or from Πj+1 to Πj\nwhich is compatible with ρj , ρj+1, and, moreover, is of one of the following four\ntypes:\n50 SHINICHI MOCHIZUKI\n(a) Type \u0004: In this case, the elementary operation Πj \u0003 Πj+1 consists of\nan open immersion of profinite groups φ : Πj+1 ↪→ Πj .\n(b) Type \u0005: In this case, the elementary operation Πj \u0003 Πj+1 consists of\nan open immersion of profinite groups φ : Πj ↪→ Πj+1.\n(c) Type •: This type of elementary operation is only defined if X is a\nhyperbolic orbicurve. In this case, the elementary operation Πj \u0003 Πj+1\nconsists of a surjection of profinite groups φ : Πj \u0002 Πj+1, such that Ker(φ)\nis topologically normally generated by a cuspidal decomposition group C\nin Δj such that C is contained in some normal open torsion-free subgroup\nof Δj .\n(d) Type \u0006: This type of elementary operation is only defined if X is a\nhyperbolic orbicurve and Σ = Primes [which forces the characteristic of k\nto be zero]. In this case, the elementary operation Πj \u0003 Πj+1 consists\nof a surjection of profinite groups φ : Πj \u0002 Πj+1, such that Ker(φ) is\ntopologically normally generated by a finite closed subgroup of Δj .\nThus, any Π-chain determines a sequence of symbols ∈ {\u0004,\u0005, •,\u0006} [corresponding\nto the types of elementary operations in the Π-chain], which we shall refer to as the\ntype-chain associated to the Π-chain.\n(iv) An isomorphism between two Π-chains with identical type-chains [hence\nof the same length]\n(Π0 \u0003 . . . \u0003 Πn) ∼→ (Ψ0 \u0003 . . . \u0003 Ψn)\nis defined to be a collection of isomorphisms of profinite groups Πj\n∼→ Ψj [for\nj = 0, . . . , n] that are compatible with the rigidifying homomorphisms. [Here, we\nnote that the condition of compatibility with the rigidifying homomorphisms implies\n[since all of the profinite groups involved are slim] that every automorphism of a Π-\nchain is given by the identity, and that every isomorphism of Π-chains is compatible\nwith the respective operation homomorphisms.] Thus, one obtains a category\nChain(Π)\nwhose objects are the Π-chains [with arbitrary associated type-chain], and whose\nmorphisms are the isomorphisms between Π-chains [with identical type-chains].\nA terminal homomorphism between two Π-chains [with arbitrary associated type-\nchains]\n(Π0 \u0003 . . . \u0003 Πn) → (Ψ0 \u0003 . . . \u0003 Ψm)\nis defined to be an open outer homomorphism Πn → Ψm that is compatible [up to\ncomposition with an inner automorphism] with the open homomorphisms Πn → G,\nΨm → G. Thus, one obtains a category\nChaintrm(Π)\nwhose objects are the Π-chains [with arbitrary associated type-chain], and whose\nmorphisms are the terminal homomorphisms between Π-chains; write\nChainiso-trm(Π) ⊆ Chaintrm(Π)\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 51\nfor the subcategory determined by the terminal isomorphisms [i.e., the isomor-\nphisms of Chaintrm(Π)]. Thus, it follows immediately from the definitions that we\nobtain natural functors Chain(Π) → Chainiso-trm(Π) → Chaintrm(Π).\n(v) We shall use the notation\nChainiso-trm(∼){−} ⊆ Chainiso-trm(∼); Chaintrm(∼){−} ⊆ Chaintrm(∼)\n— where “(∼)” is either equal to “(X̃\/X)” or “(Π)”, and “{−}” contains some\nsubset of the set of symbols {\u0004,\u0005, •,\u0006}— to denote the respective full subcategories\ndetermined by the chains whose associated type-chain only contains the symbols\nthat belong to “{−}”. In particular, we shall write:\nDLoc(X̃\/X)\ndef\n= Chaintrm(X̃\/X){\u0004, •}; DLoc(Π) def= Chaintrm(Π){\u0004, •}\nÉtLoc(X̃\/X)\ndef\n= Chainiso-trm(X̃\/X){\u0004,\u0005}; ÉtLoc(Π) def= Chainiso-trm(Π){\u0004,\u0005}\n[cf. the theory of [Mzk9], §2; Remark 4.7.1 below].\nRemark 4.2.1. Thus, it follows immediately from the definitions that if, in the\nnotation of Definition 4.2, (i),\nX0 \u0003 X1 \u0003 . . . \u0003 Xn−1 \u0003 Xn\nis an X̃\/X-chain, then the resulting profinite groups Πj determine a Π-chain\nΠ0 \u0003 Π1 \u0003 . . . \u0003 Πn−1 \u0003 Πn\nwith the same associated type-chain. In particular, we obtain natural functors\nChain(X̃\/X) → Chain(Π)\nChainiso-trm(X̃\/X) → Chainiso-trm(Π); Chaintrm(X̃\/X) → Chaintrm(Π)\nwhich are compatible with the natural functors of Definition 4.2, (ii), (iv).\nRemark 4.2.2. Note that in the situation of Definition 4.2, (i), Gj is a slim\nprofinite group; 1 → Δj → Πj → Gj → 1 is an extension of GSAFG-type that\nadmits base-prime partial construction data (kj , Xj ,Σ), where Xj is a hyperbolic\norbicurve whenever X0 is a hyperbolic orbicurve; α, ρj determine [in light of the\nslimness of Πj ] a scheme-theoretic envelope αj : π\ntame\n1 (Xj) \u0002 Πj . That is to say,\nwe obtain, for each j, similar data to the data introduced at the beginning of\nDefinition 4.2. Here, relative to issue of verifying that Δj admits an open subgroup\nthat corresponds to a scheme-like covering of Xj , it is useful to recall, in the case\nof de-cuspidalization operations, i.e., “•”, the condition [cf. Definition 4.2, (i), (c);\nDefinition 4.2, (iii), (c)] that the cuspidal decomposition group under consideration\nbe contained in a normal open torsion-free subgroup [cf. Lemma 4.1, (iv)]; in the\ncase of de-orbification operations, i.e., “\u0006”, it is useful to recall the assumption\nthat Σ = Primes, together with the equivalence of definitions of the notion of a\n“hyperbolic orbicurve” discussed in §0.\n52 SHINICHI MOCHIZUKI\nProposition 4.3. (Re-ordering of Chains) In the notation of Definition 4.2,\nsuppose that Σ = Primes; let X0 \u0003 . . . \u0003 Xn be an X̃\/X-chain. Then there\nexists a terminally isomorphic X̃\/X-chain Y0 \u0003 . . . \u0003 Ym whose associated\ntype-chain is of the form\n\u0004, •, •, . . . , •,\u0005,\u0006,\u0006, . . . ,\u0006,\u0005\n— i.e., consists of the symbol \u0004, followed by a sequence of the symbols •, followed\nby the symbol \u0005, followed by a sequence of the symbols \u0006, followed by the symbol\n\u0005. Moreover, Ym−1 → Ym may be taken to arise from an extension of the base field\n[where we recall that this base field will always be a finite extension of k].\nProof. Indeed, let us first observe that by taking Ym−1 → Ym to arise from an\nappropriate extension of the base field, we may ignore the “kj-rationality” issues\nthat occur in Definition 4.2, (i), (c), (d). Next, let us observe that it is immediate\nfrom the definitions that we may always “move the symbol \u0004 to the top of the type-\nchain”. This completes the proof of Proposition 4.3 when X is not a hyperbolic\norbicurve. Thus, in the remainder of the proof, we may assume without loss of\ngenerality that X is a hyperbolic orbicurve, and that the symbols indexed by j ≥ 1\nof the type-chain are ∈ {\u0005, •,\u0006}. Next, let us observe that the operation morphisms\nindexed by j ≥ 1 always have domain indexed by j and codomain indexed by j+1.\nThus, by composing these operation morphisms, we obtain a morphism X1 →\nXn−1. Here, we may assume, without loss of generality, that X1 is a hyperbolic\ncurve, and that X1 → Xn−1 induces a Galois extension of function fields and an\nisomorphism of base fields. Also, we may assume that the morphism X1 → Xn−1\nfactors through a connected finite étale covering Z → Xn−1, where Z is a hyperbolic\ncurve. Thus, by considering the extension of function fields determined by X1 →\nXn−1, it follows immediately that Xn−1 may be obtained from X1 by applying\nde-cuspidalization operations [i.e., “•”] to X1 at the cusps of X1 that map to points\nof Xn−1, then forming the stack-theoretic quotient by the action of Gal(X1\/Xn−1)\n[i.e., “\u0005”], and finally applying suitable de-orbification [i.e., “\u0006”] operations to this\nquotient to recover Xn−1. This yields a type-chain of the desired form. ©\nAs the following example shows, the issue of permuting the symbols “\u0005”, “\u0006”\nis not so straightforward.\nExample 4.4. Non-permutability of Étale Quotients and De-orbifications.\nIn the notation of Definition 4.2, let us assume further Σ = Primes [so k is of char-\nacteristic zero]. Then there exists an X̃\/X-chain X0 \u0003 X1 \u0003 X2 of length 2 with\nassociated type-chain ∗0, ∗1, where ∗0, ∗1 ∈ {\u0005,\u0006}, ∗0 \u000b= ∗1, which is not terminally\nisomorphic to any X̃\/X-chain Y0 \u0003 Y1 \u0003 Y2 of length 2 with associated type-chain\n∗1, ∗0. Indeed:\n(i) The case of type-chain \u0005,\u0006: Let X be a hyperbolic curve of type (g, r)\nover k equipped with an automorphism σ of the k-scheme X of order 2 that has\nprecisely one fixed point x ∈ X(k); X0 = X \u0003 X1 the elementary operation of type\n\u0005 given by forming the stack-theoretic quotient of X by the action of σ; x1 ∈ X1(k)\nthe image of x in X1; X1 \u0003 X2 the nontrivial elementary operation of type \u0006 [i.e.,\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 53\nsuch that the corresponding operation morphism X1 → X2 is a non-isomorphism]\ndetermined by the point x1 ∈ X1(k). Thus, we assume thatX2 is a hyperbolic curve,\nwhose type we denote by (g2, r2). On the other hand, since X is a scheme, any\nchain Y0 \u0003 Y1 \u0003 Y2 of length 2 with associated type-chain \u0006,\u0005 satisfies Y0 ∼→ Y1\n[compatibly with X̃]. Thus, if Y2\n∼→ X2 over k, then the covering X = X0 → X2,\nwhich is ramified, of degree 2, together with the covering X\n∼→ Y0 ∼→ Y1 → Y2,\nwhich is unramified, of some degree d, yields equations\n2 · χ2 + 1 = χ = d · χ2\n[where we write χ\ndef\n= 2g − 2 + r, χ2 def= 2g2 − 2 + r2] — which imply [since d, χ,\nχ2 are positive integers] that d− 2 = χ2 = 1, hence that d = 3, χ2 = 1, χ = 3. In\nparticular, by choosing X so that χ is > 3 [e.g., X such that g ≥ 3], we obtain a\ncontradiction.\n(ii) The case of type-chain \u0006,\u0005: Let X be a proper hyperbolic orbicurve over\nk; X → C the coarse space associated to the algebraic stack X. Let us assume\nfurther that C is a [proper] hyperbolic curve over k; that the morphism X → C\nis a non-isomorphism which restricts to an isomorphism away from some point\nc ∈ C(k); and that there exists a finite étale covering \u0007 : C → D of degree 2 [so D\nis also a proper hyperbolic curve over k, which is not isomorphic to C]. [It is easy\nto construct such objects by starting from D and then constructing C, X.] Now\nwe take X0 = X \u0003 X1 def= C to be the elementary operation of type \u0006 determined\nby the unique point of x ∈ X(k) lying over c ∈ C(k); C = X1 \u0003 X2 def= D to be the\nelementary operation of type \u0005 determined by the finite étale covering \u0007 : C → D.\nWrite ex ≥ 2 for the ramification index of X → C at x; gD ≥ 2 for the genus of\nD; χD\ndef\n= 2gD − 2 ≥ 2. On the other hand, let us suppose that Y0 \u0003 Y1 \u0003 Y2\nis a chain of length 2 with associated type-chain \u0005,\u0006 such that X2 ∼→ Y2 over k.\nThen since D = X2\n∼→ Y2 is a scheme, it follows that the hyperbolic orbicurve Y1\nadmits a point y1 ∈ Y1(k) such that Y1 is a scheme away from y1. Write ey1 for\nthe ramification index of the operation morphism Y1 → Y2 at y1. Note that if Y1 is\na scheme, then the finite étale covering X = Y0 of Y1 is as well — a contradiction.\nThus, we conclude that Y1 is not a scheme at y1, i.e., ey1 ≥ 2. Next, let us observe\nthat if the finite étale morphism Y0 → Y1 is not an isomorphism, i.e., of degree\nd ≥ 2, then the morphisms X → C → D and X = Y0 → Y1 give rise to a relation\n2χD + (ex − 1)\/ex = d(χD + (ey1 − 1)\/ey1)\n— i.e., 1 > (ex − 1)\/ex = (d − 2)χD + d(ey1 − 1)\/ey1 ≥ d(ey1 − 1)\/ey1 ≥ d\/2 ≥ 1,\na contradiction. Thus, we conclude that d = 1, i.e., that the operation morphism\nX = Y0 → Y1 is an isomorphism. But this implies that Y2 is isomorphic to the\ncoarse space associated to X, i.e., that we have an isomorphism Y2\n∼→ C, hence an\nisomorphism D = X2\n∼→ Y2 ∼→ C — a contradiction.\nNext, we recall the group-theoretic characterization of the cuspidal decomposi-\ntion groups of a hyperbolic [orbi]curve given in [Mzk12].\nLemma 4.5. (Cuspidal Decomposition Groups) Let G be a slim profinite\ngroup;\n1 → Δ → Π → G → 1\n54 SHINICHI MOCHIZUKI\nan extension of GSAFG-type that admits base-prime [cf. Definition 2.1, (iv)]\npartial construction data (k, k̃,X,Σ), where X is a hyperbolic orbicurve; α :\nπtame1 (X) \u0002 Π a scheme-theoretic envelope; l ∈ Σ a prime such that the\ncyclotomic character χ\ncyclo\nG : G → Z×l [i.e., the character whose restriction\nto πtame1 (X) via α and the surjection Π \u0002 G is the usual cyclotomic character\nπtame1 (X) \u0002 Gal(k̃\/k) → Z×l ] has open image [i.e., in the terminology of [Mzk12],\n“the outer action of G on Δ is l-cyclotomically full”]. We recall from [Mzk12]\nthat a character χ : G → Z×l is called Q-cyclotomic [of weight w ∈ Q] if there\nexist integers a, b, where b > 0, such that χb = (χ\ncyclo\nG )\na, w = 2a\/b [cf. [Mzk12],\nDefinition 2.3, (i), (ii)]. Then:\n(i) X is non-proper if and only if every torsion-free pro-Σ open subgroup of\nΔ is free pro-Σ.\n(ii) Let M be a finite-dimensional Ql-vector space equipped with a continuous\nG-action. Then we shall say that this action is quasi-trivial if it factors through\na finite quotient of G [cf. [Mzk12], Definition 2.3, (i)]. We shall write τ(M)\nfor the quasi-trivial rank of M [cf. [Mzk12], Definition 2.3, (i)], i.e., the sum\nof the Ql-dimensions of the quasi-trivial subquotients M\nj\/M j+1 of any filtration\nMn ⊆ . . . ⊆ M j ⊆ . . .M0 = M of M by Ql[G]-modules such that each M j\/M j+1 is\neither quasi-trivial or has no nontrivial subquotients. If χ : G → Z×l is a character,\nthen we shall write\ndχ(M)\ndef\n= τ(M(χ−1))− τ(HomQl(M,Ql))\n[where “M(χ−1)” denotes the result of “twisting” M by the character χ−1]. We\nshall say that two characters G → Z×l are power-equivalent if there exists a\npositive integer n such that the n-th powers of the two characters coincide. Then\ndχ(M), regarded as a function of χ, depends only on the power-equivalence\nclass of χ.\n(iii) Suppose that X is not proper [cf. (i)]. Then the character G → Z×l\narising from the determinant of the G-module Hab⊗Ql, where H ⊆ Δ is a torsion-\nfree pro-Σ characteristic open subgroup such that Hab ⊗Ql \u000b= 0, is Q-cyclotomic\nof positive weight. Moreover, for every sufficiently small characteristic open\nsubgroup H ⊆ Δ, the power-equivalence class of the cyclotomic character χcycloG\nmay be characterized as the unique power-equivalence class of characters χ : G →\nZ×l of the form χ = χ\n∗ · χ∗, where χ∗ : G → Z×l (respectively, χ∗ : G → Z×l )\nis a Q-cyclotomic character χ• of maximal (respectively, minimal) weight such\nthat τ(M(χ−1• )) \u000b= 0 for some subquotient G-module M of (Hab ⊗Ql)⊕Ql [where\nthe final direct summand Ql is equipped with the trivial G-action]. Moreover, in\nthis situation, if χ = χ\ncyclo\nG , then the divisor of cusps of the covering of X ×k k̃\ndetermined by H is a disjoint union of dχ(H\nab ⊗Ql) + 1 copies of Spec(k̃).\n(iv) Suppose that X is not proper [cf. (i)]. Let H ⊆ Δ be a torsion-free\npro-Σ characteristic open subgroup; H \u0002 H∗ the maximal pro-l quotient of H.\nThen the decomposition groups of cusps ⊆ H∗ may be characterized [“group-\ntheoretically”] as the maximal closed subgroups I ⊆ H∗ isomorphic to Zl which\nsatisfy the following condition: We have\ndχcyclo\nG\n(Jab ⊗Ql) + 1 = [I · J : J ] · dχcyclo\nG\n((I · J)ab ⊗Ql) + 1\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 55\n[i.e., “the covering of curves corresponding to J ⊆ I · J is totally ramified at\nprecisely one cusp”] for every characteristic open subgroup J ⊆ H∗.\n(v) Let X, H, H∗ be as in (iv). Then the set of cusps of the covering of X×k k̃\ndetermined by H is in natural bijective correspondence with the set of conjugacy\nclasses in H∗ of decomposition groups of cusps [as described in (iv)]. Moreover,\nthis correspondence is functorial in H and compatible with the natural actions\nby Π on both sides. In particular, by allowing H to vary, this yields a [“group-\ntheoretic”] characterization of the decomposition groups of cusps in Π.\n(vi) Let I ⊆ Π be a decomposition group of a cusp. Then I = CΠ(I\n⋂\nΔ)\n[cf. §0].\nProof. Assertion (i) may be reduced to the case of hyperbolic curves via Lemma 4.1,\n(iv), in which case it is well-known [cf., e.g., [Mzk12], Remark 1.1.3]. Assertion (ii)\nis immediate from the definitions. Assertion (iii) follows immediately from [Mzk12],\nProposition 2.4, (iv), (vii); the proof of [Mzk12], Corollary 2.7, (i). Assertion (iv) is\n[in light of assertion (iii)] precisely a summary of the argument of [Mzk12], Theorem\n1.6, (i). Finally, assertions (v), (vi) follow immediately from [Mzk12], Proposition\n1.2, (i), (ii). ©\nDefinition 4.6.\n(i) Let V (respectively, F; S) be a set of isomorphism classes of algebraic\nstacks (respectively, set of isomorphism classes of fields; set of nonempty subsets of\nPrimes);\nD ⊆ V× F× S\na subset of the direct product set V × F × S, which we shall think of as a set of\ncollections of partial construction data. In the following discussion, we shall use\n“[−]” to denote the isomorphism class of “−”. We shall say that D is chain-full if\nfor every extension 1 → Δ → Π → G → 1 of GSAFG-type, where G is slim, that\nadmits base-prime partial construction data (X, k,Σ) such that ([X], [k],Σ) ∈ D [cf.\nDefinition 4.2], it follows that every “Xj , kj” [cf. Definition 4.2, (i)] appearing in\nan X̃\/X-chain [where X̃ → X is the pro-finite étale covering of X determined by\nsome scheme-theoretic envelope for Π] determines an element ([Xj ], [kj ],Σ) ∈ D.\n(ii) Let D be as in (i); suppose that D is chain-full. Then we shall say that\nthe rel-isom-DGC holds [i.e., “the relative isomorphism version of the Grothendieck\nConjecture for D holds”] (respectively, the rel-hom-DGC holds [i.e., “the relative\nhomomorphism version of the Grothendieck Conjecture for D holds”]), or that, the\nrel-isom-GC holds for D (respectively, the rel-hom-GC holds for D) if the following\ncondition is satisfied: For i = 1, 2, let\n1 → Δi → Πi → Gi → 1\nbe an extension of GSAFG-type, where Gi is slim, that admits base-prime partial\nconstruction data (ki, Xi,Σi) such that ([Xi], [ki],Σi) ∈ D; αi : πtame1 (Xi) \u0002 Πi a\n56 SHINICHI MOCHIZUKI\nscheme-theoretic envelope; ζk : k1\n∼→ k2 an isomorphism of fields that induces, via\nthe αi, an outer isomorphism ζG : G1\n∼→ G2. Then the natural map\nIsomk1,k2(X1, X2) → IsomoutG1,G2(Π1,Π2)\n(respectively, Homdomk1,k2(X1, X2) → Homout-openG1,G2 (Π1,Π2))\ndetermined by the αi from the set of isomorphisms of schemes X1\n∼→ X2 lying over\nζk : k1\n∼→ k2 (respectively, the set of dominant morphisms of schemes X1 → X2\nlying over ζk : k1\n∼→ k2) to the set of outer isomorphisms of profinite groups Π1 ∼→ Π2\nlying over ζG : G1\n∼→ G2 (respectively, the set of open outer homomorphisms of\nprofinite groups Π1 → Π2 lying over ζG : G1 ∼→ G2) is a bijection.\nRemark 4.6.1. Of course, in a similar vein, one may also formulate the notions\nthat “the absolute isomorphism version of the Grothendieck Conjecture holds for\nD”, “the absolute homomorphism version of the Grothendieck Conjecture holds for\nD”, “the semi-absolute isomorphism version of the Grothendieck Conjecture holds\nfor D”, “the semi-absolute homomorphism version of the Grothendieck Conjecture\nholds for D”, etc. Since we shall not use these versions in the discussion to follow,\nwe leave the routine details of their formulation to the interested reader.\nTheorem 4.7. (Semi-absoluteness of Chains of Elementary Operations)\nLet D be a chain-full set of collections of partial construction data [cf. Def-\ninition 4.6, (i)] such that the rel-isom-DGC holds [cf. Definition 4.6, (ii)]. For\ni = 1, 2, let Gi be a slim profinite group;\n1 → Δi → Πi → Gi → 1\nan extension of GSAFG-type that admits base-prime [cf. Definition 2.1,\n(iv)] partial construction data (ki, k̃i, Xi,Σi) such that ([Xi], [ki],Σi) ∈ D; αi :\nπtame1 (Xi) \u0002 Πi a scheme-theoretic envelope. Also, let us suppose further that\nthe following conditions are satisfied:\n(a) if either X1 or X2 is a hyperbolic orbicurve, then both X1 and X2\nare hyperbolic orbicurves;\n(b) if either X1 or X2 is a non-proper hyperbolic orbicurve, then there\nexists a prime number l ∈ Σ1\n⋂\nΣ2 such that for i = 1, 2, the cyclotomic\ncharacter Gi → Z×l [i.e., the character whose restriction to πtame1 (Xi)\nvia αi and the surjection Πi \u0002 Gi is the usual cyclotomic character\nπtame1 (Xi) \u0002 Gal(k̃i\/ki) → Z×l ] has open image.\nLet\nφ : Π1\n∼→ Π2\nbe an isomorphism of profinite groups that induces isomorphisms φΔ : Δ1\n∼→ Δ2,\nφG : G1\n∼→ G2. Then:\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 57\n(i) The natural functors [cf. Remark 4.2.1]\nChain(X̃i\/Xi) → Chain(Πi); Chainiso-trm(X̃i\/Xi) → Chainiso-trm(Πi)\nÉtLoc(X̃i\/Xi) → ÉtLoc(Πi)\nare equivalences of categories that are compatible with passing to type-chains.\n(ii) The isomorphism φ induces equivalences of categories\nChain(Π1)\n∼→ Chain(Π2); Chainiso-trm(Π1) ∼→ Chainiso-trm(Π2)\nÉtLoc(Π1)\n∼→ ÉtLoc(Π2)\nthat are compatible with passing to type-chains and functorial in φ.\n(iii) Suppose further that the rel-hom-DGC holds [cf. Definition 4.6, (ii)],\nand that for i = 1, 2, Xi is a hyperbolic orbicurve. Then the natural functors\n[cf. Remark 4.2.1]\nChaintrm(X̃i\/Xi) → Chaintrm(Πi); DLoc(X̃i\/Xi) → DLoc(Πi)\nare equivalences of categories that are compatible with passing to type-chains.\n(iv) In the situation of (iii), the isomorphism φ induces equivalences of cat-\negories\nChaintrm(Π1)\n∼→ Chaintrm(Π2); DLoc(Π1) ∼→ DLoc(Π2)\nthat are compatible with passing to type-chains and functorial in φ.\nProof. First, we consider the natural functor\nChain(X̃i\/Xi) → Chain(Πi)\nof Remark 4.2.1. To conclude that this functor is an equivalence of categories, it\nfollows immediately from the definitions of the categories involved that it suffices to\nverify that the X̃i\/Xi-chain and Πi-chain versions of the four types of elementary\noperations \u0004, \u0005, •, \u0006 described in Definition 4.2, (i), (iii), correspond bijectively\nto one another. This is immediate from the definitions (respectively, the cuspidal\nportion of Lemma 4.1, (i), (v); the “closed point of X” portion of Lemma 4.1, (iii),\n(v)) for \u0004 (respectively, •; \u0006). [Here, we note that in the case of •, \u0006, the “kj+1-\nrationality” [in the notation of Definition 4.2, (i), (c), (d)] of the cusp or possibly\nnon-scheme-like point in question follows immediately from Lemma 4.1, (vi), by\ntaking “xB” to be the various Galois conjugates of this point.] Finally, the desired\ncorrespondence for \u0005 follows from our assumption that the rel-isom-DGC holds\nby applying this “rel-isom-DGC” as was done in the proofs of [Mzk7], Theorem\n2.4; [Mzk9], Theorem 2.3, (i). This completes the proof that the natural functor\nChain(X̃i\/Xi) → Chain(Πi) is an equivalence. A similar application of the “rel-\nisom-DGC” then yields the equivalences Chainiso-trm(X̃i\/Xi)\n∼→ Chainiso-trm(Πi),\n58 SHINICHI MOCHIZUKI\nÉtLoc(X̃i\/Xi)\n∼→ ÉtLoc(Πi). In a similar vein, the “rel-hom-DGC” [cf. asser-\ntion (iii)] implies the equivalences of categories Chaintrm(X̃i\/Xi)\n∼→ Chaintrm(Πi),\nDLoc(X̃i\/Xi)\n∼→ DLoc(Πi) of assertion (iii). This completes the proof of assertions\n(i), (iii).\nFinally, to obtain the equivalences of assertions (ii), (iv), it suffices to observe\nthat the definitions of the various categories involved are entirely “group-theoretic”.\nHere, we note that the “group-theoreticity” of the elementary operations of type \u0004,\n\u0005, \u0006 is immediate; the “group-theoreticity” of the elementary operations of type\n• follows immediately from Lemma 4.5, (v) [in light of our assumptions (a), (b)].\nAlso, we observe that whenever the Xi [for i = 1, 2] are hyperbolic orbicurves, Σi\nmay be recovered “group-theoretically” from Δi [i.e., as the unique minimal subset\nΣ′ ⊆ Primes such that Δi is almost pro-Σ′]. This completes the proof of assertions\n(ii), (iv). ©\nRemark 4.7.1. The portion of Theorem 4.7 concerning the categories “ÉtLoc(−)”\n[cf. also Example 4.8 below; Corollary 2.8, (ii)] and “DLoc(−)” allows one to relate\nthe theory of the present §4 to the theory of [Mzk9], §2 [cf., especially, [Mzk9],\nTheorem 2.3].\nExample 4.8. Hyperbolic Orbicurves. Let p be a prime number; S the set\nof subsets of Primes containing p; V the set of isomorphism classes of hyperbolic\norbicurves over fields of cardinality ≤ the cardinality of Qp.\n(i) Let F be the set of isomorphism classes of generalized sub-p-adic fields [i.e.,\nsubfields of finitely generated extensions of the quotient field of the ring of Witt\nvectors with coefficients in an algebraic closure of Fp — cf. [Mzk5], Definition 4.11];\nD = V× F× S. Then let us observe that:\nThe hypotheses of Theorem 4.7, (i), (ii), are satisfied relative to this D.\nIndeed, it is immediate that D is chain-full; the rel-isom-DGC follows from [Mzk5],\nTheorem 4.12; the prime p clearly serves as a prime “l” as in the statement of\nTheorem 4.7. Moreover, we recall from [Mzk5], Lemma 4.14, that the absolute\nGalois group of a generalized sub-p-adic field is always slim.\n(ii) Let F be the set of isomorphism classes of sub-p-adic fields [i.e., subfields of\nfinitely generated extensions of Qp — cf. [Mzk3], Definition 15.4, (i)]; D = V×F×S.\nThen let us observe that:\nThe hypotheses of Theorem 4.7, (iii), (iv), are satisfied relative to this D.\nIndeed, it is immediate that D is chain-full; the rel-hom-DGC follows from [Mzk3],\nTheorem A; the prime p clearly serves as a prime “l” as in the statement of Theorem\n4.7. Moreover, we recall from [Mzk3], Lemma 15.8, that the absolute Galois group\nof a sub-p-adic field is always slim.\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 59\nExample 4.9. Iso-poly-hyperbolic Orbisurfaces.\n(i) Let k be a field of characteristic zero. Then we recall from [Mzk3], Definition\na2.1, that a smooth k-scheme X is called a hyperbolically fibred surface if it admits\nthe structure of a family of hyperbolic curves [cf. §0] over a hyperbolic curve Y\nover k. If X is a smooth, generically scheme-like, geometrically connected algebraic\nstack over k, then we shall say that X is an iso-poly-hyperbolic orbisurface [cf. the\nterm “poly-hyperbolic” as it is defined in [Mzk4], Definition 4.6] if X admits a finite\nétale covering which is a hyperbolically fibred surface over some finite extension of\nk.\n(ii) Let p be a prime number; S\ndef\n= {Primes} [where we regard Primes as the\nunique non-proper subset of Primes]; F the set of isomorphism classes of sub-p-adic\nfields; V the set of isomorphism classes of iso-poly-hyperbolic orbisurfaces [cf. (i)]\nover sub-p-adic fields; D = V× F× S. Then let us observe that:\nThe hypotheses of Theorem 4.7, (i), (ii), are satisfied relative to this D.\nIndeed, it is immediate that D is chain-full; the rel-isom-DGC, as well as the slim-\nness of the Δi [for i = 1, 2], follows immediately from [Mzk3], Theorem D. Moreover,\nwe recall from [Mzk3], Lemma 15.8, that the absolute Galois group of a sub-p-adic\nfield is always slim.\n(iii) Let k be a sub-p-adic field; X the moduli stack of hyperbolic curves of type\n(0, 5) [i.e., the moduli stack of smooth curves of genus 0 with 5 distinct, unordered\npoints] over k; X̃ → X a “universal” pro-finite étale covering of X; k the algebraic\nclosure of k determined by X̃ → X. Then one verifies immediately that X is an\niso-poly-hyperbolic orbisurface over k. Write 1 → Δ → Π → G → 1 for the GSAFG-\nextension defined by the natural surjection π1(X) = Gal(X̃\/X) \u0002 Gal(k\/k) [which\nwe regard as equipped with the tautological scheme-theoretic envelope given by the\nidentity] and Loc(X̃\/X) ⊆ Chaintrm(X̃\/X) (respectively, Loc(Π) ⊆ Chaintrm(Π))\nfor the subcategory determined by the terminal morphisms (respectively, homomor-\nphisms) which are finite étale (respectively, injective). Then it follows immediately\nfrom (ii); Theorem 4.7, (i), that we have an equivalence of categories\nLoc(X̃\/X)\n∼→ Loc(Π)\n[cf. [Mzk7], Theorem 2.4; [Mzk9], Theorem 2.3, (i)]. Moreover, the objects of these\ncategories “Loc(−)” determined by X, Π [i.e., by the unique chain of length 0] is\nterminal [cf. [Mzk2], Theorem C] — i.e., a “core” [cf. the terminology of [Mzk7],\n§2; [Mzk8], §2].\nFinally, we observe that the theory of the present §4 admits a “tempered ver-\nsion”, in the case of hyperbolic orbicurves over MLF’s. We begin by recalling basic\nfacts concerning tempered fundamental groups. Let k be an MLF of residue char-\nacteristic p; k an algebraic closure of k; X a hyperbolic orbicurve over k. We shall\nuse a subscript k to denote the result of a base-change from k to k. Write\nπ\ntp\n1 (X); π\ntp\n1 (Xk)\n60 SHINICHI MOCHIZUKI\nfor the tempered fundamental groups of X, Xk [cf. [André], §4; [Mzk10], Exam-\nples 3.10, 5.6]. Thus, the profinite completion of π\ntp\n1 (X) (respectively, π\ntp\n1 (Xk))\nis naturally isomorphic to the usual étale fundamental group π1(X) (respectively,\nπ1(Xk)). If H ⊆ πtp1 (Xk) is an open subgroup of finite index, then recall that the\nminimal co-free subgroup of H\nHco-fr ⊆ H\n[cf. §0] is precisely the subgroup of H with the property that the quotient H \u0002\nH\/Hco-fr corresponds to the tempered covering of Xk determined by the universal\ncovering of the dual graph of the special fiber of a stable model of Xk — cf. [André],\nproof of Lemma 6.1.1.\nProposition 4.10. (Basic Properties of Tempered Fundamental Groups)\nIn the notation of the above discussion, suppose further that φ : X → Y is a mor-\nphism of hyperbolic orbicurves over k. For Z = X,Y , let us write\nΠ\ntp\nZ\ndef\n= π\ntp\n1 (Z); Δ\ntp\nZ\ndef\n= π\ntp\n1 (Zk)\nand denote the profinite completions of Π\ntp\nZ , Δ\ntp\nZ by Π̂\ntp\nZ , Δ̂\ntp\nZ , respectively; in\nthe following, all “co-free completions” [cf. §0] of open subgroups of finite index\nin Π\ntp\nX (respectively, Δ\ntp\nX ) will be with respect to [the intersections of such open\nsubgroups with] the subgroup Δ\ntp\nX ⊆ ΠtpX (respectively, ΔtpX ⊆ ΔtpX ). Then:\n(i) The natural homomorphism Π\ntp\nX → Π̂tpX ∼→ π1(X) (respectively, ΔtpX →\nΔ̂\ntp\nX\n∼→ π1(Xk)) is injective. In fact, if H ⊆ ΔtpX is any characteristic open\nsubgroup of finite index, then Π\ntp\nX\/H\nco-fr, Δtp\/Hco-fr inject into their respective\nprofinite completions. In particular, π\ntp\n1 (X) (respectively, π\ntp\n1 (Xk)) is naturally\nisomorphic to its π1(X)-co-free completion (respectively, π1(Xk)-co-free com-\npletion) [cf. §0].\n(ii) Π\ntp\nX (respectively, Δ\ntp\nX ) is normally terminal in Π̂\ntp\nX (respectively, Δ̂\ntp\nX ).\n(iii) Suppose that φ is either a de-cuspidalization morphism [i.e., an open\nimmersion whose image is the complement of a single k-valued point of Y — cf.\nDefinition 4.2, (i), (c)] or a de-orbification morphism [i.e., a partial coarsification\nmorphism which is an isomorphism over the complement of a single k-valued point\nof Y — cf. Definition 4.2, (i), (d)]. Then the natural homomorphism Π\ntp\nX → ΠtpY\n(respectively, Δ\ntp\nX → ΔtpY ) may be reconstructed — “group-theoretically” —\nfrom its profinite completion Π̂\ntp\nX \u0002 Π̂\ntp\nY (respectively, Δ̂\ntp\nX \u0002 Δ̂\ntp\nY ) as the natural\nmorphism from Π\ntp\nX (respectively, Δ\ntp\nX ) to the co-free completion of Π\ntp\nX with\nrespect to Π̂\ntp\nY (respectively, Δ̂\ntp\nY ) [cf. §0].\n(iv) Let l ∈ Primes. If J ⊆ ΔtpX is an open subgroup of finite index, write\nJ → J [l] for the co-free completion of J with respect to the maximal pro-l\nquotient of the profinite completion of J . Let H ⊆ ΔtpX be an open subgroup of\nfinite index. Suppose that l \u000b= p. Then the dual graph ΓH of the special fiber of a\nstable model of the covering of Xk corresponding to H determines verticial and\nedge-like subgroups of H [l] [i.e., decomposition groups of the vertices and edges\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 61\nof ΓH — cf. [Mzk10], Theorem 3.7, (i), (iii)]. The verticial (respectively, edge-\nlike) subgroups of H [l] may be characterized — “group-theoretically” — as the\nmaximal compact subgroups (respectively, nontrivial intersections of two\ndistinct maximal compact subgroups) of H [l]. In particular, the graph ΓH\nmay be reconstructed — “group-theoretically” — from the verticial and edge-\nlike subgroups of H [l], together with their various mutual inclusion relations.\n(v) The prime number p may be characterized — “group-theoretically”\n— as the unique prime number l such that their exist open subgroups H, J ⊆ ΔtpX\nof finite index, together with distinct prime numbers l1, l2, satisfying the following\nproperties: (a) H is a normal subgroup of J of index l; (b) for i = 1, 2, the outer\naction of J on H [li] [cf. (iv)] fixes [the conjugacy class in H [li] of] and induces\nthe trivial outer action on some maximal compact subgroup of H [li] [cf.\n(iv)].\n(vi) Let l be a prime number \u000b= p [a “group-theoretic” condition, by (v)!];\nH ⊆ ΔtpX an open subgroup of finite index. Then the set of cusps of the covering\nof Xk corresponding to H may be characterized — “group-theoretically” — as\nthe set of conjugacy classes in H [l] of the commensurators in H [l] of the images in\nH [l] of edge-like subgroups of J [l] [cf. (iv)], where J ⊆ H is an open subgroup of\nfinite index, which are not contained in edge-like subgroups of H [l]. In particular,\nby allowing H to vary, this yields a [“group-theoretic”] characterization of the\ndecomposition groups of cusps in Δ\ntp\nX , Π\ntp\nX [i.e., a “tempered version” of\nLemma 4.5, (v)].\nProof. Assertion (i) follows immediately from the discussion at the beginning of\n[Mzk10], §6 [cf. also the discussion of [André], §4.5]. Assertion (ii) is the content\nof [Mzk10], Lemma 6.1, (ii), (iii) [cf. also [André], Corollary 6.2.2]. Assertion (iii)\nfollows immediately from assertion (i). Assertion (iv) follows immediately from\n[Mzk10], Theorem 3.7, (iv); [Mzk10], Corollary 3.9 [cf. also the proof of [Mzk10],\nCorollary 3.11]. Assertions (v), (vi) amount to summaries of the relevant portions\nof the proof of [Mzk10], Corollary 3.11. Here, in assertion (v), we observe that at\nleast one of the li is \u000b= p; thus, for this choice of li, the action of J fixes and induces\nthe trivial outer action on some verticial subgroup of H [li]. ©\nRemark 4.10.1. It is not clear to the author at the time of writing how to prove\na version of Proposition 4.10, (vi), for decomposition groups of closed points which\nare not cusps [i.e., a “tempered version” of Lemma 4.1, (iii)].\nRemark 4.10.2. A certain fact applied in the portion of the proof of [Mzk10],\nCorollary 3.11, summarized in Proposition 4.10, (vi), is only given a somewhat\nsketchy proof in loc. cit. A more detailed treatment of this fact is given in [Mzk15],\nCorollary 2.11.\nNow we are ready to state the tempered version of Definition 4.2.\nDefinition 4.11. In the notation of the above discussion, let\n1 → Δ → Π → G → 1\n62 SHINICHI MOCHIZUKI\nbe an extension of topological groups that is isomorphic to the natural extension\n1 → πtp1 (Xk) → πtp1 (X) → Gal(k\/k) → 1 via some isomorphism α : πtp1 (X)\n∼→ Π,\nwhich we shall refer to as a scheme-theoretic envelope. Write Π̂ for the profinite\ncompletion of Π; X̃ → X for the pro-finite étale covering of X determined by\nthe completion of α [so Π̂ = Gal(X̃\/X)]; k̃ for the resulting field extension of k.\nIn a similar vein, we shall write Π̃ for the projective system of topological groups\ndetermined by the open subgroups of finite index of Π [cf. Definition 4.2]. Then:\n(i) We shall refer to as an [Π-]chain [of length n] [where n ≥ 0 is an integer]\nany finite sequence\nΠ0 \u0003 Π1 \u0003 . . . \u0003 Πn−1 \u0003 Πn\nof topological groups Πj [for j = 0, . . . , n] with slim profinite completions Π̂j , each\nequipped with a “rigidifying homomorphism” ρj : Π̃ → Πj which is of DOF-type\n[i.e., which maps some member of the projective system Π̃ onto a dense subgroup of\nan open subgroup of finite index of Πj — cf. §0] satisfying the following conditions:\n(0tp) Π0 = Π [equipped with its natural rigidifying homomorphism Π̃ → Π].\n(1tp) There exists a [uniquely determined] surjection Πj \u0002 Gj , where Gj ⊆ G\nis an open subgroup, that is compatible with ρj and the natural composite\nmorphism Π̃ → Π \u0002 G.\n(2tp) Each kernel\nΔj\ndef\n= Ker(Πj \u0002 Gj ↪→ G)\nhas a slim, nontrivial profinite completion Δ̂j .\n(3tp) The topological groups Πj , Δj are residually finite. Also, we shall refer\nto as a cuspidal decomposition group in Δ̂j any Δ̂j-conjugate of the com-\nmensurator in Δ̂j of a nontrivial image via ρj of the inverse image in Π̃ of\nthe decomposition group in Δ [determined by α] of a cusp of X.\n(4tp) Each “Πj \u0003 Πj+1” [for j = 0, . . . , n − 1] is an “elementary operation”,\nas defined below.\nHere, an elementary operation “Πj \u0003 Πj+1” is defined to consist of the datum of\nan “operation homomorphism” φ of DOF-type either from Πj to Πj+1 or from Πj+1\nto Πj which is compatible with ρj , ρj+1, and, moreover, is of one of the following\nfour types:\n(a) Type \u0004: In this case, the elementary operation Πj \u0003 Πj+1 consists of\nan immersion of OF-type [cf. §0] φ : Πj+1 ↪→ Πj .\n(b) Type \u0005: In this case, the elementary operation Πj \u0003 Πj+1 consists of\nan immersion of OF-type [cf. §0] φ : Πj ↪→ Πj+1.\n(c) Type •: In this case, the elementary operation Πj \u0003 Πj+1 consists\nof a dense homomorphism φ : Πj → Πj+1 which is isomorphic to the\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 63\nco-free completion of Πj with respect to the induced profinite quotient\nφ̂ : Π̂j \u0002 Π̂j+1 [and the subgroup Δj ], such that Ker(φ̂) is topologically\nnormally generated by a cuspidal decomposition group C in Δ̂j such that\nC is contained in some normal open torsion-free subgroup of Δ̂j .\n(d) Type \u0006: In this case, the elementary operation Πj \u0003 Πj+1 consists\nof a dense homomorphism φ : Πj → Πj+1 which is isomorphic to the\nco-free completion of Πj with respect to the induced profinite quotient\nφ̂ : Π̂j \u0002 Π̂j+1 [and the subgroup Δj ], such that Ker(φ̂) is topologically\nnormally generated by a finite closed subgroup of Δ̂j .\nThus, any Π-chain determines a sequence of symbols ∈ {\u0004,\u0005, •,\u0006} [corresponding\nto the types of elementary operations in the Π-chain], which we shall refer to as the\ntype-chain associated to the Π-chain.\n(ii) An isomorphism between two Π-chains with identical type-chains [hence of\nthe same length]\n(Π0 \u0003 . . . \u0003 Πn) ∼→ (Ψ0 \u0003 . . . \u0003 Ψn)\nis defined to be a collection of isomorphisms of topological groups Πj\n∼→ Ψj [for\nj = 0, . . . , n] that are compatible with the rigidifying homomorphisms. [Here, we\nnote that the condition of compatibility with the rigidifying homomorphisms implies\n[since all of the topological groups involved are residually finite with slim profinite\ncompletions] that every automorphism of a Π-chain is given by the identity, and that\nevery isomorphism of Π-chains of the same length is compatible with the respective\noperation homomorphisms.] Thus, one obtains a category\nChain(Π)\nwhose objects are the Π-chains [with arbitrary associated type-chain], and whose\nmorphisms are the isomorphisms between Π-chains [with identical type-chains].\nA terminal homomorphism between two Π-chains [with arbitrary associated type-\nchains]\n(Π0 \u0003 . . . \u0003 Πn) → (Ψ0 \u0003 . . . \u0003 Ψm)\nis defined to be an outer homomorphism of DOF-type [cf. §0; [Mzk10], Theorem\n6.4] Πn → Ψm that is compatible [up to composition with an inner automorphism]\nwith the open homomorphisms Πn → G, Ψm → G. Thus, one obtains a category\nChaintrm(Π)\nwhose objects are the Π-chains [with arbitrary associated type-chain], and whose\nmorphisms are the terminal homomorphisms between Π-chains; write\nChainiso-trm(Π) ⊆ Chaintrm(Π)\nfor the subcategory determined by the terminal isomorphisms [i.e., the isomor-\nphisms of Chaintrm(Π)]. Thus, it follows immediately from the definitions that we\n64 SHINICHI MOCHIZUKI\nobtain natural functors Chain(Π) → Chainiso-trm(Π) → Chaintrm(Π). Finally, we\nhave (sub)categories\nChainiso-trm(Π){−} ⊆ Chainiso-trm(Π); Chaintrm(Π){−} ⊆ Chaintrm(Π)\nDLoc(Π)\ndef\n= Chaintrm(Π){\u0004, •}; ÉtLoc(Π) def= Chainiso-trm(Π){\u0004,\u0005}\n[cf. Definition 4.2, (v)].\nRemark 4.11.1. Just as in the profinite case [i.e., Remark 4.2.1], we have natural\nfunctors\nChain(X̃\/X) → Chain(Π) → Chain(Π̂)\nChainiso-trm(X̃\/X) → Chainiso-trm(Π) → Chainiso-trm(Π̂)\nChaintrm(X̃\/X) → Chaintrm(Π) → Chaintrm(Π̂)\n— where we apply Proposition 4.10, (iii), in the construction of the first arrow in\neach line; the second arrow in each line is the natural functor obtained by profinite\ncompletion; the various composite functors of the two functors in each line are the\nnatural functors of Remark 4.2.1.\nRemark 4.11.2. A similar remark to Remark 4.2.2 applies in the present tem-\npered case.\nTheorem 4.12. (Tempered Chains of Elementary Operations) For i =\n1, 2, let ki be an MLF of residue characteristic pi; ki an algebraic closure of ki;\nXi a hyperbolic orbicurve over ki;\n1 → Δi → Πi → Gi → 1\nan extension of topological groups that is isomorphic to the natural exten-\nsion 1 → πtp1 ((Xi)ki) → π\ntp\n1 (Xi) → Gal(ki\/ki) → 1 via some scheme-theoretic\nenvelope αi : π\ntp\n1 (Xi)\n∼→ Πi. Let\nφ : Π1\n∼→ Π2\nbe an isomorphism of topological groups. Then:\n(i) The natural functors [cf. Remark 4.11.1]\nChain(X̃i\/Xi) → Chain(Πi); Chainiso-trm(X̃i\/Xi) → Chainiso-trm(Πi)\nÉtLoc(X̃i\/Xi) → ÉtLoc(Πi)\nChaintrm(X̃i\/Xi) → Chaintrm(Πi); DLoc(X̃i\/Xi) → DLoc(Πi)\nare equivalences of categories that are compatible with passing to type-chains.\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 65\n(ii) We have p1 = p2; the isomorphism φ induces isomorphisms φΔ : Δ1\n∼→ Δ2,\nφG : G1\n∼→ G2, as well as equivalences of categories\nChain(Π1)\n∼→ Chain(Π2); Chainiso-trm(Π1) ∼→ Chainiso-trm(Π2)\nÉtLoc(Π1)\n∼→ ÉtLoc(Π2)\nChaintrm(Π1)\n∼→ Chaintrm(Π2); DLoc(Π1) ∼→ DLoc(Π2)\nthat are compatible with passing to type-chains and functorial in φ.\nProof. In light of Proposition 4.10, (iii), together with the “tempered anabelian\ntheorem” of [Mzk10], Theorem 6.4, the proof of Theorem 4.12 is entirely similar\nto the proof of Theorem 4.7. [Here, we note that in the case of de-cuspidalization\noperations, instead of applying the de-cuspidalization portion of Proposition 4.10,\n(iii), one may instead apply the “group-theoretic” characterization of Proposition\n4.10, (vi).] Also, we recall that the portion of assertion (ii) concerning, “p1 = p2”,\n“φΔ”, “φG” follows immediately [by considering the profinite completion of φ] from\nTheorem 2.14, (i). ©\nRemark 4.12.1. A similar remark to Remark 4.7.1 applies in the present tem-\npered case [cf. [Mzk10], Theorem 6.8].\n66 SHINICHI MOCHIZUKI\nAppendix: The Theory of Albanese Varieties\nIn the present Appendix, we review the basic theory of Albanese varieties [cf.,\ne.g., [NS], [Serre1], [Chev], [BS], [SS]], as it will be applied in the present paper.\nOne of our aims here is to present the theory in modern scheme-theoretic language\n[i.e., as opposed to [NS], [Serre1], [Chev]], but without resorting to the introduction\nof motives and derived categories, as in [BS], [SS]. Put another way, although there\nis no doubt that the content of the present Appendix is implicit in the literature,\nthe lack of an appropriate reference that discusses this material explicitly seemed\nto the author to constitute sufficient justification for the inclusion of a detailed\ndiscussion of this material in the present paper.\nIn the following discussion, we fix a perfect field k, together with an algebraic\nclosure k of k. The result of base-change [of k-schemes and morphisms of k-schemes]\nfrom k to k will be denoted by means of a subscript “k”. Write Gk\ndef\n= Gal(k\/k) for\nthe absolute Galois group of k.\nWe will apply basic well-known properties of commutative group schemes of\nfinite type over k without further explanation. In particular, we recall the following:\n(I) The category of such group schemes is abelian [cf., e.g., [SGA3-1], VIA,\n5.4]; subgroup schemes are always closed [cf., e.g., [SGA3-1], VIB, 1.4.2];\nreduced group schemes over k are k-smooth [cf., e.g., [SGA3-1], VIA, 1.3.1].\n(II) Every connected reduced subquotient of a semi-abelian variety over k\n[i.e., an extension of an abelian variety by a torus] is itself a semi-abelian\nvariety over k. [Indeed, this may be verified easily by applying a well-\nknown theorem of Chevalley [cf., e.g., [Con], for a treatment of this result\nin modern language; [Bor], Theorems 10.6, 10.9], to the effect that any\nsmooth connected commutative group scheme over k may be written as\nan extension of a semi-abelian variety by a successive extension of copies\nof the additive group (Ga)k.]\n(III) Let φ : B → A be a connected finite étale Galois covering of a semi-\nabelian variety A over k, with identity element 0A ∈ A(k), such that\n(φ−1(0A))(k) \u000b= ∅, and the degree of φ is prime to the characteristic of\nk. Then each element of b ∈ (φ−1(0A))(k) determines on B a unique\nstructure of semi-abelian variety over k on B such that b is the identity\nelement of the group B(k), and φ is a homomorphism of group schemes\nover k. [Indeed this may be verified easily by applying the theorem of\nChevalley quoted in (II) above.] Note, moreover, that in this situation,\nif k = k, then we obtain an inclusion Gal(B\/A) ↪→ B(k), which implies,\nin particular, that the covering φ is abelian, and, moreover, appears as\na subcovering of a covering A → A given by multiplication by some n\ninvertible in k.\nDefinition A.1.\n(i) A variety over k, or k-variety, is defined to be a geometrically integral\nseparated scheme of finite type over k. A k-variety will be called complete if it is\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 67\nproper over k. We shall refer to a pair (V, v), where V is a k-variety and v ∈ V (k),\nas a pointed variety over k; a morphism of pointed varieties over k, or pointed k-\nmorphism, (V, v) → (W,w) [which we shall often simply write V → W , when v, w\nare fixed] is a morphism of k-varieties that maps v \u0002→ w. Any reduced group scheme\nG over k has a natural structure of pointed variety over k determined by the identity\nelement 0G ∈ G(k). If G, H are group schemes over k, then we shall refer to a\nk-morphism G → H as a [k-]trans-homomorphism if it factors as the composite of a\nhomomorphism of group schemes G → H over k with an automorphism of H given\nby translation by an element of H(k). If V is a k-variety, then we shall use the\nnotation π1(V ) to denote the étale fundamental group [relative to an appropriate\nchoice of basepoint] of V . Thus, we have a natural exact sequence of fundamental\ngroups 1 → π1(Vk) → π1(V ) → Gk → 1. Let Σk ⊆ Primes [cf. §0] be the set of\nprimes invertible in k; use the superscript “(Σk)” to denote the maximal pro-Σk\nquotient of a profinite group; if V is a k-variety, then we shall write\nΔV\ndef\n= π1(Vk)\n(Σk); ΠV\ndef\n= π1(V )\/Ker(π1(Vk) \u0002 π1(Vk)\n(Σk))\nfor the resulting geometrically pro-Σk fundamental groups, so we have a natural\nexact sequence of fundamental groups 1 → ΔV → ΠV → Gk → 1.\n(ii) Let C be a class of commutative group schemes of finite type over k. If A\nis a group scheme over k that belongs to the class C, then we shall write A ∈ C. If\n(V, v) is a pointed k-variety, then we shall refer to a morphism of pointed k-varieties\nφ : V → A\nas a C-Albanese morphism if A ∈ C [so A is equipped with a point 0A ∈ A(k), as\ndiscussed in (i)], and, moreover, for any pointed k-morphism φ′ : V → A′, where\nA′ ∈ C, there exists a unique homomorphism ψ : A → A′ of group schemes over k\nsuch that φ′ = ψ ◦ φ. In this situation, A will also be referred to as the C-Albanese\nvariety of V . We shall write Cabk for the class of abelian varieties over k and Cs-abk for\nthe class of semi-abelian varieties over k. When C = Cs-abk , the term “C-Albanese”,\nwill often be abbreviated “Albanese”.\n(iii) If X is a k-variety (respectively, noetherian scheme) which admits a log\nstructure such that the resulting log scheme X log is log smooth over k [where we\nregard Spec(k) as equipped with the trivial log structure] (respectively, log regular\n[cf. [Kato]]), then we shall refer to X as k-toric (respectively, absolutely toric)\nand to X log as a torifier, or torifying log scheme, for X. [Thus, “k-toric” implies\n“absolutely toric”.]\n(iv) If k is of positive characteristic, then, for any k-scheme X and integer\nn ≥ 1, we shall write XFn for the result of base-changing X by the n-th iterate\nof the Frobenius morphism on k; thus, we obtain a k-linear relative Frobenius\nmorphism ΦnX : X → XF\nn\n. If k is of characteristic zero, then we set XF\nn def\n= X,\nΦnX\ndef\n= idX , for integers n ≥ 1. If φ : X → Y is a morphism of k-schemes, then\nwe shall refer to φ as a sub-Frobenius morphism if, for some integer n ≥ 1, there\nexists a k-morphism ψ : Y → XFn such that ψ ◦ φ = ΦnX , φF\nn ◦ψ = ΦnY . [Thus, in\ncharacteristic zero, a sub-Frobenius morphism is simply an automorphism.]\n68 SHINICHI MOCHIZUKI\nRemark A.1.1. As is well-known, if V is a k-variety, then ΦnV induces an\nisomorphism ΠV\n∼→ ΠV Fn , for all integers n ≥ 1. Note that this implies that every\nsub-Frobenius morphism V → W of k-varieties induces isomorphisms ΠV ∼→ ΠW ,\nΔV\n∼→ ΔW .\nBefore proceeding, we review the following well-known result.\nLemma A.2. (Morphisms to Abelian and Semi-abelian Schemes) Let\nS be a noetherian scheme; X an S-scheme whose underlying scheme is absolutely\ntoric; A an abelian scheme over S (respectively, a semi-abelian scheme over\nS which is an extension of an abelian scheme B → S by a torus T → S); V ⊆ X\nan open subscheme whose complement in X is of codimension ≥ 1 (respectively,\n≥ 2) in X. Then any morphism of S-schemes V → A extends uniquely to X.\nProof. First, we consider the case where A is an abelian scheme. If X is regular,\nthen Lemma A.2 follows from [BLR], §8.4, Corollary 6. When X is an arbitrary\nabsolutely toric scheme with torifier X log, we reduce immediately to the case where\nX is strictly henselian, hence admits a resolution of singularities [cf., e.g., [Mzk4],\n§2]\nY log → X log\n— i.e., a log étale morphism of log schemes which induces an isomorphism UY\n∼→ UX\nbetween the respective interiors such that Y log arises from a divisor with normal\ncrossings in a regular scheme Y . Since the “regular case” has already been settled,\nwe may assume that UX ⊆ V ; also, it follows that the restriction UY → A to\nUY of the resulting morphism UX → A extends uniquely to a morphism Y → A.\nThe graph of this morphism determines a closed subscheme Z ⊆ AY def= A ×S Y .\nMoreover, by considering the image of Z under the morphism AY → AX def= A×S\nX of proper X-schemes, we conclude from “Zariski’s main theorem” [since X is\nnormal] that to obtain the [manifestly unique, since V is schematically dense in\nX] desired extension X → A, it suffices to show that the fibers of Y → X map\nto points of A. On the other hand, as is observed in the discussion of [Mzk4], §2,\neach irreducible component of the fiber of Y → X at a point x ∈ X is a rational\nvariety over the residue field k(x) at x, hence maps to a point in the abelian variety\nAx\ndef\n= A ×S k(x) [cf., e.g., [BLR], §10.3, Theorem 1, (b), (c)]. This completes the\nproof of Lemma A.2 in the non-resp’d case. Thus, to complete the proof of Lemma\nA.2 in the resp’d case, we may assume that A = T is a torus over S. In fact, by étale\ndescent, we may even assume that T is a split torus over S. Then it suffices to show\nthat if L is any line bundle on X that admits a generating section sV ∈ Γ(V,L),\nthen it follows that sV extends to a generating section of L over X. But since X is\nnormal, this follows immediately from [SGA2], XI, 3.4; [SGA2], XI, 3.11. ©\nProposition A.3. (Basic Properties of Albanese Varieties) Let C ∈\n{Cabk , Cs-abk }; φV : V → A, φW : W → B C-Albanese morphisms. Then:\n(i) (Base-change) Let k′ be an algebraic field extension of k; denote the\nresult of base-change [of k-schemes and morphisms of k-schemes] from k to k′ by\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 69\nmeans of a subscript “k′”. If C = Cabk (respectively, C = Cs-abk ), then set C′ = Cabk′\n(respectively, C′ = Cs-abk′ ). Then (φV )k′ : Vk′ → Ak′ is a C′-Albanese morphism.\n(ii) (Functoriality) Given any k-morphism βV : V → W , there exists a\nunique k-trans-homomorphism βA : A → B such that φW ◦ βV = βA ◦ φV . If,\nmoreover, βV is pointed, then βA is a homomorphism.\n(iii) (Relative Frobenius Morphisms) For any integer n ≥ 1, φFnV : V F\nn →\nAF\nn\nis a C-Albanese morphism. If, moreover, in (ii), φW = φFnV , βV = ΦnV ,\nthen βA = Φ\nn\nA.\n(iv) (Sub-Frobenius Morphisms) If, in (ii), βV is a sub-Frobenius mor-\nphism, then so is βA.\n(v) (Toric Open Immersions) Suppose, in (ii), that βV is an open im-\nmersion, that W is k-toric, and that if C = Cabk (respectively, C = Cs-abk ), then\nthe codimension of the complement of the image of βV in W is ≥ 1 (respectively,\n≥ 2). Then βA is an isomorphism.\n(vi) (Dominant Quotients) If, in (ii), βV is dominant, then βA is surjec-\ntive.\n(vii) (Surjectivity of Fundamental Groups) The [outer] homomorphisms\nΠφV : ΠV → ΠA, ΔφV : ΔV → ΔA induced by φV are surjective.\n(viii) (Semi-abelian versus Abelian Albanese Morphisms) Suppose that\nC = Cs-abk . Write A \u0002 Aab for the maximal quotient of group schemes over k such\nthat Aab ∈ Cabk . Then the composite morphism V → A \u0002 Aab is a Cabk -Albanese\nmorphism.\n(ix) (Group Law Generation) For integers n ≥ 1, write\nζn : V ×k . . .×k V → A\n(v1, . . . , vn) \u0002→\n∑n\nj=1 vj\nfor the morphism from the product over k of n copies of V to A given by adding\nthe images under φV of the points in the n factors. Then there exists an integer N\nsuch that ζn is surjective for all n ≥ N . In particular, if V is proper, then so is\nA.\nProof. To verify assertion (i), we may assume that k′ is a finite [hence necessarily\nétale, since k is perfect] extension of k. Then assertion (i) follows immediately\nby considering the Weil restriction functor Wk′\/k(−) from k′ to k. That is to\nsay, it is immediate that Wk′\/k(−) takes objects in C′ to objects in C. Thus, to\ngive a k′-morphism Vk′ → A′ (respectively, Ak′ → A′) is equivalent to giving a\nk-morphism V → Wk′\/k(A′) (respectively, A → Wk′\/k(A′)). This completes the\nproof of assertion (i). Assertions (ii), (iii) follow immediately from the definition of\na “C-Albanese morphism”; assertion (iv) follows immediately from assertion (iii).\nAssertion (v) follows immediately from the definition of a “C-Albanese morphism”,\nin light of Lemma A.2.\n70 SHINICHI MOCHIZUKI\nAssertion (vi) follows from the definition of a “C-Albanese morphism”, by ar-\nguing as follows: First, we observe that βV is an epimorphism in the category of\nschemes. Also, we may assume without loss of generality that βV is pointed. Now\nconsider the composite β ◦ φW : W → B\/C of φW : W → B with the natural quo-\ntient morphism β : B \u0002 B\/C, where we write C def= Im(βA) ⊆ B [so C ∈ C]. Since\nβ ◦ φW has the same restriction [via βV ] to V as the constant pointed morphism\nW → B\/C, we thus conclude that β ◦ φW is constant, i.e., that Im(φW ) ⊆ C.\nBut, by the definition of a “C-Albanese morphism”, this implies the existence of\na section B → C of the natural inclusion C ↪→ B [i.e., such that the composite\nB → C ↪→ B is equal to the identity], hence that B = C, as desired. In a similar\nvein, assertion (vii) follows from the definition of a “C-Albanese morphism”, by ob-\nserving that if ΠφV : ΠV → ΠA fails to surject, then [after possibly replacing k by a\nfinite extension of k, which is possible, by assertion (i)] it follows that φV : V → A\nfactors V → C → A, where the morphism C → A is a nontrivial finite étale Galois\ncovering, with C geometrically connected over k, so C ∈ C. But this implies, by\nthe definition of a “C-Albanese morphism”, the existence of a section A → C of the\nsurjection C \u0002 A [i.e., such that the composite A → C \u0002 A is the identity], hence\nthat this surjection is an isomorphism C\n∼→ A, a contradiction.\nNext, we observe that assertion (viii) follows immediately from the definitions,\nin light of the well-known fact that any homomorphism G → H of group schemes\nover k, where G is a torus and H is an abelian variety, is trivial [cf., e.g., [BLR],\n§10.3, Theorem 1, (b), (c)].\nFinally, we consider assertion (ix). First, let us observe that we may assume\nwithout loss of generality that k = k. Next, let us observe that since the image\nof φV contains 0A ∈ A(k), it follows that for n ≥ m, the image In ⊆ A(k) of ζn\ncontains the image Im of ζm. Write Fn ⊆ A for the [reduced closed subscheme given\nby the] closure of In. Since the domain of ζn is irreducible, it follows immediately\nthat Fn is irreducible. Thus, the ascending sequence . . . ⊆ Fm ⊆ . . . ⊆ Fn ⊆ . . .\nterminates, i.e., we have Fn = Fm for all n,m ≥ N ′, for some N ′; write F def= FN ′ .\nSince IN ′ is constructible, it follows that IN ′ contains a nonempty open subset U\nof [the underlying topological space of] F ; let u ∈ U(k). Now let us write I ′n for\nthe union of the translates of U by elements of In; thus, one verifies immediately\nthat I ′n is open in F , that I\n′\nn ⊆ In+N ′ , and that u+ In ⊆ I ′n. Since F is noetherian,\nit thus follows that the ascending sequence . . . ⊆ I ′m ⊆ . . . ⊆ I ′n ⊆ . . . terminates,\ni.e., that for some N ′′ > N ′, we have I ′n = I\n′\nm for all n,m ≥ N ′′; write I ⊆ F\nfor the resulting open subscheme. Thus, for n ≥ N ′′, u + I ⊆ u + In+N ′ ⊆ I. On\nthe other hand, again since F is noetherian, it follows that the ascending sequence\nI ⊆ I − u ⊆ I − 2u ⊆ . . . terminates, hence that u+ I = I. In particular, for some\nN ′′′ > N ′′, we have In = I, for n ≥ N ′′′. Next, let us observe that for any j ∈ I(k),\nit follows from the definition of the In that j + I ⊆ I, hence [as in the case where\nj = u], we have j + I = I. Since 0A ∈ I, it thus follows that I is closed under\nthe group operation on A, as well as taking inverses in A. Thus, it follows that\nI is a subgroup scheme of A, hence that I is a closed subscheme of A [so I = F ].\nBut this implies, by the definition of a “C-Albanese morphism”, the existence of a\nhomomorphism A → I whose composite with the inclusion I ↪→ A is the identity\non A. Thus, we conclude that the inclusion I ↪→ A is a surjection, i.e., that I = A,\nas desired. ©\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 71\nA proof of the following result may be found, in essence, in [NS] [albeit in\nsomewhat archaic language], as well as in [FGA], 236, Théorème 2.1, (ii) [albeit\nin somewhat sketchy form]. Various other approaches [e.g., via Weil divisors] to\nthis result are discussed in [Klei], Theorem 5.4, and the discussion following [Klei],\nTheorem 5.4.\nTheorem A.4. (Properness of the Identity Component of the Picard\nScheme) The identity component of the Picard scheme\nPic0V\/k\n[cf., e.g., [BLR], §8.2, Theorem 3; [BLR], §8.4] associated to a complete normal\nvariety V over a field k is proper.\nProof. Write G for the reduced group scheme (Pic0V\/k)red over k. Then by a\nwell-known theorem of Chevalley [cf., e.g., [Con], for a treatment of this result\nin modern language; [Bor], Theorems 10.6, 10.9], it follows that to show that G\n[hence also Pic0V\/k] is proper, it suffices to show that G does not contain any copies\nof the multiplicative group (Gm)k or the additive group (Ga)k. On the other hand,\nsince (Gm)k, (Ga)k may be thought of as open subschemes of the affine line A\n1\nk,\nthis follows immediately from Lemma A.5 below [i.e., by applying the functorial\ninterpretation of Pic0V\/k — cf., e.g., [BLR], §8.1, Proposition 4]. ©\nLemma A.5. (Rational Families of Line Bundles) Let V be a normal\nvariety over k; U ⊆ A1k a nonempty open subscheme of the affine line A1k. Then\nevery line bundle LU on V ×k U arises via pull-back from a line bundle Lk on V .\nProof. In the following, let us regard A1k as an open subscheme A\n1\nk ⊆ P1k of the\nprojective line [obtained in the standard way by removing the point at infinity\n∞k ∈ P1k(k)]. First, let us verify Lemma A.5 under the further hypothesis that\nV is smooth over k. Then it follows immediately that V ×k P1k is smooth over k,\nhence locally factorial [cf., e.g., [SGA2], XI, 3.13, (i)]. Thus, LU extends to a line\nbundle LP on P def= V ×k P1k (⊇ V ×k A1k ⊇ V ×k U). Moreover, by tensoring\nwith line bundles associated to multiples of the divisor on P arising from ∞k, we\nmay assume that the degree of LP on the fibers of the trivial projective bundle\nf : P → V is zero. Thus, the natural morphism f∗f∗LP → LP is an isomorphism,\nwhich exhibits LP , hence also LU , as a line bundle Lk pulled back from V .\nNow we return to the case of an arbitrary normal variety V . As is well-known,\nV contains a dense open subscheme W ⊆ V which is smooth over k and such that\nthe closed subscheme F\ndef\n= V \\ W [where we equip F with the reduced induced\nstructure] is of codimension ≥ 2 in V [cf., e.g., [SGA2], XI, 3.11, applied to the\ngeometric fiber of V → Spec(k)]. Thus, by the argument given in the smooth case,\nwe conclude that MU def= LU |W×kU arises from a line bundle Mk on W . Next, let\nus write ιk : W ↪→ V , ιU : W ×k U ↪→ V ×k U for the natural open immersions.\nSince U is k-flat, it follows immediately that we have a natural isomorphism\n((ιk)∗Mk)|V×kU ∼→ (ιU )∗MU\n72 SHINICHI MOCHIZUKI\n[arising, for instance, by computing the right-hand side by means of an affine cov-\nering of W ×k U obtained by taking the product over k with U of an affine cov-\nering of W ]. On the other hand, since V ×k U is normal and F ×k U ⊆ V ×k U\nis a closed subscheme of codimension ≥ 2, it follows from the definition of MU\nthat (ιU )∗MU ∼→ LU [cf., e.g., [SGA2], XI, 3.4; [SGA2], XI, 3.11], i.e., that\n((ιk)∗Mk)|V×kU is a line bundle on V ×kU . On the other hand, since the morphism\nU → Spec(k), hence also the projection morphism V ×k U → V , is faithfully flat,\nwe thus conclude that Lk def= (ιk)∗Mk is a line bundle on V whose pull-back to\nV ×k U is isomorphic to LU , as desired. ©\nProposition A.6. (Duals of Picard Varieties as Albanese Varieties)\nLet V be a complete normal variety over k; Pic0V\/k the identity component of\nthe associated Picard scheme; A the dual abelian variety to G\ndef\n= (Pic0V\/k)red\n[which is an abelian variety by Theorem A.4]; v ∈ V (k). Then the universal\nline bundle PV [cf., e.g., [BLR], §8.1, Proposition 4] on V ×k G relative to the\nrigidification determined by v [i.e., such that PV |{v}×G is trivial] determines [by\nthe definition of A] a morphism of pointed k-varieties\nφ : V → A\nsuch that the pull-back of the Poincaré bundle PA on A ×k G via φ ×k G : V ×k\nG → A ×k G is isomorphic to PV [in a fashion compatible with the respective\nrigidifications]. Moreover:\n(i) The morphism φ is a Cabk -Albanese morphism.\n(ii) Suppose, in the situation of Proposition A.3, (ii), that W is also complete\nand normal, and that βV is pointed and birational. Then the dual morphism\nβG : H → G to βA : A → B is a closed immersion. In particular, βA is an\nisomorphism if and only if dimk(A) ≤ dimk(B).\n(iii) The morphism φ induces an injection H1(A,OA) ↪→ H1(V,OV ).\n(iv) The morphism φ induces an isomorphism Δab-tV\n∼→ ΔA [where we refer\nto §0 for the notation “ab-t”].\nProof. First, we consider assertion (i). Let ψV : V → C be a morphism of pointed\nk-varieties, where C ∈ Cabk . Now by the functoriality of “Pic0(−)\/k”, ψV induces a\nmorphism D\ndef\n= Pic0C\/k → Pic0V\/k [so D is the dual abelian variety to C], hence a\nmorphism ψD : D → G, whose dual gives a morphism ψA : A → C. The fact that\nψV = ψA◦φ : V → C follows by thinking of morphisms as classifying morphisms for\nline bundles and considering the following [a priori, not necessarily commutative]\ndiagram of morphisms between varieties equipped with [isomorphism classes of] line\nbundles:\n(V ×k D,L) id−→ (V ×k D,L) V×kψD−→ (V ×k G,PV )⏐⏐\u0005ψV ×kD\n⏐⏐\u0005φ×kD\n⏐⏐\u0005φ×kG\n(C ×k D,PC) ψA×kD←− (A×k D,M) A×kψD−→ (A×k G,PA)\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 73\n—where we write L def= (ψV ×kD)∗PC ; M def= (ψA×kD)∗PC ∼= (A×kψD)∗PA. That\nis to say, the desired commutativity of the left-hand square follows by computing:\n(φ×k D)∗(ψA ×k D)∗PC ∼= (φ×k D)∗(A×k ψD)∗PA\n∼= (V ×k ψD)∗(φ×k G)∗PA\n∼= (V ×k ψD)∗PV\n∼= (ψV ×k D)∗PC\n— which implies that ψV = ψA ◦ φ. Finally, the uniqueness of such a “ψA” follows\nimmediately by applying “Pic0(−)\/k” to the condition “ψV = ψA ◦φ : V → A → C”.\nThis completes the proof of assertion (i).\nNext, we consider assertion (ii). First, observe that there exists a k-smooth\nopen subscheme U ⊆ W such that W \\ U has codimension ≥ 2 in W [cf., e.g.,\n[SGA2], XI, 3.11, as it was applied in the proof of Lemma A.5], and, moreover,\nβV : V → W admits a section σ : U → V over U . Note, moreover, that if S is any\nlocal artinian finite k-scheme, and we write ιS : US\ndef\n= U ×k S ↪→ WS def= W ×S k\nfor the natural inclusion, then for any line bundle L on WS , we have a natural\nisomorphism (ιS)∗(ι∗SL) ∼→ L [cf., e.g., [SGA2], XI, 3.4; [SGA2], XI, 3.11]. Thus, by\napplying this natural isomorphism, together with the section σ, we conclude that\nthe map Pic0W\/k(S) → Pic0V\/k(S) [induced by βV ] is an injection, which implies\nthat the kernel group scheme of βG : H → G is trivial, hence that βG is a closed\nimmersion, as desired. This completes the proof of assertion (ii).\nNext, we consider assertion (iii). The morphism H1(A,OA) → H1(V,OV ) in\nquestion may be interpreted as the morphism induced by φ on tangent spaces to the\nPicard scheme, i.e., as the morphism\nG(k[\u0007]\/(\u00072)) = Pic0A\/k(k[\u0007]\/(\u0007\n2)) → Pic0V\/k(k[\u0007]\/(\u00072))\n[cf., e.g., [BLR], §8.4, Theorem 1, (a)]. But, by the definition of G, this morphism\narises from the natural closed immersion G ↪→ Pic0V\/k, hence is an injection, as\ndesired.\nFinally, we consider assertion (iv). The surjectivity portion of assertion (iv)\nfollows immediately from Proposition A.3, (vii). To verify the fact that the sur-\njection Δab-tV \u0002 ΔA is an isomorphism, we reason as follows: First, we recall that\nif n ≥ 1 is an integer invertible in k, then a line bundle L on V such that L⊗n\nis trivial may be interpreted [via the Kummer exact sequence in étale cohomol-\nogy] as a continuous homomorphism ΔV → (Z\/nZ)(1) [where the “(1)” denotes\na “Tate twist”]. On the other hand, by [BLR], §8.4, Theorem 7, there exists an\ninteger m ≥ 1 such that for every integer n ≥ 1, the cokernel of the inclusion\nnG(k) ↪→ nPicV\/k(k) [where the “n” preceding an abelian group denotes the kernel\nof multiplication by n] is annihilated by m. In light of the functorial interpretation\nof the inclusion G ↪→ Pic0V\/k ⊆ PicV\/k, this implies that the cokernel of the homo-\nmorphism Hom(ΔA,Q\/Z) → Hom(ΔV ,Q\/Z) is annihilated by m. But, by applying\nHom(−,Q\/Z), this implies that the induced homomorphism ΔabV → ΔA has finite\nkernel, hence [in light of the surjectivity already verified] induces an isomorphism\nupon passing to “ab-t”. ©\n74 SHINICHI MOCHIZUKI\nRemark A.6.1. The content of Proposition A.6, (i), is discussed in [FGA], 236,\nThéorème 3.3, (iii).\nRemark A.6.2. Suppose that we are in the situation of Proposition A.6, (ii).\nThen it is not necessarily the case that the induced morphism βA is an isomorphism.\nThis phenomenon already appears in the work of Chevalley — cf. [Chev]; the\ndiscussion of [Klei], p. 248; Example A.7 below.\nExample A.7. Albanese Varieties and Resolution of Singularities. For\nsimplicity, suppose that k = k. Write P2k = Proj(k[x1, x2, x3]) [i.e., where we\nconsider k[x1, x2, x3] as a graded ring, in which x1, x2, x3 are of degree 1]. Let\nf ∈ k[x1, x2, x3] be a homogeneous polynomial that defines a smooth plane curve\nX ⊆ P2k of genus ≥ 1. Thus, any x ∈ X(k) determines an embedding X ↪→ J , where\nJ is the Jacobian variety of X. Set Y\ndef\n= Spec(k[x1, x2, x3]\/(f)); write y ∈ Y (k)\nfor the origin, UY\ndef\n= Y \\ {y}. Thus, we have a natural morphism Y ⊇ UY → X;\nUY → X is a Gm-torsor over X. In particular, UY is k-smooth. Thus, since Y\nis clearly a local complete intersection [hence, in particular, Cohen-Macaulay], it\nfollows from Serre’s criterion of normality [cf., e.g., [SGA2], XI, 3.11] that Y is\nnormal. Let Z → Y be the blow-up of Y at the origin y. Thus, we obtain an\nisomorphism UZ\ndef\n= Z ×Y UY ∼→ UY . Moreover, one verifies immediately that the\nmorphism UZ\n∼→ UY → X extends to a morphism Z → X which has the structure\nof an A1-bundle, in which E\ndef\n= Z ×Y {y} ⊆ Z forms a “zero section” [so E ∼→ X].\nThus, Z admits a natural compactification Z ↪→ Z∗ to a P1-bundle Z∗ → X.\nMoreover, by gluing Z∗ \\ E to Y along Z \\ E = UZ ∼→ UY ⊆ Y , we obtain a\ncompactification Y ↪→ Y ∗ such that the blow-up morphism extends to a morphism\nZ∗ → Y ∗ [which may be thought of as the blow-up of Y ∗ at y ∈ Y (k) ⊆ Y ∗(k)].\nOn the other hand, note that the composite Z∗ → X ↪→ J determines a closed\nimmersion Z∗ ⊇ E ∼→ X ↪→ J . Thus, the restriction UY ∼→ UZ → J of this\nmorphism Z∗ → J to UY ∼→ UZ does not extend to Y or Y ∗. In particular, it follows\nthat if we write Y ∗ → AY , Z∗ → AZ for the Cabk -Albanese varieties of Proposition\nA.6, (i), then the surjection AZ \u0002 AY induced by Z∗ → Y ∗ [cf. Proposition A.6,\n(ii)] is not an isomorphism.\nProposition A.8. (Albanese Varieties of Complements of Divisors with\nNormal Crossings) Let Z be a smooth projective variety over k; D ⊆ Z a\ndivisor with normal crossings; Y\ndef\n= Z \\D ⊆ Z; y ∈ Y (k);\nD =\nr⋃\nn=1\nDn\n[for some integer r ≥ 1] the decomposition of D into irreducible components; M\nthe free Z-module [of rank r] of divisors supported on D; P ⊆ M the submodule\nof divisors that determine a line bundle ∈ Pic0Z\/k(k). Then:\n(i) (Y, y) admits an Albanese morphism Y → AY .\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 75\n(ii) Suppose that each of the Dn is geometrically irreducible. Then the AY\nof (i) may be taken to be an extension of the abelian variety AZ given by the\ndual to GZ\ndef\n= (Pic0Z\/k)red [cf. Propositions A.3, (viii); A.6, (i)] by a torus whose\ncharacter group is naturally isomorphic to P .\n(iii) The morphism Y → AY of (i) induces an isomorphism Δab-tY ∼→ ΔAY .\nProof. By étale descent [with respect to finite extensions of k], it follows immedi-\nately that to verify assertion (i), it suffices to verify assertion (ii). Next, we consider\nassertion (ii). Again, by étale descent, we may assume without loss of generality\nthat k = k. Note that the tautological homomorphism P → GZ(k) determines an\nextension\n0 → TY → AY → AZ → 0\nof AZ by a split torus TY with character group P . Now the fact that AY serves\nas an Albanese variety for Y is essentially a tautology: Indeed, since any pointed\nmorphism from Y to an abelian variety C extends [cf. Lemma A.2] to a pointed\nmorphism Z → C, and, moreover, we already know that AZ is a Cabk -Albanese\nvariety for Z [cf. Proposition A.6, (i)], it follows that it suffices to consider pointed\nmorphisms Y → B, where B is an extension of AZ by a [split] torus, and the\ncomposite morphism Y → B \u0002 AZ coincides with the morphism that exhibits\nAZ as a Cabk -Albanese variety for Y . In fact, for simplicity, we may even assume\nthat this torus is simply (Gm)k. Thus, it suffices to consider pointed morphisms\nY → B, where B is an extension of AZ by (Gm)k, determined by some extension\nclass κB ∈ GZ(k), and the composite morphism Y → B \u0002 AZ coincides with\nthe morphism that exhibits AZ as a Cabk -Albanese variety for Y . Then the datum\nof such a morphism Y → B corresponds precisely to an invertible section of the\nrestriction to Y of the line bundle L on Z given by pulling back the Gm-torsor\nB → AZ via the Albanese morphism Z → AZ . Note that such an invertible section\nof L|Y may be thought of as the datum of an isomorphism OZ(E) ∼→ L for some\ndivisor E supported on D. That is to say, since the isomorphism class of L is\nprecisely the class determined by the element κB ∈ GZ(k) ⊆ PicZ\/k(k), it thus\nfollows that E ∈ P , and that κB is the image of E ∈ P in Pic0Z\/k(k) = GZ(k).\nThus, in summary, the datum of a pointed morphism Y → B, where B is an\nextension of AZ by a [split] torus, and the composite morphism Y → B \u0002 AZ\ncoincides with the morphism that exhibits AZ as a Cabk -Albanese variety for Y , is\nequivalent [in a functorial way] to the datum of a homomorphism AY → B lying\nover the identity morphism of AZ . In particular, the identity morphism AY → AY\ndetermines a morphism Y → AY . This completes the proof of assertion (ii).\nFinally, we consider assertion (iii). We may assume without loss of generality\nthat k = k [cf. Proposition A.3, (i)]. Let F ⊆ D be a closed subscheme of\ncodimension ≥ 1 in D such that Z ′ def= Z \\ F ⊆ Z, D′ def= D \\ F ⊆ D are k-smooth.\nThen one has the associated Gysin sequence in étale cohomology\n0 → H1ét(Z ′,Zl(1)) → H1ét(Y,Zl(1)) → M ⊗ Zl → H2ét(Z ′,Zl(1))\nfor l ∈ Σk [cf. [Milne], p. 244, Remark 5.4, (b)]. Moreover, we have natural isomor-\nphismsH\nj\nét(Z\n′,Zl(1))\n∼→Hjét(Z,Zl(1)), for j = 1, 2. [Indeed, by applying noetherian\n76 SHINICHI MOCHIZUKI\ninduction, it suffices to verify these isomorphisms in the case where F is k-smooth,\nin which case these isomorphisms follow from [Milne], p. 244, Remark 5.4, (b).]\nNote, moreover, that the morphism M ⊗ Zl → H2ét(Z ′,Zl(1)) ∼→ H2ét(Z,Zl(1)) is\nprecisely the “fundamental class map”, hence factors through the natural inclusion\nPicZ\/k(k)\n∧ ↪→ H2ét(Z,Zl(1))\n[where the “∧” denotes the pro-l completion] arising from the Kummer exact se-\nquence on Z. On the other hand, since Pic0Z\/k(k) is l-divisible, and the quotient\nPicZ\/k(k)\/Pic\n0\nZ\/k(k) is finitely generated [cf. [BLR], §8.4, Theorem 7], it follows\nthat we have an isomorphism\n(PicZ\/k(k)\/Pic\n0\nZ\/k(k))⊗ Zl ∼→ PicZ\/k(k)∧\n— i.e., that the kernel of the morphism M⊗Zl → H2ét(Z ′,Zl(1)) is precisely P ⊗Zl.\nIn particular, the isomorphism Δab-tZ\n∼→ ΔAZ of Proposition A.6, (iv), implies [in\nlight of the above exact sequence] that H1ét(Y,Zl(1)) [i.e., Hom(Δ\nab-t\nY ,Zl(1))], hence\nalso Δab-tY ⊗Zl, is a free Zl-module of the same rank as ΔAY ⊗Zl. Thus, we conclude\nthat the surjection Δab-tY \u0002 ΔAY of Proposition A.3, (vii), is an isomorphism, as\ndesired. ©\nRemark A.8.1. A sharper version [in the sense that it includes a computation of\nthe torsion subgroup of ΔabY ] of Proposition A.8, (iii), is given in [SS], Proposition\n4.2. The discussion of [SS] involves the point of view of 1-motives. On the other\nhand, such a sharper version may also be obtained directly from the Gysin sequence\nargument of the above proof of Proposition A.8, (iii), by working with torsion\ncoefficients.\nThe following result is elementary and well-known.\nLemma A.9. (Descending Chains of Subgroup Schemes) Let G be a\n[not necessarily reduced] commutative group scheme of finite type over k;\n. . . ⊆ Gn ⊆ . . . ⊆ G1 ⊆ G0 = G\na descending chain of [not necessarily reduced!] subgroup schemes of G, indexed\nby the nonnegative integers. Then there exists an integer N such that Gn = Gm\nfor all n,m ≥ N .\nProof. First, let us consider the case where all of the Gn, for n ≥ 0, are reduced and\nconnected. Then since all of the Gn are closed integral subschemes of G, it follows\nimmediately that if we take any integer N such that dimk(Gn) = dimk(Gm) for\nall n,m ≥ N , then Gn = Gm for all n,m ≥ N . Now we return to the general\ncase. By what we have done so far, we may assume without loss of generality that\n(G0)red = (Gn)red for all n ≥ 0. Thus, by forming the quotient by (G0)red, we\nmay assume that all of the Gn are finite over Spec(k). Then Lemma A.9 follows\nimmediately. ©\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 77\nBefore proceeding, we recall the following result of de Jong.\nLemma A.10. (Equivariant Alterations) Suppose that k = k; let V be\na variety over k. Then there exists a smooth projective variety Z over k, a\nfinite group Γ of automorphisms of Z over k, a divisor with normal crossings\nD ⊆ Z stabilized by Γ, and a Γ-equivariant [relative to the trivial action of Γ on V ]\nsurjective, proper, generically quasi-finite morphism\nY\ndef\n= Z \\D → V\nsuch that if we write k(Z), k(V ) for the respective function fields of Z, V , then the\nsubfield of Γ-invariants k(Z)Γ ⊆ k(Z) forms a purely inseparable extension of\nk(V ).\nProof. This is the content of [deJong], Theorem 7.3. ©\nWe are now ready to prove the main result of the present Appendix, the first\nportion of which [i.e., Corollary A.11, (i)] is due to Serre [cf. [Serre1]].\nCorollary A.11. (Albanese Varieties of Arbitrary Varieties)\n(i) Every pointed variety (V, v) over k admits an Albanese morphism\nV → A.\n(ii) Let φ : V → A be an Albanese morphism, where (V, v) is a k-toric\npointed variety. Then φ induces an isomorphism Δab-tV\n∼→ ΔA.\nProof. First, we consider assertion (i). By applying étale descent, we may assume\nwithout loss of generality that k = k. Let Z ⊇ Y → V be as in Lemma A.10,\ny ∈ Y (k) a point that maps to v ∈ V (k) [where we observe that, as is easily verified,\nthe existence of an Albanese morphism as desired is independent of the choice of\nv]. Then by Proposition A.8, (i), it follows that Y admits an Albanese morphism\nY → B. Thus, every pointed morphism ν : V → C, where C ∈ Cs-abk , determines,\nby restriction to Y , a homomorphism B → C, whose kernel is a subgroup scheme\nHν ⊆ B. In particular, the collection of such pointed morphisms ν : V → C\ndetermines a projective system of subgroup schemes Hν ⊆ B which is filtered [a\nfact that is easily verified by considering product morphisms V → C1 ×k C2 of\npointed morphisms ν1 : V → C1, ν2 : V → C2]. Moreover, by Lemma A.9, this\nprojective system admits a cofinal subsystem which is constant, i.e., given by a\nsingle subgroup scheme H ⊆ B. Now it is a tautology that the composite morphism\nY → B \u0002 B\/H factors uniquely [where we observe that uniqueness follows from\nthe fact that Y → V is dominant] through a morphism V → B\/H which serves as\nan Albanese morphism for V .\nNext, we consider assertion (ii). First, let us observe that, by Proposition A.3,\n(i), we may assume without loss of generality that k = k. Next, let Z ⊇ Y → V ,\nΓ be as in Lemma A.10; write Y → V ′ → V for the factorization through the\nnormalization V ′ → V of V in the purely inseparable extension k(Z)Γ of k(V ). Let\n78 SHINICHI MOCHIZUKI\nφ′ : V ′ → A′ be an Albanese morphism [which exists by assertion (i)]. Since V is\nnormal, it follows immediately that V ′ → V is a sub-Frobenius morphism. Thus,\nby Proposition A.3, (iv) [cf. also Remark A.1.1], it follows that V ′ → V induces\nisomorphisms Δab-tV ′\n∼→ Δab-tV , ΔA′ ∼→ ΔA. In particular, to complete the proof of\nassertion (ii), it suffices to verify that φ′ induces an isomorphism Δab-tV ′\n∼→ ΔA′ .\nNext, let Y → B be an Albanese morphism for Y [cf. Proposition A.8, (i)].\nThen, by Proposition A.3, (ii), the action of Γ on Y extends to a compatible action\nof Γ on B by k-trans-homomorphisms. This action of Γ on B may be thought of\nas the combination of an action of Γ on the group scheme B [i.e., via group scheme\nautomorphisms], together with a twisted homomorphism χ : Γ → B(k) [where Γ\nacts on B(k) via the group scheme action of Γ on B]. Write B \u0002 C ′ for the quotient\nsemi-abelian scheme of B by the group scheme action Γ, i.e., the quotient of B by\nthe subgroup scheme generated by the images of the group scheme endomorphisms\n(1− γ) : B → B, for γ ∈ Γ. Thus, χ determines a homomorphism χ′ : Γ → C ′(k);\nwrite C ′ → C for the quotient semi-abelian scheme of C ′ by the finite subgroup\nscheme of C ′ determined by the image of χ′. Note that every trans-homomorphism\nof semi-abelian schemes B → D which is Γ-equivariant with respect to the trivial\naction of Γ on D and the trans-homomorphism action of Γ of B factors uniquely\nthrough B \u0002 C.\nNow I claim that the composite Y → B \u0002 C factors uniquely through V ′.\nIndeed, this is clear generically; write ξ′ : ηV ′ → C for the resulting morphism.\nHere, we use the notation “η(−)” to denote the spectrum of the function field of\n“(−)”. Since the morphism V ′ → V is a sub-Frobenius morphism, it thus follows\nthat for some integer n ≥ 1, the composite ηV ′ → C → CFn of ξ′ with ΦnC factors\nthrough the natural morphism ηV ′ → ηV , thus yielding a morphism ξ : ηV → CFn .\nNow since V is normal, it follows from the properness of Y → V that ξ extends\nuniquely to points of height 1 of V ; thus, since V is k-toric, it follows from Lemma\nA.2 that ξ extends uniquely to the entire scheme V . Finally, by the definition of\nV ′ → V as a normalization morphism, it follows that from the fact that ΦnC is finite\nand surjective that ξ′ extends uniquely to the entire scheme V ′. This completes the\nproof of the claim.\nNext, let us observe that it is a tautology that the morphism V ′ → C resulting\nfrom the above claim is an Albanese morphism for V ′. In particular, we may\nassume without loss of generality that φ′ : V ′ → A′ is V ′ → C. Next, let us\nobserve that it follows immediately from the description of finite étale coverings\nof semi-abelian schemes reviewed at the beginning of the present Appendix that\nthe functor “(−) \u0002→ Δ(−)” transforms exact sequences of semi-abelian schemes\ninto exact sequences of profinite groups. Thus, if follows immediately from the\nconstruction of C (= A′) from B that, for l ∈ Σk, the surjection ΔB⊗Ql \u0002 ΔA′⊗Ql\ninduces an isomorphism\n(ΔB ⊗Ql)\/Γ ∼→ ΔA′ ⊗Ql\n[where the “\/Γ” denotes the maximal quotient on which Γ acts trivially].\nOn the other hand, by Proposition A.8, (iii), it follows that we have a natural\nisomorphism Δab-tY\n∼→ ΔB , hence, in particular, a natural isomorphism\n(Δab-tY ⊗Ql)\/Γ ∼→ (ΔB ⊗Ql)\/Γ ∼→ ΔA′ ⊗Ql\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 79\nfor l ∈ Σk. Moreover, since the morphism Y → V ′ is dominant, it induces an open\nhomomorphism ΔY → ΔV ′ , hence a surjection Δab-tY ⊗ Ql \u0002 Δab-tV ′ ⊗ Ql which is\nΓ-equivariant [with respect to the trivial action of Γ on Δab-tV ′ ⊗Ql]. In particular,\nwe obtain that the natural isomorphism (Δab-tY ⊗Ql)\/Γ ∼→ ΔA′ ⊗Ql factors as the\ncomposite of surjections\n(Δab-tY ⊗Ql)\/Γ \u0002 Δab-tV ′ ⊗Ql \u0002 ΔA′ ⊗Ql\n[cf. Proposition A.3, (vii)]. Thus, we conclude that these surjections are isomor-\nphisms, hence that the surjection Δab-tV ′ \u0002 ΔA′ induced by φ′ [cf. Proposition A.3,\n(vii)], is an isomorphism, as desired. ©\nRemark A.11.1. In fact, given any variety V over k, one may construct an\n“Albanese morphism” V → A, where A is a torsor over a semi-abelian variety over\nk, by passing to a finite [separable] extension k′ of k such that V (k′) \u000b= ∅, applying\nCorollary A.11, (i), over k′, and then descending back to k. This morphism V → A\nwill then satisfy the universal property for morphisms V → A′ to torsors A′ over\nsemi-abelian varieties over k [i.e., every such morphism V → A′ admits a unique\nfactorization V → A → A′, where the morphism A → A′ is a k-morphism that base-\nchanges to a trans-homomorphism over k]. In the present Appendix, however, we\nalways assumed the existence of rational points in order to simplify the discussion.\nRemark A.11.2. One may further generalize Remark A.11.1, as follows. If V is\na generically scheme-like [cf. §0] geometrically integral separated algebraic stack of\nfinite type over k that is obtained by forming the quotient, in the sense of stacks,\nof some variety W over k by the action of a finite group of automorphisms Γ ⊆\nAut(W ), then, by applying Remark A.11.1 to W to obtain an Albanese morphism\nW → B for W , one may construct an “Albanese morphism”\nV → A\nfor V [i.e., which satisfies the universal property described in Remark A.11.1] by\nforming the quotient B → A of B as in the proof of Corollary A.11, (ii): That\nis to say, after reducing, via étale descent, to the case k = k, the action of Γ on\nW induces an action of Γ by k-trans-homomorphisms on B, hence an action of\nΓ by group scheme automorphisms on B, together with a twisted homomorphism\nχ : Γ → B(k). Then we take B \u0002 A′ to be the quotient by the images of the\ngroup scheme endomorphisms [arising from the group scheme action of Γ on B]\n(1 − γ) : B → B, for γ ∈ Γ, and A′ \u0002 A to be the quotient by the image of the\nhomomorphism χ′ : Γ → A′(k) determined by χ. If, moreover, V [i.e., W ] is k-toric,\nthen just as in the proof of Corollary A.11, (ii), we obtain a natural isomorphism\nΔab-tV\n∼→ ΔA\n[where we use the notation “Δ(−)” to denote the evident stack-theoretic general-\nization of this notation for varieties].\nThe content of more classical works [cf., e.g., [NS], [Chev]] written from the\npoint of view of birational geometry may be recovered via the following result.\n80 SHINICHI MOCHIZUKI\nCorollary A.12. (Albanese Varieties and Birational Geometry)\n(i) Let βV : V\n′ → V be a proper birational morphism of normal varieties\nover k which restricts to an isomorphism βU : U\n′ def= V ′ ×V U ∼→ U over some\nnonempty open subscheme U ⊆ V ; βA : A′ → A the induced morphism on Al-\nbanese varieties [cf. Corollary A.11, (i)]; W ⊆ V a k-toric open subscheme.\nThen the composite morphism U\n⋂\nW ↪→ U ∼→ U ′ ↪→ V ′ → A′ extends uniquely\nto a morphism W → A′ which induces a surjection ΔW \u0002 ΔA′ .\n(ii) Let\n. . . → Vn → . . . → V1 → V0 = V\nbe a sequence [indexed by the nonnegative integers] of birational morphisms of\ncomplete normal varieties over k. Then there exists an integer N such that\nfor all n,m ≥ N , where n ≥ m, the induced morphism on Albanese varieties\nAn → Am is an isomorphism. If V is k-toric, then one may take N = 0.\nProof. First, we consider assertion (i). We may assume without loss of generality\nthat U ⊆ W . Then since V ′ → V is proper, and W is normal, it follows that\nthe morphism U\n∼→ U ′ ↪→ V ′ extends uniquely to an open subset W \\ F ⊆ W ,\nwhere F is a closed subscheme of codimension ≥ 2 in W . Thus, the fact that the\nresulting morphism W \\ F → V ′ → A′ extends uniquely to W follows immediately\nfrom Lemma A.2. To verify the surjectivity of ΔW → ΔA′ , it suffices to verify the\nsurjectivity of ΔU → ΔA′ , i.e., of ΔU ′ → ΔV ′ → ΔA′ . On the other hand, this\nfollows from the surjectivity of ΔV ′ → ΔA′ [cf. Proposition A.3, (vii)], together\nwith the surjectivity of ΔU ′ → ΔV ′ [cf. the fact that U ′ ⊆ V ′ is a nonempty open\nsubscheme of the normal variety V ′].\nNext, we consider assertion (ii). By Proposition A.6, (i), (ii) [cf. also Propo-\nsition A.3, (viii), (ix); Corollary A.11, (i)], each induced morphism on Albanese\nvarieties An → Am, for n ≥ m, is a surjection of abelian varieties which is an\nisomorphism if and only if dimk(An) ≤ dimk(Am). On the other hand, if W ⊆ V\nis any nonempty k-toric [e.g., k-smooth] open subscheme, whose Albanese mor-\nphism [cf. Corollary A.11, (i)] we denote by W → AW , then assertion (i) yields a\nmorphism W → An that induces a surjection ΔW \u0002 ΔAn , hence, in particular, a\nmorphism AW → An that induces a surjection ΔAW \u0002 ΔAn . But this implies that\ndimk(An) ≤ dimk(AW ), hence that for some integer N , dimk(An) = dimk(Am), for\nall n,m ≥ N . In particular, if W = V , then dimk(An) ≤ dimk(A0), for all n ≥ 0.\n©\nTOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 81\nBibliography\n[BS] L. Barbieri-Viale, V. Srinivas, Albanese and Picard 1-motives,Mem. Soc. Math.\nFr. (N.S.) 87 (2001).\n[Bor] A. Borel, Linear Algebraic Groups, Second Enlarged Edition, Graduate Texts\nin Mathematics 126, Springer-Verlag (1991).\n[BLR] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Ergebnisse der Math-\nematik und ihrer Grenzgebiete 21, Springer-Verlag (1990).\n[Chev] C. 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Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku\nMath J. 59 (2007), pp. 455-479.\n[Mzk13] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves,\nJ. Math. Kyoto Univ. 47 (2007), pp. 451-539.\n[Mzk14] S. Mochizuki, The Étale Theta Function and its Frobenioid-theoretic Manifes-\ntations, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349.\n[Mzk15] S. Mochizuki, Topics in Absolute Anabelian Geometry II: Decomposition Groups\nand Endomorphisms, RIMS Preprint 1625 (March 2008).\n[MT] S. Mochizuki, A. Tamagawa, The algebraic and anabelian geometry of config-\nuration spaces, Hokkaido Math. J. 37 (2008), pp. 75-131.\n[Mumf] D. Mumford, Abelian Varieties, Oxford Univ. Press (1974).\n[NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields,Grundlehren\nder Mathematischen Wissenschaften 323, Springer-Verlag (2000).\n[NS] A. Néron and P. Samuel, La variété de Picard d’une variété normale, Ann.\nInst. 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Grothendieck, eds., Séminaire de Géométrie Algébrique du\nBois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1, Lecture notes\nin mathematics 151, Springer-Verlag (1970).\n[Tama1] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio\nMath. 109 (1997), pp. 135-194.\n[Tama2] A. Tamagawa, Resolution of nonsingularities of families of curves, Publ. of\nRIMS 40 (2004), pp. 1291-1336.\n[Tate] J. Tate, p-Divisible Groups, Proceedings of a Conference on Local Fields,\nDriebergen, Springer-Verlag (1967), pp. 158-183.\n[Yoshi] T. Yoshida, Finiteness theorems in the class field theory of varieties over local\nfields, J. of Number Theory 101 (2003), pp. 138-150.\n"}
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{"id":"plroz116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/The%20Intrinsic%20Hodge%20Theory%20of%20p-adic%20Hyperbolic%20Curves%20(ICM98).pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":125,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"The%20Intrinsic%20Hodge%20Theory%20of%20p-adic%20Hyperbolic%20Curves%20(ICM98)","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["MANU.DVI"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 12:25:29 UTC 2011","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":2008612308453129513,"exact_signature_unique_b":true,"fuzzy_signature_l":-6267149552075975884,"fuzzy_signature_unique_b":true,"wordcount_i":4116,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:45 UTC 2022","fresh_date_dt":"Thu Sep 23 19:01:53 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":108784,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086032557178880,"text_t":"The Intrinsic Hodge Theory of p-adic Hyperbolic Curves\nby Shinichi Mochizuki\n§1. Introduction\n(A.) The Fuchsian Uniformization\nA hyperbolic curve is an algebraic curve obtained by removing r points from a smooth,\nproper curve of genus g, where g and r are nonnegative integers such that 2g−2+r > 0. If\nX is a hyperbolic curve over the field of complex numbers C, then X gives rise in a natural\nway to a Riemann surface X . As one knows from complex analysis, the most fundamental\nfact concerning such a Riemann surface (due to Köbe) is that it may be uniformized by\nthe upper half-plane, i.e.,\nX ∼= H\/Γ\nwhere H\ndef\n= {z ∈ C | Im(z) > 0}, and Γ ∼= π1(X ) (the topological fundamental group of X )\nis a discontinuous group acting on H. Note that the action of Γ on H defines a canonical\nrepresentation\nρX : π1(X ) → PSL2(R) def= SL2(R)\/{±1} = AutHolomorphic(H)\nThe goal of the present manuscript is to survey various work ([Mzk1-5]) devoted to gener-\nalizing Köbe’s uniformization to the p-adic case.\nFirst, we observe that it is not realistic to expect that hyperbolic curves over p-\nadic fields may be literally uniformized by some sort of p-adic upper half-plane in the\nfashion of the Köbe uniformization. Of course, one has the theory of Mumford ([Mumf]),\nbut this theory furnishes a p-adic analogue not of Köbe’s Fuchsian uniformization (i.e.,\nuniformization by a Fuchsian group), but rather of what in the complex case is known\nas the Schottky uniformization. Even in the complex case, the Fuchsian and Schottky\nuniformizations are fundamentally different: For instance, as the moduli of the curve vary,\nits Schottky periods vary holomorphically, whereas its Fuchsian periods vary only real\nanalytically. This fact already suggests that the Fuchsian uniformization is of a more\narithmetic nature than the Schottky uniformization, i.e., it involves\nreal analytic structures ⇐⇒ complex conjugation ⇐⇒ Frobenius at the infinite prime\nThus, since one cannot expect a p-adic analogue in the form of a literal global uniformiza-\ntion of the curve, the first order of business is to reinterpret the Fuchsian uniformization\nin more abstract terms that generalize naturally to the p-adic setting.\n1991 Mathematics Subject Classification: 14F30\nKeywords and Phrases: hyperbolic curves, p-adic, fundamental group, Hodge theory\n1\n(B.) The Physical Interpretation\nThe first and most obvious approach is to observe that the Fuchsian uniformization\ngives a new physical, geometric way to reconstruct the original algebraic curve X. Namely,\none may think of the Fuchsian uniformization as defining a canonical arithmetic structure\nρX : π1(X ) → PSL2(R) on the purely topological invariant π1(X ). Alternatively (and\nessentially equivalently), one may think of the Fuchsian uniformization as the datum of a\nmetric (given by descending to X ∼= H\/Γ the Poincaré metric on H) – i.e., an arithmetic\n(in the sense of arithmetic at the infinite prime) structure – on the differential manifold\nunderlying X (which is a purely topological invariant). Then the equivalence\nX ⇐⇒ SO(2)\\PSL2(R)\/Γ\nbetween the algebraic curve X and the physical\/analytic object SO(2)\\PSL2(R)\/Γ ob-\ntained from ρX is given by considering modular forms on H = SO(2)\\PSL2(R), which\ndefine a projective (hence, algebraizing) embedding of X .\n(C.) The Modular Interpretation\nNote that ρX may also be regarded as a representation into PGL2(C) = GL2(C)\/C×,\nhence as defining an action of π1(X ) on P1C. Taking the quotient of H×P1C by the action\nof π1(X ) on both factors then gives rise to a projective bundle with connection on X .\nIt is immediate that this projective bundle and connection may be algebraized, so we\nthus obtain a projective bundle and connection (P → X,∇P ) on X. This pair (P,∇P )\nhas certain properties which make it an indigenous bundle (terminology due to Gunning).\nMore generally, an indigenous bundle on X may be thought of as a projective structure\non X , i.e., a subsheaf of the sheaf of holomorphic functions on X such that locally any\ntwo sections of this subsheaf are related by a linear fractional transformation. Thus, the\nFuchsian uniformization defines a special canonical indigenous bundle on X.\nIn fact, the notion of an indigenous bundle is entirely algebraic. Thus, one has a\nnatural moduli stack Sg,r → Mg,r of indigenous bundles, which forms a torsor (under\nthe affine group given by the sheaf of differentials on Mg,r) – called the Schwarz torsor\n– over the moduli stack Mg,r of hyperbolic curves of type (g, r). Moreover, Sg,r is not\nonly algebraic, it is defined over Z[12 ]. Thus, the canonical indigenous bundle defines a\ncanonical real analytic section\ns : Mg,r(C) → Sg,r(C)\nof the Schwarz torsor at the infinite prime. Moreover, not only does s “contain” all\nthe information that one needs to define the Fuchsian uniformization of an individual\nhyperbolic curve (indeed, this much is obvious from the definition of s!), it also essentially\n“is” (interpreted properly) the Bers uniformization of the universal covering space (i.e.,\n“Teichmüller space”) of Mg,r(C) (cf. the discussions in the Introductions of [Mzk1,4]).\n2\nThat is to say, from this point of view, one may regard the uniformization theory of\nhyperbolic curves and their moduli as the study of the canonical section s. Alternatively,\nfrom the point of view of Teichmüller theory, one may regard the uniformization theory of\nhyperbolic curves and their moduli as the theory of (so-called) quasi-fuchsian deformations\nof the representation ρX .\n(D.) The Notion of “Intrinsic Hodge Theory”\nNote that both the physical and modular approaches to the Fuchsian uniformization\nassert that there is a certain equivalence\nalgebraic geometry ⇐⇒ topology endowed with an arithmetic structure\nThat is, on the algebraic geometry side, we have the scheme (respectively, stack) given\nby the curve X itself in the physical approach (respectively, its moduli Mg,r in the mod-\nular approach), whereas on the “topology plus arithmetic structure” side, we have the\ntheory of the canonical representation ρX of π1(X ) (i.e., SO(2)\\PSL2(R)\/Γ in the phys-\nical approach; quasi-fuchsian deformations of ρX in the modular approach). This sort of\nequivalence is reminiscent of that given by classical or p-adic Hodge theory between the\nde Rham or Hodge cohomology of an algebraic variety (on the algebraic geometry side),\nand the singular or étale cohomology (equipped with Galois action) on the topology plus\narithmetic side. In our case, however, instead of dealing with the cohomology of the curve,\nwe are dealing with “the curve itself” and its moduli. It is for this reason that we refer to\nthis sort of theory as the intrinsic Hodge theory of the curve X.\nFinally, we note that this formal analogy with classical\/p-adic Hodge theory is by no\nmeans merely philosophical. Indeed, even in the classical theory reviewed in (B.) and (C.)\nabove, the methods of classical Hodge theory play an important technical role in the proofs\nof the main theorems. Similarly, in the theory of [Mzk1-5] – which constitute our main\nexamples of intrinsic Hodge theory for hyperbolic curves – the more recently developed\ntechniques of p-adic Hodge theory play a crucial technical role in the proofs of the main\nresults.\n§2. The Physical Approach in the p-adic Case\n(A.) The Arithmetic Fundamental Group\nLet K be a field of characteristic zero. Let us denote by K an algebraic closure of\nK. Let ΓK\ndef\n= Gal(K\/K). Let XK be a hyperbolic curve over K; write XK\ndef\n= X ×K K.\nThen one has an exact sequence\n1 → π1(XK) → π1(XK) → ΓK → 1\nof algebraic fundamental groups. (Here, we omit the base-points from the notation for the\nvarious fundamental groups.)\n3\nWe shall refer to π1(XK) as the geometric fundamental group of XK . Note that the\nstructure of π1(XK) is determined entirely by (g, r) (i.e., the “type” of the hyperbolic curve\nXK). In particular, π1(XK) does not depend on the moduli of XK . Of course, this results\nfrom the fact that K is of characteristic zero; in positive characteristic, on the other hand,\npreliminary evidence ([Tama2]) suggests that the fundamental group of a hyperbolic curve\nover an algebraically closed field (far from being independent of the moduli of the curve!)\nmay in fact completely determine the moduli of the curve.\nOn the other hand, we shall refer to π1(XK) (equipped with its augmentation to ΓK)\nas the arithmetic fundamental group of XK . Although it is made up of two “parts” – i.e.,\nπ1(XK) and ΓK – which do not depend on the moduli of XK , it is not unreasonable to\nexpect that the extension class defined by the above exact sequence, i.e., the structure of\nπ1(XK) as a group equipped with augmentation to ΓK , may in fact depend quite strongly\non the moduli of XK . Indeed, according to the anabelian philosophy of Grothendieck\n(cf. [LS]), for “sufficiently arithmetic” K, one expects that the structure of the arithmetic\nfundamental group π1(XK) should be enough to determine the moduli of XK . Although\nmany important versions of Grothendieck’s anabelian conjectures remain unsolved (most\nnotably the so-called Section Conjecture (cf., e.g., [LS], p. 289, 2)), in the remainder of\nthis §, we shall discuss various versions that have been resolved in the affirmative. Finally,\nwe note that this anabelian philosophy is a special case of the notion of “intrinsic Hodge\ntheory” discussed above: indeed, on the algebraic geometry side, one has “the curve itself,”\nwhereas on the topology plus arithmetic side, one has the arithmetic fundamental group,\ni.e., the purely (étale) topological π1(XK), equipped with the structure of extension given\nby the above exact sequence.\n(B.) The Main Theorem\nBuilding on earlier work of H. Nakamura and A. Tamagawa (see, especially, [Tama1]),\nthe author applied the p-adic Hodge theory of [Falt2] and [BK] to prove the following result\n(cf. Theorem A of [Mzk5]):\nTheorem 1. Let p be a prime number. Let K be a subfield of a finitely generated field\nextension of Qp. Let XK be a hyperbolic curve over K. Then for any smooth variety SK\nover K, the natural map\nXK(SK)\ndom → HomopenΓK (π1(SK), π1(XK))\nis bijective. Here, the superscripted “dom” denotes dominant (⇐⇒ nonconstant) K-\nmorphisms, while Hom\nopen\nΓK\ndenotes open, continuous homomorphisms compatible with the\naugmentations to ΓK , and considered up to composition with an inner automorphism aris-\ning from π1(XK).\nNote that this result constitutes an analogue of the “physical aspect” of the Fuchsian\nuniformization, i.e., it exhibits the scheme XK (in the sense of the functor defined by\n4\nconsidering (nonconstant) K-morphisms from arbitrary smooth SK to XK) as equivalent\nto the “physical\/analytic object”\nHom\nopen\nΓK\n(−, π1(XK))\ndefined by the topological π1(XK) together with some additional canonical arithmetic\nstructure (i.e., π1(XK)).\nIn fact, the proof of Theorem 1 was also motivated by this point of view: That is\nto say, just as one may regard the algebraic structure of a hyperbolic curve over C as\nbeing defined by certain (a priori) analytic modular forms on H, the proof of Theorem\n1 proceeds by considering certain p-adic analytic representations of differential forms on\nXK . In the p-adic case, however, the domain of definition of these analytic forms (i.e., the\nanalogue to the upper half-plane) is the spectrum of the p-adic completion of the maximal\ntame extension of the function field of XK along various irreducible components of the\nspecial fiber of a stable model X → Spec(OK) of XK (where OK is the ring of integers of\na finite extension K of Qp). It turns out that this object is, just like the upper half-plane,\nindependent of the moduli of XK .\nIn fact, various slightly stronger versions of Theorem 1 hold. For instance, instead of\nthe whole geometric fundamental group π1(XK), it suffices to consider its maximal pro-p\nquotient π1(XK)\n(p). Another strengthening allows one to prove the following result (cf.\nTheorem B of [Mzk5]), which generalizes a result of Pop ([Pop]):\nCorollary 2. Let p be a prime number. Let K be a subfield of a finitely generated field\nextension of Qp. Let L and M be function fields of arbitrary dimension over K. Then the\nnatural map\nHomK(Spec(L),Spec(M)) → HomopenΓK (ΓL,ΓM )\nis bijective. Here, Hom\nopen\nΓK\n(ΓL,ΓM ) is the set of open, continuous group homomorphisms\nΓL → ΓM over ΓK , considered up to composition with an inner homomorphism arising\nfrom Ker(ΓM → ΓK).\n(C.) Comparison with the Case of Abelian Varieties\nNote that there is an obvious formal analogy between Theorem 1 above and Tate’s\nconjecture on homomorphisms between abelian varieties (cf., e.g., [Falt1]). Indeed, in\ndiscussions of Grothendieck’s anabelian philosophy, it was common to refer to statements\nsuch as that of Theorem 1 as the “anabelian Tate conjecture,” or the “Tate conjecture for\nhyperbolic curves.” In fact, however, there is an important difference between Theorem 1\nand the “Tate conjecture” of, say, [Falt1]: Namely, the Tate conjecture for abelian varieties\nis false over local fields (i.e., finite extensions of Qp). Moreover, until the proof of Theorem\n5\n1, it was generally thought that, just like its abelian cousin, the “anabelian Tate conjecture”\nwas essentially global in nature. That is to say, it appears that the point of view of the\nauthor, i.e., that Theorem 1 should be regarded as a p-adic version of the “physical aspect”\nof the Fuchsian uniformization of a hyperbolic curve, does not exist in the literature (prior\nto the work of the author).\n§3. The Modular Approach in the p-adic Case\n(A.) The Example of Shimura Curves\nAs discussed in §1, (C.), classical complex Teichmüller theory may be formulated as\nthe study of the canonical real analytic section s of the Schwarz torsor Sg,r → Mg,r.\nThus, it is natural suppose that the p-adic analogue of classical Teichmüller theory should\nrevolve around some sort of canonical p-adic section of the Schwarz torsor. Then the\nquestion arises:\nHow does one define a canonical p-adic section of the Schwarz torsor?\nPut another way, for each (or at least most) p-adic hyperbolic curves, we would like to\nassociate a (or at least a finite, bounded number of) canonical indigenous bundles. Thus,\nwe would like to know what sort of properties such a “canonical indigenous bundle” should\nhave.\nThe model that provides the answer to this question is the theory of Shimura curves.\nIn fact, the theory of canonical Schwarz structures, canonical differentials, and canonical\ncoordinates on Shimura curves localized at finite primes has been extensively studied by Y.\nIhara (see, e.g., [Ihara]). In some sense, Ihara’s theory provides the prototype for the “p-adic\nTeichmüller theory” of arbitrary hyperbolic curves ([Mzk1-4]) to be discussed in (B.) and\n(C.) below. The easiest example of a Shimura curve is M1,0, the moduli stack of elliptic\ncurves. In this case, the projectivization of the rank two bundle on M1,0 defined by the\nfirst de Rham cohomology module of the universal elliptic curve on M1,0 gives rise (when\nequipped with the Gauss-Manin connection) to the canonical indigenous bundle on M1,0.\nMoreover, it is well-known that the p-curvature (a canonical invariant of bundles with\nconnection in positive characteristic which measures the extent to which the connection is\ncompatible with Frobenius) of this bundle has the following property:\nThe p-curvature of the canonical indigenous bundle on M1,0 (reduced mod p) is\nsquare nilpotent.\nIt was this observation that was the key to the development of the theory of [Mzk1-4].\n(B.) The Stack of Nilcurves\nLet p be an odd prime. Let Ng,r ⊆ (Sg,r)Fp denote the closed algebraic substack of\nindigenous bundles with square nilpotent p-curvature. Then one has the following key\nresult ([Mzk1], Chapter II, Theorem 2.3):\n6\nTheorem 3. The natural map Ng,r → (Mg,r)Fp is a finite, flat, local complete in-\ntersection morphism of degree p3g−3+r. Thus, up to “isogeny” (i.e., up to the fact that\nthis degree is not equal to one), Ng,r defines a canonical section of the Schwarz torsor\n(Sg,r)Fp → (Mg,r)Fp in characteristic p.\nIt is this stack Ng,r of nilcurves – i.e., hyperbolic curves in characteristic p equipped with\nan indigenous bundle with square nilpotent p-curvature – which is the central object of\nstudy in the theory of [Mzk1-4].\nOnce one has the above Theorem, next it is natural to ask if one can say more about\nthe fine structure of Ng,r. Although many interesting and natural questions concerning\nthe structure of Ng,r remain unsolved at the time of writing, a certain amount can be\nunderstood by analyzing certain substacks, or strata, of Ng,r defined by considering the\nloci of nilcurves whose p-curvature vanishes to a certain degree. For instance, nilcurves\nwhose p-curvature vanishes identically are called dormant. The locus of dormant nilcurves\nis denoted Ng,r[∞] ⊆ Ng,r. If a nilcurve is not dormant, then its p-curvature vanishes on\nsome divisor in the curve. We denote by Ng,r[d] ⊆ Ng,r the locus of nilcurves for which\nthis divisor is of degree d. The zeroes of the p-curvature are referred to as spikes. Now we\nhave the following result (cf. Theorems 1.2, 1.6 of the Introduction of [Mzk4]):\nTheorem 4. The Ng,r[d] are all smooth over Fp and either empty or of dimension\n3g − 3 + r. Moreover, Ng,r[0] is affine.\nIt turns out that this affineness of Ng,r[0], interpreted properly, gives a new proof of the\nconnectedness of (Mg,r)Fp (for p large relative to g). This fact is interesting (relative to\nthe claim that this theory is a p-adic version of Teichmüller theory) in that one of the\nfirst applications of classical complex Teichmüller theory is to prove the connectedness\nof Mg,r. Also, it is interesting to note that F. Oort has succeeded in giving a proof of\nthe connnectedness of the moduli stack of principally polarized abelian varieties by using\naffineness properties of certain natural substacks of this moduli stack in characteristic p.\nDespite the fact that the Ng,r[d] are smooth and of the same dimension as Ng,r, we\nremark that in most cases Ng,r is not reduced at Ng,r[d]. In fact, roughly speaking, the\nlarger d is, the less reduced Ng,r is at Ng,r[d]. In order to give sharp quantitative answers\nto such questions as:\nHow reduced is Ng,r at the generic point of Ng,r[d]? Or, what is the generic degree\nof Ng,r[d] over (Mg,r)Fp?\nit is necessary to study what happens to a nilcurve as the underlying curve degenerates to\na totally degenerate stable curve (i.e., a stable curve each of whose irreducible components\nis P1, with a total of precisely three marked points\/nodes). To do this, one must formulate\nthe theory (using “log structures”) in such a way that it applies to stable curves, as well.\nOnce one formulates the theory for stable curves, one sees that the answers to the\nquestions just posed will follow as soon as one:\n7\n(i.) Classifies all molecules – i.e., nilcurves whose underlying curve is a totally\ndegenerate stable curve.\n(ii.) Understands how molecules deform.\nThe answer to (i.) and (ii.) depends on an extensive analysis of molecules (cf. [Mzk2-4]),\nand, although combinatorially quite complicated, is, in some sense, complete. Although\nwe do not have enough space here to discuss this answer in detail, we pause to remark the\nfollowing: It turns out that the answer to (i.) consists of regarding molecules as concatena-\ntions of atoms – i.e., toral nilcurves (a slight generalization of nilcurves) whose underlying\ncurve is P1 with three marked points – and then classifying atoms. The difference be-\ntween a toral nilcurve and a (nontoral) nilcurve is that unlike the nontoral case, where the\n“radii” at the three marked points are assumed to be zero, in the toral case, one allows\nthese radii to be arbitrary elements of Fp\/{±1} (i.e., the quotient of the set Fp obtained\nby identifying λ and −λ for all λ ∈ Fp). Then it turns out that considering the three radii\nof an atom defines a natural bijection between the isomorphism classes of atoms and the\nset of (ordered) triples of elements of Fp\/{±1}.\nThe reason that we digressed to discuss the theory of atoms is that it is interesting\n(relative to the analogy with classical complex Teichmüller theory) in that it is reminiscent\nof the fact that a Riemann surface may be analyzed by decomposing it into pants (i.e.,\nRiemann surfaces which are topologically isomorphic to P1 − {0, 1,∞}). Moreover, the\nisomorphism class of a “pants” is completely determined by the radii of its three holes.\n(C.) Canonical Liftings\nSo far, we have been discussing the characteristic p theory. Ultimately, however, we\nwould like to know if the various characteristic p objects discussed in (B.) lift canonically\nto objects which are flat over Zp. Unfortunately, it seems that it is unlikely that Ng,r\nitself lifts canonically to some sort of natural Zp-flat object. If, however, we consider the\nopen substack – called the ordinary locus – (N ordg,r )Fp ⊆ Ng,r which is the étale locus of\nthe morphism Ng,r → (Mg,r)Fp , then (since the étale site is invariant under nilpotent\nthickenings) we get a canonical lifting, i.e., an étale morphism\nN ordg,r → (Mg,r)Zp\nof p-adic formal stacks. Over N ordg,r , one has the sought-after canonical p-adic splitting of\nthe Schwarz torsor (cf. Theorem 0.1 of the Introduction of [Mzk1]):\nTheorem 5. There is a canonical section N ordg,r → Sg,r of the Schwarz torsor over N ordg,r\nwhich is the unique section having the following property: There exists a lifting of Frobenius\nΦN : N ordg,r → N ordg,r such that the indigenous bundle on the tautological hyperbolic curve\nover N ordg,r defined by the section N ordg,r → Sg,r is invariant with respect to the Frobenius\naction defined by ΦN .\n8\nMoreover, it turns out that the Frobenius lifting ΦN : N ordg,r → N ordg,r (i.e., morphism\nwhose reduction modulo p is the Frobenius morphism) has the special property that 1p ·dΦN\ninduces an isomorphism Φ∗NΩN ordg,r\n∼= ΩN ordg,r . Such a Frobenius lifting is called ordinary. It\nturns out that any ordinary Frobenius lifting (i.e., not just ΦN ) defines a set of canonical\nmultiplicative coordinates in a formal neighborhood of any point α valued in an alge-\nbraically closed field k of characteristic p, as well as a canonical lifting of α to a point\nvalued in W (k) (Witt vectors with coefficients in k). Moreover, there is a certain analogy\nbetween this general theory of ordinary Frobenius liftings and the theory of real analytic\nKähler metrics (which also define canonical coordinates). Relative to this analogy, the\ncanonical Frobenius lifting ΦN on N ordg,r may be regarded as corresponding to the Weil-\nPetersson metric on complex Teichmüller space (a metric whose canonical coordinates are\nthe coordinates arising from the Bers uniformization of Teichmüller space). Thus, ΦN is,\nin a very real sense, a p-adic analogue of the Bers uniformization in the complex case.\nMoreover, there is, in fact, a canonical ordinary Frobenius lifting on the “ordinary locus”\nof the tautological curve over N ordg,r whose relative canonical coordinate is analogous to the\ncanonical coordinate arising from the Köbe uniformization of a hyperbolic curve.\nNext, we observe that Serre-Tate theory for ordinary (principally polarized) abelian\nvarieties may also be formulated as arising from a certain canonical ordinary Frobenius\nlifting. Thus, the Serre-Tate parameters (respectively, Serre-Tate canonical lifting) may be\nidentified with the canonical multiplicative parameters (respectively, canonical lifting to\nthe Witt vectors) of this Frobenius lifting. That is to say, in a very concrete and rigorous\nsense, Theorem 5 may be regarded as the analogue of Serre-Tate theory for hyperbolic\ncurves. Nevertheless, we remark that it is not the case that the condition that a nilcurve\nbe ordinary (i.e., defines a point of (N ordg,r )Fp ⊆ Ng,r) either implies or is implied by the\ncondition that its Jacobian be ordinary. Although this fact may disappoint some readers,\nit is in fact very natural when viewed relative to the general analogy between ordinary\nFrobenius liftings and real analytic Kähler metrics discussed above. Indeed, relative to\nthis analogy, we see that it corresponds to the fact that, when one equips Mg with the\nWeil-Petersson metric and Ag (the moduli stack of principally polarized abelian varieties)\nwith its natural metric arising from the Siegel upper half-plane uniformization, the Torelli\nmap Mg → Ag is not isometric.\nNext, we remark that (N ordg,r )Fp ⊆ Ng,r[0]. Thus, the other Ng,r[d]’s are left out of\nthe theory of canonical liftings arising from Theorem 5. Nevertheless, in [Mzk2,4], a more\ngeneral theory of canonical liftings is developed that includes arbitrary Ng,r[d]. In this\nmore general theory, instead of getting local uniformizations by multiplicative canonical\nparameters, i.e., uniformizations by ̂Gm, we get uniformizations by more general types\nof Lubin-Tate groups, or twisted products of such groups. Roughly speaking, the more\n“spikes” in the nilcurves involved – i.e., the larger the d of Ng,r[d] – the more Lubin-Tate\nthe uniformization becomes.\nFinally, we remark that once one develops these theories of canonical liftings, one\nalso gets accompanying canonical (crystalline) Galois representations of the arithmetic\nfundamental group of the tautological curve over N ordg,r (and its Lubin-Tate generalizations)\ninto PGL2 of various complicated rings with Galois action. It turns out that these Galois\n9\nrepresentations are the analogues of the canonical representation ρX (of §1, (A.)) – which\nwas the starting point of our entire discussion.\nBibliography\n[BK] Bloch, S. and Kato, K., L-Functions and Tamagawa Numbers of Motives in The\nGrothendieck Festschrift, Volume I, Birkhäuser (1990), pp. 333-400.\n[Falt1] Faltings, G., Endlichkeitssätze für Abelschen Varietäten über Zahlkörpern, Inv. Math.\n73 (1983), pp. 349-366.\n[Falt2] Faltings, G., p-adic Hodge Theory, Journal of the Amer. Math. Soc. 1, No. 1 (1988),\npp. 255-299.\n[Ihara] Ihara, Y., On the Differentials Associated to Congruence Relations and the Schwarzian\nEquations Defining Uniformizations, Jour. Fac. Sci. Univ. Tokyo, Sect. IA Math. 21\n(1974), pp. 309-332.\n[LS] Lochak, P. and Schneps, L., Geometric Galois Actions: 1. Around Grothendieck’s\nEsquisse d’un Programme, London Math. Soc. Lect. Note Ser. 242, Cambridge Univ.\nPress, 1997.\n[Mumf] Mumford, D., An Analytic Construction of Degenerating Curves over Complete Local\nRings, Comp. Math. 24 (1972), pp. 129-174.\n[Mzk1] Mochizuki, S., A Theory of Ordinary p-adic Curves, RIMS Preprint 1033 (September\n1995); Publ. of RIMS 32, No. 6 (1996), pp. 957-1151.\n[Mzk2] Mochizuki, S., The Generalized Ordinary Moduli of p-adic Hyperbolic Curves, RIMS\nPreprint 1051 (December 1995); 281 pp.\n[Mzk3] Mochizuki, S., Combinatorialization of p-adic Teichmüller Theory, RIMS Preprint\n1076 (April 1996); 32 pp.\n[Mzk4] Mochizuki, S., Foundations of p-adic Teichmüller Theory, in preparation.\n[Mzk5] Mochizuki, S., The Local Pro-p Anabelian Geometry of Curves, RIMS Preprint 1097\n(August 1996); 84 pp.\n[Pop] Pop, F., On Grothendieck’s conjecture of birational anabelian geometry II, Preprint\n(1995).\n[Tama1] Tamagawa, A., The Grothendieck conjecture for affine curves, Compositio Math. 109,\nNo. 2 (1997), pp. 135-194.\n[Tama2] Tamagawa, A., On the fundamental groups of curves over algebraically closed fields\nof characteristic > 0, RIMS Preprint 1182 (January 1998).\nShinichi Mochizuki\nResearch Institute for Mathematical Sciences\nKyoto University\nKyoto 606-01, JAPAN\nmotizuki@kurims.kyoto-u.ac.jp\n10\n"}
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{"id":"u3Lg-116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/The%20Local%20Pro-p%20Anabelian%20Geometry%20of%20Curves.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":103,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"The%20Local%20Pro-p%20Anabelian%20Geometry%20of%20Curves","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["The Local Pro-p Anabelian Geometry of Curves.pdf"],"author":"<965D8C8E8140905688EA>","author_sxt":["<965D8C8E8140905688EA>"],"last_modified":"Mon Nov 07 12:26:40 UTC 2011","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":6158402056961508941,"exact_signature_unique_b":true,"fuzzy_signature_l":2653549190742099624,"fuzzy_signature_unique_b":true,"wordcount_i":43410,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:49 UTC 2022","fresh_date_dt":"Thu Sep 23 19:01:22 UTC 2027","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 10.1.1 (Windows)","language_s":"en","size_i":718609,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086036496678912,"text_t":"The Local Pro-p Anabelian Geometry of Curves\nby Shinichi Mochizuki\nINTRODUCTION\nThe Anabelian Geometry of Grothendieck:\nLet X be a connected scheme. Then one can associate (after Grothendieck) to X\nits algebraic fundamental group π1(X). This group π1(X) is a profinite group which is\nuniquely determined (up to inner automorphisms) by the property that the category of\nfinite, discrete sets equipped with a continuous π1(X)-action is equivalent to the category\nof finite étale coverings of X. Moreover, the assignment X \u0002→ π1(X) is a functor from the\ncategory of connected schemes (and morphisms of schemes) to the category of profinite\ntopological groups and continuous outer homomorphisms (i.e., continuous homomorphisms\nof topological groups, where we identify any two homomorphisms that can be obtained from\none another by composition with an inner automorphism).\nNow let K be a field. Let ΓK be the absolute Galois group of K. Then π1(Spec(K))\nmay be identified with ΓK . Let XK be a variety (i.e., a geometrically integral separated\nscheme of finite type) over K. Then the structure morphism XK → Spec(K) defines a\nnatural augmentation π1(XK) → ΓK . The kernel of this morphism π1(XK) → ΓK is a\nclosed normal subgroup of π1(XK) – called the geometric fundamental group of XK –\nwhich may be identified with π1(XK) (where XK\ndef\n= XK ⊗K K). If, moreover, one fixes\na prime number p, then one can form the maximal pro-p quotient π1(XK)\n(p) of π1(XK).\nSince the quotient π1(XK) → π1(XK)(p) is characteristic, it follows that the kernel of this\nquotient is, in fact, a normal subgroup of π1(XK). The quotient of π1(XK) by this normal\nsubgroup will be denoted ΠXK . Thus, ΠXK inherits a natural augmentation ΠXK → ΓK\nfrom that of π1(XK).\nNow let us consider the assignment\nπ1(−)K : {XK → Spec(K)} \u0002→ {π1(XK) → ΓK}\nThis assignment defines a functor from the category CK of K-varieties (whose morphisms\nare K-linear morphisms of varieties) to the category GK whose objects are profinite topo-\nlogical groups equipped with an augmentation to ΓK , and whose morphisms are continu-\nous outer homomorphisms of topological groups that lie over ΓK . It was the intuition of\nGrothendieck (see [Groth]) that:\n1\nFor certain types of K, if one replaces CK and GK by “certain appro-\npriate” subcategories C′K and G′K (such that π1(−)K still maps C′K into\nG′K), then π1(−)K should be fully faithful.\nHere, the “certain appropriate” subcategories C′K for which this piece of intuition was to\nhold true were tentatively assigned the appellation anabelian, while the piece of intuition\nitself came to be referred to as Grothendieck’s Conjecture of Anabelian Geometry (or, sim-\nply, the “Grothendieck Conjecture,” for short). Roughly speaking, the sorts of varieties\nthat were thought to be likely to be “anabelian” were varieties that are “sufficiently hyper-\nbolic.” (Note that, as one can see in the case of curves, hyperbolic varieties tend to have\nhighly nonabelian fundamental groups, hence the term “anabelian.” In higher dimensions,\nhowever, things are not so simple – see, e.g., [IN].)\nA variant of the above “profinite” Grothendieck Conjecture is the following “pro-\np” Grothendieck Conjecture: Namely, instead of considering π1(−)K , one considers the\nfunctor that assigns to the K-variety XK the augmented group ΠXK → ΓK . The “pro-p”\nGrothendieck Conjecture then asserts that, in certain situations, this functor should be\nfully faithful. The present paper is concerned with proving various versions of the profinite\nand pro-p Grothendieck Conjectures under various conditions.\nStatement of the Main Results:\nLet p be a prime number. Let K be a sub-p-adic field (cf. Definition 15.4 (i)), i.e., a\nsubfield of a finitely generated field extension of Qp. In this paper, we prove a pro-p version\nof the Grothendieck Conjecture (Theorem A) for dominant morphisms between “smooth\npro-varieties” and “hyperbolic pro-curves” over K. (Here, by a “smooth pro-variety”\n(respectively, “hyperbolic pro-curve”) over K, we mean a K-scheme which can be written\nas a projective limit of smooth varieties (respectively, hyperbolic curves) over K in which\nthe transition morphisms are birational – cf. Definitions 15.4 (ii) and 16.4.) We then give\nvarious versions of Theorem A (namely, Theorems A′ and A′′) for truncated fundamental\ngroups. From Theorem A, we also derive a profinite version of the Grothendieck Conjecture\n(Theorem B) for morphisms between function fields (of arbitrary dimension) over K. Next,\nwe apply Theorem A to prove the “injectivity part” (Theorem C) of the pro-p “Section\nConjecture.” Finally, in an Appendix, we derive from Theorem A an isomorphism version\nof the Grothendieck Conjecture for certain hyperbolic surfaces (Theorem D).\nNotation: If K is a field and XK is a K-scheme, we denote by Π\nprf\nXK\ndef\n= π1(XK) the\nfundamental group of XK (for some choice of base-point), and by ΓK the absolute Galois\ngroup of K. Then we have a natural morphism Π\nprf\nXK\n→ ΓK whose kernel is the geometric\nfundamental group Δ\nprf\nX ⊆ Π\nprf\nXK\n. Let ΔX be the maximal pro-p quotient of Δ\nprf\nX . Then\nthe kernel of Δ\nprf\nX → ΔX is a normal subgroup of Π\nprf\nXK\n, so by forming the quotient of Π\nprf\nXK\nby this normal subgroup, we obtain a group ΠXK , together with a morphism ΠXK → ΓK\nwhose kernel is ΔX . Next, if Δ is a topological group, we let: Δ{0}\ndef\n= Δ; for i ≥ 1,\nΔ{i} def= [Δ{i− 1}, Δ{i− 1}] (here we mean “the closed subgroup generated by the purely\n2\ngroup-theoretic commutator subgroup”). In the specific case of fundamental groups, let\nus write “Πi” (respectively, “Δi”) for Π\/Δ{i} (respectively, Δ\/Δ{i}).\nOur first main theorem is the following:\nTheorem A. Let p be a prime number. Let K be sub-p-adic (cf. Definition 15.4\n(i)). Let XK be a smooth pro-variety over K. Let YK be a hyperbolic pro-curve over\nK. Let HomdomK (XK , YK) be the set of dominant K-morphisms from XK to YK. Let\nHom\nopen\nΓK\n(ΠXK ,ΠYK ) (respectively, Hom\nopen\nΓK\n(Π\nprf\nXK\n,Π\nprf\nYK\n)), be the set of open, continuous\ngroup homomorphisms ΠXK → ΠYK (respectively, Π\nprf\nXK\n→ ΠprfYK ) over ΓK , considered up\nto composition with an inner automorphism arising from ΔY (respectively, Δ\nprf\nY ). Then\nthe natural maps\nHomdomK (XK , YK) → HomopenΓK (Π\nprf\nXK\n,Π\nprf\nYK\n)\n→ HomopenΓK (ΠXK ,ΠYK )\nare bijective.\nThis Theorem is given as Theorem 16.5 (cf. also the Remark following Theorem 16.5) in\nthe text. It is from Theorem A that all of the other major results of this paper are derived.\nWe remark that:\n(1) In fact, really, the main portion of Theorem A is the bijectivity of the\nfirst and third Hom’s. That is to say, the bijectivity of the first and\nsecond Hom’s follows formally from the bijectivity of the first and third\nHom’s. See the Remark following Theorem 16.5 for more details.\n(2) The notion of a “smooth pro-variety” (respectively, “hyperbolic pro-\ncurve”) has as special cases: (i) a smooth variety (respectively, hyper-\nbolic curve) over K; (ii) the spectrum of a function field of arbitrary\ndimension (respectively, function field of dimension one) over K.\n(3) There exists a substantial body of people who, when they speak of\n“the Grothendieck Conjecture” (respectively, “the pro-p Grothendieck\nConjecture”), refer to the following rather specific statement (which\nis a special case of the general philosophy discussed in the preceding\nsubsection): If K is a finitely generated extension of Q, and XK and\nYK are either hyperbolic curves over K or the spectra of one-dimensional\nfunction fields over K, then the isomorphisms of XK with YK are in\nnatural bijective correspondence with the outer isomorphisms over ΓK of\nΠ\nprf\nXK\n(respectively, ΠXK ) with Π\nprf\nYK\n(respectively, ΠYK ). This statement\nis manifestly a special case of the profinite version – see (1) above –\n(respectively, pro-p version, i.e., version as stated above) of Theorem A.\n3\nWe also have truncated versions of Theorem A (Theorems 18.1 and 18.2 in the text):\nTheorem A′. Let K be sub-p-adic. Let XK be a smooth variety over K. Let YK be a\nhyperbolic curve over K. Let n ≥ 5. Then every continuous open homomorphism\nθ : ΠnXK → Π\nn\nYK\nover ΓK induces a dominant morphism μ : XK → YK whose induced morphism on fun-\ndamental groups coincides (up to composition with an inner automorphism arising from\nΔYK ) with the morphism Δ\nn−3\nXK\n→ Δn−3YK defined by considering θ “modulo Δ{n − 3}.”\nTheorem A′′. Let K be sub-p-adic. Let XK be a smooth pro-variety over K. Let YK\nbe a hyperbolic pro-curve over K. Let n′0 be the minimum transcendence degree over Qp\nof all finitely generated field extensions of Qp that contain K. Let n\n′′\n0 be the transcendence\ndegree over K of the function field of XK . Let n0\ndef\n= n′0 + 2(n′′0 − 1) + 1.\nLet n ≥ 3n0 + 5. Then every continuous open homomorphism\nθ : ΠnXK → Π\nn\nYK\nover ΓK induces a dominant morphism μ : XK → YK whose induced morphism on\nfundamental groups coincides (up to composition with an inner automorphism arising\nfrom ΔYK ) with the morphism Δ\nn−3−3n0\nXK\n→ Δn−3−3n0YK defined by considering θ “mod-\nulo Δ{n − 3 − 3n0}.”\nOur second main theorem (Theorem 17.1 in the text) is the following:\nTheorem B. Let p be a prime number. Let K be sub-p-adic. Let L and M be\nfunction fields of arbitrary dimension over K. (In particular, we assume that K is alge-\nbraically closed in L and M .) Let HomK(Spec(L),Spec(M)) be the set of K-morphisms\nfrom M to L. Let Hom\nopen\nΓK\n(ΓL,ΓM ) be the set of open, continuous group homomorphisms\nΓL → ΓM over ΓK , considered up to composition with an inner automorphism arising\nfrom Ker(ΓM → ΓK). Then the natural map\nHomK(Spec(L),Spec(M)) → HomopenΓK (ΓL,ΓM )\nis bijective.\nNote that in characteristic zero, this generalizes the results of [Pop1], [Pop2], where a\nsimilar result to Theorem B is obtained, except that the morphisms ΓL → ΓM are required\nto be isomorphisms, and K is required to be finitely generated over Q.\n4\nOur third main theorem (Theorem 19.1 in the text) is the following:\nTheorem C. Let p be a prime number. Let K be sub-p-adic. Let XK be a hyperbolic\ncurve over K. Let XK(K) be the set of K-valued points of XK . Let Sect(ΓK ,ΠXK ) be\nthe set of sections ΓK → ΠXK of ΠXK → ΓK , considered up to composition with an inner\nautomorphism arising from ΔX . Then the natural map\nXK(K) → Sect(ΓK ,ΠXK )\nis injective.\nFinally, in the Appendix, we use Theorem A to derive the following result:\nTheorem D. Let p be a prime number. Let K be sub-p-adic. Let XK and YK be\nhyperbolically fibred surfaces (see Definition a2.1 in the Appendix for a precise definition\nof this term) over K. Let IsomK(XK , YK) be the set of K-isomorphisms (in the category\nof K-schemes) between XK and YK . Let IsomΓK(Π\nprf\nXK\n,Π\nprf\nYK\n) be the set of continuous\ngroup isomorphisms Π\nprf\nXK\n→ ΠprfYK over ΓK, considered up to composition with an inner\nautomorphism arising from Δ\nprf\nY . Then the natural map\nIsomK(XK , YK) → IsomΓK (ΠprfXK ,Π\nprf\nYK\n)\nis bijective.\nRecent Work on the Grothendieck Conjecture:\nIn this subsection, we would like to take a brief look at recent work on the Grothendieck\nConjecture. We will concentrate only on major and recent (since the early 1980’s) develop-\nments that relate to the present paper, and we make no pretense of giving a complete history\nof work on the Grothendieck Conjecture. Before beginning, it is worth pointing out that\nuntil the appearance of the present paper (and its earlier version [Mzk2]), it was widely\nassumed that the base field “K” that appears in the Grothendieck Conjecture should be\nassumed to be finitely generated over Q. It is not clear precisely why this came to be as-\nsumed by most people working in the field, but one possible cause is that the Grothendieck\nConjecture appears to have originated as an alternative approach to diophantine geometry\n([Groth], [Pop3]). Thus, if one’s ultimate aim is applications to diophantine geometry, it\nis quite natural only to look at global fields (i.e., finitely generated extensions of Q), and\nnot at p-adic local fields, for instance.\nAnother reason for the fixation on finitely generated extensions of Q appears to have\nbeen that many people conceived of the Grothendieck Conjecture as an anabelian version\n5\nof the Tate Conjecture (Faltings’ theorem – see [Falt3]) for abelian varieties over number\nfields. In fact, this point of view became so deeply engrained that it gave rise to a tendency\nfor people to try to prove the Grothendieck Conjecture for hyperbolic curves over number\nfields by deducing it from the Tate Conjecture for abelian varieties over number fields (see,\ne.g., [Naka3]).\nIn fact, even important early work – such as [Naka1] on the Grothendieck Conjecture\nfor genus zero hyperbolic curves over number fields, or [Pop1], [Pop2] which treat the\nbirational case – which does not try to deduce the Grothendieck Conjecture from the\nTate Conjecture still had a distinctly global flavor, and relied extensively on essentially\nglobal techniques. These global techniques ultimately proved to be rather irrelevant to the\nproof of the main results of the present paper. Nevertheless, what was important (from\nthe point of view of the author) about the work of H. Nakamura was that it established a\nhost of basic techniques for studying the outer Galois action on the (nonabelian) geometric\nfundamental group of a hyperbolic curve, which was unknown territory to most arithmetic\ngeometers, who were only familiar with the Galois action on the abelian fundamental group\nof an abelian variety. It was this culture of basic techniques (due to H. Nakamura) which\npermitted the subsequent development of more powerful approaches to the Grothendieck\nConjecture itself, as discussed below.\nThe next important development (in early 1995) was the work of A. Tamagawa\n([Tama]) in which it was shown that the isomorphism class of any affine hyperbolic curve\nover an absolutely finitely generated field is functorially determined by the outer Galois ac-\ntion on its profinite geometric fundamental group. (In fact, by a relatively straightforward\nargument ([Mzk1]), it is possible to remove the “affineness” hypothesis in Tamagawa’s\nresult, at least in characteristic zero.) Unlike the results of the present paper, Tamagawa’s\nresult holds in positive characteristic as well as in characteristic zero. Moreover, Tama-\ngawa’s ideas on characterizing those sections of π1(XK) → ΓK (where XK is a hyperbolic\ncurve over a field K) that arise from geometric points of XK played a pivotal role in inspir-\ning the author to prove the results of the present paper. Concretely speaking, the influence\nof these ideas of Tamagawa can be seen in the argument of Section 9 of the present paper.\nFinally, we turn to discussing the results of the present paper in a historical context.\nAs observed previously, the results of the present paper are (unlike Tamagawa’s results)\nonly valid in characteristic zero. On the other hand, the main advances of Theorem A of\nthe present paper relative to [Tama] are as follows:\n(1) The hyperbolic curves involved are allowed to be proper. In fact, we\neven allow what we call hyperbolic pro-curves.\n(2) Instead of dealing with profinite geometric fundamental groups, we\ndeal with pro-p geometric fundamental groups. Pro-p results tend to\nbe stronger than profinite results in the sense that profinite results usu-\nally follow immediately from pro-p results (see the Remark following\nTheorem 16.5 for more details).\n(3) Instead of just considering isomorphisms, we allow arbitrary dominant\nmorphisms between the varieties involved.\n6\n(4) The variety “on the left” (in the “Hom(−,−)” of Theorem A) is allowed\nto be higher-dimensional, and need not even be hyperbolic.\n(5) The base field can be any subfield of a finitely generated extension of\nQp, whereas for Tamagawa (in characteristic zero) the base field must\nbe a finitely generated extension of Q.\nFrom the point of view of the author, (5) is the most fundamental and important advance\nas it is associated with the fact that the techniques of the present paper are fundamentally\ndifferent from those of both Nakamura and Tamagawa in that they are couched in the\nworld of p-adic Hodge theory. That is to say, the proof of Theorem A may be regarded\nas an application of the theory of [Falt1] and [BK]. It is precisely the use of these p-adic\ntechniques that allowed the author to prove a result which was much stronger (in the above\nfive senses) than the result of [Tama] (in characteristic zero) or [Mzk1]. Moreover, it is the\nopinion of the author that:\nThe reason that it took so long for Theorem A to be discovered was the\noverwhelming prejudice of most people in the field that the Grothendieck\nConjecture for hyperbolic curves is an essentially global result, akin to\nthe Tate Conjecture for abelian varieties over number fields. In fact,\nhowever, it is much more natural to regard the Grothendieck Conjecture\nfor hyperbolic curves as an essentially local, p-adic result that belongs\nto that branch of arithmetic geometry known as p-adic Hodge theory.\nMoreover, it is the feeling of the author that, more than the technical details of the state-\nment of Theorem A, it is this fact – i.e., that the Grothendieck Conjecture for hyperbolic\ncurves is best understood not as a global, number-theoretic result, but rather as a result in\np-adic Hodge theory – that is the central discovery of this paper.\nThe Structure of the Proof:\nFirst, we remark that most of the paper (Sections 1 through 14) is devoted to proving\nthe following technical result:\n(∗)tech Suppose that K is a finite extension of Qp. Let XK and YK\nbe proper hyperbolic curves over K. Let UK be the spectrum of the\nfunction field of XK. Then every continuous surjective homomorphism\nθ : ΠUK → ΠYK over ΓK arises from some geometric morphism UK →\nYK .\nSection 15 is devoted to the rather standard technicalities necessary to generalize (∗)tech\nto the case where K is any subfield of a finitely generated extension of Qp. Section 16\nis devoted to carrying out a standard “cutting by hyperplane sections argument” which\nallows one to replace UK by a higher-dimensional smooth pro-variety (which thus completes\n7\nthe proof of Theorem A). Section 17 is devoted to an induction argument used to derive\nTheorem B from Theorem A. Section 18 discusses how the techniques used to prove\nTheorem A in fact also give rise to the truncated versions Theorem A′ and Theorem\nA′′. Finally, Section 19 reviews a standard argument (already present in the work of H.\nNakamura) which allows one to derive Theorem C from Theorem A. Thus, the rest of this\nsubsection will be devoted to outlining the proof of (∗)tech in Sections 1 through 14.\nIn order to simplify the discussion, let us consider first the following slightly modified\nversion of (∗)tech:\n(∗)prop Suppose that K is a finite extension of Qp. Let XK and YK\nbe proper hyperbolic curves over K. Then every continuous surjective\nhomomorphism θ : ΠXK → ΠYK over ΓK arises from some geometric\nmorphism XK → YK.\n(Here, “prop” stands for “proper” – i.e., since, unlike in (∗)tech, where UK appeared, in\n(∗)prop only proper curves appear.) Thus, (∗)tech implies (∗)prop. In other words, (up to\nsome standard general nonsense) one may think of (∗)tech as the concatenation of (∗)prop\nwith the following assertion:\n(∗)iner Any θ : ΠUK → ΠYK as in (∗)tech necessarily factors through the\nquotient ΠUK → ΠXK .\n(Here, “iner” stands for “inertia” – since the kernel of ΠUK → ΠXK is generated by\ninertia groups, so (∗)iner is the assertion that θ : ΠUK → ΠYK always sends inertia groups\nin ΠUK to the identity.) We will come back to the issue of (∗)iner later, but for now, let us\nconcentrate on outlining the proof of (∗)prop, since it is this which is the core of the paper.\nThus, let us assume that we have been given a continuous surjective homomorphism:\nθ : ΠXK → ΠYK\nover ΓK . We would like to somehow manufacture out of θ a morphism from XK to YK . A\nnatural, naive way to start is the following: First, by replacing XK and YK by appropriate\nétale coverings, we may assume that neither is hyperelliptic (see, e.g., Lemma 10.4 (4)).\nLet us write DX (respectively, DY ) for H\n0(XK , ωXK\/K) (respectively, H\n0(YK , ωYK \/K)),\nthe space of global differentials on XK (respectively, YK). Then it is well-known that XK\n(respectively, YK) embeds naturally in the projective space PX\ndef\n= P(DX ) (respectively,\nPY\ndef\n= P(DY )). Moreover, since the p-adic étale cohomology of XK or YK can be computed\nas the group cohomology of the respective pro-p geometric fundamental group, it follows\nthat the surjection θ induces a ΓK-equivariant injection\nθH : H\n1(YK ,Qp) ↪→ H1(XK ,Qp)\n8\nMoreover, it is well-known (see, e.g., [Falt1], [Tate]) that the ΓK -module H\n1(XK ,Qp) is\nHodge-Tate, and, moreover, that if we tensor H1(XK ,Qp) over Qp with Cp(1) (where the\n“(1)” is a “Tate twist”), and take ΓK-invariants, we naturally recover the space DX of\ndifferentials. Thus, since θH is ΓK -equivariant, if we tensor θH over Qp with Cp(1) and\ntake ΓK-invariants, we obtain from θ (in a natural way) a K-linear injection\nθD : DY ↪→ DX\nand hence a rational map from PX to PY .\nThus, in some sense, without doing anything terribly new (i.e., we have only just\napplied results known to Tate since the 1960’s), we have already come relatively close to\nconstructing a morphism from XK to YK . Indeed, what we have done is to construct a\nmorphism θD from differentials on YK to differentials on XK , which we would like to hope\narises as the pull-back map on differentials associated to a morphism from XK to YK .\nMoreover, since XK (respectively, YK) is canonically embedded in PX (respectively, PY ),\nit follows from elementary algebraic geometry that the “only” thing we need to show is\nthat θD preserves relations: That is, if i is a positive integer, let D\ni\nX\ndef\n= H0(XK , ω\n⊗i\nXK\/K\n).\nSimilarly, we have DiY . Note that multiplication of differential forms defines a natural\nmorphism\ni⊗\nDY → DiY\nLet us denote the kernel of this morphism by Ri. Thus, Ri is the set of relations (of degree\ni) defining YK as a subvariety of PY . Moreover, by composing the i\nth tensor power of θD\nwith the natural morphism from the ith tensor power of DX to D\ni\nX , we obtain a morphism\nκi :\ni⊗\nDY → DiX\nThen we shall say that θ : ΠXK → ΠYK preserves relations if κi(Ri) = 0 for all positive\nintegers i. As stated earlier, once we know that θ preserves relations, it follows from\nelementary algebraic geometry (see [Harts], Chapter II) that the rational map from PX to\nPY defined by θD induces a (dominant) morphism XK → YK , as desired. (Once one has\nthis morphism XK → YK , the fact that the map that it induces on fundamental groups\ncoincides with θ is a matter of general nonsense – for details, we refer to the argument\npreceding Theorem 14.1.)\nThus, to review what we have done so far, we have reduced the main problem to\nshowing that any θ : ΠXK → ΠYK as in (∗)prop preserves relations. It should be emphasized\nat this point, that so far everything that we have done has been painless general nonsense\n– the substantive mathematical core of the argument is yet to come (after some more\ngeneral nonsense in the next few paragraphs). The next step is to introduce a field L,\n9\nas follows: Let us assume for simplicity that XK extends to a stable curve X over OK .\nLet p ∈ X be an irreducible component of the special fiber of X . Then the completion\nof the local ring OX ,p is a p-adically complete discrete valuation ring OL, whose quotient\nfield we denote by L. The space ΩL of p-adically continuous differentials of L over K is\nthen a one-dimensional L-vector space. Thus, L is, in some sense, one-dimensional over\nK. Alternatively, one may think of Spec(L) roughly as some sort of small p-adic open set\nin XK . At any rate, from the construction of L, it follows that we have a natural L-valued\npoint ξX : Spec(L) → XK . Since L is “one-dimensional,” it is easy to check that the\nrestriction map on differentials\nDiX → Ω⊗iL\n(where “Ω⊗iL ” denotes the tensor product of i copies of ΩL over L) is injective. Thus,\ninstead of checking that κi(Ri) = 0, it suffices to check that the composite\nκiL :\ni⊗\nDY → Ω⊗iL\nof κi with the above restriction map vanishes on Ri. In other words, we would like to\ncompute κiL.\nTo “compute” κiL, we need to go back to looking at fundamental groups. First of all,\nlet us observe that by functoriality of the fundamental group, the natural L-valued point\nξX : Spec(L) → XK defines a morphism\nαLX : ΓL → ΠXK\nwhose composite with the augmentation ΠXK → ΓK is the morphism ΓL → ΓK on Galois\ngroups induced by the inclusion of fields K ⊆ L. Moreover, if we compose αLX with θ, we\nobtain a morphism\nαLY : ΓL → ΠYK\nNow let us suppose that we know that αLY is geometric, i.e., that it arises from some L-\nvalued point ξY : Spec(L) → YK of YK . Then it follows immediately from the theory of\n[Falt1] that κiL may be computed as the restriction map on differentials associated to ξY .\nBut it is clear that this restriction map on differentials annihilates Ri. Thus, to summarize:\nin order to show that θ preserves relations, it suffices to show that αLY is geometric.\nAs remarked earlier, the above prefatory remarks are just “general nonsense.” The\nmathematical core of the present paper lies in showing that given a geometric αLX , together\nwith a θ : ΠXK → ΠYK as in (∗)prop, the resulting αLY is again geometric. The proof that\nαLY is geometric is long and intricate, and can be divided roughly into four parts:\n10\n(1) First, we consider a K-valued point x ∈ XK(K) of XK , and its asso-\nciated map on fundamental groups αKX : ΓK → ΠXK . It is elementary\nto show that the process of passing from the point x ∈ XK(K) to the\narithmetic first Chern class of its associated line bundle\nηKX\ndef\n= c1(OXK (x)) ∈ H2et(XK ,Zp(1)) = H2(ΠXK ,Zp(1))\ncan be carried out just by working with αKX (see the first half of Section 7\nfor details on this “group-theoretic recipe” for concocting ηKX out of α\nK\nX ).\nOn the other hand, by composing αKX with θ, we get an α\nK\nY : ΓK → ΠYK .\nThe first step then is to show that: If one carries out this recipe for\nαKY (which may or may not arise geometrically) so as to obtain a class\nηKY ∈ H2(ΠYK ,Zp(1)), then ηKY is the arithmetic first Chern class of a\nline bundle (necessarily of degree prime to p) on YK . This is the topic\nof Sections 1 through 5.\n(2) Next, we go back to L-valued points, and their associated αLX , α\nL\nY .\nThen, just as in the case of K-valued points, one can form ηLY , and\none would like to know that ηLY is the arithmetic first Chern class of\na line bundle (necessarily of degree prime to p) on YL\ndef\n= YK ⊗K L.\nThis is technically much more intricate than the K-valued case, and\nis done (roughly speaking) by thinking about the “difference” between\nthe Chern class of a K-valued point and of an L-valued point, and\nshowing that this difference has special properties that are preserved by\nθ. This allows one to derive the assertion in the L-valued case from\nthe corresponding assertion in the K-valued case, which was already\nhandled in (1) above. This reduction is the topic of Sections 6 and 7.\n(3) Recall that we would like to show that αLY is geometric. We know from\n(2) above that at least there exists a line bundle on YL of degree prime\nto p. Then an elementary algebraic geometry argument shows that this\nimplies the existence of a rational point of YL defined over a tamely\nramified (this will be crucial in Step (4)!) extension L′ of L. Although\nthis portion of the proof is technically rather trivial, its discovery was a\nkey step in the creation of the proof of Theorem A. This portion of the\nproof is discussed in Section 8.\n(4) Finally, by applying (3) to all the curves in a certain tower of coverings of\nthe original YL, we obtain a collection of rational points of the curves of\nthis tower that are defined over tamely ramified extensions of L. Then\nby using Faltings’ p-adic Hodge theory ([Falt1]), we show that these\nrational points necessarily converge p-adically to a single L-valued point\nof YL = YK ⊗K L whose associated ΓL → ΠYK is necessarily equal to\n11\nαLY . This completes the proof of the geometricity of α\nL\nY . This portion\nof the proof is given in Sections 9 and 10 (and applied in Section 13).\nAs discussed above, once one knows the geometricity of αLY , one can conclude the preser-\nvation of relations – this is discussed in Sections 11 through 13. Finally, the proof of (∗)iner\n(i.e., the difference between (∗)tech and (∗)prop) is given in Section 14. Unfortunately, in\norder to show (∗)iner, it is necessary to go through all the steps discussed so far for a given\nθ : ΠUK → ΠYK , and then to conclude (∗)iner from the preservation of relations. This\nmakes the proof much more technically intricate than it would be if one could prove (∗)iner\nfrom some sort of a priori argument, and then prove preservation of relations only for\nθ : ΠXK → ΠYK as in (∗)prop.\nWe would like to close this outline of the proof of Theorem A by discussing Step (1)\nabove (i.e., the content of Sections 1 through 5) in greater detail. The reason for this is\nthat Step (1) is what allowed us to generalize the “isomorphisms only” result of [Mzk2] to\nthe homomorphism result of the present paper. First of all, let us recall the notion of the\nMalčev completion (cf., e.g., [Del], §9) of ΔX . In fact, we shall only need the “smallest\nnontrivial part of Malčev completion”: concretely, a unipotent algebraic group over Qp,\nwhich we denote by MX , whose representations are the same as continuous representations\nof ΔX\/[ΔX , [ΔX ,ΔX ]] on some Qp-vector space V which is equipped with a ΔX-invariant\nfiltration on whose subquotients ΔX acts trivially. In fact, since unipotent algebraic groups\nare equivalent to their Lie algebras, we shall consider instead the Lie algebra MX of MX .\nNext, observe that any section αKX : ΓK → ΠXK defines (by conjugation) a true action\n(i.e., not just an action up to inner automorphisms) of ΓK on ΔX , hence on MX . Relative\nto this action, their exists a unique “weight zero quotient” MX⊗QpCp → ZX . Here, we call\nthe quotient “weight zero” because it is the maximal quotient MX ⊗Qp Cp → Q for which\nthe action of ΓK on Q is such that Q has a filtration by ΓK-submodules whose subquotients\nare ΓK-equivariantly isomorphic to Cp. Moreover, this quotient MX ⊗Qp Cp → ZX is\nindependent of the choice of section αKX .\nStep (1) is based on the following pair of observations:\n(i) On the one hand, if αKX arises geometrically, then ZX is “Hodge-\nTate.” (By abuse of terminology, we use the term “Hodge-Tate” here\nto mean that ZX has a Cp-basis of ΓK -invariant elements.) This is the\ncontent of Proposition 3.5.\n(ii) On the other hand, if ZX is “Hodge-Tate,” then the class η\nK\nX is the\nfirst Chern class of a line bundle. This is essentially the content of the\ncalculation performed in Proposition 4.4 (see also Lemma 7.3).\nMoreover, if one starts with a surjective θ : ΠXK → ΠYK , then ZX maps naturally to,\nand in fact, surjects onto ZY . Thus, if one starts with an α\nK\nX that arises geometrically,\nthen one knows from Observation (i) above that ZX is Hodge-Tate, but any quotient of a\nHodge-Tate representation of ΓK – i.e., such as ZY – is always Hodge-Tate, so Observation\n(ii) above thus allows one to complete Step (1).\n12\nAt this point, the reader may wonder how the author stumbled upon the two key\nobservations of the preceding paragraph. In fact, the author first realized what was going\non by considering the ordinary case (see Section 1 for a detailed discussion). In this\ncase, the “weight zero quotient” exists at the level of groups, without passing to Malčev\ncompletions or tensoring with Cp: namely, (in the notation of Section 1) it is the quotient\nΔX → ΔetX (which thus gives rise to a quotient ΠXK → ΠetXK ). Moreover, as is shown in\nSection 1, it is elementary to show that ΔX → ΔetX can be recovered group-theoretically,\nand that every section ΓK → ΠXK that arises geometrically (from a point of XK) induces\na fixed, group-theoretically constructible section θX : ΓK → ΠetXK of ΠetXK → ΓK . These\nobservations are enough to complete Step (1) (in the ordinary case). Thus, one may\nregard the discussion in the preceding paragraphs (which is valid in the nonordinary case)\nas simply the result of generalizing the observations discussed in this paragraph in the\nordinary case to the possibly nonordinary case by means of the technical machinery of the\nMalčev completion and p-adic Hodge theory.\nFinally, let us make the following observation: As one can see from the key argument\ndiscussed above, in fact, really, one does not need all of ΔX . That is to say, in the\nabove argument (concerning the issue of when ZX is Hodge-Tate), one actually only uses\nthe quotient ΔX\/[ΔX , [ΔX ,ΔX ]]. It is this observation that is behind the truncated\ngeneralizations (Theorems A′ and A′′) of Theorem A.\nAcknowledgements:\nI would like to thank A. Tamagawa: (i) for inspiring me by means of his paper\n[Tama]; (ii) for numerous discussions during which I presented the proof of the main\nresults of the present paper; (iii) for advice on surmounting two technical problems (see\nthe Remark at the end of Section 12; Lemmas 15.6 and 15.8); (iv) for explaining to me\nthe “general nonsense argument” preceding Theorem 14.1; (v) for explaining to me basic\nfacts concerning the Malčev completion, which were of immense importance in developing\nthe theory discussed in this paper. Also, I would like to thank T. Saito and G. Faltings\nfor pointing out various minor errors in earlier versions of this manuscript.\nSection 0: Preliminaries and Notations\nLet p be a prime number. Throughout this paper the symbol “∧” over or to the upper-\nright of an object will denote the p-adic completion of that object. Let K be a Qp-algebra.\nLet Ω be an algebraically closed field. Then given a base-point b ∈ HomRing(K,Ω), we can\nform the algebraic fundamental group\nπ1(Spec(K), b)\nTypically, the choice of base-point b will not be important for us, so we shall write ΓK for\nπ1(Spec(K), b). Suppose that Spec(K) is a “universal covering space” for Spec(K) such\n13\nthat b arises from a ring homomorphism K → Ω. Then we may think of ΓK as Gal(K\/K).\nWe shall denote the étale cohomology of Spec(K) by Hi(K,−). If M is a profinite abelian\ngroup with a continuous ΓK -action, then M naturally defines an inverse system {Fα} of\nlocally constant sheaves on the étale site of Spec(K), and we shall write\nHi(K,M)\nfor the inverse limit (over α) of the Hi(K,Fα). Note that for each r ∈ Z, Zp(r) (where\nthe “(r)” is a Tate twist) has a natural structure of ΓK -module.\nDefinition 0.1. We shall call K a p-adic field if it is the quotient field of a p-adically\ncomplete, mixed characteristic discrete valuation ring OK . We shall denote the residue\nfield (respectively, maximal ideal) of OK by k (respectively, mK). We shall call K a p-adic\nlocal field if k is a finite field.\nIf K is a p-adic field, and K is an algebraic closure of K, then K ↪→ Ω def= K determines\na base-point “b,” and we have ΓK = Gal(K\/K). In this case, H\ni(K,−) is equal to the\ncontinuous group cohomology of the profinite group ΓK . If K is a p-adic local field, then\nK is, in fact, a finite extension of Qp.\nNow (without any assumptions on the Qp-algebra K), let us assume that we are given\na hyperbolic curve XK → Spec(K) over K of type (g, r). (By “curve,” we shall always\nmean a smooth, one-dimensional, geometrically connected scheme over the base. By “type\n(g, r),” we mean that XK ⊗K K is obtained by removing r mutually nonintersecting K-\nvalued points from a proper curve over K of genus g. By “hyperbolic,” we mean that\n2g− 2 + r ≥ 1.) When XK is proper, we shall denote its Jacobian (an abelian scheme over\nK) by JX (or JXK when several bases are in use and it is necessary to specify the base in\nquestion).\nLet us assume that XK is equipped with a base-point x ∈ XK(Ω) (which is compatible\nwith the base-point b of Spec(K)). Then we can form Π\nprf\nX\ndef\n= π1(XK , x) and Δ\nprf\nX\ndef\n=\nπ1(XK , x). (The use of “Δ” to denote the geometric fundamental group may be new to\nsome readers. Here, we use “Δ” partly because “Π” is already used for the arithmetic\nfundamental group and partly to conform to the notations of [Falt1], a reference on which\nthe present paper depends heavily.) Occasionally, to avoid confusion, we shall also use the\nnotation Δ\nprf\nXK\nfor Δ\nprf\nX . Let ΔX be the maximal pro-p quotient of Δ\nprf\nX . Since the kernel\nof Δ\nprf\nX → ΔX is normal in Π\nprf\nXK\n, we may form the quotient of Π\nprf\nXK\nby this kernel, and\ncall the resulting quotient group ΠXK . Thus, we have an exact sequence\n1 → ΔX → ΠXK → ΓK → 1\nMoreover, this exact sequence induces a representation\n14\nρX : ΓK → Out(ΔX)\ndef\n= Aut(ΔX)\/Inn(ΔX)\ninto the outer automorphism group of ΔX . (Here, “Aut(ΔX)” (respectively, “Inn(ΔX)”)\ndenotes the group of continuous automorphisms (respectively, inner automorphisms) of\nΔX .) Conversely, it is well-known (see, e.g., [Tama], §7, A.) that the above exact sequence\ncan be recovered from ρX .\nNext, we would like to introduce some terminology particular to pro-p groups. Let Δ\nbe a pro-p group (i.e., a topological group obtained by taking an inverse limit of finite groups\nof p-power order). Let Δ′ ⊆ Δ be the unique normal subgroup of Δ with the following\nproperty: Δ → Δ\/Δ′ is the maximal (topologically) Hausdorff abelian quotient of Δ which\nis annihilated by p. For i ≥ 0, let Δ<0> def= Δ; Δ<i+1> def= (Δ<i>)′. Thus, we obtain a\ndescending series of closed normal (even characteristic!) subgroups . . . ⊆ Δ<i> ⊆ . . . ⊆ Δ.\nNote that since Δ, being a pro-p group, is “pro-solvable,” it follows that the intersection\nof all the Δ<i> is {1}. Moreover, if Δ is topologically finitely generated, it follows that\nthe Δ\/Δ<i> are all finite groups.\nDefinition 0.2. We shall refer to any one of the Δ<i> as a p-derivate of Δ.\nThus, in particular, if Δ = ΔX , then it follows (by the structure of the fundamental group\nof an algebraic curve in characteristic zero) that Δ is topologically finitely generated, so\nthe Δ\/Δ<i> are all finite groups.\nNext, let us consider the Kummer sequence on XK , i.e., the exact sequence of étale\nsheaves on XK given by 0 → Z\/pnZ(1) → Gm → Gm → 0 (for n ≥ 1). (Here, the\n“(1)” is a “Tate twist,” and the morphism from Gm to Gm is given by raising to the\n(pn)th power.) The connecting morphism induced on étale cohomology by the Kummer\nsequence then gives us a morphism H1(XK ,Gm) → H2(XK , (Z\/pnZ)(1)). Now suppose\nthat L is a line bundle on XK . Applying the connecting morphism just considered to L\n(which defines an element of H1(XK ,Gm)), we obtain a compatible system of classes in\nH2(XK , (Z\/p\nnZ)(1)) (for each n ≥ 1), hence a class c1(L) ∈ H2(XK ,Zp(1)).\nDefinition 0.3. We shall refer to c1(L) as the arithmetic first Chern class of L.\nFinally, we have the following elementary technical result, which states that the étale\ncohomology of a hyperbolic curve may be computed as the group cohomology of its fun-\ndamental group:\nLemma 0.4. Assume that K is a field. For all integers i, r, the natural morphisms\nHi(ΔX ,Zp(r)) → Hi(XK ,Zp(r)); H\ni(ΔXK×KXK ,Zp(r)) → H\ni(X\nK\n×\nK\nX\nK\n,Zp(r))\n15\nand\nHi(ΠXK ,Zp(r)) → H\ni(XK ,Zp(r)); H\ni(ΠXK×KXK ,Zp(r)) → H\ni(XK ×K XK ,Zp(r))\nare isomorphisms.\nProof. By the Leray-Serre spectral sequence, it suffices to prove that the morphisms of\nthe first line are isomorphisms. Let Z be X\nK\nor X\nK\n×\nK\nX\nK\n. Then it follows by general\nnonsense that it suffices to check that for any finite étale Galois covering Y → Z of p-power\norder, and any cohomology class η ∈ Hi(Y,Fp) (where i > 0), there exists a finite étale\ncovering Y ′ → Y of p-power order such that η|Y ′ = 0. By the Künneth formula, it suffices\nto do the case Z = X\nK\n. Then what we must check is trivial for i > 2 (since then η = 0 to\nbegin with), and clear for i = 1 (by the relationship between étale coverings and H1). If\ni = 2, then it suffices to take Y ′ → Y such that Y ′ → Y has degree p over every connected\ncomponent of Y . This completes the proof. ©\nSection 1: The Ordinary Case\nLet p be a prime number. Let K be a p-adic field with algebraically closed residue\nfield. Then, as discussed in Section 0, the absolute Galois group of K will be denoted\nΓK . Let XK → Spec(K) be a hyperbolic curve over K of type (g, r). In this Section,\nlet us also assume that XK admits a stable extension X → Spec(OK) over OK . By this,\nwe mean that there exists a (necessarily unique) r-pointed stable (in particular, proper)\ncurve X → Spec(OK) of genus g such that X is the complement in X of the images of the\nr marking sections of X. Let us write Xk (respectively, Xk) for X ⊗OK k (respectively,\nX ⊗OK k).\nRemark. Recall that an r-pointed stable curve of genus g (where 2g−2+r ≥ 1) is a proper,\nflat morphism f : C → S, together with r mutually disjoint sections σ1, . . . , σr : S → C ,\nsuch that\n(1) The geometric fibers of f are connected, reduced, of arithmetic genus\ng, and have at most nodes as singularities.\n(2) The sheaf ωC\/S(σ1 + . . .+σr) (i.e., the sheaf of sections of the dualizing\nbundle of C over S with poles of order ≤ 1 at the divisors defined by\nthe images of the sections σ1, . . . , σr) is relatively ample over S.\nWe refer to [DM], [Knud] for more details.\n16\nNow we make the following\nDefinition 1.1. We shall call XK ordinary if the Jacobian of every connected component\nof the normalization of the curve Xk is an ordinary abelian variety.\nIn this Section, we would like to assume that XK is ordinary. Under this assumption, it is\nwell-known that π1(Xk)\n(p) (where the “(p)” denotes the maximal pro-p quotient) is a free\npro-p group of rank g. (Indeed, this follows from the fact that H2et(Xk,Fp) = 0 (which may\nbe shown by using the long exact sequence in étale cohomology obtained by considering\nFp as the kernel of “1− Frobenius” acting on OXk), plus Proposition 2.3 of Chapter III,\n§3, of [Sha].) Moreover, since étale coverings of Xk lift uniquely to characteristic zero, it\nfollows that we have a continuous surjection:\n\fX : ΠXK → π1(Xk)\n(p)\nIf we restrict \fX to ΔX , we obtain a surjection ΔX → π1(Xk)(p). In the following, we\nshall regard π1(Xk)\n(p) as a quotient of ΔX via this surjection. Let us denote this quotient\nby ΔetX . Note that the kernel of the surjection ΔX → ΔetX is normal as a subgroup of ΠXK .\nThus, by taking the quotient of ΠXK by this kernel, we obtain a quotient ΠXK → ΠetXK .\nIn other words, we have an exact sequence\n1 → ΔetX → ΠetXK → ΓK → 1\ntogether with a surjection\nζX : Π\net\nXK\n→ ΔetX\nwhich is the identity on ΔetX .\nNow observe that the kernel of ζX projects isomorphically to ΓK . Thus, we obtain a\nsection\nθX : ΓK → ΠetXK\nNext, let us observe that every element of Im(θX ) commutes with every element of Δ\net\nX .\nIndeed, this follows from the fact that such commutation relations hold after projection\nby ΠetXK → ΓK , plus the fact that Im(θX ) = Ker(ζX) is normal in ΠetXK (and maps\nisomorphically to ΓK via the projection Π\net\nXK\n→ ΓK). On the other hand, since, as\nis well-known (see, e.g., [Tama], §1, Propositions 1.1, 1.11), pro-p free groups (of rank\n≥ 2) have trivial centers, it thus follows that we obtain the following “group-theoretic”\ncharacterization of θX (when g ≥ 2):\n17\nLemma 1.2. Suppose that g ≥ 2. Then the section θX : ΓK → ΠetXK of ΠetXK → ΓK is\nthe unique section whose image commutes with every element of ΔetX ⊆ ΠetXK .\nOn the other hand, the quotient ΔX → ΔetX (and hence also the quotient ΠXK →\nΠetXK ) can also be reconstructed group-theoretically by means of the following condition\non an open normal subgroup H ⊆ ΔX :\n(∗)et Let N ⊆ ΔX be any subgroup such that H ⊆ N and N\/H is cyclic.\nThen there exists a surjection Nab\ndef\n= N\/[N, N ] → Q, where Q is a free\nZp-module of rank one, with the following properties: (i) there exists\nan open subgroup Γ′ ⊆ ΓK that stabilizes N and Nab → Q and acts\ntrivially on Q; (ii) the surjection N → N\/H factors through Q.\nThen we have the following “group-theoretic” characterization of the quotient ΔX → ΔetX :\nLemma 1.3. The kernel of ΔX → ΔetX is the intersection of all open normal subgroups\nH ⊆ ΔX such that H satisfies the condition (∗)et.\nProof. The proof is entirely the same as that of Sections 3 and 8 of [Mzk1]. The basic\nidea is that if the covering corresponding to ΔX\/H is not étale (over OK), then it has\nnontrivial inertia subgroups. Such inertia subgroups have nontrivial cyclic subgroups. If\nwe then apply (∗)et to the case where N\/H is one of these nontrivial cyclic subgroups,\nthen we have a contradiction, since the quotient N → N\/H factors through the quotient\nN → Q; moreover, the quotient N → Q necessarily corresponds to a covering which is\nétale over OK because of the assumption concerning the action of Γ′ ⊆ ΓK on Q. ©\nLet us review what we have done so far. So far, we have:\n(1) constructed quotients ΠXK → ΠetXK and ΔX → ΔetX , as well as a section\nΓK → ΠetXK via various geometric considerations concerning XK ;\n(2) shown (when g ≥ 2) that the above quotients and section may be\nreconstructed entirely “group-theoretically.”\nHere, we pause to make the following\nRemark Concerning the Term “Group-Theoretic.” In [Mzk1] and [Mzk2], we imparted\nmathematical rigor to the term “group-theoretic” (cf. the remark on this issue in Section\n14 of [Mzk2]) by specifying that it meant “preserved by isomorphism.” In the present\npaper, however, we would like to consider homomorphisms which are not necessarily iso-\nmorphisms. Thus, in the present paper, when we wish to state that a certain property or\nobject is preserved by such homomorphisms, we shall state this explicitly without using\n18\nthe term “group-theoretic.” In the above discussion, however, we recommend the reader to\nsimply accept this term at the level of “common sense,” since we will not use the “group-\ntheoreticity” stated in any of the results of this Section in the proof of any of the main\ntheorems of this paper.\nFinally, before continuing, we make the following important observation: Let αx :\nΓK → ΠXK denote the section of ΠXK → ΓK determined (up to composition with an\ninner automorphism induced by an element of ΔX) by a point x ∈ XK(K). Denote by\nαetx : ΓK → ΠetXK the composite of αx with ΠXK → ΠetXK .\nLemma 1.4. We have αetx = θX .\nProof. It suffices to show that the composite of αetx with the surjection ζX : Π\net\nXK\n→\nΔetX is trivial. Interpreted geometrically, this simply means that “the pull-back of any\nétale covering Y → X (i.e., étale over OK) to Spec(K) via x is the trivial étale covering\nof Spec(K).” But this assertion follows immediately from the fact that OK is strictly\nhenselian. ©\nSection 2: Review of Galois Cohomology\nLet K be a p-adic field whose residue field is perfect. Let S → Spec(OK) be a\ngeometrically connected smooth morphism, and let D ⊆ S be a relative (over OK) divisor\nwith normal crossings. Let us write S log for the log scheme obtained by equipping S with\nthe log structure defined by D (as in [Kato]). Let us also assume that S is small (in the sense\nof [Falt1], [Falt2]): Recall that this simply means that S is affine, say, equal to Spec(R),\nand, moreover, étale over some OK [X1, . . . ,Xd] in such a way that D is schematically the\ninverse image of the zero locus of the function X1 · . . . ·Xd. The reason we wish to deal with\nsmall (S,D) is that in [Falt1], certain Galois cohomology groups associated to such (S,D)\nare computed explicitly by means of the theory of almost étale extensions. The purpose\nof this Section is to review certain consequences of the theory of [Falt1] and [Falt2] that\nare of relevance to us in this paper.\nLet RK\ndef\n= R ⊗OK K. Let RK → RK be the maximal extension of RK which is étale\noutside of DK . Let R be the normalization of R in RK . Let R̂ (respectively, R̂) be the\np-adic completion of R (respectively, R); let R̂K\ndef\n= R̂⊗OK K, R̂K\ndef\n= R̂⊗OK K. Let R̂OK\nbe the p-adic completion of R ⊗OK OK ; R̂K\ndef\n= (R̂O\nK\n) ⊗OK K. Let ΓR\ndef\n= Gal(RK\/RK).\nThus, we have a natural surjection ΓR → ΓK whose kernel we denote by ΔprfR . This gives\nus a natural exact sequence:\n1 → ΔprfR → ΓR → ΓK → 1\n19\nNow we would like to compute some Galois cohomology groups:\nLemma 2.1. For all j ∈ Z, we have:\n(1) H0(Δ\nprf\nR , R̂K(j)) = R̂K(j);\n(2) H1(Δ\nprf\nR , R̂K(j)) = ΩSlog\/OK ⊗R R̂K(j − 1).\n(Here the “(j)” is a Tate twist.)\nProof. This follows from Theorem 4.4 of Section I of [Falt1]. Indeed, (1) follows from\n[Falt1], Section I, Theorem 4.4, (i) (applied to the case i = 0, and tensored with K(j)),\nwhile (2) follows from [Falt1], Section I, Theorem 4.4, (iv) (tensored with K(j)). ©\nLemma 2.2. We have: (i) H0(ΓK , R̂K) = R̂K; (ii) H\nn(ΓK , R̂K(j)) = 0 (for j \f= 0;\nn = 0, 1); (iii) H1(ΓK , R̂K) = R̂K.\nProof. This result is due to [Tate] in the case R = OK . (Indeed, in this case, (i) and (iii)\nfollow from [Tate], §3.3, Theorem 1; while (ii) – at least in the case j = 1 (the proof for\narbitrary j \f= 0 is entirely similar) – follows from [Tate], §3.3, Theorem 2.) The slightly\nmore general result stated here (i.e., for R not necessarily equal to OK) follows by the\nsame argument as that employed by Tate in the case R = OK (the point being that in\ngeneral, R is OK-flat). Alternatively, the cohomology groups in the Lemma can also be\ncomputed using almost étale extensions as in [Falt1]. ©\nLemma 2.3. For n = 0, 1, we have: (i) Hn(ΓR, R̂K) = R̂K; (ii) H\nn(ΓR, R̂K(−1)) = 0.\nProof. This follows from the Leray-Serre spectral sequence (applied to the exact sequence\nof groups that appears directly before Lemma 2.1), plus the preceding two Lemmas. ©\nNow let us assume that we are given an r-pointed stable log-curve f log : X log → S log\nof genus g, where 2g − 2 + r ≥ 1. (By this we mean that X log is obtained by pulling\nback the universal log-curve Clog → Mlogg,r via some log morphism S log → M\nlog\ng,r . The log\nstructure of the universal log curve is defined by the divisor with normal crossings which\nis the union of the marked points and the singular fibers. See, e.g., [Mzk3], Section 3, for\nmore details; cf. also the Remark preceding Definition 1.1 of this paper for a review of\nthe notion of a “pointed stable curve.”) Let U\ndef\n= S − D. Then taking the relative first\ncohomology module of f |UK in the étale topology with coefficients in Zp gives rise to a\nlocal system over UK , hence a ΓR-module H\n∨. As a Zp-module, H∨ is free of rank = 2g\n20\n(if r = 0) and = 2g + r − 1 (if r > 0). Let H be the ΓR-module given by HomZp(H∨,Zp).\nNow let\nH0 def= (R1f∗OX)∨\n(a vector bundle over S, or alternatively, a projective R-module, of rank g) and\nH1 def= (f∗ωX log\/Slog )∨\n(a vector bundle over S, or alternatively, a projective R-module, whose rank is = g if r = 0\nand = g + r − 1 if r > 0).\nProposition 2.4. There is a natural exact sequence\n0 → H1 ⊗R R̂K(1) → H ⊗Zp R̂K → H0 ⊗R R̂K → 0\nwhich is compatible with the natural action of ΓR on all three terms.\nProof. This is an immediate consequence of the “Comparison Theorem” (Theorem 6.2 of\n[Falt2]). The validity of the proof of this Theorem given in [Falt2] has been disputed by\nvarious mathematicians. However, since Proposition 2.4 is a relatively weak consequence\nof the “Comparison Theorem,” it already follows from the portion of [Falt2] that is not in\ndispute. (That is to say, we need only that M is “Hodge-Tate,” not that it is “crystalline.”)\nAlternatively, although the sort of parametrized (i.e., over a base S) Hodge-Tate decom-\nposition that we need here is not stated in [Falt1], it follows immediately from the theory\nof [Falt1] by exactly the same proof as that of the main result of [Falt1]. Yet another proof\nof this sort of result is given in [Hyodo] (the final Theorem – i.e., the “relative version”\n– in [Hyodo], §0.3), although here we need the (relatively straightforward) logarithmic\ngeneralization of [Hyodo]’s result. ©\nSection 3: The Weight Zero Quotient\nIn this Section, we maintain the notation of the preceding Section. The purpose of\nthis Section is to give a (rather weak) nonabelian analogue of Proposition 2.4. In fact,\na much stronger “nonabelian comparison theorem” for the fundamental group of a curve\ncan be proven, but since I do not know of any place where such a result has been written\nup, and, moreover, in this paper, only a relatively weak nonabelian comparison theorem is\nneeded, I decided instead to give a rather short ad hoc treatment of this issue which will,\nnonetheless, be sufficient for the purposes of this paper.\n21\nLet us denote the fundamental group of X\nlog\nK by Π\nprf\nXK\n. Thus, we have a natural\nsurjection Π\nprf\nXK\n→ ΓR. As usual, we denote the kernel of this surjection by ΔprfX ⊆ Π\nprf\nXK\n,\nand the maximal pro-p quotient of Δ\nprf\nX by ΔX . Moreover, by forming the quotient of\nΠ\nprf\nXK\nby the kernel of Δ\nprf\nX → ΔX , we obtain ΠXK . Thus, just as in Section 1, we have an\nexact sequence\n1 → ΔX → ΠXK → ΓR → 1\nMoreover, any section σ : S → X of X → S whose image avoids the marked points and\nnodes defines a section πσ : ΓR → ΠXK of the above exact sequence. Such a section πσ\nthen defines an action of ΓR on ΔX by conjugation. Note that until one specifies the\nsection ασ : ΓR → ΠXK , one only has an outer action of ΓR on ΔX (i.e., an action defined\nonly up to inner automorphisms); that is to say, in general, there is no natural action (in\nthe usual, non-outer sense) of ΓR on ΔX until one specifies a section of ΠXK → ΓR.\nThe next step is to introduce the Malčev completion of ΔX . We refer to [Del], [NT] for\nmore details. In fact, for our purposes, it will be sufficient to consider a truncated form of\nthe Malčev completion of ΔX . This truncated form may be defined as follows: Let C be the\ncategory of finite dimensional Qp-vector spaces V equipped with a continuous ΔX -action\nthat factors through ΔX\/[ΔX , [ΔX ,ΔX ]] and which admits a ΔX-invariant filtration on\nwhose subquotients ΔX acts trivially. (The morphisms of this category are the Qp[ΔX ]-\nlinear morphisms V → V ′.) Then C is a Tannakian category over Qp, hence gives rise to\nan algebraic group MX over Qp. Moreover, this algebraic group MX is unipotent, hence\ncorresponds to a nilpotent Lie algebra MX . For any Qp-algebra A, we shall write (MX)A\n(respectively, (MX)A) for MX ⊗Qp A (respectively, MX ⊗Qp A).\nLet us write MX [1] for the commutator [MX ,MX ] of this Lie algebra, and MX [0] for\nthe quotient MX\/MX [1]. Then MX [0] may be identified with HQp\ndef\n= H⊗Zp Qp (where H\nis as in the discussion following Lemma 2.3 in Section 2). Thus, we get an exact sequence\n0 → MX [1] → MX → MX [0] = HQp → 0\nMoreover, the commutator [−,−] defines a surjection\ncX : ∧2 HQp → MX [1]\nNow observe that although there is no natural action of ΓR on MX (unless one chooses a\nsection of ΠXK → ΓR), there is nonetheless a natural action of ΓR on MX [0] and MX [1]\nwith respect to which cX is equivariant. Now we have the following classical result\nLemma 3.1. The kernel of cX is zero if r > 0. If r = 0, then the kernel of cX is one-\ndimensional, and equal to the image of the dual to the intersection form ∧2 H∨Qp → Qp(−1)\n(defined by the cup product pairing on the cohomology of ΔX).\n22\nProof. If r > 0, then ΔX is a free pro-p group. Thus, in this case the result follows by\nconsideration of the fact that a representation on a vector space V of the free group on n\ngenerators is the same as giving n endomorphisms of V . If r = 0, then ΔX is the quotient\nof a free pro-p group by a single relation; it is this relation which generates the kernel of\ncX . Moreover, it is easy to see from the well-known explicit form of this relation that its\nimage in ∧2 HQp is precisely as specified in the statement of the Lemma (cf. the discussion\nin §2 of [NT]). ©\nNext, we would like to construct a certain special quotient ZX of (MX)\nR̂K\n. By the\nwell-known categorical equivalence between unipotent algebraic groups and nilpotent Lie\nalgebras (cf. [Del], §9), this quotient will define a quotient ZX of (MX)\nR̂K\n. The construc-\ntion of ZX from (MX)\nR̂K\nconsists of two steps. The first step is as follows: Consider the\nsurjection\n∧2 H ⊗Zp R̂K → ∧2 H0 ⊗R R̂K\ndefined by projecting by means of the surjection in the short exact sequence of Proposition\n2.4. It follows from Lemmas 2.3 (ii) (in the case n = 0) and 3.1 that this surjection factors\nthrough (MX [1])\nR̂K\n. Thus, we obtain a surjection\n(MX [1])\nR̂K\n→ ∧2 H0 ⊗R R̂K\nBy pushing forward the exact sequence 0 → MX [1] → MX → MX [0] → 0 (tensored over\nQp with R̂K) via this surjection, we thus obtain a Lie algebra UX (over R̂K). Thus, we\nhave a surjection of Lie algebras (MX)\nR̂K\n→ UX , together with an exact sequence\n0 → UX [1] → UX → UX [0] → 0\nwhere UX [1]\ndef\n= [UX ,UX ] = ∧2 H0 ⊗R R̂K , and UX [0] = (MX [0])\nR̂K\n.\nNow we come to the second step in the construction of ZX . First let us denote by\nUX [0] → UX [0, 0] def= H0 ⊗R R̂K the surjection defined by the surjection in the short exact\nsequence of Proposition 2.4. Let UX [0, 1] ⊆ UX [0] be the kernel of this surjection. Let\nBX ⊆ UX denote the inverse image of UX [0, 1] ⊆ UX [0] under the surjection UX → UX [0].\nThen it follows from the definition of UX that BX is an abelian Lie algebra which, in fact,\nlies in the center of the Lie algebra of UX . (Indeed, this will follow as soon as we show the\nvanishing of [BX ,UX ], which is equal to the image of UX [0, 1]∧UX [0] ⊆ ∧2 HQp under the\ncomposite of cX : ∧2 HQp → MX [1] with the projection MX [1] → UX [1] = ∧2 H0 ⊗R R̂K ;\nbut this image is zero, by the definition of UX [0, 1].) This observation implies, in particular,\nthat although, a priori, we have only an outer action of ΓR on UX , hence on BX , in fact,\nwe get a natural (non-outer) action of ΓR on BX . (That is, the point is that the various\n23\nactions of ΓR on BX arising from different sections of ΠXK → ΓR differ only by an\nautomorphism on BX induced by conjugation by some element of ΔX ; but since BX lies\nin the center of UX , it follows that such an automorphism of BX is always equal to the\nidentity.)\nNext, let us note that we have exact sequences\n0 → BX → UX → UX [0, 0] → 0\nand\n0 → UX [1] → BX → UX [0, 1] → 0\nwhere the latter exact sequence is an exact sequence of ΓR-modules (a statement which has\nmeaning as a result of the observation at the end of the preceding paragraph). Now observe\nthat Lemma 2.3 (ii) (for n = 0, 1 – note that here we use that UX [1] = ∧2 H0 ⊗R R̂K\nis “of weight zero,” while UX [0, 1] = H1 ⊗R R̂K(1) is “of weight one”) implies that the\nlatter exact sequence admits a unique ΓR-equivariant splitting: BX → UX [1]. Moreover,\nsince BX is (as observed above) contained in the center of the Lie algebra UX , it follows\nthat the kernel of this splitting BX → UX [1] forms a Lie ideal in UX . Thus, if we then\npush forward the former exact sequence via this surjection BX → UX [1], we obtain a Lie\nalgebra ZX . As usual, this Lie algebra fits into an exact sequence\n0 → ZX [1] → ZX → ZX [0] → 0\nMoreover, one has natural identifications: ZX [1] = ∧2 ZX [0] (via the commutator map);\nand ZX [0] = H0 ⊗R R̂K (induced by the surjection of the short exact sequence of Propo-\nsition 2.4). In particular, ZX [0] is of rank g over R̂K . Finally, as noted above, ZX defines\na unipotent algebraic group ZX .\nDefinition 3.2. We shall refer to ZX as the weight zero quotient of ΔX (even though\nit is not literally a quotient). (Here, the “Z” of ZX stands for the “zero” of “weight zero\nquotient.”)\nNow let us fix a (continuous) section\nα : ΓR → ΠXK\nThen α induces an action of ΓR on ΔX . Since ZX was formed naturally – and, for that\nmatter, group-theoretically – from ΔX , we thus obtain an action of ΓR on ZX which, in\n24\ngeneral, will depend on the choice of α. The next issue we would like to consider is to what\nextent the sequence of ΓR-modules\n0 → ZX [1] → ZX → ZX [0] → 0\nadmits a ΓR-equivariant section. At any rate, this exact sequence defines an extension\nclass\nηα ∈ H1(ΓR, (ZX [0])∨ ⊗\nR̂K\nZX [1])\nBy Lemma 2.3 (i) (for n = 1 – here we use that both ZX [0] = H0 ⊗R R̂K and ZX [1] =\n∧2 ZX [0] are “of weight zero”), it follows that this Galois cohomology group may be\nidentified naturally with\n(H∨0 ⊗R ∧2 H0) ⊗R R̂K\nThat is to say, one may think of ηα as a section of a certain vector bundle over R̂K .\nProposition 3.3. Suppose that α arises as the ασ associated to some section σ : S → X\n(whose image avoids the marked points and nodes – cf. the discussion at the beginning of\nthis Section). Then ηα = 0.\nProof. Since everything is functorial, one reduces immediately to the universal case, as\nfollows: In the present context, the essential data that we begin with is a r-pointed stable\ncurve of genus g (i.e., f log : X log → S log), plus a section (i.e., σ). The moduli stack for\nthis data is (a certain dense open substack of) the tautological curve C → Mg,r over the\nmoduli stack Mg,r of r-pointed stable curves of genus g over Zp. Thus, by “restriction,”\nit suffices to prove the Proposition in the case where S is étale over the algebraic stack\nC. Also, since everything involved commutes with base-extension, it is easy to see that we\nmay assume that k is algebraically closed. Then over some dense open T ⊆ (S−D)⊗OK k,\nthe Jacobian of X ⊗OK k will be ordinary. Thus, every point of β ∈ S(OK) that maps\nSpec(k) ⊆ Spec(OK) into T defines an ordinary (in the sense of Definition 1.1) hyperbolic\ncurve Yβ → Spec(K) over K (by restricting (the complement of the marking sections in)\nX → S to β).\nNext, observe that that if we restrict the section σ to the point β, we get a section\nσβ : Spec(K) → Yβ which induces an action of ΓK on ΔetYβ (i.e., the quotient ΔYβ → ΔetYβ\nconsidered in Section 1). By Lemmas 1.2 and 1.4, this action is, in fact, the trivial action.\nNow I claim that in the case of such an ordinary curve:\nThe quotient ΔYβ → ΔetYβ (considered in Section 1) induces a natural\n(in particular, ΓK -equivariant) isomorphism of the weight zero quotient\n25\nZYβ of ΔYβ (= ΔX) with the Malčev completion of Δ\net\nYβ\n(truncated at\nthe second step and tensored over Zp with K̂).\nIndeed, it follows immediately from the construction of ZYβ that the surjection ΔYβ → ΔetYβ\ninduces a morphism from ZYβ to the “weight zero quotient” (i.e., constructed in a fashion\nanalogous to the construction of ZYβ ) of Δ\net\nYβ\n. On the other hand, since the action of\nΓK on Δ\net\nYβ\nis trivial, it follows that the “weight zero quotient” of ΔetYβ is just the Malčev\ncompletion of ΔetYβ (truncated at the second step and tensored over Zp with K̂). Moreover,\nit is clear that the map induced on abelianizations by this morphism from ZYβ to this\nweight zero quotient of ΔetYβ is an isomorphism. (Indeed, surjectivity follows from the fact\nthat we are dealing (by definition) with various quotients of a single object; injectivity\nthen follows from Proposition 2.4 and Lemma 2.3 (ii) (for n = 0), and the fact that both\nabelianizations are of rank g over K̂.) Thus, (since ΔetYβ is a free pro-p group of rank g\n– cf. the discussion following Definition 1.1) we conclude that we get an isomorphism as\nstated in the “claim.” This completes the verification of the claim.\nThus, any minimal choice of generators of ΔetYβ (which will necessarily be fixed by ΓK)\ndefines a splitting of the sequence of ΓK-modules\n0 → ZX [1] → ZX → ZX [0] → 0\nHere the ΓK-action is given by composing the ΓR-action considered above with the mor-\nphism – well-defined up to composition with an inner automorphism (which does not\nbother us since this inner automorphism corresponds to a coboundary in the computation\nof the cohomology class ηα that we are interested in) – ΓK → ΓR defined by β. But from\nthe definition of the extension class ηα, this means that the restriction\nηα|β ∈ (H∨0 ⊗R ∧2 H0) ⊗R,β K\nof ηα to the point β is zero. On the other hand, if ηα is zero when restricted to any such\nβ (i.e., any β ∈ S(OK) that maps Spec(k) ⊆ Spec(OK) into T ), it is clear that ηα itself\nmust be zero. This completes the proof of the Proposition. ©\nNote that so far, in this Section and the last, we have been dealing with families\nof curves, parametrized over a base S. Before continuing on to the next Section, it is\nworthwhile to go back to the case of “a single curve” over K in order to make explicit the\nconsequences for such single curves of the theory developed thus far.\nThus, let XK → Spec(K) be a hyperbolic curve. As in Section 1, we have an exact\nsequence\n1 → ΔX → ΠXK → ΓK → 1\n26\nThen let us first note that the theory reviewed in Section 2 can be applied in the present\nnonparametrized situation as well to produce (following precisely the same recipe as in the\nparametrized situation considered as above) a Lie algebra ZX , together with a group ZX\n(both over K̂).\nDefinition 3.4. We shall refer to ZX as the weight zero quotient of ΔX .\nLet x ∈ XK(K). Then x defines a section αx : ΓK → ΠXK (well-defined up to composition\nwith an inner automorphism arising from ΔX). Moreover, αx defines an action of ΓK on\nZX .\nProposition 3.5. Relative to the action of ΓK on ZX defined by αx, the exact sequence\nof ΓK-modules\n0 → ZX [1] → ZX → ZX [0] → 0\nsplits.\nProof. Note that this Proposition is not, strictly speaking, a special case of Proposition\n3.3 (i.e., where we take S = OK), since Proposition 3.3 only addresses the case where\nthe divisor of bad reduction is flat over OK . However, by means of specialization, one\ncan reduce the present Proposition to the “universal case” considered during the proof of\nProposition 3.3. Moreover, in this case, Proposition 3.3 already tells us that the relevant\nexact sequence is split. This completes the proof. ©\nRemark. Let us write (ΔX)Qp for the full (i.e., not truncated as above) Malčev completion\n(cf. [Del], [NT]) of ΔX . Then in some sense, Proposition 3.5 above is a truncated version\nof a theorem that states “relative to the action of ΓK on (ΔX)Qp defined by αx, (ΔX)Qp is\nHodge-Tate.” Here, since (ΔX)Qp is an inverse limit of unipotent algebraic groups, one can\ninterpret “Hodge-Tate” to mean that the Lie algebras of each of these unipotent algebraic\ngroups are Hodge-Tate representations. In fact, it is not difficult to prove that (ΔX)Qp\nis Hodge-Tate (even for arbitrary higher-dimensional smooth XK → Spec(K)) as follows:\nOne reduces the higher-dimensional case to the curve case by cutting with hyperplane\nsections. Then for curves, by considering the universal case, one can reduce to the case of\ncurves smooth over OK . Finally, for curves smooth over OK , one can apply the techniques\nof [Falt1] (by considering cohomology spaces with coefficients valued in unipotent algebraic\ngroups over Qp). This shows that (ΔX)Qp is Hodge-Tate. Moreover, one can also construct\na nontruncated “weight zero quotient (ΔX)Qp → Z∞X ” (that is to say, if XK is a curve of\ngenus g, then Z∞X will be (ΓK -equivariantly) isomorphic to the Malčev completion over K̂\nof the free group on g generators). In fact, it is even possible to show that (ΔX)Qp is de\nRham, but we shall not pursue such issues here since they are not relevant to the proof of\n27\nthe main results of this paper. Nevertheless, we remark that this sort of theorem has been\nverified by A. Shiho ([Shiho]), in a manuscript in preparation.\nSection 4: J-Geometric Sections\nWe maintain the notations of the latter portion of Section 3. Moreover, we assume\nthat our curve XK → Spec(K) is proper (hence of genus ≥ 2) and that XK(K) \f= ∅. In\nthis Section, we would like to consider a continuous homomorphism\nα : ΓK → ΠXK\nwhich defines a section of the surjection ΠXK → ΓK . The section α defines an action of\nΓK on ΔX by conjugation, and hence also an action of ΓK on ZX . In particular, we would\nlike to consider the significance of the following condition on α:\n(∗)spl The exact sequence of ΓK-modules (relative to the action defined\nby α)\n0 → ZX [1] → ZX → ZX [0] → 0\nsplits.\nThe main result of this Section is to show that the group-theoretic condition (∗)spl is\nequivalent to the statement that α is “J -geometric” (a term which means that α acts in\nsome respects as if it came from a geometric point x ∈ XK(K) – see Definition 4.3 for a\nprecise definition).\nTo do this, first we need to recall certain facts concerning Jacobians and their funda-\nmental groups. For d ∈ Z, let J (d)X → Spec(K) be the Picard scheme of line bundles on XK\nof degree d. Thus, J\n(d)\nX is a torsor over JX = J\n(0)\nX , the Jacobian of XK . Note that J\n(d)\nX is\ndefined even if XK does not admit any K-rational points, as the scheme representing the\nétale sheafification of the usual Picard functor of degree d line bundles. Note that a base-\npoint x ∈ XK(Ω) (where Ω is the algebraically closed field of Section 0) may be regarded\nas a degree one divisor on XΩ = XK ⊗K Ω, hence (by multiplying this divisor by d) we\nobtain a point xd ∈ J (d)X (Ω). This allows us to define the arithmetic fundamental group\nπ1(J\n(d)\nX , xd), as well as its geometric counterpart π1(J\n(d)\nX\nK\n, xd). If we form the quotient\nof π1(J\n(d)\nX , xd) by the kernel of π1(J\n(d)\nX\nK\n, xd) → Δ\nJ\n(d)\nX\n(i.e., the projection to the maximal\npro-p quotient of the geometric fundamental group), then we obtain a topological group\nΠ\nJ\n(d)\nX\n. Moreover, we have a natural exact sequence\n28\n1 → Δ\nJ\n(d)\nX\n→ Π\nJ\n(d)\nX\n→ ΓK → 1\nFinally, the natural embedding XK ↪→ J (1)X induces a natural surjection ΠXK → ΠJ(1)\nX\n,\nwhose kernel is the commutator subgroup of ΔX .\nNow let us observe that we can reconstruct Π\nJ\n(d)\nX\nfrom ΠXK → ΓK , as follows. First of\nall, we can reconstruct Π\nJ\n(1)\nX\nas the quotient of ΠXK by the commutator subgroup of ΔX .\nNext, let us observe that, since (for all d ∈ Z) J (d)X is a JX -torsor, considering the action\nof JX on J\n(d)\nX allows one to identify ΔJ(d)\nX\nwith ΔJX . Thus, we have an exact sequence\n1 → ΔJX → ΠJ(1)\nX\n→ ΓK → 1\nIf we consider (for nonzero d), the morphism J\n(1)\nX → J\n(d)\nX given by multiplication by d, we\nsee that the result of pushing forward this sequence by means of the morphism ΔJX\nd→ΔJX\n(i.e., multiplication by d) gives rise to an exact sequence which can be naturally identified\nwith the exact sequence\n1 → Δ\nJ\n(d)\nX\n= ΔJX → ΠJ(d)\nX\n→ ΓK → 1\nThis completes our review of Jacobians and their fundamental groups.\nNow recall from the definition of ZX [0] in Section 3 that ZX [0] may be identified with\nthe weight zero portion of ΔJX ⊗Zp K̂. In particular, one has a natural ΓK-equivariant\nmorphism ΔJX → ZX [0]. Thus, by pushing forward the exact sequence of the preceding\nparagraph (for d = 1) by means of this morphism, we obtain a morphism of exact sequences\nof topological groups as follows:\n1 −→ ΔJX −→ ΠJ(1)\nX\n−→ ΓK −→ 1⏐⏐\u0005 ⏐⏐\u0005 ⏐⏐\u0005 1\n1 −→ ZX [0] −→ ΠZ\nJ\n(1)\nX\n−→ ΓK −→ 1\nNext, we would like to consider sections of Π\nJ\n(1)\nX\n→ ΓK and ΠZ\nJ\n(1)\nX\n→ ΓK . Recall that the\ndifference between any two sections of Π\nJ\n(1)\nX\n→ ΓK (respectively, ΠZ\nJ\n(1)\nX\n→ ΓK) is given by\nan element of H1(K,ΔJX ) (respectively, H\n1(K,ZX [0])). Moreover, if θL, θM : Γ → Π\nJ\n(1)\nX\nare sections that arise from geometric points L,M ∈ J (1)X (K), then the difference\nδ\ndef\n= θL − θM ∈ H1(K,ΔJX )\n29\nmaps to zero in H1(K,ZX [0]) ([BK], Example 3.11 – cf. the proof of Lemma 4.1 for some\nmore details). Thus, it follows that the section θL[0] : ΓK → ΠZ\nJ\n(1)\nX\nobtained from θL\nby composing θL with Π\nJ\n(1)\nX\n→ ΠZ\nJ\n(1)\nX\nis independent (up to composition with an inner\nautomorphism of ΠZ\nJ\n(1)\nX\narising from ZX [0] ⊆ ΠZ\nJ\n(1)\nX\n) of the choice of L. In particular, as\nlong as J\n(1)\nX (K) is nonempty (which is the case here, since XK(K) has been assumed to\nbe nonempty), we thus obtain a canonical section (well-defined up to composition with an\ninner automorphism of ΠZ\nJ\n(1)\nX\narising from ZX [0] ⊆ ΠZ\nJ\n(1)\nX\n)\nθgeom : ΓK → ΠZ\nJ\n(1)\nX\nof ΠZ\nJ\n(1)\nX\n→ ΓK .\nNow let θ : ΓK → Π\nJ\n(1)\nX\nbe an arbitrary section, and let θ[0] : ΓK → ΠZ\nJ\n(1)\nX\ndenote\nthe section induced by θ. Then taking the difference between θ[0] and θgeom defines a\ncohomology class\nδθ ∈ H1(K,ZX [0]) = H0\n(where the last equality follows from Lemma 2.3, applied in the case RK = K). Now we\nhave the following result:\nLemma 4.1. Suppose that the residue field k is finite. Then there exists a geometric\npoint L ∈ J (1)X (K) such that θ = θL if and only if δθ = 0.\nProof. This follows immediately from the theory of [BK], §3, especially Example 3.11,\nplus the following observation: If H is (as in Section 3) the ΓK -module that arises as the\nabelianization of ΔX , then\nKer{H1(K,H) → H1(K,H ⊗ B+DR)}\n(i.e., by Lemma 3.8.1 of [BK], “H1g ” in the notation of [BK]) is equal to\nKer{H1(K,H) → H1(K,H ⊗ K̂) = H1(K,ZX [0])}\n(where the last equality follows from Proposition 2.4 and Lemma 2.2 (ii) (for n = 1, j = 1)).\nIndeed, this equality of kernels follows by using the natural filtration on B+DR (whose\nsubquotients are equal to K̂(i), i ≥ 0), together with the fact that H1(K,H ⊗ K̂(i)) = 0\nfor i > 0 (by Proposition 2.4, Lemma 2.2 (ii) (for n = 1, j > 0)). ©\n30\nNow, we return to considering the section α : ΓK → ΠXK . Let β : ΓK → ΠXK be any\nsection that arises from some geometric point x ∈ XK(K). We would like to compare α\nand β, and, in particular, show that, under the assumption (∗)spl, α and β are, in some\nsense (to be specified precisely below), relatively close. Let ζ : ΓK → ΔX be the continuous\nfunction (not necessarily a group homomorphism) such that\nα(γ) = ζ(γ) · β(γ) ∈ ΠXK\nfor all γ ∈ ΓK .\nNow let φ ∈ ZX(K̂). If γ ∈ ΓK , then let us write γα(φ) ∈ ZX(K̂) (respectively,\nγβ(φ) ∈ ZX(K̂)) for the result of letting γ act on φ by means of the action defined by α\n(respectively, β). Note that it follows from the construction of Section 3 that we have a\nnatural morphism ΔX → ZX(K̂). For \f ∈ ΔX , let us denote by \fZ ∈ ZX(K̂) the image of\n\f in ZX(K̂). Then we have the following:\nLemma 4.2. We have: γα(φ) = ζ(γ)Z · γβ(φ) · ζ(γ)−1Z .\nProof. Indeed, this follows immediately from the fact that the respective actions of ΓK\non ZX(K̂) are induced by conjugation by α(γ) and β(γ) inside ΠXK . ©\nLet α′ : ΓK → ΠXK be any section of ΠXK → ΓK , and let α′Z : ΓK → ΠZJ(1)\nX\nbe the\nsection obtained by composing α′ with ΠXK → ΠZJ(1)X\n. Then we make the following\nDefinition 4.3. We shall call α′ J -geometric if α′Z coincides with θgeom (up to compo-\nsition with an inner automorphism of ΠZ\nJ\n(1)\nX\narising from ZX [0] ⊆ ΠZ\nJ\n(1)\nX\n).\nThe following result is the main technical observation that made it possible to substantially\nstrengthen the result of [Mzk2].\nProposition 4.4. Let α : ΓK → ΠXK be a continuous group homomorphism that defines\na section of ΠXK → ΓK . Then α satisfies (∗)spl if and only if α is J -geometric.\nProof. Let φ ∈ ZX(K̂) be β-invariant, i.e., invariant under the action of ΓK on ZX(K̂)\ndefined by β. We would like to calculate the action of ΓK on φ that is induced by α. Thus,\nfor γ ∈ ΓK , we have, by Lemma 4.2:\nγα(φ) = ζ(γ)Z · γβ(φ) · ζ(γ)−1Z\n= ζ(γ)Z · φ · ζ(γ)−1Z\n31\nThus, γα(φ) · φ−1 = [ζ(γ)Z , φ] (where the brackets denote the “commutator”). Note that\n[ζ(γ)Z , φ] ∈ ZX [1](K̂) = ZX [1] = ∧2 ZX [0] = (∧2 H0) ⊗K K̂\nNow it remains to reinterpret the calculation just performed in terms of cohomology classes.\nRecall the class\nηα ∈ H∨0 ⊗K ∧2H0 = HomK(H0,∧2H0)\ndiscussed in Section 3. Let φ[0] be the image of φ ∈ ZX(K̂) in ZX [0](K̂) = ZX [0]. Since\nφ is β-invariant, it follows that φ[0] is a ΓK -invariant element of ZX [0], hence belongs to\nH0 ⊆ H0⊗K K̂ = ZX [0]. Let δα def= αZ−θgeom ∈ H1(K,ZX [0]) = H0. Then it is immediate\nfrom the definitions that the calculation of the preceding paragraph, interpreted in terms\nof cohomology classes, becomes:\nηα(φ[0]) = δα ∧ (φ[0])\n(Note that since θgeom may be computed – cf. the discussion preceding Lemma 4.1 – using\nthe geometric section β, it follows that δα is precisely the cohomology class defined by the\nimage of the cocycle ζ(−)Z : ΓK → ZX(K̂) in ZX [0](K̂) = ZX [0].) Next observe that this\nequation holds for all β-invariant φ, and that every element of H0 ⊆ H0 ⊗K K̂ = ZX [0] =\nZX [0](K̂) lifts to a β-invariant φ ∈ ZX(K̂). (Indeed, this follows from Proposition 3.5\nsince β arises from a point of XK(K).) Thus, it follows that ηα : H0 → H0 ∧H0 is simply\nthe map “δα∧.” In particular, ηα = 0 if and only if δα = 0. This completes the proof of\nthe Proposition. ©\nRemark. So far here we have been dealing with the truncated weight zero quotient ΔX →\nZX , but it would be interesting also to see what happens in the case of the full nontruncated\nweight zero quotient ΔX → Z∞X (as in the Remark at the end of Section 3). For instance,\nif the action of α on Z∞X is such that Z\n∞\nX has “enough invariants” (i.e., there exists a pro-\nalgebraic group G over K such that G⊗K K̂ is ΓK -equivariantly isomorphic to Z∞X ), does\nit follow that α itself automatically comes from a geometric point x ∈ XK(K)? Although\nsuch questions are beyond the scope of this paper, it is the opinion of the author that\nsuch questions deserve further study. Note that this sort of issue is closely related to the\nso-called Section Conjecture – cf. the Remark following Theorem 19.1 for more details on\nthis conjecture.\n32\nSection 5: The J-Geometricity of K-Valued Points\nLet K be a p-adic field with perfect residue field. Let XK and YK be proper hyperbolic\ncurves over K. Let UK be the K-scheme obtained by localizing XK at its generic point.\nThus, the underlying topological space of UK consists of one point, and the ring of functions\non UK is the function field KX of XK . Note in particular that we can consider ΠUK , ΠYK .\nIn particular, ΠUK is a certain quotient of the absolute Galois group of KX . Let us assume\nthat we are given a continuous open homomorphism\nθ : ΠUK → ΠYK\nover ΓK . In this Section, we would like to begin the proof of the main theorem of this\npaper by showing that any such θ necessarily “maps geometric sections to J -geometric\nsections.” (Naturally, we will explain below precisely what is meant by the expression in\nquotes.)\nFirst observe that if x ∈ XK(K) is any K-valued point, then we can form the com-\npletion (KX )x of the field KX with respect to the valuation defined by x. Moreover, we\nhave a natural morphism Π\n(KX )x\n→ ΠUK (well-defined up to composition with conjugation\nby an element of ΔU ) whose image is “the” (more rigorously: any of the various conju-\ngate) decomposition group associated to x. Moreover, as is well-known (see, e.g., [Ser2]),\n(KX )x is (noncanonically) isomorphic to K((t)) (where t is an indeterminate), so Δ(KX)x\nmay be identified with Zp(1). In particular, by forming, relative to some isomorphism\n(KX )x ∼= K((t)), the field extension of (KX )x corresponding to adjoining a compatible\nsystem of p-power roots of t to K((t)), one sees immediately that Π\n(KX)x\n→ ΓK admits\nmany sections.\nDefinition 5.1. We shall refer to as geometric any section ΓK → ΠUK of ΠUK → ΓK\nobtained by composing a section of Π\n(KX)x\n→ ΓK with (any one of the conjugate natural\nhomomorphisms) Π\n(KX)x\n→ ΠUK .\nNote, in particular, that Π\n(KX )x\n→ ΠXK factors through ΓK , so in particular, the com-\nposite αX : ΓK → ΠXK with ΠUK → ΠXK of any geometric section αU : ΓK → ΠUK is\ninduced by some point x : Spec(K) → XK .\nLet αU : ΓK → ΠUK be a geometric section. By composing αU with θ, we obtain a\nsection αY : ΓK → ΠYK . In this Section, we would like to prove that αY is necessarily\nJ -geometric in the sense of Definition 4.3. To do this, let us first observe that we have a\ndiagram of continuous morphisms\nΔU −→ ZX(K̂)⏐⏐\u0005θ|ΔU\nΔY −→ ZY (K̂)\n33\nwhich are compatible with the ΓK -actions defined by αU and αY on the all the groups\ninvolved. (Here, the upper horizontal morphism is obtained by composing ΔU → ΔX with\nΔX → ZX(K̂).) Then we have the following\nLemma 5.2. There exists a (natural) surjective ΓK -homomorphism ZX(K̂) → ZY (K̂)\nthat makes the above diagram commute.\nProof. Observe that the kernel of ΔU → ΔX is generated by inertia groups (i.e., images of\nthe various Δ\n(KX)x\n→ ΔU ), and the action of ΓK on an inertia group is via the cyclotomic\ncharacter, i.e., (in the language of Hodge-Tate Galois representations) of weight one. It\nthus follows (from Lemma 2.2 (ii), for n = 0, j = −1) that the morphism ΔU → ZY (K̂)\n(obtained from the diagram above) factors through ΔX . Hence, we obtain a morphism\nΔX → ZY (K̂). But now, it follows immediately (by the universal property of the truncated\nMalčev completion, plus “weight arguments”) from the construction of ZX(K̂) from ΔX\nthat this morphism ΔX → ZY (K̂) factors naturally through ZX(K̂). This shows the\nexistence of a morphism as claimed in the statement of the Lemma. The fact that this\nmorphism is surjective follows from the fact that θ is open (which implies that the induced\nmorphism ΔabU → ΔabY on abelianizations is open, hence surjective after tensoring with\nQp). ©\nLemma 5.3. The action of ΓK on ZY (K̂) defined by αY satisfies the condition (∗)spl\ndiscussed in Section 4.\nProof. Indeed, that the action of ΓK on ZX(K̂) defined by αX satisfies (∗)spl follows\nfrom Proposition 3.5. Thus, Lemma 5.3 follows from the surjectivity of the morphism of\nLemma 5.2. ©\nProposition 5.4. Let θ : ΠUK → ΠYK be a continuous open homomorphism over ΓK ,\nαU : ΓK → ΠUK a geometric (Definition 5.1) section, and αY : ΓK → ΠYK the composite\nof αU with θ. Then αY is J -geometric (Definition 4.3).\nProof. This follows by combining Lemma 5.3 with Proposition 4.4. ©\nSection 6: F-Geometricity and FI-Geometricity\nLet K be a finite extension of Qp. Let S\ndef\n= Spec(R), where R is an OK -algebra\nnoncanonically isomorphic to OK [[t]] (and t is an indeterminate). Let ηS be the generic\n34\npoint of S (regarded as a scheme). Let ΓS\ndef\n= π1(SK) (where SK\ndef\n= S ⊗OK K, and the\n“π1” is with respect to some base-point which we omit to simplify notation); let ΓηS be\nthe absolute Galois group of K(ηS) (the function field of S). Thus, ΓS can be naturally\nregarded as a quotient of ΓηS . If H ⊆ ΓηS is an open subgroup corresponding to some\nfinite étale covering ηS′ → ηS , then we shall write ΓηS′ for H; S ′ = Spec(R′) for the\nnormalization of S in ηS′ ; and ΓS′ for the quotient of ΓηS′ corresponding to étale coverings\nof S ′K\ndef\n= S ′ ⊗OK K. Thus, S ′ is finite and flat (since S is regular of dimension 2, and R′\nis normal, hence an R-module of depth 2) over S.\nRemark. For the reader familiar with [Mzk2], we remark that this Section and the next\nare, in some sense, a generalization of Section 5 of [Mzk2], and play a comparable role in\nthe present paper to that of Section 5 in [Mzk2]. For the reader not familiar with [Mzk2],\nwe remark that, nevertheless, we do not assume any knowledge of [Mzk2] in the following\ndiscussion.\nWe begin by considering a continuous Zp[ΓηS ]-module V , where V , as a Zp-module,\nis a finite and free.\nDefinition 6.1. We shall refer to V as potentially geometric if there exists some open\nsubgroup ΓηS′ ⊆ ΓηS such that the ΓηS′ -module obtained by restricting the ΓηS -action\non V to ΓηS′ arises as the Tate module of some p-divisible group G → Spec(OK′ ), where\nK ′ ⊆ R′K is a finite extension of K.\nNote, in particular, that the ΓηS′ -action on V then factors through the quotient ΓηS′ → ΓS′ .\nMoreover, if V is potentially geometric, then we may make the following construction: Let\nus write G for the (not necessarily connected) formal group (over OK′) associated to the\np-divisible group G → Spec(OK′ ) of Definition 6.1. Thus, we have a natural isomorphism\n(of finite flat group schemes over OK′ ) between the kernels\nG[pn] ∼= G[pn]\nof multiplication by pn (for all n ≥ 0) on G and G. In particular, if we consider the exact\nsequence (generalizing the Kummer sequence, which corresponds to the case where G is\nthe formal group associated to the multiplicative group Gm)\n0 −→ G[pn] −→ G p\nn\n−→ G −→ 0\nas an exact sequence of sheaves on the finite flat site of S ′, then we get a natural map\nG(S′) → H1flat(S′,G[pn])\n35\n(where the cohomology group is relative to the finite flat topology on S ′) for n ≥ 0. Since\nétale morphisms are quasi-finite and flat, we also have a natural morphism\nH1flat(S\n′,G[pn]) → H1et(S′,G[pn]K′ )\nThus, if we compose the above two morphisms, take the inverse limit with respect to n,\nand use the fact that H1et(S\n′,−) ∼= H1(ΓS′ ,−), we get a natural morphism\nκG : G(S′) → H1(ΓS′ , V )\nwhich one may regard as a generalization of the Kummer map (from units of a field to a\ncertain Galois cohomology group of the field).\nRemark. Note that when the formal group G arises from an abelian variety over OK′ ,\nthen the cohomology class that one obtains by applying κG to a point of G coincides with\nthe cohomology class that one obtains (cf. the discussion of Section 4) by looking at the\nmorphism induced on arithmetic fundamental groups by the corresponding point of the\nabelian variety. Indeed, this is a matter of general nonsense – cf., e.g., [Naka2], Claim\n(2.2); [NTs], Lemma (4.14).\nDefinition 6.2. Suppose that V is potentially geometric, and that ΓηS′ ⊆ ΓηS is as in\nDefinition 6.1. Then we define\nH1f (S\n′\nK , V ) ⊆ H1(S′K , V )\ndef\n= H1(ΓS′ , V )\nto be the subset of elements ζ ∈ H1(S′K , V ) such that some nonzero multiple of ζ lies in\nthe image of G(S′) under the morphism κG.\nNow let XηS → ηS be a proper hyperbolic curve over ηS such that for some open\nΓηS′ ⊆ ΓηS , and some proper, hyperbolic curve Z → Spec(K ′) (where K ′ ⊆ R′K is a finite\nextension of K, and we assume that Z(K′) \f= ∅), we have an isomorphism of ηS′ -curves\nXηS ×ηS ηS′ ∼= Z ×K′ ηS′\nThen we make the following technical\nDefinition 6.3. We shall call XηS irreducibly splittable if for some open ΓηS′ ⊆ ΓηS ,\nand some proper, hyperbolic curve Z → Spec(K ′) (where K ′ ⊆ R′K is a finite extension of\nK, and we assume that Z(K′) \f= ∅), we have an isomorphism of ηS′ -curves\nXηS ×ηS ηS′ ∼= Z ×K′ ηS′\n36\nand, moreover, the (induced) map S ′ → Spec(OK′ ) satisfies: (i) S ′ → Spec(OK′ ) admits\na section; (ii) the morphism S ′ → Spec(OK′ ) has geometrically irreducible fibers.\nAs usual, we have an exact sequence\n1 → ΔXηS = HX → ΠXηS → ΓηS → 1\nwhere we write HX for the abelianization of ΔXηS\n. Thus, HX has a natural structure of\ncontinuous ΓηS -module. Moreover, as a Zp-module, it is free of rank 2gX (where gX is the\ngenus of XηS). Let us write\nHX → HIX\nfor the quotient of HX by all elements of HX on which some open subgroup of ΓηS acts via\nthe cyclotomic character. (The “I” comes from the fact that HIX is obtained by forming\nthe quotient of HX by all of its “inertia-like” subgroups.)\nNext, let\nHFX ⊆ HX\nbe the largest ΓηS -submodule of HX which has no nonzero torsion-free quotients H\nF\nX → Q\nsuch that some open subgroup of ΓηS acts trivially on Q. (Here, the “F” stands for\n“finite.” This is because HFX corresponds to the portion of HX that (potentially) extends\nto a p-divisible group – i.e., a direct limit of finite flat group schemes – over OK′ .) Then\nit is well-known (see, e.g., [FC], Chapter III) that HFX and H\nI\nX are Cartier-dual to one\nanother, and, moreover, that HFX is potentially geometric (Definition 6.1). (Indeed, in [FC],\nChapter III, one finds a discussion of how one may obtain (in a natural fashion) abelian\nvarieties over K ′ as “quotients” (by some group of periods) of semi-abelian varieties that\n(potentially) extend over OK′ in such a way that the dimensions of their toral parts are the\nsame over the generic and special points of OK′ . If we apply this theory to the Jacobian JZ\nof the curve Z, then HFX is the Tate module of this semi-abelian variety (that potentially\nextends over OK′).) Since HIX is the Cartier dual of HFX , it thus follows that HIX is also\npotentially geometric.\nLet\nHFIX ⊆ HIX\nbe the image of HFX in H\nI\nX . Thus, we have a surjection H\nF\nX → HFIX , and HFIX is also\npotentially geometric. Finally, let\nHMX\ndef\n= Ker(HX → HIX) ⊆ HFX\n37\n(Here, the “M” stands for “multiplicative.” This is because HMX arises from the portion\nof HFX that corresponds to twisted copies of (the Tate module of) the multiplicative group\nGm.) H\nM\nX is also potentially geometric.\nNext, we would like to consider a section α : ΓηS → ΠXηS of ΠXηS → ΓηS . Now recall\nthe isomorphism\nXηS ×ηS ηS′ ∼= Z ×K′ ηS′\nBy composing α|Γη\nS′\n: ΓηS′ → ΠXηS with the projection ΠXηS → ΠZ induced by this\nisomorphism, we thus obtain a morphism ΓηS′ → ΠZ , whose composite with ΠZ → ΓK′ is\nthe natural morphism ΓηS′ → ΓK′ . Now let us make the following assumption on α:\n(∗)S′ This morphism ΓηS′ → ΠZ factors through ΓS′ .\nLet us denote the resulting morphism by β : ΓS′ → ΠZ .\nNow let γ : ΓS′ → ΠZ be any morphism obtained by composing the natural morphism\nΓS′ → ΓK′ with some section ΓK′ → ΠZ arising from a geometric point ∈ Z(K′). Let\nβJ , γJ : ΓS′ → Π\nJ\n(1)\nZ\nbe the morphisms obtained by composing β and γ, respectively, with\nΠZ → ΠJ(1)\nZ\n. Then the difference βJ − γJ defines an element\nδZ ∈ H1(S′K ,HZ )\nhence an element\nδX ∈ H1(S′K ,HX )\nwhose image in H1(S′K ,H\nI\nX ) we denote by δ\nI\nX .\nDefinition 6.4. Suppose that α : ΓηS → ΠXηS is a section that satisfies the assump-\ntion (∗)S′ above. Then we shall call α F -geometric (respectively, FI-geometric) if some\nnonzero multiple of δX (respectively, δ\nI\nX) lies in the image of H\n1\nf (S\n′\nK ,H\nF\nX) (respectively,\nH1f (S\n′\nK ,H\nFI\nX )) in H\n1(S′K ,HX) (respectively, H\n1(S′K ,H\nI\nX)).\nNote that if α is F -geometric, it is also automatically FI-geometric. Also, let us observe\nthat the definition of “F -geometric” or “FI-geometric” is independent of the choice of γ\n(cf. Lemma 4.1 and the discussion preceding it). Indeed, to see this, it suffices to verify\nthat some nonzero multiple of the class ∈ H1(S′K ,HX) arising from the difference of two\nγ’s lies in H1f (S\n′\nK ,H\nF\nX). But the difference of two γ’s (both of which arise from a geometric\npoint ∈ Z(K′)) defines a point ∈ JZ(K′) (where JZ is the Jacobian of Z). Moreover, since\nthe residue field of K ′ is finite, it follows that some nonzero multiple of this point in JZ(K′)\n38\nextends to an OK′ -valued point of the Néron model JZ of JZ over OK′ which maps the\nspecial point of Spec(OK′ ) to the identity of JZ . On the other hand, since GF – i.e., the\n“G” (cf. the discussion following Definition 6.1) for the potentially geometric module HFX\n– is simply the formal group defined by JZ , it thus follows that such a point ∈ JZ(OK′ )\ndefines a point in GF (OK′ ) ⊆ GF (S′). Thus, by Definition 6.2, we see that the difference\nbetween the two γ’s lies in H1f (S\n′\nK ,H\nF\nX ), as desired.\nProposition 6.5. Suppose that XS is irreducibly splittable (cf. Definition 6.3). Then\nany FI-geometric α is also F -geometric.\nProof. First, let us state that throughout the proof, “S ′” will be a fixed S ′ satisfying the\nconditions of Definition 6.3. Next, let us observe that the cokernel of the natural morphism\nH1f (S\n′\nK ,H\nF\nX) → H1f (S′K ,HFIX )\nis torsion. Indeed, this follows from Definition 6.2 and the fact that if GF and GFI are\nthe respective formal groups as in Definition 6.2, then the natural morphism GF → GFI is\nformally smooth (hence surjective on S ′-valued points).\nThus, if we start with a class ζ ∈ H1(S′K ,HX ) whose image ζI ∈ H1(S′K ,HIX) lies\nin the image of H1f (S\n′\nK ,H\nFI\nX ), then (after replacing ζ by a nonzero multiple of ζ), we\nmay assume that there exists a ζ ′ ∈ H1(S′K ,HX) such that: (a.) ζ ′ lies in the image\nof H1f (S\n′\nK ,H\nF\nX ); (b.) ζ\n′ − ζ maps to 0 in H1(S′K ,HIX). Moreover, by the definition of\nHMX\ndef\n= Ker(HX → HIX), it follows that (b.) may be rewritten in the form: “ζ ′ − ζ lies in\nthe image of H1(S′K ,H\nM\nX ) in H\n1(S′K ,HX).”\nWith these observations in hand, it follows that it suffices to prove that\n(∗)im The image of H1(S′K ,HMX ) in H1(S′K ,HX) is contained up to\ntorsion (i.e., up to multiplication by a nonzero integer) in the image of\nH1f (S\n′\nK ,H\nF\nX ) in H\n1(S′K ,HX).\nNote first that by replacing K ′ by a finite extension of K ′ (and thus also enlarging S ′ –\nnote that this does not affect the validity of the conditions (i) and (ii) of Definition 6.3),\nwe may assume that the action of ΓS′ on H\nM\nX (−1) (where the “(−1)” is a Tate twist) is\ntrivial. (Thus, in particular, as a ΓS′ -module, H\nM\nX is isomorphic to a direct sum of a finite\nnumber of copies of Zp(1).) On the other hand, by Lemma 6.6 below (and the Kummer\nexact sequence), it follows that H1(S′K ,Zp(1)) = {(R′K)×}∧ (where the “∧” denotes p-adic\ncompletion) is generated up to torsion by H1f (S\n′\nK ,Zp(1)) = {(R′)×}∧ and (the image of)\nH1(K′,Zp(1)) = {(K′)×}∧. Thus, it follows that H1(S′K ,HMX ) is generated up to torsion\nby H1f (S\n′\nK ,H\nM\nX ) and (the image of) H\n1(K′,HMZ ).\nNext, observe that the image of H1(K′,HMZ ) in H\n1(K′,HZ ) is contained up to torsion\nin the image of H1f (K\n′,HFZ ) in H\n1(K′,HZ ). Indeed, this follows from the theory of [BK],\n39\n§3 (cf. the proof of Lemma 4.1). Namely, since HMZ is “of weight one,” the image of\nHMZ ⊆ HZ in the weight zero component of HZ ⊗ K̂ is zero, so any cohomology classes\nin H1(K′,HMZ ) go to zero in H\n1(K′,−) of the weight zero component of HZ ⊗Zp K̂. But\nthis means (as we saw in Lemma 4.1), that such cohomology classes arise geometrically,\nas desired. (Note that, by the above argument involving Néron models (in the discussion\nimmediately preceding Proposition 6.5), it follows that (relative to pulling back cohomology\nclasses over K ′ to cohomology classes over S ′K), the notation “H\n1\nf (S\n′\nK ,−)” of Definition\n6.2 is consistent with the notation “H1f (K\n′,−)” of [BK], §3.)\nPutting everything together, we thus see that (∗)im has been verified. This completes\nthe proof of the Proposition. ©\nLemma 6.6. For S ′ satisfying condition (ii) of Definition 6.3, we have that (R′K)\n× is\ngenerated up to torsion by (R′)× and (K′)×.\nProof. Indeed, condition (ii) of Definition 6.3 implies that if π is a uniformizing element\nof OK′ ⊆ R′, then the ideal π ·R′ is contained in a unique prime ideal of height one of R′.\nLet us denote this prime ideal by ℘. Let x ∈ R′ be an element which becomes a unit in\nR′K . Then x is invertible at every height one prime of R\n′ except (possibly) ℘. Moreover,\nby replacing x by some xn (where n is independent of x), we may assume that x has the\nsame valuation as πm (for some nonnegative integer m) in the discrete valuation ring R′℘.\nIt thus follows that π−m · x is a unit at every height one prime of R′. Since R′ is normal,\nthis implies that π−m · x ∈ (R′)×. This completes the proof of the Lemma. ©\nSection 7: From F-Geometricity to Line Bundles\nIn this Section, we use some elementary algebraic geometry (Lemmas 7.1 and 7.2)\nto translate the rather abstract and technical condition of “F -geometricity” into a more\ntractable existence criterion (Proposition 7.4) for line bundles. Let S be the spectrum of a\nfield of characteristic zero. Let X → S be a proper hyperbolic curve of genus g over S. Let\nN be a fixed positive integer. We would like to consider the natural morphism X → J (N )X\n(given by mapping a point x of X to the line bundle OX(N · x)). Taking the product of\nthis morphism with X (on the right), we obtain a morphism ξ : X ×S X → J (N )X ×S X.\nLemma 7.1. There exists a line bundle on J\n(N )\nX ×S X whose pull-back via ξ is a nonzero\ntensor power of the line bundle D def= OX×SX(Δ) (where Δ ⊆ X ×S X) on X ×S X.\nProof. First, let us take the product of ξ with one more copy of X on the right, to\nobtain a morphism ξ′ : X ×S X ×S X → J (N )X ×S X ×S X. For i, j = 1, 2, 3 such that\n40\ni \f= j, let Δij ⊆ X ×S X ×S X denote the diagonal given by setting equal the ith and\njth components of the triple product. Let Dij denote the line bundle OX×SX×SX(Δij).\nIn the triple product of X, we consider the second X to be the “true curve X”; the first\nX to be a “parameter space for the family of line bundles D12” (regarded as line bundles\non the “true curve” given by the second factor); and the third X to be a “base extension\nX → S.” Over this extended base, the “true curve” acquires a “section” Δ23. Thus, it\nfollows from the general theory of the Picard functor of a family of curves for which a\nsection exists that there exists a line bundle L′ on J (N )X ×S X ×S X whose pull-back via ξ′\nis equal to E ′ def= (D12 ⊗D−113 )⊗N .\nNow let us consider the “determinant of the higher direct image sheaves” (cf. [MB],\n§1, for an exposition of this notion) of L′ and E ′ for the morphisms X×SX×S X → X×S X\nand J\n(N )\nX ×S X ×S X → J\n(N )\nX ×S X given by forgetting the third factor. We denote the\nrespective “determinants of the higher direct image sheaves” of L′ and E ′ by L and E. Thus,\n(by the functoriality of forming the “determinant of the higher direct image sheaves”) L\nis a line bundle on J\n(N )\nX ×S X such that ξ∗(L) = E. Moreover, I claim that we can write\nE = D⊗N ⊗ F\nwhere F is a line bundle on X ×S X obtained by pulling back some ω⊗mX\/S (for m ∈ Z)\nfrom the first factor of X ×S X, and tensoring with the pull-back to X ×S X of some\nline bundle M on S. Indeed, the “D⊗N -term” arises from the fact that D12 is the pull-\nback of D via the projection X ×S X ×S X → X ×S X under consideration. Then\nthe difference between the “determinants of the higher direct image sheaves” of the line\nbundles D−N13 = OX×SX×SX(−N ·Δ13) and OX×S×SX may be computed using the natural\ninclusion\nD−N13 = OX×SX×SX(−N · Δ13) ⊆ OX×S×SX\nBy using the fact that the “determinant of the higher direct image sheaves” is multiplicative\non exact sequences, we thus obtain that this difference is equal to some power of the pull-\nback to X ×S X (via the projection to the first factor) of OX×SX(−Δ)|Δ=X = ωX\/S . On\nthe other hand, the “determinant of the higher direct image sheaves” of the trivial line\nbundle on X ×S X ×S X is clearly the pull-back (to X ×S X) of a line bundle M on S.\nThis gives us a line bundle F of the form discussed above, hence completes the proof of\nthe claim.\nBy replacing L with L ⊗OS M−1, we may assume that M is trivial. Moreover, by\nLemma 7.2 below, there exists a line bundle P on J (N )X whose pull-back to X (via the\nnatural morphism X → J (N )X ) is some nonzero tensor power of ωX\/S . Thus, replacing L\nby a tensor product of appropriate powers of L and (P|\nJ\n(N)\nX\n×SX) completes the proof of\nthe Lemma. ©\n41\nLemma 7.2. There exists a line bundle P on J (N )X whose pull-back to X is a nonzero\ntensor power of ωX\/S.\nProof. Let us first consider the case N = g − 1. In this case, we take P to be the line\nbundle on J\n(g−1)\nX defined by the natural theta divisor on J\n(g−1)\nX (i.e., the image of the\nnatural morphism from the (g − 1)-fold product of X to J (g−1)X ). Then the fact that the\npull-back of P to X is equal to ω⊗\n1\n2 g(g−1)\nX\/S\nis an immediate consequence of [MB], Corollary\n2.5: in the notation of loc. cit., we are interested here in the case where one takes n = 0;\nS = X; and a : S → X to be the identity; then specializing the formula of loc. cit. to\nthe zero section of the Jacobian (note that the restriction of “U (a)n ” in loc. cit. to the zero\nsection of the Jacobian is trivial) proves the assertion concerning the pull-back of P to X.\nNext, let us consider the case where N is divisible by g − 1. Then the morphism\nX → J (N )X factors through J\n(g−1)\nX → J\n(N )\nX . Moreover, it follows from the basic theory of\nline bundles on abelian varieties (see, e.g., [AV], §23, the Corollary to Theorem 2 on p.\n231) that a nonzero tensor power of the “P” considered in the previous paragraph descends\nfrom J\n(g−1)\nX to J\n(N )\nX . This completes the proof in the case where N is divisible by g − 1.\nFinally, we consider the case of arbitrary positive N . In this case, we have a natural\nmap J\n(N )\nX → J\n(N (g−1))\nX (multiplication by g − 1). But, by the preceding paragraph, we\nalready have a suitable “P” on J (N (g−1))X . Thus, by pulling this line bundle back to J\n(N )\nX ,\nwe obtain a suitable “P” on J (N )X . This completes the proof of the Lemma. ©\nNext, we would like to consider a section α : ΓS\ndef\n= π1(S) → ΠX of ΠX → ΓS . By\ncomposing α with π1(−) applied to X → J (N )X , we obtain a section αNJ : ΓS → ΠJ(N)\nX\n.\nLemma 7.3. Suppose that αNJ arises from a geometric section ∈ J\n(N )\nX (S). Then there\nexists a line bundle on X of degree prime to p.\nProof. By taking the fibered product (over ΓS) of α with the identity on ΠX , we obtain a\nmorphism αX : ΠX → ΠX×SX . Since S is the spectrum of the field, the group cohomology\nof ΠX×SX computes the p-adic étale cohomology of X ×S X (cf. Lemma 0.4). Thus,\nwe can form the arithmetic first Chern class of the line bundle D of Lemma 7.1 (cf.\nDefinition 0.3): c1(D) ∈ H2(ΠX×SX ,Zp(1)). Let ζ\ndef\n= α∗X(c1(D)) ∈ H2(ΠX ,Zp(1)). On\nthe other hand, by composing αX with π1(−) of the morphism ξ, we obtain a morphism\nαNξ : ΠX → ΠJ(N)\nX\n×SX\n. Moreover, by assumption, αNξ arises from a geometric morphism\nX → J (N )X ×S X. Thus, if L is a line bundle on J\n(N )\nX ×S X, we obtain that (αNξ )∗(c1(L)) ∈\nH2(ΠX ,Zp(1)) can be written as c1(M), for some line bundle M on X. Now recall that\nby Lemma 7.1, there exists a line bundle L on J (N )X ×S X such that ξ∗L is a nonzero tensor\npower of D. Thus, putting everything together, it follows that some nonzero multiple of\n42\nthe abstract cohomology class ζ ∈ H2(ΠX ,Zp(1)) can be written in the form c1(M) for\nsome line bundle M on X, i.e., m · ζ = c1(M) (for some nonzero m ∈ Z). Next, let\nus recall that since ζ was constructed from D, the image of ζ under the “degree map”\nH2(ΠX ,Zp(1)) → H2(ΔX ,Zp(1)) ∼= Zp is equal to 1, so deg(M) = m. Thus, we can\nargue as follows (cf. [Mzk2], Lemma 6.1): Write m = a · pb, where a is an integer prime to\np, and b is a nonnegative integer. Thus, c1(M) vanishes in H2(ΠX , (Z\/pbZ)(1)). But, by\nthe Kummer exact sequence, this implies that there exists a line bundle P on X such that\nP⊗pb ∼= M. In particular, deg(P) = p−b · deg(M) = a, so the existence of P completes\nthe proof of the Lemma. ©\nNow let us consider the following situation: Let K be a finite extension of Qp. Let\nXK → Spec(K) and YK → Spec(K) be proper hyperbolic curves over K. Let UK be the\ngeneric point of XK . Moreover, let us assume that we have been given a continuous open\nhomomorphism over ΓK\nθ : ΠUK → ΠYK\nNow let S, ηS be as at the beginning of Section 6. Write XηS for XK ×K ηS (and similarly,\nfor UηS , YηS). Then by base-change (note that ΠUηS\n= ΠUK ×ΓK ΓηS ; ΠYηS = ΠYK ×ΓK\nΓηS), θ induces a continuous open homomorphism\nθηS : ΠUηS\n→ ΠYηS\nLet Y ′ηS → YηS be a finite étale covering induced by some open subgroup of ΠYηS that\nsurjects onto ΓηS . Note that θηS allows us to define the pull-back of Y\n′\nηS\n→ YηS to\nUηS . Let U\n′\nηS\n→ UηS be a connected component of this pull-back which is geometrically\nconnected over ηS. Note that U\n′\nηS\n→ UηS extends to a finite (possibly ramified) covering\nX ′ηS → XηS . Thus, X ′ηS is a proper hyperbolic curve over ηS. Let us suppose that Y ′ηS\nis irreducibly splittable (Definition 6.3). Then it follows immediately that X ′ηS is also\nirreducibly splittable. (Indeed, the definition of “irreducibly splittable” only involves the\nbase (i.e., S, S ′, etc.), plus finite étale coverings of the curve, i.e., it may be phrased\nentirely in terms of (the base plus) fundamental groups. Thus, the fact that X ′ηS and\nY ′ηS are related by θηS is enough to guarantee that “Y\n′\nηS\nirreducibly splittable =⇒ X ′ηS\nirreducibly splittable.”)\nNext, we would like to consider a section\nαU : ΓηS → ΠU ′ηS\nof ΠU ′ηS\n→ ΓηS . Composing αU with π1(−) of the natural morphism U ′ηS → X ′ηS gives rise\nto a section αX : ΓηS → ΠX ′ηS . Composing αU with the morphism θ\n′\nηS\n: ΠU ′ηS\n→ ΠY ′ηS\ninduced by θηS gives a section αY : ΓηS → ΠY ′ηS .\n43\nThe following key result is the culmination of our efforts in Sections 1 through 7:\nProposition 7.4. Under the circumstances just described above, let us assume that αX\nis FI-geometric (Definition 6.4). Then it follows that Y ′ηS admits a line bundle of degree\nprime to p.\nProof. First observe that the morphism ΔU ′ηS\n→ ΔY ′ηS → H\nI\nY ′ηS\n(where HIY ′ηS\nis as in the\ndiscussion preceding Definition 6.4) factors through HIX ′ηS\n. (Indeed, this follows by observ-\ning, as in the proof of Lemma 5.2, that the inertia groups of ΔU ′ηS\nmap to zero in HIY ′ηS\n.)\nThus, we obtain a ΓηS -equivariant morphism H\nI\nX ′ηS\n→ HIY ′ηS . In particular, it follows by\nusing Tate’s theorem (i.e., Theorem 4 of [Tate] – that morphisms between Tate modules\nof p-divisible groups induce morphisms between the p-divisible groups, hence morphisms\nbetween the respective formal groups) that H1f (S\n′\nK ,H\nFI\nX ′ηS\n) maps to H1f (S\n′\nK ,H\nFI\nY ′ηS\n), for any\nfinite étale covering ηS′ → ηS of ηS as in Definition 6.2.\nNext, I claim that αY is FI-geometric. Indeed, this follows from Proposition 5.4,\nplus the observation of the preceding paragraph. In words: This claim amounts to the\nclaim (cf. Definition 6.4) that the “J -portion” of αY (i.e., the result of composing αY with\nΠY ′ηS\n→ Π\nJ\n(1)\nY ′\n, where J\n(1)\nY ′ is the Picard scheme of line bundles of degree 1 on Y\n′\nηS\n) differs\nfrom the “J -portion” of a constant (i.e., arising from a point defined over a finite extension\nof K of the curve “Z” of the discussion of Section 6) geometric section of ΠY ′ηS\nby a class\nin H1f (S\n′\nK ,H\nFI\nY ′ηS\n). On the other hand, since we are operating under the assumption that\nαX is FI-geometric, we know that the J -portion of αX differs from the J -portion of a\nconstant geometric section of ΠX ′ηS\nby a class in H1f (S\n′\nK ,H\nFI\nX ′ηS\n). Moreover, by Proposi-\ntion 5.4, constant geometric sections of ΠU ′ηS\nmap to constant J -geometric (which is as\ngood as “geometric” for us, since we are only interested in “J -portions” here) sections of\nΠY ′ηS\n. Thus, since (by the observation of the preceding paragraph) H1f (S\n′\nK ,H\nFI\nX ′ηS\n) maps\nto H1f (S\n′\nK ,H\nFI\nY ′ηS\n), we conclude that αY is FI-geometric, as desired. This completes the\nproof of the claim. Now since Y ′ηS is also assumed to be irreducibly splittable, it follows\nfrom Proposition 6.5 that αY is, in fact, F -geometric.\nNow let us consider the morphism αNJ : ΓηS → ΠJ(N)\nY ′\n(where J\n(N )\nY ′ is the Picard\nscheme of line bundles of degree N on Y ′ηS), for some N > 0, induced by composing αY\nwith ΠY ′ηS\n→ Π\nJ\n(N)\nY ′\n. Note that the F -geometricity of αY is, by definition, a property\nconcerning α1J . By the definition of F -geometricity (Definition 6.4), it follows immediately\nthat if N is large enough, then αNJ arises from a geometric section J\n(N )\nY ′ (SK). Indeed, the\nlarge N is to take care of the phrase “nonzero multiple of” in Definitions 6.2 and 6.4, plus\nthe fact that a priori the geometric point whose existence is guaranteed by Definition 6.2\nis only defined over ηS′ , so we may need to apply the norm map (for ηS′ → ηS) to get a\n44\ngeometric point over ηS, which may cause N to increase again. Now, applying Lemma 7.3\ncompletes the proof of the Proposition. ©\nRemark. Note that Sections 6 and 7 are made much more technically difficult by the\nfact that we start with a morphism ΠUK → ΠYK rather than a morphism ΠXK → ΠYK .\nIndeed, the whole business of distinguishing “F -geometric” from “FI-geometric” arises\nbecause one does not have a proper theory of “p-divisible groups of infinite rank,” which\nis the sort of object that one must deal with if one tries to work directly with HU (the\nabelianization of ΔU ) without passing to H\nI\nU . Similar technical problems (arising from the\nfact that HU is of infinite rank) also are the reason behind the lengthiness of Sections 11\nand 12.\nSection 8: From Line Bundles to Tame Points\nIn this Section, which is something of an appendix to that portion of the paper con-\nstituted by Sections 1 through 7, we again apply some elementary algebraic geometry, this\ntime to pass from line bundles to rational points of the curve defined over tamely ramified\nextensions of the given field. Thus, let M be a p-adic field whose residue field kM is iso-\nmorphic to k((t)), where k is a finite field, and t is an indeterminate. Let XM be a proper\nhyperbolic curve over M . Then we have the following result:\nProposition 8.1. Suppose that XM admits a line bundle L of degree prime to p. Then\nthere exists a tamely ramified extension M ′ of M such that XM (M ′) is nonempty.\nProof. By replacing L by an appropriate (prime to p) tensor power of the original L, we\nmay assume that L is very ample. Thus, by Bertini’s Theorem, it follows that there exists\na divisor D ⊆ XM such that L ∼= OXM (D), and D is étale over M . Let us write\nD =\nr⋃\ni=1\nDi\nwhere each Di = Spec(Mi). Thus, Mi is a finite field extension of M . Since deg(D) is\nprime to p, it follows that at least one of the Mi (say, M1) is such that [M1 : M ] is prime\nto p. Let Mc be the Galois closure of M1 over M . Then it follows from the elementary\ntheory of p-adic fields (in particular, the fact that the wild inertia subgroup of ΓM is a\nnormal pro-p subgroup of ΓM , hence contained in any Sylow-pro-p subgroup of ΓM – see,\ne.g., [Ser2], Chapter IV, §2) that Mc is tamely ramified over M . Thus, in particular, M1\nis tamely ramified over M , and XM (M1) is nonempty. This completes the proof of the\nProposition. ©\n45\nNext, before continuing, we would like to discuss an auxiliary Lemma that will be\nuseful when we apply Proposition 8.1. Let us suppose that there exists a p-adic field\nL ⊆ M satisfying the following properties:\n(1) We have an inclusion OL ⊆ OM , relative to which mL · OM = mM .\n(2) If kL is the residue field of L, then k ⊆ kL, and kL is a function field in\none variable over k.\n(3) The inclusion kL ⊆ kM is obtained by completing the function field kL\nat one of its k-valued points.\nLet us suppose, moreover, that there exists a curve XL → Spec(L) such that XM =\nXL ⊗L M . Then we have the following result:\nLemma 8.2. Suppose that there exists a tamely ramified extension M ′ of M of ramifica-\ntion index e such that XM (M\n′) is nonempty. Then there exists a tamely ramified extension\nL′ of L of ramification index e such that XL(L′) is nonempty.\nProof. Since tamely ramified extensions of M descend to tamely ramified extensions\nof L, it is easy to see that without loss of generality, we may assume that e = 1, and\nM ′ = M . Thus, we must show that XL admits a rational point over some unramified\nextension of L. Note that since OL is a discrete valuation ring, it is well-known that XL\nadmits a regular model XL → Spec(OL). Let XM def= XL ⊗OL OM . Let us consider the\nmorphism φ : Spec(OM ) → Spec(OL). Since finitely generated algebras over a finite field\nare excellent, it follows that kM is separable over kL (i.e., the morphism Spec(kM ) →\nSpec(kL) is geometrically regular). In particular, it follows that φ is geometrically regular.\nSince the natural morphism XM → XL is obtained from φ by base-change, it thus follows\nthat it, too, is geometrically regular. Thus, the regularity of XL implies that of XM . Now\nthe fact that XM (M) is nonempty (by assumption) implies that there exist points in the\nspecial fiber of XM at which XM is smooth over OM . But this implies (by descent) that\nthere exist points in the special fiber of XL at which XL is smooth over OL, which, in turn,\nimplies that for some unramified extension L′ of L, XL(L′) is nonempty, as desired. ©\nSection 9: Convergence via p-adic Hodge Theory\nWhereas Sections 1 through 8 formed a unit devoted essentially to showing Propo-\nsition 7.4, Sections 9 and 10 form a new unit, devoted to showing, by means of p-adic\nHodge theory, that certain types of sequences of points converge p-adically to a uniquely\ndetermined point defined over a relatively small field. Thus, in particular, we shall start\nwith fresh hypotheses, as follows:\n46\nLet K be a p-adic local field, with residue field k. Let L be a p-adic field containing\nK whose residue field kL is a function field in one variable over k. Let us also assume that\nk is algebraically closed in kL, and that mK · OL = mL. Let us denote by\nΓL\/K\ndef\n= Ker(ΓL → ΓK) = ΓL·K\nthe “geometric” fundamental group of L. Let HΩL be the quotient of H1(ΓL\/K , ÔL(1)) by its\ntorsion submodule. Let XK → Spec(K) be a proper hyperbolic curve. Let XL def= XK⊗KL.\nThus, as usual, we have a group extension (obtained by pulling back 1 → ΔX → ΠXK →\nΓK → 1 via ΓL → ΓK)\n1 → ΔX → ΠXL → ΓL → 1\nLet us assume that we are given a continuous, group homomorphism\nα : ΓL → ΠXK\nwhose composite with the projection to ΓK is the natural morphism ΓL → ΓK . As usual, α\ndefines a section αs : ΓL → ΠXL of ΠXL → ΓL. Let us also assume that α is nondegenerate\nin the sense that it satisfies the following group-theoretic condition:\n(∗)non The natural morphism H1(ΔX ,Zp(1)) → HΩL induced by α is\nnonzero.\nLet Ltm be a maximal tamely ramified extension of L. Let M\ndef\n= (Ltm)∧ be its p-adic\ncompletion. Since Ltm · K is unramified over L · K, and Galois cohomology “ignores\nunramified extensions,” we have that\nH1(ΓL\/K , ÔL(1)) = H1(ΓL·K , ÔL(1)) = H1(ΓLtm·K , ÔL(1))\nIn particular, since HΩL is a quotient of this cohomology module, it follows from assumption\n(∗)non that α|Γ\nLtm·K\n: Γ\nLtm·K → ΔX is nontrivial.\nFor n ≥ 0, let InL\ndef\n= Im(αs) · (ΔX)<n> ⊆ ΠXL (cf. the discussion preceding Definition\n0.2 for an explanation of the notation “< n >”). Let ψn : X\nn\nL → XL be the corresponding\nfinite étale covering. Let X∞L → XL be the inverse limit of the XnL . In this Section, we\nwould like to prove the following assertion:\n(∗)con Suppose that we have a sequence of points {x̃n}, where x̃n ∈\nXnL(L\ntm). Let xn\ndef\n= ψn(x̃n) ∈ XL(Ltm). Then there exists a subse-\nquence of {xn} which converges p-adically in XL(M).\n47\n(Note that by choosing any proper model over OL of XL, we obtain a p-adic topology on\nXL(L\ntm) which is easily seen to be independent of the model chosen.)\nBy replacing K be a finite extension of K, we may make the following simplifying\nassumptions:\n(1) XK has stable reduction. We denote by X → Spec(OK) the unique\nstable extension of XK over OK . We denote by J → Spec(OK) the\nunique semi-abelian scheme over OK whose generic fiber is the Jacobian\nof XK .\n(2) XK(K) is nonempty.\nNote that replacing K by a finite extension of K does not affect the validity of the con-\nvergence assertion (∗)con.\nLet us fix an n ≥ 1. Let us consider the morphism xn : Spec(Ltm) → XK . This\nmorphism induces a morphism\nβn : ΓLtm → ΠXK → ΠXK \/(ΔX)\n<n>\nwhich is well-defined up to conjugation by an element of ΔX . Then it follows immediately\nfrom the definitions (and the fact that xn arose from an L\ntm-rational point of XnL) that:\nThis morphism βn is equal to the composite of α|ΓLtm : ΓLtm → ΠXK\nwith the natural projection ΠXK → ΠXK \/(ΔX)<n>.\nThus, in particular, βn is independent of the choice of x̃n. Moreover, by (∗)non, it follows\nthat by taking n to be sufficiently large, we may assume that the restriction of βn to\nΓ\nLtm·K (⊆ ΓLtm) is nontrivial. This implies, in particular, that xn does not factor through\nany finite extension of K.\nNote that by properness, xn extends to a morphism ξn : Spec(OLtm) → X . Let Ktm\nbe the algebraic closure of K in Ltm. Thus, Ktm is a maximal tamely ramified extension\nof K. Let us denote by ΩOLtm\/OKtm the OLtm-module of p-adically continuous differentials\nof OLtm over OKtm . Thus, ΩOLtm\/OKtm is a free OLtm-module of rank one.\nLemma 9.1. By differentiating ξn, we obtain a natural morphism dξn : ξ\n∗\nnωX\/OK →\nΩOLtm\/OKtm .\nProof. Suppose that xn is defined over some finite tamely ramified extension L\n′ of L.\nSuppose, moreover, that L′ contains a finite tamely ramified extension K ′ of K such\nthat mK′ · OL′ = mL′ . Then, if we equip Spec(OK), Spec(OK′ ) and Spec(OL′) with\nthe log structures defined by the special points, and X with the log structure whose\nmonoid is the sheaf of functions invertible on the generic fiber, we obtain a log morphism\n48\nSpec(OL′)log → X log compatible with ξn and Spec(OK′ )log → Spec(OK)log. Moreover,\nsince ωX\/OK = ΩX log\/Spec(OK)log , and ΩOLtm\/OKtm = OLtm ⊗OL′ ΩSpec(OL′)log\/Spec(OK′)log ,\nwe thus obtain dξn by means of the functoriality of logarithmic differentials. The naturality\nof dξn is easily checked by looking at the generic fibers. ©\nLet ΓΩX\ndef\n= H0(X , ωX\/OK ). Then we propose to prove in the present situation that\n(∗)ind There exists a number n0 depending only on K such that for all\nn ≥ n0, the natural morphism ΓΩX → ΩOLtm\/OKtm ⊗Z\/p\nn−n0Z induced\nby dξn is independent of xn (i.e., depends only on α).\nTo keep the notation simple, let us note that by replacing K by a tamely ramified extension\nof K, and then replacing L by an unramified extension of L, we may assume that xn is, in\nfact, defined over L. (Note that replacing L and K by extensions in this fashion will not\naffect the validity of (∗)ind.) Thus, we shall regard xn and ξn as morphisms Spec(L) → XK\nand Spec(OL) → X , respectively.\nLet P\ndef\n= P(ΓΩX ). (That is, P is the projective space over OK defined by ΓΩX .)\nThen there is (by the definition of ΓΩX) a natural finite morphism λK : XK → PK . Let\nN → Spec(OK) be the Néron model of JX . Thus, J ⊆ N is an open subscheme of N .\nSince XK(K) is nonempty, we can use a K-rational point of XK to identify JX with J\n(1)\nX .\nThus, we obtain a morphism XK ↪→ NK = JX . By the defining property of the Néron\nmodel, and the fact that OL is geometrically regular over OK , it follows that composing\nthis morphism with xn gives rise to a morphism ζL : Spec(OL) → N . Moreover, (after\npossibly replacing K by a finite unramified extension of K), we may assume that there\nexists some ζK : Spec(OK) → N such that ζK and ζL map the special points of Spec(OK)\nand Spec(OL), respectively, to the same geometric connected component of the special fiber\nof N . Thus, by translating by ζK , we may assume that our identification of JX with J (1)X is\nsuch that ζL maps into J ⊆ N . We denote the resulting morphism by jn : Spec(OL) → J .\nNow let us consider the ΓK-module H\n1(X\nK\n,Zp(1)) = H\n1(J\nK\n,Zp(1)). Here, the iden-\ntification of H1(X\nK\n,Zp(1)) with H\n1(J\nK\n,Zp(1)) is the identification induced by XK ↪→\nJ\n(1)\nX\n∼= JX . This identification is the same as the standard identification since translation\nby K-valued points of J induces the identity on H1(J\nK\n,Zp(1)). Let H\ncb ⊆ H1(J\nK\n,Zp(1))\nbe the maximal Zp-submodule such that ΓK acts on H\ncb(−1) through a finite, unram-\nified quotient. (Here, “cb” stands for “combinatorial.” Note that this inclusion Hcb ⊆\nH1(J\nK\n,Zp(1)) is Cartier-dual to the quotient HX → HX\/HFX (in the notation of the dis-\ncussion following Definition 6.3 in Section 6).) By the theory of [FC], Chapter III, we know\nthat CX def= H1(JK ,Zp(1))\/Hcb may be (ΓK -equivariantly) identified with H1(GK ,Zp(1))\nfor some semi-abelian scheme G → Spec(OK) which can be represented as an extension\n0 → T → G → A → 0\n49\nof an abelian scheme A → Spec(OK) by a torus T → Spec(OK). Moreover, the p-adic\ncompletion Ĝ of G is equal to the p-adic completion Ĵ of J . Indeed, relative to the\nidentification Ĝ = Ĵ , the identification CX = H1(GK′ ,Zp(1)) is obtained as follows: Given\na finite étale covering Q\nK\n→ J\nK\n, we let Q be the normalization of J in Q\nK\n, so Q is\nfinite over J . Thus, by p-adically completing, we obtain a formal morphism Q̂ → Ĵ = Ĝ\n(whose relative differentials are annihilated by a power of p) which can be algebrized to\nsome finite R → G (whose relative differentials are annihilated by a power of p). Thus,\nR\nK\n→ G\nK\nis finite étale. In other words, the correspondence between coverings given by\nQ\nK\n\u0002→ R\nK\ninduces a morphism H1(J\nK\n,Zp(1)) → H1(GK ,Zp(1)) whose kernel is Hcb,\nand thus allows us to identify H1(G\nK\n,Zp(1)) with CX .\nNow by p-adically completing jn : Spec(OL) → J , applying Ĝ = Ĵ , and then alge-\nbrizing, we thus obtain a morphism gn : Spec(OL) → G. Moreover, gn induces a morphism\non cohomology groups\nH1(gn) : H\n1(G\nK\n,Zp(1)) → H1(ΓL\/K ,Zp(1))\nThus, we have a commutative diagram of ΓK-modules:\nH1(J\nK\n,Zp(1))\nH1(jn)−→ H1(ΓL\/K ,Zp(1)) −→ HΩL ⊗ (Z\/pnZ)⏐⏐\u0005 ⏐⏐\u0005id ⏐⏐\u0005id\nH1(G\nK\n,Zp(1))\nH1(gn)−→ H1(ΓL\/K ,Zp(1)) −→ HΩL ⊗ (Z\/pnZ)\nwhere the vertical morphism on the left is the natural projection discussed in the preceding\nparagraph, and the horizontal morphisms on the right are the natural ones. Recall that\nHΩL is, by definition, the quotient of H1(ΓL\/K , ÔL(1)) by its torsion submodule. Now,\nby sorting through the definitions (cf. the discussion preceding Lemma 9.1), it is clear\nthat the composite morphism on the upper row is completely determined by α (i.e., is\nindependent of the particular choice of xn). Thus, it follows that the composite morphism\non the bottom row gives rise to a morphism\n\fn : H\n1(G\nK\n, Ô\nK\n(1)) → HΩL ⊗ Z\/pnZ\nwhich is completely determined by α. Moreover, since HΩL is p-adically separated (see Con-\nsequence (2) of [Falt1] below), it follows (from (∗)non) that \fn is nonzero for n sufficiently\nlarge.\nNow, note that G is a smooth OK-scheme with an obvious compactification G such\nthat G − G is a divisor with normal crossings. (Indeed, G is a product of Gm-torsors\nover the OK-proper scheme A, so we simply compactify each of these Gm-torsors to a\nP1-bundle.) In particular, it follows that we can apply the theory of [Falt1], in the case of\ngood reduction.\n50\nRemark. The argument of [Falt1] in the bad reduction case is known to have gaps, which\nis one reason why we took pains to avoid applying [Falt1] to JK . Another reason is that in\nthe bad reduction case, one only has “Qp-results,” not integral results (say, over Z\/p\nnZ),\nas we need here.\nLet n′′0 be an integer such that the p-adic valuation of the different of K over Qp is ≤\nthe p-adic valuation of pn\n′′\n0 . Let n′0\ndef\n= n′′0 + ordp(g!). Let n0\ndef\n= n′0 + 2. The consequences\nof the theory of [Falt1] that we use are as follows:\n(1) The ΓK -module H\n1(G\nK\n,K\n∧\n) is Hodge-Tate. Let us denote by HΩG the\nquotient of H1(G\nK\n, Ô\nK\n(1)) by the submodule that is contained in the\nportion of H1(G\nK\n,K\n∧\n(1)) of weight 1. Thus, HΩG ⊗Zp Qp is of weight\n0. Let ΩinvG be the OK-module of invariant differentials (over OK) on\nthe group scheme G. Then there is a natural ΓK -equivariant morphism\n(Theorems 2.4 and 3.1 of II. of [Falt1])\nΩinvG ⊗OK {(pn0 · OK)\/(pn0+n · OK)} → HΩG ⊗ Z\/pnZ\nwith an inverse “up to a factor of pn0 .” Here the point of the n0 is to\ntake care of the “g! · ρ” of [Falt1]: the 2 = 1 + 1 added to n′0 is to take\ncare of the 1\np−1 ≤ 1 that always arises, plus the additional e−1e < 1 that\narise from the tame ramification which we allow (see [Fo], Theorem 1′\nfor more details on the computation of [Falt1]’s “ρ”).\n(2) HΩL is p-adically separated. Moreover, there is a natural ΓK -equivariant\nmorphism (Theorem 4.2 of I. of [Falt1])\nΩOL\/OK ⊗OK {(pn0 · OK)\/(pn0+n · OK)} → HΩL ⊗ Z\/pnZ\nwith an inverse “up to a factor of pn0 .” In particular, by consid-\nering weights, it follows that \fn factors through the quotient HΩG of\nH1(G\nK\n, Ô\nK\n(1)).\n(3) The morphisms (and inverses up to a factor) of (1) and (2) are com-\npatible with each other, \fn, and the natural morphism Ω\ninv\nG → ΩOL\/OK\ninduced by differentiating gn. That is to say, we have a commutative\ndiagram:\nΩinvG ⊗OK {(pn0 · OK)\/(pn0+n · OK)} −→ HΩG ⊗ Z\/pnZ⏐⏐\u0005 ⏐⏐\u0005\nΩOL\/OK ⊗OK {(pn0 · OK)\/(pn0+n · OK)} −→ HΩL ⊗ Z\/pnZ\n51\n(where the horizontal morphisms are the morphisms of (1) and (2), re-\nspectively, the vertical morphism on the left is induced by differentiating\ngn, and the vertical morphism on the right is induced by \fn).\nThese assertions allow us to immediately conclude (∗)ind, as follows: Since \fn is determined\nby α, it thus follows that the morphism (induced by differentiating gn) Ω\ninv\nG → ΩOL\/OK ⊗\n(Z\/pn−n0Z) is determined by α. On the other hand, ΩinvG can naturally be identified with\nΩinvJ . Moreover, Ω\ninv\nJ can naturally be identified with Γ\nΩ\nX . This completes the proof of\n(∗)ind.\nLemma 9.2. For n sufficiently large, the morphism HΩG → HΩL ⊗ (Z\/pnZ) induced by\n\fn is nonzero.\nProof. This follows formally from (∗)non. ©\nNow we are ready to tackle the convergence assertion (∗)con. Thus, we go back to\nworking over Ltm and Ktm. In particular, by composing xn with λK and applying the\nvaluative criterion for properness, we obtain a morphism λn : Spec(OLtm) → P . Moreover,\nthis OLtm-valued point λn of the projective space P is precisely the point defined by the\nmorphism ΩinvG → ΩOL\/OK ⊗OL OLtm = ΩOLtm\/OKtm (obtained by differentiating gn).\n(Note that here we use the fact remarked above that ΩOLtm\/OKtm is a free OLtm -module\nof rank one.) Thus, (∗)ind and Lemma 9.2 imply that the points λn of P (OLtm) converge\np-adically to a point λ∞ ∈ P (ÔLtm) = P (M). On the other hand, by “Krasner’s Lemma”\n(see, e.g., [Kobl], p. 70), since λK : XK → PK is finite, it follows that some subsequence\n{xni} of {xn} converges to a point x∞ ∈ XK(M). This proves the assertion (∗)con, as\ndesired.\nIn fact, we can say more. For m ≥ 1, let x̃mn be the image of x̃n+m in XmL (Ltm) under\nthe natural morphism Xn+mL → XmL . Thus, we obtain a sequence {x̃mn } (in the index n) of\npoints of XmL (L\ntm). Since XmL → XL is finite, then by appying “Krasner’s Lemma” again,\nsee that a subsequence of the sequence {x̃mn } converges p-adically in XmL (M). Moreover,\nby the process of “Cantor diagonalization” (of elementary analysis), we thus see that we\nhave proven the following key result (which is the main result of this Section):\nLemma 9.3. We assume notation as in the first paragraph of this Section. Suppose that\nwe have a sequence of points {x̃n}, where x̃n ∈ XnL(Ltm). Then there exists a subsequence\n{x̃ni} of this sequence with the following property: For each m ≥ 0, the sequence obtained\nby projecting those x̃ni with ni ≥ m to XmL (M) converges to some x̃m∞ ∈ XmL (M). In\nparticular, X∞L (M) is nonempty.\nProof. The last assertion is proven by noting that the x̃m∞ ∈ XmL (M) form a compatible\nsystem, hence define a point of x̃∞∞ ∈ X∞L (M). ©\n52\nFinally, before proceeding, we make the following technically trivial, but important\nobservation: Since the arithmetic fundamental group of X∞L is just ΓL (i.e., Im(αs)), it\nfollows that the morphism\nΓM → ΠXK\ninduced on fundamental groups by the image in XK(M) of any point (e.g., x̃\n∞∞) of X∞L (M)\nis none other than the restriction of α : ΓL → ΠXK to ΓM .\nSection 10: Uniqueness and Rationality of the Limit Point\nWe continue with the notation of the preceding Section. In this Section, we would\nlike to show that the point x̃∞∞ ∈ X∞L (M) constructed in Lemma 9.3 is the unique element\nof X∞L (M), and that it in fact descends to a point of X\n∞\nL (L). We remark that much of\nthe material of this Section is not absolutely logically necessary for the proof of the main\nresults of this paper, but is included partly for aesthetic reasons, and partly because its\ninclusion (in the opinion of the author) makes the proof of the main results of the paper\nmore transparent.\nFirst, let us make the trivial observation that any point in X∞L (M) arises as a x̃\n∞∞ (as\nin Lemma 9.3) for some sequence of x̃n ∈ XnL(Ltm). (Indeed, this follows by using the fact\nthat M = (Ltm)∧ to approximate the images of the given point in the various XnL(M)’s.)\nThus, without loss of generality, we can apply the results of the discussion of Section 9 to\nour analysis of an arbitrary point of X∞L (M).\nNow recall the limit point λ∞ ∈ P (M) of the discussion following Lemma 9.2. This\nlimit point is clearly independent of the choice of the particular sequence {x̃n} under\nconsideration – i.e., it depends only on α. More precisely, λ∞ ∈ P (M) ⊆ P (L\n∧\n) is the\npoint defined by the surjection\nΓΩX ⊗OK L\n∧\n= ΩinvG ⊗OK L\n∧\n= HΩG ⊗Ô\nK\nL\n∧ → HΩL ⊗Zp Qp = ΩOL\/OK ⊗OL L\n∧\ninduced by α (cf. the discussion preceding Lemma 9.2). Note that ΓL acts naturally on\nall of these modules, and that this surjection is ΓL-equivariant. Indeed, this follows by\ntransport of structure from the fact that α is defined on ΓL, not just on ΓLtm·K . Thus, we\nconclude the following\nLemma 10.1. The image λ∞ ∈ P (M) of x̃∞∞ ∈ X∞L (M) under the natural morphism\nX∞L → XK → PK is independent of x̃∞∞ and, moreover, λ∞ is defined over L.\n53\nThe next step is to observe that this argument can be modified so as to show that\nthe image of x̃∞∞ in the projective space defined by the differentials on XnL (for n ≥ 1) is\nindependent of x̃∞∞ and defined over L. To see this, we need to introduce some new objects.\nFirst, let us fix n. Now observe that (after possibly enlarging K) there exists a finite étale\ncovering ZnK → Spec(K) of XK with stable reduction Zn → Spec(OK), together with a\nfinite Galois extension L̃ of L such that\nXn\nL̃\n= XnL ⊗L L̃ ∼= ZnL̃ = Z\nn\nK ⊗K L̃\nNow let m ≥ n, and consider x̃m ∈ XmL (Ltm) ⊆ XmL (L̃tm), where L̃tm is the composite\nof Ltm and L̃ (as extensions of L). (Note that L̃tm is then a maximal tamely ramified\nextension of L̃.) Thus, by projecting x̃m to a point of Z\nn\nK , we obtain a morphism\nzm : Spec(OL̃tm) → Z\nn\nLet us assume that we have chosen K large enough so that ZnK(K) is nonempty, so that\nwe can identify JZn\nK\n(i.e., the Jacobian of ZnK) with J\n(1)\nZn\nK\n. Let NZn\nK\n→ Spec(OK) be the\nNéron model of ZnK over OK . More generally, in the following discussion we shall denote\nby N(−) the Néron model of any proper hyperbolic curve “(−)” over a discretely valued\nfield. In fact, we shall even use this notation for proper hyperbolic curves over inductive\nlimits of discretely valued fields in which case this notation is to be taken to mean the\ncorresponding inductive limit of the Néron models. Also, we shall denote by\nComp(N(−))\nthe (inductive limit of) finite abelian (étale) group scheme(s) of connected components of\nthe special fiber of the Néron model N(−).\nThe main technical difficulty that we must overcome in order to apply the argument\nof Section 9 (for XK) to the curves Z\nn\nK is the following: The points that we are interested\nin are the points zm of Z\nn\nK(L̃\ntm) = Zn(O\nL̃tm\n). These points define points of J\n(1)\nZn\nK\n∼= JZn\nK\n,\nhence points\nzNm : Spec(OL̃tm) → NZn\nL̃tm\nThen, in order to apply the argument of Section 9, we must show that:\n(After replacing K by a finite extension of K which is independent of\nm), zNm maps the special point of Spec(OL̃tm) to the same component\nof Comp(NZn\nL̃tm\n) as some Ktm-valued point of NZn\nL̃tm\n.\n54\n(see Lemma 10.2 below for an alternate formulation.) To show this, we reason as follows:\nLet us write L̃unr for the maximal unramified extension of L̃ in L̃tm. Thus, L̃tm is a totally\ntamely ramified extension of L̃unr. In particular, L̃tm is a union of finite Galois extensions\nof L̃unr which are of degree (over L̃unr) prime to p. In particular, it follows that the cokernel\nof the inclusion\nComp(NZn\nL̃unr\n) ↪→ Comp(NZn\nL̃tm\n)\nis annihilated by integers prime to p. (Indeed, this follows by applying the “trace map”\n(for subextensions of L̃tm over L̃unr) to the second “Comp,” and observing that (since\nGal(L̃tm\/L̃unr) clearly acts trivially on Comp(NZn\nL̃tm\n)) this trace map is just multiplication\nby some integer prime to p.) On the other hand, since L̃unr is an unramified extension of\nL̃, it follows (essentially from the definition of the Néron model) that NZn\nL̃unr\n= NZn\nL̃\n. In\nparticular, since L̃ is a discretely valued field, it follows that Comp(NZn\nL̃\n) is finite. In other\nwords, we conclude that: the p-torsion of Comp(NZn\nL̃tm\n) is annihilated by some (finite)\npower of p. We are now ready to prove the following Lemma:\nLemma 10.2. After possibly enlarging K by a larger field that is independent of m, we\nmay arrange that the morphism zJm : Spec(L̃\ntm) → J (1)Zn\nK\n∼= JZn\nK\ninduced by zm extends to a\nmorphism Spec(O\nL̃tm\n) → NZn\nKtm\n, for all m ≥ n.\nProof. Indeed, without enlarging K, it follows from our observation above concerning the\np-torsion of Comp(NZn\nL̃tm\n) that a · zJm extends to (zN )′m : Spec(OL̃tm) → NZnK for some\npositive integer a whose order at the prime p is independent of m. Now let us assume\n(by enlarging K – independently of m) that all of the p-torsion points of JZn\nK\nannihilated\nby a are rational over K. Since all prime-to-p torsion points of JZn\nK\nare rational over\nKtm, it thus follows that the morphism “multiplication by a” on NZn\nKtm\nis finite over a\nneighborhood of the image of (zN )′m⊗OK OKtm in NZnK ⊗OK OKtm ⊆ N\n′\nZn\nKtm\n. Since (zN )′m\nwas constructed by multiplying zJm by a, it thus follows that z\nJ\nm extends to a morphism\nSpec(O\nL̃tm\n) → NZn\nKtm\n, as desired. ©\nNow that we have Lemma 10.2, we can use the extended morphism of Lemma 10.2\nto construct the analogue of the morphism “gn” of Section 9 (over some tamely ramified\nextension of K). Then the rest of the argument of Section 9 goes through without difficulty.\nMore precisely:\nLet ΓΩZn\ndef\n= H0(Zn, ωZn\/OK ). Let Pn\ndef\n= P(ΓΩZn ). (That is, P\nn is the projective space\nover OK defined by ΓΩZn .) Then there is (by the definition of ΓΩZn ) a natural finite morphism\nλnK : Z\nn\nK → PnK . Let λnm : Spec(OL̃tm ) → P\nn be the morphism obtained by composing zm\nwith λnK . Then, just as in Section 9, one sees via p-adic Hodge theory that as m → ∞ (and\n55\nn remains fixed), the λnm converge to a point in λ\nn∞ ∈ Pn(ÔL̃tm) = P\nn((L̃tm)∧). Moreover,\nλn∞ is independent of the original sequence {x̃m}. Finally, just as we saw for Lemma 10.1,\nby transport of structure and the fact that α (and, here, XnL) are defined over L, it follows\nthat if we regard λn∞ as a point of PX\nn\n((L̃tm)∧), where PX\nn\nis the projective space over\nL associated to\nH0(XnL, ωXnL\/L)\n(for the original L), then λn∞ ∈ PX\nn\n(L). Thus, we see that we obtain the following\nanalogue of Lemma 10.1:\nLemma 10.3. The image λn∞ ∈ PX\nn\n(M) of x̃∞∞ ∈ X∞L (M) under the natural morphism\nX∞L → XnL → PX\nn\nis independent of x̃∞∞ and, moreover, λn∞ is defined over L.\nNow let us recall some basic facts on hyperelliptic curves:\nLemma 10.4. Let Q be a proper hyperbolic curve over an algebraically closed field Ω of\ncharacteristic zero. Then:\n(1) If Q is hyperelliptic (i.e., admits a “g12,” or linear system of dimension\n1 and degree 2), then the g12 is unique.\n(2) If Q is non-hyperelliptic, then the canonical morphism from Q into the\nprojective space associated to H0(Q,ωQ\/Ω) is an embedding.\n(3) Suppose that W → Q is a finite étale covering, where W is connected.\nThen if Q is non-hyperelliptic, so is W .\n(4) Suppose that W → Q (where W is connected) is a cyclic étale covering\nof degree m > 2. Then W is non-hyperelliptic.\nThus, in particular, none of the XnL is hyperelliptic, for n ≥ 2.\nProof. For proofs of assertions (1) and (2), we refer to [Harts], Chapter IV, §5, Propo-\nsitions 5.2 and 5.3. As for (3), the push-forward of a g12 on W via W → Q is a g12 on Q,\nso (3) follows immediately. Now let us consider assertion (4). Let σ be a generator of the\nGalois group of W over Q. If W admits a g12, it is unique, hence stabilized by σ. But\nthis means that there exists some rational function fW on W in this g\n1\n2 which satisfies\nσ−1(fW ) = T (fW ), where T is some linear fractional transformation with coefficients in\nΩ. Note that since σ has finite order, so does T . Thus, if we diagonalize T (by choosing\na different fW ), we may assume that σ\n−1(fW ) = λ · fW , for some nonzero λ ∈ Ω. (Note\nthat T cannot be parabolic (i.e., a transformation of the form fW \u0002→ fW + λ, for some\nnonzero λ ∈ Ω) since T is of finite order and Ω is of characteristic zero.) Thus, fmW (which\n56\nis a constant multiple of the norm (relative to W → Q) of fW ) defines a rational function\nfQ on Q. Now fQ is contained in some g\n1\n2 on Q, so its zeroes have order equal to 1 or 2.\nOn the other hand, fQ has an m\nth root in the function field of W , so it follows from the\nassumption that m > 2 that the covering W → Q must be ramified at the zeroes of fQ,\nwhich is absurd. This contradiction completes the proof. ©\nWe summarize our efforts in Sections 9 and 10 as follows:\nCorollary 10.5. Let M be the p-adic completion of a maximal tamely ramified extension\nLtm of L. Suppose that we are given a nondegenerate α : ΓL → ΠXK , which thus gives\nrise to XnL, X\n∞\nL . Suppose that for each n ≥ 0, XnL(Ltm) is nonempty. Then it follows that\nthe set X∞L (M) consists of precisely one point, which is, in fact, contained in X\n∞\nL (L) ⊆\nX∞L (M).\nIn particular, there exists a unique L-valued point of Spec(L) → XK whose induced\nmorphism on fundamental groups (for an appropriate choice of base-points) is the mor-\nphism α : ΓL → ΠXK .\nProof. Indeed, if x̃∞∞ ∈ X∞L (M), then by Lemma 10.4, if n ≥ 2, the image of x̃∞∞ in\nXnL(M) is determined by the image λ\nn∞ of x̃∞∞ in PX\nn\n(M). But, by Lemma 10.3, λn∞ is\nindependent of x̃∞∞ and defined over L. Thus, it follows that the image of x̃∞∞ in XnL(M)\n(for n ≥ 2) is independent of x̃∞∞ and defined over L. But, by the definition of X∞L , this\nmeans that x̃∞∞ itself is “independent of x̃∞∞” and defined over L. (In other words, X∞L (M)\nconsists of precisely one point, which is, in fact, defined over L.) The last sentence is a\nformal consequence of the rest of the Corollary. This completes the proof. ©\nSection 11: Hodge-Tate Representations of Infinite Rank\nLet K be a finite extension of Qp. Let XK → Spec(K) be a proper hyperbolic curve\nover K. Let UK be its generic point. Let HU be the abelianization of ΔU . Thus, HU is\na Zp-flat topological ΓK -module of infinite rank. Note that we have a natural surjection\nHU → HX → HIX (cf. the discussion following Definition 6.3 for the definition of HIX).\nLet HMU ⊆ HU be the kernel of this surjection. Note that the inertia groups (defined by\nclosed points of XK) of ΔU all map into H\nM\nU . Let H\nP\nU ⊆ HMU denote the closure of the\nimage of all these inertia groups. (Here the “P” stands for “points.”) Thus, HU\/H\nP\nU may\nbe identified with HX , hence is of finite rank over Zp. In this Section, we would like to\nanalyze HU , and, in particular, H\nP\nU in greater detail.\nFor simplicity, let us assume (by replacing K by a finite extension of K) that XK\nhas stable reduction over OK, and that there exists a K-valued point x ∈ XK(K). Let\n℘ ∈ XK be a closed point of XK . Then there are finitely many points y1, . . . , yr ∈\n57\nXK(K) = Hom(Spec(K),XK) that map to ℘. For i = 1, . . . , r, let Iyi ⊆ ΔU denote “the”\ninertia group (well-defined up to conjugation by an element of ΔU ) associated to yi. Thus,\nprojection to the quotient ΔU → HU yields a map\nI℘\ndef\n=\nr⊕\ni=1\nIyi → HPU\nNote, moreover, that I℘ has a natural structure of ΓK-module (given, for instance, by\nconjugating by the image of a section ΓK → ΠUK (cf. Definition 5.1) induced by x ∈\nXK(K) – note, however, that the ΓK -action is independent of the choice of x). Relative\nto this ΓK -action on I℘, the above morphism is ΓK-equivariant. Moreover, by letting ℘\nrange over all closed points of XK other than that defined by x, we obtain a continuous\nΓK -equivariant morphism\nΞ : Mx\ndef\n=\n∏\n℘\b=x\nI℘ → HPU\nIn fact,\nLemma 11.1. The morphism Ξ is an isomorphism.\nProof. Note that ΠUK is equal to the inverse limit of ΠVK , where the limit is taken over\nall open subsets VK ⊆ XK −{x}. For such a VK , we can define HPV just as we defined HPU ,\nand it is well-known that HPV is the direct sum of the I℘, where the sum is taken over all\n℘ ∈ XK −{x}−VK – indeed, this follows from the well-known (from elementary algebraic\ntopology) structure of the homology group of a Riemann surface obtained by removing a\nfinite number of points from a compact Riemann surface. Passing to the limit proves the\nLemma. ©\nNext, we would like to define a quotient I℘ → IT℘ as follows: Note that if K℘ is the\nresidue field of XK at ℘, then the ΓK-module I℘(−1) (where the “(−1)” is a Tate twist)\nmay be naturally identified with Zp[HomK(K℘,K)]. (Here, Zp[a set] denotes the free Zp-\nmodule generated by the elements of the set.) Thus, there exists a unique nonzero quotient\nI℘(−1) → Q℘ stabilized by ΓK such that ΓK acts trivially on Q℘. Moreover, Q℘ is a free\nZp-module of rank one. Let I℘ → IT℘ be the quotient obtained by tensoring I℘(−1) → Q℘\nwith Zp(1). Moreover, by taking the product of these quotients, we obtain a quotient\nMx → MTx\ndef\n=\n∏\n℘\b=x\nIT℘\nLet HPU → HTU be the quotient corresponding to this quotient under the isomorphism\nΞ. Thus, the ΓK -action on H\nT\nU (−1) is trivial. Moreover, HPU → HTU has the following\nuniversal property:\n58\nLemma 11.2. Let N be a Zp-flat topological ΓK -module such that ΓK acts trivially on\nN(−1). Then any continuous ΓK -morphism HPU → N factors through HTU .\nProof. This follows immediately from the construction of HTU . ©\nNow let us define the subquotient HTX of HU as follows: First, let H\nF\nU ⊆ HU denote the\ninverse image of HFX ⊆ HX under the projection HU → HX (cf. the discussion following\nDefinition 6.3 for the definition of HFX ). Now let H\nT\nX be the quotient of H\nF\nU by the kernel\nof HPU → HTU . Thus, we have an exact sequence of topological ΓK-modules\n0 → HTU → HTX → HFX → 0\nLet us take the continuous dual HomcontZp (−,Zp) of this exact sequence. This gives us a\nnew exact sequence of topological ΓK -modules\n0 → CFX → CTX → CTU → 0\n(Here, one may think of the “H’s” as standing for “homology,” and the “C ’s” as standing\nfor “cohomology.”) In particular, we have\nCTU =\n(⊕\n℘ \b=x\nZT℘\n)∧\nwhere ZT℘ is defined to be the dual of I\nT\n℘ . If we pull-back this last exact sequence by\nZT℘ ⊆ CTU , we obtain exact sequences of topological ΓK -modules\n0 → CFX → CT℘ → ZT℘ → 0\nall of which are of finite rank over Zp. Moreover, it is clear that that C\nT\nX is obtained by\nsumming these extensions (over ℘) and then p-adically completing.\nWe would like to show that CTX is “Hodge-Tate”. Unfortunately, typically the theory\nof Hodge-Tate Galois representations only goes through for modules of finite rank over Zp.\nOn the other hand, CT℘ is of finite rank over Zp. Thus, our approach in the following will\nbe to use the CT℘ to show that C\nT\nX is, in some sense, like a Hodge-Tate representation.\nFirst, let X℘ be the singular curve obtained from XK by considering the subsheaf of\nOXK of functions f such that f(x) = f(℘). Let J ′℘ be the generalized Jacobian associated\nto X℘ (see, e.g., [Ser1], Chapitre I, §1). Thus, J ′℘ is an extension of the abelian variety JX\nby some torus. Moreover, the p-adic Tate module T (J ′℘) of J\n′\n℘ fits into an exact sequence\nof ΓK-modules\n59\n0 → I℘ → T (J ′℘) → HX → 0\nIn particular, one sees immediately that there exists a quotient J ′℘ → J℘ whose p-adic Tate\nmodule fits into the exact sequence\n0 → IT℘ → Tp(J℘) → HX → 0\nobtained by pushing forward the preceding exact sequence by I℘ → IT℘ . Moreover, CT℘\nmay be identified with the dual of the pull-back of this exact sequence by HFX ↪→ HX .\nAlternatively, CT℘ may be identified with a certain quotient Tp(J℘)\n∨ = H1(J℘,Zp) → CT℘\n(whose kernel is the dual of HX\/H\nF\nX). Note, in particular, that J℘ is an extension of JX\nby the trivial torus Gm. Thus, J℘ extends to a semi-abelian variety over OK (cf. the\nparagraph following Theorem 2.6 of Chapter I of [FC]).\nNext, let us recall (from the theory of [FC], Chapters I through III) that JX may\nbe constructed as a certain rigid analytic quotient of a semi-abelian variety J̃X which has\ngood reduction (i.e., the dimension of the torus part is constant) over OK by some discrete\ngroup. The kernel of the induced pull-back map on cohomology H1(JX ,Zp) → H1(J̃X ,Zp)\nis precisely the dual of HX\/H\nF\nX . If we pull-back the extension J℘ → JX from JX to J̃X ,\nwe then obtain an extension J̃℘ of J̃X by the trivial torus of dimension one. Moreover, J̃℘\nhas good reduction over OK . Finally, the quotient H1(J℘,Zp) → H1(J̃℘,Zp) is precisely\nthe quotient H1(J℘,Zp) → CT℘ referred to in the preceding paragraph.\nNow, just as in Section 9, we can apply the theory of [Falt1] (in the case of good\nreduction) to J̃℘. Let F0 def= H1(X,OX), where X → Spec(OK) is the stable extension of\nXK over OK . Let F℘ be the space of invariant differentials on J̃℘, where J̃℘ → Spec(OK)\nis the extension of J̃℘ to a semi-abelian variety over OK . Thus, F0 and F℘ are both free\nof finite rank over OK . Then it follows from the theory of [Falt1] (specifically, Theorems\n2.4 and 3.1 of II. of [Falt1]) that\n(∗)℘ There exists a natural ΓK-equivariant, ÔK-linear morphism\nCT℘ ⊗Zp ÔK → (F0 ⊗OK ÔK) ⊕ (F℘ ⊗OK ÔK(−1))\nwhich has an inverse up to pn, where n is a nonnegative integer that is\nindependent of ℘.\nNote that the fact that n is independent of ℘ is of profound importance in what follows.\nMoreover, if, instead of J̃℘, we had worked with J℘ (which, in general, has “bad (though\nstable) reduction”), or J ′℘ (which has only potentially stable reduction – i.e., for general\n℘, one needs to enlarge K in order to get stable reduction), we would have been unable to\nobtain this crucial “independence of ℘” from the theory of [Falt1].\n60\nNow observe that F℘ fits into a natural exact sequence\n0 → FFX → F℘ → Z℘ → 0\nwhere FFX\ndef\n= H0(X,ωX\/OK ), and Z℘ is defined so as to make the above sequence exact.\nNow let FTX be the sum of these extensions over all ℘; let FTU be the direct sum of the Z℘\nover all ℘. Thus, we have an exact sequence of flat OK-modules\n0 → FFX → FTX → FTU → 0\nThus, by taking the p-adic completion of the sum of the morphisms in (∗)℘, and using the\nfacts that:\n(i) the “n” of (∗)℘ is independent of ℘;\n(ii) both CTX and FTX are sums of extensions which are of finite rank;\nwe obtain the following result:\nProposition 11.3. We have a natural, Ô\nK\n-linear morphism of exact sequences of\ntopological ΓK-modules\n0 −→ CFX ⊗̂ZpÔK −→ CTX ⊗̂ZpÔK⏐⏐\u0005 ⏐⏐\u0005\n0 −→ (FFX ⊗̂OK ÔK(−1)) ⊕ (F0 ⊗̂OKÔK) −→ (FTX ⊗̂OKÔK(−1)) ⊕ (F0 ⊗̂OKÔK)\n−→ CTU ⊗̂ZpÔK −→ 0⏐⏐\u0005\n−→ (FTU ⊗̂OKÔK(−1)) −→ 0\nwhich become isomorphisms when tensored with Qp.\nNext, we would like to relate FTX to a certain space of differentials on X, as follows:\nLet E be a finite set of closed points of XK . Let S\ndef\n= Spec(O\nK\n), and let us endow S\nwith the log structure defined by the monoid OS − {0}; denote the resulting log scheme\nby S log. Let X log[E] → S log be the pointed stable log-curve (see, e.g., [Mzk3], Section 3,\nfor more details on this terminology) extending XK ⊗K K, equipped with all the points\n61\nof XK(K) that map to primes in E as marked points. Let L[E] def= ωX log[E]\/Slog . Then if\nE ⊆ E′, we get a natural morphism X log[E′] → X log[E], which lies under a morphism of\nsheaves L[E] → L[E′]. Thus, by taking the projective limit of the (X log[E],L[E]) as E\nranges over all finite sets of closed points of XK , we obtain (X\nlog[∞],L[∞]). Moreover,\nnote that H0(X[∞],L[∞]) is a flat, p-adically separated O\nK\n-module.\nIf E contains x and ℘, then it is easy to see that we get a natural inclusion F℘ ↪→\nH0(X[E],L[E]), which extends to a morphism F℘ ⊗OK OK ↪→ H0(X[E],L[E]) whose\ncokernel is torsion free. Taking the limit with respect to E, we thus obtain a map F℘ ⊗OK\nO\nK\n↪→ H0(X[∞],L[∞]). In fact, it is not difficult to see that, by summing over the F℘\nfor all ℘ \f= x, we get a morphism\nFTX ⊗OK OK ↪→ F∞\ndef\n= H0(X[∞],L[∞])\nwhose cokernel is torsion free. Moreover, this morphism remains injective after p-adic\ncompletion. Thus, we see from Proposition 11.3 that we get a morphism\nΦ : CTX ⊗̂ZpÔK → F∧∞(−1)\n(where the “∧” denotes p-adic completion). Moreover, the kernel of Φ is of weight zero.\nFinally, if N is a positive integer, let G∞[N ] def= H0(X[∞],L[∞]⊗N ). Thus, G∞[N ] is a free\nO\nK\n-module of infinite rank (such that G∞[1] = F∞), and we have a natural multiplication\nmorphism\n⊗NO\nK\nF∞ → G∞[N ]\nSection 12: The Preservation of Relations\nThe purpose of this Section is to show how the material of Section 11 will be applied\nin Section 13 to show the “preservation of relations.” We maintain the notation of Section\n11. Moreover, let us assume that we have been given a continuous open homomorphism\nover ΓK\nθ : ΠUK → ΠYK\n(where YK → Spec(K) is a proper hyperbolic curve over K). For simplicity, let us assume\n(by enlarging K) that YK extends to a stable curve Y → Spec(OK) over OK , and that\nthe ΓK -action on H\nM\nY (−1) (cf. the discussion following Definition 6.3 for an explanation\nof the notation “HM”) is trivial. Next, observe that θ induces a continuous morphism\nof ΓK -modules HU → HY whose cokernel is torsion. Note that this morphism induces\n62\na map HFU → HFY (cf. the discussions following Definition 6.3 and Lemma 11.2 for an\nexplanation of the notation “HF ”) whose cokernel is torsion. Moreover, the restriction of\nthis morphism to HPU (cf. the discussion at the beginning of Section 11 for an explanation\nof the notation “HPU ”) maps into H\nM\nY , hence (by Lemma 11.2 and the fact that ΓK acts\ntrivially on HMY (−1)), we get a morphism HTX → HFY whose cokernel is torsion. Taking\nthe dual of this morphism yields an injection CFY ↪→ CTX (where CFY is the dual of HFY ).\nTopologically tensoring with O\nK\n, dividing out by the “weight zero part,” and applying the\nmorphism Φ considered at the end of Section 11, we thus obtain (after further tensoring\nwith Zp(1)) an injection\nH0(Y, ωY\/OK ) ⊗OK ÔK ↪→ F∧∞\nIn this Section, we shall be concerned with the issue of whether or not this morphism\n“preserves relations.” By this, we mean the following: Let N be a positive integer. By\ntaking the N th tensor power of this morphism, and then composing with the completion\nof the multiplication morphism ⊗N F∞ → G∞[N ] considered at the end of Section 11, we\nobtain a morphism\nΨN : ⊗N (H0(Y, ωY\/OK ) ⊗OK ÔK) → (G∞[N ])∧\nWe would like to know whether or not ΨN annihilates the elements of\nRN def= Ker(⊗N (H0(Y, ωY\/OK )) → H0(Y, ω⊗NY\/OK ))\ni.e., the “relations.” In the following, we would like to state a certain assertion that will\nbe proven in the next Section, and explain why this assertion implies that ΨN “preserves\nrelations.”\nFirst, let us introduce the notation necessary to state the assertion. Suppose that E\nis a finite set of points of XK(K). Then the pointed stable curve X\nlog[E] → Spec(O\nK\n)\nintroduced at the end of Section 11 is defined over K, i.e., we have X log[E]OK → Spec(OK).\nLet p ∈ X[E]OK be a generic point of the special fiber of X[E]OK . Let Spf(OL) be the\ncompletion of the localization of X[E]OK at p. Thus, L (the quotient field of OL) is a p-adic\nfield whose residue field is k(p) (i.e., the residue field of X[E]OK at the prime p), and we\nhave a canonical L-valued point ξL ∈ XK(L). Let ΩL\/K denote the module of p-adically\ncontinuous differentials of L over K. Thus, ΩL\/K is an L-vector space of dimension one.\nMoreover, note that we have a natural restriction morphism (induced by ξL)\nF∧∞ → ΩL\/K⊗̂KK̂\nThus, if we compose this morphism with the morphism\nH0(Y, ωY\/OK ) ⊗OK ÔK ↪→ F\n∧\n∞\n63\nconsidered above, we get a morphism\nκL : H\n0(Y, ωY\/OK ) ⊗OK ÔK → ΩL\/K⊗̂KK̂\nNext, let us observe that the coverings of XK defined by subgroups of ΠUK are étale\nat the point ξL. Thus, it follows that ξL induces a natural morphism αLU : ΓL → ΠUK\n(well-defined up to conjugation by an element of ΔU ) whose composite with ΠUK → ΓK\nis the natural morphism ΓL → ΓK . Let αLY : ΓL → ΠYK be the composite of αLU with θ.\nThen the assertion that will be proven in Section 13 is the following:\n(∗)L−pt Suppose that p satisfies the condition that κL is not identically\nzero. Then it follows that there exists a point (ξ′)L ∈ YK(L) such that\nαLY arises from (ξ\n′)L.\nNote in particular, that if αLY arises from (ξ\n′)L, then it follows immediately that the group-\ntheoretically constructed morphism κL coincides (cf. the application of the theory of [Falt1]\ndiscussed in Section 9) with the natural restriction map on differentials induced by (ξ′)L.\nThus, if ρ ∈ RN is a relation, then it follows that the restriction of ΨN(ρ) ∈ (G∞[N ])∧ to\nΩ⊗N\nL\/K\n⊗̂KK̂ is zero. This argument already leads one to believe that there should be some\nsort of connection between (∗)L−pt and the “preservation of relations.” In fact, we have\nthe following:\nProposition 12.1. Suppose that (∗)L−pt always holds (i.e., for all data of the sort\ndiscussed above). Then ΨN(RN ) = 0, for all positive integers N .\nProof. Suppose that ρ ∈ RN is such that ψ′ def= ΨN (ρ) \f= 0. By dividing ψ′ by a suitable\nelement of O\nK\n, we obtain an element ψ ∈ (G∞[N ])∧ such that ψ \f≡ 0 (mod mK) (where\nm\nK\n⊆ O\nK\nis the maximal ideal). Let us write k for the residue field O\nK\n\/m\nK\n. Write ψ for\nψ considered modulo m\nK\n. Then ψ is a section of L[E]⊗N ⊗ k for some E. By enlarging\nK, we may assume that E consists solely of K-valued points of XK . Moreover, since ψ is\nnonzero, there exists some irreducible component of the special fiber of X[E]⊗k over which\nψ is nonzero. If we choose p (in the above discussion) to be such that p ⊗OK OK is this\nirreducible component, then it follows that the restriction of ψ ∈ (G∞[N ])∧ to Ω⊗NL\/K⊗̂KK̂\nwill be nonzero. Now I claim that p satisfies the condition that κL is not identically zero:\nIndeed, if κL were identically zero, then it would follow that κ\n⊗N\nL would be identically\nzero. But the restriction of ψ to Ω⊗N\nL\/K\n⊗̂KK̂ (which is assumed to be nonzero) is a nonzero\nK-multiple of κ⊗NL (ρ) (which would have to be zero). This contradiction proves the claim.\nThus, we are in a position to apply (∗)L−pt. As discussed in the paragraph preceding\nthis Proposition, it then follows that the restriction of ψ to Ω⊗N\nL\/K\n⊗̂KK̂ is zero. Thus, we\nget a contradiction. This completes the proof of the Proposition. ©\n64\nRemark. In [Mzk2], where one only considers isomorphisms of π1’s, as opposed to homo-\nmorphisms as we do here, there is no need to place (as we did in (∗)L−pt) the condition\non p that κL be not identically zero. Because in the present context it is necessary to\ninclude such a condition on p in (∗)L−pt, the author at first did not see how it would be\npossible to prove the “preservation of relations” in the present context. However, in fact,\nin order to prove the preservation of relations (Proposition 12.1), it suffices to consider\nonly p for which one knows already that κL is not identically zero. This observation arose\nin discussions between the author and A. Tamagawa.\nSection 13: The Preservation of L-Points\nThe purpose of this Section is to verify the assertion (∗)L−pt discussed in Section\n12. The technique is similar to that employed in [Mzk2] (although we do not assume any\nknowledge of [Mzk2] in the following discussion). We continue with the notation of the\npreceding Section. In particular, we assume that we have been given a continuous open\nhomomorphism over ΓK\nθ : ΠUK → ΠYK\nMoreover, we assume that XK extends to a stable curve X → Spec(OK). Let p be as in\n(∗)L−pt. Thus, we have a blow-up X̃ → X (that is, X̃ is what we denoted by X[E]OK\nin Section 12), and p is a generic point of the special fiber of X̃ → Spec(OK). Choose a\nsmooth, affine, geometrically connected open neighborhood W of p in the special fiber of\nX̃. Let T be the affine scheme whose coordinate ring RT is such that Spf(RT ) (where we\nequip RT with the p-adic topology) is the completion of X̃ along W . Thus, T → Spec(OK)\nis (p-adically) formally smooth, and T ⊗k = W . By abuse of notation, we shall write p for\nthe prime of T that maps to the original p under T → X̃. Let us write ηT for the generic\npoint of T (regarded as a scheme). Let L be the quotient field of the p-adic completion of\n(RT )p. Thus, L is a p-adic field whose residue field is k(W ) (the function field of W ).\nNote that we have natural morphisms ξTX : T → X; ξ\nηT\nX : ηT → XK ; ξLX : Spec(L) →\nXK ; ξ\nηT\nU : ηT → UK ; ξLU : Spec(L) → UK . Let α\nηT\nU : ΓηT → ΠUK be the morphism\ndetermined (up to conjugation by an element of ΔU ) by ξ\nηT\nU . Similarly, we have α\nL\nU :\nΓL → ΠUK . Composing α\nηT\nU , α\nL\nU with θ gives α\nηT\nY : ΓηT → ΠYK , αLY : ΓL → ΠYK .\nNow let ΔY ′ ⊆ ΔY be a p-derivate (see Definition 0.2) of ΔY . Thus, ΔY ′ is an open,\ncharacteristic subgroup of ΔY . Note that α\nηT\nY defines a section a\nηT\nY : ΓηT → ΠYηT of\nΠYηT\n→ ΓηT . Write ΠY ′ηT for the open subgroup Im(a\nηT\nY ) ·ΔY ′ ⊆ ΠYηT . Thus, ΠY ′ηT gives\nrise to a finite étale covering\nY ′ηT → YηT\n65\nMoreover, there exists some finite étale covering ηT ′ → ηT such that\nY ′ηT ×ηT ηT ′ ∼= ZYK′ ×K′ ηT ′\nfor some curve ZYK′ → Spec(K ′), where K ′ is a finite extension of K contained in K(ηT ′).\nLet T ′ be the normalization of T in ηT ′. Then T ′ → T is finite (since ηT is of characteristic\nzero) and generically étale.\nBy means of θ, we can pull-back the above covering to UηT : Thus, we obtain an\nopen subgroup ΠU ′ηT\n⊆ ΠUηT , which corresponds to some finite étale covering U\n′\nηT\n→ UηT\n(which is geometrically connected over ηT since ΠU ′ηT\ncontains the graph of α\nηT\nU in ΠUηT\n=\nΓηT ×ΓK ΠUK ). This covering extends to a finite, possibly ramified covering X ′ηT → XηT .\nMoreover, we have an isomorphism\nX ′ηT ×ηT ηT ′ ∼= Z\nX\nK′ ×K′ ηT ′\nfor some curve ZXK′ → Spec(K ′). By enlarging K ′, we may assume that this curve has a\nstable extension ZX → Spec(OK′ ).\nNext, let us go back to considering the morphism ξ\nηT\nU : ηT → UK . Now it follows\ntautologically from the way we defined α\nηT\nY (i.e., the fact that it comes originally from ξ\nηT\nU )\nthat ξ\nηT\nU lifts naturally to a point ξ\nηT\nU ′ : ηT → U ′ηT . Let ξ\nηT\nX ′ : ηT → X ′ηT be the result of\ncomposing this morphism with U ′ηT → X ′ηT . Moreover, by restricting to ηT ′, we get a point\nξ\nηT ′\nX ′ : ηT ′ → X ′ηT ′ . Projecting to Z\nX\nK′ , we thus get a point ξ\nηT′\nZX\n: ηT ′ → ZXK′ . Moreover, it\nis easy to see that this morphism extends to a morphism ξT\n′′\nZX\n: T ′′ → ZX , where T ′′ ⊆ T ′\nis an open subscheme obtained as the “D(f)” (i.e., the complement of V (f) ⊆ T ′) for\nsome f ∈ Γ(T ′,OT ′) with the property that f is nonzero at every generic point of the\nspecial fiber of T ′ → Spec(OK′ ). Indeed, this follows from the following two facts: (i) we\nalready have an extension to T ′K′ , since Z\nX\nK′ → XK is finite, and ξ\nηT\nX : ηT → XK extends\nto TK ; (ii) to extend from T\n′\nK′ to some T\n′′, it suffices to apply the valuative criterion for\nproperness (since ZX → Spec(OK′ ) is proper).\nNext, let us consider the morphism T ′′ ⊆ T ′ → T . Clearly (after possibly enlarging\nK ′) there exists a closed point t ∈ T such that if we let S be the completion of T at t, then\nthe morphism S ′′ → S obtained by base-changing T ′′ → T by S → T has the following\nproperties: (i) S ′′ → S is finite and flat; (ii) S ′′ is a finite disjoint sum of connected\ncomponents which are geometrically irreducible over OK′ ; (iii) each of these components\nadmits a section over OK′ . Let S ′ be any one of these connected components. Then we\nhave the following:\nLemma 13.1. The scheme S ′ is normal.\nProof. (Note that this is not entirely obvious since it is not clear that T is excellent.) First,\nI claim that the (flat) morphism S → T is geometrically regular. Indeed, over the generic\n66\nfiber, this follows from the fact that S is regular and generically of characteristic zero;\nover a closed point of TK or T , either the fiber of the morphism S → T is (schematically)\nisomorphic to the given closed point, or it is empty. Thus, it remains to check what happens\nover the height one prime of T which is the special fiber of T → Spec(OK). But over this\nprime, the geometric regularity of S → T follows from the excellence of T ⊗ k = W (which\nis finitely generated over a finite field). This completes the proof of the claim. Thus, it\nfollows that S ′ → T ′ is geometrically regular. Moreover, T ′ is normal; hence, we obtain\nthat S ′ is normal, as desired. ©\nThus, S ′ → S satisfies the properties listed in Definition 6.3. Moreover, if we apply the\nbase-change S → T to all the objects in the above discussion, it is easy to see that we are\nin the situation discussed in Proposition 7.4. Note, in particular, that the existence of the\nmorphism ξT\n′′\nZX\n: T ′′ → ZX shows that the condition (∗)S′ (cf. the discussion preceding\nDefinition 6.4) is satisfied by α\nηS\nX ′ : ΓηS → ΠX ′ηS . In fact, we also have the following:\nLemma 13.2. α\nηS\nX ′ is F -geometric, hence also FI-geometric.\nProof. Indeed, first observe that restricting ξT\n′′\nZX\ngives rise to a morphism ξ1 : S\n′ → ZX .\nSince S ′ was constructed so as to admit a section over OK′ , let σ : Spec(OK′ ) → S ′\nbe such a section. Then we may form the composite ξσ : S\n′ → ZX of the structure\nmorphism S ′ → Spec(OK′ ) with ξ1 ◦ σ. Now observe that ξ1 and ξσ both define sections\nof ZXS′\ndef\n= ZX ×OK′ S ′ → S ′ (which coincide over Im(σ)). Let D1,Dσ ⊆ ZXS′ be the Weil\ndivisors which are the images of these two sections. Next, let us observe that by Lemma\n13.3 below, these two Weil divisors are Q-Cartier. Hence it follows that there exists a\npositive integer N such that N · D1 and N · Dσ are, in fact, Cartier divisors, so we may\nform the line bundle L def= OZX\nS′\n(N ·D1−N ·Dσ). Note that the line bundle L is trivial over\nIm(σ) ⊆ S ′, hence over the closed point of S ′. Let JZX → Spec(OK′ ) be the unique semi-\nabelian scheme whose generic fiber is the Jacobian of ZXK′ . Then it follows that L defines\nan S ′-valued point of JZX which arises from an S ′-valued point of the formal completion\nof JZX at the identity. But, sorting through the definitions (in particular, Definitions 6.2,\n6.4) reveals that this implies that α\nηS\nX ′ is F -geometric, hence also FI-geometric (cf. the\nparagraph following Defintion 6.4). ©\nLemma 13.3. The scheme ZXS′\ndef\n= ZX ×OK′ S ′ is normal, and, moreover, any Weil\ndivisor D ⊆ ZXS′ arising as the graph of an OK′-morphism ξD : S ′ → ZX is Q-Cartier.\nProof. By Lemma 13.1 above, S ′ is normal. Moreover, since ZXS′ is a generically smooth\nstable curve over S ′, one checks easily that the conditions “R1” and “S2” of Serre (see, e.g.,\n[Mats], Chapter 7, §17.I, Theorem 39) are satisfied by ZXS′ , so ZXS′ is normal. Now recall\nthat to say that a Weil divisor is “Q-Cartier” simply means that some nonzero multiple of\nthat Weil divisor is Cartier. Next, let us observe that since the graph of ξD is defined by a\n67\nsingle equation in a neighborhood of any point of the smooth locus of ZX → Spec(OK′ ), it\nsuffices to check that D is Q-Cartier in a neighborhood of the nodes of the special fiber of\nZX → Spec(OK′ ). In fact, it even suffices to check that D is Q-Cartier over the spectrum\nof the completion of the local ring of ZXS′ at such a node.\nThus, let us write\nA def= Γ(S′,OS′); B def= OK′ [[x, y]]\/(xy − πn); C def= A[[x, y]]\/(xy − πn)\nwhere π ∈ OK′ is a uniformizer, and n is a positive integer. Here, we think of B (respec-\ntively, C) as the result of completing ZX (respectively, ZXS′) at a node (respectively, at\nthe inverse image of this node via the projection ZXS′ → ZX). Recall that S ′ is affine, so\nS ′ = Spec(A). Now ξD is given by some morphism B → A. Write sx, sy ∈ A for the images\nof x and y, respectively, under this morphism. Note that sx · sy = πn ∈ A. Moreover, the\nrestriction DC of the subscheme D to Spec(C) is defined by the equations x − sx, y − sy,\ni.e., DC = V (x − sx, y − sy). Now let us consider the regular function x − sx on Spec(C).\nI claim that the ideals (x − sx) and (x − sx, y − sy) in C coincide in C[ 1π ]. Indeed, this\nfollows from the following equation:\ny − sy =\n−y · sy\nπn\n(x − sx) ∈ C[\n1\nπ\n]\nThus, we conclude that on Spec(C), the Weil divisor DC is linearly equivalent to a Weil\ndivisor E on Spec(C) which is supported on the special fiber FC of Spec(C) → Spec(OK′ ).\nOn the other hand, the fact that S ′ → Spec(OK′) is geometrically irreducible implies that\n(FC)red has precisely two irreducible components, namely, V (x)red and V (y)red. Thus, it\nfollows that any divisor supported on FC is Q-Cartier, as desired. This completes the proof\nof the Lemma. ©\nThus, (by Lemma 13.2) we may apply Proposition 7.4 to conclude that Y ′ηS admits a\nline bundle of degree prime to p. Moreover, by Proposition 8.1 and Lemma 8.2, it thus\nfollows that:\nY ′L\ndef\n= Y ′ηT ×ηT Spec(L) has a rational point over some tamely ramified\nextension L′ of L.\nIt is this key result that will allow us to conclude the proof of (∗)L−pt.\nLet us review what we have done so far, from the point of view of objects over YL.\nFirst, we have a section aLY : ΓL → ΠYL of ΠYL → ΓL (defined by αLY ). Moreover, given\nany p-derivate ΔY ′\nL\n⊆ ΔY , we obtain a finite étale covering Y ′L → YL defined by ΠY ′\nL\ndef\n=\nIm(aLY ) · ΔY ′\nL\n⊆ ΠYL . Under these circumstances, we just showed that Y ′L necessarily has\n68\na rational point over some tamely ramified extension of L. Moreover, tracing through the\ndefinitions, it is easy to see that the assumption placed on p in (∗)L−pt that κL be not\nidentically zero means precisely (in the language introduced at the beginning of Section\n9) that aLY is nondegenerate. It thus follows that we can apply Corollary 10.5 to conclude\nthat:\naLY : ΓL → ΠYL arises from some geometric point (ξ′)L ∈ YL(L).\nIn other words, the proof of (∗)L−pt has been completed. Thus, by Proposition 12.1, we\nconclude the following:\nCorollary 13.4. Let θ : ΠUK → ΠYK be a continuous open homomorphism over ΓK .\nThen θ “preserves relations,” i.e., (in the notation of Section 12), we have ΨN (RN ) = 0,\nfor all positive integers N .\nNote that in [Mzk2] (where we essentially dealt with the case where θ is an isomor-\nphism), the preservation of relations is already enough to conclude the proof of the main\ntheorem. In the present context, however, because of the fact that HU is of infinite rank,\nit is necessary to go through one more intermediate technical step before we can complete\nthe proof of the (first part of the) first main theorem. This step essentially amounts to\nshowing that any θ as in Corollary 13.4 necessarily factors through ΠXK . The proof of\nthis next step is the main topic of the following Section. Note that if this fact (i.e., that θ\nfactors through ΠXK ) could be proven more directly, then this paper could be simplified con-\nsiderably. (For instance, Section 11, as well as the rather technical notions of “irreducibly\nsplittable” and “FI-geometric” would be unnecessary.) Unfortunately, however, the proof\nof this fact in Section 14 relies heavily on the “preservation of relations” (Corollary 13.4).\nSection 14: The Annihilation of Inertia\nIn this Section, we prove that any continuous surjective homomorphism θ : ΠUK →\nΠYK over ΓK necessarily (acts as though it) factors through ΠXK . In the process of doing\nthis, we complete the proof of the (first part of) the first main theorem of this paper, in the\ncase where the base field is a local field. Throughout most of this Section (except for the\nvery end – i.e., from the statement of Theorem 14.1 on), we continue to use the notation of\nthe preceding three Sections. Let us assume, moreover, that YK is not hyperelliptic. Thus,\nit follows (e.g., from Lemma 10.4 (3)) that any connected finite étale covering of YK is also\nnon-hyperelliptic.\nLet us first consider the morphism\nH0(Y, ωY\/OK ) ⊗OK ÔK ↪→ F\n∧\n∞\n69\nconstructed in Section 12. We would like to show first of all that\n(∗)FFX This morphism factors through FFX ⊗OK ÔK ⊆ F∧∞, where FFX\ndef\n=\nH0(X,ωX\/OK ) (cf. the discussion preceding Proposition 11.3).\nTo do this, we argue as follows: Let x ∈ XK(K). Let\n\fx : F∧∞ → ÔK\nbe the morphism induced by restriction: i.e., elements of F∞ are sections of L[∞], hence\ndifferentials on X[∞]; thus, by taking the residue of such a differential at x, we get a\nresidue map F∞ → OK ; then taking the p-adic completion of this residue map gives us\n\fx. We would like to show in the following that the restriction\nζx : H\n0(Y, ωY\/OK ) → ÔK\nof \fx to H\n0(Y, ωY\/OK ) is zero. If we show this, then this will be also hold for all x\n′ ∈\nXK(K\n′) (where K ′ is a finite extension of K), hence (∗)FFX will follow immediately (from\nthe fact that FFX consists precisely of all those differentials whose residues at every point\nare zero).\nThus, let us assume that ζx \f= 0. Let Ix ⊆ ΔU denote the inertia group (well-defined\nup to conjugation) corresponding to x. Then observe that the restriction Ix ⊆ ΔU → ΔY\nof θ to ΔU is nontrivial. Indeed, if this restriction were zero, then it is clear from the way\nthat ζx was constructed (cf. the “comparison theorem” of Proposition 11.3) that ζx would\nbe zero. Thus, θ(Ix) \f= {1}. In particular, it follows (by possibly enlarging K) that there\nexists an open normal subgroup Π\nY ′\nK\n⊆ ΠYK (corresponding to a covering Y ′K → YK) such\nthat: (i) Π\nY ′\nK\nsurjects onto ΓK ; and (ii) if we let U\n′\nK → UK be the result of pulling back\nY ′K → YK via θ, then U ′K → UK is a connected Galois covering which is ramified over x.\nLet G\ndef\n= ΠYK \/ΠY ′K\n. Thus, G is a finite group, and G = Gal(U ′K\/UK) = Gal(Y\n′\nK\/YK).\nNote that U ′K → UK extends to a ramified covering X ′K → XK . Let x′ ∈ XK(K) (where\nwe enlarge K if necessary) be a point lying above x.\nNow let us denote with primes the objects corresponding to U ′K , X\n′\nK , and Y\n′\nK that\nare analogous to the objects already constructed for UK , XK , and YK . Thus, we have\n\f′x′ : (F ′∞)∧ → ÔK\nand\nζ ′x′ : H\n0(Y ′, ωY ′\/OK ) → ÔK\n70\nNote that the fact that ζx \f= 0 implies (since ζ ′x′|H0(Y,ωY\/OK ) = ζx) that ζ\n′\nx′ \f= 0. Let σ ∈ G\nbe such that σ(x′) = x′, σ \f= 1. Then the fact that σ fixes x′ implies that σ fixes \f′x′ ,\nwhich, in turn, implies (since the inclusion H0(Y ′, ωY ′\/OK ) ↪→ (F ′∞)∧ is G-equivariant)\nthat σ fixes ζ ′x′.\nOn the other hand, by Corollary 13.4 (“preservation of relations”) and the fact\nthat Y ′K is not hyperelliptic, it follows that the point of the projective space P\n′\nK\ndef\n=\nP(H0(Y ′, ωY ′\/OK )) defined by ζ\n′\nx′ lies on the canonically embedded curve Y\n′\nK ⊆ P ′K .\nThus, the fact that ζ ′x′ is fixed by σ implies that Y\n′\nK admits a K̂-valued point which is fixed\nby σ. Since Y ′K → YK = Y ′K\/G is étale, however, this is absurd. This contradiction thus\ncompletes the proof that ζx = 0, and hence also the proof of (∗)F\nF\nX .\nLet us review what we have done so far. Given a surjective continuous homomorphism\nθ : ΠUK → ΠYK over ΓK , we have seen that θ induces an injection\nH0(YK , ωYK \/K) ↪→ H0(XK , ωXK\/K)\nthat preserves relations. Thus, it follows (by using the canonical embedding of YK) that\nwe get a dominant morphism θU : UK → YK which extends to a morphism\nθX : XK → YK\nby properness. Moreover, given any finite Galois étale covering Y ′K → YK whose geometric\npart has p-power order, we can pull-back this covering via θ to obtain U ′K → UK (finite\nétale), X ′K → XK (finite and possibly ramified), together with θ′ : ΠU ′\nK\n→ Π\nY ′\nK\n. If\nwe then repeat the argument just applied to θ for θ′, we see that we get a morphism\nθ′X ′ : X\n′\nK → Y ′K which lies over θX . If we then continue this procedure for arbitrary\nfinite Galois étale coverings (whose geometric parts have p-power order), the well-known\ncorrespondence between fundamental groups and categories of étale coverings thus shows\nthat the morphism induced by θU on fundamental groups coincides with θ up to composition\nwith an inner automorphism induced by an element of ΠYK . On the other hand, since both\nθ and the morphism induced by θU on fundamental groups have the property that they\nlie over ΓK , it follows that the element of ΠYK in question must map to the the center\nof ΓK . Since, however, ΓK is center-free (see Lemma 15.6 below – one checks easily that\nthere are no “vicious circles” in the reasoning), it follows that the element in question\nmust therefore be an element of ΔY ⊆ ΠYK . That is to say, we have essentially proven the\nfollowing result:\nTheorem 14.1. Let K be a finite extension of Qp. Let YK be a hyperbolic curve (not\nnecessarily proper) over K. Let UK be the spectrum of a one-dimensional function field\nover K. Let HomdomK (UK , YK) be the set of dominant K-morphisms from UK to YK. Let\nHom\nopen\nΓK\n(ΠUK ,ΠYK ) be the set of open, continuous group homomorphisms ΠUK → ΠYK\nover ΓK, considered up to composition with an inner automorphism arising from ΔY .\nThen the natural map\n71\nHomdomK (UK , YK) → HomopenΓK (ΠUK ,ΠYK )\nis bijective.\nProof. We begin by proving that this map is injective. First, observe that by replacing UK\nand YK by coverings defined by “p-derivates” (as in Definition 0.2) of ΔU and ΔY (where\nwe use p-derivates since they are natural), we may assume that both UK and YK admit\nhyperbolic compactifications UK and Y K (i.e., UK and Y K are proper hyperbolic curves\nover K). Since dominant K-morphisms UK → YK are the same as dominant K-morphisms\nUK → Y K , it suffices to show that such a morphism UK → Y K is determined by its\ninduced morphism Δab\nU\n→ Δab\nY\n(on abelianizations of geometric fundamental groups). But\nthis follows from the fact that this morphism Δab\nU\n→ Δab\nY\nclearly determines the morphism\nbetween all p-power torsion points of the Jacobians of UK and Y K , hence it determines\nthe induced morphism between the Jacobians of UK and Y K . Moreover, since UK and\nY K are both proper hyperbolic (i.e., of genus ≥ 2), they both embed in their Jacobians,\nso we conclude that the original morphism UK → Y K is uniquely determined, as desired.\nNext, we consider surjectivity. Let us first consider the case where YK is proper. By\nreplacing UK by a finite étale covering of UK (and then descending at the end, which is\npossible since (by the preceding paragraph) the natural map in the Theorem is injective),\nwe can assume that the proper model XK of UK is hyperbolic. Note, moreover, that if\nθ : ΠUK → ΠYK is open, then its image is of finite index, so by replacing YK by a finite\nétale covering of YK , it is clear that we may assume that θ is surjective. Finally, by Lemma\n10.4 (4), by replacing YK by a finite étale covering of YK , it is clear that we may assume\nthat YK is non-hyperelliptic. Now we are in the circumstances considered above, and so\nwe see that θ arises from a geometric morphism UK → YK , as desired. This completes the\nproof of the Theorem when YK is proper.\nNow let us consider the case when YK is not proper. First note that by replacing YK\nby a finite étale covering of YK , we may assume that the compactification ZK of YK is\nhyperbolic. Now the point is to compose the given θ : ΠUK → ΠYK with the morphism\nΠYK → ΠZK arising from the compactification inclusion YK ⊆ ZK . Since we know the\nTheorem to be true for morphisms between UK and ZK , we thus obtain that ΠUK → ΠZK\narises from some dominant UK → ZK (which necessarily factors – since UK is the spectrum\nof a field – through YK , thus yielding a UK → YK). Thus it remains only to see that the\nmorphism induced on π1’s by this UK → YK is the same as the given θ. But this is done\n(as usual) by considering a finite étale covering Y ′K → YK , and applying the argument just\ndescribed to U ′K and Y\n′\nK to obtain a U\n′\nK → Y ′K which lies over the UK → YK constructed\npreviously. As usual, this is enough to show (cf. the argument directly preceding the\nstatement of Theorem 14.1) that the morphism induced on fundamental groups by the\nUK → YK that we constructed is the same as the given θ. This completes the proof of the\nTheorem. ©\nIn fact, we have the following:\n72\nCorollary 14.2. Let K be a finite extension of Qp. Let XK (respectively, YK) be either\na hyperbolic curve (not necessarily proper!) over K or the spectrum of a one-dimensional\nfunction field over K. Let HomdomK (XK , YK) be the set of dominant K-morphisms from\nXK to YK. Let Hom\nopen\nΓK\n(ΠXK ,ΠYK ) be the set of open, continuous group homomorphisms\nΠXK → ΠYK over ΓK , considered up to composition with an inner automorphism arising\nfrom ΔY . Then the natural map\nHomdomK (XK , YK) → HomopenΓK (ΠXK ,ΠYK )\nis bijective.\nProof. By an argument analogous to that of the final paragraph of the proof of Theorem\n14.1, it follows that it suffices to consider the case where YK is a hyperbolic curve. If UK is\nthe generic point (considered as a scheme) of XK , then given any open θX : ΠXK → ΠYK ,\ncomposing θX with the natural surjection ΠUK → ΠXK induced by UK → XK gives us an\nopen θU : ΠUK → ΠYK . Applying Theorem 14.1 to θU thus gives us a dominant morphism\nXK → Y K (where Y K is the compactification of the curve YK). To see that this morphism\nfactors through YK , it suffices to apply this construction to Y\n′\nK → YK , where Y ′K is a finite,\ngeometrically connected (over K), Galois étale covering (of p-power order) of YK which\nis ramified over all the points of Y K − YK . In fact, we may even choose Y ′K so that the\nramification indices over all the points of Y K − YK are larger than the degree of XK over\nY K . Then the fact that we get some X\n′\nK → Y\n′\nK (where X\n′\nK is étale over XK) lying\nover the morphism XK → Y K obtained previously shows that this morphism XK → Y K\nfactors through YK , as desired. This completes the proof of the Corollary. ©\nRemark. Corollary 14.2 is thus a special case of Theorem A, the first main theorem of this\npaper. In fact, Corollary 14.2 holds even in the case where K is only finitely generated over\nQp. However, unlike the situation in [Mzk2], deriving this finitely generated case from the\nlocal field case is not so trivial, again (cf. the discussion at the end of Section 13) because\nof the fact that ΔU is so large. Thus, we save the derivation of the finitely generated case\nfor the following Section.\nSection 15: Base Fields Finitely Generated over the p-adics\nIn this Section, we let L be a finitely generated extension of Qp. We would like to\nprove versions of Theorem 14.1 and Corollary 14.2 in the case where the base field is L\n(as opposed to a finite extension of Qp). Thus, by induction on the transcendence degree\nof L over Qp, we may assume that Theorem 14.1, for instance, is known for all base fields\nwhose transcendence degree over Qp is < that of L.\nLet XL and YL be proper hyperbolic curves over L. Let UL be the generic point of\nXL (regarded as a scheme). Moreover, let us assume that we have been given a continuous\nsurjective homomorphism over ΓL\n73\nθ : ΠUL → ΠYL\nOnce it is shown that θ factors though ΠXL , it is relatively easy to derive that θ arises\ngeometrically from Theorem 14.1. Thus, the first order of business is to show that θ factors\nthrough ΠXL . To achieve this, we assume that this is not the case. Then (cf. the argument\nemployed in Section 14) by replacing YL and UL by finite étale coverings Y\n′\nL → YL and\nU ′L → UL (the latter obtained by pulling back the former via θ), we may assume that we\nare in the following situation: There is a finite cyclic group G (with generator σ) acting\nfaithfully on UL (hence also XL) and YL – via L-linear automorphisms – in such a way that\nσ fixes a point x0 ∈ XL(L), but acts without fixed points on YL. Moreover, we assume\nthat σ ◦ θ ◦ σ−1 coincides with θ up to composition with an inner automorphism defined\nby an element of ΔY . If we can show that these assumptions lead to a contradiction, it\nwill follow immediately that θ factors through ΠXL .\nLet (as usual) HU (respectively, HY ) be the abelianization of ΔU (respectively, ΔY )\nregarded as a ΓK-module. Let H\nP\nU ⊆ HU be the closure of the image of all the inertia\ngroups (cf. the beginning of Section 11). Note that θ induces a ΓK -morphism HU → HY .\nSince HY is a finitely generated Zp-module, it follows that (by possibly replacing L by a\nfinite extension of L) there exist points x1, . . . , xr ∈ XL(L) such that the images of the\ncorresponding inertia groups θ(Ii) (for i = 1, . . . , r) in HY is equal to the image of θ(H\nP\nU )\nin HY .\nNow let K ⊆ L be a subfield such that L is a one-dimensional function field over\nK (hence, in particular, we assume that K is algebraically closed in L). Note that such\na K always exists (as long as L is not a finite extension of Qp). Thus, there exists a\nsmooth affine model M → Spec(K) of L such that XL and YL extend to smooth curves\nXM → M and YM → M over M . Moreover, we may also assume (by shrinking M) that\nx1, . . . , xr extend to sections s1, . . . , sr : XM → M whose images are disjoint from one\nanother, and that σ acts on XM and YM . The point x0 ∈ XL(L) then extends (by the\nvaluative criterion for properness) to a section s0 : M → XM . Finally, let us observe that\n(by further shrinking M), we may assume that σ acts without fixed points on YM .\nThe first thing that we would like to get our hands on is a morphism like θ, except for\nobjects over K, so that we can apply Theorem 14.1 over K (which we know to be true via the\ninduction hypothesis). To construct such a morphism, we argue as follows. Let m ∈ M(K)\nbe a point (which, after possibly enlarging K, always exists). Let D be the spectrum of the\ncompletion of OM,m. (Here, one should think of the “D” as standing for “disk.”) Thus, we\nmay choose an isomorphism D ∼= Spec(K[[t]]) (where t is an indeterminate). Let D∞ → D\nbe the ramified covering of infinite degree obtained by adjoining all t\n1\nN (for N a positive\ninteger) to OD. Let p ∈ XM be the prime which is the fiber Xm of XM → M over M . Let\nUD be the spectrum of the completion of OXM ,p. Thus, UD is the spectrum of a complete\ndiscrete valuation ring with residue field equal to K(Um), the function field of Xm. Let ηD\n(respectively, ηD∞) be the generic point of D (respectively, D∞). Let UηD\ndef\n= UD ×D ηD.\nThus, UηD is the spectrum of a complete, discretely valued field. In particular, it follows\n74\nfrom the well-known theory of such fields (see, e.g., [Ser2]) that if ΓUηD is the absolute\nGalois group of this field, then we have an exact sequence\n1 → Ẑ(1) → ΓUηD → ΓK(Um) → 1\nHere the covering corresponding to the Ẑ(1) is given by adjoining all t\n1\nN (for N a positive\ninteger) – cf. the covering D∞ → D.\nNow let us denote by ΠUηD the result of replacing the geometric portion (i.e., relative\nto the morphism UηD → ηD) of the fundamental group of UηD by its maximal pro-p\nquotient. Then we get an exact sequence\n1 → ΔUm → ΠUηD → ΓηD → 1\nIf we pull this exact sequence back via ΓηD∞ = ΓK → ΓηD , we thus get an exact sequence\n1 → ΔUm → ΠUηD∞ → ΓηD∞ = ΓK → 1\nMoreover, it is easy to see that this last exact sequence can be identified with\n1 → ΔUm → ΠUm → ΓK → 1\nOn the other hand, if we pull back ΠYL → ΓL via ΓK = ΓηD∞ → ΓL, we get an exact\nsequence\n1 → ΔYm → ΠYm → ΓK → 1\nThus, if we pull-back θ via ΓK = ΓηD∞ → ΓL to obtain a morphism ΠUηD∞ → ΠYηD∞\n(where the subscripted ηD∞ denotes “⊗LηD∞”) and compose with “π1” of the natural\nmorphism UηD∞ → UηD∞ , we get a continuous homomorphism\nθm : ΠUm → ΠYm\nover ΓK (where K we regard here as the residue field of M at m).\nLemma 15.1. The morphism θm is surjective.\nProof. It suffices to show that the restriction of θm to ΔUm surjects onto ΔYm = ΔY . In\nfact, by the basic theory of p-groups, it suffices to show that θm induces a surjection of\nΔUm onto HYm = HY . To see this, it suffices to consider (after replacing M by a finite\n75\nétale covering of M , and enlarging K if necessary) a finite abelian covering Y ′L → YL\n(where Y ′L → L is geometrically connected) of degree a power of p which is > 1. Let\nus pull-back this covering to some covering U ′L → UL via θ. Then we must show that\nthe pull-back U ′ηD∞ → UηD∞ of this covering to UηD∞ is not the trivial covering. Thus,\nsuppose that it is the trivial covering. Then the covering U ′m → Um that it induces (cf.\nthe exact sequences above) of Um is again trivial. Since the images of s1, . . . , sr in Xm\nare disjoint this implies first of all that U ′L → UL is unramified at x1, . . . , xr. But because\nof the way in which x1, . . . , xr were chosen, this implies that the covering Y\n′\nL → YL was\nobtained from a quotient of HY \/Im(θ(H\nP\nU )). Thus, it follows that U\n′\nL → UL extends to\na finite étale covering X ′L → XL. Moreover, since XM is smooth over M at m, it follows\nthat X ′L → XL is split if and only if the induced X ′m → Xm is split. But this X ′m → Xm\nextends the covering U ′m → Um which we already saw to be trivial. Thus, we obtain that\nX ′L → XL, hence U ′L → UL is trivial. Since θ is surjective, however, this implies that\nY ′L → YL is trivial. This contradiction completes the proof of the Lemma. ©\nNow by the induction hypothesis on the transcendence degree of L, it follows that θm\narises from some geometric morphism Um → Ym which is compatible with the action of\nσ on both sides (since σ is compatible with θ). By the valuative criterion for properness,\nthis morphism extends to a morphism Xm → Ym. Moreover, σ fixes s0(m) ∈ Xm(K), so\ns0(m) ∈ Xm(K) is mapped to a fixed point of Ym(K), which is absurd, since σ acts on Ym\nwithout fixed points. This contradiction completes the proof of the following result:\nLemma 15.2. Let L be a finitely generated extension of Qp. Let XL and YL be\nproper hyperbolic curves over L. Let UL be the generic point of XL. Then any continuous\nsurjective homomorphism θ : ΠUL → ΠYL over ΓL necessarily factors through ΠXL .\nNow we can conclude that θ arises geometrically, as follows. Consider the M-scheme\nHM def= HomM (XM , YM ) → M . Since YM is hyperbolic, it is well-known that HM is\nfinite and unramified over M . (Indeed, that HM → M is unramified (respectively, quasi-\nfinite; proper) follows since the pull-back of the tangent bundle of Ym to Xm (for any\nm ∈ M) has no global sections (respectively, follows from the Hurwitz formula, which\nallows one to bound the degree of a morphism Xm → Ym; follows via the same argument\nas that used in Lemma 8.3 of [Mzk1]).) By shrinking M , we may assume that HM is finite\nétale over M . Then the fact that θm arises geometrically (from some Xm → Ym) shows\nthat over some finite extension L′ of L, we have a morphism XL′ → YL′ that specializes\nto Xm → Ym. Moreover, if M ′ is the normalization of M in L′, then since the pro-p\ngeometric fundamental groups of XM ′ and YM ′ form local systems over M\n′, it follows (by\nchecking what happens over the point m) that the morphism on Δ’s induced by XL′ → YL′\nis the same (up to composition with an inner automorphism) as that induced by θ. In\nparticular, the morphism HX → HY induced by XL′ → YL′ is the same as that induced\nby θ. On the other hand, a morphism from XL to YL (over any field) is determined by\nthe morphism it induces from HX to HY (cf. the proof of the injectivity part of Theorem\n14.1). Thus, since the HX → HY in question is ΓL- (not just ΓL′ -) equivariant, we obtain\n76\nthat XL′ → YL′ descends to a XL → YL. By repeating this construction (as usual) for\nall finite étale coverings of YL, we also obtain that the morphism induced on fundamental\ngroups by this XL → YL is the original θ.\nThat is to say, in summary, we have proven that any θ : ΠUL → ΠYL as in Lemma\n15.2 necessarily arises geometrically from some morphism UL → YL. Thus, by arguments\nformally analogous to those of the proofs of Theorem 14.1 and Corollary 14.2, we obtain\nthe following analogue of Corollary 14.2:\nCorollary 15.3. Let K be a finitely generated field extension of Qp. Let XK (re-\nspectively, YK) be either a hyperbolic curve over K or the spectrum of a one-dimensional\nfunction field over K. Let HomdomK (XK , YK) be the set of dominant K-morphisms from\nXK to YK. Let Hom\nopen\nΓK\n(ΠXK ,ΠYK ) be the set of open, continuous group homomorphisms\nΠXK → ΠYK over ΓK , considered up to composition with an inner automorphism arising\nfrom ΔY . Then the natural map\nHomdomK (XK , YK) → HomopenΓK (ΠXK ,ΠYK )\nis bijective.\nIt turns out that it is most natural to generalize Corollary 15.3 to the case where the\nlong and unwieldy expression “hyperbolic curve\/spectrum of a one-dimensional function\nfield defined over a finitely generated field extension of Qp” is replaced by the much shorter\nexpression “pro-hyperbolic curve over a sub-p-adic field.” Thus, we make the following\ndefinition: Let K be a field. Let XK be a K-scheme.\nDefinition 15.4. (i) We shall call a field K a sub-p-adic field if there exists a prime num-\nber p, together with a finitely generated field extension L of Qp such that K is isomorphic\nto a subfield of L.\n(ii) We shall call XK a hyperbolic pro-curve (over K) if XK can be written as the projec-\ntive limit of a projective system of hyperbolic curves over K such that all the transition\nmorphisms in the projective system are birational.\nThus, the following are all examples of sub-p-adic fields:\n(1) finitely generated (in particular, finite) extensions of Qp\n(2) number fields (i.e., finite extensions of Q)\n(3) the subfield of Q which is the composite of all number fields of degree\n≤ n over Q (for some fixed integer n).\n77\nAnother way to think of a pro-hyperbolic curve XK is as the result of removing some set\n(possibly infinite, possibly empty) of closed points from some hyperbolic curve. In partic-\nular, the notion of a “hyperbolic pro-curve” generalizes both the notion of a hyperbolic\ncurve and the case of the spectrum of a function field of dimension one. Then we have the\nfollowing result:\nCorollary 15.5. Let K be sub-p-adic. Let XK (respectively, YK) be a hyperbolic\npro-curve over K. Let HomdomK (XK , YK) be the set of dominant K-morphisms from XK\nto YK. Let Hom\nopen\nΓK\n(ΠXK ,ΠYK ) be the set of open, continuous group homomorphisms\nΠXK → ΠYK over ΓK , considered up to composition with an inner automorphism arising\nfrom ΔY . Then the natural map\nHomdomK (XK , YK) → HomopenΓK (ΠXK ,ΠYK )\nis bijective.\nProof. First, let us observe that if K is a finitely generated field extension of Qp, then\nCorollary 15.5 follows immediately from Corollary 15.3 by arguments formally analogous\nto those of the proofs of Theorem 14.1 and Corollary 14.2. Thus, the only (slightly)\n“new” phenomenon here is the fact that we allow K to be a subfield of a finitely generated\nextension of Qp. The argument for such subfields is as follows: Let K be a subfield of a\nfinitely generated extension field L of Qp. Suppose that we have been given XK and YK\nas in the statement of Corollary 15.5, as well as an open θ : ΠXK → ΠYK over ΓK . Let us\ndenote by θgeom the morphism ΔX → ΔY induced by θ on geometric fundamental groups.\nBy base-changing to L and applying the Corollary 15.5 over L, we obtain that there exists\na morphism φL : XL → YL whose induced morphism on geometric fundamental groups\ncoincides with that defined by θ. On the other hand, since morphisms between hyperbolic\ncurves (hence also hyperbolic pro-curves), clearly have no moduli (cf. the fact that the\nscheme HM of the discussion preceding Corollary 15.3 was finite and unramified over M),\nit follows that φL descends to a finite Galois extension K\n′ of K. Thus, we have a morphism\nφK′ : XK′ → YK′ . It remains to descend φK′ to K. But this follows from the fact that\nif σ ∈ Gal(K ′\/K), then conjugating φK′ by σ gives a morphism φσK′ : XK′ → YK′ whose\ninduced morphism on geometric fundamental groups is the result of conjugating θgeom by\nσ. On the other hand, θgeom arises from θ which lies over ΓK , so θgeom is fixed (up to\ncomposition with an inner automorphism defined by an element of ΔY ) by conjugation\nby σ. Thus, φK′ and φ\nσ\nK′ induce the same morphism on geometric fundamental groups,\nhence coincide (cf. the argument of the discussion preceding Corollary 15.3). This shows\nthat φK′ descends to a morphism φK : XK → YK . Repeating this construction (as usual\n– cf. the argument preceding the statement of Theorem 14.1) for coverings of XK and YK\nshows that the morphism induced by φK on fundamental groups coincides with θ (up to\ncomposition with an inner automorphism defined by an element of ΔY ). ©\nRemark. Note that in Corollaries 15.3 and 15.5, in fact, we implicitly used the fact that\n78\nfor K as in those Corollaries, ΓK is center-free. This may be proven as follows. First, if\nK is a finite extension of Qp, then we have the following:\nLemma 15.6. If K is a finite extension of Qp, then the group ΓK is center-free.\nProof. (The argument given here is “well-known,” but was related to the author by A.\nTamagawa.) Since one knows explicitly (see, e.g., [Ser2], Chapter IV, §2) the structure of\nGal(Ktm\/K), it is clear that the quotient Gal(Ktm\/K) of ΓK is center-free. Thus, the\ncenter of ΓK must lie in ΓKtm ⊆ ΓK . On the other hand, ΓKtm is a pro-p group. Moreover,\nit follows immediately from the facts that\n(1) H2(K,Fp(1)) = Fp;\n(2) if K ′ is a finite unramified extension of K of degree p (which always\nexists), then the natural morphism H2(K,Fp(1)) → H2(K′,Fp(1)) is\nzero;\nthat H2(ΓKtm ,Fp) = 0. But by [Shatz], Chapter III, §3, Proposition 2.3, this is enough\nto imply that ΓKtm is free (as a pro-p group), which implies that its center is trivial (see,\ne.g., [Tama], §1, Propositions 1.1, 1.11). ©\nNext, we consider the case of a finitely generated extension of Qp:\nLemma 15.7. Suppose that K0 is a field of characteristic zero with the property that\nevery open subgroup of ΓK0 is center-free. Then any finitely generated field extension K\nof K0 also has this property. In particular, if K is a finitely generated field extension of\nQp, then ΓK is center-free.\nProof. The last statement follows from the first plus Lemma 15.6. Thus, let us prove\nthe first statement. Note that an extension of a center-free group by a center-free group\nis center-free. Thus, it suffices to prove that if L is a function field (of arbitrary finite\ndimension) over K0, then ΓL is center-free. Next, note that the projective limit of a\nprojective system of center-free groups in which all the transition morphisms are surjective\nis center-free. Moreover, as is well-known, Spec(L) is a projective limit of “hyperbolic Artin\nneighborhoods” over K0 (i.e., successive fibrations of hyperbolic curves – see [SGA4], XI\n3.3). Thus, it suffices to prove that the fundamental group of such an Artin neighborhood\nis center-free. But this then reduces to showing that the fundamental group of a hyperbolic\ncurve is center-free, which is well-known (see, e.g., [Tama], §1, Proposition 1.11). ©\nThis is already enough for Corollary 15.3. Now we can conclude the result for arbitrary\nK as in Corollary 15.5 by means of the following Lemma (due to A. Tamagawa):\n79\nLemma 15.8. Let K be sub-p-adic. Then ΓK is center-free.\nProof. Let K ′ be a finitely generated field extension of Qp that contains K. Suppose\nthat σ ∈ ΓK lies in the center of ΓK , but is not equal to the identity. Then there exists\na finite Galois extension L of K such that σ maps to an element of Gal(L\/K) other than\nthe identity. Write L = K(α), for some α ∈ L. Let EL be an elliptic curve over L with\nj-invariant α. Let XL be the complement of “0” in EL. Thus, XL is a hyperbolic curve\nover L. Let YL be the result of base-changing XL by σ : L → L. Thus, it follows that\nXL and YL are isomorphic as schemes over K. Moreover, conjugating by σ defines an\nisomorphism ΠXL\n∼= ΠYL which lies over ΓL because σ is in the center of ΓK . Base-\nchanging this isomorphism to L′ (the composite of K ′ and L over K), we get a ΓL′ -\nisomorphism ΠXL′\n∼= ΠYL′ . By Corollary 15.3 (in fact, really, this follows already from the\nresults of [Mzk2]), we obtain that this isomorphism arises from an L′-isomorphism of XL′\nwith YL′ . But this implies that the j-invariants of the compactifying elliptic curves of XL′\nand YL′ are the same, i.e., that α = α\nσ ∈ L ⊆ L′, which is absurd. This contradiction\ncompletes the proof of the Lemma. ©\nSection 16: Maps From Higher-Dimensional Function Fields to Curves\nLet K be sub-p-adic (cf. Definition 15.4 (i)). Let UK be the spectrum of a function\nfield over K. (Note that here, we do not assume that the dimension of the function field is\none.) Let XK be a smooth projective model of UK (which exists by [Hiro]). Thus, UK is\nthe generic point of XK . Let n be the dimension of XK . By abuse of terminology, we shall\nalso say that n is the “dimension of UK over K.” Since the one-dimensional case has been\ndealt with previously, we assume here that n ≥ 2. The purpose of this Section is to prove\na result like Corollary 15.5, except for morphisms between UK and a hyperbolic pro-curve.\nLet L be a very ample line bundle on XK . Let\nV\ndef\n= Γ(XK ,L)\nSince the dimension of XK is ≥ 2, and L is very ample, it follows that dimK(V ) ≥ 3.\nLet W ′ ⊆ V be a one-dimensional (over K) subspace, generated by a section of L whose\nzero locus forms a smooth, connected subvariety of XK . (Note that by Bertini’s theorem\n(see, e.g., Theorem 6.3, pp. 66, of [Jou]), it is well-known that such a W ′ exists.) Let\nW ⊆ V be a two-dimensional subspace containing W ′. Choose a basis {e1, e2} for W such\nthat e1 ∈ W ′. Let e3 ∈ V be an element whose image in V\/W is nonzero. Let us also\nassume that the common zero locus in XK of the three sections e1, e2, e3 is a subscheme\nof XK of codimension ≥ 3. (This can always be achieved by choosing e1, e2, e3 sufficiently\ngenerically.) Let K ′ = K(t) (where t is an indeterminate). Let s1\ndef\n= t; s2\ndef\n= t−1. Then we\ncan construct a new two-dimensional subspace W̃ ⊆ VK′ def= V ⊗K K ′, over K ′, as follows:\n80\nWe let W̃ be the subspace generated by the vectors ẽ1\ndef\n= e1 + s1 · e3, ẽ2 def= e2 + s2 · e3.\nThus, to summarize, we have the following situation:\nW ′ ⊆ W ⊆ K · e1 + K · e2 + K · e3 ⊆ V\n‖ ‖ ‖\nK · e1 ⊆ K · e1 + K · e2 ⊆ K · e1 + K · e2 + K · e3\nW̃ = K ′ · ẽ1 + K ′ · ẽ2 ⊆ VK′ = V ⊗K K ′; K ′ = K(t)\nẽ1 = e1 + s1 · e3; ẽ2 def= e2 + s2 · e3; s1 = t; s2 = t−1\nLemma 16.1. Let Ω be an algebraically closed field containing K ′. Then there do not\nexist any one-dimensional subspaces W ′′ of W̃Ω\ndef\n= W̃ ⊗K′ Ω such that W ′′ is defined (as\na subspace of V ⊗K Ω) over a finite extension of K.\nProof. Indeed, if such a W ′′ existed, then there would exist elements a, b, c ∈ Ω (with\nc \f= 0, and at least one of a, b nonzero), together with α1, α2, α3 ∈ K such that\na · ẽ1 + b · ẽ2 = c · (α1 · e1 + α2 · e2 + α3 · e3)\nBy dividing a and b by c, we may assume that c = 1. Then, equating the coefficients of\ne1, e2, e3, we obtain that a = α1; b = α2; s1 · a + s2 · b = α3. But this implies that t, t−1,\nand 1 (as elements of K ′) are linearly dependent over K, which is absurd. This completes\nthe proof of the Lemma. ©\nBefore proceeding, let us pause to interpret what this Lemma means. Note that the\n3-dimensional K-subspace of V generated by e1, e2, e3 defines a rational morphism ψ from\nXK to QK\ndef\n= P(K · e1 + K · e2 + K · e3). Thus, ψ is defined outside of some closed\nsubscheme of XK of codimension ≥ 2. Moreover, note that the inverse image via ψ of any\nclosed point of QK is a subscheme of XK of codimension ≥ 2. (Indeed, this follows from\nthe fact that the common zero locus of e1, e2, e3 has codimension ≥ 3 in XK .) Now let Ω\nbe a finite extension of K ′, and let us consider a one-dimensional (over Ω) subspace W ′′\nof W̃Ω. Thus, W\n′′ corresponds to a line LΩ ⊆ QΩ def= QK ⊗K Ω, and the zero locus (in\nXΩ\ndef\n= XK ⊗K Ω) of a nonzero section of W ′′ is equal to the closure of ψ−1Ω (LΩ) ⊆ XΩ.\nNow we are ready to interpret Lemma 16.1: Namely, I claim that Lemma 16.1 implies\nthat no irreducible component of the (closure of the) divisor ψ−1Ω (LΩ) is defined over K.\n81\nIndeed, if there were a divisor D ⊆ XK such that ψΩ(DΩ) ⊆ LΩ, then by spreading out Ω\n(respectively, LΩ) to a smooth affine curve CK over K (respectively, a family of lines in QK\nparametrized by CK), we obtain that for every closed point c ∈ CK , we have ψ(D) ⊆ Lc.\nBut since (by Lemma 16.1) LΩ is not defined over K, it follows that for two sufficiently\ngeneric closed points c, c′ ∈ CK , dimK(Lc\n⋂\nLc′) = 0, which implies that D is contained\nin the inverse image of a finite set of closed points of QK . But we saw in the preceding\nparagraph that the inverse image via ψ of a closed point of QK is of codimension ≥ 2.\nSince D is a divisor, this is absurd. This completes the proof of the claim stated at the\nbeginning of this paragraph.\nNext, let us base-change UK and XK to K\n′: this gives rise to UK′ and XK′ . The\ntwo-dimensional subspace W̃ ⊆ VK′ defines a rational map from XK′ to PK′ def= P(W̃ ). Let\nX ′K′ ⊆ XK′ be the complement of the indeterminacy locus of this rational map. Thus, the\ncomplement of X ′K′ in XK′ is of codimension ≥ 2, and, moreover, we obtain a dominant\nmorphism (a “pencil”)\nξ : X ′K′ → PK′\nLet ηP be the generic point of PK′ . Let ηP be the spectrum of an algebraic closure of\nK(ηP ) (the function field of PK′ ). Let\nFηP ⊆ XηP\ndef\n= XK ×K ηP\nbe the divisor which is the zero locus of the section of L⊗K K(ηP ) defined by the generic\npoint ηP of PK′ . (Thus, FηP → ηP is proper.) Put another way, FηP is the (closure in XηP\nof the) fiber (⊆ X ′ηP ⊆ XηP ) of the morphism ξ over the generic point ηP ∈ PK′ (whence\nthe use of the letter “F”). Since the pencil PK′ contains ẽ1, which is a genericization of e1\n(i.e., e1 is the specialization of ẽ1 at s1 = t = 0), and the zero locus of e1 is (geometrically)\nsmooth and connected, it follows that FηP will also be geometrically smooth and connected\nover ηP . Moreover, by the Lefshetz hyperplane theorem (see, e.g., [SGA2]), it follows (since\ndimK(XK) ≥ 2) that the natural morphism\nπ1(FηP ) → π1(XηP ) = π1(XK′) = π1(XK)\nis surjective. Let\nGηP\ndef\n= FηP ×XηP UηP = FηP ×XK UK\nSince UK is a projective limit of dense open subschemes of XK , it thus follows that GηP\nis a projective limit of open subschemes of FηP . Moreover, since FηP (thought of as (the\nclosure of) the fiber of the dominant morphism ξ over the generic point ηP of PK′ ) contains\n(as a dense open subset) a scheme which is a projective limit of dense open subschemes of\nXK′ , we conclude that GηP is an integral, nonempty scheme.\n82\nAt any rate, we get a natural morphism\nπ1(GηP ) → π1(UηP ) = π1(UK′) = π1(UK)\nNow we have the following important\nLemma 16.2. This morphism π1(GηP ) → π1(UK) is surjective.\nProof. It suffices to take a finite, connected étale covering Ũ\nK\n→ U\nK\n(of degree > 1),\npull it back to a covering G̃ηP → GηP over GηP , and show that this pulled back covering\ncan never be split. Indeed, suppose that G̃ηP → GηP is split. Now observe that since ξ is\ndominant, it follows that FηP → XK′ and GηP → UK′ are dominant. In fact, FηP → XK′\nand GηP → UK′ are even birational isomorphisms. Moreover, I claim that the divisors (in\nXK′ ) that were thrown out of XK′ to create FηP – let us call these divisors F -divisors –\nare different from the divisors that were thrown out of XK′ to create UK′ – which we shall\ncall U-divisors. Indeed, as divisors of XK′ = XK ⊗K K ′, the U-divisors are all defined\nover finite extensions of K, whereas the F -divisors (which are just fibers of ξ over closed\npoints of PK′ ) are, by Lemma 16.1 (see also the interpretation of Lemma 16.1 in the two\nparagraphs following the proof of Lemma 16.1), never defined over finite extensions of K.\nThis proves the claim.\nNext, let us observe that the morphism FηP → FηP is ramified only over F -divisors.\nThus, it is unramified over U -divisors. Since GηP → FηP is birational, the fact that\nG̃ηP → GηP splits (hence, in particular, extends to a finite étale covering over FηP ) thus\nimplies that the original Ũ\nK\n→ U\nK\nis unramified over the U-divisors, i.e., Ũ\nK\n→ U\nK\narises\nfrom a covering X̃\nK\n→ X\nK\n. (Here we use “purity of the branch locus” for regular local\nrings – see, e.g., [SGA2], Exposé X, p. 118, Théorème 3.4.) But then the fact that the\npull-back F̃ηP → FηP of this covering to FηP does not split follows from the surjectivity\nobserved immediately before the statement of this Lemma. Thus, since GηP → FηP is\nbirational, it follows that G̃ηP → GηP cannot split either. This contradiction completes\nthe proof of the Lemma. ©\nNow let us assume that we are given a surjective continuous homomorphism over ΓK\nθ : ΠUK → ΠYK\nwhere YK is a proper hyperbolic curve over K. Let YηP\ndef\n= YK ×K ηP . Thus, YηP is a\nproper hyperbolic curve over ηP . Suppose, moreover, that we know that morphisms like\nθ necessarily arise geometrically (i.e., from a dominant morphism UK → YK) for UK\nof dimension < n. (We shall refer to this assumption as the “Induction Hypothesis.”\n83\nNote that by Corollary 15.5, we already know that this induction hypothesis is true for\ndimK(UK) = 1.) Then the morphism\nΠGηP\n→ ΠYK\nobtained by composing (“π1 of”) the natural morphism GηP → UK with θ lies over ΓηP →\nΓK , and by Lemma 16.2, is such that it induces a surjection between the (maximal pro-\np quotients of the) respective geometric fundamental groups. Moreover, this morphism\nnaturally defines a morphism\nθG : ΠGηP\n→ ΠYηP\n(since ΠYηP\nis the fibered product of ΠYK and ΓηP over ΓK). Thus, to summarize, θG is\na surjective continuous homomorphism over ΓηP .\nLet UGηP be the generic point of GηP . Thus, by composing (“π1 of”) U\nG\nηP\n→ GηP with\nθG, we obtain a continuous surjective homomorphism\nθUG : ΠUGηP\n→ ΠYηP\nover ηP . Moreover, the dimension of U\nG\nηP\nover ηP is < n. Thus, by the induction hypothesis,\nit follows that θUG arises geometrically, from some unique dominant morphism U\nG\nηP\n→ YηP .\nProjecting to YK′ then gives a dominant morphism U\nG\nηP\n→ YK′ .\nWe would like to observe that this morphism UGηP → YK′ extends to UK′ . Indeed,\nto see this, observe that UGηP and UK′ are both projective limits of open subsets of the\nprojective K ′-variety XK′ . Moreover, UK′ can be written as the projective limit of K ′-\nsmooth open subsets of XK′ . Thus, the fact that we get a morphism\nUK′ → YK′\nfollows from the following\nLemma 16.3. Let L be a field; YL be a proper, smooth, geometrically connected curve\nof nonzero genus over L; and ZL be a smooth L-variety. Suppose that we have a rational\nmap φ from ZL to YL. Then φ is defined over all of ZL.\nProof. This Lemma is a well-known consequence of the classical theory of the Albanese\nvariety (see, e.g., [Lang]): Namely, let UL ⊆ ZL be a (nonempty) open over which φ is\ndefined. Assume (without loss of generality) that YL(L) \f= ∅. Let AL be the Jacobian of\nYL, and let BL be its dual. Then a point of YL(L) defines an embedding YL ↪→ AL which\nwe can compose with φ to obtain a rational map φA from ZL to AL. It suffices to show\n84\nthat φA extends to a morphism on ZL. But note that the portion of φA which already is\na morphism (from UL to AL) defines (by pulling back the Poincaré bundle on AL ×L BL)\na line bundle on UL ×L BL. Since ZL ×L BL is a regular scheme, it follows that this line\nbundle extends to ZL ×L BL. Then taking the classifying morphism associated to this\nextended line bundle gives a morphism ZL → AL, as desired. ©\nLet us review what we have done so far. We started with a continuous surjective\nhomomorphism\nθ : ΠUK → ΠYK\nover ΓK . Moreover, we have shown that if we base-change θ from K to K\n′ to obtain\nθK′ : ΠUK′ → ΠYK′\nthen θK′ arises geometrically from some dominant UK′ → YK′ . Note, moreover, that this\nmorphism UK′ → YK′ is the unique morphism that gives rise to (the geometric portion\nof) θ. (This follows, for instance, from the inductive hypothesis on n. Moreover, this\nuniqueness also holds, of course, over any finite extension of K ′.) Thus, if we specialize\nthe indeterminate t ∈ K ′ to some element of a finite extension L of K that is sufficiently\ngeneric so that UK′ → YK′ specializes to UL → YL, and then base-change this UL → YL\nback up to a morphism UL′ → YL′ (where L′ is the composite of K ′ and L), then this\nmorphism UL′ → YL′ coincides with the morphism obtained by base-changing the original\nUK′ → YK′ via K ′ ⊆ L′. But this means that the original UK′ → YK′ is, in fact, defined\nover L. Finally, since K ′ and L are linearly disjoint over K, it follows that the original\nUK′ → YK′ is defined over K. Thus, we get a morphism UK → YK which clearly induces\nthe original θ (since, for instance, ΠUK′\n→ ΠUK is surjective).\nLet us step back now and take stock of what we have done so far in this Section. We\nstarted with UK , the spectrum of a function field over K, and a proper hyperbolic curve\nYK over K. Then given any surjective continuous homomorphism\nθ : ΠUK → ΠYK\nover ΓK , we showed that θ necessarily arises geometrically. Now let us pause for a defini-\ntion:\nDefinition 16.4. Let QK be a K-scheme. We shall call QK a smooth pro-variety if it\nis the projective limit of a projective system of smooth (geometrically connected) varieties\nover K such that the transition morphisms are all birational.\nNow Lemma 16.3, plus the techniques of the proofs of Theorem 14.1 and Corollary 14.2\nshow that we have, in fact, proven the following (our first main theorem – Theorem A in\nthe Introduction):\n85\nTheorem 16.5. Let K be sub-p-adic (cf. Definition 15.4 (i)). Let XK (respectively, YK)\nbe a smooth pro-variety (respectively, hyperbolic pro-curve) over K. Let HomdomK (XK , YK)\nthe set of dominant K-morphisms from XK to YK. Let Hom\nopen\nΓK\n(ΠXK ,ΠYK ) be the set of\nopen, continuous group homomorphisms ΠXK → ΠYK over ΓK, considered up to composi-\ntion with an inner automorphism arising from ΔY . Then the natural map\nHomdomK (XK , YK) → HomopenΓK (ΠXK ,ΠYK )\nis bijective.\nRemark. Finally, we make the following important observation:\nNote that given any pro-p result such as Theorem 16.5, one can always\nimmediately derive a corresponding profinite result from it.\n(Here by “corresponding profinite result,” we mean the same result, except that “Π”\n(respectively, “Δ”) is replaced by “Πprf” (respectively, “Δprf”).) Indeed, suppose that K,\nXK , and YK are as in the statement of Theorem 16.5, and let\nθ : Π\nprf\nXK\n= π1(XK) → ΠprfYK = π1(YK)\nbe an open homomorphism over ΓK . Then note that θ immediately induces an open\nhomomorphism θp : ΠXK → ΠYK . Applying Theorem 16.5 shows that θp arises from\nsome φ : XK → YK . Thus, it remains only to show that the morphism induced by φ\non fundamental groups coincides with θ. But this follows from the argument preceding\nTheorem 14.1: namely, we consider an arbitrary finite étale covering Y ′K → YK . Pulling\nthis covering back via θ gives a finite étale covering X ′K → XK . Moreover, θ induces a\nmorphism between the full profinite fundamental groups of X ′K and Y\n′\nK . Next, observe that\nthis morphism gives us a φ′ : X ′K → Y ′K (by the same argument as that used to construct\nφ from θ) which (by naturality of the constructions involved) lies over φ. Thus, if we\ncontinue this procedure for arbitrary finite étale coverings, the well-known correspondence\nbetween fundamental groups and categories of étale coverings shows that the morphism\ninduced by φ on fundamental groups coincides with θ up to composition with an inner\nautomorphism induced by an element of Δ\nprf\nY . This completes the proof of the “profinite\nanalogue of Theorem 16.5.”\nSection 17: Maps Between Higher-Dimensional Function Fields\nLet K be sub-p-adic (cf. Definition 15.4 (i)). Let L and M be function fields (of\narbitrary dimension) over K. (Thus, in particular, we assume here that K is algebraically\nclosed in L and M). We denote by ΓL and ΓM the absolute Galois groups of L and\n86\nM , respectively. In this Section, we would like to show how to derive a result (Theorem\n17.1, which is stated as Theorem B in the Introduction) like Theorem 16.5 for morphisms\nbetween L and M (over K).\nTheorem 17.1. Let K be sub-p-adic (cf. Definition 15.4 (i)). Let L and M be func-\ntion fields of arbitrary dimension over K. Let HomK(Spec(L),Spec(M)) be the set of\nK-morphisms from M to L. Let Hom\nopen\nΓK\n(ΓL,ΓM ) be the set of open, continuous group\nhomomorphisms ΓL → ΓM over ΓK , considered up to composition with an inner automor-\nphism arising from Ker(ΓM → ΓK). Then the natural map\nHomK(Spec(L),Spec(M)) → HomopenΓK (ΓL,ΓM )\nis bijective.\nProof. First, recall (cf. the Remark following Theorem 16.5) that any pro-p result such\nas Theorem 16.5 always implies a corresponding profinite result. Now we use induction\non the transcendence degree – which we shall henceforth denote by dimK(M) – of M over\nK. When dimK(M) = 0, the result is vacuous. When dimK(M) = 1, the result follows\nfrom the profinite version of Theorem 16.5; thus, we may assume that dimK(M) > 1. Now\nsuppose that we know Theorem 17.1 to be true for maps to functions fields of transcendence\ndegree < dimK(M). Suppose that we are given an open continuous homomorphism\nθ : ΓL → ΓM\nover ΓK . Observe that just as previously (e.g., in the proof of Theorem 14.1), we can\nassume without loss of generality that θ is surjective.\nNext, observe that there exists a function field P ⊆ M such that 0 < dimK(P ) <\ndimK(M). Moreover, we may assume that P is algebraically closed in M . Thus, we\nget a surjection ΓM → ΓP . Composing θ with this surjection, we obtain a surjection\nθP : ΓL → ΓP , which, by the induction hypothesis on dimK(M), we know arises geo-\nmetrically from some P ↪→ L (over K). Moreover, since θP is surjective, it follows that\nP is algebraically closed in L. Thus, we may regard L and M as function fields over P .\nMoreover, dimP (M) < dimK(M). Thus, since θ : ΓL → ΓM is (by the definition of θP )\na morphism over ΓP , it follows from the induction hypothesis on dimK(M) that θ arises\nfrom some M ↪→ L over P (hence also over K). This completes the proof of the Theorem.\n©\nRemark. Note that in this case, we needed to work with profinite (not pro-p) fundamental\ngroups because the operation of taking the maximal pro-p quotient is not well-behaved\nwith respect to fibrations: i.e., if we had replaced Ker(ΓM → ΓK) with its maximal\npro-p quotient, we would run into trouble because it is not clear that the maximal pro-p\nquotient of Ker(ΓM → ΓP ) injects into the maximal pro-p quotient of Ker(ΓM → ΓK).\n87\nAlso, we needed to work with function fields (as opposed to varieties) because, if, for\ninstance, Spec(L) and Spec(P ) had been varieties, the fact that the geometric (over K)\nfundamental group of Spec(L) surjects onto that of Spec(P ) does not necessarily imply that\nthe morphism Spec(L) → Spec(P ) has geometrically connected fibers. Moreover, if the\ngeneric geometric fiber of Spec(L) → Spec(P ) has several distinct connected components, it\nis not necessarily the case that the fundamental group of any of these connected components\nsurjects onto the geometric (over P ) fundamental group of Spec(M). Indeed, by replacing\nSpec(M) by a smooth projective variety YK (of dimension ≥ 3), and Spec(L) by the\nresult – call it XK – of cutting this variety with a generic hyperplane section, then the\ninclusion XK ↪→ YK induces an isomorphism on geometric fundamental groups despite\nthe fact that XK is not isomorphic to YK . Moreover, note that such a counterexample to\na “variety version” of Theorem 17.1 exists even if YK is “hyperbolic” (say, a product of\nproper hyperbolic curves), in which case, one would expect XK (at least if the ample line\nbundle used to cut YK to form XK has sufficiently high degree) to be “quite hyperbolic,”\nas well.\nSection 18: Truncated Fundamental Groups\nIf Δ is a topological group, let us introduce the following notation: Δ{0} def= Δ; for\ni ≥ 1, Δ{i} def= [Δ{i−1}, Δ{i−1}]. Also, let us write “Πi” (respectively, “Δi”) for Π\/Δ{i}\n(respectively, Δ\/Δ{i}). The purpose of this Section is to observe that much of the theory\nof this paper continues to hold to a large extent even when we consider morphisms not\nbetween the full Π’s, but between certain truncated versions of the Π’s. Our first main\nresult is the following:\nTheorem 18.1. Let K be sub-p-adic (cf. Definition 15.4 (i)). Let XK be a smooth\nvariety over K. Let YK be a hyperbolic curve over K. Let n ≥ 5. Then every continuous\nopen homomorphism\nθ : ΠnXK → Π\nn\nYK\nover ΓK induces a dominant morphism μ : XK → YK whose induced morphism on fun-\ndamental groups coincides (up to composition with an inner automorphism arising from\nΔYK ) with the morphism Δ\nn−3\nXK\n→ Δn−3YK defined by considering θ “modulo Δ{n − 3}.”\nProof. First let us consider the case where: (i) K is a finite extension of Qp; (ii) XK\nis a curve; (iii) YK is a non-hyperelliptic proper hyperbolic curve; (iv) n = 3. Then note\nthat the theory of Sections 1 through 5 manifestly only involves Δ2XK . It thus follows\nimmediately (via the help of a technical lemma – Lemma 18.4 below – necessary in order\nto assure that Lemma 7.3 goes through in the truncated context) that to prove Proposition\n7.4, we really only used Δ2Y ′ηS\n. Thus, in summary, as long as the covering Y ′ηS → YηS (in\n88\nthe notation of the discussion preceding Proposition 7.4) arises from an open subgroup of\nΠ1YK , we can apply Proposition 7.4 to Y\n′\nηS\nto conclude that Y ′ηS admits a line bundle of\ndegree prime to p.\nNow we would like to conclude the “preservation of relations.” Note that Section 11\nmanifestly only involves the abelianization of the geometric fundamental group. More-\nover, Section 12 is just formal manipulation. In Section 13, the only thing from Sections\n1 through 7 that is used is Proposition 7.4 (whose applicability under the present circum-\nstances was discussed in the preceding paragraph). Thus, by using Propositions 7.4 and\n8.1, Lemma 8.2, and the theory of Section 9, we may conclude “preservation of relations”\nfor the morphism H0(YK , ωYK\/K) → H0(XK , ωXK\/K) induced by Δ1XK → Δ1YK . (Note,\nhowever, that this time, in the application of Section 9, we take for our tower of coverings\n“XnL” not the coverings corresponding to the p-derivates of the whole geometric fundamen-\ntal group Δ, but rather the p-derivates of Δ1 – which is enough to carry out the argument\nof Section 9.) Hence we get a morphism μ : XK → YK .\nNow let us lift the hypothesis that n = 3. Then we obtain the following: If Y ′K → YK\nis a covering arising from an open subgroup of Πn−3YK , and X\n′\nK → XK is a connected\ncomponent of the pull-back of this covering to XK via θ, then we get a natural morphism\nμ′ : X ′K → Y ′K . Moreover, the naturality of the construction of this morphism means that\nit always lies over the morphism XK → YK . Thus, it follows by the usual argument (i.e.,\nthe one preceding Theorem 14.1) that the morphism induced by μ on fundamental groups\nis equal to θ modulo Δ{n − 3} (up to composition with an inner automorphism arising\nfrom ΔYK ).\nSo far we have not used that n ≥ 5. (In fact, we have only used that n ≥ 3.)\nThe purpose of assuming that n ≥ 5 is to lift hypothesis (iii) (in the first paragraph of\nthis proof). Namely, given any hyperbolic curve YK , it is elementary to show that there\nalways exists a covering Y ′K → YK defined by an open subgroup of Π2YK such that the\ncompactification Y\n′\nK of Y\n′\nK is hyperbolic and non-hyperelliptic, and such that Y\n′\nK → Y K\nhas arbitrarily large (specified) ramification over all the points of Y K −YK . Thus, we get a\nmap X ′K → Y\n′\nK which descends to a map XK → Y K . Moreover, the fact that X ′K → XK\nis étale, while Y\n′\nK → Y K is ramified over the points of Y K −YK (with ramification indices\narbitrarily large) implies that the map XK → Y K factors through YK . Thus, we get a\nmap XK → YK , as desired. Then arguing as in the preceding paragraph completes the\nproof, albeit still under the assumptions (i) and (ii) (of the first paragraph of the proof).\nThe extension to the case of fields K that are subfields of finitely generated extensions\nof Qp then follows via the same “specialization argument” as that employed previously\nin the nontruncated case (following Lemma 15.2). Thus, we can also lift assumption (i).\nFinally, the “cutting with a hyperplane argument” of Section 16 extends immediately to\nthe truncated case. This allows us to lift assumption (ii), thus completing the proof of the\nTheorem. ©\nTheorem 18.2. Let K be a subfield of a finitely generated field extension of Qp. Let\nXK be a smooth pro-variety over K. Let YK be a hyperbolic pro-curve over K. Let n\n′\n0 be\n89\nthe minimum transcendence degree over Qp of all finitely generated field extensions of Qp\nthat contain K. Let n′′0 be the transcendence degree over K of the function field of XK .\nLet n0\ndef\n= n′0 + 2(n\n′′\n0 − 1) + 1.\nLet n ≥ 3n0 + 5. Then every continuous open homomorphism\nθ : ΠnXK → Π\nn\nYK\nover ΓK induces a dominant morphism μ : XK → YK whose induced morphism on\nfundamental groups coincides (up to composition with an inner automorphism arising\nfrom ΔYK ) with the morphism Δ\nn−3−3n0\nXK\n→ Δn−3−3n0YK defined by considering θ “mod-\nulo Δ{n − 3 − 3n0}.”\nProof. The reason for the inclusion of the extra “3n0” (i.e., a “price” of three steps for\nevery additional transcendence degree that is used, plus an extra “tax” of three steps for\nallowing the prefix “pro”) is the following: in order to reduce to the situation discussed\nin the proof of Theorem 18.1, we need to show (in the present “pro” context) that inertia\ngroups are annihilated. Moreover, to apply the first “inertia annihilation argument” (at\nthe beginning of Section 14), we needed to know that after one passes to some covering\nY ′K → YK arising from an open subgroup of Πn−3YK (i.e., we wish to apply the arguments\nof Theorem 18.1 for n − 3), one knows “preservation of relations” for Y ′K . Thus, already\none needs some extra padding – to the tune of three steps (necessary, as we saw in the\nfirst two paragraphs of the proof of Theorem 18.1, to derive “preservations of relations”\nfor Y ′K). (This accounts for the 1 in the definition of n0.) Moreover, each time one adds a\ntranscendence degree, one needs to apply the “inertia annihilation argument” of the first\nhalf of Section 15. Thus, by the same line of reasoning, one needs extra padding consisting\nof three steps. (This accounts for the n′0 in the definition of n0.) In Section 16, one uses not\nonly the transcendence degrees inherent in K, but also 2(n′′0 − 1) auxiliary transcendence\ndegrees – here the “2 = 1 + 1” comes from the transcendence degree of “K ′” over “K,”\nplus the transcendence degree of the pencil “ξ.” (This accounts for the 2(n′′0 − 1) in the\ndefinition of n0.) Finally, we remark that although in Sections 14, 15, and 16, we assumed\nthat the morphism of fundamental groups “θ” was surjective, it is easy to see that this\nassumption is merely cosmetic, i.e., is inessential and serves only to simplify the discussion.\nThis completes the proof of the Theorem. ©\nRemark. Thus, the essential difference between Theorems 18.1 and 18.2 is that in Theo-\nrem 18.2, we allow “pro-objects,” at the cost of having to apply “annihilation of inertia”\narguments, which require us to use larger quotients of ΔXK , ΔYK (i.e., each application of\nannihilation of inertia costs three units of “n”).\nRemark. We do not mean to pretend that the estimates in the above two Theorems (e.g.,\nthe “5’s,” “n0,” etc.) are the best possible. Especially if one is willing to add hypotheses\n90\nto YK , it should not be so difficult to improve these estimates. The point of the above two\nTheorems is simply to illustrate the principle involved.\nWe conclude this Section with two technical lemmas that were used in the proofs of\nthe above two Theorems.\nLemma 18.3. Let XK be a proper hyperbolic curve over a field K of characteristic zero.\nLet ΔX be (as usual) the maximal pro-p quotient of its geometric fundamental group. Let\nΞX\ndef\n= ΔX\/[ΔX , [ΔX ,ΔX ]]. Then the natural morphism\nH2(ΞX ,Zp(1))\nΓK → H2(ΔX ,Zp(1))ΓK\n(induced by the quotient map ΔX → ΞX) is surjective. Here, the superscripted “ΓK”\ndenotes “the submodule of ΓK-invariants.”\nProof. Let us first consider the morphism\nH2(ΞX ,Zp(1)) → H2(ΔX ,Zp(1))\nTo do this, we shall use the Hochschild-Serre spectral sequence for the quotient ΞX → HX\n(where, as usual, we write HX for the abelianization of ΔX). Write ΨX ⊆ ΞX for the kernel\nof ΞX → HX . Thus, ΨX may be identified with a certain well-understood (cf. Lemma\n3.1) quotient of ∧2 HX . Consideration of the E··2 -term of this spectral sequence shows\nthat there is a natural ΓK -equivariant injection of the cokernel of the natural morphism\n(induced by the quotient ∧2 HX → ΨX)\nH1(ΨX ,Zp(1)) = HomZp(ΨX ,Zp(1)) → H2(HX ,Zp(1)) = HomZp(∧2HX ,Zp(1))\ninto H2(ΞX ,Zp(1)). (In other words, the above morphism is the differential (from E\n0,1\n2 to\nE\n2,0\n2 ) of the “E\npq\n2 = H\np(HX ,H\nq(ΨX ,Zp(1)))”-term of the spectral sequence.) Moreover,\nit follows from Lemma 3.1 that the cokernel of this natural morphism is precisely the\nquotient of H2(HX ,Zp(1)) = (∧2 H1(HX ,Zp))(1) given by the (surjective) cup-product\nmap (∧2 H1(HX ,Zp))(1) → H2(ΔX ,Zp)(1). Thus, in summary, we have a ΓK-equivariant\ndiagram:\nH2(ΞX ,Zp(1)) ⊇ Image(H2(HX ,Zp(1))) → H2(ΔX ,Zp(1))\nin which the arrow is bijective. This implies the assertion stated in the Lemma. ©\n91\nLemma 18.4. Let K, XK , ΔX , and ΞX be as in Lemma 18.3. Since the kernel of\nΔX → ΞX is a normal subgroup not only of ΔX , but also of ΠXK , write ΠXK → ΠΞXK for\nthe quotient of ΠXK by this subgroup. Then the natural morphisms\nH2(ΠΞXK ,Zp(1)) → H\n2(ΠXK ,Zp(1))\nand\nH2(ΠXK ×ΓK Π\nΞ\nXK\n,Zp(1)) → H2(ΠXK ×ΓK ΠXK = ΠXK×KXK ,Zp(1))\nare surjective. In particular, if η ∈ H2(ΠXK×KXK ,Zp(1)) denotes the first Chern class of\nthe diagonal in XK ×K XK , then η lies in the image of H2(ΠXK ×ΓK ΠΞXK ,Zp(1)).\nProof. The assertions of this Lemma follow by considering the consequences of the sur-\njectivity assertion of Lemma 18.3 for the Hochschild-Serre spectral sequences associated\nto ΠΞXK → ΓK ; ΠXK → ΓK ; ΠXK ×ΓK ΠΞXK → ΓK ; and ΠXK ×ΓK ΠXK → ΓK . ©\nSection 19: Injectivity Result\nIn this Section, we prove the following “pro-p injectivity part of the so-called Section\nConjecture”:\nTheorem 19.1. Let K be sub-p-adic (cf. Definition 15.4 (i)). Let XK be a hyperbolic\ncurve over K. Let XK(K) be the set of K-valued points of XK . Let Sect(ΓK ,ΠXK ) be\nthe set of sections ΓK → ΠXK of ΠXK → ΓK , considered up to composition with an inner\nautomorphism arising from ΔXK . Then the natural map\nXK(K) → Sect(ΓK ,ΠXK )\nis injective.\nRemark. The so-called “Section Conjecture” states that (for instance when XK is proper)\nthe natural map in Theorem 19.1 is bijective. (When XK is affine, the statement must be\nmodified slightly.) At the time of writing (July 1996), the Section Conjecture has not yet\nbeen proven.\nProof. (of Theorem 19.1) Let HX be the abelianization of ΔX . First, note that by\nreplacing XK by some finite Galois étale covering of XK of p-power order, we may assume\n92\nthat the compactification XK of XK is itself hyperbolic and non-hyperelliptic. Let r be\nthe number of points in (XK − XK)(K). Let ZX be the complement of the diagonal in\nXK ×K XK . By taking the second projection ZX → XK , we may regard ZX as a family\n(parametrized by XK) of smooth hyperbolic curves obtained by removing r + 1 distinct\npoints from some compactification.\nLet ηX be the generic point of XK . Let ηX be the spectrum of some algebraic closure\nof K(ηX). Let ZηX\ndef\n= ZX ×X ηX . Then (since ZX → XK is a family of smooth hyperbolic\ncurves obtained by removing precisely r + 1 distinct points from some compactification)\nwe obtain an exterior Galois representation\nρZ : Π\nprf\nXK\n→ Out(ΔZηX )\nNow I claim that ρZ factors through ΠXK . Indeed, to see this, note that if HZηX\nis the\nabelianization of ΔZηX\n, then since ΔZηX\nis a pro-p group, the kernel of\nΞ : Out(ΔZηX\n) → Aut(HZηX )\nis itself a pro-p group. Thus, it suffices to show that Ξ◦ρZ maps ΔprfX into a pro-p subgroup\nof Aut(HZηX\n). But now observe that one has a Galois-equivariant surjection HZηX\n→ HX\nwhose kernel is either 0 or Zp(1) (depending on whether r = 0 or r > 0). Moreover,\nthe actions of Δ\nprf\nX on HX and Zp(1) are trivial. Thus, it follows immediately that the\nΔ\nprf\nX -action on HZηX\nis by unipotent matrices, so Ξ ◦ ρZ maps ΔprfX into a pro-p subgroup\nof Aut(MZ ), as desired. This completes the proof of the claim.\nThus, we have a representation\nρZ : ΠXK → Out(ΔZηX )\nNow suppose that we have two points α, β ∈ XK(K) that induce the same element φ ∈\nSect(ΓK ,ΠXK ). Let Zα (respectively, Zβ) be the pull-back of ZX → X via α (respectively,\nβ). Thus, Zα and Zβ are hyperbolic curves over K. Moreover, since the action of ΓK on\nΔZα (respectively, ΔZβ) is determined by composing φ with ρZ , it follows that there exists\nsome isomorphism ψ : ΔZα\n∼= ΔZβ such that (i) ψ is compatible with the respective outer\nΓK -actions; (ii) ψ preserves and induces the identity between the quotients ΔZα → HX ,\nΔZβ → HX . Thus, by Theorem 16.5 (in fact, really, all we need is Theorem A of [Mzk2]),\nit follows that ψ arises from some isomorphism Zα ∼= Zβ which is compatible with and\ninduces the identity on the inclusions Zα ↪→ XK , Zβ ↪→ XK . But this clearly implies that\nα = β, thus completing the proof of the Theorem. ©\nRemark. Note that in fact, in the proof of Theorem 19.1, we really only used the weaker\nresults of [Mzk2] – i.e., we did not need to use Theorem A of the present paper. On the\n93\nother hand, at the time that [Mzk2] was written, the author was unaware of Theorem 19.1,\nwhich is the primary reason that Theorem 19.1 did not appear in [Mzk2]. Note also that\nalthough Theorem 19.1 implies a corresponding profinite result, the profinite result can\nbe proven much more easily, by using the “Kummer exact sequence” for the Jacobian of\nXK . Finally, it should be remarked that the argument employed in the proof of Theorem\n19.1 (as well as the Kummer sequence argument just mentioned) have been well-known\nfor some time (see, e.g., [Naka2]). The main reason that Theorem 19.1 was included in\nthis paper was that the author just wanted to make explicit that a pro-p injectivity result\n(such as Theorem 19.1) could now be proven.\nAPPENDIX: A Grothendieck Conjecture-Type Result for\nCertain Hyperbolic Surfaces\nSection a0: Introduction\nIn this Appendix, we show how Theorem A (cf. the Introduction) can be used to\nprove a Grothendieck Conjecture-type result for certain types of surfaces. The surfaces\nconsidered are families of (smooth) hyperbolic curves that are parametrized by hyperbolic\ncurves (cf. Definition a2.1). We call such surfaces hyperbolically fibred. Our notation is\nsimilar to that of the rest of the paper, except that since here we consider only profinite\n(i.e., not pro-p) fundamental groups, in this Appendix, we will write ΠXK for the profinite\nfundamental group of XK :\nNotation: If K is a field and XK is a K-scheme, we denote by ΠXK\ndef\n= π1(XK) the\nfundamental group of XK (for some choice of base-point), and by ΓK the absolute Galois\ngroup of K. Then we have a natural morphism ΠXK → ΓK whose kernel ΔX ⊆ ΠXK is\nthe geometric fundamental group π1(XK ⊗K K) (where K is an algebraic closure of K).\nOur main theorem is the following:\nTheorem D. Let K be sub-p-adic (cf. Definition 15.4 (i)). Let XK and YK be\nhyperbolically fibred surfaces over K. Let IsomK(XK , YK) be the set of K-isomorphisms\n(in the category of K-schemes) between XK and YK. Let IsomΓK (ΠXK ,ΠYK ) be the set of\ncontinuous group isomorphisms ΠXK → ΠYK over ΓK , considered up to composition with\nan inner automorphism arising from ΔY . Then the natural map\nIsomK(XK , YK) → IsomΓK (ΠXK ,ΠYK )\n94\nis bijective.\nThis Theorem is given as Theorem a2.4 in the text.\nWe remark that:\n(1) Theorem D above is (modulo a certain technical result – Lemma a1.1)\nessentially derived from Theorem A. Note that one needs the full power\nof the “Hom version” of the Grothendieck Conjecture for hyperbolic\ncurves (i.e., Theorem A) in order to prove an “Isom-type result” for\nsurfaces (i.e., Theorem D above). That is to say, the “Isom version”\nof Theorem A of the present paper – i.e., Theorem A of [Mzk2] – is\nnot sufficient to prove Theorem D above. Moreover, even with the\n“Hom version” of the Grothendieck Conjecture for hyperbolic curves\n(i.e., Theorem A), at the present time, I am unable to prove a “Hom\nversion” of Theorem D.\n(2) One might ask whether Theorem D can be extended to hyperbolically\nfibered varieties (i.e., varieties obtained as successive fibrations of hy-\nperbolic curves) of higher dimension. The problem is that just as we\nneeded (see Remark (1)) the “Hom version” in dimension one to prove\nan “Isom-type result” in dimension two, we would need a “Hom-type\nresult” in dimension two – which is currently not available – in order\nto prove an “Isom-type result” in, say, dimension three. Thus, at the\npresent time, we are unable to advance beyond dimension two.\n(3) Since Theorem A is valid in the pro-p case as well, one might ask why one\ncannot prove a pro-p version of Theorem D above. The problem is that\nthe process of passing to the maximal pro-p quotient is not well-behaved\nwith respect to fibrations. That is to say, if XK → X ′K is a family of\nhyperbolic curves parametrized by a hyperbolic curve (as in Definition\na2.1), then the maximal pro-p quotient of the fundamental group of\nthe geometric generic fiber of XK → X ′K does not map injectively (in\ngeneral) to the maximal pro-p quotient of ΔX . Thus, any attempt to\nprove a pro-p version of Theorem D by means of the techniques employed\nhere would result in a rather unnatural theorem.\n(4) Since Theorem A admits various truncated versions (cf. the Introduc-\ntion) as well, one might ask why one cannot prove a truncated version\nof Theorem D. The problem here is the same as the problem that arises\nwhen one tries to prove a pro-p result: i.e., truncating is not well-behaved\nwith respect to fibrations.\n(5) To a slight extent, the content of Theorem D above overlaps with recent\nresults of H. Nakamura and N. Takao (see Theorem A and Corollary B\nof [NT]).\n95\nSection a1: A Key Lemma\nIn this Section, we prove a simple technical lemma which will be the key technical\ningredient that allows us to extend Theorem A to the case of surfaces. Let K be an\nalgebraically closed field of characteristic zero. Let X and Y be hyperbolic curves over K\n(cf. Section 0 for a definition of this term).\nLemma a1.1. Let φ : X → Y be a finite K-morphism. Let ψ : ΠXK → ΠYK be the\ninduced morphism on fundamental groups. Suppose that Ker(ψ) is topologically finitely\ngenerated. Then φ is étale.\nProof. By replacing Y by the finite étale covering of Y corresponding to Im(ψ) ⊆ ΠYK ,\nwe may assume that ψ is surjective. Under this assumption, φ is étale if and only if φ is\nan isomorphism. Thus, it suffices to assume that:\n(1) Ker(φ) is topologically finitely generated; and\n(2) φ is not an isomorphism (hence has degree > 1)\nand derive a contradiction.\nLet Y ′ → Y be a finite étale covering Y , where Y ′ is connected. Let X ′ def= X ×Y Y ′.\nThen X ′ → X is finite étale, and it follows from the assumption that ψ is surjective that\nX ′ is connected. Since Y is hyperbolic, it follows that for a suitable choice of Y ′ → Y , Y ′\nwill have genus ≥ 2. Thus, by replacing our original X → Y by X ′ → Y ′, we may assume\nthat X ′ and Y ′ have genus ≥ 2.\nSince X → Y is assumed to have degree > 1, it follows that gX (the genus of X)\nis strictly greater than gY (the genus of Y ). Let Y\n′ → Y be a finite étale covering of\ndegree d such that Y ′ is connected, and Y ′ → Y extends to an étale covering over some\ncompactification of Y . Then the Riemann-Hurwitz formula implies that, as Y ′ → Y varies,\ngY ′ (which we think of as a function of d) is equal to d(gY − 1) + 1. On the other hand,\nif X ′ def= X ×Y Y ′, then (again by the Riemann-Hurwitz formula) gX ′ = d(gX − 1) + 1.\nThus, it follows that as d → ∞, the difference gX ′ − gY ′ → ∞.\nNow let HX ′ denote the first homology group of the compactification of X\n′ with\ncoefficients in Zp (for some prime number p which will be fixed throughout the discussion).\nThen ψ induces a surjection H ′ψ : HX ′ → HY ′ . Thus, Ker(H ′ψ) is a free Zp-module of rank\n2(gX ′ − gY ′). On the other hand, any topological generators of Ker(ψ) clearly define\na set of Zp-generators of Ker(H\n′\nψ). But this implies that rankZp(Ker(H\n′\nψ)) is bounded,\nindependently of Y ′ → Y , which contradicts the fact that rankZp(Ker(H ′ψ)) = 2(gX ′ −\ngY ′) → ∞ as d → ∞. This contradiction completes the proof. ©\n96\nSection a2: The Main Theorem\nLet K be a field of characteristic zero. Let XK be a surface over K, by which we\nmean that XK is a smooth (geometrically connected) variety over K of dimension two.\nDefinition a2.1. We shall say that XK is a hyperbolically fibred surface if the following\ncondition holds: There exists a hyperbolic curve X ′K over K, together with a smooth,\nproper, connected morphism XK → X ′K of relative dimension one such that XK embeds\nas an open subvariety of XK satisfying the following conditions: (i) XK −XK is a divisor\nin XK which is étale over X\n′\nK ; (ii) the geometric fibers of XK ⊆ XK → X ′K are hyperbolic\ncurves.\nIf XK is a hyperbolically fibred surface, then we shall refer to XK → X ′K (as above)\nas a parametrizing morphism for XK .\nNote that given a hyperbolically fibered surface XK , in general, there can exist more than\none parametrizing morphism for XK . If XK → X ′K is a parametrizing morphism for\nXK , and FΩ ⊆ XK is a fiber of this morphism (over some point of XK(Ω) valued in an\nalgebraically closed field Ω), then we have an exact sequence of fundamental groups:\n1 → ΠFΩ → ΠXK → ΠX ′K → 1\nNote that since FΩ and X\n′\nK are hyperbolic curves, it follows that ΠFΩ and ΠX ′K\n(hence\nalso ΠXK ) are topologically finitely generated.\nNow let K be sub-p-adic (cf. Definition 15.4 (i)) (for some prime number p), and let\nXK and YK be hyperbolically fibred surfaces over K. Let\nφ : ΠXK → ΠYK\nbe a continuous group isomorphism over ΓK . We would like to show that φ arises geomet-\nrically.\nLet ζY : YK → Y ′K be a parametrizing morphism for YK . Then by composing φ\nwith π1(ζY ), we obtain a continuous surjection ΠXK → ΠY ′K . By Theorem A, it follows\nthat this surjection arises geometrically, from some morphism XK → Y ′K . Thus, we may\nregard XK and YK as objects over Y\n′\nK . Let ZK be the normalization of Y\n′\nK in XK . Thus,\nthe morphism XK → Y ′K factors through δ : ZK → Y ′K . Observe that ZK is a smooth,\ngeometrically connected (since XK is geometrically connected over K) curve over K, and\nthat ZK → Y ′K is finite. (In particular, since Y ′K is hyperbolic, it follows that ZK is also\nhyperbolic.) Moreover, it follows from the definition of ZK that the morphism ΠXK →\nΠZK induced by XK → ZK is surjective. This implies that Ker(π(δ)) is a quotient of\nKer(ΠXK → ΠY ′K )\n∼= Ker(ΠYK → ΠY ′K ), which (by the definition of a hyperbolically fibred\n97\nsurface) is topologically finitely generated. Thus, it follows that Ker(π(δ)) is topologically\nfinitely generated. But then Lemma a1.1 implies that ZK → Y ′K is étale. On the other\nhand, since ΠXK → ΠY ′K is surjective, we thus see that ZK → Y\n′\nK must be an isomorphism.\nIn particular, we thus conclude that the generic fiber of XK → Y ′K is smooth, geometrically\nconnected, and of dimension one.\nIn fact, the argument of the preceding paragraph can be applied more generally to\ncoverings of XK , as follows: Let Ỹ\n′\nK → Y ′K be a finite étale covering such that Ỹ ′K is\ngeometrically connected over K. Let ỸK → YK ×Y ′\nK\nỸ ′K be a finite étale covering such that\nỸK → Ỹ ′K is geometrically connected. Let X̃K → XK be the covering corresponding (via\nφ) to ỸK → YK . Thus, X̃K → XK factors through X̃K → XK ×Y ′\nK\nỸ ′K . Then I claim that\nX̃K → Ỹ ′K is geometrically connected. Indeed, this follows by the same argument as that\nemployed in the preceding paragraph: Namely, we simply observe that Π\nX̃K\n→ Π\nỸ ′\nK\nis\nisomorphic to Π\nỸK\n→ Π\nỸ ′\nK\n, which is surjective with topologically finitely generated kernel.\nThis proves the claim. Let L be the function field of Y ′K ; let L be its algebraic closure.\nNote that the natural map Spec(L) → Y ′K factors through Ỹ ′K . Moreover, the above claim\nimplies that if we base-change X̃K → Ỹ ′K via Spec(L) → Ỹ ′K , the resulting X̃K×Ỹ ′\nK\nSpec(L)\nis connected.\nNow let us reinterpret the conclusions of the preceding paragraph in terms of funda-\nmental groups. Let ηY ′ be the generic point of Y\n′\nK . Let YηY ′\ndef\n= YK ×Y ′\nK\nηY ′ ; XηY ′\ndef\n=\nXK ×Y ′\nK\nηY ′ . Then what we did in the preceding paragraph implies precisely that the\nmorphism\nKer(π1(XηY ′ ) → π1(ηY ′)) → Ker(π1(YηY ′ ) → π1(ηY ′)) = Ker(ΠYK → ΠY ′K )\n(induced by φ) is surjective. Thus, we see that by composing the natural surjection of\nπ1(XηY ′ ) onto ΠXK with φ, we obtain a continuous surjective group homomorphism\nφηY ′ : ΠXη\nY ′\n→ ΠYη\nY ′\nover ΓηY ′ (where we regard XηY ′ and YηY ′ as curves over ηY ′). Now we would like to\napply Theorem A again. This Theorem tells us that φηY ′ arises geometrically from some\nηY ′ -morphism XηY ′ → YηY ′ .\nRemark. Note that in the argument of the preceding three paragraphs, it was absolutely\nessential to invoke Lemma a1.1 before we could apply Theorem A. In fact, if the variety\n“on the left” (i.e., in this case, XηY ′ ) is not geometrically connected over the base field\n(i.e., in this case, L), then it is not difficult to see that Theorem A does not hold. Indeed,\nto construct such a counterexample, suppose that YK is proper, and let HK ⊆ YK be a\n98\nhyperplane section (with respect to some projective embedding of YK) such that HK is\nsmooth and geometrically connected over K, and π1(HK) → π1(YK) is surjective. (Such\nan HK exists by the Lefshetz hyperplane theorem.) Write ηH for the generic point of\nHK , and take for our “XηY ′ ” any hyperbolic curve over ηH . Then note that (if HK is\nsufficiently generic so that every fiber of HK → Y ′K contains at least one point at which\nHK → Y ′K is étale, then) the induced morphism π1(ηH) → π1(YηY ′ ) is also surjective, so we\nget a surjective morphism π1(XηY ′ ) → π1(YηY ′ ) (by composing the (necessarily surjective)\nstructure morphism π1(XηY ′ ) → π1(ηH) with π1(ηH) → π1(YηY ′ )). But this surjective\nmorphism does not arise from a dominant morphism XηY ′ → YηY ′ .\nThis completes the portion of the proof which is an application of (the nontrivial, sur-\njectivity part of) Theorem A. The remainder of the proof will consists of using elementary\nalgebraic geometry to show that the morphism XηY ′ → YηY ′ constructed above extends to\na morphism XK → YK .\nLemma a2.2. The morphism XηY ′ → YηY ′ above extends to a morphism XK → YK .\nProof. First, observe that there exists a finite étale covering VK → YK with the following\nproperties:\n(1) VK is a hyperbolically fibered surface that admits a parametrizing mor-\nphism VK → V ′K that fits into a commutative diagram:\nVK −→ V ′K⏐⏐\u0005 ⏐⏐\u0005\nYK −→ Y ′K\n(2) The fibers of VK → V ′K have genus ≥ 2.\nLet UK → XK be the finite étale covering that corresponds (via φ) to VK → YK . Let\nV K → V ′K be the family of proper hyperbolic curves that compactifies VK → V ′K (as in\nDefinition a2.1). Write ηV ′ for the generic point of V\n′\nK . Note that there exists a natural map\nUK → V ′K covering XK → Y ′K , and that both UK and VK are geometrically connected over\nηV ′ (cf. the argument used above to show that X̃K → Ỹ ′K is geometrically connected).\nThus, we may form (geometrically connected) ηV ′ -curves UηV ′ , VηV ′ . Moreover, by the\ndefinition of UK , we have a commutative diagram\nUηV ′ −→ VηV ′ −→ V ηV ′⏐⏐\u0005 ⏐⏐\u0005\nXηY ′ −→ YηY ′\n99\nThus, in particular, we obtain a morphism UηV ′ → V ηV ′ (over ηV ′). I claim that this\nmorphism extends to a morphism UK → V K . Indeed, by the elementary theory of surfaces\n– “elimination of indeterminacy” (see, e.g., Theorem 5.5 of Chapter V of [Harts]) – it follows\nthat there exists some ŨK → UK (obtained by successively blowing up smooth points) such\nthat the birational transformation from UK to V K defined by UηV ′ → V ηV ′ extends to a\nmorphism ŨK → V K . On the other hand, any exceptional curve in ŨK necessarily maps\ninto a fiber of V K → V ′K (this follows since UK → V ′K is already a morphism). Thus, any\nexceptional P1 in ŨK maps quasi-finitely into a fiber of V K → V ′K (which will always be a\nsmooth, proper, hyperbolic curve) – which is clearly absurd. This contradiction completes\nthe proof of the claim.\nThus, we have a morphism UK → V K . Let Y K → Y ′K compactify YK → Y ′K (as in\nDefinition a2.1). Then clearly VK → YK extends to a finite (in general, ramified) morphism\nV K → Y K . Thus, to summarize, we have a rational map from XK to Y K which is covered\nby a morphism UK → V K , where UK (respectively, V K) is finite over XK (respectively,\nY K). By elementary algebraic geometry, this implies that we get a morphism XK → Y K\n(covered by UK → V K).\nTo complete the proof of the Lemma, we must verify that this morphism XK → Y K\nfactors through YK . To do this, we simply choose VK → YK in the above argument\nsuch that VK is ramified, with very large (say, compared to the degree of XK → Y K)\nramification index, over all of the divisor Y K − YK . (Note that such coverings exist, by\nthe exact sequence of fundamental groups following Definition a2.1, plus the well-known\nfact from topology that the fundamental group of a compact surface with finitely many\npunctures has coverings with arbitrarily large ramification index at those punctures.) Then\nsince XK → Y K is covered by UK → V K , and UK → XK is finite étale, we see that we\nobtain a contradiction, unless XK → Y K maps into YK . Thus, we conclude that we have\na morphism XK → YK , as desired. ©\nLet us summarize what we have done so far. We started with a continuous group\nisomorphism\nφ : ΠXK → ΠYK\nover ΓK . From φ, we constructed a K-morphism XK → YK , which we denote by α. Since\nφ is invertible, we thus see that we have also constructed a K-morphism β : YK → XK\nfrom φ−1. Since everything we have been doing is functorial, it follows that π1(β ◦ α)\nis the identity on ΠXK (up to composition with an inner automorphism arising from an\nelement of ΔX). Thus, by Lemma a2.3 below, we conclude that β ◦ α is the identity on\nXK . Similarly, α ◦ β is the identity on YK .\nLemma a2.3. Let γ : XK → XK be a K-morphism such that π1(γ) is the identity\non ΠXK (up to composition with an inner automorphism arising from an element of ΔX).\nThen γ is the identity.\n100\nProof. First observe that π1(γ) is trivially compatible (up to composition with a geometric\ninner automorphism) with the map induced by π1’s by XK → X ′K . Thus, (the injectivity\npart of) Theorem A tells us that γ is compatible with XK → X ′K . Let ηX ′ be the\ngeneric point of X ′K . Then γ defines a morphism γηX′ : XηX′ → XηX′ which induces the\nidentity (up to composition with a geometric inner automorphism) on π1’s. Applying (the\ninjectivity part of) Theorem A again then tells us that γηX′ is the identity. Thus, it follows\nthat γ is the identity, as desired. ©\nThus, α and β are isomorphisms. 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Shiho, manuscript in preparation.\n[Tama] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio Math. 109,\nNo. 2 (1997), pp. 135-194.\n[Tate] J. Tate, p-Divisible Groups, Proceedings of a Conference on Local Fields, Driebergen,\nSpringer Verlag (1967), pp. 158-183.\n103\n"}
{"index":{}}
{"id":"uUZPG116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Absolute%20Anabelian%20Geometry.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":78,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Absolute%20Anabelian%20Geometry","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["absanab.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Sat Nov 28 12:47:52 UTC 2015","dates_in_content_count_i":3,"dates_in_content_dts":["Mon Mar 01 00:00:00 UTC 2010","Tue Mar 01 00:00:00 UTC 2011","Thu Mar 01 00:00:00 UTC 2012"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-6250677665875785164,"exact_signature_unique_b":true,"fuzzy_signature_l":705254436867562237,"fuzzy_signature_unique_b":true,"wordcount_i":15171,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:50 UTC 2022","fresh_date_dt":"Fri Sep 12 18:50:48 UTC 2025","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":349099,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086037333442560,"text_t":"The Absolute Anabelian Geometry of\nHyperbolic Curves\nShinichi Mochizuki\nContents:\n§0. Notations and Conventions\n§1. Review of Anabelian Geometry\n§1.1. The Anabelian Geometry of Number Fields\n§1.2. The Anabelian Geometry of p-adic Local Fields\n§1.3. The Anabelian Geometry of Hyperbolic Curves\n§2. Reconstruction of the Logarithmic Special Fiber\nAppendix: Terminology of Graph Theory\nIntroduction\nLet XK be a hyperbolic curve (cf. §0 below) over a field K of characteristic 0.\nDenote its algebraic fundamental group by ΠXK . Thus, we have a natural surjection\nΠXK \u0002 GK\nof ΠXK onto the absolute Galois group GK of K.\nWhen K is a finite extension of Q or Qp, and one holds GK fixed, then it is known\n(cf. [Tama1], [Mzk6]; Theorem 1.3.4 of the present manuscript) that one may recover\nthe curve XK in a functorial fashion from ΠXK . This sort of result may be thought of\nas a “relative result” (i.e., over GK). Then the question naturally arises:\nTo what extent are the “absolute analogues” of this result valid — i.e., what\nif one does not hold GK fixed?\nIf K is a number field, then it is still possible to recover XK from ΠXK (cf. Corollary\n1.3.5), by applying the theorem of Neukirch-Uchida (cf. Theorem 1.1.3). On the other\n1\n2 SHINICHI MOCHIZUKI\nhand, when K is a p-adic local field (i.e., a finite extension of Qp), the analogue of the\ntheorem of Neukirch-Uchida fails to hold, and indeed, it is the opinion of the author at\nthe time of writing that it is unlikely (in the p-adic local case) that one can recover XK\nin general (i.e., in the fashion of Corollary 1.3.5) from ΠXK .\nIn the present manuscript, we begin by reviewing\/surveying in §1 the anabelian\ngeometry of number fields, p-adic local fields, and hyperbolic curves from the point of\nview of the goal of understanding to what extent the anabelian geometry of hyperbolic\ncurves over p-adic local fields can be made “absolute”. Our main result (Theorem\n2.7), given in §2, states that when K is a p-adic local field, (although we may be unable\nto recover XK itself) one may recover (in a functorial fashion) the special fiber of\nXK , together with its natural log structure, in an absolute fashion, i.e., solely from\nthe isomorphism class of the profinite group ΠXK .\nAcknowledgements: I would like to thank A. Tamagawa for the time that he so gen-\nerously shared with me in numerous stimulating discussions, and especially for the\nfollowing: (i) informing me of the arguments used to prove Lemma 1.1.4 in §1.1; (ii) ex-\nplaining to me the utility of a theorem of Raynaud in the context of §2 (cf. Lemma 2.4).\nAlso, I would like to thank F. Oort, as well as the referee, for various useful remarks.\nSection 0: Notations and Conventions\nNumbers:\nWe will denote by N the set of natural numbers, by which we mean the set of\nintegers n ≥ 0. A number field is defined to be a finite extension of the field of rational\nnumbers Q.\nTopological Groups:\nLet G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us write\nZG(H)\ndef\n= {g ∈ G | g · h = h · g, ∀ h ∈ H}\nfor the centralizer of H in G;\nNG(H)\ndef\n= {g ∈ G | g · H · g−1 = H}\nfor the normalizer of H in G; and\nCG(H)\ndef\n= {g ∈ G | (g · H · g−1)\n⋂\nH has finite index in H, g · H · g−1}\nABSOLUTE ANABELIAN GEOMETRY 3\nfor the commensurator of H in G. Note that: (i) ZG(H), NG(H) and CG(H) are\nsubgroups of G; (ii) we have inclusions\nH, ZG(H) ⊆ NG(H) ⊆ CG(H)\nand (iii) H is normal in NG(H).\nNote that ZG(H), NG(H) are always closed in G, while CG(H) is not necessarily\nclosed in G. Indeed, one may construct such an example as follows: Let\nM\ndef\n=\n∏\nN\nZp\nendowed with the product topology (of the various copies of Zp equipped with their usual\ntopology). Thus, M is a Hausdorff topological group. For n ∈ N, write Fn(M) ⊆ M\nfor the sub-topological group given by the product of the copies of Zp indexed by\nm ≥ n. Write AutF (M) for the set of automorphisms of the topological group M\nthat preserve the filtration F ∗(M) on M . If α ∈ AutF (M), then for every n ∈ N,\nα induces a continuous homomorphism αn : M\/F\nn(M) → M\/Fn(M) which is clearly\nsurjective, hence an isomorphism (since M\/Fn(M) is profinite and topologically finitely\ngenerated — cf. [FJ], Proposition 15.3). It thus follows that α induces an isomorphism\nFn(M)\n∼→ Fn(M), hence that the inverse of α also lies in AutF (M). In particular, we\nconclude that AutF (M) is a group. Equip AutF (M) with the coarsest topology for which\nall of the homomorphisms AutF (M) → Aut(M\/Fn(M)) (where Aut(M\/Fn(M)) ∼=\nGLn(Zp) is equipped with its usual topology) are continuous. Note that relative to\nthis topology, AutF (M) forms a Hausdorff topological group. Now define G to be the\nsemi-direct product of M with AutF (M) (so G is a Hausdorff topological group), and\nH to be ∏\nn∈N\npn · Zp ⊆\n∏\nN\nZp = M\n(so H ⊆ G is a closed subgroup). Then CG(H) is not closed in G. For instance, if one\ndenotes by en ∈\n∏\nN Zp the vector with a 1 in the n-th place and zeroes elsewhere, then\nthe limit A∞ (where\nA∞(en)\ndef\n= en + en+1\nfor all n ∈ N) of the automorphisms Am ∈ CG(H) (where Am(en) def= en +en+1 if n ≤ m,\nAm(en)\ndef\n= en if n > m) is not contained in CG(H).\nDefinition 0.1.\n(i) Let G be a profinite group. Then we shall say that G is slim if the centralizer\nZG(H) of any open subgroup H ⊆ G in G is trivial.\n4 SHINICHI MOCHIZUKI\n(ii) We shall say that a continuous homomorphism of profinite groups G → H is\nrelatively slim if the centralizer in H of the image of every open subgroup of G is trivial.\n(iii) We shall say that a closed subgroup H ⊆ G of a profinite group G is commen-\nsurably (respectively, normally) terminal if the commensurator CG(H) (respectively,\nnormalizer NG(H)) is equal to H.\nRemark 0.1.1. Thus, a profinite group G is slim if and only if the identity morphism\nG → G is relatively slim. Moreover, if H ⊆ G is a closed subgroup such that the\ninclusion H ↪→ G is relatively slim, then both H and G are slim.\nRemark 0.1.2. It is a formal consequence of the definitions that:\ncommensurably terminal =⇒ normally terminal\nand that (if H ⊆ G is a closed subgroup of a profinite group G, then):\nH ⊆ G commensurably terminal, H slim =⇒\nthe inclusion H ↪→ G is relatively slim\nRemark 0.1.3. It was pointed out to the author by F. Oort that a profinite group is\nslim if and only if every open subgroup of G has trivial center. (Indeed, the necessity\nof this condition is clear. Its sufficiency may be shown as follows: If H ⊆ G is an open\nsubgroup, then for any h ∈ ZG(H), let H ′ ⊆ G by the (necessarily open) subgroup\ngenerated by H and h. Thus, h lies in the center of H ′, which is trivial by assumption.)\nThis property of a profinite group — i.e., that every open subgroup be center-free —\nwas investigated in detail in [Naka1](cf., e.g., Corollaries 1.3.3, 1.3.4).\nCurves:\nSuppose that g ≥ 0 is an integer. Then a family of curves of genus g\nX → S\nis defined to be a smooth, proper, geometrically connected morphism X → S whose\ngeometric fibers are curves of genus g.\nSuppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denote the\nmoduli stack of r-pointed stable curves of genus g (where we assume the points to be\nunordered) by Mg,r (cf. [DM], [Knud] for an exposition of the theory of such curves;\nstrictly speaking, [Knud] treats the finite étale covering of Mg,r determined by ordering\nABSOLUTE ANABELIAN GEOMETRY 5\nthe marked points). The open substack Mg,r ⊆ Mg,r of smooth curves will be referred\nto as the moduli stack of smooth r-pointed stable curves of genus g or, alternatively, as\nthe moduli stack of hyperbolic curves of type (g, r).\nA family of hyperbolic curves of type (g, r)\nX → S\nis defined to be a morphism which factors X ↪→ Y → S as the composite of an open\nimmersion X ↪→ Y onto the complement Y \\D of a relative divisor D ⊆ Y which is finite\nétale over S of relative degree r, and a family Y → S of curves of genus g. One checks\neasily that, if S is normal, then the pair (Y,D) is unique up to canonical isomorphism.\n(Indeed, when S is the spectrum of a field, this fact is well-known from the elementary\ntheory of algebraic curves. Next, we consider an arbitrary connected normal S on which\na prime l is invertible (which, by Zariski localization, we may assume without loss of\ngenerality). Denote by S ′ → S the finite étale covering parametrizing orderings of the\nmarked points and trivializations of the l-torsion points of the Jacobian of Y . Note that\nS ′ → S is independent of the choice of (Y,D), since (by the normality of S), S ′ may be\nconstructed as the normalization of S in the function field of S ′ (which is independent\nof the choice of (Y,D) since the restriction of (Y,D) to the generic point of S has\nalready been shown to be unique). Thus, the uniqueness of (Y,D) follows by considering\nthe classifying morphism (associated to (Y,D)) from S ′ to the finite étale covering of\n(Mg,r)Z[ 1l ] parametrizing orderings of the marked points and trivializations of the l-\ntorsion points of the Jacobian [since this covering is well-known to be a scheme, for l\nsufficiently large].) We shall refer to Y (respectively, D; D; D) as the compactification\n(respectively, divisor at infinity; divisor of cusps; divisor of marked points) of X. A\nfamily of hyperbolic curves X → S is defined to be a morphism X → S such that the\nrestriction of this morphism to each connected component of S is a family of hyperbolic\ncurves of type (g, r) for some integers (g, r) as above.\nSection 1: Review of Anabelian Geometry\n§1.1. The Anabelian Geometry of Number Fields\nIn this §, we review well-known anabelian (and related) properties of the Galois\ngroups of number fields and (mainly p-adic) local fields.\nLet F be a number field. Fix an algebraic closure F of F and denote the resulting\nabsolute Galois group of F by GF . Let p be a (not necessarily nonarchimedean!) prime\nof F . Write Gp ⊆ GF for the decomposition group (well-defined up to conjugacy)\n6 SHINICHI MOCHIZUKI\nassociated to p and Fp for the completion of F at p. Then we have the following result\n(cf. also Corollary 1.3.3 below for a generalization of the slimness of Gp, GF ):\nTheorem 1.1.1. (Slimness and Commensurable Terminality) Suppose that p\nis nonarchimedean. Then:\n(i) The closed subgroup Gp ⊆ GF is commensurably terminal.\n(ii) The inclusion Gp ↪→ GF is relatively slim. In particular, Gp, GF are slim.\nProof. Assertion (i) is a formal consequence of [NSW], Corollary 12.1.3. As for assertion\n(ii), the slimness of Gp follows from local class field theory (cf., e.g., [Serre2]). (That is,\nif σ ∈ Gp commutes with an open subgroup H ⊆ Gp, then σ induces the trivial action\non the abelianization Hab. But, by local class field theory, Hab may be identified with\nthe profinite completion of K×, where K is the finite extension of Fp determined by\nH. Thus, σ acts trivially on all sufficiently large finite extensions K of Fp, so σ = 1,\nas desired.) Relative slimness thus follows formally from the slimness of Gp and (i) (cf.\nRemark 0.1.2). ©\nTheorem 1.1.2. (Topologically Finitely Generated Closed Normal Sub-\ngroups) Every topologically finitely generated closed normal subgroup of GF is trivial.\nProof. This follows from [FJ], Theorem 15.10. ©\nTheorem 1.1.3. (The Neukirch-Uchida Theorem on the Anabelian Nature\nof Number Fields) Let F1, F2 be number fields. Let F 1 (respectively, F 2) be an\nalgebraic closure of F1 (respectively, F2). Write Isom(F 2\/F2, F 1\/F1) for the set of field\nisomorphisms F 2\n∼→ F 1 that map F2 onto F1. Then the natural map\nIsom(F 2\/F2, F 1\/F1) → Isom(Gal(F 1\/F1),Gal(F 2\/F2))\nis bijective.\nProof. This is the content of [NSW], Theorem 12.2.1. ©\nRemark 1.1.3.1. It is important to note, however, that the analogue of Theorem\n1.1.3 for finite extensions of Qp is false (cf. [NSW], p. 674). Nevertheless, by considering\nisomorphisms of Galois groups that preserve the higher ramification filtration, one may\nobtain a partial analogue of this theorem for p-adic local fields (cf. [Mzk5]).\nABSOLUTE ANABELIAN GEOMETRY 7\nNext, we would like to consider a situation that arises frequently in anabelian\ngeometry. Suppose that G is equal to GF or Gp (where we assume now that p is\nnonarchimedean!), and that we are given an exact sequence of profinite groups:\n1 → Δ → Π → G → 1\nSuppose, moreover, that this sequence splits over some open subgroup of G, and that Δ\nis topologically finitely generated. The following result was related to the author by A.\nTamagawa:\nLemma 1.1.4. (Intrinsic Characterization of Arithmetic Quotients)\n(i) Suppose that G = GF . Let Π\n′ ⊆ Π be an open subgroup. Then the kernel of\nthe homomorphism Π′ → G may be characterized as the unique maximal closed normal\nsubgroup of Π′ which is topologically finitely generated.\n(ii) Suppose that G = Gp. Assume further that for every open subgroup Π\n′′ ⊆ Π,\nthe abelianization (Δ′′)ab of Δ′′ (where Δ′′ def= Π′′\n⋂\nΔ) satisfies the following property:\n(∗) The maximal torsion-free quotient (Δ′′)ab \u0002 Q′′ of (Δ′′)ab on which the\naction of G′′ def= Π′′\/Δ′′ (by conjugation) is trivial is a finitely generated free\nẐ-module.\nLet Π′ ⊆ Π be an arbitrary open subgroup. Then:\n[G : G′] · [Fp : Qp] = dimQp((Π′)ab ⊗\u0000Z Qp) − dimQl((Π′)ab ⊗\u0000Z Ql)\n(where Δ′ def= Δ\n⋂\nΠ′; G′ def= Π′\/Δ′; p is the rational prime that p divides; and l is\nany prime number distinct from p). (In fact, p may also be characterized as the unique\nprime number for which the difference on the right is nonzero for infinitely many prime\nnumbers l.) In particular, the subgroup Δ ⊆ Π may be characterized as the intersection\nof those open subgroups Π′ ⊆ Π such that:\n[G : G′] = [Π : Π′]\n(i.e., such that [G : G′] · [Fp : Qp] = [Π : Π′] · ([G : G] · [Fp : Qp])).\nProof. Assertion (i) is a formal consequence of Theorem 1.1.2.\nNow we turn to assertion (ii). Denote by K ′ the finite extension of Fp determined\nby G′. Then:\n[G : G′] · [Fp : Qp] = [K ′ : Qp]\n8 SHINICHI MOCHIZUKI\nNext, let us observe that we have an exact sequence\n0 → Im(Δ′) → (Π′)ab → (G′)ab → 0\n(where Im(Δ′) is the image of Δ′ in (Π′)ab) of finitely generated Ẑ-modules. Note,\nmoreover, that we have a natural surjection Q′ \u0002 Im(Δ′)\/(torsion) (where Q′ is as\nin (∗)). On the other hand, it follows formally from our assumption that Π′ \u0002 G′\nsplits over an open subgroup of G′ (together with the well-known fact that the group\ncohomology of a finite group is always annihilated by the order to the group) that this\nnatural surjection is, in fact, an isomorphism. In particular, it follows formally from (∗)\nthat:\ndimQp((Π\n′)ab⊗\n\u0000Z\nQp)−dimQl((Π′)ab⊗\u0000ZQl) = dimQp((G′)ab⊗\u0000ZQp)−dimQl((G′)ab⊗\u0000ZQl)\nThus, to complete the proof of Lemma 1.1.4, it suffices to prove that:\n[K ′ : Qp] = dimQp((G\n′)ab ⊗\n\u0000Z\nQp) − dimQl((G′)ab ⊗\u0000Z Ql)\nBut this is a formal consequence of local class field theory (cf., e.g., [Serre2]; §1.2 below),\ni.e., the fact that (G′)ab is isomorphic to the profinite completion of (K′)×. ©\nTypically, in applications involving hyperbolic curves, one shows that the condition\n(∗) of Lemma 1.1.4 is satisfied by applying the following:\nLemma 1.1.5. (Tate Modules of Semi-abelian Varieties) Let K be a finite\nextension of Qp. Fix an algebraic closure K of K; write GK\ndef\n= Gal(K\/K). Let A be a\nsemi-abelian variety over K. Denote the resulting (profinite) Tate module of A by:\nT (A)\ndef\n= Hom(Q\/Z, A(K))\nThen the maximal torsion-free quotient T (A) \u0002 Q on which GK acts trivially is a\nfinitely generated free Ẑ-module.\nProof. A semi-abelian variety is an extension of an abelian variety by a torus. Thus,\nT (A) is the extension of the Tate module of an abelian variety by the Tate module of a\ntorus. Moreover, since (after restricting to some open subgroup of GK) the Tate module\nof a torus is isomorphic to the direct sum of a finite number of copies of Ẑ(1), we thus\nconclude that the image of the Tate module of the torus in Q is necessarily zero. In\nparticular, we may assume for the remainder of the proof without loss of generality that\nA is an abelian variety.\nABSOLUTE ANABELIAN GEOMETRY 9\nNow it follows from the theory of [FC](cf., in particular, [FC], Chapter III, Corollary\n7.3), that T (A) fits into exact sequences (of GK-modules)\n0 → Tgood → T (A) → Tcom → 0\n0 → Ttrc → Tgood → T (B) → 0\nwhere T (B) is the Tate module of an abelian variety B over K with potentially good\nreduction; and Tcom = Mcom ⊗Z Ẑ, Ttrc = Mtrc ⊗Z Ẑ(1) for finitely generated free Z-\nmodules Mcom, Mtrc on which GK acts via a finite quotient. [Here, “com” (respectively,\n“trc”) stands for “combinatorial” (respectively, “toric”).] It is thus evident that Ttrc\nmaps to 0 in Q. Moreover, by [Mzk4], Lemma 8.1 (the proof of which is valid for\narbitrary B, even though in loc. cit., this result is only stated in the case of a Jacobian),\nand the Riemann Hypothesis for abelian varieties over finite fields (cf., e.g., [Mumf], p.\n206), it follows that T (B) also maps to 0 in Q. Thus, we conclude that Q is equal to\nthe maximal torsion-free quotient of Tcom on which GK acts trivially. Since Ẑ is Z-flat,\nhowever, this implies that Q is equal to the result of applying ⊗ZẐ to the maximal\ntorsion-free quotient of Mcom on which GK acts trivially. But this last quotient is\nmanifestly finitely generated and free over Z. This completes the proof. ©\n§1.2. The Anabelian Geometry of p-adic Local Fields\nIn this §, we review certain well-known “group-theoretic” properties of Galois groups\nof p-adic local fields, i.e., properties preserved by arbitrary isomorphisms between such\nGalois groups.\nFor i = 1, 2, let pi be a prime number. Let Ki be a finite extension of Qpi . We\ndenote the ring of integers (respectively, maximal ideal; residue field) of Ki by OKi\n(respectively, mKi ; ki). Also, we assume that we have chosen an algebraic closures Ki\nof Ki and write\nGKi\ndef\n= Gal(Ki\/Ki)\nfor the corresponding absolute Galois group of Ki. Thus, by local class field theory (cf.,\ne.g., [Serre2]), we have a natural isomorphism\n(K×i )\n∧ ∼→ GabKi\n(where the “∧” denotes the profinite completion of an abelian group; “×” denotes the\ngroup of units of a ring; and “ab” denotes the maximal abelian quotient of a topological\ngroup). In particular, GabKi fits into an exact sequence\n0 → O×Ki → GabKi → Ẑ → 0\n10 SHINICHI MOCHIZUKI\n(arising from a similar exact sequence for (K×i )\n∧). Moreover, we obtain natural inclu-\nsions\nk×i ↪→ O×Ki ⊆ K×i ↪→ GabKi\nK×i \/O×Ki\n∼→ Z ↪→ GabKi\/Im(O×Ki)\n(where “\n∼→ ” denotes the morphism induced by the valuation on K×i ) by considering the\nTeichmüller representatives of elements of k×i and the Frobenius element, respectively.\nAlso, in the following discussion we shall write:\nμ\n\u0000Z\n(Ki)\ndef\n= Hom(Q\/Z,K\n×\ni ); μ\u0000Z′(Ki)\ndef\n= μ\n\u0000Z\n(Ki) ⊗\u0000Z Ẑ′;\nμQ\/Z(Ki)\ndef\n= μ\n\u0000Z\n(Ki) ⊗\u0000Z Q\/Z\n(where Ẑ′ def= Ẑ\/Zp). Finally, we denote the cyclotomic character of GKi by:\nχi : GKi → Ẑ×\nProposition 1.2.1. (Invariants of Arbitrary Isomorphisms of Galois Groups\nof Local Fields) Suppose that we are given an isomorphism of profinite groups:\nα : GK1\n∼→ GK2\nThen:\n(i) We have: p1 = p2. Thus, (in the remainder of this proposition and its proof)\nwe shall write p\ndef\n= p1 = p2.\n(ii) α induces an isomorphism IK1\n∼→ IK2 between the respective inertia sub-\ngroups of GK1 , GK2 .\n(iii) The isomorphism αab : GabK1\n∼→ GabK2 induced by α preserves the images\nIm(O×Ki), Im(k×i ), Im(K×i ) of the natural morphisms discussed above.\n(iv) The morphism induced by α between the respective quotients GabKi\/Im(O×Ki)\npreserves the respective Frobenius elements.\n(v) [K1 : Qp] = [K2 : Qp]; [k1 : Fp] = [k2 : Fp]. In particular, the ramification\nindices of K1, K2 over Qp coincide.\n(vi) The morphisms induced by α on the abelianizations of the various open sub-\ngroups of the GKi induce an isomorphism\nμQ\/Z(K1)\n∼→ μQ\/Z(K2)\nABSOLUTE ANABELIAN GEOMETRY 11\nwhich is Galois-equivariant with respect to α. In particular, α preserves the cyclotomic\ncharacters χi.\n(vii) The morphism H2(K1,μQ\/Z(K1))\n∼→ H2(K2,μQ\/Z(K2)) induced by α (cf.\n(vi)) preserves the “residue map”\nH2(Ki,μQ\/Z(Ki))\n∼→ Q\/Z\nof local class field theory (cf. [Serre2], §1.1).\nProof. Property (i) follows by considering the ranks of GabKi over various Zl (cf. Lemma\n1.1.4, (ii)). Property (iii) for Im(k×i ) follows from the fact that Im(k\n×\ni ) may be recovered\nas the prime-to-p torsion subgroup of GabKi . Property (v) follows for [Ki : Qp] by\nconsidering the Zp-rank of G\nab\nKi\n(minus 1) and for [ki : Fp] by considering the cardinality\nof Im(k×i ) (plus 1) — cf. (i), (iii). Property (ii) follows from the fact that whether or not\na finite extension is unramified may be determined group-theoretically by considering\nthe variation of the ramification index over Qp (cf. (v)). Property (iii) for Im(O×Ki)\nfollows formally from (ii) (since this image is equal to the image in GabKi of IKi). Property\n(iv) follows by applying (iii) for Im(k×i ) to the various open subgroups of GKi that\ncorrespond to unramified extensions of Ki and using the fact the Frobenius element\nis the unique element that acts on k\n×\ni (where ki denotes the algebraic closure of ki\ninduced by Ki) as multiplication by |k1| = |k2| [where, relative to our use of the term\n“multiplication”, we think of the abelian group k\n×\ni additively]. Here, we note that if Li\nis a finite extension of Ki, then the inclusion\nGabKi\n∼→ (K×i )∧ ↪→ (L×i )∧ ∼→ GabLi\nmay be reconstructed group-theoretically by considering the Verlagerung, or transfer,\nmap (cf. [Serre2], §2.4). Property (iii) for Im(K×i ) follows formally from (iv). Property\n(vi) follows formally from (iii). Finally, property (vii) follows (cf. the theory of the\nBrauer group of a local field, as exposed, for instance, in [Serre2], §1) from the fact that\nthe morphism H2(Ki,μQ\/Z(Ki))\n∼→ Q\/Z may be constructed as the composite of the\nnatural isomorphism\nH2(Ki,μQ\/Z(Ki)) = H\n2(GKi ,μQ\/Z(Ki))\n∼→ H2(GKi ,K\n×\ni )\n— which is group-theoretic, by (iii) — with the inverse of the isomorphism\nH2(Gal(Kunri \/Ki), (K\nunr\ni )\n×) ∼→ H2(GKi ,K\n×\ni )\n(where Kunri denotes the maximal unramified extension of Ki) — which is group-\ntheoretic, by (ii), (iii) — followed by the natural isomorphism\nH2(Gal(Kunri \/Ki), (K\nunr\ni )\n×) ∼→ H2(Gal(Kunri \/Ki),Z) = H2(Ẑ, Z) = Q\/Z\n12 SHINICHI MOCHIZUKI\n— which is group-theoretic, by (ii), (iii), (iv). ©\nBefore proceeding, we observe that Proposition 1.2.1, (i), may be extended as\nfollows: Write\nAQ\ndef\n= |Spec(Z)|\n⋃\n{∞}\n(where “| − |” denotes the underlying set of a scheme) for the set of “all arithmetic\nprimes of Q”. If v ∈ AQ is equal to (0) ∈ |Spec(Z)| (respectively, ∞), set Gv def= GQ\n(respectively, Gv\ndef\n= Gal(C\/R)). If v ∈ |Spec(Z)| ⊆ AQ is equal to the prime determined\nby a prime number p, set Gv\ndef\n= GQp .\nProposition 1.2.2. (Intrinsicity of Arithmetic Types) For i = 1, 2, let vi ∈ AQ.\nSuppose that Hi is an open subgroup of Gvi . Then H1\n∼= H2 implies v1 = v2.\nProof. Indeed, open subgroups of GQ may be distinguished by the fact that their\nabelianizations fail to be topologically finitely generated. (Indeed, consider the abelian\nextensions arising from adjoining roots of unity.) By contrast, abelianizations of open\nsubgroups of GR or GQp (cf. the above discussion) are topologically finitely generated.\nNext, open subgroups of GR may be distinguished from those of any GQp by the fact\nthey are finite. The remainder of Proposition 1.2.2 follows from Proposition 1.2.1, (i).\n©\nNext, let us write Spec(OKi)log for the log scheme obtained by equipping the scheme\nSpec(OKi) with the log structure defined by the divisor V (mKi). Thus, by pulling back\nthis log structure via the natural morphism Spec(ki) ↪→ Spec(OKi), we obtain a log\nscheme Spec(ki)\nlog, which we denote by\nklogi\nfor short. Note that the “étale monoid” that defines the log structure on klogi “admits a\nglobal chart” in the sense that it is defined by a single constant monoid (in the Zariski\ntopology of Spec(ki)) Mklogi\n, which fits into a natural exact sequence (of monoids):\n1 → k×i → Mklogi → N → 0\nThus, the k×i -torsor Ui determined by considering the inverse image of 1 ∈ N in this\nsequence may be identified with the k×i -torsor of uniformizers ∈ mKi considered modulo\nm2Ki .\nNext, let us write\nGKi \u0002 Glogki\nABSOLUTE ANABELIAN GEOMETRY 13\nfor the quotient defined by the maximal tamely ramified extension Ktamei of Ki. Thus,\nGlogki may also be thought of as the “logarithmic fundamental group” π1(k\nlog\ni ) of the log\nscheme klogi . Moreover, G\nlog\nki\nfits into a natural exact sequence:\n1 → μ\n\u0000Z\n(ki) → Glogki → Ẑ → 1\nwhere, just as in the case of Ki, we write:\nμ\n\u0000Z\n(ki) = μ\u0000Z′ (ki)\ndef\n= Hom(Q\/Z, k\n×\ni );\nμQ\/Z(ki)\ndef\n= μ\n\u0000Z\n(ki) ⊗\u0000Z Q\/Z\nThe “abelianization” of this exact sequence yields an exact sequence:\n1 → k×i → (Glogki )ab → Ẑ → 1\nNow we have the following:\nProposition 1.2.3.\n(i) Any isomorphism α : GK1\n∼→ GK2 (as in Proposition 1.2.1) induces an isomor-\nphism Glogk1\n∼→ Glogk2 between the respective quotients.\n(ii) There is a natural isomorphism of k×i -torsors between the torsor of uni-\nformizers Ui discussed above and the H\n1(Ẑ,μ\n\u0000Z\n(ki)) = k\n×\ni -torsor H\n1(Glogki ,μ\u0000Z(ki))\n[1]\nof elements of H1(Glogki ,μ\u0000Z(ki)) that map to the identity element in H\n1(μ\n\u0000Z\n(ki),μ\u0000Z(ki)) =\nHom\n\u0000Z′(μ\u0000Z(ki),μ\u0000Z(ki)). This isomorphism is defined by associating to a uniformizer\nπ ∈ Ui the μ\u0000Z(ki)-torsor over klogi determined by the roots π1\/N , as N ranges (multi-\nplicatively) over the positive integers prime to pi.\n(iii) The profinite group Glogki is slim.\nProof. Property (i) follows from Proposition 1.2.1, (ii), together with the fact that\nthe quotient GKi \u0002 Glogki may be identified with the quotient of GKi by the (unique)\nmaximal pro-p subgroup of IKi . Next, since any morphism of k\n×\ni -torsors is necessarily an\nisomorphism, property (ii) follows by observing that the stated association of coverings\nto uniformizers is indeed a morphism of k×i -torsors — a tautology, which may by verified\nby considering the case N = qi − 1 (where qi is the cardinality of ki), in which case\nthis tautology amounts to the computation: (ζ1\/N)qi = ζ · (ζ1\/N ) (for ζ ∈ k×i ). Finally,\nproperty (iii) follows formally from the fact that the quotient Glogki \/Im(μ\u0000Z(ki)) is infinite\nand acts faithfully on all open subgroups of the closed subgroup Im(μ\n\u0000Z\n(ki)) ⊆ Glogki . ©\nIn the following, let us denote by (klogi )\n∼ → klogi the “universal covering” of klogi\ndefined by the extension Ktamei of Ki. Thus, G\nlog\nki\nacts naturally as the group of covering\ntransformations of (klogi )\n∼ → klogi .\n14 SHINICHI MOCHIZUKI\nProposition 1.2.4. (The “Grothendieck Conjecture” for the Logarithmic\nPoint) Suppose that we are given an isomorphism of profinite groups:\nλ : Glogk1\n∼→ Glogk2\nThen:\n(i) We have: |k1| = |k2|; p1 = p2. Thus, (in the remainder of this proposition and\nits proof) we shall write p\ndef\n= p1 = p2.\n(ii) λ preserves the subgroups Im(μ\n\u0000Z\n(ki)) ⊆ Glogki as well as the Frobenius elements\nin the quotients Glogki \/Im(μ\u0000Z(ki)).\n(iii) Assume further that the morphism\nk\n×\n1\n∼→ k×2\ninduced by λ (by thinking of k\n×\ni as μQ\/Z(ki)) arises from an isomorphism of fields\nσ : k1\n∼→ k2. Then there exists a unique commutative diagram\n(klog1 )\n∼ (σlog)∼−→ (klog2 )∼⏐⏐\t ⏐⏐\t\nklog1\nσlog−→ klog2\nof log schemes, compatible with the natural action of Glogki on (k\nlog\ni )\n∼ (for i = 1, 2), in\nwhich the vertical morphisms are the natural morphisms, and the horizontal morphisms\nare isomorphisms for which the morphisms on the underlying schemes are those in-\nduced by σ.\nProof. Property (i) follows by observing that pi is the unique prime number such\nthat 1 plus the cardinality of the torsion subgroup of (Glogki )\nab — i.e., the cardinality\nof ki — is equal to a power of pi. Property (ii) follows by thinking of the quotients\nGlogki \/Im(μ\u0000Z(ki)) as the quotients of G\nlog\nki\nobtained by forming the quotient of (Glogki )\nab\nby its torsion subgroup, and then using that the Frobenius element is the unique element\nthat acts on the abelian group Im(μ\n\u0000Z\n(ki)) via multiplication by |k1| = |k2|. As for (iii),\nthe morphism σlog is the unique logarithmic extension of σ whose induced morphism\nU1\n∼→ U2 is the morphism obtained (cf. Proposition 1.2.3, (ii)) by considering the\nmorphism induced by λ between the H1(Ẑ,μ\n\u0000Z\n(ki)) = k\n×\ni -torsors H\n1(Glogki ,μ\u0000Z(ki))\n[1]\n(for i = 1, 2) — which are preserved by λ, by (ii). Note that here we also use (cf. (ii))\nthat the Frobenius element ∈ Ẑ is preserved, since this element is necessary to ensure\nthe compatibility of the identifications\nH1(Ẑ,μ\n\u0000Z\n(ki)) = k\n×\ni\nABSOLUTE ANABELIAN GEOMETRY 15\n(cf. Proposition 1.2.3, (ii)). The morphism (σlog)∼ is obtained by applying this con-\nstruction of “σlog” to the various finite log étale coverings of klogi obtained by considering\nvarious open subgroups of Glogki . Here, the transition morphisms among coverings are\ninduced by the Verlagerung, as in the proof of Proposition 1.2.1. Finally, the uniqueness\nof the lifting (σlog)∼ of σlog is a formal consequence of the fact that Glogki is center-free\n(cf. Proposition 1.2.3, (iii)). ©\n§1.3. The Anabelian Geometry of Hyperbolic Curves\nCharacteristic Zero:\nLet K be a field of characteristic 0 whose absolute Galois group is slim. Let X be\na hyperbolic curve of type (g, r) over S\ndef\n= Spec(K). Fix an algebraic closure K of K\nand write s : Spec(K) → S; GK def= Gal(K\/K). Let x ∈ X(K) be a K-valued point of\nX lying over s. Then, setting ΠX\ndef\n= π1(X,x), we obtain an exact sequence\n1 → ΔX → ΠX → GK → 1\nwhich determines a well-defined continous homomorphism\nGK → Out(ΔX)\nto the group of outer automorphisms Out(ΔX) of ΔX .\nLemma 1.3.1. (Slimness of Geometric and Arithmetic Fundamental\nGroups) The profinite groups ΔX, ΠX are slim.\nProof. The slimness of ΠX is a formal consequence of the slimness of ΔX and our\nassumption that GK is slim. Thus, it remains to prove that ΔX is slim. One approach\nto proving this fact is given in [Naka1], Corollary 1.3.4. Another approach is the fol-\nlowing: Let H ⊆ ΔX be an open normal subgroup for which the associated covering\nXH → XK def= X ×K K is such that XH is a curve of genus ≥ 2. Thus, Hab may be\nthought of as the profinite Tate module associated to the generalized Jacobian of the\nsingular curve obtained from the unique smooth compactification of XH by identifying\nthe various cusps (i.e., points of the compactification not lying in XH) to a single point.\nIn particular, if conjugation by an element δ ∈ ΔX induces the trivial action on Hab,\nthen we conclude that the image of δ in ΔX\/H induces the trivial action on the gen-\neralized Jacobian just discussed, hence on XH itself. But this implies that δ ∈ H. By\ntaking H to be sufficiently small, we thus conclude that δ = 1. ©\n16 SHINICHI MOCHIZUKI\nIn particular, it follows formally from Lemma 1.3.1 that:\nCorollary 1.3.2. (A Natural Exact Sequence) We have a natural exact sequence\nof profinite groups:\n1 → ΔX → Aut(ΔX) → Out(ΔX) → 1\nIn particular, by pulling back this exact sequence, one may recover the exact sequence\n1 → ΔX → ΠX → GK → 1 entirely group-theoretically from the outer Galois represen-\ntation GK → Out(ΔX).\nOne example of the sort of “K” under consideration is the case of a “sub-p-adic\nfield”:\nCorollary 1.3.3. (Slimness of Sub-p-adic Fields) The absolute Galois group of\na sub-p-adic field (i.e., a field isomorphic to a subfield of a finitely generated field\nextension of Qp, for some prime number p) is slim.\nProof. This fact is implied by the argument of the proof of [Mzk6], Lemma 15.8. ©\nIn [Mzk6], the author (essentially) proved the following result (cf. [Mzk6], Theorem\nA, for a stronger version that treats arbitrary dominant morphisms and open group\nhomomorphisms [i.e., which are not necessarily isomorphisms]):\nTheorem 1.3.4. (“Sub-p-adic Profinite Grothendieck Conjecture”) Suppose\nthat K is a sub-p-adic field, and that X and Y are hyperbolic curves over K. Denote\nby IsomK(X,Y ) the set of K-isomorphisms X\n∼→ Y ; by IsomOutGK (ΔX ,ΔY ) the set of\nouter isomorphisms between the two profinite groups in parentheses that are compatible\nwith the respective outer actions of GK . Then the natural map\nIsomK(X,Y ) → IsomOutGK (ΔX ,ΔY )\nis bijective.\nThus, by combining Theorems 1.1.3; 1.3.4; Lemma 1.1.4, (i), we obtain the follow-\ning:\nCorollary 1.3.5. (Absolute Grothendieck Conjecture over Number Fields)\nLet K, L be number fields; X (respectively, Y ) a hyperbolic curve over K (re-\nspectively, L). Denote by Isom(X, Y ) the set of scheme isomorphisms X\n∼→ Y ; by\nABSOLUTE ANABELIAN GEOMETRY 17\nIsomOut(ΠX ,ΠY ) the set of outer isomorphisms between the two profinite groups in\nparentheses. Then the natural map\nIsom(X, Y ) → IsomOut(ΠX ,ΠY )\nis bijective.\nRemark 1.3.5.1. Since the analogue of Theorem 1.1.3 in the p-adic local case is false,\nit seems unlikely to the author at the time of writing that the analogue of Corollary\n1.3.5 should hold over p-adic local fields.\nRemark 1.3.5.2. In fact, the statement of Corollary 1.3.5 holds when K is taken\nto be an arbitrary finitely generated field extension of Q. This follows by applying a\ntheorem of F. Pop (in place of the Neukirch-Uchida theorem) — cf. the discussion on\n[NSW], p. 677, for more details.\nOne interesting result in the present context is the following, due to M. Matsumoto\n(cf. [Mtmo], Theorems 2.1, 2.2):\nTheorem 1.3.6. (Kernels of Outer Galois Representations) Let X be an\naffine hyperbolic curve over a sub-complex field K — i.e., a field isomorphic to\na subfield of the field of complex numbers. Then the kernel of the resulting outer Galois\nrepresentation\nGK → Out(ΔX)\nis contained in the kernel of the natural homomorphism GK → GQ.\nRemark 1.3.6.1. Thus, in particular, when K is a number field, the homomorphism\nGK → Out(ΔX) is injective. This injectivity was first proven by Belyi in the case of\nhyperbolic curves of type (g, r) = (0, 3). It was then conjectured by Voevodskii to be\ntrue for all (hyperbolic) (g, r) and proven by Voevodskii to be true for g = 1. Finally,\nit was proven by Matsumoto to hold for all (g, r) such that r > 0. To the knowledge\nof the author, the proper case remains open at the time of writing. We refer to the\ndiscussion surrounding [Mtmo], Theorem 2.1, for more details on this history.\nRemark 1.3.6.2. One interesting aspect of the homomorphism appearing in Theorem\n1.3.6 is that it allows one to interpret Theorem 1.3.4 (when X = Y ) as a computation\nof the centralizer of the image of this homomorphism GK → Out(ΔX). This point of\nview is surveyed in detail in [Naka2].\n18 SHINICHI MOCHIZUKI\nNext, we would like to discuss various properties of the inertia groups of the cusps\nof a hyperbolic curve. For every cusp x of XK\ndef\n= X ×K K — i.e., point of the unique\nsmooth compactification of XK over K that does not lie in XK — we have an associated\ninertia group (abstractly isomorphic to Ẑ)\nIx ⊆ ΔX\n(well-defined, up to conjugation). If l is any prime number, then let us denote the\nmaximal pro-l quotient of a profinite group by means of a superscript “(l)”. Thus, we\nalso obtain an inertia group I\n(l)\nx ⊆ Δ(l)X (abstractly isomorphic to Zl).\nLemma 1.3.7. (Commensurable Terminality of Inertia Groups) The sub-\ngroups I\n(l)\nx ⊆ Δ(l)X , Ix ⊆ ΔX are commensurably terminal.\nProof. Indeed, let σ be an element of the commensurator. If the asserted commensu-\nrable terminality is false, then by projecting to a finite quotient, we may assume that\nwe have a finite Galois covering\nZ → XK\n(of degree a power of l in the pro-l case), together with a cusp z of Z such that: (i) z\nmaps to x; (ii) z \u000f= zσ; (iii) z, zσ have conjugate inertia groups in ΔZ. We may also\nassume (by taking ΔZ ⊆ ΔX to be sufficiently small) that Z has genus ≥ 2 and admits\na cusp z′ \u000f= z, zσ . Then it is easy to see that Z admits an infinite abelian (pro-l, in the\npro-l case) covering which is totally ramified at z, z′, but not at zσ. But this contradicts\nproperty (iii). ©\nRemark 1.3.7.1. As was pointed out to the author by the referee, in the case r ≥ 2,\nthe assertion of Lemma 1.3.7 is a formal consequence of Theorem B′ of [HR].\nNow, let us assume that we are given two hyperbolic curves (Xi)Ki (for i = 1, 2),\neach defined over a finite extension Ki of Qpi. Let us write qi for the cardinality of\nthe residue field of Ki. Choose basepoints for the (Xi)Ki and denote the resulting\nfundamental groups by Π(Xi)Ki . Also, let us denote the unique proper smooth curve\nover Ki that compactifies (Xi)Ki by (Yi)Ki . Suppose, moreover, that we are given an\nisomorphism\nαX : Π(X1)K1\n∼→ Π(X2)K2\nof profinite groups.\nLemma 1.3.8. (Group-Theoreticity of Arithmetic Quotients) The isomor-\nphism αX is necessarily compatible with the quotients Π(Xi)Ki \u0002 GKi .\nABSOLUTE ANABELIAN GEOMETRY 19\nProof. This follows formally from Lemmas 1.1.4, 1.1.5. ©\nThus, Lemma 1.3.8, Proposition 1.2.1, (v), imply that q1 = q2.\nLemma 1.3.9. (Group-Theoreticity of the Cusps) The types (gi, ri) of the\nhyperbolic curves (Xi)Ki coincide. In particular, for any prime number l, αX maps\ninertia groups of cusps in ΔX1 (respectively, Δ\n(l)\nX1\n) to inertia groups of cusps in ΔX2\n(respectively, Δ\n(l)\nX2\n).\nProof. Whether or not ri = 0 may be determined by considering whether or not ΔXi\nis free as a profinite group. When ri > 0, one may compute ri by considering the weight\n— i.e., the number w such that the eigenvalues of the action are algebraic numbers of\narchimedean absolute value qwi — of the action of the Frobenius element ∈ Gki (cf.\nProposition 1.2.1, (iv)) as follows: First, we observe that (as is well-known) the weights\nof the action of Frobenius on ΔabXi ⊗ Ql (where l is a prime number distinct from p1,\np2) belong to the set {0, 1, 12}. (Here, we compute weights by choosing a lifting of the\nFrobenius element ∈ Gki to GKi . Since (as is well-known) the action of the inertia\nsubgroup of GKi on Δ\nab\nXi\n⊗Ql is quasi-unipotent (i.e., unipotent on an open subgroup of\nthis inertia subgroup), it follows immediately that the weights are independent of the\nchoice of lifting.) Now if M is a Ql-vector space on which Frobenius acts, let us write\nMwt w\nfor the Ql-subspace of M on which Frobenius acts with eigenvalues of weight w. Then,\nsetting Ii\ndef\n= Ker(ΔabXi ⊗ Ql \u0002 ΔabYi ⊗ Ql), we have:\nri − 1 = dimQl(Ii) = dimQl(Iwt 1i )\n= dimQl(Δ\nab\nXi\n⊗ Ql)wt 1 − dimQl(ΔabYi ⊗ Ql)wt 1\n= dimQl(Δ\nab\nXi ⊗ Ql)wt 1 − dimQl(ΔabYi ⊗ Ql)wt 0\n= dimQl(Δ\nab\nXi ⊗ Ql)wt 1 − dimQl(ΔabXi ⊗ Ql)wt 0\n(where the fourth equality follows from the auto-duality (up to a Tate twist) of ΔabYi ⊗\nQl; and the second and fifth equalities follow from the fact that Frobenius acts on Ii\nwith weight 1). On the other hand, the quantities appearing in the final line of this\nsequence of equalities are all “group-theoretic”. Thus, we conclude that r1 = r2. Since\ndimQl(Δ\nab\nXi\n⊗ Ql) = 2gi − 1 + ri (respectively, = 2gi) when ri > 0 (respectively, when\nri = 0), this implies that g1 = g2, as desired.\nFinally, the statement concerning preservation of inertia groups follows formally\nfrom the fact that “ri is group-theoretic” (by applying this fact to coverings of Xi).\n20 SHINICHI MOCHIZUKI\nIndeed, let l be a prime number (possibly equal to p1 or p2). Since ri may be recovered\ngroup-theoretically, given any finite étale coverings\nZi → Vi → Xi\nsuch that Zi is Galois, of degree a power of l, over Vi, one may determine group-\ntheoretically whether or not Zi → Vi is “totally ramified at a single point of Zi and\nunramified elsewhere”, since this condition is easily verified to be equivalent to the\nequality:\nrZi = deg(Zi\/Vi) · (rVi − 1) + 1\nMoreover, the group-theoreticity of this condition extends immediately to the case of\npro-l coverings Zi → Vi. Thus, by Lemma 1.3.7, we conclude that the inertia groups of\ncusps in (ΔXi)\n(l) (i.e., the maximal pro-l quotient of ΔXi) may be characterized (group-\ntheoretically!) as the maximal subgroups of (ΔXi)\n(l) that correspond to (profinite)\ncoverings satisfying this condition.\nIn particular, (by Lemma 1.3.7) the set of cusps of Xi may be reconstructed (group-\ntheoretically!) as the set of (ΔXi)\n(l)-orbits (relative to the action via conjugation) of\nsuch inertia groups in (ΔXi)\n(l). Thus, by applying this observation to arbitrary finite\nétale coverings of Xi, we recover the inertia subgroups of the cusps of ΔXi as the\nsubgroups that fix some cusp of the universal covering X̃i → Xi of Xi determined by\nthe basepoint in question. This completes the proof. ©\nRemark 1.3.9.1. As was pointed out to the author by the referee, the argument\ngiven in the second paragraph of the proof of Lemma 1.3.9 may be replaced by the\nmore group-theoretic argument of [Tama2], Proposition 2.4.\nPositive Characteristic:\nFor i = 1, 2, let ki be a finite field of characteristic p; Xi a hyperbolic curve over ki.\nChoose a universal tamely ramified (i.e., at the punctures of Xi) covering X̃i → Xi of\nXi; write\nΠtameXi\ndef\n= Gal(X̃i\/Xi)\nfor the corresponding fundamental groups. Thus, we obtain exact sequences:\n1 → ΔtameXi → ΠtameXi → Gki → 1\n(where Gki is the absolute Galois group of ki determined by X̃i). As is well-known, the\nFrobenius element determines a natural isomorphism Ẑ ∼= Gki .\nLemma 1.3.10. (Slimness of Fundamental Groups) For i = 1, 2, the profinite\ngroups ΔtameXi , Π\ntame\nXi\nare slim.\nABSOLUTE ANABELIAN GEOMETRY 21\nProof. The slimness of ΔtameXi follows by exactly the same argument — i.e., by consid-\nering the action of ΔtameXi on abelianizations of open subgroups — as that given in the\nproof of Lemma 1.3.1. [Alternatively, as was pointed out to the author by the referee,\nthe slimness of ΔtameXi follows from [Tama1], Proposition 1.11 (cf. Remark 0.1.3).]\nBy a similar argument, the slimness of ΠtameXi follows formally from:\n(i) the slimness of ΔtameXi ;\n(ii) some positive power of the character of Gki arising from the determinant of\nthe prime-to-p portion of (ΔtameXi )\nab coincides with some positive power of the\ncyclotomic character.\n(Here, we note (ii) is sufficient to deal with both the “l-primary portion” of Ẑ ∼= Gki\n(for l \u000f= p) and the “p-primary portion” of Ẑ ∼= Gki .) ©\nRemark 1.3.10.1. Property (ii) in the proof of Lemma 1.3.10 was substantially\nsimplified by a suggestion made to the author by the referee.\nThe following fundamental result is due to A. Tamagawa (cf. [Tama1], Theorem\n4.3):\nTheorem 1.3.11. (The Grothendieck Conjecture for Affine Hyperbolic\nCurves over Finite Fields) Assume, for i = 1, 2, that Xi is affine. Then the natural\nmap\nIsom(X̃1\/X1, X̃2\/X2) → Isom(ΠtameX1 ,ΠtameX2 )\n(from scheme-theoretic isomorphisms X̃1\n∼→ X̃2 lying over an isomorphism X1 ∼→ X2\nto isomorphisms of profinite groups ΠtameX1\n∼→ ΠtameX2 ) is bijective.\nFinally, we observe that, just as in the characteristic zero case, inertia groups of\ncusps are commensurably terminal: If xi is a cusp of (Xi)ki\ndef\n= Xi ×ki ki, then we have\nan associated inertia group (abstractly isomorphic to Ẑ′)\nIxi ⊆ ΔtameXi\n(well-defined, up to conjugation). If l is any prime number distinct from p, then we also\nobtain an inertia group I\n(l)\nxi ⊆ (ΔtameXi )(l) (abstractly isomorphic to Zl).\nLemma 1.3.12. (Commensurable Terminality of Inertia Groups) The\nsubgroups I\n(l)\nxi ⊆ (ΔtameXi )(l), Ixi ⊆ ΔtameXi are commensurably terminal.\n22 SHINICHI MOCHIZUKI\nProof. The proof is entirely similar to that of Lemma 1.3.7. [Alternatively, as was\npointed out to the author by the referee, the assertion concerning Ixi ⊆ ΔtameXi follows\nfrom [Tama2], Lemma 2.1, while the assertion concerning I\n(l)\nxi ⊆ (ΔtameXi )(l) follows, at\nleast in the case r ≥ 2, formally from Theorem B′ of [HR].] ©\nSection 2: Reconstruction of the Logarithmic Special Fiber\nFor i = 1, 2, let Ki be a finite extension of Qpi (cf. §1.2), and suppose that we are\ngiven a hyperbolic curve (Xi)Ki over Ki. Let us fix a Ki-valued basepoint for (Xi)Ki\nand denote the resulting fundamental group π1((Xi)Ki) by Π(Xi)Ki . Suppose, moreover,\nthat we are given an isomorphism αX : Π(X1)K1\n∼→ Π(X2)K2 , which, by Lemma 1.3.8,\nnecessarily fits into a commutative diagram\nΠ(X1)K1\nαX−→ Π(X2)K2⏐⏐\t ⏐⏐\t\nGK1\nαK−→ GK2\nwhere the vertical morphisms are the natural ones, and the horizontal morphisms are\nassumed to be isomorphisms. Note that by Proposition 1.2.1, (i), this already implies\nthat p1 = p2; set p\ndef\n= p1 = p2. That such a diagram necessarily arises “geometrically”\nfollows from the main theorem of [Mzk6] (cf. Theorem 1.3.4) — if one assumes that αK\narises geometrically (i.e., from an isomorphism of fields K1\n∼→ K2). In this §, we would\nlike to investigate what one can say in general (i.e., without assuming that αK arises\ngeometrically) concerning this sort of commutative diagram. In some sense, all the\nkey arguments that we use here are already present in [Mzk4], except that there, these\narguments were applied to prove different theorems. Thus, in the following discussion,\nwe explain how the same arguments may be used to prove Theorem 2.7 below.\nLet us denote the type of the hyperbolic curve (Xi)Ki by (gi, ri). Also, we shall\ndenote the geometric fundamental group by\nΔXi\ndef\n= Ker(Π(Xi)Ki → GKi)\nand the unique proper smooth curve over Ki that compactifies (Xi)Ki by (Yi)Ki .\nLemma 2.1. (Group-Theoreticity of Stability) (X1)K1 has stable reduction if\nand only if (X2)K2 does.\nProof. This follows (essentially) from the well-known stable reduction criterion: That\nis to say, (Xi)Ki has stable reduction if and only if the inertia subgroup of GKi acts\nABSOLUTE ANABELIAN GEOMETRY 23\nunipotently on ΔabYi ⊗ Ẑ′ and trivially on the (finite) set of conjugacy classes of inertia\ngroups of cusps in ΔXi (i.e., the set of cusps of (Xi)Ki ⊗Ki Ki — cf. Lemma 1.3.9).\n[Note that this condition is group-theoretic, by Proposition 1.2.1, (ii).] ©\nNow let us assume that (Xi)Ki has stable reduction over OKi . Denote the stable\nmodel of (Xi)Ki over OKi by:\n(Xi)OKi → Spec(OKi)\nHere, in the case where ri > 0, we mean by the term “stable model” the complement of\nthe marked points in the unique stable pointed curve over OKi that extends the pointed\ncurve over Ki determined by (Xi)Ki . Then, by the theory of [Mzk4], §2, 8, there exists\na well-defined quotient\nΠ(Xi)Ki \u0002 Π\nadm\n(Xi)Ki\nwhose finite quotients correspond to (subcoverings of) admissible coverings of the result\nof base-changing (Xi)OKi to rings of integers of tamely ramified extensions of Ki. In\nparticular, we have a natural exact sequence:\n1 → ΔadmXi → Πadm(Xi)Ki → G\nlog\nki\n→ 1\n(where ΔadmXi is defined so as to make the sequence exact). Moreover, Π\nadm\n(Xi)Ki\nitself\nadmits a natural quotient (cf. [Mzk4], §3)\nΠ(Xi)Ki \u0002 Π\nadm\n(Xi)Ki\n\u0002 Πet(Xi)Ki\nwhose finite quotients correspond to coverings of (Xi)Ki that extend to finite étale\ncoverings of (Xi)OKi which are tamely ramified at the cusps. In particular, we have a\nnatural exact sequence:\n1 → ΔetXi → Πet(Xi)Ki → Gki → 1\n(where ΔetXi is defined so as to make the sequence exact).\nLemma 2.2. (Admissible and Étale Quotients)\n(i) The profinite groups Π(Xi)Ki , Π\nadm\n(Xi)Ki\n, and ΔadmXi are all slim.\n(ii) The morphism αX is compatible with the quotients\nΠ(Xi)Ki \u0002 Π\nadm\n(Xi)Ki\n\u0002 Πet(Xi)Ki\nof Π(Xi)Ki .\n24 SHINICHI MOCHIZUKI\nProof. We begin with assertion (i). The slimness of Π(Xi)Ki follows formally from\nTheorem 1.1.1, (ii); Lemma 1.3.1. Next, let us assume that ΔadmXi has already been\nshown to be slim. Then it follows formally from Proposition 1.2.3, (iii), that Πadm\n(Xi)Ki\nis\nslim.\nThus, to complete the proof of assertion (i), it suffices to show that ΔadmXi is slim.\nWe argue as in Lemma 1.3.1, i.e., we consider the covering associated to an open normal\nsubgroup H ⊆ ΔadmXi . To simplify notation, we assume that (Xi)ki is “that covering”;\nthen it suffices to show (cf. the proof of Lemma 1.3.1) that any automorphism (ki-linear)\nσ of (Xi)ki which acts trivially on (ΔadmXi )ab is itself trivial. We may assume without\nloss of generality [i.e., by taking H to be sufficiently small] that (Xi)ki is sturdy (cf.\n[Mzk4], Definition 1.1) — i.e., that the normalizations of all the geometric irreducible\ncomponents of (Xi)ki have genus ≥ 2. Then by [Mzk4], Proposition 1.3, it follows that\nσ acts trivially on the set of geometric irreducible components of (Xi)ki . In particular,\nσ acts on each such geometric irreducible component individually. Thus, by considering\nthe action of σ on the torsion points of the Jacobians of each these geometric irreducible\ncomponents of (Xi)ki (cf. the proof of Lemma 1.3.1), we conclude that σ acts as the\nidentity on each of these geometric irreducible components, as desired.\nNext, we turn to assertion (ii). For Πadm\n(Xi)Ki\n, this follows (essentially) from Propo-\nsition 8.4 of [Mzk4] (together with Lemma 1.3.9, which shows that αX preserves the\npro-p inertia groups associated to the cusps in Π(Xi)Ki ). Of course, in [Mzk4], K1 = K2\nand αK is the identity, but in fact, the only property of αK necessary for the proof of\n[Mzk4], Proposition 8.4 — which is, in essence, a formal consequence of [Mzk4], Lemma\n8.1 (concerning unramified quotients of the p-adic Tate module of a semi-abelian vari-\nety over a p-adic local field) — is that αK preserve the inertia and wild inertia groups\n(which we know, by Proposition 1.2.1, (ii); Proposition 1.2.3, (i), of the present paper).\nSimilarly, the portion of assertion (ii) concerning Πet\n(Xi)Ki\nfollows (essentially) from\n[Mzk4], Proposition 3.2, at least in the case ri = 0. That is to say, even though αK is\nnot necessarily the identity in the present discussion, the only properties of αK that are\nnecessary for the proof of [Mzk4], Proposition 3.2, are Proposition 1.2.3, (i); Proposition\n1.2.4, (ii) (of the present paper).\nFinally, to treat the case of arbitrary ri > 0, we argue as follows: Consider an\nopen normal subgroup H ⊆ Πadm\n(Xi)Ki\n. Then observe that there exists an open normal\nsubgroup H ′ ⊆ Πadm\n(Xi)Ki\nsuch that the orders of the finite groups Πadm\n(Xi)Ki\n\/H, Πadm\n(Xi)Ki\n\/H ′\nare relatively prime and such that the covering of (Xi)ki determined by H ′ is sturdy.\nMoreover, [by the assumption of relative primeness] it follows easily that the covering\ndetermined by H is “of étale type” (i.e., arises from a quotient of Πet\n(Xi)Ki\n) if and only\nif it becomes a covering “of étale type” after base-change via the covering determined\nby H ′. Thus, we conclude that we may assume without loss of generality that (Xi)ki is\nsturdy.\nABSOLUTE ANABELIAN GEOMETRY 25\nNow, let us observe that since (Xi)ki is sturdy, it admits admissible coverings “of\nétale type” with arbitrarily large prime-to-p ramification at the cusps. Thus, [since the\ncondition that an admissible covering be “of étale type” amounts to the condition that\nthere be no ramification at the nodes or at the prime of OKi ] we conclude that the\ncovering of (Xi)ki determined by H is of étale type if and only if, for every open normal\nH ′′ ⊆ Πadm\n(Xi)Ki\nwhich has ramification index divisible by the prime-to-p portion of the\norder of the finite group Πadm\n(Xi)Ki\n\/H at all of the cusps, it becomes a covering of étale\ntype after base-change by the covering determined by H ′′. [Indeed, the necessity\nof this condition is clear; the sufficiency of this condition follows from the existence of\ncoverings of étale type with arbitrarily large prime-to-p ramification at the cusps. Also,\nwe note that by Lemma 1.3.9, this condition is preserved by αX .] On the other hand,\nsuch base-changed coverings are [by construction] necessarily unramified at the cusps.\nThus, the issue of whether or not this base-changed covering is of étale type reduces\n— by Lemma 1.3.9, which shows that αX preserves the quotient Π(Xi)Ki \u0002 Π(Yi)Ki\n— to the “ri = 0” case, which has already been resolved. This completes the proof of\nassertion (ii) in the case ri > 0. ©\nLemma 2.3. (Group-Theoreticity of Dual Semi-Graphs of the Special\nFiber) The morphism αX induces an isomorphism\nαX,Γc : Γ\nc\n(X1)k1\n∼→ Γc(X2)k2\nbetween the “dual semi-graphs with compact structure” (i.e., the usual dual graphs\nΓ(Xi)ki , together with extra edges corresponding to the cusps — cf. the Appendix) of\nthe special fibers (Xi)ki of (Xi)OKi . Moreover, αX,Γc is functorial with respect to pas-\nsage to finite étale coverings of the (Xi)Ki .\nProof. Indeed, if one forgets about the “compact structure”, then this is a formal\nconsequence of Lemma 1.3.9 [which shows that αX preserves the quotient Π(Xi)Ki \u0002\nΠ(Yi)Ki ]; Lemma 2.2, (ii), and the theory [concerning the ri = 0 case] of [Mzk4], §1 – 5,\nsummarized in [Mzk4], Corollary 5.3. Even though αK is not necessarily the identity\nin the present discussion, the only properties of αK that are necessary for the proof of\n[Mzk4], Corollary 5.3 are Proposition 1.2.3, (i); Proposition 1.2.4, (ii) (of the present\npaper). That is to say, the point is that the Frobenius element is preserved, which means\nthat the weight filtrations on l-adic cohomology (where l is a prime distinct from p) are,\nas well.\nThe compatibility with the “compact structure” follows from the pro-l (where l \u000f= p)\nportion of Lemma 1.3.9, together with the easily verified fact (cf. the proof of Lemma\n1.3.7) that the inertia group of a cusp in Πadm\n(Xi)Ki\nis contained (up to conjugacy) in the\ndecomposition group of a unique irreducible component of (Xi)ki . ©\n26 SHINICHI MOCHIZUKI\nNext, we would like to show that αX is necessarily “of degree 1”. This is essentially\nthe argument of [Mzk4], Lemma 9.1, but we present this argument in detail below since\nwe are working here under the assumption that αK is arbitrary. For simplicity, we\nassume until further notice is given that:\n(∗CSSN) ri = 0 [i.e., the curves are compact] and, moreover, the special fiber\n(Xi)ki of (Xi)OKi is singular and sturdy (cf. [Mzk4], Definition 1.1) — i.e.,\nthe normalizations of all the geometric irreducible components of (Xi)ki have\ngenus ≥ 2 — and has a noncontractible dual semi-graph Γc\n(Xi)ki — i.e., this\nsemi-graph is not a tree.\n(These conditions may always be achieved by replacing (Xi)Ki by a finite étale covering\nof (Xi)Ki — cf. [Mzk4], Lemma 2.9; [Mzk4], the first two paragraphs of the proof of\nTheorem 9.2.)\nWe begin by introducing some notation. Write:\nVi\ndef\n= ΔabXi ;\nHi\ndef\n= Hsing1 (Γ\nc\n(Xi)ki , Z) = H\nsing\n1 (Γ(Xi)ki , Z)\n(where “Hsing1 ” denotes the first singular homology group). Thus, by considering the\ncoverings of (Xi)OKi induced by unramified coverings of the graph Γ(Xi)ki , we obtain\nnatural (group-theoretic!) “combinatorial quotients”:\nVi \u0002 (Hi)\u0000Z\ndef\n= Hi ⊗ Ẑ\nLemma 2.4. (Ordinary New Parts, after Raynaud) For a “sufficiently\nlarge prime number l” (where “sufficiently large” depends only on p, gi), and after\npossibly replacing Ki by a finite unramified extension of Ki, there exists a cyclic étale\ncovering (Zi)OKi → (Xi)OKi of (Xi)OKi of degree l such that the “new part” V newi\ndef\n=\nΔab\n(Zi)Ki\n\/Δab\n(Xi)Ki\nof the abelianized geometric fundamental group of (Zi)Ki satisfies the\nfollowing:\n(i) We have an exact sequence:\n0 → V mlti → (V newi )Zp def= V newi ⊗\u0000Z Zp → V etli → 0\n— where V etli is an unramified GKi -module, and V\nmlt\ni is the Cartier dual\nof an unramified GKi -module.\n(ii) The “combinatorial quotient” of Δab\n(Zi)Ki\n(arising from the coverings of\n(Zi)OKi induced by unramified coverings of the dual semi-graph of the special\nfiber of (Zi)OKi ) induces a nonzero quotient V newi \u0002 (Hnewi )\u0000Z of V newi .\nABSOLUTE ANABELIAN GEOMETRY 27\nHere, the injection Δab\n(Xi)Ki\n↪→ Δab\n(Zi)Ki\nis the injection induced by pull-back via (Zi)Ki →\n(Xi)Ki and Poincaré duality (or, alternatively, by the “Verlagerung”).\nProof. Note that since both conditions (i), (ii) are group-theoretic, they may be realized\nsimultaneously for i = 1, 2. Now to satisfy condition (i), it suffices — cf., e.g., the\ndiscussion in [Mzk4], §8, of “VG”, “VGord” — to choose the covering so that the “new\nparts” of the Jacobians of the irreducible components of the special fiber of (Zi)OKi are\nall ordinary. That this is possible for l sufficiently large is a consequence of a theorem of\nRaynaud (as formulated, for instance, in [Tama1], Lemma 1.9). Then let us observe that,\nso long as we choose the étale covering (Zi)OKi → (Xi)OKi so that it is nontrivial over\nevery irreducible component of (Xi)ki , condition (ii) is automatically satisfied: Indeed,\nif we write hi\ndef\n= rkZ(Hi) — so hi > 0 since Γ\nc\n(Xi)ki is assumed to be noncontractible —\nthen to assert that condition (ii) fails to hold — i.e., that there are “no new cycles in\nthe dual graph” — is to assert that we have an equality of Euler characteristics:\n(∑\nj\ngZ,j\n)\n+ hi − 1 = l\n{(∑\nj\ngX,j\n)\n+ hi − 1\n}\n(where the first (respectively, second) sum is the sum of the genera of the irreducible\ncomponents of the geometric special fiber of (Zi)OKi (respectively, (Xi)OKi )). But, since\n∑\nj\n(gZ,j − 1) =\n∑\nj\nl(gX,j − 1)\nwe thus conclude that (l − 1) =\n{ ∑\nj (l − 1)\n}\n+ hi(l − 1), hence that 1 = (\n∑\nj 1) + hi\n— which is absurd, since both the sum and hi are ≥ 1. This completes the proof. ©\nRemark 2.4.1. The author would like to thank A. Tamagawa for explaining to him\nthe utility of Raynaud’s theorem in this sort of situation.\nIn the following discussion, to keep the notation simple, we shall replace (Xi)Ki by\nsome (Zi)Ki as in Lemma 2.4. Thus, V\nnew\ni is a GKi -quotient module of Vi. Moreover,\nwe have a surjection\nV newi \u0002 (Hnewi )\u0000Z\nsuch that the quotient (Hi)\u0000Z \u0002 (Hnewi )\u0000Z is defined over Z, i.e., arises from a quo-\ntient Hi \u0002 Hnewi . (Indeed, this last assertion follows from the fact that the quotient\nHi \u0002 Hnewi arises as the cokernel (modulo torsion) of the morphism induced on first sin-\ngular cohomology modules by a finite (ramified) covering of graphs — i.e., the covering\ninduced on dual graphs by the covering (Zi)OKi → (Xi)OKi of Lemma 2.4.)\n28 SHINICHI MOCHIZUKI\nOn the other hand, the cup product on group cohomology gives rise to a nondegen-\nerate (group-theoretic!) pairing\nV ∨i ⊗\u0000Z V ∨i ⊗\u0000Z μ\u0000Z(Ki) → Mi\ndef\n= H2(ΔXi ,μ\u0000Z(Ki)) (\n∼= Ẑ)\n(where we think of V ∨i\ndef\n= Hom(Vi, Ẑ) as H\n1(ΔXi , Ẑ)), hence, by restriction to\n(V newi )\n∨ ↪→ V ∨i , a pairing\n(V newi )\n∨ ⊗\n\u0000Z\n(V newi )\n∨ ⊗\n\u0000Z\nμ\n\u0000Z\n(Ki) → Mi def= H2(ΔXi ,μ\u0000Z(Ki)) (∼= Ẑ)\nwhich is still nondegenerate (over Q), since it arises from an ample line bundle — namely,\nthe restriction of the polarization determined by the theta divisor on the Jacobian of\n(Xi)Ki to the “new part” of (Xi)Ki . This pairing determines an “isogeny” (i.e., a\nmorphism which is an isomorphism over Q):\n(V newi )\n∨ ⊗\n\u0000Z\nμ\n\u0000Z\n(Ki) ⊗\u0000Z M∨i ↪→ V newi\nThus, if we take the dual of the surjection discussed in the preceding paragraph, then\nwe obtain an inclusion\n(Hnewi )\n∨\n\u0000Z\n⊗ μ\n\u0000Z\n(Ki) ⊗ M∨i ↪→ (V newi )∨ ⊗ μ\u0000Z(Ki) ⊗ M∨i ↪→ V newi\nwhich (as one sees, for instance, by applying the fact that μ\n\u0000Z\n(K i)\nGKi = 0) maps into\nthe kernel of the surjection V newi \u0002 (Hnewi )\u0000Z.\nNext, let us observe that the kernel Ni of the surjection of unramified GKi -modules\n(i.e., Gki -modules)\nV etli \u0002 (Hnewi )Zp\nsatisfies:\nH0(Gki , Ni ⊗ Qp) = H1(Gki , Ni ⊗ Qp) = 0\n(Indeed, Ni arises as a submodule of the module of p-power torsion points of an abelian\nvariety over ki, so the vanishing of these cohomology groups follows from the Riemann\nHypothesis for abelian varieties over finite fields (cf., e.g., [Mumf], p. 206), i.e., the\nfact that (some power of) the Frobenius element of Gki acts on Ni with eigenvalues\nwhich are algebraic numbers with complex absolute values equal to a nonzero rational\npower of p.) In particular, we conclude that the above surjection admits a unique\nGKi -equivariant splitting (H\nnew\ni )Zp ↪→ (V etli )Qp . Similarly, (by taking Cartier duals)\nthe injection (Hnewi )\n∨\nZp\n⊗μZp(Ki) ⊗M∨i ↪→ V mlti also admits a unique GKi -equivariant\nsplitting over Qp. Thus, by applying these splittings, we see that the GKi -action on\n(V newi )Zp determines a p-adic extension class\n(ηi)Zp ∈ {(Hnewi )∨Qp}⊗2⊗M∨i ⊗(H1(Ki,μ\u0000Z(K i))\/H1f (Ki,μ\u0000Z(Ki))) = {(Hnewi )∨Qp}⊗2⊗M∨i\nABSOLUTE ANABELIAN GEOMETRY 29\nwhere (by Proposition 1.2.1, (vii)) H1(Ki,μ\u0000Z(Ki)) may be identified with (K\n×\ni )\n∧, and\nwe define\nH1f (Ki,μ\u0000Z(Ki))\ndef\n= O×Ki ⊆ (K×i )∧\n∼→ H1(Ki,μ\u0000Z(Ki))\nso the quotient group (H1(Ki,μ\u0000Z(K i))\/H\n1\nf (Ki,μ\u0000Z(Ki))) may be identified with Ẑ.\nNext, let us observe that the kernel N ′i of (V\nnew\ni )\u0000Z′ \u0002 (Hnewi )\u0000Z′ is an unramified\nrepresentation of GKi (since it arises from the module of prime-to-p-power torsion points\nof a semi-abelian variety over ki). Moreover, the injection of unramified GKi -modules\n(Hnewi )\n∨\n\u0000Z′ ⊗ μ\u0000Z′ (Ki) ⊗ M∨i ↪→ N ′i\nsplits uniquely over Q, since (by the Riemann Hypothesis for abelian varieties over finite\nfields — cf., e.g., [Mumf], p. 206) the Frobenius element of Gki acts on the smaller\nmodule (respectively, quotient by this smaller module) with weight 1 (respectively, 1\n2\n).\nThus, just as in the p-adic case, we may construct a prime-to-p-adic extension class\n(ηi)\u0000Z′ from the GKi -action on (V\nnew\ni )\u0000Z′ , which, together with (ηi)Zp , yields an extension\nclass (cf. [FC], Chapter III, Corollary 7.3):\nηi ∈ {(Hnewi )∨\u0000Z}⊗2⊗M∨i ⊗{H1(Ki,μ\u0000Z(K i))\/H1f (Ki,μ\u0000Z(Ki))} ⊗ Q = {(Hnewi )∨\u0000Z}⊗2⊗M∨i ⊗ Q\nThat is to say, ηi may be thought of as a (group-theoretically reconstructible!) bilinear\nform:\n〈−,−〉i : (Hnewi )⊗2\n\u0000Z\n→ (M∨i )Q def= M∨i ⊗ Q\nMoreover:\nLemma 2.5. Assume that (Xi)OKi arises as some “(Zi)OKi ” as in Lemma 2.4.\nThen:\n(i) (Positive Rational Structures) The image of (Hnewi )\n⊗2 under the morphism\n(Hnewi )\n⊗2\n\u0000Z\n→ (M∨i )Q forms a rank one Z-submodule of (M∨i )Q. Moreover, for any two\nnonzero elements a, b ∈ Hi, 〈a, a〉i differs from 〈b, b〉i by a factor in Q>0 (i.e., a positive\nrational number). In particular, this image determines a “Q>0-structure” on (M\n∨\ni )Q,\ni.e., a Q-rational structure on (M∨i )Q, together with a collection of generators of this\nQ-rational structure that differ from one another by factors in Q>0. Finally, this Q>0-\nstructure is the same as the Q>0-structure on M\n∨\ni determined by the first Chern class\nof an ample line bundle on (Xi)Ki in Mi = H\n2(ΔXi,μ\u0000Z(K i)).\n(ii) (Preservation of Degree) The isomorphism\nM1 = H\n2(ΔX1 ,μ\u0000Z(K1))\n∼→ H2(ΔX2 ,μ\u0000Z(K2)) = M2\ninduced by αX preserves the elements on both sides determined by the first Chern class\nof a line bundle on (Xi)Ki of degree 1.\n30 SHINICHI MOCHIZUKI\nProof. Indeed, assertion (i), follows formally from [FC], Chapter III, Corollary 7.3,\nand Theorem 10.1, (iii) (by considering “new part” of the Jacobian of (Xi)Ki equipped\nwith the polarization induced by the theta polarization on the Jacobian).\nAs for assertion (ii), the elements in question are the unique elements that, on the\none hand, are rational and positive with respect to the structures discussed in assertion\n(i), and, on the other hand, generate Mi as a Ẑ-module. ©\nRemark 2.5.1. Note that the conclusion of Lemma 2.5, (ii), is valid not just for\n(Xi)Ki , but for any finite étale cover of the original (Xi)Ki , i.e., even if this cover does\nnot arise as some “(Zi)OKi ” as in Lemma 2.4. Indeed, this follows from the fact that\nthe crucial “Q>0-structure” of Lemma 2.5, (i), is preserved by pull-back to such a cover,\nwhich just multiplies the Chern class at issue in Lemma 2.5, (ii), by the degree of the\ncover (an element of Q>0!).\nRemark 2.5.2. In the discussion of [Mzk4], §9, it was not necessary to be as careful\nas we were in the discussion above in constructing the p-adic class (ηi)Zp (i.e., “μp” in\nthe notation of loc. cit.). This is because in loc. cit., we were working over a single\np-adic base-field “K”. In this more restricted context, the extension class (ηi)Zp may be\nextracted much more easily from Vi by simply forming the quotient by the submodule\nof H1(Ki,Ker((Vi)Zp \u0002 (Hi)Zp)) generated by the elements which are “crystalline”,\nor, more simply, of “geometric origin” (i.e., arise from OK-rational points of the for-\nmal group associated to the p-divisible group determined — via “Tate’s theorem” (cf.\nTheorem 4 of [Tate]) — by the GKi -module Ker((Vi)Zp \u0002 (Hi)Zp)). Unfortunately, the\nauthor omitted a detailed discussion of this aspect of the argument in the discussion of\n[Mzk4], §9.\nRemark 2.5.3. Relative to Remark 2.5.2, we note nevertheless that even in the\ndiscussion of [Mzk4], §9, it is still necessary to work (at least until one recovers the\n“Q>0-structure” — cf. Remark 2.5.1) with (Xi)Ki such that the dual graph of the\nspecial fiber (Xi)ki is noncontractible. This minor technical point was omitted in the\ndiscussion of [Mzk4], §9.\nNext, let us write (X logi )OKi for the log scheme obtained by equipping (Xi)OKi\nwith the log structure determined by the monoid of regular functions ∈ O(Xi)OKi which\nare invertible on the open subscheme (Xi)Ki ⊆ (Xi)OKi . Thus, in the terminology of\n[Kato2], (X logi )OKi is log regular. Also, let us write (X\nlog\ni )ki for the log scheme obtained\nby equipping (Xi)ki with the log structure determined by restricting the log structure\nof (X logi )OKi . Thus, the quotient Π(Xi)Ki \u0002 Πadm(Xi)Ki determines a “universal admissible\ncovering”\n(X̃ logi )ki → (X logi )ki\nABSOLUTE ANABELIAN GEOMETRY 31\nof (X logi )ki .\nNow let us choose a connected component Ĩi of the ki-smooth locus (i.e., the com-\nplement of the nodes) of (X̃ logi )ki . Write Ii ⊆ (Xi)ki for the image of Ĩi in (Xi)ki .\nThus,\nĨi → Ii\nis a “tame universal covering” of Ii (i.e., a universal covering of the hyperbolic curve\nIi among those finite étale coverings that are tamely ramified at the “cusps” of this\nhyperbolic curve). In the following discussion, we shall also assume, for simplicity,\nthat Ii is geometrically connected over ki (a condition that may always be achieved by\nreplacing Ki by a finite unramified extension of Ki).\nNow the Galois group ΠIi of this covering may also be thought of as the quotient\nof the decomposition group in Πadm\n(Xi)Ki\nof Ĩi by its inertia group. In particular, since\nΠIi is formed by taking the quotient by this inertia group, it follows that the surjection\nΠadm\n(Xi)Ki\n\u0002 Glogki induces a natural surjection\nΠIi \u0002 Gki\nwhose kernel is the geometric (tame) fundamental group πtame1 ((Ii)ki) of Ii.\nFinally, we observe that it makes sense to speak of Ĩ1 and Ĩ2 as corresponding\nvia αX . Indeed, by Lemma 2.3, αX induces an isomorphism between the pro-graphs\ndetermined by the (X̃ logi )ki . Thus, the Ĩi may be said to correspond via αX when the\nvertices that they determine in these pro-graphs correspond. Moreover, when the Ĩi\ncorrespond via αX , it follows (by considering the stabilizer of the vertex determined\nby Ĩi) that αX induces a bijection between the respective decomposition groups Di in\nΠadm\n(Xi)Ki\nof Ĩi, as well as between the respective inertia subgroups of these decomposi-\ntion groups Di (which may be characterized group-theoretically as the centers of the\nsubgroups Di\n⋂\nKer(Πadm\n(Xi)Ki\n\u0002 Gki), since πtame1 ((Ii)ki) is center-free — cf. Lemma\n1.3.10). Thus, in summary, αX induces a commutative diagram:\nΠI1\n∼→ ΠI2⏐⏐\t ⏐⏐\t\nGk1\n∼→ Gk2\nWe are now ready (cf. [Mzk4], §7) to apply the main result of [Tama1]. This result\nstates that commutative diagrams as above are in natural bijective correspondence with\ncommutative diagrams\nĨ1 ∼→ Ĩ2⏐⏐\t ⏐⏐\t\nI1 ∼→ I2\n32 SHINICHI MOCHIZUKI\nlying over commutative diagrams\nk1\n∼→ k2⏐⏐\t ⏐⏐\t\nk1\n∼→ k2\n(cf. Theorem 1.3.11). In particular, these commutative diagrams induce an isomorphism\n(Ẑ′ ∼=) H2c ((I1)k1 ,μ\u0000Z′(k1))\n∼→ H2c ((I2)k2 ,μ\u0000Z′(k2)) (∼= Ẑ′)\n(where “H2c ” denotes étale cohomology with compact supports — cf. [Milne], Chapter\nIII, Proposition 1.29; Remark 1.30) which maps the element “1” (i.e., the element\ndetermined by the first Chern class of a line bundle of degree 1) on the left to the\nelement “1” on the right. (Indeed, this follows from the fact that the morphism I1 ∼→ I2\nappearing in the above commutative diagram is an isomorphism, hence of degree 1.)\nNote that the isomorphism μ\n\u0000Z′(k1)\n∼→ μ\n\u0000Z′(k2) that we use here is that obtained from\nthe commutative diagram above, i.e., that provided by Theorem 1.3.11.\nLemma 2.6. (Compatibility of Isomorphisms Between Roots of Unity)\nAssume that (Xi)OKi arises as some “(Zi)OKi ” as in Lemma 2.4. Then the following\ndiagram\nμ\n\u0000Z′(k1)\n∼→ μ\n\u0000Z′(k2)⏐⏐\t ⏐⏐\t\nμ\n\u0000Z′(K1)\n∼→ μ\n\u0000Z′(K2)\n— in which the vertical morphisms are the natural ones (obtained by considering Te-\nichmüller representatives); the upper horizontal morphism is the morphism determined\nby Theorem 1.3.11; and the lower horizontal morphism is the morphism determined by\nProposition 1.2.1, (vi) — commutes.\nProof. Indeed, the diagram in the statement of Lemma 2.6 induces a diagram:\nH2c ((I1)k1 ,μ\u0000Z′(k1))\n∼→ H2c ((I2)k2 ,μ\u0000Z′(k2))⏐⏐\t ⏐⏐\t\nH2c ((I1)k1,μ\u0000Z′ (K1))\n∼→ H2c ((I2)k2 ,μ\u0000Z′ (K2))\n. . . (∗1)\nMoreover, we have a diagram\nH2c ((I1)k1 ,μ\u0000Z′ (K1))\n∼→ H2c ((I2)k2 ,μ\u0000Z′ (K2))⏐⏐\t ⏐⏐\t\nH2((X1)K1,μ\u0000Z′ (K1))\n∼→ H2((X2)K2 ,μ\u0000Z′ (K2))\n. . . (∗2)\nABSOLUTE ANABELIAN GEOMETRY 33\nwhere the horizontal morphisms are induced by αK (cf. Proposition 1.2.1, (vi)), and\nthe vertical morphisms are induced “group-theoretically” as follows: First, observe that\n[since we continue to operate under the assumption (∗CSSN)]\nH2((Xi)Ki ,μ\u0000Z′ (Ki))\n∼= H2(ΔXi ,μ\u0000Z′ (Ki)) ∼= H2(ΔadmXi ,μ\u0000Z′(Ki))\nwhile\nH2c ((Ii)ki ,μ\u0000Z′ (Ki)) ∼= H2((Ii)ki ,μ\u0000Z′(Ki)) ∼= H2(π1((Ii)ki),μ\u0000Z′ (Ki))\n(where we write Ii for the unique nonsingular compactification of Ii). Moreover, since\nwe continue to operate under the assumption (∗CSSN), it follows (cf. the discussion\nof “Second Cohomology Groups” in the Appendix) that the natural “push-forward”\nmorphism in étale cohomology\n(H2(π1((Ii)ki),μ\u0000Z′ (Ki)) ∼=) H2c ((Ii)ki ,μ\u0000Z′(Ki))\n→ H2((Xi)ki ⊗ki ki,μ\u0000Z′ (Ki)) (∼= H2(ΔetXi ,μ\u0000Z′(K i)))\nmay be reconstructed group-theoretically (by using the various natural homomorphisms\nπ1((Ii)ki) → ΔetXi [well-defined up to composite with an inner automorphism]). Thus,\nthe desired vertical morphisms of diagram (∗2) may be obtained by composing these\n“push-forward” morphisms with the morphisms\nH2(ΔetXi ,μ\u0000Z′ (Ki)) → H2(ΔadmXi ,μ\u0000Z′ (Ki))\ninduced by the surjections ΔadmXi \u0002 Δ\net\nXi\n[which are group-theoretic by Lemma 2.2,\n(ii)]. In particular, we thus see that the vertical morphisms of diagram (∗2) are group-\ntheoretic, i.e., (in other words) diagram (∗2) commutes.\nNow let us compose the above two diagrams (∗1), (∗2) to form a single diagram:\nH2c ((I1)k1 ,μ\u0000Z′(k1))\n∼→ H2c ((I2)k2 ,μ\u0000Z′(k2))⏐⏐\t ⏐⏐\t\nH2((X1)K1,μ\u0000Z′ (K1))\n∼→ H2((X2)K2 ,μ\u0000Z′ (K2))\n. . . (∗3)\nNote that this diagram (∗3) commutes, since, by Lemma 2.5, (ii) [applied to the lower\nhorizontal morphism of (∗3)]; the discussion immediately preceding the present Lemma\n2.6 [applied to the upper horizontal morphism of (∗3)]; and the discussion of “Second\nCohomology Groups” in the Appendix [applied to the vertical morphisms of (∗3)], all\nof the morphisms of this diagram are compatible with the elements “1” determined\nby the first Chern class of a line bundle of degree 1. But this implies that diagram\n(∗1) commutes (since diagram (∗2) has already been shown to be commutative, and\nall the arrows in both of these diagrams (∗1), (∗2) are isomorphisms between rank one\n34 SHINICHI MOCHIZUKI\nfree Ẑ′-modules). On the other hand, since diagram (∗1) was obtained by applying\nthe functors H2c ((Ii)ki ,−) (which are manifestly faithful, at least when restricted to\nconstant coefficient modules which are finitely generated and free over Ẑ′ — as in the\npresent situation) to the diagram appearing in the statement of Lemma 2.6, we thus\nconclude that the diagram appearing in the statement of Lemma 2.6 is commutative,\nas desired. ©\nThe significance of Lemma 2.6 from our point of view is the following: Lemma 2.6\nimplies that αK induces an isomorphism\nαGlogk\n: Glogk1\n∼→ Glogk2\nwhich satisfies the hypothesis of Proposition 1.2.4, (iii). Thus, we conclude from Propo-\nsition 1.2.4, (iii), that αGlogk\narises geometrically. In particular, it follows that we may\napply [Mzk4], Theorem 7.2, to the commutative diagram\nΠadm\n(X1)K1\n∼→ Πadm\n(X2)K2⏐⏐\t ⏐⏐\t\nGlogk1\n∼→ Glogk2\n(where we note that Lemma 2.6 also implies — when translated into the terminology\nof [Mzk4], §7 — that the “RT-degree” associated to this commutative diagram is 1, as\nis necessary for the application of [Mzk4], Theorem 7.2). In particular, we conclude\nthat the above commutative diagram of fundamental groups arises geometrically from\na commutative diagram:\n(X log1 )k1 ∼→ (X log2 )k2⏐⏐\t ⏐⏐\t\nklog1\n∼→ klog2\nMoreover, the isomorphism exhibited in the upper horizontal arrow of this commutative\ndiagram is easily seen to be functorial with respect to arbitrary finite étale coverings of\nthe (Xi)Ki (i.e., not just coverings that arise from finite étale coverings of the (Xi)OKi ).\nIndeed, this functoriality follows formally from the uniqueness assertion in Proposition\n1.2.4, (iii), and the fact that dominant (i.e., not just finite étale) morphisms between\nproper hyperbolic curves in characteristic p may be distinguished by considering the\nmorphisms that they induce between the respective Jacobians, hence, in particular, by\nthe morphisms that they induce between the l-power torsion points (where l \u000f= p) of\nthe respective Jacobians. [Note also that the morphisms between log structures at the\nnodes are uniquely determined by the morphisms between the various normal bundles\non the branches at the nodes — cf. [Mzk3], §3.7, 3.8, 3.10.] Thus, in summary:\nABSOLUTE ANABELIAN GEOMETRY 35\nTheorem 2.7. (Group-Theoretic Reconstruction of the Logarithmic Special\nFiber of a p-adic Hyperbolic Curve) Let p be a prime number. For i = 1, 2, let Ki\nbe a finite extension of Qp, and (Xi)Ki a hyperbolic curve over Ki whose associated\npointed stable curve has stable reduction over OKi . Denote the resulting “stable model”\nof (Xi)Ki over OKi by (Xi)OKi . Assume that we have chosen basepoints of the (Xi)Ki\n(which thus induce basepoints of the Ki). Then every isomorphism of profinite groups\nΠ(X1)K1\n∼→ Π(X2)K2 induces commutative diagrams:\nΠ(X1)K1\n∼→ Π(X2)K2⏐⏐\t ⏐⏐\t\nGK1\n∼→ GK2\nΠadm\n(X1)K1\n∼→ Πadm\n(X2)K2⏐⏐\t ⏐⏐\t\nGlogk1\n∼→ Glogk2\nMoreover, the latter commutative diagram (of admissible quotients Πadm\n(Xi)Ki\nof the\nΠ(Xi)Ki lying over the tame Galois groups G\nlog\nki\nof the Ki) necessarily arises from unique\ncommutative diagrams of log schemes\n(X̃ log1 )k1 ∼→ (X̃ log1 )k1⏐⏐\t ⏐⏐\t\n(X log1 )k1 ∼→ (X log2 )k2\n(klog1 )\n∼ ∼→ (klog2 )∼⏐⏐\t ⏐⏐\t\nklog1\n∼→ klog2\nwhere the commutative diagram on the left lies over the commutative diagram on the\nright (which is as in Proposition 1.2.4, (iii)). Here, we equip Spec(OKi) (respectively,\nSpec(ki); (Xi)OKi ; (Xi)ki) with the log structure determined by the closed point (re-\nspectively, determined by restricting the log structure on Spec(OKi); determined by the\nmonoid of functions invertible on the open subscheme (Xi)Ki ; determined by restricting\nthe log structure on (Xi)OKi ) and denote the resulting log scheme by O\nlog\nKi\n(respectively,\nklogi ; (X logi )OKi ; (X\nlog\ni )ki); the vertical morphisms in the above commutative diagrams\nof log schemes are the universal coverings induced by the various basepoints chosen.\nProof. First, note that the additional assumptions that were used in the course of\nthe above discussion — e.g., “(∗CSSN)”; the assumption that (Xi)OKi arise as some\n“(Zi)OKi ” as in Lemma 2.4 — were applied only to show that the hypotheses of Propo-\nsition 1.2.4, (iii) (and [Mzk4], Theorem 7.2) are satisfied. Moreover, we observe that\nalthough [Mzk4], Theorem 7.2, is only stated in the proper singular case, it extends im-\nmediately to the affine singular, as well as affine nonsingular, cases. Thus, (cf. Remark\n2.5.1; the discussion surrounding “(∗CSSN)”) one concludes that — except when (Xi)Ki\nis proper, with good reduction — one may reconstruct the logarithmic special fiber in a\nfunctorial fashion (i.e., with respect to finite étale coverings of the (Xi)Ki ), as desired.\nIn the case that (Xi)Ki is proper, but has good reduction over OKi , we may still\nreconstruct its logarithmic special fiber (despite the fact that [Tama1], Theorem 4.3,\n36 SHINICHI MOCHIZUKI\nis only stated in [Tama1] for affine hyperbolic curves!) by arguing as follows: First\nof all, we observe that in the case of good reduction, the log structure of the special\nfiber of the curve is obtained by simply pulling back the log structure of klogi . Thus,\nit suffices to construct the (non-logarithmic, scheme-theoretic) special fiber. Next, we\nobserve that (after possibly enlarging Ki) there exist — cf., e.g., [Mzk4], the first two\nparagraphs of the proof of Theorem 9.2 — corresponding finite Galois étale coverings\n(Zi)Ki → (Xi)Ki (for i = 1, 2), where (Zi)Ki is a hyperbolic curve over Ki with bad\nstable reduction (Zi)OKi over OKi . Thus, by applying Theorem 2.7 to (Zi)Ki allows\nus to reconstruct the logarithmic special fiber (Z logi )ki , together with the action of the\nGalois group Gi\ndef\n= Gal((Zi)Ki \/(Xi)Ki). Note that irreducible components of (Zi)ki\nthat dominate (Xi)ki may be distinguished (group-theoretically!) by the fact that their\ngeometric fundamental groups map surjectively onto open subgroups of the geometric\nfundamental group of (Xi)ki . Let us choose corresponding (closed, proper) irreducible\ncomponents\nCi ⊆ (Zi)ki\nthat dominate (hence surject onto) (Xi)ki . Denote the decomposition (respectively, in-\nertia) group associated to Ci by Di ⊆ Gi (respectively, Ii ⊆ Di ⊆ Gi). Thus, Di\/Ii\nacts faithfully on Ci, and the order |Ii| of Ii is a power of p, equal to the degree of\ninseparability of the function field of Ci over the function field of (Xi)ki . Then we may\nreconstruct (Xi)ki as a finite flat quotient of Ci by considering the subsheaf\n(O|Ii|Ci )Di ⊆ OCi\n(i.e., the Di-invariants of the subalgebra O|Ii|Ci ⊆ OCi, where we use that |Ii| is a power\nof p). By applying the functoriality with respect to finite étale coverings of (Xi)Ki\nobserved in the discussion immediately preceding the statement of Theorem 2.7, we\nconclude that this construction of (Xi)ki is independent of the choice of (Zi)Ki , Ci, and\nitself functorial with respect to finite étale coverings of (Xi)Ki .\nThis completes our reconstruction of the logarithmic special fibers of the (Xi)Ki , in\na fashion that is functorial with respect to finite étale coverings of the (Xi)Ki . Thus,\nwe conclude, in particular, (from this functoriality, applied to covering transformations;\nthe slimness of Lemma 2.2, (i)) that the morphism induced on admissible fundamental\ngroups by the isomorphism constructed between logarithmic special fibers coincides with\nthe original given morphism between admissible fundamental groups. This completes\nthe proof of Theorem 2.7. ©\nRemark 2.7.1. Given data as in Theorem 2.7, one may consider the outer Galois\nrepresentation\nGKi → Out(ΔXi)\nABSOLUTE ANABELIAN GEOMETRY 37\nwhich is known to be injective if ri > 0 (cf. Theorem 1.3.6). Thus, at least in the case\nri > 0, it is natural to ask:\nWhat is the commensurator of Im(GKi ) in Out(ΔXi)?\nAlthough Theorem 2.7 does not give a complete explicit answer to this question, it\ntells us that, at any rate, elements of this commensurator (which define isomorphisms\nof the sort that are treated in Theorem 2.7) preserve the logarithmic special fiber. In\nparticular, (although one does not know whether or not elements of this commensurator\ninduce “self-isogenies” of Ki, i.e., are “geometric”) one obtains that elements of this\ncommensurator do induce “self-isogenies” of klogi . [Here, by a “self-isogeny of an object”,\nwe mean an isomorphism between two finite étale coverings of the object.] Moreover,\nsince it follows from Theorem A of [Mzk6] (cf. Theorem 1.3.4, Remark 1.3.6.2) that\nthe centralizer of Im(GKi) in Out(ΔXi) consists precisely of those (finitely many) au-\ntomorphisms that arise geometrically (i.e., from automorphisms of (Xi)Ki), it follows\nthat a “self-isogeny” of GKi induced by an element of this commensurator corresponds\nto (up to finitely many well-understood possibilities) an essentially unique element of\nthis commensurator. This motivates the point of view that:\nThe “self-isogenies” of GKi defined by elements of this commensurator —\nwhich we shall refer to as quasi-conformal self-isogenies of GKi — are\nnatural objects to study in their own right.\nThe reason for the choice of the terminology “quasi-conformal” is that those self-\nisogenies that are “of geometric origin” — i.e., “conformal” — are (by the main theo-\nrem of [Mzk5]) precisely those which preserve the higher ramification filtration, which\nis closely related to the “canonical p-adic metric” on the local field in question. Thus,\nquasi-conformal self-isogenies do not preserve the “metric (or conformal) structure” but\ndo preserve the “logarithmic special fiber” which one may think of as a sort of p-adic\nanalogue of the “topological type” of the objects in question.\nRemark 2.7.2. Note that isomorphisms\nklog1\n∼→ klog2\n(such as those arising from “quasi-conformal isomorphisms” GK1\n∼→ GK2 as in Theorem\n2.7) need not be “geometric” from the point of view of characteristic zero (i.e., induced\nby an isomorphism of fields K1\n∼→ K2). For instance, such an isomorphism might take\nthe section of the log structure corresponding to p to some multiple of this section by\na root of unity (a situation which could never occur if the isomorphism arose from an\nisomorphism K1\n∼→ K2). Whether or not, however, this sort of phenomenon actually\n38 SHINICHI MOCHIZUKI\ntakes place in the case of “quasi-conformal isomorphisms” as in Theorem 2.7 is not clear\nto the author at the time of writing.\nRemark 2.7.3. The theory of the present § prompts the question:\nDo isomorphisms Π(X1)K1\n∼→ Π(X2)K2 as in Theorem 2.7 only preserve the\nlogarithmic special fiber or do they preserve other information as well con-\ncerning the liftings (Xi)Ki of the respective logarithmic special fibers?\nAlthough the author is unable to give a complete answer to this question at the time of\nwriting, it does appear that when the lifting in question is in some sense “canonical”,\nthen this canonicality is preserved by isomorphisms as in Theorem 2.7. In a future paper,\nwe hope to discuss this sort of phenomenon — which may be observed, for instance, in\nthe following cases:\n(1) Serre-Tate canonical liftings;\n(2) “arithmetic hyperbolic curves”, i.e., hyperbolic curves isogenous to\na Shimura curve [that is to say, curves which admit a finite étale\ncovering which is isomorphic to a finite étale covering of some Shimura\ncurve];\n(3) canonical liftings in the sense of “p-adic Teichmüller theory” (cf.\n[Mzk1], [Mzk2])\n— in more detail. Perhaps this phenomenon should be regarded as a natural extension\nof the phenomenon of preservation of the logarithmic fiber in the sense that canonical\nliftings are, in some sense, liftings that are “defined over F1” — i.e., a hypothetical (but,\nof course, fictional!) absolute field of constants sitting inside Zp.\nAppendix: Terminology of Graph Theory\nThe Notion of a Semi-Graph:\nWe shall refer to as a semi-graph Γ the following collection of data:\n(1) a set V — whose elements we refer to as “vertices”;\n(2) a set E — whose elements we refer to as “edges” — each of whose elements\ne is a set of cardinality 2 satisfying the property “e \u000f= e′ ∈ E =⇒ e⋂ e′ = ∅”;\n(3) a collection ζ of maps ζe [one for each edge e] — which we refer to as\nthe “coincidence maps” — such that ζe : e → V\n⋃{V} [where we note that\nV ⋂{V} = ∅ since V \/∈ V ] is a map from the set e to the set V ⋃{V}.\nABSOLUTE ANABELIAN GEOMETRY 39\nWe shall refer to the subset ζ−1e (V) ⊆ e [i.e., the inverse image of the subset V ⊆ V\n⋃{V}\nof elements \u000f= V ] as the verticial portion of an edge e; to the restriction of ζe to the\nverticial portion of e as the verticial restriction of ζe; and to the cardinality of the\nverticial portion of e as the verticial cardinality of e. A graph Γ is a semi-graph Γ for\nwhich every e ∈ E has verticial cardinality precisely 2. We will say that a graph or semi-\ngraph is finite if its sets of vertices and edges are finite. We shall refer to an element\nb ∈ e as a branch of the edge e.\nLet Γ = {V , E, ζ} be a semi-graph. If e ∈ E is an edge of Γ of verticial cardinality 2\nwhose image via ζe consists of two (not necessarily distinct) elements v1, v2 of V , then\nwe shall say that e joins v1 to v2. If v is any vertex in the image of ζe, then we shall\nsay that e meets or abuts to v. Thus, an edge of a graph always abuts to at least one\nvertex, while an edge of a semi-graph may abut to no vertices at all.\nBy thinking of vertices as points and edges as line segments that join points to\npoints or are “open” at one or both ends, we may think of semi-graphs as defining\ntopological spaces. Thus, it makes sense to speak of a semi-graph as being contractible\n(in the sense of algebraic topology). Such a semi-graph will be referred to as a tree.\nFinally, a morphism between semi-graphs\nΓ = {V , E, ζ} → Γ′ = {V ′, E ′, ζ ′}\nis a collection of maps V → V ′; E → E ′; and for each e ∈ E mapping to e′, an injection\ne ↪→ e′ — all of which are compatible with the verticial restrictions of the respective\ncoincidence maps. Thus, here, we allow an edge that abuts to no (respectively, precisely\none) vertex to map to an edge that abuts to any number ≥ 0 (respectively, ≥ 1) of\nvertices.\nSemi-Graphs of Profinite Groups:\nWe shall refer to the following data G:\n(i) a finite semi-graph Γ;\n(ii) for each vertex v of Γ, a profinite group Gv;\n(iii) for each edge e of Γ, a profinite group Ge, together with, for each branch\nb ∈ e mapping to a vertex v via ζe, a continuous homomorphism b∗ : Ge → Gv\nas a semi-graph of profinite groups. When Γ is a graph, we shall refer to this data G as\na graph of profinite groups.\nSuppose that we are given a semi-graph of profinite groups G. Then to G, one\nmay associate (in a natural, functorial fashion) a profinite group ΠG — namely, the\n40 SHINICHI MOCHIZUKI\nprofinite analogue of the well-known construction of the fundamental group associated\nto a (semi-)graph of groups (cf. [Serre1], I, §5.1). One way to describe this profinite\nanalogue is via Galois categories (cf., e.g., [SGA1] for an exposition of the theory of\nGalois categories) as follows: If H is an arbitrary profinite group, then let us write\nB(H)\nfor the category of finite sets with continuous H-action. Then H may be recovered as the\n“fundamental group” of the Galois category B(H). Also, we note that any continuous\nhomomorphism of profinite groups H → H ′ determines a natural functor B(H′) → B(H)\n[in the reverse direction]. In particular, the homomorphisms b∗ determine functors:\nb∗ : Cv def= B(Gv) → Ce def= B(Ge)\nAt any rate, to define ΠG, it suffices to define a Galois category CG [which we would\nlike to think of as “B(ΠG)”]. We define CG to be the category of collections of data:\n{Sv;αe}\nHere, “v” (respectively, “e”) ranges over the vertices (respectively, edges of verticial\ncardinality 2) of Γ; Sv is an object of Cv; and αe : b∗1(Sv1) ∼= b∗2(Sv2) (where e = {b1, b2};\nvi\ndef\n= ζe(bi), for i = 1, 2) is an isomorphism in Ce. Morphisms between such collections\nof data are defined in the evident way. One then verifies easily that this category CG is\nindeed a Galois category, as desired.\nPointed Stable Curves:\nLet k be an algebraically closed field. Let g, r ≥ 0 be integers such that 2g−2+r > 0.\nLet (X → Spec(k),D ⊆ X) be an r-pointed stable curve of genus g (where D ⊆ X is\nthe divisor of marked points) over k, and set:\nX\ndef\n= X\\D\nAlso, let us write X ′ ⊆ X for the complement of the nodes in X.\nNext, we define the dual graph\nΓX\nof X. The set of vertices of ΓX is the set of irreducible components of X. To avoid\nconfusion, we shall write Xv for the irreducible component corresponding to a vertex\nv. The set of edges of ΓX is the set of nodes of X. To avoid confusion, we shall write\nνe for the node corresponding to an edge e. The node νe has two branches νe[1] and\nνe[2], each of which lies on some well-defined irreducible component of X. We take\nABSOLUTE ANABELIAN GEOMETRY 41\ne\ndef\n= {νe[1], νe[2]} [so e is of verticial cardinality 2], and let ζe be the map that sends\nνe[i] to the irreducible component on which the branch νe[i] lies.\nLet us write X ′v\ndef\n= Xv\n⋂\nX ′ ⊆ X, and (for i = 1, 2) X ′νe[i] for the scheme-theoretic\nintersection with X ′ of the completion of the branch νe[i] at the node νe. Thus, X ′νe[i]\nis noncanonically isomorphic to Spec(k[[t]][t−1]) (where t is an indeterminate).\nIn the following discussion, we would like to fix (k-linear) isomorphisms:\nX ′νe[1]\n∼= X ′νe[2]\nvia which we shall identify X ′νe[1] with X\n′\nνe[2]\nand denote the resulting object by X ′e. In\nparticular, we have natural morphisms X ′e → X ′v1 , X ′e → X ′v2 (where vi\ndef\n= ζe(νe[i]),\nfor i = 1, 2). One verifies immediately that the induced morphism on tame algebraic\nfundamental groups [i.e., corresponding to coverings which are tamely ramified at the\nnodes and marked points] is independent of the choice of isomorphism. Thus, the dual\ngraph ΓX , together with the result of applying “π\ntame\n1 (−)” [more precisely: the tame\nalgebraic fundamental group functor, together with choices of basepoints for the various\nX ′v, X ′e, and choices of paths to relate these basepoints via the natural morphisms\nX ′e → X ′vi discussed above] to the data {X ′v;X ′e;X ′e → X ′v} determines a graph of\nprofinite groups GX associated to the stable curve X.\nWhen considering the case of a curve with marked points (i.e., r > 0), it is useful\nto consider the following slightly modified “data with compact structure”: Let us denote\nby ΓcX the semi-graph obtained from ΓX by appending to ΓX , for each marked point\nx ∈ X , the following:\nan edge ex\ndef\n= {x}, where x ∈ ex is sent by ζex to the vertex vx corresponding\nto the irreducible component of X that contains x.\nWe shall refer to the new edges “ex” that were added to ΓX to form Γ\nc\nX as the marked\nedges of ΓcX and to Γ\nc\nX itself as the dual graph with compact structure associated to X.\nIf, moreover, we associate to ex the scheme X\n′\nx (which is noncanonically isomorphic\nto Spec(k[[t]][t−1])) obtained by removing x from the completion of X at x, and apply\n“πtame1 (−)” to the natural morphism X ′x → X ′vx , then we obtain a natural semi-graph\nof profinite groups GcX with underlying semi-graph ΓcX . Moreover, one checks easily that\nwhen k is of characteristic 0, the profinite group associated (as described above) to GX\nor GcX is isomorphic to “Π̂g,r”, i.e., the profinite completion of the fundamental group of\na Riemann surface of genus g with r points removed.\nIn fact, if we take k = C, and we think of the X ′v as Riemann surfaces and of the\nX ′e, X ′x as “copies of the circle S1”, then we see that this construction corresponds quite\ngeometrically to gluing Riemann surfaces with boundary along copies of the circle.\n42 SHINICHI MOCHIZUKI\nFinally, let us observe the following (valid for k of arbitrary characteristic): The\nprofinite group associated to GX or GcX may be identified with the admissible funda-\nmental group\nΠadmX\nof (X,D) (cf. [Mzk4], §2, 8; [Mzk3], §3). Moreover, if we define semi-graphs of groups\nGetX , Gc,etX by taking the profinite group at a vertex v to be the tame fundamental group\nof Xv [i.e., instead of X\n′\nv] and the profinite group at all of the edges to be trivial, then\nthe resulting profinite group may be identified with the fundamental group\nΠetX\nassociated to the Galois category of finite étale coverings of X which are tamely ramified\nat the cusps. In particular, we have a natural surjection ΠadmX \u0002 ΠetX .\nSecond Cohomology Groups:\nWe continue with the notation of the above discussion. Let l be a prime number dis-\ntinct from the characteristic of k. In the following discussion, we shall use the notation\n“Hj(−)” (respectively, “Hjc (−)”) to denote the j-th étale cohomology module (respec-\ntively, j-th étale cohomology module with compact supports — cf. [Milne], Chapter III,\nProposition 1.29; Remark 1.30) with coefficients in Zl.\nIf v is a vertex of ΓX , write Xv for the normalization of the corresponding irre-\nducible component of X, so X ′v = X ′ ×X Xv. Then we have a natural “push-forward”\nisomorphism\nH2c (X\n′\nv)\n∼→ H2(Xv)\nbetween free Zl-modules of rank 1. Moreover, the natural morphisms Xv → X (as v\nranges over the vertices of ΓX) determine natural restriction morphisms whose direct\nsum is easily verified to be an isomorphism:\nH2(X)\n∼→\n⊕\nv\nH2(Xv)\nOn the other hand, the composite of the natural “push-forward” morphism H2c (X\n′\nv) →\nH2(X) with a restriction morphism H2(X) → H2(Xw), where w \u000f= v, is easily verified\nto be zero. [Indeed, this follows by considering the Chern class of the line bundle on\nX associated to a closed point of X ′v: This Chern class is the image of a generator of\nH2c (X\n′\nv) and, moreover, vanishes upon restriction to Xw since the restriction to Xw of\nthis line bundle is clearly trivial.] In particular, we conclude that the direct sum of the\nnatural “push-forward” morphisms H2c (X\n′\nv) → H2(X) yields an isomorphism⊕\nv\nH2c (X\n′\nv)\n∼→ H2(X)\nABSOLUTE ANABELIAN GEOMETRY 43\nwhich, if we identify H2c (X\n′\nv), H\n2(Xv) via the isomorphism H\n2\nc (X\n′\nv)\n∼→ H2(Xv), is\ninverse to the isomorphism H2(X)\n∼→ ⊕v H2(Xv).\nPut another way, if we identify the cohomology modules H2c (X\n′\nv), H\n2(Xv) via\nthe isomorphism H2c (X\n′\nv)\n∼→ H2(Xv), then the natural “push-forward” morphism\nH2c (X\n′\nv) → H2(X) may be recovered as the composite\nH2(Xv) ↪→\n⊕\nv\nH2(Xv)\n∼→ H2(X)\nof the natural inclusion of a direct summand with the inverse of the isomorphism\nH2(X)\n∼→ ⊕v H2(Xv) determined by the restriction morphisms.\nFinally, let us observe that if X is sturdy (cf. [Mzk4], Definition 1.1) — i.e., every\nXv has genus ≥ 2 — then the natural morphisms from profinite group cohomology to\nétale cohomology give rise to a commutative diagram\nH2(πtame1 (X) = π1(X),Zl) −→ H2(X)⏐⏐\t ⏐⏐\t\nH2(πtame1 (Xv) = π1(Xv),Zl) −→ H2(Xv)\nin which the horizontal morphisms are isomorphisms [since arbitrary étale cohomology\nclasses with finite l-power torsion coefficients vanish upon restriction to some finite étale\ncovering of X or Xv].\nReferences\n[FC] G. Faltings and C.-L. Chai, Degenerations of Abelian Varieties, Springer-Verlag\n(1990).\n[FJ] M. Fried and M. Jarden, Field Arithmetic, Springer-Verlag (1986).\n[HR] W. Herfort and L. Ribes, Torsion elements and centralizers in free products of\nprofinite groups, J. Reine Angew. Math. 358 (1985), pp. 155-161.\n[Kato1] K. Kato, Logarithmic Structures of Fontaine-Illusie, Proceedings of the First JAMI\nConference, Johns Hopkins Univ. Press (1990), pp. 191-224.\n[Kato2] K. Kato, Toric Singularities, Amer. J. Math. 116 (1994), pp. 1073-1099.\n[Knud] F. F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, II, Math.\nScand. 52 (1983), 161-199.\n[Mtmo] M. Matsumoto, Galois representations on profinite braid groups on curves, J. Reine\nAngew. Math. 474 (1996), pp. 169-219.\n44 SHINICHI MOCHIZUKI\n[Milne] J. S. Milne, Étale Cohomology, Princeton Mathematical Series 33, Princeton Uni-\nversity Press (1980).\n[Mzk1] S. Mochizuki, A Theory of Ordinary p-adic Curves, Publ. of RIMS 32 (1996), pp.\n957-1151.\n[Mzk2] S. Mochizuki, Foundations of p-adic Teichmüller Theory, AMS\/IP Studies in\nAdvanced Mathematics 11, American Mathematical Society\/International Press\n(1999).\n[Mzk3] S. Mochizuki, The Geometry of the Compactification of the Hurwitz Scheme, Publ.\nof RIMS 31 (1995), pp. 355-441.\n[Mzk4] S. Mochizuki, The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves\nover Number Fields, J. Math. Sci. Univ. Tokyo 3 (1996), pp. 571-627.\n[Mzk5] S. Mochizuki, A Version of the Grothendieck Conjecture for p-adic Local Fields,\nThe International Journal of Math. 8 (1997), pp. 499-506.\n[Mzk6] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138\n(1999), pp. 319-423.\n[Mumf] D. Mumford, Abelian Varieties, Oxford Univ. Press (1974).\n[Naka1] H. Nakamura, Galois rigidity of pure sphere braid groups and profinite calculus, J.\nMath. Sci. Univ. Tokyo 1 (1994), pp. 71-136.\n[Naka2] H. Nakamura, Galois rigidity of profinite fundamental groups [translation of Sugaku\n47 (1995), pp. 1-17], Sugaku Expositions 10 (1997), pp. 195-215.\n[NTM] H. Nakamura, A. Tamagawa, and S. Mochizuki, The Grothendieck Conjecture on\nthe Fundamental Groups of Algebraic Curves, Sugaku Expositions 14 (2001), pp.\n31-53.\n[NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields, Grundlehren\nder Mathematischen Wissenschaften 323, Springer-Verlag (2000).\n[Serre1] J.-P. Serre, Trees, Springer-Verlag (1980).\n[Serre2] J.-P. Serre, Local Class Field Theory in Algebraic Number Theory, ed. J.W.S. Cas-\nsels and A. Fröhlich, Academic Press (1967).\n[SGA1] Revêtement étales et groupe fondamental, Séminaire de Géometrie Algébrique du\nBois Marie 1960-1961 (SGA1), dirigé par A. Grothendieck, augmenté de deux ex-\nposés de M. Raynaud, Lecture Notes in Mathematics 224, Springer-Verlag (1971).\n[Tama1] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio Math.\n109 (1997), pp. 135-194.\n[Tama2] A. Tamagawa, On the Fundamental Groups of Curves over Algebraically Closed\nFields of Characteristic > 0, International Mathematics Research Notices 16 (1999),\npp. 853-873.\nABSOLUTE ANABELIAN GEOMETRY 45\n[Tate] J. Tate, p-Divisible Groups, Proceedings of a Conference on Local Fields, Drieber-\ngen, Springer Verlag (1967), pp. 158-183.\n\u0000\u0002\u0003\u0002\u0004\u0005\u0006\u0007 \b\t\u0003\n\u000b\n\f\n\u0002 \r\u000e\u0005 \u000f\u0004\n\u0007\u0002\u0010\u0004\n\u000b\u0006\u0004\u0011 \u0012\u0006\u000b\u0002\t\u0006\u0002\u0003\u0013\n\u0014\u0015\u000e\n\u000e \u0016\t\u000b\u0017\u0002\u0005\u0003\u000b\n\u0015\u0013\n\u0014\u0015\u000e\n\u000e \u0018\u0019\u0018\u001a\u001b\u001c\u0019\u001d\u0013 \u001e\u0004\u001f\u0004\t\nE-mail address: motizuki@kurims.kyoto-u.ac.jp\n"}
{"index":{}}
{"id":"F1YCZ116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Arithmetic%20Elliptic%20Curves%20in%20General%20Position%20(comments).pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":116,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Arithmetic%20Elliptic%20Curves%20in%20General%20Position%20(comments)","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["gen-ell-comments-2016-07-16.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Tue Jul 12 17:13:20 UTC 2016","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-4027827257030440691,"exact_signature_unique_b":true,"fuzzy_signature_l":-2455012660958979319,"fuzzy_signature_unique_b":true,"text_t":"COMMENTS ON “ARITHMETIC ELLIPTIC CURVES\nIN GENERAL POSITION”\nShinichi Mochizuki\nJuly 2016\n(1.) The notation “ordv(−) : Fv → Z” in the final sentence of the first paragraph\nfollowing Definition 1.1, should read “ordv(−) : F×v → Z”.\n(2.) In Definition 1.2, (ii), the non-resp’d and first resp’d items in the display should\nbe reversed! That is to say, the notation “α \u0002F β” corresponds to “α(x)− β(x) ≤\nC”; the notation “α \u0003F β” corresponds to “β(x)− α(x) ≤ C”.\n(3.) The first portion of the first sentence of the statement of Corollary 4.4, should\nread: “Let Q be an algebraic closure of Q; . . . ”.\n(4.) The “log-diffMell([EL]))” in the second inequality of the final display of the\nstatement of Corollary 4.4, should read “log-diffMell([EL])”.\n(5.) The equality\nhtE ≈ (deg(E)\/deg(ωX)) · htωX\nimplicit in the final “≈” of the final display of the proof of Theorem 2.1, should be\nreplaced by an inequality\nhtE \u0002 2 · (deg(E)\/deg(ωX)) · htωX\n[which follows immediately from Proposition 1.4, (ii)], and the expression “deg(E)\/\ndeg(ωX)” in the inequality imposed on the choice of \u0005\n′ should be replaced by the\nexpression “2 · (deg(E)\/deg(ωX))”.\n(6.) The phrase “Corollary 2.1” in the first paragraph of §2, should read “Theorem\n2.1”.\n(7.) The inequality “xS◦ ≤ xS◦” at the end of the second to last sentence of the\nproof of Corollary 4.3, should read “xS◦ ≤ xS•”.\n(8.) Suppose that we are in the situation of Example 1.3, (ii). Let U ⊆ X be an\nopen subscheme. Then a “compactly bounded subset”\nKV ⊆ U(Q) (⊆ X(Q))\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\nof U(Q) is to be understood as a subset which forms a compactly bounded subset\nof X(Q) [i.e., in the sense discussed in Example 1.3, (ii)] and, moreover, satisfies\nthe property that for each v ∈ V arc def= V ⋂V(Q)arc (respectively, v ∈ V non def=\nV\n⋂\nV(Q)non), the compact domain Kv ⊆ Xarc (respectively, Kv ⊆ X(Qv)) is, in\nfact, contained in\nU(C) ⊆ X(C) = Xarc (respectively, U(Qv) ⊆ X(Qv)).\nIn particular, this convention should be applied to the use of the term “compactly\nbounded subset” in the statements of Theorem 2.1, Lemma 3.7, Theorem 3.8, Corol-\nlary 4.4. Although this convention was not discussed explicitly in Example 1.3, (ii),\nit is, in effect, discussed explicitly in the discussion of “compactly bounded subsets”\nat the beginning of the Introduction. Moreover, this convention is implicit in the\narguments involving compactly bounded subsets in the proof of Theorem 2.1.\n(9.) In the discussion following the second display of Example 1.3, (ii), the phrase\n“(respectively, X(Qv))” should read “(respectively, X(Qv))”.\n(10.) The first display of the paragraph immediately following Remark 3.3.1, should\nread as follows:\n|α|2 def=\n∣∣∣\n∫\nEv\nα ∧ α\n∣∣∣\n[i.e., the integral should be replaced by the absolute value of the integral].\n(11.) In the discussion following the final display of Example 1.3, (ii), the phrase\n“to a subset” should read “to a subset of”.\n(12.) In the text immediately following the first display of Definition 1.5, (iv), the\nphrase “conductor of x” should read “conductor of x [with respect to D]”.\n(13.) The following phrase should be added to the end of the third sentence [i.e.,\n“Here, the condition of ...”] of §3:\n[cf. the conclusion concerning the condition “(b)” in Lemma 3.7].\n(14.) In the discussion in the paragraph immediately preceding Lemma 3.2 of\nthe extension class associated to the exact sequence of GK-modules of the sec-\nond display of this paragraph, it should be recalled that there is a natural iso-\nmorphism between the group of extensions Ext1GK (Fl,Fl(1)) and the cohomology\nmodule H1(GK ,Fl(1)).\n(15.) In the proof of Theorem 3.8, the text “[which has the effect of ...” should\nread “[so that choosing a level structure on the 3 · 5-torsion points has the effect of\n...”.\n(16.) In the proof of Theorem 3.8, the text “finite primes of L” should read “finite\nprimes of L′”.\n","wordcount_i":598,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:50 UTC 2022","fresh_date_dt":"Thu May 22 04:38:05 UTC 2025","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":56654,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086037395308544}
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{"id":"GxkaPd16muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/en\/seminar\/seminar-arakawa.html","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":64,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["en","seminar"],"url_file_name_s":"seminar-arakawa","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Representation Theory Seminar｜RIMS"],"last_modified":"Wed Jun 08 05:17:37 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Thu Jan 13 00:00:00 UTC 2011","Wed Dec 08 00:00:00 UTC 2010"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-9021752674804467906,"exact_signature_unique_b":true,"fuzzy_signature_l":2829501377746569565,"fuzzy_signature_unique_b":true,"h1_txt":["Representation Theory Seminar","Colloquium\/Seminars"],"h2_txt":["Representation Theory Seminar"],"bold_txt":["Langlands duality for real groups","see the Japanese page","Relaxed highest-weight modules over affine VOAs","Classical weight modules over i-quantum groups","W-algebras and their representations","量子ベキ単部分群と双対標準基底（Quantum Unipotent Subgroup and dual canonical basis）","Instanton and W-algebras (review)","Borcherds products in monstrous moonshine","Hypertoric (and other) categories O (joint w\/ Braden, Licata and Proudfoot)","Shifted Yangians and finite W-algebras (survey)","共形場理論、モノドロミー保存変形とAGT予想","Lie Superalgebras and Enveloping Algebras","Quantum queer superalgebra and crystal bases.","Ding-Iohara algebra and K-theoretic AGT conjecture","A finite analog of the AGT relation (survey)","Two-parameter deformation of multivariate hook product formulae","Maulik-Okounkovの理論の紹介 - 応用として、AGT予想の証明","Induction and Restriction functors for cyclotomic q-Schur algebras.","Harish-Chandra bimodules for quantized Slodowy slices (survey)","Cyclotomic quiver Hecke algebras and categorifications of highest weight module","Pseudo-trace functions for orbifold models associated with symplectic fermions","Weyl module, Demazure moduleとfundamental representationのテンソル積のcrystal basisとの関係について","Saga of Dunkl elements.","A COMPARISON OF q-DECOMPOSITION NUMBERS IN THE q-DEFORMED FOCK SPACES OF HIGHER LEVELS","Johnson準同型と外Galois表現","曲面の写像類群に付随するJohnson余核のSp-加群構造について（佐藤隆夫 氏（東京理科大）との共同研究）","On the universal sl_2 invariant of bottom tangles","線形常微分方程式とルート系","Fourier-Laplace変換とKatz-Deligne-Arinkinの定理","An application of the classification of visible linear actions to nilpotent orbits","Fourier-Laplace変換とKatz-Deligne-Arinkinの定理 その2","Localization of affine W-algebras at the critical level","Quiver Hecke superalgebras","Quantization of Quasimaps' Spaces (joint work with M. Finkelberg)","形式次数とテータ対応","コホモロジカル誘導の局所化","Finite W-algebras","対称空間のコンパクト化によるJacquet加群の実現","グリーン関数のホモロジー論的側面について","Coproduct on Yangian","次数付き箙多様体と量子クラスター代数","A型アファインシュプリンガーファイバーとdiagonal coinvariantの組み合わせ論","Self-extensions and prime factorizations for simple $U_q(L\\mathfrak{sl}_2)$-modules","Linear recurrence relations for cluster variables","Parameters of quiver Hecke algebras","cyclotomic q-Schur 代数の Drinfeld 型の表示について","On Hall algebra of complexes","環積が作用する確率モデルと多変数 Krawtchouk 多項式","行列型１階線形常微分方程式と箙の表現","BC型量子ループ代数のminimal affinizationについて","Drinfeld twists and finite groups","Quantum Lakshmibai-Seshadri paths and Ram-Yip's combinatorial formula for Macdonald polynomials","Quantum Painleve systems from hypergeometric integrals of Euler type","Representation theory of finite W-algebras","Symmetric quiver Hecke algebras and R-matirce of quantum affine algebras","Derived subalgebras of centralizers and completely prime primitive ideals.","The structures of the centers of Hecke 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level-zero extremal weight modules over quantum affine algebras","Cluster algebras, dilogarithm, and Y-systems","Springer多様体のトーラス同変コホモロジー環 (Torus equivariant cohomology ring of Springer varieties)","Gaudin subalgebras and stable rational curves","Superconformal Blocks","Elliptic Quantum Groups, Drinfeld Coproduct and Deformed W-algebras","Elliptic stable bases and applications","1) COMMUTATORS IN ALGEBRAIC GROUPS 2) K-THEORY OF EXCEPTIONAL GROUPS","Local Fourier transform and blowing up","Coulomb branches of 3d N=4 gauge theories and the affine Grassmannian","量子アフィン展開環のレベル・ゼロ表現に対するギャラリー模型","Geometric Satake, Springer correndence, and small representations","Toroidal and affine Yangian algebras, and their commutative subalgebras","Representations of quantized function algebras the transition matrices from Canonical bases to PBW bases","アフィン・スーパー・リー環の表現と モック・テータ函数","A deformation of affine Hecke algebra and integrable stochastic particle system","New realization of cyclotomic q-Schur algebras","RoCK blocks, wreath products and KLR algebras","Finite W-superalgebras and existence of Kac-Weisfeiler modules for basic Lie superalgebras in positive chatacteristic","Positive Casimir and Central Characters of Split Real Quantum Groups","Calabi-Yau圏の安定性条件の空間とフロベニウス多様体","Sheaves of asymptotic chiral differential operators on symplectic resolutions","Banach Algebraic Geometry","Classifying simple modules at admissible levels symmetric polynomials","Parabolic analogue of periodic Kazhdan-Lusztig polynomials","Kazama-Suzuki coset construction and logarithmic extensions of weight modules","A Fock sheaf for Givental quantization","中心拡大されたリースーパー代数sl(2|2)に付随する 量子アファイン代数について (Quantum affine algebra associated with the centrally extended Lie superalgebra sl(2|2))","Gauge Theory in 2 and 3 dimensions and categorical representations.","On a general Schur's partition identity","K-theoretic AGT relation","Screening operators for W-algebras","Local theta correspondence of tempered representations and Langlands parameters","Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A","Homological description of crystal structures on quiver varieties","Combinatorial formulas expressing periodic R-polynomials and periodic Kazhdan-Lusztig polynomials","Kraskiewicz-Pragacz modules and positivity properties of Schubert polynomials","Visible actions of compact Lie groups on complex spherical varieties","A description of principal series representations of SU(1,1) over ${\\mathbb{Z}}\\left[1\/2\\right]$-algebras","Classical and orientifold Donaldson-Thomas invariants as Chow groups","PARABOLIC PRESENTATION OF THE SUPER YANGIAN $Y_{M|N}$ AND ITS APPLICATION","Fukaya category in the Koszul duality theory","Tilting modules of affine quasi-hereditary algebras","The Hard Lefschetz Theorem in Positive Characteristic for the Flag Varieties","The p-canonical basis of Hecke algebras","Positivity for the dual of the Temperley-Lieb basis","Generalized Weyl modules and nonsymmetric Macdonald polynomials","Determinant formula for parabolic Verma modules of Lie superalgebras","Hessenberg varieties and hyperplane arrangements","The universal quantum invariant and colored ideal triangulation","D-modules on partial flag varieties and intertwining functors","Cluster realization of Uq(g) and factorization of universal R matrix","非対称Macdonald多項式の特殊化の間の双対性について","Homogeneous spaces that do not model any compact manifold","拡大カスピダルワイル群の不変式論と軌道空間上のフロベニウス構造","Faithful actions from hyperplane arrangements","Specht modules for the KLR algebras of type C","DAHA approach to plane curve singularities","Weyl modules: one variable vs. multivariable","Affinoids in the Lubin-Tate perfectoid space and special cases of the local Langlands correspondence","Coproducts for W-algebras in A-type","The Chamber Ansatz formulae for quantum unipotent cells","The Universal Differential Operator","Affine highest weight categories for quantum loop algebras of Dynk in types","Representation theory of quantum symmetric pairs and Kazhdan-Lusztig bases","Springer theory for symmetric spaces","Elliptic Hall algebra over $\\mathbb{F}_1$","Quasimaps counts and Bethe eigenfunctions","Double-Bosonization and dual bases of quantum groups ℂq[SL( 2)] and ℂq[SL(3)]","Mirror symmetry for some homogeneous spaces","Mackey’s formula for cyclotomic Hecke algebras and rational Cher ednik algebras of type G(r,1,n)","Periodic modules for hypertoric varieties","Monoidal categorification of cluster algebras","Positive representations of quantum groups & modular functor","Cluster algebra realization of quantum groups and their positive representations","An introduction to higher Teichmuller theory","The universal Poisson deformation space of hypertoric varieties.","Geometric construction of integrable Hamiltonian hierarchies associated with the classical affine W-algebras.","Koszul duality for real groups","Vertex algebras and coordinate rings of semi-infinite flags","S-duality in the example of the large N=4 superconformal algebra","Positive Peter-Weyl Theorem","Relaxed modules over affine vertex operator algebras","Affine osp(1|2) and its coset construction","Singular vectors of DIM algebra and generalized Macdonald functions arising from AGT conjecture","Jacobian factors in any ranks and DAHA superpolynomials","Gaudin model and crystals (joint work with Iva Halacheva, Joel Kamnitzer and Alex Weekes)","A prescient SGA3: Applications of reductive group schemes to infinite dimensional Lie theory","Coulomb branches, shifted quantum algebras and modified q-Toda systems","Representations of Lie algebras of vector fields on affine algebraic varieties.","Drinfeld-Gaitsgory interpolation Grassmannian and geometric Satake equivalence","Double graded Ginzburg Calabi-Yau algebras, q-deformations of root lattices and q-stability conditions on their derived categories","Modular linear differential equations of fourth order and minimal W-algebras","Toward double affine flag varieties and Grassmannians","A local characterization of $B_2$ regular crystals","Laurent phenomenon and simple modules over symmetric quiver Hecke algebras","Quasireductive supergroups and their representations","Elliptic Quantum Group and Elliptic Stable Envelopes","Quantum Grothendieck ring isomorphisms for quantum affine algebras of type A and B","Two families if commutation classes and categories attached to them","Rigged Configurations and parabolic Kostka polynomials","Localization for quiver Hecke algebras","Minimal Models, Arc spaces, and Rogers-Ramanujan Identities","A twofold generalization of Gabriel's theorem","17:00-18:00 Bernard Leclerc","Graded Lie algebras, character sheaves, and DAHA representations","15:30-16:30 Ting Xue","Character sheaves for graded Lie algebras, geometric theory","14:00-15:00 Kari Vilonen","An approach to the invariant theory for the elliptic Weyl groups","16:30-18:00 Ikuo Satake (Kagawa University)","[CANCEL] 14:30-16:00 Kazuya Kawasetsu (RIMS)","Semibricks in $\\tau$-tilting theory","Deformed Hecke algebras and quantum cohomology of graph varieties","Integral representations of GKZ hypergeometric functions: Gauss-Manin connection, intersection theory, and quadratic relations","Hurwitz numbers of Riemann sphere and integrable hierarchies.","Peter-Weyl, Howe and Schur-Weyl theorems for current groups","Affine Super Yangian","Hall algebras of toric varieties over F_1","On the Feigin-Tipunin VOA","On module categories of preprojective algebras with symmetrizers","Demazure slices of type $A_{2l}^{(2)}$","Dualities of W-algebras and the W_{\\infty}-algebra","On representation of the finite W-superalgebras","Irregular vertex algebras","On Gaiotto-Rapcak's dualities in W-superalgebras and their affine cosets","Algebraic entropy of sign-stable mutation loops","Derived gluing construction of chiral algebras","Difference equations arising from cluster algebras","Affine Yangians 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forms","Systematic construction of isomorphisms among quantum Grothendieck rings","From W-algebras to W-superalgebras","Sheaf quantization: example and construction","Generalized Fateev-Zamolodchikov-Zamolodchikov dualities and Gaiotto- Rapcak's VOAs","Hypertoric Hitchin systems and Kirchoff polynomials","Chiral homology of elliptic curves","Vertex Algebras and Factorization Algebras","Reduction cohomology on Riemann surfaces","Code conformal field theory and framed algebra","Quantum difference equations and elliptic stable envelopes","Categorification of DAHA and Macdonald polynomials","The small quantum group and certain affine Springer fibers","Whittaker vectors for W-algebras from topological recursion","The quantum group dual of the first-row modules for the generic Virasoro VOA","Feigin-Semikhatov Duality in W-superalgebras","Nakayama functors for Frobenius tensor categories","p-DG structures in higher representation theory","A new proof of positivity of DT invariants of 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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Representation Theory Seminar. TOP 　　. Seminars 　　Representation Theory Seminar. Representation Theory Seminar. Date. June 24th (Fri), 14:30--16:00, 2022. Room. Zoom. Speaker. Yoshiyuki Kimura (Osaka Metropolitan University). Title. Cluster twist automorphisms and compatible Poisson structures. Abstract. 　 We introduce cluster twist automorphisms for (upper) cluster algebras and cluster Poisson algebras (with coefficients). We study their existence and their compatibility with Poisson structures and we also show that the cluster twist automorphisms always permute well-behaved bases for cluster algebras. This is a joint work with Fan Qin and Qiaoling Wei. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Ryo SATO, Kota MURAKAMI. Date. June 16 (Thu), 10:30--12:00, 2022. Room. Zoom (access information will appear in. https:\/\/kyoto-u-edu.zoom.us\/j\/81651182256 meeting ID：816 5118 2256. passcode：109268. Speaker. Yuchen Fu (RIMS). Title. Kazhdan-Lusztig Equivalence at the Iwahori Level. Abstract. 　 We construct an equivalence between Iwahori-integrable representations of affine Lie algebras and representations of the mixed quantum group, thus confirm a conjecture by Gaitsgory. Our proof utilizes factorization methods: we show that both sides are equivalent to algebraic\/topological factorization modules over a certain factorization algebra, which can then be compared via Riemann-Hilbert. On the quantum group side this is achieved via general machinery of homotopical algebra, whereas the affine side requires inputs from the theory of (renormalized) ind-coherent sheaves as well as compatibility with global Langlands over P1. This is joint work with Lin Chen. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Ryo SATO, Kota MURAKAMI. Date. February 3 (Thu), 10:00--11:30, 2022. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4811 Speaker. Angela Gibney (University of Pennsylvania). Title. Vector bundles on the moduli space of curves from vertex operator algebras. Abstract. 　 Algebraic structures like vector bundles, their sections, ranks, and characteristic classes, give information about spaces on which they are defined. The stack parametrizing families of stable n-pointed curves of genus g, and  the space that (coarsely) represents it, give insight into curves and their degenerations, are prototypes for moduli of higher dimensional varieties, and are interesting objects of study in their own right. Vertex operator algebras and their representation theory, have had a profound influence on mathematics and mathematical physics, playing a particularly important role in understanding conformal field theories, finite group theory, and invariants in topology.   In this talk I will discuss vector bundles on moduli of curves defined by certain representations of VOAs. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO. Date. January 27 (Thu), 10:30--12:00, 2022. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4806 Speaker. Zijun Zhou (Kavli IPMU). Title. Virtual Coulomb branch and quantum K-theory. Abstract. 　 In this talk, I will introduce a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N\/\/\/G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under variation of GIT. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO. Date. January 13 (Thu), 17:00--18:30, 2022. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4793 Speaker. Vladimir Dotsenko (University of Strasbourg). Title. A new proof of positivity of DT invariants of symmetric quivers. Abstract. 　 I shall talk about a new interpretation of refined Donaldson-Thomas invariants of symmetric quivers, in particular re-proving their positivity (conjectured by Kontsevich and Soibelman, and proved by Efimov). This interpretation has two key ingredients. The first is a certain Lie (super-)algebra, for which we have two interpretations, in the context of Koszul duality theory and in the context of vertex Lie algebras. The second is an action of the Weyl algebra of polynomial differential operators on that Lie algebra, for which the characters of components of the space of generators give precisely the refined DT invariants. This is a joint project with Evgeny Feigin and Markus Reineke, partially relying on my recent work with Sergey Mozgovoy. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO. Date. December 9 (Thu), 21:00--22:30, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4760 Speaker. Joshua Sussan　(City University of New York). Title. p-DG structures in higher representation theory. Abstract. 　 One of the goals of the categorification program is to construct a homological invariant of 3-manifolds coming from the higher representation theory of quantum groups.  The WRT 3-manifold invariant uses quantum groups at a root of unity. p-DG theory was introduced by Khovanov as a means to categorify objects at prime roots of unity.  We will review this machinery and show how to construct categorifications of certain representations of quantum sl(2) at prime roots of unity. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO. Date. November 25 (Thu), 16:00--17:30, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4762 Speaker. Kenichi Shimizu (Shibaura Institute of Technology). Title. Nakayama functors for Frobenius tensor categories. Abstract. 　 This talk is based on my joint work with Taiki Shibata. The Nakayama functor is an important notion in the representation theory of finite-dimensional algebras. Fuchs, Schaumann, and Schweigert pointed out that the Nakayama functor has a certain universal property and, by using this property, defined the Nakayama functor for finite abelian categories. As they also pointed out, such an abstract treatment of the Nakayama functor turned out to be very useful for proving general results on finite tensor categories. In this talk, I will explain how and when one can define the Nakayama functor for a locally finite abelian category. Let A be a locally finite abelian category. Technical difficulty is that there is no endofunctor on A satisfying the same universal property as in the finite case. Such an endofunctor on A exists if, for example, A is the category of finite-dimensional comodules over a semiperfect coalgebra. This observation allows us to define the Nakayama functor for Frobenius tensor categories in the sense of Andruskiewitsch, Cuadra and Etingof. Using the Nakayama functor, one can prove some general results on Frobenius tensor categories in the same way as the finite case. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO. Date. November 11 (Thu), 16:00--17:30, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4757 Speaker. Ryo Sato (RIMS). Title. Feigin-Semikhatov Duality in W-superalgebras. Abstract. 　 W-superalgebras are a large class of vertex superalgebras which generalize affine Lie superalgebras and the Virasoro algebras. Recently, D. Gaiotto and M. Rapčák found dualities between certain hook-type W-superalgebras in relation to certain four-dimensional supersymmetric gauge theories. A large part of their conjecture is proved by T. Creutzig and A. Linshaw, and a specific subclass (Feigin-Semikhatov duality) is done by T. Creutzig, N. Genra, and S. Nakatsuka in a different way. In this talk I will talk about a monoidal correspondence of representations induced by the latter duality and relative semi-infinite cohomology. This talk is based on a joint work with T. Creutzig, N. Genra, and S. Nakatsuka. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO. Date. July 29 (Thu), 15:00--16:30, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4695 Speaker. Shinji Koshida (Chuo University). Title. The quantum group dual of the first-row modules for the generic Virasoro VOA. Abstract. 　 In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. In this talk, we discuss such a duality in the case of the Virasoro VOA at generic central charge. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules and the analyticity of compositions of intertwining operators. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the 6j-symbols of the quantum group Uq(sl2). This talk is based on a joint work with Kalle Kytölä. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. July 15 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4669 Speaker. Gaëtan Borot　(Humboldt-Universität zu Berlin). Title. Whittaker vectors for W-algebras from topological recursion. Abstract. 　 Inspired by Alday-Gaiotto-Tachikawa conjecture in physics, Schiffman-Vasserot and Braverman-Finkelberg-Nakajima showed that, if G is a simple simply-laced Lie group, the partition function of pure N = 2 supersymmetric gauge theories with gauge group G can be reconstructed as the norm of certain Whittaker vectors of principal W(g)-algebras. After reviewing the context, I will explain how such Whittaker vectors (and in principle many more \"Whittaker-like\" vectors) can be computed by a topological recursion a la Eynard-Orantin, and potential consequences. This is based on joint works with Bouchard, Chidambaram, Creutzig and Noshchenko. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. July 8 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4683 Speaker. Pablo Boixeda Alvarez (Institute for Advanced Study). Title. The small quantum group and certain affine Springer fibers. Abstract. 　 In this talk I will discuss several connections between the small quantum group and a certain affine Springer fiber. In particular I will mainly discuss some relation of the center of the small quantum group and the cohomology of the affine Springer fiber, part of ongoing joint work with R. Bezrukavnikov, P. Shan and E. Vasserot. I will also mention some description of the category in terms of microlocal sheaves as part of ongoing work with R.Bezrukavnikov, M. McBreen and Z. Yun and certain connections of cohomology of the affine Springer fiber and the Hilbert scheme of points of C^2 as part of joint work with O. Kivinen and I. Losev. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. July 1 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4680 Speaker. Syu Kato (Kyoto University). Title. Categorification of DAHA and Macdonald polynomials. Abstract. 　 We exhibit a categorification of the double affine Hecke algebra (DAHA) associated with an untwisted affine root system (except for type G) and its polynomial representation by using the (derived) module category of some Lie superalgebras associated to the root system. This particularly yields a categorification of symmetric Macdonald polynomials. This is a joint work with Anton Khoroshkin and Ievgen Makedonskyi. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. June 24 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4663 Speaker. Andrey Smirnov (University of North Carolina at Chapel Hill). Title. Quantum difference equations and elliptic stable envelopes. Abstract. 　 A remarkable class of differential and q-difference equations emerges naturally in the study of enumerative geometry of quiver varieties. This class includes Knizhnik-Zamolodchikov equations, quantum dynamical equations and other important objects in representation theory. In my talk I overview a geometric approach to these equations based on the theory of elliptic stable envelopes and three-dimensional mirror symmetry. In this approach we use geometric methods to constrain the monodromy of the associated q-difference equations. Then, the equations can be reconstructed from the monodromy via a simple limiting procedure. The three-dimensional mirror symmetry of the elliptic stable envelopes relates the equations associated to a quiver variety with those of symplectic dual variety. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. June 17 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4662 Speaker. Yuto Moriwaki (RIMS). Title. Code conformal field theory and framed algebra. Abstract. 　 It is known that there are 48 Virasoro algebras acting on the \"monster conformal field theory\". We call conformal field theories with such a property, which are not necessarily chiral, code conformal field theories. Recently, we introduce a notion of a framed algebra, which is a finite-dimensional non-associative algebra, and showed that the category of framed algebras and the category of code conformal field theories are equivalent. We have also constructed a new family of conformal field theories using this equivalence. These conformal field theories are expected to be useful for the study of moduli spaces of conformal field theories. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. June 10 (Thu), 16:00--17:30, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4649 Speaker. Alexander Zuevsky (Czech Academy of Sciences). Title. Reduction cohomology on Riemann surfaces. Abstract. 　 We define and compute a cohomology of the space of Jacobi forms based on precise analogues of Zhu reduction formulas derived by Bringmann-Krauel-Tuite. It is shown that the reduction cohomology for Jacobi forms is given by the cohomology of $n$-point connections over a deformed vertex algebra bundle defined on the torus. The reduction cohomology for Jacobi forms for a vertex algebra is determined in terms of the space of analytical continuations of solutions to Knizhnik-Zamolodchikov equations. A counterpart of the Bott-Segal theorem for the reduction cohomology of Jacobi forms on the torus is proven. Algebraic, geometrical, and cohomological meanings of reduction formulas is clarified. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. May 27 (Thu), 15:00--16:30, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4639 Speaker. Daniel Bruegmann (Max Planck Institute for Mathematics). Title. Vertex Algebras and Factorization Algebras. Abstract. 　 Vertex algebras and factorization algebras are two approaches to chiral conformal field theory. Costello and Gwilliam describe how every holomorphic factorization algebra on the plane of complex numbers satisfying certain assumptions gives rise to a Z-graded vertex algebra. They construct some models of chiral conformal theory as factorization algebras. We attach a factorization algebra to every Z-graded vertex algebra. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. May 20 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4633) Speaker. Jethro van Ekeren (Universidade Federal Fluminense). Title. Chiral homology of elliptic curves. Abstract. 　 In this talk I will discuss results of an ongoing project (joint with Reimundo Heluani) on the chiral homology of elliptic curves with coefficients in a conformal vertex algebra. Since the work of Y. Zhu it is clear that this homology has important applications to the representation theory of vertex algebras. We construct a flat connection on the first chiral homology over the moduli space, and relate the nodal curve limit with the Hochschild homology of the Zhu algebra. We construct flat sections from self-extensions of modules. Along the way we find interesting links between these structures, associated varieties of vertex algebras, and classical identities of Rogers-Ramanujan type (this last part joint work with George Andrews). Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. May 13 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4624) Speaker. Michael McBreen (The Chinese University of Hong Kong). Title. Hypertoric Hitchin systems and Kirchoff polynomials. Abstract. 　 I will present joint work with Michael Groechenig, which associates a degenerating family of abelian varieties to a graph. On the one hand, it is an algebraisation of the `Dolbeault' hypertoric spaces originally defined by H ausel and Proudfoot. On the other, it is an approximation to the relative compactified Jacobian of a family of curves degenerating to a nodal curve with specified dual graph. We then consider our construction over the p-adic numbers, and compute the p-adic volumes of the fibers. We find they are given by the Kirchoff polynomial of the graph. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. May 6 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4616) Speaker. Yasuaki Hikida (YITP, Kyoto University). Title. Generalized Fateev-Zamolodchikov-Zamolodchikov dualities and Gaiotto- Rapcak's VOAs. Abstract. 　 About two decades ago, Fateev, Zamolodchikov and Zamolodchikov (FZ Z) conjectured a strong\/weak duality between two dimensional conformal field theories, SL(2)\/U(1) coset model and sine-Liouville theory. Recently, we ha ve succeeded to generalize the FZZ-duality by extending its original derivat ion done with Schomerus. The generalized FZZ-dualities can be regarded as a conformal field theoretic realization of dualities among VOAs conjectured by Gaiotto and Rapcak via brane junctions in string theory. In this talk, I wi ll explain how to derive the generalized FZZ-dualities and relation to the G aiotto-Rapcak's dualities. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. April 22 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4595) Speaker. Tatsuki Kuwagaki (Osaka University). Title. Sheaf quantization: example and construction. Abstract. 　 Constructible sheaves have played an important role in the development of representation theory. The topic of this talk is sheaf quantization, which is a geometric refinement of the notion of constructible sheaf (“constructible sheaf (or local system) of 21st century”). I will give an introduction to sheaf quantization and how it is difficult (at present) to construct it in general; I’d like to explain how the ideas from exact WKB analysis, resurgent analysis, and Fukaya category come into the story. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. April 15 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4590) Speaker. Thomas Creutzig (University of Alberta). Title. From W-algebras to W-superalgebras. Abstract. 　 A W-superalgebra is a vertex superalgebra associated to a Lie superalgebra, g, an invariant bilinear form on g and an even nilpotent element in g. If g is a Lie algebra and f is principal nilpotent then one obtains the principal W-algebra of g. Feigin-Frenkel duality are isomorphisms between principal W-algebras. These isomorphisms somehow generalize to non principal nilpotent elements, however the isomorphism is only between coset subalgebras of W-algebras and W-superalgebras. In my talk I will first introduce the isomorphisms that generalize Feigin-Frenkel duality. I then want to outline a program on how to use the dualities to get correspondences between tensor categories of W-algebra modules and dual W-superalgebra modules. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI. Date. February 18 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4575) Speaker. Hironori Oya (Shibaura Institute of Technology). Title. Systematic construction of isomorphisms among quantum Grothendieck rings. Abstract. 　 A quantum Grothendieck ring of the monoidal category of finite-dimensional modules over a quantum loop algebra Uq(Lg) is a one parameter deformation of the usual Grothendieck ring. It is introduced by Nakajima and Varagnolo-Vasserot in the case when Uq(Lg) is of simply-laced type through a geometric method, and subsequently by Hernandez when Uq(Lg) is of arbitrary untwisted affine type through an algebraic method. In the simply-laced case, quantum Grothendieck rings are known to give an algorithm for calculating q-characters of simple modules, which is an analogue of Kazhdan-Lusztig algorithm. In this talk, we present a collection of algebra isomorphisms among quantum Grothendieck rings, which respect the (q,t)-characters of simple modules. As a corollary, we obtain new positivity results for the simple (q,t)-characters of non-simple-laced types. Moreover, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we show that an analogue of Kazhdan-Lusztig algorithm for computing simple q-characters is available when g is of type B. This result is a vast generalization of our previous work [Hernandez-O, Adv. Math. 347 (2019), 192--272]. Hence, besides the summary of main results, I will explain some details of the proof, focusing on the tools and results obtained after [HO]. This talk is based on a joint work (arxiv:2101.07489) with Ryo Fujita, David Hernandez, and Se-jin Oh. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. February 4 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4565) Speaker. Xuanzhong Dai (Fudan University). Title. Chiral de Rham complex on the upper half plane and modular forms. Abstract. 　 Chiral de Rham complex constructed by Malikov, Schechtman and Vaintrob in 1998, is a sheaf of vertex algebras on a complex manifold. For any congruence subgroup $\\Gamma$, we consider the $\\Gamma$-invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic at all the cusps. We show that it contains an $N = 2$ superconformal structure and we give an explicit lifting formula from modular forms to it. As an application, the vertex algebra structure modifies the Rankin-Cohen bracket, and the modified bracket with the Eisenstein series involved becomes nontrivial between constant modular forms. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. January 28th (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4535) Speaker. Xuhua He (The Chinese University of Hong Kong). Title. Flag varieties over semifields. Abstract. 　 In 1994, Lusztig developed the theory of total positivity for arbitrary split real reductive groups and their flag manifolds. Later the theory has found important applications in different areas: cluster algebras, higher Teichmuller theory, the theory of amplituhedron in physics, etc. Recently, Lusztig initiated the study of Kac-Moody monoids over arbitrary semifield and their flag manifolds. In the case where the Kac-Moody datum comes from a real reductive group and the semifield is $R_{0}$, the Kac-Moody monoid over $R_{0}$ is exactly the totally nonnegative part of the real reductive group. In this talk, I will discuss my joint work with Huanchen Bao on the flag manifolds B(K) over arbitrary semifield K and associated to any Kac-Moody? datum G. We show that B(K) admits a natural action of the Kac-Moody monoid G(K) and admits a decomposition into cells. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. January 21 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4564) Speaker. Kota Murakami (Kyoto University). Title. PBW parametrizations and generalized preprojective algebras. Abstract. 　 Kashiwara-Saito realized crystal bases of quantum enveloping algebras on irreducible components of varieties of nilpotent modules over preprojective algebras for simply-laced types. Recently, Geiss-Leclerc-Schroer generalized these realizations to non-simply laced types by developing representation theory of a class of 1-Iwanaga-Gorenstein algebras and their preprojective algebras associated with symmetrizable GCMs (=generalized Cartan matrices) and their symmetrizers. In this talk, we relate representation theory of the generalized preprojective algebras with numerical data about the dual canonical bases, so called Lusztig data, for symmetrizable GCMs of finite types. In particular, we realize Mirkovic-Vilonen polytopes from some generic modules over generalized preprojective algebras as a generalization of the work of Baumann-Kamnitzer-Tingley. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. January 14 (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4552) Speaker. Haruhisa Enomoto (Nagoya University). Title. ICE-closed subcategories of module categories. Abstract. 　 In the representation theory of algebras, the study of subcategories of module categories has been one of the main topics, and is related to many areas. Among them, torsion classes and wide subcategories are important and have been studied by many people. In this talk, I will introduce the notion of ICE-closed subcategories of module categories, which are closed under taking Images, Cokernels and Extensions. This class contains both torsion classes and wide subcategories. In the representation category of a Dynkin quiver, they bijectively correspond to rigid representations. For a general finite-dimensional algebra, I will explain how to classify ICE-closed subcategories using the poset structure of torsion classes, or using $\\tau$-tilting theory. This talk is based on my joint work with Arashi Sakai (Nagoya). Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. January 7th (Thu), 10:30--12:00, 2021. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4544) Speaker. Toshiya Yurikusa (Tohoku University). Title. Denseness of g-vector fans for tame algebras. Abstract. 　 The g-vector fan of a finite dimensional algebra is a simplicial polyhedral fan whose rays are the g-vectors of the indecomposable 2-term presilting complexes. We consider the property that the g-vector fan is dense. We prove that gentle algebras satisfy it by using their surface model (based on a joint work with Toshitaka Aoki). The main ingredients of our proof are the g-vectors of the laminations and their asymptotic behavior under Dehn twists. More generally, using the generic decompositions and twist functors instead of them, we can prove it for tame algebras (based on a joint work with Pierre-Guy Plamondon). Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. December 24th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4536) Speaker. Sven Moeller (RIMS). Title. Schellekens' VOAs, Generalised Deep Holes and the Very Strange　Formula. Abstract. 　 In this talk I will summarise recent results regarding the classification of strongly rational, holomorphic VOAs (or CFTs) of central charge 24 (based on joint works with Jethro van Ekeren, Gerald Höhn, Ching Hung Lam, Nils Scheithauer and Hiroki Shimakura). First, we show that there is an abstract bijection (without classifying either side) between these VOAs and the generalised deep holes of the Leech lattice VOA. The proof uses a dimension formula obtained by pairing the VOA character with a vector-valued Eisenstein series and an averaged version of Kac's Lie theoretic very strange formula. This is a quantum analogue of the result by Conway, Parker and Sloane (and Borcherds) that the deep holes of the Leech lattice are in bijection with the Niemeier lattices. Then, we explain how this can be used to classify the (exactly 70) strongly rational, holomorphic VOAs of central charge 24 with non-zero weight-one space. (The case of zero weight-one space, which includes the Moonshine module, is more difficult and still open.). Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. December 3rd (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4509) Speaker. Takahiro Nishinaka (Ritsumeikan University). Title. The Chiral Algebra of Genus-Two Class S Theory. Abstract. 　 Vertex operator algebras (VOAs) of class S are those closely related to four-dimensional N=2 superconformal field theories in physics, and give an interesting functor from the category of 2-bordisms to a category of VOAs. In particular, gluing two bordisms corresponds to a certain BRST reduction of VOAs. VOAs in this category are generally associated with a semi-simple Lie algebra and a 2-manifold. In this talk, I will discuss the one associated with sl(2) and genus-two manifold without boundary, arguing that its automorphism group contains an SU(2) sub-group which is unexpected even from physics. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. November 26th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4505) Speaker. Dinakar Muthiah (Kavli IPMU). Title. Equations for affine Grassmannians and their Schubert varieties. Abstract. 　 I will discuss work on a conjectural moduli description of Schubert varieties in the affine Grassmannian and proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on equations defining type A affine Grassmannians. As an application of our ideas, we prove a conjecture of Pappas and Rapoport about nilpotent orbit closures. This involves work with Joel Kamnitzer, Alex Weekes, and Oded Yacobi. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. November 12th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4475) Speaker. Yuto Moriwaki (Kavli IPMU). Title. Two dimensional conformal field theory, current-current deformation and Mass formula. Abstract. 　 This talk deals with a deformation of a two dimensional conformal field theory. We introduce a notion of a full vertex algebra, which is a mathematical formulation of a compact two dimensional conformal field theory on $R^2$. We also give examples of full vertex algebras and discuss the relation between vertex algebras and full vertex algebras. Then, we construct a deformation of a full vertex algebra, which serves as a current-current deformation of the conformal field theory in physics. As an application, we consider the deformation of a tensor product of a vertex algebra and some full vertex algebra. Such deformation may produce new vertex algebras. We give a formula which counts a weighted sum of the number of vertex algebras appearing in the deformation. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. November 5th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4476) Speaker. Yoshinori Namikawa (RIMS). Title. Universal coverings of nilpotent orbits and birational geometry. Abstract. 　 The normalization of a nilpotent orbit closure of a complex semisimple Lie algebra is a symplectic variety. Its symplectic resolution or Q-factorial terminalization has been extensively studied. In this lecture, we take a symplectic variety associated with the universal covering of a nilpotent orbit and consider similar problems. When the Lie algebra is classical, we will give an explicit algorithm for constructing a Q-factorial terminalization of such a symplectic variety. Moreover, we can give an explicit formula how many different Q-factorial terminalizations it has. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE. Date. August 6th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4426) Speaker. Ryo Kanda (Osaka City University). Title. Feigin-Odesskii's elliptic algebras. Abstract. 　 This talk is based on joint work with Alex Chirvasitu and S. Paul Smith. Feigin and Odesskii introduced a family of noncommutative graded algebras, which are parametrized by an elliptic curve and some other data, and claimed a number of remarkable results in their series of papers. The family contains all higher dimensional Sklyanin algebras, which have been widely studied and recognized as important examples of Artin-Schelter regular algebras. In this talk, I will explain some properties of Feigin-Odesskii's algebras, including the nature of their point schemes and algebraic properties obtained by using the quantum Yang-Baxter equation. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Ryo FUJITA, Hideya WATANABE. Date. July 30th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/4418) Speaker. Liron Speyer (OIST). Title. Semisimple Specht modules indexed by bihooks. Abstract. 　 I will first give a brief survey of some previous results with Sutton, in which we found a large family of decomposable Specht modules for the Hecke algebra of type B indexed by `bihooks'. We conjectured that outside of some degenerate cases, our family gave all decomposable Specht modules indexed by bihooks. There, our methods largely relied on some hands-on computation with Specht modules, working in the framework of cyclotomic KLR algebras. I will then move on to discussing a new project with Muth and Sutton, in which we have studied the structure of these Specht modules. By transporting the problem to one for Schur algebras via a Morita equivalence of Kleshchev and Muth, we are able to show that in most characteristics, these Specht modules are in fact semisimple, and give all composition factors (including their grading shifts). In some other small characteristics, we can explicitly determine the structure, including some in which the modules are `almost semisimple'. I will present this story, with some running examples that will help the audience keep track of what's going on. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. July 16th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4413) Speaker. Koichi Harada (Department of Physics, The University of Tokyo). Title. Quantum deformation of Feigin-Semikhatov's W-algebras from quantum toroidal gl(1). Abstract. 　 Quantum toroidal algebras contain many q-W algebras as truncations, but most of them have not been studied in detail. The typical examples among them are the subregular W-algebras of type A. The screening charges were proposed by Feigin and Semikhatov and their q-deformation was also found in the study of quantum toroidal gl(1). Further, the recent work by Gaiotto and Rapcak provides a clue to obtain many q-W algebras by gluing quantum toroidal gl(1)s. In this talk, I will discuss quantum deformation of Feigin-Semikhatov's W-algebras by using quantum toroidal gl(1) and Gaiotto-Rapcak's framework. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Ryo FUJITA, Hideya WATANABE. Date. July 9th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4395) Speaker. Wataru Yuasa (RIMS, Kyoto University). Title. The tail of the one-row colored sl(3) Jones polynomial and the Andrews-Gordon type identity. Abstract. 　 I will review my works on the one-row colored sl(3) tail of knots and links. The tail is a q-series obtained as a limit of the colored Jones polynomial. The first topic is the existence of tails of the one-row colored sl(3) colored Jones polynomials for oriented \"adequate\" links. In the case of sl(2), it showed by Armond and Garoufalidis-Le independently. The second topic is the Andrews-Gordon type identities for (false) theta series obtained from the tail of (2,m)-torus knots and links. It is known that our formula of one-row colored sl(3) tail coincides with the diagonal part of the sl(3) false theta function obtained by Bringmann-Kaszian-Milas. In this talk, I will also give a quick review on quantum invariants of knots and links. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. July 2nd (Thu), 14:00--15:30, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4399) Speaker. Kanam Park (Kobe University). Title. A certain generalization of $q$-hypergeometric functions and its related monodromy preserving deformation. Abstract. 　 We define a series $\\mathcal{F}_{M,N}$ as a certain generalization of $q$-hypergeometric functions. We also study the system of $q$-difference nonlinear equations which admits particular solutions in terms of $\\mathcal{F}_{N,M}$. The function $\\mathcal{F}_{N,M}$ is a common generalization of $q$-Appell-Lauricella function $\\varphi_D$ and the generalized $q$-hypergeometric function ${}_{N+1}\\varphi_N$. We construct a Pfaffian system which the function $\\mathcal{F}_{N,M}$ satisfies. We derive from the Pfaffian system a monodromy preserving deformation which admits particular solutions in terms of $\\mathcal{F}_{N,M}$. In this talk, we will introduce the function $\\mathcal{F}_{N,M}$ and its fundamental properties and the system derived from a Pfaffian system which $\\mathcal{F}_{M,N}$ satisfies. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. June 25th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4382 ) Speaker. Ryosuke Kodera (Kobe University). Title. Affine Yangians and rectangular W-algebras. Abstract. 　 W-algebra is a class of vertex algebras attached to a complex reductive Lie algebra, a nilpotent element in the Lie algebra, and a complex number. We consider the case of the general linear Lie algebra $\\mathfrak{gl}_N$ with $N=l \\times n$ and a nilpotent element whose Jordan form corresponds to the partition $(l^n)$. We call it rectangular W-algebra. Its current algebra (or enveloping algebra) is defined as the associative algebra generated by the Fourier modes of generating fields. The goal of this talk is to construct an algebra homomorphism from the affine Yangian of type A to the current algebra of the rectangular W-algebra. We use the coproduct and the evaluation map of the affine Yangian to construct it. We hope that the homomorphism will be applied to the study of the AGT correspondence for parabolic sheaves and of integrable systems associated with the W-algebra. The talk is based on a joint work with Mamoru Ueda. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. June 18th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4366 ) Speaker. Yuma Mizuno (Tokyo Institute of Technology). Title. Difference equations arising from cluster algebras. Abstract. 　 The theory of cluster algebras gives powerful tools for systematic studies of discrete dynamical systems. Given a sequence of quiver mutations that preserves the quiver, we obtain a finite set of algebraic relations, yielding a discrete dynamical system. Such a set of algebraic relations is called a T-system. In this talk, I will explain that T-systems are characterized by pairs of matrices that have a certain symplectic property. This generalize a characterization of period 1 quivers, which was given by Fordy and Marsh, to arbitrary mutation sequences. I will also explain the relation between T-systems and Nahm's problem about modular functions, which is one of the main motivations of our study. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. June 11th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4359 ) Speaker. Shintarou Yanagida (Nagoya). Title. Derived gluing construction of chiral algebras. Abstract. 　 We discuss the gluing construction of class S chiral algebras in derived setting. The gluing construction in non-derived setting was introduced by Arakawa to construct a family of vertex algebras of which the associated varieties give genus 0 Moore-Tachikawa symplectic varieties. Motivated by the higher genus case, we introduce a dg vertex algebra version of the category of Moore-Tachikawa symplectic varieties, where taking associated schemes gives a functor to the category of derived Moore-Tachikawa varieties. This talk is based on the preprint arXiv:2004.10055. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. June 4th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4345 ) Speaker. Tsukasa Ishibashi (RIMS). Title. Algebraic entropy of sign-stable mutation loops. Abstract. 　 In the theory of cluster algebra, a mutation loop is a certain equivalence class of a sequence of seed mutations and permutations of indices. They form a group called the cluster modular group, which can be regarded as a combinatorial generalization of the mapping class groups of marked surfaces. 　We introduce a new property of mutation loops which we call the “sign stability”, as a generalization of the pseudo-Anosov property of a mapping class. A sign-stable mutation loop has a numerical invariant which we call the “cluster stretch factor”, in analogy with the stretch factor of a pA mapping class. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two are estimated by the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. May 28th (Thu), 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4331 ) Speaker. Shigenori Nakatsuka (Kavli IPMU). Title. On Gaiotto-Rapcak's dualities in W-superalgebras and their affine cosets. Abstract. 　 The principal affine W-algebras enjoy the Feigin-Frenkel duality a nd the Goddard-Kent-Olive construction if associated with simply-laced Lie a lgebras. Recently, Gaiotto and Rapcak proposed a generalization of this tria lity (or dualities) among W-superalgebras（and their affine cosets) by using the 4-dimensional N=4 super Yang-Mills theories. In this talk, we prove the Feigin-Frenkel type duality between the Heisenberg cosets of the subregular W-algebras and the principal W-superalgebras for A and B-types. This is a j oint work with T. Creutzig and N. Genra. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. Date. May 14th (Thu) 10:30--12:00, 2020. Room. Zoom (access information will appear in. https:\/\/www.math.kyoto-u.ac.jp\/en\/event\/seminar\/4328 ) Speaker. Hideya Watanabe (RIMS). Title. Classical weight modules over i-quantum groups. Abstract. 　 i-quantum groups are certain coideal subalgebras of quantum groups appearing in the theory of quantum symmetric pairs. Many results concerning quantum groups have been generalized to i-quantum groups. However, representation theory of i-quantum groups is much more difficult than that of quantum groups due to the lack of Chevalley-like generators. In this talk, I introduce the notion of classical weight modules over i-quantum groups, and explain that they can be thought of as counterparts of the weight modules over quantum groups. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE. --- CANCEL ---. Date. March 9th. (Mon). 10:30--12:00, 2020. Room. RIMS 006. Speaker. Hideya Watanabe (RIMS). Title. Classical weight modules over i-quantum groups. Abstract. 　 i-quantum groups are certain coideal subalgebras of quantum groups appearing in the theory of quantum symmetric pairs. Many results concerning quantum groups have been generalized to i-quantum groups. However, representation theory of i-quantum groups is much more difficult than that of quantum groups due to the lack of Chevalley-like generators. In this talk, I introduce the notion of classical weight modules over i-quantum groups, and explain that they can be thought of as counterparts of the weight modules over quantum groups. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. January 23th (Thu) 14:00–15:30, 2020. Room. RIMS 006. Speaker. Yota Shamoto (IPMU). Title. Irregular vertex algebras. Abstract. 　 Mainly motivated by Alday-Gaiotto-Tachikawa correspondence and their applications, irregular singularities in conformal field theory are investigated in mathematical physics. We shall initiate an attempt to give a mathematical language of such theory by introducing the notions of coherent state modules and irregular vertex algebras. In this talk, we shall explain the definitions, examples and some fundamental properties. This talk is based on a joint work with Akishi Ikeda at Osaka university. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. January 16th (Thu) 10:30–12:00, 2020. Room. RIMS 006. Speaker. Husileng Xiao (Harbin Engineering University). Title. On representation of the finite W-superalgebras. Abstract. 　 In the last decade, Losev found a Poisson geometric realization of the finite W-algebras . This provides him a powerful tool to study the representations of the finite W-algebras. In this talk, I will first introduce my joint work with Bin Shu, which generalize the above realization to the finite W-superalgebra case. Then I will discuss its application to the representations of the finite W-superalgebras. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. December 12th (Thu) 10:30–12:00, 2019. Room. RIMS 006. Speaker. Andrew Linshaw (Denver). Title. Dualities of W-algebras and the W_{\\infty}-algebra. Abstract. 　 We classify one-parameter vertex algebras that arise as extensions of affine gl_m tensored with a vertex algebra of type W(2,3,\\dots), where the extension is generated by 2m fields in a fixed conformal weight which transform as the standard representation of gl_m and its dual. As a consequence, we obtain some new dualities between families of W-algebras and W-superalgebras. We also give a new proof of the coset realization of principal W-algebras of type A that was obtained in my recent work with Arakawa and Creutzig. This is a joint work with T. Creutzig. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. November 14th (Thu) 10:30–12:00, 2019. Room. RIMS 006. Speaker. Masahiro Chihara (Kyoto U). Title. Demazure slices of type $A_{2l}^{(2)}$. Abstract. 　 Demazure slices are associated graded pieces of infinite-dimensional version of Demazure modules for affine Lie algebras. In this talk, we review (1) Demazure slices for other types and (2) usual finite dimensional Demazure modules of type $A_{2l}^{(2)}$. Then we explain a relation between graded characters of Demazure slices for $A_{2l}^{(2)}$ and specialized nonsymmetric Macdonald-Koornwinder polynomials. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. November 7th (Thu) 10:30–12:00, 2019. Room. RIMS 006. Speaker. Kota Murakami (Kyoto U). Title. On module categories of preprojective algebras with symmetrizers. Abstract. 　 Geiss-Leclerc-Schröer has introduced preprojective algebras for symmetrizable GCMs and their symmetrizers. They are expected to generalize some Lie theoretical aspects of Gelfand-Ponomarev's preprojective algebras. In this talk, we will discuss module categories and some combinatorical invariants of these algebras from a viewpoint of tilting theory. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. Oct. 17th (Thu) 10:30–12:00, 2019. Room. RIMS 006. Speaker. Shoma Sugimoto (RIMS Kyoto U). Title. On the Feigin-Tipunin VOA. Abstract. 　 The triplet VOA ($A_1$ type Feigin-Tipunin VOA) is one of the most famous examples of $C_2$-cofinite and irrational VOA. However, there are not much known about the Feigin-Tipunin VOA $W(p)_Q$, the $ADE$ type generalization of the triplet VOA. In this talk, we will give the geometric construction and character formulas of $W(p)_Q$ that conjectured in the paper of Feigin-Tipunin. Moreover, we will give a $W$-algebraic conditioning of $C_2$-cofiniteness of $W(p)_Q$ under the expectable assumptions of simpleness. In the case of $A_2$ type with a fixed $p$, this conditioning enables us to prove the $C_2$-cofiniteness of $W(p)_Q$ much easier than direct calculation. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. Oct. 10 (Thu) 10:30–12:00, 2019. Room. RIMS 006. Speaker. Matt Szczesny (Boston University). Title. Hall algebras of toric varieties over F_1. Abstract. 　 Hall algebras of categories of quiver representations and coherent sheaves on smooth projective curves over F_q recover interesting representation-theoretic objects such as quantum groups and their generalizations. I will define and describe the structure of the Hall algebra of coherent sheaves on a projective variety over F_1, with P^2 as the main example. Examples suggest that it should be viewed as a degenerate q-1 limit of its counterpart over F_q. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. Oct. 3rd (Thu) 13:30--14:30, 2019. Room. RIMS 006. Speaker. Mamoru Ueda (RIMS Kyoto). Title. Affine Super Yangian Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. September 5th (Thu) 10:30--12:00, 2019. Room. RIMS 006. Speaker. Ievgen Makedonskyi (Kyoto). Title. Peter-Weyl, Howe and Schur-Weyl theorems for current groups. Abstract. 　 The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems is provided by the (current version of) Howe duality. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. May 23 (Thu) 10:30--12:00, 2019. Room. RIMS 006. Speaker. Kanehisa Takasaki. Title. Hurwitz numbers of Riemann sphere and integrable hierarchies. Abstract. 　 The Hurwitz numbers count the topologically non-equivalent types of finite ramified coverings of a given Riemann surface. When the base Riemann surface is the Riemann sphere, these numbers are known to be related to intersection numbers of the Hodge classes and the psi classes on the moduli space of stable complex curves. On the other hand, the same numbers can be expressed in a genuinely combinatorial form in terms of symmetric groups. The latter expression reveals that the generating functions of a particular class of Hurwitz numbers of the Riemann sphere become tau functions of the KP hierarchy and its relatives. I will review these facts for non-experts of integrable systems. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI. Date. May 10 (Fri) 16:30--18:00, 2019. Room. RIMS 402. Speaker. Saiei-Jaeyeong Matsubara-Heo (Kobe). Title. Integral representations of GKZ hypergeometric functions: Gauss-Manin connection, intersection theory, and quadratic relations. Abstract. 　 GKZ(Gelfand, Kapranov, Zelevinsky) system is a holonomic system which describes classical hypergeometric systems in a unified manner. In this talk, we realize GKZ system as a Gauss-Manin connection where we treat Euler integral and Laplace integral at the same time. Focusing on regular holonomic case, we give a method of reinterpreting the combinatorics of regular triangulations to the construction of the basis of twisted cycles at \"toric infinity\". This naturally gives rise to an orthogonal decomposition of the twisted homology group with respect to the intersection pairing. As an application, we give a general quadratic relation of GKZ hypergeometric functions associated to a unimodular triangulation. We also discuss an algorithm of computing cohomology intersection numbers based on a joint work with Nobuki Takayama. The techniques above produce several new quadratic relations of hypergeometric functions of several variables. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. April 25 (Thurs) 10:30--12:00, 2019. Room. RIMS 006. Speaker. Anatol Kirillov. Title. Deformed Hecke algebras and quantum cohomology of graph varieties. Abstract. 　 For any finite graph I define certain algebras, including deformed Hecke type ones. All these algebras contain commutative subalgebras generated by either additive or multiplicative, or elliptic Dunkl elements. I'm planing to explain why these commutative subalgebras can be identified with classical and (small) quantum cohomology (and that of K-theory) of certain graph varieties, including partial flag and Hessenberg varieties. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Apr 19 (Fri), 14:30--16:00, 2019. Room. RIMS 402. Speaker. Kazuya Kawasetsu (RIMS). Title. Relaxed highest-weight modules over affine VOAs. Abstract. 　 The (chiral) symmetry algebras of 2d conformal field theory are described as vertex operator algebras (VOAs). Among them, C_2-cofinite ones with semisimple module categories correspond to rational conformal field theory, whose characters span a SL_2(\\Z)-invariant vector space and whose fusion rules in module categories satisfy the celebrated Verlinde formula. The affine VOAs are those construted by (and almost the same as) affine Kac-Moody algebras. Among them, non-integrable affine VOAs are important examples of non-C_2-cofinite VOAs. The Verlinde formula for non-integrable (admissible) affine VOAs proposed by T. Creutzig and D. Ridout involves in relaxed highest-weight modules over affine Kac-Moody Lie algebras, which are modules parabolically induced from weight modules over the associated finite-dimensional simple Lie algebras. In this talk, we briefly review the Creutzig-Ridout theory and discuss recent progress on character formulas and classification of relaxed highest-weight modules. This talk is based on joint works with David Ridout. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Apr 18 (Thu), 10:30--12:00, 2019. Room. RIMS 006. Speaker. Sota Asai (RIMS). Title. Semibricks in $\\tau$-tilting theory. Abstract. 　 This talk is based on my paper \"Semibricks\" in IMRN. In representation theory of a finite-dimensional algebra $A$ over a field $K$, bricks and semibricks are fundamental and useful notions. Here, a brick means an $A$-module whose endomorphism ring is a division $K$-algebra, and a semibrick means a set of bricks which are pairwise Hom-orthogonal, so (semi)bricks are a generalization of (semi)simple modules. I study semibricks from the point of view of $\\tau$-tilting theory. I proved that there is a one-to-one correspondence between the support $\\tau$-tilting modules and the semibricks satisfying a certain condition called left finitess. Also, I introduced brick labels for the exchange quiver of the support $\\tau$-tilting modules by using this bijection. I would like to explain these results and the new perspective of $\\tau$-tilting theory given by them. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Apr 11 (Thu), 14:30--16:00, 16:30--18:00, 2019. Room. RIMS 006. Speaker. [CANCEL] 14:30-16:00 Kazuya Kawasetsu (RIMS). Title. Relaxed highest-weight modules over affine VOAs. Abstract. 　 The (chiral) symmetry algebras of 2d conformal field theory are described as vertex operator algebras (VOAs). Among them, C_2-cofinite ones with semisimple module categories correspond to rational conformal field theory, whose characters span a SL_2(\\Z)-invariant vector space and whose fusion rules in module categories satisfy the celebrated Verlinde formula. The affine VOAs are those construted by (and almost the same as) affine Kac-Moody algebras. Among them, non-integrable affine VOAs are important examples of non-C_2-cofinite VOAs. The Verlinde formula for non-integrable (admissible) affine VOAs proposed by T. Creutzig and D. Ridout involves in relaxed highest-weight modules over affine Kac-Moody Lie algebras, which are modules parabolically induced from weight modules over the associated finite-dimensional simple Lie algebras. In this talk, we briefly review the Creutzig-Ridout theory and discuss recent progress on character formulas and classification of relaxed highest-weight modules. This talk is based on joint works with David Ridout. ---. Speaker. 16:30-18:00 Ikuo Satake (Kagawa University). Title. An approach to the invariant theory for the elliptic Weyl groups. Abstract. 　 We have 3 kinds of generators of the invariant ring for the elliptic Weyl groups: (1) the fundamental characters of the affine Lie algebras, (2) the (generalized weak) Jacobi forms, (3) the flat invariants for the Frobenius structure. For the D4 case, we could construct (2) by (1) by a determinant of a matrix whose entries are fundamental characters. By these Jacobi forms, we could give the explicit description of the Frobenius structure, since the Frobenius structure has the modular invariance. Then we could construct (3) by (2). So we could construct (3) by (1) (arxiv:1708.03875). Afterwards we find a characterization of (3) by using the behavior on the fixed points of the modified Coxeter transformation for the elliptic root system. Combining these results, we give some new properties of the fundamental characters and the Weyl denominator for the affine Lie algebra of type D4. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Apr 4 (Thu), 14:00--18:00, 2019. Room. RIMS 006. Speaker. 14:00-15:00 Kari Vilonen. Title. Character sheaves for graded Lie algebras, geometric theory. Abstract. 　 In recent joint work with Grinberg and Xue we studied a nearby cycle construction which forms a basis for Springer theory in the symmetric space setting. In consequent work with Xue we applied this theory to classify character sheaves in the symmetric space setting. In this talk we extend the geometric theory to the more general case of graded Lie algebras. ---. Speaker. 15:30-16:30 Ting Xue. Title. Graded Lie algebras, character sheaves, and DAHA representations. Abstract. 　 In their recent work, Lusztig and Yun construct representations of certain graded double affine Hecke algebras (DAHA) using geometry of graded Lie algebras. In joint work (in progress) with Vilonen we study the geometry of graded Lie algebras from another point of view. More precisely, we classify character sheaves in the setting of graded Lie algberas, where representations of Hecke algebras associated with complex reflection groups enter the story. We will explain some conjectures arising from the connection between the above two works, which relate finite dimensional irreducible representations of graded DAHA to irreducible representations of Hecke algebras. If time permits, we will also explain a Schur-Weyl duality conjecture arising from the geometric construction of rational Cherednik algebra modules of Oblomkov and Yun using affine Springer fibres. ---. Speaker. 17:00-18:00 Bernard Leclerc. Title. A twofold generalization of Gabriel's theorem. Abstract. 　 (joint work with C. Geiss and J. Schröer). By a celebrated theorem of Gabriel, isomorphism classes of indecomposable representations of an A-D-E quiver are in one-to-one correspondence with positive roots of the corresponding root system. Let C be a Cartan matrix (of type A, B, C, ..., F4, G2), and let D be a symmetrizer for C (i.e. a diagonal matrix with positive integer entries such that DC is symmetric). Fix an orientation of the Dynkin diagram corresponding to C, and an arbitrary field F. To this datum, in joint work with Geiss and Schröer we have introduced an F-algebra H and studied its representation theory. When C is of A-D-E type and the symmetrizer is equal to k times the identity matrix, H is isomorphic to the path algebra over the truncated polynomial ring F[t]\/(t^k) of the quiver corresponding to C and the fixed orientation. I will present a twofold generalization of Gabriel's theorem in this situation. Namely there are two bijections:. (1) between isoclasses of indecomposable rigid locally free modules in rep(H) and positive roots of C;. (2) between isoclasses of bricks in rep(H) and positive roots of the tranpose of C. I will also sketch how this generalizes to symmetrizable generalized Cartan matrices C. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. [changed] Jan 31 (Thu), 15:30--17:00, 2019. Room. RIMS 006. Speaker. Jethro van Ekeren (Universidade Federal Fluminense - Brazil). Title. Minimal Models, Arc spaces, and Rogers-Ramanujan Identities. Abstract. 　 (Joint work with Reimundo Heluani.) With each vertex algebra one may canonically associate two affine schemes: its 'singular support' and its 'associated scheme'. Let us call a vertex algebra 'classically free' if its singular support coincides with the arc space of its associated scheme. Motivated by questions from the theory of chiral algebras and the geometric Langlands program, we study classical freeness of Virasoro minimal models and simple affine vertex algebras. We show that the minimal models of type (2, 2k+1), the so called 'boundary minimal models', are classically free and all others are not. The coordinate rings of the two schemes in question are naturally graded and the isomorphism yields an equality of graded dimensions which recovers the celebrated Rogers-Ramanujan identity. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Jan 31 (Thurs), 10:30--12:00, 2019. Room. RIMS 006. Speaker. Euiyong Park (University of Seoul). Title. Localization for quiver Hecke algebras. Abstract. 　 In this talk, I explain my recent work with Masaki Kashiwara, Myungho Kim and Se-jin Oh on a generalization of the localization procedure for monoidal categories developed in [S.-J. Kang, M. Kashiwara and M. Kim, Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, Invent. Math. 211 (2018), no. 2, 591-685]. Let $R$ be a quiver Hecke algebra of arbitrary symmetrizable type and $R$-gmod the category of finite-dimensional graded $R$-modules. For an element w of the Weyl group, $C_w$ is the subcategory of $R$-gmod which categorifies the quantum unipotent coordinate algebra $A_q(n(w))$. We introduce the notions of braiders and a real commuting family of braiders, and produce a localization procedure which is applicable more general cases. We then construct the localization $\\tilde{C}_w$ of $C_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q(n(w))$. The localization $\\tilde{C}_w$ is left rigid and it is conjectured that $\\tilde{C}_w$ is rigid. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Jan 25 (Fri), 16:30--18:00, 2019. Room. RIMS 402. Speaker. Anatol Kirillov. Title. Rigged Configurations and parabolic Kostka polynomials. Abstract. 　 I'm planning to explain several applications of a Fermionic Formula for parabolic Kostka polynomials discovered by the author of this talk in 1982\/83, to give proofs of a number of conjectures (and their generalization) in Combinatorics and Representation Theory (for type A ) concerning q-Kostka polynomials, such as the Gupta--Brylinski , Generalized Saturation, Combinatorial Unimodality, Polynomiality Conjectures, and new interpretation of Littlewood--Richardson and Kronecker numbers. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. CANCELLATION. Date. Jan 24 (Thu), 10:30--12:00, 2019. Room. RIMS 006. Speaker. Ikuo Satake (Kagawa University). Title. see the Japanese page. Abstract. 　 see the Japanese page. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Jan 24 (Thu), 14:00--15:30, 2019. Room. RIMS 006. Speaker. Se-jin Oh (Ewha Woman's University). Title. Two families if commutation classes and categories attached to them. Abstract. 　 In this talk, I will introduce two families of commutation classes of the same simply-laced finite type and their combinatorial features. The one is called \"adapted classes\" and the another is called \"twisted adapted classes\". Surprisingly, they encode information of categries of finite dimensional modules over Langlads dual quantum affine algebras in an interesting way. Using the generalized Schur-Weyl daulity constructed by Kang-Kashiwara-Kim, we can construct simplicity-preserving correspondences between \"heart\" subcategories of finite dimensional modules over Langlads dual quantum affine algebras in various ways. The correspondece between Langlads dual quantum affine algebras was obervated by Frenkel-Hernandez and is not well-understood yet. This is joint work with Kashiwara, Kim, Suh and Scrimshaw. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Dec 21 (Fri), 17:00--18:30, 2018. Room. RIMS 402. Speaker. Hironori Oya (Shibaura Institute of Technology). Title. Quantum Grothendieck ring isomorphisms for quantum affine algebras of type A and B. Abstract. 　 In this talk, I present a ring isomorphism between ``$t$- deformed'' Grothendieck rings (=quantum Grothendieck rings) of finite- dimensional module categories of quantum affine algebras of type $\\ mathrm{A}_{2n-1}^{(1)}$ and $\\mathrm{B}_n^{(1)}$. This isomorphism implies several new positivity properties of $(q, t)$-characters of simple modules of type $\\mathrm{B}_n^{(1)}$. Moreover, it specializes at $t = 1$ to the isomorphism between usual Grothendieck rings obtained by Kashiwara, Kim and Oh via generalized quantum affine Schur-Weyl dualities. This coincidence gives an affirmative answer to Hernandez's conjecture (2002) for type $\\mathrm{B}_n^{(1)}$ : the $(q, t)$- characters of simple modules specialize to their actual $q$-characters. Hence, in this case, the multiplicities of simple modules in standard modules are given by the evaluation of certain analogues of Kazhdan- Lusztig polynomials whose coefficients are positive. If time permits, we discuss a refinement of description of our isomorphism. This talk is based on a joint work with David Hernandez. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Nov 29 (Thu), 16:30--18:00, 2018. Room. Room 110 Department of Mathematics, Kyoto Universtity. Speaker. Hitoshi Konno (Tokyo University of Marine Science and Technology). Title. Elliptic Quantum Group and Elliptic Stable Envelopes. Abstract. 　 See the Japanese page. References; H.Konno, ``Elliptic Weight Functions and Elliptic q-KZ Equation'', J.Int.Systems 2 (2017), H.Konno, ``Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group'', J.Int.Systems 3 (2018). Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Nov 16 (Fri), 16:30--18:00, 2018. Room. RIMS 402. Speaker. Taiki Shibata (Okayama University of Science). Title. Quasireductive supergroups and their representations. Abstract. 　 An algebraic supergroup is a group-valued functor on the category of commutative superalgebras represented by a finitely generated commutative Hopf superalgebra. It has been known that representations of algebraic supergroups can be applied in non-super (modular) representation theory. In 2011, V. Serganova introduced the notion of quasireductive supergroups as a super version of the notion of split reductive groups. This is an interesting and forms an important class of algebraic supergroups including Chevalley supergroups (introduced by R. Fioresi and F. Gavarini) and queer supergroups $Q(n)$ (whose Lie superalgebra is a queer superalgebra $q(n)$). She constructed irreducible representations of quasireductive supergroups over an algebraically closed field of characteristic zero, in terms of their Lie superalgebras. In this talk, I will explain a Hopf-algebraic approach to the study of quasireductive supergroups $G$, and give a category equivalence between rational $G$-supermodules and \"locally finite\" $hy(G)$-supermodules, where $hy(G)$ is a certain cocommutative Hopf superalgebra, called the super-hyperalgebra of $G$ (due to M. Takeuchi). As an application, we get (1) a generalization of Serganova's result to the case when the base field is arbitrary, and (2) a super-analogue of Kempf's cohomology vanishing theorem when $G$ has a \"good\" parabolic super-subgroup. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Room. Kyoto University North Campus. Maskawa Building for Education and Research Room 507. (North direction from the RIMS main building). Date. Nov 8 (Thu), 13:00--14:15, 2018. Speaker. Myungho Kim (Kyung Hee University). Title. Laurent phenomenon and simple modules over symmetric quiver Hecke algebras. Abstract. 　 The unipotent quantum coordinate ring $A_q(n(w))$ is isomorphic to the Grothendieck ring of a monoidal category $C_w$ consisting of some finite dimensional graded modules over a quiver Hecke algebra. Moreover $A_q(n(w))$ has a quantum cluster algebra structure, and it is shown that the cluster monomials are classes of real simple modules in $C_w$. In this talk, I will present some interesting consequences of this \"monoidal categorification\" of $A_q(n(w))$ with the Laurent phenomenon of the cluster algebras. This is a joint work with Masaki Kashiwara. ---. Date. Nov 8 (Thu), 14:30--16:15, 2018. Speaker. Shunsuke Tsuchioka (Tokyo Institute of Technology). Title. A local characterization of $B_2$ regular crystals. Abstract. 　 Stembridge characterizes regular crystals associated with a simply-laced GCM in terms of local graph-theoretic quantities. We give a similar axiomatization for $B_2$ regular crystals and thus for regular crystals of finite GCM except $G_2$ and affine GCM except $A^{(1)}_1, G^{(1)}_2, A^{(2)}_2, D^{(3)}_4$. As we will explain in detail the previous studies, finding a set of local axioms that characterizes $B_2$ regular crystals has been an open problem. ---. Date. Nov 8 (Thu), 16:30--18:00, 2018. Speaker. Toshiyuki Tanisaki (Osaka City University). Title. see the Japanese page. Abstract. 　 see the Japanese page. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Oct 26 (Fri), 16:30--18:00, 2018. Room. RIMS 402. Speaker. Dinakar Muthiah (IPMU). Title. Toward double affine flag varieties and Grassmannians. Abstract. 　 Recently there has been a growing interest in double affine Grassmannians, especially because of their relationship with Coulomb branches of quiver gauge theories. However, not much has been said about double affine flag varieties. I will discuss some results toward understanding double affine flag varieties and Grassmannians (and their Schubert subvarieties) from the point of view of $p$-adic Kac-Moody groups. I will discuss Hecke algebras, Bruhat order, and Kazhdan-Lusztig polynomials in this setting. This includes work joint with Daniel Orr and joint with Manish Patnaik. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Oct 19 (Fri), 16:30--18:00, 2018. Room. RIMS 402. Speaker. Kazuya Kawasetsu (Melbourne). Title. Modular linear differential equations of fourth order and minimal W-algebras. Abstract. 　 Modular linear differential equations are differential equations invariant under modular transformations. They play important roles in the study of 2D conformal field theory, vertex operator algebras and modular forms. For example, characters of lisse (C_2-cofinite) vertex algebras and more generally, those of quasi-lisse vertex algebras, satisfy modular linear differential equations. Moreover, they have also used in the attempt to classify lisse vertex algebras from their characters. In this talk, we study a certain family of modular linear differential equations of fourth order and discuss which vertex operator algebras satisfy the differential equations. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Oct 15 (Mon), 16:30--18:00, 2018. Room. Room 507, Maskawa Building for Eduation and Research. Speaker. Akishi IKEDA (IPMU). Title. Double graded Ginzburg Calabi-Yau algebras, q-deformations of root lattices and q-stability conditions on their derived categories. Abstract. 　 In this talk, first we introduce a double graded version of the Ginzburg's Calabi-Yau algebra for a quiver Q and show that the derived category of dg-modules over this algebra gives a categorification of the q-deformed root lattice associated with the quiver Q. It is also shown that the action of Seidel-Thomas spherical twists on the q-deformed root lattice factors the Hecke algebra of Q. In the case that Q is of type affine ADE, this construction describes the derived category of C^*-equivariant coherent sheaves on the corresponding Kleinian singularity and also relates to the graded preprojective algebra. Next we introduce a q-stability condition on this derived category which is a Bridgeland stability condition with the additional axiom. We state the conjecture in the case Q is of type ADE (theorem for type A) that central charges of q-stability conditions give the horizontal section of the Cherednik's KZ type connection (whose monodromy is described by the Hecke algebra) on the Cartan subalgebra. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Oct 3 (Wed), 10:30--12:00, 2018. Room. Room 108 3rd building, Department of Mathematics. Speaker. Vasily Kryov. Title. Drinfeld-Gaitsgory interpolation Grassmannian and geometric Satake equivalence. Abstract. 　 This talk is a review of the paper arxiv.org\/abs\/1805.07721. Let G be a reductive complex algebraic group. Recall that a geometric Satake isomorphism is an equivalence between the category of G(O)-equivariant perverse sheaves on the affine Grassmannian for G and the category of finite dimensional repre- sentations of the Langlands dual group Gˇ. They are equivalent as Tannakian categories, the fiber functor sends a perverse sheaf to its global cohomology. It follows from the above that for any perverse sheaf P there exists an action of the Lie algebra of Gˇ on the global cohomology of P. We will explain how to construct this action explicitly. To do so, we will describe a geometric construction of the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra U(nˇ). Using this construction, we will provide the desired action. It will be obtained via a cospecialization morphism for a certain one-parametric deformation of the affine Grassmannian of G. If time permits, we will discuss some possible generalizations of our constr uction of the action, in particular, we will discuss the relation of the def ormation mentioned above with the Drinfeld-Gaitsgory deformation considered in their paper on Braden’s theorem. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Oct 5 (Fri), 16:30--18:00, 2018. Room. RIMS 402. Speaker. Yuri Billig (Carleton). Title. Representations of Lie algebras of vector fields on affine algebraic varieties. Abstract. 　 Very little has been known about representation theory of Lie algebras of polynomial vector fields on affine algebraic varieties beyond the cases of affine space and a torus. We study a category of representations of the Lie algebras of vector fields on a smooth algebraic variety X that admit a compatible action of the algebra of polynomial functions on X. We construct simple modules in this category and state a conjecture on the general structure of such modules. This is a joint work with Slava Futorny and Jonathan Nilsson. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. Aug 17 (Fri), 16:30--18:00, 2018. Room. RIMS 402. Speaker. Oleksandr Tsymbaliuk (Yale). Title. Coulomb branches, shifted quantum algebras and modified q-Toda systems. Abstract. 　 In the recent series of papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d N=4 quiver gauge theories was proposed (the latter are supposed to be symplectic dual to the corresponding well-understood Higgs branches). They can be also realized as slices in the affine Grassmannian and therefore admit a multiplication. In the current talk, we shall discuss the quantizations of these Coulomb branches and their K-theoretic analogues, and the (conjectural) down-to-earth realization of these quantizations via shifted Yangians and shifted quantum affine algebras. Those admit a coproduct quantizing the aforementioned multiplication of slices. In type A, they also act on equivariant cohomology\/K-theory of parabolic Laumon spaces. As another interesting application, the shifted quantum affine algebras in the simplest case of sl(2) give rise to a new family of 3^{n-2} q-Toda systems of sl(n), generalizing the well-known one due to Etingof and Sevostyanov. If time permits, we shall explain how to obtain 3^{rk(g)-1} modified q-Toda systems for any simple Lie algebra g. This talk is based on the joint works with M. Finkelberg and R. Gonin. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. July 16 (Mon), 10:30--12:00, 2018. Room. RIMS 402. Speaker. Arturo Pianzola. Title. A prescient SGA3: Applications of reductive group schemes to infinite dimensional Lie theory. Abstract. 　 Many interesting infinite dimensional Lie (super conformal) algebras can be thought as being \"finite dimensional\" when viewed, not as algebras over the given base field, but rather as algebras over their centroids (usually a Laurent polynomial ring). From this point of view, the algebras in question look like \"twisted forms\" of simpler objects which with one is familiar. The quintessential example of this type of behaviour is given by the affine Kac-Moody Lie algebras. Once the twisted form point of view is embraced,the theory of torsors and reductive group schemes developed by Demazure and Grothendieck [SGA3] arises naturally. The talk will explain these concepts and connections. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. June 18 (Mon), 16:00--18:00, 2018. Room. Room 108, Department of Math, Kyoto Univ. Speaker. Leonid Rybnikov (Higher School of Economics \/ Tokyo). Title. Gaudin model and crystals (joint work with Iva Halacheva, Joel Kamnitzer and Alex Weekes). Abstract. 　 Drinfeld-Kohno theorem relates the monodromy of KZ equation to the braid group action on a tensor product of $U_q(\\mathfrak{g})$-modules by R-matrices. The KZ equation depends on the parameter $\\kappa$ such that $q=\\exp(\\frac{\\pi i}{\\kappa})$. We describe the limit Drinfeld-Kohno correspondence when $\\kappa\\to 0$ along the imaginary line. On the KZ side this limit is the Gaudin integrable magnet chain while on the quantum group side the limit is a $\\mathfrak{g}$-crystal. Namely, we construct a bijection between the set of solutions of the algebraic Bethe ansatz for the Gaudin model and the corresponding $\\mathfrak{g}$-crystal, which preserves the natural cactus group action on these sets. This can be regarded as the $\\kappa\\to 0$ limit of the Drinfeld-Kohno theorem. If time allows I will also dicuss some conjectural generalizations of this result relating it to works of Losev and Bonnafe on cacti and Kazhdan-Lusztig cells. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. June 8 (Fri), 16:30--18:00, 2018. Room. Room 402, RIMS. Speaker. Ivan Cherednik (Chapel Hill \/ Kyoto). Title. Jacobian factors in any ranks and DAHA superpolynomials. Abstract. 　 The theory of the moduli spaces of torsion free sheaves in any ranks over singular curves is quite a challenge, including nodal curves and rk=2 (Gieseker, Bertram, others). Its local counterpart is the theory of affine Springer fibers for non-reduced (germs of) singular curves, which is unsettled too. For type A and in the nil-elliptic case, these fibers can be identified with the Jacobian factors, which are simple to define projective(!) varieties, though this approach was not extended to higher ranks as well. For plane curve singularities (spectral curves are of this kind in type A), there is a strong support: the corresponding geometric superpolynomials are expected to coincide with the DAHA superpolynomials colored by columns, and through them to be connected with any other theories of superpolynomials, including the original (uncolored) ones due to Khovanov-Rozansky. I will define in this talk Jacobian factors in any ranks and state their connection with the DAHA superpolynomials. This is joint with Ian Philipp. The connection conjecture was checked in many cases and a general proof seems doable (at least in the motivic setting, to be explained). Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. June 1 (Fri), 13:00--14:30, 2018. Room. Room 402, RIMS. Speaker. Yusuke Ohkubo (Tokyo). Title. Singular vectors of DIM algebra and generalized Macdonald functions arising from AGT conjecture. Abstract. 　 The Ding-Iohara-Miki algebra (DIM algebra) is a Hopf algebra regarded as q-deformation of the W-infinity algebra. In the free field representation of the DIM algebra, a certain algebra can be obtained and plays an essential role in a q-deformed version of the AGT correspondence. In this talk, I will explain the representation of this algebra, AGT correspondence and properties of generalized Macdonald functions (used in AGT correspondence and also called AFLT basis or fixed point basis). In particular, I will describe the coincidence between singular vectors of that algebra and generalized Macdonald functions with some specific N-tuples of Young diagrams. This coincidence can be regarded as a sort of generalization of the one between ordinary Macdonald functions and singular vectors of the deformed W-algebra. -----. Date. June 1 (Fri), 15:00--16:00, 2018. Room. Room 402, RIMS. Speaker. Tianshu Liu (Melbourne). Title. Affine osp(1|2) and its coset construction. Abstract. 　 Conformal field theory is an essential tool of modern mathematical physics with applications to string theory and to the critical behaviour of statistical lattice models. The symmetries of a conformal field theory include all angle-preserving transformations. In two dimensions, these transformations generate the Virasoro algebra, a powerful symmetry that allows one to calculate observable quantities analytically. The construction of one family of conformal field theories from the affine Kac-Moody algebra sl(2) were proposed by Kent in 1986 as a means of generalising the coset construction to non-unitary Virasoro minimal models, these are known as the Wess-Zumino-Witten models at admissible levels. This talk aims to illustrate, with the example of the affine Kac-Moody superalgebra osp(1|2) at admissible levels, how the representation theory of a vertex operator superalgebra can be studied through a coset construction. The method allows us to determine key aspects of the theory, including its module characters, modular transformations and fusion rules. -----. Date. June 1 (Fri), 16:30--18:00, 2018. Room. Room 402, RIMS. Speaker. David Ridout (Melbourne). Title. Relaxed modules over affine vertex operator algebras. Abstract. 　 Some of the most important non-rational VOAs are the admissible level affine ones. Their simple modules in category O have been classified, but this omits many physically necessary simple modules. Motivated by modularity (and physical) consistency, we report on work towards a classification of simple modules in category R, the category of \"relaxed\" highest-weight modules. Joint work with Kazuya Kawasetsu. Organizers. 荒川知幸、加藤周、疋田辰之. Date. 5月18日 (Fri), 16:30--18:00, 2018. Room. Room 402, RIMS. Speaker. Ivan Ip (Kyoto). Title. Positive Peter-Weyl Theorem. Abstract. 　 I will explain the Peter-Weyl Theorem for split real quantum groups of type An, generalizing the previous result in the case of Uq(sl(2,R)). I will talk about the necessary ingredients needed to state and proof the theorem, including the GNS representation of C*-algebra, Heisenberg double construction, and cluster realization of positive representations. This is a joint work with Gus Schrader and Alexander Shapiro. Organizers. Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA. Date. 5月17日 (Thu), 10:30--12:00, 2018. Room. Room 006, RIMS. Speaker. Thomas Creutzig (Alberta\/RIMS). Title. S-duality in the example of the large N=4 superconformal algebra. Abstract. 　 There is a very rich interplay between certain supersymmetric four-dimensional gauge theories, the quantum geometric Langlands program and vertex algebras. The key ingredient for all three are master chiral algebras that serve as functors between representation categories. I will present a few theorems concerning a family of vertex superalgebras called the large N=4 superconformal algebras at central charge -6 and explain how they confirm some conjectures originating from physics and geometry. Organizers. 荒川知幸、加藤周、疋田辰之. Date. 4月19日 (Thu), 10:30--12:00, 2018. Room. Room 006, RIMS. Speaker. Ievgen Makedonskyi (Kyoto U). Title. Vertex algebras and coordinate rings of semi-infinite flags. Abstract. 　 The direct sum of irreducible level one integrable representations of affne Kac-Moody Lie algebra of (affne) type ADE carries a structure of P\/Q-graded vertex operator algebra. There exists a fi ltration on these modules due to Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-in finite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of P\/Q-graded vertex operator algebra. We use the vertex algebra approach to derive semiinfi nite Pluecker-type relations in the homogeneous coordinate ring. Organizers. 荒川知幸、加藤周、疋田辰之. Date. 4月6日 (Fri), 16:30--18:00, 2018. Room. Room 402, RIMS. Speaker. Kari Vilonen氏 (Melbourne). Title. Koszul duality for real groups. Abstract. 　 I will begin by explaining the Langlands\/Vogan duality for real groups. After that I will discuss its categorical version. This is joint work with Roman Bezrukavnikov. Organizers. 荒川知幸、加藤周、疋田辰之. Date. 3月9日 (Fri), 16:30--18:00, 2018. Room. Room 402, RIMS. Speaker. Shigenori Nakatsuka氏 (U. Tokyo). Title. Geometric construction of integrable Hamiltonian hierarchies associated with the classical affine W-algebras. Abstract. 　 The Drinfeld-Sokolov hierarchies are integrable Hamiltonian hierarchies associated with the principal classical affine W-algebras. Feigin-Frenkel realized these hierarchies geometically using a geometric interpretation of the free field realization of principal classsical affine W-algebras. In this talk, we consider a generalization of this result to some classical affine W-algebras which are not principal. We also obtain a characterization of the corresponding Hamiltonians as the set of conservative quantities of some differential equations which are the affine Toda equations in the principal cases. Organizers. 中島啓、加藤周、疋田辰之. Date. 2月23日 (Fri), 16:30--18:00, 2018. Room. Room 402, RIMS. Speaker. Takahiro Nagaoka氏 (Kyoto U). Title. The universal Poisson deformation space of hypertoric varieties. Abstract. 　 Hypertoric variety $Y(A, \\alpha)$ is a (holomorphic) symplectic variety, which is defined as Hamiltonian reduction of complex vector space by torus action. By definition, there exists projective morphism $\\pi:Y(A, \\alpha) \\to Y(A, 0)$, and for generic $\\alpha$, this gives a symplectic resolution of affine hypertoric variety $Y(A, 0)$. In general, for conical symplectic variety and it's symplectic resolution, Namikawa showed the existence of universal Poisson deformation space of them. We construct universal Poisson deformation space of hypertoric varieties $Y(A, \\alpha)$, $Y(A, 0)$. We will explain this construction and concrete description of Namikawa-Weyl group action in this case. If time permits, We will also talk about some classification results of affine hypertoric variety. This talk is based on my master thesis. Organizers. 中島啓、加藤周、疋田辰之. Date. 2月16日 (Fri), 14:00--15:30, 2018. Room. Room 402, RIMS. Speaker. Ian Le氏 (Perimeter Institute). Title. An introduction to higher Teichmuller theory. Abstract. 　 Let S be a topological surface. Teichmuller space parameterizes the different ways of giving S the structure of a Riemann surface. Uniformization tells us that any Riemann surface can realized as a quotient of the upper-half-plane by a subgroup of PSL(2,R). Thus Teichmuller space to be viewed as a space of representations of the fundamental group of S into PSL(2,R). We will explain how cluster algebras and the theory of total positivity give an approach to Teichmuller theory which recovers classical ideas (like hyperbolic geometry, measured laminations, and quadratic differentials) while also permitting a generalization to PSL(n,R). ---. Date. 2月16日 (Fri), 15:45--16:45, 2018. Room. Room 402, RIMS. Speaker. Gus Schrader氏 (Columbia University). Title. Cluster algebra realization of quantum groups and their positive representations. Abstract. 　 I will speak about recent joint work with Alexander Shapiro in which we develop a cluster realization of the quantum group U_q(sl(n)) using quantized moduli spaces of framed local systems on marked surfaces. I will also discuss the notion of a positive representation of a quantum cluster algebra, and explain how the positive representations of U_q(sl(n)) introduced by Frenkel and Ip can be studied in the cluster algebra framework. ---. Date. 2月16日 (Fri), 17:00--18:00, 2018. Room. Room 402, RIMS. Speaker. Alexander Shapiro氏 (UC Berkeley). Title. Positive representations of quantum groups & modular functor. Abstract. 　 Igor Frenkel and Ivan Ip conjectured that positive representations of quantum groups are closed under tensor products. This conjecture happens to be closely related to the so-called modular functor conjecture by Fock and Goncharov. I will speak about joint works with Gus Schrader (some of which are works in progress) where we prove the above conjectures. Organizers. 中島啓、加藤周、疋田辰之. Date. 2月2日 (Fri), 16:30--18:00, 2018. Room. Room 402, RIMS. Speaker. Myungho Kim氏 (Kyung Hee University). Title. Monoidal categorification of cluster algebras. Abstract. 　 In this talk, I will explain our work on the monoidal categorification of the quantum coordinate ring $A_q(n(w))$ of the unipotent subgroup associated with a symmetric Kac-Moody algebra $g$ and an element $w$ of the Weyl group. This is a joint work with Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh. The notion of monodical categorification of cluster algebras was introduced by Hernandez and Leclerc: an abelian monodical category $C$ is called a monodical categorification of a cluster algebra $A$ if the Grothendieck ring of $C$ is isomorphic to $A$ and the cluster monomials of $A$ belong to the classes of real simple objects of $C$. The existence of a monodical categorification of a cluster algebra $A$ implies several nice properites of $A$ in a natural way, for example, the positivity of the coefficients of the expansion of cluster monomials with respect to an arbitrary cluster. Our main result is that a subcategory $C_w$ of category of finite-dimensional graded modules over the symmetric quiver Hecke algebra is a monodical categorification of the (quantum) cluster algebra $A_q(n(w))$. Combining the results of Khovanov-Lauda, Rouquier and Varagnolo-Vasserot, we conclude that the cluster monomials of $A_q(n(w))$ belongs to the upper global basis (dual canonical basis). It answers the conjecture by Kimura and Geiss-Leclerc-Schröer, which can be also regarded as a sharpened version of a question asked by Fomin-Zelevinsky. Organizers. 中島啓、加藤周、疋田辰之. Date. 12月22日 (Fri), 10:30--12:00, 2017. Room. Room 402, RIMS. Speaker. 疋田辰之氏 (RIMS). Title. Periodic modules for hypertoric varieties. Abstract. 　 Lusztig defined certain representations of affine Hecke alg ebras called periodic modules using some periodic hyperplane arrangements an d constructed canonical bases for them. Also he gave a geometric interpretat ion using equivariant K-theory of Slodowy varieties. I will explain an analo gue of such combinatorics for hypertoric varieties and give some geometric a nd representation theoretic applications. ---. Date. 12月22日 (Fri), 13:00--14:30, 2017. Room. Room 402, RIMS. Speaker. 桑原敏郎氏 (筑波大). Title. Mackey’s formula for cyclotomic Hecke algebras and rational Cher ednik algebras of type G(r,1,n). Abstract. 　 The restriction\/induction functors play an important role f or the representation theory of cyclotomic Hecke algebras and rational Cherednik algebras of type G(r,1,n). In this talk, we discuss an analog of Mackey’s formula for two parabolic su balgebras of the cyclotomic Hecke algebras and the rational Cherednik algebras. ---. Date. 12月22日 (Fri), 14:45--16:15, 2017. Room. Room 402, RIMS. Speaker. Konstanze Rietsch氏 (King’s College London). Title. Mirror symmetry for some homogeneous spaces. Abstract. 　 I will give an overview of results on mirror symmetry for G \/P, including Grassmannians and Dubrovin\/Givental style mirror symmetry in t he presence of a torus action. ---. Date. 12月22日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. Shahn Majid氏 (Queen Mary University of London). Title. Double-Bosonization and dual bases of quantum groups ℂq[SL( 2)] and ℂq[SL(3)]. Abstract. 　 The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finit e-dimensional braided-Hopf algebra B in the category of comodules of a coqua sitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra coDA(B). As an application we find new generators for ℂq[SL(2)] reduc ed at q a primitive odd root of unity with the remarkable property that thei r monomials are essentially a dual basis to the standard PBW basis of the re duced Drinfeld-Jimbo quantum enveloping algebra uq(픰픩(2)). O ur methods apply in principle for general ℂq[G] as we demonstrate for ℂq[SL(3)] at certain odd roots of unity. Organizers. 中島啓、加藤周、疋田辰之. Date. 12月1日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. Andrei Okounkov氏 (Columbia University\/Kyoto University). Title. Quasimaps counts and Bethe eigenfunctions. Abstract. 　 I will explain several aspects of what is done in a paper with the same title, joint with Mina Aganagic. Organizers. 中島啓、加藤周、疋田辰之. Date. 11月17日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 柳田　伸太郎氏 (名古屋大学). Title. Elliptic Hall algebra over $\\mathbb{F}_1$. Abstract. 　 This talk is motivated by the recent work of Morton and Samuelson which states that the Turaev skein algebra for torus is isomorphic to a specialization of the elliptic Hall algebra. In this talk we introduce the category $B_q$ which is an $\\mathbb{F}_1$-analogue of the category of coherent sheaves over an elliptic curve. Although our category is not an abelian category, even nor an additive category, it is an example of so-called belian and quasi-exact category in the sense of Deitmar. Then we can consider the Hall algebra $U_q$ associated to $B_{q}$ using Szczesny's construction of Hall algebra for monoid representations. The main statement is that $U_{q}$ is isomorphic to the Turaev skein algebra of torus. Thus our construction gives the `B-side counter-part' of the torus skein algebra directly, not replying on an `bi-hand' specialization process on Hall algebra.　. Organizers. 中島啓、加藤周、疋田辰之. Date. 10月27日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. Kari Vilonen氏 (Melbourne). Title. Springer theory for symmetric spaces. Abstract. 　 A year ago I spoke on Springer theory for symmetric spaces in the special case of SL(n,R). In this talk I will discuss the general case. We start with a detailed discussion of a nearby cycle construction which plays a crucial role in the theory. This part is joint work with Grinberg and Xue. After that I will explain how using the geometric input one obtains the Springer theory in general. This part is joint work with Xue. Organizers. 中島啓、加藤周、疋田辰之. Date. 10月20日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 渡邉 英也氏 (東工大). Title. Representation theory of quantum symmetric pairs and Kazhdan-Lusztig bases. Abstract. 　 In 2013, Huanchen Bao and Weiqiang Wang discovered the Schur -Weyl-type duality between some quantum symmetric pair coideal subalgebras $U^{\\jmath}$ and the Hecke algebra $H$ (with unequal parameter) of type B. Namely, they equipped the $d$-th tensor power of the vector representation $V$ of $U_q(\\mathfrak{sl}_n)$ with a $(U^{\\jmath},H)$-bimodule structure which satisfies the double centralizer property. In this talk, we investigate the bimodule structure of $V^{\\otimes d}$ and see that its $\\jmath$-canonical basis (introduced by Bao and Wang) coincides with its (parabolic) Kazhdan-Lusztig basis. Time permitting, we will see how this result relates to the Lusztig's periodic $W$-graphs. This talk is partially based on a joint work with Bao and Wang. Organizers. 中島啓、加藤周、疋田辰之. Date. 10月6日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 藤田 遼氏 (京都大学). Title. Affine highest weight categories for quantum loop algebras of Dynk in types. Abstract. 　 For a Dynkin quiver $Q$ (i.e. Dynkin graph of a simple Lie a lgebra $\\mathfrak{g}$ of type ADE with an orientation), Hernandez-Leclerc defined a monoidal subcategory $\\mathcal{C}_{Q}$ of the category of finite-dimensional modules over the quantum loop algebra associated with $\\mathfrak{g}$. They proved that its Grothendiek ring is isomorphic to the coordinate algebra of the maximal unipotent subgroup associated with $\\mathfrak{g}$ and that the classes of simple modules correspond to the dual canonical basis elements. In this talk, we see that a \"central completion\" of the category $\\mathcal{C}_{Q}$ has a structure of affine highest weight category. We rely on Nakajima's geometric method using the equivariant K-theory of graded quiver varieties. As an application, we conclude that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives a monoidal equivalence between the Hernandez-Leclerc category $\\mathcal{C}_{Q]$ and the category of finite-dimensional modules over the quiver Hecke (KLR) algebra associated with $Q$, assuming the simpleness of poles of normalized R-matrices for type E. Organizers. 中島啓、加藤周、疋田辰之. Date. 8月4日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. Prof. Evgeny Mukhin (IUPUI). Title. The Universal Differential Operator. Abstract. 　 In this expository talk I will review various facts around Gaudin model associated to gl(n). I will focus on the universal differential operator which plays the central role in many important constructions and which was neglected for many years. Organizers. 中島啓、加藤周、疋田辰之. Date. 7月28日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 大矢浩徳氏 (東京大学・数理). Title. The Chamber Ansatz formulae for quantum unipotent cells. Abstract. 　 Berenstein, Fomin and Zelevinsky introduced biregular automorphisms, called twist automorphisms, on unipotent cells in their study of total positivity criteria. These automorphisms are essentially used for describing the inverses of specific embeddings of tori into unipotent cells. The resulting descriptions are called the Chamber Ansatz. In this talk, we consider a quantum analogue of their setting. First, we construct the twist automorphisms on arbitrary quantum unipotent cells and study their compatibility with the dual canonical bases. Next, we provide quantum analogues of the Chamber Ansatz formulae. We also discuss the relation between our results and the quantum cluster algebra structures on quantum unipotent cells introduced by Geiss-Leclerc-Schröer and Goodearl-Yakimov. A part of this talk is based on joint work with Yoshiyuki Kimura. Organizers. 中島啓、加藤周、疋田辰之. Date. 7月21日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 元良直輝氏 (RIMS). Title. Coproducts for W-algebras in A-type. Abstract. 　 The (affine) W-algebras are vertex algebras defined by generalized Drinfeld-Sokolov reductions associated with Lie algebras and nilpotent orbits. Using Zhu's functor, they can be associated with finite W-algebras, in particular, with trancations of shifted Yangians in A-type due to Brundan-Kleshchev. We will introduce the \"coproduct\" structure on W-algebras in A-type, which is analogue of finite case. In our construction, we use the Wakimoto representations. Organizers. 中島啓、加藤周、疋田辰之. Date. 6月16日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 時本一樹氏 (RIMS). Title. Affinoids in the Lubin-Tate perfectoid space and special cases of the local Langlands correspondence. Abstract. 　 Let F be a non-archimedean local field. The non-abelian Lubin-Tate theory asserts that the local Langlands correspondence for GL_n(F) and the local Jacquet-Langlands correspondence are realized in the cohomology of the Lubin-Tate tower. Motivated by this theory, Boyarchenko-Weinstein and Imai-Tsushima constructed affinoid subspaces of the Lubin-Tate perfectoid space (a certain limit space of the tower) and proved that the cohomology of the reduction of each affinoid realizes the two correspondences for certain representations. In this talk, I will discuss a similar result for some other representations. Organizers. 中島啓、加藤周、疋田辰之. Date. 6月9日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. Sergey Loktev氏 (Higher School of Economics). Title. Weyl modules: one variable vs. multivariable. Abstract. 　 Weyl modules over current algebras are defined as universal highest weight modules. They are well-defined for current algebras in any number of variables. We discuss which properties of one-variable Weyl modules can be generalized for the multivariable case. Organizers. 中島啓、加藤周、疋田辰之. Date. 5月26日 (Fri),. 15:30--unfixed,. 2017. Room. Room 402, RIMS. Speaker. Ivan Cherednik氏 (Chapell Hill\/RIMS). Title. DAHA approach to plane curve singularities. Abstract. 　 I will present a recent conjecture that connects the geometry of compactified Jacobians of unibranch plane singularities with the DAHA-superpolynomials of algebraic knots. This is directly related to p-adic orbital integrals (in Fundamental Lemma) and theory of affine Springer fibers (the anisotropic case, type A). This is based on certain partition of the flagged Jacobian factors (new objects, to be defined from scratch); there are connections with the Kazhdan-Lusztig dimension formulas (their 1988 paper), and the works by Bezrukavnikov, Lusztig-Smelt and Piontkowski. The DAHA-superpolynomials are expected to coincide with the stable Khovanov-Rozansky polynomials of algebraic knots. They depend on the paarameters a,q,t; for instance, a=-1, q=t result in Alexander polynomials, which can be directly expressed via the corresponding singularities (without any Jacobian factors). When a=0, q=1, t=1\/p the DAHA-superpolynomials conjecturally coincide with the p-adic orbital integrals. Our conjecture readily implies that the orbital integrals in type A depend only on the topological (not just analytic!) type of singularity. This is joint with Ivan Danilenko and Ian Philipp. I may skip some details concerning the DAHA construction (the first hour), but the geometric superpolynomials will be defined in full and from scratch. This is an entirely local theory; the definitions are not too involved. Organizers. 中島啓、加藤周、疋田辰之. Date. 5月19日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. Liron Speyer氏 (大阪大学・情報). Title. Specht modules for the KLR algebras of type C. Abstract. 　 The KLR algebras were introduced almost a decade ago to cate gorify the negative half of a quantum group. In type A, Brundan and Kleshchev showed that cyclot omic quotients of KLR algebras are isomorphic to cyclotomic Hecke algebras, which has spurred on the development of their graded representation theory, in particular with a theory of Specht modules. We will report on recent joint work with Susumu Ariki and E uiyong Park, in which we have defined a family of Specht modules for the KLR algebras in type C. We will outline some of their basic properties and explain why they are int eresting objects to study. We will finally discuss how we used these Specht modules to classify which cyclotomic quotients of the KLR algebras of type C are semisimple. Organizers. 中島啓、加藤周、疋田辰之. Date. 5月12日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 平野 雄貴氏 (京都大学). Title. Faithful actions from hyperplane arrangements. Abstract. 　 Donovan--Wemyss associates to a flopping contraction $f:X\\to Y$ of 3-folds a group action from the fundamental group of the complement of a complex hyperplane arrangement on the derived category of coherent sheaves on $X$. We show that this action is faithful when $f$ is a crepant resolution by studying tilting modules over noncommutative crepant resolutions of $Y$. This is a joint work with M.Wemyss. Organizers. 中島啓、加藤周、疋田辰之. Date. 4月21日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 白石勇貴氏 (京都大学). Title. 拡大カスピダルワイル群の不変式論と軌道空間上のフロベニウス構造. Abstract. 　 本研究は高橋篤史先生との共同研究です。 拡大カスピダルワイル群と呼ばれる、星状コクセター ディンキン図形に付随したワイル群をアフィン化し、 更に1次元分とある方向への作用で拡大した群を 考えます。. ルーイエンガの不変式論により、この群による 軌道空間は複素多様体の構造を持ち、とある予想的 条件（性質Pと呼ぶ）の下で、ドゥブロヴィンの フロベニウス構造が入る事を紹介します。. また時間が有れば、このフロベニウス構造と、 ある種の特異点や正則関数に対する齋藤の原始形式の 理論から構成されるフロベニウス構造との同型について 説明します。. Organizers. 中島啓、加藤周、疋田辰之. Date. 4月14日 (Fri),. 16:20--17:50. , 2017. Room. Room 402, RIMS. Speaker. Yosuke Morita氏 (Kyoto U). Title. Homogeneous spaces that do not model any compact manifold. Abstract. 　 A manifold is said to be locally modelled on a homogeneous space G\/H if it is obtained by patching open sets of G\/H by left translations of elements of G. A typical example is a Clifford-Klein form, namely, a quotient of G\/H by a discrete subgroup of G acting properly and freely on G\/H. Since T. Kobayashi's work in the late 1980s, a number of obstructions to the existence of compact manifolds locally modelled on G\/H (or compact Clifford-Klein forms of G\/H) has been found. I will explain an obstruction arising from the comparison of relative Lie algebra cohomology and de Rham cohomology. Organizers. 中島啓、加藤周、疋田辰之. Date. 4月7日 (Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 加藤周氏 (京大・理). Title. 非対称Macdonald多項式の特殊化の間の双対性について. Abstract. 　 $ADE$型のルート系に対しては対応するアフィン・リー環の レベル$1$の(適切な)Demazure加群の指標と非対称Macdonald多項式の$t = 0$ への特殊化が一致することが知られている(Sanderson-Ion)。. この講演では$ADE$型のルート系の非対称Macdonald多項式の$t = \\infty$への 特殊化も類似の解釈を持つことを説明し、$t = 0$への特殊化との関係を議論 する。講演内容はEvgeny Feigin, Ievgen Makedonskyi氏との共同研究arXiv:1703.04 108の第５節とAppendix (及びよく知られた結果)に基づく。. Organizers. 中島啓、加藤周、疋田辰之. Date. 1月20日 January 20(Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. Ivan Ip氏 (京都大学). Title. Cluster realization of Uq(g) and factorization of universal R matrix. Abstract. 　 For each simple Lie algebra g, I will talk about a new　 presentation of an embedding of Uq(g) into certain quantum torus algebra, described by a quiver diagram, using the previous construction of positive representations of split real quantum groups. We will discuss its relation to cluster structure of G-local system described recently by Le, and a factorization of the universal R matrix which corresponds to a sequence of quiver mutations giving the half-Dehn twist of the triangulation of a twice- punctured disk with two marked points. This generalizes the well-known result of Faddeev for type A1 and the recent work of Schrader-Shapiro for type An. Organizers. 中島啓、加藤周、疋田辰之. Date. 1月13日 January 13(Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 八尋耕平氏 (東大数理). Title. D-modules on partial flag varieties and intertwining functors. Abstract. 　 Beilinson and Bernstein provided a relationship between the category of D-modules on the full flag variety and a category of representations of semisimple Lie algebras. They introduced intertwining functors for D-modules on the full flag variety and gave a proof of Casselman submodule theorem using them. In this talk, we discuss the case of partial flag varieties. We show that in some cases intertwining functors are equivalences of derived categories. We also discuss the behavior of global sections under the intertwining operators. Organizers. 中島啓、加藤周、疋田辰之. Date. 1月6日 January 6(Fri), 16:30--18:00, 2017. Room. Room 402, RIMS. Speaker. 鈴木咲衣氏 (RIMS). Title. The universal quantum invariant and colored ideal triangulation. Abstract. 　 The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix. On the other hand, R. Kashaev showed that the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the S-tensor) satisfying the pentagon relation. In this talk we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant for colored singular triangulations of topological spaces, especially for colored ideal triangulations of tangle complements. In this construction, a copy of the S-tensor is attached to each tetrahedron, and the invariance under the colored Pachner (2; 3) moves is shown by the pentagon relation of the S-tensor. Organizers. 中島啓、加藤周、疋田辰之. Date. 12月9日 December 9(Fri), 16:30--18:00, 2016. Room. Room 402, RIMS. Speaker. 佐藤 敬志氏 (京大理). Title. Hessenberg varieties and hyperplane arrangements. Abstract. 　 A Hessenberg variety is a subvariety of a flag variety determined by a \"good\" subset of the positive root system. By the way, a subset of the positive root system gives a hyperplane arrangement in the Lie algebra of a maximal torus. Similarly to a flag variety, the chambers of this arrangement denote a cell decomposition of the (regular nilpotent) Hessenberg variety. By this relation between a Hessenberg variety and a hyperplane arrangement, we describe the cohomology ring of the (regular nilpotent) Hessenberg variety in terms of the subarrangement and show that its Poincaré polynomial has two expressions like the Borel's work on flag varieties. This is a joint work with Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, and Satoshi Murai. Organizers. 中島啓、加藤周、疋田辰之. Date. 11月25日 November 25(Fri), 16:30--18:00, 2016. Room. Room 402, RIMS. Speaker. 大島 芳樹氏 (IPMU). Title. Determinant formula for parabolic Verma modules of Lie superalgebras. Abstract. 　 We give a determinant formula for parabolic Verma modules of contragredient finite-dimensional Lie superalgebras assuming that the Levi component is contained in the even part. Our formula generalizes previous results of Jantzen for parabolic Verma modules of (non-super) Lie algebras, and of Kac concerning (non-parabolic) Verma modules for Lie superalgebras. This is a joint work with Masahito Yamazaki. Organizers. 中島啓、加藤周、疋田辰之. Date. 11月18日 November 18(Fri), 16:30--18:00, 2016. Room. Room 402, RIMS. Speaker. Evgeny Feigin 氏(Higher School of Economics). Title. Generalized Weyl modules and nonsymmetric Macdonald polynomials. Abstract. 　 We define a family of modules over the Iwahori subalgebra of an affine Kac-Moody Lie algebra, generalizing classical Weyl modules. The modules in the family are labeled by integral weights of the underlying finite-dimensional algebra. We describe the representation theoretical and combinatorial properties of the generalized Weyl modules. In particular, we show that they serve as categorification of various specializations of the nonsymmetric Macdonald polynomials and of the Orr-Shimozono combinatorial formula. Organizers. 中島啓、加藤周、疋田辰之. Date. 11月11日 November 11(Fri), 16:30--18:00, 2016. Room. Room 402, RIMS. Speaker. Benoit Collins氏（京大・理). Title. Positivity for the dual of the Temperley-Lieb basis. Abstract. 　 A problem raised by Vaughan Jones is to consider the basis dual to the canonical basis of the Temperley-Lieb algebra for non-degenerate loop values, and investigate the coefficients of this basis element in the original basis. For example, the dual of the identity element is a multiple of the Jones Wenzl projection, and computing it is an important problem for which some formulas have been given recently (e.g. by Morrisson). The goal of this talk is to describe a new combinatorial formula for all of these coefficients. As a byproduct, we solve one question of Jones and prove that all these coefficients are never zero for real parameters \\ge 2, and we compute their sign. Our strategy relies on identifying these coefficients with the Weingarten function of the free orthogonal quantum group, and on developing quantum integration techniques. I will spend some time on recalling definitions and properties of some objects that are less well-known, such as Weingarten functions and free orthogonal quantum groups. This talk is based on joint work with Mike Brannan, arXiv:1608.03885. Organizers. 中島啓、加藤周、疋田辰之. Attention. いつもと曜日・お部屋が異なります. Date and room are different from the usual. Date. 10月27日 October 27(Thu). , 16:30--18:00, 2016. Room. Dept. of Math. 305. Speaker. Thorge Jensen氏(MPI\/RIMS). Title. The p-canonical basis of Hecke algebras. Abstract. 　 Motivated by open problems in modular representation theory, we describe a positive characteristic analogue of the Kazhdan-Lusztig basis of the Hecke algebra of a crystallographic Coxeter system and investigate some of its properties. After giving several examples, we will mention recent results about p-cells. Organizers. 中島啓、加藤周、疋田辰之. Date. 10月21日 October 21(Fri), 16:30--18:00, 2016. Room. RIMS 402. Speaker. Leonardo Patimo氏（MPI）. Title. The Hard Lefschetz Theorem in Positive Characteristic for the Flag Varieties. Abstract. 　 Hodge theoretic properties of the Flag Varieties (in characteristic 0) are a fundamental ingredient in the proof of the Kazhdan-Lusztig conjectures. Investigating Hodge theoretic properties in positive characteristic could lead to a better understanding of Lusztig's conjecture on algebraic groups in positve characteristic. As a first step in this direction, in this talk we prove, for any flag variety, that the Hard Lefschetz Theorem holds in characteristic p if p is larger than the number of positive roots. Organizers. 中島啓、加藤周、疋田辰之. Date. 10月7日 October 7(Fri), 16:30--18:00, 2016. Room. RIMS 402. Speaker. 藤田遼氏（京大・理). Title. Tilting modules of affine quasi-hereditary algebras. Abstract. 　 We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor gives a fully faithful embedding of a block of the deformed BGG category of glm into the module category of a suitable completion of degenerate affine Hecke algebra of $\\mathop{GL}_n$. Organizers. 中島啓、加藤周、疋田辰之. Date. 9月30日 September 30(Fri),. 15:00--18:00,. 2016. Room. RIMS 402. Speaker. 杉山 聡 氏（東大数理）. Title. Fukaya category in the Koszul duality theory. Abstract. 　 We compute all the Ext groups, composition of them, and their higher structure of the simple modules of a path algebra with relations over a tree type quiver $A$ by using the Fukaya categories of some exact Riemann surfaces. This is nothing but a computation of an $A_\\infty$- Koszul dual $A^!$ of $A$. In this talk, (i) we review the theory of Fukaya categories, (which appeared in symplectic geometry, mainly in the context of Homological Mirror Symmetry), then (ii) we study the \"abstract\" method of computing Koszul dual via Fukaya categories, finally (iii) we see the three examples of the computation (which are enough convincing so that they let us imagine the full proof). Organizers. 中島啓、加藤周、疋田辰之. Date. 9月23日 September 23(Fri), 16:30--18:00, 2016. Room. RIMS 402. Speaker. 小寺 諒介氏 (京大・理). Abstract. 　 We prove that the quantized Coulomb branches associated with framed quiver gauge theory of Jordan type are isomorphic to spherical Cherednik algebras. Organizers. 中島啓、加藤周、疋田辰之. Attention. いつもと曜日・お部屋が異なります. Date and room are different from the usual. Date. 8月8日 August 8(Mon), 16:30--18:00, 2016. Room. Room 307,Research Building No.4, Kyoto University. （京都大学総合研究4号館307号室）. Speaker. Yung-Ning Peng氏（National Central University）. Title. PARABOLIC PRESENTATION OF THE SUPER YANGIAN $Y_{M|N}$ AND ITS APPLICATION. Abstract. 　 The super Yangian Y_{M|N} associated to the general linear Lie superalgebra gl_{M|N}, defined by Nazarov, is a super analogue of the classical Yangian algebra Y_{N} associated to gl_{N}. It can be described in two different manners: the RTT presentation and the Drinfeld's presentation. In this talk, we introduce a series of presentations of Y_{M|N}, depending on a composition \\mu of M +N and a 0^{M}1^{N} -sequence s. Our presentation covers both the RTT presentation and Drinfeld's presentation as special examples by taking \\mu = (M + N) or \\mu = (1^{M+N}), and a lot of new presentations that never appeared before. Moreover, we will discuss about some application of our result. Organizers. 中島啓、疋田辰之. Date. 7月15日 July 15(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. Hans Franzen氏（Bonn）. Title. Classical and orientifold Donaldson-Thomas invariants as Chow groups. Abstract. 　 We show that the primitive part of Kontsevich-Soibelman's Cohomological Hall algebra of a quiver can be identified with Chow groups of moduli spaces of stable quiver representations. This shows that the Donaldson-Thomas invariants agree with the dimensions of these Chow groups. A similar method also applies for Young's orientifold DT invariants which are an analog of classical DT invariants for orthogonal\/symplectic groups. We show that these invariants can be identified with Chow groups of moduli spaces of $\\sigma$-stable self-dual representations. Organizers. 中島啓、疋田辰之. Date. 7月8日 July 8(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. 林拓磨氏（東大数理）. Title. A description of principal series representations of SU(1,1) over ${\\mathbb{Z}}\\left[1\/2\\right]$-algebras. Abstract. 　 Principal series representations are Hilbert representations of real reductive Lie groups obtained by parabolic inductions. Their associated (${\\mathfrak{g}},K$)-modules are known to be obtained from the corresponding parabolic induction of ($\\mathfrak{g},K$)-modules. Hence they satisfy a universal property. If we start with the Lie group SU(1,1) the associated (${\\mathfrak{g}},K$)-modules to the principal series representations have an explicit description. In particular, they are defined over commutative rings. In this talk, I will prove that these ($ {\\mathfrak{g}},K$)-modules over ${\\mathbb{Z}}\\left[1\/2\\right]$-algebras enjoy a similar universal property as well. Organizers. 中島啓、疋田辰之. Date. 6月17日 June 17(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. 田中雄一郎氏（東大数理）. Title. Visible actions of compact Lie groups on complex spherical varieties. Abstract. 　 With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds. In this talk we consider visible actions of a compact real form U of a connected complex reductive algebraic group G on spherical varieties. Here a connected complex G-variety X is said to be spherical if a Borel subgroup of G has an open orbit on X. The sphericity implies the multiplicity-freeness property of the space of polynomials on X. We firstly give a proof of the visibility for affine homogeneous spherical varieties, and then show the visibility for general spherical varieties by using the method of induction of visible actions. A prototype of the method of induction was introduced by Kobayashi (2005) for the case of complex spherical nilpotent orbits of type A, and recently extended by A. Sasaki (2016) to the case of arbitrary type. Our proof is highly motivated by those earlier results. Organizers. 中島啓、疋田辰之. Date. 6月3日 June 3(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. 渡部正樹氏（東大数理）. Title. Kraskiewicz-Pragacz modules and positivity properties of Schubert polynomials. Abstract. 　 Kraskiewicz-Pragacz modules are certain family of modules over the upper triangular Lie algebra whose characters are Schubert polynomials. Due to this property, some problems on Schubert-positivities of polynomials are closely related with the class of modules having filtrations with successive quotients being KP modules. In this talk I will explain my result which give a characterization of such modules, in terms of certain Ext groups, using the methods of highest weight categories. As applications of such a characterization we obtain a representation-theoretic proof (other than the classical geometric proof) for the positivity of the products of Schubert polynomials, as well as a new result generalizing the positivity of plethysms of Schur functions to Schubert polynomials. Organizers. 中島啓、疋田辰之. Date. 5月20日 May 20(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. 渡邉英也氏（東工大）. Title. Combinatorial formulas expressing periodic R-polynomials and periodic Kazhdan-Lusztig polynomials. Abstract. 　 Periodic Kazhdan-Lusztig polynomials naturally appear in the representation theory of affine Hecke algebras and affine quantum groups. They are computed from periodic R-polynomials. In this talk, we will give a combinatorial formula expressing periodic R-polynomials by using the \"doubled\" Bruhat graph. Then, a combinatorial formula for periodic KL-polynomials can be constructed from this formula. Time permitting, we will briefly explain how periodic Kazhdan-Lusztig polynomials appear in the representation theory of affine quantum groups. Organizers. 中島啓、疋田辰之. Date. 5月13日 May 13(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. Bea Schumann氏 (Cologne and Tokyo). Title. Homological description of crystal structures on quiver varieties. Abstract. 　 We explain a crystal isomorphism in finite types between the explicit crystal structure on Lusztig's parametrisation of the canonical basis obtained by Reineke in terms of representations of quivers and the geometric construction of crystal bases obtained by Kashiwara and Saito in terms of quiver varieties. Using the interplay between the representation theory of the Dynkin quiver and the representation theory of the preprojective algebra, we thereby compute the actions of the Kashiwara operators on the irreducible components of the quiver varieties. Organizers. 中島啓、疋田辰之. Date. 5月6日 May 6(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. Hiraku Nakajima 氏（RIMS）. Title. Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A. Abstract. 　 Cherkis bow varieties are found in the ADHM type description of instantons on the Taub-NUT space. They were originally given in terms of Nahm's equations, but I will give their quiver description, which are useful for analysis of their properties. As an application, I will explain that they are Coulomb branches of quiver gauge theories of affine type A. This is a joint work with Yuuya Takayama. Organizers. 中島啓、疋田辰之. Date. 4月22日 April 22(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. 跡部発 氏（京都大学）. Title. Local theta correspondence of tempered representations and Langlands parameters. Abstract. 　 局所テータ対応について、2つの問題がある。. 一つは与えられた既約表現に対して、そのテータリフトはいつ nonzero になるか という問題。もう一つは、テータリフトが nonzero の時、その唯一の既約商は どのような表現かという問題。本講演では、p 進体上の unitary dual pair に おいて、与えられた既約表現が緩増加である時に、これら2つの問題に Langlands 対応の言葉で答えを与える。. なお、本研究は Wee Teck Gan 氏との共同研究である。. Organizers. 中島啓、疋田辰之. Date. 4月15日 April 15(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. Naoki Genra氏 (RIMS). Title. Screening operators for W-algebras. Abstract. 　 We show that the (affine) W-algebras for generic levels are constructed as intersections of kernels of screening operators. As applications, we prove two conjectures. First, Fateev-Lukyanov's WBn-algebras are isomorphic to the W-algebras for osp(1,2n) and its regular nilpotent element. Second, Feigin-Semikhatov's W^(2)_{n}-algebras are isomorphic to the W-algebras for sln and its subregular nilpotent element. Organizers. 中島啓、疋田辰之. Date. 4月7日 April 7. (Thu). , 16:30--18:00, 2016. Room. RIMS Room. 109. Speaker. Shintaro Yanagida氏 (Nagoya). Title. K-theoretic AGT relation. Abstract. 　 We sutdy a geometric action of the deformed Virasoro algebra on the torus equivariant K groups of instantont moduli spaces on the complex plane. The main ingredient is the analysis of K-theoretic stable envelopes. As a consequence, we can prove some parts of the conjectures on K-theoretic AGT correspondence proposed in the previous collaboration with Awata, B. Feigin, Hoshino, Kanai and Shiraishi. Organizers. 中島啓、疋田辰之. Date. 1月22日 January 22(Fri), 16:30--18:00, 2016. Room. RIMS Room 402. Speaker. 土岡俊介氏（東大数理）. Title. On a general Schur's partition identity. Abstract. 　 We will talk on a generalization of Schur's partition identity which is a kind of Rogers-Ramanujan type identity. Our identity comes from K\\\"{u}lshammer-Olsson-Robinson theory of generalized blocks and the Fock space representations of quantum affine algebras due to Kashiwara-Miwa-Petersen-Yung. This is a joint work with Masaki Watanabe (University of Tokyo). Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. いつもと曜日・お時間・お部屋が異なります. Date and room are different from the usual. Date. 12月21日 December 21(Mon), 15:00--16:30, 2015. Room. RIMS Room 110. Speaker. Constantin Teleman 氏 (UC Berkeley). Title. Gauge Theory in 2 and 3 dimensions and categorical representations. Abstract. 　 I will introduce the notion of a categorified (topological) representation of a compact Lie group G, which is the mathematical counter-part to a topological boundary condition for (pure) 3-dimensional gauge theory. The main examples come from the Gromov-Witten theories of compact symplectic manifolds with Hamiltonian group action. The character theory of these representations is captured, in the spirit of quantum mechanics, by the holomorphic symplectic geometry of a certain manifold, now recognised as the `Coulomb branch’ of the pure 3D gauge theory. Twisted versions of Gromov-Witten theory relate to gauge theory with `matter’. The theory gives a clean account of some aspects of the gauged (non-linear!) Sigma-model and the appearance of the Toda integrable system. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. いつもと曜日・お部屋が異なります. Date and room are different from the usual. Date. 12月3日 December 3(Thu), 16:30--18:00, 2015. Room. RIMS Room 206 [changed!]. Speaker. 松本 拓也 氏 (名古屋大学). Title. 中心拡大されたリースーパー代数sl(2|2)に付随する 量子アファイン代数について (Quantum affine algebra associated with the centrally extended Lie superalgebra sl(2|2)). Abstract. 　 リースーパー代数sl(2|2)は全ての超リー代数の中で唯一2次元の 普遍中心拡大を持つ特殊なものであるが， 超弦理論におけるゲージ\/重力対応や1次元ハバード模型における 対称性として現れ，様々な物理的文脈で重要な役割を果たしている． よって，その代数構造を詳しく調べることは重要であると思われる． 今回は，その中心拡大されたsl(2|2)代数に付随する量子群と 無限次元への拡張である量子アファイン代数を紹介したい． また，これまで知られていたヤンギアン代数への退化も議論したい． 本講演はN. Beisert氏(ETH)とW. Galleas氏(DESY)との共同研究 http:\/\/arxiv.org\/abs\/1102.5700 に基づきます．. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 11月13日 November 13(Fri), 15:30--16:30, 16:45 - 17:45, 2015. Room. RIMS Room 402. Speaker. 入谷 寛 氏 (京都大学). Title. A Fock sheaf for Givental quantization. Abstract. 　 Genus-zero Gromov-Witten (GW) theory defines a generalized variation of Hodge structure (sometimes called semi-infinite Hodge structure). For a given generalized variation of Hodge structure, we define a sheaf of Fock spaces on the base of the Hodge variation. The Fock sheaf is locally modeled on the quantization formalism of Givental, and higher-genus GW potentials can be regarded as a section of the Fock sheaf. This formalism gives a framework to discuss the modularity and the crepant transformation conjecture in higher-genus GW theory; for example, we observe that the total descendant GW potentials of compact toric orbifolds X are \"modular\" with respect to a certain subgroup of the group of autoequivalences of D(X). This is based on joint work with Tom Coates. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 11月6日 November 6(Fri), 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. 佐藤僚氏 (東大). Title. Kazama-Suzuki coset construction and logarithmic extensions of weight modules. Abstract. 　 Kazama-Suzukiコセット構成とは，アフィンLie代数のsmoothな加群と荷電フェル ミオンFock加群をテンソルした空間に互いに可換な$\\mathcal{N}=2$超共形代数 とHeisenberg Lie代数の作用を構成する手法である．この手法によっ て，$A_{1}^{(1)}$型アフィンLie代数の既約ユニタリ（=可積分）最高ウェイト 表現から$\\mathcal{N}=2$超共形代数の全ての既約ユニタリ最高ウェイト加群が 得られることはよく知られている．本講演では，非ユニタリな場合にもこの構成 が適切な加群圏の間にアーベル圏としての圏同値を与えることを解説する．特 に，Virasoro代数の$L_{0}$作用素が非対角に作用する（対数的）な加群につい て取り扱う．. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 10月2日 October 2(Fri), 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. 渡邉英也(Hideya Watanabe)氏 (東工大). Title. Parabolic analogue of periodic Kazhdan-Lusztig polynomials. Abstract. 　 We construct a parabolic analogue of so-called periodic modules, which are modules of Hecke algebra associated with an affine Weyl group. These modules have a basis similar to Kazhdan-Lusztig basis. Our construction enables us to see the relation between periodic KL-polynomials and parabolic ones. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 9月25日 September 25(Fri), 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. Simon Wood氏 (Australian National University). Title. Classifying simple modules at admissible levels symmetric polynomials. Abstract. 　 From infinite dimensional Lie algebras such as the Virasoro algebra or affine Lie (super)algebras one can construct universal vertex operator algebras. These vertex operator algebras are simple at generic central charges or levels and only contain proper ideals at so called admissible levels. The simple quotient vertex operator algebras at these admissible levels are called minimal model algebras. In this talk I will present free field realisations of the universal vertex operator algebras and show how they allow one to elegantly classify the simple modules over the simple quotient vertex operator algebras by using a deep connection to symmetric polynomials. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 8月7日 August 7(Fri), 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. Oren Ben-Bassat 氏 (Oxford). Title. Banach Algebraic Geometry. Abstract. 　 I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry. Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in these various areas of math and how it is characterised categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces. The categorical characterization that we need uses homological algebra in these 'quasi-abelian' categories which is work of Schneiders and Prosmans. In fact, we work with the larger category of Ind-Banach spaces for reasons I will explain. This gives us a way to establish foundations of analytic geometry and to compare with the standard notions such as the theory of affinoid algebras, Grosse-Klonne's theory of dagger algebras (over-convergent functions), the theory of Stein domains and others. I will explain how this extends to a formulation of derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi. This is joint work with Federico Bambozzi (Regensburg) and Kobi Kremnizer (Oxford). Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 7月10日 July 10(Fri), 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. 桑原敏郎 Toshiro Kuwabara 氏 (High School of Economics, Russia). Title. Sheaves of asymptotic chiral differential operators on symplectic resolutions. Abstract. 　 In this seminar, we discuss sheaves of (h-adic) vertex algebras on symplectic manifolds, which give quantization of vertex Poisson algebras of their Jet bundles. On each formal coordinate, these sheaves are isomorphic to the vertex algebra of a formal beta-gamma system and we can determine the Lie algebra of derivations. Using Harish-Chandra extensions, we consider the classification of such sheaves. Such sheaves include localization of affine W-algebras which were constructed by Arakawa, Malikov and the speaker. Moreover, they include quantization of Jet bundles of hypertoric varieties and Nakajima quiver varieties. We also discuss construction of such quantization by semi-infinite reduction. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 6月12日 June 12(Fri), 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. 池田 曉志 氏 (東大カブリIPMU). Title. Calabi-Yau圏の安定性条件の空間とフロベニウス多様体. Abstract. 　 三角圏に対してBridgelandは安定性条件の概念を導入し, 安定性条件全体の成す空間は複素多様体になることを示した. この空間はフロベニウス多様体と関連があることが期待されていたが, 最近, A型箙のGinzburgのdg代数の導来圏上の安定性条件の空間と A型特異点に付随するフロベニウス多様体の関係性が明らかになった. 本講演ではその結果をきっかけとして, Calabi-Yau圏の安定性条件の空間とフロベニウス多様体の間に 期待される関係性について説明する. また, 安定性条件の空間の中心電荷とフロベニウス多様体の 周期の間の関係性についての予想を述べる. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. いつもとお時間・場所が違います. Date and Place are different from the usual. Date. 6月4日 June 4(Thu) 13:30--15:00, 2015. Room. RIMS Room 006(basement). Speaker. Ivan Ip氏 (Kyoto University SGU). Title. Positive Casimir and Central Characters of Split Real Quantum Groups. Abstract. 　 The notion of the positive representations was introduced in a joint work with Igor Frenkel as a new research program devoted to the representation theory of split real quantum groups. Explicit construction of the these irreducible representations have been made corresponding to classical Lie type. In this talk, I will discuss the action of the generalized Casimir operators, which is important to understand the tensor product decomposition of these representations. These operators are shown to admit positive eigenvalues, and that their image defines a semi-algebraic region bounded by real points of the discriminant variety. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 5月29日 May 29(Fri), 16:30--18:00, 2015. Room. Room RIMS Room 402. Speaker. Bin Shu氏 (East China Normal University). Title. Finite W-superalgebras and existence of Kac-Weisfeiler modules for basic Lie superalgebras in positive chatacteristic. Abstract. 　 In this talk, we will introduce finite W-superalgebras for basic Lie superalgebras associated with even nilpotent elements. We will then present the PBW theorem and other properties fo them. We will finally discuss the existence of the so-called Kac-Weisfeiler modules for basic Lie superalgebras in positive characteristic. This is a joint work with Yang Zeng. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 5月22日 May 22(Fri), 16:30--18:00, 2015. Room. Room RIMS Room 402. Speaker. Anton Evseev氏 (University of Birmingham). Title. RoCK blocks, wreath products and KLR algebras. Abstract. 　 The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\\wr S_d$ of symmetric groups, where $d$ is the \"weight\" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. いつもと時間・場所が違います. Date. 5月14日 May 14(Thu), 13:30--15:00, 2015. Room. Room RIMS Room 006. Speaker. Kentaro Wada氏 (Sinsyu). Title. New realization of cyclotomic q-Schur algebras. Abstract. 　 G(r,1,n)型の複素鏡映群に付随した cyclotomic q-Schur 代数は Ariki-Koike 代数の quasi-hereditary cover の1つである。r=1の場合，cyclotomic q-Schur 代数は古典的な q-Schur 代数そのものであり, q-Schur 代数は一般線形リー代 数に付随する量子群の商代数であることが知られている。量子群の Hopf代数と しての構造と普遍R-行列によって，(nを全て動かした)q-Schur 代数の加群圏上 にはモノイダル構造が定まる。r1の場合には，そのような構造は知られていな いが，Rouquier-Shan- Varagnolo-Vasserot によって得られているアファイン一 般線形リー代数のアファイン放物型圏O との関係等によって，cyclotomic q-Schur 代数の加群圏上にもモノイダル構造が定まることが期待される。この講 演ではその可能性の1つについてお話ししたい。 まず，r個に分けられた一般線形リー代数のカルタンデータに付随してリー代 数 g を導入する。r=1 の場合は一般線形リー代数に付随したカレントリー代数 そのものであり，r1 の場合，リー代数 g はカレントリー代数の filtered deformation になっている。すると q=1 の場合の cyclotomic q-Schur 代数 は，このリー代数 g の普遍包絡代数の商代数となる。次に，リー代数 g の普遍 包絡代数の q-類似として代数 U_q を導入し，cyclotomic q-Schur 代数を U_q の商代数として実現する。その後，（まだ分かっていないことが多いが) リー代 数 g やその q-類似 U_q の表現論について可能な限り説明したいと思います。. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 5月8日 May 8(Fri), 16:30--18:00, 2015. Room. Room RIMS Room 402. Speaker. Yoshihiro Takeyama氏 (Tsukuba University). Title. A deformation of affine Hecke algebra and integrable stochastic particle system. Abstract. 　 We introduce a deformation of the affine Hecke algebra of type $GL$ with four parameters. Making use of its representation on the space of polynomials, we can construct a discrete analogue of integration operators satisfying the braid relations. It determines a difference operator which can be regarded as a discretization of the Hamiltonian of the one-dimensional delta Bose gas. By specializing the parameters of the discrete Hamiltonian, we get the transition rate matrix of an integrable stochastic particle system called (a continuous time limit of ) the $q-$Hahn system. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. 今回は連続講演でいつもと時間・場所が違います. Date. 4月16日 Apr 16(Thu) 10:30--12:00, 13:30--15:00, 2015. Room. RIMS Room 006(地階). Date. 4月17日 Apr 17(Fri) 15:00-18:00. Room. RIMS Room 402. Speaker. 脇本実氏 (九州大学). Title. アフィン・スーパー・リー環の表現と モック・テータ函数. Abstract. 　 アブストラクト： アフィン・スーパー・リー環の指標のモジュラー性質が どのようなものかを調べるのは難しい問題であった。S. Zwegers は 学位論文（２００３年）において，アフィン・スーパー・リー環 $\\widehat sl(2|1)$ の レベル１の表現 $L(\\Lambda_0)$ のスーパー指標に非正則な補正項を付加する ことにより実解析的なモジュラー函数が得られることを示した （S.P. Zwegers: Mock Theta Functions, ArXiv:0807.4834）。 Zwegers の方法を適用することによって，すべての basic classical アフィン・スーパー・リー環の maximally atypical 表現について， それらの（スーパー）指標を実解析的なモジュラー函数に拡張し， そのモジュラー変換行列を計算することが出来る。このセミナーでは これについて V.G. Kac との共同研究で得られた最近の成果を解説する。. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 2月6日 Feb 6(Fri) 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. Hironori Oya氏 (University of Tokyo). Title. Representations of quantized function algebras the transition matrices from Canonical bases to PBW bases. Abstract. 　 Let $G$ be a connected simply connected simple complex algebraic group of type $ADE$ and $\\mathfrak{g}$ the corresponding simple Lie algebra. In this talk, I will explain our new algebraic proof of the positivity of the transition matrices from the canonical basis to the PBW bases of $U_q(\\mathfrak{n}^+)$. Here, $U_q(\\mathfrak{n}^+)$ denotes the positive part of the quantized enveloping algebra $U_q(\\mathfrak{g})$. We use the relation between $U_q(\\mathfrak{n}^+)$ and the specific irreducible representations of the quantized function algebra $\\mathbb{Q}_q[G]$. This relation has recently been pointed out by Kuniba, Okado and Yamada (SIGMA. 9 (2013)). Firstly, we study it taking into account the right $U_q(\\mathfrak{g})$-algebra structure of $\\mathbb{Q}_q[G]$. Next, we calculate the transition matrices from the canonical basis to the PBW bases using the result obtained in the first step. I mention also some remarks which have recently been perceived. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 1月30日 Jan 30(Fri) 16:30--18:00, 2015. Room. RIMS Room 402. Speaker. Oleksandr Tsymbaliuk氏 (Simons Center and RIMS). Title. Toroidal and affine Yangian algebras, and their commutative subalgebras. Abstract. 　 We will recall the construction of certain families of representations of the toroidal algebras of $sl_n$, due to [Feigin-Jimbo-Miwa-Mukhin]. We explain how to adapt those to the setting of the affine Yangians. We will generalize the result of [Gautam-Toledano Laredo] to the toroidal setting. Using an alternative realization of Fock representations, due to [Saito], we recover a functional realization of certain commutative subalgebras in the toroidal\/affine quantum algebras. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. Date and Place are different from the usual. Date. 1月27日 Jan 27(Tue) 16:30--18:00, 2015. Room. Room 110, Building No.3. Speaker. Anthony Henderson氏 (University of Sydney). Title. Geometric Satake, Springer correndence, and small representations. Abstract. 　 Let $G$ be a connected reductive group and $W$ its Weyl group. Consider the functor $\\Phi$ from representations of $G$ to representations of $W$ defined by taking the zero weight space. This functor contains important information, but is hard to describe in general. Note that when $G = GLn$, the restriction of $\\Phi$ to the subcategory of representations whose weights $(a_{1},\\cdots ,a_{n})$ satisfy $a_{1}+\\cdots +a_{n}=0$ and $a_{i} \\ge -1$ is essentially the famous Schur functor. In particular, this restriction is of the form $Hom_{GL_{n}}(E, - )$ where $E$ is a tilting module that carries a commuting $S_{n}$-action. For general $G$, the analogous subcategory to consider is that of small representations, and the restriction of $\\Phi$ to this subcategory was studied by Broer and Reeder in the complex case. However, there is no representation analogous to $E$ in other types. In joint work with Pramod Achar (Louisiana State University) and Simon Riche (Universit\\'e Blaise Pascal - Clermont-Ferrand II), we describe the restriction of $\\Phi$ geometrically, in terms of the perverse sheaves on the affine Grassmannian of the complex dual group $G^{\\lor}$ that correspond to small representations under geometric Satake; this makes sense for any ground field. As we show, the correct substitute for Eis the Springer sheaf on the nilpotent cone of $G^{\\lor}$, with its $W$-action that gives rise to the Springer correspondence. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 1月9日 Jan 9(Fri) 16:30--18:00, 2015. Room. RIMS 402. Speaker. 石井基裕氏 (東北大・情報). Title. 量子アフィン展開環のレベル・ゼロ表現に対するギャラリー模型. Abstract. 　 Gaussent-LittelmannのLakshmibai-Seshadriギャラリー模型を自然に 拡張することに よって、量子アフィン展開環の端ウェイト加群、及び基本加群（のテンソル積）の結 晶基底に対する実現が得られることについてお話しする。Gaussent-Littelmannの研 究目的はLakshmibai-Seshadriパス模型とアフィンGrassmann多様体 （Mirkovic-Vilonenサイクル）との間の関係を記述することであったが、 我々の設定では半無限Lakshmibai-Seshadriパス模型とFeigin-Frenkelの 半無限旗多様体との間の関係が観察されることについても述べる。. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. Time and Place are different from the usual. Date. 11月27日 Nov 27(Thu) 16:30--18:00, 2014. Room. Dept. Math. 3rd build. 109. Speaker. 中島啓氏 (京大・数理研). Title. Coulomb branches of 3d N=4 gauge theories and the affine Grassmannian. Abstract. 　 We propose a mathematically rigorous definition of Coulomb branch of a 3d N=4 SUSY gauge theory, as an affine algebraic variety, based on the homology group of a variant of the affine Grassmannian. In particular, coordinate rings of various hyper-Kaehler manifolds, such as instanton moduli spaces on ALE spaces, nilpotent orbits, etc, are conjecturally given by such a construction. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 11月21日 Nov 21(Fri) 17:00--18:30, 2014. Room. Room 402, RIMS. Speaker. 廣惠一稀氏(Kazuki Hiroe) (城西大学). Title. Local Fourier transform and blowing up. Abstract. 　 We study linear ordinary differential equations with ramified irregular singularies with the help of the theory of singularities of plane curve germs. Especially we shall see analogies between. - Komatsu-Malgrange irregularities of ODEs and intersection numbers and Milnor numbers of curves,. - Local Fourier transform of ODEs and blow up of curves,. - Stokes structures of ODEs and iterated torus knots of curves. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. Time is different from the usual. Date. 10月24日 Oct 24(Fri) 16:00--18:00, 2014. Room. Room 402, RIMS. Speaker. Nikolai Vavilov 氏 (State University of Saint-Petersburg). Title. 1) COMMUTATORS IN ALGEBRAIC GROUPS. 2) K-THEORY OF EXCEPTIONAL GROUPS. Abstract. 　 1) As we teach our students, in an abstract group an element of the commutator subgroup is not necessarily a commutator. However, the famous Ore conjecture, recently completely settled by Ellers---Gordeev and Liebeck---O'Brien---Shalev--Tiep, asserts that any element of a finite simple group, or, more generally, of an adjoint elementary Chevalley group over a field, is a single commutator. On the other hand, from the work of van der Kallen, Dennis and Vaserstein it was known that nothing like that can possibly hold in general, for commutators in classical groups over rings. Actually, these groups do not even have bounded width with respect to commutators. Using new versions of localisation methods, Stepanov, partly in cooperation with myself, Hazrat and Zhang, succeeded in showing that there is finiteness on the other end. Namely, it turned our that commutators have bounded width with respect to elementary generators. Morally, these amazing results show that in algebraic groups over rings there are very few commutators. The only reason, why it appears that there are many commutators in the groups of points over zero-dimensional rings (such as fields or local rings) is that in these cases there exist very short expressions of arbitrary elements in terms of elementary generators. Also, I plan to discuss some further applications of our methods, such as multiple commutator formulae, etc., as well as some further related asymptotic problems. 2) Let $\\Phi$ be a reduced irreducible root system, $R$ be a commutative ring with $1$. We study the following three closely related groups, associated to $(\\Phi,R)$. * The (simply-connected) Chevalley group $G(\\Phi,R)$. * The (simply-connected) elementary Chevalley group $E(\\Phi,R)$. * The Steinberg group $\\St(\\Phi,R)$. We set $K_1(\\Phi,R)=G(\\Phi,R)\/E(\\Phi,R)$ and denote by $K_2(\\Phi,R)$ the kernel of the natural projection $\\St(\\Phi,R)\\map E(\\Phi,R)$. For the classical groups, the initial groundbreaking contributions to the study of these groups were made by Bass, Steinberg, Milnor in the early 1960-ies, followed by the monumental works by Bak, Suslin, Dennis, Vaserstein, van der Kallen, and many others. But for exceptional groups, apart from the very important work of Matsumoto, Stein, and their followers, in particular Plotkin and myself, very little was known until recently. We are mainly interested in the four large exceptional groups of types $\\E_6$, $\\E_7$, $\\E_8$ and $\\F_4$, but actually many of the outstanding problems first stated some 50 years ago still remain open even for classical groups, apart from the linear case. I plan to discuss recent progress towards solution of these probelms, including. * Nilpotent structure of relative $K_1$ (Bak--Vavilov--Hazrat, and recent generalisations due to Hazrat--Vavilov--Zhang and Stepanov),. * Centrality of $\\K_2$, where the first major progress in 30 years (after the solution of linear case by van der Kallen and Tulenbaev) was recently achieved by Lavrenov, who solved the symplectic case,. * Stability for $K_1$ and $K_2$, where Sinchuk has succeeded to improve stability results obtained by Stein and Plotkin for exceptional embeddings (as also recent versions of stability results for classical groups by Bak--Petrov--Tang). I will also survey some of the background and history, some of the methods used, and relevance of these results in other branches of the algebraic group theory. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. Time and Place are different from the usual. Date. 7月17日 July 17(Thu) 16:00--18:00, 2014. Room. Room 108, Building No.3, Kyoto University. Speaker. Andrei Okounkov氏 (Columbia University). Title. Elliptic stable bases and applications. Abstract. 　 This will be a report on a joint work in progress with Mina Aganagic. Our goal is to produce an elliptic generalization of the stable envelopes in K-theory (which will be briefly reviewed). Elliptic stable envelopes depend on an additional parameter $z$ in the complexification of Pic(X). Elliptic stable envelopes limit to K-theoretic stable envelopes with slope $s$ as the elliptic curve degenerates and the Kahler\/dynamical parameter $z$ goes to infinity so that the ratio $z\/\\tau$ has a finite limit $s$. Application awaiting such elliptic generalization include:. (1) geometric construction of elliptic R-matrices,. (2) monodromy of the K-theoretic quantum difference equation,. (3) precise correspondence of boundary conditions in dual 3-dimensional susy gauge theories, and others. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. There will be two informal talks on the elliptic quantum groups by the same speaker at (1) July 11(Fri) 14:30 - 15:30 and (2) July 12(Sat) 10:00 - 13:00. Date. 7月11日 July 11(Fri) 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. Hitoshi Konno 氏 (Tokyo University of Marine Science and Technology). Title. Elliptic Quantum Groups, Drinfeld Coproduct and Deformed W-algebras. Abstract. 　 We first discuss a quantum Z-algebra structure of the elliptic algebra U_{q,p}(g) associated with an untwisted affine Lie algebra g, and show that the irreducibility of the level-k representation of the U_{q,p}(g)-module is governed by the corresponding Z-algebra module. The level-1 examples for g=A_l^{(1)}, B_l^{(1)}, D_l^{(1)} show that the irreducible U_{q,p}(g)-modules are decomposed as a direct sum of the irreducible W-algebra modules. We secondly introduce the Drinfeld coproduct to U_{q,p}(g) and discuss the intertwining operators (vertex operators) with respect to this new coproduct. Constructing the vertex operators for the level-1 U_{q,p}(g)-modules with g=A_l^{(1)}, B_l^{(1)}, D_l^{(1)} explicitly, we show that these vertex operators factor the generating functions of the known deformed W-algebras associated with A_l^{(1)}, D_l^{(1)}, and further obtain a conjectural expression for the B_l^{(1)} case corresponding to a deformation of Fateev-Lukyanov's WB_l-algebra. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. !Caution There are two talks and the first talk starts at 13:00! Date. 6月20日(金) June 20(Fri) 13:00--14:30, 2014. Room. Room 402 of RIMS, Kyoto University. Speaker. Jethro Van Ekeren氏 (Technische Universität Darmstadt). Title. Superconformal Blocks. Abstract. 　 Let V be a chiral algebra (associated to a vertex algebra) over a family X of complex curves. An important collection of objects associated to V are the spaces of conformal blocks. Roughly speaking these are spaces of sections of V over fibres, whose dependence on the moduli yields a bundle with flat connection over the family. Understanding the structure of conformal blocks in particular cases leads to interesting theorems. Examples include nonabelian theta functions, and Zhu's theorem on modular invariance of vertex algebra characters. In this talk I will describe joint work with R. Heluani in which we construct superconformal blocks associated to N=2 SUSY vertex algebras living on super-analogues of elliptic curves. The family of supercurves is described as a quotient by the classical Jacobi group, and equivariance of normalised superconformal blocks under this group establishes their transformation under this group as Jacobi forms. Date. 14:45 - 16:15, 2014. Speaker. Alexander P. Veselov氏 (Loughborough, UK and Tokyo, Japan). Title. Gaudin subalgebras and stable rational curves. Abstract. 　 Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n, associated to A-type hyperplane arrangement. It turns out that Gaudin subalgebras form a smooth algebraic variety isomorphic to the Deligne-Mumford moduli space \\bar M_{0,n+1} of stable genus zero curves with n+1 marked points. A real version of this result allows to describe the moduli space of separation coordinates on the unit sphere in terms of geometry of Stasheff polytope. The talk is based on joint works with L. Aguirre and G. Felder and with K. Schoebel. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 6月13日 June 13(Fri) 17:00--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. Hiraku Abe氏 (Osaka City University). Title. Springer多様体のトーラス同変コホモロジー環. (Torus equivariant cohomology ring of Springer varieties). Abstract. 　 谷崎俊之氏によるA型Springer多様体のコホモロジー環の表示は旗多様体のコホモロ ジー環のBorel表示を自然に一般化するものであった.本講演ではこの表示のトーラス 同変版を解説する．すなわち, A型Springer多様体がもつ自然なトーラス作用に関し てその同変コホモロジー環の表示を与える．この際，Springer多様体の同変コホモロ ジーに対称群の表現を構成することが鍵となる．同変コホモロジー理論でよく用いら れる局所化と呼ばれる手法を用いてこの表現を構成する．本研究は大阪市立大学の堀 口達也氏との共同研究である．. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 6月6日 June 6(Fri) 14:45--16:15 & 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. Tomoki Nakanishi氏 (Nagoya University). Title. Cluster algebras, dilogarithm, and Y-systems. Abstract. 　 Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an underlying combinatorial structure in Lie theory. They also (often quite unexpectedly) appear in several branches of mathematics besides representation theory, e.g., hyperbolic geometry and Teichm\\\"uller theory, Poisson geometry, discrete dynamical systems, exact WKB analysis, etc. In this talk I review the application of cluster algebras to the dilogarithm and Y-systems, based on joint works over the recent years with R. Inoue, O. Iyama, R. Kashaev, B. Keller, A. Kuniba, R. Tateo, J. Suzuki, and S. Stella. The talk consists of two parts. In the first part, after reviewing some basic properties of cluster algebras, I present the dilogarithm identity associated with any period of seeds in a cluster algebra. In the second part, I explain that this identity is related to the longstanding conjectures on the periodicities of Y-systems and the associated dilogarithm identities in conformal field theory, which arose through the thermodynamic Bethe ansatz approach in 90's. Then, I show how efficiently cluster algebra theory proves these conjectures. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 5月23日 May 23(Fri) 16:30--18:00, 2014. Room. RIMS 402, Kyoto University. Speaker. Daisuke Sagaki氏 (Tsukuba University). Title. Demazure subcrystals of crystal bases of level-zero extremal weight modules over quantum affine algebras. Abstract. 　 We give a characterization of the crystal bases of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra. This characterization is given in terms of the initial directions of semi-infinite Lakshmibai-Seshadri paths (SiLS paths), and is established under a suitably normalized isomorphism between the crystal basis of the level-zero extremal weight module and the crystal of SiLS paths. This talk is based on a joint work with Satoshi Naito (arXiv:1404.2436). Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 5月16日 May 16(Fri) 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. 瀧雅人氏 (理研). Title. Seiberg Duality, 5d SCFTs and Nekrasov Partition Functions. Abstract. 　 We propose an equality between five-dimensional (5d) Nekrasov partition functions that are associated with local del Pezzo surfaces, namely the generating functions of the refined Gopakumar-Vafa invariants. It is known that M-theory compactified on a local del Pezzo Calabi-Yau 3-fold leads to a 5d superconformal field theory (SCFT), and their BPS partition functions are these Nekrasov partition functions. One can expect that the \"Picard-Lefschetz transformation\" of the 3-folds implies the duality between the compactified M-theories and the resulting 5d SCFTs. This stringy argument yields conjectural relations between the corresponding Nekrasov partition functions. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. !開始時間がいつもとは異なりますので、ご注意ください! Date. 5月9日 May 9(Fri) 14:30--16:00, 2014. Room. Rims 402, Kyoto University. Speaker. Chul-Hee Lee氏 (SNU). Title. Kirillov-Reshetikhin modules and the WZW fusion ring. Abstract. 　 The Kirillov-Reshetikhin modules form a special class of finite dimensional representations of quantum groups. Their characters are known to satisfy some functional relations called T-systems and Q-systems. In an attempt to calculate the central charges of certain conformal field theories using the dilogarithm function based on the Thermodynamic Bethe Ansatz method, some conjectures about solutions of level restricted version of Q-systems have been proposed. In this talk, I will explain how the WZW fusion ring can be used to answer them and discuss their status and related problems. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. !今回は2コマ講演でいつもより開始時間が早くなっていますのでご注意ください! Date. 4月25日(金) April 25(Fri) 14:45--16:15, 2014. Room. Room 402 of RIMS, Kyoto University. Speaker. Andrei Negut氏(RIMS, Columbia University). Title. Quantum toroidal $gl_n$ and its shuffle presentation. Abstract. 　 We will discuss the Feigin-Odesskii shuffle algebra presentation of the quantum toroidal gl_n algebra. This will allow us to identify many copies of quantum affine gl_n sitting inside the quantum toroidal, each corresponding to a choice of rational slope. In particular, the universal R-matrix of quantum toroidal gl_n decomposes as a product of universal R-matrices for quantum affine gl_n, in a way reminiscent of the Khoroshkin-Tolstoy factorization for affine types. The role of positive roots in the direction of the affinization is played by these rational slopes. Date. 4月25日(金) April 25(Fri) 16:30--, 2014. Room. Room 402 of RIMS, Kyoto University. Speaker. Ivan Cherednik氏(RIMS, UNC at Chapel Hill). Title. Generalized Rogers-Ramanujan identities and Nil-DAHA. Abstract. 　 The core application of Nil-DAHA so far is the construction of the global Q-Whittaker functions and Dunkl operators in the Q-Toda theory and its nonsymmetric variant, including a surprising application to the PBW-filtration (counting the minimal number of creation opertors). As Boris Feigin and the speaker demonstrated, this new theory is closely related to coset algebras and can be used to define Rogers-Ramanujan sums of modular type associated with any root systems. The sums we obtain quantize the constant Y-systems (of type $RxA_n$ for any reduced root systems R). This involves dilogarithms, the so-called Nahm Conjecture and a lot of interesting RT, arithmetic and physics, though the talk will be mainly focused on the main construction (practically from scratch). Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. !通常セミナーは金曜日ですが、この回は変則的に木曜日となります! Date. 4月17日 April 17(Thu) 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. 名古屋創氏 (立教大). Title. On the tau function of the sixth Painleve equation from Virasoro conformal field theory. Abstract. 　 An explicit asymptotic expansion of the tau function of the sixth Painleve equation was discovered by Gamayun, Iorgov and Lisovyy [arxiv:1207.0787]. I will explain that their series expansion of the tau function can be derived from Virasoro conformal field theory. I note that the same approach was done by Iorgov, Lisovyy and Teschner [arxiv:1401.6104]. I will begin by reviewing known results of the tau function, fundamentals of Virasoro conformal field theory, the connection problem of conformal blocks, and then I will explain how to obtain the fundamental solution to the linear problem of PVI and the tau function from Virasoro conformal field theory. My talk is based on a joint work with Hiroe, Jimbo and Sakai. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 4月11日 April 11(Fri) 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. Kari Vilonen氏 (Northwestern University). Title. Langlands duality for real groups. Abstract. 　 For real reductive groups the Langlands duality, as refined by Vogan, acquires a symmetry and both sides of the duality can be viewed as representations of reductive groups. Lifting this duality to the level of categories is a conjecture of Soergel. I will discuss this conjecture and its proof in the case when on one side of the duality the group is quasi-split. This is joint work with Roman Bezrukavnikov. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 4月4日 April 4(Fri) 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. Seok-Jin Kang氏 (Seoul National University). Title. Cyclotomic categorification theorem and 2-representation theory. Abstract. 　 The khovanov-Lauda-Rouquier algebras and their cyclotomic quotients provide categorification of the negativehalf of quantum groups and their integrable highest weight modules. We will discuss the motivation and basic ideas of these categorification theorems and possible future developments. Most materials are based on the joint work with Masaki Kashiwara. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 2月14日 February 14(Fri) 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. Myngho Kim氏 (KIAS). Title. Symmetric quiver Hecke algebras and R-matrices for quantum affine algebras. Abstract. 　 In this talk, I will introduce a family of functors between the category of finite-dimensional graded $R(n)-$modules and the category of finite-dimensional $U_q'(g)-$modules. Here, $R(n)$ is a symmetric quiver Hecke algebra and $U_q'(g)$ is a quantum affine algebra. We call these functors the ``quantum affine Schur-Weyl duality functors''. As an example, I will explain how one can lift the ring homomorphism discovered by Hernandez-Leclerc to a categorical level. This is a joint work with Seok-Jin Kang and Masaki Kashiwara. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 2月7日 February 7(Fri) 16:30--18:00, 2014. Room. Rims 402, Kyoto University. Speaker. 池田岳氏 (岡山理科大). Title. Pfaffian sum formula for the symplectic Grassmannians. Abstract. 　 The classical Giambelli formula for the Grassmannian expresses a Schubert class as the determinant of a matrix whose entries are Chern classes of the universal quotient bundle. We seek for a Giambelli-type formula for the isotropic Grassmannians of a symplectic vector space. For the Lagrangian case, i.e. the case when the maximal dimensional isotropic subspaces are considered, P.Pragacz proved a formula which expresses a Schubert class as a single Pfaffian, which is nothing but Schur's Q-function. The torus equivariant analogue of Pragacz's formula has been proved by M. Kazarian, and myself in different context. The non-maximal and non-equivariant cases were studied by A.Buch, A.Kresch, and H.Tamvakis. They proved a Giambelli-type formula for any Schubert class written in terms of Young's raising operators. We study the non-maximal and equivariant cases. Our formula expresses any torus equivariant Schubert class as a *sum* of the Pfaffians whose entries are equivariantly modified Chern classes of the quotient bundle. As a corollary, we obtain a proof of E. Wilson's conjectural formula. This is joint work with Tomoo Matsumura. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 11月1日 November 1(Fri) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. 廣惠一希氏 (城西大学). Title. Riemann球面上の線型微分方程式と箙の表現. Abstract. 　 Crawley-BoeveyはFuchs型微分方程式と星型箙の表現が対応 することを用いて、加法的Deligne-Simpson問題を解決した。 これを拡張して不確定特異点を持つ方程式と箙の表現との対応が Boalchによって特別な場合に与えられ、さらに山川大亮氏（東工大） と講演者との共同研究で一般化された。. これらを元にして、本講演では加法的Delinge-Simpson問題を 不確定特異点をもつ微分方程式に対して良い条件下で定式化し、 微分方程式と箙の表現との対応のこの問題への応用についてお話ししたい。. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 10月11日 October 11(Fri) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. 山川大亮氏 (東京工業大学). Title. 有理型接続のモジュライ空間と箙多様体. Abstract. 　 本講演では，Boalchによってアナウンスされ， 廣惠一希氏（城西大学）との共同研究によって証明が得られた， ある種の（射影直線上定義された）有理型接続のモジュライ空間が 特別な場合に箙多様体と複素シンプレクティック多様体として 同型になるという結果を紹介する． これは対数型接続に関するCrawley-Boeveyの結果を拡張するものであり， 現れる箙はより複雑なものになる． また時間が許せば，関連する話題として モノドロミー保存変形のWeyl群対称性についても触れる．. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 10月4日 October 4(Fri) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. Anne Moreau氏 (Universit\\'{e} de Poitiers). Title. The arc space of spherical varieties and motivic integration. Abstract. 　 In this talk, we will be interested in the motivic integral over the arc space of a complex Q-Gorensein spherical G-variety X where G is a reductive connected group. We gave a formula for the stringy E-function of X in term of its colored fan, which generalizes that of Batyrev for the toric case. As an application, we obtain a smoothness criterion for locally factorial horospherical varieties and we conjecture that this criterion still holds for any locally factorial spherical variety. All this is based on joint works with Victor Batyrev. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 7月26日 July 26(Fri) 15:30--17:00, 2013. Room. Rims 402, Kyoto University. Speaker. Toshiro Kuwabara氏 (Higher School of Economics). Title. BRST cohomologies for rational Cherednik algebras. Abstract. 　 Quantization of Kleinian singularities can be realized as two different quantum Hamiltonian reductions. They are known as rational Cherednik algebras (symplectic reflection algebras) and finite W-algebras. Losev showed that these two quantizations are isomorphic by using realization of these algebras in terms of deformation-quantization. One can define a cohomology theory associated with Hamiltonian reduction, which is known as BRST cohomologies. In this talk, we see that higher BRST cohomologies corresponding to the rational Cherednik algebras do not vanish, while ones corresponding to the finite W-algebras vanish. Moreover, we see that the higher BRST cohomologies can be determined explicitly. To determine the higher cohomologies, we use the realization as deformation-quantization algebras and affinity properties of these sheaves of deformation-quantization algebras. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 7月19日 July 19(Fri) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. 奥村将成氏(東大数理）. Title. 頂点代数と同変リー亜代数コホモロジー. Abstract. 　 Lian-Linshaw は，Malikov-Schechtman-Vaintrob が導入したカイラルドラーム 複体を可微分多様体の場合に詳しく取り扱った．その後 Lian-Linshaw-Song は， カイラルドラーム複体のある部分複体を用いて，リー群の作用を持つ可微分多様 体の同変コホモロジーの頂点代数類似物を構成した．本講演では，彼らが用いた 複体を，リー亜代数を用いて一般化し，同変リー亜代数コホモロジーの頂点代数 類似物を構成する．同変リー亜代数コホモロジーは，リー群の作用を持つ可微分 多様体の同変コホモロジーだけでなく，同変ポアソンコホモロジーも含む概念で あり，その頂点代数類似物も得られている．また，ある特別な複体を導入し，そ の性質を調べる．そこで得られた性質を用いて，変形リー亜代数と呼ばれる，リ ー環の多様体上の無限小作用を反映して得られるリー亜代数に対し計算を行う. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 6月21日 June 21(Fri) 14:45--16:15, 2013. [attention]お時間が変更になりました. Room. Rims 402, Kyoto University. Speaker. 井上玲氏(千葉大). Title. クラスター代数と結び目の複素体積. Abstract. 　 クラスター代数を用いて結び目の複素体積を定式化 する方法を紹介する。 特に、クラスター代数の特徴的な操作であるmutationを使ってR 作用素を構成し、 結び目補空間の理想四面体分割を調べる。 本講演は樋上和弘氏(九州大学)との共同研究に基づく。. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 5月31日 May 31(Fri) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. 森真樹氏(東大・数理). Title. 非整数次数のHecke代数の表現圏とセルラー構造. Abstract. 　 Hecke代数のスーパー化であるHecke-Cliffordスーパー代数のモジュラー既約表 現は、 BrundanとKleshchev及び土岡によりLie代数の圏化を用いて分類された。 一方講演者は一般化されたセルラー代数の構造を用いて、より具体的かつ初等的 に これらの既約表現を構成することに成功した。この証明の中で、 「非整数次数のHecke代数の表現圏」が有効に使われたのでそれを紹介したい。 これはDeligneが構成した、自然数とは限らないtに対する「t次対称群の表現圏」 の自然な拡張である。. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 5月22日(Wed)(May 22) 14:45--16:15, 2013. [attention]いつもとお時間が違います. Room. Rims 006, Kyoto University. [attention]いつもとお部屋が違います. Speaker. 中筋麻貴氏(上智大). Title. Hecke algebraとIwahori fixed vector. Abstract. 　 $p$-進群の不分岐主系列表現のintertwining 作用素の明示公式を得るために， Hecke algebraを用いる手法がRogawski(1985)によって報告されている． Rogawskiの目的はHecke algebraの既約表現の分類であったが，これは Casselman基底にIntertwining作用素を作用させたIwahori fixed vectorをもつ$p$-進群の既約表現の分類と同値である． 本講演では，Rogawskiのアイデアから得られるいくつかのIwahori部分群の特性関数に関する結果と， これらを応用することによって得られるIwahori fixed vectorの基底の明示公式について話す．. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 4月26日(Fri)(April 26) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. 岡田　崇 氏 (小山高専). Title. Quintic periods and stability conditions via homological mirror symmetry. Abstract. 　 For the Fermat quintic Calabi-Yau threefold and the theory of stability conditions [Bri07], there have been two natural aims. One is that we should define central charges of stability conditions by quintic periods involving Gamma functions [CdGP] without losing quantum corrections. The other is that for well-motivated stability conditions on a derived Fukaya-type category, stable objects should be Lagrangians. For the Fermat quintic Calabi-Yau threefold, we discuss these aims with the simplest homological mirror symmetry in [Oka09,FutUed], taking advantages of derived categories of representations of tensor products of quivers. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 4月19日(Fri)(April 19) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. 尾角正人氏 (大阪市立大学). Title. PBW bases of the nilpotent subalgebra of U_q(g) and quantized algebra of functions. Abstract. 　 For a finite-dimensional simple Lie algebra g, let U^+_q(g) be the positive part of the quantized universal enveloping algebra, and A_q(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U^+_q(g) coincides with the intertwiner between the irreducible A_q(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A_2 related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C_2. Our proof is based on a realization of U^+_q(g) in a quotient ring of A_q(g). This is a joint work with Atsuo Kuniba and Yasuhiko Yamada. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 2月22日(Fri)(February 22) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. Weiqiang Wang氏 (Virginia). Title. The structures of the centers of Hecke algebras. Abstract. 　 We will establish a precise connection between the centers of Hecke algebras associated to the symmetric groups and the ring of symmetric functions, quantizing the classical Frobenius characteristic map. This leads to an answer to a question of Lascoux on identification of several remarkable bases of the centers with bases of symmetric functions. In addition, we will describe a remarkable filtered algebra structure on such a center, which in its classical limit is intimately related to the cohomology ring of Hilbert scheme of points on the affine plane. This is based on joint work with Jinkui Wan (Beijing) and Andrew Francis (Sydney). Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 2月8日(Fri)(February 15) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. Alexander Premet氏 (Manchester). Title. Derived subalgebras of centralizers and completely prime primitive ideals. Abstract. 　 Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In my talk I am going to explain how to classify the primitive ideals I of U(g) whose associated variety occurs with multiplicity 1 in the associated cycle AC(I). The classification is based on the detailed study of the abelian quotients g_e\/[g_e, g_e] where g_e is the centraliser of a nilpotent element e in g. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 2月1日(Fri)(February 1) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. 柏原正樹氏 (京大・数理研). Title. Symmetric quiver Hecke algebras and R-matirce of quantum affine algebras. Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 1月18日(Fri)(January 18) 16:30--18:00, 2013. Room. Rims 402, Kyoto University. Speaker. Simon Goodwin氏 (Birmingham). Title. Representation theory of finite W-algebras. Abstract. 　 There has been a great deal of recent research interest in finite W-algebras motivated by important connection with primitive ideals of universal enveloping algebras and applications in mathematical physics. There have been significant breakthroughs in the rerpesentation theory of finite W-algebras due to the research of a variety of mathematicians. In this talk, we will give an overview of the representation theory of finite W-algebras focussing on W-algebras associated to classical Lie algebras (joint with J. Brown) and W-algebras associated to general linear Lie superalgebras (joint with J. Brown and J. Brundan). Organizers. 荒川知幸、加藤周、柳田伸太郎. Date. 12月7日(Fri)(December 7) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 名古屋創氏 (神戸大). Title. Quantum Painleve systems from hypergeometric integrals of Euler type. Abstract. 　 Euler 型積分表示から, Lie 環 sl_n に付随する超幾何積分の系列を定め, この系列の満たす Schroedinger 方程式として, 量子パンルヴェ系 （モノドロミー保存変形の量子化）が得られるという予想について話す. この話は, 共形場理論の Knizhnik-Zamolodchikov 方程式が超幾何積分の 系列を解として持つが、逆に超幾何積分の系列から KZ 方程式が復元する という話(Schechtman-Varchenko, Looijenga)の類似である. KZ 方程式と Euler 型積分表示から得られる量子パンルヴェ系との 関係は, Lie 環が sl_2 のときには分かっている(N). 講演では, 例として Gauss の超幾何から量子 PVI を導出する方法, 量子 PVI と KZ 方程式や BPZ 方程式との関係について話した後, 一般の場合の予想と具体例について話す. Organizers. 荒川知幸、加藤周、柳田伸太郎. Attention. 今回11\/30は2コマ講演でいつもより開始時間が早くなっていますのでご注意ください. Date. 11月30日(Fri) (November 30) 14:45--16:15, 2012. Room. Room 402 of RIMS, Kyoto University. Speaker. Satoshi Naito (Tokyo Institute of Technology). Title. Quantum Lakshmibai-Seshadri paths and Ram-Yip's combinatorial formula for Macdonald polynomials. Abstract. 　 First, I will explain Ram-Yip's combinatorial formula for Macdonald polynomials, which is described in terms of the so-called alcove walks. Then, I will explain what happens in this formula when we specialize the parameter \"t\" to $0$. Finally, I will mention the relation between the specialized Macdonald polynomials above and the graded characters of tensor products of level-zero fundamental representations, which can be described in terms of quantum Lakshmibai-Seshadri paths. Date. 11月30日(Fri) (November 30) 16:30--18:00, 2012. Room. Room 402 of RIMS, Kyoto University. Speaker. Christian Kassel (Strasbourg). Title. Drinfeld twists and finite groups. Abstract. 　 Drinfeld twists were introduced by Drinfeld in his work on quasi-Hopf algebras. In joint work with Pierre Guillot (published in IRMN 10 (2010), 1894-1939), after observing that the invariant Drinfeld twists on a Hopf algebra form a group, we determine this group when the Hopf algebra is the algebra of a finite group. The proofs use quantum group techniques and Tannakian theory. Organizers. 荒川知幸、加藤周. Date. 11月16日(Fri)(November 16) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 直井克之氏(東大・Kavli IPMU). Title. BC型量子ループ代数のminimal affinizationについて. Abstract. 　 同じ最高ウェイトを持つ量子ループ代数$U_q(L\\mathfrak{g})$の有限次元加群の 中で、（適当な半順序に関して）極小なものをminimal affinizationと呼ぶ。. 量子ループ代数の有限次元加群について、その有限型部分代 数$U_q(\\mathfrak{g})$加群構造を調べることは、古典極限を調べることに帰着 される。. 本講演ではBC型のminimal affinizationについて、その古典極限を少し変形して 得られる次数付き極限を用いることで、 指標公式などが得られることについて述べたいと思う。. Organizers. 荒川知幸、加藤周. Date. 6月29日(Fri)(June 29) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 廣惠一希氏(京大・数研). Title. 行列型１階線形常微分方程式と箙の表現. Abstract. 　 W. Crawley-Boeveyは確定特異点型１階行列線形常微分方程式と箙の表現との対 応を発見し、 方程式を既約に実現しうる留数行列の共約類を決定した（加法的Deligne- Simpson問題）。. さらにこの対応は不確定特異点を1点のみ許した方程式に P. Boalchによって拡張されている（正確には高々極の位数３の不分岐不確定特 異点）。. 本講演では極の位数が一般の不分岐不確定特異点を任意個許した微分方程式を考え、 これと箙の表現との対応を与える。 またこの応用として、方程式の既約性と箙のルート系との関係を述べたい。. Organizers. 荒川知幸、加藤周. Date. 6月22日(Fri)(June 22) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 水川裕司氏(防衛大). Title. 環積が作用する確率モデルと多変数 Krawtchouk 多項式. Abstract. 　 多変数 Krawtchouk 多項式は R. C. Griffiths によって1971年に定義された離 散直交多項式である． 表現論的な枠組では，この直交多項式は複素鏡映群のなすゲルファントペアの帯 球関数として 得られることが知られている．また，最近特殊関数論サイドからの研究とし て，Grunbaum と Rahman によって直交性を与える必要十分条件が考えられたり，Ilievにより差分方程式 へのLie環論からのアプローチなどが行われている．. 本公演では，この多項式によって記述されるいくつかの確率モデルの表現論的な 解釈を試みる． 具体的には Rahamn と Hoare によって考案された Poker dice ゲームや，古典 的な Ehrenfest の拡散モデルを取り上げたい．. Organizers. 荒川知幸、加藤周. Date. 6月8日(Fri)(June 8) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 柳田伸太郎氏(RIMS). Title. On Hall algebra of complexes. Abstract. 　 The topic of my talk is the Hall algebra of 2-periodic complexes, which is recently introduced by T. Bridgeland. I will discuss its properties and relation to auto-equivalences of derived category. I shall also mention the connection of this theory and the notion of stabilities. Organizers. 荒川知幸、加藤周. Date. 6月1日(Fri)(June 1) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 和田 堅太郎氏(信州). Title. cyclotomic q-Schur 代数の Drinfeld 型の表示について. Abstract. 　 cyclotomic q-Schur 代数は，Ariki-Koike 代数の quasi-hereditary cover の１つである。講演では，cyclotomic q-Schur 代数 （加算無限個の)生成元とその間の関係式を与え，（可能な限り) その表現論へ の応用をお話ししたいと思います。. Organizers. 荒川知幸、加藤周. Date. 4月20日(Fri)(April 20) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. Masaki Kashiwara氏 (RIMS). Title. Parameters of quiver Hecke algebras. Abstract. 　 Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis. In this talk I will show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case. Organizers. 荒川知幸、加藤周. Date. 4月13日(Fri)(April 13) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. Sarah Scherotzke氏 (Univ. Bonn). Title. Linear recurrence relations for cluster variables. Abstract. 　 In a recent paper by Asssem Reitenaure and Smith, frieze sequences were associated to acyclic quiver. They are a natural generalization of the Coxeter-Convey frieze pattern. Using categorification of cluster algebras, we show that frieze sequences associated to acyclic quivers satisfy linear recurrence relations if and only if the quiver is an affine quiver. Organizers. 荒川知幸、加藤周. Date. 2月10日(Fri)(Februrary 10) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 小寺諒介氏(東大・数理). Title. Self-extensions and prime factorizations for simple $U_q(L\\mathfrak{sl}_2)$-modules. Abstract. 　 In the category of finite-dimensional modules over a quantum loop algebra, it often occurs that a simple module is factorized into a tensor product of smaller simple modules. Hence the notion of prime simple module (a simple module which does not admit a nontrivial factorization) was introduced and prime factorizations of simple modules have been studied. For the quantum loop algebra of $\\mathfrak{sl}_2$, prime simple modules exactly coincide with the evaluation modules. Chari-Moura-Young proposed in a recent paper arXiv:1112.6376 a study of prime simple modules from a homological point of view. They conjectured that a simple module is prime if and only if it has the one-dimensional self-extension group, and proved it in the $\\mathfrak{sl}_2$ case. In this talk, I will review Chari-Moura-Young's paper and explain that my previous result on nonself-extensions can be applied to refining their result. It establishes a relation between the dimension of the self-extension group and the number of factors in the prime factorization for a simple $U_q(L\\mathfrak{sl}_2)$-module. Organizers. 荒川知幸、加藤周. Date. 2月3日(Fri)(Februrary 3) 16:30--18:00, 2012. Room. Rims 402, Kyoto University. Speaker. 疋田辰之氏(京大・理). Title. A型アファインシュプリンガーファイバーとdiagonal coinvariantの組み合わせ論. Abstract. 　 Diagonal coinvariant ringのbigraded Frobenius seriesに関してHaglund, Haiman, Loehr, Remmel, Ulyanovはそれを記述する組み合わせ論的公式を予想した。講演ではこの公式がA 型のアファインシュプリンガーファイバーのホモロジーを用いることで幾何的に 現れることを説明したい。. Organizers. 荒川知幸、加藤周. Date. 12月16日(Fri)(December 16) 15:00--18:00, 2011. Room. Rims 402, Kyoto University. Speaker. 木村嘉之氏(京大・数研). Title. 次数付き箙多様体と量子クラスター代数. Abstract. 　 クラスター代数の正値性予想とは、任意のクラスター変数の任意の 種におけるローラン展開(クラスター展開)に関する正 値性に関する予想である。 今回、非輪状の箙を種として含むようなクラスター代数における正 値性予想が解決された。 中島啓氏によるbipartite quiverに付随する(量子) クラスター代数のモノイダル圏論化の証明の手法に従い、 「非輪状型(acyclic quiver)に適合した」次数付き(アフィン) 箙多様体上の同変偏屈層の量子表現環による量子クラ スター代数の実現により証明される。 講演では、(クラスター代数のモノイダル圏論化の中心的な問 題である)(量子)クラスター単項式が``双対標準基底'' に含まれることを説明したい。 本研究は、パリ第７大学のFan Qin氏との共同研究に基づく。. Organizers. 荒川知幸、加藤周. Date. 12月2日(金)(December 2) 16:30--18:00, 2011. Room. Rims 402, Kyoto University. Speaker. 中島啓氏(京大・数研). Title. Coproduct on Yangian. Abstract. 　 Consider the Yangian $Y$ associated with an affine Lie algebra, which is not of type $A^{(1)}_1$ nor $A^{(2)}_2$. We define a coproduct $\\Delta$, which takes value in a certain completion of $Y\\otimes Y$. This is a work in progress, with Nicolas Guay. Organizers. 荒川知幸、加藤周. Date. 11月18日(金)(November 18) 16:30--18:00, 2011. Room. Rims 402, Kyoto University. Speaker. 加藤周氏(京大・理). Title. グリーン関数のホモロジー論的側面について. Abstract. 　 グリーン関数は簡約群の冪単指標の一般化として得られる直交関数系であり、二 つの複素鏡映群の既約指標の組を添字にもつ。この話では一般の複素鏡映群(と 良い付加データ)に対してコストカ系と呼ぶ(一般には存在するかどうか分からな いが、一旦存在するとよい性質を満たす)加群の族を導入し、(標数が良い時に) 簡約群の冪単指標に付随するグリーン関数は常にコストカ系として実現される事 を説明する。この事は特に任意の簡約群に付随するグリーン関数が直既約加群の 次数付き指標という解釈を許す事を意味する。この解釈を用いると小ワイル群を BC型として等しくする任意の簡約群の冪単指標に付随するグリーン関数達がどの ように互いに移りあうか等も見る事ができる。. Organizers. 荒川知幸、加藤周. Date. 11月11日(金)(November 11) 16:30--18:00, 2011. Room. Rims 402, Kyoto University. Speaker. 阿部紀行氏(北大・創成研究機構). Title. 対称空間のコンパクト化によるJacquet加群の実現. Abstract. 　 Beilinson-Bernstein対応は，実半単純Lie群の表現の旗多様 体上における幾何学 的な実現を与えるが，一方で対称空間上での幾何学的な実現も与え る．三枝洋一 氏との共同研究により，対称空間の境界に向けてうまく極限をとる と，Jacqeut 加群が実現されることがわかったので，それについて話をする．. Organizers. 荒川知幸、加藤周. Attention. いつもとは時間・場所が異なりますのでご注意ください。. Date. 10月31日(月), 11月1日(火),2日(水),4日(金). (October 31, November 1, 2, 4 (4days)) 13:15--14:15, 2011. Room. Room 475 of Research Bldg. No.2 , Kyoto University. http:\/\/www.kyoto-u.ac.jp\/en\/access\/campus\/main.htm. Speaker. Ivan Losev氏 (Northeastern). Title. Finite W-algebras. Abstract. 　 Finite W-algebras are associative algebras that can be thought as generalizations of universal enveloping algebras of semisimple Lie algebras. Each W-algebra is constructed from a pair of a semisimple Lie algebra and its nilpotent orbit. These algebras first appeared in the work of Kostant in the late 70's in a special case. In the whole generality they were defined by Premet in the beginning of 2000's. In my lectures I am going to emphasize connections between W-algebras and universal enveloping algebras. I will start by giving two definitions of W-algebras, one due to Premet and one due to myself. Then I will introduce various functors between the representation categories for W-algebras on one side and semisimple Lie algebras on the other side. Using this functors I will explain an interplay between primitive ideals in the universal enveloping algebras and irreducible finite dimensional modules for W-algebras. Organizers. 荒川知幸、加藤周. Date. 10月28日(金) (October 28) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 大島芳樹氏(東大・数理). Title. コホモロジカル誘導の局所化. Abstract. 　 コホモロジカル誘導は(g,K)-加群に対して代数的に定義さ れ、半単純リー群の離 散系列表現、主系列表現(のHarish-Chandra加 群)、 Zuckerman加群などを生成する。 Borel部分代数の1次元表現からの誘導の場合、誘導された表 現は旗多様体上のD 加群を用いて実現できることが、Hecht, Milicic, Schmid, Wolfにより示されている。 講演では、より一般の表現からの誘導についてこの結果を拡張する ことを考える。. Organizers. 荒川知幸、加藤周. Date. 10月21日(金) (October 21) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 市野篤史氏(京大・理). Title. 形式次数とテータ対応. Abstract. 　 局所テータ対応とは, (p進体上の)古典群の(複素数係数)表現から, 別の古典群 の表現を, ある種の分岐則を用いて構成する方法である. この構成の下での, 表現の解析的不変量の振る舞いについて述べ, これを表現の 分類(局所Langlands対応)を用いて, 数論的に解釈する. この講演は, Wee Teck Gan氏との共同研究に基づく. Organizers. 荒川知幸、加藤周. Date. 10月14日(金) (October 14) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. Leonid Rybnikov氏　(HSE). Title. Quantization of Quasimaps' Spaces (joint work with M. Finkelberg). Abstract. 　 Quasimaps' space $Z_d$ (also known as Drinfeld's Zastava space) is a remarkable compactification of the space of based degree d maps from the projective line to the flag variety of type A. The space $Z_d$ has a natural Poisson structure, which goes back to Atiyah and Hitchin. We describe the Quasimaps' space as some quiver variety, and define the Atiyah-Hitchin Poisson structure in quiver terms. This gives a natural way to quantize this Poisson structure. The quantization of the coordinate ring of the Quasimaps' space turns to be some natural subquotient of the Yangian of type A. I will also discuss some generalization of this result to the BCD types. Organizers. 荒川知幸、加藤周. Date. 7月22日(金) (July 22) 15:00--16:30, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 土岡俊介 (Shunsuke Tsuchioka)氏　(IPMU). Title. Quiver Hecke superalgebras. Abstract. 　 We introduce two families of superalgebras $R_n$ and $RC_n$ which are weakly Morita superequivalent each other. The quiver Hecke superalgebra $R_n$ is a generalization of the Khovanov-Lauda-Rouquier algebras. We show that, after suitable specialization and completion, the quiver Hecke-Clifford superalgebra $RC_n$ is isomorphic to the affine Hecke-Clifford superalgebras and its rational degeneration. This is a joint work with Seok-Jin Kang and Masaki Kashiwara. Organizers. 荒川知幸、加藤周. Date. 7月8日(金) (July 8) 15:00--16:30, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 荒川 知幸氏 (RIMS). Title. Localization of affine W-algebras at the critical level. Abstract. 　 We localize the simple affine W-algebras at the critical level on the infinite jet schemes of Slodowy varieties, by introduction a chiral analogue of the Kashiwara-Rouquier deformation quantization algebra. This is a joint work with Toshiro Kuwabara and Fyodor Malikov. Organizers. 荒川知幸、加藤周. Attention. 今回はいつもより開始時間が早めになり、. 講演時間も長くなっていますのでご注意ください. Date. 7月1日(金) (July 1) 13:30--15:00, 15:15--16:45, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 山川大亮氏（神戸大）. Title. Fourier-Laplace変換とKatz-Deligne-Arinkinの定理 その2. Abstract. 　 6\/17に同セミナーにて行った講演では， 射影直線上の有理型接続に対するFourier-Laplace変換を ある条件下で接続の係数行列に対する変換として初等的に書き下し， それを利用したKatz-Deligne-Arinkinの定理（genericな場合） の別証明を駆け足で紹介した．. 今回はもう少し踏み込んで，Fourier-Laplace変換が誘導する 接続の（ナイーヴな）モジュライ空間の間のシンプレクティック同型写像や， モジュライ空間と箙多様体との関係等，幾何学的側面も紹介したい． なお，前回の講演の内容は仮定せず話をする．. Organizers. 荒川知幸、加藤周、榎本直也、木村嘉之. Date. 6月24日(金) (June 24) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 笹木集夢氏（東海大）. Title. An application of the classification of visible linear actions to nilpotent orbits. Abstract. 　 小林俊行氏によって導入された複素多様体における可視的作用という概念は， 無重複表現の統一理論において重要な役割を果たすことが近年明らかになってき ている． 作用が線型な場合は可視的作用の分類は与えられ， それはKacやBenson-Ratcliff, Leahyによるmultiplicity-free作用の分類に一致する． さらに最近，線型な可視的作用の分類を用いることで， 複素リー環の冪零軌道における(線型でない)作用が 可視的であることと冪零軌道がsphericalであることが同値であることが分かった．. 本講演では，線型な可視的作用の研究結果について概説した後， この結果を冪零軌道における可視的作用の研究に応用する様子を解説する予定で ある．. Organizers. 荒川知幸、加藤周、榎本直也、木村嘉之. Attention. !今回6\/17は2コマ講演でいつもより開始時間が早めですのでご注意ください! 第1報から時間が変更になりました. また,1コマ目はいつもと部屋が異なります. Date. 6月17日(金) (June 17) 11:30--13:00, 2011. Room. Room 204 of RIMS, Kyoto University. Speaker. 山川大亮氏（神戸大）. Title. Fourier-Laplace変換とKatz-Deligne-Arinkinの定理. Abstract. 　 Katz-Deligne-Arinkinの定理は，rigidと呼ばれる性質を満たす射影直線上の有 理型接続を， 座標変換・階数1の有理型接続によるテンソル積・Fourier-Laplace変換，の3つ の操作を有限回繰り返す事によって， 必ず階数1の有理型接続にする事ができると主張する． これはもともとKatzによって有理型接続が確定特異点のみを持つ場合に示され （Katzの定理）， 後にDeligneとArinkinによって不確定特異点の場合に拡張された．. この講演では，Dettweiler-ReiterによるKatzの定理の別証明が， 自然な形で不確定特異点の場合（ただしgenericな仮定を課す）に拡張される事 を紹介する．. 特に，Dettweiler-Reiterの議論では明示されていないFourier-Laplace変換の具 体的な記述に焦点を当てる．. Date. 6月17日(金) (June 17) 14:45--16:15, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 廣恵一希氏（東京大）. Title. 線形常微分方程式とルート系. Abstract. 　 直既約な箙の表現とルートとの対応を示すKacの定理は箙の表現論では基本的な 定理だが， Fuchs型の線形常微分方程式においてこのKacの定理の不思議なアナロジーが知ら れている． すなわち微分方程式に対してあるルート系とルート格子の元が決まり，方程式の 既約性とルートの条件が対応する． さらに微分方程式のある種のモジュライ空間の次元がルートの長さによって決定 されるというのである． 本講演ではこの対応をFuchs型でない場合にも拡張することを目標として今まで に得られている結果を報告する．. Organizers. 荒川知幸、加藤周、榎本直也、木村嘉之. Date. 6月10日(金) (June 10) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 鈴木咲衣氏 （京大数理研）. Title. On the universal sl_2 invariant of bottom tangles. Abstract. 　 Jones多項式の発見を機に量子トポロジーと呼ばれる分野が誕生し，現在までに急速 な発展を遂げてきた． まず単純リー環gの量子群とその表現を用いて定義される絡み目の「量子g不変 量」，次に量子群のみを用いて定義され，表現に関して量子g不変量に普遍性を 持つ「普遍量子g不変量」，さらにはリー環の関係式を用いて定義され，量子群 に関して量子g不変量に普遍性をもつ「Kontsevich不変量」の構成である．そし てその３つの段階の不変量に対応して，絡み目に沿った手術の理論を経由して３ 次元多様体の不変量が構成される．. この講演では「底タングル」を用いた普遍量子sl_2不変量の研究の枠組みを説明し， 講演者の結果を簡単に紹介する．. Organizers. 荒川知幸、加藤周、榎本直也、木村嘉之. Attention. !今回5\/27は2コマ講演でいつもより開始時間が早めですのでご注意ください! Date. 5月27日(金) (May 27) 14:30--16:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 榎本直也氏（京大理）. Title. 曲面の写像類群に付随するJohnson余核のSp-加群構造について（佐藤隆夫 氏（東京理科大）との共同研究）. Abstract. 　 境界を1つ持つ種数gの向き付けられたコンパクトリーマン面$\\Sigma_{g,1}$の写像類 群$M_{g,1}$は、 $H_1(\\Sigma_{g,1},Z)$に自明に作用するTorelli部分群とよばれる部分群を持ち、そ の商はSp(2g,Z)と同型になる。 Torelli部分群のJohnson filtrationの次数商を自由Lie代数の微分代数へ移すJohnson準同型は、 Torelli部分群の構造を調べるための重要な道具のひとつであり，言わばTorelli群の 近似物を記述していると考えられる。 その後、森田茂之氏によってJohnson準同型の像がある次数付き部分Lie代数$\\mf{h}_ {g,1}$に埋め込まれることが示され、 その余核の次数k-部分（k：奇数）にSp-既約表現[k]が含まれることがわかった。こ れは森田障害と呼ばれている。 本講演では、自由群の自己同型群におけるJohnson準同型とその余核のGL（あるいはS p）-構造をもとに、 写像類群のJohnson余核に現れる既約成分のあるクラスについて紹介し、 具体的に、Sp-既約表現[1^k]が次数k-部分（$k \\equiv 1 (mod 4）$に重複度1で現れることを述べたい。. Date. 5月27日(金) (May 27) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 高尾尚武氏（京大数理研）. Title. Johnson準同型と外Galois表現. Abstract. 　 Johnson準同型の定義は一般の双曲型Riemann面に対して一般化されます。 その余核の次数k-部分（$k \\equiv 2 (mod 4）$には有理数体上の絶対Galois群が「現れる」ことが、 織田孝幸氏によって予想され、現在ほぼ解決されています。 Deligne-伊原予想の解決を合わせると、より精確な定量的な評価も可能になりま した。. 講演では、織田予想を中心に、Johnson準同型の余核をめぐる数論側からの進展につ いて紹介します。. Organizers. 荒川知幸、加藤周、榎本直也、木村嘉之. Date. 5月20日(金) (May 20) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 飯島和人氏 (名古屋大学). Title. A COMPARISON OF q-DECOMPOSITION NUMBERS IN THE q-DEFORMED FOCK SPACES OF HIGHER LEVELS. Abstract. 　 The q-deformed Fock spaces of higher levels were introduced by Jimbo-Misra-Miwa-Okado. The q-decomposition matrix is a transition matrix from the standard basis to the canonical basis defined by Uglov in the q-deformed Fock space. In this talk, we show that parts of q-decomposition matrices of level $\\ell$ coincides with that of level $\\ell$ － 1 under certain conditions of multicharge. (This talk will be given in Japanese, but the slides in English.). Organizers. 荒川知幸、加藤周、榎本直也、木村嘉之. Date. 5月13日(金) (May 13) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. Anatol Kirillov (RIMS)氏**. Title. Saga of Dunkl elements. Abstract. 　 Dunkl operators has been introduced in the middle of 80's by Charles Dunkl to solve certain problems in the theory of orthogonal polynomials. Later it was observed a close connection of Dunkl operators with the theory integrable systems, as well as construction of different kinds of generalizations. Connection of (truncated) Dunkl operators with the coinvariant algebra of a finite Coxeter group has been observed by C. Dunkl and clarified by Y. Bazlov. In my talk I introduce a certain quadratic algebra and a distinguish set of mutually commuting elements in it (Dunkl elements). It appears that different kind of Dunkl operators (rational, trigonometric, elliptic, multiplicative,...) are images of the Dunkl elements in the corresponding representation of the quadratic algebra in question. The main goal of my talk is to relate the algebra generated by Dunkl elements with generalized cohomology theories of complete flag varieties of type A . Applications to other fields of Mathematics will be presented. My talk partly is based on joint works with T.Maeno. Organizers. 荒川知幸、加藤 周、榎本直也. Date. 4月22日(金) (April 22) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 直井克之(Katsuyuki Naoi)氏 (東大数理). Title. Weyl module, Demazure moduleとfundamental representationのテンソル積のcrystal basisとの関係について. Abstract. 　 Weyl moduleは生成元と関係式によって定義されるcurrent algebra（単純リー代 数と多項式環のテンソル積で定義される無限次元リー代数）の有限次元表現である。 一方fundamental representationはcrystal basisを持つ重要なquantum affine algebraの有限次元表現である。. これらは一見それほど関係がなさそうであるが、Demazure加群およびその crystalにおける対応物(Demazure crystal)を用いることで二つの対象の間に非 常に強い関係があることを示すことができる。今回の公演ではこの結果について 紹介する。. また、上で述べた結果とX=M予想との関係についても紹介する。ここでX=M予想と は、1-dimensional sumと呼ばれる有限crystalのテンソル積から定義される多項 式とfermionic formulaと呼ばれる多項式が一致する、という予想である。. Organizers. 荒川知幸、加藤 周. Date. 4月15日(金) (April 15) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 有家雄介(Yusuke Arike)氏(大阪大). Title. Pseudo-trace functions for orbifold models associated with symplectic fermions. Abstract. 　 頂点作用素代数Vに付随する楕円曲線上の一点関数は, Vの元に上半平面上の正則 関数を対応させるある性質をもつ写像です. 頂点作用素代数VがC_2有限かつ有理 的(加群の圏が半単純性であること)であるとき, 一点関数の空間はVの単純加群 上のtrace functionと呼ばれるもので張られることがZhuにより示されています. またtrace functionの真空ベクトルでの値は加群の指標と一致します. 有理性を仮定しない頂点作用素代数に付随する一点関数の空間は, pseudo-trace functionと呼ばれるもので生成されることが示されています.しかしpseudo- trace functionの定義は高次のZhu代数と呼ばれる結合代数を用いるもので, 具 体的な例に対して一点関数を構成することは非常に困難です. 本講演ではまず, Zhu代数を用いずにpseudo-trace functionを定義する方法を解 説し,次にsymplectic fermionic頂点作用素超代数のeven partとして得られる頂 点作用素代数の直既約加群上のpseudo-trace functionを構成します.さらに得ら れたpseudo-trace functionの真空ベクトルでの値を調べます. その結果として, 一点関数の真空での値で定義される一般化された指標の空間と一点関数の空間の 次元が異なる場合があることを示します. Organizers. 荒川知幸、加藤 周. Date. 4月8日(金) (April 8) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 柏原正樹(Masaki Kashiwara)氏 (RIMS). Title. Cyclotomic quiver Hecke algebras and categorifications of highest weight module. Organizers. 荒川知幸、加藤 周. Date. 4月1日(金) (April 1) 16:30--18:00, 2011. Room. Room 402 of RIMS, Kyoto University. Speaker. 加藤周(Syu Kato)氏 (Kyoto). Title. Harish-Chandra bimodules for quantized Slodowy slices (survey). Abstract. 　 Ginzburgの同名の論文[Represent. Theory 13 (2009), 236-271]の サーベイをします。. Organizers. 荒川知幸、加藤 周. Date. 3月25日(金) (March 25), 16:30--18:00, 2011. Room. Room 204 of RIMS, Kyoto University. Speaker. 和田堅太郎(Kentaro Wada)氏 (RIMS). Title. Induction and Restriction functors for cyclotomic q-Schur algebras. Abstract. 　 異なるランクの cyclotomic q-Schur 代数の有限次元加群のなす圏の間に関する 誘導, 制限関手を定義し，その性質を調べます。特に，Weyl 加群 (既約加群) に対する誘導，制限関手の性質を調べることによって (部分的にはまだ予想の段 階ですが) higher level の Fock 空間との関係を考えます。これは，Ariki- Koike 代数に対する LLT-有木理論の quasi-hereditary cover 版の１部分とな ります。　さらに，Rouquier による quasi-hereditary cover の同値定理に よって，cyclotomic q-Schur 代数の有限次元加群のなす圏が, 有理 Cherednik 代数の圏 O と同値になる場合には，今回の誘導，制限関手は，有理 Cherednik 代数に対する Bezrukavnikov-Etingof の誘導，制限関手と (cover の同値を通 じて) 同値な関手となり，Shan, Gordon-Martino によって得られている Fock 空間の圏化の \"ドミナント版\" が得られることになります。. 講演では，これらの概略を話してみようと思います。. Organizers. 荒川知幸、加藤 周. Date. March 16 (Wed) 14:00--（３時間程度）. Room. Room 204 of RIMS, Kyoto University. Speaker. 中島 啓氏(RIMS). Title. Maulik-Okounkovの理論の紹介 - 応用として、AGT予想の証明. Organizers. 荒川知幸、加藤 周. Date. February 16 (Wed), 14:45-16:15, 2011. Room. Room 204 of RIMS, Kyoto University. Speaker. Kari Vilonen (Northwestern University). Title. Langlands duality for real groups. Abstract. 　 In the case of real groups Langlands duality acquires a symmetry as both sides can be interpreted as (derived) categories of representations. We explain this duality and its proof in the case of quasi-split groups. The result was also known as the Soergel conjecture. This is joint work with R. Bezrukavnikov. Organizers. 荒川知幸、加藤 周. Date. Feb. 9 (Wed) 14:45--16:15, 2011. Room. Room 204 of RIMS, Kyoto University. Speaker. 岡田 聡一 (Soichi Okada)氏　(名大多元). Title. Two-parameter deformation of multivariate hook product formulae. Abstract. 　 The hook product formula due to Frame, Robinson, and Thrall gives the number of standard tableaux of a given shape, which is equal to the dimension of the irreducible representation of the symmetric group. The FRS hook product formula is obtained from Gansner's multivariate hook product formula for the trace generating function of reverse plane partitions. In this talk, we give another proof and a (q,t)-deformation of Gansner's formula by using operator calculus on the ring of symmetric functions. Also we present a conjectural deformation of Peterson-Proctor's hook product formula for P-partitions on d-complete posets. Organizers. 荒川知幸、加藤 周. Date. Jan. 28 (Fri) 14:30--16:15, 2011. Room. Room 204 of RIMS, Kyoto University. Speaker. 柳田伸太郎氏(神戸大・理). Title. A finite analog of the AGT relation (survey). Abstract. 　 Braverman-Feigin-Finkelberg-Rybnikovの仕事 (arXiv:1008.3655)のサーベイをしま す.特にshifted YangianのGelfand-Tsetlin基底を詳しく扱います. Organizers. 荒川知幸、加藤 周. Date. Jan. 28 (Fri) 16:45--18:00, 2011. Room. Room 204 of RIMS, Kyoto University. Speaker. 柳田伸太郎氏(神戸大・理). Title. Ding-Iohara algebra and K-theoretic AGT conjecture. Abstract. 　 K理論的AGT予想はインスタントンのモジュライ空間の同変K理論に変形Virasoro代数( ないし変形W代数)が作用することを示唆します.講演ではDing-Iohara代数という量子 アフィン環の類事物とK理論的AGT予想(及び通常のAGT予想)の関係について述べます. Organizers. 荒川知幸、加藤 周. Date. Jan. 19 (Wed) 14:45--16:15, 2011. Room. Room 204 of RIMS, Kyoto University. Speaker. Seok-Jin Kang (Seoul National University). Title. Quantum queer superalgebra and crystal bases. Abstract. 　 We will give a brief survey of recent developments in the crystal basis theory for the quantum queer superalgebra $U_q(q(n))$. The odd Kashiwara operators and 'queer' tensor product rule will be introduced. We will also discuss their combinatorial realization in terms of semistandard decomposition tableaux. Organizers. 荒川知幸、加藤 周. Date. Jan. 12 (Wed) 15:00--16:00, 2011. Jan. 13 (Thu) 11:00--12:00,13:30--14:30, 2011. Room. Jan.12:Room 204 of RIMS, Kyoto University. Jan.13:Room 110 of RIMS, Kyoto University. Speaker. Ian M. Musson（The University of Wisconsin-Milwaukee）. Title. Lie Superalgebras and Enveloping Algebras. Abstract. 　 I will give 3 lectures mainly about enveloping algebras of classical simple Lie superalgebras. The second of these will concern the center of the enveloping algebra, and the third will contain material about primitive ideals. The first lecture will contain some background material. Organizers. 荒川知幸、加藤 周. Date. December 10 (Fri), 16:30--18:00, 2010. Room. Room 204 of RIMS, Kyoto University. Speaker. 山田泰彦氏(神戸大・理). Title. 共形場理論、モノドロミー保存変形とAGT予想. Abstract. 　 ４次元超対称ゲージ理論の分配関数と２次元共形場理論の相関関数が一致するという 興味深い現象(Alday-Gaiotto-Tachikawa予想）について、モノドロミー保存変形の 量子化の観点から考察する。応用として、ゲージ理論のある分配関数が満たすと期待 される微分方程式を定式化する。. Organizers. 荒川知幸、加藤 周. Date. December 1 (Wed), 14:45--16:15, 2010. December 8 (Wed), 14:45--16:15, 2010. Room. Room 204 of RIMS, Kyoto University. Speaker. 土岡俊介氏(京大・数理研). Title. Shifted Yangians and finite W-algebras (survey). Abstract. 　 Brundan-Kleschevの同名の論文(Adv. Math. 200 (2006), 136--195, arXiv:math\/040 7012)の内容を二回に分けてサーベイします。. Organizers. 荒川知幸、加藤 周. Date. November 22 (Mon), 16:30--18:00, 2010. Room. Room 111 of RIMS, Kyoto University. Speaker. Ben Webster (Oregon). Title. Hypertoric (and other) categories O. (joint w\/ Braden, Licata and Proudfoot). Abstract. 　 The category O defined by Bernstein, Bernstein and Gelfand has been an active area of representation theory for over 30 years now. I'll explain how this construction is a special case of a more general picture, and explain how things like Koszulity, cells, the localization theorem, and the action by shuffling and twisting functors generalize. A particularly well-developed special case is hypertoric category O, which arises from torus invariant differential operators on a vector space. In this case, we can find an analogue of almost any theorem about the Lie theoretic category O, though sometimes with subtle and interesting changes. Perhaps most interestingly, results on Koszul duality in this picture point the way toward a notion of duality between certain symplectic singularities, as I will explain. Organizers. 荒川知幸、加藤 周. Date. November 17 (Wed), 14:45--16:15, 2010. Room. Room 204 of RIMS, Kyoto University. Speaker. Scott Carnahan (IPMU). Title. Borcherds products in monstrous moonshine. Abstract. 　 During the 1980s, Koike, Norton, and Zagier independently found an infinite product expansion for the difference of two modular j-functions on a product of half planes. Borcherds showed that this product identity is the Weyl denominator formula for an infinite dimensional Lie algebra that has an action of the monster simple group by automorphisms, and used this action to prove the monstrous moonshine conjectures. I will describe a more general construction that yields an infinite product identity and an infinite dimensional Lie algebra for each element of the monster group. The above objects then arise as the special cases assigned to the identity element. Time permitting, I will attempt to describe a connection to conformal field theory. Organizers. 荒川知幸、加藤 周. Date. October 27 (Wed), 14:45--16:15, 2010. Room. Room 204 of RIMS, Kyoto University. Speaker. 中島啓 (RIMS). Title. Instanton and W-algebras (review). Abstract. 　 物理学者の Alday-Gaiotto-Tachikawa の最近の研究から、R. 4. 上のインスタントンのモジュライ空間の同変交叉ホモロジーに W-代数の表現の構造が入ることが期待されている。これについて、概観する。. Organizers. 荒川知幸、加藤 周. Date. 2010年10月13日(水曜日) 14:50--16:20. Room. 数理解析研究所204号室. Speaker. 木村嘉之氏(京大・数理研). Title. 量子ベキ単部分群と双対標準基底（Quantum Unipotent Subgroup and dual canonical basis）. Abstract. 　 (量子)クラスター代数構造は、Berenstein-Fomin-Zelevinskyらによって、双対標準 基底の乗法的性質の研究のため導入された組み合わせ的な構造である。 Weyl群の元wに付随したKac-Moody群の冪単部分群N(w)の座標環のクラスター代数構造 は、Berenstein-Fomin-Zelevinskyらにより予想され、前射影多元環(preprojective algebra)に関するGeiss-Leclerc-Schr\\\"{o}erらによる研究により、双対準標準基底( dual semicanonical basis)との整合性が知られている。 Geiss-Leclerc-Schr\\\"{o}erらによる結果の量子化として、冪単部分群N(w)の座標環 の量子変形O_q[N(w)]には量子クラスター構造が存在し、双対標準基底との整合性が 予想される(量子化予想)。. 本講演では、問題の背景と量子化予想とそのいくつかの帰結について述べ、量子化予 想の準備と言えるいくつかの結果について紹介する。 これらは、Calderoの有限ADE型の結果の一般化である。. Organizers. 荒川知幸、加藤 周. Date & Room. 2010年10月6日(水) 14:45--16:15 数理解析研究所204号室. 2010年10月8日(金) 16:30--18:00 理学部３号館数学教室109号室. Speaker. 荒川知幸氏. Title. W-algebras and their representations. Abstract. 　 本講演では（アフィン)W代数とその表現論についてのreviewを行いたいと思います。. Organizer. 加藤 周. 過去の表現論セミナー Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Access Joint Usage\/Research. 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Since its establishment, RIMS has served the purpose of accelerating research in the mathematical sciences. In 2010, RIMS was certified as a Joint Usage \/ Research Center by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), providing researchers in the mathematical sciences with various opportunities for undertaking joint research projects. In November 2018, RIMS was certified as an International Joint Usage \/ Research Center by MEXT, making it an international hub for other research institutes inside and outside Japan, with the aim of leading international joint research activities and strengthening the research ability of domestic researchers. RIMS implements five types of joint research activity as an International Joint Usage \/ Research Center, which are provided in combination with any of the following programs. RIMS Workshops (Type A,B,C) RIMS Symposia RIMS Satellite Seminars RIMS Review Seminars RIMS Long-Term Researchers(only domestic). and. \"RIMS Research Projects\" In RIMS Research Projects, leading researchers from abroad stay at RIMS to conduct research activities for a medium- to long-term duration. These programs aim to promote international collaborative research activities and the development of young researchers. RIMS publicly solicits proposals for joint research activities from inside and outside Japan once a year or throughout the year depending on the types of program. Joint research activities proposed are examined and approved by the Technical Committee and the Advisory Board. RIMS may designate some urgent and important projects as special projects to expedite their efficient implementation. Operational Organizations. The Advisory Board discusses important matters relating to the implementation of Joint Research Activities, at the request of the Director of RIMS. The Technical Committee examines publicly solicited proposals for Joint Research Activities, at the request of the Director. Both committees are comprised of RIMS faculty, persons who were appointed by the Director from within Kyoto University, and internal and external specialists recommended by specialty committees of the Science Council of Japan and appointed by the RIMS Director. A majority of these committee members are specialists from outside the University. RIMS also appoints International Advisors who have a broad view and high insight into mathematics and the mathematical sciences. They provide advice concerning the organization of RIMS. List of Advisory Board Members List of Technical Committee Members List of International Advisory Board Members Overview of International Joint Usage\/Research Center Outline of the Joint Research Activities & Application forms Operational Organizations Advisory Board Technical Committee International Advisory Board ←　BACK TO THE TOP Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Access. TOP 　　. Access 　　Access. Access. Address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 JAPAN. Transportation to Kyoto Map of Kyoto City Map of North Campus A. Transportation to Kyoto. Kyoto has no airport! A1. From Kansai International Airport (KIX) to JR Kyoto station A1-1. It is convenient to take a. Haruka limited express train bound for Kyoto. Trains depart every 30 minutes in the morning and evening, with an average journey time of approximately 75 minutes. They depart once every hour in the period 10:00--16:00. You can see a \"Timetable & Route Finder\". here . The fare is 3,430 yen, including a seat reservation (subject to change). After arriving at Kyoto station, proceed along the platform in the direction of your train, and you will find Karasuma central exit on the left-hand side. A1-2. In the daytime, it may be more convenient to take an airport bus\/coach from KIX to JR Kyoto Station. A timetable is. here. A2. From Tokyo--Narita (or Tokyo-Haneda) International Airport to JR Kyoto station by trains. By rail:. Narita Express links Narita airport to Tokyo station, with an average journey time of approximately 60 minutes. Haneda and Tokyo station are linked by trains. From Tokyo station, Shinkansen Nozomi and Hikari superexpress trains frequently depart to Kyoto. The average journey time is 2.5 and 3 hours, respectively. We recommend you buy all train tickets at Narita airport in advance. The total fare for a one-way ticket will be 16,000 -- 18,000 yen (subject to change), including seat reservation. You can choose a non-smoking seat. Shinkansen platforms are located on the third floor of Kyoto station. After going down to the second floor, go out the Shinkansen central exit and walk to the north along a corridor. Then you will find bus stops on the right-hand side. A3. From Tokyo--Narita (or Tokyo-Haneda) airport through Itami Airport to JR Kyoto Station. You may take a domestic flight from Narita to Itami Airport, which is another airport in Osaka. An airport bus\/coach takes you from Itami Airport to JR Kyoto Station in 50--60 minutes. The bus fare is 1,340 yen. B. From JR Kyoto station to RIMS. (map of Kyoto). There are three ways to get to RIMS from JR Kyoto station. B1. By bus:. By bus: There is a bus terminal in front of Karasuma central exit. Take city bus. No. 17. at bus stop A2, then get off at Kitashirakawa. The fare is 230 yen, and the average journey time is around 30 min. B2. By taxi:. There is a taxi stand on the right-hand side of Karasuma central exit. Please show. this slip to the taxi driver. No tip is necessary. It costs 2,200--2,500 yen. B3. By subway:. There is a subway Kyoto station (station no. K11) in the basement of JR Kyoto station. Take a subway bound for Kokusai Kaikan and get off at Imadegawa station (station no. K06). There is a bus stop on the left-hand side of exit no. 3. Take bus. No.203. bound for Ginkakuji\/Kinrinshako and get off at Kitashirakawa. Access \/ Contact. Access Accommodation Contact ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor About RIMS. TOP 　　. About RIMS 　　Message from the Director. Message from the Director. RIMS Director. Kaoru Ono. The Research Institute for Mathematical Sciences (RIMS) is an institute that conducts comprehensive research in mathematics and the mathematical sciences. Since being established in 1963, RIMS has been recognized as one of the world's leading research institutes in these fields. Mathematics is, of course, an important basic discipline located at the deepest part of human knowledge and forms the basis of science. It is a mystery of mathematics, and at the same time fundamentally accounts for its existence, that the structure of the real world can be captured profoundly and beautifully by pure mathematical thoughts. In modern times, mathematics plays a central role in supporting science and technology. Especially since the beginning of the 21st century, vast changes to society and industry have led to mathematics gaining significant importance in creating new ideas, giving these foundations, and leading them to realization. 1. . However, mathematics has not solely been developed to meet the demands of applied fields. Indeed, looking back in time, one can find many examples where the mathematics that is widely applied for practical purposes today was created purely from a mathematical perspective. For example, Galois theory, which was introduced about 200 years ago to address the solvability of equations, supports the codes and ciphers of modern network communications, and non-Euclidean geometry is applied to present-day GPS technology through the theory of general relativity. Moreover, as a newer example, the theory of stochastic analysis, which was founded by Dr. Kiyosi Itô, a director of RIMS in the mid-20th century, is widely used in mathematical finance. Through such integration with applications, the cultivation of a deep and broad generality of mathematics has a strong power to change the world we live in. On the other hand, applications of mathematics have played a role in evoking new possibilities inherent inside mathematics and motivating further progress in mathematics from diverse viewpoints. The pursuit of mathematics based purely on mathematical interests and its applications to science can thus be compared to the roots and branches of a tree. One cannot be established without the other, and the mutual support of the two leads to the global development of mathematics. Since its inception, RIMS has had the mission to comprehensively promote fundamental research in mathematics and research on the applications of mathematics to science. To this end, there are three pillars to the activities of RIMS. The first is research by faculty members. Over the half-century of RIMS existence, this has led to many globally-recognized achievements. Currently, RIMS has around 38 faculty members and is organized into three major divisions (Fundamental Mathematics, Infinite Analysis, and Applied Mathematics), as well as a Computer Laboratory, making it a flexible organization. In addition, RIMS has established the “Center for Research in Next-Generation Geometry” to strengthen research in this area, the “Center for Research Interaction in Mathematical Sciences” to promote international research collaboration, and the “Liaison Center in Mathematics” to promote collaborative research with researchers from a wide range of scientific fields. It has always been a fundamental policy of RIMS to maintain an excellent research environment where members can concentrate on research, with space being given so that researchers may pursue their mathematical endeavors above all else, and the research time of younger members in particular being highly valued. The second pillar is the implementation of joint usage projects aimed at contributing to the research of a wide range of mathematicians and mathematical scientists. Since its foundation, RIMS has served as a national joint research institute, and since 2010 it has been certified as a Joint Usage\/Research Center (JU\/RC). At present, RIMS hosts around 80 RIMS Workshops annually, with a combined total of more than 4000 participants (of which about 400 are from abroad). RIMS Kokyuroku, a collection of reports on these projects, has now exceeded 2000 volumes, and the number of accesses of these exceeds 1.6 million annually (a little fewer than 40% being from abroad). Since November 2018, RIMS has been certified as an “International Joint Usage\/Research Center”, which has enabled it to expand its joint usage projects and make international open calls. In this direction, RIMS is promoting the internationalization of its core projects, setting up international collaborative research support mechanisms and starting large-scale international collaborative “RIMS Research Projects”. In particular, for RIMS Research Projects, we select specific research themes, offer positions of project fellows, and pursue research progress through collaborations with leading international researchers, who are invited to make mid- to long-term stays at RIMS. Moreover, by establishing new projects that support individual-scale international collaborations, recruiting projects all year round, and establishing a framework to support young researchers, RIMS will continue to play a role in supporting research in mathematics and the mathematical sciences both within Japan and beyond. The third pillar is the education of graduate students. 2. . So as to foster young researchers, RIMS has had a graduate school since 1970. Currently, it accepts 10 Master students and 10 PhD students every year. Under the guidance of outstanding faculty members, students can learn contemporary mathematics in an environment where leading researchers visit from all over the world and collaborate on a daily basis. In this way, RIMS has produced many excellent young researchers. In the activities of RIMS, the three pillars described above are not separate, rather they are a trinity that strengthens each other, which is extremely important for the development of the institute. From a practical perspective, one of the current challenges of RIMS is to secure building space that fits the variety of its activities in a cohesive manner, and efforts to realize this will continue in the future. The role of the institute is to provide a high-level research environment where researchers can discuss with each other actively and think deeply about mathematics. Through its activities, RIMS is making efforts to contribute to further developments in mathematics and the mathematical sciences in the 21st century. Footnotes. In the 5th Science and Technology Basic Plan (FY2016-2020), mathematical science is positioned as a \"transverse science and technology that supports basic technology\". For graduate education, we are conducting the KTGU (Kyoto Top Global University) project in collaboration with the Department of Mathematics, Graduate School of Science. About RIMS. Message from the Director Members Organization Graduate Education and Conferment of Degrees ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Number Theory \/ Arithmetic Geometry Seminar. TOP 　　. Seminars 　　Number Theory \/ Arithmetic Geometry Seminar. Number Theory \/ Arithmetic Geometry Seminar. Title. Galois action on the etale fundamental group of the Fermat curve. Date. October 7 (Mon), 2019, 16:30-17:30. Room. Room 110, RIMS. Speaker. Pries, Rachel (Colorado State University). Abstract. If $X$ is a curve defined over a number field $K$, then we are motivated to understand the action of the absolute Galois group of $K$ on the etale fundamental group of $X$. When $X$ is the Fermat curve of degree $p$ and $K$ is the cyclotomic field generated by a pth root of unity, Anderson proved several theorems about this action on the etale homology of $X$. In earlier work, we made Anderson's results more explicit, when the homology has coefficients modulo $p$. More recently, we use a cup product in cohomology to determine the action on the lower central series of the fundamental group of the Fermat curve with coefficients modulo $p$. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren. Organizer. Akio Tamagawa, Naotake Takao (RIMS, Kyoto Univ.). Date. March 11 and 12, 2019. Place. Room 110, RIMS Monday, March 11, 16:00-17:00. Speaker. Rasmussen, Christopher (Wesleyan University). Title. Improvements on Bounds for Heavenly Abelian Varieties. Abstract. For a rational prime $\\ell 2$, a natural question arising from the study of the arithmetic of certain Galois representations attached to fundamental groups is to consider the collection of abelian varieties over a fixed number field $K$ whose $\\ell$-power torsion generates an extension of $K(\\mu_{\\ell})$ which is both pro-$\\ell$ and unramified away from $\\ell$. (We say such an abelian variety is heavenly at $\\ell$.) Conjecturally, the set of $K$-isomorphism classes of heavenly varieties of fixed dimension $g$ is finite -- even when the prime $\\ell$ is permitted to vary. In past joint work with Tamagawa, we have provided a proof of the Conjecture under GRH, and settled the Conjecture positively in a number of cases. Such results imply the existence of a bound $L = L(K,g)$ such that $\\ell. Tuesday, March 12 10:30-11:30. Speaker. Rasmussen, Christopher (Wesleyan University). Title. Algorithms for Solving S-Unit Equations. Abstract. Many enumerative problems in arithmetic geometry and number theory take the form ``Find, up to isomorphism, all objects $\\mathcal{O}$ with arithmetic property $\\mathcal{P}(S)$,'' where the data $S$ is a finite set of places of a fixed number field $K$. Such problems often encounter a common computational obstacle; namely the solution of the $S$-unit equation $x + y = 1$ over the ring of $S$-integers of $K$. In this talk, we describe joint work with several mathematicians to create an open-source implementation of such an algorithm for general choices of $K$ and $S$. (Emphasis of this discussion will be on the mathematical, rather than computational, aspects of the project.) In addition, we provide some new partial results to various problems, including Asymptotic Fermat's Last Theorem over certain cubic number fields. Tuesday, March 12 13:30-15:00. Speaker. Sakugawa, Kenji (RIMS). Title. On Jannsen's conjecture for modular forms. Abstract. Let M be a pure motive over Q and let M_p denotes its p-adic etale realization for each prime number p. In 1987's paper, Jannsen proposed a conjecture about a range of integers r such that the second Galois cohomology of M(r)_p vanishes for any prime number p. Here, M(r) denotes the rth Tate twist of M. When M is an Artin motive, Jannsen's conjecture had already essentially proven a half by Soule aroud early 1980's. In this talk, we consider the case when M is a motive associated to elliptic modular forms. I will explain my ongoing work about an approach to the conjecture used the p-weighted fundamental groups of modular curves. (Sakugawa's lecture is given in Japanese.). ** We plan to have a meal for welcoming Prof. Rasmussen after the seminar of March 11. If you join the meal, could you inform Takao (e-mail: takao_at_kurims.kyoto-u.ac.jp) by March 4? Organizer. Akio Tamagawa, Naotake Takao (RIMS, Kyoto Univ.). Date and Place. Lecture 1, October 1st (Mon), 10:00-12:00, Room 110. Lecture 2, October 1st (Mon), 16:30-18:30, Room 110. Lecture 3, October 5th (Fri), 13:30-15:30, Room 110. Lecture 4, October 5th (Fri), 16:30-18:30, Room 110. Lecture 5, October 12th (Fri), 16:30-18:30, Room 111, 2018. Speaker. Collas, Benjamin (University of Bayreuth). Title. Homotopical Arithmetic Geometry of Stacks. Abstract. Moduli spaces of curves possess properties which make them ideal spaces where to concretely study fundamental abstract theories of arithmetic geometry: they give geometric Galois representations that can be computed explicitly, furnish examples of anabelian spaces, and in genus zero generate the category of mixed Tate motives. They also possess a dual nature, being either considered as schemes or algebraic stacks. The goal of this series of talks is to provide a basic introduction to these aspects by covering various fundamental geometric and arithmetic properties. It is intended for graduate students in algebraic geometry and non-specialists researchers. Elementary notions will be either recalled or illustrated with pictures\/examples. I- Algebraic & Deligne-Mumford Stacks (lectures 1 and 2) Taking the functor of points for schemes as initial motivation, we introduce the notion of stacks as lax functors in groupoids with descent conditions and show how to recover Laumon-Moret-Bailly's original definition. We present how the Artin and Deligne-Mumford algebraic versions -- that admit topological coverings by schemes -- allow to ``push'' algebraic geometry properties in this context. Keywords: diagrams of groupoids, Grothendieck topology and etale\/ffpf\/smooth morphisms, examples of global quotient and inertia stacks. II - Moduli Problems & Moduli Spaces of Curves (lecture 3) We present how the scheme-stack structures and the geometry of curves lead to two solutions for building classifying spaces. Having introduced the notion of functor of moduli, we present Gieseker and Deligne-Mumford constructions of the moduli space of curves: the former follows Mumford G.I.T-theory and give a quasi-projective scheme, the latter produces a smooth algebraic Deligne-Mumford global stack with a nice stable compactification. Keywords: Hilbert scheme, explicit examples in low genus, stable compactification, formal neighbourhood. III - Fundamental Group & Arithmetic (lecture 4) We follow Grothendieck construction of the etale fundamental group that leads to Geometric Galois actions of the absolute Galois group of rational on the geometric fundamental group of moduli stack of curves. We adapt this approach in the case of Deligne-Mumford stacks and show how it leads to a divisorial and a stack arithmetic of the spaces. Following the seminal work of Ihara, Matsumoto and Nakamura, we present explicit results and properties of the former, then recent similar results in the case of cyclic inertia for the latter. Keywords: properties of the etale fundamental group, explicit computations in low dimensions, tangential base points and representations. IV - Motivic Theory for Moduli Stack of Curves (lecture 5) We present recent progress on an ongoing project on the construction of a category of motives for the moduli stacks of curves, whose main property is to reflect the arithmetic properties of the cyclic stack inertia. Having recalled briefly what the category of mixed motives should be, we first present Morel-Voevodski stable\/unstable motivic homotopy categories, then how their homotopical-simplicial approach is well adapted to our goal. Keywords: Quillen model category, Artin-Mazur etale topological type, Mixed Tate motives and loop space. Organizer. Akio Tamagawa, Naotake Takao (RIMS, Kyoto Univ.). Emiliano Ambrosi's successive lectures. Title. p-adic cohomology theories with a view toward l-adic applications. Date. Lecture 1, August 24th (Thu), 13:30-15:00. Lecture 2, August 24th (Thu), 15:30-17:00. Lecture 3, August 28th (Mon), 13:30-15:00. Lecture 4, August 28th (Mon), 15:30-17:00. Lecture 5, September 1st (Fri), 13:30-15:00. Lecture 6, September 1st (Fri), 15:30-17:00, 2017. Room. Room 006, RIMS. Speaker. Emiliano Ambrosi 氏 (Ecole Polytechinique). Organizer. Akio Tamagawa (RIMS, Kyoto Univ.). Chieh-Yu Chang 氏連続講義. Title. On Hilbert's seventh problem and transcendence theory. Date. October 9th (Wed), 23rd (Wed) and 28th (Mon), 2013, 10:30-12:00. [change in the date]. October 9th (Wed) 10:30-12:00,. October 23rd (Wed) 10:30-12:00, 13:30-15:00, 2013. Room. Room 006, RIMS. Speaker. Chieh-Yu Chang 氏 (National Tsing Hua University). Abstract. Hilbert's seventh problem is about the linear independence question of two logarithms of algebraic numbers, which was solved by Gelfond and Schneider in the 1930s. Later on, it was generalized to several logarithms of algebraic numbers by Baker in the 1960s and generalized to general abelian logarithms of algebraic points by Wuestholz in the 1980s. This phenomenon can be also asked for multiple zeta values, but it is still open. In the first talk, we will give a survey on the classical theory and report recent progress on the parallel questions for function fields in positive characteristic. Current methods and tools of transcendence theory using t-motives will be discussed in the second and third talks. Organizer. Akio Tamagawa (RIMS, Kyoto Univ.). Mini-Workshop on Number Theory \/ Arithmetic Geometry. Date. Thursday, January 31, 2013. Room. Room 206, RIMS, Kyoto University. 10:00 -- 10:20. Arata Minamide (RIMS, M2). Elementary Anabelian Properties of Graphs. 10:30 -- 10:50. Yang Yu (RIMS, M2). Arithmetic Fundamental Groups and Geometry of Curves over a Discrete Valuation Ring. 11:00 -- 11:20. Takeshi Okada (RIMS, M2). On Finiteness of Twists of Abelian Varieties. 13:00 -- 13:30. Yu Iijima (RIMS, D1). Galois Action on Mapping Class Groups. 13:45 -- 14:45. Chia-Fu Yu (Academia Sinica). Density of the Ordinary Locus in the Hilbert-Siegel Moduli Spaces. Organizer. Akio Tamagawa (RIMS, Kyoto Univ.). Title. Resolution of nonsingularities for Mumford curves. Date. December 15 (Thu), 2011, 14:15-15:45. Room. Room 206, RIMS. Speaker. Emmanuel Lepage 氏 (Institut Mathematique de Jussieu). Abstract. Let $X$ be a hyperbolic curve over $\\overline Q_p$. I am interested in the following property: for every semistable model $\\mathcal X$ of $X$ and every closed point $x$ of the special fiber there exists a finite covering $Y$ of $X$ such that the minimal semistable model $\\mathcal Y$ of $Y$ above $\\mathcal X$ has a vertical component above $x$. I will try to explain why hyperbolic Mumford curves satisfy this property. I will give anabelian appplications of this to the tempered fundamental group. Mini-Workshop ``Rational Points on Modular Curves and Shimura Curves''. Date. Monday, October 26th, 2009. Room. Room 307, Research Building No. 4, Kyoto University（京都大学 総合研究４号館 ３０７号室） 13:30--14:30. Keisuke Arai (Univ. Tokyo). Points on $X_0^+(N)$ over quadratic fields. (joint work with F. Momose). Abstract: Momose (1987) studied the rational points on the modular curve $X_0^+(N)$ for a composite number $N$. He showed that the rational points on $X_0^+(N)$ consist of cusps and CM points under certain conditions on a prime divisor $p$ of $N$. But $p=37$ was excluded. For $37$ is peculiar because $X_0(37)$ is a hyperelliptic curve and $w_{37}$ is not the hyperelliptic involution. We show that the rational points on $X_0^+(37M)$ consist of cusps and CM points. We also show that the $K$-rational points on $X_0^+(N)$ consist of cusps and CM points for a quadratic field $K$ under certain conditions (both $p=37$ and $p\\ne 37$ allowed). 14:45--15:45. Fumio Sairaiji (Hiroshima International Univ.). Takuya Yamauchi (Osaka Prefecture Univ.). On rational torsion points of central $\\mathbb{Q}$-curves. Abstract: Let $E$ be a central $\\mathbb{Q}$-curve over a polyquadratic field $k$. In this talk we give an upper bound for prime divisors $p$ of the order of the $k$-rational torsion subgroup of $E$. For example, $p$ is less than or equal to 13, if the scalar restriction of $E$ from $k$ to $\\mathbb{Q}$ is of GL$_2$-type with real multiplications. Our result is a generalization of the result of Mazur on elliptic curves over $\\mathbb{Q}$, and it is a precision of the upper bounds of Merel and Oesterl\\'{e}. 16:00--17:00. Pierre Parent (Univ. Bordeaux 1). Rational points on Shimura curves. Abstract: For $B$ a rational quaternion algebra, the Shimura curve associated with $B$ (or more precisely its quotient by certain Atkin-Lehner involutions) is a moduli space, in a certain sense, for abelian surfaces with potential multiplication by $B$. Proving that those curves almost never have rational points would therefore allow a small step towards the conjecture, attributed to Coleman and Mazur, which predicts the scarcity of endomorphism algebras for abelian varieties of GL$_2$-type over $\\mathbb{Q}$. We will present a method to study such rational points, developped by A. Yafaev and myself, and recently improved by F. Gillibert. Organizers. Akio Tamagawa (RIMS, Kyoto Univ.). Marco Boggi 氏連続講演. Title. ``Profinite curve complexes and the congruence subgroup problem for the mapping class group''. Date. September 27th (Thu) and 28th (Fri), 2007. Room. at Room 206, RIMS, Kyoto University. 27th (Thu). 10:00--12:00 Boggi. lunch. 14:00--16:00 Boggi. 16:00-- free discussion. dinner. 28th (Fri). 10:00--12:00 Boggi. lunch. 14:00--16:00 free discussion. Abstract. 別紙 のとおり. 連絡先：. 松本眞（広島大）、玉川安騎男、望月新一（京大数理研）. Title. Arithmetic from Geometry on Elliptic Curves. Date. June 2 (Fri), 2006, 16:30-17:30. Room. Room 202, RIMS. Speaker. Christopher Rasmussen (Rice Univ.). Abstract. One of the philosophies of arithmetic geometry made popular by Grothendieck was the notion that the structure of the absolute Galois group of $\\mathbf{Q}$, could be determined from geometric (or even combinatoric) data. In a related vein, one finds that the arithmetic properties of a curve are sometimes determined by its geometry. Specifically, the structure of a curve as a cover of the projective line can have arithmetic consequences for the Jacobian of the curve. We will discuss this situation in the case of elliptic curves, where this connection between arithmetic and geometry can be seen very clearly. Title. Arithmetic Algebraic Geometry Lecture（集中講義）. Date. May 8 (Mon)- May 19, 2006. Room. こちらをご覧ください. Speaker. 加藤和也（京大 理）. Abstract. Weil が 1949 年に提出した Weil 予想は、 1970 年代のはじめに最終的に証明されるまで、 代数幾何学の大きな進展の原動力となり、 とくに Grothendieck が Weil 予想の証明をめざして導入した エタールコホモロジーは、現在の整数論の重要な道具となった。 その経緯をふり返りつつ、エタールコホモロジーの解説をおこなう。. Comments. 詳細はこちら. Title. Algebraic dynamical systems (preperiodic points, Mahler measures, equidistribution of small points). Date. May 1 (Mon), 2006, 16:30-. Room. Room 202, RIMS. Speaker. Lucien Szpiro (City Univ. New York). Abstract. Reference: (available at http:\/\/math.gc.cuny.edu\/faculty\/szpiro\/People_Faculty_Szpiro.html) --Joint papers with T. Tucker. --Joint paper with E. Ullmo and S. Zhang. Comments. Date. April 10 (Mon), 2006, 14:00-17:00. Room. Room 202, RIMS. (14:00-15:15). Speaker. Michel Matignon (Univ. Bordeaux 1\/Chuo Univ.). Title. Wild monodromy groups and automorphisms groups of curves. (16:00-16:45). Speaker. Barry Green (Univ. Stellenbosch\/Chuo Univ.). Title. Selected results on liftings of Galois covers of smooth curves from char. p to char. 0. Mini-Workshop ``Arithmetic Geometry of Covers of Curves and Related Topics''. Date. September 12 (Mon), 13(Tue), 2005. Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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Associate Professor. IRIE, Kei. Associate Professor. ISHIMOTO, Kenta. Associate Professor. ISONO, Yusuke. Associate Professor. KAJINO, Naotaka. Associate Professor. KAWAI, Toshiya. Associate Professor. KAWAKITA, Masayuki. Associate Professor. KAWAMURA, Akitoshi. Associate Professor. KISHIMOTO, Nobu. Lecturer. KOBAYASHI, Yusuke. Associate Professor. MAKINO, Kazuhisa. Professor. MOCHIZUKI, Shinichi. Professor. MOCHIZUKI, Takuro. Professor. NAKANISHI, Kenji. Professor. NAKAYAMA, Noboru. Associate Professor. NAMIKAWA, Yoshinori. Professor. OHKITANI, Koji. Professor. OHTSUKI, Tomotada. Professor. ONO, Kaoru. Professor. OZAWA, Narutaka. Professor. TAKEHIRO, Shin-ichi. Associate Professor. TAMAGAWA, Akio. Professor. TAN, Fucheng. Lecturer. TERUI, Kazushige. Associate Professor. YAMASHITA, Go. Lecturer. FUJITA, Ryo. Assistant Professor. HELMKE, Stefan. Assistant Professor. HIKITA, Tatsuyuki. Assistant Professor. ISHIKAWA, Katsumi. Assistant Professor. ISHIKAWA, Suguru. Assistant Professor. KOSHIKAWA, Teruhisa. Assistant Professor. MUROYA, Koko. Assistant Professor. OOURA, Takuya. Assistant Professor. TSUJIMURA, Shota. Assistant Professor. YAMASHITA, Mayuko Assistant Professor. MINAMIDE, Arata. Project Assistant Professor. SAWADA, Koichiro. Project Assistant Professor. YANG, Yu. Project Assistant Professor. HIRUTA, Yoshiki. Project Assistant Professor. Visiting Professor. SAIDI, Mohamed. LESCHE, Katrin. BOROS, Endre. PEDIT, Franz. FESENKO, Ivan. ANDRES, Sebastian. Center for Research Interaction in Mathematical Sciences. Project professor. MUKAI, Shigeru. YAMADA, Michio. SAITO, Morihiko. Project associate professor. KAWANOUE, Hiraku. Liaison Center in Mathematics. Project professor. KUMAGAI, Takashi. KOKUBU, Hiroshi. SAKAJO, Takashi. MATSUDA, Fumihiko. HIRAOKA, Yasuaki. Center for Research in Next-Generation Geometry. Project professor. KASHIWARA, Masaki. MORI, Shigefumi. NAKAJIMA, Hiraku. Project assistant professor. MINAMIDE, Arata. SAWADA, Koichiro. YANG, Yu. Members. 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Computer Facilities. Japanese Enter your login name, your password, select 'English', click [Login] button. * This link directly leads you to the target page. * If you receive the message \"ERROR You must be logged in to access this page.\", please check if cookies are enabled. Web Mail Usage Web Mail Guide for off-RIMS Wireless Network Key * Wireless Network Name is. kurims. . Changing Password Forwarding Email (includes spam filtering and vacation responder settings). RIMS Adjusting Spam Filter RIMS Shared Documents Guide for off-RIMS (no login version) Scheduled Workshops I. RIMS AV Systems II. RIMS Unix System -. Creating home page -. How to print from PC * see also. Printer Name IV. KUINS-III guide for opened (non-staff) space IMPORTANT ANNOUNCEMENT. Please adopt the. Loss\/Theft Prevention. measures. [Limited on-Campus] Preventing Information Leakage (Jan.18.2022). EOL Softwares. Adobe Acrobat 2017 and Reader 2017. Jun.6.2022 EOL. Windows 8.1. Jan.10.2023 EOL. Office 2013. Apr.11.2023 EOL. Apple OS X v10.14 Mojave or older. **EXPAND** **CLOSE** The security update is not released for a long time, so it seems to be unsupported, it has high security risks. Please upgrade. Finding the OS X version ,. How to upgrade Adobe Flash Player, Office 2010, Microsoft Office for Mac 2016, Adobe Acrobat DC(2015) and Reader DC(2015), Adobe Acrobat XI and Reader XI, Microsoft Office 2007, Microsoft Office for Mac 2011, Microsoft Word Viewer, Windows 7, Vista, Windows XP, Office 2003 and Windows Server 2003. EOL. Do not connect to the network. See also:. this page Oracle Java new licensing. : see. The security update including Java in Oracle Inc = please uninstall, use an alternative as needed (e.g. AdoptOpenJDK ,. jdk.java.net ). (Aug.26.2019). The license is updated. (Oct.1.2021). For RIMS members: Please take an. information security e-Learning once a academic year. (Apr.18.2016). \"PLEASE BEWARE OF PHISHING SITE\/E-MAIL\" :. * The administrator. never. lead you to the page that requires entering your password on the ground of system maintenance. * The administrator. never. ask your password. About P2P file sharing software:. Please refrain from using the P2P file sharing software such as Winny, Share, BitTorrent, etc(*1). *1: Winny, Share, BitTorrent, BitComet, LimeWire, Cabos, WinMX, PerfectDark, Kazaa, eMule, Edonkey, Gnutella, etc. SECURITY. :. Be careful: USB media virus:. A lot of PC were infected by the computer virus via USB media (USB flash memory, USB hard disk, SD card, and so on ). Please keep your computer secure, please do not insert unreliable USB media into your PC, please do not insert your USB media into unreliable PC. If you get your RIMS account after Apr.1.2007, spam filtering is enbaled by default. For the details:. here ANNOUNCEMENT. SECURITY. :. Execute. Microsoft Update . (May.11.2022). Vulnerabilities Information ... SECURITY. :. Update your Google Chrome. (May.11.2022). Chrome Releases: Stable Channel Update ... SECURITY. :. Update your Firefox. (May.11.2022). Vulnerabilities Information ... SECURITY. :. Update your Thunderbird. (May.11.2022). Vulnerabilities Information... SECURITY. :. Update your Adobe Reader, Adobe Acrobat. Vulnerabilities Information ... (Apr.13.2021). PC. :. KUINS-III NAT settings will be updated on Jan.26(Wed). (Jan.21.2022). RIMS WIRELESS. The network key of 'kurims' has been updated at 9:00am. (Jan.4.2022). UNIX. : The Mathematica on Ubuntu is upgraded to the version 12.0.0 (from 11.0.1). (Nov.30.2021). see also:. Softwares List (in RIMS only) UNIX. : Maple are upgraded to the version 2020 (from 2015). (Nov.18.2021). see also:. Softwares List (in RIMS only) SECURITY. :. Update your Apple software. (Sep.15.2021). Vulnerabilities Information ... macOS ,. Safari ,. iOS, iPadOS UNIX. : changing the anti-spam system on Aug.26, see. here for the details. (Aug.25.2021). RECOVERED. Because of the maintenance, the kurims wireless and the RIMS NAT service had been unavailable for about 10 minutes at 8:30am on May.18(Tue.). 9:20--11:43, the problem that the kurims wireless clients may not be able to obtain an IP address occured. (May.18.2021). RECOVERED. The network printers had been unavailable May.15 because of the trouble. (May.17.2021). UNIX. : The RIMS mail server starts to scan the messages. see also. Action for SPAM (Apr.27.2021). UNIX. :. The RIMS https web pages will be available (Mar.29.2021). How to set the longer password (temporarily) (Mar.30.2021). The server maintenance since Feb.26 have been finished. Thank you for your cooperation. (Mar.5.2021). RIMS WIRELESS. The network key of 'kurims' has been updated at 9:00am. (Jan.4.2021). UNIX. :. Rejecting on the mail server : updated. (Dec.22.2020). RECOVERED. The network had been unavailable Nov.8 8:35pm--Nov.9 9:00am because of the trouble. (. information page ) (Nov.9.2020). RECOVERED. The network had been unavailable 4:50pm--8:30pm on Oct.22 because of the trouble. (. information page ) (Oct.23.2020). SECURITY. :. Update your. Adobe Flash Player. Vulnerabilities Information ... (Oct.14.2020). ref. Adobe Flash Player:Version Information RECOVERED. The e-mail delivery to off-RIMS had been delayed for about 90 minutes since around 10am on Sep.23. Now it is recovered. (Sep.24.2020). RIMS WIRELESS. The network key of 'kurims' has been updated at 9:00am. (Jan.6.2020). UNIX. : MATLAB was updated (to 2019b). (Dec.17.2019). see also. SOFTWARES LIST (rimsonly) SECURITY. :. Symantec Endpoint Protection may cause frequent crashes and failures of the definition update,. see the. this page . Please manually download and update the \"IPS\" definition,. see. this page . (Oct.21.2019). SECURITY. :. Oracle Java new licensing. : see. The security update including Java in Oracle Inc = please uninstall, use an alternative as needed (e.g. AdoptOpenJDK ,. jdk.java.net ). (Aug.26.2019). UNIX. : Intel\/PGI Compiler on Ubuntu servers were updated. (Jun.27.2019). see also. SOFTWARES LIST (rimsonly) I. RIMS AV Systems. USB camera, USB speakerphone [New] (May.17.2019). RECOVERED. Because of the RIMS server trouble, mail and some services had been unavailable since yesterday Jan.7 6:44pm. We restarted the server this morning, the services are available now. We are sorry for causing so much trouble. (Jan.8.2019). RIMS WIRELESS. The network key of 'kurims' has been updated at 9:00am. (Jan.7.2019). UNIX. :. Mathematica is unavailable because of the upgrade of Ubuntu. Please use math7 as a temporary. Because of the system trouble,. sapporo,. kagome will be available in this week. (Nov.7.2018). Google English is available by. this link . (Sep.10.2018). The black out occurred during around 9:10am--9:45am on Aug.20(Mon). Please let us know any problems. (Aug.20.2018.). RECOVERED. The black out occurred during around 9:10am--9:45am on Aug.20(Mon). Please let us know any problems. (Aug.20.2018.). On Aug.6.(Mon.), we will introduce the SPF (Sender Policy Framework) into the kurims e-mail system because recently Gmail rejects our e-mail less frequently and due to the prevention for email spoofing. (Aug.2.2018). RECOVERED. : The imaps server had been unstable. By the workaround, 2:13pm, it starts to recover. We're sorry for causing so much trouble. (Jun.21.2018.). RECOVERED. Jun.11(Mon.) 9:36am--9:39am: The wireless kurims in Bldg. No.15 is not available because of the software trouble, now it is available. (Jun.11.2018). RECOVERED. Because of the network trouble in Kyoto Univ., many information services are unavailable now. (May.24.2018.). RECOVERED. : The imaps server had been unstable. By the workaround, 3:40am, it starts to recover. We're sorry for causing so much trouble. (May.10.2018.). UNIX. :. RIMS UNIX System : The information of the common scanner at Bldg. No.15 is added. (Apr.25.2018). UNIX. :. Printers : common printers at Research Bldg. No.15 are added. (Apr.3.2018). RIMS WIRELESS. The network key of 'kurims' has been updated at 9:00am. (Jan.4.2018). UNIX. : The path and time of the backup of public is changed. see. backup of public (rimsonly) (Nov.29.2017). E-Journals. : RIMS proxy configuration ( http:\/\/wpad.kurims.kyoto-u.ac.jp\/proxy.pac ) was updated. (Nov.14.2017). UNIX. :. I want to use a newer TeX . (Jul.14.2017.). RECOVERED. : The ANNEX wireless has been unstable since Jan.1 9:00am, it is recovered at around 9:30am on Jan.4. (Jan.4.2017). RIMS WIRELESS. The network key of 'kurims' will be updated at 9:00am on Jan.4.2017. After Jan.4.2017, if your computer does not ask the network key, please. forget or delete. the 'kurims' wireless. (Dec.28.2016). RIMS WIRELESS. The wireless authentication \"PLEASE GO TO ...\" page will be shown automatically on iOS\/macos(since Dec.21), Android\/Windows(since Dec.27). * Please note some Android\/Windows does not show automatically. (Dec.28.2016). UNIX. : Intel\/PGI Compiler on Ubuntu servers were updated. (Nov.16.2016). see also. SOFTWARES LIST (rimsonly) UNIX. : The Mathematica on Ubuntu is upgraded to the version 11.0.1 (from 10.2). (Nov.7.2016). see also:. Softwares List (in RIMS only) UNIX. :. How to Export Plot in CLI math (rimsonly) (Oct.14.2016). UNIX. : \"RIMS Adjusting Spam Filter\" setting is available. (see. left side of this screen). You can change the threshold value and so on. see. SPAM filtering ... (rimsonly) (Oct.04.2016). UNIX. :. Emacs default version is changed to the version 22 for the new user. (Sep.27.2016). SECURITY. :. Uninstall your QuickTime for Windows. Apple ends support QuickTime for Windows. (Apr.21.2016). Vulnerabilities Information... RIMS WIRELESS. The network key of 'kurims' will be updated at 9:00am on Jan.4.2015. (Dec.28.2015). UNIX. : The Mathematica on Ubuntu(kirin,asahi,sapporo,kagome) are upgraded to the version 2015.1 (from 17.02). (Nov.18.2015). see also:. Softwares List (in RIMS only) UNIX. : The Mathematica on Ubuntu(kirin,asahi,sapporo,kagome) are upgraded to the version 10.2 (from 10.0.2). (Nov.4.2015). SECURITY. :. Update your. PuTTY , and. WinSCP . (Aug.23.2013). Vulnerabilities Information ... SECURITY. :. Update your Ghostscript. (Sep.1.2010). SECURITY. :. Update your xpdf. Vulnerabilities information ... (Oct.21.2009). SECURITY. :. Update your xpdf, poppler. Vulnerabilities information ... (Apr.22.2009). SECURITY. : Please update or configure your ssh client software. See. here for the details. (Jan.7.2009). SECURITY. : If you use ssh key(s) generated by Debian GNU\/Linux, see. http:\/\/www.kb.cert.org\/vuls\/id\/925211 , if your key(s) are vulnerable, please update them. (May.16.2008). older annoouncements * Some older security announcements still exist in the top page. Contact to:admin. kurims.kyoto-u.ac.jp (Computer Manager Office)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Call for proposals. Call for proposals Research Institute for. Mathematical Sciences,. Kyoto University. International Joint Usage \/ Research Center Workshop Colloquium\/Seminars What's New. View logs. Procedures to be followed at RIMS. If a Member, Student, Family Member or Visitor is Confirmed as Infected with COVID-19 Health status log Manual of Measures to Prevent the Spread of the Novel Coronavirus (COVID-19): Precautions to be Taken Regarding Classes in the Frist Semester of 2022 (Ver.8) 2022\/5\/23. Announcement of Recruitment of a Female Program-Specific Assistant Professor 2022\/4\/28. AY2024\/2023 Call for proposals of RIMS Joint Research Activities --- Int.JU\/RC. deadline:2022\/8\/31. 2024 RIMS Research Projects 2023 RIMS Workshops(Type C)* 2023 RIMS Satellite Seminars 2023 RIMS Review Seminars* *We have introduced a \"Promotion of Women's Participation\" workshop\/seminar program. 2022\/4\/1. AY2022 List of RIMS Joint Research Activities --- Int.JU\/RC 2022\/3\/18. Introducing RIMS 2022\/2\/28. France-RIMS collaboration: beyond the mathematical research frontier 2021\/10\/1. COVID-19 updates for organizers and participants (Lv.1 from October 1st) --- Int.JU\/RC 2021\/6\/29. 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Admission decisions for the Doctoral Program are based on oral examinations about the applicants' research achievements, especially their master's theses. For details of the entrance examination, please see the website of. the Graduate School of Science. • Research Students. RIMS accepts Research Students (Kenkyusei) who conduct research under the supervision of RIMS faculty members; however, they are not eligible for academic credits or degrees from Kyoto University. The enrolment period typically begins in April or October. Applicants should apply at the latest four months before the desired enrolment period, and contact. the Admissions Assistance Office (AAO) before the application. For details about becoming a Research Student. at RIMS, please see the website of. Kyoto University. • Others. General information about admission is available on the website of. Kyoto University. • Before the application. 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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Colloquium\/Seminars. TOP 　　. Seminars 　　Seminar on Geometry and Related Topics. Seminar on Geometry and Related Topics. Date and Time. February 21, 2020 15:00-16:30. Room. Room 111, RIMS, Kyoto University. Speaker. Cheol-Hyun Cho (Seoul National University). Title. Homological mirror symmetry for invertible curve singularities. Abstract. Given an invertible curve singularity, namely one of $x^p+y^q, x^p+xy^q, x^py+xy^q$, we introduce a constructive method to describe its homological mirror symmetry ( to its Berglund-Hubsch mirror). For the case of ADE singularites, we find interesting relationship between Auslander-Reiten quiver of the mirror singularity and Lagrangian surgery. This is a work in progress with Dongwook Choa and Wonbo Jung. Organizers. Kaoru Ono. Date. December 2 -- 4, 2019. Room. Room 110, RIMS, Kyoto University. RIMS & IBS-CGP Joint Workshop |. Abstract Organizers. Yong-Geun Oh (IBS-CGP), Kaoru Ono (RIMS). Date. October 8 (Tuesday) 15:00-16:30, 2019. Room. Room 110, RIMS, Kyoto University. Speaker. Pengfei Guan (McGill University). Title. The Weyl problem and isometric embedding of surfaces in 3-manifolds. Abstract. The classical Weyl problem concerns isometric embedding of positively curved compact surface in to $R^3$. The solution by Nirenberg (in 1950s) and Pogorelov's subsequent solution for hyperbolic space $H^3$ play important roles in the definition of quasi-local masses in general relativity, e.g., the Brown-York mass. The recent works of Liu-Yau and Wang-Yau generated some renewed interests on this classical problem: isometric embedding of a given surface $S^2,g)$ to general $3$-manifolds. Of particular interest is the anti de sitter-Schwarzchild space. We will discuss some progress in this direction. When the ambient space is replaced by a general $3$-manifold, local solvablity ( Li-Wang) and a priori estimates (Guan-Lu) can be established from recent works. The existence of such isometric embeddings can be obtained if the topology of the ambient space is trivial. We will discuss some applications and open problems. Organizer. K. Ono. Date. February 15, 10:30-12:00, 2019. Room. Room 110, RIMS, Kyoto University. Speaker. Xiaobo Liu (BICMR, Beijing University). Title. Connecting Hodge integrals to Gromov-Witten invariants by Virasoro operators. Abstract. Kontsevich-Witten tau function and Hodge tau functions are important tau functions for KP hierarchy which arise in geometry of moduli space of curves. Alexandrov conjectured that these two functions can be connected by Virasoro operators. In a joint work with Gehao Wang, we have proved Alexandrov's conjecture. In a joint work with Haijiang Yu, we show that this conjecture can also be generalized to Gromov-Witten invariants and Hodge integrals over moduli spaces of stable maps to smooth projective varieties. Organizer. K. Ono. Date. March 15, 15:00-16:30, 2018. Room. Room 111, RIMS, Kyoto University. Speaker. 小川　竜 氏 (東海大学). Title. Levi-flat CR manifolds and the embedding problems. Abstract. A Levi-flat CR manifold is an odd dimensional real manifold foliated by complex hypersurfaces. In this talk, we consider the embedding problem of Levi-flat CR manifolds into complex manifolds. Also we focus on the Barrett's non-embeddability theorem and its higher dimensional analogue. This is based on a joint work with Takayuki Koike. Organizer. K. Ono. Date. February 16, 15:00-16:30, 2018. Room. Room 110, RIMS, Kyoto University. Speaker. Jack Smith氏 (University College of London). Title. A monotone Lagrangian menagerie. Abstract. Apart from a few striking rigidity results, relatively little is known in general about the possible topologies of monotone Lagrangian submanifolds. I will survey some interesting examples, and a selection of techniques for computing or constraining their self-Floer cohomology, some old and some new. A common theme is subtle dependence of the Floer theory on the characteristic of the coefficient ring. Organizer. K. Ono. Date. December 20, 14:00-16:00, 2017. Room. Room 111, RIMS, Kyoto University. Speaker. Vivek Shende氏 (UC Berkeley). Title. The conormal torus is a complete knot invariant. Organizer. K. Ono. Date. December 19, 10:30-12:00, 14:30-16:30, 2017. December 20, 10:00-12:00. Room. Room 111, RIMS, Kyoto University. Speaker. Vivek Shende氏 (UC Berkeley). Title. Localization of Fukaya category. Organizer. K. Ono. Date. June 2, 15:00-16:30, 2017. Room. Room 006, RIMS, Kyoto University. Speaker. Tian-Jun Li氏 (University of Minnesota). Title. The symplectomorphism group of rational surfaces. Abstract. I will talk about the joint work with Jun Li and Weiwei Wu on the topology ofthe symplectomorphism group of a symplectic rational surface. We will illustrate our approach with the 5 point blowup of the projective plane. For an arbitrary symplectic form on this rational surface, we are able to determine the symplectic mapping class group and describe the answer in terms of the Dykin diagram of Lagrangian sphere classes. In this case, we are also able to compute the fundamental group for an open region of the symplectic cone. Organizer. K. Ono. Date. April 3, 15:00-16:30, 2017. Room. Room 110, RIMS, Kyoto University. Speaker. Tobias Ekholm 氏 (Uppsala University). Title. Legendrian surgery formulas and duality between Lagrangian and Legendrian invariants. Abstract. We give a Legendrian surgery formula for partially wrapped Floer cohomology and study (Koszul) duality between wrapped Floer cohomology and ordinary Floer cohomology using the surgery isomorphism. The talk reports on joint work with Y. Lekili. Organizer. K. Ono. Date. April 3, 13:00-16:30, 2017. Room. Room 110, RIMS, Kyoto University. Speaker. Naichung Conan Leung氏 (Chinese University of Hong Kong). Title. SYZ for coisotropic A-branes. Organizer. K. Ono. Date. March 21, 15:00-16:30, 2017,. Room 110. March 22, 10:30-12:00, 2017,. Room 402. Room. Room 110, 402, RIMS, Kyoto University. Speaker. 大場貴裕氏 (東京工業大学). Title. Higher-dimensional contact manifolds with infinitely many Stein fillings. Organizer. K. Ono. Date. March 6,7, 13:30-15:00, 2017. Room. Room 110, RIMS, Kyoto University. Speaker. 池　祐一氏　(東大数理). Title. Sheaf-theoretic approaches to symplectic geometry in cotangent bundles. Organizer. K. Ono. Date. February 21, 10:30-12:00, 2017. Room. Room 111, RIMS, Kyoto University. Speaker. Georgios Dimitroglou-Rizell 氏 (Uppsala University). Title. The wrapped Fukaya category of a Weinstein manifold is generated by the Lagrangian cocore discs. Abstract. In a joint work with B. Chantraine, P. Ghiggini, and R. Golovko we decompose any object in the wrapped Fukaya category as a twisted complex built from the cocores of the critical (i.e. half-dimensional) handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), and relations between them. Note that exact Lagrangians admit Legendrian lifts, and that appropriate Lagrange surgeries can be seen to give rise to an exact Lagrangian cobordisms of this type. Organizer. K. Ono. Date & Room. 2017年2月15日 15:00-16:30, Room 402. 2017年2月16日. 10:30-12:00. , Room 110. Speaker. Georgios Dimitroglou-Rizell 氏 (Uppsala University). Title. Classification of two-dimensional Lagrangian embeddings using pseudoholomorphic foliations. Abstract. We describe how the techniques of pseudoholomorphic foliations and neck stretching can be used to derive several strong classification results for Lagrangian embeddings in the four-dimensional setting. Notably, in a joint work with E. Goodman and A. Ivrii, we establish the nearby Lagrangian conjecture in the cotangent bundle of the two-torus. In an ongoing work we also provide a classification up to Hamiltonian isotopy of Maslov-0 Lagrangian tori, as well as spheres with one transverse double point, in the complement of a binodal cubic in the projective plane. Organizer. K. Ono. Date. 7月29日 15:00-16:30, 2016. Room. Room 111, RIMS, Kyoto University. Speaker. Egor Shelukhin氏 (IAS). Title. Lagrangian cobordisms and measurements. Abstract. We describe a natural way of measuring the distance between two (possibly non-isotopic!) Lagrangian submanifolds based on the notion of a Lagrangian cobordism, and study the non-degeneracy or degeneracy properties of the resulting pseudo-metrics, reflecting the rigidity or flexibility of the class of cobordisms considered. This is a joint work with Octav Cornea. Organizer. K. Ono. Date. 7月19日 14:00-16:00, 2016. Room. Room 402, RIMS, Kyoto University. Speaker. Jake Solomon氏 (Hebrew University). Title. TBA. Organizer. K. Ono. Date. 5月20日 16:00-17:00, 2016. Room. Room 204, RIMS, Kyoto University. Speaker. Viktor Ginzbrug氏 (UC Santa Cruz). Title. Random chain complexes. Organizer. K. Ono. Date. 3月29日 14:00-15:30, 2016. Room. Room 006 &#8594 111,. RIMS, Kyoto University. Speaker. Emmy Murphy 氏 (MIT). Title. Weinstein manifolds and flexibility of loose Legendrians. Abstract. Weinstein manifolds are a class of open symplectic manifolds, including all smooth affine varieties. By definition they admit a Morse decomposition compatible with the symplectic geometry, which allows us to completely describe their geometry by a collection of Legendrian spheres. For example, the Legendrian contact homology of these spheres gives a computation of the wrapped Fukaya category. On the other hand, the flexibility phenomenon of loose Legendrians gives a partial classification of Weinstein manifolds up to symplectomorphism. We will discuss this flexibility, which is proven using a resolution of certain Legendrian singularities. Date. 3月30日 10:30-12:00, 2016. Room. Room 006 &#8594 111,. RIMS, Kyoto University. Speaker. Emmy Murphy 氏 (MIT). Title. Applications of flexibility to constructions of exotic Lagrangian embeddings and immersions. Abstract. The h-principle for loose Legendrians tells us that we have far more freedom to isotope loose Legendrians than what one would expect for non-loose Legendrians. For example, a loose Legendrian can be isotoped to become disjoint from any given subset, as long as it is possible topologically. Since the graph of a Legendrian isotopy is a Lagrangian embedding, this gives new methods for constructing Lagrangians which unusual properties. For example, we build compact Lagrangian embeddings in $C^n$ which are not uniruled, and have infinite relative Gromov width. We also show that the number of self-intersections of exact Lagrangian immersions fail to conform to the Arnol'd conjecture philosophy, for example every 3-manifold admits an exact Lagrangian immersion in $C^3$ with a single self-intersection, regardless of total homological rank. Date. 3月31日 10:30-12:00, 2016. Room. Room 111,. RIMS, Kyoto University. Speaker. Emmy Murphy 氏 (MIT). Title. Lefschetz fibrations from the perspective of Legendrian Kirby diagrams. Abstract. Many Weinstein manifolds are naturally presented as the total space of a Lefschetz fibration. We give an explicit algorithm to translate the Lefschetz fibration picture into a presentation of a Legendrian Kirby diagram. This in turn gives an efficient method to visualize and study the total space of Lefschetz fibrations. On the flexibility side this can be used to demonstrate a number of surprising symplectomorphisms, for example the affine hypersurfaces ${xy^k + z^2 + w^2 = 1}$ are all symplectomorphic, independent of k. On the rigidity side this gives an efficient combinatorial method to compute wrapped Fukaya categories, which can then be applied to homological mirror symmetry. The main advantage of this perspective is that the contact geometry picture has more geometric freedom, which allows us to make a number a geometric simplifications before making pseudo-holomorphic computations. Organizer. K. Ono. Date. 2月25日 14:00-15:30, 2016. Room. Room 110, RIMS, Kyoto University. Speaker. Ailsa Keating氏 (Columbia University). Title. Homological mirror symmetry for $T_{pqr}$ singularities. Organizer. K. Ono. Date. 11月20日 15:00-16:30, 2015. Room. Room 110, RIMS, Kyoto University. Speaker. Yakov Eliashberg 氏 (Stanford University). Title. TBA. Organizer. K. Ono. Date. 10月9日 15:00-16:30, 2015. Room. Room 111, RIMS, Kyoto University. Speaker. Patrick Bernard 氏 (ENS, Paris). Title. Variational and viscosity solutions of the Hamilton-Jacobi equation. Organizer. K. Ono. Date. 7月16日, July 16, 2015. Room. Room 006, RIMS, Kyoto University. Speaker. 13:30-15:00 石田裕昭氏 (京大数理研). Hiroaki Ishida (RIMS, Kyoto University). Title. Equivariant basic cohomology (in progress). Speaker. 15:30-17:00 江孟蓉 (River Chiang) 氏 (国立成功大学, 台湾). River Chiang (National Cheng Kung University, Taiwan). Title. TBA. Organizer. K. Ono. Date. 6月11日 15:00-16:30, 2015. Room. Room 420, RIMS, Kyoto University. Speaker. Garrett Alston 氏 (University of Oklahoma). Title. Legendrian DGA as Immersed Floer Theory. Abstract. An embedded Legendrian in a contact manifold of the form P x R can be interpreted as an immersed Lagrangian in P. I will explain how the Floer theory of the immersed Lagrangian contains the information of the Legendrian dga. I will also give some conjectural SFT-type applications to non-compact Lagrangians. Organizer. K. Ono. Date. 4月1日 15:00-16:30 2015. Room. Room 006, RIMS, Kyoto University. Speaker. Sobhan Seyfaddini (MIT). Title. Toward a dynamical interpretation of Hamiltonian spectral invariants on surfaces. Organizer. K. Ono. Date & Place. March 12, RIMS Room 206, 2015. Time & Speaker. 15:00-16:30 Thomas Schick (Gottingen). Title. Differential K-theory. Date & Place. March 25, RIMS Room 110. Time & Speaker. 10:30-11:30 Sobhan Seyfaddini (MIT). Title. Coisotropic submanifolds and $C^0$-symplectic geometry. Time & Speaker. 13:30-14:30 Otto van Koert (Seoul National University). Title. Contact homology and some of its problem. Time & Speaker. 16:30-17:30 Erkao Bao (UCLA). Title. Definition of contact homology in dimension three (I). Date & Place. March 26, RIMS Room 110. Time & Speaker. 10:30-11:30 Otto van Koert (Seoul National University). Title. Nonfillable contact manifolds. Time & Speaker. 13:30-14:30 Erkao Bao (UCLA). Title. Definition of contact homology in dimension three (II). Date & Place. March 27, RIMS Room 110. Time & Speaker. 10:30-11:30 Otto van Koert (Seoul National University). Title. Symplectic and equivariant symplectic homology. Time & Speaker. 13:30-14:30 Erkao Bao (UCLA). Title. Definition of contact homology in dimension three (III). Organizer. Kaoru Ono. Date. 2014年11月18日 （火）15:00-16:30. Room. Room 206, RIMS, Kyoto University. Speaker. Manabu Akaho (Tokyo Metropolitan University). Title. Symplectic displacement energy for exact Lagrangian immersions. Summary. We give an inequality of the displacement energy for exact Lagrangian immersions and the symplectic area of punctured holomorphic discs. Our approach is based on Floer homology for Lagrangian immersions and Chekanov's homotopy technique of continuations. Moreover, we discuss our inequality and the Hofer--Zehnder capacity. Organizer. Kaoru Ono. Date. 2014年11月18日 （火）13:00-14:30. Room. Room 206, RIMS, Kyoto University. Speaker. Dingyu Yang (Institut de Mathématiques de Jussieu). Title. Deriving maximality and 4 applications of level-1 structures from FOOO Kuranishi structures. Summary. In this talk, I will sketch elements of polyfold--Kuranishi correspondence which perfectly identifies Fukaya-Oh-Ohta-Ono's Kuranishi structures with Hofer-Wysocki-Zehnder's polyfold Fredholm structures up to germs in two geometric ways intertwining associated perturbation theories. These two aforementioned structures are abstract geometric structures appearing in the foundational theory of moduli spaces in Lagrangian Floer theory and symplectic field theory respectively. Polyfold Fredholm theory generalizes the notion of smoothness, and underpins the ideal of making domain reparametrization smooth and equipping smooth structures to general moduli spaces in symplectic geometry while making almost no extra choice. As a finite dimensional approach, a Kuranishi structure is a global way to consistently fit together finite dimensional reductions constructed from local neighborhoods of moduli spaces, and provides a framework to globally perturb moduli spaces in spite of possible dimension jumps when switching to nearby local models. I defined a notion of Kuranishi structures by adding maximality and topological matching conditions to FOOO's Kuranishi structures; from this, one naturally obtains a Hausdorff level-1 good coordinate system, which is a coherent system of structured tubular neighborhoods built upon an ordered finite cover and serves as a device to overcome dimension jumps (in 4 applications below). I also defined a notion of a (global) germ via common refinements possibly increasing local dimensions, which is applicable to any version of Kuranishi structures. I will show how to functorially obtain these two conditions from any representative (and thus all of its shrinkings) within each germ of FOOO's Kuranishi structures, and thus level-1 structures become available to use to show the following: (i) A germ of FOOO's Kuranishi structures is an equivalence class. (ii) We can define and compose maps between germs (where level-1 structures define left inverses to chart embeddings), and obtain a category of FOOO Kuranishi structure germs. (iii) We can lift local perturbations using level-1 structures to globally perturb a Kuranishi structure, and can lift small such global perturbation through level-1 refinement to show that perturbation theory descends to germs; and combining (i) to (iii), we do not need to compare different perturbations constructed using choices for truncated moduli space of different finite energy levels and can do a single perturbation for the entire moduli space, just as in polyfold theory. (iv) Level-1 structures provide sc-smooth retract models for gluing across dimension-jumping coordinate changes and thus convert FOOO Kuranishi structures to polyfold Fredholm structures, and a further polyfold--Kuranishi correspondence for analysis involved in constructing these geometric structures would merge methods and results from these two fields and make construction of Legendrian symplectic field theory easier and as a common extension. Organizer. Kaoru Ono. Date. 2014年11月17日 (月) 10:30-12:00. Room. Room 110, RIMS, Kyoto University. Speaker. Naichung Conan Leung (Chinese University of Hong Kong). Title. Witten deformation, A_infinity Morse category and scattering. Date. 2014年11月17日 (月) 13:30-15:00. Room. Room 110, RIMS, Kyoto University. Speaker. Naichung Conan Leung (Chinese University of Hong Kong). Title. Donaldson-Thomas invariants for Calabi-Yau fourfolds. Organizer. Kaoru Ono. Date. 2013年7月18日 (木) 15:00-16:30. Room. Room 006, RIMS, Kyoto University. Speaker. Sheel Ganatra (Stanford University). Title. Symplectic cohomology and duality for the wrapped Fukaya category. Abstract. To an exact symplectic manifold M, one can two important Floer-theoretic invariants: symplectic cohomology SH^*(M) and the wrapped Fukaya category W(M). We will explain how, when M contains enough Lagrangians, the natural geometric open-closed string maps between the Hochschild homology of W(M), symplectic cohomology, and the Hochschild cohomology of W(M) are all isomorphisms. The induced isomorphism between Hochschild homology and cohomology is an instance of a new self-duality for the wrapped Fukaya category, a non-compact version of a Calabi-Yau structure. Organizer. Kaoru Ono. Date. 2013年7月11日（木）15:00--16:30. Room. Room 006, RIMS, Kyoto University. Speaker. Bai-Ling Wang (Australian National University). 微分トポロジーセミナーと共同開催. Title. K-theory virtual fundamental classes for moduli spaces of stable maps. Abstract. In this talk, I will propose K-theory virtual fundamental classes for moduli spaces of stable map, based on joint work in progress with Bohui Chen and Jianxun Hu in an attempt to define quantum K-theory for general symplectic manifolds.　. Organizer. K. Ono. Date. 2013年5月30日（木）15:30－17:00　. Room. Room 006, RIMS, Kyoto University. Speaker. Andrei Pajitnov (Universite Nantes). Title. Twisted Novikov homology, jump loci and non-abelian Hodge theory. Abstract. We begin with a brief introduction to the Novikov homology and circle-valued Morse theory. Then I will explain our joint work with Toshitake Kohno about the applications of the Novikov homology to the jump loci in the homology with local coefficients, in particular on compact Kaehler manifolds. Organizer. K. Ono. Date. April 2 (Tue), 2013, 11:00－12:00, 13:30－14:30, 15:00-16:00. Room. Room 204, RIMS, Kyoto University. Speaker. Naichung Conan Leung (Chinese University of Hong Kong). Title. SYZ mirror transformation. Organizer. K. Ono. Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Colloquium\/Seminars. TOP 　　. Colloquium 　　Colloquium. Colloquium. Title. Japanese version only. Date. 2022.6.22 (Wed)　16:45-17:45 　　. Place. Zoom. Speaker. Kei Irie (RIMS, Kyoto University). Comment. Registration：. Please register using the form (. https:\/\/forms.gle\/VZ3MNRE9AQJRuZfE7 ). by 6:00 p.m. on Monday 20th June. The colloquium will take place online, and can be accessed via a course site on PandA. The access to the course site will be granted once your registration is confirmed. Title. Fundamental groups of curves in positive characteristic, following Pop, Raynaud, Saïdi, Tamagawa, and myself. Date. 2022.6.15 (Wed)　16:45-17:45 　　. Place. Rm110, Research Institute for Mathematical Sciences, Kyoto University. Speaker. Yu Yang (RIMS, Kyoto University). Abstract. [pdf] Comment. Registration：. This colloquium talk will take place face-to-face. Please register using the form (. https:\/\/forms.gle\/BB7G3iqdDX8QVMJN6 ). The number of participants might be limited. * Participants are encouraged to wash hands frequently, wear a mask. * Note that under our health policy, persons with some typical symptoms (continued cough, fatigue or temperature 1℃ higher than normal) are not allowed in the building. * Please fill in the participant list at the venue. * No beverage or snacks should be served in meetings. Title. Gauge theory and the diffeomorphism groups of 4-manifolds. Date. 2022.6.1 (Wed)　16:45-17:45 　　. Place. Rm110, Building No.3, Faculty of Science, Kyoto University. Speaker. Hokuto Konno (The University of Tokyo). Comment. Registration：. This colloquium talk will take place face-to-face. Please register using the form (. https:\/\/forms.gle\/YqMMBGUUdFMaXobH6 ). The number of participants might be limited. * Participants are encouraged to wash hands frequently, wear a mask. * Note that under our health policy, persons with some typical symptoms (continued cough, fatigue or temperature 1℃ higher than normal) are not allowed in the building. * Please fill in the participant list at the venue. * No beverage or snacks should be served in meetings. Title. Oka manifolds and the dual Levi problem. Date. 2022.4.27 (Wed)　16:45-17:45 　　. Place. Rm110, Building No.3, Faculty of Science, Kyoto University. Speaker. Yuta Kusakabe (Kyoto University). Comment. Registration：. This colloquium talk will take place face-to-face. Please register using the form (. https:\/\/forms.gle\/voEjeQGAoBsd9eJB9 ). by 6:00 p.m. on Friday 22th April. The number of participants might be limited. * Participants are encouraged to wash hands frequently, wear a mask. * Note that under our health policy, persons with some typical symptoms (continued cough, fatigue or temperature 1℃ higher than normal) are not allowed in the building. * Please fill in the participant list at the venue. * No beverage or snacks should be served in meetings. Title. Medians, rearrangements and their maximal functions. Date. 2022.4.20 (Wed)　16:45-17:45 　　. Place. Zoom. Speaker. Yohei Tsutsui (Kyoto University). Comment. Registration：. Please register using the form (. https:\/\/forms.gle\/QwTWpVgUxKB3SwKv9 ). by 6:00 p.m. on Monday 18th April. The colloquium will take place online, and can be accessed via a course site on PandA. The access to the course site will be granted once your registration is confirmed. 2021 |. 2020 |. 2019 |. 2018 |. 2017 |. 2016 |. 2015 |. 2014 |. 2013 |. 2012 |. 2011 |. 2010 |. 2009 |. 2008 |. 2007 |. 2006 |. 2005 |. 2004 |. 2003 |. 2002 |. 2001 |. 2000 |. 1999 |. Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Colloquium\/Seminars. TOP 　　. Seminars 　　セミナー -- Computer Science Seminar. セミナー -- Computer Science Seminar. Title. （１） Quantum Logic, Quantum Computation and the Proof Theory of Compact Categories. （２） Complementarity in Categorical Quantum Mechanics. Date. ２０１０年２月１２日（金）. （１）１０：３０より２時間程度. （２）１４：４５より２時間程度. Room. 京都大学理学部３号館（数学教室）３０５号室. Speaker. Prof. Ross Duncan (Oxford University Computing Laboratory). Abstract. 講演（１）. Quantum Logic, Quantum Computation and the Proof Theory of Compact Categories. Quantum computation has reawakened interest in the underlying logic of quantum mechanics. We briefly examine the \"old school\" Birkhoff-von Neumann style quantum logic, and discuss why it is not a suitable basis for quantum computation. We then discuss a newer approach, initiated by Abramsky and Coecke and based on a view of logic informed by the Curry-Howard correspondence between proofs and programs. Abstract: Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a system of proof-nets. This logic captures much of the necessary structure needed to represent quantum processes under classical control, while remaining agnostic to the fine details. We demonstrate how to represent quantum processes as proof-nets, and show that the dynamic behaviour of a quantum process is captured by the cut-elimination procedure for the logic. We show that the cut elimination procedure is strongly normalising: that is, that every legal way of simplifying a proof-net leads to the same, unique, normal form. Finally, taking some initial set of operations as non-logical axioms, we show that that the resulting category of proof-nets is a representation of the free compact closed category with biproducts generated by those operations. The main part of this talk is based on book chapter. http:\/\/arxiv.org\/abs\/0903.5154 . ================================== 講演（２）. Complementarity in Categorical Quantum Mechanics. The fundamental feature of quantum mechanics is the existence of incompatible observables. This is usually phrased in a negative way: operators are NOT commuting; the lattice of observables is NON-distributive etc. In this talk I will present the positive content to be found in the existence of complementary quantum observables. We will see that each such pair gives rise to an incredibly rich zoo of algebraic structures all of which can be presented graphically. The resulting graphical language is expressive enough to denote any quantum physical state of an arbitrary number of qubits, and any processes thereof. The rules for manipulating these result in very concise and straightforward computations with elementary quantum gates, translations between distinct quantum computational models, and simulations of quantum algorithms such as the quantum Fourier transform. They enable the description of the interaction between classical and quantum data in quantum informatic protocols. I will describe this formalism in detail, and if time permits explain some applications in the study of measurement based quantum computation and the local hidden variable models. The main part of this talk can be found in. http:\/\/arxiv.org\/abs\/0906.4725 . ==================================. Organizer. M. Hasegawa. Title. Behavioural Equivalence and Indistinguishability in Higher-Order Typed Languages. Date. August 8 (Fri), 2003, 16:00-17:00. Room. Room 115, RIMS, Kyoto University. Speaker. 勝股 審也. Abstract. We extend the study of the relationship between behavioural equivalence and indistinguishability relation to the simply typed lambda calculus, where higher-order types are available. The main technical tool of this study is pre-logical relations, which give a precise characterisation of behavioural equivalence. We establish a relationship between behavioural equivalence and indistinguishability in terms of factorisability. We then consider a higher-order logic to reason about models of the simply typed lambda calculus, and relate the resulting standard satisfaction relation to behavioural satisfaction. Comments. Organizer. M. Hasegawa. Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. 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The Poincare Conjecture, that every simply connected 3-manifold is homeomorphic to the sphere, has been a central question in topology since its formulation in 1904. In the early 1980s Thurston gave a vast generalization and reformulation of this question that goes under the name of the `Geometrization Conjecture' It differed in two essential ways: (i) it covered all 3-manifolds, not just simply connected ones, and (ii) it was not a purely topological conjecture; it mixed geometric and topological notions. This conjecture suggested that every 3-manifold could be cut in a canonical way so that the resulting pieces all have especially nice Riemannian metrics, metrics derived from Lie groups. It easily implies the Poincare Conjecture. Richard Hamilton suggested a method to attack the problem of constructing the metrics required by Thurston's conjecture. His method was to begin with any Riemannian metric on the 3-manifold under study and deform this metric by a non-linear parabolic equation (a flow equation) which is a non-linear heat-type equation. It is called the Ricci flow equation. The intuition is that just as heat flow distributes the temperature evenly, this equation should distribute the curvature evenly over the manifold and hence produce the homogeneous metrics that the Geometrization Conjecture posits. There is an essential difficulty in this approach coming from the fact that the evolution equation is non-linear. This means that singularities can develop in finite-time stopping the evolution before it reaches its homogeneous limit at time = +infinty. In 2002\/2003 Perelman posed a series of preprints explaining how to deal with these finite-time singularities and continue the flow past them, constructing what is known as a Ricci flow with surgery. He went on to justify Hamilton's vision and complete the proof of the Geometrization Conjecture. In this talk I will discuss the various types of geometric 3-manifolds, state Thurston's conjecture, give an overview of the work of Hamilton, and finally a brief introduction to Perelman's proof. 連絡先. 大槻知忠（京大数理研、tomotada@kurims.kyoto-u.ac.jp ）. Back to Seminars Research Institute for Mathematical Sciences , Kyoto University Kyoto, 606-8502 JAPAN FAX: +81-75-753-7272. Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. 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Visual Presentator (document camera\/LCD projector). Blackboard\/screen (1). Whiteboard (1). *For hybrid (face-to-face and online) meetings. Rear Camera: SONY BRC-X400W. Front Camera: Panasonic AW-UE4WGN. Swiching Camera → USB. Indoor microphone → USB. USB audio output → Indoor speaker. Wireless display: MAX HUB 75inch. Room 110 (48 people). Visual Presentator (document camera\/LCD projector). Blackboard\/screen (1). Whiteboard (1). *For hybrid (face-to-face and online) meetings. Camera: SONY BRC-X400B. Mic+Speaker: YAMAHA speaker microphone YVC-300. Wireless display: Microsoft Surface HUB 2S 85inch. Room 201 in Building No.15 (54 people). Access: Research Building No. 15, ANNEX Visual Presentator (document camera\/LCD projector). Blackboard\/screen (1). Whiteboard (1). *For hybrid (face-to-face and online) meetings. Camera: Panasonic AW-UE70. Mic+Speaker: YAMAHA speaker microphone YVC-300. Lobby. Sofa, tables (3),. Tea machine (hot water, cold water, green tea, brown rice tea). Photocopier (1). Desk top computers (2). Printer (1). OS: Windows 10. Installed software: Internet Explorer, SSH. Service hours: 9:00～17:00. If you want to use a computer, ask the staff member in charge of joint use. Past RIMS Joint Research Activities. →. Past RIMS Research Projects →. Past RIMS Satellite Seminars →. Past RIMS Review Seminars →. Past RIMS Workshops\/Symposia Joint Research Activities Report. Since 2004, to publicly communicate the results of Joint Research Activities conducted at RIMS, project results have been compiled into a Joint Research Activities Report every year. Report forms are available for download on the following page. RIMS Symposia. PDF WORD RIMS Workshops (Type A). PDF WORD RIMS Workshops (Type B). PDF WORD RIMS Workshops (Type C). PDF WORD RIMS Satellite Seminars. PDF WORD RIMS Review Seminars. PDF WORD Preparation of Kôkyûroku and Kôkyûroku Bessatsu. 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Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities. →. here COVID-19 updates for organizers and participants. →. here Processes of a Joint Research Activities Seminar rooms and facilities for joint research Past RIMS Joint Research Activities Joint Research Activities Report Preparation of the Kôkyûroku and the Kôkyûroku Bessatu Call for proposals About childcare services during Joint Research Activities Initiatives for Gender Equality and Diversity in Joint Research Activities at RIMS COVID-19 updates for organizers and participants ←　BACK TO THE TOP About Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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Skip to content English. English 日本語 Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. TEST YOUR. JAPANESE LEVEL. Name. *. First. Last. Email. *. Enter Email. Confirm Email. Did you submit an Application Form? *. Yes. No. Which course did you apply to? *. 2 years. 1 year. 10 weeks. 4 weeks. Summer course. JLPT. Private \/ Semi-private. Others. Please, specify. *. Please, specify. When are you planning to start the course? *. 2022. 2023. Year. Session. *. January. April. July. October. Session. Save and Continue Later Have you ever studied Japanese before? *. Yes. No. Select \"Yes\" if you have more than 50 hours of experience studying Japanese in a school or by yourself. Japanese study detail. *. Study type. Institution Name. Institution Location. Start Date. End Date. Self-Study. High School. University. Language School. Did you use any specific textbook? *. Minna no Nihongo. Genki. Fun & Easy Japanese. No, I didn't use any textbook. Other. Other textbook. About how many hours in total have you studied? 50-100. 101-200. 201-400. 401-600. 600+. Do you know any of the following? Hiragana. Katakana. Kanji. How many kanji are you able to read and understand? 20 to 50. 51 to 200. 201 to 400. 400+. Have you passed any level of the JLPT? Yes. No. I have passed a different test (J-Test, NAT test, etc.). What is the highest level you passed? N5. N4. N3. N2. N1. When did you pass it? DD slash MM slash YYYY. Other test details. Test name, level etc. Save and Continue Later Let's get started! このシャツは、二千五百えんでした。. にせんごぴゃく. にぜんごびゃく. にぜんごぴゃく. にせんごひゃく. 出口はあちらです。. しゅつくち. いりぐち. にゅうくち. でぐち. わたしの弟は、だいがくせいです。. おとうと. あに. あね. いもうと. 明日は、都合がわるいので、行けません。. とごう. ぐあい. つごう. とあい. ここは、えきから遠くて不便です。. ふびん. ふうべん. ふうびん. ふべん. 正月は、かぞくといっしょにハワイへ行きます。. せいつき. せいげつ. しょうつき. しょうがつ. パーティーに出席する人は、ここに名前を書いてください。. しゅっせき. しゅつせい. しゅせき. しゅうせい. ここにあるパソコンは、自由につかってください。. じゅう. しゆう. じゆう. しゅう. きのう、とても悲しい本を読んだ. かなしい. やさしい. くわしい. むずかしい. 来月ヨーロッパへ行くので、インターネットでホテルの予約をした。. よやくう. よやく. よあく. ようやく. 昨日デパートで、小学校のときの友だちに偶然会った。. とつぜん. ぐうぜん. ぜんぜん. とうぜん. 旅行中、景色がいいところの写真をたくさんとった。. けいしき. けしき. けしょく. けいしょく. スーパーで、ぎゅうにゅうをかいました。. 買いました. 員いました. 貴いました. 貝いました. ともだちとでんわではなしました。. 読みました. 言いました. 語しました. 話しました. がっこうのまえに、ゆうびんきょくがあります。. 後. 前. 外. 中. かいぎで、わたしのいけんをいいました。. 意見. 発見. 発言. 意思. 日本のぶんがくを勉強するために来ました。. 文楽. 文字. 文学. 文化. パーティーのかいじょうはあちらです。. 会所. 会場. 開場. 開所. にもつをたくさん入れたら、紙のふくろがやぶれてしまった。. 被れて. 破れて. 涙れて. 彼れて. この魚は海のふかいところにすんでいます。. 温い. 浅い. 涼い. 深い. 朝からずっとおなかがいたいので、くすりを飲みました。. 痒い. 疲い. 痛い. 病い. 帰りがおそくなる時は、電話でれんらくしてください。. 練絡. 練結. 連結. 連絡. 彼のてきせつな説明のおかげで、問題点がよくわかった。. 適切. 的節. 練絡. 適設. こんな天気の悪い日に、山に登るのはあぶないからやめなさい。. 険ない. 恐ない. 危ない. 怖ない. 「すみません、このペンは（　　　）ですか。」「350円です。」. いくつ. いかか. いつ. いくら. コーヒーは（　　　）すきじゃありません。. すぐ. あまり. よく. とても. わたしはまいあさ、シャワーを（　　　）。. いれます. あります. あびます. はいります. （　　　）ぜひ、うちへ　あそびに来てください。. ずっと. さいきん. このごろ. こんど. 先生がびょうきなので、（　　　）に行きました。. おみまい. おれい. おいわい. おまつり. かぜをひいて、りょこうに行けなくて（　　　）だった。. ねっしん. ざんねん. あんぜん. むり. 今回の旅行ではあまりお金をつかわないように（　　　）安いホテルをさがそう。. とつぜん. なるべく. しだいに. やっと. ４月からのしごとの（　　　）がきまった。. シリーズ. コース. ダイヤ. スケジュール. わたしの（　　　）は、いつも元気なところです。. 強所. 明所. 良所. 長所. さいきんは家でDVDを見ることが多く、映画館へは（　　　）行かない。. めったに. たまには. とうとう. どんどん. 時間はまだ（　　　）あるから、そんなにあわてなくてもいいですよ。. たっぷり. ゆっくり. さっぱり. ぎっしり. 開館時間は10時から18時までです。（　　　）土曜日は14時までです。. それで. さて. ただし. つまり. きょねんのなつは（　　　）。. あつくでした. あつぃかったです. あついでした. あつかったです. あのレストランは、（　　　）やすいです。. おいしくて. おいしいと. おいしい. おいしいで. きのう、こうえん（　　　）テニスをしました。. を. の. で. に. ひるごはんを（　　　）まえに、てをあらいました。. たべた. たべる. たべて. たべ. デパートへいきましたが、（　　　）かいませんでした。. なにを. なにも. なにが. なにか. これは、わたしのははが（　　　）ケーキです。. つくって. つくり. つくった. つくります. かってきたビール、れいぞうこにいれて（　　　）ね。. おいて. あって. ないで. いよう. おいしいすしを（　　　）、あの店がいいよ。. 食べながら. 食べるなら. 食べたり. 食べたい. 私は夏休みに、レストランでアルバイトをする（　　　）だ。. つもり. ばかり. こと. ところ. 新聞によると、きのう中国でじしんが（　　　）。. あるそうでした. あるそうです. ありそうです. あったそうです. 木村さんはアメリカに10年すんでいたから、英語がじょうずな（　　　）。. はずだ. だろう. ところだ. らしい. 「山田さんはいますか。」「山田さんはさっき（　　　）ところです。」. 帰っている. 帰った. 帰りたい. 帰る. すみませんが、こちらにお名前を（　　　）ください。. お書きして. お書き. お書きされて. お書きいただいて. あそこに小さい男の子が立っている（　　　）が見えますか。. の. もの. ところ. こと. 子どものころ、わたしは母にそうじを（　　　）。. てつだわされた. てつだいた. てつだわれた. てつだわせた. スーパーでお金を（　　　）としてかばんを開けたら、さいふがなかった。. はらいたい. はらおう. はらえる. はらう. タバコは体に悪いと知り（　　　）やめることができない。. ながら. ても. ないで. にくく. 暗くならない（　　　）、うちへ帰ろう。. あいだ. うちに. まえに. までに. あたたかくなる（　　　）、外へあそびに出かける人がふえてきた。. につれて. ところ. としても. にしては. ぜひ（　　　）たいのですが、今週のご予定はいかがでしょうか。. お目にかかり. ごらんになり. お会いになり. お目にかけ. 妹は体が弱くて、学校を休み（　　　）。. 気味だ. っぽい. だらけだ. がちだ. あのまじめな川上さん（　　　）そんないいかげんな仕事をするわけがない。. に伴って. に関して. に限って. に応じて. 図書館へ行く（　　　）、スーパーで買い物をしてこよう。. ついでに. につれて. うえに. ばかりか. 母にはだいじょうぶだと言った（　　　）、本当はとても不安だった。. からは. とおり. ものの. つつも. Additional Comments. You can include any additional information, notes or questions you may have.You can include any additional information, notes or questions you may have. Comments. This field is for validation purposes and should be left unchanged. Save and Continue Later Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. Privacy Policy Designed by Souzou Web Scroll To Top."}
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Choose from Home Stay, Guest House or find your perfect apartment in Tokyo."],"last_modified":"Wed Jun 08 16:52:13 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":8444864816389132750,"exact_signature_unique_b":true,"fuzzy_signature_l":-8382443486528300293,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h2_txt":["STUDENT ACCOMMODATION","HOME STAY","SHARE HOUSE","APARTMENT"],"h4_txt":["Price","Application","Meals","Location","Duration","Rent","Initial cost","Utilities and Internet","Location","Discounts","Rent","Initial cost","Utilities and Internet","Location","Duration","Student Voice","Courses","Contact us"],"bold_txt":["BORDERLESS HOUSE","J&F Plaza","https:\/\/www.sakura-house.com\/contact\/shibuyajp","Shibuya Gaigo Gakuin - Exclusive Contact Form","SAKURA HOUSE - Home Away from Home.","TIERRA","TIERRA ","Borderless House","English English 日本語"],"imagescount_i":7,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/06\/Japan-Homestay-1.jpg","shibuyajp.net\/wp-content\/uploads\/2018\/06\/Japan-Sharehouse-1.jpg","shibuyajp.net\/wp-content\/uploads\/2018\/11\/tierra.jpg","shibuyajp.net\/wp-content\/uploads\/2018\/11\/TIERRA.png","shibuyajp.net\/wp-content\/uploads\/2020\/01\/Sakura-House-Logo201912.png","shibuyajp.net\/wp-content\/uploads\/2020\/01\/JF_LOGO-01.png","shibuyajp.net\/wp-content\/uploads\/2022\/04\/bordelesshouse_en.jpg"],"images_alt_sxt":["Japan Homestay","Japan Sharehouse","tierra","","","",""],"images_height_val":[400,400,400,97,33,100,33],"images_width_val":[600,600,600,225,261,133,221],"images_text_t":"shibuyajp net wp content uploads 2018 06 Japan Homestay 1 jpg Japan Homestay Sharehouse Japan Sharehouse 11 tierra TIERRA png 2020 01 Sakura House Logo201912 Logo 201912 JF LOGO 2022 04 bordelesshouse en","responsetime_i":78,"text_t":"Student Housing Options in Tokyo &ndash; Shibuya Gaigo Gakuin. Skip to content English. English 日本語 Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. STUDENT ACCOMMODATION. Finding accommodation in Japan can be hard for foreigners. Through our partners, we have several accommodation options available from which you can choose. We can present options for Home Stay, Guest House accommodations as well as provide Realtor recommendation if you are looking to rent an apartment. HOME STAY. Staying with a Japanese family in a Homestay program provides a better understanding and appreciation of the Japanese family daily life, while also having the opportunity to practice your Japanese. You will enjoy family meals and outings, as well as introduce your own culture to your host family. Price. 4,536 yen per night. Application. 10,800 yen fee. Meals. includes breakfast and dinner. Location. Within 60 minutes from the school. Duration. 3-month maximum per family. SHARE HOUSE. Living in a Share House or Guest House will provide you with the opportunity to widen your circle of friends when you go home as well. Live with Japanese and Foreign nationals in a multicultural environment, most share houses provide both private and shared rooms depending on your preference, some share houses have women-only properties available as well. Sample Guesthouse at. Borderless House. . Rent. Starting at 40,000 yen per month. Initial cost. Starting at 50,000 yen. Utilities and Internet. Usually included in the rent. Location. Share houses are available throughout Tokyo area. Discounts. Our Students may receive discounts. Photo by. TIERRA   . APARTMENT. Apartments come in all shapes and sizes. There are unfurnished, semi-furnished, fully furnished, serviced apartments and many more to choose from. Apartments can be relatively expensive compared to homestay or share house but if you are staying for a long period, looking for complete privacy or wish to share an apartment with a friend then this might be right for you. Rent. Starting at 70,000 Yen. Initial cost. Starting with the equivalent of 3 month rent. Utilities and Internet. Usually not included. Location. Anywhere in Tokyo. Duration. Usually a 2 years contract but you may terminate the contract whenever you wish. TIERRA. - Real estate agent with English and Italian speaking staff. SAKURA HOUSE - Home Away from Home. Provides furnished apartments, share houses, dormitories to foreign students since 1992. Accommodation near the school available. Shibuya Gaigo Gakuin - Exclusive Contact Form. https:\/\/www.sakura-house.com\/contact\/shibuyajp. J&F Plaza.  -Welcome to a comfortable share house. BORDERLESS HOUSE. - A Multicultural Share House with Locals. Student Voice Courses APPLY NOW. LEVEL CHECK. 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I'm confident that with your motivation and willingness to put in the work necessary, you will find both the teaching methods and the school environment, to be a very useful medium for you to learn Japanese, start conversing in a relatively short amount of time and to experience the different facets of the Japanese culture. Please continue browsing our website, follow our social media accounts and discover the unique experiences that await you at Shibuya Gaigo Gakuin. If you have any questions, our staff members are just an email or phone call away. Looking forward to meeting you in Japan! WHY SHIBUYA GAIGO GAKUIN. LOCATION. *The school has temporarily moved to a new building in Takadanobaba. Moving back to Shibuya is expected to take place between spring and fall of 2023. We are located in the heart of Tokyo. SHIBUYA. is one of the busiest districts in Tokyo and is considered the center of youth fashion, culture and one of the prime entertainment neighborhoods in Tokyo. 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Extension of Online Classes for April 2020 Term （日本語訳準備中）. 4月 28, 2020. お知らせ一覧. 渋谷で約50年続く語学学校. 渋谷外語学院は、50年にわたり日本語や外国語を教えてきた歴史ある語学学校です。. 同じ校内には外国語を学ぶ日本人のクラスもあり、. 身近に触れ合うことができるので、. 習ったことをすぐに試すことができます。. 少人数でアットホームなクラス、会話力に特に重点をおいた指導方法が好評で、. 「使える日本語」の習得をするのにとても理想的な環境です。. コースについて. オンライン完結の申し込みプロセス. 安全で便利なオンライン完結のお申し込みプロセスをご用意しました。. もちろん学生ビザが必要なコースのお申込みにもご利用いただけます。. 留学ビザ取得の流れ. SUBMIT APPLICATION. Application formを送信する. APPLICATION FEE. 選考料のお支払い. RECEIVE CoE. 在留資格認定証明書（CoE）を受け取る. VISA APPLICATION. 留学ビザの申請・交付を受ける。. START SCHOOL. 日本での生活がスタート！. お申し込みはこちら. 入学前にわからないことがあっても、スタッフが親切に答えてくれました。. Nathaly. コロンビア. 先生方の質の高い指導方法とサポートが渋谷外語学院をお勧めする理由です。. Anthony. イタリア. ただ学校に通うというより、一緒に勉強をしに友達に会いに行くような感じ。. Nicole. ドイツ. 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. SHIBUYA GAIGO GAKUIN. NEWS. State of Emergency. April 26, 2021. Facebook. Twitter. In response to the current State of Emergency that was recently announced, we are continuing to ensure our best efforts to be compliant to Japan Government regulations. To prevent the spread of infection, please refrain from coming to the school if possible. In order for us to offer the proper service to our students, we would like to kindly remind you, for those who need to visit the school building if necessary, you must make an appointment or notify us via email a day in advance minimum before your arrival. Most of our staff are continuing to work remotely during this time to ensure the safety of our staff, teachers and other students. Therefore, please contact the school directly with the date and time you wish to visit the school so we may properly prepare. Our office hours and email response times are Weekdays from 9:00 – 17:00. We appreciate your understanding and cooperation. Posts navigation. &larr; April 2021 Term Online Immigration Updates and COVID-19 VACCINE &rarr; Recent News. Change of address and contact information Immigration Updates and COVID-19 VACCINE State of Emergency April 2021 Term Online School Visits Require Advance Appointment! Tokyo State of Emergency Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. Privacy Policy Designed by Souzou Web Scroll To Top.","wordcount_i":257,"linkscount_i":39,"linksnofollowcount_i":0,"inboundlinkscount_i":28,"outboundlinkscount_i":11,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https","018-https","019-https","020-https","021-https","022-https","023-https","024-https","025-https","026-https","027-https"],"inboundlinks_urlstub_sxt":["shibuyajp.net\/state-of-emergency\/","shibuyajp.net\/wp-content\/plugins\/bb-plugin\/fonts\/fontawesome\/5.15.4\/webfonts\/fa-brands-400.woff2","shibuyajp.net\/wp-json\/","shibuyajp.net\/wp-json\/wp\/v2\/posts\/5331","shibuyajp.net\/xmlrpc.php?rsd=","shibuyajp.net\/wp-includes\/wlwmanifest.xml","shibuyajp.net\/?p=5331","shibuyajp.net\/wp-json\/oembed\/1.0\/embed?url=https%3A%2F%2Fshibuyajp.net%2Fstate-of-emergency%2F","shibuyajp.net\/wp-json\/oembed\/1.0\/embed?url=https%3A%2F%2Fshibuyajp.net%2Fstate-of-emergency%2F&","shibuyajp.net\/site.webmanifest","shibuyajp.net\/","shibuyajp.net\/category\/news\/","shibuyajp.net\/school\/","shibuyajp.net\/japanese-course\/","shibuyajp.net\/schedule\/","shibuyajp.net\/student-housing\/","shibuyajp.net\/learn-japanese-online-application\/","shibuyajp.net\/shibuya-gaigo-gakuin\/","shibuyajp.net\/application","shibuyajp.net\/april-2021-term-online\/","shibuyajp.net\/covid-19-vaccine\/","shibuyajp.net\/change-of-address-and-contact-information\/","shibuyajp.net\/visiting-school-requires-advance-appointment\/","shibuyajp.net\/tokyo-state-of-emergency\/","shibuyajp.net\/students-interviews\/","shibuyajp.net\/application\/","shibuyajp.net\/level-check\/","shibuyajp.net\/privacy-policy\/"],"inboundlinks_anchortext_txt":["","","","","","","","","","","","News","School","Courses","Schedule","Accommodation","Application","Contact us","APPLY NOW","&larr; April 2021 Term Online","Immigration Updates and COVID-19 VACCINE &rarr;","Change of address and contact information","School Visits Require Advance Appointment!","Tokyo State of Emergency","Student Voice","APPLY NOW","LEVEL CHECK","Privacy Policy"],"outboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https"],"outboundlinks_urlstub_sxt":["gmpg.org\/xfn\/11","fonts.googleapis.com\/","s.w.org\/","fonts.gstatic.com\/","www.facebook.com\/Shibuya.Gaigo.Gakuin\/","twitter.com\/shibuya_gaigo?lang=en","www.instagram.com\/shibuyagaigo\/","www.facebook.com\/sharer.php?u=https%3A%2F%2Fshibuyajp.net%2Fstate-of-emergency%2F","twitter.com\/share?url=https%3A%2F%2Fshibuyajp.net%2Fstate-of-emergency%2F","twitter.com\/shibuya_gaigo","souzouweb.com\/"],"outboundlinks_anchortext_txt":["","","","","","","","Facebook","Twitter","","Designed by Souzou Web"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:52:16 UTC 2022","fresh_date_dt":"Wed Jun 08 16:52:16 UTC 2022","referrer_id_s":"-amSNs9l2MMx","language_s":"en","size_i":65709,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086190430781440}
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Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Swaziland (renamed to Eswatini). Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Togo. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Tuvalu. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Zambia. Zimbabwe. Inquiry Categories. *. Courses. Student Visa. Application Procedure. Accommodation. Other. Message. *. Shibuya Gaigo Gakuin. Address. Yakō Bldg. 2F, 3F. 1-34-6 Takadanobaba,. Shinjuku-ku, Tokyo. 169-0075 JAPAN. Telephone (Temporary). (+81) 70-1371-8866 Fax (Temporarily Unavailable). -. LINE Official Account. @shibuyajp Office Hours. Monday to Friday. 9:00~17:30. How to get to Shibuya Gaigo Gakuin. 1 minute walk from Toyama exit of the Takadanobaba Station on the JR Yamanote Line. Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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{"id":"N3KGAd16muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/en\/seminar\/seminar-shimizu.html","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":64,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["en","seminar"],"url_file_name_s":"seminar-shimizu","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Kyoto Young Topologists Seminar｜RIMS"],"last_modified":"Mon May 23 22:23:52 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-945419988143787691,"exact_signature_unique_b":true,"fuzzy_signature_l":-4470049805742695356,"fuzzy_signature_unique_b":true,"h1_txt":["Colloquium\/Seminars","Colloquium\/Seminars"],"h2_txt":["Kyoto Young Topologists Seminar"],"bold_txt":["Anosov representations and their deformation spaces","Factorizations of the universal R matrix and the universal quantum invariant for framed 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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Colloquium\/Seminars. TOP 　　. Seminars 　　Kyoto Young Topologists Seminar. Kyoto Young Topologists Seminar. Date. 22th (Mon) July, From 2:00 p.m., 2019. Speaker. David Leturcq (Institut Fourier, Universite Grenoble Alpes). Title. Knot invariants from counting diagrams. Abstract. A long knot is an embedding from R into R^3 which follows the z-axis outside a ball. Two such long knots psi and psi' are isotopic if there exists a smooth family (psi_t) of long knots such that psi_0= psi and psi_1=psi'. In this talk, we will explain how to define a long knot isotopy invariant z_2 counting some diagrams with four vertices. The obtained invariant is a Vassiliev invariant and is then connected to the Alexander polynomial. We will explain how to get this relation directly from the diagram counting definition. In the remaining time, we will present the generalization of these objects, methods, and results to higher dimensions or to higher number of vertices. Venue. Room 206. Organizer. Organizers: Takahiro Oba (RIMS) , Kenta Okazaki (RIMS), Tatsuro Shimizu (RIMS). HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Date. 19th (Fri.) July, From 3:00 p.m., 2019. Speaker. Tomonori Fukunaga (Kyushu Sangyo University). Title. On convexity of simple closed frontals. Venue. Room 006. Organizer. Organizers: Takahiro Oba (RIMS) , Kenta Okazaki (RIMS), Tatsuro Shimizu (RIMS). HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Date. 3rd July (Wed.) From 1:30 p.m., 2019. Speaker. Toru Yoshiyasu (Kyoto). Title. On Lagrangian embeddings of closed non-orientable 3-manifolds. Abstract. I will give new examples of closed non-orientable Lagrangian submanifolds in the standard symplectic 6-space. These Lagrangians are constructed by concatenating a Lagrangian filling and a Lagrangian cap. The existence of a Lagrangian cap is a consequence of Eliashberg-Murphy’s h-principle. The main part of the proof is to construct a Lagrangian filling of a loose Legendrian torus in the standard contact 5-space. After reviewing basics on the h-principle, I will explain the construction. Venue. Room 402. Organizer. Organizers: Takahiro Oba (RIMS) , Kenta Okazaki (RIMS), Tatsuro Shimizu (RIMS). HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Date. 28th Feb. and 1st Mar., 2019. Speaker. Koji Yamazaki (Tokyo Tech). Title. Engel Manifolds and Contact Structures. Abstract. A completely non-integrable 2-distribution on a 4-manifold is called an Engel structure. An Engel manifold is a manifold equiped with an Engel structure. Engel manifolds are very similar and closely related to contact manifolds. In this seminar, we follow R. Montgomery's tequniques with the characteristic foliation, the Cartan prolongation and the development map. Moreover, we give the later application of that including my result. Speaker. Nobuo Iida(univ. of Tokyo). Title. Bauer-Furuta type refinement of Kronheimer-Mrowka's invariant for four-manifolds with contact boundary. Abstract. The Seiberg-Witten invariant is an invariant for closed 4-manifolds and there are many variants of it.I construct a new variant of the Seiberg-Witten invariant based on two previous works. First, Bauer and Furuta refined the Seiberg-Witten invariant, and made an invariant called the stable cohomotopy invariant, which is an S^1-equivariant stable homotopy map obtained by Furuta's finite dimensional approximation of the Seiberg-Witten map. Second, Kronheimer and Mrowka defined a variant of the Seiberg-Witten invariant for 4-manifolds with contact boundary. I combine these two variants of the Seiberg-Witten invariant; that is, using Furuta's finite dimensional approximation, I refine Kronheimer-Mrowka's invariant for 4-manifolds with contact boundary. Timetable:. 28th Feb. 14:00-15:30 Yamazaki(1). 16:00-17:30 Iida(1). 1st Mar. 10:00-11:30 Yamazaki (2). 13:00-14:30 Iida (2). Venue. Room 110. Organizer. Organizers: Takahiro Oba (RIMS) , Kenta Okazaki (RIMS), Tatsuro Shimizu (RIMS). HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Date. 13th Feb.(Wed.) From 1:00 p.m., 2019. Speaker. Katsumi Ishikawa (RIMS). Title. Quandle coloring conditions and zeros of the Alexander polynomials of Montesinos links. Abstract. We give a simple condition for the existence of a nontrivial quandle coloring on a Montesinos link, which describes the distribution of the zeros of the Alexander polynomial. By this condition, we show the existence of infinitely many counterexamples for Hoste's conjecture. Venue. Room 206. Organizer. Organizers: Takahiro Oba (RIMS) , Kenta Okazaki (RIMS), Tatsuro Shimizu (RIMS). HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Date. 7th (Fri.) Dec. From 3:00 p.m., 2018. Speaker. Hironobu Naoe (Tohoku univ.). Title. Shadows and Milnor fibrations of divides. Abstract. A’Campo introduced a divide as a generalization of real morsified curves of complex plane curve singularities. A’Campo showed that the link of a connected divide is fibered, moreover such a fibration comes from the “boundary” of a Lefschetz fibration. We interpret them in terms of Turaev' s shadows. This is a joint work with Masaharu Ishikawa. Venue. Room 110. Organizer. Organizers: Takahiro Oba (RIMS) , Kenta Okazaki (RIMS), Tatsuro Shimizu (RIMS). HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Date. 27th (Mon.) August, 2:30 p.m.-- , 2018. Speaker. Delphine Moussard (RIMS). Title. Torsions of 4-manifolds from trisection diagrams. Abstract. We will see how to compute the (non-)twisted homology, the (non-)twisted intersection form and the abelian torsions of a 4-manifold from a trisection diagram. This is a joint work with Vincent Florens. HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Venue. #478(4th floor of the Research Building no.2 in Yoshida Main campus). Organizer. Delphine Moussard(RIMS), Kenta Okazaki(RIMS), Tatsuro Shimizu(RIMS). Date. 23th (Mon.) July, 2:15 p.m.-- , 2018. Speaker. Sakie Suzuki (Tokyo Tech). Title. Factorizations of the universal R matrix and the universal quantum invariant for framed 3-manifolds. Abstract. Take the Drinfeld double D(B) of the Borel subalgebra B of the quantized enveloping algebra Uq(sl2) of sl2. We consider two embeddings of D(B) as an algebra, into a double of Heisenberg double and into a quantum torus algebra. With both embeddings, each image of the universal R matrix has a factorization into a product of four elements each satisfying a pentagon relation. This setting leads us to the Kashaev invariant of links and to quantum Teichmuller* theory . In this talk I will explain these situations and show our trials to unify these studies in a view point of the universal S tensor and framed 3-manifolds. This talk includes a joint work with Y. Terashima. (*: \"u\" is u-Umlaut.). HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Venue. 110 @RIMS main building. Organizer. Delphine Moussard(RIMS), Kenta Okazaki(RIMS), Tatsuro Shimizu(RIMS). Date. 25th(Mon.) June, 13:00 -- , 2018. Speaker. Yusuke Inagaki(Osaka). Title. Anosov representations and their deformation spaces. Abstract. Anosov representations are representations of Gromov hyperbolic groups into higher rank Lie groups with a dynamical property. These representations have been introduced by Labourie, and studied in the viewpoint of a generalization of Kleinian groups. In this talk, we review the definition and remarkable properties of Anosov representations, and discuss examples of their deformation spaces, called Hitchin components. HP：http:\/\/www.kurims.kyoto-u.ac.jp\/~shimizu\/YTS.html Venue. Room #478(4th floor of the Research Building no.2 in Yoshida Main campus). Organizer. Delphine Moussard(RIMS), Kenta Okazaki(RIMS), Tatsuro Shimizu(RIMS). Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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{"id":"OjhfDd16muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/en\/seminar\/seminar-motizuki.html","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":65,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["en","seminar"],"url_file_name_s":"seminar-motizuki","url_file_ext_s":"html","content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["特別講演 \/ Lecture｜RIMS"],"last_modified":"Mon May 23 22:23:45 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":2358398812730008979,"exact_signature_unique_b":true,"fuzzy_signature_l":-4079802848791921767,"fuzzy_signature_unique_b":true,"h1_txt":["Colloquium\/Seminars","Colloquium\/Seminars"],"h2_txt":["特別講演 \/ Lecture"],"bold_txt":["On p-adic differential equations, q-difference equations, Galois representations and Hasse-Arf filtrations","The Hitchin-Mochizuki Morphism"],"imagescount_i":2,"images_protocol_sxt":["000-https","001-https"],"images_urlstub_sxt":["www.kurims.kyoto-u.ac.jp\/en\/files\/top\/images\/sitelogo.png","www.kurims.kyoto-u.ac.jp\/ja\/common\/images\/fb-icon.png"],"images_alt_sxt":["京都大学数理解析研究所｜RIMS-Research Institute for Mathematical Sciences Kyoto University",""],"images_height_val":[-1,-1],"images_width_val":[-1,-1],"images_text_t":"www kurims kyoto u ac jp en files top images sitelogo png 京都大学数理解析研究所｜RIMS-Research Institute for Mathematical Sciences Kyoto University ja common fb icon","responsetime_i":858,"text_t":"特別講演 \/ Lecture｜RIMS. HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Colloquium\/Seminars. TOP 　　. Seminars 　　特別講演 \/ Lecture. 特別講演 \/ Lecture. Title. The Hitchin-Mochizuki Morphism. Date. June 3 (Fri), 2011, 11:00-12:00. Room. RIMS, Room 206. Speaker. Kirti JOSHI (University of Arizona, USA). Abstract. S. Mochizuki in his foundational work on p-adic uniformization of curves initiated the study of indigenous vector bundles of rank two in positive characteristic and defined a natural morphism from the space of such bundles to the space of quadratic differentials. In this talk I will explain a part of my recent joint work with C. Pauly where we provide a natural generalization of this morphism (which we call the Hitchin-Mochizuki morphism) to opers, which are natural generalizations of indigenous bundles to higher ranks. Organizer. S. Mochizuki. Slides. PDF Title. On p-adic differential equations, q-difference equations, Galois representations and Hasse-Arf filtrations. Date. January 31 (Mon), 2005, 15:00-17:00. Room. RIMS, Room 206. Speaker. Yves ANDRE (Ecole Normale Superieure, France). Abstract. We shall first sketch two relatively well-known analogies: between differential equations and q-difference equations on one hand, and between Galois representations of local fields of characteristic p and complex linear differential equations over a punctured disk on the other hand. In the p-adic world, it turns out that both analogies can be substantially strengthened, and we shall explain how they eventually lead to some equivalences of categories. The three categories under consideration carry natural filtrations of ``Hasse-Arf type\", which turn out to correspond to each other despite their very different natures. A byproduct of the theory is that every p-adic differential module defined over an infinitesimally thin annulus of outer radius one, and which is invariant (up to isomorphism) under raising the variable x to the p-th power, can be canonically deformed into a family of q-difference modules over the annulus, parametrized by a p-adic number q close to 1. 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University. HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Workshop for Principal Researcher Call for proposals. Call for proposals（RIMS Research Projects） Call for proposals（RIMS Workshops\/Symposia） Call for proposals（RIMS Satellite Seminars） Call for proposals（RIMS Review Seminars） Workshop. TOP 　　 Workshop. Symposia (open). Various aspects of multiple zeta values. Location:. Hybrid Meeting. Period:. 2022-05-16--2022-05-20. Organizer:. TASAKA Koji(School of Information Science and Technology, Aichi Prefectural University). New trends of transformation groups. Location:. Hybrid Meeting. Period:. 2022-05-24--2022-05-27. Organizer:. FUJITA Hajime(Faculty of Science, Japan Women’s University). Intelligence of Low-dimensional Topology Location:. Hybrid Meeting. Period:. 2022-05-25--2022-05-27. Organizer:. OHTSUKI Tomotada(RIMS, Kyoto University). Regularity and asymptotic analysis for critical cases of partial differential equations Location:. Hybrid Meeting. Period:. 2022-06-01--2022-06-03. Organizer:. OGAWA Takayoshi (Gradute Scrhool of Science, Tohoku University). MISAWA Masashi (Faculty of Science, Kumamoto University). Set-Theoretic and Geometric Topology, and their applications to related fields Location:. Online via Zoom. Period:. 2022-06-06--2022-06-08. Organizer:. MATSUHASHI Eiichi(Interdisciplinary Faculty of Science and Engineering, Shimane University). Development and Applications of Dynamical Systems. Location:. Hybrid Meeting. Period:. 2022-06-06--2022-06-10. Organizer:. OBAYASHI Ippei(Cypher, Okayama University). Japan forum associated with ICM 2022 Location:. Hybrid Meeting. Period:. 2022-06-13--2022-06-14. Organizer:. KOZONO Hideo (Faculty of Science and Engineering, Waseda University･RACMaS, Tohoku University). SHIMIZU Senjo (Graduate School of Human and Environmental Studies, Kyoto University). Mathematical Logic and its Applications. Location:. Hybrid Meeting. Period:. 2022-06-21--2022-06-24. Organizer:. KADA Masaru(Graduate School of Science, Osaka Metropolitan University). New developments in submanifold theory and geometric analysis. Location:. Hybrid Meeting. Period:. 2022-06-27--2022-06-29. Organizer:. HONDA Atsufumi (Faculty of Engineering, Yokohama National University). Mathematical Analysis in Fluid and Gas Dynamics Location:. Rm 420. Period:. 2022-07-06--2022-07-08. Organizer:. UEDA Yoshihiro(Faculty of Maritime Sciences, Kobe University). Harmonic Analysis and Nonlinear Partial Differential Equations. Location:. Rm 420. Period:. 2022-07-11--2022-07-13. Organizer:. TARASAWA Yutaka (Graduate School of Mathematics, Nagoya University). MASAKI Satoshi (School of Engineering Science , Osaka University). Predictability and controllability of turbulence. Location:. Rm 420. Period:. 2022-07-20--2022-07-22. Organizer:. MATSUMOTO Takeshi (Graduate School of Science, Kyoto University). TOH Sadayoshi (Graduate School of Science, Kyoto University). Various Issues on Representation Theory and Related Topics. Location:. Rm 111. Period:. 2022-07-26--2022-07-29. Organizer:. NAOI Katsuyuki(Faculty of Engineering, Tokyo University of Agriculture and Technology). Mathematical structures of integrable systems, their developments and applications. Location:. Rm 420. Period:. 2022-08-08--2022-08-10. Organizer:. MATSUYA Keisuke (Faculty of Engineering, Musashino University). Probability and Analysis on Random Structures and Related Topics Location:. Rm 111. Period:. 2022-08-08--2022-08-10. Organizer:. KAJINO Naotaka(RIMS, Kyoto University). Mathematical Optimization: Models, Theory, and Algorithms. Location:. Rm 111. Period:. 2022-08-29--2022-08-30. Organizer:. GOTOH Jun-ya(Faculty of Science and Engineering, Chuo University). Study on Nonlinear Analysis and Convex Analysis. Location:. Rm 110. Period:. 2022-08-29--2022-08-31. Organizer:. TOYODA Masashi (Faculty of Science, Toho University). AOYAMA Koji (Faculty of Law, Politics and Economics, Chiba University). Study of Mathematical Software and Its Effective Use for Mathematics Education. Location:. Maskawa Hall Period:. 2022-08-29--2022-08-31. Organizer:. KANEKO Masataka(Faculty of pharmaceutical sciences, Toho University). Symposium of Algebra. Location:. Rm 420. Period:. 2022-08-30--2022-09-02. Organizer:. MORIWAKI Atsushi(Graduate School of Science, Kyoto University). Recent Developments in Operator Algebras. Location:. Rm 111. Period:. 2022-09-05--2022-09-07. Organizer:. OZAWA Narutaka(RIMS, Kyoto University). Study of the History of Mathematics. Location:. Rm 110. Period:. 2022-09-05--2022-09-08. Organizer:. OGAWA Tsukane(Seki Kowa Institute of Mathematics …, Yokkaichi University). Financial Modeling and Analysis. Location:. Doshisha University Imadegawa Campus Period:. 2022-09-07--2022-09-09. Organizer:. TSUJIMURA Motoh(Graduate School of Commerce, Doshisha University). [Promotion of Women's Participation] Women in Mathematics Location:. Rm 420. Period:. 2022-09-07--2022-09-09. Organizer:. ITO Yukari(Kavli IPMU, The University of Tokyo). Analysis, Geometry and Stochastics on Metric Spaces. Location:. Rm 111. Period:. 2022-09-26--2022-09-28. Organizer:. TUSCHMANN Wilderich (IAG, Karlsruhe Institute of Technology). SHIMIZU Senjo (Graduate School of Human and Environmental Studies, Kyoto University). Workshop on Non-compact Variational Problems and Related Topics. Location:. Rm 420. Period:. 2022-09-26--2022-09-28. Organizer:. TAKAHASHI Futoshi (Graduate School of Science, Osaka Metropolitan University). ISHIWATA Michinori (Graduate School of Engineering Science, Osaka University). Perspectives of studies on random\/non-autonomous dynamical systems: theory and applications. Location:. Rm 201,Research Building No.15. Period:. 2022-09-26--2022-09-30. Organizer:. SATO Yuzuru(Research Institute for Electronic Sciences, Hokkaido University). [Promotion of Women's Participation]. Nonlinear and Random Waves. Location:. Rm 110. Period:. 2022-10-03--2022-10-05. Organizer:. FUKUIZUMI Reika (GSIS, Tohoku University). Anne de Bouard (CMAP, Ecole Polytechnique). Singularity theory and its applications. Location:. Rm 420. Period:. 2022-10-03--2022-10-05. Organizer:. SAJI Kentaro (Graduate School of Science, Kobe University). Prospects in microlocal analysis and asymptotic analysis. Location:. Rm 111. Period:. 2022-10-03--2022-10-07. Organizer:. HONDA Naofumi(Faculty of Science, Hokkaido University). Analytic Number Theory and Related Topics. Location:. Rm 420. Period:. 2022-10-11--2022-10-14. Organizer:. YAMASAKI Yoshinori (Graduated School of Science and Engineering, Ehime University). Numerical Analysis for Future Information Society: from Edge to Fugaku. Location:. Maskawa Hall Period:. 2022-10-12--2022-10-14. Organizer:. SUDA Reiji(Graduate School of Information Science and Technology, The University of Tokyo). Innovation of the theory for evolution equations: developments via cross-disciplinary studies. Location:. Rm 420. Period:. 2022-10-17--2022-10-19. Organizer:. AKAGI Goro(Gradute Scrhool of Science, Tohoku University). Singularity theory and geometric topology. Location:. Rm 420. Period:. 2022-10-24--2022-10-27. Organizer:. SAJI Kentaro (Graduate School of Science, Kobe University). New Developments on Forcing and Cardinal Arithmetic. Location:. Rm 111. Period:. 2022-10-25--2022-10-28. Organizer:. SAKAI Hiroshi(Graduate School of System Informatics, Kobe University). Research on Real, Complex and Functional Analysis from the Perspective of Reproducing Kernel Hilbert Spaces. Location:. Rm 110. Period:. 2022-10-26--2022-10-28. Organizer:. SETO Michio (Department of Mathematics, National Defense Academy). SAWANO Yoshihiro (Faculty of Science and Technology, Chuo University). Mathematical Decision Making Under Uncertainty and Related Topics. Location:. Rm 111. Period:. 2022-10-31--2022-11-02. Organizer:. TAMURA Yoshinobu(Graduate School of Sciences and Technology for Innovation, Yamaguchi University). Operator means and related topics. Location:. Rm 420. Period:. 2022-10-31--2022-11-02. Organizer:. YANAGIDA Masahiro(Faculty of Science, Tokyo University of Science). Singularity theory and differential geometry. Location:. Not decided yet. Period:. 2022-11-07--2022-11-09. Organizer:. SAJI Kentaro (Graduate School of Science, Kobe University). Recent developments in Combinatorial Representation Theory. Location:. Rm 420. Period:. 2022-11-07--2022-11-11. Organizer:. IKEDA Takeshi(School of Fundamental Science and Engineering, Waseda University). Mathematical Analysis on pattern formation, propagation and interfacial phenonema. Location:. Rm 420. Period:. 2022-11-14--2022-11-16. Organizer:. MATSUZAWA Hiroshi (Faculty of Science, Kanagawa University). NINOMIYA Hirokazu (School of Interdisciplinary Mathematical Sciences, Meiji University). Qualitative Theory on ODEs and its Applications to Mathematical Sciences. Location:. Rm 111. Period:. 2022-11-14--2022-11-16. Organizer:. MATSUNAGA Hideaki(Graduate School of Science, Osaka Metropolitan University). Rigorous statistical mechanics and related topics. Location:. Rm 110. Period:. 2022-11-15--2022-11-18. Organizer:. SASAMOTO Tomohiro(School of Science, Tokyo Institute of Technology). Time-delay systems and mathematical sciences: new development of theory and applications. Location:. Rm 201,Research Building No.15. Period:. 2022-11-16--2022-11-18. Organizer:. NISHIGUCHI Junya(Advanced Institute for Materials Research , Tohoku University). Mathematical analysis of viscous incompressible fluid. Location:. Rm 420. Period:. 2022-11-30--2022-12-02. Organizer:. HISHIDA Toshiaki(Graduate School of Mathematics , Nagoya University). Spectral and Scattering Theory and Related Topics. Location:. Rm 111. Period:. 2022-11-30--2022-12-02. Organizer:. ISHIDA Atsuhide(Institute of Arts and Sciences, Tokyo University of Science). Mathematical Aspects of Quantum Fields and Related Topics. Location:. Rm 110. Period:. 2022-12-05--2022-12-07. Organizer:. HIROSHIMA Fumio (Faculty of Mathematics, Kyushu University). Geometric Aspects of Partial Differential Equations. Location:. Rm 111. Period:. 2022-12-05--2022-12-07. Organizer:. ONODERA Michiaki(School of Science, Tokyo Institute of Technology). Research on finite groups, algebraic combinatorics, and vertex algebras. Location:. Rm 420. Period:. 2022-12-05--2022-12-08. Organizer:. MIEZAKI Tsuyoshi(Faculty of Science and Engineering, Waseda University). Model theoretic aspects of the notion of independence and dimension. Location:. Rm 110. Period:. 2022-12-12--2022-12-14. Organizer:. IKEDA Koichiro(Faculty of Business Administration, Hosei University). Classification problems in singularity theory and their applications. Location:. Not decided yet. Period:. 2022-12-12--2022-12-15. Organizer:. SAJI Kentaro (Graduate School of Science, Kobe University). Algebraic Number Theory and Related Topics. Location:. Rm 420. Period:. 2022-12-12--2022-12-16. Organizer:. KASHIO Tomokazu(Faculty of Science and Technology, Tokyo University of Science). Complex dynamics and related topics. Location:. Rm 111. Period:. 2022-12-12--2022-12-16. Organizer:. KAWAHIRA Tomoki(Graduate School of Economics, Hitotsubashi University). Computer Algebra – Foundations and Applications. Location:. Rm 111. Period:. 2022-12-19--2022-12-21. Organizer:. KITAMOTO Takuya (Faculty of Education, Yamaguchi University). HAMADA Tatsuyoshi (College of Bioresource Sciences, Nihon University). Probability Symposium. Location:. Rm 420. Period:. 2022-12-19--2022-12-22. Organizer:. HORA Akihito (Department of Mathematics, Hokkaido University). FUKUSHIMA Ryoki (Faculty of Pure and Applied Sciences, University of Tsukuba). Inverse Problems, Medical Imaging, and Related Topics. Location:. Rm 420. Period:. 2023-01-11--2023-01-13. Organizer:. MACHIDA Manabu (Preeminent Medical Photonics Education & Research Center, Hamamatsu University). SAMULI Siltanen (Dept. of Math. & Statistics, Univ. of Helsinki). Analytic and arithmetic aspects of automorphic representations. Location:. Rm 420. Period:. 2023-01-23--2023-01-27. Organizer:. MIYAZAKI Tadashi(College of Liberal Arts and Sciences, Kitasato University). Foundations of Computer Science and their New Trends. Location:. Rm 420. Period:. 2023-01-30--2023-02-01. Organizer:. YASUNAGA kenji(School of Computing, Tokyo Institute of Technology). Workshop on Variational Methods and Dispersive Equations. Location:. Rm 420. Period:. 2023-02-06--2023-02-08. Organizer:. TAKAHASHI Futoshi (Graduate School of Science, Osaka Metropolitan University). Theory of function spaces and related topics. Location:. Rm 420. Period:. 2023-02-13--2023-02-15. Organizer:. OHWADA Tomoyoshi (Faculty of Education, Shizuoka University). MATSUOKA Katsuo (College of Economics, Nihon University). Workshop on Variational Methods and Functional Inequalities. Location:. Osaka Metropolitan University, Department of Science, Large Lecture Room (E408) Period:. 2023-02-13--2023-02-15. Organizer:. TAKAHASHI Futoshi (Graduate School of Science, Osaka Metropolitan University). NAKANISHI Kenji (RIMS, Kyoto University). Characteristic Polynomials of Hyperplane Arrangements and Ehrhart Polynomials of Convex Polytopes. Location:. Rm 110. Period:. 2023-02-14--2023-02-17. Organizer:. NAKASHIMA Norihiro(Faculty of Engineering, Nagoya Institute of Technology). Group, Ring, Language and Related Areas in Computer Science. Location:. Rm 111. Period:. 2023-02-15--2023-02-17. Organizer:. NISHINAKA Tsunekazu(School of Economics and Management, University of Hyogo). [Promotion of Women's Participation]. Zeta functions and their representations. Location:. Rm 420. Period:. 2023-03-06--2023-03-08. Organizer:. NAKASUJI Maki(Faculty of Science and Technology, Sophia University). Aspects of Educational Mathematics -- Diversity and universality in higer education of mathematics --. Location:. Rm 111. Period:. 2023-03-06--2023-03-09. Organizer:. SHIMIZU Yuji (College of Liberal Arts, International Christian University). Workshops (Type A). Functional and Logic Programming. Location:. Hybrid Meeting. Period:. 2022-05-10--2022-05-12. Organizer:. IGARASHI Atsushi (Graduate School of Informatics, Kyoto University). HANUS Michael (Kiel University). Multifaceted Research on Linear and Nonlinear Dispersive Equations. Location:. Hybrid Meeting. Period:. 2022-05-16--2022-05-19. Organizer:. MIYAZAKI Hayato(Faculty of Education, Kagawa University). Applications of Harmonic Maps and Higgs Bundles to Differential Geometry. Location:. Hybrid Meeting. Period:. 2022-05-28--2022-05-31. Organizer:. PEDIT Franz(University of Massachusetts Amherst). Mathematics of multi-dimensional quantum walks. Location:. Hybrid Meeting. Period:. 2022-06-20--2022-06-22. Organizer:. ASAHARA Keisuke (Data Science and AI Innovation Research Promotion Center, Shiga University). WADA Kazuyuki (Department of General Science and Education, Hachinohe National College of Technology). [Canceled]. [Promotion of Women's Participation]. Generalizing the Knutson-Miller program for determinantal-type ideals. Location:. Rm 111. Period:. 2022-07-01--2022-07-15. Organizer:. HARADA Megumi (Faculty of Science Department of Mathematics and Statistics, McMaster University). Mathematical geographical modelling for environmental humanities Location:. Hybrid Meeting. Period:. 2022-07-06--2022-07-07. Organizer:. FUJIWARA Naoya(Graduate School of Information Sciences, Tohoku University). Seminar on Combinatorial Optimization. Location:. Rm 420. Period:. 2022-07-25--2022-07-27. Organizer:. MAKINO Kazuhisa(RIMS, Kyoto University). Forward and inverse scattering problems in quantum mechanics. Location:. Rm 420. Period:. 2022-09-20--2022-09-22. Organizer:. TADANO Yukihide(Faculty of Science, Tokyo University of Science). Summer School on Variational Problems and Functional Inequalities. Location:. Osaka Metropolitan University, Department of Science, Large Lecture Room (E408) Period:. 2022-09-21--2022-09-22. Organizer:. TAKAHASHI Futoshi (Graduate School of Science, Osaka Metropolitan University). Animals’ motions in relation to biofluid mechanics. Location:. Rm 110. Period:. 2022-10-31--2022-11-02. Organizer:. IIMA Makoto(Graduate School of Integrated Sciences for Life, Hiroshima University). Mathematics of tilings and quasiperiodic systems. Location:. Rm 111. Period:. 2022-11-07--2022-11-09. Organizer:. YAMAGISHI Yoshikazu(Faculty of advanced science and technology, Ryukoku University). Recent Progress in Ergodic Theory. Location:. Rm 110. Period:. 2022-11-09--2022-11-11. Organizer:. NATSUI Rie(Faculty of Science, Japan Women’s University). Group actions on operator algebras and mathematical physics. Location:. Rm 111. Period:. 2023-01-23--2023-01-25. Organizer:. SATO Yasuhiko(Faculty of Mathematics, Kyushu University). Extensibility and non-extensibility of graphs to edge-colored graphs. Location:. Rm 420. Period:. 2023-02-27--2023-03-03. Organizer:. YASHIMA Takamasa(Department of Computer and Information Science, Seikei University). New developments of nonlinear problems with precision analyses. Location:. Rm 201,Research Building No.15. Period:. 2023-03-06--2023-03-08. Organizer:. TANAKA Satoshi (Faculty of Science, Tohoku University). NAITO Yuki (Graduate School of Advanced Science and Engineering, Hiroshima University). Efficiencies of Estimation Procedures in Various Statistical Models. Location:. Rm 110. Period:. 2023-03-06--2023-03-08. Organizer:. KOIKE Ken-ichi(College of Commerce, Nihon University). Workshops (Type A) - Tandem-workshop -. Workshops (Type B). Postponed (new date:TBA). Statistical nature of Hall MHD turbulence. Location:. Rm 110. Period:. 2022-07-25--2022-08-06. Organizer:. MIURA Hideaki (Department of Helical Plasma Research, National Institute for Fusion Science). MATSUMOTO Takeshi (Graduate School of Science, Kyoto University). Analysis of fluid dynamical PDEs. Location:. Rm 110. Period:. 2023-03-13--2023-03-15. Organizer:. YAMAZAKI Kazuo (Texas Tech University). OHKITANI Koji (RIMS, Kyoto University). Workshops (Type C). Satellite Seminars. Homogenization and\/or non local operators. Location:. Toyoko Inn Hokkaido Sapporo-eki Kita-guchi Period:. 2022-09-05--2022-09-09. Organizer:. MASAMUNE Jun(Graduate School of Science, Tohoku University). The hypoelliptic Laplacian and tempered unitary representations. Location:. Nara Prefectural Convention Center Period:. 2022-11-21--2022-11-25. Organizer:. KATO Tsuyoshi(Graduate School of Science, Kyoto University). Mathematical modelling of turbulent flows in astrophysical and geophysical phenomena. Location:. Kansai Seminar House Period:. 2023-03-15--2023-03-18. Organizer:. TOBIAS Steven(Department of Applied Mathematics, University of Leeds). Review Seminars. Fluid Dynamics of superrotation phenomena observed in the atmospheres of slowly rotating planets （FDEPS 2022）. Location:. Kansai Seminar House Period:. 2022-11-29--2022-12-02. Organizer:. HAYASHI Yoshiyuki(Graduate School of Science, Kobe University). Long-term Visiting Researchers. Symposia (open) Workshops (Type A) Workshops (Type A) - Tandem-workshop - Workshops (Type B) Workshops (Type C) Satellite Seminars Review Seminars Long-term Visiting Researchers Past Workshops. AY2021 AY2020 AY2019 AY2018 AY2017- ←　BACK TO THE TOP About Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Follow on. Research Institute for Mathematical Sciences (RIMS)."}
{"index":{}}
{"id":"U29_Jg16muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~kenkyubu\/RIMSreport.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":57,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~kenkyubu"],"url_file_name_s":"RIMSreport","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["RIMSreport.pdf"],"author":"pdfTeX-1.40.21","author_sxt":["pdfTeX-1.40.21"],"last_modified":"Tue Mar 30 07:40:19 UTC 2021","dates_in_content_count_i":2,"dates_in_content_dts":["Fri Dec 16 00:00:00 UTC 2022","Sat Dec 16 00:00:00 UTC 2023"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":8663842673760574590,"exact_signature_unique_b":true,"fuzzy_signature_l":8644357348599312951,"fuzzy_signature_unique_b":true,"wordcount_i":4380,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:28 UTC 2022","fresh_date_dt":"Thu Jan 12 09:24:02 UTC 2023","referrer_id_s":"eOG-td16muho","publisher_t":"pdfTeX-1.40.21","language_s":"en","size_i":109966,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":28,"references_internal_i":28,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086013987946496,"text_t":"RIMS REVIEW 2021\nDAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI,\nAND TAKASHI TSUBOI\nCONTENTS\n1. Introduction 1\n2. Summary of Recommendations 2\n3. Assessment of RIMS organization and activities 3\n4. Facilities and Resources 6\n5. Internationalization 10\n6. Outreach 11\n7. Appendix A: Members of the Review Committee 12\n8. Appendix B. Description of the Evaluation Process 12\n1. INTRODUCTION\nMathematics is the foundation on which science and technology\nare built: The global pandemic has put the use of statistics for epi-\ndemiology, and for public health generally, in the spotlight. The in-\nteraction of mathematics and theoretical physics is currently very\nstrong. The very possibility of internet commerce depends on results\nin number-theory. These examples could be multiplied indefinitely.\nRIMS is a jewel of Japanese mathematics. It is widely known and\nhighly respected by foremost mathematicians around the world, and\na great many of them have visited RIMS for conferences or for longer\nstays. Intellectually, and in fame, RIMS operates on the highest in-\nternational plane.\nFounded in 1963 as a national hub for mathematics, RIMS has\nbroadened its activity to include graduate training at the Masters\nand Doctoral levels, and serves as an International Joint Usage Re-\nsearch Center.\nRIMS runs an extraordinarily large number of workshops and con-\nferences every year. It contributes broadly to the community by pro-\nviding outreach activities and helping to diversifying the population\nof mathematicians in Japan.\n1\n2 DAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI, AND TAKASHI TSUBOI\nThis review was initiated by RIMS Director Professor Takashi Ku-\nmagai. The review committee could not travel to RIMS because\nof the COVID-19 global pandemic, so RIMS prepared not only the\nusual documentation of its activities and needs, and the report of\nthe visiting committee of 2018, but also a sequence of video presen-\ntations and video interviews with students and junior and senior\nfaculty to inform the committee. As RIMS is a part of Kyoto Uni-\nversity, a meeting of the committee with Kyoto University President\nMinato around the crucial question of space was also arranged.\n!\nDavid Eisenbud, Chair of the Review Committee, March 29, 2021\n2. SUMMARY OF RECOMMENDATIONS\n(1) RIMS’ current physical facilities are not appropriate to its mis-\nsion as an international center of mathematics. To keep its\nplace as a beacon of Japanese mathematics in the world, it\nwill be important to find larger and better adapted space. A\nmajor part of the function of a research institute is to promote\ncollaboration, so if RIMS must be spread out over different\nbuildings, it is very important that they be close to one an-\nother. We strongly recommend that the available space be\nincreased and that the space be co-located.\n(2) We applaud the progress made since the 2018 review in se-\ncuring outside grants. It will be important to build on this\nsuccess, and ensure a multiplicity of funding sources and a\nmultiplicity of projects.\n(3) We note that two women researchers have been hired since\nthe 2018 review, a good start. It is important that RIMS con-\ntinue on this path, and actively develop the talent pool so\nthat the number of applications from women researchers can\ngrow; see Section 3.6 for particular suggestions.\n(4) To fully exploit the possibilities of international collaboration,\nRIMS should advertise more effectively and more widely in\nmedia with international reach; See Section 5 for particular\nsuggestions.\n(5) As Journal prices increase, RIMS and other institutions face\na growing problem of having sufficient journal access. We\nrecommend that Kyoto University cover the cost of electronic\njournal expenses, as it is of benefit not only to RIMS but also\nRIMS REVIEW 2021 3\nto other departments. If some sort of consortium arrange-\nment for electronic journal access is possible, this may be de-\nsirable.\n(6) As a government-funded agency, RIMS has a responsibility\nto help educate the public on the beauty, power and impor-\ntance of mathematics. We applaud the once-a-year event with\nlectures for a wide audience, and we recommend looking for\nfurther opportunities, especially those that can be promoted\non a national scale.\n3. ASSESSMENT OF RIMS ORGANIZATION AND ACTIVITIES\n3.1. RIMS as a leading International Mathematical Center. Since\nits establishment almost 60 years ago, RIMS has played an important\nrole in the organization and acceleration of mathematical research in\nJapan and worldwide. The certification of RIMS in 2020 as a Joint\nUsage \/ Research Center, and in 2018 as an International Joint Usage\n\/ Research Center underscored the ever increasing importance and\nimpact of RIMS activities.\nThe scope and the scale of RIMS activities is very impressive. In\n2018 and 2019 RIMS organized 125 international symposia (confer-\nences) and smaller meetings (workshops) intended to promote new\ninternational collaboration and advance certain particular problems.\nRIMS also enables satellite seminars devoted to focused research\non a particular topic, or to surveying an area of Mathematics and\nMathematical Sciences. RIMS has a successful visitors program that\nhosted a total of almost 8000 participants, of whom 925 were interna-\ntional, in the 2018 and 2019. 162 of these stayed at RIMS for at least\n2 weeks.\n3.2. Research at RIMS. The mathematical research produced by the\nfull and associate professors at RIMS is outstanding. They publish in\nhighly rated journals, and receive invitations to deliver named lec-\ntures and colloquia across the mathematical sciences. Three mem-\nbers of the RIMS faculty were invited speakers at the recent Inter-\nnational Congresses of Mathematicians, which took place in 2014 in\nSeoul, and in 2018 in Rio de Janeiro. In the last 4 years, RIMS full pro-\nfessors delivered over 20 invited lectures at major meetings around\nthe world. The area of expertise of the current 12 full professors of\nRIMS covers a broad area of mathematics: these include traditional\nstrengths in algebraic geometry, representation theory and number\n4 DAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI, AND TAKASHI TSUBOI\ntheory, but also areas of analysis, differential geometry, symplec-\ntic geometry, topology, operator algebras, probability, fluid dynam-\nics, discrete mathematics, logic and computer science, mathematical\nphysics, Quantum Field Theory.\nOperating at this level of mathematical achievement requires high\ncalibre researchers, who are given time, opportunity and support\nto develop their research. RIMS currently employs 12 full profes-\nsors, 10 associate professors, 3 lecturers and 11 assistant professors.\nPositions at all levels except Full Professor are filled after openly\nadvertised searches. Those at full, associate professor and lecturer\nlevels have permanent appointments, while assistant professors are\nappointed for 7 years with the possibility of renewal for further 3\nyears. Most of the assistant professors in recent years have moved\nto permanent faculty positions elsewhere in Japan well before their\nterms expired; this underscores the high quality of RIMS junior fac-\nulty. The number of appointments at the top two levels is limited by\nKyoto University. There are also project appointments and postdoc-\ntoral appointments that are dependent on externally funded grants.\nThe researchers interviewed for this review were very satisfied\nwith their position at RIMS. However, some expressed concern that\noffices are located across 5 different buildings on campus, making\nspontaneous conversations and collaborations difficult, especially for\nthe younger researchers. This problem has been exacerbated during\nthe COVID-19 pandemic, although seminars and workshops have\ncontinued online.\n3.3. Workshops, Conferences and Symposia. In 2019 alone, RIMS\nhosted 19 Workshops, 64 Symposia, 5 Satellite Seminars and 2 Re-\nview Seminars, serving a total of 4103 participants, of whom 542\nwere from abroad. The series seem to be well organized and well\nmanaged by a system of internal and external review committees.\n3.4. RIMS’ Graduate Program. As noted in previous reviews, the\nattraction of graduate students to RIMS is constrained by longstand-\ning circumstances. Students in Japan tend to stay in the same ge-\nographic location, rather than moving elsewhere to pursue under-\ngraduate or graduate study. RIMS researchers typically do not teach\nundergraduate students, so the recruitment of graduate students re-\nlies on reputation rather than personal knowledge.\nAlthough it is possible in principle for RIMS to recruit students\ninternationally, very few scholarships are available. The review com-\nmittee recommends that RIMS considers seeking philanthropic fund-\ning to overcome this problem.\nRIMS REVIEW 2021 5\nStudents choose their advisors early in the program. Although\nthis often works well, it has the potential to cause isolation and may\nlead to students acquiring a narrow view of the field. The review\ncommittee recommends that RIMS encourage networking among\nstudents in different groups, including joint seminars with the De-\npartment of Mathematics at Kyoto University and informal seminars\nwhere students could learn about each other’s research.\nDespite some effort, the number of female students in the doc-\ntoral program remains very low, and in the current cohort of stu-\ndents there are no female PhD students at all. The pool of female\napplicants is very small. To help change this unacceptable situation,\nRIMS should incorporate additional programs that will increase the\npool; some possibilities are suggested in Section 3.6.\nRIMS students are generally quite successful in finding jobs af-\nter completing their doctoral studies. In 2018 and 2019, 3 students\nobtained jobs as faculty members at Japanese universities, 7 found\npostdoctoral research positions and 3 were employed in industry.\nIt is excellent that RIMS, together with the Department of Math-\nematics, operates the Mathematics unit of the Japan Gateway: Ky-\noto University Top Global (KTGU) Program. Through KTGU, RIMS\nenables international experiences for graduate students by inviting\ntop-ranked international researchers to give intensive lecture series,\nselecting and inviting researchers to act as additional supervisors,\nand providing opportunities for students to visit abroad.\nRIMS also has a vibrant research program for young postdoctoral\nmathematicians. In 2017-2020 RIMS had 42 researchers at this level,\nand also hired 2 graduate students in the framework of the ”Sendan\nproject”. Furthermore, three postdocs per year are hired as “ RIMS\nResearch Fellows”, a group of young researchers is hired as “RIMS\nProject Fellows” associated with a RIMS Research Project, and RIMS\naccepts post-doctoral fellows funded by the Japan Society for the\nPromotion of Science (JSPS).\nRIMS budget problems adversely affect the PhD and postdoctoral\nprograms. An urgent issue for RIMS is to secure an increase in the\nPhD and post-doctoral researcher employment budget.\n3.5. International Joint Usage Research Center. International co-\noperation in mathematical research has always been an important\npart of RIMS’ activity. With its designation in 2018 as International\n6 DAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI, AND TAKASHI TSUBOI\nJoint Usage Research Center, internationalization is even more cen-\ntral. There are currently 4 JU\/RC in Japan that are related to Mathe-\nmatics and Mathematical Sciences, but RIMS is the only one whose\nmission includes all core mathematical subjects.\nThe number of international participants grew from 257 in 2017\nto 383 in 2018 and 543 in 2019. RIMS has international academic\nresearch agreements with a large number of leading mathematical\norganizations around the world. Further, RIMS created an Inter-\nnational Advisory Board which whose members are Professors J.-P.\nBourguignon, M. Barlow, M. Reid and M. Kotani.\n3.6. Diversity. The development and growth of the mathematical\ncommunity relies on diversity. However, the entry pipeline into\nmathematical sciences in Japan is dominated by men and the propor-\ntion of women entering undergraduate and graduate study in this\narea has remained very low for decades. While this makes recruit-\nment to RIMS heavily skewed towards male candidates, the review\ncommittee was of the opinion that RIMS could do more to develop\nthe applicant pool and broaden the entry pipeline across Japan. We\nsuggest the following actions.\nWhile the number of women in mathematics may be very low in\nJapan, this is not the case internationally. RIMS should put in place\nguidelines for all applications for workshops and research programs\nto require that the organizing committees include more than one\ngender and that the participant list include the names of an appro-\npriate number of women who will be invited to participate.\nAll organizers of events at RIMS should be required to subscribe\nto a code of conduct before their applications can be considered and\nall participants should also agree to such a code before they can reg-\nister to attend any RIMS-endorsed event. RIMS should provide a\nstatement of procedures for dealing with breaches of collegiality, in-\ncluding the name and contact details of an external ombudsperson\nwho will consider appeals. (If such an office already exists at Kyoto\nUniversity, it could simply be referenced.) Examples of codes and\nprocedures documents are available at cognate institutions such as\nMSRI and Banff International Research Station (BIRS).\nRIMS should consider holding events featuring women speakers,\nincluding international speakers (since there are relatively few women\nmathematicians in Japan). Male mathematicians should be invited\nto participate in such events, but the opportunity to speak should be\nprimarily reserved for women and people of diverse gender.\nRIMS REVIEW 2021 7\nSuch events for women in mathematics could create mentoring\nnetworks involving students and researchers that could continue be-\nyond the event. They might include the introduction of undergrad-\nuate students to possible mentors, and might provide platforms to\nenable research projects between mentor and mentee. A good model\nis the network formed by the ”Women in Number Theory” work-\nshops at BIRS.\n4. FACILITIES AND RESOURCES\n4.1. Space. With its three missions, RIMS needs lecture halls, offices\nfor faculty, short and long term visitors, postdoctoral fellows and\nPh.D. students. It also needs seminar rooms, and meeting\/discussion\nrooms. The reports of the three previous external review committes\nemphasized that RIMS is seriously handicapped by the space avail-\nable. The offices of faculty members are not appropriate for their\nrange of activities, which include work in education, research and\nadministration. There is a shortage of discussion spaces and semi-\nnar rooms that could facilitate collaboration and discussion among\nthe young researchers, and there are not enough individual offices\nfor the international visitors.\nAfter the previous assessment in 2018, RIMS took a 5-year lease\non 8 rooms in the Maskawa Building. The rented space is used for\nnormal activity of RIMS and we hope that the University will find a\nway to provide space to RIMS without charging rent.\nEven with the added space, visitors are frequently asked to share\nrooms and the average office space in square meters is small. The\nspace allotted to RIMS is just a fraction of the government standard\nfor institutions of this size (71% without the rented space and 77%\nwith the rented space—and this is apparently without taking into\naccount the large number of senior visitors, both long and short-\nterm). In particular, the office space for long-term senior visitors\nfrom abroad is inadequate.\nThe review report of 2018 pointed to the seriousness of the prob-\nlem created by RIMS offices being split into four distinct buildings,\nsome at considerable distance from the others. The fact that RIMS\nbegan renting space in Maskawa Building relieved some of the pres-\nsure but worsened the splitting problem. This problem causes a loss\nof RIMS’ cohesion. For mathematical research, face-to-face discus-\nsions are of paramount importance. The splitting problem reduces\nopportunities for members and visitors to meet and discuss outside\n8 DAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI, AND TAKASHI TSUBOI\ntheir own research groups, which is of essential importance to pro-\nduce unexpected encounters of knowledge and ideas. Such an inap-\npropriate situation tends to cause isolation of researchers, and post-\ndoctoral fellows. The geographic separation of the faculty lessens\ntheir interaction.\nOn Wednesday, March 10 (JST) the Review Committee, accompa-\nnied by Director Takashi Kumagai and Deputy Director Narutaka\nOzawa, met with Kyoto University President Nagahiro Minato to\ndiscuss possible solutions to the space problem at RIMS. The com-\nmittee explained that RIMS is a major source of prestige for Kyoto\nUniversity, known and esteemed around the world in the company\nof such venerable institutions as the Institute for Advanced Study\nin Princeton, USA or the Isaac Newton Institute in Cambridge, Eng-\nland; but that visitors to RIMS must note that the physical facilities\nmade available to RIMS do not measure up to those at other insti-\ntutes of the same standing. As RIMS is very much an icon of Japan-\nese mathematics, the committee feels that this problem deserves ur-\ngent University and even national attention.\nPresident Minato explained some of the difficulties in correcting\nthis situation, but also explained that there is a major planning exer-\ncise going on in the University that could result in an improvement\nfor RIMS; and he promised to keep RIMS’ need in mind as that plan-\nning goes forward. He also told the committee of a beautiful new\nguesthouse that will become available, and predicted that visitors to\nRIMS would be among the frequent users of that facility.\n4.2. RIMS activities in the time of COVID. RIMS activities in 2020\nwere severely affected by the COVID-19 pandemic. The individual\nresearch programs of RIMS faculty continued but, following govern-\nmental and Kyoto university regulations, RIMS had to cancel its vis-\nitors program and many planned meetings, while converting other\nmeetings to online or hybrid formats. RIMS prepared guidelines for\nthe organizers and participants for all types of meetings in these cir-\ncumstances.\n4.3. Administrative Staff. RIMS’ staff takes care of around 4000 par-\nticipants and nearly 500 foreign visitors each year. The staff coordi-\nnates the schedules of parallel conferences, prepares for visas and\ntravel and housing for foreign visitors, and processes the large vol-\nume of paperwork corresponding to all the services rendered. Sup-\nporting the level of activity of the International JU\/RC requires a\nhigh level of administrative skill. This differs from the standard ex-\npectation of administrative staff in a university. Kyoto University\nRIMS REVIEW 2021 9\nmay wish to consider a career-path for staff with such skill to main-\ntain the current high quality.\nProviding an excellent research environment and maintaining RIMS’\nreputation requires an excellent staff. The current quality of support\nis highly appreciated by both faculty members and visitors.\nRIMS’ IT staff maintains the computer system and organises com-\nputer access for visitors. In the COVID-19 situation, the RIMS staff\nsupports online and hybrid conferences and workshops, and made\nhandbooks for running hybrid meetings. This activity is also very\nmuch appreciated by the community.\n4.4. Financial Resources. RIMS’ revenue comes from three sources:\nManagement expenses grants, International JU\/RC funds, and com-\npetitive external grants (Grant-in-aid of JSPS) won by individual mem-\nbers. There are several pressing problems:\nWhen it was selected as International JU\/RC in 2018, RIMS’ bud-\nget increased—but is now less than it was in 2012. Moreover, Kyoto\nUniversity funds and International JU\/RC funds are gradually de-\ncreasing. The International JU\/RC budget is not sufficient for its\nactivity, and the deficit is covered by RIMS’ management money.\nRIMS fosters young researchers, but this program is not financially\nsound. The employment of postdocs is partly paid by Kashiwara’s\nChern Medal Award, which will end in this year.\nPart of International JU\/RC funds are earned in competitions, but\nRIMS has to organise its program of meetings before the outcome\nof the competition is known. RIMS has had a high success rate so\nfar, but this is an additional threat to RIMS’ operations. It would be\nvery important to reduce the financial risks so that RIMS can plan\neffectively.\nModern mathematics finds applications in industry so it may be\npossible for RIMS or its affiliated centers to obtain some corporate\nfunding. Professor Bourguignon, a member of the International Ad-\nvisory Committee, has had success in doing this for the I.H.E.S near\nParis, and may be able to help RIMS start such programs.\n4.5. Computing Facilities. An Institute of RIMS stature must have\na high quality computer network, providing fast and reliable con-\nnectivity, and this need will only increase in the future. High quality\nIT management is also required, especially for hybrid meetings. The\nmagnitude and the success of RIMS meetings under pandemic con-\nditions shows that this requirement is currently fulfilled. However,\nthe committee was told the Wifi may not be strong enough in some\nrooms.\n10DAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI, AND TAKASHI TSUBOI\nMost researchers at RIMS have relatively modest computing needs,\nwhich are met by desktop and laptop computers purchased using\nthe individual researchers’ Grants-in-Aid. However, some groups,\nsuch as the numerical analysis group, have much greater needs; some\nof these are fulfilled by a computer in RIMS’ basement, by university\nsupercomputing facilities, and by applying to use the Fugaku com-\nputer at RIKEN. RIMS has a terminal room for the use of visitors\nand students. Computing facilities seems to be sufficient even in\nthis COVID-19 situation.\nIt should be noted that researchers’ own computers and the pro-\ncessor in the RIMS basement need to be renewed every 5 years and\nsoftware must be updated periodically.\n4.6. Library and Publications. In mathematics, it is still necessary\nto maintain a good library, and RIMS’ has several challenges in do-\ning so: The library of RIMS is an important national resource, since\nmany smaller universities do not subscribe to a full range of journals.\nThus it would be reasonable for the library to be funded nationally,\nor directly by Kyoto University, but RIMS must presently fund its li-\nbrary largely by itself. Second, there is a space problem in the library\nbecause of the increased number of academic books and journals.\nFinally, the escalation in prices of electronic journals can no longer\nbe addressed by an individual university but should be handled na-\ntionally.\nRIMS also supports research by its publication activities. It issues\nthree kinds of publications: the journal Publications of the Research\nInstitute for Mathematical Sciences, the series Kokyuroku, and the\nRIMS Kokyuroku Bessatsu series. Each of the three publications has\nits own importance. It is important in mathematics to continue the\npublication activity.\n5. INTERNATIONALIZATION\nIn its capacity as an International Joint Usage Research Institute,\nRIMS has a remarkably large and varied program of workshops and\nconferences of several types. Most of these workshops are organized\nby Japanese mathematicians, and we believe that information about\nthe possibilities is relatively widely known in Japan. However, this\nseems not yet to be the case abroad.\nGiven RIMS international reputation, we believe that the possibil-\nity of organizing events at RIMS should be very attractive to a broad\ninternational community. To encourage more applications from abroad,\nwe feel that RIMS should advertise the possibilities more broadly.\nRIMS REVIEW 2021 11\nThe first and simplest way to do this is to add information to the\nwebsite, which is presently very sparse. The different types of work-\nshop listed should be more clearly distinguished, and the website\nshould say what kinds of events are appropriate for each category.\nLinks to examples of successful workshops of each type should be\ngiven, showing things like the number of national and international\nparticipants. The available facilities should be made clear: how much\nspace is available (with pictures), how much financial support for\ndomestic and international travel should be spelled out. The process\nfor judging applications should also be clarified: who will make the\nfinal decisions, and when they will be made. The data should be\navailable in HTML, not just through downloaded PDFs.\nWe recommend that the calls for proposal include requirements\nor at least strong recommendations for national and gender diver-\nsity. In particular, women on an organizing committee tend to know\nwomen in the field, and can help attract them, which leads to more\ndiverse workshops.\nA second step in encouraging international applications and par-\nticipation is to advertise internationally. The Notices of the American\nMathematical Society, the European Mathematical Society Newslet-\nter, the London Mathematical Society Newsletter and other similar\npublications would all be appropriate venues; some may be willing\nto accept a short article about the possibilities, in addition.\n6. OUTREACH\nAn institution such as RIMS can significantly add to the public\nawareness–and thus support–of mathematics. RIMS currently hosts\na program of lectures for the general public each year in which sev-\neral RIMS faculty each give a few lectures on some topic of mathe-\nmatical interest, and these are well-received. However, they are local\nin character, not fully reflecting RIMS standing as a national and in-\nternational resource.\nWe recommend broadening this program in several ways. First,\nspeakers for the public could be chosen from a broader pool, not just\nfrom RIMS, and could include speakers from the domains of applica-\ntion of mathematics—it is probably more effective to hear someone\nfrom the “outside” say how important mathematics is than to hear a\nmathematician say the same thing.\nNow that recording is easy, RIMS could aspire to present lectures\nthat would be seen and appreciated by a large national audience.\nThe ease of adding captions would even make it possible to include\n12DAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI, AND TAKASHI TSUBOI\nnon-Japanese speakers. YouTube, or something similar, might be a\nuseful platform.\nSocial media have become important not only for reaching the\npublic, but also for reaching young researchers. Relatively few se-\nnior mathematicians are at home with these tools, so RIMS might\nconsider employing a communications specialist who would intro-\nduce RIMS in these formats.\nAnother medium that can have wide distribution is film, made ei-\nther for television documentaries or for commercial theatres. RIMS\ncould consider producing films about mathematicians or about im-\nportant societal applications.\n7. APPENDIX A: MEMBERS OF THE REVIEW COMMITTEE\n• Yakov Eliashberg eliash@stanford.edu\nDepartment of Mathematics\nStanford University\n450 Jane Stanford Way\nStanford, CA 94305-2125\n• David Eisenbud (Chair)\nde@msri.org\nMSRI\n17 Gauss Way\nBerkeley CA 94720\n• Nalini Joshi\nnalini.joshi@sydney.edu.au\nSchool of Mathematics and Statistics F07,\nThe University of Sydney,\nSydney, NSW 2006, Australia\n• Takashi Tsuboi\nmail@tsuboi-takashi.sakura.ne.jp\nFaculty of Engineering, Musashino University\n3-3-3 Ariake Koto-ku, Tokyo 135-8181\nRIKEN iTHEMS\n2-1 Hirosawa Wako-shi, Saitama 351-0198\nRIMS REVIEW 2021 13\n8. APPENDIX B. DESCRIPTION OF THE EVALUATION PROCESS\nAlready when the Review Committee was organized it was clear\nthat an on-site review would not be realized because of the COVID-\n19 pandemic. The evaluation was thus performed online, using re-\nmote meeting systems.\nIn early December, 2020, the members of the Review Commit-\ntee received the self-evaluation report document produced by the\nRIMS Self-evaluation committee (A. Tamagawa, M. Hasegawa, S.\nMochizuki and N. Ozawa). The Review Committee met in Zoom\non December 16 to discuss what further information it would like,\nand sent the resulting list of questions to Professor Takashi Kumagai,\nDirector of RIMS, at the beginning of January 2021. These questions\nwere answered a week later, and the committee also received links to\nthe information RIMS had prepared to help the organizers of hybrid\nmeetings.\nIn the place of on-site review, the Review Committee was sent a\nseries of videos prepared by RIMS that was accessible from mid Jan-\nuary to the end of evaluation process. The contents of the videos\nwere:\n• Welcome message from Professor Shigefumi Mori [3min];\n• Presentation by the Director of RIMS, Professor Takashi Ku-\nmagai [1hr 5min] with [Slides PDF];\n• Tour of RIMS buildings (guided by Professor Kaoru Ono)[11min];\n• Presentation on the conduct of hybrid Workshops by Profes-\nsor Takashi Kumagai [11min];\n• Interviews with senior faculty members, part 1: Interviews\nof Professor Tomoyuki Arakawa, Associate Professor David\nCroydon, Associate Professor Masayuki Kawakita, and Asso-\nciate Professor Kazushige Terui [46min];\n• Interviews with junior faculty members, part 2: Interviews of\nLecturer Go Yamashita, Assistant Professor Suguru Ishikawa,\nAssistant Professor Mayuko Yamashita, and Project Assistant\nProfessor Yu Yang [29min];\n• Interviews with other researchers: Collective interview of Dr.\nMorimichi Kawasaki, Dr. Hideya Watanabe, Dr. Sven Moller\nand Dr. Kazumi Higashiyama [32min];\n• Interviews with students: Collective interview of Kohei Sasaya,\nHiroaki Karuo, Youji Fukihara, Zeming Sun and Ryoga Ma-\nhara [19min],\nwhere the interviewer was always Professor Narutaka Ozawa.\n14DAVID EISENBUD, YAKOV ELIASHBERG, NALINI JOSHI, AND TAKASHI TSUBOI\nOn Thursday February 4, 8AM-10 AM JST, Professor Takashi Ku-\nmagai, Director of RIMS, accompanied by members of the RIMS\nExecutive Board, made an extended presentation, by Zoom, of the\nself-evaluation report. This was followed by questions from the re-\nviewers. Then Members of the Review Committee had discussion\non additional data necessary for the evaluation. Recognizing that\nthe problem of space was a major issue for RIMS, the review com-\nmittee decided to write a letter to the President of Kyoto University.\nThis was also discussed at the summarizing session with the RIMS\nmembers.\nMembers of the Review Committee discussed the body of the let-\nter to the President of Kyoto University on February 6, JST, and gath-\nered additional questions and a request for additional data, which\nwere sent to Professor Takashi Kumagai.\nOn February 9, 8AM-10 AM JST, RIMS members answered the\nquestions and explained on supplementary data by Zoom. Then\nMembers of the Review Committee discussed the structure and prepa-\nration of the evaluation report. In the summarizing discussions, the\nmembers committed to delivering the report in March. Professor\nTakashi Kumagai, Director of RIMS, expressed his gratitude to the\njob of the Review Committee.\nOn Tuesday February 16 and again on Tuesday February 22 (JST),\nthe Review Committee met to work on the report.\nProfessor Nagahiro Minato, President of Kyoto University, accepted\nthe Review Committee’s request to meet, and the date was set for\nWednesday March 10, 9:45-10:30 JST. Members of the Review Com-\nmittee first meet with RIMS’ Director Takashi Kumagai and Deputy\nDirector Narutaka Ozawa, who accompanied them at the meeting\nwith President Minato. The discussions concerned RIMS’s space\nproblem, and the review more generally.\n"}
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HOME ACCESS Japanese English 研究所について. — 研究所について — 研究所紹介 研究所紹介ビデオ 行動規範 メンバー 組織について 社会との連携 教育 受賞 Publ.RIMS 出版物 数理解析研究所だより 京大数理解析研究基金 — 共同利用事業 — 概要 提案募集・公募 運営委員 専門委員 国際アドバイザー 外部評価書 — 設備とアクセス — 図書室 コンピュータ設備 リンク アクセス 共同利用事業. 国際共同利用・共同研究拠点 概要 共同研究リスト 訪問滞在型研究リスト 提案募集 研究代表者の方へ 会場設備について 論文への謝辞記載に関するお願いについて 拠点事業 論文アンケート 講究録 講究録別冊 共同利用研究報告書 国際交流 RIMS共同研究における男女共同参画・ダイバーシティ等に関する取り組みについて 新型コロナウイルスに関するRIMS共同研究の対応について 過去の訪問滞在型研究 過去の合宿型セミナー 過去の総合研究セミナー 過去のRIMS共同研究 大学院教育・講義. 大学院教育と入試案内 全学共通科目「現代の数学と数理解析」 談話会・セミナー. 談話会 セミナー 公募案内. 特定助教（女性限定） 共同利用研究公募 現在、公募はありません。. 共同利用研究公募 一般の方へ. 高校生向けの講座等 一般向け公開講座等 講義ノート 数理解析研究基金 　　芳名録 所内情報. 北部構内共通事務部ホームページ 北部構内研究助成等公募情報「鎗（やり）」京都大学外部資金公募情報サイト コンピュータ設備 部局行動計画等 所内資料 リンク 談話会・セミナー. TOP 　　. 談話会・セミナー 　　セミナー. 談話会. 談話会 6\/15 陽 煜 氏 （京大・数理研）. 6\/22 [Zoom] 入江 慶 氏 （京大・数理研）. セミナー. Representation Theory Seminar (Arakawa, Kato, Hikita, Moriwaki, Sato) 6\/16 [Zoom] Yuchen Fu (RIMS). 6\/24 [Zoom] Yoshiyuki Kimura (Osaka Metropolitan University). Algebraic differential geometry seminar (T.Mochizuki) Low-dimensional Topology Seminar (Ohtsuki, Habiro) Seminar on geometry and related topics(Ono) 講演会 (T. Mochizuki) Number Theory \/ Arithmetic Geometry Seminar (Tamagawa) Kyoto Young Topologists Seminar (Oba, Okazaki, Shimizu) Kyoto Operator Algebra Seminar (Ozawa) 特別講演 (S. Mochizuki) Computer Science Seminar (Hasegawa) Morgan 氏講演 (Ohtsuki) 過去セミナー予定表はこちら 談話会・セミナー. 談話会 セミナー ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. 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HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Kyoto University International Service Office Link Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Graduate School. Graduate School Colloquium\/Seminars. Colloquium Seminars Recruitment. Female Program-Specific Assistant Professor Lecture. TOP 　　. Seminars 　　Algebraic differential geometry seminar. Algebraic differential geometry seminar. Title. Wall-crossing for framed quiver moduli. Date. May 12 (Thu), 17:00-18:30, 2022. Room. Zoom. Speaker. Ryo Ohkawa (Osaka Metropolitan University). Organizer. T. Mochizuki. Title. Stokes filtered sheaves and differential-difference modules. Date. September 10 (Thu), 13:30-15:30, 2020. Room. Zoom. Speaker. Yota Shamoto (Kavli IPMU). Organizer. T. Mochizuki. Title. Sheaves and symplectic geometry of cotangent bundles. Date. February 19 (Mon), 10:00-12:00, 15:00-17:00, 2018. February 20 (Tue), 10:00-12:00, 15:00-17:00, 2018. Room. RIMS, Rm 110. Speaker. Stéphane Guillermou (Université Grenoble Alpes Institut Fourier). Abstract. We will explain how the microlocal theory of sheaves, introduced by Kashiwara and Schapira in the 80's, is used in symplectic geometry (mainly of cotangent bundles), since works of Nadler-Zaslow and Tamarkin in 2008. In particular we will sketch a proof of the following result: a compact exact Lagrangian submanifold of a cotangent bundle $T^*M$ is homotopically equivalent to the base $M$, through the natural projection from $T^*M$ to $M$. The lectures will be a more detailed exposition of the topics introduced in the first talk (Feb. 2):. -- (constructible) sheaves and microsupport; operations. -- sheaves associated with a Hamiltonian isotopy of a cotangent bundle; first applications (maybe the non squeezing lemma). -- Sato's microlocalization and the muhom functor. -- study of compact exact Lagrangian submanifolds of a cotangent bundle. Organizer. T. Mochizuki. Title. Sheaves and symplectic geometry of cotangent bundles. Date. February 2 (Fri), 15:00-17:00, 2018. Room. RIMS, Rm 110. Speaker. Stéphane Guillermou (Université Grenoble Alpes Institut Fourier). Abstract. We will explain how the microlocal theory of sheaves, introduced by Kashiwara and Schapira in the 80's, is used in symplectic geometry (mainly of cotangent bundles), since works of Nadler-Zaslow and Tamarkin in 2008. In particular we will sketch a proof of the following result: a compact exact Lagrangian submanifold of a cotangent bundle $T^*M$ is homotopically equivalent to the base $M$, through the natural projection from $T^*M$ to $M$. In the course of the talk we will quickly recall the main notions of the microlocal theory of sheaves and give examples. Organizer. T. Mochizuki. Title. Period maps, spectral numbers and Stokes matrices of isolated hypersurface singularities. Date. July 6 (Thu), 15:00-17:00, 2017. July 7 (Fri), 15:00-17:00, 2017. Room. RIMS, Rm 111. Speaker. Claus Hertling (University of Mannheim). Abstract. Holomorphic function germs with isolated singularities have been studied since the end of the 1960ies. A lot is known about their topology, their Milnor lattices, their Gauss-Manin connections and induced Hodge structures, and the behaviour of these data in families of functions. But also a lot is still not well understood, basic properties of the integral monodromy and Seifert form, a mysteriously well working interplay between lattice data and period maps for families, the spectral numbers. The talk will give an introduction to the playing characters in the theory of isolated hypersurface singularities and will present a bouquet of old and new conjectures and related results. The first talk will focus on single singularities and mu-constant families, the second talk on universal unfoldings and Stokes structures of generic members. For unmarked as well as marked singularities, moduli spaces, period maps, Torelli conjectures and Torelli results will be presented. An old conjecture on the integral monodromy of quasihomogeneous singularities will be recalled. Results and conjectures on the spectral numbers will be given. This will be useful for a conjectural characterization of the Stokes matrices and Coxeter-Dynkin diagrams of singularities. Organizer. T. Mochizuki. Title. Topological recursion, WKB analysis and Painlevé equations. Date. June 22 (Thu), 13:30-15:30, 2017. Room. RIMS, Rm 006. Speaker. Kohei Iwaki (Nagoya University). Organizer. T. Mochizuki. Title. Differential equations and algebraic points on transcendental varieties. Date. April 6 (Thu), 13:30-15:00, 2017. Room. RIMS, Rm 110. Speaker. Gal Binyamini (Weizmann Institute). Abstract. The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate for a very wide class of transcendental sets. This result plays a key role in Pila-Zannier's proof of the Manin-Mumford conjecture, Pila's proof of the Andre-Oort conjecture for modular curves, Masser-Zannier's work on torsion anomalous points in elliptic families, and many more recent developments. I will briefly sketch the Pila-Wilkie theorem and the way it enters into the arithmetic applications. I will then discuss recent work on an effective form of the Pila-Wilkie theorem (for certain sets) which leads to effective versions of many of the applications. I will also discuss a joint work with Dmitry Novikov on sharpening the asymptotic from sub-polynomial to poly-logarithmic for certain structures, leading to a proof of the restricted Wilkie conjecture. The structure of the systems of differential equations satisfied by various transcendental functions play the main role for both of these directions. Organizer. T. Mochizuki. Title. Filtered holonomic D-modules in dimension one. Date. December 1 (Thu), 15:00-17:00, 2016. December 2 (Fri), 15:00-17:00, 2016. Room. RIMS, Rm 110. Speaker. Claude Sabbah. Abstract. Holonomic D-modules on the affine complex line offer a simple prototype of various properties also occurring in higher dimensions. We will focus on filtered holonomic D-modules from various points of view. In the first lecture, we start from a filtered regular holonomic D-module M underlying a Hodge module, and we explain the construction an some properties of the associated Deligne filtration on the D-module obtained from M by applying an exponential twist. This is strongly related to considering the Laplace transform of M. We will give motivations for considering such a filtration. In the second lecture, we focus on rigid irreducible holonomic D-modules on the affine line. Generic (possibly confluent) hypergeometric differential equations give naturally rise to examples of such objects. After having explained the Katz algorithm (respectively the Arinkin-Deligne algorithm) for reducing the regular (respectively possibly irregular) such D-modules to ones having generic rank one, we will consider the behaviour of Hodge (respectively Deligne) filtrations along the algorithm (joint work with M. Dettweiler) and we will explain some examples. Organizer. T. Mochizuki. Back to Seminars Colloquium\/Seminars. Colloquium Seminars ←　BACK TO THE TOP ←　BACK TO THE TOP Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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{"id":"cZGLXg16muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~kenkyubu\/koubo\/2022\/FemaleProgram-SpecificAssistantProfessor-e.pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":100,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["~kenkyubu","koubo","2022"],"url_file_name_s":"FemaleProgram-SpecificAssistantProfessor-e","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["FemaleProgram-SpecificAssistantProfessor-e.pdf"],"author":"Akio Tamagawa","author_sxt":["Akio Tamagawa"],"last_modified":"Mon May 23 05:16:54 UTC 2022","dates_in_content_count_i":4,"dates_in_content_dts":["Fri May 20 00:00:00 UTC 2022","Thu Sep 01 00:00:00 UTC 2022","Fri Mar 31 00:00:00 UTC 2023","Fri Mar 31 00:00:00 UTC 2028"],"http_unique_b":false,"www_unique_b":true,"exact_signature_l":-4452222812541564248,"exact_signature_unique_b":true,"fuzzy_signature_l":1013056593344525211,"fuzzy_signature_unique_b":true,"wordcount_i":663,"linkscount_i":1,"linksnofollowcount_i":0,"inboundlinkscount_i":1,"outboundlinkscount_i":0,"inboundlinks_protocol_sxt":["000-https"],"inboundlinks_urlstub_sxt":["www.kurims.kyoto-u.ac.jp\/~kenkyubu\/koubo\/2022\/CV-English-20220520.doc"],"inboundlinks_anchortext_txt":[""],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:38 UTC 2022","fresh_date_dt":"Thu Jun 16 22:36:01 UTC 2022","referrer_id_s":"eOG-td16muho","publisher_t":"Adobe PDF Library 20.5.126","language_s":"en","size_i":47014,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":22,"references_internal_i":22,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086024971780096,"text_t":"May 20, 2022 \nTo whom it may concern: \nAnnouncement of Recruitment of  \na Female Program-Specific Assistant Professor, \nResearch Institute for Mathematical Sciences (RIMS) \nThe Research Institute for Mathematical Sciences (RIMS), Kyoto University announces \nthe recruitment of a female Program-Specific Assistant Professor as per below. Selection \nwill be done through screening of documents and possibly through interviews. Promising \nfemale researchers regardless of any nationality are encouraged to apply for the \nposition. \nProgram-Specific Assistant Professorship position: One vacancy. \nExpected starting date of appointment: September 1, 2022. \nEligibility: Each applicant must be female. Each applicant must have obtained a Ph.D. \ndegree by the starting date of the appointment, or must demonstrate equivalent ability \nas a researcher.  \nDuties: The successful applicant will belong to the International Research Center for \nNext-Generation Geometry at RIMS, work on management of research exchanges and \ninternational workshops with a focus on young researchers and support for RIMS \n“Promotion of Women’s Participation” workshop\/symposium programs at the Diverse \nHuman Resources Development Unit of the Center, and, in addition, work on \ninternational dissemination of results of new mathematics originating from Japan at the \nInternational Information Dissemination Unit of the Center. \nSalary and fringe benefits: Determined on the basis of Kyoto University Regulations \nRegarding Salaries of Faculty and Staff. \nTerm of employment: Until March 31, 2023. Reappointment may be granted every year \nuntil March 31, 2028.  \nProbation period: Six months. \nWorking conditions: Working hours: 38 hours 45 minutes per week under the flexible- \nhours system. Days off: Saturdays, Sundays, public holidays, Year-end and New Year \nholidays, Foundation Day, and summer vacation. \nSocial insurance: Eligible for MEXT* mutual aid association membership, employees' \npensions, employment insurance, and workers’ compensation insurance. \n*Japan’s Ministry of Education, Culture, Sports, Science and Technology\nThe following documents should be sent preferably as PDF files by email. They should \nbe written in Japanese or English on A4 or US letter format. \n1. Curriculum Vitae. (Format (alternative format is acceptable)) \n2. List of publications.\n3. Copies of a few major publications.\n4. Summary of achievements and plans for future research of about 3 pages long in total.\n5. Reference list, including the names, affiliations, and email addresses of three\nresearchers who are familiar with the applicant’s research.\n6. At least one reference letter from the above three researchers, who are supposed to\nsend the letter directly to the account for application <rims.application[at]kurims.kyoto-\nu.ac.jp>.\n7. A document stating the expected date of the applicant’s obtaining a Ph.D. degree from\nthe advisor or a substitute, in case the applicant has not yet obtained a Ph.D. degree.\nApplications should be sent by email to <rims.application[at]kurims.kyoto-u.ac.jp>, and \nshould be entitled “Job Application (Female Program-Specific Assistant Professor)”. In \nresponse to each Job Application email, an acknowledgment is to be emailed within two \ndays. If the applicant does not receive an email of receipt within two days, she is asked \nto contact the email account for inquiry below. \nIf the applicant still prefers to send the documents by conventional mail, a suitable \nmethod should be used, including EMS, DHL, Fedex, in such a way that she can trace \nthe delivery status herself. In this case, \"Job Application (Female Program-Specific \nAssistant Professor)\" should be written in red on the front page of the envelope, which \nshould be sent to the following address: General Affairs Section, Research Institute for \nMathematical Sciences, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ward, \nKyoto-city, Kyoto, Japan 606-8502. \n \nApplication documents must be received by Thursday, June 23, 2022, 5:00 p.m. (JST). \nApplication documents will not be returned. \n \nAny Inquiries should be directed to General Affairs Section, RIMS, Kyoto University \nby email or by FAX (Email : rims.recruitment [at]kurims.kyoto-u.ac.jp Fax.: +81-75-\n753-7272). \n \nWe do not disclose any personal information to any third party without a legitimate \nreason.  \n \nKyoto University promotes equal opportunities in gender, and this recruitment is \nbased on the provisions of Article 8 of the Equal Employment Opportunity Act.  \n \nAt Kyoto University, smoking is prohibited indoors on all campuses, and outdoors \nexcept in designated smoking areas in order to prevent passive smoking. \n"}
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visitors list"],"outboundlinks_protocol_sxt":["000-https","001-https"],"outboundlinks_urlstub_sxt":["www.en.kikin.kyoto-u.ac.jp\/funds-in-operation\/mathematical\/","w10.kurims.kyoto-u.ac.jp\/kyodo\/paper\/en\/","www.math.kobe-u.ac.jp\/HOME\/saji\/math\/conf2022\/spmon\/index.html"],"outboundlinks_anchortext_txt":["KU Math Sci Res Fund","Questionnaire to RIMS workshop participants","Singularity theory special months"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:40 UTC 2022","fresh_date_dt":"Thu Jun 09 11:22:25 UTC 2022","referrer_id_s":"eOG-td16muho","language_s":"ja","size_i":31864,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":33,"references_internal_i":33,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086026316054528,"text_t":"RIMS-Research Institute for Mathematical Sciences Kyoto University. HOME ACCESS Japanese English About RIMS. -- About RIMS -- Message from the Director Introducing RIMS Code of Conduct Members Organization Graduate Education and Conferment of Degrees Publ.RIMS Publications Brochure KU Math Sci Res Fund -- International Joint Usage\/Research Center -- Overview Call for Proposals Advisory Board Technical Committee International Advisors Assessment Report -- Facilities and Access Information -- Library Computer Facilities Access Joint Usage\/Research. International Joint Usage\/Research Center Overview Symposia\/Workshops RIMS Research Projects Call for proposals Information for Principal Researcher Seminar rooms and facilities for joint research Request regarding Research Paper Acknowledgements Questionnaire to RIMS workshop participants RIMS Kôkyôroku RIMS Kôkyôroku Bessatsu International exchange Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities COVID-19 updates for organizers and participants Past RIMS Research Projects Past RIMS Satellite Seminars Past RIMS Review Seminars Past RIMS Workshops\/Symposia Workshop for Principal Researcher Call for proposals. Call for proposals（RIMS Research Projects） Call for proposals（RIMS Workshops\/Symposia） Call for proposals（RIMS Satellite Seminars） Call for proposals（RIMS Review Seminars） International Joint Usage \/. Research Center. International Center for Collaborative Study in Mathematical Sciences. COVID-19 updates for organizers and participants (June 1st, 2022) On November 2018, RIMS was certified as one of the. International Joint Usage\/Research Centers. by the Ministry of Education, Culture, Sports, Science and Technology (MEXT). Request regarding Research Paper Acknowledgements Childcare Services during Joint Research Activities. It is possible to set up a temporary nursery on campus during research meetings for the use of researchers with children. For details, please contact the organizer of the meeting you are attending or the Cooperative Research Service Section at RIMS. Overview Workshop Call for proposals List of RIMS Research Projects. 2022. Deepening and applications of variational problems Modern variational methods have a wide range of theories; the direct method, the critical point theory represented by the minimax method and the mountain pass theorem, Morse-Conley theory, and so on. These theories have been extensively studied not only in the field of partial differential equations, such as finding solutions to nonlinear elliptic partial differential equations as the stationary point of some energy functional, but also in connection with various mathematical fields, such as Riemannian geometry, symplectic geometry, mathematical physics, and so on. We carry out this project based on the following two subjects:. (A) Deepening the insight of the lack of compactness appearing in non-compact variational problems;. (B) Applications of variational methods to evolution equations. “Non-compact variational problem” in subject (A) is the variational problem such that the approximate sequence (such as minimizing sequence, or Palais-Smale sequence) is not always compact by itself, and it often appears in the application to the fields mentioned in the introduction. By virtue of the concentration-compactness principle, introduced by Prof. Pierre-Louis Lions in 1980s, the mechanism of the lack of compactness is generally understood and this theory has been widely applied to the study of non-compact variational problems. Subject (A) aims to refine the understanding of the mechanism and give new applications for non-compact variational problems. Besides, by the use of the concentration-compactness principle, there are various studies of the asymptotic behavior of global-in-time solutions to nonlinear evolution equations, such as nonlinear parabolic, wave, dissipative equations. The purpose of subject (B) is to develop the study of evolution equations using variational methods, represented by these applications. In order to achieve the two subjects, we will invite Prof. Bernhard Ruf (University of Milan) and Prof. Luca Martinazzi (University of Rome ``La Sapienza\") for a long-term to give a series of lectures and hold three workshops. For subject (B), in particular, we will hold a workshop in order to identify current issues and open problems based on the progress since the RIMS workshop “Progress in Variational Problems -Variational Methods in the Study of Evolution Equations-”, which was held under the same theme. In addition, we will hold a school mainly for young researchers. At these workshops and schools, we will invite young researchers from abroad for a short-term. This project will be carried out in close collaboration with the Osaka Central Advanced Mathematics Institute (OCAMI). Summer School on Variational Problems and Functional Inequalities (RIMS Research Project 2022)【RIMS Workshops(Type A)】. Location: Osaka Metropolitan University, Department of Science, Large Lecture Room (E408) Period: 2022-09-21〜2022-09-22. Organizer: Futoshi Takahashi(Graduate School of Science, Osaka Metropolitan University). Workshop on Non-compact Variational Problems and Related Topics (RIMS Research Project 2022)【RIMS Symposia】. Location: Room 420 Period: 2022-09-26〜2022-09-28. Organizer: Futoshi Takahashi(Graduate School of Science, Osaka Metropolitan University)\/Michinori Ishiwata(Graduate School of Engineering Science, Osaka University). 　　　　　　　　　　　　　　　　. Workshop on Variational Methods and Dispersive Equations (RIMS Research Project 2022)【RIMS Symposia】. Location: Room 420 Period: 2023-02-06〜2023-02-08. Organizer: Futoshi Takahashi(Graduate School of Science, Osaka Metropolitan University)\/Kenji Nakanishi(RIMS, Kyoto University). Workshop on Variational Methods and Functional Inequalities (RIMS Research Project 2022)【RIMS Symposia】. Location: Osaka Metropolitan University, Department of Science, Large Lecture Room (E408) Period: 2023-02-13〜2023-02-15. Organizer: Futoshi Takahashi(Graduate School of Science, Osaka Metropolitan University). 2022. Singularity theory special months The singularity theory of smooth maps describes how the singular points of such maps behave. Originating with the pioneering works of Thom, Morse, and Whitney up until about 1960, the basic tools were then developed by Mather in the late 1960s. It continued to develop since then, and continues to be actively researched today. In addition, both inside and outside of mathematics, it has been applied to various areas, particularly in geometric contexts, by considering singular phenomena as singular points of smooth mapsand analyzing them in detail using singularity theory. A basic tool of singularity theory and its applications is the classification of singularities. For this, the structures of quotient rings of various commutative rings play important roles, but these structures can become very complicated, and investigating them cannot be done by hand, so the use of software becomes necessary. However, well-known general-purpose software is of limited help for this. In recent years, in addition to the significant development of computer performance, a new tool for classification using computer algebra has been developed by Japanese researchers, and the above limits have been greatly exceeded and great progress has been made. Additionally, there has been remarkable development of criteria for singularities that should be studied hand-in-hand with classification. Furthermore, new kinds of applications have been discovered, such as the analytic Hamiltonian appearing in topological insulators and continuum mechanics, with possibilities for these applications to develop rapidly. These recently emerging new theories are multidisciplinary and worthy of earnest study. In view of this situation, meetings will be held to present the latest results of singularity theory and to provide lectures by experts in these new theories. Holding such meetings over the course of several months and interacting with young researchers, singularity theory researchers, and researchers who are looking to apply singularity theory as well, we can mutually develop both pure singularity theory and its applications. We can also promote training the next generation of researchers. Designating October to December 2022 as \"Singularity Theory Special Months\", and inviting two leading researchers from Brazil and Spain, and representative Japanese researchers as well, we will regularly hold schools and research meetings. In addition, the Mathematical Society of Japan Seasonal Institute (MSJ-SI2022) \"Deepening and developing applied singularity theory\" will be held during this period. This project will be linked with this event, with each complementary to the other. Singularity theory and its applications (RIMS Research Project 2022)【RIMS Symposia】. Location: Room 420 Period: 2022-10-03〜2022-10-05. Organizer: Kentaro Saji(Graduate School of Science Division of Mathematics, Kobe University). Singularity theory and geometric topology (RIMS Research Project 2022)【RIMS Symposia】. Location: Room 420 Period: 2022-10-24〜2022-10-27. Organizer: Kentaro Saji(Graduate School of Science Division of Mathematics, Kobe University). 　　　　　　　　　　　　　　　　. Singularity theory and differential geometry (RIMS Research Project 2022)【RIMS Symposia】. Location: Not decided yet Period: 2022-11-07〜2022-11-09. Organizer: Kentaro Saji(Graduate School of Science Division of Mathematics, Kobe University). Classification problems in singularity theory and their applications (RIMS Research Project 2022)【RIMS Symposia】. Location: Not decided yet Period: 2022-12-12〜2022-12-15. Organizer: Kentaro Saji(Graduate School of Science Division of Mathematics, Kobe University). 2023. Stochastic Processes and Related Fields. Since the founding of the mathematical theory of stochastic differential equations by Kiyosi Itô in 1942, research on stochastic calculus in Japan has had a great influence on the development of probability theory throughout the world. Probability theory itself has grown remarkably over the last century, and now encompasses an extremely wide range of research topics. Within mathematics, it has strong interactions with fields such as partial differential equations, potential theory and geometry. Moreover, the vast expansion of probability research is directly linked to applications across the sciences, in areas such as statistical mechanics, biology, economics and the analysis of big data. As a result of probability theory now having so many different aspects, it is difficult for individual researchers to grasp the subject in its entirety. With the keywords \"stochastic processes and stochastic analysis\", this RIMS Research Project aims to further promote international joint research in probability by disseminating results in the area that originate in Japan. Centering on the following three themes, the program will provide a bird's-eye view of modern probability theory. (For each theme, a conference is planned, the title of which is shown in parentheses.). (i) Stochastic Processes Related to Stochastic Partial Differential Equations (Stochastic Partial Differential Equations and Stochastic Calculus). (ii) Analysis of Stochastic Models Motivated by Statistical Mechanics (Stochastic Analysis on Large Scale Interacting Systems). (iii) Random Matrix Theory, Combinatorial Probability and Quantum Information (Random Matrices and Their Applications). Furthermore, to inspire exchanges between these themes, in September 2023, we will organize a large-scale conference at the Research Institute for Mathematical Sciences, inviting researchers in stochastic processes and stochastic analysis who are active at the front line of probability research internationally. Mid-career researchers will take responsibility for the practical organization of conferences on each theme and, in doing so, will promote the participation of young researchers, including as speakers, so that the next generation of researchers plays an active role. We encourage the participation of female researchers in all aspects of the program, seeing it as an opportunity to promote gender diversity in research on probability theory in Japan. Past RIMS Research Projects Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Foreign visitors list Access and Facility Information ←　BACK TO THE TOP About Call for proposals Kôkyûroku\/Kôkyûroku Bessatsu International Exchange Workshop Access and Facility Information Foreign visitors list Information for Principal Researcher COVID-19 updates for organizers and participants Initiatives for Gender Equality and Diversity in RIMS Joint Research Activities Follow on. Research Institute for Mathematical Sciences (RIMS)."}
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Skip to content English. English 日本語 Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. Experience the. SHIBUYA GAIGO GAKUIN. Shibuya Gaigo Gakuin is a Japanese Language School located in the heart of Shibuya, Tokyo. *The school has temporarily moved to a new building in Takadanobaba. Moving back to Shibuya is expected to take place between spring and fall of 2023. With students from around the globe, Japanese courses from beginner to advanced levels, cultural events and Japanese students studying foreign languages under the same roof. We provide a unique environment to study Japanese in Japan and explore both modern and traditional Japan. Discover a Unique Learning Experience. LATEST NEWS. Change of address and contact information. April 1, 2022. Immigration Updates and COVID-19 VACCINE. July 20, 2021. State of Emergency. April 26, 2021. Change of address and contact information. April 1, 2022. Immigration Updates and COVID-19 VACCINE. July 20, 2021. State of Emergency. April 26, 2021. News. A UNIQUE JAPANESE LANGUAGE SCHOOL. Established in 1970, Shibuya Gaigo Gakuin is a language school with. more than 45 years of history teaching a variety of languages. Located in. Shibuya, Tokyo, students have the opportunity to experience Japanese. culture and explore the latest trends. During the course, students can. participate in a variety of our extracurricular activities in some of Tokyo's. most popular spots. Our strength is our unique teaching style and study. environment which allows students to learn Japanese and its culture. while interacting with the native Japanese students who are studying. other languages in our school. DISCOVER OUR COURSES. 100% ONLINE APPLICATION PROCESS. We recognize that it is inconvenient for students living abroad to send their application form and all the required documents by post mail as is requested by most schools in Japan. We have developed a 100% online application process that is both secure and convenient where you are not required to do so, even when applying for a course that requires a Japanese Student Visa! The Process for Getting a Student Visa is easy! SUBMIT APPLICATION. Submit application form. APPLICATION FEE. Pay the application fee. RECEIVE CoE. Receive your Certificate of Eligibility ( CoE ). VISA APPLICATION. Using the CoE, apply & receive your Student Visa. START SCHOOL. Welcome to Japan! APPLY NOW. Everyone in this school is very kind and you won't feel alone! Nathaly. from Colombia. Shibuya Gaigo Gakuin is a unique experience! Anthony. from Italy. Shibuya Gaigo Gakuin makes studying so much fun! Nicole. from Germany. Students Voice. Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Flywire allows you to pay securely from any country and any bank, generally in your home currency. By making your payment with Flywire you can track your payments from start to finish, save on bank fees and exchange rates, and connect with their multilingual customer support team with any questions, day or night. . MAKE A PAYMENT. REFUND POLICY. Once the school notifies your application has been accepted, there is an 8 day cooling off period, Japan Standard Time. Any refund transfer fees will be the responsibility of the student. Any bank transfers processed other than via Flywire will have a special handling fee. Refunds will be made only to a bank account owned by the student, or a relative, or to the bank account from which the payment was originally made. ①Cancellation after submitting the Certificate of Eligibility (COE) application. :. Regardless if the COE is issued or not, the Application fee is non-refundable. ②Cancellation after Immigration Bureau issued the Certificate of Eligibility (COE). and before. the start of. the course. : The Application fee and Admission fee are non-refundable. The tuition. refund will be processed once the COE and Certificate of Admission has been returned. Moreover, in case the Visa has been issued, our staff will also have to confirm that the visa has not been used. ③Student Visa denied by Japan Embassy or Consulate. : The Application fee is non-refundable. The Admission fee and Tuition will be processed once confirming that the. visa has been denied and the COE and Certificate of Admission has been returned. ④Cancellation after the start of the course. : The Application fee and Admission fee. are non-refundable. There will be a 20% (up to 50,000 yen) Cancellation Fee charged. on the remaining days of the Tuition. 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Skip to content English. English 日本語 Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. Privacy Policy. Effective date: July 17, 2018. Shibuya Gaigo Gakuin (\"us\", \"we\", or \"our\") operates the https:\/\/shibuyajp.net website (the \"Service\"). This page informs you of our policies regarding the collection, use, and disclosure of personal data when you use our Service and the choices you have associated with that data. We use your data to provide and improve the Service. By using the Service, you agree to the collection and use of information in accordance with this policy. Unless otherwise defined in this Privacy Policy, terms used in this Privacy Policy have the same meanings as in our Terms and Conditions, accessible from https:\/\/shibuyajp.net. Information Collection And Use. We collect several different types of information for various purposes to provide and improve our Service to you. Types of Data Collected. Personal Data. While using our Service, we may ask you to provide us with certain personally identifiable information that can be used to contact or identify you (\"Personal Data\"). Personally identifiable information may include, but is not limited to:. Email address. First name and last name. Phone number. Address, State, Province, ZIP\/Postal code, City. Cookies and Usage Data. Usage Data. We may also collect information about how the Service is accessed and used (\"Usage Data\"). This Usage Data may include information such as your computer's Internet Protocol address (e.g. IP address), browser type, browser version, the pages of our Service that you visit, the time and date of your visit, the time spent on those pages, unique device identifiers and other diagnostic data. Tracking & Cookies Data. We use cookies and similar tracking technologies to track the activity on our Service and hold certain information. Cookies are files with small amount of data which may include an anonymous unique identifier. Cookies are sent to your browser from a website and stored on your device. Tracking technologies also used are beacons, tags, and scripts to collect and track information and to improve and analyze our Service. You can instruct your browser to refuse all cookies or to indicate when a cookie is being sent. However, if you do not accept cookies, you may not be able to use some portions of our Services. Examples of Cookies we use:. Session Cookies.  We use Session Cookies to operate our Service. Preference Cookies.  We use Preference Cookies to remember your preferences and various settings. Security Cookies.  We use Security Cookies for security purposes. Use of Data. Shibuya Gaigo Gakuin uses the collected data for various purposes:. To provide and maintain the Service offered. To notify you about changes to our Service. To allow you to participate in interactive features of our Service when you choose to do so. To provide customer care and support. To provide analysis or valuable information so that we can improve the Service. To monitor the usage of the Service. To detect, prevent and address technical issues. Transfer Of Data. Your information, including Personal Data, may be transferred to — and maintained on — computers located outside of your state, province, country or other governmental jurisdiction where the data protection laws may differ than those from your jurisdiction. If you are located outside of Japan and choose to provide information to us, please note that we transfer the data, including Personal Data, to Japan and process it there. Your consent to this Privacy Policy followed by your submission of such information represents your agreement to that transfer. Shibuya Gaigo Gakuin will take all steps reasonably necessary to ensure that your data is treated securely and in accordance with this Privacy Policy and no transfer of your Personal Data will take place to an organization or a country unless there are adequate controls in place including the security of your data and other personal information. Disclosure Of Data. Legal Requirements. Shibuya Gaigo Gakuin may disclose your Personal Data in the good faith belief that such action is necessary to:. To comply with a legal obligation. To protect and defend the rights or property of Shibuya Gaigo Gakuin. To prevent or investigate possible wrongdoing in connection with the Service. To protect the personal safety of users of the Service or the public. To protect against legal liability. Security Of Data. The security of your data is important to us, but remember that no method of transmission over the Internet, or method of electronic storage is 100% secure. While we strive to use commercially acceptable means to protect your Personal Data, we cannot guarantee its absolute security. Service Providers. We may employ third party companies and individuals to facilitate our Service (\"Service Providers\"), to provide the Service on our behalf, to perform Service-related services or to assist us in analyzing how our Service is used. These third parties have access to your Personal Data only to perform these tasks on our behalf and are obligated not to disclose or use it for any other purpose. Analytics. We may use third-party Service Providers to monitor and analyze the use of our Service. Google Analytics. . Google Analytics is a web analytics service offered by Google that tracks and reports website traffic. Google uses the data collected to track and monitor the use of our Service. This data is shared with other Google services. Google may use the collected data to contextualize and personalize the ads of its own advertising network. You can opt-out of having made your activity on the Service available to Google Analytics by installing the Google Analytics opt-out browser add-on. The add-on prevents the Google Analytics JavaScript (ga.js, analytics.js, and dc.js) from sharing information with Google Analytics about visits activity.For more information on the privacy practices of Google, please visit the Google Privacy & Terms web page: . https:\/\/policies.google.com\/privacy?hl=en Advertising or Targeting Cookies. . These cookies are usually placed and used by advertising companies to develop a profile of your browsing interests and serve advertisements on other websites that are related to your interests. You will see less targeted and more irrelevant advertising if you disable these cookies. We may also use cookies for:. Identifying the areas of our website that you have visited. Personalizing content that you see on our website. Our website analytics. Remarketing our products or services to you. Remembering your preferences, settings, and login details. Targeted advertising and serving ads relevant to your interests. Affiliate marketing. Allowing you to post comments. Allowing you to share content with social networks. Most web browsers can be set to disable the use of cookies. However, if you disable cookies, you may not be able to access some features on our website correctly or at all. Google Remarketing. . Why am I seeing ads by Google for products I’ve viewed? Our website uses a remarketing advertising service. Our remarketing service is provided by Google and other companies that show our ads on websites across the Internet. With remarketing you may see ads for our products you have previously looked at. As an example, suppose you visit a website that sells computers, but you do not buy a computer on your first visit to that website. The website’s owner might like to encourage you to revisit his\/her site and buy a computer by showing you his\/her ads again on other websites that you visit. We use remarketing for similar purposes. For this to happen, Google will read a cookie that is already in your browser, or they place a cookie in your browser when you visit our site or other sites using remarketing. You can opt out of Google’s use of cookies and re-marketing at this link: . https:\/\/support.google.com\/ads\/answer\/2662922?hl=en  or you can opt out using the Network Advertising Initiative opt-out page at . http:\/\/optout.networkadvertising.org\/#!\/ Facebook Remarketing. . Third parties, including Facebook, may use cookies, web beacons, and other storage technologies to collect or receive information from our website and elsewhere on the internet, and use that information to provide measurement services and target ads. With Facebook remarketing, you may see our ads on Facebook after you have visited our site. For this to happen, Facebook uses a Custom Audience Pixel that is activated when a visitor lands on a webpage and a unique “cookie” is placed in their browser. Facebook lookalike audience targeting allows us to show ads on Facebook to people who are similar to those who have already visited or made a purchase from our website. To opt out of Facebook’s collection and use of information for ad targeting visit: . https:\/\/www.facebook.com\/help\/568137493302217 Links To Other Sites. Our Service may contain links to other sites that are not operated by us. If you click on a third party link, you will be directed to that third party's site. We strongly advise you to review the Privacy Policy of every site you visit. We have no control over and assume no responsibility for the content, privacy policies or practices of any third party sites or services. Changes To This Privacy Policy. We may update our Privacy Policy from time to time. We will notify you of any changes by posting the new Privacy Policy on this page. We will let you know via email and\/or a prominent notice on our Service, prior to the change becoming effective and update the \"effective date\" at the top of this Privacy Policy. You are advised to review this Privacy Policy periodically for any changes. Changes to this Privacy Policy are effective when they are posted on this page. Contact Us. If you have any questions about this Privacy Policy, please contact us:. By sending us a message via our. contact form . Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. Privacy Policy Designed by Souzou Web Scroll To Top."}
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{"id":"WWKLx116muho","host_id_s":"16muho","sku":"https:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/Absolute%20Anabelian%20Cuspidalizations%20(comments).pdf","host_s":"www.kurims.kyoto-u.ac.jp","host_organization_s":"kyoto-u","url_chars_i":99,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["~motizuki"],"url_file_name_s":"Absolute%20Anabelian%20Cuspidalizations%20(comments)","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["cuspidalizations-comments-2014-02-20.pdf"],"author":"<905688EA>","author_sxt":["<905688EA>"],"last_modified":"Sat Feb 15 15:18:24 UTC 2014","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":true,"exact_signature_l":6510873264224049719,"exact_signature_unique_b":true,"fuzzy_signature_l":-7807800164727783089,"fuzzy_signature_unique_b":true,"wordcount_i":738,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:49:59 UTC 2022","fresh_date_dt":"Tue Aug 04 05:35:47 UTC 2026","referrer_id_s":"7ctS0116muho","publisher_t":"Acrobat Distiller 11.0 (Windows)","language_s":"en","size_i":56783,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086047173279744,"text_t":"COMMENTS ON “ABSOLUTE ANABELIAN\nCUSPIDALIZATIONS OF PROPER HYPERBOLIC CURVES”\nShinichi Mochizuki\nFebruary 2014\n(1.) The argument given in Remark 1.18.2 [i.e., Remark 10 in the LaTeX Version],\n(iii), is somewhat sketchy and a bit misleading. This argument should read as\nfollows:\n(iii) Nevertheless, as was pointed out to the author by A. Tamagawa, even\nif X, Y are not necessarily Σ-separated, it is still possible to conclude, essentially\nfrom the arguments of [Tama], Corollary 2.10, Proposition 3.8, that:\nAny Frobenius-preserving isomorphism α is quasi-point-theoretic.\nIndeed, it suffices to give a “group-theoretic” characterization of the quasi-sections\nD ⊆ ΠX which are decomposition groups of points ∈ Xcl. Suppose, for simplicity,\nthat X is proper. Write\nX̃ → X\nfor the pro-finite étale covering corresponding to ΠX . If E ⊆ ΠX is a closed\nsubgroup whose image in GkX is open, then let us write kE for the finite extension\nfield of kX determined by this image. If J ⊆ ΠX is an open subgroup, then let us\nwrite XJ → X for the covering determined by J and JΔ def= J\n⋂\nΔX . If J ⊆ ΠX\nis an open subgroup such that JΔ is a characteristic subgroup of ΔX , then we\nshall say that J is geometrically characteristic. Now let J ⊆ ΠX be a geometrically\ncharacteristic open subgroup. Let us refer to as a descent-group for J any open\nsubgroup H ⊆ ΠX such that J ⊆ H, JΔ = HΔ. Thus, a descent-group H for\nJ may be thought of as an intermediate covering XJ → XH → X such that\nXH ×kH kJ ∼= XJ . Write\nXJ(kJ )\nfld-def ⊆ XJ(kJ)\nfor the subset of kJ -valued points of XJ that do not arise from points ∈ XH(kH) for\nany descent-group H \u0006= J for J — i.e., the kJ -valued points whose field of definition\nis kJ with respect to all possible “descended forms” of XJ . Thus, if x̃ is a closed\npoint of X̃ that maps to x ∈ XJ (kJ), and we write Dx̃ ⊆ ΠX for the stabilizer\nin ΠX [i.e., “decomposition group”] of x̃, then it is a tautology that x maps to a\npoint ∈ XHx(kHx) for Hx def= Dx̃ · JΔ (⊇ J) [so Hx forms a descent-group for J ]; in\nparticular, it follows immediately that:\nTypeset by AMS-TEX\n1\n2 SHINICHI MOCHIZUKI\nx ∈ XJ (kJ)fld-def ⇐⇒ Dx̃ ⊆ J ⇐⇒ Hx = J .\nNow it follows immediately from this characterization of “fld-def” that if J1 ⊆ J2 ⊆\nΠX are geometrically characteristic open subgroups such that kJ1 = kJ2 , then the\nnatural map XJ1(kJ1) → XJ2(kJ2) induces a map XJ1(kJ1)fld-def → XJ2(kJ2)fld-def.\nMoreover, these considerations allow one to conclude [cf. the theory of [Tama]]\nthat:\nA quasi-section D ⊆ ΠX is a decomposition group of a point ∈ Xcl if and\nand only if, for every geometrically characteristic open subgroup J ⊆ ΠX\nsuch that D · JΔ = J , it holds that XJ(kJ)fld-def \u0006= ∅.\nThus, to render this characterization of decomposition groups “group-theoretic”, it\nsuffices to give a “group-theoretic” criterion for the condition thatXJ (kJ)\nfld-def \u0006= ∅.\nIn [Tama], the Lefschetz trace formula is applied to compute the cardinality of\nXJ (kJ). On the other hand, if we use the notation “| − |” to denote the cardinality\nof a finite set, then one verifies immediately that\n|XJ (kJ)| =\n∑\nH\n|XH(kH)fld-def|\n— where H ⊇ J ranges over the descent-groups for J . In particular, by applying\ninduction on [ΠX : J ], one concludes immediately from the above formula that\n|XJ (kJ)fld-def| may be computed from the |XH(kH)|, as H ranges over the descent-\ngroups for J [while |XH(kH)| may be computed, as in [Tama], from the Lefschetz\ntrace formula]. This yields the desired “group-theoretic” characterization of the\ndecomposition groups of ΠX .\n(2.) In the proof of Theorem 1.16, (iii) [i.e., Theorem 1.1, (iii), in the LaTeX\nversion], the phrase “Z ′X → X, Z ′Y → Y are diagonal coverings” that appears at\nthe beginning of this proof should read “Z ′X → X ×X, Z ′Y → Y × Y are diagonal\ncoverings”.\n(3.) In the proof of Proposition 2.2, (i), it is asserted that one has a “natural isomor-\nphism H1(k,ΔabX )\n∼→ J(k)∧ (⊇ J(k))”. In fact, however, at least from an a priori\npoint of view, one only has a natural injection (J(k) ⊆) J(k)∧ ↪→ H1(k,ΔabX ).\nThis does not have any effect on the proof, however, since the proof only requires\nthe use of this natural injection.\n(4.) In the proof of Proposition 1.6, (i), the phrase “fact that G†k acts faithfully on\nMX via the cyclotomic character” should read “fact that the image in Aut(MX) of\nthe action of G†k on MX via the cyclotomic character is infinite”.\n"}
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Google アナリティクスの利用によるデータの収集および処理につきましては、「. ユーザーがGoogleパートナーのサイトやアプリを使用する際のGoogleによるデータ使用 」をご確認ください。. また、Google アナリティクスの以下の広告機能を利用しております。. この広告機能を利用することによって、お客様に最適な広告を提供しています。. Google アナリティクス リマーケティング. Google ディスプレイ ネットワークの表示回数レポート. Google アナリティクスのユーザーの分布とインタレスト カテゴリに関するレポート. Google アナリティクスを使用して広告クッキーと識別子を使ったデータを収集する統合的なサービス. 外部サービスの利用停止. Googleによるインタレストベース広告はGoogle社の提供する. 広告設定 より、表示されないようにオプトアウトしていただくことが可能です。. Google アナリティクスは、ブラウザのアドオン設定で無効にすることにより、情報の収集を停止することが可能です。詳しくは「. Google アナリティクス オプトアウトアドオン 」をご覧ください。. 当校のウェブサイトについてのご意見、お問い合わせは、. お問い合わせフォーム からお願いいたします。. 個人情報の取り扱いについて. 受講申し込みや資料請求にあたり、お客様からご提供いただいた個人情報（住所・氏名等のお客様の個人的な情報）につきましては、下記の通り取扱わさせていただきます。. 個人情報の利用目的. 当校では、ご提供いただいた個人情報を、当該講座の実施や資料送付に必要な範囲で利用する他、次の目的のみに利用し、他の目的には利用いたしません。例外的に他の目的に利用する場合は、別途その利用目的をお客様に通知し、お客様の同意を得たうえで行います。. (1)　他の関連講座や資料に関連するご案内. (2)　顧客層の分析等、各種統計資料の作成. (3)　アンケート等へのご協力のお願い. 個人情報の提供. 当校では、次の場合を除き、お客様の同意を得ずに、お客様の個人情報を第三者に提供することはいたしません。. (1)　前記１の利用目的達成のために、当校の業務委託先等に提供することが必要な場合. (2)　法令に基づく場合. (3)　お客様個人を識別できないように加工した状態（各種統計資料等）で第三者に提供する場合. 個人情報の安全管理. 当校では、ご提供いただいた個人情報を、外部への個人情報の漏洩等がなされないように、当校の取扱い基準に従い適切な安全対策を講じて保管いたします。. 照会・訂正・削除等. お問い合わせ. 当校の個人情報保護方針、その他個人情報に関するご質問は、. お問い合わせフォーム からお願いいたします。. 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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Skip to content English. English 日本語 Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. STUDENT VISA. JAPANESE COURSE. If you wish to live in Japan under a Student Visa and immerse yourself in the Japanese way of life for a long period of time, if you plan to attend a vocational school or university in Japan, work in Japan or if you are already living in Japan and wish to study full time then the Long-term Japanese Course is made for you. You will study Japanese 3 hours a day 5 days a week for up to 2 years in order to fully immerse yourself in the Japanese Language, Culture and gain the linguistic skills required to effectively use Japanese in your daily life. The Full-time course is available from beginner to advanced level. COURSE LEVELS. Basic. Pre-intermediate. Intermediate. Advanced. ADMISSION PERIOD. JANUARY. APRIL. JULY. OCTOBER. CLASSES PERIOD. Morning 9:30-13:00. Afternoon 13:30-17:00. 4 lessons per day (45 minutes per lesson). *. Morning or Afternoon classes are determined by the students level. TOTAL STUDY HOURS. 1 Year:. 800 lessons. 2 Years:. 1600 lessons. APPLICATION DEADLINE. For July 2022 session:. March 1st, 2022. For October 2022 session:. May 15th, 2022. For January 2023 session:. August 30th, 2022. For April 2023 session:. October 31st, 2022. TEXT MATERIALS. Dekiru Nihongo. Kanji Tamago. \"TRY!\" Series. Ryugakusei no tame no Kanji no Kyokasyo. *. Textbooks may change. ADMISSION REQUIREMENTS. 1 ).  . Must be 18 years old or over. 2 ).  . 12 years of formal school education or equivalent. 3 ). For 2-year course, 150 hours or more of Japanese language studies, or language proficency equal to JLPT N5 or higher. For 1-year course, language proficency equal to JLPT N3 or higher. *. A level check test will be given in advance. FEES. 2 YEARS. 1,430,000 JPY. Application:. 20,000 JPY. Admission:. 50,000 JPY. Tuition:. 1,360,000 JPY. *. Textbooks fees are not included. APPLY NOW. 1 YEAR. 750,000 JPY. Application:. 20,000 JPY. Admission:. 50,000 JPY. Tuition:. 680,000 JPY. *. Textbooks fees are not included. APPLY NOW. Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. SHIBUYA GAIGO GAKUIN. NEWS. January 2021 Term Continues Online – Overseas Participation Available! November 16, 2020. Facebook. Twitter. As we approach the end of the year 2020 year, we had hopes our school will resume in the school building by the 2021 New Year.  However, it is with our deepest sincerities that we decided to continue the lessons online for our January 2021 term.  Therefore, there will be no lessons held at the school building until further notice. As many of you may already know, the Japanese Government has lifted the Entry Ban on some countries.  Unfortunately, some of these countries may be entering lockdown once more.  In response to this, the school would like to offer the possibility to students who are overseas to participate in our online lessons in the levels that are available.  Depending on the time zone and the amount of students who opt for this choice, details and further information will be sent later on. In the meantime, please refer to our 2021 schedule for the year including lesson days and holidays. Posts navigation. &larr; Online Lessons Will Continue School Closed for New Years! &rarr; Recent News. Change of address and contact information Immigration Updates and COVID-19 VACCINE State of Emergency April 2021 Term Online School Visits Require Advance Appointment! Tokyo State of Emergency Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Skip to content 日本語. English 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. コーススケジュール. 留学ビザ対象コース. オリエンテーション. フルタイムコース休講日. 留学ビザ対象コース スケジュール. 1年コース. ２年コース. 1年コース. Session. 2022. 2023. April. Apr. 5, 2022 - Mar. 23, 2023. Apr. 5, 2023 - Mar. 2024. October. Oct. 12, 2022 - Sep. 27, 2023. Oct. 11, 2023 - Sep. 2024. ２年コース. Session. 2022. 2023. JANUARY. N\/A. JAN. 11, 2023 - DEC. 2024. April. Apr. 5, 2022 - Mar. 2024. Apr. 5, 2023 - Mar. 2025. JULY. JUL. 5, 2022 - JUN. 2024. JUL. 4, 2023 - JUN. 2025. October. Oct. 12, 2022 - Sep. 2024. Oct. 11, 2023 - Sep. 2025. オリエンテーション 日程. Session. 2022. 2023. January. THURSDAY, JAN. 7, 14:00～. TUESDAY, JAN. 10, 14:00～. April. MONDAY, APR. 4, 14:00～. TUESDAY, APR. 4, 14:00～. July. MONDAY, JUL. 4, 14:00～. MONDAY, JUL. 3, 14:00～. October. TUESDAY, OCT. 11, 14:00～. TUESDAY, OCT. 10, 14:00～. フルタイムコース 休講日. Session. 2022. 2023. Spring. MAR. 24 - APR. 4. MAR. 24 - APR. 4. Golden Week. APR. 29 - MAY 8. APR. 29 - MAY 7. Summer. JUN. 22 - JUL. 4 & AUG. 6 – AUG. 21. JUN. 21 - JUL. 3 & AUG. 5 – AUG. 20. Fall. SEP. 30 - OCT. 11. SEP. 28 - OCT. 10. Winter. DEC. 23, 2022 - JAN. 10, 2023. DEC. 22, 2023 - JAN. 2024. 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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{"id":"q3UlDs9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/student-interview-nathaly-colombia\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":57,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["student-interview-nathaly-colombia"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Student Interview Nathaly from Colombia &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:52:26 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":1854076227147985361,"exact_signature_unique_b":true,"fuzzy_signature_l":-581165261279718947,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["Nathaly from Colombia","Nathaly from Colombia"],"h4_txt":["Q:   Why did you decide to live in Japan?","Q:   Why did you choose Shibuya Gaigo Gakuin?","Q:   In your opinion, what is the best thing about studying in Japan at Shibuya Gaigo Gakuin?","Q:   What are your plans after graduating from Shibuya Gaigo Gakuin?","Q:   What would you tell to others who wish to study at Shibuya Gaigo Gakuin?","Student Voice","Courses","Contact us"],"bold_txt":["Q:  ","A:  ","A:","English English"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/07\/Shibuya-Gaigo-Gakuin-Logo-black.svg"],"images_alt_sxt":["Japanese Language School"],"images_height_val":[2000],"images_width_val":[2000],"images_text_t":"shibuyajp net wp content uploads 2018 07 Shibuya Gaigo Gakuin Logo black svg Japanese Language School","responsetime_i":65,"text_t":"Student Interview Nathaly from Colombia &ndash; Shibuya Gaigo Gakuin. 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When I first applied as well, I had so many questions about the visa process and requirements, so I decided to contact the school through both email and by phone and every time the staff was so kind and helpful. Q:.  . In your opinion, what is the best thing about studying in Japan at Shibuya Gaigo Gakuin? A:.  . I love that our teachers teach us grammar and vocabulary that we can actually use in our everyday lives here in Japan. The teachers really do care about us and they always find a way to make us enjoy our lessons and have us practice as much as possible. Q:.  . What are your plans after graduating from Shibuya Gaigo Gakuin? A:.  . I would like to find a full-time job here in Japan and make this country my permanent home. Q:.  . What would you tell to others who wish to study at Shibuya Gaigo Gakuin? A:.  . Come! haha. Everyone in this school is very kind and you won't feel alone when are you are in this school. 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Skip to content 日本語. English 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ. 2022-06-01. Facebook. Twitter. この度、渋谷外語学院は一時的に仮校舎へ移転いたしました。. 仮校舎の住所は以下の通りです。. 169-0075. 東京都新宿区. 高田馬場1-34-6. 八光ビル2階・3階. Yakō Bldg. 2F, 3F. 1-34-6 Takadanobaba,. Shinjuku-ku, Tokyo. 169-0075 JAPAN. 行政上のやむを得ない事情により、一時的な措置として高田馬場への移転となりました。. 私たちは「渋谷外語学院」として、今年度中に渋谷の新校舎への移転準備を整えていきます。. 渋谷の新校舎への移転は2023年春から秋の間に行う予定です。. また、今回の仮校舎への移転に伴い、電話番号も変更となります。. 当面の連絡先は以下の通りです。. 電話番号：(+81) 70-1371-8866. 上記は固定電話が開通するまでの一時的な連絡先となっております。固定電話開通までの間、FAXはご利用いただけませんのでご了承ください。. 新しい電話番号及びFAX番号は、4月中に改めてお知らせいたします。. 数々の変更に伴いまして関係者の方々にはご不便をおかけいたしますが、何卒ご了承くださいますようお願い申し上げます。. ご質問やご不明な点がございましたら、お気軽にお問い合わせください。. 渋谷外語学院. Posts navigation. &larr; Online Lessons Will Continue 最近のニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ Online Lessons Will Continue Extension of Online Classes for April 2020 Term （日本語訳準備中） Students on a Student Visa Leaving Japan （日本語訳準備中） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 オンラインレッスンの延長について 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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Skip to content English. English 日本語 Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. Study in Japan with Shibuya Gaigo Gakuin. Start by applying to Shibuya Gaigo Gakuin using the Application Form below and you will be on your way to study in Japan. . The Application Form is separated into several steps, the number of steps required will change dynamically depending on your chosen course. You can. SAVE. the form at any time using the. Save and Continue Later. link at the bottom of the page. Your progress will be saved and you will be sent a URL link by email which you can use when you are ready to resume. Course Information. Which course would you like to attend? *. Long-term. Do you require a Student Visa? *. Yes. No. Year. *. 2022. 2023. 2024. Year. *. 2022. 2023. 2024. Session. *. January. April. July. October. Period of study. *. 1 Year. 2 years. Period of study. *. 1 Month. 10 Weeks. Are you applying to the courses made in collaboration with \"Università Ca'Foscari di Venezia”？. *. Yes. No. JLPT Session. *. JLPT Summer - N1. JLPT Summer - N2. When would you like to start? *. DD slash MM slash YYYY. Course Level. *. Beginner\/Elementary. Pre-intermediate. Intermediate. Save and Continue Later Personal Information. Name. *. First. Last. Email. *. Enter Email. Confirm Email. Date of Birth. *. YYYY slash MM slash DD. Gender. *. Male. Female. Marital Status. *. Single. Married. Nationality. *. Albania. American Samoa. Antigua and Barbuda. Argentina. Armenia. Australia. Austria. Bahamas. Bahrain. Barbados. Belarus. Belgium. Belize. Bermuda. Bhutan. Bolivia. Bosnia and Herzegovina. Brazil. Brunei. Bulgaria. Canada. Cayman Islands. Chile. Colombia. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Denmark. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Estonia. Faroe Islands. Fiji. Finland. France. French Polynesia. Georgia. Germany. Greece. Greenland. Grenada. Guam. Guatemala. Haiti. Honduras. Hong Kong. Hungary. Iceland. Iran. Ireland. Israel. Italy. Jamaica. Japan. Jordan. Kazakhstan. North Korea. South Korea. Kosovo. Kuwait. Kyrgyzstan. Latvia. Lebanon. Liechtenstein. Lithuania. Luxembourg. Macedonia. Malaysia. Maldives. Malta. Marshall Islands. Mauritius. Mexico. Micronesia. Moldova. Monaco. Montenegro. Netherlands. New Zealand. Nicaragua. Northern Mariana Islands. Norway. Oman. Palau. Palestine, State of. Panama. Paraguay. Peru. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Saint Kitts and Nevis. Saint Lucia. Saint Vincent and the Grenadines. Saint Martin. Samoa. San Marino. Saudi Arabia. Serbia. Seychelles. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Spain. Suriname. Sweden. Switzerland. Taiwan. Tajikistan. Thailand. Tonga. Trinidad and Tobago. Turkey. Turkmenistan. Tuvalu. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Venezuela. Virgin Islands, British. Virgin Islands, U.S. Same as written on your passport. Nationality. *. Afghanistan. Albania. Algeria. American Samoa. Angola. Antigua and Barbuda. Argentina. Armenia. Australia. Austria. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bosnia and Herzegovina. Botswana. Brazil. Brunei. Bulgaria. Burkina Faso. Burundi. Cabo Verde. Cambodia. Cameroon. Canada. Cayman Islands. Central African Republic (CAR). Chad. Chile. China. Colombia. Comoros. Costa Rica. Cote d'Ivoire. Croatia. Curaçao. Cyprus. Czech Republic. Democratic Republic of the Congo. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (formerly Swaziland). Ethiopia. Faroe Islands. Fiji. Finland. France. French Polynesia. Gabon. Gambia. Georgia. Germany. Ghana. Greece. Greenland. Grenada. Guam. Guatemala. Guinea. Guinea-Bissau. Haiti. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Israel. Italy. Jamaica. Japan. Jordan. Kazakhstan. Kenya. Kosovo. Kuwait. Kyrgyzstan. Laos. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Mauritania. Mauritius. Mauritius. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Morocco. Mozambique. Myanmar. Namibia. Nepal. Netherlands. New Zealand. Nicaragua. Niger. Nigeria. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Paraguay. Peru. Philippines. Poland. Portugal. Puerto Rico. Qatar. Republic of the Congo. Romania. Russia. Rwanda. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Swaziland (renamed to Eswatini). Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Togo. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Tuvalu. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Zambia. Zimbabwe. Same as written on your passport. Occupation. *. Phone Number (Home Country). *. Permanent Address (Home Country). *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Where do\/will you live in Japan. *. Same as the permanent address above. Different from the permanent address above. I would like a homestay with a Japanese family. I haven't decided. Address in Japan. *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Phone Number in Japan. Save and Continue Later Company Information. Contact Name. *. First. Last. Email. *. Enter Email. Confirm Email. Company Name. *. Company Address. *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Company Phone. *. Website. Save and Continue Later Emergency Contact and Guarantor. Name (Emergency Contact). *. First. Last. Email (Emergency Contact). Enter Email. Confirm Email. Relation to applicant. *. Phone Number (Emergency Contact). *. Address (Emergency Contact). *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Guarantor - for those who are 17 years old or younger. Are you 18 years old or older? *. Yes. No. Who is your guarantor. *. Same as the emergency contact. Different than the emergency contact. Guarantor's Name. *. First. Last. Relation to applicant. *. Phone Number (Guarantor). *. Address (Guarantor). *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Save and Continue Later Past Japanese Study Experience. Have you ever studied Japanese before? *. Yes. No. Select \"Yes\" if you have more than 50 hours of experience studying Japanese in a school or by yourself. Japanese study detail. *. Click on the. button to add an entry. Study type. Institution Name. Institution Location. Start Date. End Date. Self-Study. High School. University. Language School. Other. Have you passed any level of the JLPT? *. Yes. No. I have passed a different test (J-Test, NAT test, etc.). What is the highest level you passed? *. N5. N4. N3. N2. N1. When did you pass it? *. DD slash MM slash YYYY. Other test details. *. Test name, level etc. Save and Continue Later Educational Background. Please provide the list of schools that you have attended. If you do not remember the exact day, please use 01 for the day. Educational background. *. Click on the. button to add an entry. Please specify type of institution if you choose. \"Other\". under School Type. School Type. \"Other\" School Type. School Name. City\/Country. Start Date. Graduation Date. Elementary. Junior High School. High School. University. Other. Save and Continue Later Past (or current) entry into \/ departure from Japan. Have you ever been to Japan before? *. Yes. No. Are you currently in Japan? *. Yes. No. Number of times. *. Entries\/Departures Details. *. Click on the. button to add an entry. Entry Date. Departure Date. Purpose of Visit. Do you have a Criminal Record in Japan or Overseas? *. Yes. No. Please provide details. *. Have you ever been deported from Japan? *. Yes. No. Number of times. *. Last departure by deportation. *. YYYY slash MM slash DD. Deportation Details. *. Have you ever received a Visa for Japan and\/or a Certificate of Eligibility? *. Yes. No. Visa Type. *. Student Visa. Details. *. Have you ever been declined a Certificate of Eligibility and\/or a Visa from the Immigration Bureau of Japan? Have you ever been declined the permission of landing, the change of status and\/or the extension of a Japanese Visa or permit? *. Yes. No. Visa Type. *. Student Visa. Details. *. Save and Continue Later Family in Japan. Do you have any family members living in Japan? *. Yes. No. Details of the your family member(s) in Japan. *. Click on the. button to add an entry. Name. Relationship. Date of Birth. Nationality. Will you live together? Place of employment\/school. Residence Card number. Yes. No. Save and Continue Later Reason for Studying Japanese. This isn't an essay, but it is a formality that the Japanese Immigration Bureau requires. Write at least 150 words in English or 300 characters in Japanese where you explain why you wish to study Japanese in Japan, if you have any previous experience studying Japanese, please mention that. Also include your future plans after graduating Shibuya Gaigo Gakuin. Pro Tip: Focus more on things like education, international relations, work etc and less on anime, manga & games. Details. *. Plans after graduation from Shibuya Gaigo Gakuin. *. Return to home country. Enter school of higher education in Japan. Find work in Japan. Save and Continue Later Work History. What is your current occupation? *. Student. Full time employee. Part time employee. Unemployed \/ Preparing for Japan. Business Owner. Other. Occupation. *. Company name. *. Position title. *. Start Date. *. YYYY slash MM slash DD. Company phone number. *. Company address. *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Past Work Experience. Have you had any full-time or part-time jobs in the past? *. Yes. No. Work History. *. Click on the. button to add an entry. Location Format: City\/Country. Company Name. Location. Start Date. End Date. Save and Continue Later Financial Support. Who will finance your stay in Japan? *. I will myself. My financial supporter living outside Japan. My financial supporter living in Japan. Scholarship. Other. Details. Select yourself if you have a stable income since 2 years back or more and if you're able to provide a bank statement with enough funds to pay your school tuition and living expenses in Japan. Otherwise you will have to use a financial supporter. We do not recommend using a financial supporter in Japan as the immigration office is much more strict in such case. Though, if you would still like to indicate someone in Japan for this role, please send by e-mail the following documents as well: 1. 身分証明書 2. 在職証明書 3. 収入証明書（過去3年分）源泉徴収票など 4. 銀行残高証明書 5. 銀行預金通帳の写し（過去3年分） 6. 世帯全員の記載のある住民票の写し 7. 住民税の課税（又は非課税）証明書及び納税証明書（１年間の総所得及び納税状況が記載されたもの）（過去3年分）. Financial Supporter Name. *. First. Last. Financial Supporter's relation to you? *. Father. Mother. Grandfather. Grandmother. Husband. Wife. Brother. Sister. Other. we strongly recommend limiting this role to a close family member. Supporter's home address. *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Supporter's phone number. *. Reason for support. *. Supporter's employment information. Occupation. *. Position title. *. Company Name. *. Company Phone Number. *. Company Address. *. Street Address. Address Line 2. City. State \/ Province \/ Region. ZIP \/ Postal Code. Afghanistan. Albania. Algeria. American Samoa. Andorra. Angola. Anguilla. Antarctica. Antigua and Barbuda. Argentina. Armenia. Aruba. Australia. Austria. Azerbaijan. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bonaire, Sint Eustatius and Saba. Bosnia and Herzegovina. Botswana. Bouvet Island. Brazil. British Indian Ocean Territory. Brunei Darussalam. Bulgaria. Burkina Faso. Burundi. Cambodia. Cameroon. Canada. Cape Verde. Cayman Islands. Central African Republic. Chad. Chile. China. Christmas Island. Cocos Islands. Colombia. Comoros. Congo, Democratic Republic of the. Congo, Republic of the. Cook Islands. Costa Rica. Croatia. Cuba. Curaçao. Cyprus. Czech Republic. Côte d'Ivoire. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (Swaziland). Ethiopia. Falkland Islands. Faroe Islands. Fiji. Finland. France. French Guiana. French Polynesia. French Southern Territories. Gabon. Gambia. Georgia. Germany. Ghana. Gibraltar. Greece. Greenland. Grenada. Guadeloupe. Guam. Guatemala. Guernsey. Guinea. Guinea-Bissau. Guyana. Haiti. Heard and McDonald Islands. Holy See. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Isle of Man. Israel. Italy. Jamaica. Japan. Jersey. Jordan. Kazakhstan. Kenya. Kiribati. Kuwait. Kyrgyzstan. Lao People's Democratic Republic. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macau. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Martinique. Mauritania. Mauritius. Mayotte. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Montserrat. Morocco. Mozambique. Myanmar. Namibia. Nauru. Nepal. Netherlands. New Caledonia. New Zealand. Nicaragua. Niger. Nigeria. Niue. Norfolk Island. North Korea. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Papua New Guinea. Paraguay. Peru. Philippines. Pitcairn. Poland. Portugal. Puerto Rico. Qatar. Romania. Russia. Rwanda. Réunion. Saint Barthélemy. Saint Helena. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Pierre and Miquelon. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Georgia. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Svalbard and Jan Mayen Islands. Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Timor-Leste. Togo. Tokelau. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Turks and Caicos Islands. Tuvalu. US Minor Outlying Islands. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Vanuatu. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Wallis and Futuna. Western Sahara. Yemen. Zambia. Zimbabwe. Åland Islands. Country. Financial Information. Annual income. *. Currency. *. Bank Balance. *. You will be requested to upload a bank statement showing the amount above. Balance required: 1,200,000 JPY for 6 months; 2,000,000 JPY for 1 year course. Currency. *. Monthly Support. Method of Support. *. International money transfer. Carrying cash from outside Japan. Name of the individual carrying cash. *. Date and time of carrying cash. *. Monthly support\/self-support amount (in Japanese yen). *. The recommended amount is 80,000 JPY or more. Save and Continue Later Required Documents. Photo. A picture of yourself on a white background, similar to one taken for a passport or ID. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Photo. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Passport. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. For Student Visa Applicants:. If you have been to Japan before, please upload a copy of the visa stamp pages. Picture page. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Past Japan Immigration Stamps Pages. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Japan Residence Card. If you have a Japanese Residence Card, please upload scans of the front and back of your card. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Japanese Residence Card. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Education Certificates. Please upload Graduation certificates you may have. You may upload a Certificate of Attendance if you are currently attending a school. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Graduation Certificates. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Latest Official Transcripts. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Document that proves the relationship between you and your financial supporter. If the financial supporter is different from yourself, we will need a proof of your relation: “birth certificates”, “wedding contracts” and similar are all valid documents. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Proof of Relationship. *. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Financials. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Employment Certificate. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Tax Returns. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Bank Account Statement. Drop files here. Select files. Accepted file types: jpg, jpeg, png, tiff, pdf, Max. file size: 512 MB. Save and Continue Later Additional Comments. You can include any additional information, notes or questions you may have. The data collected will be used for the sole purpose of the school admission process, Student Visa request process and\/or Accommodation request. Please read the details of our. Privacy Policy. . *. I have read and agree with the Privacy Policy. Future Contact. *. You may also contact me with discounts, special offers, events and other promotions. I don't need discounts, You may ONLY contact me in regards to my application. *. By checking this box and providing your signature below, you declare that all the information you have provided is true and correct! Signature. *. Comments. This field is for validation purposes and should be left unchanged. Save and Continue Later ※Applicants who need to apply for a student visa: the next available session is January 2023. ※Applicants who wish to transfer from another school: the next available session is July 2022. (The transfer procedure needs to be complete by June 21st, 2022). ※Applicants who wish to change their visa status to \"Student Visa\" (excluding foreigners in possess of \"short-term permit\" or \"designated activities\" visa): the next available session is July 2022. (Please inquire at your local Immigration Bureau if you are allowed to change your visa status to \"Student\"). Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. Privacy Policy Designed by Souzou Web Scroll To Top."}
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{"id":"ELXxxs9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/april-2021-term-online\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":45,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["april-2021-term-online"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["April 2021 Term Online &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:52:42 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Mar 30 00:00:00 UTC 2021"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-3670622101285526306,"exact_signature_unique_b":true,"fuzzy_signature_l":-8682418350460376916,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["April 2021 Term Online"],"h2_txt":["SHIBUYA GAIGO GAKUIN NEWS","Posts navigation"],"h4_txt":["Recent News","Student Voice","Courses","Contact us"],"h5_txt":["March 30, 2021","Change of address and contact information","Immigration Updates and COVID-19 VACCINE","State of Emergency","April 2021 Term Online","School Visits Require Advance Appointment!","Tokyo State of Emergency"],"bold_txt":["English English"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/07\/Shibuya-Gaigo-Gakuin-Logo-black.svg"],"images_alt_sxt":["Japanese Language School"],"images_height_val":[2000],"images_width_val":[2000],"images_text_t":"shibuyajp net wp content uploads 2018 07 Shibuya Gaigo Gakuin Logo black svg Japanese Language School","responsetime_i":126,"text_t":"April 2021 Term Online &ndash; Shibuya Gaigo Gakuin. Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. SHIBUYA GAIGO GAKUIN. NEWS. April 2021 Term Online. March 30, 2021. Facebook. Twitter. Our April 2021 term is right around the corner with the first day this Friday the 2nd! It has been a full year of transitioning our lessons from in-person to online. To all our current and new students, we would like to announce a couple of reminders:. The entire April term (April, May, June) is held online until further notice. Therefore, it is mandatory to have compatible Wifi, microphone and camera on any smart device in order to be enrolled in our school. It is the student’s responsibility to make the necessary arrangements prior to the online lessons and to maintain connection throughout the course. Please note, BOTH your camera and microphone must be on and active for the entire duration of the lesson in order to receive full attendance for that day. We appreciate your cooperation and look forward to seeing you all soon! Posts navigation. &larr; School Visits Require Advance Appointment! State of Emergency &rarr; Recent News. Change of address and contact information Immigration Updates and COVID-19 VACCINE State of Emergency April 2021 Term Online School Visits Require Advance Appointment! Tokyo State of Emergency Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Skip to content 日本語. English 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. TEST YOUR. JAPANESE LEVEL. Name. *. 名. 姓. Email. *. メールアドレスの入力. メールアドレスを確認. Did you submit an Application Form? *. Yes. No. Which course did you apply to? *. 2 years. 1 year. 10 weeks. 4 weeks. Summer course. JLPT. Private \/ Semi-private. Others. Please, specify. *. Please, specify. When are you planning to start the course? *. 2022. 2023. Year. Session. *. January. April. July. October. Session. Save and Continue Later Have you ever studied Japanese before? *. Yes. No. Select \"Yes\" if you have more than 50 hours of experience studying Japanese in a school or by yourself. Japanese study detail. *. Study type. Institution Name. Institution Location. Start Date. End Date. Self-Study. High School. University. Language School. Did you use any specific textbook? *. Minna no Nihongo. Genki. Fun & Easy Japanese. No, I didn't use any textbook. Other. Other textbook. About how many hours in total have you studied? 50-100. 101-200. 201-400. 401-600. 600+. Do you know any of the following? Hiragana. Katakana. Kanji. How many kanji are you able to read and understand? 20 to 50. 51 to 200. 201 to 400. 400+. Have you passed any level of the JLPT? Yes. No. I have passed a different test (J-Test, NAT test, etc.). What is the highest level you passed? N5. N4. N3. N2. N1. When did you pass it? DD スラッシュ MM スラッシュ YYYY. Other test details. Test name, level etc. Save and Continue Later Let's get started! このシャツは、二千五百えんでした。. にぜんごびゃく. にせんごぴゃく. にせんごひゃく. にぜんごぴゃく. 出口はあちらです。. いりぐち. でぐち. しゅつくち. にゅうくち. わたしの弟は、だいがくせいです。. あに. あね. いもうと. おとうと. 明日は、都合がわるいので、行けません。. とごう. とあい. ぐあい. つごう. ここは、えきから遠くて不便です。. ふうべん. ふうびん. ふべん. ふびん. 正月は、かぞくといっしょにハワイへ行きます。. せいつき. しょうがつ. しょうつき. せいげつ. パーティーに出席する人は、ここに名前を書いてください。. しゅっせき. しゅつせい. しゅうせい. しゅせき. ここにあるパソコンは、自由につかってください。. しゆう. じゅう. じゆう. しゅう. きのう、とても悲しい本を読んだ. かなしい. むずかしい. くわしい. やさしい. 来月ヨーロッパへ行くので、インターネットでホテルの予約をした。. よやくう. よあく. ようやく. よやく. 昨日デパートで、小学校のときの友だちに偶然会った。. とうぜん. ぐうぜん. とつぜん. ぜんぜん. 旅行中、景色がいいところの写真をたくさんとった。. けいしょく. けしき. けいしき. けしょく. スーパーで、ぎゅうにゅうをかいました。. 員いました. 貝いました. 貴いました. 買いました. ともだちとでんわではなしました。. 話しました. 読みました. 語しました. 言いました. がっこうのまえに、ゆうびんきょくがあります。. 前. 中. 後. 外. かいぎで、わたしのいけんをいいました。. 発見. 発言. 意見. 意思. 日本のぶんがくを勉強するために来ました。. 文化. 文楽. 文学. 文字. パーティーのかいじょうはあちらです。. 開所. 会場. 開場. 会所. にもつをたくさん入れたら、紙のふくろがやぶれてしまった。. 破れて. 彼れて. 涙れて. 被れて. この魚は海のふかいところにすんでいます。. 浅い. 温い. 涼い. 深い. 朝からずっとおなかがいたいので、くすりを飲みました。. 痛い. 痒い. 疲い. 病い. 帰りがおそくなる時は、電話でれんらくしてください。. 連結. 練結. 練絡. 連絡. 彼のてきせつな説明のおかげで、問題点がよくわかった。. 適設. 練絡. 的節. 適切. こんな天気の悪い日に、山に登るのはあぶないからやめなさい。. 険ない. 危ない. 恐ない. 怖ない. 「すみません、このペンは（　　　）ですか。」「350円です。」. いかか. いつ. いくら. いくつ. コーヒーは（　　　）すきじゃありません。. よく. とても. すぐ. あまり. わたしはまいあさ、シャワーを（　　　）。. あびます. いれます. はいります. あります. （　　　）ぜひ、うちへ　あそびに来てください。. このごろ. こんど. ずっと. さいきん. 先生がびょうきなので、（　　　）に行きました。. おいわい. おまつり. おれい. おみまい. かぜをひいて、りょこうに行けなくて（　　　）だった。. ねっしん. あんぜん. むり. ざんねん. 今回の旅行ではあまりお金をつかわないように（　　　）安いホテルをさがそう。. とつぜん. なるべく. しだいに. やっと. ４月からのしごとの（　　　）がきまった。. シリーズ. コース. ダイヤ. スケジュール. わたしの（　　　）は、いつも元気なところです。. 長所. 良所. 強所. 明所. さいきんは家でDVDを見ることが多く、映画館へは（　　　）行かない。. たまには. とうとう. めったに. どんどん. 時間はまだ（　　　）あるから、そんなにあわてなくてもいいですよ。. ぎっしり. たっぷり. さっぱり. ゆっくり. 開館時間は10時から18時までです。（　　　）土曜日は14時までです。. さて. つまり. それで. ただし. きょねんのなつは（　　　）。. あつぃかったです. あつかったです. あついでした. あつくでした. あのレストランは、（　　　）やすいです。. おいしい. おいしいと. おいしくて. おいしいで. きのう、こうえん（　　　）テニスをしました。. で. の. に. を. ひるごはんを（　　　）まえに、てをあらいました。. たべる. たべ. たべて. たべた. デパートへいきましたが、（　　　）かいませんでした。. なにが. なにか. なにを. なにも. これは、わたしのははが（　　　）ケーキです。. つくります. つくった. つくり. つくって. かってきたビール、れいぞうこにいれて（　　　）ね。. おいて. ないで. あって. いよう. おいしいすしを（　　　）、あの店がいいよ。. 食べるなら. 食べながら. 食べたい. 食べたり. 私は夏休みに、レストランでアルバイトをする（　　　）だ。. ところ. つもり. ばかり. こと. 新聞によると、きのう中国でじしんが（　　　）。. あるそうです. ありそうです. あるそうでした. あったそうです. 木村さんはアメリカに10年すんでいたから、英語がじょうずな（　　　）。. ところだ. らしい. だろう. はずだ. 「山田さんはいますか。」「山田さんはさっき（　　　）ところです。」. 帰った. 帰っている. 帰りたい. 帰る. すみませんが、こちらにお名前を（　　　）ください。. お書きいただいて. お書き. お書きして. お書きされて. あそこに小さい男の子が立っている（　　　）が見えますか。. もの. の. こと. ところ. 子どものころ、わたしは母にそうじを（　　　）。. てつだわれた. てつだわされた. てつだわせた. てつだいた. スーパーでお金を（　　　）としてかばんを開けたら、さいふがなかった。. はらおう. はらえる. はらいたい. はらう. タバコは体に悪いと知り（　　　）やめることができない。. にくく. ないで. ながら. ても. 暗くならない（　　　）、うちへ帰ろう。. あいだ. まえに. うちに. までに. あたたかくなる（　　　）、外へあそびに出かける人がふえてきた。. にしては. ところ. につれて. としても. ぜひ（　　　）たいのですが、今週のご予定はいかがでしょうか。. お目にかけ. お会いになり. ごらんになり. お目にかかり. 妹は体が弱くて、学校を休み（　　　）。. っぽい. がちだ. だらけだ. 気味だ. あのまじめな川上さん（　　　）そんないいかげんな仕事をするわけがない。. に伴って. に関して. に限って. に応じて. 図書館へ行く（　　　）、スーパーで買い物をしてこよう。. につれて. うえに. ばかりか. ついでに. 母にはだいじょうぶだと言った（　　　）、本当はとても不安だった。. つつも. とおり. ものの. からは. Additional Comments. You can include any additional information, notes or questions you may have.You can include any additional information, notes or questions you may have. Name. このフィールドは入力チェック用です。変更しないでください。. Save and Continue Later 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 個人情報保護方針 情報公開 Designed by Souzou Web Scroll To Top."}
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Skip to content 日本語. 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. Extension of Online Classes for April 2020 Term （日本語訳準備中）. 2020-04-28. Facebook. Twitter. Unfortunately, due to the increasing amount of COVID-19 cases in Japan, our school has decided to extend the online lessons until the end of the current April term.  Therefore, class will continue online even after the Golden Week\/ Spring Break May 11th and onward.  Details regarding the exams and Graduation will be given later on. Please note the extension may be shortened or prolonged depending on how the situation progresses.  In the event there are any changes, we will inform all students accordingly.  Please continue to stay safe and stay healthy. We apologize for any inconvenience and as always, if you have any questions or concerns, please feel free to contact us. Posts navigation. &larr; Students on a Student Visa Leaving Japan （日本語訳準備中） Online Lessons Will Continue &rarr; 最近のニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ Online Lessons Will Continue Extension of Online Classes for April 2020 Term （日本語訳準備中） Students on a Student Visa Leaving Japan （日本語訳準備中） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 オンラインレッスンの延長について 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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{"id":"gsBpXs9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/tokyo-state-of-emergency\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":47,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["tokyo-state-of-emergency"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Tokyo State of Emergency &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:52:29 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Wed Jan 13 00:00:00 UTC 2021"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":8155514133532352424,"exact_signature_unique_b":true,"fuzzy_signature_l":-403786388371799230,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["Tokyo State of Emergency"],"h2_txt":["SHIBUYA GAIGO GAKUIN NEWS","Posts navigation"],"h4_txt":["Recent News","Student Voice","Courses","Contact us"],"h5_txt":["January 13, 2021","Change of address and contact information","Immigration Updates and COVID-19 VACCINE","State of Emergency","April 2021 Term Online","School Visits Require Advance Appointment!","Tokyo State of Emergency"],"bold_txt":["English English"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/07\/Shibuya-Gaigo-Gakuin-Logo-black.svg"],"images_alt_sxt":["Japanese Language School"],"images_height_val":[2000],"images_width_val":[2000],"images_text_t":"shibuyajp net wp content uploads 2018 07 Shibuya Gaigo Gakuin Logo black svg Japanese Language School","responsetime_i":370,"text_t":"Tokyo State of Emergency &ndash; Shibuya Gaigo Gakuin. Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. SHIBUYA GAIGO GAKUIN. NEWS. Tokyo State of Emergency. January 13, 2021. Facebook. Twitter. We hope you are enjoying your first lesson of the January 2021 term! As you all may already know, the Japanese Government has confirmed and declared a State of Emergency. Please kindly refer to the following article for your reference:. https:\/\/www3.nhk.or.jp\/nhkworld\/en\/news\/20210107_42\/ Also, as a reminder regarding the United Kingdom, it is not possible to enter Japan if you are in or visited the UK in the last 14 days prior to your arrival in Japan. Since 1st October 2020, foreign nationals have been permitted to newly enter Japan from (in principle) any country\/region provided they meet the requirements of the Global Residence Track, i.e. provided a host company\/organization can guarantee that disease control measures will be followed. However, from 24th December (Japan Standard Time), foreign nationals will not be permitted to travel from the UK and newly enter Japan under this scheme, for the time being. Specifically, anyone holding a visa for Japan which says “EX-R” in the Remarks section will not be able to enter Japan using this visa if they have been in the UK in the previous 14 days. Until further notice, we will not be able to accept visa applications based on the Global Residence Track scheme, and any applications which we have accepted will take a considerable amount of time to be processed. If you are planning to visit the UK or currently reside there or are departing from there to enter Japan, we advise you to check the Japanese Embassy in the UK to be well informed and updated. Embassy website:. https:\/\/www.uk.emb-japan.go.jp\/itprtop_en\/index.html We appreciate everyone’s patience and cooperation. Posts navigation. &larr; UPDATED 2021 Year Schedule! School Visits Require Advance Appointment! &rarr; Recent News. Change of address and contact information Immigration Updates and COVID-19 VACCINE State of Emergency April 2021 Term Online School Visits Require Advance Appointment! Tokyo State of Emergency Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. SHIBUYA GAIGO GAKUIN. NEWS. School Closed for New Years! December 18, 2020. Facebook. Twitter. As our challenging 2020 year comes to an end, we look forward to see you all in the New Year 2021! Our school will be closed from December 28th (Monday) 2020 through January 4th (Monday) 2021.  Emails received during this time will be responded on a first come, first serve basis when we resume our normal business\/ school hours on January 5th (Tuesday) 2021. We hope you all have a wonderful Winter Break and safe holidays! Posts navigation. &larr; January 2021 Term Continues Online – Overseas Participation Available! UPDATED 2021 Year Schedule! &rarr; Recent News. Change of address and contact information Immigration Updates and COVID-19 VACCINE State of Emergency April 2021 Term Online School Visits Require Advance Appointment! Tokyo State of Emergency Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Skip to content 日本語. English 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. お申し込み方法はシンプル. 全てのコースはオンラインでお申し込みが可能です。もちろん学生ビザが必要なコースのお申込みにもご利用いただけます。. 留学ビザ取得までの流れ. 申し込む. 申し込みフォームを送信する. 選考料. 選考料のお支払い. 在留資格認定証明書の受け取り. 在留資格認定証明書（CoE）を受け取る. 留学ビザ申請. 留学ビザの申請・交付を受ける。. 入学. 日本での生活がスタート！. お申し込み. STEP 1:. 申し込みフォームの送信. お申し込み. STEP 2:. 日本語レベルチェック. レベルチェック. 支払い方法. 渋谷外語学院は海外送金の手段として提携先であるFlywire（フライワイヤー）を強くお勧めします。世界のあらゆる国から少ない手数料でスピーディな手続きが可能です。. お支払い. 返金規定. 　. 本人が申込みの意思を学校に伝え、学校が承諾したメールまたは文書をだした日から8日間であればクーリングオフが可能です。（日本時間が基準）. 返金に伴う銀行振込手数料は受取人負担です。Flywire以外で返金する場合は別途事務手数料を申し受けます。. 返金先の口座は原則本人名義か親族名義(または当初の送金人名義)に限ります。. ①. 在留資格認定証明書交付申請書類提出後のキャンセルの場合. ： 「在留資格認定証明書（COE）」の交付、不交付に関わらず、選考料は返金しません。. ②. 出入国在留管理局から「在留資格認定証明書（COE）」交付後で授業開始前のキャンセルの場合. ：. 理由の如何に関わらず選考料および入学金は返金しません。授業料は在留資格認定証明書および入学許可証と引き換えの上、全額返金いたします。また、入国ビザ発給後のキャンセルの場合、. 入国ビザが未使用であることが当校職員によって確認される必要となる。. ③. 日本大使館、領事館によってビザ発給が拒否された場合. ： 選考料は返金しません。入学金および授業料はビザ発給拒否が当校職員によって確認されたた後、在留資格認定証明書および入学許可証と引き換えの上、全額を返金いたします。. ④. 授業開始後のキャンセルの場合 ：. 選考料および入学金は返金しません。50,000円を上限に残りのレッスン分の授業料の20%に相当する金額をキャンセル料として申し受けます。在留カードが無効になったことが当校職員によって確認された後、残金を返金致します。. 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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Skip to content 日本語. English 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. Online Lessons Will Continue. 2020-09-04. Facebook. Twitter. Due to the ongoing pandemic, our lessons will continue online until further notice.  Therefore including our October 2020 term, there will be no lessons held at the school building. As many of you may already know, the Japanese Government has yet to allow certain countries to enter Japan.  In response to this, the school would like to offer the possibility to students who are overseas to participate in our online lessons in the levels that are available.  Depending on the time zone and the amount of students who opt for this choice, details and further information will be sent later on. We are grateful for your dedication, willingness and flexibility as we make these necessary adjustments during this time. Please continue to stay safe and healthy. 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Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. Alex. from England. Alex. from England. The Japanese we are taught is not standard textbook situations, but real life scenarios. The Japanese we are taught is not standard textbook situations, but real life scenarios. Q:.  . Why did you decide to live in Japan? A:.  . My dream is to open a restaurant in Tokyo, I decided to make another step towards my dream by studying Japanese in Japan. I figured that learning Japanese in Japan would be much more effective as you are constantly immersed in and forced to use Japanese. I couldn't have made a better choice as I had studied Japanese in my own country for 3 years, needless to say, 2 hours per week was not very effective. Q:.  . Why did you choose Shibuya Gaigo Gakuin? A:.  . Shibuya Gaigo Gakuin boosts focusing the curriculum more towards speaking the language which I think is the most important topic in any language. From what I have heard from students who have attended other language schools, Shibuya Gaigo Gakuin's curriculum is as they say  ( more oriented around speaking ) which is much more suited to me as writing essays and constantly copying from a textbook like other schools just isn't suitable for me as a practical learner. . Q:.  . In your opinion, what is the best thing about studying in Japan at Shibuya Gaigo Gakuin? A:.  . For me, the best thing is first being able to use what I learn in class out in the real world, as the Japanese we are taught are not standard textbook situations, but real life scenarios. Being able to experience life in Japan is great but being able to take what I learn in the classroom and apply it to my everyday life here is the best thing for me. . Q:.  . What are your plans after graduating from Shibuya Gaigo Gakuin? A:.  . After graduating I will apply for a working holiday visa so that I can travel around the country and explore the wonders it has to offer, while still being able to work to fund my living costs. After this, I will enter a business school here in Japan to take me one step closer to my dream. Q:.  . What would you tell to others who wish to study at Shibuya Gaigo Gakuin? A:.  . If you consider yourself a practical learner and value the importance of speaking then I would highly recommend Shibuya Gaigo Gakuin. Listening, reading and writing are still all taught effectively, however, speaking practice is where Shibuya Gaigo Gakuin is really ahead above the rest! Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Suriname. Sweden. Switzerland. Taiwan. Tajikistan. Thailand. Tonga. Trinidad and Tobago. Turkey. Turkmenistan. Tuvalu. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Venezuela. Virgin Islands, British. Virgin Islands, U.S. Same as written on your passport. Nationality. *. Afghanistan. Albania. Algeria. American Samoa. Angola. Antigua and Barbuda. Argentina. Armenia. Australia. Austria. Bahamas. Bahrain. Bangladesh. Barbados. Belarus. Belgium. Belize. Benin. Bermuda. Bhutan. Bolivia. Bosnia and Herzegovina. Botswana. Brazil. Brunei. Bulgaria. Burkina Faso. Burundi. Cabo Verde. Cambodia. Cameroon. Canada. Cayman Islands. Central African Republic (CAR). Chad. Chile. China. Colombia. Comoros. Costa Rica. Cote d'Ivoire. Croatia. Curaçao. Cyprus. Czech Republic. Democratic Republic of the Congo. Denmark. Djibouti. Dominica. Dominican Republic. Ecuador. Egypt. El Salvador. Equatorial Guinea. Eritrea. Estonia. Eswatini (formerly Swaziland). Ethiopia. Faroe Islands. Fiji. Finland. France. French Polynesia. Gabon. Gambia. Georgia. Germany. Ghana. Greece. Greenland. Grenada. Guam. Guatemala. Guinea. Guinea-Bissau. Haiti. Honduras. Hong Kong. Hungary. Iceland. India. Indonesia. Iran. Iraq. Ireland. Israel. Italy. Jamaica. Japan. Jordan. Kazakhstan. Kenya. Kosovo. Kuwait. Kyrgyzstan. Laos. Latvia. Lebanon. Lesotho. Liberia. Libya. Liechtenstein. Lithuania. Luxembourg. Macedonia. Madagascar. Malawi. Malaysia. Maldives. Mali. Malta. Marshall Islands. Mauritania. Mauritius. Mauritius. Mexico. Micronesia. Moldova. Monaco. Mongolia. Montenegro. Morocco. Mozambique. Myanmar. Namibia. Nepal. Netherlands. New Zealand. Nicaragua. Niger. Nigeria. Northern Mariana Islands. Norway. Oman. Pakistan. Palau. Palestine, State of. Panama. Paraguay. Peru. Philippines. Poland. Portugal. Puerto Rico. Qatar. Republic of the Congo. Romania. Russia. Rwanda. Saint Kitts and Nevis. Saint Lucia. Saint Martin. Saint Vincent and the Grenadines. Samoa. San Marino. Sao Tome and Principe. Saudi Arabia. Senegal. Serbia. Seychelles. Seychelles. Sierra Leone. Singapore. Sint Maarten. Slovakia. Slovenia. Solomon Islands. Somalia. South Africa. South Korea. South Sudan. Spain. Sri Lanka. Sudan. Suriname. Swaziland (renamed to Eswatini). Sweden. Switzerland. Syria. Taiwan. Tajikistan. Tanzania. Thailand. Togo. Tonga. Trinidad and Tobago. Tunisia. Turkey. Turkmenistan. Tuvalu. Uganda. Ukraine. United Arab Emirates. United Kingdom. United States. Uruguay. Uzbekistan. Venezuela. Vietnam. Virgin Islands, British. Virgin Islands, U.S. Zambia. Zimbabwe. Same as written on your passport. Occupation. *. Phone Number (Home Country). *. Permanent Address (Home Country). *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Where do\/will you live in Japan. *. Same as the permanent address above. Different from the permanent address above. I would like a homestay with a Japanese family. I haven't decided. Address in Japan. *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Phone Number in Japan. Save and Continue Later Company Information. Contact Name. *. 名. 姓. Email. *. メールアドレスの入力. メールアドレスを確認. Company Name. *. Company Address. *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Company Phone. *. Website. Save and Continue Later Emergency Contact and Guarantor. Name (Emergency Contact). *. 名. 姓. Email (Emergency Contact). メールアドレスの入力. メールアドレスを確認. Relation to applicant. *. Phone Number (Emergency Contact). *. Address (Emergency Contact). *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Guarantor - for those who are 17 years old or younger. Are you 18 years old or older? *. Yes. No. Who is your guarantor. *. Same as the emergency contact. Different than the emergency contact. Guarantor's Name. *. 名. 姓. Relation to applicant. *. Phone Number (Guarantor). *. Address (Guarantor). *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Save and Continue Later Past Japanese Study Experience. Have you ever studied Japanese before? *. Yes. No. Select \"Yes\" if you have more than 50 hours of experience studying Japanese in a school or by yourself. Japanese study detail. *. Click on the. button to add an entry. Study type. Institution Name. Institution Location. Start Date. End Date. Self-Study. High School. University. Language School. Other. Have you passed any level of the JLPT? *. Yes. No. I have passed a different test (J-Test, NAT test, etc.). What is the highest level you passed? *. N5. N4. N3. N2. N1. When did you pass it? *. DD スラッシュ MM スラッシュ YYYY. Other test details. *. Test name, level etc. Save and Continue Later Educational Background. Please provide the list of schools that you have attended. If you do not remember the exact day, please use 01 for the day. Educational background. *. Click on the. button to add an entry. Please specify type of institution if you choose. \"Other\". under School Type. School Type. \"Other\" School Type. School Name. City\/Country. Start Date. Graduation Date. Elementary. Junior High School. High School. University. Other. Save and Continue Later Past (or current) entry into \/ departure from Japan. Have you ever been to Japan before? *. Yes. No. Are you currently in Japan? *. Yes. No. Number of times. *. Entries\/Departures Details. *. Click on the. button to add an entry. Entry Date. Departure Date. Purpose of Visit. Do you have a Criminal Record in Japan or Overseas? *. Yes. No. Please provide details. *. Have you ever been deported from Japan? *. Yes. No. Number of times. *. Last departure by deportation. *. YYYY スラッシュ MM スラッシュ DD. Deportation Details. *. Have you ever received a Visa for Japan and\/or a Certificate of Eligibility? *. Yes. No. Visa Type. *. Student Visa. Details. *. Have you ever been declined a Certificate of Eligibility and\/or a Visa from the Immigration Bureau of Japan? Have you ever been declined the permission of landing, the change of status and\/or the extension of a Japanese Visa or permit? *. Yes. No. Visa Type. *. Student Visa. Details. *. Save and Continue Later Family in Japan. Do you have any family members living in Japan? *. Yes. No. Details of the your family member(s) in Japan. *. Click on the. button to add an entry. Name. Relationship. Date of Birth. Nationality. Will you live together? Place of employment\/school. Residence Card number. Yes. No. Save and Continue Later Reason for Studying Japanese. This isn't an essay, but it is a formality that the Japanese Immigration Bureau requires. Write at least 150 words in English or 300 characters in Japanese where you explain why you wish to study Japanese in Japan, if you have any previous experience studying Japanese, please mention that. Also include your future plans after graduating Shibuya Gaigo Gakuin. Pro Tip: Focus more on things like education, international relations, work etc and less on anime, manga & games. Details. *. Plans after graduation from Shibuya Gaigo Gakuin. *. Return to home country. Enter school of higher education in Japan. Find work in Japan. Save and Continue Later Work History. What is your current occupation? *. Student. Full time employee. Part time employee. Unemployed \/ Preparing for Japan. Business Owner. Other. Occupation. *. Company name. *. Position title. *. Start Date. *. YYYY スラッシュ MM スラッシュ DD. Company phone number. *. Company address. *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Past Work Experience. Have you had any full-time or part-time jobs in the past? *. Yes. No. Work History. *. Click on the. button to add an entry. Location Format: City\/Country. Company Name. Location. Start Date. End Date. Save and Continue Later Financial Support. Who will finance your stay in Japan? *. I will myself. My financial supporter living outside Japan. My financial supporter living in Japan. Scholarship. Other. Details. Select yourself if you have a stable income since 2 years back or more and if you're able to provide a bank statement with enough funds to pay your school tuition and living expenses in Japan. Otherwise you will have to use a financial supporter. We do not recommend using a financial supporter in Japan as the immigration office is much more strict in such case. Though, if you would still like to indicate someone in Japan for this role, please send by e-mail the following documents as well: 1. 身分証明書 2. 在職証明書 3. 収入証明書（過去3年分）源泉徴収票など 4. 銀行残高証明書 5. 銀行預金通帳の写し（過去3年分） 6. 世帯全員の記載のある住民票の写し 7. 住民税の課税（又は非課税）証明書及び納税証明書（１年間の総所得及び納税状況が記載されたもの）（過去3年分）. Financial Supporter Name. *. 名. 姓. Financial Supporter's relation to you? *. Father. Mother. Grandfather. Grandmother. Husband. Wife. Brother. Sister. Other. we strongly recommend limiting this role to a close family member. Supporter's home address. *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Supporter's phone number. *. Reason for support. *. Supporter's employment information. Occupation. *. Position title. *. Company Name. *. Company Phone Number. *. Company Address. *. 住所 (丁目、番地、号). 住所2. 区市町村. 州 \/ 都道府県. 郵便番号. アイスランド. アイルランド. アゼルバイジャン. アフガニスタン. アメリカ合衆国. アメリカ領ヴァージン諸島. アラブ首長国連邦. アルジェリア. アルゼンチン. アルバ. アルバニア. アルメリア. アンギラ. アンゴラ. アンチグアバーブーダ. アンドラ. イエメン. イギリス. イギリス領インド洋地域. イギリス領ヴァージン諸島. イスラエル. イタリア. イラク. イラン. インド. インドネシア. ウォリス・フツナ. ウガンダ. ウクライナ. ウズベキスタン. ウルグアイ. エクアドル. エジプト. エストニア. エスワティニ. エチオピア. エリトリア. エルサルバドル. オマーン. オランダ. オーストラリア. オーストリア. オーランド諸島. カザフスタン. カタール. カナダ. カメルーン. カンボジア. カーボヴェルデ. ガイアナ. ガボン. ガンビア. ガーナ. ガーンジー. キプロス. キュラソー島. キューバ. キリバス. キルギスタン. ギニア. ギニアビサウ. ギリシャ. クウェート. クック諸島. クリスマス島. クロアチア. グアテマラ. グアドループ. グアム. グリーンランド. グレナダ. ケイマン諸島. ケニア. ココス諸島. コスタリカ. コモロ. コロンビア. コンゴ共和国. コンゴ民主共和国. コートジボワール. サウジアラビア. サウスジョージア. サモア. サントメ・プリンシペ. サンピエール島・ミクロン島. サンマリノ. サン・バルテルミー. ザンビア. シエラレオネ. シリア. シンガポール. ジブチ. ジブラルタル. ジャマイカ. ジャージー島. ジョージア. ジンバブエ. スイス. スウェーデン. スペイン. スリナム. スリランカ. スロバキア. スロベニア. スヴァールバル諸島およびヤンマイエン島. スーダン. セイシェル. セネガル. セルビア. セントクリストファー・ネイビス. セントビンセントおよびグレナディーン諸島. セントヘレナ. セントルシア. セント・マーチン. セント・マーチン. ソマリア. ソロモン諸島. タイ. タジキスタン. タンザニア. タークス・カイコス諸島. チェコ共和国. チャド. チュニジア. チリ. ツバル. デンマーク. トケラウ諸島. トリニダード・トバゴ. トルクメニスタン. トルコ. トンガ. トーゴ. ドイツ. ドミニカ. ドミニカ共和国. ナイジェリア. ナウル. ナミビア. ニウエ. ニカラグア. ニジェール. ニューカレドニア. ニュージーランド. ネパール. ノルウェー. ノーフォーク島. ハイチ. ハンガリー. ハード島とマクドナルド諸島. バヌアツ. バハマ. バミューダ. バルバドス. バングラデシュ. バーレーン. パキスタン. パナマ. パプアニューギニア. パラオ. パラグアイ. パレスチナ自治政府. ピトケアン諸島. フィジー. フィリピン. フィンランド. フェロー諸島. フォークランド諸島. フランス. フランス南部領土. フランス領ギアナ. フランス領ポリネシア. ブラジル. ブルガリア. ブルキナファソ. ブルネイ・ダルサラーム国. ブルンジ. ブータン. ブーベ島. プエルトリコ. ベトナム. ベナン. ベネズエラ. ベラルーシ. ベリーズ. ベルギー. ペルー. ホンジュラス. ボスニア・ヘルツェゴビナ. ボツワナ. ボネール、シント・ユースタティウスおよびサバ. ボリビア. ポルトガル. ポーランド. マカオ. マケドニア. マダガスカル. マヨット. マラウイ. マリ. マルタ. マルティニーク. マレーシア. マン島. マーシャル諸島. ミクロネシア. ミャンマー. メキシコ. モザンビーク. モナコ. モルディブ. モルドバ. モロッコ. モンゴル. モンテネグロ. モントセラート. モーリシャス. モーリタニア. ヨルダン. ラオス人民民主共和国. ラトビア. リトアニア. リヒテンシュタイン. リビア. リベリア. ルクセンブルク. ルワンダ. ルーマニア. レソト. レバノン. レユニオン. ロシア. 中国. 中央アフリカ共和国. 北マリアナ諸島連邦. 北朝鮮. 南アフリカ. 南スーダン. 南極大陸. 台湾. 合衆国領有小離島. 日本. 東ティモール. 米サモア. 聖座. 西サハラ. 赤道ギニア共和国. 韓国. 香港. 国. Financial Information. Annual income. *. Currency. *. Bank Balance. *. You will be requested to upload a bank statement showing the amount above. Balance required: 1,200,000 JPY for 6 months; 2,000,000 JPY for 1 year course. Currency. *. Monthly Support. Method of Support. *. International money transfer. Carrying cash from outside Japan. Name of the individual carrying cash. *. Date and time of carrying cash. *. Monthly support\/self-support amount (in Japanese yen). *. The recommended amount is 80,000 JPY or more. Save and Continue Later Required Documents. Photo. A picture of yourself on a white background, similar to one taken for a passport or ID. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Photo. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Passport. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. For Student Visa Applicants:. If you have been to Japan before, please upload a copy of the visa stamp pages. Picture page. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Past Japan Immigration Stamps Pages. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Japan Residence Card. If you have a Japanese Residence Card, please upload scans of the front and back of your card. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Japanese Residence Card. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Education Certificates. Please upload Graduation certificates you may have. You may upload a Certificate of Attendance if you are currently attending a school. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Graduation Certificates. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Latest Official Transcripts. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Document that proves the relationship between you and your financial supporter. If the financial supporter is different from yourself, we will need a proof of your relation: “birth certificates”, “wedding contracts” and similar are all valid documents. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Proof of Relationship. *. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Financials. The following formats are accepted: JPG, JPEG, PNG, PDF, TIFF. Employment Certificate. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Tax Returns. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Bank Account Statement. ファイルをここにドロップするか、. ファイルを選択. サポートしているファイルタイプ: jpg, jpeg, png, tiff, pdf, 最大ファイルサイズ：512 MB. Save and Continue Later Additional Comments. You can include any additional information, notes or questions you may have. The data collected will be used for the sole purpose of the school admission process, Student Visa request process and\/or Accommodation request. Please read the details of our. Privacy Policy. . *. I have read and agree with the Privacy Policy. Future Contact. *. You may also contact me with discounts, special offers, events and other promotions. I don't need discounts, You may ONLY contact me in regards to my application. *. By checking this box and providing your signature below, you declare that all the information you have provided is true and correct! Signature. *. Name. このフィールドは入力チェック用です。変更しないでください。. Save and Continue Later ※学生ビザの申請が必要な方： 次回は2022年10学期のコースのお申し込みを受付けます。. ※他校から転校を希望する方： 次回は2022年7学期のコースのお申し込みを受付けます。. ※現在お持ちのビザを学生ビザに変更したい方： 次回は2022年7学期のコースのお申し込みを受付けます。(現在のビザの種類が変更可能か、まず入国管理局にお尋ねください). 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 個人情報保護方針 情報公開 Designed by Souzou Web Scroll To Top."}
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Skip to content 日本語. 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. オンラインレッスンの延長について. 2020-04-10. Facebook. Twitter. これまで皆様にご協力をいただき感謝申し上げます。. ご存知のとおり、感染拡大の状況はまだ改善されておりません。. 当校は日本政府の最近の要請に応じて、また、学生のみなさま及び講師やスタッフの健康を最優先に検討した結果、. 4月学期のオンラインによるレッスン実施を延長することを決定しました。. オンラインレッスンは4月30日(木)まで、. 校舎での授業再開は、ゴールデンウィーク休み（春休み）明けの5月11日(月)を予定しております。. この期間に必要な調整を行うため、皆様の献身、意欲、柔軟性に感謝いたします。. 私たちは、急速な感染拡大を防ぐための協力的な取り組みとして、「社会的距離」を実践し続けることを全員に助言します。. ご質問やご不明な点がございましたら、いつでもお気軽にお問い合わせください。. ーーーーーーーー. We appreciate everyone’s sincere cooperation thus far. Moving forward, after careful consideration and many discussions, the school has officially decided to extend the online lessons for the April term. As many of you may already know, the situation has yet to improve.  In response to the Japanese Government’s recent updates, we feel this is the best decision for not only our students, but our teachers and staff as well. The online lesson change has been extended from April 10. th. (Friday) to the new date of April 30. th. (Thursday). Therefore, lessons held at the school building will resume. May 11. th. (Monday). after the Golden Week (Spring Break) holiday. We are grateful for your dedication, willingness and flexibility as we make these necessary adjustments during this time. We advise everyone to continue to practice social distancing as a collaborated effort to prevent the rapid spread. May you have any questions or concerns, please do not hesitate to contact us anytime. Posts navigation. &larr; 2020年4月学期に関する情報– 新型コロナウイルス関連（COVID-19） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 &rarr; 最近のニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ Online Lessons Will Continue Extension of Online Classes for April 2020 Term （日本語訳準備中） Students on a Student Visa Leaving Japan （日本語訳準備中） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 オンラインレッスンの延長について 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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Skip to content 日本語. English 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. 2019年サマーコースの申込締切を延長しました. 2019-07-17. Facebook. Twitter. 2019年サマーコースの申込締切を下記の通り延長しました。. 締切：2019年7月31日. ※2019年度のサマーコースは、入門\/初級レベルのみの募集となります。. Posts navigation. &larr; 【復旧しました】緊急連絡先 電話番号のご案内 JR渋谷駅 西口からのアクセス方法　詳しい道順を写真付きでご紹介 &rarr; 最近のニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ Online Lessons Will Continue Extension of Online Classes for April 2020 Term （日本語訳準備中） Students on a Student Visa Leaving Japan （日本語訳準備中） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 オンラインレッスンの延長について 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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{"id":"diWjDs9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/online-lessons-will-continue\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":51,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["online-lessons-will-continue"],"content_type":["text\/html"],"crawldepth_i":3,"collection_sxt":["user"],"title":["Online Lessons Will Continue &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:53:18 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Fri Aug 21 00:00:00 UTC 2020"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2543856608806356091,"exact_signature_unique_b":true,"fuzzy_signature_l":6554968583818798762,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["Online Lessons Will Continue"],"h2_txt":["SHIBUYA GAIGO GAKUIN NEWS","Posts navigation"],"h4_txt":["Recent News","Student Voice","Courses","Contact us"],"h5_txt":["August 21, 2020","Change of address and contact information","Immigration Updates and COVID-19 VACCINE","State of Emergency","April 2021 Term Online","School Visits Require Advance Appointment!","Tokyo State of Emergency"],"bold_txt":["English English 日本語"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/07\/Shibuya-Gaigo-Gakuin-Logo-black.svg"],"images_alt_sxt":["Japanese Language School"],"images_height_val":[2000],"images_width_val":[2000],"images_text_t":"shibuyajp net wp content uploads 2018 07 Shibuya Gaigo Gakuin Logo black svg Japanese Language School","responsetime_i":55,"text_t":"Online Lessons Will Continue &ndash; Shibuya Gaigo Gakuin. Skip to content English. English 日本語 Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. SHIBUYA GAIGO GAKUIN. NEWS. Online Lessons Will Continue. August 21, 2020. Facebook. Twitter. Due to the ongoing pandemic, our lessons will continue online until further notice.  Therefore including our October 2020 term, there will be no lessons held at the school building. As many of you may already know, the Japanese Government has yet to allow certain countries to enter Japan.  In response to this, the school would like to offer the possibility to students who are overseas to participate in our online lessons in the levels that are available.  Depending on the time zone and the amount of students who opt for this choice, details and further information will be sent later on. We are grateful for your dedication, willingness and flexibility as we make these necessary adjustments during this time. Please continue to stay safe and healthy. Posts navigation. &larr; SGG TV is now LIVE on YouTube! January 2021 Term Continues Online – Overseas Participation Available! &rarr; Recent News. Change of address and contact information Immigration Updates and COVID-19 VACCINE State of Emergency April 2021 Term Online School Visits Require Advance Appointment! Tokyo State of Emergency Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. JR渋谷駅 西口からのアクセス方法. January 24, 2020. 〒150-0031東京都渋谷区桜丘町15-17 NKG東京ビル. JR渋谷駅 南改札 西口から日伊学院までのアクセス方法を写真付きでご紹介いたします。. 東急東横線・副都心線 渋谷駅をご利用の方も、ヒカリエ改札(地下3階)を出て14番出口よりJR渋谷駅 南改札を経由してお越しいただけます。. ◆Google mapはこちら まずは南改札から、JR駅構内にある案内に沿って「西口」を目指してください。. . . JR渋谷駅の西口です。正面にバスロータリー、左手に「緑の窓口」があります。西口に出たら左方向へ。. . . 「緑の窓口」の角を左に曲がり、すぐ正面に見えてくる歩道橋を上がります。（首都高速道路・246号の方向）. . . . . 桜丘方面を目指して歩道橋を渡ります。「Ikebe MUSIC」の黒い建物が目印です。. . . . . 「Ikebe MUSIC」のある方向を目指して階段を降ります。. . . 「Ikebe MUSIC」の文字のある黒い建物の方へ横断歩道を渡り、左方向へ進みます。. . . 「Ikebe MUSIC」を右にして進むと、その並びに「キリンシティ」が見えてきます。. . . 「キリンシティ」の先に道路を挟んで「すき屋」が見えるのでそれに向かって横断歩道を渡ります。. . . 「すき家」を左にして坂を上ります。. . . 「グランベルホテル」の建物（１Fはピザ店「CONA」）のすぐ先を左に入ります。. . . 通りから少し奥まったところにある薄い黄色のビルが渋谷ホール＆スタジオのある「NKG東京ビル」です。. . . １F駐車場内に入ってすぐ右手のエレベーターをご利用いただけます。. ※当施設の建物は「NKG東京ビル」ですが、一部の地図で「NKG東京本社ビル」と表示されることがございます。. ※隣の茶色の建物は「NKG東京 第2ビル」です。お越しの際はお間違えのないようご注意ください。. Share with your friends! Facebook. Twitter. Discover other blogs in. Uncategorized category. Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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Skip to content 日本語. 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. Students on a Student Visa Leaving Japan （日本語訳準備中）. 2020-04-28. Facebook. Twitter. For students on a Student Visa, we would like to reiterate that if you decide to officially withdraw from our school or you complete your studies and leave Japan, you have to be sure to cancel your Student Visa at airport Immigration.  Please fill out the Embarkation Card for Reentrant form permit located at the tables near the Immigration windows.  On the 【DEPARTURE】 side, you must check the box #2 stating . “I do not plan to re-enter Japan while my re-entry permit is valid”. .   If you are asked by the Immigration officer whether you are planning to come back to Japan in the future, even if you plan to do so you cannot do it with the same Residence Card, so be sure to answer that you want to cancel your current visa.   When you submit this to Immigration along with your Passport and Residence Card, Immigration will then stamp a hole in your Residence Card and place a Departure stamp into your Passport.  If you are withdrawing, please send us a picture of your voided Residence Card front and back, as well as a picture of your Departure Stamp so that we can complete your Withdrawal.   Failure to void your Residence Card\/ Student Visa may result in issues or additional procedures if you wish to return to Japan in the future. Posts navigation. &larr; 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 Extension of Online Classes for April 2020 Term （日本語訳準備中） &rarr; 最近のニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ Online Lessons Will Continue Extension of Online Classes for April 2020 Term （日本語訳準備中） Students on a Student Visa Leaving Japan （日本語訳準備中） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 オンラインレッスンの延長について 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. SHIBUYA GAIGO GAKUIN. NEWS. Orientation for New Students! July 2, 2019. Facebook. Twitter. Today June 2nd (Tuesday) we will have Orientation for incoming new students both Short-Term and Long-Term.  It will be held 5th floor from 14:00 and will be about an hour to an hour and a half.  See you soon! . Posts navigation. &larr; Spring Term Break Schedule! Summer Course 2019 Application Deadline Extended! &rarr; Recent News. Change of address and contact information Immigration Updates and COVID-19 VACCINE State of Emergency April 2021 Term Online School Visits Require Advance Appointment! Tokyo State of Emergency Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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{"id":"3OyOGu9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/ja\/april-2020-term-information-covid-19\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":62,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","april-2020-term-information-covid-19"],"content_type":["text\/html"],"crawldepth_i":3,"collection_sxt":["user"],"title":["2020年4月学期に関する情報– 新型コロナウイルス関連（COVID-19） &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:53:05 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Fri Apr 10 00:00:00 UTC 2020"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":8042466987361283414,"exact_signature_unique_b":true,"fuzzy_signature_l":-7047822335898090145,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["2020年4月学期に関する情報– 新型コロナウイルス関連（COVID-19）"],"h2_txt":["渋谷外語学院 ニュース","Posts navigation"],"h4_txt":["最近のニュース","学生の声","日本語コース","お問い合わせ"],"h5_txt":["2020-04-10","仮校舎への移転に伴う住所及び電話番号変更のお知らせ","Online Lessons Will Continue","Extension of Online Classes for April 2020 Term （日本語訳準備中）","Students on a Student Visa Leaving Japan （日本語訳準備中）","新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言","オンラインレッスンの延長について"],"bold_txt":["最近のニュース","Online Classes","Orientation","Travel Restrictions","オンラインレッスン","オリエンテーション","渡航制限","日本語 日本語"],"imagescount_i":1,"images_protocol_sxt":["000-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/07\/Shibuya-Gaigo-Gakuin-Logo-black.svg"],"images_alt_sxt":["Japanese Language School"],"images_height_val":[2000],"images_width_val":[2000],"images_text_t":"shibuyajp net wp content uploads 2018 07 Shibuya Gaigo Gakuin Logo black svg Japanese Language School","responsetime_i":71,"text_t":"2020年4月学期に関する情報– 新型コロナウイルス関連（COVID-19） &ndash; Shibuya Gaigo Gakuin. Skip to content 日本語. 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. 2020年4月学期に関する情報– 新型コロナウイルス関連（COVID-19）. 2020-04-10. Facebook. Twitter. 在籍者及び新入生に、これまでのアウトブレイクに関する情報が届きました。以下は、情報の概略です。. 渡航制限. 日本到着日の14日前に次のリンクに記載されているいずれかのエリアに行った場合、出入国在留管理庁は入国を拒否する可能性があります。. http:\/\/www.moj.go.jp\/nyuukokukanri\/kouhou\/nyuukokukanri01_00136.html 詳細については、次のリンクでも確認できます。. https:\/\/www.mhlw.go.jp\/index.html オリエンテーション. なお、新入生オリエンテーションは、4月1日（水）14:00からです。これは、この期間にオンラインレッスンがどのように行われるかを新入生に説明する機会です。当校では、Google ClassroomとZOOMを活用します。コンピューター、タブレット、またはスマートフォンを使用してアカウントを設定できます。入学に必要な情報や資料の共有、基本的な手続きを行います。. オンラインレッスン. 新型コロナウイルス感染拡大対策として、最初の7回のレッスンを4月2日から4月10日までオンラインで行う予定です。以前のメールで説明されているように、Google Classroom \/ ZOOMからログインしてください。注：出入国在留管理庁からの指示に応じて、この日程を延長する必要がある可能性があります。. 繰り返しになりますが、感染症対策への皆様のご理解・ご協力に感謝申し上げます。. ーーーーーーーー. To all current and new arriving students, by now you have received an influx of emails regarding the outbreak updates.  Below is a summary of the information as we prepare for the new term this week:. Travel Restrictions. If you have been to any of the areas listed in the following link within 14 days of your arrival to Japan, Japanese Immigration may refuse your entry into the country:   . http:\/\/www.moj.go.jp\/EN\/nyuukokukanri\/kouhou\/m_nyuukokukanri01_00003.html You can find more information on the following link as well in English: . https:\/\/www.mhlw.go.jp\/english\/ Orientation. As a reminder, Orientation for new students is this Wednesday April 1. st. (Wednesday) at 14:00.  This will be an opportunity for new students to learn how online lessons are held during this time.  Our school will be utilizing Google Classroom and ZOOM.  You can use a computer, tablet or your smart phone to set up an account.  We will be going through important information and materials as well as basic procedures for your enrollment. Online Classes. As a measure of social distancing, our plans are to hold the first 7 lessons online, April 2. nd. – April 10. th. .   Please log in through Google Classroom \/ ZOOM as explained in the previous emails.  Note:  There is the possibility we may need to extend this date in response to the spread and instructions\/ updates from Immigration. Again, we thank everyone for their patience and understanding as we manage under this global pandemic. Posts navigation. &larr; JR渋谷駅 西口からのアクセス方法　詳しい道順を写真付きでご紹介 オンラインレッスンの延長について &rarr; 最近のニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ Online Lessons Will Continue Extension of Online Classes for April 2020 Term （日本語訳準備中） Students on a Student Visa Leaving Japan （日本語訳準備中） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 オンラインレッスンの延長について 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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{"id":"GkQobs9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/extension-of-online-classes-for-april-2020-term\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":70,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["extension-of-online-classes-for-april-2020-term"],"content_type":["text\/html"],"crawldepth_i":3,"collection_sxt":["user"],"title":["Extension of Online Classes for April 2020 Term &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:53:28 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Mon Apr 27 00:00:00 UTC 2020"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-1560895786088899116,"exact_signature_unique_b":true,"fuzzy_signature_l":9040048375542297853,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["Extension of Online Classes for April 2020 Term"],"h2_txt":["SHIBUYA GAIGO GAKUIN NEWS","Posts navigation"],"h4_txt":["Recent News","Student Voice","Courses","Contact us"],"h5_txt":["April 27, 2020","Change of address and contact information","Immigration Updates and COVID-19 VACCINE","State of Emergency","April 2021 Term Online","School Visits Require Advance Appointment!","Tokyo State of Emergency"],"bold_txt":["English English"],"imagescount_i":2,"images_protocol_sxt":["000-https","001-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/07\/Shibuya-Gaigo-Gakuin-Logo-black.svg","shibuyajp.net\/wp-content\/uploads\/2020\/04\/19-600x200.png"],"images_alt_sxt":["Japanese Language School",""],"images_height_val":[2000,200],"images_width_val":[2000,600],"images_text_t":"shibuyajp net wp content uploads 2018 07 Shibuya Gaigo Gakuin Logo black svg Japanese Language School 2020 04 19 600x200 png 600x 200","responsetime_i":133,"text_t":"Extension of Online Classes for April 2020 Term &ndash; Shibuya Gaigo Gakuin. 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{"id":"FnDocs9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/we-will-be-moving-in-early-november-2018\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":63,"url_protocol_s":"https","url_paths_count_i":1,"url_paths_sxt":["we-will-be-moving-in-early-november-2018"],"content_type":["text\/html"],"crawldepth_i":3,"collection_sxt":["user"],"title":["We will be moving 校舎移転のお知らせ &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:53:28 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Thu Oct 25 00:00:00 UTC 2018"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-2819225762102599021,"exact_signature_unique_b":true,"fuzzy_signature_l":-6465322221431520392,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["We will be moving 校舎移転のお知らせ"],"h2_txt":["SHIBUYA GAIGO GAKUIN NEWS","Posts navigation"],"h4_txt":["Recent News","Student Voice","Courses","Contact us"],"h5_txt":["October 25, 2018","Change of address and contact information","Immigration Updates and COVID-19 VACCINE","State of Emergency","April 2021 Term Online","School Visits Require Advance Appointment!","Tokyo State of Emergency"],"bold_txt":["新住所地：渋谷区桜丘町15-17 NKG東京ビル( Googleマップ ) JR渋谷駅 南改札西口より徒歩3分","=>2018年11月1日より新校舎での授業が始まります。詳しくは こちら","3 minutes walk away from JR Shibuya station west exit.","NEW ADDRESS：NKG Tokyo building 15-17 Sakuragaokachō, Shibuya-ku, Tokyo( Google Maps )","→　We will start classes at our new location on Wednesday November 1st, 2018. 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SHIBUYA GAIGO GAKUIN. NEWS. We will be moving 校舎移転のお知らせ. October 25, 2018. Facebook. Twitter. Japanese follows the English \/ 日本語のご案内は英文の後に続きます. Due to the redevelopment of the area around Shibuya station, we will be moving from our current location to the address below. Based on the results of the student questionnaire, we are establishing a better learning environment, and there will be no closure due to relocation. We will be informing everyone of the exact date of the transition as soon as the decision has been made. →　We will start classes at our new location on Wednesday November 1st, 2018. View Detail NEW ADDRESS：NKG Tokyo building 15-17 Sakuragaokachō, Shibuya-ku, Tokyo(. Google Maps ). 3 minutes walk away from JR Shibuya station west exit. The new building is between Café Renoir and the Granbell Hotel. We are taking this unique opportunity to start a new business. We will offer a music hall and music studio equipped with grand pianos(. SHIBUYA HALL ), and a kitchen studio which will offer cooking classes(. SHIBUYA KITCHEN STUDIO ). In addition, other events will be added to our service line-up in the near future. It will be our great pleasure, to offer the space not only for professionals but also for hobbyists as well. 渋谷駅周辺地区再開発に伴い、本校は現在の建物からの下記の住所へ移転をすることとなりました。. 皆さまからのアンケート等に基づき、より良い学習環境の準備を進めているところです。. 移転に伴う休講は予定しておりませんが、具体的な移転日については決定次第ご案内いたします。. =2018年11月1日より新校舎での授業が始まります。詳しくは. こちら 新住所地：渋谷区桜丘町15-17 NKG東京ビル(. Googleマップ ). JR渋谷駅 南改札西口より徒歩3分. 喫茶室ルノアールのビルと渋谷グランベルホテルの間の少し奥まった建物が新校舎です。. 移転を機に、新規事業もスタートします。グランドピアノが備わった音楽ホール・音楽スタジオ(. 渋谷ホール )、また料理講習なども行えるキッチンスタジオ(. 渋谷キッチンスタジオ )が、新サービスとして加わります。プロの方から趣味の方まで幅広く使っていただける空間を提供できればと願っています。. 株式会社 渋谷外語学院 日伊学院 欧日協会ドイツ語ゼミナール　. (11月より 株式会社シブガイ に社名変更予定). Posts navigation. &larr; Website Renewal We will start classes at our new location on Wednesday November 1st, 2018. &rarr; Recent News. 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{"id":"SnFFSu9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/ja\/covid-19-update-japan-state-of-emergency\/","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":66,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["ja","covid-19-update-japan-state-of-emergency"],"content_type":["text\/html"],"crawldepth_i":3,"collection_sxt":["user"],"title":["新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 &ndash; Shibuya Gaigo Gakuin"],"last_modified":"Wed Jun 08 16:53:23 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Fri Apr 10 00:00:00 UTC 2020"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-8397041870087236829,"exact_signature_unique_b":true,"fuzzy_signature_l":2631239197677081046,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https","003-https"],"icons_urlstub_sxt":["shibuyajp.net\/favicon-16x16.png","shibuyajp.net\/apple-touch-icon.png","shibuyajp.net\/favicon-32x32.png","shibuyajp.net\/safari-pinned-tab.svg"],"icons_rel_sxt":["icon","apple-touch-icon","icon","mask-icon"],"icons_sizes_sxt":["16x16","180x180","32x32",""],"h1_txt":["新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言"],"h2_txt":["渋谷外語学院 ニュース","Posts navigation"],"h4_txt":["最近のニュース","学生の声","日本語コース","お問い合わせ"],"h5_txt":["2020-04-10","仮校舎への移転に伴う住所及び電話番号変更のお知らせ","Online Lessons Will Continue","Extension of Online Classes for April 2020 Term （日本語訳準備中）","Students on a Student Visa Leaving Japan （日本語訳準備中）","新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言","オンラインレッスンの延長について"],"bold_txt":["最近のニュース","日本語 日本語"],"imagescount_i":3,"images_protocol_sxt":["000-https","001-https","002-https"],"images_urlstub_sxt":["shibuyajp.net\/wp-content\/uploads\/2018\/07\/Shibuya-Gaigo-Gakuin-Logo-black.svg","shibuyajp.net\/wp-content\/uploads\/2020\/04\/000059528-01-425x600.jpg","shibuyajp.net\/wp-content\/uploads\/2020\/04\/Covid-1-424x600.jpg"],"images_alt_sxt":["Japanese Language School","",""],"images_height_val":[2000,600,600],"images_width_val":[2000,425,424],"images_text_t":"shibuyajp net wp content uploads 2018 07 Shibuya Gaigo Gakuin Logo black svg Japanese Language School 2020 04 000059528 01 425x600 jpg 425x 600 Covid 1 424x600 424x","responsetime_i":101,"text_t":"新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 &ndash; Shibuya Gaigo Gakuin. Skip to content 日本語. 日本語 ホーム. ニュース. 当校について. 日本語コース. スケジュール. 日本での住居. お申し込み. お問い合わせ. お申し込みはこちら. 渋谷外語学院. ニュース. 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言. 2020-04-10. Facebook. Twitter. ご存知のように、4月7日(火)、日本政府は緊急事態宣言を発出しました。. 詳しくは、東京都防災ホームページをご覧ください。. https:\/\/www.bousai.metro.tokyo.lg.jp\/1007617\/index.html 緊急事態宣言の期間中、できるだけ外出しないことが重要です。登校のレッスンは、来月のゴールデンウィーク明けまでオンラインでの実施を継続します。. 国内外問わず旅行はしないことを強くお勧めします。やむを得ず、ゴールデンウィークの休暇中に旅行（移動）する場合は、移動の制限に関する最新情報にご注意ください。. 新型コロナウイルス感染症に関する情報:. （英語）. http:\/\/www.moj.go.jp\/EN\/nyuukokukanri\/kouhou\/m_nyuukokukanri01_00003.html （日本語）. http:\/\/www.moj.go.jp\/hisho\/kouhou\/20200131comment.html COVID-19に関する一般的なニュースと更新については、予防策と症状\/原因を含め、WHOのWebサイトを参照してください. （英語）：. https:\/\/www.who.int\/emergencies\/diseases\/novel-coronavirus-2019 ーーーーーーーー. As most of you already know, this past Tuesday April 7th, Japan’s Prime Minister had officially declared a State of Emergency. You can find more details here from the Tokyo Disaster Prevention Agency:. https:\/\/translation2.j-server.com\/LUCTBOUSAI\/ns\/tl.cgi\/https:\/\/www.bousai.metro.tokyo.lg.jp\/1007617\/index.html?SLANG=ja&TLANG=en&XMODE=0&XCHARSET=utf-8&XJSID=0 It is important to continue to remain at home as much as possible during this time.  Our lessons will continue to be online until the end of Golden Week next month. Although not restricted, traveling is strongly advised against.  If you happen to travel during the Golden Week break, please kindly refer to the updated information regarding the travel restrictions:. (English)  . http:\/\/www.moj.go.jp\/EN\/nyuukokukanri\/kouhou\/m_nyuukokukanri01_00003.html (Japanese) . http:\/\/www.moj.go.jp\/hisho\/kouhou\/20200131comment.html For general news and updates about COVID-19, including preventative measures and symptoms\/ causes, please refer to the WHO website:  . https:\/\/www.who.int\/emergencies\/diseases\/novel-coronavirus-2019 Posts navigation. &larr; オンラインレッスンの延長について Students on a Student Visa Leaving Japan （日本語訳準備中） &rarr; 最近のニュース. 仮校舎への移転に伴う住所及び電話番号変更のお知らせ Online Lessons Will Continue Extension of Online Classes for April 2020 Term （日本語訳準備中） Students on a Student Visa Leaving Japan （日本語訳準備中） 新型コロナウイルス（COVID-19）関連の最新情報：日本の緊急事態宣言 オンラインレッスンの延長について 学生の声 日本語コース お申し込みは. レベルチェック. お問い合わせ © 2018 Shibuya Gaigo Gakuin. 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蔭山陽太さんは、京都市立の劇場「ロームシアター京都」で支配人をしながら地元の消防団にも所属し、大文字の送り火などに出動していた=蔭山さん提供 日本政府の核禁条約参加を訴える広島の被爆者団体代表ら=3月22日、広島市中区、岡田将平撮影 骨太の方針を決めた経済財政諮問会議・新しい資本主義実現会議合同会議で、発言する岸田文雄首相（左から2人目）=2022年6月7日午後5時41分、首相官邸、上田幸一撮影 ronza 論座 準決勝で天野倉優臣さんに勝った川島滉生さん=6月5日午前11時32分、東京都港区、村瀬信也撮影 11年ぶりに発売されたシャチホコ型のシュークリーム「シャチボン」=2022年6月8日、名古屋市中村区、高橋俊成撮影 ギネスの認定証を持つ安達健太さん=2022年5月26日午前10時29分、山口県周南市、垣花昌弘撮影 東北新幹線、懐かし200系カラー再現　開業40周年でE2系が変身 体育のリレー中に12人熱中症か、1人重症　神戸の小学校 and andW png W &amp;w andM M &amp;M andTravel Travel &amp;TRAVL andTime Time &amp;TIME andHuman Human &amp;Human asm アエラスタイルマガジン AgingGracefully Aging Gracefully Aging Gracefully globePlus globe Plus GLOBE＋ 名古屋市役所 愛知県警は「CAN（キャン）インベーダー」に使用したとみられる装置などを押収した=2022年6月8日午後7時15分、愛知県刈谷市、藤牧幸一撮影 警視庁本部 地層処分での考慮事項について議論する原子力規制委員会の委員ら=2022年6月8日午前10時49分、東京都港区、佐々木凌撮影 仮想空間に立ち上がった「しまね縁結び商店街」のPRイベント。組み立て式の日本庭園なども出品された=2022年6月8日午後3時2分、福岡市のJR博多駅前、藤原慎一撮影 北方領土周辺海域で安全操業でロシア側の臨検を受け、漁獲したタコなどを調べられる日本漁船の乗員ら=2019年12月、サハリン・インフォ提供 日本銀行本店=東京都中央区 ジョーシン日本橋店のエアコン売り場には、入荷時期が未定の商品もあった=大阪市浪速区 昨年11月、第1回全世代型社会保障構築会議・第1回公的価格評価検討委員会の合同会議に臨む岸田文雄首相（左から2人目）=首相官邸、上田幸一撮影 すかいらーくホールディングスに切り捨てられた賃金の支払いを求めてきたアルバイトの須賀大文さん（左）ら=2022年6月8日、東京都千代田区、橋本拓樹撮影 細田博之衆院議長に対する不信任決議案を岡田憲治衆院事務総長（右）に提出する立憲民主党の馬淵澄夫国対委員長（中央）=2022年6月8日午後5時12分、国会内、上田幸一撮影 日本維新の会の藤田文武幹事長 「平和でも、くらしでも、希望がもてる日本に」とのキャッチフレーズを冠した参院選の公約を発表する共産党の志位和夫委員長=2022年6月8日午後2時1分、国会 参院改革協議会座長の世耕弘成・参院自民幹事長から報告書を受け取る山東昭子参院議長=2022年6月8日午後、国会内、楢崎貴司撮影 神奈川県立瀬谷西高校=2022年6月8日午後5時30分、横浜市瀬谷区、阿部育子撮影 ウクライナ東部ドネツク州で2022年6月7日、医療支援地点に担架で運ばれる負傷した男性=AP トルコの首都アンカラで8日、会談後に記者会見に臨むチャブシュオール外相（右）とロシアのラブロフ外相=ロイター ヤンゴンで2021年2月4日、クーデターに対する抗議デモに備える警察官=ロイター 香港で発行を続けていた頃の「リンゴ日報」本社=2021年6月、香港、奥寺淳撮影 ベルリン中心部で8日、店舗に突っ込んだ乗用車のそばで通話をする警察官。少なくとも1人が死亡した=ロイター フリーアナウンサーの竹内由恵さん=西岡臣撮影 昨年の日本選手権男子5000メートルで優勝した遠藤日向 1回戦に勝ち、校歌を歌う弘前学院聖愛の選手たち=福島県営あづま 今夏の世界選手権（米・オレゴン州）代表内定をめざし、練習する遠藤日向=2022年6月、長野県東御市、加藤秀彬撮影 はやぶさ２　生命起源に迫る 京都府立医科大学 半田病院へのサイバー攻撃と、判明した問題点 アップルのiPhone向け新OS「iOS　16」の画面。時間表示のフォントや色などが好みのものに変えられる 中島さち子さん=steAm提供 亀裂が広がった状態（左側面）=文化庁提供 小島よしおさん=ユーチューブから 岡野弘彦さん=門間新弥撮影 記者会見をする大樹孝啓・第258世天台座主=大津市坂本4丁目の天台宗務庁 金賞に輝いた「栗黒」=2022年5月30日午後4時26分、大畠正吾撮影 採卵に使用する孵化から16年が経過した巨大なイトウ=2022年5月11日、青森県鰺ケ沢町一ツ森町、安田琢典撮影 宿泊施設が立ち並ぶ東京・山谷の街角に立つ末並俊司さん。「マイホーム山谷」で小学館ノンフィクション大賞を受賞した=5月、東京都台東区 「僕は中途半端でええ。どっちかに偏らんかったら、道の真ん中歩いていける。それが僕には生きやすいねん。そうやって、長い長い人生の宿題を果たして死ぬねん」=2022年5月26日、大阪市西成区、槌谷綾二撮影 道教育委員会が入る道庁別館の建物=2021年5月、札幌市中央区 大阪観光大の理事長に就いた山本健慈さん=大阪府熊取町 大学入学共通テストを受ける受験生たち=1月15日、東京都文京区 「いつも想っているよ」と刻まれたガラス細工のオブジェと本郷優希さんの写真 ベネッセ海外留学センターの責任者、赤木麻美さん=同社提供 米マサチューセッツ州のモデルナ本社=AP 厚生労働省に新型コロナ対策を助言する専門家組織の会合=2022年6月8日午後4時58分、東京・霞が関の厚生労働省、市野塊撮影 新型コロナウイルスのオミクロン株の電子顕微鏡写真=国立感染症研究所提供 東京都の新規感染者 第33期女流王位戦第4局で勝ち、防衛を決めた里見香奈女流王位（左）と、敗れた挑戦者の西山朋佳女流二冠=2022年6月7日午後、徳島市、日本将棋連盟提供 第48期棋王戦の予選・決勝で古森悠太五段（手前右）に勝ち、女性初の8タイトル戦の本戦入りを決めた里見香奈女流四冠=2022年5月27日午後、大阪市福島区の関西将棋会館、佐藤圭司撮影 最年少で公式戦100勝を達成して笑顔を見せる仲邑菫二段=6日、東京・市ケ谷の日本棋院 引退した小林宏七段。「棋士になっていなければ？　人生は分からないが、山の登り方は変わっていた」=東京都渋谷区、北野新太撮影 ソニーのペットロボ「アイボ」 元ソニーCEO（最高経営責任者）の出井伸之氏=2019年1月21日、東京・赤坂、迫和義撮影 石井一さん 特別鑑賞会に駆け付けた黒柳徹子さん（右）と田沼武能さん=2004年 コリア国際学園中等部・高等部=2022年6月8日午後、大阪府茨木市豊川2丁目、甲斐江里子撮影 国際・外交 タイムライン 【速報中】米大統領、G7サミットで訪独へ　ウクライナ支援など協議 【詳報】ウクライナ、イスラエルを批判「最低限の支援すらせず」 【詳報】ウクライナ侵攻22、5月24～30日の動き いま読む「音楽展望」　吉田秀和 没後10年 吉田秀和、97歳の「楽観」を打ち砕いた東日本大震災 吉田秀和「音楽展望」　中原中也との日々　音楽と言葉の柔らかな融合 吉田秀和、最初で最後の「相撲展望」　生身の肉体を文章修行の礎に 2.24後の世界で 戦争映像に「感動」する私たち　単純な善悪、止まる思考　大塚英志氏 記憶の動員目指すロシアとウクライナ　民衆が受け継ぐ独ソ戦の記憶 ロシアとウクライナの「記憶」をめぐる戦争　対立の根元にあるもの 中小企業の経営に役立つ情報を優先配信！無料会員募集中 朝デジ10周年記念　サッカー日本代表ユニフォームを贈呈 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IT・科学 はやぶさ2の砂に生命の源　「科学史に残る大発見」から何がわかるか. はやぶさ2の砂に生命の源　「科学史に残る大発見」から何がわかるか 2022年6月8日 19時00分. 週末の夜更かし、何時間まで大丈夫？　社会的時差ぼけに着目して解析. 週末の夜更かし、何時間まで大丈夫？　社会的時差ぼけに着目して解析 2022年6月8日 15時40分. 「閉じられたネット」病院を取り巻く安全神話　身代金ウイルス被害. 「閉じられたネット」病院を取り巻く安全神話　身代金ウイルス被害 2022年6月8日 8時30分. アップル、iPhoneの新OS発表　新型MacBook Airも. アップル、iPhoneの新OS発表　新型MacBook Airも 2022年6月7日 18時00分. ピアニストで数学研究者の中島さち子さん　「文系・理系の枠超えて」. ピアニストで数学研究者の中島さち子さん　「文系・理系の枠超えて」 2022年6月7日 15時00分. 文化・芸能 藤ノ木古墳の国宝出土品を破損 修理前の調査中. 藤ノ木古墳の国宝出土品を破損 修理前の調査中 2022年6月8日 19時00分. 消防団にも入った劇場支配人が語る「公金は時に怖いもの」. 消防団にも入った劇場支配人が語る「公金は時に怖いもの」 2022年6月8日 17時00分. 「関係ねぇって言わないで」 小島よしおさんが小学生に田植えを説く. 「関係ねぇって言わないで」 小島よしおさんが小学生に田植えを説く 2022年6月8日 13時01分. いい歌はふうっと心に入ってくる　97歳の歌人岡野弘彦さんの言葉. いい歌はふうっと心に入ってくる　97歳の歌人岡野弘彦さんの言葉 2022年6月8日 13時00分. 「説法で命果てても、心残りはない」 97歳の天台座主が初めて会見. 「説法で命果てても、心残りはない」 97歳の天台座主が初めて会見 2022年6月8日 12時41分. ライフ レシピ残ってなく再現にひと苦労 「シャチボン」11年ぶりに復活. レシピ残ってなく再現にひと苦労 「シャチボン」11年ぶりに復活 2022年6月8日 19時30分. 世界2大コンテスト制したクラフトビール　その意外な副原料とは. 世界2大コンテスト制したクラフトビール　その意外な副原料とは 2022年6月8日 16時00分. 「幻の魚」イトウの養殖、レア感が売り　味わうには「来るしかない」. 「幻の魚」イトウの養殖、レア感が売り　味わうには「来るしかない」 2022年6月8日 11時17分. 挫折した山谷の「ホスピス」創設者と失踪した妻…取材続け見えたこと. 挫折した山谷の「ホスピス」創設者と失踪した妻…取材続け見えたこと 2022年6月8日 11時00分. 私生児、同性愛…93歳の新天地は大阪・西成 「矛盾は値打ちや」. 私生児、同性愛…93歳の新天地は大阪・西成 「矛盾は値打ちや」 2022年6月8日 8時00分. 教育・子育て いじめ被害の子ども、道教委がネットで相談. いじめ被害の子ども、道教委がネットで相談 2022年6月8日 21時00分. 21億円着服事件で揺れた経営難の私大、新理事長に「物言う学長」. 21億円着服事件で揺れた経営難の私大、新理事長に「物言う学長」 2022年6月8日 19時00分. 相次ぐ大学入試不正　共通テストはイヤホン禁止に　文科省も対策通知. 相次ぐ大学入試不正　共通テストはイヤホン禁止に　文科省も対策通知 2022年6月8日 18時32分. 「いつも想っているよ」　付属池田小事件21年、父は娘に語りかける. 「いつも想っているよ」　付属池田小事件21年、父は娘に語りかける 2022年6月8日 16時00分. 海外大への進学準備、高校からでは遅い？　コロナ3年目の留学事情. 海外大への進学準備、高校からでは遅い？　コロナ3年目の留学事情 2022年6月8日 15時30分. 医療・健康 モデルナがオミクロン株特化のワクチン開発中「秋のブースター候補」. モデルナがオミクロン株特化のワクチン開発中「秋のブースター候補」 2022年6月9日 0時00分. 「頻繁な消毒は不要」　専門家組織、医療介護現場での負担軽減を提言. 「頻繁な消毒は不要」　専門家組織、医療介護現場での負担軽減を提言 2022年6月8日 21時00分. 全国で新たに1万8416人が感染　死者は21人 新型コロナ. 全国で新たに1万8416人が感染　死者は21人 新型コロナ 2022年6月8日 20時36分. 賃上げの「呼び水」のはずが…やる気を阻害　対象外の看護師の思いは. 賃上げの「呼び水」のはずが…やる気を阻害　対象外の看護師の思いは 2022年6月8日 19時00分. 東京都内はコロナ感染1935人、前週の同じ曜日から大幅減少. 東京都内はコロナ感染1935人、前週の同じ曜日から大幅減少 2022年6月8日 18時44分. 将棋・囲碁. 「藤井聡太世代」の川島さん　「絶対王者」を破って将棋・学生名人に. 「藤井聡太世代」の川島さん　「絶対王者」を破って将棋・学生名人に 2022年6月8日 20時00分. 里見女流王位が防衛、4連覇　棋士編入試験「ギリギリまで考えたい」. 里見女流王位が防衛、4連覇　棋士編入試験「ギリギリまで考えたい」 2022年6月7日 20時30分. 藤井聡太竜王の師匠が感じた　弟子たちの成長. 藤井聡太竜王の師匠が感じた　弟子たちの成長 2022年6月7日 12時00分. 囲碁の仲邑菫二段、史上最年少で公式戦100勝　50年ぶり記録更新. 囲碁の仲邑菫二段、史上最年少で公式戦100勝　50年ぶり記録更新 2022年6月6日 19時00分. ある棋士の引退　夢を見た山　遠かった頂. ある棋士の引退　夢を見た山　遠かった頂 2022年6月4日 13時00分. おくやみ VAIOもAIBOもなくなった　それでも芽吹いた出井氏がまいた種. VAIOもAIBOもなくなった　それでも芽吹いた出井氏がまいた種 2022年6月7日 21時44分. 14人抜き出井氏、ソニー黄金の10年　見誤った経営、皮肉な退陣劇. 14人抜き出井氏、ソニー黄金の10年　見誤った経営、皮肉な退陣劇 2022年6月7日 16時08分. ソニー元社長の出井伸之さん死去　「VAIO」など発売　84歳. ソニー元社長の出井伸之さん死去　「VAIO」など発売　84歳 2022年6月7日 12時52分. 元自治相の石井一さん死去　ダッカ事件で交渉役　87歳. 元自治相の石井一さん死去　ダッカ事件で交渉役　87歳 2022年6月6日 18時34分. 黒柳徹子さんが追悼　田沼武能さんは「子どもを撮ったら世界一」. 黒柳徹子さんが追悼　田沼武能さんは「子どもを撮ったら世界一」 2022年6月2日 14時34分. 地域 北海道・東北 「細かい情報入らず不安」　日ロ漁業協定停止、地元漁業者らに波紋. 「細かい情報入らず不安」　日ロ漁業協定停止、地元漁業者らに波紋 2022年6月8日 21時30分. 関東 菅前首相の高校講演が中止に　参院選直前、18歳への影響懸念する声. 菅前首相の高校講演が中止に　参院選直前、18歳への影響懸念する声 2022年6月8日 20時18分. 東海・甲信越 名古屋市教育長に坪田氏　河村市長が6月議会に提案方針固める. 名古屋市教育長に坪田氏　河村市長が6月議会に提案方針固める 2022年6月8日 22時00分. 近畿・北陸 コリア国際学園に火付けた疑いで男を逮捕　辻元氏事務所侵入でも起訴. コリア国際学園に火付けた疑いで男を逮捕　辻元氏事務所侵入でも起訴 2022年6月8日 19時29分. 中国・四国 博多駅前に島根の「見えない商店街」出現　メタバースでローン相談も. 博多駅前に島根の「見えない商店街」出現　メタバースでローン相談も 2022年6月8日 21時00分. 九州・沖縄 博多駅前に島根の「見えない商店街」出現　メタバースでローン相談も. 博多駅前に島根の「見えない商店街」出現　メタバースでローン相談も 2022年6月8日 21時00分. 連載 【速報中】米大統領、G7サミットで訪独へ　ウクライナ支援など協議. 【速報中】米大統領、G7サミットで訪独へ　ウクライナ支援など協議 2022年6月9日 0時16分. 【詳報】ウクライナ、イスラエルを批判「最低限の支援すらせず」. 【詳報】ウクライナ、イスラエルを批判「最低限の支援すらせず」 2022年6月8日 5時38分. 【詳報】ウクライナ侵攻22、5月24～30日の動き. 【詳報】ウクライナ侵攻22、5月24～30日の動き 2022年5月30日 23時29分. 吉田秀和、97歳の「楽観」を打ち砕いた東日本大震災. 吉田秀和、97歳の「楽観」を打ち砕いた東日本大震災 2022年6月8日 21時43分. 吉田秀和「音楽展望」　中原中也との日々　音楽と言葉の柔らかな融合. 吉田秀和「音楽展望」　中原中也との日々　音楽と言葉の柔らかな融合 2022年5月30日 14時00分. 吉田秀和、最初で最後の「相撲展望」　生身の肉体を文章修行の礎に. 吉田秀和、最初で最後の「相撲展望」　生身の肉体を文章修行の礎に 2022年5月27日 11時00分. 戦争映像に「感動」する私たち　単純な善悪、止まる思考　大塚英志氏. 戦争映像に「感動」する私たち　単純な善悪、止まる思考　大塚英志氏 2022年6月8日 17時30分. 記憶の動員目指すロシアとウクライナ　民衆が受け継ぐ独ソ戦の記憶. 記憶の動員目指すロシアとウクライナ　民衆が受け継ぐ独ソ戦の記憶 2022年6月7日 11時00分. ロシアとウクライナの「記憶」をめぐる戦争　対立の根元にあるもの. ロシアとウクライナの「記憶」をめぐる戦争　対立の根元にあるもの 2022年6月7日 11時00分. 最低限知っておきたい「相続のキホン」とは？. すべての人の「学び」を応援！【寺子屋朝日】. 中小企業の経営に役立つ情報を優先配信！無料会員募集中. ツギノジダイ 中小企業の経営に役立つ情報を優先配信！無料会員募集中 朝デジ10周年記念　サッカー日本代表ユニフォームを贈呈. 朝日IDのサービス 朝デジ10周年記念　サッカー日本代表ユニフォームを贈呈 あなたの時事力を測ってみませんか. ニュース検定 あなたの時事力を測ってみませんか 朝日新聞社の関連サイト. 論座 &M &w &Travel EduA Moovoo NewsVideo withnews CNET Japan CNN.co.jp Huffpost AERA dot. bouncy telling, sippo GLOBE＋ 好書好日 なかまぁる ４years. かがみよかがみ 相続会議 ツギノジダイ みらのび 各国語サイト. 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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT LIFE. 2022.06.08. ＆w. 東京の台所２ 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 2022.06.06. ＆w. 久保純子 LIFE in N.Y. Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 2022.06.02. ＆w. 花のない花屋 ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを 2022.06.01. ＆w. リノベーション・スタイル 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” 2022.05.26. ＆w. 花のない花屋 スペインから父へ。直接その手を握る代わりに 2022.05.25. ＆w. 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 2022.05.19. ＆w. 花のない花屋 今も私を励ますのは、先生がくれた「大丈夫」の言葉 2022.05.18. ＆w. リノベーション・スタイル 〈243〉住む人に合った動線で、暮らしも、家事も、片づけもスムーズに 2022.05.12. ＆w. 花のない花屋 母となべちゃん、最期まで続いた友情に。私からもありがとう FOR YOU. あなたにおすすめの記事. 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祇園祭 西日本豪雨 Ｇ２０大阪サミット 宝塚歌劇団 まだまだ勝手に関西遺産 2019統一地方選挙 ＳＴＵ瀬戸リスト 沖縄1935 よみがえる戦前 記者襲撃、あの夜から 阪神支局襲撃30年を越えて 「酷道」をいく 戦火の残響 三浦雄一郎の挑戦 全日本小中学生ダンスコンクール オウムを暴走させた３つの転機 100回つなぐ始球式リレー 環境破壊、報いの現場を歩く 平成とは がんとともに 再生！太陽の塔 日本代表レジェンドカード 「13」最高のスタジアムを巡る旅 2018サッカーワールドカップ 孤高の星 羽生結弦-SPIN THE DREAM SPIN THE DREAM 羽生結弦 LEGEND 葛西紀明 平昌オリンピック 平昌パラリンピック 金正恩とトランプ 「トランプ王国」熱狂のあと 災害大国 被害に学ぶ 河田防災塾 あのときの、ラジオ 国際女性デー特集「Dear Girls」 小さないのち「児相の現場から」 SDGs フィギュアスケート特集 ＨＡＫＵＴＯの挑戦 築地市場の「音」 平成とは いじめと君 女子組 朝日・東大共同調査 ドローンズアイ Ｔｈｅ　Ｒｏａｄ 日本初「監獄ホテル」誕生へ 景気１００社アンケート 衆院の区割り変更 オバマ米大統領、広島へ 南極観測６０年 トワイライトエクスプレス瑞風 政治資金特集 タイムライン 奄美群島国立公園 復興支援音楽祭 いきもの目線 全日本合唱コンクール ノーベル賞 もっと！ロケット HUG STA（ハグスタ） 子ども食堂 大学入試センター試験 築地－時代の台所 真田太平記 皇室とっておき （各駅停話）東北新幹線特別編 待機児童問題 どうぶつ新聞 家庭遺産 朝デジスペシャル. 1.17 再現／阪神・淡路大震災 120日後のフェス　アラバキ 2.24後の世界で　ウクライナ侵攻から考える 4 years from ICHIRO｜イチローが伝えたかったこと GHOSN（ゴーン）日本最大の司法取引 GRIM-成田緑夢（ぐりむ） HAKUTO-Rの挑戦　月への旅路 HAYABUSA2 地球帰還 LEGEND 葛西紀明 MINAMATA　ユージン・スミスの伝言 SDGs｜国谷裕子さんと考える SDGｓ｜下川町編 SPIN THE DREAM　夢をつむぐ羽生結弦 The Days ウクライナ侵攻 VR｜築地市場の360度動画 「13」最高のスタジアムを巡る旅 「オウム」を暴走させた3つの転機 「会いたい、会わせたい　東日本大震災10年」 「祈りの旅」 あなたは誰ですか　ルポ孤独死 あのときの、ラジオ ３．１１震災特集 ありがとう普門館 いきもの目線 いまの政治と選挙は　朝日新聞社世論調査 いま憲法をどう考える　朝日新聞社世論調査 そのとき官邸は　新たな調書 ふたつのアメリカ　分極社会 みえない交差点　全国68万件の交通事故マップ イチローMLB3000安打特集 イラン抵抗の三日月　謎の武装組織を追う オバマ大統領、広島へ カルロス・ゴーン　もたらした光と影 コロナ後の世界を語る　現代の知性たちの視線 コロナ第２波　東京100days チャレンジド wonder athletes デレク＆マイケル データで見る「就活」の歴史 ドーピングの深い闇 ナガサキノート あの日、人々の足取り ノモンハン　大戦の起点と終止符 マー君の410球を徹底分析 ヤンキース田中打者の目で解析 ラストダンス ワールドアスリートボイス ワールドカップの地球 三浦雄一郎 南米最高峰に挑む 主役なきゴーン法廷 五輪TVゲーム年代記 住み続けたい島へ　SDGs｜島の人たちと考えた 冬季オリンピック 「持続可能性」を巡る旅 動物たちはどこへ　変わりゆく動物園 北陸新幹線　駆ける　つながる 医療危機　コロナ東京100days 吉田調書 名人への道 藤井聡太 地殻変動 自公vs.非自公 声　語り継ぐ戦争 夢を力に2014 大谷翔平 -G.O.A.T- 2021年を象徴する5シーン 失踪村　ベトナム人技能実習生 孤高の星　羽生結弦-SPIN THE DREAM 学校の死角 宇宙へ　苦難と栄光、終わらない旅 安倍×石破　二人がみる日本―自民党総裁選２０１８ 対立の海 平成人語 待機児童問題 憲法をどう考える2022 戦後７０年　ビジュアル年表 手塚治虫文化賞２０周年 拓く　北海道新幹線 政党の現在地　与野党はどう動いた　朝日・東大調査15年 新型コロナ　みんなの考えは？ 新型コロナウイルス感染　世界の１年 新型コロナウイルス感染　日本の１年 日本代表 レジェンドカード 日米開戦80年 朝デジ１０周年特集 未完の最長政権 未来のデザイン　コロナで変わる 李香蘭が語るアジア 東京100days 新型コロナウイルスの記録 枯れる五輪 永世七冠 羽生善治 沖縄1935　写真でよみがえる戦前 沖縄1972　写真でたどる日本復帰50年 沖縄を語る、沖縄から考える 沖縄復帰50年、基地はどこへ 海からみた被災地　東日本大震災10年 琉球から日本へ　よみがえる戦前の沖縄 生きる、未来へ　東日本大震災10年 疑惑の島「パラダイス文書」 石川祐希　世界へ 砂上の国家　満州のスパイ戦 私、がんになるの？　ゲノムが迫る選択 空からみた被災地　11年後の現在 空から見た熱海土石流 空襲１９４５ 築地　時代の台所 米中争覇 裁かれる工藤会　全公判傍聴記 追跡 メイド・イン・ジャパン 和牛編 追跡 メイド・イン・ジャパン 腕時計編 逆さ地図で見る米中最前線の中の沖縄 里海てんぷら　東京の天然を揚げる 金正恩とトランプ 金正男暗殺を追う　2017\/2\/13 飽くなき挑戦 ロータリーエンジンの半世紀 高校野球 夏１００回 高橋源一郎 伯父さんは戦場へ行った １６年リオ五輪　超人１６人に注目 ＥＸＰＯ‘７０　ＯＳＡＫＡ ＮＩＣＫ　私がたどりついた日本 地域一覧. 北海道・東北. 北海道 青森 秋田 岩手 山形 宮城 福島 関東. 群馬 栃木 茨城 埼玉 千葉 東京 神奈川 東海・甲信越. 静岡 岐阜 愛知 三重 新潟 山梨 長野 近畿・北陸. 滋賀 京都 奈良 和歌山 大阪 兵庫 富山 石川 福井 中国・四国. 鳥取 島根 岡山 広島 山口 香川 愛媛 徳島 高知 九州・沖縄. 福岡 大分 宮崎 鹿児島 佐賀 長崎 熊本 沖縄 朝日新聞デジタルの基本機能. ログイン画面 朝刊 夕刊 ｂｅ 紙面ビューアー 地域面紙面 スクラップブック スクラップブック MYキーワード MYキーワード 記者イベント 動画・音声・写真 コメント閲覧 天気 数独 12星座占い 交通情報 アプリ プッシュ通知設定 アクセスランキング ソーシャルランキング トピックス デジタル限定 サポート. 総合ガイド お客さまサポート Q&A（よくある質問） サービス紹介 新聞宅配申し込み デジタル申し込み ウェブマガジン. &w &M &Travel 朝日新聞デジタルの関連サイト. 論座 朝日新聞社のサイト. 朝日新聞社　会社案内 朝日新聞SHOP 注目の連載記事 もっと見る いま読む「音楽展望」　吉田秀和 没後10年 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All rights reserved. No reproduction or republication without written permission.","language_s":"ja","size_i":89339,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":0,"references_external_i":1,"references_exthosts_i":1,"host_extent_i":-1,"_version_":1735087022064795648,"text_t":"モバイル向けサービス一覧：朝日新聞デジタル. メインメニューをとばして、このページの本文エリアへ お使いのブラウザはJavaScriptに対応していないか、または無効になっています。詳しくは サイトポリシーのページ をご覧ください。 06月09日朝日新聞デジタル朝刊記事一覧へ（朝5時更新） 新着記事一覧へ ウクライナ情勢 速報 朝刊. 記事一覧 紙面ビューアー 夕刊. 記事一覧 紙面ビューアー 連載 ランキング コメント その他. 特集 動画・音声・写真 土曜別刷り「be」 記者イベント 天気 数独 12星座占い サイトマップ 検索. ヘルプ. 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4f 92035b 05ab 2d 39cf 3e 沢口靖子さん、「キャリア優先」の30代で出会えたもの 06 eye031 eye 031 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” 967f1782eff2941bc6ece6f0beb1b1a9 967f 1782eff 2941bc 6ece 6f 0beb 1b 1a 9 まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet ba8f917ee7123fad357ba9b8c4a32d6c ba 8f 917ee 7123fad 357ba 9b 8c 4a 32d 6c 08a1bee912777ead2fbfc80b46b32292 08a 1bee 912777ead 2fbfc 80b 46b 32292 「何を着るか」にマンネリを感じたら、冒険しませんか？ L1230304 L 1230304 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき bcc8bb0048ff6549f30e4f867752d6a5 bcc 8bb 0048ff 6549f 30e 867752d 6a スペインから父へ。直接その手を握る代わりに 255 5680 2 〈255〉50歳で選んだ別居婚「第二幕」 8fc46c88600a6570b8deddfa41ebf978 8fc 46c 88600a 6570b 8deddfa 41ebf 978 Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 6b50c0fffa5bdf413e677626e7da782d 6b 50c 0fffa 5bdf 413e 677626e 7da 782d ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを DSC5449 DSC 5449 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った SANJI AND SPOON final animation 猫に学べ。by サジとスプン（飼い主・三宅和歌子さん） 60d4af5fa755cfac59f4dd7726706631 60d 4af 5fa 755cfac 59f 4dd 7726706631 NY発　子どもたちの夏の過ごし方 256 5874 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 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PAGE","ログアウト","新規登録","","","","","","","","","こちら"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:05:29 UTC 2022","fresh_date_dt":"Wed Jun 08 17:05:30 UTC 2022","referrer_id_s":"RTQakgEvDQH4","language_s":"ja","size_i":135790,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087022300725248,"text_t":"寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 | 朝日新聞デジタルマガジン＆[and]. TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 30歳からのコンパス 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT 30歳からのコンパス 連載をフォローする 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」. ARTS & CULTURE 2022.05.27. 0. 撮影／小山幸佑. S.S. 坂口さゆり ライター. 生命保険会社のOLから編集者を経て、1995年からフリーランスライターに。映画評や人物インタビュ …. 昨年11月、作家で僧侶の瀬戸内寂聴さんが亡くなった（享年99）。この10年、寂聴さんの秘書としていつも傍らで寄り添っていたのが、秘書の瀬尾まなほさん（34）だ。年齢差は66歳。大好きな寂聴さんから多くのことを学びながらも軽口をたたき合い、時に瀬尾さんから寂聴さんに今の女性たちの働き方を教える。はたから見てもほほえましく羨（うらや）ましい関係を築いてきた。. たまたまの縁から始まった. 「瀬戸内との出会いが人生の転機だった」. そう瀬尾さんが言う通り、すばらしい出会いは人生を変える。瀬尾さんにとって、寂聴さんはまさに人生のコンパスだったに違いない。. 撮影／小山幸佑. 学生時代から秘書を目指していたわけでも、寂庵に就職しようと決めていたわけでもない。それどころか、結局、大学生活4年の間に「したいこと」が見つからず、ただ、ふつうに就職して結婚して子どもをもうけて……という人生を“ふつう”に送るものだと思っていた。ところが、就職活動は思うようにはいかず、「本当にたまたま」縁があったのが、寂庵だった。. 「瀬戸内については、『あの尼さん』くらいにしか思っていませんでした。私たち世代だと、作家ということ自体を知らない人がたくさんいただろうと思います。私もその一人でした」. 尼僧だと思っていた寂聴さんは、波瀾（はらん）万丈の人生を送ってきたベストセラー作家。日常生活から対人関係、思想……と、共に過ごした10年で寂聴さんから学んだことは数知れない。. 誰に対しても平等 「おもてなしの心」. 例えば、歳時。旬の果物や旬を意識した和菓子があること、暖簾（のれん）などのしつらえも季節に応じたものを準備する。日本の四季や作法は寂庵で学んだ。そして、客人に対する心遣いやおもてなし。寂聴さんの人を幸せな気持ちにさせるような思いやりにはいつも圧倒された。. 提供／©2022「瀬戸内寂聴 99年生きて思うこと」製作委員会. 「例えば、テレビの収録など取材が終わると、お土産として瀬戸内の本を必ずお渡しするんです。当時の私は25、6歳、本はプロデューサーの方など代表者にお渡しすればよいと思っていました。でも、瀬戸内は『アシスタントの方たちにもあげてね』と言う。『先生の本なんか読みそうにないよ』と伝えても、『読む読まないは関係ない。私から本をもらったことが、彼にとったら良い思い出や経験になるかもしれないでしょ』とおっしゃった。そう聞いた時に、そこまで考えていない自分の浅はかさを思い知らされました。『瀬戸内寂聴がアシスタントの自分も気にかけてくれて本をくれた』ということは、確かに、おばあちゃんに自慢できることかもしれない。その人が何か感じるものがあるかもしれない。そこまで瀬戸内は考えている、それも自然に。やっぱり器が全然違うなと思い知らされました」. 寂聴さんは誰に対しても平等で、偉い人もアシスタントさんも分け隔てせず、誰に対してもやさしかった。そんなやさしさは身近な人へも伝播（でんぱ）する。. 提供／©2022「瀬戸内寂聴 99年生きて思うこと」製作委員会. 「自分のことばかりではダメ」の後押し. 2016年に設立された「若草プロジェクト」という団体をご存じだろうか。貧困、虐待、DV、いじめ、薬物依存などさまざまな問題に苦しみ、生きづらさを抱える若い女性たちを支援する団体だ。瀬尾さんは現在その団体の理事を務める。寂聴さんの「自分のことばかり考えていてはダメだよ。宇宙と自分、世界と自分、日本と自分ということを意識しなさい」という言葉に後押しされた。. 「私はボランティア活動をするならその前に、まず私の家族が一番幸せで、友人たちが幸せで、と周囲の人たちを優先的に考えていたんです。でも、瀬戸内が反原発や反戦などで、自分の意見を口に出したり、デモに参加したりする姿を見るにつけ、私も自分の身近なことばかり見ている自分はもうそろそろ卒業なのではないか、そろそろ世界や社会のために自分が何かするべきなのではないかと思い始めるようになりました。そんな時に、瀬戸内にそう言われたんです。ちょうどそのタイミングで若草プロジェクトの話があり、瀬戸内から『まなほ、良かったら参加しなさい』と勧められました」. そこで知った現実は悲惨だった。瀬尾さんが当たり前に過ごした青春時代を、過ごせない女性たちがいた。父親から虐待を受け、自分に価値を見出せず、体を売り物にしている女の子がいた。. 「すごくショックでした。呑気（のんき）に生きてきた私がこの問題に関わってもいいのか、当事者じゃないのに何がわかるのだろうとすごく思いました」. そんな瀬尾さんを寂聴さんは距離を置いて見てくれた。研修会に快く送り出し、学んできたことを報告すると、「まなほ、行くたびに頭が良くなっているね」「学んでいるね」と声をかけてくれた。. 「多分、私がスポンジのようにいろいろなことを吸収していくことを良く思ってくれていたのかなって思います」. NEXT PAGE. 人生に縛りはいらない. 1. 2 REACTION. LIKE. COMMENT. 0. 連載をフォローする TAGS. # インタビュー # キャリア # 映画・舞台・ドラマ SHARE. ゲスト MY PAGE ログアウト 新規登録 ログイン LIKE. COMMENT. 0. 他の記事へ 表現して伝えること。デビュー盤リリース、演奏家Cocomiが描く夢 2022.04.26. ゲスト 0. ログイン. 利用規約 送信する. &MEMBER（登録無料）としてログインすると. コメントを書き込めるようになります。. 評価の高い順 ▼. 最新順 ▼. 古い順 ▼. BACK NUMBER. この連載をもっとみる. 香里奈さん、自分らしくいられる時間から見つけたこと（後編） 2022.04.01. モデルであり俳優。香里奈さんの、全力疾走し続けた20代（前編） 2022.03.25. 「挑戦することを我慢しない」 板谷由夏さんの自分に正直な生き方 2022.03.18. 30代で謎の不調に。羽田美智子さん「へこむ時はチャンスと今なら思える」 2021.10.04. 沢口靖子さん、「キャリア優先」の30代で出会えたもの 2021.08.23. FOR YOU. あなたにおすすめの記事. 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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT CREATORS. 絞り込む ALL. アーティスト. フォトグラファー. ライター・編集者. 俳優・タレント. 学者・研究者. 料理家・ソムリエ. 俳優・写真家. 永瀬正敏 1966年生まれ、宮崎県出身。1983年、映画「ションベン・ライダー」でデビュー。「息子」(91)で日本アカデミー賞新人俳優賞・最優秀助演男優賞他、計8つの映画賞を受賞。その後日本アカデミー賞は、優秀主演男優賞1回、優秀助演男優賞2回受賞。海外作品にも多数出演しカンヌ国際映画祭・最優秀芸術貢献賞「ミステリー・トレイン」(89)、ロカルノ国際映画祭・グランプリ「オータム・ムーン」(92)、リミニ国際映画祭グランプリ、トリノ映画祭審査員特別賞「コールド・フィーバー」(95)では主演を務めた。台湾映画「KANO」(2014)では、金馬奨で中華圏以外の俳優で初めて主演男優賞にノミネートされ、「あん」(15)、「パターソン」(16)、「光」(17)でカンヌ国際映画祭に3年連続で公式選出された初のアジア人俳優となった。近作は「赤い雪」(19)、「ある船頭の話」(19)、「最初の晩餐」(19)、「カツベン！」(19)、「ファンシー」(20)、「星の子」(20)、「茜色に焼かれる」(21)、「名も無い日」(21)、「ホテルアイリス」(22)ほか。写真家としても多くの個展を開き、20年以上のキャリアを持つ。2018年芸術選奨文部科学大臣賞を受賞。. デザイナー／アーティスト. 篠原ともえ 1995年歌手デビュー。文化女子大学（現・文化学園大学）短期大学部服装学科デザイン専攻卒。映画、ドラマ、舞台など歌手・俳優活動を経て、現在はイラストレーター、テキスタイルデザイナーなどさまざまな企業ブランドとコラボレーションするほか、衣装デザイナーとしても松任谷由実コンサートツアー、嵐ドームコンサートなどアーティストのステージ・ジャケット衣装を多数手がける。2020年、アートディレクター・池澤樹と共にクリエーティブスタジオ「STUDEO」を設立。. 作詞家・音楽プロデューサー. いしわたり淳治 1997年にロックバンドSUPERCARのメンバーとしてデビューし、オリジナルアルバム7枚、シングル15枚を発表。そのすべての作詞を担当する。2005年のバンド解散後は、作詞家として、Superfly『愛をこめて花束を』、Little Glee Monster『世界はあなたに笑いかけている』他、SMAP、関ジャニ∞、矢沢永吉、布袋寅泰、今井美樹、JUJU、少女時代、SHINee、EXO、大原櫻子、中島美嘉など、音楽プロデューサーとして、チャットモンチー、9mm Parabellum bullet、flumpool、ねごと、NICO Touches the Walls、GLIM SPANKYなど、ジャンルを問わず数多くのアーティストを手掛ける。現在までに700曲以上の楽曲制作に携わり、数々の映画、ドラマ、アニメの主題歌も制作している。2017年には映画『SING\/シング』の日本語歌詞監修を行い、国内外から高い評価を得る。音楽活動のかたわら、映画・音楽雑誌等での執筆活動も行っている。著書の短編小説集『うれしい悲鳴をあげてくれ』（筑摩書房）は20万部を刊行、ほかに『次の突き当たりをまっすぐ』（筑摩書房）がある。本連載『いしわたり淳治のWORD HUNT』に掲載されたコラムをまとめた書籍『言葉にできない想いは本当にあるのか』（筑摩書房）を発売中。. フリーアナウンサー. 宇賀なつみ 東京都出身。テレビ朝日「報道ステーション」の気象キャスターとしてデビュー。同番組スポーツキャスターとして、トップアスリートへのインタビューやスポーツ中継等を務めた後、「グッド！モーニング」「羽鳥慎一モーニングショー」など、情報・バラエティー番組を幅広く担当。2019年にフリーランスとなり、現在はテレビ朝日「池上彰のニュースそうだったのか!!」や、自身初の冠番組「川柳居酒屋なつみ」を担当。TBSラジオ「テンカイズ」やTOKYO FM「SUNDAYʼS POST」など、ラジオパーソナリティーにも挑戦している。2020年4月からは、MCを務める、関西テレビ・フジテレビ系「土曜はナニする！？」がスタートした。. フリーアナウンサー. 久保純子 1972年、東京都生まれ。小学校時代をイギリス、高校時代をアメリカで過ごす。大学卒業後、NHKに10年間勤め、「ニュース11」「おはよう日本」「紅白歌合戦」「プロジェクトＸ」など、キャスターやインタビュアーとして幅広く活躍。2004年からは、フリーアナウンサーとして活動し、テレビやラジオに出演する傍ら、執筆活動や絵本の読み聞かせ、翻訳も手がける。日本ユネスコ協会連盟による世界寺子屋運動の広報特使を務め、2014年にはモンテッソーリ教育の資格を取得するなど、「子ども」と「言葉」、そして「教育」をキーワードに活動の場を広げている。現在は、夫と2人の娘（20歳と13歳）とともに、ニューヨーク在住。著書に、翻訳を手がけた絵本『クレムとカニさん うみのためにできること』『あしたはクリスマス』（共に文化出版局）など。. 料理人／「eatrip」主宰. 野村友里 おもてなし教室を開く、母・野村紘子さんの影響を受けて料理の道に。主な活動に、レセプションパーティーなどのケータリングフードの演出、料理教室、雑誌の連載、ラジオ番組など。2009年、初の監督作品『eatrip』を公開。11年、「シェ・パニース」のシェフたちとともに、参加型の食とアートのイベント「OPEN harvest」を開催。その経験を経て日本のシェフたちとともに「nomadic kitchen」プロジェクトをスタート。12年、東京・原宿に「restaurant eatrip」をオープン、19年、表参道・GYREに「eatrip soil」をオープン。著書に『eatlip gift』『春夏秋冬 おいしい手帖』（マガジンハウス）、『Tokyo Eatrip』（講談社）、共著に『TASTY OF LIFE』（青幻舎）がある。. ミュージシャン. 小宮山雄飛 東京・原宿生まれ。1996年ホフディランのVo&Keyとしてデビュー。音楽活動以外にもラジオ・テレビ・雑誌など活躍の場を広げ、カルチャー・流行面で同世代へ大きな影響力を持つ一人である。食通としても知られ“音楽界のグルメ番長”の異名を持つ。特にカレーのオリジナルレシピは200を超える。おひとりディズニーがライフワーク。. 歌手. UA 1972年大阪生まれ。母方の故郷は奄美大島。1995年デビュー。1996年発表のシングル『情熱』が大ヒット。以降、浅井健一（元ブランキー・ジェット・シティ）らと組んだ「AJICO」、ジャズサックスプレーヤー菊地成孔とのコラボ、映画主演、NHK教育テレビでの歌うお姉さんなど、活動は多岐にわたる。ライブ、フェス、楽曲制作と精力的に活動している。また、2005年より都会を離れ、田舎で農的暮らしを実践中。現在はカナダに居住。4人の母でもある。α-STATION（FM京都）の番組「FLAG RADIO」にレギュラー出演中。2020年6月21日、デビュー25年を迎えた。. 俳優・タレント. 渡辺早織 モデルとして2007年デビュー。その後、ドラマ\/映画\/CM\/テレビ番組にてレギュラー出演をするなど俳優・タレントとして活躍中。酒好きが高じて、唎酒師の資格を持っている。. はな モデル。12月21日生まれ、神奈川県横浜市出身。2才から横浜のインターナショナルスクールに通い、17才からモデル活動を始める。上智大学に進学後も、学業と並行してモデル活動を続け、その後テレビやラジオ、ナレーター、エッセイの執筆など活動の範囲を広げる。現在もファッション誌で活躍するかたわら、FMヨコハマ「Lovely Day〜hana金〜」（毎週金曜）のナビゲーターや、日本テレビ「夢の通り道」（毎週日曜）のナレーションも務めている。英語・フランス語に堪能で、その語学力を活かした絵本も多数。趣味はお菓子作りや茶道、仏像鑑賞。2017年9月国宝応援大使、2019年4月奈良国立博物館評議員に就任。著書に、『はな、茶の湯に出会う』（淡交社）、『hana’s style book』（宝島社）、『ちいさいぶつぞう　おおきいぶつぞう』（幻冬舎文庫）、『おくるおかし』（集英社）ほか多数。 Instagram：https:\/\/www.instagram.com\/hanalovestaco\/. 小説家. 白岩玄 1983年生まれ。京都市出身。2004年、小説『野ブタ。をプロデュース』で文藝賞を受賞し、小説家デビュー。同作は芥川賞候補作になり、テレビドラマ化。70万部のベストセラーになった。著書に『空に唄う』『愛について』『未婚30』『ヒーロー！』『たてがみを捨てたライオンたち』など. デザイナー. 小倉ヒラク 1983年、東京都生まれ。早稲田大学文学部で文化人類学を学ぶ。企業でデザイナーとして活動した後、東京農業大学で研究生として発酵学を学び、山梨県甲州市の山の上に発酵ラボを創設。「発酵デザイナー」を肩書として、全国の醸造家たちと商品開発や絵本・アニメの制作、ワークショップを開催。絵本＆アニメ『てまえみそのうた』でグッドデザイン賞2014受賞。15年より絵本『おうちでかんたん　こうじづくり』とともに「こうじづくり講座」をスタート。海外でも発酵文化の伝道師として活動するほか、雑誌、ラジオ、テレビでも活躍。18年～19年、47都道府県を旅し、日本の超ローカルな発酵文化を発掘。渋谷ヒカリエでキュレーターを務めた発酵食の展示会は大盛況。20年、下北沢に「発酵デパートメント」をオープン。著作に『発酵文化人類学 微生物から見た社会のカタチ』（木楽舎）、 『日本発酵紀行』（d47 MUSEUM）、『発酵する日本』（Aoyama Book Cultivation）。. フラワースタイリスト. 平井かずみ ikanika主宰。草花がより身近に感じられるような「日常花」の提案をしている。東京を拠点に「花の会」や「リース教室」を全国各地で開催。雑誌や広告などでのスタイリングのほか、ラジオやテレビに出演。著書『フラワースタイリングブック』『ブーケとリース』『あなたの暮らしに似合う花』ほか多数。. 経営共創基盤（IGPI）共同経営者. 塩野誠 ハングリーだけどノーブルな感じの大人に憧れるコンサルタント。外資系投資銀行、IT企業などを経て、現職。政府機関やグローバル企業へのコンサルティング、M&Aアドバイザリー、欧州、ロシアでの投資などに従事。著書に『20代のための「キャリア」と「仕事」入門』『世界で活躍する人は、どんな戦略思考をしているのか？』『デジタルテクノロジーと国際政治の力学』など。本の匂いやレコードのノイズを感じながら、社会とビジネスについて考える日々。「明日は我が身、困った時はお互い様という世の中が良い気がします」. ストリート・スタイル・フォトグラファー／ジャーナリスト. シトウレイ 毎シーズン、世界各国のコレクション取材を行い、独自の審美眼で綴（つづ）られる言葉と写真が人気を博している。ファッションにおける感性の高さと分析力で講演や執筆、テレビやラジオ出演、商品プロデュースやコンサルタント等、ジャンルを超えて活動。ストリートスタイルの随一の目利きであり、「東京スタイル」の案内人。また彼女自身のセンスも、ストリートスナップの権威あるブログ「The Sartorialist」の書籍で特集を組まれる等、ファッション・インフルエンサーとしても活躍中。STYLEfromTOKYO 著書：　東京ストリート写真集「STYLE from TOKYO」（discover21刊）　東京ガイド「日々是東京百景」（文化出版局刊）. 作家. 大道珠貴 1966年福岡市生まれ。2003年、『しょっぱいドライブ』で第128回芥川賞。2005年、『傷口にはウオッカ』で第15回Bunkamuraドゥマゴ文学賞受賞。小説に『ケセランパサラン』『ショッキングピンク』『煩悩の子』など多数。エッセーに『東京居酒屋探訪』。神奈川県鎌倉市在住。. 文筆家. 大平一枝 長野県生まれ。失われつつあるが失ってはいけないもの・こと・価値観をテーマに各誌紙に執筆。著書に『東京の台所』『男と女の台所』『ただしい暮らし、なんてなかった。』（平凡社）、『届かなかった手紙』（角川書店）、『紙さまの話』（誠文堂新光社）、『昭和式もめない会話帖』（中公文庫）など多数。HP『暮らしの柄』. フラワーアーティスト. 東信 1976年生まれ。2002年より花屋を営み続け、現在は東京・南青山にてオートクチュールの花屋「JARDINS des FLEURS」を構える。2005年よりフラワーアーティストとして、ニューヨーク、パリ、ドイツ、ブラジル等、国内外で精力的な活動を展開。独自の視点から花や植物の美を表現し続けている。近著に作品集『ENCYCLOPEDIA OF FLOWERS 植物図鑑V』（青幻舎）など。. 旅行作家. 下川裕治 1954年生まれ。「12万円で世界を歩く」（朝日新聞社）でデビュー。おもにアジア、沖縄をフィールドに著書多数。近著に「週末ちょっとディープなベトナム旅」（朝日文庫）、「ディープすぎるシルクロード中央アジアの旅」（中経の文庫）、「世界最悪の鉄道旅行　ユーラシア大陸横断2万キロ」（朝日文庫）など。最新刊は、「台湾の秘湯迷走旅」（双葉文庫）。. モデル. 高山都 1982年生まれ。モデル、俳優、ラジオパーソナリティーなど幅広く活動。趣味は料理とマラソン。 「#みやれゴハン」として料理やうつわなどを紹介するインスタグラムが人気。 趣味のマラソンでは、横浜マラソン2016を3時間41分で完走の記録を持っている。著書に『高山都の美 食 姿』『高山都の美 食 姿 2』『高山都の美 食 姿 3』『高山都の美 食 姿 4』（双葉社）がある。. FOR YOU. あなたにおすすめの記事. LIFE 2022.06.01. ＆w. リノベーション・スタイル 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” INTERESTS 2022.05.31. SPONSORED. ＆Travel. 世界の街へ 旅先で最高の「地元体験」を提供　 高級ホテル「マリオット」のMarriott Bonvoyプログラム ESSAY 2022.05.31. ＆Travel. 宇賀なつみ わたしには旅をさせよ 長崎の島で教えられたこと　宇賀なつみがつづる旅(33) FOOD 2022.05.31. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet FOOD 2022.05.31. ＆M. 渡辺早織の思い出ちょっぴり、つまみぐい。 益子陶器市×びわの白あえ　うつわ探しで向き合う心の声 GALLERY 2022.05.31. ＆Travel. 上から撮るか、下から撮るか サングラス大仏、アートな佐久島　愛知県 POPULAR. 人気記事. アクセス LIKE コメント 小川フミオのモーターカー 強烈なダッシュ力とデザインのよさ　ヤマハ「RZ250」 0. ＆M. インタビュー 『機動戦士ガンダム　ククルス・ドアンの島』アムロ役・古谷徹さん　第一線で活躍を続ける原動力 0. ＆M. 渡辺早織の思い出ちょっぴり、つまみぐい。 益子陶器市×びわの白あえ　うつわ探しで向き合う心の声 3. ＆M. 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. 京都ゆるり休日さんぽ 京都にたたずむフランス　日仏学館の小さなビストロ「レ・ドゥ・ギャルソン」 0. ＆Travel. 東京の台所２ 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った 0. ＆w. 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. 国境なき衣食住 覚悟を背負って赴任したアフガニスタン　宿舎のガーデンで出会った子猫 0. ＆M. あの街の素顔 今こそ「よみがえりの道」熊野古道　山上の「伝説の宿」から 6. ＆Travel. スペイン カミーノ・デ・サンティアゴ ～星に導かれ、巡礼のスペイン旅へ 0. ＆Travel. ニッポン銭湯風土記 沖縄最後の「ゆーふるやー」中乃湯再訪・コザの街歩き 11. ＆Travel. 欧州おいしい旅 美しい5月に食べたくなる魚は？　オーストリア、リヒテンシュタイン、ドイツ、ノルウェー 7. ＆Travel. あの街の素顔 今こそ「よみがえりの道」熊野古道　山上の「伝説の宿」から 6. ＆Travel. 楽しいひとり温泉 地酒と響き合うミシュランの美食　新潟・松之山「酒の宿 玉城屋」 5. ＆Travel. 久保純子 LIFE in N.Y. NY発　子どもたちの夏の過ごし方 5. ＆w. ※アクセスは過去7日間、LIKE、コメントは過去30日間で集計しています。. RECOMMEND. おすすめの記事. 日々にちょっとずつ、彩りを。. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト サイトポリシー 利用規約 広告事例・ガイド 個人情報 アクセスデータの利用 著作権 お問い合わせ 朝日新聞デジタルに掲載の記事・写真の無断転載を禁じます。すべての内容は日本の著作権法並びに国際条約により保護されています。. Copyright © The Asahi Shimbun Company. All rights reserved. No reproduction or republication without written permission. ＆MEMBER限定の機能です. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. ＆MEMBER登録 ログイン ＆MEMBERログイン. 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SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT WORK. 2022.06.03. ＆M. 会議のチカラ 自信を失って発言をしないベテラン世代の活性化策 2022.05.20. ＆M. 会議のチカラ なぜ、ベテラン世代は自分たちの「武勇伝」を語りたがるのか 2022.05.06. ＆M. 会議のチカラ 部長の手帳は会議でびっしり　「アリバイ会議」の弊害 2022.04.22. ＆M. 会議のチカラ 本当に話した方がいいこと　コロナで浮上した雑談の価値 2022.04.08. ＆M. 会議のチカラ 会議の席順　いつもの席でないと落ち着かない 2022.03.25. ＆M. 不条理とのつきあい方 自分にやさしく、周りにやさしくするべき時があるとすれば今 2022.03.25. ＆M. 会議のチカラ 「私は聞いていない」　若手・中堅社員が巻き込まれる「報告会議」 2022.03.11. ＆M. 不条理とのつきあい方 「ミッドライフクライシス」の処方箋　損得のないコミュニティー探し 2022.02.18. ＆M. 不条理とのつきあい方 どのように「天職」を見つけたらいいのでしょうか？　 FOR YOU. あなたにおすすめの記事. WORK 2022.05.20. ＆M. 会議のチカラ なぜ、ベテラン世代は自分たちの「武勇伝」を語りたがるのか WORK 2022.05.06. ＆M. 会議のチカラ 部長の手帳は会議でびっしり　「アリバイ会議」の弊害 WORK 2022.04.22. ＆M. 会議のチカラ 本当に話した方がいいこと　コロナで浮上した雑談の価値 WORK 2022.04.08. ＆M. 会議のチカラ 会議の席順　いつもの席でないと落ち着かない WORK 2022.03.25. ＆M. 会議のチカラ 「私は聞いていない」　若手・中堅社員が巻き込まれる「報告会議」 LIFE 2022.03.25. ＆M. 不条理とのつきあい方 自分にやさしく、周りにやさしくするべき時があるとすれば今 POPULAR. 人気記事. アクセス LIKE コメント 小川フミオのモーターカー 強烈なダッシュ力とデザインのよさ　ヤマハ「RZ250」 0. ＆M. インタビュー 『機動戦士ガンダム　ククルス・ドアンの島』アムロ役・古谷徹さん　第一線で活躍を続ける原動力 0. ＆M. 渡辺早織の思い出ちょっぴり、つまみぐい。 益子陶器市×びわの白あえ　うつわ探しで向き合う心の声 3. ＆M. 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. 京都ゆるり休日さんぽ 京都にたたずむフランス　日仏学館の小さなビストロ「レ・ドゥ・ギャルソン」 0. ＆Travel. 東京の台所２ 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った 0. ＆w. 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. 国境なき衣食住 覚悟を背負って赴任したアフガニスタン　宿舎のガーデンで出会った子猫 0. ＆M. あの街の素顔 今こそ「よみがえりの道」熊野古道　山上の「伝説の宿」から 6. ＆Travel. スペイン カミーノ・デ・サンティアゴ ～星に導かれ、巡礼のスペイン旅へ 0. ＆Travel. ニッポン銭湯風土記 沖縄最後の「ゆーふるやー」中乃湯再訪・コザの街歩き 11. ＆Travel. 欧州おいしい旅 美しい5月に食べたくなる魚は？　オーストリア、リヒテンシュタイン、ドイツ、ノルウェー 7. ＆Travel. あの街の素顔 今こそ「よみがえりの道」熊野古道　山上の「伝説の宿」から 6. ＆Travel. 楽しいひとり温泉 地酒と響き合うミシュランの美食　新潟・松之山「酒の宿 玉城屋」 5. ＆Travel. 久保純子 LIFE in N.Y. 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祇園祭 西日本豪雨 Ｇ２０大阪サミット 宝塚歌劇団 まだまだ勝手に関西遺産 2019統一地方選挙 ＳＴＵ瀬戸リスト 沖縄1935 よみがえる戦前 記者襲撃、あの夜から 阪神支局襲撃30年を越えて 「酷道」をいく 戦火の残響 三浦雄一郎の挑戦 全日本小中学生ダンスコンクール オウムを暴走させた３つの転機 100回つなぐ始球式リレー 環境破壊、報いの現場を歩く 平成とは がんとともに 再生！太陽の塔 日本代表レジェンドカード 「13」最高のスタジアムを巡る旅 2018サッカーワールドカップ 孤高の星 羽生結弦-SPIN THE DREAM SPIN THE DREAM 羽生結弦 LEGEND 葛西紀明 平昌オリンピック 平昌パラリンピック 金正恩とトランプ 「トランプ王国」熱狂のあと 災害大国 被害に学ぶ 河田防災塾 あのときの、ラジオ 国際女性デー特集「Dear Girls」 小さないのち「児相の現場から」 SDGs フィギュアスケート特集 ＨＡＫＵＴＯの挑戦 築地市場の「音」 平成とは いじめと君 女子組 朝日・東大共同調査 ドローンズアイ Ｔｈｅ　Ｒｏａｄ 日本初「監獄ホテル」誕生へ 景気１００社アンケート 衆院の区割り変更 オバマ米大統領、広島へ 南極観測６０年 トワイライトエクスプレス瑞風 政治資金特集 タイムライン 奄美群島国立公園 復興支援音楽祭 いきもの目線 全日本合唱コンクール ノーベル賞 もっと！ロケット HUG STA（ハグスタ） 子ども食堂 大学入試センター試験 築地－時代の台所 真田太平記 皇室とっておき （各駅停話）東北新幹線特別編 待機児童問題 どうぶつ新聞 家庭遺産 朝デジスペシャル. 1.17 再現／阪神・淡路大震災 120日後のフェス　アラバキ 2.24後の世界で　ウクライナ侵攻から考える 4 years from ICHIRO｜イチローが伝えたかったこと GHOSN（ゴーン）日本最大の司法取引 GRIM-成田緑夢（ぐりむ） HAKUTO-Rの挑戦　月への旅路 HAYABUSA2 地球帰還 LEGEND 葛西紀明 MINAMATA　ユージン・スミスの伝言 SDGs｜国谷裕子さんと考える SDGｓ｜下川町編 SPIN THE DREAM　夢をつむぐ羽生結弦 The Days ウクライナ侵攻 VR｜築地市場の360度動画 「13」最高のスタジアムを巡る旅 「オウム」を暴走させた3つの転機 「会いたい、会わせたい　東日本大震災10年」 「祈りの旅」 あなたは誰ですか　ルポ孤独死 あのときの、ラジオ ３．１１震災特集 ありがとう普門館 いきもの目線 いまの政治と選挙は　朝日新聞社世論調査 いま憲法をどう考える　朝日新聞社世論調査 そのとき官邸は　新たな調書 ふたつのアメリカ　分極社会 みえない交差点　全国68万件の交通事故マップ イチローMLB3000安打特集 イラン抵抗の三日月　謎の武装組織を追う オバマ大統領、広島へ カルロス・ゴーン　もたらした光と影 コロナ後の世界を語る　現代の知性たちの視線 コロナ第２波　東京100days チャレンジド wonder athletes デレク＆マイケル データで見る「就活」の歴史 ドーピングの深い闇 ナガサキノート あの日、人々の足取り ノモンハン　大戦の起点と終止符 マー君の410球を徹底分析 ヤンキース田中打者の目で解析 ラストダンス ワールドアスリートボイス ワールドカップの地球 三浦雄一郎 南米最高峰に挑む 主役なきゴーン法廷 五輪TVゲーム年代記 住み続けたい島へ　SDGs｜島の人たちと考えた 冬季オリンピック 「持続可能性」を巡る旅 動物たちはどこへ　変わりゆく動物園 北陸新幹線　駆ける　つながる 医療危機　コロナ東京100days 吉田調書 名人への道 藤井聡太 地殻変動 自公vs.非自公 声　語り継ぐ戦争 夢を力に2014 大谷翔平 -G.O.A.T- 2021年を象徴する5シーン 失踪村　ベトナム人技能実習生 孤高の星　羽生結弦-SPIN THE DREAM 学校の死角 宇宙へ　苦難と栄光、終わらない旅 安倍×石破　二人がみる日本―自民党総裁選２０１８ 対立の海 平成人語 待機児童問題 憲法をどう考える2022 戦後７０年　ビジュアル年表 手塚治虫文化賞２０周年 拓く　北海道新幹線 政党の現在地　与野党はどう動いた　朝日・東大調査15年 新型コロナ　みんなの考えは？ 新型コロナウイルス感染　世界の１年 新型コロナウイルス感染　日本の１年 日本代表 レジェンドカード 日米開戦80年 朝デジ１０周年特集 未完の最長政権 未来のデザイン　コロナで変わる 李香蘭が語るアジア 東京100days 新型コロナウイルスの記録 枯れる五輪 永世七冠 羽生善治 沖縄1935　写真でよみがえる戦前 沖縄1972　写真でたどる日本復帰50年 沖縄を語る、沖縄から考える 沖縄復帰50年、基地はどこへ 海からみた被災地　東日本大震災10年 琉球から日本へ　よみがえる戦前の沖縄 生きる、未来へ　東日本大震災10年 疑惑の島「パラダイス文書」 石川祐希　世界へ 砂上の国家　満州のスパイ戦 私、がんになるの？　ゲノムが迫る選択 空からみた被災地　11年後の現在 空から見た熱海土石流 空襲１９４５ 築地　時代の台所 米中争覇 裁かれる工藤会　全公判傍聴記 追跡 メイド・イン・ジャパン 和牛編 追跡 メイド・イン・ジャパン 腕時計編 逆さ地図で見る米中最前線の中の沖縄 里海てんぷら　東京の天然を揚げる 金正恩とトランプ 金正男暗殺を追う　2017\/2\/13 飽くなき挑戦 ロータリーエンジンの半世紀 高校野球 夏１００回 高橋源一郎 伯父さんは戦場へ行った １６年リオ五輪　超人１６人に注目 ＥＸＰＯ‘７０　ＯＳＡＫＡ ＮＩＣＫ　私がたどりついた日本 地域一覧. 北海道・東北. 北海道 青森 秋田 岩手 山形 宮城 福島 関東. 群馬 栃木 茨城 埼玉 千葉 東京 神奈川 東海・甲信越. 静岡 岐阜 愛知 三重 新潟 山梨 長野 近畿・北陸. 滋賀 京都 奈良 和歌山 大阪 兵庫 富山 石川 福井 中国・四国. 鳥取 島根 岡山 広島 山口 香川 愛媛 徳島 高知 九州・沖縄. 福岡 大分 宮崎 鹿児島 佐賀 長崎 熊本 沖縄 朝日新聞デジタルの基本機能. ログイン画面 朝刊 夕刊 ｂｅ 紙面ビューアー 地域面紙面 スクラップブック スクラップブック MYキーワード MYキーワード 記者イベント 動画・音声・写真 コメント閲覧 天気 数独 12星座占い 交通情報 アプリ プッシュ通知設定 アクセスランキング ソーシャルランキング トピックス デジタル限定 サポート. 総合ガイド お客さまサポート Q&A（よくある質問） サービス紹介 新聞宅配申し込み デジタル申し込み ウェブマガジン. &w &M &Travel 朝日新聞デジタルの関連サイト. 論座 朝日新聞社のサイト. 朝日新聞社　会社案内 朝日新聞SHOP 注目の連載記事 もっと見る いま読む「音楽展望」　吉田秀和 没後10年 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Twitter. 記者ページ. 朝日新聞デジタルから 紙面ビューアーに北海道支社版を追加 紙面ビューアーの公開期間拡大について 注目コンテンツ. ご案内 【＆ｗ】 「何を着るか」に困ったら シトウレイの好きな服似合う服 【＆M】 強烈なダッシュ力 ヤマハ「RZ250」 【＆Travel】 サングラスかけた？大仏 上から撮るか、下から撮るか 【＆M】 裏路地の店、ゆったり美味！ 口福のカレー　東京・下北沢 【＆ｗ】 50歳の別居婚「第二幕」 東京の台所２ 好書好日 Z世代のNEO川柳に注目 暮田真名「ふりょの星」刊行 論座 株ブーム去り「宴の後始末」 今日の編集長おすすめ記事 アエラスタイルマガジン アートで蘇る街とは？ 高崎 丈さんが目指す未来 Aging Gracefully Aging Gracefully５周年 メイキング風景を大公開！ GLOBE＋ 二極化で日本がとるべき道は ウクライナ侵攻は長期化の様相 sippo ごめんねしか言えなかった 亡き愛犬が教えてくれたこと 働き方・就活 転職情報　朝日求人ウェブ 動画ランキング 動画一覧 1:03. 1. 東北新幹線、懐かし200系カラー再現　開業40周年... 2:29. 2. 「いつもロシアは攻撃される」今も引きずるナポレオン... 2:05. 3. 赤字覚悟のワンコイン刺し盛りも　金沢・近江町市場、... 1:40. 4. 電車内で事件に遭遇…その時記者は　疑似体験して初め... 3:47. 5. 響くサイレン、廊下駆ける不審者…緊迫の付属池田小訓練. 前へ. 次へ. 写真 フォトギャラリー 遠藤、守田、田中のデュエル. 震災から11年、当時と今. 小室圭さん眞子さんNYへ. 瀬戸内寂聴さんのあゆみ. 美しきエンゼルスタジアム. 上皇后美智子さまと眞子さま. 平成の怪物　松坂の軌跡. 長嶋茂雄さん　不滅の歩み. 前へ. 次へ. ページトップへ戻る 朝日新聞デジタルの関連サイト 朝日新聞社から 会社案内 サステナビリティ 採用情報 記事のご利用案内 新聞広告ガイド デジタル事業から デジタルサービス一覧 携帯サービス Astand(コンテンツ販売) 法人向け配信 写真の購入案内 記事データベース案内 朝日ID グループ企業 朝日新聞出版の本 朝日新聞出版(AERA dot.) 朝日インタラクティブ 朝日学生新聞社 各国語サイト (News in various languages). The Asahi Shimbun Asia&Japan Watch (ENGLISH) Asahi Weekly (ENGLISH\/JAPANESE) ハフポスト日本版 (JAPANESE) CNN.co.jp (JAPANESE) サイトポリシー サイトマップ 利用規約 web広告ガイド リンク 個人情報 著作権 ご意見・お問い合わせ 朝日新聞デジタルに掲載の記事・写真の無断転載を禁じます。すべての内容は日本の著作権法並びに国際条約により保護されています。. Copyright © The Asahi Shimbun Company. All rights reserved. No reproduction or republication without written permission."}
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357ba 9b 8c 4a 32d 6c 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 08a1bee912777ead2fbfc80b46b32292 08a 1bee 912777ead 2fbfc 80b 46b 32292 「何を着るか」にマンネリを感じたら、冒険しませんか？ EYE L1230304 L 1230304 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき スペインから父へ。直接その手を握る代わりに 255 5680 2 〈255〉50歳で選んだ別居婚「第二幕」 8fc46c88600a6570b8deddfa41ebf978 8fc 46c 88600a 6570b 8deddfa 41ebf 978 Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 6b50c0fffa5bdf413e677626e7da782d 6b 50c 0fffa 5bdf 413e 677626e 7da 782d ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを DSC5449 DSC 5449 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った 750 SANJI AND SPOON final animation 猫に学べ。by サジとスプン（飼い主・三宅和歌子さん） 60d4af5fa755cfac59f4dd7726706631 60d 4af 5fa 755cfac 59f 4dd 7726706631 NY発　子どもたちの夏の過ごし方 256 5874 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 051fe2b145281cf904496fa5388fcc0f 051fe 2b 145281cf 904496fa 5388fcc 0f 毎日食べたくなる！　花も酵母も慈しむ店主の軽快なごほうび感／ベーグルスタンダード footer ＆Human corporate 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PAGE","ログアウト","新規登録","","","","","","","","","こちら"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:05:24 UTC 2022","fresh_date_dt":"Wed Jun 08 17:05:25 UTC 2022","referrer_id_s":"RTQakgEvDQH4","language_s":"ja","size_i":133025,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087017163751424,"text_t":"〈244〉適度な距離感でいい関係に。“家族でシェアハウス” | 朝日新聞デジタルマガジン＆[and]. TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. リノベーション・スタイル 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT リノベーション・スタイル 連載をフォローする 〈244〉適度な距離感でいい関係に。“家族でシェアハウス”. LIFE 2022.06.01. 石井健 建築家. 1969年生まれ。日本のリノベーション・シーンの創成期から手がけてきた「ブルースタジオ」クリエイ …. Nさんご家族（夫40代、妻40代、子2人）. 東京都 \/ 築37年 \/ 65.84㎡ \/ 総工費1230万円. 子どもが2人いるNさんご家族。子どもが小さい頃に購入したマンションを、リノベーションすることにしました。今回は、成長期の子どもたちと気持ちよく暮らすための家作りです。コンセプトは、適度な距離感が保てる“家を家族でシェアする”です。. 元の家は3LDKでしたが、思い切って1LDKにし、そのほかに用途を考えたスペースを作りました。テレワーク用のワークスペース、家族の洋服が全て収納できるファミリークローゼット、そして多目的に使える小上がり。小上がりは、夜はご夫婦の寝室にもなります。. 引き戸で区切られた小上がりの前は縁側のような空間となり、読書スペースとしても活用している. 残した個室は子ども部屋です。元は和室を子ども部屋として使っていたところを、洋室にして中央に収納家具を設置。窓が一つの部屋しかなく、二部屋にはできないので、緩く仕切り、それぞれのテリトリーを作ることにしました。子どもたちは自分の居場所ができ、好きなものを飾ったり、自分のものを整理整頓したりするようになりました。. 2人の子ども部屋前の廊下には、ホワイトボードやうんていを取り付けて、アクティビティースペースとして活用. LDKは広めにしましたが、以前のように家族全員が一緒に過ごすことは少なくなりました。それぞれの予定があって、4人そろってごはんを食べることも減ったので、ダイニングテーブルはフレキシブルに使えるタイプのものを特注。真ん中から細長く二つに分けられ、キッチンカウンター風に配置することも可能です。. 忙しい平日は、カウンターとして使うことが多いようで、配膳もラクだとか。食事で使う以外にも、勉強や仕事に使うこともあります。ゆっくりできる休日は、4人で座れるダイニングテーブルに変更し、家族でごはんを食べています。. ゆっくりと食事を楽しむときは、カウンターを動かしてダイニングテーブルとして使える. 用途を考えたスペースを作りましたが、それ専用ということではなく、家族それぞれが使うことができます。ワークスペースで子どもが勉強したり、子ども部屋でNさんが仕事をしたりなど、貸し借りは自由です。家族で暮らしているけれど、家全体に、1人になれるパーソナルな場所が色々あるというのが特徴です。. そんなスペースをつないでいるのが、廊下です。通路としての役割だけでなく、洗面台を設置したり、キャンプなどのアウトドア用品の収納場所にしたりと、共有スペースとして活用。子ども部屋の前は少し広めにし、みんなが集まる交差点にしました。懸垂ができるエクササイズ用のバー、連絡事項が書き込めるホワイトボードと、ユニークなアイテムが家族の足を止めるきっかけを作ります。. 家の中に通路と交差点があり、そこを通って家族がそれぞれの場所に移動し、自分の仕事、勉強、趣味などをしています。適度な距離感を保ちながら、それぞれの気配を感じられる、“家族でシェアハウス”です。どんどん成長していく子どもたちと、いい関係を保つヒントになる事例です。. 10. Nさんご夫婦に聞く. リノベーションQ&A. Q１　リノベーションで一番大切にしたことはなんですか？. 収納スペースを確保し、片づけやすいお部屋にすることです。. Q２　リノベーション後、想像していた以上のことはありましたか？. 広くない部屋ですが、目的に応じた様々な空間を作ることができ、個の時間、家族の時間のメリハリがつけやすく、家での時間が快適になりました。. スチールドア越しに見えるのはワークスペース。夜は子どもが勉強スペースとして活用することも. Q３　リノベーションのプロセスで楽しかったことは？. 間取りを決める際に、色々なご提案を頂きながら、子どもたちも一緒に、家族で考えられたことが楽しかったです。. 構成・大橋史子. 関連記事. 〈231〉予定は変更しても目的は達成！　住みながら必要な部分だけリノベーション Sさんご家族（夫・妻40代　長男6歳　長女4歳）東京都大田区 \/ 68.01㎡ \/ 築41年 …. 2021.06.16. REACTION. LIKE. 連載をフォローする TAGS. # インテリア # おうち時間 # デザイン SHARE. ゲスト MY PAGE ログアウト 新規登録 ログイン LIKE. 他の記事へ スペインから父へ。直接その手を握る代わりに 2022.05.26. ゲスト 0. ログイン. 利用規約 BACK NUMBER. この連載をもっとみる. 〈243〉住む人に合った動線で、暮らしも、家事も、片づけもスムーズに 2022.05.18. 〈242〉夫婦それぞれのワークスペースがある、家族と個人を両立できる間取り 2022.05.04. 〈241〉日常を忘れられる、郊外の広めの家での一人暮らし 2022.04.20. 〈240〉より広さを感じる余白のある家。アートや香りで居心地良く 2022.03.23. 〈239〉家作りの新しい視点、眺めがいい家に注目！　窓を生かして明るく 2022.02.16. FOR YOU. あなたにおすすめの記事. 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SPONSORED. ＆w. ＆BRAND VOICE 六本木ヒルズを開業からずっと「育ててきた」 ――優しさにあふれる人の街づくり # インタビュー # キャリア # 働き方 ARTS & CULTURE 2022.03.25. ＆w. 30歳からのコンパス モデルであり俳優。香里奈さんの、全力疾走し続けた20代（前編） # インタビュー # キャリア # 心と体 ARTS & CULTURE 2022.03.18. ＆w. 30歳からのコンパス 「挑戦することを我慢しない」 板谷由夏さんの自分に正直な生き方 # インタビュー # キャリア # 映画・舞台・ドラマ ARTS & CULTURE 2022.03.15. ＆w. 明日のわたしに還る Charaさん「小さくてもいい、可能性を試したい」　心機一転のオーケストラ公演 # インタビュー # キャリア # 子育て・家族 # 音楽 FASHION 2022.02.04. ＆w. ファッションニュース 生き方が魅力をつくる　冨永愛ならではの美を追い続け # キャリア # ファッション FASHION 2022.01.21. ＆w. ファッションニュース 鳥居ユキ60年　オシャレに自由に # キャリア # ファッション ARTS & CULTURE 2022.05.27. ＆w. 30歳からのコンパス 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 # インタビュー # キャリア # 映画・舞台・ドラマ ARTS & CULTURE 2022.04.01. ＆w. 30歳からのコンパス 香里奈さん、自分らしくいられる時間から見つけたこと（後編） # インタビュー # キャリア # 心と体 INTERESTS 2022.03.25. 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祇園祭 西日本豪雨 Ｇ２０大阪サミット 宝塚歌劇団 まだまだ勝手に関西遺産 2019統一地方選挙 ＳＴＵ瀬戸リスト 沖縄1935 よみがえる戦前 記者襲撃、あの夜から 阪神支局襲撃30年を越えて 「酷道」をいく 戦火の残響 三浦雄一郎の挑戦 全日本小中学生ダンスコンクール オウムを暴走させた３つの転機 100回つなぐ始球式リレー 環境破壊、報いの現場を歩く 平成とは がんとともに 再生！太陽の塔 日本代表レジェンドカード 「13」最高のスタジアムを巡る旅 2018サッカーワールドカップ 孤高の星 羽生結弦-SPIN THE DREAM SPIN THE DREAM 羽生結弦 LEGEND 葛西紀明 平昌オリンピック 平昌パラリンピック 金正恩とトランプ 「トランプ王国」熱狂のあと 災害大国 被害に学ぶ 河田防災塾 あのときの、ラジオ 国際女性デー特集「Dear Girls」 小さないのち「児相の現場から」 SDGs フィギュアスケート特集 ＨＡＫＵＴＯの挑戦 築地市場の「音」 平成とは いじめと君 女子組 朝日・東大共同調査 ドローンズアイ Ｔｈｅ　Ｒｏａｄ 日本初「監獄ホテル」誕生へ 景気１００社アンケート 衆院の区割り変更 オバマ米大統領、広島へ 南極観測６０年 トワイライトエクスプレス瑞風 政治資金特集 タイムライン 奄美群島国立公園 復興支援音楽祭 いきもの目線 全日本合唱コンクール ノーベル賞 もっと！ロケット HUG STA（ハグスタ） 子ども食堂 大学入試センター試験 築地－時代の台所 真田太平記 皇室とっておき （各駅停話）東北新幹線特別編 待機児童問題 どうぶつ新聞 家庭遺産 朝デジスペシャル. 1.17 再現／阪神・淡路大震災 120日後のフェス　アラバキ 2.24後の世界で　ウクライナ侵攻から考える 4 years from ICHIRO｜イチローが伝えたかったこと GHOSN（ゴーン）日本最大の司法取引 GRIM-成田緑夢（ぐりむ） HAKUTO-Rの挑戦　月への旅路 HAYABUSA2 地球帰還 LEGEND 葛西紀明 MINAMATA　ユージン・スミスの伝言 SDGs｜国谷裕子さんと考える SDGｓ｜下川町編 SPIN THE DREAM　夢をつむぐ羽生結弦 The Days ウクライナ侵攻 VR｜築地市場の360度動画 「13」最高のスタジアムを巡る旅 「オウム」を暴走させた3つの転機 「会いたい、会わせたい　東日本大震災10年」 「祈りの旅」 あなたは誰ですか　ルポ孤独死 あのときの、ラジオ ３．１１震災特集 ありがとう普門館 いきもの目線 いまの政治と選挙は　朝日新聞社世論調査 いま憲法をどう考える　朝日新聞社世論調査 そのとき官邸は　新たな調書 ふたつのアメリカ　分極社会 みえない交差点　全国68万件の交通事故マップ イチローMLB3000安打特集 イラン抵抗の三日月　謎の武装組織を追う オバマ大統領、広島へ カルロス・ゴーン　もたらした光と影 コロナ後の世界を語る　現代の知性たちの視線 コロナ第２波　東京100days チャレンジド wonder athletes デレク＆マイケル データで見る「就活」の歴史 ドーピングの深い闇 ナガサキノート あの日、人々の足取り ノモンハン　大戦の起点と終止符 マー君の410球を徹底分析 ヤンキース田中打者の目で解析 ラストダンス ワールドアスリートボイス ワールドカップの地球 三浦雄一郎 南米最高峰に挑む 主役なきゴーン法廷 五輪TVゲーム年代記 住み続けたい島へ　SDGs｜島の人たちと考えた 冬季オリンピック 「持続可能性」を巡る旅 動物たちはどこへ　変わりゆく動物園 北陸新幹線　駆ける　つながる 医療危機　コロナ東京100days 吉田調書 名人への道 藤井聡太 地殻変動 自公vs.非自公 声　語り継ぐ戦争 夢を力に2014 大谷翔平 -G.O.A.T- 2021年を象徴する5シーン 失踪村　ベトナム人技能実習生 孤高の星　羽生結弦-SPIN THE DREAM 学校の死角 宇宙へ　苦難と栄光、終わらない旅 安倍×石破　二人がみる日本―自民党総裁選２０１８ 対立の海 平成人語 待機児童問題 憲法をどう考える2022 戦後７０年　ビジュアル年表 手塚治虫文化賞２０周年 拓く　北海道新幹線 政党の現在地　与野党はどう動いた　朝日・東大調査15年 新型コロナ　みんなの考えは？ 新型コロナウイルス感染　世界の１年 新型コロナウイルス感染　日本の１年 日本代表 レジェンドカード 日米開戦80年 朝デジ１０周年特集 未完の最長政権 未来のデザイン　コロナで変わる 李香蘭が語るアジア 東京100days 新型コロナウイルスの記録 枯れる五輪 永世七冠 羽生善治 沖縄1935　写真でよみがえる戦前 沖縄1972　写真でたどる日本復帰50年 沖縄を語る、沖縄から考える 沖縄復帰50年、基地はどこへ 海からみた被災地　東日本大震災10年 琉球から日本へ　よみがえる戦前の沖縄 生きる、未来へ　東日本大震災10年 疑惑の島「パラダイス文書」 石川祐希　世界へ 砂上の国家　満州のスパイ戦 私、がんになるの？　ゲノムが迫る選択 空からみた被災地　11年後の現在 空から見た熱海土石流 空襲１９４５ 築地　時代の台所 米中争覇 裁かれる工藤会　全公判傍聴記 追跡 メイド・イン・ジャパン 和牛編 追跡 メイド・イン・ジャパン 腕時計編 逆さ地図で見る米中最前線の中の沖縄 里海てんぷら　東京の天然を揚げる 金正恩とトランプ 金正男暗殺を追う　2017\/2\/13 飽くなき挑戦 ロータリーエンジンの半世紀 高校野球 夏１００回 高橋源一郎 伯父さんは戦場へ行った １６年リオ五輪　超人１６人に注目 ＥＸＰＯ‘７０　ＯＳＡＫＡ ＮＩＣＫ　私がたどりついた日本 地域一覧. 北海道・東北. 北海道 青森 秋田 岩手 山形 宮城 福島 関東. 群馬 栃木 茨城 埼玉 千葉 東京 神奈川 東海・甲信越. 静岡 岐阜 愛知 三重 新潟 山梨 長野 近畿・北陸. 滋賀 京都 奈良 和歌山 大阪 兵庫 富山 石川 福井 中国・四国. 鳥取 島根 岡山 広島 山口 香川 愛媛 徳島 高知 九州・沖縄. 福岡 大分 宮崎 鹿児島 佐賀 長崎 熊本 沖縄 朝日新聞デジタルの基本機能. ログイン画面 朝刊 夕刊 ｂｅ 紙面ビューアー 地域面紙面 スクラップブック スクラップブック MYキーワード MYキーワード 記者イベント 動画・音声・写真 コメント閲覧 天気 数独 12星座占い 交通情報 アプリ プッシュ通知設定 アクセスランキング ソーシャルランキング トピックス デジタル限定 サポート. 総合ガイド お客さまサポート Q&A（よくある質問） サービス紹介 新聞宅配申し込み デジタル申し込み ウェブマガジン. &w &M &Travel 朝日新聞デジタルの関連サイト. 論座 朝日新聞社のサイト. 朝日新聞社　会社案内 朝日新聞SHOP 注目の連載記事 もっと見る いま読む「音楽展望」　吉田秀和 没後10年 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Skip to content English. English Home. News. School. Courses. Schedule. Accommodation. Application. Contact us. APPLY NOW. JR渋谷駅 西口からのアクセス方法. January 27, 2020. 〒150-0031東京都渋谷区桜丘町15-17 NKG東京ビル. JR渋谷駅 南改札 西口から日伊学院までのアクセス方法を写真付きでご紹介いたします。. 東急東横線・副都心線 渋谷駅をご利用の方も、ヒカリエ改札(地下3階)を出て14番出口よりJR渋谷駅 南改札を経由してお越しいただけます。. ◆Google mapはこちら まずは南改札から、JR駅構内にある案内に沿って「西口」を目指してください。. . . JR渋谷駅の西口です。正面にバスロータリー、左手に「緑の窓口」があります。西口に出たら左方向へ。. . . 「緑の窓口」の角を左に曲がり、すぐ正面に見えてくる歩道橋を上がります。（首都高速道路・246号の方向）. . . . . 桜丘方面を目指して歩道橋を渡ります。「Ikebe MUSIC」の黒い建物が目印です。. . . . . 「Ikebe MUSIC」のある方向を目指して階段を降ります。. . . 「Ikebe MUSIC」の文字のある黒い建物の方へ横断歩道を渡り、左方向へ進みます。. . . 「Ikebe MUSIC」を右にして進むと、その並びに「キリンシティ」が見えてきます。. . . 「キリンシティ」の先に道路を挟んで「すき屋」が見えるのでそれに向かって横断歩道を渡ります。. . . 「すき家」を左にして坂を上ります。. . . 「グランベルホテル」の建物（１Fはピザ店「CONA」）のすぐ先を左に入ります。. . . 通りから少し奥まったところにある薄い黄色のビルが渋谷ホール＆スタジオのある「NKG東京ビル」です。. . . １F駐車場内に入ってすぐ右手のエレベーターをご利用いただけます。. ※当施設の建物は「NKG東京ビル」ですが、一部の地図で「NKG東京本社ビル」と表示されることがございます。. ※隣の茶色の建物は「NKG東京 第2ビル」です。お越しの際はお間違えのないようご注意ください。. Share with your friends! Facebook. Twitter. Discover other blogs in. Uncategorized category. Student Voice Courses APPLY NOW. LEVEL CHECK. Contact us © 2018 Shibuya Gaigo Gakuin. All Rights Reserved. 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{"id":"hRZMBS9l2MMx","host_id_s":"9l2MMx","sku":"https:\/\/shibuyajp.net\/wp-content\/uploads\/2018\/08\/73ca0f97207fb165f0509c079af4d0b6.pdf","host_s":"shibuyajp.net","host_organization_s":"shibuyajp","url_chars_i":85,"url_protocol_s":"https","url_paths_count_i":4,"url_paths_sxt":["wp-content","uploads","2018","08"],"url_file_name_s":"73ca0f97207fb165f0509c079af4d0b6","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":5,"collection_sxt":["user"],"title":["73ca0f97207fb165f0509c079af4d0b6.pdf"],"author":"Adobe PDF Library 10.0.1","author_sxt":["Adobe PDF Library 10.0.1"],"last_modified":"Fri Mar 23 01:44:29 UTC 2018","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-1590075205152530391,"exact_signature_unique_b":true,"fuzzy_signature_l":-2032229320822311726,"fuzzy_signature_unique_b":true,"wordcount_i":1525,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 16:54:26 UTC 2022","fresh_date_dt":"Sun May 05 09:21:58 UTC 2024","referrer_id_s":"tBr8hu9l2MMx","publisher_t":"Adobe PDF Library 10.0.1","language_s":"ja","size_i":1097702,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735086326727835648,"text_t":"日本語科\n2018 - 2019\n一般財団法人　日本語教育振興協会　認定校\n1 2\n「大量生産的規模ではなく、小さくても家庭的な規模で」というの\nが私たちの教育方針です。当校の経験豊かな講師陣は、長い間この\n精神を忘れずに指導に当たって参りました。\n総合語学学校である渋谷外語学院では、外国語を学ぶ日本人学生と\nの交流ができることが大きな魅力のひとつです。この国際的な環境\nのもと、学生たちが楽しく学ぶと同時に、異なる文化、歴史、ライ\nフスタイルをお互いに理解し合って頂けることを望みます。\nご挨拶 目次\n■渋谷外語学院とは\n■コース案内　\n　全日制コース   ( 長期：留学生ビザ申請 )\n　　　　　　　   ( 短期 )\n　サマーコース\n　日本語能力試験対策コース\n　プライベート・セミプライベートコース\n　出張レッスンコース\n■レベル表と教材\n■オプション\n　課外活動\n　生活（ホームステイ、住居の紹介）\n■入学手続き\n■入学に際しての注意\n■受講生の声\n■アクセス\n3-4\n5 -7\n8\n9\n10\n10\n11-12\n13\n14\n15-17\n18-19\n20-21\n22\n当校の日本語コースで、「聞く」、「話す」、「読む」、「書く」をまん\nべんなく訓練し、特に会話力のスキルアップを最大の目標と定め、\nそのため多くの練習や活動を様々に盛り込んでいます。 \nコースが始まればあなたも会話力がいかに上達したか、ほどなく実\n感していただけるはずです。 \n日本文化が大好きな当校の生徒たちは、世界の色々な国からやって\nきます。是非あなたも私たちと一緒に日本語を勉強する仲間に入り\nませんか？\nあなたと会えることを楽しみにお待ちしています。\nPresident\nT. Mastumoto\n3 4\n■外国語を学ぶ日本人学生と交流ができる\n日本人が外国語を学ぶクラスが同じフロアにあり、身近に触れ合うことができま\nす。習ったことをすぐに試せる理想的な環境が私たちの強みです。\n■実用会話力を第一の目的とし、\n　直接法とコミュニカティブアプローチでの指導法が好評\n会話を中心として学ぶため、すぐに生活で使うことができます。もちろん読む、\n書くなどの練習もしますので、最終的には総合的に日本語力が上がります。\n■様々な国籍の学生が混在する空間\nヨーロッパを中心に世界各国の生徒が集まっています。\n（グラフは 2017 年 4 月現在）\n　　■ヨーロッパ　■北米　　■アジア\n　　■中南米　　　■その他\n■こじんまりとして居心地の良い雰囲気\n   （1 クラス６～ 16 人 )\n■日本生活のサポート\n■文化的教育\n■熱心で優秀な講師陣\n1970 年開設の渋谷外語学院（一般財団法人　日本語教育振興協会の認定\n校）は 45 年以上続く歴史ある学校です。\n学校では日本語を含む 10 か国語の外国語を学ぶことができます。\n渋谷は日本を代表する街の一つで『若者の街』、『カルチャーの街』、\n『ビジネスの街』という様々な側面を持つ魅力的な街です。\n東京の主要都市やベッドタウンへのアクセスも抜群。ショッピングや食事\nはもちろんのこと、映画館やカラオケなどのあらゆる娯楽を体験でき、文\n化の発信地とても有名です。街を歩くだけで様々な刺激を受けます。\n渋谷外語学院は渋谷駅から歩いて７分の抜群の立地です。\n綺麗な桜並木が有名な通りの近くで、都会にありながら日本の四季を感じ\nることもできます。\n学校が終わったら、渋谷の街を歩いて日本の文化も体験しましょう。\n最新のものや多くの人々が集まる渋谷で素敵な出会いがあなたを待ってい\nることでしょう。\n渋谷外語学院とは 渋谷\n15%\n6%\n8%\n11%\n60%\n・Free wi-fi\n・コーヒーマシーン * 有料\n・図書コーナー\n・課外活動あり\n  * 任意参加、p.13 参照\n5 6\n毎日勉強する全日制コースは入門から会話に重点をおいたカリキュラムが特\n徴です。また、ことばの勉強だけではなく、文化体験も多く取り入れ、日本\n語の聞く、話す、読む、書くの 4 つのコミュニケーションスキルを鍛えると\nともに中級以上は活動科目で実習も行います。日本での就職や、日本の大学\n院や専門学校へ入学できるレベルを目指します。\n授業は１期 10 週間で、１月、４月、７月、１０月の年４期制です。\n長期コースは時間をかけ確実に日本語力をつけることに適したコースで\nす。留学生ビザを申請する必要があります。事前の申請が必要なため、申\n込締切日が決まっていますのでご注意ください。\n短期コースは 10 週間と１ヶ月があり、短期間で日本語を学びたい方のた\nめのコースです。大学生や企業人の日本語研修にも適しています。\nまた部分受講も可能ですので、お気軽にご相談ください。\nカレンダー\n長期コース\nコース 学期 2018 2019\n２年\n４月 4 月 4 日 - 2020 年 3 月 4 月 9 日 - 2021 年 3 月\n10 月  10 月 10 日 - 2020 年 9 月  10 月 10 日 - 2021 年 9 月\n1 年\n1 月 1 月 11 日 – 12 月 19 日 1 月 8 日 – 2019 年 12 月\n4 月 4 月 4 日 – 2019 年 3 月 20 日 4 月 4 日 – 2020 年 3 月\n7 月 7 月 3 日 – 2019 年 6 月 25 日 7 月 3 日 – 2020 年 6 月\n10 月 10 月 10 日 – 2019 年 9 月 27 日 10 月 10 日 – 2020 年 9 月\n6 ヶ月\n1 月 1 月 11 日 - 6 月 19 日 1 月 8 日 - 6 月 19 日\n4 月 4 月 4 日 - 9 月 27 日 4 月 9 日 - 9 月 27 日\n7 月 7 月 3 日 - 12 月 19 日 7 月 3 日 - 12 月 19 日\n10 月 10 月 11 日 - 2019 年 3 月 20 日 10 月 10 日 - 2020 年 3 月\n短期コース\nコース 学期 2018 2019\n10 週間\n1 月 1 月 11 日 - 3 月 23 日 1 月 8 日 - 3 月 20 日\n4 月 4 月 4 日 - 6 月 19 日 4 月 9 日 - 6 月 25 日\n7 月 7 月 3 日 - 9 月 27 日 7 月 3 日 - 9 月 27 日\n10 月 10 月 10 日 - 12 月 19 日 10 月 10 日 - 12 月 20 日\n1 ヶ月\n1 月 1 月 11 日 - 2 月 7 日 1 月 8 日 - 2 月 5 日\n4 月 4 月 4 日 - 5 月 8 日 4 月 9 日 - 5 月 14 日\n7 月 7 月 3 日 - 7 月 31 日 7 月 3 日 - 7 月 31 日\n10 月 10 月 10 日 - 11 月 6 日 10 月 10 日 - 11 月 8 日\n全日制コース\n長期 短期\n期間 ２年／１．５年／１年／６ヶ月 １０週間／１ヶ月\nレベル\n（１～１０）\n初級：１・２　初中級：３・４　中級：５・６・７\n中上級：８・９　上級：１０　　　　レベル表は p.11-12 参照\n授業時間 一日 45 分× 4 レッスン\n月 - 金 9:30-13:00 または 13:30-17:00\n入学に際し、レベルチェックを行い、クラスが決定します。\nレベルチェックにより、午前か午後のクラスに分かれます。\n総授業時間 2 　年 :1600 時間 \n1.5 年 :1200 時間\n1　 年 :800 時間\n6 ヵ月 :400 時間\n10 週間：200 時間\n1 ヶ月：80 時間\n課外活動 日本文化体験を主体に、学期毎に設定されています。\n課外活動の詳細については p.13 参照。\nクラス人数 ６〜１６名\n入学申込\n申込詳細は\np.15-17\n入学手続き書類が締切日に揃うよ\nう、お申し込みください。\n短期コースを希望される方は、\nクラス開講日の 1 か月前までに\nお申し込みください。\nまた、稼働中のクラスのレベル\nに適合し、かつ定員に空きがあ\nれば編入することが可能です。\n入学資格 ①原則１６才以上\n②原則として 12 年以上の正規\n学校教育または相当と認められ\nる教育を修了していること。\n③日本語学習歴があること、平\n仮名とカタカナの読み書きでき\nること。\n④大学院進学希望者は本国で学\n士を取得しており、英語能力が\nあること。\n①原則１６才以上\n②入学に際しては、平仮名とカ\nタカナの読み書きができること\nが必要。\n入学を予定されている方で、初\nめて日本語を勉強される場合は、\nコースが始まるまでに準備をし\nておいてください。\n2018 年\n学期 締　切\n7 月期 2018 年 3 月 5 日\n10 月期 2018 年 5 月 26 日\n2019 年\n学期 締　切\n1 月期 2018 年 9 月 9 日\n4 月期 2018 年 11 月 17 日\n7 月期 2019 年 3 月上旬\n10 月期 2019 年 5 月下旬\n授業説明会（全日制）\n学期 2018 2019\n1 月 1 月 10 日（木）　午後 2 時〜 1 月 7 日（月）　午後 2 時〜\n4 月 4 月 3 日（火）　午後 2 時〜 4 月 8 日（月）　午後 2 時〜\n7 月 7 月 2 日（月）　午後 2 時〜 7 月 2 日（火）　午後 2 時〜\n10 月 10 月 9 日（火）　午後 2 時〜 10 月 9 日（水）　午後 2 時〜\n学費\n長期 短期\n1 年 ６ヶ月 10 週間 1 ヶ月\n選考料 20,000 円 なし\n入学金 50,000 円 ( ビザ申請あり )10,800 円 ( ビザ申請なし ) 10,800 円\n授業料 680,000 円 350,000 円\n162,000 円 66,960 円\n部分受講可能\n1 日 3,348 円\n×連続する参加日数\n教材費 レベル表 p.11-12 を参照\n合計 750,000 円 420,000 円 172,800 円 77,760 円\n7 8\n渋谷外語学院のサマーコースでは、少人数クラスで、集中的に目的に\n合った日本語が学べます。午前中は日本語の学習、午後はオプション\nで、日本文化に触れる様々なアクティビティのプログラムが用意され\nています。興味のあるものに参加し、体験することができます。 世界\n各国からの参加者と一緒に、日本の夏を楽しく過ごしましょう！サマー\nコースには３つの目的別のコースがあります。\n1. 会話コース：\n日本滞在中に出会う様々なシーンで、すぐに使える会話を練習します。 \nパーティーで、買い物で、レストランで、さっそく使ってみたくなる\n表現を学びましょう。\n2.JLPT コース：\n日本語能力試験、N1、N2 の受験を目標とする方のための対策クラス\nです。 短期間で集中的に、試験内容のすべての分野を実践的に練習し\nながら苦手を克服するとともに、実力を養成します。\nサマーコース\n会話 JLPT\n日程 2018 年：8 月 6 日 - 8 月 17 日　2019 年：8 月 5 日 - 8 月 16 日\nレベル 入門 & 初級・初中級 中級\nN1・N2\n授業時間\n課外活動\n月〜金 9:30-13:00　全 30h の授業\n+ 文化体験や課外活動（平日午後もしくは休日）\n課外活動は任意参加。\nクラス人数 ２〜１０名\n入学申込 申込詳細は p.17 を参照。\n入学準備\n入学の条件は、平仮名とカタカナの読み書きができること。\n入学を予定されている方で、初めて日本語を勉強される場合は、\nコースが始まるまでに準備をしておいてください。\nクラスレベルは事前に受けていただくレベルチェックで決まります。\n学費\n会話・JLPT\n入学金 なし\n授業料 64,800 円\n教材費 実費\n合計 64,800 円\n休校日（全日制）\n2018 2019\n春 3 月 24 日 - 4 月 3 日 3 月 22 日 - 4 月 8 日\nゴールデン\nウィーク 4 月 28 日 - 5 月 6 日 4 月 29 日 - 5 月 6 日\n夏 6 月 20 日 - 7 月 2 日8 月 4 日 - 8 月 19 日\n6 月 26 日 - 7 月 2 日\n8 月 5 日 - 8 月 16 日\n秋 9 月 28 日 - 10 月 9 日 9 月 30 日 - 10 月 9 日\n冬 12 月 20 日 - 2019 年 1 月 12 月 23 日 - 2020 年 1 月\n9 10\n渋谷外語学院の JLPT 対策は前期 10 日間と後期 10 日間の 2 期にわたっ\nての日曜日開講のクラスとサマークラスをご用意しています。7 月ま\nたは 12 月実施の試験合格を目指し、それぞれのご計画に合わせて受\n講していただけます。\n前期はより基本的な内容を含んだプログラムです。「文字・語彙」「文法」\nの分野では間違いやすい項目を中心に説明を受け、理解を深めてから\n練習問題をします。「聴解」「読解」は比較的簡単なものからはじめて\nJLPT の形式に慣れるための練習を行います。\n後期はより実践的な演習問題を中心としたプログラムです。「文字・語\n彙」「文法」は JLPT 形式の練習を数多くこなしながら知識の不足を補\nい、試験範囲を網羅していきます。「聴解」「読解」は数多くの問題を\nこなすことで時間配分、解答テクニックなど「合格するための練習」\nに特化してレベルアップを目指します。\n前期、後期とも毎回練習問題が宿題として出され、次の授業でチェッ\nクしますので、時間を有効に使って学習することが可能です。前期ま\nたは後期のみの受講でも十分にレベルアップを感じていただけますが、\n2 期連続、半年をかけて受講していただくことにより、さらなる手ご\nたえ、合格への自信をつけていただけるはずです。 \nプライベート＆セミプライベートコースでは語学習得の中でその方が\n特に必要とされている部分に焦点を当てながらレッスンを行います。\n個々のペースに合わせながら、個別のカリキュラム作りと教師の細や\nかなケアにより、より速い学習効果を望むことができます。 詳しくは\n学校にご相談ください。レッスン時間やクラスタイプを選択肢よりお\n選びいただけます。（1 レッスン 45 分〜）\nセミプライベートレッスンはお知り合いの方同士でお申込みください。\n会社やご家庭など、ご自分の学びやすい環境での学習を希望される場\n合は講師が出張してレッスンを行います。外国人従業員のためのビジ\nネス日本語トレーニングなどや、カリキュラムはご相談に応じてアレ\nンジいたします。詳しくはお問合せください。\n日本語能力試験対策コース プライベート・セミプライベートコース\n出張レッスンコース\n日曜クラス サマークラス\n期間 2018 年：4 月 15 日 - 6 月 24 日\n　　　　　*4 月 29 日はナシ\n　　　　　9 月 30 日 - 12 月 2 日\n2019 年：4 月 21 日 - 6 月 30 日\n　　　　　*4 月 29 日はナシ\n　　　　　9月15日 - 11月24日\n2018 年：8 月 6 日 - 8 月 17 日\n2019 年：8 月 5 日 - 8 月 16 日\nレベル N1・N2\n授業時間 日曜クラス　 : 1 期 10:00-15:30 計 45 時間 (1 時間休憩あり )\nサマークラス : 9:00-13:00   計 30 時間\n教材 主教材：『パターン別徹底ドリル日本語能力試験 』( アルク）N1 \/N2\n副教材：前期『ドリル＆ドリル』\n　　　　後期『日本語能力試験問題集 N1\/N2 聴解スピードマスター』\nクラス人数 ２〜１０名\n入学申込 申込詳細は p.17 を参照。\n学費\n日曜クラス サマークラス\n入学金 10,800 円 なし\n授業料 86,400 円 64,800 円\n教材費 実費 実費\n合計 97,200 円 64,800 円\n個人・セミプライベート（1 名分の値段）\n開始時期：随時　レベル：全レベル　１期 10 レッスン（有効期限 6 ヶ月）\n入学金 10,800 円\n１名 ２名 ３名\n45 分 X 10 45,360 円 28,080 円 21,600 円\n60 分 X 10 60,480 円 37,368 円 28,728 円\n90 分 X 10 88,460 円 54,760 円 42,120 円\n出張レッスン\n開始時期：随時　レベル：全レベル　授業時間等はアレンジ可能。\n企業一般会話レッスン 個人レッスン\n入学金 なし 10,800 円\n60 分当たりの料金 10,800 円 – 8,640 円 - \n教材費 実費\n11 12\nレベル表\nレベル 到達目標 漢字数 語彙数 JLPT CEFR 教材例\nレベル 1\n初級 1\n挨拶、自分や他人の紹介ができ、日常生活で多用される基本表現を理解し、　それらを用いて、\n短い簡単なコミュニケーションをとることができる。短い会話文や 400　字程度の文を読ん\nで理解することができる。自分の生活などの身近な話題についての 200　字程度の作文が書\nける。　\n約 70 字 約 500 語 A1\n『みんなの日本語　初級 I』\n1~13 課\n2,700 円\nレベル 2\n初級 2\n家族、友人といった個人的範囲に加え、駅、店、病院といった公的な場　面においても、目\n的を達するために質問、依頼などが表現でき、指示や説明などを理解で　き、コミュニケー\nションをとることができる。具体的な話題についての 400~500 字程度　の文章が理解でき\nる。身近な話題についてのさまざまな形式での 250 字程度の作文を書く　ことができる。\n約 150 字 ~1000 語 N5 A1\n『みんなの日本語　初級 I』\n14~25 課　\n2,700 円\nレベル 3\n初中級 1\n日常生活の身近な場面の話やニュースの内容をある程度理解し、説明を　求めたり自分の状\n況や物事の状態を表現したりして情報交換に応ずることができる。500　~600 字程度の具\n体的な話題についての文章を理解し、意見や感想を表明することができ　る。趣味や身の回\nりの出来事など関心を持つ話題について具体例、経験、印象などに言及　しながらまとまっ\nた内容の作文を書くことができる。\n約 230 字 ~1500 語 N5 A2\n『みんなの日本語　初級 II』\n26~37 課\n2,700 円\nレベル４\n初中級 2\nより自然な速さ、話し方の場合でも内容を理解し、適切に受け答え　ることができる。\n600~800 字程度の具体的な話題についてのテキストを理解し、意見や　感想を表明するこ\nとができる。場面や相手に応じてフォーマル、カジュアルのスタイルを　使い分けて会話が\nできる。身の回りの関心事に加え、過去の経験や、将来の予定などのにつ　いて目的や理由、\n意見を述べながらまとまりのある作文を書くことができる。\n約 300 字 ~2000 語 N4 A2\n『みんなの日本語　初級 II』\n38~50 課\n2,700 円\nレベル５\n中級 1\n仕事、学校、街頭などの普段遭遇するような場面で、一般の日本人の標　準的な話し方であ\nれば要点を理解してコミュニケーションをとることができる。場面に応　じて初歩的な敬語\nを交えるなど、初級文法を踏まえた会話ができる。自分の専門分野や関　心が及ぶ範囲の話\n題についてのテキストの内容を十分理解でき、それについての意見、感　想を口頭や作文で\n表明することができる。\n約 440 字 ~2500 語 N4 B1\n『みんなの日本語　中級 I』\n1~4 課\n3,024 円\nレベル６\n中級 2\nあまり複雑ではないテーマについて、一般の日本人が言語コントロール　なしに話すことを\n十分に理解し、質問、応答ができる。調査やインタビューなどの結果を　まとめて発表した\nり、話し合ったりすることができる。ある程度の長さのテキストの内容　理解に加え、広告、\n説明書、パンフレットなどの資料から必要な情報を探し出して課題を遂　行することができ\nる。接続表現などを用いて、段落ごとに脈絡のあるまとまった文章を書　くことができる。\n約 580 字 ~3000 語 N3 B1\n『みんなの日本語　中級 I』\n5~8 課\n3,024 円\nレベル 7\n中級３\n身近な話題についての会話なら、事前の準備なしで参加することができ　る。日本国内での\n生活、旅行などで起こりうるたいていの事態に対処することができる。　社会的な問題につ\nいてのテキストを読んで筆者の視点や意見を理解することができる。また　それに対しての\n賛成、反対の自分の意見を理由を説明しながら口頭、または文章で表現す　ることができる。\n約 720 字 ~3500 語 N3 B1\n『みんなの日本語　中級 I』\n9~12 課\n3,024 円\nレベル８\n中上級 1\n言葉や表現を探しながらではあっても、長い休止なしに話し続けること　ができる。異なっ\nた意見を持つ相手と論理的で明瞭な理由を示しながら討論をすることが　できる。専門分野\nや社会的テーマについて、具体的文章のみならず、多少複雑で抽象的な　テキストの内容も\n理解することができる。多様な話題についてさまざまな情報や議論をま　とめて首尾一貫し\nた主張のある文章を書くことができる。\n約 860 字 ~4500 語 N2 B2\n『中級を学ぼう　中級中期』\n1~4 課\n2,592 円\nレベル９\n中上級 2 ~1000 字 ~5500 語 N2 B2\n『中級を学ぼう　中級中期』\n5 〜 8 課\n2,592 円\nレベル 10\n上級\n一般の日本人とお互いに緊張せずにやり取りできるくらい流暢かつ自然　な会話ができる。\nさまざまなテーマについての複雑な内容を明瞭かつ詳細に述べることが　できる。未習の語\n彙の意味を推測しながら比較的広範なテーマについてのテキストを独力　で読み進め、理解\nすることができる。\n約 1300 字 ~7000 語 N2 B2\n『中級を学ぼう　中級中期』\n9、10 課\nその他 N2 相当教材\n2,592 円\n13 14\nオプション 生活（ホームステイ、住居の紹介）\n課外活動\n日本語科受講生は課外活動にオプションで参加することができます。\n勉強以外にも日本の文化に触れ、体験するチャンスです。参加は任意で別\n途参加費がかかります。１学期毎に約８〜１０ほどの活動が企画されます。\n課外活動例：茶道、華道、書道、風呂敷包み体験、風鈴絵付け体験などの\n日本文化体験。歌舞伎鑑賞、相撲観戦。神社や寺院、浅草や鎌倉のような\n伝統的な街並み、大江戸博物館、お台場などの探訪。\n■ホームステイ\nホームステイプログラムは日本人の家庭で日本人のように生活をするまたと\nない機会です。また、ホストファミリーと日本語で会話をすれば、日本語を\nブラッシュアップすることもできるでしょう。\n・ 一泊当たり 4,536 円（朝食・夕食付き）\n・ 手配料：19,440 円\n・ 一家庭当たり 3 ヶ月までの滞在\n■ゲストハウス・アパート\nゲストハウスまたアパートは、ご自身でお探しください。不動産会社は下記\nの例以外にもあります。\n【 BORDERLESS HOUSE 】（ 渋谷外語学院特別価格で提供！）\nURL : http:\/\/www.borderless-house.jp\/partner\/languageschool\/\n・1 ヶ月当たり 39,000 円から日本人と外国人と一緒に暮らせる\n・初期費用は 1 回だけ 30,000 円\n・渋谷外語学院の学生からの評判：「Borderless House に住むとコミュ\nニケーションスキルがもっと上達し、大勢の日本人の友達ができた！」\n・渋谷外語学院の学生向けのスペシャルオファーあり\n【 TOKYO STAY 】\nURL : http:\/\/www.tokyostay.co.jp\/\n・山手線から歩いて 5 分以内の小規模で国際的なゲストハウス（2 － 6\n部屋）もしくはプライベートアパートで暮らせます。\n・一ヶ月当たり 57,000 円（ゲストハウス）、128,000 円（アパート）から。\n家具、光熱費、インターネット付きで、敷金は返金可能です。\n・渋谷外語学院の学生だと、毎月 3,000 円の割引もらえます！お申し込\nみの際に、ぜひ当校の名前を忘れないでください！\n・TOKYO STAYの国際的なスタッフがいつも近くにいてお手伝いします。\nSPEAKEASY という色々な国からお客さんが来るバーも運営しており、\n住居者は毎月 2 ドリンク無料で飲めます！\n【 TIERRA 】\nURL : http:\/\/www.tierra.jp\/\n・渋谷外語学院がおすすめする英語・イタリア語が対応ができる不動産\n会社です\n【 GaijinPot 】\nURL：http:\/\/apartments.gaijinpot.com\/en\/rent\/listing\n15 16\n5. 東京入国管理局より在留資格認定証明書 (COE) 交付後、学\n費請求書発行\n※申請内容によっては在留資格認定証明書 (COE) が交付され\nない場合があります。\n6. 学費納入確認後、在留資格認定証明書を申込者に送付\n学費は銀行振込または現金でご納入ください。\n7. 在留資格認定証明書 (COE) を持って、現地の日本大使館ま\nたは領事館で留学ビザ申請\n※何らかの理由で査証の発行を受けられなかった場合、その\n　証明書を付けて学校に必ず連絡してください。納入済みの\n　学費（入学金、授業料、その他）を返金する際に必要とな\n　ります。ただし、選考料 20,000 円は返金されません。\n8. 大使館もしくは領事館に申請後、VISA が発行されます。\n3. 学校スタッフが東京入国管理局に代理申請\n4. 在留資格認定の結果発表は、申請後約２ヶ月半〜３ヶ月か\nかります。※入国管理局から関連書類の追加提出を求められる\n場合があります\n入学手続き\n長期滞在希望\n（長期コース・在留資格認定証明書(COE)交付申請）\n１．渋谷外語学院ウェブサイトのオンライン申込フォームから\n下記内容を提出。\n・入学申込 ・就学理由 ・過去の出入国歴 \/ 在日親族 ・職歴 \/ 経費支弁者\n・学歴 \/ 日本語学習歴 ・オンラインレベルチェック\nhttp:\/\/shibuya-gaigo.com\/sls2\/jp\/app\/index.html\n2. 選考料支払（20,000 円）\nPaypal 決済（推奨）、銀行振込（手数料送金人負担）または現\n金でご納入ください。納入後の選考料は返金できません。\n日本に入国。\n学校生活がスタートします。\n2018 年\n学期 締　切\n7 月期 2018 年 3 月 5 日\n10 月期 2018 年 5 月 26 日\n2019 年\n学期 締　切\n1 月期 2017 年 9 月 9 日\n4 月期 2017 年 11 月 17 日\n7 月期 2019 年 3 月上旬\n10 月期 2019 年 5 月下旬\n留学ビザ申請のためには、まず在留資格認\n定証明書 (COE) を取得する必要があります。\nこれは渋谷外語学院が学生のかわりに東京\n入国管理局で申請します。\n交付された証明書は本国の日本大使館または\n領事館で留学ビザを申請する際に必要です。\n申込締切期限が決まっています。\n右記期日までに関連書類の提出と選考料の\n支払いを済ませてください。\n申込手順\n17 18\n短期滞在希望（全日制長期以外のコース）\n申込期限\n短期コースを希望される方は、クラス開講日の 1 か月前まで\nにお申し込みください。ただし、初級者で 50 時間未満の学習\n経験がある方で、稼働中のクラスのレベルに適合し、かつ定員\nに空きがあれば編入することが可能です。\n申込方法\n１. 渋谷外語学院ウェブサイトのオンライン申込フォームおよ\n　  び、関連フォームから下記内容を提出。\n・入学申込　・オンラインレベルチェック\nhttp:\/\/shibuya-gaigo.com\/sls2\/jp\/app\/short_term.html\n2. 送られた入学願書とレベルチェックテストを審査した後、\n　 こちらよりクラス開講 2 週間前 までに請求書と参加クラス\n　 の詳しいスケジュールをお届けいたします。\n \n　 学費支払\n 　学費は銀行振込または現金でご納入ください。\n　 Flywire 使えます。\n２\n3\n入学に際しての注意\n①初級クラス対象者でも入学前にひらがな、カタカナの読み書きを\n　できるように準備してください。\n②オンラインレベルチェックで事前にお知らせしたクラスレベル\n　は、入学後の会話チェックや授業進行具合、個々の学習理解度等\n　で変更になる場合があります。\n③全日制クラスはレベルにより開講時間が決められています。受講\n　生が選択することはできませんのでご注意ください。\n④特定レベルについて定員に満たない場合は、開講が中止となるこ\n　とがあります。\n⑤学費の納入は銀行振込または直接学校の受付でご納入いただけま\n　す。銀行送金手数料はご負担ください。\n⑥短期コースの方、特にホームステイを希望される方は来日前に必\n　ず旅行保険への加入をお願いします。長期（留学ビザ対象）コー\n　スの方は、来日後所定の保険に加入していただきます。\n返金規定【長期（VISA 申請）・短期共通】\n①本人が申込みの意思を学校に伝え、学校が承諾したメール（文書）\n　をだした日から 8 日間であればクーリングオフが可能です。（日\n　本時間が基準）\n②学費総額 50,000 円未満かつ受講期間 2 ヶ月以内のコースについ\n　ては中途解約による返金制度の対象となりません。\n③返金に伴う、銀行振込手数料は受講生負担です。\n返金規定【長期（VISA 申請）】p.18-19\n①ビザ申請書類提出後のキャンセルの場合\n　「在留資格認定証明書（COE）」の交付、不交付に関わらず、選考\n　料は返金しません。\n②入国管理局から「在留資格認定証明書（COE）」交付後で授業料\n　支払い前のキャンセルの場合\n　選考料は返金しません。また開講 2 週間を切っている場合は別途\n　キャンセル料 50,000 円を申し受けます。\n③入国管理局から「在留資格認定証明書（COE）」交付後で授業料\n　支払い後、入国前かつ授業開始前のキャンセルの場合\n　理由の如何に関わらず選考料および入学金は返金しません。\n　授業料は在留資格認定証明書および入学許可証と引き換えの上、\n　全額返金いたします。\n※返金規定【長期（VISA 申請）】p.19 に続きます\n19 20\n受講生の声\n当校を選んだ理由はなんですか？\nI visited a few schools, I found Shibuya Gaigo Gakuen to be \nmore friendly atmosphere and relaxed attitude.\n学校のどんなところが好きですか？\nI like it, I have been here for 2 years. The teachers get to \nknow your good & bad points and work on them accordingly. \nAll the teachers want you too do well.\n今まで受けた授業で好きなレッスンは何ですか？\nReading is my favorite lesson, I can re-read it till I understand \nit.\n学校の感じはどうですか？　\nThe schools facility is good, reception \nstaffs are very friendly they have helped \nme a few times.\nTheresia さん　国籍：Sweden\n全日制コース初級クラス修了\nPeter さん　国籍：Scotland\n全日制コース初級クラス修了\n当校を選んだ理由はなんですか？　\nI was looking for a language school that was located in \nShibuya, as it is my favourite place in Tokyo. I chose to study \nat Shibuya Language School because I had heard a lot of good \nthings about it.\n学校のどんなところが好きですか？\nIt is a small school but I like the atmosphere of the school, \neveryone is friendly. I have been studying here for 6 months \nand I have made so many friends and my Japanese has \nimproved a lot. The teachers are very supporting, they always \nwant you to do well and help you out.\n今まで受けた授業で好きなレッスンは何ですか？　\nI enjoyed studying kanji the most, it is not only fun but it is \nalso very necessary for living in Japan. \nI want to continue to study kanji, \nfor example I want to be able to read books \nin Japanese.\n返金規定【長期（VISA 申請）】p.18 のつづき\n④入国ビザを取得した後，来日以前に入学を辞退した場合\n　理由の如何に関わらず選考料および入学金は返金しません。授業\n　料は入国ビザが未使用であることが当校職員によって確認された\n　後、在留資格認定証明書および入学許可証と引き換えの上、全額\n　返金いたします。\n⑤入国ビザを取得し来日後に入学を辞退した場合\n　理由の如何に関わらず選考料および入学金は返金しません。授業\n　料は留学ビザが無効になったことが当校職員によって確認された\n　後、入学許可証と引き換えの上、全額返金いたします。\n⑥日本大使館、領事館によってビザ発給が拒否された場合\n　選考料は返金しません。入学金および授業料はビザ発給拒否が当\n　校職員によって確認されたた後、在留資格認定証明書および入学\n　許可証と引き換えの上、全額を返金いたします。\n⑦授業開始後のキャンセルの場合\n　選考料および入学金は返金しません。キャンセル料として\n　50,000 円を上限に残りのレッスン分の授業料の 20% に相当す\n　る金額をキャンセル料として申し受けます。留学ビザが無効になっ　\n　たことが当校職員によって確認された後、残金を返金致します。\n返金規定【短期】\n①授業開始 2 週間以上前のキャンセルの場合\n　納入学費を全額返金致します。\n②授業開始前 2 週間以内のキャンセルの場合\n　キャンセル料として 15,000 円申し受け、残金を返金致します。\n③授業開始日以降のキャンセルの場合\n　キャンセル料として 50,000 円を上限に入学金＋残りのレッスン\n　分の授業料の 20％に相当する金額をキャンセル料として申し受\n　け、残金を返金致します。\"\nご自分の受付状況についてお問い合わせを希望される場合は、日本\n時間の月〜金9:00〜21:00の間に学校担当者までご連絡ください。\n21 22\nアクセス\n成田空港から渋谷へ\n行き先 所要時間 運賃\nリムジンバス セルリアンタワー東急ホテル 75-105 分 3,100 円\n成田エクスプレス JR 渋谷駅 80 分 3,190 円\n京成スカイライナー\n+ 山手線 JR 渋谷駅 80 分 2,659 円\n在来線 JR\/Metro 渋谷駅 100-120 分 1,210 〜 1,473 円\n羽田空港から渋谷へ\n行き先 所要時間 運賃\nリムジンバス セルリアンタワー東急ホテル 35-65 分 1,030 円\n東京モノレール\n+ 山手線 JR 渋谷駅 35-45 分 677 円\n京急線 + 山手線 JR 渋谷駅 35-45 分 572 円\n当校を選んだ理由はなんですか？\n渋谷外語学院を選んだ理由は、友達がすすめてくれたことと学費が\n他の学校より安く、入学手続きも簡単だったためです。\n渋谷という立地や、駅から近くかった事も私にとって大変便利だと\n思いました。\n学校のどんなところが好きですか？\nやはり、色々な国の人と交流出来ること。そして学校雰囲気が非常\nに楽しいことや、事務スタッフが親切で生徒達の悩みを心広く聞い\nてくれることも良いと思います。\n今まで受けた授業で好きなレッスンは何ですか？\n「会話」、「読む」、学外活動などが良かったです。その理由は、そういっ\nた勉強や体験のお陰で非常に短い時間で日本語が上達し、更に日本\n人への理解が深まったと思ったからです。\n学校の感じはどうですか？　\n事務スタッフも含めて皆が仲良く、学校全体が\n家族のような雰囲気です。また、学校で行われる\n行事や遠足を通して日本人の友達が増えた事も\n学校の魅力の一つです。\nLabib さん　国籍：Egypt\n全日制コース上級クラス修了\nTamara さん　国籍：Germany\n全日制コース中級クラス修了\n当校を選んだ理由はなんですか？　\nRecommendation.\n学校のどんなところが好きですか？\nAtmosphere,close to station,good schedule and teachers \nlessons.\n今まで受けた授業で好きなレッスンは何ですか？　\nGrammar:Always something to learn, good and easy \nexplanations,many examples and how to use in different \nsituations.\n学校の感じはどうですか？　\nEverything, there what you need \nfor a normal, comfortable school life.\nshibuya-gaigo.com\/sls2\/\nT E L :03-3461-8854\n　　　  海外から :+81-3-3461-8854\nF A X :03-3463-4901\n　　　  海外から :+81-3-3463-4901\nE-mail:nihongo@shibuya-gaigo.com\n〒 150-0031  　渋谷区桜丘町 8-18 プレム桜丘ビル 4 階\n受付時間（年末年始・祝日を除く）\n平日 10:00-20:00 土曜日 10:00-18:00 日曜日 10:00-16:00\n□ 案内記載事項有効期限は 2019 年 12 月 31 日までとなり、内容はやむを得ず変更となる\n 　場合があります。予めご了承ください。\n工事中\n"}
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asahicom jp and wp content themes assets img logo w 01 svg ＆w asahi com ico search user guest png badge heart LIKE comment COMMENT mile bg sns fb tw ig yt 02 ＆ m ＆M travel ＆Travel time ＆Time human bookmark uploads 2019 05 0523 re jpg 日本語教師：福島真衣さん（28歳） 03 04 06 07 08 twitterシェア facebookシェア line LINEでシェア hatena このエントリーをはてなブックマークに追加 750 0725 化粧品会社勤務／横山優さん（35歳） i 0704 パーソナルファッションアドバイザー：モトコさん（39歳） DSC 1079 旅行系SNS運営会社:堀真菜実さん（33歳） 1130 引っ越しスタッフ：大谷明奈さん（23歳） アロマセラピスト：湯澤紗世子さん（36歳） data 2022 eye031 eye 031 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” 967f1782eff2941bc6ece6f0beb1b1a9 967f 1782eff 2941bc 6ece 6f 0beb 1b 1a 9 まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet ba8f917ee7123fad357ba9b8c4a32d6c ba 8f 917ee 7123fad 357ba 9b 8c 4a 32d 6c 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 08a1bee912777ead2fbfc80b46b32292 08a 1bee 912777ead 2fbfc 80b 46b 32292 「何を着るか」にマンネリを感じたら、冒険しませんか？ EYE L1230304 L 1230304 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき bcc8bb0048ff6549f30e4f867752d6a5 bcc 8bb 0048ff 6549f 30e 4f 867752d 6a 5 スペインから父へ。直接その手を握る代わりに 255 5680 2 〈255〉50歳で選んだ別居婚「第二幕」 8fc46c88600a6570b8deddfa41ebf978 8fc 46c 88600a 6570b 8deddfa 41ebf 978 Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 6b50c0fffa5bdf413e677626e7da782d 6b 50c 0fffa 5bdf 413e 677626e 7da 782d ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを DSC5449 5449 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った SANJI AND SPOON final animation 猫に学べ。by サジとスプン（飼い主・三宅和歌子さん） 60d4af5fa755cfac59f4dd7726706631 60d 4af 5fa 755cfac 59f 4dd 7726706631 NY発　子どもたちの夏の過ごし方 256 5874 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 051fe2b145281cf904496fa5388fcc0f 051fe 2b 145281cf 904496fa 5388fcc 0f 毎日食べたくなる！　花も酵母も慈しむ店主の軽快なごほうび感／ベーグルスタンダード footer ＆Human corporate 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PAGE","ログアウト","新規登録","","","","","","","","","","こちら"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:05:22 UTC 2022","fresh_date_dt":"Wed Jun 08 17:05:22 UTC 2022","language_s":"ja","size_i":143310,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":0,"references_internal_i":0,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087014055772160,"text_t":"日本語教師：福島真衣さん（28歳） | 朝日新聞デジタルマガジン＆[and]. TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 東京ではたらく 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT 東京ではたらく 連載をフォローする 日本語教師：福島真衣さん（28歳）. WORK 2019.05.23. Y.K. 小林百合子 ライター・編集者. 1980年兵庫県生まれ。出版社勤務を経て独立。山岳や自然、動物、旅などにまつわる雑誌、書籍の編集 …. K.N. 野川かさね 写真家. 1977年神奈川県生まれ。山や自然の写真を中心に作品を発表する。クリエイティブユニット〈kvin …. 職業：日本語教師. 勤務地：東京都千代田区. 仕事歴：２年目. 勤務時間：曜日でまちまち. 休日：土・日. 　　　　◇. 英語、中国語、韓国語。さまざまな言語が飛び交うクラスルーム。扉が開き、「せんせい」が入ってくるとおしゃべりはピタリとやみ、ここから話されるのは日本語のみ。たどたどしい部分はあるけれど、覚えたての単語を一生懸命に組み合わせてコミュニケーションを取ろうとする生徒たちの姿を見ていると、学生時代にしばらく通ったアメリカの語学学校時代の記憶がよみがえってくる。母国語以外の言語を習得するのはけっして簡単なことじゃない。. このクラスの「せんせい」こと、福島真衣さんは28歳。東京・千代田区にある九段日本語学院で日本語教師として働いて２年になる。受け持つクラスは初級者向けの会話中心のクラスと、話す、聞く、書く、読むという語学における４技能すべてを教える中級者向けのクラス。生徒は北欧やヨーロッパ諸国を中心に、韓国、中国などアジア圏からの留学生も多い。. 「年齢や目的は本当にさまざまで、日本での大学進学や就職を目指している学生もいれば、旦那さんの転勤についてきた奥様たちもいらっしゃいますし、ちょっとだけ日本語を勉強してみたいという短期留学の人も。夏休みのシーズンになると学生さんがぐっと増えてにぎやかになりますね」. 漢字について学ぶクラス。会話に困って英語で質問する生徒がいても、福島さんはやさしく日本語での会話を促していた. 基本的には毎日授業を受け持つが、半日は授業を担当せず、学内でテストの採点や授業用の教材を用意する。クラスによっては学校共通のテキストが用意されているものもあるが、テキスト以外で学生の理解の手助けとなる教材は自分で作成するというから驚く。. 「授業中は原則、日本語以外の言語を使いませんので、文法ももちろん日本語で教えます。初級クラスではまだ日本語をそこまで理解できない生徒が多いので、教材にはイラストを多く使ったりして理解度を高められるよう工夫しています。イラストや写真探しも自分でやるので大変ですが、ベテランの先生にアドバイスをもらったりしてなんとかやっています」. ちなみにこの日準備していたのは、中級クラス向けのプリント。「～しておけばよかった」という、英語文法でいうところの「仮定法過去」（後悔）の表現を学ぶ回なのだが、この微妙なニュアンスを教えるのがなかなか難しい。. 「表現が複雑になればなるほど、言葉で説明するハードルは当然高くなります。文法の構造を解説することも重要ですが、まず大切なのは生徒たちにその言い回しのニュアンスを理解してもらうこと。そのためにできるだけ日常生活でよくあるようなシチュエーションを設定して例文を作ったり、クラスで身ぶり手ぶりを交えて会話を再現したりしています。熱が入りすぎるとひとり寸劇みたいになって、生徒たちに笑われることもあるんですけど（笑）」. 手作りの教材にはできるだけ日常でよくあるシチュエーションを盛り込んでいる。会話のシーンを想像しやすいよう、イラストも使う. この日福島さんが用意した例文は、「もっとバイトのシフトを少なくすればよかった……」「別の味のパスタを注文すればよかった……」など、確かに日々の生活でよくありそうなシーンを想定したものばかり。例文の横には、シフトの入れすぎでクタクタになったバイトちゃんのイラストがある。. 「ベテランの先生になると、この例文の引き出しがものすごく多いんです。私なんてまだまだで……。ちょっとポイントがずれていると、どれだけ説明しても生徒たちの顔に「？」という表情が出るんです。その時は申し訳ないやら悔しいやらで、もっと頑張らなくちゃと思います。反対に、用意した例文を見て生徒たちが「うん、うん」という顔をしていると、ものすごくうれしくなりますね」. 幼稚園から高校まで、将来の目標はずっと「先生になること」だった。「ずっといい先生に恵まれてきたから、自然と憧れるようになったんだと思う」と振り返る。そしてもうひとつ、物心つく前から抱いていた夢が「世界一周」だったというから、なんともスケールの大きい子どもだ。. 「小学校低学年の頃、将来の夢を描きましょうという授業があったんですけど、そこで私、地球の上に自分が乗っかっている絵を描いたんです。何に影響されたのかは覚えていないんですけど、その夢もずっとどこかに持っていましたね」. 授業がない時間帯は教員控室で教材作り。イラストを切りとってプリントに貼るなど、丁寧な作業. 先に芽を出したのは、「世界一周」の夢だった。地元を離れ、長崎の大学に通っていた３年生の終わり。街角でピースボート（船で世界一周をするツアー）の広告を見かけ、「これこそ私のやりたいことだ！」と確信。１年の休学届を出し、子供の頃からの夢をかなえることにした。. 「日本を出発して、ベトナム、シンガポール、インド、トルコ、ヨーロッパ、エジプト、南米と、３カ月間かけて世界をめぐる旅でした。本来は高額なツアーですが、ボランティアスタッフとしてポスター貼りなどを手伝うと手伝った分だけ割引になる制度があると知って、それを利用して全額割引にして無料で参加しました。部屋はもちろん一番安い４人部屋。同世代はほとんどがひとり参加だったので、友達もたくさんできました。忘れられない体験でしたね」. ツアーに参加する前には、語学を磨くために２カ月間、オーストラリアとニュージーランドに語学留学もした。それまで知らなかった世界に出会うのは刺激的で、「いつかは海外で働いてみたい」という新しい夢がまたひとつできた。. 「大学に復学してからは、就職活動をしながら日本語教師を養成するために学校に通って資格を取りました。小さい頃から憧れていた教師と、海外に関わる仕事、そのふたつの要素をドッキングさせたら、自然と日本語教師という選択肢が出てきたという感じで」. 「これは教材に使えそう」と思ったイラスト素材は日頃からストックしておく。先輩教師からアイデアを分けてもらうこともしばしば. が、大学卒業後に福島さんが選んだのは、意外な道だった。. 「いろいろな企業から委託を受けて、化粧品や健康食品の営業をするコールセンターに就職したんです。海外と関われる仕事に就きたいという気持ちはもちろんありましたが、自分は一度も仕事というものをしたことがない身。まずは日本の会社で『働く』とはどういうことかを知ろうと思って。最初はしばらく東京で働いて、その後は福岡で２年ほど勤務しました。でもやっぱりまた新たにやりたいことが出てくるんですよね（笑）。それで上司に『私、やっぱり海外で働きたいので辞めます！』ってついに伝えたんです」. ここからが意外な展開。なんとその会社にはフィリピンのセブ島にも支社があったのだ。. 「私にとっては夢みたいな話でしたけど、海しかないようなところなので、誰も転勤したがらなかったみたいで。『それなら私、行きます！』って、すぐに転勤させてもらいました。憧れの海外暮らしは本当に楽しかったですし、日本に強い憧れを持っている現地の人も多くて、少し日本語を教えたりもしていました。そんな風に過ごしてたら、やっぱり日本語教師っていいなと思い始めて」. １年半ほどセブ島で働いた後、福島さんは心を決めた。2017年の９月に帰国し、大みそかには東京へ。家も仕事も何も決まらないままの上京だったが、「日本語教師として働く」ということだけは決めていた。. 授業前に生徒たちとおしゃべり。「休日に行くおすすめスポット」など、東京案内を買って出ることも. 「2018年の元旦から東京に住む！と心に決めていたんです。それくらいしないと飛び込めないかなと思って（笑）。しばらくは友人の家に居候をさせてもらって、知人の紹介で学習塾の講師をしながら日本語教師の職を探しました」. 上京して半年ほど経った頃、日本語教師をしている友人から日本語教師向けの講座があると教えてもらい、参加することにした。それを開催していたのが今勤めている日本語学校だった。. 「大学４年生の頃に資格を取ったきり、４年ほどブランクがありましたから、それを埋めるつもりで受講したんです。運良く２カ月の講座が終わったタイミングで講師枠に空きが出て、このままここで働きませんかと声をかけていただいて。本当にラッキーでした」. 教壇に立ち始めてすぐの頃は生徒の顔になかなか納得の表情が浮かばず、「この教え方でいいのだろうか？」と悩むことも多かった。それでも仕事を続けてこられたのは、生徒たちのおかげだ。. 「楽しいな、うれしいなと思うのは、当時も今も生徒たちが『せんせい！』と声をかけてくれる瞬間。私も海外で生活したことがあるので想像できるのですが、みんな慣れない外国での暮らしで、少なからず心細い思いをしているはずです。私は教師ですが、それでも日頃から顔を合わせて、気兼ねなくおしゃべりができる存在になれていたら、それはすごくうれしいことだなと思うんです」. 手製の漢字書き取りクイズカード。生徒は1枚ずつカードを引き、ひらがなの部分の漢字をホワイトボードに書いていくルール. できることなら語学を習得するだけでなく、日本での楽しい思い出もたくさん作って欲しい。福島さんのそんな思いは、授業にも色濃く表れている。. 「授業の冒頭、会話のウォーミングアップとしてちょっとした雑談というか近況報告をし合うんです。例えば『週末は何してた？』とか。でも中にはまだ会話に自信がなくて、なかなか話に入ってこられない生徒もいて、そんな時はできるだけ自然な形で会話を促して、同じ趣味を持っている生徒同士をつなげてあげられるように工夫しています」. 例えば『日本のアニメ映画を見に行った』という生徒がいたら、『あら、あなたもアニメが好きだったわよね？』と別の子に話を振る。そうすれば放課後に自然と生徒同士の会話が弾み、休日に一緒に出かけるきっかけになるかもしれない。. 「そんな風に雑談の時間も大切にしていると、最初はほとんど日本語を話せなかった生徒が、『コンビニ行きませんか？』なんてさらっと話したりしていて驚くこともあります（笑）。タームが終わる３カ月後には見違えるほど上手になって、そんな姿を見ると、やっぱり日本語教師になってよかったなと思いますね」. 「子供の頃から、やりたいと思ったことはやるタイプ」という福島さん。大学時代に参加した世界一周ツアーも両親の反対を押し切って参加したのだとか. 将来の夢は海外で日本語教師として働くこと。. 「どこか特定の国というよりは、フリーランスとしていろいろな国を渡り歩きたいなと思っています。日本語を教えるスキルがあれば世界中どこでも働けるんじゃないかって（笑）。実際はそんなに甘くないかもしれませんが、いつか実現させたい夢なんです」. 新しい世界に飛び込むたびに新しい夢を見つけてきた福島さん。日本語教師になる目標をかなえた今も、まだまだ「やりたいこと」は尽きない。それは東京という街の単位を飛び越えて、世界へ。「この先、どこへ向かうのか、自分でもわからないんです」と本人は苦笑いするけれど、そんな人生、最高に面白いじゃないか。行き先未定の旅ほどワクワクするものはないのだから。. 東京ではたらく女性へ、パーソナルな５つの質問. ◎休日はどう過ごしている？. 面白そうなイベントに行ったり、友達とランチしたり、飲んだり……。基本誰かと会っていて、あまり家にはいないです。.  . ◎東京で好きな場所はどこ？（理由も）. 田舎から出てきた私にとってどの町も面白くてひとつに絞れませんが、なんだかんだ渋谷です。とにかく人が多く、海外の方も多いので。あとは行きつけのお店もあるので気づいたら渋谷にいることが多いです。.  . ◎最近うれしかったことは？（できれば仕事以外で）. 星座占いで私の星座の運命を変える3daysの一つが私の誕生日だと書いてあったこと(笑)。6月が誕生日なので何があるか楽しみです。.  . ◎100万円あったら何に使う？. 海外旅行。.  . ◎今日の朝ごはん、何食べた？. 納豆＋卵かけご飯とお味噌汁。. https:\/\/www.kudan-japanese-school.com\/jp\/ REACTION. 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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT ENTERTAINMENT. 2022.06.08. ＆w. ドウェイン・ジョンソンの“破壊神”登場『ブラックアダム』特報解禁 12月に公開 記事提供：cinemacafe.net. 2022.06.08. ＆w. 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asahicom jp and wp content themes assets img logo w 01 svg ＆w asahi com ico search user guest png badge heart LIKE comment COMMENT mile bg sns fb tw ig yt 02 ＆ m ＆M travel ＆Travel time ＆Time human bookmark data uploads 2022 05 CT L1230304 jpg L 1230304 2019 04 hiyamizu i 1 冷水希三子 re L1230247 1230247 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき L1230262 2 1230262 L1230277 3 1230277 L1230283 1230283 end paragraph 06 MG 9164 twitterシェア facebookシェア line LINEでシェア hatena このエントリーをはてなブックマークに追加 051fe2b145281cf904496fa5388fcc0f 051fe 2b 145281cf 904496fa 5388fcc 0f EYE L1230222 1230222 食卓華やぐおもてなし。季節の魚のアクアパッツァ L1220832 1220832 旬のアスパラの食感を楽しむ、アサリと豆豉の炒めもの L1220884 e1648793412159 1220884 e 1648793412159 ふんわり卵とハーブでおめかし。春のしらす丼 03 L1210206 1210206 和風出汁でやさしく。豆腐とクレソンの煮込み eye L1210107 1210107 牛乳は使わない。大人風味の小松菜ポタージュ eye031 031 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” 967f1782eff2941bc6ece6f0beb1b1a9 967f 1782eff 2941bc 6ece 6f 0beb 1b 1a 9 まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet ba8f917ee7123fad357ba9b8c4a32d6c ba 8f 917ee 7123fad 357ba 9b 8c 4a 32d 6c 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 08a1bee912777ead2fbfc80b46b32292 08a 1bee 912777ead 2fbfc 80b 46b 32292 「何を着るか」にマンネリを感じたら、冒険しませんか？ bcc8bb0048ff6549f30e4f867752d6a5 bcc 8bb 0048ff 6549f 30e 4f 867752d 6a 5 スペインから父へ。直接その手を握る代わりに 255 5680 〈255〉50歳で選んだ別居婚「第二幕」 8fc46c88600a6570b8deddfa41ebf978 8fc 46c 88600a 6570b 8deddfa 41ebf 978 Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 6b50c0fffa5bdf413e677626e7da782d 6b 50c 0fffa 5bdf 413e 677626e 7da 782d ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを DSC5449 DSC 5449 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った 750 SANJI AND SPOON final animation 猫に学べ。by サジとスプン（飼い主・三宅和歌子さん） 60d4af5fa755cfac59f4dd7726706631 60d 4af 5fa 755cfac 59f 4dd 7726706631 NY発　子どもたちの夏の過ごし方 256 5874 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 毎日食べたくなる！　花も酵母も慈しむ店主の軽快なごほうび感／ベーグルスタンダード footer ＆Human corporate 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PAGE","ログアウト","新規登録","","","","","","","","","こちら"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:05:35 UTC 2022","fresh_date_dt":"Wed Jun 08 17:05:35 UTC 2022","referrer_id_s":"RTQakgEvDQH4","language_s":"ja","size_i":139458,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":8,"references_internal_i":8,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087027914801152,"text_t":"夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき | 朝日新聞デジタルマガジン＆[and]. TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 料理家・冷水希三子の何食べたい？ 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT 料理家・冷水希三子の何食べたい？ 連載をフォローする 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき. FOOD 2022.05.26. 冷水希三子 料理家・フードコーディネーター. レストランやカフェ勤務を経て独立。季節の食材を使ったやさしい味の料理が評判を呼び、雑誌や広告など …. M.S. 関めぐみ 写真家. スポーツ誌、カルチャー誌、女性誌などで活躍。また、広告やカタログ、CDジャケット、俳優の写真集な …. Y.K. 小林百合子 ライター・編集者. 1980年兵庫県生まれ。出版社勤務を経て独立。山岳や自然、動物、旅などにまつわる雑誌、書籍の編集 …. 料理家・冷水希三子（ひやみず・きみこ）さんがリクエストに応えて料理を作ってくださるという夢の連載。今回は夏みかんをソースとトッピングに使った、爽やかで優しい食べ心地の鰹（かつお）のたたきをご紹介します。. ――　冷水先生、こんにちは。すごく過ごしやすい今日この頃ですね。. 冷水. 　気持ちがいいですね〜。. ――　まもなく梅雨だと思うと気が重いですが、心地いい季節をうんと楽しみたいです。. 冷水. 　ベランダで過ごすのも、楽しい季節ですね。. ――　ベランダといえば、今年もベランダでハーブを育てています。バジルやパクチーが元気に育ってきて、お料理も一段と楽しいです。. 冷水. 　それはいいですね。ハーブが身近な存在になると、お料理の幅もグッと広がりますよ。. ――　先生のお料理にはハーブが欠かせませんものね。じゃあ今日はたっぷりとハーブを使ったメニューはいかがでしょうか？. 冷水. 　爽やかでいいですね。どんなお料理が食べたいですか？. ――　う〜ん、旬のお魚とか？　. 冷水. 　では鰹のたたきを使ったメニューはどうでしょう？　ハーブもよく合うんですよ。. 旬の夏みかんを使って、自家製ポン酢を作ります。ハーブはディルやパクチー、木の芽、かいわれ、みょうがなど、和洋のものをたっぷりと. ――　へ〜、鰹のたたきがそんなおしゃれになるんですか！？　ぜひ教えてほしいです。. 冷水. 　今日はパクチーやディルに加えて、木の芽も使いましょう。. ――　和洋のハーブがそろい踏み！　みょうがやかいわれもあって、具材からして爽やかですね。. 冷水. 　鰹は少しクセがあって苦手……という方もいるかもしれませんが、ハーブが入ることで独特のにおいや風味も和らいで、おいしく食べられると思いますよ。. ――　居酒屋さんでは、薬味にすりおろした生姜（しょうが）とニンニクがついてくることが多いですが、ずっと食べているとちょっと刺激が強いな〜と思うことがあります。あとポン酢も風味が強いですものね。. 冷水. 　今日は、酸味を旬の夏みかんで出していきたいと思っています。. ――　鰹のたたきに夏みかん！　なんだか素敵そう〜。. 冷水. 　じゃあまずは夏みかんの準備から。少し手間ですが、夏みかんの皮をむいて、実をひとつひとつ房から取り出します。さらに丁寧に薄皮もむきましょう。. ――　ひえ〜、細かい作業（笑）。でも、おいしさのためなら頑張れます！. 夏みかんは薄皮も丁寧に取り除きます。根気がいる作業ですが、頑張って. 冷水. 　夏みかんは、半分はソースに、残りは果肉をそのまま鰹のたたきの上に乗せて使います。どちらも薄皮が残っていると苦味が出たり、食感が悪くなったりするので、ここは頑張って！. ――　はい！. 冷水. 　夏みかんの準備ができたら、半量分を手でぎゅっと握って果汁を搾ります。. ――　果肉を取り出してから搾ると、たくさん果汁が取れますね。頑張ったかいがありました……。. 冷水. 　搾った果汁と薄口しょうゆ、酢、EXVオリーブオイルを混ぜたら、ソースの完成です。. ――　お手製のポン酢ですね！. 冷水. 　そうそう、旬の柑橘（かんきつ）の果汁を使うと、香りも風味もよく仕上がるので、おすすめです。できるだけ酸味のある柑橘を使うといいですね。. ――　あれ、もしかして調理の工程は、ソースを作るだけ！？. みょうがはこんなふうに斜め薄切りに。あまり太いと食感が悪くなるので、注意してください. 冷水. 　はい（笑）。あとは鰹のたたきをお皿に盛って、その上に残りの夏みかんの果肉、千切りにしたみょうが、お好みのハーブをどっさりのせて、ソースをかければ完成です。. ――　なんという手軽さ！. 冷水. 　夏みかんの準備が大変でしたけど、その後は簡単なのでね。ぜひやってみてください。. ――　わあ、鰹のたたきの上に夏みかんの果肉がのると、なんだかすごくオシャレですね。. 冷水. 　おもてなしにもおすすめです。では召し上がってみてください。. ――　こ、これは……！　ポン酢のツンとくるあの感じがなくて、すごく優しいソースですね。夏みかんの酸味は感じるのですが、酸っぱいというより、爽やか〜という程度で。ものすごく上品な鰹のたたきです。. 冷水. 　夏みかんの果肉も一緒に食べてみてください。. ――　おお、一緒に食べると口の中で果肉とソースが混じって、また違った風味のソースになりますね。味にリズムが出て、すごく楽しいです。. 鰹のたたきの上に夏みかんをのせます。果肉が大きい場合はカットしても. 冷水. 　酸味はそれぞれ好みがあると思うので、トッピングの夏みかんの量を調整していただければいいかなと思います。. ――　あと、いろいろな種類のハーブがのっているのも、味の変化があって楽しいです。ディルが鰹のたたきに合うとは、発見でした。. 冷水. 　鰹は味がしっかりしているので、香りや風味の強いハーブと一緒に食べても風味がぼやけないんです。. ――　ハーブのおかげで鰹の風味が引き立っている感じがします。. 冷水. 　ベランダで育てたハーブを使うと、感動ひとしおですよ。. ――　ハーブって、お値段がまあまあするので、ついケチってしまいがちですが、家庭菜園のものなら気兼ねなくどっさり使えます！. 冷水. 　お財布にも優しいですね（笑）。. ――　この自家製の柑橘ポン酢、冷しゃぶとか、他のお料理にも使えそうですね。. 冷水. 　はい、ぜひ夏みかんが手に入るうちに、色々試してみてください。. 鰹のたたき　夏みかんソース. 材料（2､3人分）. 夏みかん. 1\/2個. 鰹のたたき. 適量. みょうが. 2個. かいわれ. 適量. パクチー、ディル、木の芽. 適量. A. 薄口しょうゆ. 小さじ2. 酢. 小さじ2. EXVオリーブオイル. 大さじ1. 夏みかんは実を房から外して薄皮をむく。半量を手で搾って大さじ2杯分の果汁をとる。残りはトッピング用にとっておく。. １．. の夏みかん果汁と、Ａの材料を全て混ぜ合わせ、ソースにする。. みょうがは千切りにする。. 皿に鰹のたたきを盛り、. ２．. のソースを適量かける。残りの夏みかんとみょうが、かいわれ、パクチーやディル、木の芽などをのせる。. 関連記事. ミョウガの風味が決め手。カリカリ豚の和風あんかけ 料理家・冷水希三子（ひやみず・きみこ）さんが読者と私たち編集部のリクエストに応えて料理を作ってくだ …. 2019.06.27. REACTION. LIKE. 連載をフォローする TAGS. # 料理・レシピ SHARE. ゲスト MY PAGE ログアウト 新規登録 ログイン LIKE. 他の記事へ 毎日食べたくなる！　花も酵母も慈しむ店主の軽快なごほうび感／ベーグルスタンダード 2022.05.19. ゲスト 0. ログイン. 利用規約 BACK NUMBER. この連載をもっとみる. 食卓華やぐおもてなし。季節の魚のアクアパッツァ 2022.05.12. 旬のアスパラの食感を楽しむ、アサリと豆豉の炒めもの 2022.04.21. ふんわり卵とハーブでおめかし。春のしらす丼 2022.04.07. 和風出汁でやさしく。豆腐とクレソンの煮込み 2022.03.24. 牛乳は使わない。大人風味の小松菜ポタージュ 2022.03.10. FOR YOU. あなたにおすすめの記事. 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祇園祭 西日本豪雨 Ｇ２０大阪サミット 宝塚歌劇団 まだまだ勝手に関西遺産 2019統一地方選挙 ＳＴＵ瀬戸リスト 沖縄1935 よみがえる戦前 記者襲撃、あの夜から 阪神支局襲撃30年を越えて 「酷道」をいく 戦火の残響 三浦雄一郎の挑戦 全日本小中学生ダンスコンクール オウムを暴走させた３つの転機 100回つなぐ始球式リレー 環境破壊、報いの現場を歩く 平成とは がんとともに 再生！太陽の塔 日本代表レジェンドカード 「13」最高のスタジアムを巡る旅 2018サッカーワールドカップ 孤高の星 羽生結弦-SPIN THE DREAM SPIN THE DREAM 羽生結弦 LEGEND 葛西紀明 平昌オリンピック 平昌パラリンピック 金正恩とトランプ 「トランプ王国」熱狂のあと 災害大国 被害に学ぶ 河田防災塾 あのときの、ラジオ 国際女性デー特集「Dear Girls」 小さないのち「児相の現場から」 SDGs フィギュアスケート特集 ＨＡＫＵＴＯの挑戦 築地市場の「音」 平成とは いじめと君 女子組 朝日・東大共同調査 ドローンズアイ Ｔｈｅ　Ｒｏａｄ 日本初「監獄ホテル」誕生へ 景気１００社アンケート 衆院の区割り変更 オバマ米大統領、広島へ 南極観測６０年 トワイライトエクスプレス瑞風 政治資金特集 タイムライン 奄美群島国立公園 復興支援音楽祭 いきもの目線 全日本合唱コンクール ノーベル賞 もっと！ロケット HUG STA（ハグスタ） 子ども食堂 大学入試センター試験 築地－時代の台所 真田太平記 皇室とっておき （各駅停話）東北新幹線特別編 待機児童問題 どうぶつ新聞 家庭遺産 朝デジスペシャル. 1.17 再現／阪神・淡路大震災 120日後のフェス　アラバキ 2.24後の世界で　ウクライナ侵攻から考える 4 years from ICHIRO｜イチローが伝えたかったこと GHOSN（ゴーン）日本最大の司法取引 GRIM-成田緑夢（ぐりむ） HAKUTO-Rの挑戦　月への旅路 HAYABUSA2 地球帰還 LEGEND 葛西紀明 MINAMATA　ユージン・スミスの伝言 SDGs｜国谷裕子さんと考える SDGｓ｜下川町編 SPIN THE DREAM　夢をつむぐ羽生結弦 The Days ウクライナ侵攻 VR｜築地市場の360度動画 「13」最高のスタジアムを巡る旅 「オウム」を暴走させた3つの転機 「会いたい、会わせたい　東日本大震災10年」 「祈りの旅」 あなたは誰ですか　ルポ孤独死 あのときの、ラジオ ３．１１震災特集 ありがとう普門館 いきもの目線 いまの政治と選挙は　朝日新聞社世論調査 いま憲法をどう考える　朝日新聞社世論調査 そのとき官邸は　新たな調書 ふたつのアメリカ　分極社会 みえない交差点　全国68万件の交通事故マップ イチローMLB3000安打特集 イラン抵抗の三日月　謎の武装組織を追う オバマ大統領、広島へ カルロス・ゴーン　もたらした光と影 コロナ後の世界を語る　現代の知性たちの視線 コロナ第２波　東京100days チャレンジド wonder athletes デレク＆マイケル データで見る「就活」の歴史 ドーピングの深い闇 ナガサキノート あの日、人々の足取り ノモンハン　大戦の起点と終止符 マー君の410球を徹底分析 ヤンキース田中打者の目で解析 ラストダンス ワールドアスリートボイス ワールドカップの地球 三浦雄一郎 南米最高峰に挑む 主役なきゴーン法廷 五輪TVゲーム年代記 住み続けたい島へ　SDGs｜島の人たちと考えた 冬季オリンピック 「持続可能性」を巡る旅 動物たちはどこへ　変わりゆく動物園 北陸新幹線　駆ける　つながる 医療危機　コロナ東京100days 吉田調書 名人への道 藤井聡太 地殻変動 自公vs.非自公 声　語り継ぐ戦争 夢を力に2014 大谷翔平 -G.O.A.T- 2021年を象徴する5シーン 失踪村　ベトナム人技能実習生 孤高の星　羽生結弦-SPIN THE DREAM 学校の死角 宇宙へ　苦難と栄光、終わらない旅 安倍×石破　二人がみる日本―自民党総裁選２０１８ 対立の海 平成人語 待機児童問題 憲法をどう考える2022 戦後７０年　ビジュアル年表 手塚治虫文化賞２０周年 拓く　北海道新幹線 政党の現在地　与野党はどう動いた　朝日・東大調査15年 新型コロナ　みんなの考えは？ 新型コロナウイルス感染　世界の１年 新型コロナウイルス感染　日本の１年 日本代表 レジェンドカード 日米開戦80年 朝デジ１０周年特集 未完の最長政権 未来のデザイン　コロナで変わる 李香蘭が語るアジア 東京100days 新型コロナウイルスの記録 枯れる五輪 永世七冠 羽生善治 沖縄1935　写真でよみがえる戦前 沖縄1972　写真でたどる日本復帰50年 沖縄を語る、沖縄から考える 沖縄復帰50年、基地はどこへ 海からみた被災地　東日本大震災10年 琉球から日本へ　よみがえる戦前の沖縄 生きる、未来へ　東日本大震災10年 疑惑の島「パラダイス文書」 石川祐希　世界へ 砂上の国家　満州のスパイ戦 私、がんになるの？　ゲノムが迫る選択 空からみた被災地　11年後の現在 空から見た熱海土石流 空襲１９４５ 築地　時代の台所 米中争覇 裁かれる工藤会　全公判傍聴記 追跡 メイド・イン・ジャパン 和牛編 追跡 メイド・イン・ジャパン 腕時計編 逆さ地図で見る米中最前線の中の沖縄 里海てんぷら　東京の天然を揚げる 金正恩とトランプ 金正男暗殺を追う　2017\/2\/13 飽くなき挑戦 ロータリーエンジンの半世紀 高校野球 夏１００回 高橋源一郎 伯父さんは戦場へ行った １６年リオ五輪　超人１６人に注目 ＥＸＰＯ‘７０　ＯＳＡＫＡ ＮＩＣＫ　私がたどりついた日本 地域一覧. 北海道・東北. 北海道 青森 秋田 岩手 山形 宮城 福島 関東. 群馬 栃木 茨城 埼玉 千葉 東京 神奈川 東海・甲信越. 静岡 岐阜 愛知 三重 新潟 山梨 長野 近畿・北陸. 滋賀 京都 奈良 和歌山 大阪 兵庫 富山 石川 福井 中国・四国. 鳥取 島根 岡山 広島 山口 香川 愛媛 徳島 高知 九州・沖縄. 福岡 大分 宮崎 鹿児島 佐賀 長崎 熊本 沖縄 朝日新聞デジタルの基本機能. ログイン画面 朝刊 夕刊 ｂｅ 紙面ビューアー 地域面紙面 スクラップブック スクラップブック MYキーワード MYキーワード 記者イベント 動画・音声・写真 コメント閲覧 天気 数独 12星座占い 交通情報 アプリ プッシュ通知設定 アクセスランキング ソーシャルランキング トピックス デジタル限定 サポート. 総合ガイド お客さまサポート Q&A（よくある質問） サービス紹介 新聞宅配申し込み デジタル申し込み ウェブマガジン. &w &M &Travel 朝日新聞デジタルの関連サイト. 論座 朝日新聞社のサイト. 朝日新聞社　会社案内 朝日新聞SHOP 注目の連載記事 もっと見る いま読む「音楽展望」　吉田秀和 没後10年 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All rights reserved. No reproduction or republication without written permission.","language_s":"ja","size_i":82288,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087029836840960,"text_t":"朝日・日刊スポーツ - モバイル向けサービス：朝日新聞デジタル. メインメニューをとばして、このページの本文エリアへ お使いのブラウザはJavaScriptに対応していないか、または無効になっています。詳しくは サイトポリシーのページ をご覧ください。 06月09日朝日新聞デジタル朝刊記事一覧へ（朝5時更新） 新着記事一覧へ ウクライナ情勢 速報 朝刊. 記事一覧 紙面ビューアー 夕刊. 記事一覧 紙面ビューアー 連載 ランキング コメント その他. 特集 動画・音声・写真 土曜別刷り「be」 記者イベント 天気 数独 12星座占い サイトマップ 検索. ヘルプ. 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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT LIFE. 2022.06.08. ＆w. 東京の台所２ 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 2022.06.06. ＆w. 久保純子 LIFE in N.Y. 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HuntingWorld286648composite F01 Web f1 Hunting World 286648composite F 未来の子ども達に美しい生態系を繋ぎたい。ハンティング・ワールドの新作チャリティーバッグ 87A8853shusei ok 87 8853shusei あの人に贈りたい。このお菓子と、 このうつわ。 3d139b6b199bcc2f5a3725b140027658 3d 139b 6b 199bcc 2f 5a 3725b 140027658 スペイン・アラゴン州テルエル県のアルバラシン村©スペイン政府観光局 1b505b259218ae2a970fe6a9f3503adb 1b 505b 259218ae 2a 970fe 6a 3503adb 環境にやさしいランニングシューズ。アディダスとオールバーズのコラボで、カーボンフットプリントを削減 C5A4911 4911 「奇界」の旅へいざなう高画質と携帯性　写真家・佐藤健寿がノートPCに求めるもの 7492 新しい旅の魅力をお届け「おうちで世界旅行LIVE　オーストラリア編」 「おうちで世界旅行LIVE」第2回は、オーストラリアの魅力を発信 DSC02349 02349 コロナ禍でのマスク、手洗い……、それ以外に何か大切なこと忘れていませんか？ 1b791edde4b321ca06ed55f56df06425 791edde 321ca 06ed 55f 56df 06425 森山良子さんと清水ミチコさんが絶賛。ピンでとめないウィッグ「フィーリン」がさらに進化 2966 六本木ヒルズを開業からずっと「育ててきた」&lt;br&gt;――優しさにあふれる人の街づくり d019eee18a6a9c8841133a983909d218 019eee 18a 9c 8841133a 983909d 218 大阪－奈良－京都　歴史のメインロードを走り出す。近鉄の新・観光特急「あをによし」　4月29日（金・祝）運行開始！ 6fa66a350ea3056b0a08554064ffe225 6fa 66a 350ea 3056b 0a 08554064ffe 225 新谷仁美選手、4度目のマラソン。アディゼロと共に、“ジブンを信じきる”瞬間を目指して eye031 031 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” cd0e1db01bc077195863b2e3a9bd73b1 cd 0e 1db 01bc 077195863b 2e 9bd 73b 長崎の島で教えられたこと　宇賀なつみがつづる旅(33) 967f1782eff2941bc6ece6f0beb1b1a9 967f 1782eff 2941bc 6ece 6f 0beb まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet 483037c0a46af7f52a165d121b4f202e 483037c 46af 52a 165d 121b 4f 202e 渡辺早織さん サングラス大仏、アートな佐久島　愛知県 強烈なダッシュ力とデザインのよさ　ヤマハ「RZ250」 DSCF4936 DSCF 4936 『機動戦士ガンダム　ククルス・ドアンの島』アムロ役・古谷徹さん　第一線で活躍を続ける原動力 255 5680 〈255〉50歳で選んだ別居婚「第二幕」 1287a7e9081393f63d9a416ba88a5975 1287a 9081393f 63d 9a 416ba 88a 5975 ワインが進む前菜も豊富。「ソテーしたベーコンと温かいシェーブルチーズのサラダ」（1,100円） DSC5449 5449 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った eyecat 覚悟を背負って赴任したアフガニスタン　宿舎のガーデンで出会った子猫 4e67c97046d5ab41ab31ea94b081171c 67c 97046d 5ab 41ab 31ea 94b 081171c 今こそ「よみがえりの道」熊野古道　山上の「伝説の宿」から 10 Catedral de Santiago y Plaza del Obradoiro 19930101 1500262 カミーノ・デ・サンティアゴ ～星に導かれ、巡礼のスペイン旅へ 996deb709f88183c0c842a28099c7594 996deb 709f 88183c 842a 28099c 7594 脱衣場と浴室が一体となっている中乃湯の内部＝沖縄県沖縄市 美しい5月に食べたくなる魚は？　オーストリア、リヒテンシュタイン、ドイツ、ノルウェー 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NY発　子どもたちの夏の過ごし方 5. ＆w. ※アクセスは過去7日間、LIKE、コメントは過去30日間で集計しています。. SERIAL STORY. 連載記事. ３ミリ新しい私. 大人を探求する. 特別な、旅の扉. 東京の台所２ 2022.06.08. 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 大平一枝 本城直季 連載をフォローする 隔週水曜日. バックステージ 2022.06.07. 市川染五郎さん「とにかく今はいろんな役を経験したい」　17歳の胸に宿る野望 連載をフォローする 不定期. ほんやのほん 2022.06.06. 100年前のルポが示唆すること　いま改めてのウクライナ　 蔦屋書店 コンシェルジュ 連載をフォローする 月曜日. 久保純子 LIFE in N.Y. 2022.06.06. Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 久保純子 連載をフォローする 月１回. 花のない花屋 2022.06.02. ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを 東信 S.S. 椎木俊介 連載をフォローする 木曜日. リノベーション・スタイル 2022.06.01. 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” 石井健 連載をフォローする 月2回・水曜日. このパンがすごい！ 2022.05.31. まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet 池田浩明 連載をフォローする 隔週火曜日. 30歳からのコンパス 2022.05.27. 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 S.S. 坂口さゆり 連載をフォローする 不定期. シトウレイの好きな服、似合う服。 2022.05.27. 「何を着るか」にマンネリを感じたら、冒険しませんか？ シトウレイ 連載をフォローする 月１回. 山城さくらのキャンプFM 2022.06.07. メスティンの中身を衣替え　unumPlate“至高の焼肉セット” 山城さくら 連載をフォローする 不定期. 小川フミオのモーターカー 2022.06.06. 強烈なダッシュ力とデザインのよさ　ヤマハ「RZ250」 小川フミオ 連載をフォローする 週１回. 会議のチカラ 2022.06.03. 自信を失って発言をしないベテラン世代の活性化策 古屋聡一 連載をフォローする 月２回. インタビュー 2022.06.02. 『機動戦士ガンダム　ククルス・ドアンの島』アムロ役・古谷徹さん　第一線で活躍を続ける原動力 古屋聡一 花田龍之介 連載をフォローする 随時. 渡辺早織の思い出ちょっぴり、つまみぐい。 2022.05.31. 益子陶器市×びわの白あえ　うつわ探しで向き合う心の声 渡辺早織 連載をフォローする 月１回. 小川フミオのモーターカー 2022.05.30. 車内で音楽を聴きますか？　私が「音づくり」で気に入ったクルマ 小川フミオ 連載をフォローする 週１回. いしわたり淳治のWORD HUNT 2022.05.27. くだけた言葉とやさしさと男気の三つ巴『辛』 いしわたり淳治 連載をフォローする 毎月20日前後. 口福のカレー 2022.05.26. 裏路地は、ゆったり美味！　激戦区・下北沢で見つけた南インドカレー「アンジャリ」 熊野由佳 連載をフォローする 月２回. キャンピングカーで行こう！ 2022.05.25. 快適に過ごす工夫が詰まった「8ナンバー」のコンパクトバンコン「ROBBY」 渡部竜生 連載をフォローする 水曜日. あの街の素顔 2022.06.08. 雨の日も退屈しない、グリーンパワーに癒やされる八丈島 鈴木博美 連載をフォローする 随時. 楽園ビーチ探訪 2022.06.07. いにしえのリゾート地、白砂のきらめきと古湯　南紀白浜 古関千恵子 連載をフォローする 毎月第4木曜日. 城旅へようこそ 2022.06.06. いにしえの風が吹く星ケ城　小豆島の城と石文化（1） 萩原さちこ 連載をフォローする 月2回　月曜日. 永瀬正敏フォトグラフィック・ワークス 記憶 2022.06.03. (164) 眠気を覚ましてくれた女の子　永瀬正敏が撮った台湾 永瀬正敏 連載をフォローする 金曜日. 京都ゆるり休日さんぽ 2022.06.03. 京都にたたずむフランス　日仏学館の小さなビストロ「レ・ドゥ・ギャルソン」 大橋知沙 津久井珠美 連載をフォローする 毎月第1・3金曜日. 世界の街へ 2022.05.31. SPONSORED. 旅先で最高の「地元体験」を提供　 高級ホテル「マリオット」のMarriott Bonvoyプログラム 連載をフォローする 宇賀なつみ わたしには旅をさせよ 2022.05.31. 長崎の島で教えられたこと　宇賀なつみがつづる旅(33) 宇賀なつみ 連載をフォローする 月１回. 上から撮るか、下から撮るか 2022.05.31. サングラス大仏、アートな佐久島　愛知県 野見山大地 連載をフォローする 月1回　火曜日. クルーズへの招待状 2022.05.30. ダイナミックで多彩、ホエールウォッチング　「ぱしふぃっく びいなす」小笠原の旅（後編） 上田寿美子 連載をフォローする 月末1回. VIEW ALL CREATORS. 著者・クリエーター. VIEW ALL 文筆家. 大平一枝 長野県生まれ。失われつつあるが失ってはいけないもの・こと・価…. ライター. 髙橋晃浩 福島県郡山市生まれ。ライター／グラフィックデザイナー。ライタ…. 作家. 大道珠貴 1966年福岡市生まれ。2003年、『しょっぱいドライブ』で…. 温泉ビューティー研究家・トラベルジャーナリスト. 石井宏子 日本・世界の温泉や大自然を旅して写真撮影・執筆をする旅行作家…. 編集者. 池田美樹 旅と写真とワインと猫を愛する編集者。（株）マガジンハウスにて…. タベアルキスト. マッキー牧元 1955年東京生まれ。立教大学卒。年間幅広く、全国および世界…. 歌手. UA 1972年大阪生まれ。母方の故郷は奄美大島。1995年デビュ…. 編集者・ライター. 山田静 女子旅を元気にしたいと1999年に結成した「ひとり旅活性化委…. フォトグラファー. 葛西亜理沙 横浜生まれ。サンフランシスコ州立大学芸術学部写真学科卒業後、…. 料理家. 栗原友 1975年生まれ、東京都出身。母は料理家の栗原はるみさん、父…. コラムニスト. 大矢アキオ　Akio Lorenzo OYA 音大でヴァイオリンを専攻、大学院で比較芸術学を修める。イタリ…. クルーズジャーナリスト. 上田寿美子 クルーズライター、クルーズジャーナリスト。日本旅行作家協会会…. デザイナー. 小倉ヒラク 1983年、東京都生まれ。早稲田大学文学部で文化人類学を学ぶ…. 編集者・ライター. 奈々村久生 文筆業。「キネマ旬報」「クイック・ジャパン」等にて執筆中。 …. 料理家. 室田万央里 無類の食べ物好きの両親の元、東京に生まれる。17歳でNYに移…. H.Y. 写真家. 山田秀隆 2009年、日本写真芸術専門学校卒業。2012年から石塚定人…. 俳優・写真家. 永瀬正敏 1966年生まれ、宮崎県出身。1983年、映画「ションベン・…. 釣りライター. 安田明彦 1959年大阪市生まれ。父親の影響で子供のときから釣りが大好…. ビーチライター. 古関千恵子 リゾートやカルチャー、エコなどを切り口に、国内外の海にフォー…. 作詞家・音楽プロデューサー. いしわたり淳治 1997年にロックバンドSUPERCARのメンバーとしてデビ…. T.I 校正・校閲者. 石田知之 1976年生まれ、海無し県育ち。職業はフリーランスの校正・校…. フォトグラファー. 野呂美帆 三重県出身。旅、アウトドア、ライフスタイルを中心に、雑誌、広…. ライター・出版社経営. 松本康治 1962年、大阪府生まれ。出版社勤務を経て、1987年に医療…. S.E. ジャーナリスト. 江藤詩文 トラベルジャーナリスト、フードジャーナリスト、コラムニスト。…. ライター. 山城さくら 地方で事務員として働いていたが、もっと好きなことに時間を使い…. K.F ライター. 藤原かすみ 元編集者、今は国境なき旅びと。1990年にオランダ・アムステ…. リーマントラベラー. 東松寛文 1987年、岐阜県生まれ。神戸大学卒業後、広告会社に就職。現…. フリーアナウンサー. 久保純子 1972年、東京都生まれ。小学校時代をイギリス、高校時代をア…. アマチュア写真家. 村松拓 1991年1月生まれ。川崎市出身。 2004年の夏休み、初め…. ライター＆エディター. 熊野由佳 徳島県出身。外資系ジュエリーブランドのPRを10年以上経験し…. ライター. 坪井由美子 ドイツ在住。日本では食の仕事に従事。商品開発、メディアの監修…. ライター・編集者. 小見山友子 ファッション業界に従事後、メディアディレクターを経て、201…. 薬膳アテンダント、食文化ジャーナリスト. 池田陽子 宮崎生まれ、大阪育ち。立教大学社会学部を卒業後、広告代理店を…. 「パンラボ」主宰. 池田浩明 ライター、パンの研究所「パンラボ」主宰。日本中のパンを食べま…. ライター. 阿部裕華 1992年生まれ。Webメディアのディレクター・編集を経て、…. ライター. 矢口あやは 1983年大阪生まれ。旅行誌やカルチャー誌を中心に、国内外の…. 作家・写真家. 相原恭子 慶應大学卒業。ドイツ政府観光局勤務を経て、作家＆写真家。「ド…. 旅ライター. 宮﨑健二 1958年、福岡市生まれ。新聞社を退社後、主に九州の旅とアウ…. ライター. 吉田志帆 左京区に住まう、京都歴20余年の“よそさん”。在学中より、関…. 編集者・ライター. 武田由紀子 1978年、富山県生まれ。出版社や編集プロダクション、WEB…. フードジャーナリスト. 畑中三応子 編集者、ライター、フードジャーナリスト。『シェフ・シリーズ』…. 経営共創基盤（IGPI）共同経営者. 塩野誠 ハングリーだけどノーブルな感じの大人に憧れるコンサルタント。…. 編集者・ライター. 草深早希 1983年東京生まれ。2005年よりキャリアをスタートし、カ…. ライター. こだまゆき 知らない土地での朝食がなによりも楽しみなフォト派のライター。…. 料理家・フードコーディネーター. 冷水希三子 レストランやカフェ勤務を経て独立。季節の食材を使ったやさしい…. 写真家. 高橋伸哉 人物、風景、日常スナップなど、フィルムからデジタルまでマルチ…. ライター. 西森路代 愛媛県生まれ。地元テレビ局、派遣社員、編集プロダクション勤務…. M.K ライター. カルーシオン真梨亜 1985年千葉県生まれ。幼少の頃見ていた海外のホームドラマに…. ライター. 江澤香織 旅、食、クラフトなどを中心に執筆。 著書『青森・函館めぐり　…. フラワーアーティスト. 東信 1976年生まれ。2002年より花屋を営み続け、現在は東京・…. ワインジャーナリスト. 浮田泰幸 青山学院大学文学部卒。月刊誌編集者を経てフリーに。広く海外・…. ライター・編集者. 内山さつき ライター、編集者として旅、アート、文学をテーマに、雑誌や書籍…. 国境なき医師団（MSF）手術室看護師. 白川優子 1973年埼玉県出身。高校卒業後、坂戸鶴ヶ島医師会立看護専門…. サウナユニット. 東京シティサウナボーイズ 元クラブDJ、2人によるサウナユニット。コロナ禍で自らが主催…. フードライター. 川村明子 大学卒業後、1998年よりフランス在住。ル・コルドン・ブルー…. 写真家. 石野明子 2003年、大学卒業後、新聞社の契約フォトグラファーを経て0…. 音楽ライター. 村上麗奈 2000年生まれ。2019年6月、ライブリポートの寄稿をきっ…. R.S. 写真家. 佐藤良一 物語のある街を題材に旅系の各媒体で活動中。旅先で出会った「色…. 写真家. 三井昌志 1974年生まれ。京都府出身。神戸大学工学部卒業後、機械メー…. ライター. 松田朝子 東京・銀座の料亭に生まれ、戦前から続いた料亭を2009年にた…. T.N. ライター. 中尾隆之 高校教師、出版社を経てフリーの紀行文筆業。町並み、鉄道、温泉…. 写真家. 阿部稔哉 1965年岩手県生まれ。「週刊朝日」嘱託カメラマンを経てフリ…. ライター. 小林由季 1995年よりスペイン在住。ライター、通訳、コーディネーター…. ストリート・スタイル・フォトグラファー／ジャーナリスト. シトウレイ 毎シーズン、世界各国のコレクション取材を行い、独自の審美眼で…. 写真家. 本城直季 現実の都市風景をミニチュアのように撮る独特の撮影手法で知られ…. 写真家. 津久井珠美 大学卒業後、1年間映写技師として働き、写真を本格的に始める。…. ライター. 太田瑞穂 ライター、翻訳＆通訳。旅先でその土地の日常的な暮らしやそこに…. 現代美術家. 伊藤知宏 1980年生まれ。東京育ちの画家、現代美術家、メディア・アー…. モータージャーナリスト. 小川フミオ クルマ雑誌の編集長を経て、フリーランスとして活躍中。新車の試…. 写真家. 野見山大地 1993年、日本人の父とタイ人の母の間に生まれる。幼少期をタ…. 紛争解決活動家. 永井陽右 1991年、神奈川県生まれ。NPO法人アクセプト・インターナ…. ライター／アーティスト. 熊山准 1974年、徳島県生まれ。北海道・沖縄の大学生活からリクルー…. 写真家. 中田浩資 1975年、徳島市生まれ。大学休学中の1997年に渡中。19…. 小説家. 白岩玄 1983年生まれ。京都市出身。2004年、小説『野ブタ。をプ…. 作家. 髙森玲子 1969年富山県生まれ。東京大学教養学部を卒業後、出版社に勤…. 朝日新聞記者. 西田健作 1971年、神奈川県茅ケ崎市生まれ。15年ほど前に千葉県浦安…. 旅行家. 鈴木博美 旅行業界で15年間の勤務を経てフリーの旅行家に。旅を通じて食…. 釣りライター. 木村俊一 1952年生まれ。兵庫県尼崎市で底物釣り専門店「木村商」を経…. キャンピングカージャーナリスト. 渡部竜生 会社員からフリーライターに転身後、キャンピングカーに出会って…. デザイナー／アーティスト. 篠原ともえ 1995年歌手デビュー。文化女子大学（現・文化学園大学）短期…. コミュニティープロデューサー. 水代 優 1978年生まれ。愛媛県出身。2002年より株式会社イデーに…. 俳優・タレント. 渡辺早織 モデルとして2007年デビュー。その後、ドラマ\/映画\/CM\/…. ライター・コメンテーター. 山口一臣 1961年東京生まれ。ゴルフダイジェスト社を経て89年に朝日…. デザイナー・エディター. コヤナギユウ 給食の調理員から書籍出版社で営業職ののち、渋谷やNYの路上で…. 料理人／「eatrip」主宰. 野村友里 おもてなし教室を開く、母・野村紘子さんの影響を受けて料理の道…. フリーアナウンサー. 宇賀なつみ 東京都出身。テレビ朝日「報道ステーション」の気象キャスターと…. モデル. 高山都 1982年生まれ。モデル、俳優、ラジオパーソナリティーなど幅…. 編集者・ライター. 大橋知沙 東京でインテリア・ライフスタイル系の編集者を経て、2010年…. N.S. ジャーナリスト. シュピッツナーゲル典子 英米文学部卒業後、ドイツ企業に勤務したのが運のつき。そこでド…. フォトグラファー. 花田龍之介 1979年生まれ栃木県益子町育ち。アシスタントを経て2010…. 城郭ライター. 萩原さちこ 小学2年生のとき城に魅了される。執筆業を中心に、メディア・イ…. TAGS. 旬のキーワード. #＆ふくしま #NOCOMPETITION #SDGs #TRAVEL FUTURE #アート #アウトドア #アウトドア　車 #アウトドア、車 #アナタのミライはワタシのミライ #イラスト・アニメ #インタビュー #インテリア #おうち時間 #お笑い #お酒 #ガンダム #キャリア #グルメ #ゲーム・eスポーツ #ジェンダー #ショップ＆カフェ #スポーツ #テクノロジー #デザイン #ビジネス #ビューティー #ファッション #プレゼント #ヘルスケア #マネー SHOPPING. さらっと快適、安心の日本製　洗える立体ガーゼマスク. ＆M 2021.05.11. ARTICLE SHOP NOW 糖質オフパンも甘酒も　おいしく健康、シロカのおうちベーカリー. ＆w 2021.05.11. ARTICLE SHOP NOW VIEW ALL 日々にちょっとずつ、彩りを。. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト サイトポリシー 利用規約 広告事例・ガイド 個人情報 アクセスデータの利用 著作権 お問い合わせ 朝日新聞デジタルに掲載の記事・写真の無断転載を禁じます。すべての内容は日本の著作権法並びに国際条約により保護されています。. Copyright © The Asahi Shimbun Company. All rights reserved. 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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 日々にちょっとずつ、彩りを。. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT Presented by. ＪＡ共済. 助け合いの輪 共済で一歩を踏み出せた　　 人をつなぐ助け合いの輪 SPONSORED INTERESTS 2022.03.10. LATEST. 最新記事. INTERESTS 2022.03.10. SPONSORED. ＆Human. 助け合いの輪 共済で一歩を踏み出せた　人をつなぐ助け合いの輪 INTERESTS 2022.03.10. ＆Human. 小山薫堂が撮る風景 人生の喜びは、当たり前の日常のなかにある 次回更新をお楽しみに。. FOR YOU. あなたにおすすめの記事. 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PAGE","ログアウト","新規登録","","","","","■アート引越センター","","","","","こちら"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:05:39 UTC 2022","fresh_date_dt":"Wed Jun 08 17:05:39 UTC 2022","referrer_id_s":"RTQakgEvDQH4","language_s":"ja","size_i":140668,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087032259051520,"text_t":"引っ越しスタッフ：大谷明奈さん（23歳） | 朝日新聞デジタルマガジン＆[and]. TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 東京ではたらく 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT 東京ではたらく 連載をフォローする 引っ越しスタッフ：大谷明奈さん（23歳）. WORK 2019.06.06. Y.K. 小林百合子 ライター・編集者. 1980年兵庫県生まれ。出版社勤務を経て独立。山岳や自然、動物、旅などにまつわる雑誌、書籍の編集 …. K.N. 野川かさね 写真家. 1977年神奈川県生まれ。山や自然の写真を中心に作品を発表する。クリエイティブユニット〈kvin …. 職業:引っ越しスタッフ. 勤務地:東京都港区. 仕事歴:3年目. 勤務時間:8:00～. 休日:月8～10日. 　　　　◇. 引っ越しトラックがひっきりなしに走り回っていた4月。忙しく家財の積み下ろしをする引っ越しスタッフの中に、小柄な女性の姿を見かけた。それも一度ではなく、何度も。ガタイのいい男性に混じって、なんてたくましいのだろうと感心しつつ、しかしどうしてまた彼女たちはこの仕事を選んだのかと、ずっと気になっていた。. 大谷明奈さんもまた、引っ越しの現場で働くムービングスタッフ。旭川の高校を卒業後、引っ越し大手のアート引越センターに入社、今年で3年目になる23歳だ。. 「入社して2年半ほどは地元、旭川の支店に勤めていて、昨年の10月に東京・港区の支店に異動してきました。全国でも東京は年間を通して引っ越しの依頼が多いエリア。東京のスタッフを増員するということで、移ってきました。もともと東京で働きたいと思っていたので、抵抗はなかったですね」. 東京での暮らしについて尋ねると、「楽しいです！」と目を輝かせる。. 段ボールを重ねて持つときは右手を箱が重なる部分に、左手を下段の段ボールの底面に当てる。こうすると安定感が増し、さらに受け取るスタッフとの連携もスムーズになる. 「どこが楽しいかと言われると……全部ですね(笑)。電車でどこへでも行けますし、地元にはなかったものが全部あるという感じで。とくに買い物をするのが好きですね。なんでもそろっていて、すごく楽しいんです」. プライベートの話をするときは、いたって普通の23歳。しかも身長は152cmと女性の平均よりも小柄。目の前に座る彼女が男性に混じって大きな家財を運んでいるなんて、全然想像できない。. 「もちろん入社してすぐは段ボール1個しか持てませんでした。でも持ち方のコツがちゃんとあるんですよ。あとはほぼ毎日引っ越し作業をしていますから、自然と筋力がつきましたね。今では3個はいけます」. なんと彼女、入社3年目にして現場スタッフを束ねるチームリーダーを務めている。「だいたいの場合、2年目にはリーダーになれるんですよ」と言うけれど、全現場スタッフの男女比が約8:2というなかで評価を得るには、苦労も多かっただろうと想像する。. 2トンロング車まで運転できる大谷さん。高い運転席にもひょいと飛び乗る。「都内の道は狭くて、まだまだ緊張します」. 「うちの会社は男女で作業内容に差があるということはないんです。とはいえ体力の差はもちろんあります。でも女性だからこそできる気遣いもあると思うんです。男女それぞれが得意な部分を生かし合っていくことで、よりいい仕事ができるんじゃないかなと思います」. それでも、ときにはお客様から「えっ! あなたが運ぶの?」と驚かれることもあると言う。そんなときは「はい、運びます!」と笑顔で答える。. 「引っ越しの仕事で一番楽しいなと思うのは、ひとつとして同じ現場がないこと」と言う大谷さん。「毎日違うお客様に会えますし、お部屋も家財もみんな違います。だから飽きるということがないですし、スムーズに作業が終わったときに『ありがとう、助かりました』と言っていただけると、やっぱり毎回うれしいなと思いますね」. 根っからのスポーツ少女で、空手は黒帯。体操や水泳、キックボクシングなども習っていたそう. この仕事を選んだ理由もそこにある。じつは大谷さん、高校１年生のときにアート引越センターでアルバイトを経験し、その体験が今の仕事につながっている。. 「高校で野球部のマネジャーをしていたのですが、３月に長期の遠征があって。１年生のマネジャーはそれに同行できないということだったので、その空いた期間にアルバイトでもしようかなと。引っ越しスタッフって時給がいい上に日払いなので、学生にはおいしいバイトなんです（笑）」. とはいえ、一直線に引っ越しの業界を目指していたわけではない。. 「私は父が自衛官で、その姿にずっと憧れて育ったんです。中学生くらいからは『絶対に自衛官になる！』と決めていて、高校時代は試験に向けて猛勉強していました。でも試験直前になって、身体検査で通過できない項目があることがわかってしまって……。就職は自衛官一本で考えていたので、そのときは途方に暮れてしまいました」. そんな時に思い出したのが、楽しかった引っ越しスタッフのアルバイトだった。. 専用のシューズにはつま先に鉄板が入っていて、万が一重い荷物を落としても怪我を負わないように配慮されている. 「子供の頃から体操や空手、キックボクシングを習っていて、体を動かすことが好きというのも大きかったと思います。あと私、4人兄妹の一番上で、世話をするのが好きなんです。高校で野球部のマネジャーをやってからは、誰かの役に立つことが自分は好きなんだなと改めて実感して。だから引っ越しの仕事は合うんじゃないかなと」. その選択が間違っていなかったと改めて感じた出来事があった。. 「依頼主はご年配のご夫婦で、旦那様は体が不自由でひとりでは動けない状態。しかもご事情があって、荷物の行き先が２カ所に分かれていたんです。奥様ひとりでは何かと大変だろうと、梱包など色々お手伝いをさせていただいたのですが、それをとても喜んでくださって、翌日に予定していた２カ所目の運搬もぜひお願いしたいとおっしゃってくれて。. 大型のタンスを2階から搬出。狭い階段でも壁や家財に傷がつかないよう、スタッフ同士声を掛け合って慎重に作業する。リーダーの大谷さんは上から全体を見渡し、下にいるスタッフに指示を出す. 2日目、すべての作業を終えたとき、奥様が涙ながらに感謝を伝えてくれたときには、私もウルっときてしまいました。『本当に、色々大変だったの』と、これまでの暮らしのお話も少ししてくださって。寝たきりの旦那さんと二人の暮らしで、お一人でいろいろなものを抱えてきたのだなと思ったら、少しでもお手伝いができてよかったなと思ったんです」. 引っ越しとは、ただ荷物を運ぶだけの仕事ではない。誰かの生活や人生、そうした一切を新しい場所へと送り届けることだ。そう考えると、日々、見知らぬ人のプライベートに入っていくのは、なかなか大変なことだろうとも思う。. 「お客様の私物に手を触れるわけですから、もちろん慎重になりますし、緊張もします。でも、それが辛いと思ったことはほとんどないんです。この仕事で辛いことは何かと聞かれたら、思い浮かぶのはお客様との関係ではなくて、荷物の量とか運びやすさとか、そういう物理的なことが大半ですね」. 素人考えでは、エレベーターがないとか、荷物がものすごく多いとか、そういうケースが大変なのかと思うけれど、じつは一概に言えないというから驚く。. コースによってはスタッフが梱包を行うサービスも。食器などの割れ物も、丁寧に梱包。「中には梱包に技術が必要なものもあるのですが、そのあたりはその都度先輩の動きを見ながら学んでいきます」. 「たとえばエレベーターがあるマンションでも、家の前の道が狭くてトラックが入れない場合は、エレベーターを降りてから荷物を運ぶ距離が長くなります。それならたとえエレベーターがなくても、マンションの真ん前に車を停められた方がずっと楽なんです。最高なのは家の前に車が停められて、なおかつエレベーターが部屋の前というケース（笑）。でもそれは実際にお邪魔してみないと正確なことはわかりませんから、毎回ドキドキします」. 「ひとつとして同じ引っ越しがないのが楽しい」というのは、「そういうこともひっくるめて、です」とのこと。目下の課題もまた、そんな難しい現場を担当することになっても、スムーズかつスピーディーに作業を進められるようになることだ。. 「私の所属する支店には3人の女性先輩スタッフがいるのですが、みなさん本当にすごくて、憧れます。凹凸が多くて、『これ、どうやって梱包すればいいの？』と一瞬迷ってしまうものでもあっという間に梱包してしまいますし、トラックで積み込みをしているスタッフのことを考えて、ベストな順序で荷物を搬出していきます。その判断が早くて的確で、尊敬する点ばかりです」. 仕事の必需品は、梱包したダンボールに品名を書く油性ペンと、荷ほどき用のカッターナイフ、書類用のボールペン. 引っ越し作業というのは、数人のスタッフでバケツリレーのようにして行う。部屋から荷物を出すスタッフは、どの荷物がどんな風にトラックに積み込まれているのかを実際に見ることができない。. 「とはいえ、やみくもに搬出しても、トラックにうまく積み込むことはできません。何をどんな順番で搬出すればトラックのスペースを効率よく使えるか、常にパズルのように想像しながら手を動かさないといけないんです。だからこの仕事で一番大切なのは“段取り力”。頭をフル回転させて、そこをもっともっと伸ばしていきたいなと思っています」. 引っ越しの仕事での“段取り力”とはきっと、“想像力”とも言い換えられるだろう。多く言葉を交わすことがなくても、依頼主の暮らしを想像し、苦労を思いやる。同時に、一緒に働くスタッフがいかにスムーズに作業ができるかをも考える。しかもフルスピードで。そんなことができるのは、自分以外の誰かに、自然と思いを寄せられる人だけだ。. 体力勝負だと思っていた引っ越しの仕事は、けっしてそれだけで務まるものではないと知った。多くの女性が活躍している理由もまた、すとんと腑（ふ）に落ちたのだった。. 東京ではたらく女性へ、パーソナルな５つの質問. ◎休日はどう過ごしている？. ずっと寝ています（笑）。.  . ◎東京で好きな場所はどこ？（理由も）. お台場。海を見たり、散歩をしたり、ぶらぶらしているだけで楽しいです。. ◎最近うれしかったことは？（できれば仕事以外で）. コンビニのくじでゼリーが当たったこと。. ◎100万円あったら何に使う？. 海外旅行。プーケット島とかいいですね。旭川には海がないので、きれいな海が見たいです。. ◎今日の朝ごはん、何食べた？. お水だけ. ■アート引越センター REACTION. 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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 日々にちょっとずつ、彩りを。. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT Presented by. セイコーウオッチ. ベーシスト、ハマ・オカモトさん(OKAMOTO’S)。透明なアクリルに、　 その瞬間と思いを閉じ込める SPONSORED FASHION 2022.02.04. LATEST. 最新記事. FASHION 2022.03.14. ＆Time. 発酵デザイナー・小倉ヒラクさんの大人のひとり時間。周りからの評価が気になる人こそ、ひとり時間を大切に FASHION 2022.02.04. SPONSORED. ＆Time. ベーシスト、ハマ・オカモトさん(OKAMOTO’S)。透明なアクリルに、その瞬間と思いを閉じ込める FASHION 2022.02.04. ＆Time. 好きなことに没頭する。それが最高の「大人のひとり時間」 FASHION 2022.01.14. 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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 久保純子 LIFE in N.Y. 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT 久保純子 LIFE in N.Y. 連載をフォローする ニューヨーク在住６年目の、久保純子さん。新型コロナウイルスで世界がめまぐるしく変化する中でのニューヨーク生活。家族や友人との時間、街で見かけたモノ・コト、感じたことなど、日々の暮らしを通して久保さんが見つめた「いまのニューヨーク」をつづります。. 月１回. 久保純子 フリーアナウンサー. 1972年、東京都生まれ。小学校時代をイギリス、高校時代をアメリカで過ごす。大学卒業後、N …. LIFE 2022.06.06. ＆w. Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 LIFE 2022.05.09. ＆w. NY発　子どもたちの夏の過ごし方 LIFE 2022.04.04. 変わりゆくファッション。心にも変化が LIFE 2022.03.07. 次女の小学校生活から「こんな教育を受けたかった！」 LIFE 2022.02.07. マンハッタンを抜け出して。コロナ下、冬の過ごし方 LIFE 2022.01.10. マンハッタンの住宅事情。ただ今、アパート探し中！ LIFE 2021.12.06. NY、食にまつわるエトセトラ。身体にも環境にもやさしく LIFE 2021.11.08. 黄金色に染まる、秋のNY。「Broadway is Back!」 LIFE 2021.10.11. 多様性の街・NY。誰もが自分らしくいられる社会を LIFE 2021.09.06. 「I Choose Joy」太陽のような彼女との出会いと、新しいムーブメント LIFE 2021.08.02. 夏のニューヨーカーが訪れる、緑のオアシス LIFE 2021.07.05. 「ワクチン接種の最新リポート」をお届けします！ LIFE 2021.06.14. 初夏のニューヨークに、うれしい明るさ。ビーチにプール、そして美術館！ FOR YOU. あなたにおすすめの記事. LIFE 2022.06.01. ＆w. リノベーション・スタイル 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” FOOD 2022.05.31. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet ARTS & CULTURE 2022.05.27. ＆w. 30歳からのコンパス 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 FASHION 2022.05.27. ＆w. シトウレイの好きな服、似合う服。 「何を着るか」にマンネリを感じたら、冒険しませんか？ FOOD 2022.05.26. ＆w. 料理家・冷水希三子の何食べたい？ 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき LIFE 2022.05.26. ＆w. 花のない花屋 スペインから父へ。直接その手を握る代わりに POPULAR. 人気記事. アクセス LIKE コメント 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. リノベーション・スタイル 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” 0. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet 1. ＆w. 久保純子 LIFE in N.Y. Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 4. ＆w. 花のない花屋 ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを 0. ＆w. 東京の台所２ 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った 0. ＆w. 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. 料理家・冷水希三子の何食べたい？ 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき 0. ＆w. 猫が教える、人間のトリセツ 猫に学べ。by サジとスプン（飼い主・三宅和歌子さん） 0. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet 1. ＆w. 久保純子 LIFE in N.Y. NY発　子どもたちの夏の過ごし方 5. ＆w. 久保純子 LIFE in N.Y. 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PAGE","ログアウト","新規登録","","","","","","","","","こちら"],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:05:42 UTC 2022","fresh_date_dt":"Wed Jun 08 17:05:43 UTC 2022","referrer_id_s":"RTQakgEvDQH4","language_s":"ja","size_i":140090,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":15,"references_internal_i":15,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087035939553280,"text_t":"Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 | 朝日新聞デジタルマガジン＆[and]. TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. 久保純子 LIFE in N.Y. 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT 久保純子 LIFE in N.Y. 連載をフォローする Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方. LIFE 2022.06.06. 4. 青空、白い砂浜、紺碧（こんぺき）の海、都会の喧騒（けんそう）を離れて、カリブ海のアンギラには「非日常」が待っていた。身も心も、力が抜けていくのを実感する. 久保純子 フリーアナウンサー. 1972年、東京都生まれ。小学校時代をイギリス、高校時代をアメリカで過ごす。大学卒業後、NHKに …. ニューヨーク在住6年目の、久保純子さん。新型コロナウイルスで世界がめまぐるしく変化する中での、ニューヨーク生活。家族や友人との時間、街で見かけたモノ・コト、感じたことなど、日々の暮らしを通して久保さんが見つめた「いまのニューヨーク」をつづります。. カリブ海やヨーロッパまで、ひょいひょいと. 先月の回 では、子どもたちの長〜い夏休みの過ごし方についてつづったが、今回は世界を飛び回る旅好きニューヨーカーの夏休みについてお届けしたい。. ニューヨーカーは、「旅」に全力投球だ。夏休みはもちろん、数日でもお休みが取れれば、南はカリブ、北はカナダ、西はハワイ、東はヨーロッパまで、幅広く攻める。日本にいた頃は、旅行といえば、取得できる休日数も限られていることから、国内旅行が精いっぱいで、うんと頑張って、早いうちから計画を立ててハワイに足を伸ばすことはあったが、こちらの人々は、当たり前のようにひょいひょいと南国やヨーロッパへ旅行する。. というのも、どこもかしこも数時間で行けてしまうという利点があるからだ。カリブ海へは、直行便で2、3時間。ヨーロッパへも6、7時間と、日本への飛行時間14時間を考えると断然近い。しかも、航空券も日本の国内旅行ほどの値段で購入できるのである。ニューヨーク（NY）からロンドンのチケットを調べてみると、往復＄380なんていう格安のものもある。. カリブ海には7,000以上の島々が点在する。アンギラへは、直行便はなく、セントマーティン島から船を乗り継いで行く。究極の離島だ. 私たち家族も、NYに移り住んでからというもの、あちらこちらに旅するようにしている。海好きというのもあるが、日本から行くのには相当な時間がかかるカリブ海や中米をこの機会に訪れようと意気込んでいる。コスタリカやバハマをはじめ、タークス・カイコス、バルバドス、セントマーティン、アンギラなど、これまでは恥ずかしながらどこにあるのか地図上で指し示せなかったような島々が主なデスティネーションだ。実際に訪れて目の当たりにする、それぞれの国や地域の文化、個性、人々との出会いや発見が待っているのである。. NEXT PAGE. 幸福度が高い、自然の楽園。コスタリカ. 1. 2 3 REACTION. LIKE. COMMENT. 4. 連載をフォローする TAGS. # 子育て・家族 # 海外から SHARE. ゲスト MY PAGE ログアウト 新規登録 ログイン LIKE. COMMENT. 4. 他の記事へ ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを 2022.06.02. ゲスト 4. ログイン. 利用規約 送信する. &MEMBER（登録無料）としてログインすると. コメントを書き込めるようになります。. 評価の高い順 ▼. 最新順 ▼. 古い順 ▼. ほんだ. 2022年6月6日 9:51 PM. 久保ちゃん、毎月素敵な時間をありがとう。長い夏休みって読んでいたけど、これだけ時間があればカリブ海は良いですね。日本から行くとなると時間が相当掛かりますがニューヨークからだと時間も費用も格安ですね。そして素晴らしい空間ですね。海外だなって強く感じますよね。日本にない文化。どうか今年も素晴らしい時間を！そしてまたレポートしてくださいね。待っています。. あらちん. 2022年6月6日 11:32 PM. 羨ましいの一言です✨私が海外に行ったのは2015年のニューヨークが最後です(笑)いつか長い休みを取ってカリブにも行ってみたいです。が、もうしばらく働かないと、そんな長い休暇はダメかな😅. もりもり. 2022年6月7日 7:03 AM. 日本はまだまだ海外へ行く事にはすんなり行ける状態ではありません。文化によって国民性やそこで住む感性によって随分人生の幸福感が違いますね。記事を読んで今の現状に感謝して。という事にハッと気付かされました。コロナ禍で何をもって幸せなのかを考えた人も沢山います。仕事以外でも幸福感を得られる人生を教えていただきありがとうございます。. himanaco. 2022年6月7日 9:12 AM. 毎回、久保純子さんの記事を楽しみにしています。今朝は、梅雨に入り、お天気も気持ちもどんよりしていたところにこの記事！！. 「Pura Vida!」人生楽しまなきゃ！！パワーを貰えた記事でした。まだまだ行きたい国がたくさん！. 次回も楽しみにしています。. BACK NUMBER. この連載をもっとみる. NY発　子どもたちの夏の過ごし方 2022.05.09. 変わりゆくファッション。心にも変化が 2022.04.04. 次女の小学校生活から「こんな教育を受けたかった！」 2022.03.07. マンハッタンを抜け出して。コロナ下、冬の過ごし方 2022.02.07. マンハッタンの住宅事情。ただ今、アパート探し中！ 2022.01.10. FOR YOU. あなたにおすすめの記事. LIFE 2022.06.01. ＆w. リノベーション・スタイル 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” FOOD 2022.05.31. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet ARTS & CULTURE 2022.05.27. ＆w. 30歳からのコンパス 寂聴さんとの10年間。瀬尾まなほさん「一つ一つがいまの私の羅針盤」 FASHION 2022.05.27. ＆w. シトウレイの好きな服、似合う服。 「何を着るか」にマンネリを感じたら、冒険しませんか？ FOOD 2022.05.26. ＆w. 料理家・冷水希三子の何食べたい？ 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき LIFE 2022.05.26. ＆w. 花のない花屋 スペインから父へ。直接その手を握る代わりに POPULAR. 人気記事. アクセス LIKE コメント 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. リノベーション・スタイル 〈244〉適度な距離感でいい関係に。“家族でシェアハウス” 0. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet 1. ＆w. 久保純子 LIFE in N.Y. Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 4. ＆w. 花のない花屋 ヒマワリが大好きなあなたに。これからたくさんの笑顔と幸せを 0. ＆w. 東京の台所２ 〈254〉「ワイン一杯とタバコを」。救急車を待つ彼は最期に言った 0. ＆w. 東京の台所２ 〈255〉50歳で選んだ別居婚「第二幕」 0. ＆w. 料理家・冷水希三子の何食べたい？ 夏みかん香るソースで爽やかに。ハーブたっぷりの鰹のたたき 0. ＆w. 猫が教える、人間のトリセツ 猫に学べ。by サジとスプン（飼い主・三宅和歌子さん） 0. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet 1. ＆w. 久保純子 LIFE in N.Y. NY発　子どもたちの夏の過ごし方 5. ＆w. 久保純子 LIFE in N.Y. Pura Vida！ カリブ海で触れた「人生を楽しむ」生き方 4. ＆w. 東京の台所２ 〈256〉リュックひとつで上京。毎日自炊、夢はカンヌ 1. ＆w. このパンがすごい！ まるで工芸品！　東京で味わえる北欧のシナモンロール／torpet 1. ＆w. このパンがすごい！ 毎日食べたくなる！　花も酵母も慈しむ店主の軽快なごほうび感／ベーグルスタンダード 1. ＆w. ※アクセスは過去7日間、LIKE、コメントは過去30日間で集計しています。. RECOMMEND. おすすめの記事. 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト サイトポリシー 利用規約 広告事例・ガイド 個人情報 アクセスデータの利用 著作権 お問い合わせ 朝日新聞デジタルに掲載の記事・写真の無断転載を禁じます。すべての内容は日本の著作権法並びに国際条約により保護されています。. 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{"id":"qaAhUW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/07\/leitzel.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":71,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","07"],"url_file_name_s":"leitzel","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":1,"collection_sxt":["user"],"title":["leitzel.pdf"],"author":"pdfTeX-1.40.3","author_sxt":["pdfTeX-1.40.3"],"last_modified":"Sat Jan 09 05:43:07 UTC 2010","dates_in_content_count_i":1,"dates_in_content_dts":["Fri Aug 11 00:00:00 UTC 2006"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":4478705955628426986,"exact_signature_unique_b":true,"fuzzy_signature_l":-4294491452936172358,"fuzzy_signature_unique_b":true,"wordcount_i":5282,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:21:54 UTC 2022","fresh_date_dt":"Wed Nov 22 17:01:19 UTC 2023","referrer_id_s":"w8ZqeP51bhow","publisher_t":"pdfTeX-1.40.3","language_s":"en","size_i":147327,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088054614360064,"text_t":"To appear, American Mathematical Monthly, 2010.\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES\nFRANCIS EDWARD SU\nAbstract. What does it take to turn a learner into a discoverer? Or to turn a teacher into a\nco-adventurer? A handful of experiences—from teaching a middle-school math class to doing\nresearch with undergraduates—have changed the way that I would answer these questions.\nSome of the lessons I’ve learned have surprised me.\nThe title of this article may seem a little puzzling. After all, the words teaching and research\nusually only appear in the same sentence when separated by the word and, on a list of a faculty\nmember’s obligations. Do these words belong together at all? What does teaching have to do\nwith research or discovery?\nOne of my favorite quotes about teaching comes from one we normally associate with re-\nsearch:\nThe principal aim of mathematical education is to develop certain faculties of\nthe mind, and among these intuition is not the least precious. It is through it\nthat the mathematical world remains in touch with the real world.\n—Henri Poincaré [5, p. 128]\nBut Poincaré also said:\nIt is by logic that we prove, but by intuition that we discover.\n—Henri Poincaré [5, p. 129]\nSo we teach to build intuition, and intuition enables our students to make discoveries. Notice\nthat Poincaré does not say that the principal aim of teaching is to convey facts or theorems.\nThe principal aim of mathematical teaching is to build qualities of mind that enable students\nto make discoveries.\nBut how does one do that? How does one turn a learner into a discoverer? When I was first\nstarting out as a new professor, I might have given these answers:\n• Teach the needed background.\n• Cultivate maturity.\n• Inspire them!\n• Ask good questions.\n• Select the smartest students.\n• Give open problems.\n• Advertise the thrill of research.\n• Be an expert in what you advise.\n• Encourage independence.\nNow, I shall explain why I believe every one of these pieces of advice is either plainly wrong\nor, at best, inadequate. Along the way, I will mention some concrete teaching ideas that you\nmay find useful as well.\nThis article is based on the James R. C. Leitzel Lecture, delivered by the author at the MAA MathFest,\nKnoxville TN, on Friday, August 11, 2006.\n1\n2 FRANCIS EDWARD SU\n1. The Principal Aim of Mathematical Teaching\nThere is one word that is an excellent metaphor for everything I want to say:\nYAWP.\nWhat is a yawp?\nI sound my barbaric yawp over the roof(top)s of the world!\n—John Keating, quoting a Walt Whitman poem,\nin the movie Dead Poets Society\nAs Keating explains, a yawp is a loud cry or yell. But in the poetry of Walt Whitman, the\nword yawp refers to the inner groaning inside each of us, too deep for words, that is waiting to\nbe released and experienced. In the movie Dead Poets Society, an English teacher named John\nKeating (played by Robin Williams) does more than just teach poetry—rather, he breathes\npoetry, he inhabits poetry, and he inspires his students to do the same. In one scene, his\nstudent Todd Anderson (played by Ethan Hawke) has been struggling all weekend to write a\npoem, but without success. In class, when Todd fails to deliver a poem, Keating encourages\nTodd to first find his yawp, and once he does, Keating helps him transform his yawp into\npoetry in a truly mesmerizing manner [6, 55:07-57:57].\nKeating was an exuberant teacher who exhorted his students to live extraordinary lives. He\ndid not measure this by the usual metrics of success and accomplishment; rather, for Keating,\nliving extraordinarily meant becoming who they were created to be: human beings empowered\nto express their passion through poetry, and to use their souls, spirits, hearts, and minds to\nseize the day with all its wonders.\nAs mathematicians, we have experienced one of the greatest human wonders: the thrill of\ndiscovery. And the aesthetic pleasure of an elegant proof captures our spirits just as poetry\ndoes. So what I seek to cultivate in my students is a mathematical yawp:\nDefinition. A mathematical yawp is that expression of surprise or delight at discovering the\nbeauty of a mathematical idea or argument.\nEvery student is capable of a mathematical yawp. The yawp may not be a poem when it\nis first expressed, but I believe the principal aim of mathematical teaching is to nurture the\nmathematical yawp, and help transform it into poetry.\n2. Math Fun Facts\nWhen I first began teaching calculus in graduate school, I lamented the fact that most\nstudents would leave college with a mistaken notion of what mathematics is. After all, most\nof them took math to fulfill a requirement, usually some brand of calculus or something less\nadvanced, and it was the final course they would take. As a result, the view most students\ndevelop of mathematics is that it is a cut-and-dried, 400-year-old list of theorems and appli-\ncations. They do not get the sense that mathematics is an exciting, living, and developing\nsubject.\nI wanted to change that impression. So I began to present daily “Fun Facts”, by taking\n5 minutes at the start of each lecture to show my students some mathematical idea that I\nthought was fascinating—for instance: the uncountability of the reals, the nine-point circle,\ntiling a chessboard with opposite corners removed, the ham sandwich theorem, etc. The facts I\nchose usually had nothing to do with calculus; the main point was to broaden their perspective\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 3\nof mathematics, to show them math is alive and full of interesting questions and new ways of\nthinking, and to whet their appetite for learning more.\nThe student reaction was overwhelming. They loved Fun Facts. They clamored for them at\nthe beginning of every class. Students stopped coming late! If I ever forgot to do a Fun Fact,\nthey would remind me of my obligation, as if going through Fun Fact withdrawal. Sometimes\nthey would clamor for more than one (though perhaps to postpone the calculus lecture). I had\nnurtured their mathematical yawp.\nAnd they yawped for more. They would often come back to me with questions about Fun\nFacts, or variants of Fun Facts they had studied or solved. Many of my teaching evaluations\ncited Fun Facts as a reason that they wanted to take more math, even if they didn’t like\ncalculus. I had helped them find their yawp, even if it wasn’t for calculus!\nAnd then it dawned on me that it was more important to nurture their yawp than to teach\nthem calculus.\nLesson #1. Teach the needed background?\nNo. Nurture the yawp.\nOf course, we hope that these goals are in harmony, but if there is ever a conflict, we must\nremember the principal aim. For instance, I make it a policy never to stop nurturing the yawp\nin favor of “getting through the material”.\nIf you’d like to try using Fun Facts in your courses, I have created a web archive of about\ntwo hundred of the ones that I have used. (See [7] or Google “Math Fun Facts” to find it.)\nBut the best Fun Facts are the ones that you care about the most! Give them the Fun Facts\nthat fueled your passion for mathematics.\n3. What I Learned from 8th Graders\nI found a way to use a similar idea in the K-12 arena. One of the most formative experiences\nof my career occurred in my second year as a faculty member, when I learned that a principal\nof a local middle school was looking for a professor to teach a math enrichment course to\nchallenge some bright 8th graders. Intrigued by the idea, I agreed.\nOnce a week, the 8th grade math class at this school was divided into two groups by ability.\nOne half stayed with the usual teacher to get some needed help and extra practice, and the\nother group met with me.\nWhat should I do with them? Instead of teaching them more of the same stuff that they\nmight encounter again in high school, I chose instead to teach them “enrichment material”,\nsuch as extended Fun Facts, proof techniques like induction and contradiction, fast mental\narithmetic, magic tricks using mathematical ideas, elementary knot theory and Reidemeister\nmoves, mind-boggler puzzles, elementary combinatorics, etc.\nMany things about this experience surprised me. First of all, teaching 8th graders is a\nlot harder than teaching college students! Any issue related to teaching college students is\namplified for 8th graders. If college students can’t sit still for more than fifteen minutes, 8th\ngraders can’t sit still for five. College students get bored; 8th graders noticeably fidget in their\nseats. College students give nodding winks at each other; 8th graders pass notes. If college men\nspeak up in class more than college women, 8th grade boys dominated my class discussions\nentirely.\nAs a result, for everything I would normally do to engage a college class, I had to do better\nwith these 8th graders. If my college teaching style is interactive, my 8th grade teaching style\nhad to be more so. I had to pay more attention to domineering students, give more reassurance\n4 FRANCIS EDWARD SU\nand encouragement, repeat ideas more often, ask more questions to keep them engaged and\nignite their curiosity and imagination ... and it occurred to me that doing these things was\ngreat for my college teaching!\nLesson #2. Cultivate maturity in your students?\nNo. Revive their child-like curiosity and imagination.\nI believe there isn’t a single concept in a first-year college course that an 8th grader couldn’t\nunderstand if developed well. So what will you do to develop it well, to engage your students?\nWe must find ways to revive the child-like curiosity of college students. I see that 8th graders\nare more curious, more willing to probe with a “why?”, more willing to ask questions if they\ndon’t understand, and more willing let you know your pen is leaking in your front pocket.\nSomewhere along the path to adulthood, they lose the ability to ask questions.\nWas this enrichment class a success? Did they develop a mathematical yawp? In the end-\nof-term evaluations, I asked these 8th grade students to complete the following sentence:\nI now know that mathematics is ...\nI received some responses like this:\n• ... harder than I thought (I thought it was too easy before).\nBut most of the responses were like these:\n• ... extremely challenging, yet excessively fun. Also, it’s a much broader\ntopic than I had originally thought.\n• ... can be fun and not always so boring. I also now know that sometimes,\nmathematics can have absolutely nothing to do with numbers.\n• ... a broader subject than it was to me before; this class opened up many\nnew, exciting, and challenging topics that have made me more interested\nin mathematics as a whole.\n• ... a world of mysteries waiting to be discovered.\nExactly! We seek to build intuition to enable them to make discoveries!\nRecently, I heard from a student in this class who is now just graduating from college, eight\nyears later. He said: “from what I remember I really enjoyed it ... in middle school it was the\nonly class that really opened my eyes to different thinking.” All these responses indicate that\nthey had developed a yawp.\nAnother lesson I learned from this class was how easy it was to miss hearing a yawp. There\nwas a shy girl in this class who hardly ever spoke up and seemed not to be engaged at all. I\ndidn’t notice her for several weeks until I assigned some homework on induction. I was curious\nto see what 8th graders could do, and I naively expected the best students to be the ones who\nwere answering all the questions.\nBy and large, the students seemed to grasp induction. But I’d say the homeworks looked like\nthe typical work you would get from beginning calculus students, not so good on the form and\nvery unclear in the writing. On the other hand, this student’s induction proof was astounding.\nIt had perfect form, perfect understanding, perfect writing—in complete sentences with good\nconnective phrases, journal style, with the clarity and maturity that I expect from professional\nmathematicians.\nThroughout the term she continued to impress me, always in her quiet way. What was\ninteresting to me was that her regular 8th grade math teacher did not seem to think she was\nexceptional at all! Why?\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 5\nAnd then I wondered how many students like her I was missing in my college courses because\nof the way I was teaching, and what I could be doing to look for their yawp.\nLesson #3. Identify invisible yawpers.\nAs a sad footnote, after the class was over, I told this student’s parents that if she should\nshow an interest in math later on, they should encourage it because she was truly exceptional.\nTheir reaction (or lack thereof) was as if to say: why would I want my girl to study math?\nAnd I understood then that there are still social pressures that push children out of callings\nfor which they might have a gift.\n4. Nurturing Poets\nI surveyed a dozen of my former thesis and research students and asked them to reflect\non their experiences doing research as undergraduates, whether with me or in other settings.\nI refer to them as Poets, since they have found their yawp. Nearly all of them are now in\ngraduate school or just beyond.\nOne question I asked them was what early influences caused them to think research in\nmathematics was something they wanted to try. One Poet said:\nMy parents were a huge influence on me, posing mathematical questions to me\nwhen I was young ... for example, I remember my dad asking me how many\nsteps it would take to get to the wall if on each step, I went halfway.\nShe had found her yawp because her parents nurtured it by asking probing questions, but\nthey let her experience the joy of discovery herself.\nMore than half the students cited the experience of working puzzles or doing math problem-\nsolving as early influences. Many had been part of organized problem-solving groups, or at\nsummer math programs where they, in the words of one Poet, “caught a fever of math research”.\nIn effect, these groups became their “Dead Poets Societies”, places and spaces where they could\ngather to recite poems and yawp together.\nAt Harvey Mudd College, Andy Bernoff and I run a problem-solving seminar that has become\nsomething of a phenomenon on campus. Every fall, about 70 students (that’s 10 percent of\nour student body!) gather on Tuesday nights over pizza to have fun with problem-solving.\nWe keep the atmosphere lively, and we allow them to work in groups. We introduce a few\nproblem-solving techniques at the start, then let them work on a slate of five or six problems,\nbreak for pizza, then we have students present their solutions [1, 2]. They have the space to\nmake discoveries themselves, and by letting them present their solutions, we give them the\nspace to yawp as well (just as any good Dead Poets Society would do).\nThis seminar now attracts a large fraction of students who are not even math majors, but\nwho come for the thrill of yawping, either when they solve the problems, or when they see a\nsolution and get the “aha!” experience that is at the heart of a true yawp.\nWould I still say it is important to inspire students? Of course. But I think it is even better\nto give students space for self-inspiration so they can experience and express their yawp.\nLesson #4. Inspire them?\nBetter: create spaces for self-inspiration.\n6 FRANCIS EDWARD SU\nMoore-method courses are another example of spaces where students can find their yawp\nand make discoveries themselves. I won’t say much about them here, because much has been\nsaid about them, but the best course I ever took as an undergraduate was a Moore-method\ntopology course (taught by Mike Starbird) and it had a profound impact on developing my\nyawp.\nWe all understand how important it is for a professor to ask good questions of her students.\nAfter all, a good question can prompt a discovery. But I now believe it is even better if you\ncan teach a student how to ask good questions themselves!\nLesson #5. Ask good questions?\nBetter: teach how to ask good questions.\nLet me give you an example that I learned from my colleague Lesley Ward. She has used\n“question stems” in her courses, to teach students how to ask good questions (using a suggestion\nof T. J. Mueller, and adapting questions from [3]). For instance, in an analysis course, after\ngiving part of a lecture on some topic, she asks students to form groups and she hands out\ncards which have questions like these on them, with blanks:\n• What is an example of ...?\n• How does ... affect ... ?\n• What would happen if ... ?\n• What is the difference between ... and ...?\nThe students formulate questions about the lecture based on these models, and then they\ndiscuss them in class. In addition to the benefit of teaching students the kinds of questions they\nmight think about, it has the added benefit of students often answering each others’ questions!\nAnother way of encouraging students to ask good questions is to reward them for it. On\nsome exams, I put a problem in which I ask students to write down a research question that\nthey would like to answer, based on what they have learned so far in the class. The answer to\nthe question doesn’t have to be unknown to the world, it just has to be unknown to them. I\nam happy if they have just gone through the reflection necessary to ask a question, so I give\nfull credit for any reasonable attempt at formulating a question.\nFor our students to experience true research, we should certainly give them open problems.\nBut in order to cultivate a yawping ability, I believe it is much better to give students open-\nended problems.\nLesson #6. Give open problems?\nBetter: give open-ended problems.\nI learned a terrific example of this from my colleague Michael Orrison, who has developed a\nproject that he has used successfully in a discrete mathematics course. In it he asks students\nto:\nDefine a measure of the complexity of a graph. Compute it for several examples,\nand prove some properties about your measure.\nThis is the main idea; you can find more details in [4]. What is great about this exercise is\nthat it is open-ended, and there are multiple right answers. Students are able to be creative\nabout the solution, and they are excited about investigating a concept that they have defined\nthemselves. At the end of the project, students make presentations.\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 7\nLooking at some of the slides from past presentations, one sees that students develop a wide\nvariety of notions of complexity. More interestingly, they find creative ways of expressing their\nideas, e.g., by describing their complexity measure as answers to questions: “How expensive is\nit to plow the town?”, “How many friends do you have?”, etc.\nAnd their yawps are often visible as they explore properties of their measures of complexity—\nfor instance, written on one recent memorable slide, a student exclaims “Really Cool!” and\n“Yowza!” when she realizes that her measure is invariant under taking graph complements.\nThat’s a yawp if I ever saw one!\nMy colleague Darryl Yong has even adapted Orrison’s project as an exercise for high school\nteachers, in a conference called “Imagine Math Day” [8, 9]. And there are many other project\nexamples that could follow a similar model. For instance, “What is a fair voting procedure?”\nor “Define a measure of a center of a distribution.” Last summer, I gave a couple of my\nresearch students the following open-ended problem: “List 10 generalizations of convexity.”\nAfter they came up with those ten, I asked them to focus on the most interesting ones and try\nto characterize those notions of convexity in terms of properties that people already understand.\nMy students had a lot of fun generating their list, and eventually took one of their notions and\nproved necessary and sufficient conditions that would allow someone to tell if a set satisfied\ntheir notion of convexity. This resulted in a paper that they wrote completely on their own,\nand submitted for publication. Because of one open-ended problem, they created a new idea,\nand they were motivated to study it and explore it—they yawped!\n5. Research with Undergraduates\nBut now suppose you have taken your students to the point where they can yawp. How do\nyou help them turn it into poetry? If a yawp is the thrill of discovery, a poem is a yawp that is\ncommunicated well. This is where the undergraduate research experience can play a big role.\nOne major question that every advisor faces is the question of how to choose students. I\nalways thought that the best metric for this was to choose students who were the smartest, or\ngot the best grades. However, I now think that something else is much more important—strong\nmotivation and persistence.\nLesson #7. Select the smartest students?\nNot necessarily; select motivated students!\nOne of the best research students that I have ever had was not a student that fit the mold\nof getting the best grades, and I did not notice her abilities from her coursework. She was\na B+ student, and when she approached me about summer research I had already offered\nmy research positions to other students. But she came back several times to ask, so I finally\ndecided to go out of my way to ask the dean for extra funds to support her, and he agreed.\nThis was one of the most rewarding experiences that I have had as an advisor. The persis-\ntence my student showed in inquiring about research opportunities translated into a tenacity\nin tackling research problems. She was extremely dedicated and motivated, and always came\nto research meetings having done everything that we discussed at the last meeting, and more.\nShe was the kind of student who was meticulous and took her time to carefully think things\nout, and so while taking exams was not a place where her gifts were displayed, she really shone\nin research problems where persistence and meticulousness are rewarded. And she was an\nexcellent writer. By the end of the summer, she had completed a paper that was polished and\nready to submit!\n8 FRANCIS EDWARD SU\nSo, if I had to give advice about what to look for in a research student, I would say smarts\nare important, but if you have to choose between a super-smart student and a smart motivated\nstudent, I would go for the latter. Some qualities to look for: Are they persistent? Do they\nexcel in something that shows dedication? For instance, we have a couple of research students\nin our department this summer who are B students, but one is a nationally-ranked player in\nthe computer game StarCraft, and another is a hackey-sack expert. And their advisors report\nthat these students are doing outstanding, publishable research.\nAnother question I ask about potential research students is: do they write well? I ask for a\nwriting sample, such as a proof they have written for a homework exercise. Even if you don’t\nchoose a student who can write, it is nice to know what kind of writer you are working with.\nSuch information can be useful, for instance, if you are choosing students to work in teams.\nLesson #8. Advertise the thrill of research?\nBetter: set complete expectations.\nI used to want my research students to think research was going to be really fun and exciting.\nBut now, I think that they should be aware of the whole picture. I now tell my students in\nadvance that research can be frustrating at times, and I warn them that often it hits about\nthe fourth week of the summer when they are really wrestling with their research problem. If\nI tell them in advance, then they aren’t surprised when it happens, and they understand it is\nnormal to feel that way.\nAlong these lines, one Poet says:\nThe most valuable aspect, in retrospect, was how my research jobs made re-\nsearching look far less glamorous than I had thought it might be ... in a way it\nwas demoralizing, but you likely want your bubble burst before you chase the\ndream too far.\nBut the same student also notes that understanding the expectations doesn’t diminish his\ndesire to yawp—he says: “The hope of delightful discoveries and taking part in that same cool\nelegance I recognized so long ago is still a large part of what keeps me going.”\nI also set clear expectations for my students about how to conduct their research. For\ninstance, I ask them to keep track of what they are doing in a research notebook. I explain\nthat this notebook is where they write down ideas that they want to follow through with later.\nI also tell them to start writing often and early in the process. When they get stuck or bored\nwith thinking about their problem, they can write down what they have been doing, or learn\nLaTeX and LaTeX their work. Another Poet reports:\nOne of the best things I learned was organization ... especially since none of\nmy later advisors mentioned this at all.\nWe can teach undergraduates research skills that will be useful to them later, even if, as the\nPoet suggests, graduate schools don’t teach those same skills.\nIn advising research students, I always had the impression that you should be an expert in\nwhat you advise. After all, I think it is common for us to worry: how can we nurture a yawp\nin someone else if we don’t know how it is we yawp ourselves? But now I think differently.\nLesson #9. Be an expert in what you advise?\nNo, let the student be the expert.\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 9\nWhat do I mean by this? I think it is extremely important to give my student ownership\nof the problem. I tell my students at the start of the summer: “By the time the summer is\nover, you will be the local expert on this problem. You will know more than anyone else here\nabout this problem, including me.” Doing so gives them the excitement of living up to that\nchallenge, and my job the rest of the summer is to probe, and model asking good questions\n(rather than trying to give them all the answers that I obviously don’t yet have). So they are\non an adventure to find out more than anyone else in my local neighborhood about their topic,\nand, by accompanying them on this journey, always asking or suggesting (but never forcing)\nwhere to go next, I am a “co-adventurer” with them. I listen to their yawps on this journey.\nNot surprisingly, my research with students has taken a random walk through many fields; as\nan example, our papers have been published in a diverse collection of journals: Random Struc-\ntures and Algorithms, Journal of Combinatorial Theory Series A, Discrete and Computational\nGeometry, Journal of Mathematical Analysis and its Applications, American Mathematical\nMonthly, Journal of Mathematical Biology, among others.\nIn all these examples, I was never the expert on these topics, but I picked up what I needed\nto know in a journey with my students, often with the students teaching me. I am only a co-\nadventurer. Students also learn from this journey that research often involves learning many\nnew and unfamiliar things.\n6. Building Community\nOne of the biggest pieces of advice that I hear about advising research is the adage about\nencouraging students to be independent. While I believe this should be true much later in the\nresearch process, I strongly disagree with this in the early stages.\nLesson #10. Encourage independence?\nNo! Give close guidance, and build community.\nFor instance, I think it is very important to meet often and regularly, especially in the\nbeginning of the research process. During those meetings, one should always set goals for the\nresearch students to work on for the next meeting. Doing so gives them something concrete\nto do and to report on and gives structure to the following meeting. It is especially important\nto have students write something up for each meeting. Doing so refines their thinking, and\nfocuses discussion.\nWhen it comes time to write a real paper, I always sit down with my student in front of\nthe computer and write, together with the student, at least a few pages. This not only helps\nget the paper off the ground, but doing so also models for the student the process of writing.\nStudents are often initially surprised by the care which I take to get definitions right, to make\nnotations consistent, or to ensure that attributions about prior work are correct. They get to\nsee how I think about writing and they can begin to do the same.\nThroughout their research, I want my students to see that mathematics really is a social\nenterprise. We, as a community, decide what’s important, our work depends on the work of\nothers, and we work together with others to advance the field. In one project, I included an\nundergraduate student on a project that I was doing with an economist in Germany. I included\nmy student in all the meetings (some by phone and some when my collaborator visited). By\nbeing fully part of this process, she was able to see one way in which collaboration could work,\nas well as see in this social setting how important it is to learn to communicate ideas well.\nPoets report:\n10 FRANCIS EDWARD SU\n• My research experience definitely helped me learn about writing papers,\nand perhaps more importantly, about working with others to co-write pa-\npers.\n• Working with other people besides you demonstrated the different possi-\nbilities when people with different strengths collaborated.\n• Working with another person on a research question is fundamentally dif-\nferent than any of the other group work I had been exposed to.\nI want students to see the importance of building community in their mathematical work. I\nhope that they can see that through collaboration, they can really have a lot of fun yawping\ntogether, whether that be through the sweat of persisting through a problem or the joy of\nactually solving it. As the Swedish proverb goes:\nA joy shared is twice the joy. A sorrow shared is half the sorrow.\nTo summarize, how would I answer the question: How do you turn a learner into a discoverer?\n• Nurture the yawp.\n• Revive child-like curiosity and imagination.\n• Identify invisible yawpers.\n• Create spaces for self-inspiration.\n• Teach how to ask good questions.\n• Select motivated students.\n• Give creative, open-ended problems.\n• Set complete expectations.\n• Let the student be the expert.\n• Give close guidance, and build community.\nJohn Keating did not teach his students poetry; he helped them yawp, he helped them\ninhabit the verse. He encouraged playful child-likeness. He created the space for his students\nto self-inspire in a resurrected Dead Poets Society, even though he himself never attended a\nmeeting. He taught them how to find good subjects for poems. He engaged the students\nwho were motivated, who had ears to hear. He encouraged creativity, and he encouraged\ncommunity.\nLet’s help our students find their yawp and transform it into poetry.\nAcknowledgements. I thank Art Benjamin, Jon Jacobsen, Susan Martonosi, Claus-Jochen\nHaake, Ulrike Ervig, Mike Starbird, and Bruce Palka for their advice in preparing the Leitzel\nLecture. I gratefully acknowledge NSF Grants DMS-0301129 and DMS-0701308 for supporting\nmy research with undergraduates.\nDedication. This article is dedicated to: Jim Leitzel and Christine Stevens, co-founders of\nProject NExT, who have nurtured the professional development of more than a thousand\nmathematicians and given them places and spaces to yawp; and to Michael Moody, who will\nalways be my professional Keating—O captain, my captain!\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 11\nReferences\n[1] A. J. Bernoff and F. E. Su, PCMI problem-solving references and resources (2003–2009), available at\nhttp:\/\/www.math.hmc.edu\/~ajb\/PCMI\/problem solve.html.\n[2] , Putnam, pizza & problem-solving, Math Horizons (Sept 2004) 8–9.\n[3] A. King, Inquiry as a tool in critical thinking, in Changing College Classrooms, D. F. Halpern, ed., Jossey-\nBass, San Francisco, 1994, 13–38.\n[4] M. Orrison, Graph complexity, in Resources for Teaching Discrete Mathematics: Classroom Projects, His-\ntory Modules, and Articles, B. Hopkins, ed., MAA Notes #74, Mathematical Association of America,\nWashington DC, 2009, 159–161.\n[5] H. Poincaré, Science et méthode, E. Flammarion, Paris, 1908; English trans. F. Maitland, Science and\nMethod, T. Nelson, New York, 1914.\n[6] T. Shulman, Dead Poet’s Society (dir. Peter Weir), Walt Disney Studios Home Entertainment, Burbank,\nCA, 1989.\n[7] F. E. Su, Math Fun Facts (1999–2009), available at http:\/\/www.math.hmc.edu\/funfacts\/.\n[8] D. Yong, Imagine Math Day (2009), available at http:\/\/www.math.hmc.edu\/pdo\/imd\/.\n[9] D. Yong and M. Orrison, Imagine Math Day: Encouraging secondary school students and teachers to\nengage in authentic mathematical discovery, MAA Focus 30, no. 6 (2008) 24–27.\nFRANCIS EDWARD SU earned his B.S. in mathematics at the University of Texas at Austin, and his\nPh.D. at Harvard under the direction of Persi Diaconis. He is now Professor of Mathematics at Harvey Mudd\nCollege. His research is in geometric combinatorics and applications to the social sciences, and he has co-\nauthored numerous papers with undergraduates. He has a passion for teaching and popularizing mathematics;\nhis Math Fun Facts website gets over a million hits each year. From the MAA, he received the 2001 Merten\nM. Hasse Prize for his writing and the 2004 Henry L. Alder Award for his teaching. He currently serves as\nFirst Vice-President of the MAA. In his spare time, he enjoys songwriting and gardening and is active in a\nChristian community where he can just be himself—passion not stifled by expectation, worth not derived from\naccomplishments.\nDepartment of Mathematics, Harvey Mudd College, Claremont CA 91711\nsu@math.hmc.edu\n"}
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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Writing Math Well. Francis Su. Writing Math Well. Here are a couple of handouts that I created to coach good mathematical writing:. At the lower-division level, a 1-page handout “. Writing Mathematics Well “. At the upper-division level, a 3-page handout for mathematics majors “. Guidelines for Good Mathematical Writing “. You can also download the. published version in MAA Focus. My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. 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{"id":"jwiLTW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/11\/Starbird-Su-Topology-Instructors-Resource-Nov-2019.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":114,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","11"],"url_file_name_s":"Starbird-Su-Topology-Instructors-Resource-Nov-2019","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Starbird-Su-Topology-Instructors-Resource-Nov-2019.pdf"],"author":"pdfTeX-1.40.16","author_sxt":["pdfTeX-1.40.16"],"last_modified":"Tue Nov 19 09:40:56 UTC 2019","dates_in_content_count_i":1,"dates_in_content_dts":["Sat Dec 03 00:00:00 UTC 2011"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-944495631110735123,"exact_signature_unique_b":true,"fuzzy_signature_l":-5573854848959605887,"fuzzy_signature_unique_b":true,"wordcount_i":4476,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:21:55 UTC 2022","fresh_date_dt":"Sun Sep 17 21:09:33 UTC 2023","publisher_t":"pdfTeX-1.40.16","language_s":"en","size_i":139514,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088055979606016,"text_t":"Topology Through Inquiry\nInstructors’ Resource Manual\nby\nMichael Starbird\nFrancis Su\nNovember 19, 2019\nContents\nChapter 1. Educational Philosophy 3\n1. Watching 4\n2. Keeping up 4\nChapter 2. Role in the Mathematical Education of Students 7\n1. Prerequisites 7\n2. Introduction to topology and abstract mathematics 7\n3. Advanced Transitions-to-Proof 7\n4. Future teachers 8\n5. Quasi-research in the classroom and beyond 8\nChapter 3. Practical Issues 11\n1. Sample Syllabus 11\nChapter 4. An Introduction to Inquiry Based Learning Instruction 13\n1. First Day and Early Days 13\n2. A Typical Day 14\n3. Tips 15\nTry variation 15\nExperienced students 16\nLess experienced students 16\nStudent attitudes 17\nAudience 17\nStudents like this experience 17\nResources on Inquiry Based Learning Instruction 17\nChapter 5. Possible Paths for Various Potential Courses 19\n1\n\nCHAPTER 1\nEducational Philosophy\nTopology Through Inquiry is designed to be used with an instructional technique vari-\nously called Inquiry Based Learning (IBL) or guided discovery or some other phrase that\nsuggests active investigation on the part of the learner. These styles of instruction give stu-\ndents a totally different experience compared to a standard lecture course. We will explain\nthe method in more detail later, but the essence is this: At the beginning of the semester,\nstudents are handed the Topology Through Inquiry book, which contains a motivated treat-\nment of the ideas of topology, along with a list of theorem statements as well as exercises\nwithout proofs. The standing homework assignment is for each student to prove designated\ntheorems or settle the next designated exercises in the book, thus in a sense writing his or\nher own textbook. In class, the instructor asks students to present their own proofs to the\nrest of the class. The students have devised an argument for why the assertion is true and\npresent the proof. Other students try to understand the proof. The instructor encourages\nmembers of the class to ask questions or re-phrase parts of the proof. Each student is\nresponsible for determining whether the proof is correct. The correctness is not based on\nan authority figure (the instructor) asserting it is correct, but instead on the ability of the\nindividual students to follow the argument.\nHere is the effect of this experience: Students learn to think independently. They learn\nto depend on their own minds to determine right from wrong. They develop the central,\nimportant ideas of topology on their own. From that experience, they learn that they\ncan personally create important ideas. They learn to turn to themselves when they are\nconfronted with a question, whether it’s a mathematical question or a question in other\nparts of life. They develop an attitude of personal reliance and a sense that they can think\neffectively about difficult problems. They are less likely to accept assertions on the basis of\nauthority alone if they do not personally understand why the assertions are true.\nNotice that the effect on students goes far beyond mathematics and far beyond the\nclassroom. Education can shape minds. This type of course systematically develops habits\nof thought that change students for the better. It often opens their eyes to a new way of\n3\n4 1. EDUCATIONAL PHILOSOPHY\nlooking at their whole life experience. Almost all of education teaches students that they\nare powerless pawns whose job is to learn some fraction of what some authorities know.\nBut when students daily think through questions that seem hard and create new ideas on\ntheir own and determine correctness based on their own ability to discern the soundness\nof arguments, they frequently break free from the bonds of passive learning and simply\nexplode with an exuberance of independent learning.\nA guided discovery or IBL course can liberate students to learn and think on their\nown. The effect of this type of course can strike very deep. We are not just talking about\nteaching topology or theorem-proving well. That is important, but it is not as important\nas teaching students to be better, more independent thinkers every day, in every endeavor\nof their lives. Not every class or mathematics class needs to emphasize personal discovery,\nbut that experience can be an important piece of education that can affect how students\nlearn, work, and live. A single course is a small fraction of a student’s education, but a\ntiny habit of independent analysis repeated over a lifetime can make the difference of a\nlifetime. If people like to think and like to explore new ideas, the education they acquire\nthemselves over their lifetime will be vastly greater than any amount of material covered in\na semester’s class. Attitudes far outlast facts. Knowledge comes and goes, but hatred lasts\nforever. Perhaps the joys of thought and discovery last forever too.\n1. Watching\n. We live in a world where watching is a large part of the norm of daily existence.\nWatching smart phones, watching sporting events, watching the news. Time spent watching\nconsumes a large part of our days. Schools and colleges also tend to emphasize watching.\nClasses are frequently times for students to watch someone else do something. Apparently,\ngood students are those who watch well. These habits of passivity are hard to break. Our\neducational system systematically rewards attentive passivity. A high proportion of the\nexperience that our students receive through their education encourages them to look for\nauthorities and to view learning as the acquisition of facts. Yet our society needs leaders\nwho think independently, people who can cope with novelty and find ways to solve problems\nthat have not been clearly posed or answered.\n2. Keeping up\n. Knowledge is being developed or discovered at a faster rate than ever before. In\nthe technical fields especially, the rate of increase in knowledge is exploding. Each year\n2. KEEPING UP 5\nhundreds of thousands of articles are published in research journals in mathematics, biology,\nchemistry, and computer science. How do we as educators face the daunting task of keeping\nup?\nThe answer is, we cannot keep up. Keeping up is a feeble dream and has been for\ndecades. Now it is all the more important for us to empower our students to be able to\ndeal with situations and knowledge that they do not know and that no one knows. The\naccelerating production of knowledge means we must abandon the hope of keeping up with\nnew knowledge and must instead take seriously the job of teaching students the skills of\nnavigating in a sea of change. Change is the world for which we are attempting to prepare\nstudents. Some knowledge and skills are certainly fundamental and will always be useful,\nbut many specific skills that were previously the bedrock foundation of an educated person\nmust now be looked upon with skepticism. We must try to teach life lessons that students\nwill keep and use in settings we do not foresee. Topology Through Inquiry can contribute\nto creating in our students the skills and values that will empower them to cope effectively\nwith a fast-changing world.\n\nCHAPTER 2\nRole in the Mathematical Education of Students\n1. Prerequisites\nTopology Through Inquiry is designed to foster mathematical maturity by teaching stu-\ndents a suite of attitudes and skills. Students learn to prove theorems on their own and to\nassess the correctness of proofs. They learn many techniques of proving theorems. They\ngain experience in presenting their ideas orally and in writing. They learn fundamental ideas\nof topology. This course is designed to help students adopt a mathematical perspective that\nfeatures active participation in developing ideas that are new to them and in developing\nproofs of mathematical assertions.\nThis course could theoretically be presented to students who have had little or no\nproof-oriented mathematics previously. No specific mathematical content is required as a\nprerequisite. However, in practice we have found that the abstraction of topology often\npresents an obstacle to students who are new to proof. So generally, students will be more\nlikely to succeed with this book if they have had one or more proof courses before, perhaps\nincluding an introduction to analysis.\n2. Introduction to topology and abstract mathematics\nThe text provides a solid introduction to topology. The breadth of included topics make\nit suitable for any undergraduate or first year graduate topology course. In addition, there\nare many other topics that are not generally part of standard courses and will reward the\nengaged reader with some beautiful delights.\n3. Advanced Transitions-to-Proof\nThe text can be successfully used in courses meeting many educational goals. The\nconcept of transition-to-proof really embraces a continuum of mathematical maturity. One\nstep is learning specific techniques such as inductions or proof by contradiction. However,\nlearning to prove theorems on ones own also involves additional experience with facing the\nunknown and effectively struggling to seek insight. We strongly believe that topology is an\n7\n8 2. ROLE IN THE MATHEMATICAL EDUCATION OF STUDENTS\nexcellent subject for learning the art and practice of proving theorems independently and\nfor learning the ways of mathematical thought.\n4. Future teachers\nFuture teachers can gain special benefits from this course. They can use a deeper\nunderstanding of topology to enrich their future presentations of mathematics to their own\nstudents. They will gain experience in presenting mathematics and will be able to use and\nconvey to their students their increased skills in logical thinking and proofs. Also, they will\nhave a classroom experience of a style of pedagogy that they can draw from as they develop\ntheir own methods of teaching.\n5. Quasi-research in the classroom and beyond\nThe nature of the Inquiry Based Learning presentation makes the text eminently suitable\nfor giving undergraduates rich research-like experiences. The book can be used in a class-\nroom setting and for independent studies, honors projects, inter-session courses, or other\nquasi-research experiences. In fact, we believe an inquiry based learning experience can be\njust as, if not more, valuable than a “true” undergraduate research experience. Individual\nresearch projects for undergraduates present many challenges for faculty members and for\ninstitutions. Frequently, finding and directing an individual research project is challenging,\ntime-consuming, and the resulting experiences for the students are variable. Sometimes\nthey are the life-changing, exciting, intellectually satisfying experience we seek; and some-\ntimes the project turns out to have some unexpected defect, such as not yielding sufficiently\nmany results or a sufficient variety of challenges to be of great interest. These problems are\ninherent in research, but present special obstacles as undergraduate experiences.\nIn almost all cases, finding and overseeing an individual research project is a time-\nconsuming effort for the faculty member. Most colleges and universities do not have the\nfaculty resources to provide individual undergraduate research experiences for more than a\nsmall percentage of their undergraduates. An alternative is to offer inquiry based classes\nthat systematically provide a whole class of students with a quasi-research experience by\nasking students to tackle questions that are new to them, while not being new to the world.\nIn many ways this experience is often superior to individual research projects because the\ncareful control of the challenge questions gives students a predictable range of research\nexperiences. Students can develop attitudes of self-confidence and personal reliance. They\n5. QUASI-RESEARCH IN THE CLASSROOM AND BEYOND 9\ncan develop intellectual stamina and the sense that they can personally undertake to solve\nunknown problems that arise in their academic and personal worlds.\n\nCHAPTER 3\nPractical Issues\n1. Sample Syllabus\nWhat follows is a first day handout for an introductory topology course using Topology\nThrough Inquiry. It gives the students some sense of what the class will be like, but they\nwon’t really understand what is expected of them until a couple of weeks have passed.\n11\n12 3. PRACTICAL ISSUES\nTopology\nInstructor: Emmy Noether\nOffice Hours: By appointment\nRules of the game: The class will be conducted using a method of instruction called\nInquiry Based Learning or Guided Discovery. This method fosters creativity and indepen-\ndent thinking. It is also fun. Your book contains lists of exercises and theorem statements\nwithout proofs. You will do designated exercises and prove designated theorems on your\nown and present your results to the class. These presentations are a major part of the course.\nThe ideal model to follow is to settle the questions and prove the theorems independently\nand write-up your solutions before the answers are presented in class. You may not consult\nbooks, the Internet, other people, or other sources. [Note: Some instructors encourage\ncollaboration among the students in the class. With this approach, it is useful to tell\nstudents that before collaborating they should make a serious independent attempt at\nsettling the questions or theorems before discussing them with their fellow students.\nAnd the students should be instructed not to tell one another the complete solutions,\nbut just to discuss ideas and give hints. Collaboration is an excellent thing to foster\nand both strategies (not allowing collaboration or encouraging collaboration) can work well.]\nEach day, I will select students to present their solutions in class. Your standing assign-\nment is to be prepared to present your results. When you are presenting your proofs or\nsolutions, strive to make your explanations clear and organized. When you are observing\na presentation, it is your responsibility to follow the logic of the solution and verify that\nit is correct for yourself. You may be asked during class to re-explain an argument that\nyou just heard. If you cannot follow the reasoning, it is your duty to ask a question of the\nstudent(s) presenting. If you are truly stuck on a question or proof outside of class, do not\nhesitate to ask the instructor for help. You should be working far enough ahead of the\nclassroom presentations so that there is time for this consultation.\nHomework: Your standing homework assignment is to write up solutions to the designated\nexercises and theorems in your book Topology Through Inquiry before they are presented\nin class. You will submit your solutions on a daily basis. Please make every effort to keep\nyour solutions neat and clear. You should also keep a notebook containing all your notes\nand solutions; this notebook will serve as your personal textbook for this course and will\nhelp you study for exams. Writing up the proofs and solutions is an excellent way for you\nto learn the mathematics.\nExams: There will be two hour-exams during the semester. Each will be announced at\nleast one week in advance. The final examination will be comprehensive.\nGrade: Homework and presentations: 30%\nHour exams: 20% each\nFinal: 30%\nCHAPTER 4\nAn Introduction to Inquiry Based Learning Instruction\n1. First Day and Early Days\nOn the first day of class, you might give a little speech about the method and why you\nare using it. You might mention that the fun of mathematics is doing it for yourself and\n“doing mathematics” means to prove theorems, make conjectures, and settle questions on\nyour own. Many mathematicians remember the first time they proved things on their own\nas a special moment in their own histories. Emphasize that you know the students are not\nexpected to prove every theorem. Emphasize that everyone will make mistakes and that\nmistakes are valuable sources of knowledge. Allay their fears.\nAt this point start to establish habits for presentations. If you want them to write\nthe statement before the proof, have them do that. Ask them to write complete sentences\nincluding the connective words. You might ask students to go to the board in groups of\n3 or 4 and write a joint proof of the designated theorem. You might then choose a group\nwith a good proof. After the proof is presented, ask for questions. Ask individuals in\nthe class questions such as, “Why did the presenter write the second sentence?” Answer,\n“Because that explains what the definition of the symbol is.” Etc. Then praise the students’\nwork specifically, that is, say, “This proof is just right because the first sentence stated\nthe assumption of the theorem and every sentence either used a definition to restate the\nprevious statement or used previous statements to deduce their truth.” Then go on to the\nnext theorem, again in groups.\nYour goals in the first several days are to get the students comfortable with the basic\nidea that they can prove theorems on their own. You want to establish good form, that is,\nfull sentences, neat presentations, and not leaving out steps.\nDuring these first few days, your students will make some important mistakes. Ask other\nstudents if they agree with the proof. Ask them to state whether and, if so, why various\nstatements are true. Sometimes ask them about statements that are true and sometimes\nask about those that are not true. Avoid saying that a proof is correct or not correct. But\nif it is not correct, when you ask a student to explain why some sentence is true, they will\n13\n14 4. AN INTRODUCTION TO INQUIRY BASED LEARNING INSTRUCTION\nbe unable to do so. At that point you can say one of several things. You might ask the\nclass to get together in small groups to prove a statement whose proof was omitted. Or you\nmight ask the groups to see how to fix the proof. You might ask the presenters to expand\non an idea. If it is seriously flawed, you might ask the presenters to correct the proof and\nreturn at the next class period. Or you might ask someone else to do it. Always thank the\npresenters for helping the students to understand things. Sometimes after a correct proof,\nwe will ask a student whom we know to have an incorrect proof to show the wrong proof.\nWe might say from the start that the upcoming proof is wrong, but we want the class to\nsee it because it shows an important idea. This strategy gives students the idea that wrong\nproofs can be valuable steps toward understanding and they should not feel embarrassed or\ncrushed by making a mistake.\n2. A Typical Day\nDuring a typical day, the instructor might have six or so groups of three or four students\n(if your class is that large) to go to the board where each group has a designated theorem\nor exercise to consider. Ask each group to discuss and write down a proof of the designated\ntheorem. Ideally, every student is standing up, working in a group. Often the groups work\nfor perhaps 20 minutes while the instructor walks around from group to group listening\nand sometimes giving advice. After they have all written up their proofs, ask the first\ngroup to present their proof. The instructor stands near the back of the class, watching the\npresentation and watching the other students. After the group finishes the presentation, the\ninstructor conducts the discussion. Sometimes, you might ask, “Are there any questions?”\nOther times you might say, “Maria, would you please explain the second sentence of the\nproof?” And then, “Juana, do you agree with Maria or the presenters and why?”\nThe instructor’s goal is to engage as many students as possible during the class time.\nAvoid letting the best students dominate the discussion. Generally, don’t let the strongest\nstudents speak at all during a discussion of an easier theorem. Select a student to present a\ntheorem whom you guess to be about in the middle of those students who are likely to have\ndone that particular theorem correctly. You don’t always want a perfect presentation and\nyou don’t always want a total loss. Some instructors just choose presenters randomly, but\nafter you get to know the students, it is probably better to select students who are going\nto give the best experience from the presentation. A perfect presentation is not ideal, and\nneither is a complete loss.\n3. TIPS 15\nAfter the discussion of a presentation is over and the students feel the proof is correct,\nthen you might well make this kind of statement, “One really good aspect of this presen-\ntation was this: Did you notice that they wrote down why the open set U containing p\ncontained a point q of A? They stated that p was a limit point of A and that the definition\nof limit point states that every open set around a limit point must contain a point of the\nset other than the limit point itself. Then they had a specific point to work with.”\nIf the presenters get stuck, or an error is discovered, you might ask the class to get\ntogether in groups of 3 or so and work on an example that might help. Or you might ask\nstudents to think of an easier statement that might help. Or you might suggest a hint,\nbasically a lemma or a special case. You might ask them to work in groups or you might\ngive a hint and let them work on that for homework.\nAt the end of the class session, you might say, “We’ll start next time with Theorem 1.7\nand should be able to present 1.7, 1.9, 1.10, 1.14, 1.16, 1.22, and 1.25.” This gives students\nsome sense of what theorems they should feel responsible for for the next session.\n3. Tips\nTry variation. A “typical day” is rarely typical, and it is important to vary your\ninquiry based teaching techniques to find those that work best with your specific class and\nfor you. There are several aspects of the course where choices you make influence the overall\neffectiveness.\nWill you assign problems to individuals or groups in advance or will you ask for volun-\nteers or will you simply assign individuals or groups to exercises or theorems? Assigning\ntheorems in advance speeds the pace of the class, but if Joe knows Sara has been assigned\nTheorem 2.1, he may be less inclined to work on Theorem 2.1 himself. Typically a mixture\nof these methods works best.\nWill you allow your students to take their written work to the board for their presen-\ntation or should they present “notes-free”? It may seem like a small issue, but the answer\nto this question may make quite a difference in how much time the students spend really\nlearning their proofs. Sometimes we have students use their notes in the early part of the\nsemester while encouraging more “notes-free” presentations near the end of the semester.\nWill you present any proofs? There are some hard theorems in the text and it is best to\ndecide ahead of time what strategies you will use when your class inevitably hits a wall with\na particular theorem. One solution is to present some of the most difficult proofs yourself.\nAnother is to present a proof to a student outside of class and have the student present it\n16 4. AN INTRODUCTION TO INQUIRY BASED LEARNING INSTRUCTION\nto the class. Another solution is to watch the class stew and sweat as you anxiously await\ntheir proof.\nThe pace of the course will vary with the material, as well as from class to class. If you\nfind that things are proceeding too slowly there are several things you might try. You could\nonly have a subset of the assigned proofs of theorems or exercises presented during class,\nleaving the others for written homework only. Assigning problems to individuals or groups\nin advance rather than taking volunteers (or cold calling) tends to pick up the pace. Having\nmultiple students up at the board(s) in groups writing up their proofs at the same time,\nthen going through them one by one is faster than having students present while writing as\nthey present. Using a document camera also decreases the time for actually writing words\non the board.\nExperienced students. One good way to avoid having experienced students dominate\nthe class or get bored is to give some hard theorems for students to work on if they get\nahead. For example, have them prove the Alexander Subbasis Theorem. Then ask them\neach day whether they have made progress as they walk in or leave and discuss ideas with\nthem. Just a few seconds can keep the more advanced students occupied. And when they\nsucceed with such a challenge theorem then they can present the result. Of course, the more\nsuccessful students are also a good source for doing the regular harder theorems and for\nmoving on if things start to bog down. That is, if things seem to be going too slowly, just\nask the students who can be counted on to give excellent presentations to make suggestions\nor give presentations. Using those students appropriately can help you regulate the pace of\nthe course. You don’t want to go faster than the speed at which most of the students can\nsuccessfully prove most of the theorems, but you don’t want to go too slowly.\nLess experienced students. One good way to help shy or less experienced students\nis to ask them outside of class to come to your office and present some theorem that is\ncoming up. After helping them to understand it thoroughly so you and they know that\nthey can do it and present it well, then during class assign that student to the appropriate\ngroup and say in the usual way, “Now we will do Theorem 1.24. Let’s see, Oscar would you\npresent this one.” In other words, make it seem that you have selected that student in your\nnormal way. When a struggling student has presented something, of course, praise them.\nAlso, help them with questions. In other words, if they are asked a question that they start\nto get flustered with, then you can help by re-phrasing the question in a helpful way or say,\n“By the way, in your proof you did . . . , would that answer this question?”\nRESOURCES ON INQUIRY BASED LEARNING INSTRUCTION 17\nStudent attitudes. Generally students are quite appropriate in their comments, that\nis, they do not tend to be mean or critical. There is a lot of honesty in this format of\nteaching. That is, students display their weaknesses regularly. It is not in the cultural\nnorm to show weakness or mistakes in public, but you can get the class to accept this\nreality as simply a good part of what happens by actively praising the value of making\nmistakes. Learning happens when we push against misconceptions that we are prone to\nmake when we study something new. This experience can be one of the most positive parts\nof the course, because students can realize that pretending to know more than they do is\nultimately not the most productive way to learn or be.\nAudience. At first, the students will direct their presentations to the instructor. Don’t\nlet them do this. Tell them directly that their fellow students are the audience and then\nturn your eyes toward the other students to encourage the presenter to do so also. The main\nthing is do not be a source of confirmation of the correctness of the proof. The presenters\nmust understand that they are convincing the other students, not the instructor.\nStudents like this experience. Students will enjoy this class if they get a repeated\nsense of personal accomplishment. The joy of mathematics really is in doing it—seeing\npatterns and making conjectures, devising convincing proofs that a statement is true, and\nunderstanding an argument and having the confidence to know when you know it is true.\nThese experiences are what this course can offer the students throughout the class.\nResources on Inquiry Based Learning Instruction\nThere are many resources available for IBL instruction. Attending a conference or\nworkshop on IBL is very helpful.\nHere is a great place to start to look for resources about IBL and to connect with a\ncommunity of supportive practitioners is the Academy of Inquiry Based Learning (AIBL)\nwebsite: http:\/\/www.inquirybasedlearning.org\/ . From there you can find many resources.\nAnother source for resources is the Mathematics Learning by Inquiry (MLI) website:\nhttp:\/\/www.mathlearningbyinquiry.org\/ .\nWe encourage you also to make human connections. There are many members of the\nIBL community who will welcome your inquiries and connections. If you attend an IBL\nconference or workshop, you will find an encouraging and welcoming community of dedicated\neducators.\n\nCHAPTER 5\nPossible Paths for Various Potential Courses\nTopology Through Inquiry contains far too many exercises and theorems for any semester\nlong or even year-long course. Every instructor will have to decide for yourself which\ntheorems and exercises to assign.\nThe Preface of Topology Through Inquiry contains some sample collections of theorems\nfor different potential courses. For specific sample syllabi, check out the Topology Through\nInquiry webpage, maintained on Francis Su’s website.\nIf you are an instructor using our book, we encourage you to submit your sample\nsyllabi for inclusion on this webpage, for the benefit of other instructors.\nIn every case, we encourage you to modify these collections to suit you and your students.\nA lot depends on how mathematically mature, motivated, and knowledgable your students\nare. In some cases, you will need to provide more scaffolding or hints in order for your\nstudents to be able to prove the theorems. In all cases, there are plenty of exercises and\ntheorems to keep even the keenest students happily occupied.\n19\n"}
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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Real Analysis on YouTube. Francis Su. Real Analysis on YouTube. Here you can find all 26 lectures of my Real Analysis course at Harvey Mudd College. These lectures were taped in Spring 2010 with the help of Ryan Muller and Neal Pisenti. Lectures. The entire course is assembled as a. playlist on YouTube . And below are links to individual lectures. Lecture 1 : Constructing the rational numbers. Lecture 2 : Properties of Q. Lecture 3 : Construction of R. Lecture 4 : The Least Upper Bound Property. Lecture 5 : Complex Numbers. Lecture 6 : The Principle of Induction. Lecture 7 : Countable and Uncountable Sets. Lecture 8 : Cantor Diagonalization, Metric Spaces. Lecture 9 : Limit Points. Lecture 10 : Relationship b\/t open and closed sets. Lecture 11 : Compact Sets. Lecture 12 : Relationship b\/t compact, closed sets. Lecture 13 : Compactness, Heine-Borel Theorem. Lecture 14 : Connected Sets, Cantor Sets. Lecture 15 : Convergence of Sequences. Lecture 16 : Subsequences, Cauchy Sequences. Lecture 17 : Complete Spaces. Lecture 18 : Series. Lecture 19 : Series Convergence Tests. Lecture 20 : Functions – Limits and Continuity. Lecture 21: Continuous Functions. Lecture 22 : Uniform Continuity. Lecture 23 : Discontinuous Functions. Lecture 24 : The Derivative, Mean Value Theorem. Lecture 25 : Taylor’s Theorem. Lecture 26 : Ordinal Numbers, Transfinite Induction. The text for the course was . Principles of Mathematical Analysis  by Walter Rudin, but you do not need the text to follow these lectures.  Also, I realize the board is hard to read, so I’ve supplied some linked . lecture notes  in the tab above (they may not align perfectly with these lectures, since it’s from a different semester. You can also see my. course webpage from the last time I taught the course, with homework schedule.). FAQ. Why did you make these videos? My student Ryan Muller was developing an online learning platform and asked if I’d be interested in recording my Analysis lectures as YouTube videos to embed in it. After some initial hesitancy, I agreed. But after putting them up, I began getting appreciative e-mails from around the world. So I decided to leave them on YouTube. What textbook do you use? We use the classic text . Principles of Mathematical Analysis  by Walter Rudin, and cover the first half of the textbook (Chapter 1 to 5, and part of Chapter 7).  You can see what problems I assigned from the last time I taught the course, at my course webpage, linked above. I wish the resolution were better. Me too. But that was the technology we had… the recording equipment (mounted in the ceiling of the lecture hall) only records at a lower resolution. But check out my course notes, linked above, if you would like to see some notes that loosely track with the videos. Is your course a MOOC? Technically, no. We made these videos before the word MOOC (massively open online classes) became a buzzword. Usually a MOOC has a participation and feedback component and involves a large number of students all working on the same course schedule. My real analysis videos are intended to supplement your own learning. Have you been using the videos in a ‘flipped classroom’? No, but you are welcome to. One downside to the videos is that the resolution is not that great, so the blackboard writing is hard to read. Also, the material has not been broken into the bite-size chunks that some flipped classroom models use. But I imagine some instructors have been using it this way, nonetheless — it has been neat to see how popular the videos have been in Sierra Leone. 🙂. Why do you discuss mathematical writing in this course? Learning to communicate mathematics effectively is one of the goals of my Real Analysis course at Harvey Mudd College.  This handout “. Guidelines for Good Mathematical Writing “ is my guide to help students think about their writing. Are you going to do videos for Analysis II? There are no current plans to do this. We would first need better resources for classroom video capture. But I may do other online projects… stay tuned. 🙂. How do I stay tuned to your other projects? You can . follow me on Twitter . Or check . my homepage  periodically. Or. my author webpage . Can I write you e-mail? I welcome e-mail, especially stories of how you found the videos useful in your own life situation.  If you write, please tell me the country you are writing from, and a little bit about your situation and background.  (Are you in college, or have no access to college?  Are you a non-traditional student? etc.) Although I may not be able to respond, I do read every e-mail carefully. How can I thank you? If you found the videos helpful, you can ‘like’ the videos on YouTube.  And for a good read about mathematics, you might enjoy. my book . My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Math 172: Abstract Algebra II. Francis Su. Math 172: Abstract Algebra II. Course meets:. MW 11am, Shanahan 3465. Office Hours:. Tuesdays 9am-10am and 3pm-4pm, or by appointment. Grader:. Ryan O’Dowd, ryan.o’dowd@cgu.edu. The main topics of this course will be field extensions and Galois Theory, and additional topics as time permits. Required Text. Dummit and Foote,. Abstract Algebra. (3rd edition). Doing the reading will be essential for success in this course. Also as a second-semester course, I will expect a certain level of mathematical maturity. This means that sometimes I will not prove simple statements in class; you should get in the habit of working out some details for yourself, and doing the reading. Prerequisite. Algebra I (Math 171). Grading Policy. A midterm (25%), a final (25%) and a homeworks (25%) with the lowest homework score dropped, and the strongest component of the three will be worth 50%. Honor Code. All are expected to abide by the HMC honor code. Though cooperation on homework assignments is encouraged, you are expected to. write up all your solutions individually. . It is appropriate to acknowledge the assistance of others. Part of the fun of this course is the struggle, as well as the joy of discovering the solution for yourself. Please note: using solutions found online or solutions of previous students will be regarded as a violation of the HMC Honor Code and will be handled accordingly. I encourage you instead to discuss the ideas with me or each other! Homeworks. Homeworks, assigned weekly, turn in Wednesdays (in class) or by 4pm in the bin outside my office door. Homeworks will be announced on this webpage. On Learning:. Read this. article by Carol Dweck . On Writing Mathematics Well:. Read. my handout on good mathematical writing . LaTeX homework sets. , courtesy of Luis Ramirez, may be found here: (but you are responsible for checking they are correctly transcribed):. https:\/\/github.com\/magh-marshmallow\/Galois_HM HW #1 – due 1\/29. REVIEW. 7.1–7.4 and. READ. 7.5. (R means READ but you do not have to do the problem.). DO. 7.3 ( 1, 10, R19, 26, 28 ) 7.4 ( 10 ) and Problem A:. A. (part i) Let \\(R\\) be an integral domain. As you conjectured in class, prove that the units in \\(R[x]\\) are precisely the constant polynomials \\(p(x)=u\\) where \\(u\\) is a unit in \\(R\\). (part ii) On the other hand, show that \\(p(x)=1+2x\\) is a unit in \\(R[x]\\), where \\(R= \\frac{\\mathbb Z}{4 {\\mathbb Z}}\\). (\\(\\mathbb Z\\)= integers.). HW #2 – due 2\/5. SKIM. CHAPTER 8,. READ. 9.1–9.3. DO. 7.4 ( 19 ) 7.5 ( 3 ) 8.2 ( 2, 3 ) 9.1 ( 4 )[You will probably find 7.5.3 to be the most involved problem; so you may wish to do it last. For several of the others, I encourage you to think about modding out a ring by an ideal.]. HW #3 – due 2\/12. READ. 9.4–9.5. DO. 9.1 ( R2, R6, 7 ) 9.2 ( 1, 2, 3, R4, R6 ) 9.3 ( 3, R4 ). HW #4 – due 2\/19. READ. 13.1–13.2. DO. 9.4 ( 1bcd, 2bc, R5, 6ac, R7, 11, R20 ) 13.1 ( 1 ). HW #5 – due 2\/26. READ. 13.4. DO 13.1 ( 3, 5 ) 13.2 ( R1, 3, 4, 12, 14 ). HW #6 – due 3\/4. READ. 13.4–13.6. DO. 13.3 ( 1, R2 ) 13.4 ( 2, 3, 4 ) and these problems:. Problem C. Find a real number \\(u\\) such that \\(Q(\\sqrt{3},\\sqrt{5})=Q(u)\\). Problem D. Suppose \\(K\\) is an extension of \\(F\\), and \\(\\phi: K - K\\) is an isomorphism that leaves every element of \\(F\\) fixed. Show that any polynomial in \\(F[x]\\) that has a root \\(r\\) in \\(K\\) also has \\(\\phi(r)\\) as a root. MIDTERM – due 3\/12. Since there is a midterm due Thursday 3\/12, there is no HW this week. Also, no HW over Spring Break. CHANGES DUE TO COVID-19. With the COVID-19 crisis and the disruption that it has caused and will continue to cause, here are a few changes I’m making to this course:. Principles. A higher priority than this course is your health and the health of your loved ones. We must have flexibility while planning ahead for possible further disruption, including obstacles you may have to online access or focused study while at home, and accommodations in case you (or I) get sick from COVID-19. Focusing on the ideas of this course can be an important way for you to feel normal in working towards the future, in a time of great upheaval and uncertainty. Following from these principles, here are changes to our original plan. You will have the option to take this course Pass\/No Credit. If you are an HMC student, a Pass will count towards your degree requirement (even if it would not normally do so). If you are not an HMC student, you should check with your registrar to see what your school policy is for Pass\/No Credit. You can exercise this option at any time just by sending me an e-mail—you do not have to petition the registrar. For the homework average, the lowest. two. homework scores will be dropped. Lectures. will be held as usual during the normally scheduled time at 11am Pacific. ,. but recordings will be available to watch afterwards. We will figure this out as we go and may have to improvise. HW – Do by 4\/1 [optional, not handed in]. Try out the Course Zoom Room. Assemble 3 questions you have about the course material. Discuss with a group of 2-5 others in the Su Gathering Space (in Zoom). Show up in the Su Gathering Space at. TUESDAY 6-9pm Pacific time, or self-assemble. at a different time. (Remember WEDNESDAYS are reserved for the other class.). Try out the Whiteboard feature in Zoom, or move to goboard.com. See Sakai\/email for links to Su Gathering Space. HW #7 – due 4\/8. READ. 14.1–14.2, 14.6. DO. 14.1 ( 7 ) 14.2 ( 3, 13, 14 ). All homework will now be submitted through Gradescope. Access this through Sakai, on the left tab. You will need to upload by scanning your homework, here is some guidance on how to do that:. https:\/\/gradescope-static-assets.s3-us-west-2.amazonaws.com\/help\/submitting_hw_guide.pdf . Follow these Gradescope instructions to tag which portion of your homework is associated with the four problems:. https:\/\/youtu.be\/KMPoby5g_nE HW #8 – due 4\/15. READ. 14.6. DO. 14.1 ( 1, 4 ) 14.2 ( 1, 11 ). HW #9 – due 4\/22. READ. 14.6, 14.7. DO. 14.7 ( 1 ) 14.6 ( 2ac, 22, 23 ). HW#10 – due 4\/29. SKIM. 9.6, 15.1. The main ideas are here, but I’m supplementing what’s here with my lectures. 14.6 ( 4, 37 [read the text above the problem], 38, 46 ). In problem 46, E and F are unspecified fields (that you get to specify)–it might be helpful to know that any group is isomorphic to a subgroup of some symmetric group (Cayley’s Theorem). This is your final homework! It does not depend on this week’s lectures. All late homework must be turned in by May 3 at 4pm. I will have FINAL EXAMS available by Monday afternoon May 4. HOMEWORKS BELOW NOT FINAL UNTIL THEY APPEAR ABOVE THIS LINE. – THESE HAVE NOT BEEN UPDATED FOR NEW SCHEDULE YET. My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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{"id":"QK1OaW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Phased-tilings.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":78,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Phased-tilings","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Phased-tilings.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:05 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-9141941224908950208,"exact_signature_unique_b":true,"fuzzy_signature_l":5208251283724205779,"fuzzy_signature_unique_b":true,"wordcount_i":3347,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:21:57 UTC 2022","fresh_date_dt":"Wed Dec 06 11:45:53 UTC 2023","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":174020,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088057951977472,"text_t":"appeared in: Fibonacci Quarterly 38(2000), pp. 282-288.\nPHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES\nArthur T. Benjamin\nDept. of Mathematics, Harvey Mudd College, Claremont, CA 91711\nbenjamin@hmc.edu\nJennifer J. Quinn\nDept. of Mathematics, Occidental College, Los Angeles, CA 90041\njquinn@oxy.edu\nFrancis Edward Su\nDept. of Mathematics, Harvey Mudd College, Claremont, CA 91711\nsu@math.hmc.edu\n1. Introduction\nFibonacci numbers arise in the solution of many combinatorial problems. They count the\nnumber of binary sequences with no consecutive zeros, the number of sequences of 1’s and 2’s\nwhich sum to a given number, and the number of independent sets of a path graph. Similar\ninterpretations exist for Lucas numbers. Using these interpretations, it is possible to provide\ncombinatorial proofs which shed light on many interesting Fibonacci and Lucas identities [1, 3]. In\nthis paper, we extend the combinatorial approach to understand relationships among generalized\nFibonacci numbers.\nGiven G0 and G1, a generalized Fibonacci sequence G0, G1, G2, . . . is defined recursively by\nGn = Gn−1 + Gn−2 for n ≥ 2. Two important special cases are the classical Fibonacci sequence\nFn (F0 = 0 and F1 = 1) and the Lucas sequence Ln (L0 = 2 and L1 = 1).\nThese sequences satisfy numerous relationships. Many are documented in Vajda [6], where they\nare proved by algebraic means. Our goal is to recount these identities by combinatorial means. We\nintroduce several combinatorial techniques which allow us to provide new proofs of nearly all the\nidentities in Vajda [6] involving generalized Fibonacci numbers. We show that in the framework\nof phased tilings, these identities follow naturally as the tilings are counted, represented, and\ntransformed in clever ways. These techniques are developed in the next several sections. In the\nfinal section, we discuss possible extensions.\n2. Combinatorial Interpretation\nRecall that Fn+1 counts the number of sequences of 1’s and 2’s which sum to n. Equivalently,\nFn+1 counts the number of ways to tile a 1 × n rectangle (called an n-board consisting of cells\nlabelled 1, ..., n) with 1× 1 squares and 1× 2 dominoes. For combinatorial convenience, we define\nfn = Fn+1. Then fn is the number of ways to tile an n-board with squares and dominoes.\nWhen G0 and G1 are non-negative integers, we shall obtain an analogous combinatorial inter-\npretation of the generalized Fibonacci numbers Gn. Define a phased n-tiling to be a tiling of an\nn-board by squares and dominoes in which the last tile is distinguished in a certain way. Specifi-\ncally, if the last tile is a domino, it can be assigned one of G0 possible phases, and if the last tile\nis a square it can be assigned one of G1 possible phases. For example, when G0 = 5 and G1 = 17,\nthere are G3 = 39 phased tilings of length 3 as follows: There are 5 of the form (square, phased\ndomino); 17 of the form (domino, phased square); and 17 of the form (square, square, phased\n1\n2 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES\nsquare). In general, let g0 = G0, g1 = G1, and for n ≥ 2, let gn count the number of phased\nn-tilings. By conditioning on whether the first tile is a square or domino we obtain the identity\ngn = gn−1 + gn−2\nfor n ≥ 2. Hence gn = Gn, giving the desired interpretation.\nThis combinatorial definition can be extended to n = 1 and n = 0. Clearly G1 counts the\nnumber of phased 1-tilings. It will be convenient to assign the “last” tile of a 0-board one of the\nG0 domino phases.\nNotice when there exists only one domino phase and only one square phase, we recover our\noriginal interpretation of fn.\nPrevious interpretations of the Lucas numbers Ln [1, 4, 5] counted the number of ways to tile a\n“circular” n-board by squares or dominoes. Since L0 = 2 and L1 = 1, a phased n-board tiling can\nend with a phase one domino, a phase two domino, or a phase one square. In all three cases, the\ncorresponding circular n-board tiling arises by first gluing cells n and 1 together. Tilings that end\nin a phase two domino are then rotated one cell to obtain a circular tiling with a domino covering\ncells n and 1.\n3. Elementary Identities\nBefore launching into more sophisticated techniques, we demonstrate how our combinatorial\ninterpretation of Gn yields quick proofs of some basic identities. For instance, by conditioning on\nwhether the last tile is a phased domino or a phased square, we immediately obtain for n ≥ 2,\nGn = G0fn−2 +G1fn−1.\nMore identities are obtained by conditioning on other events. Consider\nIdentity 1 (Vajda (33)).\nn∑\nk=0\nGk = Gn+2 −G1.\nThe right hand side of this equality counts all phased (n+2)-tilings containing at least one domino\n(there areG1 phased tilings consisting of all squares). The left hand side is obtained by conditioning\non the position of the first domino. If the first domino covers cells n−k+1 and n−k+2 (0 ≤ k ≤ n),\nthen the preceding cells are covered by squares and the remaining cells can be covered Gk ways.\nSimilarly there are G2n−G0 phased 2n-tilings with at least one square. By conditioning on the\nposition of the first square we obtain\nIdentity 2 (Vajda (34)).\nn∑\nk=1\nG2k−1 = G2n −G0.\nA phased (2n+ 1)-tiling must contain a first square, which leads to\nIdentity 3 (Vajda (35)).\nG1 +\nn∑\nk=1\nG2k = G2n+1.\nThe G1 term on the left hand side counts those boards that begin with n dominoes followed by a\nphased square.\nTo prove\nIdentity 4 (Vajda (8)).\nGm+n = fmGn + fm−1Gn−1\nPHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES 3\nwe consider whether or not a phased (m+ n)-tiling can be separated into an (unphased) m-tiling\nfollowed by a phased n-tiling. There are fmGn tilings breakable at cell m. The unbreakable tilings\nmust contain a domino covering cells m and m+1; the remaining board can be covered fm−1Gn−1\nways.\n4. Binomial Identities\nVajda contains several identities involving generalized Fibonacci numbers and binomial coeffi-\ncients. All of these are special cases of the following two identities:\nIdentity 5 (Vajda (46)).\nGn+p =\np∑\ni=0\n(\np\ni\n)\nGn−i,\nIdentity 6 (Vajda (66)).\nGm+(t+1)p =\np∑\ni=0\n(\np\ni\n)\nf itf\np−i\nt−1Gm+i.\nWhen n ≥ p, Identity 5 counts phased (n+p)-tilings by conditioning on the number of dominoes\nthat appear among the first p tiles. Given an initial segment of i dominoes and p− i squares,\n(\np\ni\n)\ncounts the number of ways to select the i positions for the dominoes among the first p tiles. Gn−i\ncounts the number of ways the remaining n− i cells can be given a phased tiling.\nIdentity 6 can be seen as trying to break a phased (m+(t+1)p)-tiling into p unphased segments\nof length t followed by a phased remainder. The first segment consists of the tiles covering cells\n1 through j1 where j1 = t if the tiling is breakable at cell t and j1 = t + 1 otherwise. The next\nsegment consists of the tiles covering cells j1 + 1 through j1 + j2 where j2 = t if the tiling is\nbreakable at cell j1 + t and j2 = t + 1 otherwise. Continuing in this fashion we decompose our\nphased tiling into p tiled segments of length t or t+ 1 followed by a phased remainder of length at\nleast m. Since the length t+ 1 segments must end with a domino, the term\n(\np\ni\n)\nf itf\np−i\nt−1Gm+i counts\nthe number of phased (m+ (t+ 1)p)-tilings with exactly i segments of length t.\n5. Simultaneous Tilings\nIdentities involving squares of generalized Fibonacci numbers suggest investigating pairs of\nphased tilings. The right hand side of\nIdentity 7 (Vajda (39)).\n2n∑\ni=1\nGi−1Gi = G\n2\n2n −G20\ncounts ordered pairs (A,B) of phased 2n-tilings where A or B contains at least one square. To\ninterpret the left hand side, we define the parameter kX to be the first cell of the phased tiling\nX covered by a square. If X is all dominoes, we set kX equal to infinity. Since, in this case,\nat least one square exists in (A,B), the minimum of kA and kB must be finite and odd. Let\nk = min{kA, kB +1}. When k is odd, A and B have dominoes covering cells 1 through k−1 and A\nhas a square covering cell k. Hence the number of phased pairs (A,B) with odd k is G2n−kG2n−k+1.\nWhen k is even, A has dominoes covering cells 1 through k and B has dominoes covering cells 1\nthrough k − 2 with a square covering cell k − 1. Hence the number of phased pairs (A,B) with\neven k is also G2n−kG2n−k+1. Setting i = 2n+ 1− k gives the desired identity.\nSimilarly, the next identity counts ordered pairs of phased (2n + 1)-tilings that contain an\nunphased square. Conditioning on the first unphased square yields\n4 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES\nIdentity 8 (Vajda (41)).\n2n+1∑\ni=2\nGi−1Gi = G\n2\n2n+1 −G21.\nIn the same spirit, our next identity conditions on the location of the first domino in a pair of\nphased tilings.\nIdentity 9 (Vajda (43)).\nn∑\ni=1\nGi−1Gi+2 = G\n2\nn+1 −G21\nThe right hand side counts the number of pairs (A,B) of phased (n + 1)-tilings, where A or B\ncontains at least one domino. Here we define the parameter `X to be the first cell of the phased\ntiling X covered by a domino. If X is all squares, we set `X equal to infinity. Let ` = min{`A, `B}.\nThe number of phased (n+ 1)-tiling pairs (A,B) where the `th cell of A is covered by a domino is\nGn−`Gn−`+2; the number of such pairs where the `th cell ofA is covered by a square isGn−`+1Gn−`.\nThis implies\nn∑\n`=1\nGn−`(Gn−`+2 +Gn−`+1) = G\n2\nn+1 −G21.\nSubstituting Gn−`+3 for Gn−`+2 +Gn−`+1 and letting i = n− `+ 1 yields the desired identity.\n6. A Transfer Procedure\nThe identities proved in this section all take advantage of the same technique. Before proceeding,\nwe introduce helpful notation. For m ≥ 0, define Gm to be the set of all phased m-tilings with\nG0 domino phases and G1 square phases. An element A ∈ Gm created from a sequence of e1\ndominoes, e2 squares, e3 dominoes, . . . , and ending with a phased tile can be uniquely expressed\nas A = de1se2de3se4 · · · det−1setp, where p represents the phase of the last tile. All exponents are\npositive except that e1 or et may be 0, and 2e1 + e2 + 2e3 + e4 + · · · + 2et−1 + et = m. When\net = 0, the last tile is a domino and p ∈ {1, . . . , G0}; when et ≥ 1, the last tile is a square and\np ∈ {1, . . . , G1}. Likewise, for n ≥ 0, define Hn to be the set of all phased n-tilings with H0 domino\nphases and H1 square phases. Notice that the sizes of Gm and Hn are Gm and Hn respectively.\nWe introduce a transfer procedure T to map an ordered pair (A,B) ∈ Gm ×Hn to an ordered\npair (A′, B′) ∈ Gm−1 × Hn+1 where 1 ≤ m ≤ n. T has the effect of shrinking the smaller tiling\nand growing the larger tiling by one unit. For such a pair (A,B), define k = min{kA, kB}, the\nfirst cell in A or B that is covered by a square. If the kth cell of A is covered by a square and\n1 ≤ k ≤ m − 1, then we transfer that square from A to the kth cell of B. Formally, before\nthe transfer we have A = d(k−1)\/2sa and B = d(k−1)\/2b where a ∈ Gm−k and b ∈ Hn−k+1. The\ntransfer yields A′ = d(k−1)\/2a and B′ = d(k−1)\/2sb. If the kth cell of A is covered by a domino and\n1 ≤ k ≤ m − 2, then we exchange that domino with the square in the kth cell of B. Formally,\nbefore the exchange, A = d(k+1)\/2a and B = d(k−1)\/2sb where a ∈ Gm−k−1 and b ∈ Hn−k.\nThe exchange yields A′ = d(k−1)\/2sa and B′ = d(k+1)\/2b. We abbreviate this transformation by\nT (A,B) = (A′, B′). Notice that our rules do not allow for a phased tile to be transferred or\nexchanged.\nLemma 1. For 1 ≤ m ≤ n, T establishes an almost one-to-one correspondence between Gm ×Hn\nand Gm−1 ×Hn+1. The difference of their sizes satisfies\nGmHn −Gm−1Hn+1 = (−1)m[G0Hn−m+2 −G1Hn−m+1].\nPHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES 5\nProof. Notice when T is defined, T (A,B) has the same k value as (A,B) which makes T easy\nto reverse. It remains to enumerate (A,B) ∈ Gm × Hn for which T is undefined and (A′, B′) ∈\nGm−1 ×Hn+1 that do not appear in the image of T .\nWhen m is odd, T is undefined whenever k = m. Here, A = d(m−1)\/2a and B = d(m−1)\/2b\nwhere a ∈ G1 and b ∈ Hn−m+1. Hence the domain of T contains GmHn − G1Hn−m+1 elements.\nThe elements of Gm−1×Hn+1 that do not appear in the image of T have k ≥ m and are therefore\nof the form A′ = d(m−1)\/2p and B′ = d(m−1)\/2b′ where p ∈ {1, . . . , G0} and b′ ∈ Hn−m+2. Hence\nthe image of T consists of Gm−1Hn+1−G0Hn−m+2 elements. Since T is one-to-one, we have when\nm is odd,\nGmHn −G1Hn−m+1 = Gm−1Hn+1 −G0Hn−m+2.\nWhen m is even, T is undefined whenever k ≥ m and sometimes undefined when k = m − 1.\nSpecifically, T is undefined when A = dm\/2p and B = d(m−2)\/2b where p ∈ {1, . . . , G0} and\nb ∈ Hn−m+2. Hence the domain of T contains GmHn − G0Hn−m+2 elements. The elements of\nGm−1 ×Hn+1 that do not appear in the image are of the form A′ = d(m−2)\/2a′ and B′ = dm\/2b′\nwhere a′ ∈ G1 and b′ ∈ Hn−m+1. Hence the image of T consists of Gm−1Hn+1 − G1Hn−m+1\nelements. Thus when m is even, we have\nGmHn −G0Hn−m+2 = Gm−1Hn+1 −G1Hn−m+1.\nWe specialize Lemma 1 by setting m = n and choosing the same initial conditions for Gn and\nHn to obtain\nIdentity 10 (Vajda (28)).\nGn+1Gn−1 −G2n = (−1)n(G21 −G0G2).\nAlternately, setting Gm = Fm and evaluating Lemma 1 at m+ 1, we obtain\nIdentity 11 (Vajda (9)). For 0 ≤ m ≤ n,\nHn−m = (−1)m(Fm+1Hn − FmHn+1).\nA slightly different transfer process is used to prove\nIdentity 12 (Vajda (10a)). For 0 ≤ m ≤ n,\nGn+m + (−1)mGn−m = LmGn.\nWe construct an almost one-to-one correspondence from Lm×Gn to Gm+n, where Lm denotes the\nset of Lucas tilings of length m. Let (A,B) ∈ Lm × Gn. If A ends in a (phase 1) square or a\nphase 1 domino, then we simply append A to the front of B to create an (m + n)-tiling that is\nbreakable at m. Otherwise, A ends in a phase 2 domino. In this case, before appending A to the\nfront of B, we transfer a unit from B to A by a similar process. (If the first square occurs in B,\nthen transfer it into the corresponding cell of A. Otherwise, the first square of A is exchanged\nwith the corresponding domino in B.) This creates a tiling of Gm+n that is unbreakable at m.\nWhen m is even, the transfer is undefined for the Gn−m elements of Lm × Gn where A contains\nonly dominoes, ending with a phase 2 domino, and B begins with m\/2 dominos. Otherwise the\ntransfer is one-to-one and onto Gm+n. When m is odd, the transfer is always defined, but misses\nthe Gn−m elements of Gm+n that begin with m dominoes. Identity 12 follows.\nA similar argument establishes\nIdentity 13 (Vajda (10b)). For 0 ≤ m ≤ n,\nGn+m − (−1)mGn−m = Fm(Gn−1 +Gn+1).\n6 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES\nThe transfer process T can be refined to allow us to shrink and grow pairs of phased tilings\nby more than one unit. Specifically, we construct an almost one-to-one correspondence between\nGm×Hn and Gm−h×Hn+h where 1 ≤ h ≤ m−1 < n. Let Fn denote the set of (unphased) n-tilings.\nSo |Fn| = fn. Given (A,B) ∈ Gm×Hn, define a transfer process Th as follows: If A is breakable at\ncell h, i.e., A = a1a2 where a1 ∈ Fh and a2 ∈ Gm−h, then we append segment a1 to the beginning\nof B. That is to say, Th(A,B) = (A\n′, B′) where A′ = a2 and B\n′ = a1B. If A is unbreakable at\ncell h, i.e. A = a1da2 where a1 ∈ Fh−1 and a2 ∈ Gm−h−1, then let (A′′, B′′) = T (da2, B) and\n(A′, B′) = (A′′, a1B\n′′). Notice that B′′, when defined, will necessarily begin with a domino and\ntherefore B′ will be unbreakable at h.\nDiscrepancies in Th, mapping Gm×Hn to Gm−h×Hn+h, are proportional (by a factor of fh−1)\nto the discrepancies in T mapping Gm−h+1 × Hn to Gm−h × Hn+1. Hence for 1 ≤ h ≤ m − 1,\nLemma 1 implies\nGmHn −Gm−hHn+h = (−1)m−h+1fh−1(G0Hn−m+h+1 −G1Hn−m+h).(1)\nNotice that fh−1Hn−m+h+1 counts the number of phased (n−m+ 2h)-tilings that are breakable\nat h − 1. Hence fh−1Hn−m+h+1 = Hn−m+2h − fh−2Hn−m+h. Similarly fh−1G1 = Gh − fh−2G0.\nSo equation (1) can be rewritten as\nGmHn −Gm−hHn+h = (−1)m−h+1(G0Hn−m+2h −GhHn−m+h).\nReindexing, this is equivalent to\nIdentity 14 (Vajda (18)).\nGn+hHn+k −GnHn+h+k = (−1)n(GhHk −G0Hh+k).\nThis identity is applied, directly or indirectly, by Vajda to obtain identities (19a) through (32).\n7. Binary Sequences\nThere are identities involving generalized Fibonacci numbers and powers of 2. This leads us to\ninvestigate the relationship between binary sequences and Fibonacci tilings.\nA binary sequence x = x1x2 · · ·xn can be viewed as a set of instructions for creating a Fibonacci\ntiling of length less than or equal to n. Reading x from left to right, we interpret 1’s and 01’s as\nsquares and dominoes respectively. The construction halts on encountering a 00 or the end of the\nsequence. For example, 111010110101 represents the 12-tiling s3d2sd2, 1110101110 represents the\n9-tiling s3d2s2, and 0111001011 represents the 4-tiling ds2. Binary sequences that begin with 00\ndenote the 0-tiling.\nGiven n, tilings of length n and n − 1 are represented uniquely by binary sequences of length\nn that end with a 1 or 0, respectively. For k ≤ n − 2, a k-tiling is represented by 2n−k−2 binary\nsequences of length n since the first k + 2 bits are determined by the k-tiling followed by 00; the\nremaining n− (k + 2) bits may be chosen freely. This yields the following identity:\nfn + fn−1 +\nn−2∑\nk=0\nfk 2\nn−k−2 = 2n.(2)\nBy dividing by 2n, reindexing, and employing fn+1 = fn + fn−1, we obtain\nIdentity 15 (Vajda (37a)).\nn∑\nk=2\nfk−2\n2k\n= 1− fn+1\n2n\n.\nPHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES 7\nThe same strategy can be applied to phased tilings. Here, for convenience we assume the phase is\ndetermined by the first tile (rather than the last). The phased identity corresponding to equation\n(2) is\nGn+1 +Gn +\nn−1∑\nk=0\nGk 2\nn−k−1 = 2n(G0 +G1).(3)\nThe right hand side counts the number of ways to select a length n binary sequence x and a phase\np. From this we construct a length n + 1 binary sequence. If p is a domino phase, construct the\nsequence 0x; if p is a square phase, construct the sequence 1x. Interpret this new n+ 1-sequence\nas a Fibonacci tiling in the manner discussed previously, and assign the tiling the phase p. By\nconstruction, the phase is compatible with the first tile. (Recall that empty tilings are assigned\na domino phase.) A phased tiling of length n + 1 or n has a unique (x, p) representation. For\n0 ≤ k ≤ n − 1, a phased k-tiling has 2n−k−1 representations. This establishes equation (3).\nDividing by 2n gives\nIdentity 16 (Vajda (37)).\nn−1∑\nk=0\nGk\n2k+1\n= (G0 +G1)−\nGn+1 +Gn\n2n\n.\n8. Discussion\nThe techniques presented in this paper are simple but powerful— counting phased tilings enables\nus to give visual interpretations to expressions involving generalized Fibonacci numbers. This\napproach facilitates a clearer understanding of existing identities, and can be extended in a number\nof ways.\nFor instance, by allowing tiles of length 3 or longer, we can give combinatorial interpretation\nto higher-order recurrences; however, the initial conditions do not work out so neatly, since the\nnumber of phases that the last tile admits do not correspond with the initial conditions of the\nrecurrence.\nAnother possibility is to allow every square and domino to possess a number of phases, depending\non its location. This leads to recurrences of the form xn = anxn−1 + bnxn−2. The special case\nwhere bn = 1 for all n provides a tiling interpretation of the numerators and denominators of\nsimple finite continued fractions and is treated in [2].\n9. Acknowledgment\nThe authors gratefully acknowledge the referee for many valuable suggestions.\nReferences\n[1] A. T. Benjamin and J. J. Quinn, Recounting Fibonacci and Lucas Identities, College Mathematics Journal,\nto appear.\n[2] A. T. Benjamin, J. J. Quinn, and F. E. Su, Counting on Continued Fractions, preprint.\n[3] R. C. Brigham, R. M. Caron, P. Z. Chinn, and R. P. Grimaldi, A tiling scheme for the Fibonacci numbers, J.\nRecreational Math., Vol 28, No. 1 (1996-97) 10-16.\n[4] L. Comtet, Advanced combinatorics: the art of finite and infinite expansions, D. Reidel Publishing Co.,\nDordrecht, Holland, 1974.\n[5] H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, Vol. 20 (1982) 16-21.\n[6] S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section, John Wiley and Sons, New York, 1989.\nAMS Subject Classification Numbers: 05A19, 11B39.\n"}
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Math. Monthly, 106(1999), 930-942.\nCopyright the Mathematical Association of America 1999. All rights reserved.\nRENTAL HARMONY:\nSPERNER’S LEMMA IN FAIR DIVISION\nFRANCIS EDWARD SU\nMy friend’s dilemma was a practical question that mathematics could answer,\nboth elegantly and constructively. He and his housemates were moving to a house\nwith rooms of various sizes and features, and were having trouble deciding who\nshould get which room and for what part of the total rent. He asked, “Do you\nthink there’s always a way to partition the rent so that each person will prefer a\ndifferent room?”\nAs we shall see, with mild assumptions, the answer is yes. This rent-partitioning\nproblem is really a kind of fair-division question. It can be viewed as a generaliza-\ntion of the age-old cake-cutting problem, in which one seeks to divide a cake fairly\namong several people, and the chore-division problem, posed by Martin Gardner\nin [6, p. 124], in which one seeks to fairly divide an undesirable entity, such as a list\nof chores. Lately, there has been much interest in fair division (see, for example,\nthe recent books [3] and [11]), and each of the related problems has been treated\nbefore (see [1], [4], [10]).\nWe wish to explain a powerful approach to fair-division questions that unifies\nthese problems and provides new methods for achieving approximate envy-free di-\nvisions, in which each person feels she received the “best” share. This approach\nwas carried out by Forest Simmons [13] for cake-cutting and depends on a sim-\nple combinatorial result known as Sperner’s lemma. We show that the Sperner’s\nlemma approach can be adapted to treat chore division and rent-partitioning as\nwell, and it generalizes easily to any number of players.\nFrom a pedagogical perspective, this approach provides a nice, elementary demon-\nstration of how ideas from many pure disciplines—combinatorics, topology, and\nanalysis—can combine to address a real-world problem. Better yet, the proofs can\nbe converted into constructive fair-division procedures.\n1. Sperner’s Lemma for triangles\nOur fair division approach is based on a simple combinatorial lemma, due to\nSperner [14] in 1928. However, do not be fooled—this little lemma is as powerful\nas it is simple. It can, for instance, be used to give a short, elementary proof of\nthe Brouwer fixed point theorem [7].\nAs motivation, we examine a special case of Sperner’s lemma. Consider a trian-\ngle T triangulated into many smaller triangles, called elementary triangles, whose\nvertices are labelled by 1’s, 2’s, and 3’s, as in Figure 1.\n1\n2 FRANCIS EDWARD SU\n3\n211\n3\n2\n2\n2 1 1\n3\n1\n1\n2\n21\n2\n3 2\n2\n1\nFigure 1. A Sperner labelling, with (1,2,3)-triangles marked.\nThe labelling we have chosen obeys two conditions: (1) all of the main vertices\nof T have different labels, and (2) the label of a vertex along any edge of T matches\nthe label of one of the main vertices spanning that edge; labels in the interior of T\nare arbitrary. Any labelled triangulation of T satisfying these conditions is called\na Sperner labelling. The claim:\nSperner’s Lemma for Triangles. Any Sperner-labelled triangulation of T must\ncontain an odd number of elementary triangles possessing all labels. In particular,\nthere is at least one.\nIn Figure 1, we have marked all elementary 123-triangles; their parity is indeed\nodd. An analogous statement holds in any dimension, which we develop presently.\n2. The n-dimensional Sperner’s Lemma\nWe need the concept of an n-simplex: a 0-simplex is a point, a 1-simplex is a line\nsegment, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, etc. In general, an\nn-simplex may be regarded as an n-dimensional “tetrahedron”—the convex hull of\nn+1 affinely independent points in Rm, for m ≥ n. These points form the vertices\nof the simplex. A k-face of an n-simplex is the k-simplex formed by the span of\nany subset of k + 1 vertices.\nA triangulation of an n-simplex S is a collection of (distinct) smaller n-simplices\nwhose union is S, with the property that any two of them intersect in a face\ncommon to both, or not at all. The smaller n-simplices are called elementary\nsimplices, and their vertices are called vertices of the triangulation.\nGiven an n-simplex S, any face spanned by n of the n+ 1 vertices of S is called\na facet. As examples, the facets of a line segment are its endpoints, the facets of\na triangle are its sides, and the facets of a tetrahedron are its triangular faces.\nNow number the facets of S by 1, 2, ..., n+1. Given a triangulation of S, consider\na labelling that obeys the following rule: each vertex is labelled by one of the facet\nnumbers in such a way that on the boundary of S, none of the vertices on facet j is\nlabelled j. The interior vertices can be labeled by any of the facet numbers. Such\na labelling is called a Sperner labelling of an n-simplex; it generalizes the definition\nRENTAL HARMONY: SPERNER’S LEMMA IN FAIR DIVISION 3\nwe encountered earlier for n = 2. For other low dimensions, Figures 2 and 3 show\nexamples of a Sperner-labelled 1-simplex and 3-simplex.\n211 2 12\nFigure 2. A triangulated line, with Sperner labelling.\n4\non this edge\non interior vertices\n1, 2, 3, or 4\n1 or 2\n1, 2, or 3\nfor vertices on facet #4 3\n2\n1\n2 or 3\n2, 3, or 4\nfor vertices on facet #1\nfacet #3 underneath\nfacet #2 in back\n3 or 4\n2 or 4\n1 or 3\nFigure 3. A triangulated tetrahedron, with Sperner labelling.\nA Sperner labelling may be described equivalently as one in which main vertices\nof S are assigned distinct labels, and any other vertex in the interior of some k-face\nmust be assigned one of the labels of the main vertices that span that face. In\neither description it is apparent that the Sperner labelling on S induces Sperner\nlabellings on each facet as (n− 1)-simplices.\nWe call an elementary simplex in the triangulation fully labelled if all its vertices\nhave distinct labels. Then we have:\nSperner’s Lemma. Any Sperner-labelled triangulation of a n-simplex must con-\ntain an odd number of fully labelled elementary n-simplices. In particular, there is\nat least one.\nThere are many ways to prove this lemma. The simplest proofs involve parity\narguments and are non-constructive. A constructive method for finding a fully\nlabelled simplex is based on the following induction argument; it is useful later in\nour discussion of fair-division procedures in Sections 5 and 7.\nProof. We proceed by induction on the dimension n.\nWhen n = 1, a triangulated 1-simplex is a segmented line, as in Figure 2. The\nendpoints of the line are labelled distinctly, by 1 and 2. Hence in moving from\nendpoint 1 to endpoint 2 the labelling must switch an odd number of times, i.e.,\nan odd number of (1, 2)-edges may be located in this way.\nNow assume that the theorem holds for dimensions up through (n − 1). We\nshow the theorem is true for a triangulated, Sperner-labelled n-simplex S using\n4 FRANCIS EDWARD SU\nthe labels 1 through (n+ 1). For concreteness refer to the case n = 2 as a running\nexample while following the argument. In this case, S is a triangulated triangle,\nas in Figure 4.\nThink of the n-simplex S as a “house” triangulated into many “rooms”, which\nare the elementary simplices. A facet of a room is called a “door” if that facet\ncarries the first n of the n+1 labels. In our running example, doors are (1, 2)-edges\nthat may be in the interior or on the boundary; see Figure 4. For the case n = 3,\ndoors are any room facets labelled (1, 2, 3).\n3\n211\n3\n2\n2\n2\n2 1\n3\n1\n1\n2\n21\n2\n2\n1\n1\n3\nFigure 4. House, rooms, and doors indicated by dotted lines.\nWe claim that the number of doors on the boundary of S is odd. Why? The only\nfacet that can contain doors is the (n+ 1)-st because of the Sperner labelling. But\nthat facet of S is Sperner-labelled using the labels 1, ..., n, hence by the inductive\nhypothesis there must be odd number of fully labelled (n − 1)-simplices on that\nfacet. These are boundary doors when considered in S.\nThe boundary doors can be used to locate fully labelled rooms by what we\nfondly call a “trap-door” argument. The key observation is that every room can\nhave at most 2 doors, and it has exactly 1 door if and only if the room is fully\nlabelled in S. This is true because any room with at least one door has either no\nrepeated labels (it is fully labelled), or it has one repeated label that appears twice.\nThese give rise to 2 distinct doors, one for each repeated label. As examples, verify\nthat elementary triangles in Figure 4 have either two, one, or no (1, 2)-edges. For\nn = 3, verify that a tetrahedron with labels {1, 2, 3, 3} has two doors.\nSo, start at any door on the boundary (located by the inductive step), and\n“walk” through the door into the adjoining room. Either this room is fully labelled\nor it has one other door—a “trap-door” that we can walk through. Repeat this\nprocedure, walking through doors whenever possible. Notice that this path cannot\ndouble back on itself (because each room has at most two doors), so no room is\never visited twice. Moreover the number of rooms is finite and so the procedure\nmust end, either by walking into a fully labelled room or by walking back through\nto a boundary door of S; see Figure 5.\nSince the number of boundary doors of S is odd, and trap-door paths pair up\nonly an even number of them, the number of boundary doors left over that lead to\nRENTAL HARMONY: SPERNER’S LEMMA IN FAIR DIVISION 5\n3\n2\n2\n11\n3\n2\n2\n3\n1\n1\n2\n21\n3 2\n2\n1\n1 12\nFigure 5. Walking through doors.\nfully labelled rooms must be odd. Moreover, any fully labelled rooms not reachable\nby paths from the boundary must come in pairs, matched up by their own trap-\ndoor paths, as in Figure 5. Hence the total number of fully labelled rooms in S is\nodd.\nThis proof yields a constructive method for finding such rooms in the following\nway. Trap-door paths in successive dimensions can be linked up at their endpoints,\nbecause a fully labelled room in an i-dimensional face is just a boundary door in\nan (i+ 1)-face. This creates “super-paths” with endpoints in the bottom and top\ndimensions, i.e., either (1, 2)-edges on a 1-face of S, or n-dimensional fully labelled\nrooms in the interior. The constructive procedure begins by moving along the 1-\nface of S spanned by labels 1 and 2, following any super-path that is encountered.\nBecause the number of (1, 2)-edges is odd, and super-paths can pair up only an\neven number of them, we see that at least one super-path can be followed to yield\na fully labelled room.\nThe trap-door argument to prove Sperner’s lemma constructively dates back to\nCohen [5] and Kuhn [8]. A quick non-constructive proof would note the equality\nbetween the number of doors in each room, summed over all rooms, and the\nnumber of times each door is counted, summed over all doors. Modulo two, the\nfirst sum captures the parity of the number of fully labelled rooms, and the second\nsum captures the parity of the number of boundary doors, which by the inductive\nhypothesis is odd.\n3. Simmons’ approach to cake-cutting\nNow imagine a rectangular cake to be divided among n people, who may have\ndiffering notions of what is valuable on a cake. We use n − 1 knives to cut along\nplanes parallel to the left edge of the cake, as in Figure 6.\nThe set of cuts is fully defined by the relative sizes of the pieces. Assume that the\ntotal size of the cake is 1 and denote the physical size of the i-th piece by xi; this is\nan absolute measure, unrelated to player preferences. Thus x1 + x2 + · · ·+ xn = 1\nand each xi ≥ 0. The space S of possible partitions naturally forms a standard\n6 FRANCIS EDWARD SU\nFigure 6. A cut-set of a cake.\n(n− 1)-simplex in Rn. Each point in S corresponds to a partition of the cake by\na set of cuts, which we shall call a cut-set.\nGiven a cut-set, we say that a player prefers a given piece if the player does not\nthink any other piece is better. We assume that this preference depends on the\nplayer and the entire cut-set, but not on choices made by the other players. Note\nthat, given a cut-set, a player always prefers at least one piece, and may (in case\nof ties) prefer more than one piece by our definition.\nWe make the following two assumptions:\n(1) The players are hungry. That is, players prefer any piece with mass to an\nempty piece.\n(2) Preference sets are closed. This means that any piece that is preferred for a\nconvergent sequence of cut-sets is preferred at the limiting cut-set. Note that this\ncondition rules out the existence of single points of cake with positive desirability.\nTheorem. For hungry players with closed preference sets, there exists an envy-free\ncake division, i.e., a cut-set for which each person prefers a different piece.\nWe first investigate what happens for n = 3 people. Suppose the players are\nnamed Alice, Betty, and Charlie. They are to divide a cake of total size 1, using 2\nknives. Denote the physical size of the pieces by x1, x2, x3. Since x1 + x2 + x3 = 1\nand all xi ≥ 0, the solution space S is a plane intersected with the first octant.\nThis is just a triangle.\nNow triangulate S and assign “ownership” to each of the vertices as in Figure\n7, where A stands for Alice, B for Betty, and C for Charlie. We have purposely\nassigned ownership so that each elementary triangle is an ABC triangle. Observe\nthat a similar triangulation of finer mesh can also be labelled in this way.\nWe obtain a new auxiliary labelling of the triangulation by 1’s, 2’s, and 3’s by\ndoing the following: since each point in the triangle corresponds to a set of cuts of\ncake, go to each vertex, and ask the owner of that vertex, “Which piece would you\nchoose if the cake were cut with this cut-set?” Label that vertex by the number\nof the piece that is desired.\nWe claim that this new labelling is a Sperner labelling! Why?\nAt the vertex (1, 0, 0) of S we see that one of the pieces contains the entire cake,\nand the other pieces are empty. By the hungry assumption, the owner of (1, 0, 0)\nalways chooses piece 1 no matter who the owner is. Similarly (0, 1, 0) is labelled 2\nand (0, 0, 1) is labelled 3. Next, observe that the sides of the triangle correspond\nto cuts in which one piece is devoid of any cake. Because no one would ever choose\nRENTAL HARMONY: SPERNER’S LEMMA IN FAIR DIVISION 7\n( 0, 1, 0 )\n( 0, 0, 1 )\n( 1, 0, 0 )\nA\nB C\nA\nC B\nB C\nA\nCB\nA\nC B\nA\nBA C A\nBC\nFigure 7. Labelling by Ownership.\nthis empty piece, each side of S is missing one label corresponding to the piece\nthat is empty. Hence the Sperner labelling condition is satisfied.\nBy Sperner’s lemma, there must be a (1, 2, 3)-elementary simplex in the trian-\ngulation. Since every such simplex arose from an ABC triangle, this means that\nwe have found 3 very similar cut-sets in which different people choose different\npieces of cake.\nTo show the existence of a single cut-set that would satisfy everyone with differ-\nent pieces, carry out this procedure for a sequence of finer and finer triangulations,\neach time yielding smaller and smaller (1, 2, 3)-triangles. By compactness of the\ntriangle and decreasing size of the triangles, there must be a convergent subse-\nquence of triangles converging to a single point. Such a point corresponds to a\ncut-set in which the players are satisfied with different pieces. Why?\nSince each (1, 2, 3)-triangle in the convergent subsequence arises from an ABC\ntriangle, consider the choices that the players made in each. With only finitely\nmany ways for players to choose pieces, there must be an infinite subsequence in\nwhich the choices of A, B, and C are all constant. Closed preference sets guarantee\nthat at the limit point of this subsequence of triangles, the players are satisfied\nwith distinct choices.\n4. The n-player case\nThe preceding proof generalizes easily for n players. The only issue that must be\naddressed is the choice of triangulation for S when n > 3. We need a triangulation\nin which each elementary simplex can be fully labelled by the names of the players.\nThe triangulation we proposed for n = 3 does not generalize easily. However, one\nthat works for arbitrary dimensions is a triangulation by barycentric subdivision.\nLoosely speaking, this procedure takes each elementary simplex in a triangulation\nand subdivides it by marking the barycenters of the faces in each dimension and\nconnecting them to form a new triangulation. A rigorous description of this pro-\ncedure may be found in [17]. Observe that the mesh of this triangulation can be\nmade arbitrarily small by iterating this procedure; see Figure 8.\n8 FRANCIS EDWARD SU\nFigure 8. Barycentric subdivision in a 2-simplex, iterated twice.\nSuppose we have iterated barycentric subdivision m times. The desired labelling\ncan be achieved by allowing all vertices that remain from the (m− 1)-th iteration\nto be labelled A. Any new vertices introduced in the m-th barycentric subdivision\nare barycenters of simplices of the (m−1)-th subdivision. To each class of vertices\nthat are barycenters of faces of the same dimension, assign a distinct owner from\nthe persons remaining. There are n − 1 such classes. One may verify that this\nfully labels each of the elementary simplices by owners, because each edge connects\nvertices of different classes.\nNow the proof continues almost exactly as in the case n = 3: since each point\nin S corresponds to a cut-set, we construct a new labelling of the triangulation by\nasking the owner of each vertex, “Which piece would you choose if the cake were\ncut with these cuts?” The new auxiliary labelling is a Sperner labelling and yields\nnearby cut-sets that satisfy each person differently. Because this may be done with\narbitrarily fine triangulations, by taking subsequences, one may find sequences of\ncut-sets all converging to one set of cuts in which each person chooses a different\npiece.\n5. A constructive approximate algorithm\nNotice that the preceding proof yields a constructive \u000f-approximate algorithm\nfor cake-cutting—namely, for any prespecified \u000f (such as at the level of crumbs),\none may find a set of cuts in which each person receives a piece he considers to be\nthe best up to \u000f-tolerance in the size of the pieces. Simply start the procedure with\ntriangulation mesh size less than \u000f, and then the “trap-door” argument gives a con-\nstructive method for finding a fully owner-labelled elementary simplex. Choosing\nany vertex of this simplex yields a cut-set representing the desired \u000f-approximate\nsolution.\nSuch an algorithm could be implemented on a computer, which could keep track\nof what cuts to suggest tentatively and which player to ask, by simply following\ntrap-doors through the simplex of cut-sets. Note that players do not have to\nstate their preferences on every vertex in the triangulation, but only on vertices\nnear a trap-door path, i.e., the complete auxiliary labelling may not need to be\ndetermined. So while this algorithm terminates in a number of steps bounded by\nthe number of simplices of the triangulation, it can terminate much sooner.\nWe emphasize that this notion of \u000f-approximation is based on the physical size\nof the pieces, not on any quantitative measure of player preferences. However,\nif one assumes the players’ measures are continuous over the simplex, then by\ncompactness of the simplex and the finite number of players, for any \u000f > 0 there\nRENTAL HARMONY: SPERNER’S LEMMA IN FAIR DIVISION 9\nexists a δ > 0 such that pieces of physical size less than δ are believed by each of\nthe players to be size less than \u000f.\n6. Chores and Rent-partitioning\nNow we show how Simmons’ cake-cutting method can be adapted to address\nother fair-division problems, such as chore division and rent-partitioning.\nFinding schemes for envy-free chore division has historically been a more com-\nplicated problem than cake-cutting. Most envy-free procedures for cake-cutting\ndo not carry over to chore division without significant modifications. Oskui [9]\nsolved the case for 3 people; following modifications proposed by Brams and Tay-\nlor in [2, pp. 37-39] and [3, pp. 153-55], Peterson and Su [10] gave an explicit\nchore division scheme for an arbitrary number of players. We now give a simpler\n\u000f-approximate algorithm for chore division, which falls out nicely as a special case\nof the rent-partitioning problem.\nIn this problem, n housemates have decided to rent an n-bedroom house for\nsome fixed rent. Each housemate may have different preferences—one may prefer\na large room, another may prefer a room with a view, etc. Is there a method for\nfairly dividing the rent among the rooms? We prove the following:\nRental Harmony Theorem. Suppose n housemates in an n-bedroom house seek\nto decide who gets which room and for what part of the total rent. Also, suppose\nthat the following conditions hold:\n(1) (Good House) In any partition of the rent, each person finds some room\nacceptable.\n(2) (Miserly Tenants) Each person always prefers a free room (one that costs\nno rent) to a non-free room.\n(3) (Closed Preference Sets) A person who prefers a room for a convergent\nsequence of prices prefers that room at the limiting price.\nThen there exists a partition of the rent so that each person prefers a different\nroom.\nCondition (1) ensures that the problem is well-posed—one cannot talk about\npreferences if some person finds no room acceptable, which might happen, for\ninstance, if the rent is too high for all rooms or the rooms are in poor condition.\nThe miserly condition (2) can be relaxed a bit, as we show in Section 8. The\ncondition also rules out “free closets,” i.e., rooms in which no one would live, even\nif free.\nCondition (3) merely says that in the space of all pricing schemes, preference\nsets are closed in the topological sense. Note that preference sets may overlap—if\nin some pricing scheme a person equally prefers two rooms, that person can be\nassigned to either room.\nThe rent-partitioning problem may be viewed as a generalization of the cake-\ncutting problem, in which one seeks to divide goods fairly, and the chore division\nproblem, in which one seeks to divide bads fairly. However, since the rooms (the\ngoods) are indivisible, known cake-cutting solutions cannot be applied to this prob-\nlem. And since the rental payments (the bads) are attached to specific rooms, they\n10 FRANCIS EDWARD SU\ncannot be divided into more than n pieces and reassembled, which rules out the use\nof known envy-free chore-division methods such as the discrete method proposed\nin [3, pp. 154-55] and the procedures proposed in [10]. The two other moving knife\nschemes proposed for chore division in [3, pp. 153-54] guarantee each player at\nmost 1\/n of the chores, but are not envy-free.\nAlkan, Demange, and Gale [1, pp. 1031-32] have addressed this generalization di-\nrectly and offer a solution to rent-partitioning via constrained optimization. They\nimplicitly assume conditions equivalent to our conditions (1) and (3), and use a\ncondition weaker than condition (2), but not quite as weak as the condition (2’)\nthat we give in Section 8.\nWe now show how a Sperner’s lemma approach can address the rent-partitioning\nproblem.\n7. Rental Harmony: cake-cutting with a twist\nOur proof of the Rental Harmony Theorem follows Simmons’ proof for cake-\ncutting, but with a twist, so we sketch it.\nSuppose there are n housemates, and n rooms to assign, numbered 1, ..., n. Let\nxi denote the price of the i-th room, and suppose that the total rent is 1. Then\nx1 + x2 + · · · + xn = 1 and xi ≥ 0. From this we see that the set of all pricing\nschemes S forms an (n− 1)-simplex in Rn.\nNow triangulate this simplex by barycentric subdivision of small mesh size.\nLabel it with a fully labelled vertex labelling by the names of the housemates\n(the same scheme as suggested for cake-cutting). The name at each vertex will be\nconsidered the “owner” of that vertex; recall that each vertex corresponds to some\npricing scheme for the rooms.\nConstruct a new labelling from the old by asking the owner at each vertex in\nthe triangulation: “If the rent were to be divided according to this pricing scheme,\nwhich room would you choose?” Condition (1) ensures that some answer can be\ngiven. Label the vertex by the number of the room that is answered. Let ties in\npreference be broken arbitrarily.\nHere’s the twist: the new labelling that results is quite different from the one\nthat arose in cake-cutting. It is not a Sperner-labelling. However, because of the\nmiserly condition (2), it has the property that along each (n−k)-dimensional face,\nk rooms are free and thus owners along that face prefer one of those k rooms.\nFigure 9 shows what such a labelling looks like for n = 3.\nIs there a Sperner-like combinatorial lemma that shows the existence of a fully\nlabelled elementary simplex in this triangulation?\nIf so, one could proceed as in cake-cutting, by taking finer and finer triangula-\ntions to get a sequence of fully labelled elementary simplices converging to a point,\nwhich by condition (3) yields a pricing scheme in which all housemates prefer dif-\nferent rooms. So, all that remains is to establish the Sperner-like combinatorial\nlemma with a constructive proof.\nThere are two ways one may proceed. The reader may enjoy proving a Sperner-\nlike lemma for this labelling by using a trap-door argument. The interesting thing\nthat one discovers about this labelling is that on each facet, there is only one\nRENTAL HARMONY: SPERNER’S LEMMA IN FAIR DIVISION 11\n3 only\n1 or 32 or 3\n1 only2 only\n1 or 2\nFigure 9. The dual labelling arising from rent-partitioning.\nfully labelled simplex that can be followed into the interior, so that the trap-door\nprocedure succeeds without returning to the boundary again.\nOr the reader may wish to prove the existence of a fully labelled simplex on\nthe interior by appealing directly to Sperner’s lemma. The key idea is to dualize\nthe simplex S to form a new simplex S∗. Loosely speaking, the dual of a simplex\nreverses the dimensions of k-dimensional and (n − 1 − k)-dimensional faces. For\ninstance, the corner vertices of S become the facets of S∗, and the facet barycenters\nof S become the vertices of S∗; see Figure 10.\nS S*becomes\nFigure 10. The dualization S∗ of S. Vertices, barycenters, and\none elementary simplex are marked to show how they are trans-\nformed.\nA rigorous treatment of dualization can be found in Vick [17]. Note that S∗ can\nbe triangulated—in fact, using barycentric subdivision, the vertices and elementary\nsimplices of S∗ are in 1-1 correspondence with the vertices and elementary simplices\nof S. Let the triangulation of S∗ inherit a labelling via this correspondence with\nS. One may now verify that the labelling of S∗ is a Sperner labelling! Hence\nthere exists a fully labelled elementary simplex of S∗, which corresponds to a fully\nlabelled elementary simplex of S, as desired. This “dual” Sperner lemma is due\nto Scarf [12].\n12 FRANCIS EDWARD SU\nA constructive algorithm is obtained by following “trap-doors” in Sperner’s\nlemma. Choose an \u000f smaller than the rental difference for which housemates\nwouldn’t care (a penny?). Following trap-doors corresponds to suggesting pricing\nschemes and then asking various players, “Which piece would you choose if the\nrooms were priced like this?” Once a fully labelled elementary simplex is found,\nany point inside it corresponds to an \u000f-approximate rent-partitioning. We invite\nthe reader to code a trap-door algorithm that could be implemented on a computer,\none that would propose the necessary cut-sets and question the appropriate players\nat each step.\nIt is possible to obtain the Rental Harmony Theorem without any dualization\nargument and without condition (2) if one allows the possibility of negative rents.\nSpecifically, let each person contribute a fixed amount K to a pool from which the\nrent is paid. The leftover money is used to pay “rebates” associated with each room\n(which may be larger than K). This converts the problem into a fair division of\ngoods (rebates), in which the space of rebates is a simplex that assumes a Sperner\nlabelling if players demand a non-zero rebate. For large K this is quite reasonable.\nHowever, solutions may include situations in which a housemate is being paid by\nthe others to live there. Thus allowing this possibility may not be realistic because\nin real life, paying housemates are more likely to ditch the subsidized housemate\nand use the extra room (and extra money) in other ways.\n8. Comments and Discussion\nThe Rental Harmony Theorem establishes the existence of envy-free chore divi-\nsion and a new \u000f-approximate algorithm, by simply thinking of the rent payments\nas chores and ignoring the rooms; divisibility of chores can be achieved by divid-\ning the time spent on them. When reinterpreted, the three conditions from the\nRental Harmony Theorem become: (1) all the chores must be assigned, (2) each\nperson prefers no chores to some chores, and (3) preference sets are closed. These\nare pretty reasonable assumptions. The \u000f-approximate algorithm that arises from\nthis does not involve a lot of cutting and reassembling, as do the exact methods\nproposed in [3] and [10].\nFor rent-partitioning, we point out that condition (2) may not always be a\nreasonable assumption. For instance, someone may be willing to pay a little bit of\nmoney for a room that is slightly larger than a free room. However, by inspecting\nthe proof, one sees that the Rental Harmony Theorem still holds with a weakened\nversion of condition (2):\nCondition (2’). Each person never chooses the most expensive room\nif there is a free room available. This does not require the person\nto choose the free room.\nIn particular, this will hold if a person always prefers a free room to a room\ncosting at least 1\/(n − 1) of the total rent. Hence condition (2’) is a slightly\nweaker sufficient condition than that given by [1, pp. 1031-32]. To see why the\nRental Harmony Theorem still holds, consider its proof and note that using this\ncondition gives a more complicated labelling of S, but the corresponding labelling\non S∗ still remains Sperner.\nRENTAL HARMONY: SPERNER’S LEMMA IN FAIR DIVISION 13\nWhat condition (2’) does not address is a situation in which the total rent is so\nlow, or some room so large, that one would be willing to pay for the most expensive\nroom even when some other room is free. In practice, however, housemates do not\nusually choose a house with such lopsided arrangements. Even still, condition (2’)\ncan likely be weakened further, but the extent to which it can (and still maintain\nnon-negative rents) is an open question.\nWe remark that other triangulations may be used instead of barycentric subdi-\nvision. These have better convergence properties but are harder to describe; see\n[16] for a survey and applications to fixed point algorithms.\n9. Anecdote and Acknowledgments\nMy first exposure to the Sperner argument for cake-cutting came via Michael\nStarbird, who attributed the method to a graduate student of his, Forest Sim-\nmons. Simmons had been presenting this cake-cutting scheme to math clubs and\nhigh school groups, but never formally submitted the idea for publication. His\ninspiration was the Monthly article by Stromquist [15], which made use of a\ntheorem that can be proved by Sperner’s lemma.\nMany years later, when my friend Brad Mann told me about the rent-partitioning\ndilemma that he and his housemates were facing, I was reminded of these ideas\nand realized that Sperner’s lemma could also be adapted to treat rent-partitioning,\nas well as chore division.\nI am grateful to Arthur Benjamin, Steven Brams, Brad Mann, Forest Simmons,\nand Ravi Vakil for many helpful discussions, and I thank Michael Starbird for\nintroducing me to Sperner’s lemma.\nReferences.\n1. A. Alkan, G. Demange, and D. Gale, Fair allocation of indivisible goods and criteria of\njustice, Econometrica 59 (1991) 1023-1039.\n2. S. J. Brams and A. D. Taylor, An envy-free cake division algorithm, Economic Research\nReports, C. V. Starr Center for Applied Economics, New York University, 1992.\n3. S. J. Brams and A. D. Taylor, Fair Division: from Cake-Cutting to Dispute Resolution,\nCambridge University Press, Cambridge, 1996.\n4. S. J. Brams and A. D. Taylor, An envy-free cake division protocol, Amer. Math. Monthly\n102 (1995) 9-18.\n5. D. I. A. Cohen, On the Sperner lemma, J. Combin. Theory 2 (1967) 585-587.\n6. M. Gardner, aha! Insight. W.F. Freeman and Co., New York, 1978.\n7. B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-\ndimensionale Simplexe, Fund. Math. 14 (1929) 132-137.\n8. H. W. Kuhn, Simplicial Approximation of Fixed Points, Proc. Nat. Acad. Sci. U.S.A. 61\n(1968) 1238-1242.\n9. R. Oskui, Dirty work problem, preprint.\n10. E. Peterson and F. E. Su, Exact procedures for envy-free chore division, preprint.\n11. J. M. Robertson and W. A. Webb, Cake-Cutting Algorithms: Be Fair If You Can, A K\nPeters Ltd., Natick, Mass., 1998.\n12. H. Scarf, The computation of equilibrium prices: an exposition. Cowles Foundation Dis-\ncussion Paper No. 473, Cowles Foundation for Research in Economics at Yale University,\nNov. 1977.\n13. F. W. Simmons, private communication to Michael Starbird, 1980.\n14 FRANCIS EDWARD SU\n14. E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh.\nMath. Sem. Univ. Hamburg 6 (1928) 265-272.\n15. W. Stromquist, How to cut a cake fairly, Amer. Math. Monthly 87 (1980) 640-644.\n16. M. J. Todd, The Computation of Fixed Points and Applications, Lecture Notes in Economic\nand Mathematical Systems, Springer-Verlag, New York, 1976.\n17. J. W. Vick, Homology Theory. Springer-Verlag, New York, 1994.\nFRANCIS SU earned his Ph.D. from Harvard in 1995 studying random walks, but his interest in\nfair division was a result of thesis procrastination. He now (fairly? randomly?) divides his research time\nbetween studies of random walk convergence and fair division algorithms. He is an assistant professor at\nHarvey Mudd College, a “blue dot” MAA Project NExT Fellow, and an amateur songwriter in his spare\ntime. Since the writing of this article he and HMC undergraduate Elisha Peterson wrote a web applet\nthat uses an improved trapdoor algorithm; their Fair Division Calculator works for cakes, chores, or rent\nand they invite you to try it out at http:\/\/www.math.hmc.edu\/~su\/fairdivision\/.\nHarvey Mudd College, Claremont, CA 91711\nsu@math.hmc.edu\n"}
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{"id":"4dF_zW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Teaching-Research-Encouraging-Discoveries.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":105,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Teaching-Research-Encouraging-Discoveries","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Teaching-Research-Encouraging-Discoveries.pdf"],"author":"pdfTeX-1.40.3","author_sxt":["pdfTeX-1.40.3"],"last_modified":"Sat Jan 09 05:43:07 UTC 2010","dates_in_content_count_i":1,"dates_in_content_dts":["Fri Aug 11 00:00:00 UTC 2006"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":4478705955628426986,"exact_signature_unique_b":true,"fuzzy_signature_l":-4294491452936172358,"fuzzy_signature_unique_b":true,"wordcount_i":5282,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:21:58 UTC 2022","fresh_date_dt":"Wed Dec 06 11:32:13 UTC 2023","publisher_t":"pdfTeX-1.40.3","language_s":"en","size_i":147327,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088058718486528,"text_t":"To appear, American Mathematical Monthly, 2010.\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES\nFRANCIS EDWARD SU\nAbstract. What does it take to turn a learner into a discoverer? Or to turn a teacher into a\nco-adventurer? A handful of experiences—from teaching a middle-school math class to doing\nresearch with undergraduates—have changed the way that I would answer these questions.\nSome of the lessons I’ve learned have surprised me.\nThe title of this article may seem a little puzzling. After all, the words teaching and research\nusually only appear in the same sentence when separated by the word and, on a list of a faculty\nmember’s obligations. Do these words belong together at all? What does teaching have to do\nwith research or discovery?\nOne of my favorite quotes about teaching comes from one we normally associate with re-\nsearch:\nThe principal aim of mathematical education is to develop certain faculties of\nthe mind, and among these intuition is not the least precious. It is through it\nthat the mathematical world remains in touch with the real world.\n—Henri Poincaré [5, p. 128]\nBut Poincaré also said:\nIt is by logic that we prove, but by intuition that we discover.\n—Henri Poincaré [5, p. 129]\nSo we teach to build intuition, and intuition enables our students to make discoveries. Notice\nthat Poincaré does not say that the principal aim of teaching is to convey facts or theorems.\nThe principal aim of mathematical teaching is to build qualities of mind that enable students\nto make discoveries.\nBut how does one do that? How does one turn a learner into a discoverer? When I was first\nstarting out as a new professor, I might have given these answers:\n• Teach the needed background.\n• Cultivate maturity.\n• Inspire them!\n• Ask good questions.\n• Select the smartest students.\n• Give open problems.\n• Advertise the thrill of research.\n• Be an expert in what you advise.\n• Encourage independence.\nNow, I shall explain why I believe every one of these pieces of advice is either plainly wrong\nor, at best, inadequate. Along the way, I will mention some concrete teaching ideas that you\nmay find useful as well.\nThis article is based on the James R. C. Leitzel Lecture, delivered by the author at the MAA MathFest,\nKnoxville TN, on Friday, August 11, 2006.\n1\n2 FRANCIS EDWARD SU\n1. The Principal Aim of Mathematical Teaching\nThere is one word that is an excellent metaphor for everything I want to say:\nYAWP.\nWhat is a yawp?\nI sound my barbaric yawp over the roof(top)s of the world!\n—John Keating, quoting a Walt Whitman poem,\nin the movie Dead Poets Society\nAs Keating explains, a yawp is a loud cry or yell. But in the poetry of Walt Whitman, the\nword yawp refers to the inner groaning inside each of us, too deep for words, that is waiting to\nbe released and experienced. In the movie Dead Poets Society, an English teacher named John\nKeating (played by Robin Williams) does more than just teach poetry—rather, he breathes\npoetry, he inhabits poetry, and he inspires his students to do the same. In one scene, his\nstudent Todd Anderson (played by Ethan Hawke) has been struggling all weekend to write a\npoem, but without success. In class, when Todd fails to deliver a poem, Keating encourages\nTodd to first find his yawp, and once he does, Keating helps him transform his yawp into\npoetry in a truly mesmerizing manner [6, 55:07-57:57].\nKeating was an exuberant teacher who exhorted his students to live extraordinary lives. He\ndid not measure this by the usual metrics of success and accomplishment; rather, for Keating,\nliving extraordinarily meant becoming who they were created to be: human beings empowered\nto express their passion through poetry, and to use their souls, spirits, hearts, and minds to\nseize the day with all its wonders.\nAs mathematicians, we have experienced one of the greatest human wonders: the thrill of\ndiscovery. And the aesthetic pleasure of an elegant proof captures our spirits just as poetry\ndoes. So what I seek to cultivate in my students is a mathematical yawp:\nDefinition. A mathematical yawp is that expression of surprise or delight at discovering the\nbeauty of a mathematical idea or argument.\nEvery student is capable of a mathematical yawp. The yawp may not be a poem when it\nis first expressed, but I believe the principal aim of mathematical teaching is to nurture the\nmathematical yawp, and help transform it into poetry.\n2. Math Fun Facts\nWhen I first began teaching calculus in graduate school, I lamented the fact that most\nstudents would leave college with a mistaken notion of what mathematics is. After all, most\nof them took math to fulfill a requirement, usually some brand of calculus or something less\nadvanced, and it was the final course they would take. As a result, the view most students\ndevelop of mathematics is that it is a cut-and-dried, 400-year-old list of theorems and appli-\ncations. They do not get the sense that mathematics is an exciting, living, and developing\nsubject.\nI wanted to change that impression. So I began to present daily “Fun Facts”, by taking\n5 minutes at the start of each lecture to show my students some mathematical idea that I\nthought was fascinating—for instance: the uncountability of the reals, the nine-point circle,\ntiling a chessboard with opposite corners removed, the ham sandwich theorem, etc. The facts I\nchose usually had nothing to do with calculus; the main point was to broaden their perspective\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 3\nof mathematics, to show them math is alive and full of interesting questions and new ways of\nthinking, and to whet their appetite for learning more.\nThe student reaction was overwhelming. They loved Fun Facts. They clamored for them at\nthe beginning of every class. Students stopped coming late! If I ever forgot to do a Fun Fact,\nthey would remind me of my obligation, as if going through Fun Fact withdrawal. Sometimes\nthey would clamor for more than one (though perhaps to postpone the calculus lecture). I had\nnurtured their mathematical yawp.\nAnd they yawped for more. They would often come back to me with questions about Fun\nFacts, or variants of Fun Facts they had studied or solved. Many of my teaching evaluations\ncited Fun Facts as a reason that they wanted to take more math, even if they didn’t like\ncalculus. I had helped them find their yawp, even if it wasn’t for calculus!\nAnd then it dawned on me that it was more important to nurture their yawp than to teach\nthem calculus.\nLesson #1. Teach the needed background?\nNo. Nurture the yawp.\nOf course, we hope that these goals are in harmony, but if there is ever a conflict, we must\nremember the principal aim. For instance, I make it a policy never to stop nurturing the yawp\nin favor of “getting through the material”.\nIf you’d like to try using Fun Facts in your courses, I have created a web archive of about\ntwo hundred of the ones that I have used. (See [7] or Google “Math Fun Facts” to find it.)\nBut the best Fun Facts are the ones that you care about the most! Give them the Fun Facts\nthat fueled your passion for mathematics.\n3. What I Learned from 8th Graders\nI found a way to use a similar idea in the K-12 arena. One of the most formative experiences\nof my career occurred in my second year as a faculty member, when I learned that a principal\nof a local middle school was looking for a professor to teach a math enrichment course to\nchallenge some bright 8th graders. Intrigued by the idea, I agreed.\nOnce a week, the 8th grade math class at this school was divided into two groups by ability.\nOne half stayed with the usual teacher to get some needed help and extra practice, and the\nother group met with me.\nWhat should I do with them? Instead of teaching them more of the same stuff that they\nmight encounter again in high school, I chose instead to teach them “enrichment material”,\nsuch as extended Fun Facts, proof techniques like induction and contradiction, fast mental\narithmetic, magic tricks using mathematical ideas, elementary knot theory and Reidemeister\nmoves, mind-boggler puzzles, elementary combinatorics, etc.\nMany things about this experience surprised me. First of all, teaching 8th graders is a\nlot harder than teaching college students! Any issue related to teaching college students is\namplified for 8th graders. If college students can’t sit still for more than fifteen minutes, 8th\ngraders can’t sit still for five. College students get bored; 8th graders noticeably fidget in their\nseats. College students give nodding winks at each other; 8th graders pass notes. If college men\nspeak up in class more than college women, 8th grade boys dominated my class discussions\nentirely.\nAs a result, for everything I would normally do to engage a college class, I had to do better\nwith these 8th graders. If my college teaching style is interactive, my 8th grade teaching style\nhad to be more so. I had to pay more attention to domineering students, give more reassurance\n4 FRANCIS EDWARD SU\nand encouragement, repeat ideas more often, ask more questions to keep them engaged and\nignite their curiosity and imagination ... and it occurred to me that doing these things was\ngreat for my college teaching!\nLesson #2. Cultivate maturity in your students?\nNo. Revive their child-like curiosity and imagination.\nI believe there isn’t a single concept in a first-year college course that an 8th grader couldn’t\nunderstand if developed well. So what will you do to develop it well, to engage your students?\nWe must find ways to revive the child-like curiosity of college students. I see that 8th graders\nare more curious, more willing to probe with a “why?”, more willing to ask questions if they\ndon’t understand, and more willing let you know your pen is leaking in your front pocket.\nSomewhere along the path to adulthood, they lose the ability to ask questions.\nWas this enrichment class a success? Did they develop a mathematical yawp? In the end-\nof-term evaluations, I asked these 8th grade students to complete the following sentence:\nI now know that mathematics is ...\nI received some responses like this:\n• ... harder than I thought (I thought it was too easy before).\nBut most of the responses were like these:\n• ... extremely challenging, yet excessively fun. Also, it’s a much broader\ntopic than I had originally thought.\n• ... can be fun and not always so boring. I also now know that sometimes,\nmathematics can have absolutely nothing to do with numbers.\n• ... a broader subject than it was to me before; this class opened up many\nnew, exciting, and challenging topics that have made me more interested\nin mathematics as a whole.\n• ... a world of mysteries waiting to be discovered.\nExactly! We seek to build intuition to enable them to make discoveries!\nRecently, I heard from a student in this class who is now just graduating from college, eight\nyears later. He said: “from what I remember I really enjoyed it ... in middle school it was the\nonly class that really opened my eyes to different thinking.” All these responses indicate that\nthey had developed a yawp.\nAnother lesson I learned from this class was how easy it was to miss hearing a yawp. There\nwas a shy girl in this class who hardly ever spoke up and seemed not to be engaged at all. I\ndidn’t notice her for several weeks until I assigned some homework on induction. I was curious\nto see what 8th graders could do, and I naively expected the best students to be the ones who\nwere answering all the questions.\nBy and large, the students seemed to grasp induction. But I’d say the homeworks looked like\nthe typical work you would get from beginning calculus students, not so good on the form and\nvery unclear in the writing. On the other hand, this student’s induction proof was astounding.\nIt had perfect form, perfect understanding, perfect writing—in complete sentences with good\nconnective phrases, journal style, with the clarity and maturity that I expect from professional\nmathematicians.\nThroughout the term she continued to impress me, always in her quiet way. What was\ninteresting to me was that her regular 8th grade math teacher did not seem to think she was\nexceptional at all! Why?\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 5\nAnd then I wondered how many students like her I was missing in my college courses because\nof the way I was teaching, and what I could be doing to look for their yawp.\nLesson #3. Identify invisible yawpers.\nAs a sad footnote, after the class was over, I told this student’s parents that if she should\nshow an interest in math later on, they should encourage it because she was truly exceptional.\nTheir reaction (or lack thereof) was as if to say: why would I want my girl to study math?\nAnd I understood then that there are still social pressures that push children out of callings\nfor which they might have a gift.\n4. Nurturing Poets\nI surveyed a dozen of my former thesis and research students and asked them to reflect\non their experiences doing research as undergraduates, whether with me or in other settings.\nI refer to them as Poets, since they have found their yawp. Nearly all of them are now in\ngraduate school or just beyond.\nOne question I asked them was what early influences caused them to think research in\nmathematics was something they wanted to try. One Poet said:\nMy parents were a huge influence on me, posing mathematical questions to me\nwhen I was young ... for example, I remember my dad asking me how many\nsteps it would take to get to the wall if on each step, I went halfway.\nShe had found her yawp because her parents nurtured it by asking probing questions, but\nthey let her experience the joy of discovery herself.\nMore than half the students cited the experience of working puzzles or doing math problem-\nsolving as early influences. Many had been part of organized problem-solving groups, or at\nsummer math programs where they, in the words of one Poet, “caught a fever of math research”.\nIn effect, these groups became their “Dead Poets Societies”, places and spaces where they could\ngather to recite poems and yawp together.\nAt Harvey Mudd College, Andy Bernoff and I run a problem-solving seminar that has become\nsomething of a phenomenon on campus. Every fall, about 70 students (that’s 10 percent of\nour student body!) gather on Tuesday nights over pizza to have fun with problem-solving.\nWe keep the atmosphere lively, and we allow them to work in groups. We introduce a few\nproblem-solving techniques at the start, then let them work on a slate of five or six problems,\nbreak for pizza, then we have students present their solutions [1, 2]. They have the space to\nmake discoveries themselves, and by letting them present their solutions, we give them the\nspace to yawp as well (just as any good Dead Poets Society would do).\nThis seminar now attracts a large fraction of students who are not even math majors, but\nwho come for the thrill of yawping, either when they solve the problems, or when they see a\nsolution and get the “aha!” experience that is at the heart of a true yawp.\nWould I still say it is important to inspire students? Of course. But I think it is even better\nto give students space for self-inspiration so they can experience and express their yawp.\nLesson #4. Inspire them?\nBetter: create spaces for self-inspiration.\n6 FRANCIS EDWARD SU\nMoore-method courses are another example of spaces where students can find their yawp\nand make discoveries themselves. I won’t say much about them here, because much has been\nsaid about them, but the best course I ever took as an undergraduate was a Moore-method\ntopology course (taught by Mike Starbird) and it had a profound impact on developing my\nyawp.\nWe all understand how important it is for a professor to ask good questions of her students.\nAfter all, a good question can prompt a discovery. But I now believe it is even better if you\ncan teach a student how to ask good questions themselves!\nLesson #5. Ask good questions?\nBetter: teach how to ask good questions.\nLet me give you an example that I learned from my colleague Lesley Ward. She has used\n“question stems” in her courses, to teach students how to ask good questions (using a suggestion\nof T. J. Mueller, and adapting questions from [3]). For instance, in an analysis course, after\ngiving part of a lecture on some topic, she asks students to form groups and she hands out\ncards which have questions like these on them, with blanks:\n• What is an example of ...?\n• How does ... affect ... ?\n• What would happen if ... ?\n• What is the difference between ... and ...?\nThe students formulate questions about the lecture based on these models, and then they\ndiscuss them in class. In addition to the benefit of teaching students the kinds of questions they\nmight think about, it has the added benefit of students often answering each others’ questions!\nAnother way of encouraging students to ask good questions is to reward them for it. On\nsome exams, I put a problem in which I ask students to write down a research question that\nthey would like to answer, based on what they have learned so far in the class. The answer to\nthe question doesn’t have to be unknown to the world, it just has to be unknown to them. I\nam happy if they have just gone through the reflection necessary to ask a question, so I give\nfull credit for any reasonable attempt at formulating a question.\nFor our students to experience true research, we should certainly give them open problems.\nBut in order to cultivate a yawping ability, I believe it is much better to give students open-\nended problems.\nLesson #6. Give open problems?\nBetter: give open-ended problems.\nI learned a terrific example of this from my colleague Michael Orrison, who has developed a\nproject that he has used successfully in a discrete mathematics course. In it he asks students\nto:\nDefine a measure of the complexity of a graph. Compute it for several examples,\nand prove some properties about your measure.\nThis is the main idea; you can find more details in [4]. What is great about this exercise is\nthat it is open-ended, and there are multiple right answers. Students are able to be creative\nabout the solution, and they are excited about investigating a concept that they have defined\nthemselves. At the end of the project, students make presentations.\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 7\nLooking at some of the slides from past presentations, one sees that students develop a wide\nvariety of notions of complexity. More interestingly, they find creative ways of expressing their\nideas, e.g., by describing their complexity measure as answers to questions: “How expensive is\nit to plow the town?”, “How many friends do you have?”, etc.\nAnd their yawps are often visible as they explore properties of their measures of complexity—\nfor instance, written on one recent memorable slide, a student exclaims “Really Cool!” and\n“Yowza!” when she realizes that her measure is invariant under taking graph complements.\nThat’s a yawp if I ever saw one!\nMy colleague Darryl Yong has even adapted Orrison’s project as an exercise for high school\nteachers, in a conference called “Imagine Math Day” [8, 9]. And there are many other project\nexamples that could follow a similar model. For instance, “What is a fair voting procedure?”\nor “Define a measure of a center of a distribution.” Last summer, I gave a couple of my\nresearch students the following open-ended problem: “List 10 generalizations of convexity.”\nAfter they came up with those ten, I asked them to focus on the most interesting ones and try\nto characterize those notions of convexity in terms of properties that people already understand.\nMy students had a lot of fun generating their list, and eventually took one of their notions and\nproved necessary and sufficient conditions that would allow someone to tell if a set satisfied\ntheir notion of convexity. This resulted in a paper that they wrote completely on their own,\nand submitted for publication. Because of one open-ended problem, they created a new idea,\nand they were motivated to study it and explore it—they yawped!\n5. Research with Undergraduates\nBut now suppose you have taken your students to the point where they can yawp. How do\nyou help them turn it into poetry? If a yawp is the thrill of discovery, a poem is a yawp that is\ncommunicated well. This is where the undergraduate research experience can play a big role.\nOne major question that every advisor faces is the question of how to choose students. I\nalways thought that the best metric for this was to choose students who were the smartest, or\ngot the best grades. However, I now think that something else is much more important—strong\nmotivation and persistence.\nLesson #7. Select the smartest students?\nNot necessarily; select motivated students!\nOne of the best research students that I have ever had was not a student that fit the mold\nof getting the best grades, and I did not notice her abilities from her coursework. She was\na B+ student, and when she approached me about summer research I had already offered\nmy research positions to other students. But she came back several times to ask, so I finally\ndecided to go out of my way to ask the dean for extra funds to support her, and he agreed.\nThis was one of the most rewarding experiences that I have had as an advisor. The persis-\ntence my student showed in inquiring about research opportunities translated into a tenacity\nin tackling research problems. She was extremely dedicated and motivated, and always came\nto research meetings having done everything that we discussed at the last meeting, and more.\nShe was the kind of student who was meticulous and took her time to carefully think things\nout, and so while taking exams was not a place where her gifts were displayed, she really shone\nin research problems where persistence and meticulousness are rewarded. And she was an\nexcellent writer. By the end of the summer, she had completed a paper that was polished and\nready to submit!\n8 FRANCIS EDWARD SU\nSo, if I had to give advice about what to look for in a research student, I would say smarts\nare important, but if you have to choose between a super-smart student and a smart motivated\nstudent, I would go for the latter. Some qualities to look for: Are they persistent? Do they\nexcel in something that shows dedication? For instance, we have a couple of research students\nin our department this summer who are B students, but one is a nationally-ranked player in\nthe computer game StarCraft, and another is a hackey-sack expert. And their advisors report\nthat these students are doing outstanding, publishable research.\nAnother question I ask about potential research students is: do they write well? I ask for a\nwriting sample, such as a proof they have written for a homework exercise. Even if you don’t\nchoose a student who can write, it is nice to know what kind of writer you are working with.\nSuch information can be useful, for instance, if you are choosing students to work in teams.\nLesson #8. Advertise the thrill of research?\nBetter: set complete expectations.\nI used to want my research students to think research was going to be really fun and exciting.\nBut now, I think that they should be aware of the whole picture. I now tell my students in\nadvance that research can be frustrating at times, and I warn them that often it hits about\nthe fourth week of the summer when they are really wrestling with their research problem. If\nI tell them in advance, then they aren’t surprised when it happens, and they understand it is\nnormal to feel that way.\nAlong these lines, one Poet says:\nThe most valuable aspect, in retrospect, was how my research jobs made re-\nsearching look far less glamorous than I had thought it might be ... in a way it\nwas demoralizing, but you likely want your bubble burst before you chase the\ndream too far.\nBut the same student also notes that understanding the expectations doesn’t diminish his\ndesire to yawp—he says: “The hope of delightful discoveries and taking part in that same cool\nelegance I recognized so long ago is still a large part of what keeps me going.”\nI also set clear expectations for my students about how to conduct their research. For\ninstance, I ask them to keep track of what they are doing in a research notebook. I explain\nthat this notebook is where they write down ideas that they want to follow through with later.\nI also tell them to start writing often and early in the process. When they get stuck or bored\nwith thinking about their problem, they can write down what they have been doing, or learn\nLaTeX and LaTeX their work. Another Poet reports:\nOne of the best things I learned was organization ... especially since none of\nmy later advisors mentioned this at all.\nWe can teach undergraduates research skills that will be useful to them later, even if, as the\nPoet suggests, graduate schools don’t teach those same skills.\nIn advising research students, I always had the impression that you should be an expert in\nwhat you advise. After all, I think it is common for us to worry: how can we nurture a yawp\nin someone else if we don’t know how it is we yawp ourselves? But now I think differently.\nLesson #9. Be an expert in what you advise?\nNo, let the student be the expert.\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 9\nWhat do I mean by this? I think it is extremely important to give my student ownership\nof the problem. I tell my students at the start of the summer: “By the time the summer is\nover, you will be the local expert on this problem. You will know more than anyone else here\nabout this problem, including me.” Doing so gives them the excitement of living up to that\nchallenge, and my job the rest of the summer is to probe, and model asking good questions\n(rather than trying to give them all the answers that I obviously don’t yet have). So they are\non an adventure to find out more than anyone else in my local neighborhood about their topic,\nand, by accompanying them on this journey, always asking or suggesting (but never forcing)\nwhere to go next, I am a “co-adventurer” with them. I listen to their yawps on this journey.\nNot surprisingly, my research with students has taken a random walk through many fields; as\nan example, our papers have been published in a diverse collection of journals: Random Struc-\ntures and Algorithms, Journal of Combinatorial Theory Series A, Discrete and Computational\nGeometry, Journal of Mathematical Analysis and its Applications, American Mathematical\nMonthly, Journal of Mathematical Biology, among others.\nIn all these examples, I was never the expert on these topics, but I picked up what I needed\nto know in a journey with my students, often with the students teaching me. I am only a co-\nadventurer. Students also learn from this journey that research often involves learning many\nnew and unfamiliar things.\n6. Building Community\nOne of the biggest pieces of advice that I hear about advising research is the adage about\nencouraging students to be independent. While I believe this should be true much later in the\nresearch process, I strongly disagree with this in the early stages.\nLesson #10. Encourage independence?\nNo! Give close guidance, and build community.\nFor instance, I think it is very important to meet often and regularly, especially in the\nbeginning of the research process. During those meetings, one should always set goals for the\nresearch students to work on for the next meeting. Doing so gives them something concrete\nto do and to report on and gives structure to the following meeting. It is especially important\nto have students write something up for each meeting. Doing so refines their thinking, and\nfocuses discussion.\nWhen it comes time to write a real paper, I always sit down with my student in front of\nthe computer and write, together with the student, at least a few pages. This not only helps\nget the paper off the ground, but doing so also models for the student the process of writing.\nStudents are often initially surprised by the care which I take to get definitions right, to make\nnotations consistent, or to ensure that attributions about prior work are correct. They get to\nsee how I think about writing and they can begin to do the same.\nThroughout their research, I want my students to see that mathematics really is a social\nenterprise. We, as a community, decide what’s important, our work depends on the work of\nothers, and we work together with others to advance the field. In one project, I included an\nundergraduate student on a project that I was doing with an economist in Germany. I included\nmy student in all the meetings (some by phone and some when my collaborator visited). By\nbeing fully part of this process, she was able to see one way in which collaboration could work,\nas well as see in this social setting how important it is to learn to communicate ideas well.\nPoets report:\n10 FRANCIS EDWARD SU\n• My research experience definitely helped me learn about writing papers,\nand perhaps more importantly, about working with others to co-write pa-\npers.\n• Working with other people besides you demonstrated the different possi-\nbilities when people with different strengths collaborated.\n• Working with another person on a research question is fundamentally dif-\nferent than any of the other group work I had been exposed to.\nI want students to see the importance of building community in their mathematical work. I\nhope that they can see that through collaboration, they can really have a lot of fun yawping\ntogether, whether that be through the sweat of persisting through a problem or the joy of\nactually solving it. As the Swedish proverb goes:\nA joy shared is twice the joy. A sorrow shared is half the sorrow.\nTo summarize, how would I answer the question: How do you turn a learner into a discoverer?\n• Nurture the yawp.\n• Revive child-like curiosity and imagination.\n• Identify invisible yawpers.\n• Create spaces for self-inspiration.\n• Teach how to ask good questions.\n• Select motivated students.\n• Give creative, open-ended problems.\n• Set complete expectations.\n• Let the student be the expert.\n• Give close guidance, and build community.\nJohn Keating did not teach his students poetry; he helped them yawp, he helped them\ninhabit the verse. He encouraged playful child-likeness. He created the space for his students\nto self-inspire in a resurrected Dead Poets Society, even though he himself never attended a\nmeeting. He taught them how to find good subjects for poems. He engaged the students\nwho were motivated, who had ears to hear. He encouraged creativity, and he encouraged\ncommunity.\nLet’s help our students find their yawp and transform it into poetry.\nAcknowledgements. I thank Art Benjamin, Jon Jacobsen, Susan Martonosi, Claus-Jochen\nHaake, Ulrike Ervig, Mike Starbird, and Bruce Palka for their advice in preparing the Leitzel\nLecture. I gratefully acknowledge NSF Grants DMS-0301129 and DMS-0701308 for supporting\nmy research with undergraduates.\nDedication. This article is dedicated to: Jim Leitzel and Christine Stevens, co-founders of\nProject NExT, who have nurtured the professional development of more than a thousand\nmathematicians and given them places and spaces to yawp; and to Michael Moody, who will\nalways be my professional Keating—O captain, my captain!\nTEACHING RESEARCH: ENCOURAGING DISCOVERIES 11\nReferences\n[1] A. J. Bernoff and F. E. Su, PCMI problem-solving references and resources (2003–2009), available at\nhttp:\/\/www.math.hmc.edu\/~ajb\/PCMI\/problem solve.html.\n[2] , Putnam, pizza & problem-solving, Math Horizons (Sept 2004) 8–9.\n[3] A. King, Inquiry as a tool in critical thinking, in Changing College Classrooms, D. F. Halpern, ed., Jossey-\nBass, San Francisco, 1994, 13–38.\n[4] M. Orrison, Graph complexity, in Resources for Teaching Discrete Mathematics: Classroom Projects, His-\ntory Modules, and Articles, B. Hopkins, ed., MAA Notes #74, Mathematical Association of America,\nWashington DC, 2009, 159–161.\n[5] H. Poincaré, Science et méthode, E. Flammarion, Paris, 1908; English trans. F. Maitland, Science and\nMethod, T. Nelson, New York, 1914.\n[6] T. Shulman, Dead Poet’s Society (dir. Peter Weir), Walt Disney Studios Home Entertainment, Burbank,\nCA, 1989.\n[7] F. E. Su, Math Fun Facts (1999–2009), available at http:\/\/www.math.hmc.edu\/funfacts\/.\n[8] D. Yong, Imagine Math Day (2009), available at http:\/\/www.math.hmc.edu\/pdo\/imd\/.\n[9] D. Yong and M. Orrison, Imagine Math Day: Encouraging secondary school students and teachers to\nengage in authentic mathematical discovery, MAA Focus 30, no. 6 (2008) 24–27.\nFRANCIS EDWARD SU earned his B.S. in mathematics at the University of Texas at Austin, and his\nPh.D. at Harvard under the direction of Persi Diaconis. He is now Professor of Mathematics at Harvey Mudd\nCollege. His research is in geometric combinatorics and applications to the social sciences, and he has co-\nauthored numerous papers with undergraduates. He has a passion for teaching and popularizing mathematics;\nhis Math Fun Facts website gets over a million hits each year. From the MAA, he received the 2001 Merten\nM. Hasse Prize for his writing and the 2004 Henry L. Alder Award for his teaching. He currently serves as\nFirst Vice-President of the MAA. In his spare time, he enjoys songwriting and gardening and is active in a\nChristian community where he can just be himself—passion not stifled by expectation, worth not derived from\naccomplishments.\nDepartment of Mathematics, Harvey Mudd College, Claremont CA 91711\nsu@math.hmc.edu\n"}
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{"id":"b0Bp1W51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/The-Banach-Tarski-Paradox.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":89,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"The-Banach-Tarski-Paradox","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["The-Banach-Tarski-Paradox.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 05:01:29 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":8614372990536024259,"exact_signature_unique_b":true,"fuzzy_signature_l":6557631160008515959,"fuzzy_signature_unique_b":true,"wordcount_i":7385,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:21:58 UTC 2022","fresh_date_dt":"Wed Dec 06 11:32:13 UTC 2023","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":192557,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088059079196672,"text_t":"The Banach-Tarski Paradox\nFrancis Edward Su\n(submitted in fulfillment of the “Minor Thesis” requirement for the Ph.D. program)\nHarvard University\n18 December 1990\nAuthor’s note: This paper was originally written for my Minor Thesis requirement of\nthe Ph.D. program at Harvard University. The object of this requirement is to learn about a\nbody of work that is generally not in one’s own field, in a short period of time (3 weeks), and\nwrite an article about it. I chose this topic because I thought it would be quite interesting\nto learn, and even more fun to tell others about.\nWhile I never published this expository article, I did make it available on my website:\nhttp:\/\/www.math.hmc.edu\/˜su\/papers.html.\nIf you like it, do write me and let me know. I can be reached at: Department of\nMathematics, Harvey Mudd College, Claremont, CA 91711, or by e-mail: su@math.hmc.edu.\n1\nIntroduction\nIn 1924, S. Banach and A. Tarski proved a truly remarkable theorem: given\na solid ball in R3, it is possible to partition it into finitely many pieces and\nreassemble them to form two solid balls, each identical in size to the first.\nAt first such a duplication seems patently impossible. However, a mo-\nment’s reflection reminds us that the mathematical world doesn’t always obey\nintuition.\nThe intention of this paper is to provide a self-contained exposition of the\nproof of the Banach-Tarski Paradox and to introduce related topics. We will\nshow, in fact, that the minimal number of pieces in the paradoxical decom-\nposition is five, and prove the stronger form of the Banach-Tarski Paradox :\nthat any two bounded subsets of the plane with non-empty interior are such\nthat one can be “cut” into a finite number of pieces and reassembled to form\nthe other!\nMost of the material we will cover is presented in [W], and we shall follow it\nclosely. However, we shall be less ambitious in overall scope and more mindful\nof continuity of theme. In some places we will provide slightly modified proofs.\nIf anything, the author’s hope is that this paper also gives the reader an inkling\nof how to motivate the results presented herein.\nThe Axiom of Choice, it will be noted, is indispensable to the proof of the\nBanach-Tarski paradox. However, as we shall very soon see, there are striking\nparadoxes which do not use Choice. Hence, the philosophy adopted in ths\npaper will be the unquestioned acceptance of Choice as a useful foundation in\nour work, so we will not bow to the pressure to acknowledge which results do\nor do not depend on the Axiom of Choice.\nThe reader, it will be assumed, has had a course in abstract algebra and\n2\nthe rudiments of measure theory and analysis. Nevertheless, we establish some\nterminology: Gn will denote the isometry group of R\nn, and SOn will denote\nthe group of rotations of Rn. A semigroup is a set with an associative binary\noperation and an identity. A choice set for a collection of sets (which exists\nby the Axiom of Choice) will be a set obtained by picking one element from\neach set in the collection.\nParadoxical Decompositions\nBefore we set out to prove the Banach-Tarski Paradox , we shall inves-\ntigate somewhat simpler paradoxes to understand and motivate the kinds of\napproaches that might be brought to bear on proving theorems of this sort.\nOne of the earliest paradoxes arose out of grappling with the notion of\ninfinity. The set of integers can be put in 1-1 correspondence with all integers,\nfor instance. This seems strange, at first: if set bijections are interpreted as\n“equality” in some sense, then Z is “equal” to a subset of itself. In fact, Z\nmay be partitioned into two sets (even and odd integers), each “equal” to all\nof Z. G. Cantor worked with this idea, and developed a theory of cardinality\nwhich mathematicians today have few problems understanding or accepting\nas “intuitive”.\nBecause of this theory, we know that the “number” of points in an interval\nis the same as the “number” of points in a square. We cannot be too surprised\nby this, after all, because we have allowed our points to move where they\nwish as independent “pieces” in our decomposition. Or consider “stretching”\n[0, 1) — so the number of points in [0, 1) is equal to that in [0, 2), which we\ncan see using the bijection f(x) = 2x. Again the familiar paradox emerges:\n3\n[0, 2) = [0, 1) ∪ [1, 2), the union of two sets each of which is “equal” to the\noriginal by a “stretching” bijection. But we are not very shocked that we can\nstretch [0, 1) to get [0, 2).\nHowever, when we restrict the number of pieces to be finite and the allow-\nable transformations to be isometries of the ambient space, any paradox that\npersists is hightly counterintuitive.\nNotice that the previous paradoxes depended on the set of allowable trans-\nformations. Hence we shall demand that our definition of paradoxical be de-\npendent on a group whose action on the set produces the transformations.\nDefinition 1 Let G be a group acting on a set X and suppose E ⊆ X. E is\nsaid to be G-paradoxical if for some m,n there exist g1, ..., gm and h1, ..., hn ∈\nG and pairwise disjoint A1, ..., Am and B1, ..., Bn ⊆ E such that E = ∪gi(Ai) =\n∪hj(Bj).\nWhen the isometry group is omitted, we understand it to be the isometry\ngroup of X. Note that in our definition, {Ai} ∪ {Bi} may not cover all of E,\nand the {gi(Ai)} (or {hi(Bj)}) may not be pairwise disjoint. This distinction\nis artificial, however, because we may take the Ai, Bj to be smaller to ensure\nthe pairwise disjointness, and we will prove later that we can use all of E in\nthe decomposition.\nTheorem 1 (The Banach-Tarski Paradox) Any ball in R3 is paradoxical.\nParadoxes first emerged in the study of measures. In fact, they were con-\nstructed to show the non-existence of certain kinds of measures, such as in the\nfollowing example.\nTheorem 2 S1 is countably SO2-paradoxical (i.e., paradoxical with a count-\nable number of pieces).\n4\nProof. Consider the RSO2, the subgroup of SO2 generated rotations\nof rational multiples of 2π radians. Let H be a choice set for the cosets of\nSO2\/RSO2. Now let M = {σ(1, 0) : σ ∈ H}.\nSince RSO2 is countable, we may enumerate it by ρi. Let Mi = ρi(M).\nThen {Mi} is a countable partition of S1, and moreover, all the Mi are con-\ngruent to each other by rotation.\nHence, each set in {M2,M4,M6, ...} may be individually rotated (say, by\ng2i) to yield {M1,M2,M3, ...} whose union is S1. We can clearly do the same\nfor {Mi : i odd}. Thus S1 is countably SO2-paradoxical.\nQ.E.D.\nCorollary 2.1 There is no countably additive rotation-invariant measure of\ntotal measure 1 defined for all subsets of S1.\nProof. If there were, then, in the proof above, for A = union of the even-\nindexed Mi, and B = union of odd-indexed Mi, we would have: 1 = µ(S\n1) =\nµ(A) + µ(B) = µ(S1) + µ(S1) = 2.\nCorollary 2.2 There is no countably additive, translation-invariant measure\ndefined on all subsets of Rn and normalizing [0, 1]n. Hence, there is a subset\nof [0, 1] which is not Lebesgue-measurable.\nProof. It suffices to show such a measure does not exist for R1, for any\nsuch measure µ′ in Rn would induce a measure µ in R1 by µ(A) = µ′(A ×\n[0, 1]n−1).\nBut for µ on R1, its restriction to subsets of [0, 1) would be invariant under\nG =translations mod 1 and defined on all subsets. Since SO2’s action on S\n1\nis isomorphic to G’s action on [0, 1) by the bijection j : (cos θ, sin θ) → θ\/2π,\nwe see that [0, 1) is G-paradoxical, which is a contradiction.\n5\nOne non-Lebesgue-measurable subset [0,1] is j(M), for j as above, and M\nas in the proof of the theorem.\nQ.E.D.\nEquidecomposability\nThe next example we shall discuss is not quite a paradox in the technical\nsense, but it is rather interesting and will be of use to us later. We shall\nshow that a broken circle (S1 \\ {pt}) can be partitioned into two sets, and\nreassembled to form a complete circle.\nFirst we need some language to describe this.\nDefinition 2 Let G act on a set X, and let A,B ⊆ X. A and B are said to\nbe G-equidecomposable if A and B can be partitioned into A1, ..., An and\nB1, ..., Bn such that each Ai is congruent to Bi (there is a gi ∈ G such that\ngi(Ai) = Bi).\nWe write A ∼G B, or A ∼ B.\nThis means that A can be partitioned into a finite number of subsets and\nreassembled to form B. This is easily seen to be an equivalence relation.\nMoreover, we see that if A ∼ B and A is paradoxical, so is B.\nTheorem 3 S1 \\ {pt} is equidecomposable to S1.\nBy omission of the group we understand it to be the isometry group of R2,\nand we identify S1 with {x : |x| = 1}.\nProof. Consider R2 identified with C, the complex plane. Let the pt\nbe 1 = ei0. Let A = {ein : n ∈ {1, 2, 3, ...}}, and let B = (S1 \\ {pt}) \\ A\nbe everything else. The points ein are unique, since 2π is irrational. Then,\n6\nleaving B fixed, rotate A by 1 radian— i.e., use the isometry ρ(z) = e−iz. This\nrotation sends each point “back” one radian, by which we obtain a complete\ncircle.\nQ.E.D.\nThe above construction gives us an interesting idea for the construction of\nparadoxes. Here we considered the image of a point under the subsemigroup\ngenerated by ρ (that is, only positive rotations). Because the semigroup was\nfree, we could use the inverse rotation (obtained from the group containing\nthis semigroup) to “shift” everything back.\nIn fact, this idea forms the basis of the following:\nTheorem 4 (Sierpiński-Mazurkiewicz Paradox ) There exists a subset\nof R2 which is paradoxical.\nProof. We can actually explicitly say what that subset is, namely\nE = {a0 + a1eiθ + a2e2iθ + ...+ aneniθ : n, aj ∈ N}\nfor any θ where eiθ is transcendental. (In particular, θ = 1 works.)\nOur aim is to identify two isometries τ , ρ which generate a subsemigroup S\nof the isometry group G2 of the plane (considered as C). Then we shall choose\na point x in the plane and consider E = {w(x) : w ∈ S}, all the images of this\npoint under words in S.\nIf for all a, b ∈ E, τ(a) 6= ρ(b), then we would see that E is paradoxical, for\nthen τ(E) ∩ ρ(E) = ∅, and so τ−1(τ(E)) = ρ−1(ρ(E)) = E.\nSo choose ρ(z) = rz, where r = eiθ is a transendental complex number.\nSuch a θ exists, since there are only countably many algebraic numbers on the\n7\nunit circle. Now choose τ(z) = z + 1. These are both isometries of the plane.\nLet x = 0.\nThen for any two words w1, w2 ∈ S, we shall show that τ(w1(0)) 6=\nρ(w2(0)), which will show that the words themselves are not equal.\nBut it is easy to verify that τ(w1(0)) 6= ρ(w2(0)) by explicitly writing\nout both sides of the equation as a polynomial in r, and observing that if\nthey were equal, their difference produces a non-trivial algebraic relation for\nr, contradicting the choice of r.\nQ.E.D.\nA few remarks are in order. Since the set E is countable, this theorem\nsays nothing surprising in terms of Lebesgue measure: 2(0)=0. And, since\nthe set has been explicitly defined, we have not used the Axiom of Choice\n(which supports the view that Choice is not to blame for the phenomenon of\nparadoxes).\nNotice that from τ(w1(0)) 6= ρ(w2(0)) we can conclude that S is a free\nsubsemigroup, since if distinct reduced words w1, w2 are equal, then left can-\ncellation will yield either ρ(w′1) = τ(w\n′\n2) (which must agree on 0, contradicting\nassumption) or 1 = w′ (for which one of τ(0) = w′(τ(0)) or ρ(0) = w′(ρ(0))\ncontradicts the assumption). Hence, freeness seems to play a big role in pro-\nducing paradoxes, as we shall see in the next section.\nAn open question related to the Sierpiński-Mazurkiewicz Paradox is whether\nany bounded subset of the plane is paradoxical. It is known that any such sub-\nset cannot be paradoxical using two pieces.\nWe note that the three previous examples of paradoxes involved finding one\nin a group or a subsemigroup and then in some manner “lifting” the paradox\nto set on which it acts. Thus it is natural to study paradoxical groups, where\n8\nthe group acts on itself by left multiplication.\nParadoxical Groups\nThe primary example of a paradoxical group is the free group on two gen-\nerators (sometimes called a free group of rank 2). Recall that a free group F\non two generators a, b is the group of all finite words in a±1, b±1 with no ad-\njacent pairs of inverse letters, and the operation is juxtaposition with removal\nof possibly any adjacent pairs of inverse letters.\nTheorem 5 The free group F on two generators σ, τ is F -paradoxical.\nProof. Let B(ρ) = {words beginning on the left with ρ}, where ρ may be\nσ, σ−1, τ, τ−1.\nThen F = {1}∪B(σ)∪B(σ−1)∪B(τ)∪B(τ−1), where all sets are pairwise\ndisjoint.\nBut F = B(σ) ∪ σB(σ−1) and F = B(τ) ∪ τB(τ−1), which shows that F\nis indeed paradoxical.\nWe can actually choose to partition F so that all four pieces in the para-\ndoxical decomposition cover F , i.e., {1} is included in one of the four sets. To\nsee this, put 1 in B(σ). Then we must remove σ−1 from B(σ−1) or else we\nwill duplicate 1 in σB(σ−1). Inductively we see that σ−n must be removed\nfrom B(σ−1) so as not to duplicate σ−n+1. So let our new partition be the\nsame as the old one except for: B′(σ) = B(σ) ∪ {1} ∪ {σ−n : n ∈ N+}, and\nB′(σ−1) = B(σ−1) \\ {σ−n : n ∈ N+}. These sets satisfy the same relation\nF = B′(σ) ∪ σB′(σ−1).\nQ.E.D.\n9\nTo ensure that the paradox “lifts” to disjoint well-defined sets, we need a\ncondition on G’s action, namely, that no element but the identity fixes any\npoints of the set.\nTheorem 6 Suppose G is a paradoxical group acting on X without nontrivial\nfixed points. Then X is G-paradoxical.\nProof. Let Ai, Bj ⊆ G and gi, hj ∈ G be the sets and transformations\n“witnessing” that G is paradoxical. Take a choice set M from the G-orbits in\nX. Now {g(M) : g ∈ G} partitions X because G acts without nontrivial fixed\npoints.\nLet A′i = ∪{g(M) : g ∈ Ai} and B′j = ∪{g(M) : g ∈ Bj}. Then {A′i} and\n{B′j} are all pairwise disjoint, and we can see that X = ∪{A′i} = ∪{B′j}.\nQ.E.D.\nNotice that this idea was used in a subtle way in Theorem 2, where the\ngroup acting on S1 wasRSO2, the orbits were the cosets of SO2\/RSO2, and the\nimplicitly used paradox on RSO2 was that this countable group was transitive,\nand so could be enumerated and decomposed into “even” and “odd” parts as\nwe did for the images of M in the proof of that theorem.\nWe are now faced with a very interesting question: which groups are para-\ndoxical? The answer to this lies in the investigation of groups bearing a finitely\nadditive measure of total measure 1, defined on all subsets, which is invariant\nunder left multiplication. Such groups are called amenable groups, it turns\nout that they coincide precisely with the non-paradoxical groups. We shall\ntalk a little more about amenable groups later.\nMore immediately, Theorem 6 tells us a little about which groups are para-\ndoxical.\n10\nCorollary 6.1 If a group G contains H, a paradoxical subgroup, then G is\nparadoxical.\nProof. H acts on G by left multiplication, without nontrivial fixed points.\n(Inverses prevent this from happening.) So G is H-paradoxical by Theorem 6.\nBut then G is G-paradoxical.\nQ.E.D.\nThis corollary immediately implies the next:\nCorollary 6.2 Any group with a free subgroup of rank 2 is paradoxical.\nA stronger converse of Theorem 6 is true:\nTheorem 7 If X is G-paradoxical, then G is paradoxical.\nProof. . Suppose A′i, B\n′\nj and gi, bj witness that X is G-paradoxical. Then\nconsider H, some G-orbit on X, and a point x ∈ H. Define Ai = {g : g(x) ∈\nH ∩ A′i} and Bj = {g : g(x) ∈ H ∩ B′j}. These are all pairwise disjoint, since\nthe Ai, Bj are all pairwise disjoint.\n∪gi(Ai) = ∪{gig : g(x) ∈ H ∩ A′i} = ∪{gig : gig(x) ∈ H ∩ gi(A′i)} = {g′ :\ng′(x) ∈ X} = G\nSimilarly, we obtain ∪hj(Bj) = G.\nQ.E.D.\nOur “lifting” technique is not yet overwhelmingly helpful, since group ac-\ntions often have many nontrivial fixed points. Of course, if we restrict our\nattention to only the nontrivial fixed points of a space, we can conclude the\nexistence of a paradoxical subset.\n11\nWe shall be interested in ways to applying Theorem 6 where G=F , the\nfree group of rank 2, and realizing F as a subgroup of G3, the isometry group\nof R3. Then, once we have figured out what to do with the nontrivial fixed\npoints, we shall have proved the Banach-Tarski Paradox .\nThe Hausdorff Paradox\nIn 1914, in an attempt to show the nonexistence of certain measures on the\nsphere, F. Hausdorff constructed the following paradox: Except for a countable\nnumber of points, S2 may be partitioned into three subsets A,B,C such that\nA ' B ' C ' (B ∪C), where ' denotes congruence (not to be confused with\n∼, which denotes equidecomposability).\nHis method was to find two rotations of S2 which generated a free group\nisomorphic to Z2 ∗ Z3. These were rotations of 2π\/3 and π about axes whose\ninterior angle was some θ where cos 2θ is transcendental.\nSince we are primarily interested in proving the Banach-Tarski Paradox ,\nwe shall take a slightly more direct tack, due to Swierczkowski [Sw]. It will\nsuffice to find two rotations which generate which generate a free subgroup of\nrank 2 of SO3. This will enable us to obtain a paradoxical decomposition by\nlifting to S2. (In Hausdorff’s example, Z2 ∗Z3 also contains such a subgroup.)\nTheorem 8 There exist two independent rotations φ, ρ which fix the origin in\nR3. Hence, for n ≥ 3, SOn contains a free subgroup of rank 2.\nAs it turns out, “most” pairs of rotations are independent, and in the\nfollowing proof arccos(3\/5) can be replaced by arccos(r) where r is any ra-\ntional 6= 0,±(1\/2),±1. Wagon, in [W], proves the result using an angle of\narccos(1\/3).\n12\nProof. We shall let φ and ρ be counterclockwise rotations about the z-axis\nand x-axis, respectively, each through the angle arccos 3\/5. Then\nφ±1 =\n\n\n3\/5 ∓4\/5 0\n±4\/5 3\/5 0\n0 0 1\n\n\nρ±1 =\n\n\n1 0 0\n0 3\/5 ∓4\/5\n0 ±4\/5 3\/5\n\n\n.\nWe intend to show that no nontrivial reduced word in φ±1, ρ±1 is the iden-\ntity. For if there is such a word, we may conjugate by φ (if necessary) to obtain\na word w ending in φ which equals the identity.\nWe claim that w(1, 0, 0) is of the form (a, b, c)\/5k where a, b, c are integers\nand b is not divisible by 5. This shows that w(1, 0, 0) 6= (1, 0, 0), which is a\ncontradiction.\nThe claim will be proved by induction. We start off with w of length one.\nThen w = φ±1, and w(1, 0, 0) = (3, 4, 0)\/5, and everything checks.\nNow, for longer words, if w = φ±1w′ or w = ρ±1w′, where w′ is of the form\n(a′, b′, c′)\/5k−1, then w(1, 0, 0) = (a, b, c)\/5k, where\na = 3a′ ∓ 4b′, b = 3b′ ± 4a′, c = 5c′ for w = φ±1w′,\nor a = 5a′, b = 3b′ ∓ 4c′, c = 3a′ ± 4b′ for w = ρ±1w′.\nThis shows that a, b, c will always be integers.\nTo show b is never divisible by 5, we consider what happens when w is of\nthe following four forms (where ν is any word):\nif w = φ±1ρ±1ν, then b = 3b′ ± 4a′ where 5 divides a′;\nif w = ρ±1φ±1ν, then b = 3b′ ∓ 4c′ where 5 divides c′;\nif w = φ±1φ±1ν or w = ρ±1ρ±1ν, then b = 6b′ − 25b′′,\n13\nwhere a′′, b′′, c′′ are the integers arising from ν(1, 0, 0).\nThe validity of the first two cases is obvious from the preceding equations;\nin the last two cases, the arguments are similar and go like this one for the\nthird case: b = 3b′±4a′ = 3b′± (12a′′∓16b′′) = 3b′+9b′′±12a′′−16b′′−9b′′ =\n3b′ + 3(3b′′ ± 4a′′)− 25b′′ = 6b′ − 25b′′.\nQ.E.D.\nNow every g ∈ F , our free subgroup, fixes two points on the sphere. Let\nD = { all points fixed by some g ∈ G}. Then D is countable, and to S2 \\ D\nwe can apply Theorem 6 to obtain:\nTheorem 9 (Hausdorff Paradox) There is a countable set D such that S2\\\nD is SO3-paradoxical.\nThe Banach-Tarski Paradox\nWe shall soon show, however, this countable set D doesn’t matter much.\nTheorem 10 S2 and S2 \\D are SO3-equidecomposable.\nThe proof is very similar to the proof of Theorem 3. Choose an axis\nof rotation which doesn’t fix any points of D. Then consider the set of all\nrotations around this axis for which some integer multiple of it sends a point\nof D to another point of D. This is a countable set. But there are uncountably\nmany possible angles; pick one, say θ.\nLet ρθ be the rotation around the axis by θ. Then no multiple of ρθ will\nsend a point of D to another point of D, and it follows that ρmθ (D) and ρ\nn\nθ (D)\nare disjoint for m 6= n ≥ 0.\n14\nSo let A = ∪{ρnθ (D) : n ∈ N+} and B = S2 \\ A.\nWe obtain S2 \\D = B ∪ A ∼ B ∪ ρ−1θ (A) = S2.\nQ.E.D.\nThis shows that S2 is SO3-paradoxical. Moreover, it follows that\nTheorem 11 (Banach-Tarski Paradox ) B3, the solid ball in R3, is G3-\nparadoxical.\nProof. Since S2 is paradoxical, we can obtain a paradox for any “thick-\nened” shell by producing a paradox for one sphere in this shell, and then\nputting points along a radius in the same piece in a decomposition. In partic-\nular, we see that B3 \\ {0} is paradoxical.\nIt remains to be shown that B3 \\ {0} ∼ B3, for then\nB3 = B3 \\ {0} ∪ {0}\n∼ B3 \\ {0} ∪ B3 \\ {0} ∪ {0}\n= B3 \\ {0} ∪ B3\n∼ B3 ∪ B3,\nwhich would prove the theorem.\nSo choose a broken circle C ′ ⊆ B3 \\{0} with the origin as its broken point.\nBy Theorem 3, C ′ is equidecomposable with a completed circle C. Thus,\nB3 \\ {0} = B3 \\ ({0} ∪ C ′) ∪ C ′\n∼ B3 \\ ({0} ∪ C ′) ∪ (C ′ ∪ {0})\n= B3.\nQ.E.D.\n15\nAlmost an identical proof gives\nCorollary 11.1 R3 is paradoxical.\nThe equidecomposability relation is a useful tool, as we have seen in our\nproofs, and so it deserves further investigation. It is an equivalence relation, so\nwe can define another relation between ∼-equivalence classes of sets: A \u0016 B\nif and only if A is equidecomposable with a subset of B.\nThis relation \u0016 is both reflexive and transitive, by definition. Further-\nmore, it is also anti-symmetric: if A \u0016 B and B \u0016 A, then A ∼G B. This\nfact is proved by a generalization of the well-known Schröder-Bernstein Theo-\nrem of set theory— Banach was able to generalize it for equivalence relations\nsatisfying these two properties (which ∼G satisfies):\n(i) if A \u0016 B, then there exists a bijection g : A → B such that if C ⊆ A,\nthen C ∼ g(C),\n(ii) if A1 ∩ A2 = B1 ∩ B2 = ∅, and if A1 ∼ B1 and A2 ∼ B2, then\nA1 ∪ A2 ∼ B1 ∪B2.\nTheorem 12 (Banach-Schröder-Bernstein Theorem ) Let G act on X\nand A,B ⊆ X. Then A \u0016 B and B \u0016 A implies that A ∼G B.\nProof. This proof is practically identical to the proof of the classical\ntheorem. Notice how the only information we use about ∼ are the properties\n(i) and (ii) above.\nLet f : A → B′(⊆ B) and g : B → A′(⊆ A) be bijections which exist by\nproperty (i). Now let C0 = A \\ A1, and define inductively Cn+1 = gf(Cn).\nLet C = ∪ Cn. Then (A \\ C) = g(B \\ f(C)) since if x ∈ f(C), then g(x)\nmust be in C. Hence, by property (i) A \\C ∼ B \\ f(C). But also C ∼ f(C).\nSo by property (ii), A \\ C ∪ C ∼ B \\ f(C) ∪ f(C), which means A ∼ B.\n16\nQ.E.D.\nTo show the power of this theorem we now verify that pieces in a paradox-\nical decomposition can be taken to be a partition of the set:\nTheorem 13 G acts on X, E ⊆ X. Then E is G-paradoxical if and only if\nthere exist disjoint A,B such that A ∪B = E and A ∼ B ∼ E.\nProof. We already know that if E is G-paradoxical, then we can find\ndisjoint A and B such that A ∼ E ∼ B, so all that remains to be shown is\nthat we can take A ∪B = E.\nBut E ∼ A ⊆ E \\ B ⊆ E, so we obtain A \u0016 E \\ B \u0016 E. Now since\nE ∼ A, we conclude that E \\B ∼ A. So take A′ = E \\B and B′ = B, both\nof which are equidecomposable to E, disjoint, and whose union is all of E.\nQ.E.D.\nTheorem 12 also immediately yields a much stronger form of the Banach-\nTarski Paradox :\nTheorem 14 (Banach-Tarski Paradox , Strong Form) If A and B are\nbounded subsets of R3 with nonempty interior, then A ∼ B.\nProof. We want to show A \u0016 B, for then B \u0016 A is similarly proved, and\nwe obtain by Theorem 12 that A ∼ B.\nSo choose balls K and L such that K contains A and L is contained by\nB. Then for some n large enough, K ⊆ n copies of L. Then we have that\nA ⊆ K ⊆ (n copies of L) \u0016 L ⊆ B, where the \u0016 arises from repeated Banach-\nTarski Paradox duplications of L. Hence A \u0016 B.\nSimilarly, we obtain B \u0016 A, and so A ∼ B.\n17\nQ.E.D.\nCongruence by Dissection\nA problem of classical geometry asks when a polygon can be cut up (in the\nclassical sense, ignoring boundaries) into polygonal pieces and reassembled\nto form another polygon. This kind of decomposition equivalence is called\ncongruence by dissection.\nTheorem 15 (Bolyai-Gerwain Theorem) Two polygons are congruent by\ndissection if and only if they have the same area.\nProof. The forward implication is clear. The reverse implication is shown\nby proving that any polygon can be dissected into finitely many triangles, a\ntriangle is congruent by dissection to a square, and two squares are congruent\nby dissection to a larger square. Thus every polygon is congruent by dissection\nto some square of the same area. If two polygons have the same area, then their\nassociated squares are identical, and the polygons are congruent by dissection\nwith each other.\nQ.E.D.\nHigher dimensional analogues of this theorem do not exist. For instance,\nDehn proved that a regular tetrahedron in R3 is not congruent by dissection\nwith a cube of the same volume. The Banach-Tarski Paradox , nevertheless,\nsays that they are equidecomposable.\nIn the plane, we have the following relationship between congruence by\ndissection and equidecomposability:\n18\nTheorem 16 If P1, P2 are polygons which are congruent by dissection , then\nP1 ∼ P2.\nProof. The interiors of the pieces of the polygon are obviously equidecom-\nposable. We want to show that the whole polygon is equidecomposable with\nthe interiors of the pieces, and we do this by letting the interiors “absorb” the\nboundaries of each of the pieces.\nThe boundaries consist of a finite number (and length) of intervals. So find\na disc contained in the polygon, and use a construction similar to Theorem 3,\nexcept take a broken annulus in this disc instead of a broken circle. Cutting\nup each of the intervals as necessary, we can “absorb” these (open, closed, or\nhalf-open) intervals by using an (open, closed, or half-open) annulus, which is\nequidecomposable with a broken annulus by the technique of Theorem 3, and\nthen inserting these intervals.\nSince there are only finitely many such intervals, the process terminates.\nQ.E.D.\nNote, then, that polygons of the same area are equidecomposable. The\nconverse is also true, but we shall not prove it here. The difference between\ntwo and three dimensions is highlighted in the fact that in R3, any two poly-\nhedra are equidecomposable!\nMinimizing the Pieces\nSo far, we have not paid attention to how many pieces we actually use to\neffect a duplication of a ball. In our previous discussion our duplication of the\nsphere used no more than eight pieces (four from Theorem 5 multiplied by\n19\ntwo from Theorem 10); hence, our duplication of the ball used no more than\nseventeen (by careful application of Theorem 11 we could bring this down to\nthirteen). One could question why we should even care about pieces at this\npoint, if it weren’t for the truly remarkable fact that we only need five!\nThis is surprisingly low; in fact, the minimum number needed to duplicate\nany set with any group action is four (because otherwise one of the pieces\nwould by itself be congruent to the entire set, which means it is the entire\nset). We will not eke out this minimal answer for the ball just to torture\nourselves, for the method we will use will turn out to be applicable in general\nto certain types of group actions, called locally commutative group actions.\nRecall that the stabilizer of a point x is Stab(x) = {σ ∈ G : σ(x) = x}.\nDefinition 3 Let G act on X. This action will be called locally commuta-\ntive if for every point x ∈ X the stabilizer Stab(x) is commutative, i.e., if the\nsubgroup of G which fixes x is commutative.\nWhereas earlier, fixed points of a free group’s action posed a problem, the\nidentification of a locally commutative group action will provide us with a new\nway to deal with them.\nFirst, we will need the following theorem.\nTheorem 17 Let F be the free group generated by σ and τ . Then we can\nactually partition F into four sets A1, ..., A4 such that σ(A2) = A2 ∪ A3 ∪ A4\nand τ(A4) = A1 ∪ A2 ∪ A4.\nMoreover, for any fixed w ∈ F , the partition can be chosen such that w is\nin the same piece as the identity of F .\nIn [W], Wagon seems to use an unneccesary argument to prove the theorem—\nwe shorten it below:\n20\nProof. We shall use the same four-piece partition as described in the proof\nof Theorem 5, where B(φ) denotes all words beginning on the left with φ. Fix\nsome w in F , and let ρ denote the leftmost letter in the word w.\nNow suppose, for instance, that ρ = τ−1. (The other cases are similar.)\nThen we may choose our partition as follows:\nA1 = B(σ)\nA2 = B(σ\n−1)\nA3 = B(τ) \\ {τn : n ∈ N+}\nA4 = B(τ\n−1) ∪ {τn : n ∈ N+} ∪ {1}\nThese sets have the property that 1, w are in the same piece, as well as\nsatisfy the relations above: σ(A2) = A2 ∪A3 ∪A4 and τ(A4) = A1 ∪A2 ∪A4.\nQ.E.D.\nThis yields an fruitful corollary which strengthens Theorem 6 for the par-\nticular case where G is a free group of rank 2:\nCorollary 17.1 Let F be a free group of rank 2 acting on X without nontrivial\nfixed points. Then X is F -paradoxical using four pieces.\nProof. The crucial observation is that lifting a paradox to a set preserves\nthe number of pieces.\nQ.E.D.\nWe also see, then if G is any group with a free non-Abelian subgroup, then\nit has a free subgroup of rank 2, and by virtue of Corollary 6.1, is paradoxical\nusing four pieces.\nWe are now in a position to prove the main result about locally commuta-\ntive actions:\n21\nTheorem 18 Let F be a free subgroup of rank 2 whose action on X is locally\ncommutative. Then X is paradoxical using four pieces.\nProof. This fact will be proved by defining the partition orbit by orbit\nso as to satisfy the familiar relations σ(A2) = A2 ∪ A3 ∪ A4 and τ(A4) =\nA1 ∪ A2 ∪ A4. This will give us the desired paradoxical decomposition using\nfour pieces. Let σ, τ generate F .\nNote that all F -orbits on X are either composed of nontrivial fixed points,\nor have no nontrivial fixed points (if x is fixed by w, then ρ(x) is fixed by\nρwρ−1). In the latter case, we partition the orbit as before (see Theorem 6):\ntake a choice set M from each such orbit, and let Pi = {g(M) : g ∈ Ai}, where\nAi are obtained from any four piece partition guaranteed by Theorem 17.\nFor an orbit O consisting entirely of fixed points, we must choose a point\nx carefully. Let w be a nontrivial word of minimal length which fixes a point\nin O. Then choose x to be a point which w fixes. The idea is to describe all\nthe points in this orbit as the image of a unique representative v ∈ F and, as\nbefore, to lift the paradox from the group onto this orbit, this time in a more\nrefined manner.\nLet ρ be the leftmost letter in the word w. We can see that w can not end\nin ρ−1, else conjugation of w by ρ would produce a smaller word fixing x.\nWe claim that we can write any point y ∈ O as v(x), where v does not end\nin w or in ρ−1, and that such a v is unique. Why? Take a minimal word v\nsuch that y = v(x). Then v clearly cannot end in w or w−1, and if it did end\nin ρ−1, we could take vw, which doesn’t end in w or ρ−1.\nTo show that v is unique, we use local commutativity to show that only\npowers of w can fix the point x. Else if some z fixed x, zw = wz by local\ncommutativity, and since F is free, z and w must both be powers of some\n22\nword t (else we would get a nontrivial relation on the letters of F ). This\nimmediately gives that z is a power of w, for otherwise some smaller power of\nt would fix x, contradicting the choice of w.\nThen if u and v are both words of the desired form which send x to y, we\nobtain that u−1v fixes x. So by the argument above, u−1v is a power of w. We\ncan assume it is a positive power, for if not, we can do the argument which\nfollows for v−1u. Then either u−1 begins with a ρ (meaning u ends in ρ−1) or\nu−1 cancels the first part of v (meaning v ends in a w) or u = v, as desired.\nThen we put y = v(x) in P ′i according as v ∈ Ai, where the partition is\ngiven by Theorem 17, where we ensure that w and the identity lie in the same\npiece. Notice here we are using possibly different partitions for different orbits\nO, since our partition depends on the word w, which depends on the orbit.\nIt is a straightforward, but tedious exercise to demonstrate that this con-\nstruction of P ′i satisfies σ(P\n′\n2) = P\n′\n2 ∪P ′3 ∪P ′4 and τ(P ′4) = P ′1 ∪P ′2 ∪P ′4. Really\nthe only thing we must take into account is the possibility of muliplying a\nword in the desired form by a letter and not getting a word of the desired\nform, i.e., getting something ending in w or ρ−1. But because w and 1 lie in\nthe same piece of the decomposition, we can interchange them when necessary\nto obtain a new word representation which is of the desired form, and is in the\nsame piece as before.\nTherefore, we can take Ai = Pi ∪P ′i as our four-piece decomposition of X.\nQ.E.D.\nSince the action of SO3, and in particular of its free subgroup F of rank 2,\nis locally commutative, we have the following result.\nCorollary 18.1 S2 is SO3-paradoxical using four pieces, and this is best pos-\n23\nsible.\nThe converse of Theorem 18 is true, although we shall not prove it here:\nTheorem 19 If X is G-paradoxical using four pieces, then G has a free sub-\ngroup of rank 2, whose action on X is locally commutative.\nThe proof proceeds by identifying two elements σ, τ which are independent:\nif g1(A1)∪g2(A2) = X and g3(A3)∪g4(A4) = X, then let σ = g−11 g2, and define\nτ = g−13 g4. These satisfy the stated conditions.\nIt follows that\nCorollary 19.1 A group G is paradoxical using four pieces if and only if G\nhas a free subgroup of rank 2.\nProof. This follows from Theorem 19 and Corollary 17.1, noting that a\nsubgroup acts without nontrivial fixed points on G by left multiplication.\nQ.E.D.\nLet us now apply Theorem 18 to determine the minimal decomposition of\na ball.\nTheorem 20 B3, the solid ball in R3, is paradoxical using five pieces, and\nthis is best possible.\nProof. A five piece decomposition is obtained for the unit ball B by using\nTheorem 18 to obtain a four piece decomposition of the sphere, and thickening\nthe pieces to obtain a four piece decomposition of the open ball minus the\ncenter: let C1, C2, C3, C4 be the such a partition for {x : 0 < |x| < 1}. Theorem\n24\n18 guarantees the relations: σ(C2) = C2 ∪ C3 ∪ C4 and τ(C4) = C1 ∪ C2 ∪ C4\nwhere σ, τ generate a free subgroup F of SO3.\nWe shall use the “last” spherical shell S, the boundary of B, to produce\nthe extra point to put at the center. We can choose the partition of F , the\nfree subgroup of rank 2, given by the first part of Theorem 5 and note that\nit satisfies the following relations: σ(A2) = A2 ∪ A3 ∪ A4 ∪ {1} and τ(A4) =\nA1∪A2∪A4∪{1}. (Take A1 = B(σ), A2 = B(σ−1), A3 = B(τ), A4 = B(τ−1).)\nWe apply this partition to one F -orbit O of S consisting of nonfixed points;\nfor all other orbits in S, use the same partitions as in Theorem 18. Pick any\npoint x in O, and partition O in the usual fashion with this partition: SO,i =\n{g(x) : g ∈ Ai}. We thus obtain a partition of all of S into S1, S2, S3, S4, {x},\nsatisfying σ(S2) = S2 ∪ S3 ∪ S4 ∪ {x}, and τ(S4) = S1 ∪ S2 ∪ S4 ∪ {x}.\nLet ρ be the isometry taking x to 0. Then we can piece everything together;\nlet B1 = C1 ∪ S1 ∪ {0}, and let the other Bi = Ci ∪ Si. Observe now that\nB1 ∪ σ(B2) = B and B3 ∪ τ(B4) ∪ ρ(x) = B.\nTo see that this is best possible, we assume to the contrary. Suppose we\ncould find a four piece paradoxical decomposition of the ball B, witnessed by\nA1, A2, A3, A4 such that ∪Ai = B, and σ1(A1) ∪ σ2(A2) = B and σ3(A3) ∪\nσ4(A4) = B. Without loss of generality let A1 be the piece containing {0}.\nThen not both of A3, A4 can fix the origin, else the second copy of B\nwould be missing a center point, since neither piece contains the origin. So\nsuppose that σ3(0) 6= 0. Let S denote the spherical boundary of B. Then\nσ3(A3) ∩ S ⊆ σ3(B) ∩ S ⊆ an open hemisphere of S, whence σ4(A4)\ncontains some closed hemisphere H of S.\nThis implies that A4 contains a closed hemisphere σ\n−1\n4 (H) of S, from which\nwe conclude that (A1 ∩ A2) ∩ S is contained in S \\H, an open hemisphere.\n25\nBut if either of σ1, σ2 did not fix the origin, a similar argument shows that\nthe other of the two pieces A1, A2 would contain a closed hemisphere of S,\ncontradicting the above remark.\nSo both σ1, σ2 fix the origin, meaning they map S to S. Immediately we\nhave a contradiction, for then σ1(A1) ∪ σ2(A2) should cover the entire ball B,\nincluding S, but each of σ1(A1) ∩ S, σ2(A2) ∩ S is only contained in an open\nhemisphere by the above remark, and hence cannot cover all of S.\nQ.E.D.\nm-divisibility\nWe turn now to a related topic in the study of set decompositions, namely,\ngiven a whole number m, is it possible to partition X into m congruent pieces?\nIf it is, we say that X is m-divisible. We broach this subject only because\nwe can use an argument similar in flavor to the proof of the previous theorem,\nto show the following:\nTheorem 21 Bn is not m-divisible for 2 ≤ m ≤ n.\nProof. Suppose Bn is m-divisible. Then let A1, ..., Am partition the ball\ninto congruent pieces via σi(A1) = Ai. We can assume without loss of gener-\nality that A1 is the set containing the center point {0}. This implies that the\nσi cannot fix the center point, or else the pieces would not be disjoint. Thus,\nby the argument above, we see that Ai ∩S = σi(A1)∩S ⊆ σi(Bn)∩S ⊆ some\nopen hemisphere.\nThen this is true for each Ai, including A1, because Ai∩σi(S) lies in an open\nhemisphere of σi(S). Therefore S must be coverable by m open hemispheres,\n26\nwhich is impossible if m ≤ n. This is easily seen by taking n “poles” of these\nhemispheres which determine a plane in Rn. This plane cuts S into two pieces,\nthe larger one of which contains a point more than a right angle away from any\nof the poles. Hence, this point is not covered by any of the open hemispheres.\nQ.E.D.\nIt is unknown whether Bn is m-divisible for m ≥ n. We do know a few\nother isolated results on m-divisibility: that S2 is m-divisible for any m ≥ 3,\nand that an open or closed interval is not m-divisible for m finite ≥ 1.\nAmenable Groups\nEarlier we discussed the implications that paradoxes have for the existence\nof measures. Now we shall explore a little more this relationship. We shall\npresent a quick (and not at all comprehensive) survey on the work done in this\narea in relation to paradoxes.\nRecall that an amenable group G is a group bearing a left-invariant, finitely\nadditive measure of total measure 1, defined on all subsets of G. Then clearly\nan amenable group is non-paradoxical, for otherwise the measure will produce\nthe contradiction 1 = 2.\nA somewhat surprising fact is that the converse is also true, which follows\nfrom the following result of Tarski.\nTheorem 22 (Tarski’s Theorem) Let G act on X, and let E ⊆ X. Then\nthere exists a finitely additive, G-invariant measure on X defined for all subsets\nof X and normalizing E if and only if E is not G-paradoxical.\n27\nIn particular, we see that a group G is amenable if and only if it is not\nparadoxical. For a proof of Tarski’s theorem, we refer the reader to [W, pp.\n125-128].\nThe notion of the amenability of a group seems to be quite an intrinsic one,\nfor there are many equivalent definitions of amenability in other contexts. For\ninstance, amenability is equivalent to an assertion about the cogrowth function\nof a group which is finitely presented— loosely speaking, this function counts\nthe number of words of length n which get killed, and the group is amenable if\nand only if this function in some sense asymptotically approaches the number\nof total words in the group. This characterization is due to J. Cohen. An-\nother definition of amenability, due to H. Kesten, relates this property to the\nrecurrence of random walks in a group with respect to symmetric probability\ndistributions on the group (see [W, p.157-161] for a reference, and a long list\nof other equivalent definitions.)\nOne may ask whether all groups with no free subgroup of rank 2 are\namenable. This question was only recently answered in the negative, that\nis, there is a paradoxical group with no elements of infinite order. However,\nunder certain restrictions, we do obtain that amenability is equivalent to not\nhaving a free subgroup— for instance, in the group of isometries of Euclidean\nspace.\nTherefore we can show that no paradoxes (using isometries) exist for the\nballs in R1 and R2, because these isometry groups have no free subgroups\n(since they are solvable, and solvability implies the existence of a finite relation\nfor any elements in the group derived from the commutator chain characteri-\nzation of solvability).\nThus we have the strange fact that Bn is paradoxical when n ≥ 3, and\nnot paradoxical for n = 1 or n = 2! Recall, however, that the Sierpiński-\n28\nMazurkiewicz Paradox shows that paradoxes exist in the plane for sets without\ninterior; this may be contrasted with the fact than the line has no paradoxical\nsubset.\nConclusion\nParadoxes are useful because they challenge the intuition. We have seen\nthat paradoxes were constructed to show the nonexistence of certain measures—\nbut have gained notoriety in their own right as a curiosity. We have only\nscratched the surface of the literature on paradoxes. For example, we have\nonly considered construction of paradoxes on Euclidean spaces. What about\nother spaces, such as the hyperbolic plane? Actually, it turns out a Hausdorff\nParadox for the hyperbolic plane actually uses Borel pieces! (For the Euclidean\nplane this cannot happen because of Lebesgue measure.) One open problem\nasks whether no compact metric space can be paradoxical using Borel pieces.\nAnother (Marczewski’s Problem) asks whether the S2 admits a paradoxical\ndecomposition where the pieces have the property of Baire, i.e., differs from a\nBorel set by a countable union of nowhere dense sets. Or, can a disc in the\nplane be split into three congruent pieces?\nClearly such questions will entertain mathematicians for many years to\ncome.\n29\nReferences\n[F] French, R.M., “The Banach-Tarski Theorem”, The Mathematical Intel-\nligencer, Vol. 10(1988), No. 4, pp. 21-28.\nThis expository paper is a delightful introduction to the ideas of paradox-\nical decompositions and the Hausdorff Paradox.\n[Sw] Swierczkowski, S., “On a free group of rotations of the Euclidean\nspace”, Indag. Math., Vol. 20(1958), pp. 376-378.\n[W] Wagon, S., The Banach-Tarski Paradox, Cambridge Univ. Press., 1986.\nWagon’s book will tell you anything and everything you want to know\nabout the Banach-Tarski Paradox, plus much, much more... it is truly an en-\ncyclopedic account. The bibliography is extensive.\n[W2] Wagon, S., “Partitioning intervals, spheres and balls into congruent\npieces”, Can. Math. Bull., Vol. 26(1983), pp. 337-340.\nThis short work discusses m-divisibility.\n30\n"}
{"index":{}}
{"id":"j72iZW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2020\/08\/Guidelines-for-Good-Mathematical-Writing.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":104,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2020","08"],"url_file_name_s":"Guidelines-for-Good-Mathematical-Writing","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Guidelines-for-Good-Mathematical-Writing.pdf"],"author":"pdfTeX-1.40.16","author_sxt":["pdfTeX-1.40.16"],"last_modified":"Tue Aug 18 16:16:33 UTC 2020","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Aug 18 00:00:00 UTC 2020"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":4600223512642621370,"exact_signature_unique_b":true,"fuzzy_signature_l":8610320536060035182,"fuzzy_signature_unique_b":true,"wordcount_i":1643,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:21:59 UTC 2022","fresh_date_dt":"Tue May 02 04:38:40 UTC 2023","publisher_t":"pdfTeX-1.40.16","language_s":"en","size_i":110818,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088059600338944,"text_t":"GUIDELINES FOR GOOD MATHEMATICAL WRITING\nFRANCIS EDWARD SU\nCommunicating mathematics well is an important part of doing mathematics. Whether you are speak-\ning or writing, learning to communicate effectively is not just a service to your audience; it is also an\nexercise in clarifying and structuring your own thinking. Moreover, beyond developing a basic compe-\ntence, there is an art and elegance to good writing that every writer should strive for. And writing, as a\nwork of art, can bring great personal satisfaction.\nThese guidelines may serve as a starting point for good mathematical writing.\n1. BASICS\nKnow your audience. This is the most important consideration for writers. Put yourself in your reader’s\nshoes. What background can we assume of the reader? What terminology should we define? What kind\nof “voice” do we want to project: casual or professional, serious or inviting, terse or loquacious?\nIf you are a student writing solutions for a homework set and your professor has not specified your\naudience, a good rule of thumb is to assume you are writing to another student in the course who has\nnot yet done the assignment. Though you may assume that she has attended all the same lectures and\nhas read the same textbook, it is standard courtesy to remind your reader of any relevant items that she\nhas recently learned in class or from the textbook, or things she should know but might have forgotten.\nFor instance, if the concept of a rational number was only recently learned in class, you might insert\n“Recall that a rational number can be expressed as a fraction;” before saying: “since x is rational, we can\nset x = m\/n where m and n are integers.”\nSet an invitational tone. It is traditional to create an inviting atmosphere in one’s mathematical writing.\nIn effect, we invite readers to join us in our reasoning process by writing in the present tense, using the\npronoun “we” instead of “I” (e.g., “we construct a tangent plane...”), and directing the reader with gentle\ncommands (e.g., “let n be...”, “recall that...”, or “consider the set of...”).\nUse complete sentences. All mathematics should be written in complete sentences. Open any mathe-\nmatics text and you’ll see that this is true. Equations, even displayed ones, have punctuation that help\nyou see where it fits in the context of a larger sentence. Consider this piece of writing:\n(x− 2)2 + (x− 1)2 = 52 52 = 25\n(x− 2)2 = x2 − 4x+ 4 + x2 − 2x+ 1 = 25\n2x2 − 6x− 20\n2(x+ 2)(x− 5) x = −2, 5 x > 0 x = 5\nCan you figure out what the writer is doing? What’s being assumed? What’s being proved? Where\ndoes one thought end and another begin? What’s the relationship between these phrases? Some phrases\nare dangling, and others, as statements, are not even true. The reader should not have to figure out what\nthe writer was thinking!\nNow consider the work of another writer who has attempted the same problem:\nDate: updated August 18, 2020.\n1\nProblem. Find a point on the line y = x which is distance 5 from the point (2, 1) and such\nthat x > 0.\nSolution. We wish to solve (x−2)2+(x−1)2 = 52, an equation obtained from the distance\nformula in the plane. Some algebra turns this equation into:\n2x2 − 6x− 20 = 0.\nFactoring the left side, we obtain\n2(x+ 2)(x− 5) = 0,\nwhose solutions are evidently x = −2 and x = 5. Since we assumed x > 0, we have (5, 5)\nas the desired point on the line y = x.\nHere, the writer has clearly stated the problem and described her path to a solution. She has set an\ninvitational tone, and every thought is expressed in a complete sentence. Now it is clear that x > 0\nis a condition, not a result. Notice the punctuation in equations: one ended with a period because her\nthought was complete, the other ended with a comma because she wanted to continue the thought. Since\nshe assumed her audience could do algebra, she didn’t bore them with trivial algebraic manipulation,\nwhich would only obscure the thread of her arguments. But she did show the most interesting parts: the\nresulting polynomial and its factoring. And she made sure she answered the original question.\nAvoid shorthand in formal writing. The many types of mathematical writing can be loosely grouped\ninto formal and informal writing. Informal writing includes writing on a blackboard during lecture, or\nexplaining something to a friend on a piece of scratch paper. Formal writing includes the kind of writing\nexpected on a homework assignment, or in a paper. There are differences in what is acceptable.\nFor instance, in informal writing, it is common to use shorthand for quantifiers and logical connec-\ntives: symbols such as ∀,∃, =⇒ , ⇐⇒ , or abbreviations like “iff” and “s.t.”.\nHowever, in formal writing such shorthand should generally be avoided. You should write out “for\nall”, “there exists”, “implies”, “if and only if”, and “such that”. Most other symbols are acceptable in\nformal writing, after defining them where needed. The membership symbol ∈ (“is an element of”) is\ntraditionally acceptable in formal writing, as are relations (e.g., <,>,×,+,∪,∩, etc.), variable names\n(e.g., x, y, z, etc.), and symbols for sets (Q,R,Z, etc.). Here is an acceptable use of symbols in formal\nmathematical writing.\nLet A,B be two subsets of R. We say B dominates A if for every x ∈ A there exists y ∈ B\nsuch that y > x.\nLearn the etiquette. The above example also illustrates two common conventions of mathematical eti-\nquette. It is customary to avoid starting sentences or phrases with a number or symbol, because next\nto punctuation they’re confusing. It is also customary to emphasize unfamiliar words that we are about\nto define, either by italicizing or underlining them. Other rules of etiquette can be learned through\nobserving the norms of the mathematical people or textbooks in your area of study.\n2. TOWARDS ELEGANCE\nDecide what’s important to say. Writing well does not necessarily mean writing more. If your solution\nis too wordy, it can sometimes obscure the points you are making.\nA well-written solution will present just enough details and highlight the most interesting or unex-\npected parts of the argument. What theorems or axioms were crucial in getting your solution, and where\nwere they used? Your role as a writer is not primarily to communicate details (though that can be quite\nimportant); your primary role is to give insight.\n2\nHighlight structure. If your argument is going to be a long one, with lots of technical details, then try to\nhelp the reader by summarizing the outline of your argument at the beginning. Then, throughout your\nwriting, help your reader see how you are progressing through your outline.\nUse paragraphs to emphasize blocks of ideas that are related. The role of the first sentences of para-\ngraphs are crucial: imagine a reader skimming your writing and only reading the first sentences. Will\nthey see the flow of your argument? Similarly, you might want to display only the most important\nequations. Replacing an oft-repeated argument by a good lemma can help streamline the flow of your\nargument, as well as highlight a key idea.\nChoose good examples. A difficult idea may be easier to digest if accompanied by an example. Choose\none simple enough to follow, but interesting enough to retain the salient features. A proof of a very\ngeneral idea could be preceded by an example in a specific context. A long exposition might benefit\nfrom a running example— one in which the same example is used multiple times in different contexts.\nAvoid red herrings! Omit details that have no bearing on the solution of the problem, as they may\nthrow the reader off. For example, if you say “we express the rational r as m\/n where m and n have no\ncommon factors,” you are leaving a clue that later, you will use the “no common factors” idea. So if you\nnever use that fact, you should omit saying it. It’s extraneous.\nStep back and simplify! After writing a proof, step back and ask: how can I simplify this argument?\nDid I use every tool I pulled out to solve this problem? Can I streamline this argument? For example,\nconsider this apparent proof by contradiction:\nProblem. Show that if 4 divides an integer n, then n is even.\nProof (by contradiction). Suppose n were not even.\nSince 4 divides n, we have n = 4k for some integer k. Thus n = 2(2k), which is even.\nThis contradicts our hypothesis that n is not even. \u0003\nDo you see why this is not really a proof by contradiction? The contradictory hypothesis in the first\nsentence was never used! Strip away the first and last sentence, and you have an elegant, direct proof.\nRefine, refine, refine. Good writing is a process of successive approximations. You should not expect\nyour first draft to be perfect. You will find that when you review your writing, you will see ways to\nshorten an argument, or say something in a better way. This is the part of the writing process that will\nhelp clarify your own thinking as well.\nOften, after completing a draft, a writer may notice that a particular choice of notation or a choice of\ndefinition was not optimal. A lazy writer would leave things as they are, but a thoughtful writer will\ntake the time to go back and make changes.\n3. FINALLY...\nObserve the culture. Good communication is inseparable from the culture in which it takes place. This\nrealization may unsettle budding mathematicians who are attracted to the logical absolutes of mathe-\nmatics. But even these absolutes are expressed in different ways by mathematicians of different eras, as\ncan be seen by comparing Newton’s writings to any of today’s calculus texts. The rules of mathematical\netiquette have evolved over time.\nWhile these guidelines attempt to draw up some common principles for formal writing, there will\nalways be exceptions— because some subset of mathematicians may have a slightly different norm. The\nbest rule of thumb to get a sense of what is acceptable or conventional in your cultural context is to\nbrowse several highly regarded textbooks or papers related to the document you are writing.\nAcknowledgements. It is appropriate to acknowledge any support you received. My revisions of this\npiece of writing benefited greatly from the helpful feedback of Jon Jacobsen and Lesley Ward.\n3\n"}
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All rights reserved. No reproduction or republication without written permission.","language_s":"ja","size_i":139342,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":4,"references_internal_i":4,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087036896903168,"text_t":"データでみる就活：朝日新聞デジタル. メインメニューをとばして、このページの本文エリアへ お使いのブラウザはJavaScriptに対応していないか、または無効になっています。詳しくは サイトポリシーのページ をご覧ください。 06月09日朝日新聞デジタル朝刊記事一覧へ（朝5時更新） 新着記事一覧へ ウクライナ情勢 速報 朝刊. 記事一覧 紙面ビューアー 夕刊. 記事一覧 紙面ビューアー 連載 ランキング コメント その他. 特集 動画・音声・写真 土曜別刷り「be」 記者イベント 天気 数独 12星座占い サイトマップ 検索. ヘルプ. 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Reライフ. &w. &M. &Travel. 連載. 教育・子育て. 小中高. 大学. 教育制度・話題. 教育問題. 地域の教育ニュース. 吹奏楽. 合唱. 子育て. ハグスタ. 朝日新聞EduA. 医療・健康. 新型コロナウイルス特集. 連載. 地域. 北海道・東北. 北海道. 青森. 秋田. 岩手. 山形. 宮城. 福島. 関東. 群馬. 茨城. 栃木. 埼玉. 千葉. 東京. 神奈川. 東海・甲信越. 静岡. 岐阜. 愛知. 三重. 新潟. 山梨. 長野. 近畿・北陸. 滋賀. 京都. 奈良. 和歌山. 大阪. 兵庫. 富山. 石川. 福井. 中国・四国. 鳥取. 島根. 岡山. 広島. 山口. 香川. 愛媛. 徳島. 高知. 九州・沖縄. 福岡. 大分. 宮崎. 鹿児島. 佐賀. 長崎. 熊本. 沖縄. シェア ツイート list ブックマーク 0. 朝日新聞デジタル 就活ニュース データでみる就活. 大学でても. 就職できない. 増える「大卒」. 変わる「就職先」. 学生の意識も変化. 取材：笹円香　取材協力：マイナビ、リクルートワークス研究所. 制作：米澤章憲、中西鏡子、小林由憲. 　２０１５年春に卒業した大学生の就職率（４月１日時点、就職希望者に対する就職者の割合）は９６．７％となり、４年連続で改善された。リーマン・ショック前の０８年春卒とほぼ同じ水準（９６．９％）。１６年春卒の就職活動は、さらに「売り手市場」と言われる。景気の動向や社会情勢に大きく左右される新卒採用の歴史を、データとともに振り返る。. 　. 大学でても. 「就職できない」15%. 　就職希望の大学生のうち就職した人の割合を示すのが「就職率. ※1. 」だ。１５年３月に大学を卒業した人の就職率は９６．７％で４年連続で上昇した。今期の就職活動は、景気の回復にともなって企業側が採用数を増やす姿勢を示しており、学生に有利な「売り手市場」と言われている。. ※1「就職率」は、文部科学省と厚生労働省が全国の大学、短期大学、高等専門学校、専修学校のなかから一部を抽出調査し、発表する。. 　しかし一方で、全大学の卒業者の進路を調べた文部科学省の「学校基本調査. ※2. 」をみると、卒業した人で就職も進学もしていない人が１４年春卒で約１５％（アルバイトなど一時的な仕事についている人も含む）いることがわかる。. ※2「学校基本調査」は、文部科学省が全国の学校を対象に実施する。. 大学卒業生のうち就職も進学もしなかった人の割合と人数. 文部科学省「学校基本調査」. アルバイトなど一時的な仕事についている人を含む. （％）. （人）. 　景気が回復してきたここ５年間をみてみると、毎年８万～１０万人ほどが大学を卒業後、就職も進学もしていない。いまの大学生の親世代が就活をしていた３０年前と比べると、その数は約２．５倍だ。. 増える「大卒」. 　大学をでても安定した仕事につかない人数が増えている背景には、大学卒業者そのものの増加がある。２０１４年春の大学卒業者数は約５６万５千人。この３０年で約１・５倍に増えた。. 大学卒業者と就職率の推移. 文部科学省「学校基本調査」. 卒業者数. うち就職者. （人）. 30年前. 現在. 　少子化で若者の数が減っているにもかかわらず、大学に進学する人は増え続けている。. 　就活情報を扱う「マイナビ」によると、１９８０年代後半から女性の大学進学率が上がり、女性の深夜労働や残業の規制を撤廃する９７年の男女雇用機会均等法改正などを経て、女性が４年制大学をでて働くことが一般的になった。. 　一方、２００３年の大学設置認可制度の規制緩和などで、４年制大学の新設や短期大学からの移行が続出。学生を受け入れる大学側も裾野が広がった。. 大学進学率と大学・短大の数. 文部科学省「学校基本調査」. 大学進学率. 男性. 女性. ４大学、短大の数. 大学. 短期大学. （％）. （学校数）. ４年制大学. 短期大学. 「就職先」も変化. 　雇用の場も大きく変わった。「学校基本調査」で大学卒業後の就職先を産業別に追うと、ここ３０年で第２次産業（鉱工業・建設業など）から第３次産業（運輸・通信・商業・金融・公務・サービス業など）へと大きくシフトしていることがわかる。. 学生の就職先の変遷. 文部科学省「学校基本調査」より ※グラフ内の「現在」の括弧内の数字は「84年」と比較して増減した人数. 84年. 2000年. 2014年. 1次産業. 農業. 林業・狩猟業. 漁業・水産養殖業. 1495人. 913人. 1203人. (-292). 2次産業. 鉱業. 建設業. 製造業. 9万2383人. 6万9635人. 6万4919人. (-2万7464人). 3次産業. 電気・ガス・水道業. 運輸・通信業. 卸売・小売業，飲食店. 金融・保険業. 不動産業. サービス業. 教育，学習支援業. 学術研究，専門・技術サービス業. 宿泊業，飲食サービス業. 生活関連サービス業,娯楽業. 医療，福祉. 複合サービス事業. 公務. 18万9509人. 22万3235人. 32万3239人. (+13万3730人). その他. 上記以外のもの. 2056人. 6935人. 5484人. (-3428人). メーカーは…. 　１９９０年代半ばまで大きな雇用の場だった自動車や電機といったメーカーは、グローバル化にともなって海外に工場を移転させてきた。半面、国内の求人を減らしてきた。国内自動車メーカーの場合、９０年代のバブル崩壊を起点に国内生産拠点を徐々に縮小。海外での生産を増やし、０８年のリーマン・ショック後は国内生産台数を超えた。. 日本の自動車メーカーの国内・海外生産台数の推移. 国内. 海外. リーマンショック. （08年9月）. 東日本大震災. （11年3月）. 国内. 海外. 国内と海外の生産台数が逆転. ～80年代. 現在. 銀行は…. 　文系学生の就職先として人気の高い銀行も、その数を大きく減らした。「三菱ＵＦＪフィナンシャルグループ」「三井住友フィナンシャルグループ」「みずほフィナンシャルグループ」の三つのメガバンクは、全国銀行協会によると、１３の都市銀行を含む約３０行（１９８９年時点）が、合併や再編などをへて集約されたものだ。. 主な銀行の合併. 都市銀行. 信託銀行. 地方銀行. 長期信用金庫. 第二地方銀行協会加盟行. 1989年. 1990年. 2000年. 2010年. 2014年末. 第一勧業. 富士. 日本興業. 安田信託. 三菱. 東京. 日本信託. 三菱信託. 三和. 東海. 東洋信託. 三井. 太陽神戸. 住友. 太平洋. 協和. 埼玉. 大和. 大阪. 近畿. なにわ. 福徳. 奈良. 住友信託. 三井信託. 北海道拓殖. 中央信託. みずほFG. みずほ. みずほ信託. 三菱UFJFG. 三菱東京UFJ. 三菱UFJ信託. 三井住友FG. 三井住友. りそなHD. りそな. 近畿大阪. 埼玉りそな. 三井住友トラストHD. 三井住友信託. 「ミスマッチ」も…. 　リクルートワークス研究所によると、金融業界全体では合併により極端に採用人数を減らしたわけではないものの、逆に景気回復を受けても大きく採用を増やすことは少なく、就職希望者１人あたりの求人数を示す「大卒求人倍率」は０．２％前後にとどまる「超人気業種」だ。. 　一方、深刻な人手不足に悩む業種もある。. 　同研究所の調査では、２０１６年春卒の大学生・院生への民間企業の求人数は前年比で５．４％増。就職希望者１人あたりの仕事の数を示す求人倍率は１．７３倍と、数字だけみれば就職できないという環境ではない。. 大卒求人倍率調査の推移. 各年3月卒　リクルートワークス研究所「大卒求人倍率調査」より. 求人倍率. 求人数. 民間企業就職希望者数. バブル崩壊. （91年）. リーマンショック. （08年9月）. 東日本大震災. （11年3月）. 全体. 1.73. 　しかし、東日本大震災後に人手不足が続く建設業界や、新規出店攻勢をかける小売業界は、学生側から「厳しい仕事」と見られがちで求人倍率は高止まりしている。. 業種別の大卒求人倍率. リクルートワークス研究所「大卒求人倍率調査」より. 金融業. サービス業. 全体. 製造業. 流通業. 建設業. 東日本大震災. 建設業、流通業は人手不足が続く. 就. 活. 生. 優. 位. 企. 業. 優. 位. 　企業の規模によるミスマッチもある。２０１６年３月卒の大卒求人倍率は、５千人以上の大手企業が０．５５倍。一方、３００人未満の企業では３．５９倍だ。. 　同研究所によると、学生には「景気が良くなれば大企業も採用を増やす」という意識が根強くあるものの、１９９０年代前半に新卒採用を積極的に増やした大手企業の多くが、バブル崩壊後に採用を極端に絞らざるをえなかった反省から、景気の良しあしにかかわらず新卒採用の人数を一定に保つようになっているという。. 学生の意識も変化. 　学生の求める「働きがい」や就職先の選び方も、景気の動向や世相を反映している。. 　マイナビが就職活動中の学生に、仕事に求めるものや希望する就職先を聞いた調査では、この１０年ほど、「プライドの持てる仕事」や「夢のため」に働くと答える自己実現志向の学生が減少傾向にある。一方、２０１１年の東日本大震災後には「人のためになる仕事」を求めたり「社会に貢献したい」と考えたりする学生が増えている。. 　また、好景気な時ほど仕事に「楽しさ」を求め、逆に景気が悪い時は「生活と仕事を両立させたい」とプライベートの充実を望む学生が増える傾向もうかがえる。. 就職観. 各年3月卒　マイナビ「就職意識調査」より. 楽しく働きたい. 個人の生活と仕事を両立させたい. 人のためになる仕事をしたい. 自分の夢のために働きたい. プライドの持てる仕事をしたい. 社会に貢献したい. 東日本大震災. （11年3月）. 2000～04. 就職氷河期. 04～09. 求人回復期. 09～13. 世界同時不況. 13～. アベノミクス. 企業志向. 各年3月卒　マイナビ「就職意識調査」より. ヤリガイのある仕事であれば中堅・中小企業でもよい. 自分のやりたい仕事ができるのであれば大手企業がよい. その他(公務員、Uターン志望など). 中堅・中小企業がよい. ゼッタイに大手企業がよい. 自分で会社を起こしたい. 東日本大震災. （11年3月）. 　景気は企業選びにも影響し、就職難が続くと、現実的に就職先を考え「ヤリガイのある仕事であれば中堅・中小企業でもよい」と考える学生が増えるという。. 　ただここ数年は、中小企業に目を向けるよう学生に指導する動きが大学に広がり、景気回復をうけた「大企業志向」はこれまでのようには増えないのでは、との見方もある。. 就活中の大学生向けに、お役立ちニュースやコラムを掲載しています. PR情報. 就職に関する最新ニュース. 就活ニューストップ 退職10年以内なら再び百五の行員に　「即戦力」に　三重 (6\/8). 　中途退職して10年以内の元行員の皆さん、再び一緒に働きませんか――。異業種などで経験を積んだ元行員を呼び戻そうと、百五銀行（津市）が7月から新たな正社員登用制度を始める。「即戦力」として社外で磨いた…. 成果自慢の「ガクチカ」は時代遅れ？　コロナ就活、再注目のチカラは (6\/6). 　今年も就職活動シーズンが始まった。コロナ禍で「ふつうの大学生活」を送れなかった就活生を悩ませているのが、学生時代に力を入れてきた「ガクチカ」のアピールだ。「ガクチカ」中心の就活を変えようという動きも…. 「マスク外して」に当惑、薄いガクチカ　就活生が面接で語るべきこと (6\/4). 　来春の入社に向けて就職活動をしている多くの大学4年生は、ほかの世代とはまったく違う学生生活を送ってきた。 　入学して1年もたたないうちに新型コロナウイルスの感染拡大に見舞われたためだ。授業はオンライ…. 大学1年の冬からずっとコロナ下　就活生は何に悩む (6\/1). 　政府が要請する就職活動ルールでの選考活動が1日、「解禁」された。2023年春に卒業予定の大学生らの多くは大学1年の冬ごろから新型コロナウイルスの影響を受け、オンラインでの授業や課外活動の制限を余儀な…. 就活面接「解禁」、リアルに緊張する学生も？　オンライン導入が7割 (6\/1). 　来春卒業する大学生らを対象とした企業の採用選考が1日、政府主導の「就活ルール」上で解禁された。企業の採用意欲は回復傾向で、コロナ禍で中断していた採用を再開する動きもある。オンライン面接の活用が定着し…. 日傘させますか？　「男性学」田中俊之さんが問う働き方改革の「壁」 (6\/1). 　若者の働き方や生き方を考えるうえで、常に直面するのが性別による役割の問題。本当に格差なく働き、暮らせる社会を実現するためには、どうすればいいのか。「男性学」を専門とする田中俊之さん（46）は、こう問…. 春先に消える院生「人生がかかっている」　就活長期化は誰のため (6\/1). 　毎年、春先になると大学の研究室から修士学生が消える。 　国立大学工学系大学院の男子学生（修士2年）も、「消えた院生」の一人だ。 　「3、4月は正直言って研究どころじゃなかった。就活は自分の人生がかか…. 失業率、コロナ禍前の水準に回復　求人増も、飲食業「団体客戻らず」 (5\/31). 　総務省が31日発表した4月の完全失業率（季節調整値）は前月より0・1ポイント低い2・5%だった。3カ月連続で低下し、新型コロナの感染が広がる直前の2020年3月の水準まで改善した。求人数も回復が続く…. 手話や指さしで接客する「サイニングストア」　日本生命社内に (5\/27). 　障害のある店員のみで運営するセブン―イレブンの店舗が、今月、大阪市内の日本生命の本店内に開業した。手話や指さしなどで接客対応をするため「サイニングストア」と呼ばれる。障害者の働く場を広げ、職員同士の…. 大卒就職率、2年連続で低下　ウクライナ侵攻が影響との見方も (5\/20). 　今春に卒業した大学生の就職率（4月1日時点）は95・8%で、昨年卒に比べて0・2ポイント下がった。過去最高だった一昨年卒の98・0%から2年連続で低下した。国際情勢の悪化が、中小企業の採用に影響した…. メーカーのSEだった私、50代で踏み出したセカンドキャリア (5\/20). 　長年勤めた会社が早期退職の募集を始めた。退職金は上乗せ、転職サポートもあるけれど――。終身雇用が崩れるなか、セカンドキャリアへ踏み出すかどうか、迷いや不安を感じる人もいるのでは。大手企業の社員だった…. 物理好き彼女の家で猛勉強　見習い中に結婚、2人をつなぐ赤い電車 (5\/20). 　見習い中に結婚する人は少ない。まだ何者にもなれていないのに、決断するのは難しいからだ。でも、そうでない道を選ぶこともあるらしい。 　「パンタよしっ」 　3月中旬の朝、名古屋鉄道の神宮前駅のホームに張…. コロナ禍で帰国かなわず…フランス出身26歳が日本で見つけた新天地 (5\/16). 　フランス・ボルドー市出身のガシェ・ニコラさんがこの春から、三重県志摩市の市観光協会で働いている。日本語、英語、スペイン語に加え、ウェブ制作が得意だという26歳。「志摩の良さを日本中と世界に発信したい…. 「漁師志望者」を募集、県・県漁連、育成に本腰 (5\/15). 　漁師をめざしませんか――。新米漁師を育成する研修生の募集が今年も始まった。神奈川県と県漁業協同組合連合会が、半年かけて座学と実地研修を行い、漁師の卵を育て就業先まで面倒をみる。ツテがなくても漁師にな…. 看護師をやめて猟師に　「撃っていいのか」迷いあっても続けた理由 (5\/14). 　看護師の仕事をやめ、猟師として生きると決めた。下関市菊川町の木原由紀恵さん（43）。初めて引き金を引くとき「撃っていいのか」と迷いがあったという。それでも猟を続けてきた理由がある。 　そこにいるのか…. 「ガクチカ」に悩む就活生　学生生活と就職戦線、コロナ禍で激変 (5\/12). 　学生時代、あなたはどんなことに力を入れて取り組みましたか――。就職活動でよく聞かれるいわゆる「ガクチカ」。就職活動が山場を迎えた今、就活生の間では、このガクチカへの懸念が広がっている。2年間、家にこ…. 吉田秀和、97歳の「楽観」を打ち砕いた東日本大震災. 市川染五郎さん、大河ドラマ出演で得たもの. メスティンの中身を衣替え“至高の焼肉セット”. 強烈なダッシュ力とデザインのよさ. 福島・双葉はアートとものづくりで蘇る. 「藤井聡太世代」の川島さん　「絶対王者」を破って将棋・学生名人に. リュックひとつで上京。毎日自炊、夢はカンヌ. 株安の世界で投資家が迫られる「宴の後始末」. いにしえのリゾート地、白砂の輝きと古湯　南紀白浜. 雨の日も退屈しない、東京都・八丈島の旅. アエラスタイルマガジン 福島・双葉はアートとものづくりで蘇る 2.24後の世界で 戦争映像に「感動」する私たち　単純な善悪、止まる思考　大塚英志氏 働く20代のモヤモヤ　WLBと言われても 20代から二度起業　令和トラベル篠塚社長に聞く「多様性」「若者」 注目の連載記事 もっと見る いま読む「音楽展望」　吉田秀和 没後10年 吉田秀和、97歳の「楽観」を打ち砕いた東日本大震災 国際・外交 タイムライン 【速報中】ロシア軍「制圧」宣言のヘルソン州、反撃が「一定の成果」 PR. 注目情報. 就活情報 Twitter ツイート ※Twitterのサービスが混み合っている時など、ツイートが表示されない場合もあります。. アクセスランキング もっと見る 読まれています 昨日のトップ５ なぜいま年金が減額されるの？物価高、厳しい家計が続くのに… 東京・目黒のココイチにタクシー突っ込む　男性運転手は死亡 「私は、みっともない」過食嘔吐を繰り返した　ミス日本、回復への道 摂食障害、15歳で旅立った娘　「もう笑うことはない」母は思ったが 東北新幹線、懐かし200系カラー再現　開業40周年でE2系が変身 1. 少子化の内実は？　子どもの数と親の学歴の関係、東大チームが分析 2. 「私は、みっともない」過食嘔吐を繰り返した　ミス日本、回復への道 3. 【詳報】ウクライナ、イスラエルを批判「最低限の支援すらせず」 4. 摂食障害、15歳で旅立った娘　「もう笑うことはない」母は思ったが 5. コロナ給付金不正事件、手配の男が漏らしたずさんな手口と出国理由 Facebook. 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All rights reserved. No reproduction or republication without written permission.","language_s":"ja","size_i":139342,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087038608179200,"text_t":"データでみる就活：朝日新聞デジタル. メインメニューをとばして、このページの本文エリアへ お使いのブラウザはJavaScriptに対応していないか、または無効になっています。詳しくは サイトポリシーのページ をご覧ください。 06月09日朝日新聞デジタル朝刊記事一覧へ（朝5時更新） 新着記事一覧へ ウクライナ情勢 速報 朝刊. 記事一覧 紙面ビューアー 夕刊. 記事一覧 紙面ビューアー 連載 ランキング コメント その他. 特集 動画・音声・写真 土曜別刷り「be」 記者イベント 天気 数独 12星座占い サイトマップ 検索. ヘルプ. 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Reライフ. &w. &M. &Travel. 連載. 教育・子育て. 小中高. 大学. 教育制度・話題. 教育問題. 地域の教育ニュース. 吹奏楽. 合唱. 子育て. ハグスタ. 朝日新聞EduA. 医療・健康. 新型コロナウイルス特集. 連載. 地域. 北海道・東北. 北海道. 青森. 秋田. 岩手. 山形. 宮城. 福島. 関東. 群馬. 茨城. 栃木. 埼玉. 千葉. 東京. 神奈川. 東海・甲信越. 静岡. 岐阜. 愛知. 三重. 新潟. 山梨. 長野. 近畿・北陸. 滋賀. 京都. 奈良. 和歌山. 大阪. 兵庫. 富山. 石川. 福井. 中国・四国. 鳥取. 島根. 岡山. 広島. 山口. 香川. 愛媛. 徳島. 高知. 九州・沖縄. 福岡. 大分. 宮崎. 鹿児島. 佐賀. 長崎. 熊本. 沖縄. シェア ツイート list ブックマーク 0. 朝日新聞デジタル 就活ニュース データでみる就活. 大学でても. 就職できない. 増える「大卒」. 変わる「就職先」. 学生の意識も変化. 取材：笹円香　取材協力：マイナビ、リクルートワークス研究所. 制作：米澤章憲、中西鏡子、小林由憲. 　２０１５年春に卒業した大学生の就職率（４月１日時点、就職希望者に対する就職者の割合）は９６．７％となり、４年連続で改善された。リーマン・ショック前の０８年春卒とほぼ同じ水準（９６．９％）。１６年春卒の就職活動は、さらに「売り手市場」と言われる。景気の動向や社会情勢に大きく左右される新卒採用の歴史を、データとともに振り返る。. 　. 大学でても. 「就職できない」15%. 　就職希望の大学生のうち就職した人の割合を示すのが「就職率. ※1. 」だ。１５年３月に大学を卒業した人の就職率は９６．７％で４年連続で上昇した。今期の就職活動は、景気の回復にともなって企業側が採用数を増やす姿勢を示しており、学生に有利な「売り手市場」と言われている。. ※1「就職率」は、文部科学省と厚生労働省が全国の大学、短期大学、高等専門学校、専修学校のなかから一部を抽出調査し、発表する。. 　しかし一方で、全大学の卒業者の進路を調べた文部科学省の「学校基本調査. ※2. 」をみると、卒業した人で就職も進学もしていない人が１４年春卒で約１５％（アルバイトなど一時的な仕事についている人も含む）いることがわかる。. ※2「学校基本調査」は、文部科学省が全国の学校を対象に実施する。. 大学卒業生のうち就職も進学もしなかった人の割合と人数. 文部科学省「学校基本調査」. アルバイトなど一時的な仕事についている人を含む. （％）. （人）. 　景気が回復してきたここ５年間をみてみると、毎年８万～１０万人ほどが大学を卒業後、就職も進学もしていない。いまの大学生の親世代が就活をしていた３０年前と比べると、その数は約２．５倍だ。. 増える「大卒」. 　大学をでても安定した仕事につかない人数が増えている背景には、大学卒業者そのものの増加がある。２０１４年春の大学卒業者数は約５６万５千人。この３０年で約１・５倍に増えた。. 大学卒業者と就職率の推移. 文部科学省「学校基本調査」. 卒業者数. うち就職者. （人）. 30年前. 現在. 　少子化で若者の数が減っているにもかかわらず、大学に進学する人は増え続けている。. 　就活情報を扱う「マイナビ」によると、１９８０年代後半から女性の大学進学率が上がり、女性の深夜労働や残業の規制を撤廃する９７年の男女雇用機会均等法改正などを経て、女性が４年制大学をでて働くことが一般的になった。. 　一方、２００３年の大学設置認可制度の規制緩和などで、４年制大学の新設や短期大学からの移行が続出。学生を受け入れる大学側も裾野が広がった。. 大学進学率と大学・短大の数. 文部科学省「学校基本調査」. 大学進学率. 男性. 女性. ４大学、短大の数. 大学. 短期大学. （％）. （学校数）. ４年制大学. 短期大学. 「就職先」も変化. 　雇用の場も大きく変わった。「学校基本調査」で大学卒業後の就職先を産業別に追うと、ここ３０年で第２次産業（鉱工業・建設業など）から第３次産業（運輸・通信・商業・金融・公務・サービス業など）へと大きくシフトしていることがわかる。. 学生の就職先の変遷. 文部科学省「学校基本調査」より ※グラフ内の「現在」の括弧内の数字は「84年」と比較して増減した人数. 84年. 2000年. 2014年. 1次産業. 農業. 林業・狩猟業. 漁業・水産養殖業. 1495人. 913人. 1203人. (-292). 2次産業. 鉱業. 建設業. 製造業. 9万2383人. 6万9635人. 6万4919人. (-2万7464人). 3次産業. 電気・ガス・水道業. 運輸・通信業. 卸売・小売業，飲食店. 金融・保険業. 不動産業. サービス業. 教育，学習支援業. 学術研究，専門・技術サービス業. 宿泊業，飲食サービス業. 生活関連サービス業,娯楽業. 医療，福祉. 複合サービス事業. 公務. 18万9509人. 22万3235人. 32万3239人. (+13万3730人). その他. 上記以外のもの. 2056人. 6935人. 5484人. (-3428人). メーカーは…. 　１９９０年代半ばまで大きな雇用の場だった自動車や電機といったメーカーは、グローバル化にともなって海外に工場を移転させてきた。半面、国内の求人を減らしてきた。国内自動車メーカーの場合、９０年代のバブル崩壊を起点に国内生産拠点を徐々に縮小。海外での生産を増やし、０８年のリーマン・ショック後は国内生産台数を超えた。. 日本の自動車メーカーの国内・海外生産台数の推移. 国内. 海外. リーマンショック. （08年9月）. 東日本大震災. （11年3月）. 国内. 海外. 国内と海外の生産台数が逆転. ～80年代. 現在. 銀行は…. 　文系学生の就職先として人気の高い銀行も、その数を大きく減らした。「三菱ＵＦＪフィナンシャルグループ」「三井住友フィナンシャルグループ」「みずほフィナンシャルグループ」の三つのメガバンクは、全国銀行協会によると、１３の都市銀行を含む約３０行（１９８９年時点）が、合併や再編などをへて集約されたものだ。. 主な銀行の合併. 都市銀行. 信託銀行. 地方銀行. 長期信用金庫. 第二地方銀行協会加盟行. 1989年. 1990年. 2000年. 2010年. 2014年末. 第一勧業. 富士. 日本興業. 安田信託. 三菱. 東京. 日本信託. 三菱信託. 三和. 東海. 東洋信託. 三井. 太陽神戸. 住友. 太平洋. 協和. 埼玉. 大和. 大阪. 近畿. なにわ. 福徳. 奈良. 住友信託. 三井信託. 北海道拓殖. 中央信託. みずほFG. みずほ. みずほ信託. 三菱UFJFG. 三菱東京UFJ. 三菱UFJ信託. 三井住友FG. 三井住友. りそなHD. りそな. 近畿大阪. 埼玉りそな. 三井住友トラストHD. 三井住友信託. 「ミスマッチ」も…. 　リクルートワークス研究所によると、金融業界全体では合併により極端に採用人数を減らしたわけではないものの、逆に景気回復を受けても大きく採用を増やすことは少なく、就職希望者１人あたりの求人数を示す「大卒求人倍率」は０．２％前後にとどまる「超人気業種」だ。. 　一方、深刻な人手不足に悩む業種もある。. 　同研究所の調査では、２０１６年春卒の大学生・院生への民間企業の求人数は前年比で５．４％増。就職希望者１人あたりの仕事の数を示す求人倍率は１．７３倍と、数字だけみれば就職できないという環境ではない。. 大卒求人倍率調査の推移. 各年3月卒　リクルートワークス研究所「大卒求人倍率調査」より. 求人倍率. 求人数. 民間企業就職希望者数. バブル崩壊. （91年）. リーマンショック. （08年9月）. 東日本大震災. （11年3月）. 全体. 1.73. 　しかし、東日本大震災後に人手不足が続く建設業界や、新規出店攻勢をかける小売業界は、学生側から「厳しい仕事」と見られがちで求人倍率は高止まりしている。. 業種別の大卒求人倍率. リクルートワークス研究所「大卒求人倍率調査」より. 金融業. サービス業. 全体. 製造業. 流通業. 建設業. 東日本大震災. 建設業、流通業は人手不足が続く. 就. 活. 生. 優. 位. 企. 業. 優. 位. 　企業の規模によるミスマッチもある。２０１６年３月卒の大卒求人倍率は、５千人以上の大手企業が０．５５倍。一方、３００人未満の企業では３．５９倍だ。. 　同研究所によると、学生には「景気が良くなれば大企業も採用を増やす」という意識が根強くあるものの、１９９０年代前半に新卒採用を積極的に増やした大手企業の多くが、バブル崩壊後に採用を極端に絞らざるをえなかった反省から、景気の良しあしにかかわらず新卒採用の人数を一定に保つようになっているという。. 学生の意識も変化. 　学生の求める「働きがい」や就職先の選び方も、景気の動向や世相を反映している。. 　マイナビが就職活動中の学生に、仕事に求めるものや希望する就職先を聞いた調査では、この１０年ほど、「プライドの持てる仕事」や「夢のため」に働くと答える自己実現志向の学生が減少傾向にある。一方、２０１１年の東日本大震災後には「人のためになる仕事」を求めたり「社会に貢献したい」と考えたりする学生が増えている。. 　また、好景気な時ほど仕事に「楽しさ」を求め、逆に景気が悪い時は「生活と仕事を両立させたい」とプライベートの充実を望む学生が増える傾向もうかがえる。. 就職観. 各年3月卒　マイナビ「就職意識調査」より. 楽しく働きたい. 個人の生活と仕事を両立させたい. 人のためになる仕事をしたい. 自分の夢のために働きたい. プライドの持てる仕事をしたい. 社会に貢献したい. 東日本大震災. （11年3月）. 2000～04. 就職氷河期. 04～09. 求人回復期. 09～13. 世界同時不況. 13～. アベノミクス. 企業志向. 各年3月卒　マイナビ「就職意識調査」より. ヤリガイのある仕事であれば中堅・中小企業でもよい. 自分のやりたい仕事ができるのであれば大手企業がよい. その他(公務員、Uターン志望など). 中堅・中小企業がよい. ゼッタイに大手企業がよい. 自分で会社を起こしたい. 東日本大震災. （11年3月）. 　景気は企業選びにも影響し、就職難が続くと、現実的に就職先を考え「ヤリガイのある仕事であれば中堅・中小企業でもよい」と考える学生が増えるという。. 　ただここ数年は、中小企業に目を向けるよう学生に指導する動きが大学に広がり、景気回復をうけた「大企業志向」はこれまでのようには増えないのでは、との見方もある。. 就活中の大学生向けに、お役立ちニュースやコラムを掲載しています. PR情報. 就職に関する最新ニュース. 就活ニューストップ 退職10年以内なら再び百五の行員に　「即戦力」に　三重 (6\/8). 　中途退職して10年以内の元行員の皆さん、再び一緒に働きませんか――。異業種などで経験を積んだ元行員を呼び戻そうと、百五銀行（津市）が7月から新たな正社員登用制度を始める。「即戦力」として社外で磨いた…. 成果自慢の「ガクチカ」は時代遅れ？　コロナ就活、再注目のチカラは (6\/6). 　今年も就職活動シーズンが始まった。コロナ禍で「ふつうの大学生活」を送れなかった就活生を悩ませているのが、学生時代に力を入れてきた「ガクチカ」のアピールだ。「ガクチカ」中心の就活を変えようという動きも…. 「マスク外して」に当惑、薄いガクチカ　就活生が面接で語るべきこと (6\/4). 　来春の入社に向けて就職活動をしている多くの大学4年生は、ほかの世代とはまったく違う学生生活を送ってきた。 　入学して1年もたたないうちに新型コロナウイルスの感染拡大に見舞われたためだ。授業はオンライ…. 大学1年の冬からずっとコロナ下　就活生は何に悩む (6\/1). 　政府が要請する就職活動ルールでの選考活動が1日、「解禁」された。2023年春に卒業予定の大学生らの多くは大学1年の冬ごろから新型コロナウイルスの影響を受け、オンラインでの授業や課外活動の制限を余儀な…. 就活面接「解禁」、リアルに緊張する学生も？　オンライン導入が7割 (6\/1). 　来春卒業する大学生らを対象とした企業の採用選考が1日、政府主導の「就活ルール」上で解禁された。企業の採用意欲は回復傾向で、コロナ禍で中断していた採用を再開する動きもある。オンライン面接の活用が定着し…. 日傘させますか？　「男性学」田中俊之さんが問う働き方改革の「壁」 (6\/1). 　若者の働き方や生き方を考えるうえで、常に直面するのが性別による役割の問題。本当に格差なく働き、暮らせる社会を実現するためには、どうすればいいのか。「男性学」を専門とする田中俊之さん（46）は、こう問…. 春先に消える院生「人生がかかっている」　就活長期化は誰のため (6\/1). 　毎年、春先になると大学の研究室から修士学生が消える。 　国立大学工学系大学院の男子学生（修士2年）も、「消えた院生」の一人だ。 　「3、4月は正直言って研究どころじゃなかった。就活は自分の人生がかか…. 失業率、コロナ禍前の水準に回復　求人増も、飲食業「団体客戻らず」 (5\/31). 　総務省が31日発表した4月の完全失業率（季節調整値）は前月より0・1ポイント低い2・5%だった。3カ月連続で低下し、新型コロナの感染が広がる直前の2020年3月の水準まで改善した。求人数も回復が続く…. 手話や指さしで接客する「サイニングストア」　日本生命社内に (5\/27). 　障害のある店員のみで運営するセブン―イレブンの店舗が、今月、大阪市内の日本生命の本店内に開業した。手話や指さしなどで接客対応をするため「サイニングストア」と呼ばれる。障害者の働く場を広げ、職員同士の…. 大卒就職率、2年連続で低下　ウクライナ侵攻が影響との見方も (5\/20). 　今春に卒業した大学生の就職率（4月1日時点）は95・8%で、昨年卒に比べて0・2ポイント下がった。過去最高だった一昨年卒の98・0%から2年連続で低下した。国際情勢の悪化が、中小企業の採用に影響した…. メーカーのSEだった私、50代で踏み出したセカンドキャリア (5\/20). 　長年勤めた会社が早期退職の募集を始めた。退職金は上乗せ、転職サポートもあるけれど――。終身雇用が崩れるなか、セカンドキャリアへ踏み出すかどうか、迷いや不安を感じる人もいるのでは。大手企業の社員だった…. 物理好き彼女の家で猛勉強　見習い中に結婚、2人をつなぐ赤い電車 (5\/20). 　見習い中に結婚する人は少ない。まだ何者にもなれていないのに、決断するのは難しいからだ。でも、そうでない道を選ぶこともあるらしい。 　「パンタよしっ」 　3月中旬の朝、名古屋鉄道の神宮前駅のホームに張…. コロナ禍で帰国かなわず…フランス出身26歳が日本で見つけた新天地 (5\/16). 　フランス・ボルドー市出身のガシェ・ニコラさんがこの春から、三重県志摩市の市観光協会で働いている。日本語、英語、スペイン語に加え、ウェブ制作が得意だという26歳。「志摩の良さを日本中と世界に発信したい…. 「漁師志望者」を募集、県・県漁連、育成に本腰 (5\/15). 　漁師をめざしませんか――。新米漁師を育成する研修生の募集が今年も始まった。神奈川県と県漁業協同組合連合会が、半年かけて座学と実地研修を行い、漁師の卵を育て就業先まで面倒をみる。ツテがなくても漁師にな…. 看護師をやめて猟師に　「撃っていいのか」迷いあっても続けた理由 (5\/14). 　看護師の仕事をやめ、猟師として生きると決めた。下関市菊川町の木原由紀恵さん（43）。初めて引き金を引くとき「撃っていいのか」と迷いがあったという。それでも猟を続けてきた理由がある。 　そこにいるのか…. 「ガクチカ」に悩む就活生　学生生活と就職戦線、コロナ禍で激変 (5\/12). 　学生時代、あなたはどんなことに力を入れて取り組みましたか――。就職活動でよく聞かれるいわゆる「ガクチカ」。就職活動が山場を迎えた今、就活生の間では、このガクチカへの懸念が広がっている。2年間、家にこ…. 吉田秀和、97歳の「楽観」を打ち砕いた東日本大震災. 市川染五郎さん、大河ドラマ出演で得たもの. メスティンの中身を衣替え“至高の焼肉セット”. 強烈なダッシュ力とデザインのよさ. 福島・双葉はアートとものづくりで蘇る. 「藤井聡太世代」の川島さん　「絶対王者」を破って将棋・学生名人に. リュックひとつで上京。毎日自炊、夢はカンヌ. 株安の世界で投資家が迫られる「宴の後始末」. いにしえのリゾート地、白砂の輝きと古湯　南紀白浜. 雨の日も退屈しない、東京都・八丈島の旅. アエラスタイルマガジン 福島・双葉はアートとものづくりで蘇る 2.24後の世界で 戦争映像に「感動」する私たち　単純な善悪、止まる思考　大塚英志氏 働く20代のモヤモヤ　WLBと言われても 20代から二度起業　令和トラベル篠塚社長に聞く「多様性」「若者」 注目の連載記事 もっと見る いま読む「音楽展望」　吉田秀和 没後10年 吉田秀和、97歳の「楽観」を打ち砕いた東日本大震災 国際・外交 タイムライン 【速報中】ロシア軍「制圧」宣言のヘルソン州、反撃が「一定の成果」 PR. 注目情報. 就活情報 Twitter ツイート ※Twitterのサービスが混み合っている時など、ツイートが表示されない場合もあります。. アクセスランキング もっと見る 読まれています 昨日のトップ５ なぜいま年金が減額されるの？物価高、厳しい家計が続くのに… 東京・目黒のココイチにタクシー突っ込む　男性運転手は死亡 「私は、みっともない」過食嘔吐を繰り返した　ミス日本、回復への道 摂食障害、15歳で旅立った娘　「もう笑うことはない」母は思ったが 東北新幹線、懐かし200系カラー再現　開業40周年でE2系が変身 1. 少子化の内実は？　子どもの数と親の学歴の関係、東大チームが分析 2. 「私は、みっともない」過食嘔吐を繰り返した　ミス日本、回復への道 3. 【詳報】ウクライナ、イスラエルを批判「最低限の支援すらせず」 4. 摂食障害、15歳で旅立った娘　「もう笑うことはない」母は思ったが 5. コロナ給付金不正事件、手配の男が漏らしたずさんな手口と出国理由 Facebook. 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All rights reserved. No reproduction or republication without written permission.","language_s":"ja","size_i":72246,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":0,"references_internal_i":0,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735087038815797248,"text_t":"個人情報について | 朝日新聞社の会社案内. 日本語. \/. English ご購読申込 朝日新聞デジタル 採用情報 メニュー. お知らせ サービス サービスTOP 個人のお客様 法人のお客様 教育関係のお客様 会社案内 会社案内TOP トップメッセージ 会社情報 アクセスマップ 役員・組織図 グループ企業 朝日新聞社小史 会社・工場見学 ガバナンス サステナビリティ サステナビリティTOP SDGs ジェンダー 環境 次世代 育児 健康づくり 障がい者雇用 復興支援 文化財保護 大会・コンクール・表彰 大会・コンクール・表彰TOP 大会 コンクール 表彰 朝日ID 朝日ID TOP 朝日IDとは ご利用方法 ご利用可能サイト イベント一覧 メールサービス よくあるご質問 お問い合わせ ヘルプ・お問い合わせ 日本語. \/. English トップ 個人情報について. 個人情報について. 朝日新聞社の個人情報保護方針. 個人情報の保護に関する宣言. 　株式会社朝日新聞社（以下「当社」といいます）は、個人情報の重要性を認識し、その保護の徹底を図るため、個人情報に関して適用される法令や規範を順守するとともに、個人情報（報道の用及び著述の用に供する目的で取得・利用・保有するものを除く）を「個人情報保護方針」に従って取り扱います。なお、報道・著述目的の個人情報については、法の基本精神を踏まえ、「報道・著述目的で取り扱う個人情報の保護方針」を別途定め、公表しております。個人情報の保護に関する法律（以下「個人情報保護法」といいます）第57条第1項は、報道機関等が報道・著述目的で取り扱う個人情報については、法第４章（個人情報取扱事業者等の義務等）の規定を適用しないと定めています。これを踏まえて取り決めたものです。. 報道・著述目的で取り扱う個人情報の保護方針 個人情報・特定個人情報. 　個人情報とは、お客様ご本人に関する情報で、氏名、生年月日、性別、住所、電話番号、職業、メールアドレスその他の記述等により、お客様個人を識別することができるものをいいます。. 　特定個人情報とは、マイナンバー（個人番号）を含む個人情報をいいます。. 個人情報の取得・利用. 　当社は、取得した個人情報について、利用目的の達成に必要な範囲内において、データ内容の正確性の確保に努めるとともに、当該情報への不正なアクセス及び当該情報の漏えい、滅失及び毀損などが起きないよう、適切に取り扱います。そのために、個人情報データベースを厳重に保護・管理します。個人情報を取り扱う部門ごとに責任者を定めて管理状況を把握するとともに、社内全体を統括する部門でもチェックし、遺漏のないようにします。当社は、個人の生命、身体もしくは財産の保護のために必要がある場合、またはその他法令に基づく場合などを除いて、個人情報保護法が定める「要配慮個人情報」を、原則として本人の同意を得ることなく取得いたしません。. 　また、当社、当社のグループ企業およびＡＳＡなど朝日新聞を取り扱う新聞販売所（以下「ＡＳＡ等」といい、当社、当社のグループ企業およびＡＳＡ等を総称して「朝日新聞グループ」といいます）は、取得した個人情報を、朝日新聞グループにおいて共同利用します。共同利用についての公表事項は、以下の「共同利用に関する事項」欄をご覧ください。. 　当社は、上記の共同利用またはその他法令に基づく場合などを除いて、お客様の個人情報をご本人の同意を得ることなく第三者に提供したり、開示したりすることはいたしません。. 　当社の事業目的を遂行するために個人情報を取り扱う業務を第三者に委託する場合、当社はその委託先に対し、当該個人情報を漏洩・流出させたり不正に利用したりしないよう、契約により適切な管理を義務付け、実施させております。. 　当社は、上記のほか、取得した個人情報をもとに個人を特定できない統計情報または個人情報保護法が定める「匿名加工情報」等を作成し、提携企業等を含む第三者に提供することがあります。当社は、匿名加工情報のデータベースを作成するときは、特定の個人を識別することおよび個人情報を復元することができないようにするために必要な措置をとるとともに、その作成および第三者への提供にあたって、匿名加工情報に含まれる項目を公表するなど、個人情報保護法等の定めに従い、適切にこれを取り扱います。. 　特定個人情報の取得・利用につきましては、後述の「特定個人情報（マイナンバー）の取り扱いについて」をご覧ください。. ▼新聞購読のためにご提供いただいたお客様の個人情報について. 　朝日新聞購読のためにお客様から個人情報をご提供いただくにあたっては、当社から各ＡＳＡ等に対して、個人情報を漏洩・流出させたり不正に利用したりしないよう、契約により厳正な管理を義務付けております。お客様が各ＡＳＡ等と購読契約を交わされた場合、フリーダイヤル（0120-33-0843）もしくは当社の運営するウェブサイトなどを通じて当社に直接、購読の申し込みをいただいた場合、いずれの場合においても、個人情報の利用目的のために必要な範囲で、朝日新聞グループ内にて個人情報の伝達を行います。ＡＳＡにおける個人情報の取り扱い方針については、. ＡＳＡの個人情報取り扱いについて をご覧下さい。. ▼朝日新聞デジタル購読のためにご提供いただいたお客様の個人情報について. 　朝日新聞デジタル購読のためにご提供いただいた個人情報については、当社が取得し、個人情報保護方針および朝日新聞デジタルの利用規約等に従って当社において利用するほか、新聞購読のためにご提供いただいた個人情報と同様、朝日新聞グループにおいて共同利用します。. ▼朝日ID会員の個人情報について. 　朝日ID会員の個人情報（会員情報）については、朝日ID会員規約に従って利用します。朝日ＩＤ会員規約については. こちら をご覧ください。. ▼特定個人情報（マイナンバー）の取り扱いについて. 　当社は、「行政手続における特定の個人を識別するための番号の利用等に関する法律｣、「特定個人情報の適正な取扱いに関するガイドライン」および個人情報保護法を遵守して、特定個人情報の適正な取り扱いを行います。. 　当社は、取得した特定個人情報を、以下に掲げた個人番号を取り扱う事務の範囲内でのみ取り扱います。人の生命、身体または財産の保護のために必要がある場合を除き、本人の同意があったとしても、利用目的を超えて特定個人情報を利用しません。. 　なお、特定個人情報についても、「利用目的の通知、開示、訂正等」「利用停止等」および「お問い合わせ窓口」については、各欄に記載のとおりです。ただし、当社が保有する特定個人情報に限ります。. ▼朝日新聞グループ以外からの商品案内や催し物情報などに関連してご提供いただいた個人情報について. 　朝日新聞の紙面や当社が発行する出版物、ならびに朝日新聞デジタルをはじめとするウェブサイト上で、商品案内や催し物情報などに関連して、投稿や懸賞への応募、アンケートへの回答などをお願いすることがあります。これらの中には、当社は広告や告知のためのスペースを提供するだけで、広告主や催し物の主催者が個人情報を集める主体（以下「収集者」といいます）となるケースが含まれています。その場合、個人情報の取得・管理・利用は、収集者が自らの個人情報管理方針に従って行うことになり、当社は当該個人情報を取得・管理・利用する立場にありません。当社は、紙面や出版物ならびにウェブサイト上で収集者を明示するとともに、収集者には当該個人情報を適切に管理するよう求めております。. ▼当社が運営するウェブサービス上のアクセスデータ利用について. 　朝日新聞デジタルをはじめとした当社が運営するウェブサイトや当社が提供するアプリケーションでは、クッキーやその類似技術を用いて、お客様のコンピュータ、スマートフォン、タブレット等のデバイスから、アクセスしたURL、アクセス時間、IPアドレス等の端末情報、クッキーID、広告ID等のデータを収集し、利用者の興味関心に合わせた情報の提供や利用者に適した広告配信等のため利用することがあります。. アクセスデータの利用について 「個人情報の保護に関する法律」に基づく公表事項等. 事業活動について. 新聞、雑誌、書籍、電子出版物などの制作・発行・販売と電子・電波メディアを利用した情報・サービスの提供. 前記各メディアを利用した広告の掲載・配信と、広告の企画・制作・営業. 文化事業及びスポーツ事業の企画・運営. 著作権、商標権など知的財産権の取得・管理・ライセンス（利用許諾）・譲渡・映像・音声・文字等による各種ソフトの企画・制作・販売、ならびに日用品雑貨・食料品・美術工芸品・医薬部外品・管理医療機器・化粧品の販売. 文化・スポーツ施設や住宅展示場の運営. その他、上記各号に付帯または関連する事業. 利用目的について. 　朝日新聞グループ（朝日新聞社、朝日新聞社のグループ企業およびＡＳＡなど朝日新聞を取り扱う新聞販売所）が取得した個人情報は、朝日新聞グループの事業活動に伴う（１）商品・サービスの配送・提供　（２）商品・サービス・催し物の案内　（３）既存の商品・サービスの改善や、新しい商品・サービスの開発などのための調査　（４）朝日新聞グループ以外の企業等から依頼・提供された商品・サービス・催し物の案内及びプレゼントやアンケート類の送付　（５）その他個人情報取得時に本人に明示された利用目的――などに利用いたします。. （１）商品・サービスの配送・提供. 朝日新聞グループが販売する商品などの配達・送付. イベントやセミナーの入場券などの配達・送付. 文化事業及びスポーツ事業の企画・運営. 各種会員制サービスの提供. ご協力いただいたアンケートに対する謝礼などの配達・送付. 懸賞当選者等に対する景品などの配達・送付. （２）商品・サービス・催し物の案内. 朝日新聞グループが実施するイベントやセミナーに関するご案内. 朝日新聞グループが発行する新聞、書籍、雑誌等についての、新規または継続購読に関するご案内. 朝日新聞グループが発行または運営する新聞、書籍、雑誌、ウェブサイト等における広告掲載のご案内. 会員制サービスをはじめとする各種サービスに関するご案内. 朝日新聞グループの商品・サービス・催し物に関する情報のメール・郵送物等によるお知らせや、朝日新聞グループや他の企業が運営するメディアにおける広告の配信（お客様の属性情報、ウェブサイト閲覧履歴、アプリ使用履歴、購買履歴等を分析し、分析結果に基づいて趣味・嗜好に応じたメール・郵送物等によるお知らせや広告を配信することを含みます）. （３）既存の商品・サービスの改善や、新しい商品・サービスの開発などのための調査. 朝日新聞グループが販売する商品、各種会員制サービス、朝日新聞グループが実施するイベントやセミナー、朝日新聞グループが運営するウェブサイトなどの内容を改善するための、アンケート類の配達・送付及び電話での聞き取り等による調査. 新しい商品やサービスの開発に役立てるための、アンケート類の配達・送付、及び電話での聞き取り等による調査. （４）朝日新聞グループ以外の企業等から依頼・提供された商品・サービス・催し物の案内及びプレゼントやアンケート類の送付. 朝日新聞グループ以外の企業が、朝日新聞の紙面、朝日新聞グループが発行する出版物、及び朝日新聞デジタルなどウェブサイト上に掲載された広告を利用して、希望するお客様に商品や資料などを提供する際の、当該商品や資料などの配達・送付. 朝日新聞グループ以外の企業の商品・サービス・催し物に関する情報のメール・郵送物等によるお知らせや、朝日新聞グループや他の企業が運営するメディアにおける広告の配信（お客様の属性情報、ウェブサイト閲覧履歴、アプリ使用履歴、購買履歴等を分析し、分析結果に基づいて趣味・嗜好に応じたメール・郵送物等によるお知らせや広告を配信することを含みます）. 安全管理措置について. 　当社は、個人情報の漏えい、滅失及び毀損などのリスクに対して、個人情報の安全管理が図られるよう、必要かつ適切な措置を講じているほか、従業員に対し、必要かつ適切な監督を行っています。また、当社は、個人情報の取扱いの全部または一部を委託する場合は、委託先において個人情報の安全管理が図られるよう、必要かつ適切な監督を行っています。. 　当社が個人情報の安全管理のために講じている措置の内容につきましては、下記「お問い合わせ窓口」記載の連絡先までお問い合わせください。. 共同利用に関する事項. 　朝日新聞グループにおける個人情報の共同利用に関する事項は以下のとおりです。. 共同利用する個人情報の項目. 氏名、生年月日、性別、住所、電話番号、職業、メールアドレス. およびその他お客様から朝日新聞グループにお知らせいただいた項目. 共同利用者の範囲. 朝日新聞社、朝日新聞社のグループ企業およびＡＳＡなど朝日新聞を取り扱う新聞販売所. なお、共同利用者の範囲に含まれるグループ企業の一覧は、以下の通りです。. 企業名. 住所. 連絡先（電話番号）. ㈱朝日新聞社. 東京都中央区築地5-3-2. 03-5540-7836. ㈱朝日新聞出版. 東京都中央区築地5-3-2. 03-5540-7900. 朝日新聞販売サービス㈱. 東京都中央区築地5-3-2. 03-3541-8031. ㈱朝日新聞メディアプロダクション. 東京都中央区築地5-3-2. 03-5540-7411. ㈱朝日サポートセンター. 名古屋市中区栄1-3-3. 052-221-6338. ㈱朝日販売サービスセンター. 大阪市北区中之島2-3-18. 06-6222-2200. ㈱朝日サービス. 札幌市中央区北1条西1丁目. 011-231-1237. ㈱朝日学生新聞社. 東京都中央区築地5-3-2. 03-3545-5223. ㈱朝日カルチャーセンター. 東京都新宿区西新宿2-6-1. 03-3344-1941. 利用目的. 上記の「利用目的について」に記載のとおり. 共同利用する個人情報の管理責任者. 株式会社朝日新聞社（同社の住所及び代表者名については、. こちら をご覧ください）. お客様の個人情報に関するお問い合わせについて. 利用目的の通知、開示、訂正等. 　朝日新聞グループが保有するお客様の個人情報（保有個人データ）について、お客様ご本人が、その利用目的の通知や、開示（第三者提供記録の開示を含みます。以下同じです）、または訂正・追加・削除（以下、訂正・追加・削除をまとめて「訂正等」といいます）を請求する場合は、下記の「申請方法」により、お申し出ください。申出人がご本人であることを確認したうえで必要な調査をし、請求に理由があれば、遅滞なく、利用目的の通知、開示、または訂正等の対応をし、結果をご本人に通知いたします。. 利用停止等. 　朝日新聞グループは、保有する個人情報に基づき、個別の商品・サービスの案内やアンケートなどを、ダイレクトメールやメールマガジン等によりお客様にお送りしたり、電話をおかけしたりすることがあります。不要な場合は、お送りした資料・封筒・メール等に記載されている連絡先、または下記の「お問い合わせ窓口」（個人情報係）に直接お申し出ください。. 　また、朝日新聞グループの個人情報（保有個人データ）の取得や利用もしくは提供のあり方が法律に違反していること、当該個人情報を利用する必要がなくなったこと、その他個人情報保護法の定める事由に基づいて利用停止もしくは消去または第三者提供の停止（以下「利用停止等」といいます）を請求される場合の手続きについては、下記の「申請方法」をご覧ください。申出人がご本人であることを確認したうえで必要な調査をし、請求に理由があれば、必要な限度で、遅滞なく利用停止等の措置をとります。. 申請方法. （１） ご提出いただく書面等（本人申請の場合）. 　個人情報（保有個人データ）に関して利用目的の通知、開示、訂正等、または利用停止等を当社に請求する際は、下記リンクからそれぞれのご要望に沿った申請書をダウンロードしてご利用ください。これらの申請書以外の書式を使った申請には応じかねますので、あらかじめご了承ください。. ＜各種申請書＞. 個人情報の利用目的通知申請書 個人情報の開示申請書 個人情報の訂正等申請書 個人情報の利用停止等申請書 所定の事項をすべてご記入の上、下記の本人確認のための書類を同封して、. お問い合わせ窓口（〒104-8011　朝日新聞社　個人情報係）までお送りください。. ※確実に当社に届いたことが確認できるよう、配達記録郵便や書留郵便を利用されることをお勧めします。. ※申請書がダウンロードできない場合は、同窓口までご請求ください。ファクス等でお送りします。. ＜本人確認書類＞. 　ご本人の氏名・生年月日・現住所が記載された公的書類のコピーとさせていただきます。当社で受理できる本人確認書類は以下の通りです（確認のために必要とするのは前記の３項目です。本籍など、これら三つ以外の記載欄は塗りつぶしたうえで送付いただいても結構です）。. ▼１点のみで受け付けるもの. 運転免許証のコピー（裏面に住所変更の記載があれば、裏面も）. 在留カードまたは特別永住者証明書のコピー. ▼２点のセットで受け付けるもの　（いずれも過去３カ月以内のもの）. パスポート、健康保険被保険者証、年金手帳、学生証などのコピーのいずれか１点. 本人の現住所が確認できるもの１点. （住民票、公共料金の請求書、自宅に届いた消印付き郵便物などのコピー）. 　これらの書類をご提出いただいた場合でも、当社が必要であると判断したときには、他の書類の追加提出などをご依頼させて頂く場合がございます。こうした手続きは、他人がお客様になりすまして不正に情報を取得することなどを防止するための措置ですので、ご理解ください。なお、お送りいただいた申請書と本人確認のための書類はお返しできません。あらかじめご了承ください。. （２）代理人による申請. 　個人情報に関する利用目的の通知、開示、訂正等または利用停止等を求める方が未成年者または成年被後見人である場合、法定代理人からの申請に限り受け付けをいたします。この場合、前項（１）の各書類に加え、下記（ａ）の書類を同封して申請してください。また、ご本人が弁護士など任意の代理人に委任して申請する場合は、前項（１）の各書類に加え、下記（ｂ）の書類を同封してください。これらは代理権のない者がご本人に無断で申請することを防止するための措置です。.  . （ａ）法定代理人の場合. 本人との続柄が確認できる書類（戸籍謄本、住民票、健康保険被保険者証などのコピー）. 法定代理人の本人確認書類（前項（１）と同じ）.  . （ｂ）任意代理人の場合. 本人が署名・押印した当社所定の委任状（委任状は. こちら からダウンロードできます）. 委任状に押印された本人印の印鑑証明書. 任意代理人の本人確認書類（前項（１）と同じ）.  . 　なお、追加での書類提出をお願いさせて頂く場合がございます。また、これらの書類もお返しすることはできません。あらかじめご了承ください。. （３）手数料. 　利用目的の通知及び開示を請求される場合は、１件の申請につき、５００円（当社からの回答に要する実費および消費税を含みます）を申し受けます。なお、送付書類が多くなったり、送付先が日本国外で送料が余分にかかったりする場合は、追加負担をお願いすることがあります。. 　お支払いは、下記郵便振替口座への払い込み（※払込手数料は払込ご依頼人の負担）でお願いいたします。払込ご依頼人の名義欄には、開示などを請求されるご本人のお名前をご記入ください。申請書が当社に届いてから１０日以内にご入金がない場合は、請求がなかったものとみなしますので、ご注意ください。. 　なお、訂正等（訂正・追加・削除）または利用停止等（利用停止・消去・第三者提供停止）を請求される場合、手数料はいただきません。. ▼払込先. 【口座番号】 ００１６０－７－２７９０６１. 【口座名義】 朝日新聞社　個人情報係. （４）当社からの回答方法. ＜利用目的通知・開示＞. 　前項に定める手数料のお支払いがあったことを確認した後、遅滞なく、当社が指定する方式での電磁的記録の提供または書面の郵送のうちお客様が選択された方法（ただし、利用目的の通知請求の場合には、当社が指定する方法）で回答いたします。ただし、電磁的記録の提供による開示については、開示対象などによってはお受けできない場合がございます。なお、書面の郵送による開示の場合には配達記録郵便で、送付先が日本国外の場合はそれに準ずる確実な方法でお送りいたします。. ＜訂正等・利用停止等＞. お客様の請求に正当な理由がある場合、当社は、必要な限度で、遅滞なく請求に応じます。この場合、当社がとった措置を書面やメール等の当社が適切と判断した方法でお知らせいたします。. （５）取得した個人情報の利用目的等. 　個人情報に関する利用目的の通知、開示、訂正等及び利用停止等の請求に伴い、当社が取得した申請者及び代理人の個人情報は、その請求に応じるのに必要な範囲でのみ取り扱います。お送りいただいた申請書と本人確認のための書類、及び代理申請のための書類は、後日のお問い合わせ等に対応できるよう３年間保存し、その後、適切に消去・廃棄いたします。なお、本人確認や代理申請のための書類の中に本籍の記載がある場合、当社は当該部分を黒く塗りつぶして保存します。. （６）請求に対応できない場合. 　当社で申請者の個人情報を保有していない場合や、以下の事由に該当する場合は、請求どおりの対応ができないことがあります。その場合は、請求に応じられない旨とその理由について、書面やメール等の当社が適切と判断した方法でお知らせいたします。なお、このときでも（３）に定める手数料はお返しできません。. 申請書と本人確認書類の記載内容とに相違があるなど、本人確認ができない場合. 代理人による申請に際して、代理権が確認できない場合. 申請書の記載に不備がある場合. 請求対象データが個人情報保護法に定める「保有個人データ」に該当しない場合その他請求が個人情報保護法に定められた要件を満たさない場合. 「利用目的が明らかである」「開示により、本人または第三者の生命、身体、財産その他の権利利益を害するおそれがある」「個人情報取扱事業者（当社）の業務の適正な実施に著しい支障を及ぼすおそれがある」など、個人情報保護法が定める義務免除事由にあたる場合. 利用停止等に多額の費用を要する場合など、利用停止等を行うことが困難な場合であって、お客様の権利利益を保護するために必要な代替措置を取ることができる場合. 請求対象データを、報道の用または著述の用に供する目的で当社が取り扱っている場合. お問い合わせ窓口. 　個人情報の取り扱いに関する問い合わせや苦情等にお答えする当社の総合窓口は以下の通りです。ただし、開示をはじめ個人情報保護法が定める各種請求や、それに対する当社からの回答は原則として書面やメールによるやり取りとし、電話でお答えするのは手続きに関する質問に限らせていただきます。また、当社を直接来訪されての申請や問い合わせには応じかねます。. 住　　所　〒104-8011　朝日新聞社　個人情報係. 電　　話　03-5540-7310（お客様窓口）. 受付時間　毎週月～金曜（祝日、年末年始を除く）　午前10時から午後5時まで. この記事をSNSでシェアする. 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English ツイッターで共有. ツイッターで共有. list facebookで共有. facebookで共有. 本文を読む. 動画をリプレイ. 音声 ON. 動画をリプレイ. パナマ文書の衝撃から１年半あまり。新たな秘密が再び、世界で一斉に報じられた。「パラダイス文書」。世界６７カ国の記者たちが加わる国際調査報道ジャーナリスト連合（ＩＣＩＪ）はこの１年、文書をもとに取材を重ね、タックスヘイブン（租税回避地）に隠された事実に迫ってきた。. 舞台はバミューダ. 【動画】バミューダ諸島（Google Earth）. ニューヨークから飛行機で２時間。窓の下には青い海が広がっていた。北大西洋に浮かぶ英領バミューダ諸島。「ようこそパラダイスへ」。タクシー運転手は、こんな言葉で観光客を歓待する。. ここバミューダ諸島と米フロリダ半島の先端、大西洋のプエルトリコを結ぶ三角形の海域は、「バミューダトライアングル」と呼ばれる。飛行機や船が消える魔の伝説は日本でも有名だ。. 中心都市ハミルトンの港。多くの船が浮かぶ. １０月上旬。正午すぎの気温は２６度。中心都市ハミルトンの港には無数の白いクルーザーが浮かぶ。ピンク色の砂浜、薄いパステルカラーの家々、柔らかい風……。. 米ドルが広く流通し、米国人の避寒地として人気だ。ニューヨークやアトランタなど各地から１日１０便以上が降り立つ。日本人にとっての米領グアム島のような存在だろうか。. もうひとつの顔. 我々がこの島々を訪れたのには、理由がある。. １３８の島から成るこの地域の総面積は、東京都足立区とほぼ同じ５３平方キロメートル。だが、この小さな島々には別の顔がある。. 所得税や法人税、キャピタルゲイン（金融資産の値上がり益）への課税がない「タックスヘイブン」。多くの多国籍企業がここに拠点を構え、本国の利益を移転するなどして、その恩恵に浴しているとみられている。. 中心都市ハミルトンの海岸から徒歩数分の金融街。司法省や金融庁と並んでガラス張り４階建ての近代的な建物が立つ。ここが今回の取材のターゲット。パラダイス文書の主な流出元である法律事務所「アップルビー」だ。. 「アップルビー」受付. 今回、アップルビーから流出した文書は６８０万点に及ぶ。少なくとも２万５千の法人や組合に関する情報があり、契約書や銀行口座、Ｅメールなどが含まれる。. 膨大なデータは、パナマ文書の時と同じく、南ドイツ新聞の記者が入手した。入手経路や時期は極秘。ＩＣＩＪ内の記者も知らされていない。. データは何を語るのか。それはアップルビーが通常は決して明かすことのない「人」「企業」「カネ」のつながりだ。. 仕組みはこうだ。. 打ち合わせ. 流出した文書について報じるにあたって、当事者の取材は必須だ。直接取材の予定日は１０月１０日。我々はそれまでに、同事務所に９月から３度にわたって書面で取材を申し入れていた。だが具体的な回答は得られていなかった。. 直接取材は、世界各国のメディアによる異例の協働作業となった。現地入りしたメディアは７社、約２０人。日本の朝日新聞とＮＨＫ、調査報道で知られる米新興メディアＶＩＣＥ（バイス）、オーストラリアの公共放送ＡＢＣなどだ。. バミューダ入りした各国メディアの記者たちによる取材の打ち合わせ. 「当日は何時に行く？」. 「正午だと遅すぎるし、午前１０時は早すぎる」. 「地元警察を呼ばれないか？」. 「事務所をきょう見てきた。１階の受付は行っても大丈夫だろう」. 我々は前々夜から計３時間以上かけて綿密に打ち合わせした。最も避けたいのは取材拒否とともに、記者やカメラマンが当局に拘束されることだ。「ジェントル（穏やか）であろう」と、互いに声を掛け合った。取材ルールも各国で異なる。一般人に迷惑をかけたり、取材相手を刺激したりしないよう、どう担当者と接触するか、テレビカメラはどのタイミングで建物に入るかを詰めた。. いざ突入. １０月１０日午前１０時５３分。質問状送付などを担ってきたＩＣＩＪ記者のウィル・フィッツギボン（３１）と、一番のベテランである米テレビ局「ユニビジョン」上級編集者のデイビッド・アダムス（５６）がドアを開け、各社の記者とカメラクルーが続いた。. 「メディア担当の方につないでいただけますか？」。アダムスは受付カウンターの女性職員に丁重に依頼した。ＩＣＩＪの正式名称を告げると、受付職員は「インターナショナル……？」と聞き直した後、入り口近くのソファで待つよう我々に指示をした。. 対応したのはメディア担当ではなく、施設責任者の男性だった。ひざよりやや短い半ズボン「バミューダパンツ」をはいている。この島では正装だ。アダムスが、流出文書に関する取材であることや質問状を送付したことを説明した。. 施設責任者（右）の男性に取材依頼をするアダムス記者（左）ら. ３０分後、上階から戻ってきた男性の回答は、「今日は対応できる人がいない」というものだった。. わずか１分足らずの、事実上のゼロ回答。タックスヘイブンの実態を正面から暴くことの難しさを実感させるものだった。. いかなる疑惑にも反論する. 直撃取材から半月後、アップルビーはホームページ上に相次いで声明を掲載した。. 「我々が不正行為をしたという証拠は何もない。いかなる疑惑にも反論するし、当局の適切な調査には全面的に協力する」. 「我々は合法なビジネスのアドバイスを提供しており、違法行為は容認していない」. 「違法なハッキングで文書が流出したと考えられる」. 浮かんだ疑惑. パナマ文書には、ロシア・プーチン大統領の友人や中国・習近平国家主席の義兄の名前があり、世界を驚かせた。だがパラダイス文書も、それに勝るとも劣らぬ内容だ。. 税逃れへの取り締まりを公約に掲げるカナダ・トルドー首相の資金調達者が、巨額資金をタックスヘイブンに移し、課税を逃れていた疑いが浮上した。英国のエリザベス女王や、日本の鳩山由紀夫元首相の名前もあった。ＩＣＩＪによると、文書に登場する各国の政治家・君主らの名前は、４７カ国１２７人。. その中で、ＩＣＩＪが特に注目した人物がいる。エスタブリッシュメント（既得権）批判をして大統領選を勝ち抜いたトランプ米政権の、中核を担う大物閣僚だ。. 我々はこの大物閣僚の疑惑を追及する中心人物と会うため、バミューダ諸島から米東海岸に向かった。. （敬称略）. 第２部. 影を追う （ログインまたは会員登録が必要です）. 会員登録はこちら 2017年11月6日 公開. 取材・撮影：野上 英文、吉田 美智子、高野 遼. 企画・構成：入尾野 篤彦. 動画編集：戸田 拓. デザイン：寺島 隆介. 制作：朝日新聞メディアプロダクション. サイトポリシー サイトマップ 利用規約 web広告ガイド リンク 個人情報 著作権 お問い合わせ・ヘルプ 朝日新聞デジタルに掲載の記事・写真の無断転載を禁じます。すべての内容は日本の著作権法並びに国際条約により保護されています。. 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{"id":"8kBOwW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/bio\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":28,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["su","bio"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Brief Biography – Francis Su"],"last_modified":"Wed Jun 08 17:21:49 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":5547508358856096794,"exact_signature_unique_b":true,"fuzzy_signature_l":8248898479895188798,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https"],"icons_urlstub_sxt":["i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=180%2C180&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=192%2C192&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=32%2C32&#038;ssl=1"],"icons_rel_sxt":["apple-touch-icon","icon","icon"],"icons_sizes_sxt":["","192x192","32x32"],"h1_txt":["Francis Su"],"h2_txt":["Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College"],"h3_txt":["Brief Biography"],"h4_txt":["My New Book!"],"bold_txt":["Mathematics for Human Flourishing","Social Media","Email","Department Phone","Fax","Office Phone","Mailing Address","Contact Information","book webpage","two","review","Bookshop","Amazon","The New York Times","Wired","Quanta Magazine"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2021\/03\/su-lake-louise-tight-square.jpeg?resize=300%2C300&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2021\/03\/su-lake-louise-tight-square.jpeg?resize=300%2C300&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/Flourishing20Cover_edited-e1577727952493-201x300.jpg?resize=150%2C224&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/Flourishing20Cover_edited-e1577727952493-201x300.jpg?resize=150%2C224&#038;ssl=1"],"images_alt_sxt":["","","Book Cover: Mathematics for Human Flourishing","Book Cover: Mathematics for Human Flourishing"],"images_height_val":[300,300,224,224],"images_width_val":[300,300,150,150],"images_text_t":"i0 wp com math hmc edu su content uploads sites 10 2021 03 lake louise tight square jpeg resize 300 i 0 2019 12 Flourishing20Cover edited e1577727952493 201x300 jpg 150 224 Flourishing 20 Cover e 1577727952493 201x Book Cover: Mathematics for Human Flourishing","responsetime_i":541,"text_t":"Brief Biography – Francis Su. Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Brief Biography. Francis Su. Brief Biography. Francis Edward Su is the Benediktsson-Karwa Professor of Mathematics at Harvey Mudd College and a former president of the Mathematical Association of America. He received his Ph.D. from Harvard University. His research is in geometric combinatorics and applications to the social sciences and he has co-authored numerous papers with undergraduates. He also has a passion for teaching and popularizing mathematics. From the Mathematical Association of America, he received the 2013 Haimo Award for distinguished teaching and the 2018 Halmos-Ford Award for his writing. His work has been featured in. Quanta Magazine. ,. Wired. , and. The New York Times. . His book. Mathematics for Human Flourishing (Yale University Press, 2020) won the 2021 Euler Book Prize. It is an inclusive vision of what math is, who it’s for, and why anyone should learn it. On Twitter:. @mathyawp and Instagram:. @frandogger I’m also on. Facebook (you can friend me if you know me, or follow me if you don’t). Order. Mathematics for Human Flourishing on. Amazon . My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. 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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Math 131: Real Analysis I. Francis Su. Math 131: Real Analysis I. Spring 2020. Office Hours:. WED 8:30 – 9:30am and WED 2:30–3:30pm, or by appointment. Graders:. Hanna Hoffman, Deyana Marsh. Tutoring Hours:. TUE 8-9:30pm. This course is a rigorous analysis of the real numbers, as well as an introduction to. writing and communicating mathematics well . Topics will include: construction of the real numbers, fields, complex numbers, topology of the reals, metric spaces, careful treatment of sequences and series, functions of real numbers, continuity, compactness, connectedness, differentiation, and the mean value theorem, with an introduction to sequences of functions. It is the first course in the analysis sequence, which continues in Real Analysis II. Goals of the course. Learn the content and techniques of real analysis, so that you can creatively solve problems you have never seen before. Learn to read and write rigorous proofs, so that you can convincingly defend your reasoning. Learn good mathematical writing skills and style, so that you can communicate your ideas effectively. This class is about the exciting challenge of wrestling with big ideas. I believe everyone in the class is fully capable of mastering this material. Questions are valued, especially simple ones because they can lead to profound ideas. Exploration is encouraged, especially risk-taking in trying out things that may not work, because they can lead to further areas of exploration. I expect all of us to be welcoming of the questions and explorations of others. Required Text. Walter Rudin,. Principles of Mathematical Analysis. , McGraw-Hill. We will cover Chapters 1 through 5, and part of Chapter 7. There are also many other books on analysis that you may wish to consult in the library, around the QA300 area. Homeworks, and Re-Writes. Due at my office (Shan 3416) by. 1:15 pm on Thursdays. . Please follow the. HMC Mathematics Department format for homework . Because I want you to learn from the feedback you get on your homework, as well as improve your writing skills, I will use a system of (optional) re-writes for the first few assignments, which will work as follows:. Turn in the homework on the due date. The homework will be graded and returned to you within one week. If you are not satisfied with the grade you received on the homework, you have the option of re-doing any question(s) you wish, and submitting the re-written version together with the previously graded version. (You may only re-write a question if you made a serious attempt at it on the first version.). If you choose to do a re-write, it is due at my office. two weeks. after the original due date of the assignment. Your re-write will be graded, with particular attention to whether you adopted the graders’ suggestions, and new grades will be assigned for rewritten questions. Your grade for a rewritten question will always go up or stay the same; it will never go down. Rewrites will only be accepted for Homeworks 1 through 3. See also. this guide . Rewrites should be handed in in parts. Late Homeworks. Late Homework can be accepted (with penalty) by special permission. Please ask at least 24 hours in advance. LaTeX. Some of you may find LaTeX helpful in typesetting your homework. If you’d like to learn LaTeX, or have questions about it, you can visit the. CCMS Software Lab . Midterms and Grading. There will be three exams:. Midterm 1: Take-home available Feb 24 in class, due Fri Feb 28 at 5pm. Midterm 2: Take-home available Mar 30 in class, due Apr 3. [COVID CHANGE]. Final: Take-home due during the regularly scheduled final period. Each of these and your homework average will count 25% of your course grade, with [COVID CHANGE] the highest component worth an additional 25%. The lowest two homework grades will be dropped. It is helpful to remember that course grades are just intended to assess what you have learned. But they are a not a reflection of your potential ability or your dignity! Honor Code. The HMC Honor Code applies in all matters of conduct concerning this course. Though. cooperation on homework assignments is encouraged. , you are expected to. write up all your solutions individually. . Thus copying is prohibited, and you should understand your solutions well enough to write them up yourself. It is appropriate to acknowledge the assistance of others; if you work with others on a homework question, please write their names in the margin. Part of the fun of this course is the struggle, as well as the joy of discovering a solution for yourself. Please note: using solutions found online or solutions of previous students will be regarded as a violation of the HMC Honor Code and will be handled accordingly. I encourage you instead to talk to me or the tutors or each other! Taped Youtube Lectures. These lectures were taped in 2010, and although the lectures I give this year may not be identical, they will be close enough that you may find it valuable to use them for review. Or, better yet, watch them before the class lecture, and then during class you can ask questions! I do not encourage using these lectures as a substitute for class, however, since we will be doing slightly different things and interactions with me and other students will be critical for your learning. Real Analysis Lectures, Spring 2010. Homeworks. All HW’s refer Rudin’s. Principles of Mathematical Analysis. HW #0 – Due Jan. 23. Read this. article by Carol Dweck and my. handout on good mathematical writing . Fill out. this survey. Then turn in brief answers to these questions. Keep in mind the handout and. the homework format as you write up your answers. From the article by Carol Dweck:. 1. When faced with a mathematical challenge, describe 3 ways a person with a growth mindset would respond differently than a person with a fixed mindset. Directly from the handout on good writing:. 2. What is a good rule of thumb for what you should assume of your audience as you write your homework sets? 3. Do you see why the proof by contradiction on page 3 is not really a proof by contradiction? 4. Name 3 things a lazy writer would do that a good writer wouldn’t. 5. What’s the difference in meaning between these three phrases? Let \\(A = 12.\\). So \\(A = 12.\\). \\(A = 12.\\). Now examine Section 1.1 of Rudin, showing that there is no rational \\(p\\) that satisfied \\(p^{2}=2\\). 6. There are many places in his proof where he could have used symbols to express his ideas, but he does not. (e.g., “Let \\(A\\) be the set of all positive rationals \\(p\\) such that…”) Why do you think he chooses not to use symbols? What would you change about his presentation if you were writing for a high school audience? Give a specific example. Your homework should be handed in two parts. Part A should have problems 1 to 3. Part B should have problems 4 to 6. HW #1 – Due Jan 30. Problem B. Recall that in class, we defined a rational number \\(\\frac{m}{n}\\) to be an equivalence class of pairs \\((m,n)\\) under an equivalence relation. Check that this equivalence relation is transitive: if \\((p,q)\\equiv(m,n)\\) and \\((m,n)\\equiv(a,b)\\), then \\((p,q)\\equiv(a,b)\\). Problem C. We defined addition of rational numbers in terms of representatives: \\(\\frac{a}{b} + \\frac{c}{d} = \\frac{ad+bc}{bd}\\). Show that the addition of rational numbers is well-defined. Problem D. Define a multiplication of rational numbers (corresponding to the one you are used to), and show this multiplication is well-defined. For problems B, C, D, you may assume you know properties of integers, including the cancellation law of Z. You cannot assume that you know how to divide integers (because that is really multiplication by a rational), nor how to add or multiply rationals (because that is what you are trying to show)! Do also Chapter 1 (1, 2, 3a). For these problems in Rudin, you may assume that you know familiar properties of rational numbers that we were proving in Lecture 2. Your homework should be handed in three parts. Part A should have problems B and C and D. Part B should have problems 1.1, 1.2 and 1.3a. Follow. this homework format as well as. the guidelines for good mathematical writing . HW #2 – Due Feb 6. A problem marked “R” means read, but do not do the problem. Chapter 1 (R3bcd, 4, 5, R7, 8, 9, 20) and. Problem S. . For a real number \\(a\\) and non-empty subset of reals \\(B\\), define: \\(a+B= \\{a+b:b \\in B\\} \\). Show that if \\(B\\) is bounded above, then \\(sup(a + B) = a + sup(B)\\). [For 1.20, whenever the proof is EXACTLY THE SAME as in Steps 3 and 4 of pp. 18–19, you do not need to re-write the proof. Just point out that the proof is the same as in Rudin’s. But wherever the proof differs, BE SURE to POINT OUT HOW IT DIFFERS, and VERIFY all new things.]. There will be no rewrites for HW #0. Your homework should be handed in two parts. Part A should have problems 4, 5, and 8. Part B should have problems 9, 20, and Problem S. HW #3 – Due Feb 13. Do the problems on. this handout . If you would like the LaTeX code,. it is here . Your homework should be handed in three parts. Part A consists of the first 3 problems in the order above; Part B the second 3 problems. Rewrites for HW #1 are also due. If you want to do rewrites, see. this guide . Rewrites should be handed in in parts. HW #4 – Due Feb 20. A problem marked “R” means read, but do not do the problem. There are no rewrites on this homework and forward. Do Chapter 2 (2, 3, 4, Q, T, R6, R7, R8, 11 [exclude the 2nd metric d_2]). Problem Q. Cantor’s diagonal argument shows that the real numbers are uncountable by showing that any proposed “list” of real numbers must not contain every real number. We know rationals numbers ARE countable, but why can’t Cantor’s diagonal argument be applied to show that every proposed “list” of rationals between \\([0,1]\\) can’t contain every rational number between \\([0,1]\\)? What would go wrong with this line of reasoning? Problem T. Prove that the Principle of Induction implies the Well-Ordering Principle for \\(N\\) (the natural numbers). [Hint: Let \\(L(n)\\) be the statement “if \\(A\\) is a subset of \\(N\\) that contains a number \\(. Now prove \\(L(n)\\) holds for all \\(n\\) by induction.]. In the homework, you may assume these things:. Every real number can be represented in a decimal expansion. Every rational number has a decimal expansion that eventually repeats (or terminates, which is really repeating zeroes). Every polynomial of degree \\(n\\) has \\(n\\) roots in the complex numbers. Your homework should be handed in two parts. Part A consists of 2.2 and 2.3 and 2.4, Part B consists of Problems Q and T and 2.11. Rewrites for HW #2 are also due. If you want to do rewrites, see. this guide . Save a copy since you probably won’t get the homework back in time to study for the exam. Exam #1 info. Take-home available Mon Feb 24, due Fri Feb 28 by 5pm (if you require extension to Saturday, no problem). Rewrites for HW #3 are due a week after you get it back. Rewrites should be handed in in parts. No more rewrites for HW’s after HW#3. HW #5 – Due Mar 5. A problem marked “R” means read, but do not do the problem. Do Chapter 2 ( 5, 8, 12, 9ab, 9cd, 9ef, R10, R14 ). In Problem 14, give an example of a cover that is. not. a. nested. collection of open sets. Your homework should be handed in two parts. Part A is 2.5, 2.8, 2.12. Part B is 2.9ab, 2.9cd, 2.9ef (it’s all one problem, but please group ab, bc, cd since they will be worth 3 problems). Return any outstanding rewrites for HWs 1- 3. There will be no more rewrites for HW’s after HW#3. HW #6 – Due Mar 12. A problem marked “R” means read, but do not do the problem. (I’ve split up some problems into pieces to indicate which pieces are worth a similar amount of points, and the graders will grade these parts separately.). Do:. Part A. Chapter 2 ( R15, 16, 17[countable?, dense?], 17[compact?, perfect?]). Part B. Chapter 2 (18, R19, 22, 24, R25). (R25 depends on R23 from last week’s reading.). On problem 2.17, notice a decimal expansion with only 4’s and 7’s cannot contain any 0’s. So there are no terminating decimal expansions. (So, for instance, the set does not contain the number .44 because that’s really .44000…). On problem 2.18 recall that rational numbers have decimal expansions that either terminate or eventually repeat. On problem 2.24 you should not appeal to compactness because this problem is one step of the proof that “every subset of \\(X\\) has a limit point” implies “\\(X\\) is compact”. Instead, follow the hint. I encourage you to discuss these problems with others in the class! === COURSE CHANGES DUE TO COVID-19 ===. With the COVID-19 crisis and the disruption that it has caused and will continue to cause, here are a few changes I’m making to this course:. Principles. A higher priority than this course is your health and the health of your loved ones. We must have flexibility while planning ahead for possible further disruption, including obstacles you may have to online access or focused study while at home, and accommodations in case you (or I) get sick from COVID-19. Focusing on the ideas of this course can be an important way for you to feel normal in working towards the future, in a time of great upheaval and uncertainty. Following from these principles, here are changes to our original plan. You will have the option to take this course Pass\/No Credit. If you are an HMC student, a Pass will count towards your degree requirement (even if it would not normally do so). If you are not an HMC student, you should check with your registrar to see what your school policy is for Pass\/No Credit. You can exercise this option at any time just by sending me an e-mail—you do not have to petition the registrar. There will NOT be a second midterm. , due to the disruption and loss of a week of instruction. Instead your final grade will be based on the first midterm and final exam and homework average, each worth 25% of your grade, and your highest component of these three will be worth an additional 25% of your grade (to total to 100%). For the homework average, the lowest. two. homework scores will be dropped. Lectures. will be held as usual during the normally scheduled time at 1:15am Pacific. ,. but recordings will be available to watch afterwards. We will figure this out as we go and may have to improvise. HW – Do by Apr 2 [optional, not handed in]. A problem marked “R” means read, but do not do the problem. Think about Chapter 2 ( 19 and 20 ) first, then discuss with a group of 3-5 others in the Course Zoom Room. Note on 2.20: it’s asking “is the closure of a connected set also a connected set?” and “is the interior of a connected set also a connected set?”. Show up in the Su Gathering Space at. WEDNESDAY 6-9pm Pacific time, or self-assemble. at a different time. (Remember TUESDAYS are reserved for the other class.). Try out the Whiteboard feature in Zoom, or move to goboard.com. See Sakai\/email for links to Su Gathering Space. HW #7 – Due Apr 9. A problem marked “R” means read, but do not do the problem. Do Chapter 2 ( 20 ) Chapter 3 ( 1, 3, 16 ). Note on 3.1: you should assume the sequence is in \\(\\mathbb{R}\\) or \\(\\mathbb{C}\\). Hint on 3.3: can you show the sequence is increasing? Induction may be of help here. NOTE: There’s no need to split into parts A and B anymore, since you’re submitting online through GradeScope. Access this through Sakai, on the left tab. You will need to upload by scanning your homework, here is some guidance on how to do that:. https:\/\/gradescope-static-assets.s3-us-west-2.amazonaws.com\/help\/submitting_hw_guide.pdf . Follow these Gradescope instructions to tag which portion of your homework is associated with the four problems:. https:\/\/youtu.be\/KMPoby5g_nE HW #8 – Due Apr 16. A problem marked “R” means read, but do not do the problem. Do Chapter 3 ( 20, 23, 24a, 24b, R24cde, R25 ). Note on 3.23: you are being asked to show a particular sequence converges. Note that the sequence is a sequence of distances, so they are in \\(\\mathbb{R}\\). HW #9 – Due Apr 23. A problem marked “R” means read, but do not do the problem. DO. Chapter 4 ( 1, R2, 3, 4, R6, 14 ). On the reading problem 4.6, you may assume that \\(E\\) is a subset of real numbers and \\(f\\) is a real-valued function, and the distance in \\(R^{2}\\)(where the graph lives) is the usual Euclidean metric. HW #10 – Due Apr 30. 4 (8, R10, 11[see note below], 12, 18 ). In 4.11. you do not have to do the 2nd part of the problem where it ways “Use this result to give…”. FINAL EXAMS will be available in the Sakai Resources Folder by Monday April 4, in the afternoon. === Below this line: all homeworks are tentative. —THESE HAVE NOT BEEN UPDATED FOR NEW SCHEDULE YET. ===. Below this line, all homeworks are. TENTATIVE. . This means they are likely to be assigned, but there is no guarantee that they will until you see them moved above. My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Welcome! Math Fun Facts are ideas and puzzles that will change the way you think! This archive is designed as a resource for enriching your math courses and nurturing your interest in mathematics! Each Math Fun Fact is a math. puzzle or short article that contains a cool mathematics idea—a “fun fact”. Thanks to the Mellon Foundation and Harvey Mudd College for generously supporting the development of this site! For more fun, tour the. Mathematics Department at. Harvey Mudd College . Tags:. puzzle Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved. Loading Comments... Write a Comment... Email (Required). Name (Required). 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{"id":"ORF4mW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/courses\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":32,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["su","courses"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Teaching – Francis Su"],"last_modified":"Wed Jun 08 17:21:50 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-8876022004431065357,"exact_signature_unique_b":true,"fuzzy_signature_l":-6771982075693985395,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https"],"icons_urlstub_sxt":["i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=180%2C180&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=192%2C192&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=32%2C32&#038;ssl=1"],"icons_rel_sxt":["apple-touch-icon","icon","icon"],"icons_sizes_sxt":["","192x192","32x32"],"h1_txt":["Francis Su"],"h2_txt":["Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College"],"h3_txt":["Teaching"],"h4_txt":["My New Book!"],"h5_txt":["Currently Teaching                  ","Course Pages:","Other Courses I’ve Taught [these links may no longer work]"],"bold_txt":["Social Media","Email","Department Phone","Fax","Office Phone","Mailing Address","Contact Information","book webpage","two","review","Bookshop","Amazon"],"imagescount_i":2,"images_protocol_sxt":["000-https","001-https"],"images_urlstub_sxt":["i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/Flourishing20Cover_edited-e1577727952493-201x300.jpg?resize=150%2C224&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/Flourishing20Cover_edited-e1577727952493-201x300.jpg?resize=150%2C224&#038;ssl=1"],"images_alt_sxt":["Book Cover: Mathematics for Human Flourishing","Book Cover: Mathematics for Human Flourishing"],"images_height_val":[224,224],"images_width_val":[150,150],"images_text_t":"i0 wp com math hmc edu su content uploads sites 10 2019 12 Flourishing20Cover edited e1577727952493 201x300 jpg resize 150 224 i 0 Flourishing 20 Cover e 1577727952493 201x 300 Book Cover: Mathematics for Human Flourishing","responsetime_i":471,"text_t":"Teaching – Francis Su. Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Teaching. Francis Su. Teaching. Currently Teaching                  . Calculus (Math 19). Introduction to Academic Writing (Writ 1). Course Pages:. Real Analysis (Math 131). Galois Theory (Math 172). Other Courses I’ve Taught [these links may no longer work]. The Putnam Seminar Multivariable Calculus, part I  (Math 14a). Multivariable Calculus, part II  (Math 61b). Probability and Statistics  (Math 62). Linear Algebra  . Real Analysis II  (Math 132) Topology  (Math 147). Social Choice and Decision Making  (Math 188). Algebraic Topology  (Math 189). Geometric Combinatorics and Polytopes  (Math 189). Game Theory (SPE315, for the School of Politics and Economics at the Claremont Graduate University). Calculus of One Real or Complex Variable (. Math 11 , fall). Multivariable Calculus  (Math 5, Fall 1999). Probability  (Math 151). Probability  (Math 157). Game Theory (ORIE435, at Cornell, Spring 2000). My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. 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{"id":"-_kuPW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/hymn\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":29,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["su","hymn"],"content_type":["text\/html"],"crawldepth_i":1,"collection_sxt":["user"],"title":["Light of Life – Francis Su"],"last_modified":"Wed Jun 08 17:21:52 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":5979835605874764932,"exact_signature_unique_b":true,"fuzzy_signature_l":-4250604723792091811,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https","002-https"],"icons_urlstub_sxt":["i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=180%2C180&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=192%2C192&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/cropped-Flourishing20Cover_edited-e1577727952493.jpg?fit=32%2C32&#038;ssl=1"],"icons_rel_sxt":["apple-touch-icon","icon","icon"],"icons_sizes_sxt":["","192x192","32x32"],"h1_txt":["Francis Su"],"h2_txt":["Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College"],"h3_txt":["Light of Life"],"h4_txt":["My New Book!"],"bold_txt":["Social Media","Email","Department Phone","Fax","Office Phone","Mailing Address","Contact Information","book webpage","two","review","Bookshop","Amazon","sung to the tune of “Come Thou Fount of Every Blessing” (Traditional American Melody; 1813)","I wrote the following hymn for my church (Cambridgeport Church, Cambridge, MA) as part of an Advent devotional in the Christmas season of 1995."],"imagescount_i":2,"images_protocol_sxt":["000-https","001-https"],"images_urlstub_sxt":["i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/Flourishing20Cover_edited-e1577727952493-201x300.jpg?resize=150%2C224&#038;ssl=1","i0.wp.com\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/12\/Flourishing20Cover_edited-e1577727952493-201x300.jpg?resize=150%2C224&#038;ssl=1"],"images_alt_sxt":["Book Cover: Mathematics for Human Flourishing","Book Cover: Mathematics for Human Flourishing"],"images_height_val":[224,224],"images_width_val":[150,150],"images_text_t":"i0 wp com math hmc edu su content uploads sites 10 2019 12 Flourishing20Cover edited e1577727952493 201x300 jpg resize 150 224 i 0 Flourishing 20 Cover e 1577727952493 201x 300 Book Cover: Mathematics for Human Flourishing","responsetime_i":878,"text_t":"Light of Life – Francis Su. Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Light of Life. Francis Su. Light of Life. I wrote the following hymn for my church (Cambridgeport Church, Cambridge, MA) as part of an Advent devotional in the Christmas season of 1995. Light of Life (based on John 1:1-18). sung to the tune of “Come Thou Fount of Every Blessing”. (Traditional American Melody; 1813). There’s a light for all the weary, heavy-laden souls of care. and it shines down midst the darkness, can you see it blazing there? Star of Bethle’em, star from Jacob, bright and radiant morning Star! Cast Your rays in my direction, lead me on the journey far. Though my heart feels dim and glacial, when I’m in my hour of prayer. I can feel Your fire upon it, I can see Your luster there. And Your light shines in my darkness, and the darkness cannot hide! Burn away my icy pallor, sear away my stubborn pride. In our daily toil of living with the forces of the night. the Almighty shines a beacon in the darkness of the fight—. there to cause us to gaze heavenward, there to lift our downcast eyes! from the earthly sinful struggle to the upward call of Christ. Oh, how glorious th’ living Word is, and the flesh that He became. to redeem each suff’ring soul through His own blood and sacred pain! Blessed Jesus, I am pleading— hold my bleeding heart aright! for I am needy for a Saviour, and I am blind without a Light. — by Francis Edward Su. ((Copyright December 1995 by Francis Edward Su. May be. freely distributed electronically in whole or in part, but please. keep this notice attached and do not alter the text.)). My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. 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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Topology Through Inquiry. Francis Su. Topology Through Inquiry. Topology Through Inquiry. is a comprehensive introduction to point-set, algebraic, and geometric topology, designed to support inquiry-based learning (IBL) courses for upper-division undergraduate or beginning graduate students. Order. From AMS\/MAA books:. https:\/\/bookstore.ams.org\/text-58\/ or on Amazon:. https:\/\/amzn.to\/34lbNSi Instructor’s Resource. For those teaching an Inquiry Based Learning (IBL) course, we have an. Instructor’s Resource that you may find helpful. In it, we describe the educational philosophy of Inquiry Based Learning, its role in the education of students, practical issues, and tips for how to get started. You should also avail yourself of the many resources, workshops, and conferences for IBL teaching at. The Academy of Inquiry Based Learning . LaTeX files. Students may wish to create their own book of solutions. If so, you may find these. LaTeX files of all the theorems in the book handy. (The main tex file in this folder is topology-thms.tex.). Some of you doing IBL during the pandemic are using Overleaf to do class discussion and presentations. In that case, these. LaTeX files of definitions may be handy. Sample Syllabi. The Preface to. Topology Through Inquiry. contains an outline of topics suitable for first-semester or second-semester courses. A typical first-semester course covers point-set topology with the possible inclusion of the fundamental group or the classification of surfaces. A typical second-semester course in geometric and algebraic topology would cover simplicial and\/or singular homology and other selected topics. Below are some sample syllabi from those who have used this text in their courses. Please consider submitting your own syllabus for inclusion here. Write me at “[mylastname] at-sign math.hmc.edu” and include this information:. 1. Institution. 2. What kind of course you teach [topic, semester or quarter, etc]. 3. Who the audience is [general background, prerequisites]. 4. A PDF of your syllabus and assignment schedule. 5. Link to your public webpage for the course, if you have one. Topology, Harvey Mudd College. Topology (Math 147) Harvey Mudd College, Spring 2019. Point-set topology with fundamental groups. Pre-requisite: one semester in real analysis. Prerequisites: Math 131 (analysis). Math 171 (algebra) is recommended as a co-requisite. INQUIRY BASED LEARNING:. this course will be taught in an IBL (inquiry-based learning) format. You will receive handouts containing theorems, definitions, examples. Your goal is to prove all the theorems by yourselves in a guided discovery process: with collaboration of classmates and guidance from me. You will take turns presenting proofs of theorems in class, while other students will determine if it is correct. I will provide perspective on the material, and motivating examples if you get stuck. TEXT:. The text for the course is the draft of my book with Starbird: “Topology through Inquiry”. Please do NOT distribute this book in any form to anyone without my permission. NO OUTSIDE SOURCES:. As is customary in any IBL course, you are forbidden from consulting any outside sources, including textbooks or the internet, to solve these problems. [The exceptions are sources that you are required to use for other courses.] The fun of the course is in struggling with the ideas and enjoying challenging problems that will stretch your mind. NOTEBOOK:. You should acquire a loose-leaf binder, in which you will save all course notes. As you prove theorems in this course (or see them proved in class), you will write up these proofs and add them to this notebook. In a sense, you are writing your own book on the subject, filled with your own proofs. As part of your grade for the course, these notebooks will be evaluated. More importantly, a nice notebook will be something you can be very proud of years from now, as you look back on your experience. HOMEWORK:. You will be asked to write up selected theorems to be handed in weekly. Thus it is important to pay attention to proofs of theorems presented in class, since you will write these up for credit. EXAMS:. There will be no midterm, but there will be a final examination. GRADES:. I’d rather you not worry about grades. I hope you will concentrate on learning. But since we must, grades will be determined in the following ways. 70% of your grade will be specifications-based. This means that you must meet each of the following specifications to pass the course:. Present in class 15 times over the course of the semester, including 5 productive failures. Ask 15 questions in class over the course of the semester. Contribute 15 proofs in the Common Solution Set [posted in the Resources tab]. Edit 15 proofs in the Common Solution Set (to improve the solutions of others). Keep a well-organized notebook, which will be turned in 2\/3 of way through the semester, these will be graded exceptional pass\/pass\/no pass. Homeworks, and Final Exam, will count the remaining 30% of your grade. HONOR CODE:. Cooperation is encouraged, but solutions should be written up individually. You may not consult outside mathematical sources without my permission, unless required for some other course. Expect this course to be challenging, but also very rewarding. The value of the IBL format is that when you prove theorems by yourself, you will never forget the proofs you came up with, and you will gain confidence in your abilities as mathematicians! WHAT SHOULD BE IN YOUR NOTEBOOK:. – Summary Sheet and Evaluation that tracks your progress. – The Reflection Exercise. – Proofs or sketches for Theorems and Exercises. – Any other notes you’re taking for the course. – Any returned homeworks. How you organize the notebook is up to you. The proofs in your notebook may be handwritten or electronic, either is fine. When it comes time to turn in your notebook, you may turn it in electronically if you wish. The rule of thumb is that this notebook should something you look back on with pride 10 years from now. So it should contain enough details for you to reconstruct your thoughts. I encourage you to write up theorems carefully: see my Guidelines for Good Mathematical Writing. SUMMARY SHEET and EVALUATION. Your Summary sheet should list:. 15 proofs you presented in class (including 5 productive failures) — list theorem numbers only. 15 questions you asked in class (list theorem numbers). 15 proofs you contributed you’ve made to the Common Solution Set: you’ll record them in that document but list theorem here too. 15 problems you’ve edited in the Common Solution Set: you’ll record them in that document but list theorem here too. An evaluation of your notebook according to the following rubric. Both you and I will evaluate your notebook. Use the following rubric, and include on your Summary sheet an explanation of the grade you would assign yourself with this rubric. Exceptional Pass. = every theorem and exercise discussed in class has complete correct proofs or proof sketches. There are many more proofs completed even though they were not assigned for class discussion or homework. Proofs and proof sketches are well-written. Pictures are abundant. Notation is well-chosen. Subtle points in proofs are acknowledged. Reflection Exercise included. Indicate why your Notebook should receive an Exceptional Pass. Pass. = Most theorems discussed in class have proofs or proof sketches. There are a good number of theorems completed that were not assigned for homework. Pictures are ample. If you looked at it 10 years from now, you could reconstruct most of the arguments from what you have written. Reflection Exercise included. Indicate why your Notebook should receive a Pass. Not Yet Passing. = Indicate what deficiencies you have, and what you will do to Pass. I encourage you also to seek feedback from classmates about how they would evaluate your notebook according to this rubric. REFLECTION EXERCISE. Reflect on the following question:. What have you learned in this class about the process of doing or creating mathematics? Weave in potential answers to the following:. have you experienced: joy, beauty, reward? what have you learned about the value of struggle? what have you learned about the importance of community? SUGGESTIONS FOR PROOFS. There will be many times when you are stuck on a problem. This is where the real learning occurs. Here are several ideas:. Draw pictures! For instance, unions, intersections, complements, etc. are often effectively represented by circles. Make certain you thoroughly understand the definitions involved, and work out examples! Work a special case if you cannot solve the whole problem. Is every hypothesis necessary? Construct examples to show why the theorem fails if a hypothesis is missing. This will often show you what is needed for the proof. Ask yourself if you can modify the statement to obtain a new, related theorem. When you have a proof, ask what other statements can be proved with the same proof? One of the goals of the course is to help you see mathematics more as a collection of techniques, examples, and proofs than as a collection of theorem statements. SUGGESTIONS FOR PRESENTATIONS. Sketch. Begin by giving a brief outline of the argument, before giving details. The outline should consist of a sequence of complete, true statements, whose proofs can be explained when you give the details. Be prepared to justify the details if asked. Speak loudly. Meet with others in the class, and practice presenting your proofs to each other! Knowing a proof and presenting it are two very different things! HOMEWORKS. Rule of Thumb. Stay 12 theorems ahead of where we end the previous time. For Mon 1\/28. Read the Introduction, skim Chapter 1, Read Chapter 2 introduction and Section 2.1. Due Wed 1\/30. HW#1. Hand in Chapter 2 ( 3, 5 [finite complement only], 8, 13, 14 ). For Mon 2\/4. Begin reading Chapter 3. Work up through the end of Chapter 2, especially focusing on theorems. For Wed 2\/6. HW#2. Hand in Chapter 2 ( 20, 23, 26, 30 ) and Chapter 3 ( 3 ). For Mon 2\/11. In Section 3.1, work on 9, 10. Skim Section 3.2 (subbases), work on it if you want, but we won’t spend much time on it. In Section 3.3, only work problems related to the lexicographically ordered square (20, 21). In Section 3.4, work through at least 25 – 27. For Wed 2\/13. Continue in Section 3.4 through 37, reading the introduction to product spaces. HW#3. Hand in Chapter 3 ( 8, 9, 10 (part 4), 21 (example D only), 28 ). For Mon 2\/18. Continue in Section 3.4, work 38 – 42 (think about at least 3 of them) and in Chapter 4, work 1 to 7. For Wed 2\/20. Continue in Chapter 4, work through 17, but 12 and 13 are optional. HW#4 Hand in Chapter 3 ( 31(F only), 41 ) Chapter 4 ( 1, 5, 9 ). For Mon 2\/25. Continue in Chapter 4: complete all of Section 4.1 and 4.2. Read all of Section 4.3, but focus attention on 19, 20, 23. In Section 4.4, focus attention on 29, 30, 31, 32. Read 4.5. For Wed 2\/27. Complete 4.4 and focus on 29, 30, 31, 32. Read Chapter 5 intro, and Section 5.1, and complete all exercises and theorems except 5.7 and 5.8. HW#5 Hand in Chapter 4 ( 6[part 4 only], 11, 15[all but normal], 17, 23 ). For Mon 3\/4. Work up through 5.15 (skip 7, 8, 12, 13 ). Read the rest of Chapter 5. For Wed 3\/6. In Section 5.3 only do 15 and 18. Then work in Chapter 6 through 6.9 ( 1 through 3 should be a review from analysis). HW #6 Hand in Chapter 4 ( 29, 32 ) Chapter 5 ( 6, 11, 18 ). For Mon 3\/11. Work all remaining Theorems in Section 6.1 and do all Theorems in Section 6.3 (Exercises optional.) Section 6.2 should be mostly review from Analysis, so read it, but we’ll mostly skip it. We’ll skip Sections 6.4 and 6.5. Read Section 6.6. For Wed 3\/13. Start working Chapter 7, in particular all theorems in Section 7.1 (except 8, 14). Many are straightforward to dispense with, so we’ll fly through some of them. HW#7 Hand in Chapter 6 ( 5, 9, 12, 18 ) Chapter 7 ( 7 ). For Mon 3\/25. Work these problems in Section 7.2: 15, 18, 20, 21, 22, 24, and all Theorems in Section 7.3. (You’re avoiding theorems involving Lindelöf, countably compact, 2nd countable). For Wed 3\/27. Work all Theorems in Section 7.4, and Exercise 7.41. No HW to hand in, but please be working on your Notebooks and contributing to the Online Solution Set. For Mon 4\/1. Read all of Section 7.5 and read all theorems and exercises, thinking about each one, but focus on these problems: 45 (especially part 3), 46, 48, 53, 54, 55. Read Section 7.6, 7.7 but do not do problems. Read Chapter 8 intro and Section 8.1, do 1, 2, 3, 4, 5. For Wed 4\/3. In Section 8.1, do: 6, 8, 12, 13. Read Section 8.2 (don’t do any problems). In Section 8.3 do: 18, skim rest of section. HW#8 Hand in Chapter 7 ( 21, 24, 41, 42 ) Chapter 8 ( 4 ). For Mon 4\/8. In Section 8.4 do: 35, 36, skim the rest of chapter. Skim Sections 9.1 and 9.2, especially Theorem 9.11 and 9.14. Read Section 9.3, and do the 2 theorems in that section: 24, 25. Read Section 12.1, do: 2, 3, 4, 5. For Wed 4\/10. Do Section 12.1, do: 4-14. The homework is turning in notebooks. See the Tab above on ‘Notebooks’ for the Rubric and complete the instructions found, there including the Reflection Exercise. (I’ve indicated in blue small changes to the instructions on that page.) This is a preliminary notebook check. If you get a High Pass or Pass, you will complete your notebook requirement for the semester. If you get a Not Yet Passing (or get a Pass but want a High Pass), you can turn your notebook in the final Monday of the course. For Mon 4\/15. Do 12.1 ( 11, 12, 14, 15 ) 12.2 ( 18-20, skip 21 ) 12.3 ( 22-24 ). For Wed 4\/17. Make sense of the definition of a strong deformation retract. Do 12.3 ( 25-34 ). HW#9 Hand in Chapter 12 ( 2, 3, 7, 18, 19 ). (No rewrites on this assignment.). For Mon 4\/22. Do 12.3 ( 20, 22, 24, 26, 29, 32, 33, 34, 36 ) and be ready to present. For Wed 4\/26. Work on 12.4 ( 40 – 48 ) on Van Kampen’s theorem. HW #10 Hand in Chapter 12 ( 14 [worth double], 27, 30, 40 ) (No rewrites on this). Discuss problems with each other! (In particular, if you missed class on Wed 4\/17 when we discussed problem 14, see me or others to chat about it.). For Mon 4\/29. Work on 12.4 ( 41, 42, 44 [think about why?], 45, 46, 47 ). Also, pick a few spaces of your own, and compute the fundamental group of those spaces using van Kampen’s Theorem in a couple of different ways. This should give everyone a chance to present something if you want to. Also, we will give people some time to present a question that you are now wondering about, based what you’ve seen so far of topology. You might read the last few sections 12.5 and 12.6 and 12.7 for ideas. For Wed 5\/1. Work on 12.4 ( 49, 50, 51, 52, 53 ). Also, we will give people some time to present a question that you are now wondering about, based what you’ve seen so far of topology. HW for Wednesday is to prepare your Notebooks. The criteria are the same (see Tab above on ‘Notebooks’), but update your Reflections, and hand those in separately, since I’d like to keep them! Algebraic Topology, Harvey Mudd College. Fall 2016. Simplicial and singular homology. Pre-requisite: first-semester topology course. This course used a DRAFT version of Topology Through Inquiry, so chapter and theorem numbers may not align with the published text. But you will be able to see the pacing. Course website . My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Research. Francis Su. Research. You can probably tell that my research interests have varied quite a bit. My Ph.D. was a mix of representation theory and probability used to analyze random walks on algebraic structures. More recently, I’ve been fascinated by mathematical questions arising from problems in the social sciences. I’ve been carving out a niche solving problems in geometric and topological combinatorics (e.g., triangulations of polytopes and fixed point theorems) and using them to study problems of fair division in mathematical economics, and voting problems in game theory. Be aware that preprints below may differ slightly from the published versions. Many of my papers have been jointly authored with undergraduates. Cell colors group related papers. * = undergraduate co-authors. 1. Francis Edward Su. Methods for Quantifying Rates of Convergence for Random Walks on Groups. Ph.D. Thesis, Harvard University. Advisor: Persi Diaconis. 2. Francis Edward Su. Borsuk-Ulam implies Brouwer: a direct construction. Amer. Math. Monthly,. 104(9):855–859, 1997. [PDF] 3. Francis Edward Su. Convergence of random walks on the circle generated by an irrational rotation. Trans. Amer. Math. Soc.,. 350(9):3717–3741, 1998. [PDF] 4. Francis Edward Su. Rental harmony: Sperner’s lemma in fair division. Amer. Math. Monthly,. 106(10):930–942, 1999. [PDF] This article was awarded the MAA’s. 2001 Merten M. Hasse Prize for mathematical exposition. 5. Arthur T. Benjamin, Jennifer J. Quinn, and Francis Edward Su. Phased tilings and generalized Fibonacci identities. Fibonacci Quart.,. 38(3):282–288, 2000. [PDF] 6. Arthur T. Benjamin, Francis Edward Su, and Jennifer J. Quinn. Counting on Continued Fractions. Math. Mag.,. 73(2):98–104, 2000. [PDF] 7. Francis Edward Su. A LeVeque-type lower bound for discrepancy. In. Monte Carlo and quasi-Monte Carlo methods 1998 (Claremont, CA),. pages 448–458. Springer, Berlin, 2000. [PDF] 8. Francis Edward Su. Reviews: Cake-Cutting Algorithms: Be Fair if You Can. Amer. Math. Monthly,. 107(2):185–188, 2000. [PDF] 9. Matthias G. Raith, and Francis Edward Su. Procedural support for cooperative negotiations: theory and implementation. In. Advances in Decision Technology and Intelligent Information Systems, Volume I,. pages 21–36. The International Institute for Advanced Studies in Systems Research and Cybernetics, Windsor, Canada, 2000. [PDF] 10. Francis Edward Su. Discrepancy convergence for the drunkard’s walk on the sphere. Electron. J. Probab.,. 6:no. 2, 20 pp. (electronic), 2001. [PDF] 11. Jesus A. De Loera, Elisha Peterson*, and Francis Edward Su. A polytopal generalization of Sperner’s lemma. . J. Combin. Theory Ser. A,. 100(1):1–26, 2002. [PDF] 12. Claus-Jochen Haake, Matthias G. Raith, and Francis Edward Su. Bidding for envy-freeness: a procedural approach to n-player fair-division problems. . Soc. Choice Welf.,. 19(4):723–749, 2002. [PDF] This algorithm in this paper has been implemented in. The Fair Division Calculator (New York Times Version). 13. Elisha Peterson* and Francis Edward Su. Four-Person Envy-Free Chore Division. Math. Mag.,. 75(2):117–122, 2002. [PDF] 14. Alison L. Gibbs and Francis Edward Su. On choosing and bounding probability metrics. International Statistical Review,. 70(3):419–435, 2002. [PDF] Version February 2002. 15. Arthur T. Benjamin, Christopher R. H. Hanusa*, and Francis Edward Su. Linear recurrences through tilings and Markov chains. Util. Math.,. 64:3–17, 2003. [PDF] Version June 2001. 16. Forest W. Simmons and Francis Edward Su. Consensus-halving via theorems of Borsuk-Ulam and Tucker. Math. Social Sci.,. 45(1):15–25, 2003. [PDF] 17. Doug Hensley and Francis Edward Su. Random walks with badly approximable numbers. In Unusual applications of number theory,. volume 64 of. DIMACS Ser. Discrete Math. Theoret. Comput. Sci.,. pages 95–101. Amer. Math. Soc., Providence, RI, 2004. [PDF] 18. Timothy Prescott* and Francis Edward Su. Random walks on the torus with several generators. Random Structures Algorithms,. 25(3):336–345, 2004. [PDF at the arXiv] Version April 2004. 19. Adam Bliss* and Francis Edward Su. Lower bounds for simplicial covers and triangulations of cubes. Discrete Comput. Geom.,. 33(4):669–686, 2005. [PDF at the arXiv] 20. Timothy Prescott* and Francis Edward Su. A constructive proof of Ky Fan’s generalization of Tucker’s lemma. J. Combin. Theory Ser. A,. 111(2):257–265, 2005. [PDF at the arXiv] 21. Douglas Rizzolo* and Francis Edward Su. A fixed point theorem for the infinite-dimensional simplex. J. Math. Anal. Appl.,. 332(2):1063–1070, 2007. [PDF at the arXiv] 22. Gwen Spencer* and Francis Edward Su. The LSB theorem implies the KKM lemma. Amer. Math. Monthly,. 114(2):156–159, 2007. [PDF at the arXiv] 23. Claus-Jochen Haake, Akemi Kashiwada*, and Francis Edward Su. The Shapley value of phylogenetic trees. J. Math. Biol.,. 56(4):479–497, 2008. [PDF at the arXiv] 24. Deborah Berg*, Serguei Norine, Francis Edward Su, Robin Thomas, and Paul Wollan. Voting in agreeable societies. Amer. Math. Monthly. 117:27–39, 2010. [PDF at the arXiv] 25. John Cloutier*, Kathryn L. Nyman, and Francis Edward Su. Two-player envy-free multi-cake division. Math. Social Sci. 59:26–37, 2010. [PDF at the arXiv] 26. Francis Edward Su. Teaching Research: Encouraging Discoveries. Amer. Math. Monthly,. 117:159–169, 2010. [PDF] Reprinted in. Best Writing on Mathematics 2011. (M. Pitici, ed.), Princeton University Press, 2011. 27. Francis Edward Su. The agreeable society theorem. In. Expeditions in Mathematics. (Shubin, Hayes, Alexanderson, eds.), Mathematical Association of America, 2011. [PDF at the arXiv] 28. Sanjai Gupta, Parousia Rockstroh*, and Francis Edward Su. Splitting fields and periods of Fibonacci sequences modulo primes. Math. Mag.,. Volume 85, Number 2, April 2012, 130-135. [PDF at the arXiv] Not Yet Published. 29. Rosalie Carlson*, Stephen Flood, Kevin O’Neill*, and Francis Edward Su. A Turan-type theorem for circular arc graphs. [PDF at the arXiv] 30. Claus-Jochen Haake and Francis Edward Su. Fair division procedures: why use Mathematics? In. Procedural Approaches to Conflict Resolution. (Matthias Raith, ed.), to appear. [PDF] 31. Zeph Landau and Francis Edward Su. Fair division and redistricting. [PDF] 32. Elisha Peterson* and Francis Edward Su. N-person envy-free chore division. [PDF at the arXiv] 33. Kyle E. Kinneberg*, Aaron Mazel-Gee*, Tia Sondjaja*, and Francis Edward Su. A cubical antipodal theorem. [PDF at the arXiv] 34. Andrew Niedermaier*, Douglas Rizzolo* and Francis Edward Su. Sperner and KKM-type theorems on trees and cycles. [PDF at the arXiv] 35. Tyler Seacrest* and Francis Edward Su. Minimal triangulations of simplotopes. [PDF at the arXiv] 36. Sarah Fletcher*, Christopher Hardin, and Francis Edward Su. The agreement number of tree societies. Other Writings. The Banach-Tarski Paradox. An expository paper for my. Minor Thesis. requirement, Harvard University. [PDF] My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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TOP ＆w ＆M ＆Travel ＆Time ＆Human SHOPPING EVENT & PRESENT LIKE. COMMENT. 45. MY PAGE ログアウト ゲスト ＆MEMBER. 新規登録 ログイン ＆MEMBERとは…. ＆MEMBERにご登録（無料）いただくと、気に入った記事に共感を示したり、コメントを書いたり、ブックマークしたりできます。こうしたアクションをする度にポイント「＆MILE」がたまり、限定イベントやプレゼントの当選確率が上がります。. シトウレイの好きな服、似合う服。 連載をフォローする 3ミリ新しい私. LIFE FOOD INTERESTS ARTS & CULTURE FASHION BEAUTY & HEALTH WORK ESSAY GALLERY INFO BRAND VOICE SERIAL STORY CREATORS ENTERTAINMENT NEWS PRESS RELEASE ABOUT US 朝日新聞デジタルマガジン＆[and] について ADVERTISING 広告掲載・事例紹介 FOLLOW US. SHOPPING EVENT & PRESENT ＆MEMBER登録 MY PAGE ログイン ログアウト SHOPPING EVENT & PRESENT シトウレイの好きな服、似合う服。 連載をフォローする ストリートスナップのフォトグラファーでジャーナリストのシトウレイさんの連載。世界各地で撮影したストリートスナップを通して、ちょっとした着こなしのコツだったりテクニックをご紹介します。. 月１回. シトウレイ ストリート・スタイル・フォトグラファー／ジャーナリスト. 毎シーズン、世界各国のコレクション取材を行い、独自の審美眼で綴（つづ）られる言葉と写真が人 …. 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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Photo by Mark Skovorodko for. Quanta Magazine See my creations:. I have a. new book ! Mathematics for Human Flourishing. hits bookstores Jan 7 2020! Pre-order at. Yale University Press or on. Amazon or another bookseller. See my. public webpage for reviews and press. The book is based on my speech. Mathematics for Human Flourishing , delivered Jan 6, 2017 at the Joint Math Meetings. There’s a. published version [PDF] in the American Mathematical Monthly. See also Kevin Hartnett’s article about this speech in. Quanta Magazine or. Wired . I have a. new textbook :. Topology Through Inquiry. , with Michael Starbird, which you can order through. MAA\/AMS books . There’s also resources for Instructors at. the book webpage . I have a. 24-lecture video course on Linear Algebra from the Great Courses:. Notes and Audio from my. 2013 Haimo Award lecture: “. The Lesson of Grace in Teaching “. My LA TImes op-ed:. Solve this math problem: the gender gap Some of my MAA FOCUS Columns on Race and Issues in the Classroom:. Race, Space, and the Conflict Inside Us also on. [blog] The Value of Struggle also on. [blog] Mathematical Microaggressions The Secret Mathematical Menu To the Mathematical Beach Some Guidelines for Good Mathematical Writing (See also the. handouts I use in courses). Try my iOS app:. MathFeed, the Math News App. That Makes It Invertible – Our parody of a One Direction Song. My. Math Fun Facts website or my. Math Fun Facts iPhone app ! My. Real Analysis Lectures on YouTube. Notes from my. Leitzel Lecture at the 2006 MathFest. My. Fair Division Calculator and the New York Times’. updated version and. related article . A Quote \/. Another Quote A Hymn My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. 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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Pascal Quotes. Francis Su. Pascal Quotes. Blaise Pascal (1623 – 1662). Here are some quotes from Pascal’s. Pensees. , a collection of thoughts that Pascal penned, starting in 1656, which were discovered after his death and assembled by the Jansenists in 1670. They were the beginnings of a book he was intending to write as an apologetic for the Christian faith. I present some of his thoughts here for your sampling— I think you will enjoy reading them whether or not you agree with his perspective. As a mathematician and philosopher, Pascal defended the intellectual soundness of the pursuit of faith. However, as one reads his thoughts, what one hears are the reflections of a man who struggled with the weighty issues of the soul, and his writings read like they could have been written yesterday. From. Blaise Pascal. ,. Pensees. :. #143. “Men are entrusted from infancy with the care of their honour, their property, their friends, and even with the property and honour of their friends. They are overwhelmed with business, with the study of languages, and with physical exercise; and they are made to understand that they cannot be happy unless their health, their honour, their fortune and that of their friends be in good condition, and that a single thing wanting will make them unhappy. “Thus they are given cares and business which make them bustle about from the break of day—. “It is, you will exclaim, a strange way to make them happy! What more could be done to make them miserable!— Indeed, what could be done? We should only have to relieve them from all these cares; for then they would see themselves: they would reflect on what they are, whence they came, whither they go, and thus we cannot employ and divert them too much. And this is why, after having given them so much business, we advise them, if they have some time for relaxation, to employ it in amusement, in play, and to be always fully occupied. “How hollow and full of ribaldry is the heart of man!”. #10. People are generally better persuaded by the reasons which they have themselves discovered than by those which have come in to the mind of others. #66. One must know oneself. If this does not serve to discover truth, it at least serves as a rule of life, and there is nothing better. #67. Physical science will not console me for the ignorance of morality in the time of affliction. But the science of ethics will always console me for the ignorance of the physical sciences. #81. It is natural for the mind to believe and for the will to love; so that, for want of true objects, they must attach themselves to false. #143. Men are entrusted from infancy with the care of their honour, their property, their friends, and even with the property and honour of their friends. They are overwhelmed with business, with the study of languages, and with physical exercise; and they are made to understand that they cannot be happy unless their health, their honour, their fortune and that of their friends be in good condition, and that a single thing wanting will make them unhappy. Thus they are given cares and business which make them bustle about from the break of day.– It is, you will exclaim, a strange way to make them happy! What more could be done to make them miserable!– Indeed, what could be done? We should only have to relieve them from all these cares; for then they would see themselves: they would reflect on what they are, whence they came, whither they go, and thus we cannot employ and divert them too much. And this is why, after having given them so much business, we advise them, if they have some time for relaxation, to employ it in amusement, in play, and to be always fully occupied. How hollow and full of ribaldry is the heart of man! #144. I spent a long time in the study of the abstract sciences, and was disheartened by the small number of fellow students in them. When I commenced the study of man, I saw that these abstract sciences are not suited to man and that I was wandering farther from my own state in examining them than others in not knowing them. I pardoned their little knowledge; but I thought at least to find many companions in the study of man and that it was the true study which is suited to him. I have been deceived; still fewer study it than geometry. It is only from the want of knowing how to study this that we seek the other studies. But is it not that even here is not the knowledge which man should have and that for the purpose of happiness it is better for him not to know himself? #174. Solomon and Job have best known and best spoken of the misery of man; the former the most fortunate, and the latter the most unfortunate of men; the former knowing the vanity of pleasures from experience, the latter the reality of evils. #185. The conduct of God, who disposes all things kindly, is to put religion into the mind by reason, and into the heart by grace. But to will to put it into the mind and heart by force and threats is not to put religion there, but terror;. terorrem potius quam religionem. . #187. Men despise religion; they hate it and fear it is true. To remedy this, we must begin by showing that religion is not contrary to reason; that it is venerable, to inspire respect for it; then we must make it lovable, to make good men hope it is true; finally, we must prove it is true. Venerable, because it has perfect knowledge of man; lovable because it promises the true good. #194. …Let them at least learn what is the religion they attack, before attacking it. If this religion boasted of having a clear view of God, and of possessing it open and unveiled, it would be attacking it to say that we see nothing in the world which shows it with this clearness. But since, on the contrary, it says that men are in darkness and estranged from God, that He has hidden Himself from their knowledge, that this is in fact the name which He gives Himself in the Scriptures,. Deus absconditus. ,and finally, if it endeavors equally to establish these two things: that God has set up in the Church visible signs to make Himself known to those who should seek Him sincerely, and that He has nevertheless so disguised them that He will only be perceived by those who seek Him with all their heart; what advantage can they obtain, when, in the negligence with which they make profession of being in search of the truth, they cry out that nothing reveals it to them; and since that darkness in which they are, and with which they upbraid the Church, establishes only one of the things which she affirms, without touching the other, and, very far from destroying, proves her doctrine? In order to attack it, they should have protested that they had made every effort to seek Him everywhere, and even in that which the Church proposes for their instruction, but without satisfaction. If they talked in this manner, they would in truth be attacking one of her pretensions. But I hope here to show that no reasonable person can speak thus, and I venture even to say that no one has ever done so. We know well enough how those who are of this mind behave. They believe they have made great efforts for their instruction when they have spent a few hours in reading some book of Scripture and have questioned some priests on the truths of the faith. After that, they boast of having made vain search in books and among men. But, verily, I will tell them what I have often said, that this negligence is insufferable. We are not concerned with the trifling interests of some stranger, that we should treat it in this fashion; the matter concerns ourselves and our all. The immortality of the soul is a matter which is of so great consequence to us and which touches us so profoundly that we must have lost all feeling to be indifferent as to knowing what it is…. Thus our first interest and our first duty is to enlighten ourselves on this subject, whereon depends all our conduct. Therefore among those who do not beleive, I make a vast difference between those who strive with all their power to inform themselves and those who live without troubling or thinking about it. …We do not require great education of the mind to understand that here is no real and lasting satisfaction; that our pleasures are only vanity; that our evils are infinite; and, lastly, that death, which threatens us every moment, must infallibly place us within a few years under the dreadful necessity of being for ever either annihilated or unhappy. There is nothing more real than this, nothing more terrible. Be we as heroic as we like, that is the end which awaits the noblest life in the world. Let us reflect on this and then say whether it is not beyond doubt that there is no good in this life but in the hope of another; that we are happy only in proportion as we draw near to it; and that, as there are no more woes for those who have complete assurance of eternity, so there is no more happiness for those who have no insight into it. Surely then it is a great evil thus to be in doubt, but it is at least an indispensable duty to seek when we are in such doubt; and thus the doubter who does not seek is altogether completely unhappy and completely wrong. And if besides this he is easy and content, professes to be so, and indeed boasts of it; if it is this state itself which is the subject of his joy and vanity, I have no words to describe so silly a creature. How can people hold these opinions? What joy can we find in the expectation of nothing but hopeless misery? What reason for boasting that we are in impenetrable darkness? And how can it happen that the following argument occurs to a reasonable man? Who would desire to have for a friend a man who talks in this fashion? Who would choose him out from others to tell him of his affairs? Who would have recourse to him in affliction? And indeed to what use in life could one put him? In truth, it is the glory of religion to have for enemies men so unreasonable; and their opposition to it is so little dangerous that it serves, on the contrary, to establish its truths. For the Christian faith goes mainly to establish these two facts: the corruption of nature, and redemption by Jesus Christ. Now I contend that, if these men do not serve to prove the truth of the redemption by the holiness of their behaviour, they at least serve admirably to show the corruption of nature by sentiments so unnatural. …. #221. Atheists ought to say what is perfectly evident; now it is not perfectly evident that the soul is material. #222. What reason have they for saying that we cannot rise from the dead? What is more difficult, to be born or to rise again; that what has never been should be, or that what has been should be again? Is it more difficult to come into existence than to return to it? Habit makes the one appear easy to us; want of habit makes the other impossible. A popular way of thinking! …. #253. Two extremes: to exclude reason, to admit reason only. #265. Faith indeed tells what the senses do not tell, but not the contrary of what they see. It is above them and not contrary to them. #280. The knowledge of God is very far from the love of Him. #282. We know truth, not only by the reason, but also by the heart, and it is in this last way that we know first principles; and reason, which has no part in it, tries in vain to impugn them. The sceptics, who have only this for their object, labour to no purpose. We know that we do not dream, and, however impossible it is for us to prove it by reason, this inability demonstrates only the weakness of our reason, but not, as they affirm, the uncertainty of all our knowledge. For the knowledge of first principles, as space, time, motion, number, is as sure as any of those we get from reasoning. And reasons must trust these intuitions of the heart, and must base them on every argument… And it is as useless and absurd for reason to demand from the heart proofs of her first principles, before admitting them, as it would be for the heart to demand from reason an intuition of all demonstrated propositions before accepting them. This inability ought, then, to serve only to humble reason, which would judge all, but not to impugn our certainty, as if only reason were capable of instructing us. Would to God, on the contrary, that we had never need of it, and that we knew everything by instinct and intuition! But nature has refused us this boon. On the contrary, she has given us but very little knowledge of this kind; and all the rest can be acquired only by reasoning. Therefore, those to whom God has imparted religion by intuition are very fortunate and justly convinced. But to those who do not have it, we can give it only by reasoning, waiting for God to give them spiritual insight, without which faith is only human and useless for salvation. #288. Instead of complaining that God had hidden Himself, you will give Him thanks for not having revealed so much of Himself; and you will also give Him thanks for not having revealed Himself to haughty sages, unworthy to know so holy a God. Two kinds of persons know Him: those who have a humble heart, and who love lowliness, whatever kind of intellect they may have, high or low; and those who have sufficient understanding to see the truth, whatever opposition they may have to it. #511. If we would say that man is too insignificant to deserve communion with God, we must indeed be very great to judge of it. #537. Christianity is strange. It bids man recognise that he is vile, even abominable, and bids him desire to be like God. Without such a counterpoise, this dignity would make him horribly vain, or this humiliation would make him terribly abject. #771. Jesus Christ came to blind those who saw clearly, and to give sight to the blind; to heal the sick, and leave the healthy to die; to call to repentance, and to justify sinners, and to leave the righteous in their sins; to fill the needy, and leave the rich empty. #737. So I hold my arms to my. Redeemer. , who, having been foretold for four thousand years, has come to suffer and to die for me on earth, at the time and under all the circumstances foretold. By His grace, I await death in peace, in the hope of being eternally united to Him. Yet I live with joy, whether in the prosperity which it pleases Him to bestow upon me, or in the adversity which He sends for my good, and which He has taught me to bear by His example. My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Fair Division Calculator. Francis Su. Fair Division Calculator. This program ran as a java applet which is no longer viewable with current browsers. This page contains historical information. To use the application, please access the. New York Times version. and view the. related article. . A java applet for interactive decision making to find envy-free divisions of goods, burdens, or rent. The Fair Division Calculator will soon be integrated into ARTUS, a process control system for group decision and negotiation on the Internet. ARTUS is a project directed by Matthias Raith. When the integration is complete, it will be the first . analytical.  mediation support system on the Internet! What is “Fair Division”? The theory of fair division is concerned with finding ways to divide an object among. n. people. fairly. . There are many possible notions of fairness. The notion that has interested me recently is finding. envy-free. divisions, i.e., one in which each person feels she received the. best. piece, in her own estimation. Exact vs. Approximate Algorithms. Existence of such divisions has been known for some time. Of greater interest are algorithms for achieving such divisions. For instance, in dividing a desirable object, such as cakes, a 2-person algorithm has been known since antiquity, a 3-person algorithm since at least 1960, but an exact algorithm for achieving. n. -person envy-free cake divisions was not produced until Brams and Taylor’s solution in 1995. (See the references below.) Elisha Peterson and I have recently extended these techniques to provide an. exact envy-free chore-division algorithm for. n. people. However, such methods tend to be quite complicated as. n. grows large. By contrast, I have been interested in developing . approximate.  envy-free schemes, in particular, through the use of arguments from combinatorial topology (Sperner, Tucker, and generalizations). Such methods will find envy-free divisions up to a specified tolerance for error (such as at the level of crumbs), and have the advantage of (1) being easily generalized for arbitrary numbers of players, (2) using the minimal number of cuts, (3) for a fixed number of players and a fixed error, having a bounded number of steps in which the algorithm halts. My main inspiration and partner in this endeavor has been Forest Simmons. More recently, my students Elisha Peterson, Patrick Vinograd, and Karl Mahlburg have also been working with me. To explore such algorithms, we have developed the Fair Division Calculator. The Fair Division Calculator finds approximate envy-free divisions up to any precision for the following problems:. cake-cutting (division of goods\/desirables),. chore-division (division of burdens\/undesirables). rent-partitioning (allocation of indivisible goods mediated by divisible payments). In a few weeks we’ll have a new algorithm for rent-partitioning (due to Haake-Raith-Su) encoded as an option in the Calculator. References. Some good references on subject are:. Brams, Steven J. and Taylor, Alan,. Fair Division: from cake-cutting to dispute resolution. , Cambridge Univ. Press, 1996. Robertson, Jack and Webb, William. Cake-cutting Algorithms. . AK Peters Ltd., 1998. See also the . link to my papers.  Or see also Elisha’s research page. Recognitions. Site of the Week (12-14-1999) by the Canadian Math Society! . Also featured in a 1-7-2000 article in . Science.  (p.37) and a 12-16-1999 story in the online publication . ScienceNow. . Featured in a 1-31-2000 interview for the radio show . ScienceUpdate. . Featured in a 2-26-2000 article “A Fair Deal for Housemates” in . ScienceNews. . Featured in Ivars Peterson’s MathTrek article on 3-11-2000 in . ScienceNewsOnline. . My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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{"id":"mJQ2sW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/A-LeVeque-type.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":78,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"A-LeVeque-type","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["A-LeVeque-type.pdf"],"author":"pdfTeX-0.13c","author_sxt":["pdfTeX-0.13c"],"last_modified":"Thu Jun 13 04:34:10 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-5516152060046636557,"exact_signature_unique_b":true,"fuzzy_signature_l":2458186632778338258,"fuzzy_signature_unique_b":true,"wordcount_i":3479,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:16 UTC 2022","fresh_date_dt":"Wed Dec 06 11:46:19 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13c","language_s":"en","size_i":236076,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088077750140928,"text_t":"A LeVeque-type lower bound for discrepancy\nFrancis Edward Su\nDepartment of Mathematics, Harvey Mudd College, Claremont CA 91711, USA\nsu@math.hmc.edu\nreprinted from Monte Carlo and Quasi-Monte Carlo Methods 1998,\nH. Niederreiter and J. Spanier, eds., Springer-Verlag, 2000, pp. 448-458.\nAbstract. A sharp lower bound for discrepancy on R\/Z is derived that resembles\nthe upper bound due to LeVeque. An analogous bound is proved for discrepancy on\nRk\/Zk. These are discussed in the more general context of the discrepancy of prob-\nablity measures. As applications, the bounds are applied to Kronecker sequences\nand to a random walk on the torus.\n1 Introduction\nConsider a sequence {xn}n≥1 of points in Rk modulo 1, i.e., in the k-\ndimensional torus Tk = Rk\/Zk. Let J = [a1, b1)× ...× [ak, bk) be an interval\nin Rk such that 0 < bi − ai ≤ 1 for i = 1, ..., k. Let I = J\/Zk be the cor-\nresponding set on the torus, also called an interval. The volume λk of I is\ninherited from Lebesgue measure in Rk: λk(I) =\n∏k\ni=1(bi − ai). Let AN (I)\ndenote the number of points of the first N elements of the sequence {xn} that\nfall in I, i.e., AN (I) =\n∑N\nn=1 δI(xn) where δI denotes the indicator function\nof the set I. The discrepancy DN (xn) of the sequence {xn} is defined by\nDN (xn) = sup\nI⊆Tk\n∣∣∣∣\nAN (I)\nN\n− λk(I)\n∣∣∣∣ . (1)\nLoosely speaking, this is the largest possible difference between the proportion\nof points in an interval I and the expected proportion if the points were\nuniformly distributed. Hence discrepancy is a measure of how far the sequence\nis from being uniformly distributed in Tk.\nThe purpose of this paper is to establish new lower bounds for the dis-\ncrepancy of a sequence in Tk using Fourier methods. These are similar in\nform to the upper bound established by LeVeque [7]. We illustrate the use of\nsuch bounds in several examples at the end of this paper.\nOur results will be cast in the more general form of the discrepancy of\nprobability measures, which we define presently. If P is a probability measure\non Tk, define the discrepancy of P to be\nD(P ) = sup\nI⊆Tk\n|P (I)− λk(I)|.\nThis notion of discrepancy measures the uniformity of the distribution P .\nHence discrepancy of sequences is the special case where P = AN\nN\n.\nWe work with the more general notion of discrepancy of probability mea-\nsures for two reasons: (1) our results are derived by viewing AN (I) as a\nconvolution of measures, rather than counting points in a set I, and (2) dis-\ncrepancy of probabilities is useful in its own right as a metric for quantifying\nrates of convergence of random walks on Tk to the uniform distribution. We\ngive an example at the conclusion of this paper. See [9] and [10] for further\nwork in this direction.\nLet e(z) denote the expression exp(2πiz). The Fourier coefficients of a\nmeasure Q on R\/Z are indexed by integers m ∈ Z and defined by Q̂(m) =∫ 1\n0\ne(mx) Q(dx), a Riemann-Stieltjes integral with respect to the distribution\nfunction of the measure Q. For a sequence {xn}, the Fourier coefficients\nbecome Q̂(m) = 1\nN\n∑N\ni=1 e(mxi). In [7], LeVeque proved the following well-\nknown upper bound for the discrepancy of sequences mod 1.\nLeVeque’s Inequality. If Q is any probability measure on R\/Z, then\nD(Q) ≤\n(\n6\nπ2\n∞∑\nm=1\n|Q̂(m)|2\nm2\n)1\/3\nwhere Q̂(m) denotes the m-th Fourier coefficient of Q. The constant 6\nπ2\nand\nexponent 1\/3 are best possible.\nIn this paper we establish the following lower bound for discrepancy on R\/Z,\nwhich resembles LeVeque’s inequality in form, with sharp constant.\nTheorem 1. If Q is any probability measure on R\/Z, then\nD(Q) ≥\n(\n2\nπ2\n∞∑\nm=1\n|Q̂(m)|2\nm2\n)1\/2\nwhere Q̂(m) denotes the m-th Fourier coefficient of Q. The inequality is sharp\nand the constant 2\nπ2\nis best possible.\nA similar inequality is implied in [6, Lemma 2.8] for the related star discrep-\nancy (which is smaller); with a little extra effort we have bounded the usual\ndiscrepancy and sharpened up the constant.\nUsing a related, though slightly different approach, we can establish an\nanalogous result in higher dimensions. On Tk, the Fourier coefficients are\nindexed by h ∈ Zk and are given by Q̂(h) =\n∫\nTk\ne(h·x)Q(dx). For sequences,\nthis becomes Q̂(h) = 1\nN\n∑N\ni=1 e(h · xi).\nTheorem 2. If Q is any probability measure on Tk, then\nD(Q) ≥ sup\nr\n\n ∑\n06=h∈Zk\n|Q̂(h)|2\nk∏\ni=1\n{\nsin2(2πhiri)\nπ2h2\ni\nif hi 6= 0\n4r2i if hi = 0\n} \n\n1\/2\n449\nwhere r = (r1, ..., rk) is any k-tuple such that each ri ∈ (0, 1\/2].\nNote that r may be chosen so as to optimize the bound in relation to Q̂,\ni.e., one may choose the ri so that sin(2πhiri) is large when Q̂(h) is large. To\nsee how Theorem 2 compares with Theorem 1, set each ri =\n1\n4\nin Theorem\n2. Then terms for all lattice points h that have any non-zero even coordinate\nvanish. This yields\nCorollary 3. If Q is any probability measure on Tk, then\nD(Q) ≥ 1\nπk\n\n ∑\nh6=0, hi odd or 0\n|Q̂(h)|2\nR(h)2\n\n\n1\/2\nwhere R(h) =\n∏k\ni=1 max{1, |hi|} for h = (h1, ..., hk) ∈ Zk.\nThis corollary shows that the bound of Theorem 2 is not as sharp as\nTheorem 1 when k = 1. In particular, Corollary 3 for k = 1 yields a bound\nmuch like Theorem 1, except that it omits all the even terms in the sum.\n(However, by choosing r appropriately in Theorem 2, one can obtain a bound\nlike Corollary 3 with fewer terms omitted, at the expense of having a smaller\nmultiplicative constant.)\nIn [4, p. 25], Drmota and Tichy prove a lower bound for discrepancy of\nsequences on Tk via the Koksma-Hlawka inequality. When k = 1, their bound\nis much like Theorem 1 but weaker (it has exponent 1 instead of 1\/2, and a\nsmaller constant). When k > 1, the Drmota-Tichy bound is in general not\ncomparable to our Theorem 2.\nIn practice, since all the terms of the sum in the bounds of Theorems\n1 and 2 and Corollary 3 are positive, one need only use as many terms as\none needs for a lower bound. In fact, using only the dominant term(s) may\nbe quite sufficient in some cases to yield good bounds. This occurs because,\nunlike the Drmota-Tichy bound, the dominant term in our bounds “matches”\nthe dominant term in the upper bound of Erdös-Turán-Koksma.\nWe demonstrate the use of such bounds in Section 4, where they are\napplied to Kronecker sequences and to a random walk on the torus. First, we\nprovide proofs of Theorems 1 and 2 in the next two sections of this paper.\n2 Discrepancy on R\/Z\nProof (of Theorem 1). We first prove two lemmas. Set F (x) = Q([0, x]) −\nx + K, where [0, x] is an interval in T1 = R\/Z, and K is chosen such that∫\n[0,1]\nF (x) dx = 0.\nLemma 4.\n∫\n[0,1]\nF 2(x) dx =\n1\n2π2\n∞∑\nm=1\n|Q̂(m)|2\nm2\n(2)\n450\nwhere F (x), defined as above, depends on Q.\nProof (of lemma). Compute the Fourier coefficents of F for m 6= 0:\nF̂ (m) =\n∫ 1\n0\n(Q([0, x])− x+K)e2πimx dx .\nIntegration by parts yields:\nF̂ (m) = −\n∫ 1\n0\ne2πimx\n2πim\n(Q(dx)− dx) = Q̂(m)\n−2πim\n.\nBy our choice of K, F̂ (0) = 0. Also note that Q̂(−m) = Q̂(m). These facts\nand Parseval’s identity now give the conclusion of Lemma 4. ut\nAn upper bound can be derived for\n∫\n[0,1]\nF 2(x) dx in terms of the dis-\ncrepancy. In fact, we claim\nLemma 5. ∫\n[0,1]\nF 2(x) dx ≤ D\n2\n4\nwhere D = D(Q) is the discrepancy of Q.\nWe remark that it is easy to derive a bound of the form D2; the non-trivial\npart is to improve it to D2\/4.\nProof (of lemma). Let B1 = {x : F (x) ≥ 0} and B2 = {x : F (x) < 0}. Define\nW (x) by\nW (x) =\n{\nsup[0,1] F for x ∈ B1\ninf [0,1] F for x ∈ B2.\nBy construction, |W (x)| ≥ |F (x)|. Set s = λ1(B1), t = supF . Hence 1− s =\nλ1(B2), and it follows from the definition of the discrepancy that D − t =\n| inf F |.\nWe claim that either\n∫\nB1\n|W (x)| dx or\n∫\nB2\n|W (x)| dx is bounded above\nby D\n4\n. For suppose that\n∫\nB1\n|W (x)| dx ≥ D\n4\n. It follows that st ≥ D\n4\n. Then\n∫\nB2\n|W (x)| dx = (1− s)(D − t) < (1− D\n4t\n)(D − t) = 5D\n4\n− D\n2\n4t\n− t .\nElementary calculus shows that the maximum value of the expression on the\nright is D\n4\n, which demonstrates this claim.\nTherefore, set A =\n∫\nB1\n|F (x)| dx, and note by construction of F that\nA =\n∫\nB2\n|F (x)| dx, too. Since |F (x)| ≤ |W (x)|, using the previous claim we\nsee that A ≤ D\n4\n. Hence\n∫\n[0,1]\nF 2(x) dx ≤ t\n∫\nB1\n|F (x)| dx+ (D − t)\n∫\nB2\n|F (x)| dx\n= (D − t)A+ tA = DA ≤ D\n2\n4\n.\n451\nThis is the conclusion of Lemma 5. ut\nThe bound of Theorem 1 follows as an immediate consequence of the\nprevious two lemmas.\nMoreover, the bound is sharp in the sense that the ratio of the two sides\ncan be made to approach 1. Given \u000f > 0, let Q\u000f be the measure that is uniform\non (0, 1\n2\n− \u000f) and ( 1\n2\n+ \u000f, 1) and has point mass 2\u000f at 0. This is a family of\nmeasures for which the ratio of the two sides in Lemma 5 approaches 1 as\n\u000f → 0. (Calculations show the left side is \u000f2 − 4\n3\n\u000f3, and the right side is \u000f2).\nTogether with Lemma 4, this shows that the constant 2\nπ2\nin the theorem is\nbest possible.\nThe bound is also sharp for sequences, since any of the above measures Q\u000f\ncan be approximated a closely as desired by the counting measure AN\nN\nof an\nappropriate sequence. For instance, let N be even. Consider N2 points evenly\nplaced on R\/Z, at the points 0, 1\nN2\n, 2\nN2\n, ... etc. Pick up the (2N − 1) points\nnearest the point 1\/2 and place them at 0. Calculations for this sequence of\nthe quantities in Lemma 5 show the left side is 1\nN2\n+ O( 1\nN3\n) and the right\nside is exactly 1\nN2\n. As N →∞ we see that the bound in the theorem becomes\nsharp. ut\nThe proof of Theorem 1 was motivated by the proof of LeVeque’s upper\nbound. In that proof the expression (2) was bounded below by an expression\ninvolving the discrepancy. This was accomplished by noting the existence of\ncertain “triangles” under the graph of F (x) whose height could be expressed\nin terms of the discrepancy. The main idea behind Theorem 1 is that the\nexpression (2) also can be bounded sharply from above by an expression\ninvolving the discrepancy.\n3 Discrepancy on Rk\/Zk\nWe shall take a slightly different approach to prove Theorem 2. First we\nrequire a lemma. Let Ix,r denote the interval in T\nk centered at x = (x1, ..., xk)\nwith radius vector r = (r1, ..., rk), i.e.,\nIx,r = [x1 − r1, x1 + r1)× ...× [xk − rk, xk + rk).\nEvery interval can be written in this form, with 0 < ri < 1\/2. As shorthand\nwe let Ir = I0,r denote the interval centered at 0. Let δIr denote the indicator\nfunction of the set Ir.\nLemma 6. If Q is any probability measure on Tk, then for any r, 0 < ri <\n1\/2,\nD(Q)2 ≥\n∑\n06=h∈Zk\n| Q̂(h) δ̂Ir(h) |2.\n452\nWe prove this by constructing a function, parametrized by a radius vector\nr, that measures the discrepancy only on a set of radius r centered at each\npoint; this is bounded above by the total discrepancy. It can be rewritten in\nterms of a convolution of the original measure and the indicator function of\nthe set (this was observed by J. Beck, see [2] and [3]). An appeal to Parseval’s\nidentity then gives the desired conclusion.\nProof. Define\n∆r(x) = Q(Ix,r)− λk(Ix,r).\nBy the definition of D(Q), we see that ∆r(x) ≤ D(Q), and hence for all r,\n∫\nTk\n∆2r(x) dx ≤ D(Q)2. (3)\nOn the other hand, Parseval’s identity yields\n∫\nTk\n∆2r(x) dx =\n∑\nh∈Zk\n|∆̂r(h)|2. (4)\nNotice that ∆r may be rewritten as:\n∆r(x) = Q ∗ δIr(x)− λk ∗ δIr(x) = (Q− λk) ∗ δIr(x)\nwhere δI is the indicator function of the set I, and ∗ indicates convolution of\na measure with a function: Q ∗ f(x) =\n∫\nf(x− y)Q(dy).\nThen ∆̂r may be computed as ∆̂r(h) = ( Q̂(h)− λ̂k(h) ) δ̂Ir(h).\nFor h = 0, ∆̂r(h) = 0 since\nQ̂(0)− λ̂k(0) =\n∫\nTk\nQ(dx) −\n∫\nTk\nλk(dx) = 0.\nFor h 6= 0, a trivial computation shows λ̂k(h) = 0, and thus\n∆̂r(h) = Q̂(h) δ̂Ir(h).\nCombining this with (3) and (4), we obtain the desired bound. ut\nNow we can prove Theorem 2:\nProof (of Theorem 2). We compute δ̂Ir :\nδ̂Ir(h) =\n∫\nTk\nδIr(x) e(h · x) dx\n=\n∫ r1\nx1=−r1\n· · ·\n∫ rk\nxk=−rk\ne2πih1x1 · · · e2πihkxk dx\n=\nk∏\ni=1\n[∫ ri\nxi=−ri\ne2πihixi dxi\n]\n. (5)\n453\nEach of the factors in the last expression evaluates to\nsin(2πhiri)\nπhi\nif hi 6= 0,\nand 2ri if hi = 0. This gives\nD(Q)2 ≥\n∑\n06=h∈Zk\n|Q̂(h)|2\nk∏\ni=1\n{\nsin2(2πhiri)\nπ2h2\ni\nif hi 6= 0\n4r2i if hi = 0\n}\n.\nSince we are able to choose any r for a lower bound, this yields the conclusion\nof the theorem. ut\nLemma 6 is a useful result in its own right, since it allows us to derive\nsimilar bounds for alternate variants of discrepancy, which arise from using\nsets other than intervals to evaluate the supremum in (1).\nFor example, ball-discrepancy is defined over balls under the usual Eu-\nclidean metric on Tk. Let Bx,r = {x ∈ Rk : d(x, c) ≤ r} denote the\nk-dimensional ball in Rk of radius r centered at x. (Here, d is the usual\nEuclidean metric.) This reduces to a ball in Tk which for simplicity we also\ndenote by Bx,r. If B denotes any ball in T\nk, the ball discrepancy is defined\nby\nDball(Q) = sup\nB⊆Tk\n|P (B)− λk(B)|.\nThe Fourier coefficients of δBx,r on R\nk are known (see [5]) and can be ex-\npressed in terms of Bessel functions Jk\/2:\nδ̂Bx,r (h) =\n∫\nRk\nδBx,r (x)e(h · x) dx =\n(\nr\n|h|\n)k\/2\nJk\/2(2πr|h|).\nOn Tk this result is still valid as long as r < 1\n2\n(otherwise, the ball overlaps\nitself). Therefore from Lemma 6 we obtain:\nDball(Q) ≥ sup\nr<1\/2\n\n ∑\n06=h∈Zk\n|Q̂(h)|2\n(\nr\n|h|\n)k (\nJk\/2(2πr|h|)\n)2\n\n\n1\/2\n.\nSee Holt [5] for a related upper bound.\n4 Applications\nAs a demonstration, we compute lower bounds for Kronecker sequences. (A\nKronecker sequence is the sequence of integer multiples of a given k-tuple of\nreal numbers.) The results we derive compare well with known bounds. We\nconclude by using Corollary 3 to derive a rate of convergence for a random\nwalk on the torus.\n454\nThe Kronecker sequence for k = 1.\nThis is just the sequence {nα}, n = 0, 1, ..., N − 1, on R\/Z.\nRecall that an irrational α is said to be of type η if η is the infimum of all\nnumbers τ for which there exists a positive constant C such that ‖qα‖ ≥ C\nqτ\nfor all positive integers q. Here ‖x‖ denotes the nearest distance from x to\nan integer.\nTheorem 7. Let α be an irrational of type η. The discrepancy of the se-\nquence {xn} = {nα}, n = 0, 1, 2, ..., (N − 1), satisfies, for any \u000f > 0,\nDN (xn) = Ω(N\n− 1\nη\n−\u000f\n). (6)\nHere f(x) = Ω(g(x)) means f(x) 6= o(g(x)). This result agrees with\nBehnke’s lower bound, derived in [6] by a different method.\nProof. Given \u000f > 0, there exists a δ such that 1\nη−δ =\n1\nη\n+ \u000f where 0 < δ < η.\nSince α is of type η, there exist infinitely many m such that mη−δ‖mα‖ <\n1\n4\n. For such m, let N = dmη−δe. Fix m large enough so that one still has\nN‖mα‖ < 1\n2\n. The measure Q for this sequence consists of weight 1\nN\non each\nmember of the sequence. Hence\n∣∣∣Q̂(m)\n∣∣∣ =\n∣∣∣∣∣\n1\nN\nN−1∑\nn=0\ne(nmα)\n∣∣∣∣∣ =\n1\nN\n|1− e(Nmα)|\n|1− e(mα)|\n≥ 2‖Nmα‖\nNπ‖mα‖\n(7)\nThe inequality follows from noting that 4‖x‖ ≤ |1 − e(‖x‖)| ≤ 2π‖x‖. Since\nN‖mα‖ < 1\n2\n, ‖Nmα‖ = N‖mα‖ and so (7) reduces to Q̂(m) ≥ 2\nπ\n.\nUsing only the m-th term in the lower bound of Theorem 1, and using\nthe fact that N ≥ mη−δ, we obtain:\nDN (xn) ≥\n2\n√\n2\nπ2m\n≥ 2\n√\n2\nπ2\nN\n− 1\nη\n−\u000f\n.\nThe statement holds for infinitely many choices of N (since N were derived\nfrom the choice of infinitely many m). This establishes (6). ut\nThe Kronecker sequence for k > 1.\nThis is the sequence {na}, n = 0, 1, ..., N − 1, for some a = (α1, α2, ..., αk) in\nRk\/Zk. As in the case k = 1, we shall rely only on a single dominant term\nin the lower bound of Theorem 2.\nRecall that R(h) =\n∏k\ni=1 max{1, |hi|}. Fixing any h, setting ri =\n1\n4hi\nfor\nhi 6= 0 and ri = 12π for hi = 0, and ignoring all other terms in Theorem 2,\nwe obtain:\n455\nLemma 8. For any h ∈ Zk, h 6= 0,\nD(Q) ≥ |Q̂(h)|\nπkR(h)\n.\nWe remark that a similar inequality (with weaker constant) can be derived\nvia the Koksma-Hlawka inequality applied to the function f(x) = e(h · x) of\nbounded variation V (f)\u001c R(h).\nLemma 8 can be used to establish a lower bound for Kronecker sequences\nthat depends on a type classification of k-tuples, similar to the type classifi-\ncation of irrationals defined earlier. A k-tuple a = (α1, ..., αk) of irrationals\nis said to be of type η if η is the infimum of all numbers τ for which there\nexists a positive constant C such that R(h)τ‖h ·a‖ ≥ C holds for all non-zero\nlattice points h in Zk.\nTheorem 9. Let a ∈ Rk be a k-tuple of irrationals, of type η. The discrep-\nancy of the sequence {xn} = {na}, n = 0, 1, 2, ..., (N − 1), satisfies, for any\n\u000f > 0,\nDN (xn) = Ω(N\n− 1\nη\n−\u000f\n). (8)\nThe exponent is nearly sharp; an upper bound of O(N\n− 1\nη\n+\u000f\n) may be estab-\nlished by the method of Niederreiter [8] (who carries out the case for η = 1).\nProof. Given \u000f > 0, there exists a δ such that 1\nη−δ =\n1\nη\n+ \u000f where 0 <\nδ < η. Since α is of type η, there exist infinitely many h ∈ Zk such that\nR(h)η−δ‖h · a‖ < 1\n4\n. For such h, let N = dR(h)η−δe. Fix h large enough so\nthat one still has N‖h · a‖ < 1\n2\n. It can now be shown that Q̂(h) ≥ 2\nπ\nby a\ncalculation analogous to (7) in which mα is replaced by h · a. Using this in\nLemma 8, we obtain\nDN (xn) ≥\n2\nπk+1\nN\n− 1\nη\n−\u000f\n.\nSince this holds for infinitely many N (derived from infinitely many h), (8)\nfollows. ut\nA random walk on the torus.\nConsider the random walk on Tk = Rk\/Zk which starts at the origin, and\nat each step moves to a point uniformly chosen within a box of radius A\nabout the current point. If Q is a probability measure uniformly supported\non [−A,A]k\/Zk, then Q∗n represents the n-th step probability distribution\nof this random walk, which evidently converges to the uniform distribution\non the torus. In fact, using Corollary 3, we shall show that the discrepancy\nD(Q∗n) converges to zero at a geometric rate in n, by showing that both\nupper and lower bounds converge geometrically:\n456\nTheorem 10. The discrepancy of the above random walk on the torus sat-\nisfies\nC1 β\nn < D(Q∗n) < C2 β\nn\nwhere n is the number of steps taken by the random walk, β = | sin(2πA)\n2πA\n|, and\nC1 = C1(k) and C2 = C2(k,A) are constants specified in the proof.\nProof. Since the density of Q is (2A)−k on its support, by a derivation similar\nto (5) we obtain the Fourier coefficients of Q:\nQ̂(h) =\nk∏\ni=1\n{\nsin(2πhiA)\n2πhiA\nif hi 6= 0\n1 if hi = 0\n}\n. (9)\nRecall that Q̂∗n(h) = Q̂n(h). Now for positive integers m,\nsin(2πmA)\nsin(2πA)\n=\nUm−1(cos(2πA)), where Um−1 is the Chebyshev polynomial of the second\nkind. It satisfies the inequality |Um−1(x)| ≤ m for x ∈ [−1, 1] (see [1]). Hence\nfor any hi 6= 0, | sin(2πhiA)2πhiA | ≤ |\nsin(2πA)\n2πA\n| which is less than 1 for A > 0.\nLet β = | sin(2πA)\n2πA\n|. Then |Q̂(h)| ≤ β for all h 6= 0. The dominant Fourier\ncoefficients occur when exactly one of the |hi| = 1 and all the other coordi-\nnates are 0. There are exactly 2k such coefficients. Using only these coeffi-\ncients in the lower bound from Corollary 3, we see that\nD(Q∗n) ≥ 1\nπk\n(2kβ2n)1\/2 =\n√\n2k\nπk\nβn.\nOn the other hand, an upper bound for D(Q∗n) can be obtained via\nthe Erdös-Turán-Koksma inequality (see [4]) which also involves the Fourier\ncoefficients: for any positive integer H,\nD(Q∗n) ≤ (3\/2)k\n\n 2\nH + 1\n+\n∑\n0<‖h‖∞≤H\n|Q̂n(h)|\nR(h)\n\n . (10)\nUsing |Q̂n(h)| ≤ βn−1|Q̂(h)| and equation (9),\n∑\n0<‖h‖∞≤H\n|Q̂n(h)|\nR(h)\n≤ βn−1\n∑\n0<‖h‖∞≤H\nk∏\ni=1\n{\n| sin(2πhiA)|\n2πh2\ni\nA\nif hi 6= 0\n1 if hi = 0\n}\n≤ βn−1\n\n\n(\n1 + 2\nH∑\nh=1\n1\n2πh2A\n)k\n− 1\n\n\n≤ βn−1\n(\n1 +\nπ\n6A\n)k\n.\nHence, letting H →∞ in (10), we have\nD(Q∗n) ≤\n(\n6Aπ + π2\n2 sin(2πA)\n)k\nβn.\n457\nTherefore the discrepancy convergence of this random walk is geometric in n\nwith rate β = | sin(2πA)\/2πA|, as claimed. ut\nA more refined lower bound may be obtained by taking more terms in Corol-\nlary 3 or Theorem 2. Note, however, that the extra terms in Theorem 2 and\nin the Erdös-Turán-Koksma inequality involve Q̂n(h) and thus become neg-\nligible in comparison to their dominant term(s) when the number of steps\nn is large. This is true for any random walk on the torus. The matching\ndominant terms of these bounds therefore make discrepancy an effective tool\nto measure how quickly random walks on the torus converge to the uniform\ndistribution.\nAcknowledgements\nI would like to thank Persi Diaconis and Jerry Spanier for helpful discussions,\nand Jeff Holt and Ken Ross for useful feedback.\nReferences\n1. G. Arfken, Mathematical Methods for Physicists, 3rd ed., Academic Press,\nNew York, 1985.\n2. J. Beck, On a problem of K. F. Roth concerning irregularities of point distri-\nbution, Invent. Math. 74(1983), 477-487.\n3. J. Beck and W. Chen, Irregularities of Distribution, Cambridge University\nPress, 1987.\n4. M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lec-\nture Notes in Math. 1651, Springer, 1997.\n5. J. J. Holt, On a form of the Erdös-Turán inequality, Acta. Arith. 74 (1996),\n61-66.\n6. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley,\nNew York, 1974.\n7. W. J. LeVeque, An inequality connected with Weyl’s criterion for uniform\ndistribution, Proc. Symp. Pure Math. 8(1965), 22-30.\n8. H. Niederreiter, Application of diophantine approximations to numerical in-\ntegration, in C. F. Osgood (Ed.), Diophantine Approximation and its Appli-\ncations, Academic Press, New York, 1973, pp. 129-199.\n9. F. E. Su, Methods for Quantifying Rates of Convergence for Random Walks\non Groups, Ph.D. thesis, Harvard University, 1995.\n10. F. E. Su, Convergence of random walks on the circle generated by an irrational\nrotation, Trans. Amer. Math. Soc., 350(1998), pp. 3717-3741.\n458\n"}
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{"id":"_tALgp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/bicycle\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":42,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","bicycle"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["bicycle – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:16 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Thu Jun 27 00:00:00 UTC 2019","Wed Jun 26 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-3331698066172306051,"exact_signature_unique_b":true,"fuzzy_signature_l":-5531612098117152171,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["bicycle"],"h2_txt":["Dominoes on a Chessboard","Pizza Slices","Bike with Square Wheels"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Marching.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Marching 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":775,"text_t":"bicycle – Math Fun Facts. 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{"id":"-c4oKW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Discrepancy-convergence.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":87,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Discrepancy-convergence","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["paper2.dvi"],"author":"Acrobat Distiller Command 3.02b for Solaris 2.3 and later (SPARC)","author_sxt":["Acrobat Distiller Command 3.02b for Solaris 2.3 and later (SPARC)"],"last_modified":"Thu Jun 13 04:34:17 UTC 2019","dates_in_content_count_i":2,"dates_in_content_dts":["Thu Jun 01 00:00:00 UTC 2000","Mon Feb 19 00:00:00 UTC 2001"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-5092372475016562273,"exact_signature_unique_b":true,"fuzzy_signature_l":-8037019428991407835,"fuzzy_signature_unique_b":true,"wordcount_i":7688,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:18 UTC 2022","fresh_date_dt":"Wed Dec 06 11:46:18 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"Acrobat Distiller Command 3.02b for Solaris 2.3 and later (SPARC)","language_s":"en","size_i":188163,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088079461416960,"text_t":"E l e c t r\no n\ni\nc\nJ\no\nu\nr n\na l\no\nf\nP\nr\no b a b i l i t y\nVol. 6 (2001) Paper no. 2, pages 1–20.\nJournal URL\nhttp:\/\/www.math.washington.edu\/~ejpecp\/\nPaper URL\nhttp:\/\/www.math.washington.edu\/~ejpecp\/EjpVol6\/paper2.abs.html\nDISCREPANCY CONVERGENCE FOR THE DRUNKARD’S WALK ON THE SPHERE\nFrancis Edward Su\nDepartment of Mathematics\nHarvey Mudd College\nClaremont, CA 91711\nsu@math.hmc.edu\nhttp:\/\/www.math.hmc.edu\/~su\/\nAbstract We analyze the drunkard’s walk on the unit sphere with step size θ and show that\nthe walk converges in order C\/ sin2 θ steps in the discrepancy metric (C a constant). This is an\napplication of techniques we develop for bounding the discrepancy of random walks on Gelfand\npairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group\nadmits tractable computations involving spherical functions. We advocate the use of discrepancy\nas a metric on probabilities for state spaces with isometric group actions.\nKeywords Discrepancy, random walk, Gelfand pairs, homogeneous spaces, Legendre polyno-\nmials\nAMS subject classification Primary. 60B15; Secondary. 43A85\nSubmitted to EJP on June 1, 2000. Final version accepted on February 19, 2001.\n1 Introduction\nFix θ ∈ (0, π). Consider the following random walk on the unit sphere S2 in R3, whose steps\nare geodesic arcs of length θ. (Such arcs subtend an angle of θ at the center of the sphere).\nThe random walk starts at the north pole, and at each step a uniformly random direction is\nchosen and the walk moves a geodesic distance θ in that direction. We refer to this walk as the\ndrunkard’s walk on the sphere.\nThe purpose of this paper is to develop techniques for bounding the discrepancy metric for\nrandom walks on Gelfand pairs, using the drunkard’s walk as an example. Our bounds are\nsharp enough to give a rate of convergence. Let D(k) denote the discrepancy distance (defined\nlater) between the k-th step probability distribution of the drunkard’s walk and the uniform\n(rotation-invariant) measure on S2. We show the following:\nTheorem 1 For the drunkard’s walk on the unit sphere S2 with step size θ, the discrepancy of\nthe walk after k steps satisfies, for k = C\nsin2 θ\n,\n0.4330 e−C\/2 ≤ D(k) ≤ 4.442 e−C\/8.\nThus order C\/ sin2 θ steps are both necessary and sufficient to make the discrepancy distance of\nthis walk from its limiting distribution uniformly small. The result makes intuitive sense, since\nthe number of steps to random should be large when θ is close to 0 or π, and small when θ is\nclose to π\/2. Moreover, if θ ≈ 1\/n for large n, then this result shows that order n2 steps are\nnecessary and sufficient; this is similar to nearest-neighbor random walks on Z\/nZ (e.g., see [4]).\nWe also note that given θ this walk does not exhibit a sharp cutoff phenomenon.\nWe frame our analysis in the context of a random walk on a homogeneous space, i.e., a space\nwith a transitive group action. S2 is a homogeneous space by the action of SO(3). Although\nthe drunkard’s walk is not generated by a group action, we show its equivalence with a walk\nthat is.\nFor random walks on groups, Fourier analysis is often used to obtain rates of convergence. On\nhomogeneous spaces, we can lift the walk to acting group and do Fourier analysis there, although\nfor non-commutative groups the group representations can be quite complicated. However, when\nthe homogeneous space is a Gelfand pair (as in this case), Fourier transforms of bi-invariant mea-\nsures and functions on the group simplify greatly, allowing for tractable computations involving\nthe spherical functions. In our example, the generating measure on SO(3) can be made bi-\ninvariant, and the spherical functions are Legendre polynomials.\nMuch of the literature on random walks on Gelfand pairs is limited to discrete homogeneous\nspaces. Diaconis [4] presents a survey and an annotated bibliography; applications include walks\non subspaces of vector spaces over finite fields [11] and walks on r-sets of an n-set [5]. Rates are\ngiven in the total variation metric.\nContinuous examples have been addressed by Voit, who studied families of isotropic random\nwalks on spheres [23, 25] and other homogeneous spaces [24]. Central limit theorems are obtained\nusing convergence in distribution or total variation as the dimension n → ∞. Such results differ\nfrom ours in that: (1) we work with a specific walk rather than a family, e.g., we obtain explicit\nbounds for a specific n rather than asymptotic results for large n, (2) we focus on rates of\nconvergence of the walk on the homogeneous space, rather than convergence of a central limit\n2\ntheorem on the double coset space as in [23, 24], and (3) we use the discrepancy metric to measure\nconvergence. We argue that it is a natural metric to use for walks on homogeneous spaces, and\ndevelop techniques to bound it. While we only illustrate our methods on the 2-sphere, similar\nmethods can be used to give explicit discrepancy bounds for walks on high-dimensional spheres\nand other Gelfand pairs.\nFor walks on continuous groups, we mention the work of Rosenthal [19] and Porod [17, 18],\nwho obtain total variation rates of convergence for random walks on SO(n) and other compact\ngroups where the generating measures are conjugate-invariant; this is another situation where\nthe representations simplify enough to get Fourier bounds.\nThis paper is organized as follows. Section 2 gives background on the discrepancy metric and\njustifies its use over other common metrics on probabilities. Section 3 develops several equivalent\nformulations for the drunkard’s walk. Section 4 gives a formulation with a bi-invariant generating\nmeasure. This simplifies the Fourier analysis in Section 5, where matching upper and lower\nbounds for the convergence rate of the drunkard’s walk are derived (Theorems 9 and 10). We\nsummarize our methods for handling random walks on arbitrary Gelfand pairs in Section 6.\nFor the uninitiated, Appendix A collects relevant background on Fourier analysis on groups,\nGelfand pairs, and representations of SO(3) that are needed to make this paper self-contained.\nAppendix B contains proofs of technical results that are not central to the development of the\nideas in this paper.\nAcknowledgement The author thanks Persi Diaconis for suggesting this problem, Ken Ross\nfor useful feedback, and the Cornell School of Operations Research for hospitality during a\nsabbatical where this was completed.\n2 The discrepancy metric\nLet (X, d) be a metric space with metric d. Given any two probability measures P,Q on X,\ndefine the discrepancy distance between P and Q by:\nD(P,Q) = sup\nall balls B\n|P (B) − Q(B)|\nwhere a “ball” in X denotes any subset of the form {x : d(x, x0) ≤ r} for some x0 ∈ X and real\nnumber r ≥ 0. It is easy to check that the discrepancy is a metric on probability measures.\nWhen X is the unit cube in Rn and Q is Lebesgue measure on X, this definition reduces to\nthe notion of discrepancy commonly used by number theorists to study uniform distribution of\nsequences in the unit cube (e.g., see [8, 16]). Diaconis [4] was perhaps the first to suggest the\nuse of discrepancy to measure rates of convergence of random walks. Su [20, 21, 22] explored\nproperties of this metric and obtained sharp rates of convergence for certain random walks on\nthe hypercube, circle, and torus.\nWe shall be concerned with the case where X = S2 and the metric on S2 is inherited from its\ninclusion in R3. Thus balls may be visualized as spherical “caps” on the sphere. As noted later,\nthe group of rotations SO(3) acts on S2 in a natural way and the metric on S2 is invariant under\nthis action. Thus images of balls under this action are still balls, so the discrepancy metric on\nmeasures inherits this rotation invariance.\n3\nUnlike the total variation metric (which is in more frequent use among probabilists), the dis-\ncrepancy metric recognizes both the topology and the group action on the underlying space.\nFor infinite compact state spaces this can be important. For instance, if P is a probability\nmeasure on S2 supported on a finite set of points, and Q is the uniform (rotation-invariant)\nprobability measure, then the total variation distance between P and Q remains equal to 1 no\nmatter how the points are arranged. On the other hand, the discrepancy D(P,Q) will capture\nhow “well-distributed” the points in P are. Another example is a simple random walk on the\ncircle generated by an irrational rotation (see [21]), which converges weak-* to Haar measure;\ndiscrepancy captures this convergence, but total variation is blind to it. Thus the discrepancy\nmetric is well-suited to studying random walks on continuous state spaces generated by isometric\ngroup actions.\nWe favor the use of discrepancy over other common metrics (e.g., the Prokhorov, Wasserstein\nmetrics) because there are tractable bounding techniques for discrepancy involving Fourier co-\nefficients. In fact, one of the main goals of this paper is to show that we can develop upper and\nlower bounds for discrepancy which give sharp rates of convergence in many cases because the\ndominant terms in each expression match.\nWe remark that discrepancy bounds can be used to bound other metrics by exploiting known re-\nlationships between them [10]; for instance, the discrepancy is bounded above by total variation,\nso discrepancy lower bounds also offer a way to obtain total variation lower bounds.\nA property that will be needed later is that on groups, discrepancy decreases with convolution:\nTheorem 2 If P,Q, ν are arbitrary probability measures on a compact group G, then\nD(P ∗ ν,Q ∗ ν) ≤ D(P,Q).\nHence when Q = U , the uniform (Haar) measure, we have\nD(P ∗ ν, U) ≤ D(P,U). (1)\nSee [21] for a proof.\n3 The Drunkard’s Walk and Equivalent Formulations\nThe drunkard’s walk on the sphere is not a random walk generated by a group action. However,\nwe show in this section that it is equivalent to one that is, in the sense that the two random\nwalks generate the same k-th step probability distribution even though their observed behaviors\nmay appear quite different.\nReaders familiar with hypergroups may not be surprised by the equivalence and the ensuing\nanalysis, since the associated double coset space of this walk is a commutative hypergroup, and\nmuch of our analysis can be framed in that language. We have avoided it; interested parties are\nreferred to [2].\nLet N be the isotropy subgroup of SO(3) fixing n, the north pole. Let E ⊂ SO(3) denote the set\nof all rotations which fix a point on the equator and move the north pole by geodesic distance θ\nalong the surface of the sphere. Let Q denote the probability distribution supported on E that\nis left N -invariant.\n4\nFormulation 1. The Drunkard’s Walk. This is the walk considered at the opening of this\npaper; a drunkard starts at the north pole and at each step picks a uniformly random direction\nand advances along the sphere in that direction by geodesic distance θ.\nLet Yk for k = 0, 1, 2, ... denote random variables which describe the location of the drunkard\nat time k. Thus Y0 = n. If gi is an SO(3)-valued random variable with values in E and\ndistribution Q, the position of the drunkard at time k is given by Yk = g1g2 · · · gkn. Thus, this\nwalk is not a random walk in which the next position is generated by applying group actions to\nthe current position in the walk. However, the following random walk is:\nFormulation 2. The Potted Plant. Consider a potted plant initially at the north pole. At\neach step, a rotation is chosen randomly from E according to Q and performed on the sphere.\nThus the point currently over the north pole moves a distance θ in any direction. This induces\na motion of the potted plant, wherever it currently is.\nNote that with the given generating set E, the potted plant is moved a geodesic distance less\nthan or equal to θ at every step, since for each rotation in E, n is on the equator of the rotation\naxis and hence moves the farthest.\nIf gi is a SO(3)-valued random variable with distribution Q, the position of the potted plant at\ntime k is given by Yk = gkgk−1 · · · g1n. Since the gi are independent and identically distributed,\nthis shows that Formulations 1 and 2 are equivalent and generate the same k-th step probability\ndistribution on the sphere. This may be surprising in light of the fact that the steps of the\nrandom walk in Formulation 2 are smaller than in Formulation 1.\nThe next random walk, while not essential in what follows, also generates the same k-th step\nprobability distribution and we mention it for the sake of interest.\nFormulation 3. Rotate and Spin. Fix any rotation Rθ which displaces the current north pole\nby geodesic distance θ. Start the random walk at the north pole, and at each step perform Rθ\nfollowed by a uniform spin around the north-south axis. (The uniform spin moves the random\nwalk to a random point anywhere on the same latitude.)\nThough Rθ is not necessarily contained in the set E defined earlier, it does yield the same k-th\nstep probability distribution as the previous formulations. This may be seen as follows.\nConsider the double coset space SO(3)\/\/N . Each double coset is characterized by the lat-\nitude to which it sends the north pole. Thus Rθ = n\n′g0n′′ for some n′, n′′ ∈ N and a\ng0 ∈ E. Then E = Ng0. Let ni denote an N -valued random variable distributed accord-\ning to Haar measure on N . The walk description shows that at the i-th step, nin\n′g0n′′\nacts on the random walk’s current position. Therefore its position at time k is given by\nYk = (nk n\n′g0n′′)(nk−1 n′g0n′′) · · · (n1 n′g0n′′)n. Since nin′g0 and n′′nin′g0 are identically\ndistributed according to Q and n′′n = n, the above random variable has the same k-th step\ndistribution as the other formulations above.\nOur original goal was to study Formulation 1, the drunkards’ walk. Via the above equivalence\nwe choose instead to study Formulation 2, because it is a random walk generated by a group\naction. However, the generating measure Q, while left N -invariant, is not bi-invariant. In light\nof Theorem 12, a bi-invariant generating measure would greatly simplify the ensuing Fourier\nanalysis. (See Appendix A for background material on Fourier analysis on compact groups and\nbi-invariant measures). In the next section, we remedy this problem by introducing a fourth\n5\nrandom walk (Formulation 4) which is equivalent to Formulation 2 and whose generating measure\nis bi-invariant.\n4 A Bi-invariant Formulation\nWe are interested in the discrepancy distance between the k-th step distribution of the drunkard’s\nwalk and US2, the uniform (rotation-invariant) distribution on S\n2. To simplify notation, we write\nD(k) = D(L(Yk), US2) (2)\nwhere L(Yk) denotes the distribution of the random variable Yk in Formulation 2. We investi-\ngate the behavior of D(k) as a function of the number of steps k.\nRecall that the homogeneous space S2 can be regarded as the left cosets of N in SO(3), so that\nthe quotient map SO(3) → S2 sends a rotation g to the point gn ∈ S2. A random walk on\nS2 generated by an SO(3)-action (such as Formulation 2) may then be regarded as a random\nwalk “upstairs” on SO(3) with an initial distribution UN , Haar measure on N (which is the\npre-image of the starting point n). The probability distribution upstairs evolves as usual for a\nrandom walk on a group, so that after one step the distribution is given by Q ∗ UN and after k\nsteps by Q∗k ∗ UN . The probability of finding the original walk in a ball B ⊂ S2 is the same as\nfinding the lifted walk on SO(3) in B̃ = BN ⊂ SO(3). Hence\nD(k) = sup\nB̃\n|Q∗k ∗ UN (B̃) − U(B̃)|. (3)\nwhere U is Haar measure on SO(3) and the supremum is taken over all B̃, pre-images of balls\nunder the quotient map SO(3) → S2.\nAt this point we would appeal to Fourier analysis to deal with the convolutions above. However,\nQ is left N -invariant but not bi-invariant; recall that we desire bi-invariance to simplify the\nFourier analysis.\nThe following proposition shows that for random walks on groups, averaging the generating\nmeasure Q to make it bi-invariant will affect the rate of convergence in discrepancy by at most\none step. This result is the analogue of a result of Greenhalgh [11], who obtained a similar result\nfor the total variation distance.\nProposition 3 Let Q denote any left N -invariant probability measure on a group G, let U and\nUN denote Haar measure on G and N respectively. If Q̄ = Q ∗ UN , then Q̄ is N -bi-invariant\nand\nD(Q̄∗k, U) ≤ D(Q∗k, U) ≤ D(Q̄∗(k−1), U).\nProof Left invariance for Q means UN ∗ Q = Q. We use this to establish bi-invariance for\nQ̄, which means UN ∗ Q̄ ∗ UN = Q̄. This follows from UN ∗ Q̄ ∗ UN = UN ∗ (Q ∗ UN ) ∗ UN =\n(UN ∗ Q) ∗ (UN ∗ UN ) = Q ∗ UN = Q̄.\nFor the second assertion, note that\nQ̄∗k = (Q ∗ UN )∗k = Q ∗ (UN ∗ Q)∗(k−1) ∗ UN = Q∗k ∗ UN .\nQ∗k = Q ∗ (UN ∗ Q)∗(k−1) = (Q ∗ UN )∗(k−1) ∗ Q = Q̄∗(k−1) ∗ Q.\n6\nUsing Theorem 2 we obtain D(Q∗k∗UN , U) ≤ D(Q∗k, U) and D(Q̄∗(k−1)∗Q,U) ≤ D(Q̄∗(k−1), U),\nwhich with the above equations yield the desired conclusion. \u0003\nThus a random walk on a group with generating measure Q differs by no more than one step\nfrom a random walk proceeding according to Q̄, which may be viewed as the average of the\nmeasure Q over the left cosets of N .\nHowever, for a random walk on a homogeneous space, even more can be said if the walk begins\nat the point fixed by the isotropy subgroup:\nProposition 4 Suppose X be a homogeneous G-space with isotropy subgroup N fixing x0 ∈ X,\nand Q is a left-invariant probability on G. Let D(k) denote the discrepancy of the random walk\nstarting at x0 and evolving via a group action with elements chosen according to Q. Let D̄(k)\ndenote the discrepancy of the random walk starting at x0, but evolving according to Q̄. Then\nD̄(k) = D(k).\nProof This follows from the fact shown in the previous proof, that\nQ̄∗k = Q∗k ∗ UN .\nThe right side, when regarded as a measure on S2, describes the location of the Q-generated\nwalk. But by the right invariance of Q̄, the left side is equal to Q̄∗k ∗UN , which when regarded\nas a measure on S2, describes the location of the Q̄-generated walk. \u0003\nThis shows that the following is equivalent to Formulation 2.\nFormulation 4. Let Ē denote the set of all rotations in SO(3) which move the north pole n by\na fixed angle θ. Let Q̄ = Q ∗ UN be the bi-invariant generating measure obtained by averaging\nQ from Formulation 2. Consider a potted plant which starts at n and is moved according to the\nfollowing rule: at each step, a rotation is chosen randomly from Ē according to Q̄ and performed\non the sphere. This induces a motion of the potted plant to a new location.\nObserve that we are able to throw extra rotations in “for free” and still obtain the same k-th\nstep probability distribution. This may be surprising because with the extra generating elements\nthe step size of the potted plant is no longer bounded by θ, as it was in Formulation 2. In fact,\nthe potted plant could be moved around rather wildly at each step.\nExploiting this equivalence, we shall, in the sequel, work with Formulation 4. To save notation\nwe write Q for the bi-invariant measure Q̄. Right-invariance for Q yields Q = Q ∗ UN , which\nwhen substituted into (3) gives\nD(k) = sup\nB̃\n|Q∗k(B̃) − U(B̃)|. (4)\nwhere the supremum is taken over all ball pre-images B̃. Hence, the discrepancy D(k) as defined\nin (2) can now be analyzed using expression (4).\n7\n5 A Rate of Convergence\nWe now proceed to derive a rate of convergence for the drunkard’s walk on the sphere. Sev-\neral calculations require the facts reviewed in Appendices A and B; we alert the reader with\nreferences.\nLet By,r denote a ball of geodesic radius r centered at y ∈ S2. Such balls look like spherical\n“caps” on S2. Let B̃y,r denote its pre-image “upstairs” in SO(3). To reduce notation, write\nB̃r = B̃n,r for the pre-image of a ball centered around n. Let δr denote the indicator function\nof B̃r on SO(3).\nA key observation (see [1]) for evaluating measures on balls is they can be regarded as convolu-\ntions with indicator functions on those balls, i.e., for any right-invariant measure ν on SO(3),\nν(B̃y,r) = ν ∗ δr(ỹ) (5)\nfor any ỹ ∈ yN . This follows from ν ∗ δr(ỹ) =\n∫\ng∈B̃r dν(ỹg\n−1) = ν(ỹ · B̃r).\nFrom (4) we have\nD(k) = sup\ny,r\n∣∣∣Q∗k(B̃y,r) − U(B̃y,r)∣∣∣\n= sup\ny,r\n∣∣∣Q∗k ∗ δr (ỹ) − U(B̃y,r)∣∣∣ . (6)\nWe wish to use Fourier inversion to derive bounds for these expressions in terms of the Fourier\ncoefficients. We need continuity of Q∗k ∗ δr for k ≥ 2:\nProposition 5 Let Q be defined as in Formulation 4, and let δr be denote the indicator function\nof B̃r. Then Q\n∗k ∗ δr is continuous for k ≥ 2.\nThis is proved in Appendix B.\nHereafter, assume k ≥ 2. We shall also assume for the moment that Q∗k ∗ δr has an absolutely\nconvergent Fourier series, which will be verified later in the course of our computations. Since\nQ∗k ∗ δr is a continuous function for k ≥ 2, it is exactly equal to its Fourier series (Theorem 11),\nso that from (6) and (16) we have\nD(k) = sup\ny,r\n∣∣∣∣∣\n∞∑\nn=1\n(2n + 1) Tr\n[\nQ̂k(ρn) δ̂r(ρn) ρn(ỹ)\n]∣∣∣∣∣\n≤ sup\ny,r\n∞∑\nn=1\n(2n + 1)\n∣∣∣ Tr [Q̂k(ρn) δ̂r(ρn) ρn(ỹ)]∣∣∣ (7)\nwhere ρn is the irreducible representation of SO(3) of dimension (2n + 1). The trivial represen-\ntation ρ0 does not appear here since it was cancelled in (6) by U(B̃y,r).\nRemark 6 Since Q and δr are both N -bi-invariant on SO(3), by Theorem 12 there is a basis for\nthe representations such that their transforms are identically zero except in the (1, 1)-th entry.\nAny such basis (e.g., the spherical harmonics) has its first basis element given by the Legendre\npolynomials, which are the spherical functions for the Gelfand pair (SO(3), N).\n8\nHence ρn(ỹ)(1,1) = Pn(cos γ), where γ is the geodesic distance of y from n. Since the product\nof the transforms of Q̂k and δr are identically zero except for the (1, 1)-th element, the only\ndiagonal element changed by multiplication by ρn(ỹ) is the (1, 1)-th entry. Hence the trace (7)\nreduces to\nD(k) ≤ sup\ny,r\n∞∑\nn=1\n(2n + 1)\n∣∣∣Q̂k(ρn)(1,1) δ̂r(ρn)(1,1) Pn(cos γ)∣∣∣\n≤ sup\nr\n∞∑\nn=1\n(2n + 1)\n∣∣∣Q̂k(ρn)(1,1) δ̂r(ρn)(1,1)∣∣∣ (8)\nwhere the second inequality follows from (22). Notice that the sum in (8) is precisely the sum in\nTheorem 11 that needs to be checked for convergence in verifying that Q∗k ∗δr has an absolutely\nconvergent Fourier series. Hence when we bound the above expression we will also have validated\nour use of Fourier inversion in our computations.\nFrom (20), we have\nQ̂(ρn)(1,1) = Pn(cos θ) (9)\nsince Pn is constant on the support of Q. Also, for a ball Br of geodesic radius r and n ≥ 1,\nformula (21) gives\n∣∣∣δ̂r(ρn)(1,1)∣∣∣ =\n∣∣∣∣12\n∫ 1\ncos r\nPn(x) dx\n∣∣∣∣\n=\n|Pn−1(cos r) − Pn+1(cos r)|\n2(2n + 1)\n(10)\n≤ 1\n2n + 1\n. (11)\nThe integral of Pn follows from (23) and noting that Pn(1) = 1, and the inequality follows from\n(22).\nSubstitution of (9) and (11) into (8) yields\nD(k) ≤\n∞∑\nn=1\n|P kn (cos θ)|. (12)\nTo bound the Legendre polynomials, we use the following well-known bound (see Jackson [14,\np.63]):\nProposition 7 For Pn, the n-th Legendre polynomial, and any θ,\n|Pn(cos θ)|2 ≤ 2\nπn sin2 θ\n.\nWe derive an alternate bound, suitable for small θ:\nProposition 8 For Pn, the n-th Legendre polynomial, and n sin\n2 θ ≤ .9,\n|Pn(cos θ)|2 ≤ 1 − n sin\n2 θ\n4\n.\n9\nThis bound is better than Proposition 7 when n sin2 θ < 2 −\n√\n4 − 8π ≈ .794. It is proved in\nAppendix B. Using Propositions 7 and 8 and the bound 1 − x ≤ e−x, the sum in (12) can be\nestimated:\n∞∑\nn=1\n|P kn (x)| ≤\n∑\nn≤B\/ sin2 θ\ne−nk sin\n2 θ\/8 +\n∑\nn>B\/ sin2 θ\n(\n2\nπn sin2 θ\n)k\/2\n≤ e\n−k sin2 θ\/8\n1 − e−k sin2 θ\/8 +\n(\n2\nπ sin2 θ\n)k\/2 ∑\nn>B\/ sin2 θ\n1\nnk\/2\nwhere B = .9. Note that\n∑\nn>B\/ sin2 θ\n1\nnk\/2\n≤\n∫ ∞\nB\nsin2 θ\ndx\nxk\/2\n+\n(\nsin2 θ\nB\n) k\n2\n=\n2\nk − 2\n(\nsin2 θ\nB\n) k\n2\n−1\n+\n(\nsin2 θ\nB\n) k\n2\n.\nThus\n∞∑\nn=1\n|P kn (x)| ≤\ne−k sin2 θ\/8\n1 − e−k sin2 θ\/8 +\n(\n2\nπB\n)k\/2 (\n2B\n(k − 2) sin2 θ + 1\n)\n.\nNote that\n(\n2\nπB\n)1\/2\n< e−1\/8 < e− sin2 θ\/8. For k = C\nsin2 θ\nand C ≥ 4, one sees that (k−2) sin2 θ ≥ 2\nand k ≥ 4, so that\n∞∑\nn=1\n|P kn (x)| ≤ e−k sin\n2 θ\/8\n(\n1\n1 − e−1\/2 + B + 1\n)\n≤ 4.442 e−C\/8.\nThe above bound, together with (12), proves the following theorem. (Note that the C ≥ 4\nrestriction above is not needed below because the discrepancy D(k) never exceeds 1.)\nTheorem 9 For the drunkard’s walk on the sphere with step size θ, the discrepancy after k\nsteps satisfies, for k = C\nsin2 θ\n,\nD(k) ≤ 4.442 e−C\/8.\nThus order C\nsin2 θ\nsteps are sufficient to make the discrepancy uniformly small. The following\nlower bound confirms the order is correct.\nTheorem 10 For the drunkard’s walk on the sphere with step size θ, the discrepancy after k\nsteps satisfies, for k ≥ 2,\nD(k) ≥\n√\n3\n4\n| cos θ|k.\nFor k = C\nsin2 θ\n, we have\nD(k) ≥ 0.4330 e−C\/2.\n10\nThus order C\nsin2 θ\nsteps are needed to make the discrepancy distance uniformly small. Together,\nTheorems 9 and 10 prove Theorem 1.\nOne way to obtain a lower bound for discrepancy is to evaluate the difference of Q∗k and U on\nwell-chosen ball. The same idea can be used for the total variation; one way to choose such a\nball (see [4, p.29]) is to take a set cut out by a random variable consisting of the dominant terms\nin the Fourier series of Q∗k. The mean and variance of the random variable and an appeal to\nChebyshev’s inequality yield an estimate for Q∗k on that set.\nHowever, the proof of Theorem 10 illustrates a different approach using ideas similar to those\nused in [22] for bounds on the torus. We construct a “local discrepancy” function which at each\npoint evaluates the discrepancy of the measure on a set of geodesic radius r centered at that\npoint. The function is bounded above by the total discrepancy. As before, it can be rewritten in\nterms of a convolution of the original measure and the indicator function of the set. An appeal\nto Plancherel’s identity gives a sum with only non-negative terms, so the dominant term can be\npulled out as a lower bound for discrepancy.\nWe remark that since discrepancy is a lower bound for total variation, this lower bounding\ntechnique can also be used to obtain lower bounds for random walks under total variation.\nProof Define, for g ∈ SO(3),\n∆r(g) = Q\n∗k(B̃y,r) − U(B̃y,r)\nwhere y is the image of g under the quotient map from SO(3) to S2. From (4), we see that\n∆r(x) ≤ D(k), and hence for all r,∫\nSO(3)\n∆2r(g) dµ ≤ D(k)2. (13)\nOn the other hand, Plancherel’s identity on SO(3) [9, p.256] yields∫\nSO(3)\n∆2r(g) dµ =\n∞∑\nn=0\n(2n + 1) Tr\n[\n∆̂r(ρn)∆̂r(ρn)\n∗\n]\n(14)\nwhere the ∗ denotes the conjugate transpose (here only). Notice that ∆r may be rewritten as:\n∆r(x) = Q\n∗k ∗ δB̃r (x) − U ∗ δB̃r(x) = (Q∗k − U) ∗ δB̃r (x).\nThen ∆̂r may be computed as ∆̂r(ρn) = ( Q̂\nk(ρn) − Û(ρn) ) δ̂B̃r (ρn).\nFor n = 0, ∆̂r(ρ0) = 0 since Q̂\nk(ρ0) − Û(ρ0) = 1 − 1 = 0.\nFor n 6= 0, a trivial computation shows Û(ρn) = 0, and thus\n∆̂r(ρn) = Q̂\nk(ρn) δ̂B̃r (ρn).\nRemark 6 and the computations from Equations (9) and (10), when substituted into (14), and\ncombined with (13), give\nD(k)2 ≥\n∞∑\nn=1\n(2n + 1)\n∣∣∣∣P kn (cos θ)\n(\nPn−1(cos r) − Pn+1(cos r)\n2(2n + 1)\n)∣∣∣∣2 (15)\n11\nwhere r may be chosen arbitrarily. Taking only the dominant term (n = 1) in the above\nexpression, and letting r = π\/2, we have cos r = 0, P0(0) = 1, P2(0) = −1\/2, and P1(x) = x. It\nfollows that\nD(k) ≥\n√\n3\n4\n| cos θ|k\nas was to be shown. The second inequality in the theorem follows from | cos θ|k = ek2 ln cos2 θ ≥\ne−\nk\n2\nsin2 θ, using the fact that ln(1 − x) ≥ −x for all x. \u0003\nFor a tighter lower bound, one may use more terms in (15) and adjust the choice of r; however,\nthe dominant term sufficed to obtain matching upper and lower bounds for this random walk.\n6 Conclusion\nA similar analysis can be carried out for the discrepancy convergence of any random walk on\na Gelfand pair, when the spherical functions are known. Proposition 4 shows that making\na generating measure bi-invariant will not affect the rate of convergence. The upper bound\nis obtained via (5), Fourier inversion to yield (12), and bounds on the appropriate spherical\nfunction (e.g., Prop. 8). The lower bound is obtained via Plancherel’s identity applied to the\nsquare of the local discrepancy function, e.g., equations (13) and (14), then choosing as many\nterms as needed.\nWe remark that our pair of strategies often works well for obtaining matching upper and lower\nbounds because if there is a dominant Fourier coefficient, it appears to the same order in both\nupper and lower bounds. In our example, Q̂(1) was the dominant term; compare the upper\nbound (12) and lower bound (15). See [12, 21, 22] for more examples of this phenomenon in\ndiscrepancy bounds for random walks on groups.\nAppendix A.\nThis appendix reviews material on harmonic analysis, homogeneous spaces, Gelfand pairs, rep-\nresentations of SO(3), and Legendre polynomials.\nFourier Analysis on a Compact Group\nA standard reference is the encyclopedic account by Hewitt and Ross [13]. Diaconis [4] gives\na concise introduction to Fourier analysis on finite groups. Dym and McKean [9] is a readable\nintroduction to Fourier series on SO(3).\nWe assume henceforth that all compact groups are separable and metrizable. For any compact\ngroup G there is a unique measure µ on G, called (normalized) Haar measure, such that µ is\nG-invariant and µ(G) = 1.\nLet V be a finite dimensional vector space over C, the complex numbers. Recall that a repre-\nsentation of a group G on V is a homomorphism ρ : G → GL(V ). If V has dimension n, then\nρ is said to have dimension n. A basis for V can be chosen so that the image of ρ with respect\n12\nto this basis are unitary matrices. If there is no non-trivial subspace of V invariant under the\naction of G, then ρ is said to be irreducible; otherwise ρ decomposes as a direct sum of irreducible\nrepresentations. (One can similarly define a representation ρ of G on a Hilbert space, though if\nG is compact ρ decomposes into a direct sum of unitary representations of finite dimension.)\nTwo representations ρ on V and ρ′ on V ′ are equivalent if there is an isomorphism τ : V → V ′\nsuch that τ ◦ρ = ρ′◦τ . Let Σ denote a the set of equivalence classes of irreducible representations\nof G. For a compact group, Σ is countable and furthermore, all the irreducible representations\nare finite dimensional.\nDefinition 1 The Fourier transform of a complex-valued function f on a compact group G at\na representation ρ of G is defined by\nf̂(ρ) =\n∫\ng∈G\nf(g) ρ(g−1) dµ\nwhere µ is Haar measure on G.\nSimilarly, the Fourier transform of a measure ν on G at ρ is defined by\nν̂(ρ) =\n∫\ng∈G\nρ(g−1) dν(g).\nWe show how a function may be recovered from its Fourier transforms at irreducible representa-\ntions. Let dρ denote the dimension of a representation ρ. For any operator A, let Tr[A] denote\nthe trace of A, and let ‖A‖ϕ1 denote the sum of the eigenvalues of the operator square root of\nAA∗. (Here, ∗ denotes conjugate transpose.)\nDefinition 2 For any f ∈ L1(G,µ), the series∑\nρ∈Σ\ndρ Tr\n[\nf̂(ρ) ρ(g)\n]\n(16)\nis called the Fourier series of f . (There is mild abuse of notation here: by ρ ∈ Σ, we really\nmean to choose a representative ρ from each class of irreducible representations in Σ.) If∑\nρ∈Σ\ndρ ‖f̂(ρ)‖ϕ1 < ∞ , (17)\nthen f is said to have an absolutely convergent Fourier series [13, (34.4)].\nTheorem 11 (Fourier inversion) If a function f on G has an absolutely convergent Fourier\nseries, then the Fourier series of f(g) converges uniformly to a continuous function f̄(g), and\nf(g) = f̄(g) almost everywhere on G with respect to Haar measure µ.\nProof This theorem is embedded in Hewitt and Ross [13], but obscured by their exotic notation.\nWe briefly indicate how to “prove” this theorem from results cited in [13].\nThe set of functions with absolutely convergent Fourier series is denoted in [13] by a symbol\nthat resembles R(G), defined in (34.4). Theorem (34.6) in [13] shows that any f ∈ R(G) is\nequal almost everywhere to its Fourier series. Theorem (34.5.ii) shows that this Fourier series\n13\nconverges uniformly to a continuous function that we have denoted f̄ . \u0003\nWe remark that since the notation in Hewitt and Ross [13] is cumbersome and tough to wade\nthrough, for the sake of probabilists we have simplified it by following the notation of Diaconis\n[4]. To aid the reader wishing to follow the results quoted above, we provide a “dictionary”\nbetween the two sets of notation: in Hewitt and Ross [13, (27.3)], σ denotes a class of equivalent\nirreducible representations in Σ and U is a representative of that class; we avoid reference to σ\n(to eliminate an unnecessary layer of notation) and use ρ instead of U . Hewitt and Ross denote\nan arbitrary element of a group G by x ∈ G; we use g ∈ G. Their notations Aσ and U (σ)x refer\nto operators that correspond to our f̂(ρ) and ρ(g), respectively (see [13, (34.1.i), (34.4.a)]).\nNote that if f is continuous, Theorem 11 implies that if f has an absolutely convergent Fourier\nseries, it equals its Fourier series at every point.\nHomogeneous Spaces and Gelfand Pairs\nDiaconis [4, Chap. 3F] provides an introduction to Gelfand pairs on finite groups and an an-\nnotated bibliography. Dieudonne [7] is a concise introduction to Gelfand pairs on compact and\nlocally compact groups.\nDefinition 3 Let G be a compact group and X be a topological space. An action of G on X\nis a continuous mapping from G × X → X denoted by (s, x) 7→ s · x = sx such that id · x = x\nand s · (t · x) = (st) · x.\nIf G acts transitively on X, that is, if for any x, y ∈ X there exists an s such that sx = y, we\ncall X a homogeneous space.\nGiven a point x0 ∈ X, let N denote the isotropy subgroup of G with respect to x0, i.e., the\nset of group elements which fix x0. By construction, N is a closed subset of G. The canonical\nisomorphism of X onto G\/N , the left cosets of N , respects the action of G. Thus g : xN 7→\n(gx)N .\nLet µX denote the G-invariant measure on X induced by Haar measure on G. Let L\n2(X) denote\nthe space of all complex-valued square-integrable functions on X with respect to µX . The action\nof G on X induces an action of G on L2(X) by g · f(x) = f(g−1x). This action is a 1-to-1 linear\nmapping of the vector space L2(X) into itself and so defines a representation of G.\nDefinition 4 A function f on G is said to be N -bi-invariant if f(n′gn′′) = f(g) for all n′, n′′ ∈ N\nand g ∈ G. A measure ν on G is N -bi-invariant if for any measurable set A in G, ν(n′An′′) =\nν(A) for all n′, n′′ ∈ N . Thus ν satisfies ν ∗ µ = µ ∗ ν = ν where µ is Haar measure on G.\nIn this paper bi-invariance on a homogeneous space G will be understood to mean with respect\nto the isotropy subgroup N . Note that bi-invariant functions on G are constant on double cosets\nNgN and may therefore be viewed as functions on the double coset space (denoted G\/\/N), or\nas left-invariant functions on X via its isomorphism with G\/N .\nDefinition 5 The pair (G,N) is called a Gelfand pair if the convolution algebra L2(G\/\/N) of\nN -bi-invariant functions is commutative. We sometimes say X ∼= G\/N is a Gelfand pair when\nG is understood by context.\n14\nThe next fact about Gelfand pairs is the most important for our purposes. A similar result for\nthe finite group context may be found in [4, p.54].\nTheorem 12 If (G,N) is a Gelfand pair, then for every irreducible representation ρ : G →\nGL(V ) there is a basis of V such that for all functions f (resp. measures ν) bi-invariant with\nrespect to N , the Fourier transform f̂(ρ) (resp. ν̂(ρ)) in that basis contains only zeroes except\npossibly for the (1, 1)-th entry.\nProof Dieudonne [6, (22.5.6)] shows the algebra L2(G\/\/N) is commutative if and only if the\nnumber of times the trivial representation appears in ρ|N , the the restriction of ρ to N , is zero\nor one.\nIf one, this trivial representation corresponds to a one-dimensional subspace of V fixed by N ,\ni.e., the left N -invariant functions on X; choose the unique function s(x) on X normalized so\nthat s(x0) = 1. This is sometimes called the spherical function of (G,N) corresponding to the\nrepresentation ρ. Complete s to a basis for V so that the matrices of ρ|N break into irreducible\n“blocks”. Then for a right N -invariant function f :\nf̂(ρ) =\n∫\ng∈G\nf(g) ρ(g−1) dµ\n=\n∫\nn∈N\n∫\nx∈G\/N\nf(xn) ρ(n−1x−1) dµX dµN\n=\n∫\nn∈N\nρ(n−1) dµN ·\n∫\nx∈G\/N\nf(x)ρ(x−1) dµX , (18)\nwhere µN is Haar measure on the subgroup N . The second equality is obtained by choosing a\ncoset representative x from each coset in G\/N and expressing g = xn for some x and n ∈ N ,\nand noting that Haar measure µ decomposes as a product measure µX ·µN . A similar argument\nholds for a right-invariant measure ν, noting that ν decomposes as a product measure νX · µN\nbecause of right-invariance.\nBy the orthogonality relations for matrix entries [6, (21.2.5.c)] of irreducible representations of\nN , the left-most integral of (18) produces a matrix consisting of zeroes except possibly for the\n(1, 1)-th entry. Thus f̂(ρ) (resp. ν̂(ρ)) has zero entries except possibly for the first row. A\nsimilar argument using the left-invariance of f (resp. ν) shows that f̂(ρ) (resp. ν̂(ρ)) has zero\nentries except possibly for the first column. Together, these statements imply that the only\nentry that could possibly be non-zero is the (1, 1)-th entry.\nIf the trivial representation does not appear in the the restriction of ρ to N , the argument\nabove holds by ignoring the role of s when choosing a basis for V . Orthogonality then shows\nthat the left-most integral of (18) yields a zero matrix. \u0003\nThere is thus one spherical function si(x) for every irreducible representation ρi appearing in\nL2(X). These induce N -bi-invariant functions s̃i on G. In the theorem above the s̃i(g) appears\nas the (1, 1)-th entry of ρi(g) for an appropriate basis. Hence for any measurable function f on\nG, the (1, 1)-th entry of the Fourier transform at ρi satisfies\n[f̂(ρi)](1,1) =\n∫\ng∈G\nf(g) s̃i(g) dµ. (19)\n15\nSimilarly, for a measure ν on G,\n[ν̂(ρi)](1,1) =\n∫\ng∈G\ns̃i(g) dν. (20)\nDieudonne [7] is a readable introduction to the general theory of spherical functions; Letac [15]\ncomputes them in several examples.\nThe sphere as a Gelfand pair\nThe rotation group SO(3) acts on the unit sphere S2 by the natural inclusion of S2 in R3.\nThis action is clearly transitive on S2, so S2 is a homogeneous space. In fact arises from the\nGelfand pair (G,N), where G is the rotation group SO(3), and N is the isotropy subgroup of\nrotations fixing n, the north pole. By restriction to the plane orthogonal to n ∈ R3, we see\nthat N is isomorphic to the group SO(2). The sphere S2 may then be regarded as the space\nSO(3)\/SO(2). In fact, for all n ≥ 2, Dieudonne [7] shows that Sn ∼= SO(n + 1)\/SO(n) is a\nGelfand pair.\nRepresentations of SO(3)\nFor a good reference on representations of SO(3) and other compact Lie groups, see Brocker\nand tom Dieck [3].\nLet ∆ = ∂\n2\n∂x21\n+ ∂\n2\n∂x22\n+ ∂\n2\n∂x23\nbe the Laplace operator on R3. The harmonic polynomials are the set\nof all complex-valued homogeneous polynomials f in x1, x2, x3 of degree n such that ∆f = 0;\nthe restrictions of these functions to the sphere S2 form a set Vn, the spherical harmonics of\ndegree n (one of which is the spherical function sn).\nThe action of SO(3) on Vn is induced by its action on R\n3 in the manner described earlier:\ng · f(x) = f(g−1x). Moreover, Vn is irreducible and finite-dimensional, and every irreducible\nrepresentation of SO(3) arises in this way. The dimension of Vn is 2n + 1.\nLegendre polynomials\nThe spherical functions si on S\n2 are given by the well-known Legendre polynomials Pi in the\nfollowing way: for y ∈ S2, si(y) = Pi(x) where x = cos θy ∈ [−1, 1] and θy is the geodesic\ndistance between y and n on S2. Just as S2 is (isomorphic to) the left cosets of N in G, the set\n[−1, 1] is the double coset space of this Gelfand pair.\nSince Haar measure on SO(3) induces the uniform (rotation-invariant) probability measure on\nS2 and uniform probability measure on [−1, 1], we can compute (19) as\n[f̂(ρn)](1,1) =\n∫\nSO(3)\nf(g)s̃n(g) dµ =\n1\n2\n∫ 1\n−1\nf(x)Pn(x) dx (21)\nwhere µ denotes normalized Haar measure on SO(3), dy denotes the uniform measure on S2,\nand dx is Lebesgue measure on R. See [9, p.239].\n16\nThe Legendre polynomials have the generating function\n∞∑\ni=0\nPi(x) r\ni =\n1\n(1 − 2xr + r2)1\/2\nand the first few are: P0(x) = 1, P1(x) = x, P2(x) =\n1\n2(3x\n2 − 1), P3 = 12(5x3 − 3x). Furthermore,\nfor all n ≥ 0, Pn(1) = 1 and\n|Pn(x)| ≤ 1 (22)\nfor x ∈ [−1, 1]. The following identity will be needed later. For n ≥ 1,\nPn(x) =\nP ′n+1(x) − P ′n−1(x)\n2n + 1\n(23)\nThis follows from the generating function for Pn(x). A nice account of these and other properties\nof Legendre polynomials may be found in Jackson [14].\nAppendix B.\nThis appendix contains the proofs of some technical results (Proposition 5 and Proposition 8)\nthat are not central to the development above.\nProof of Proposition 5\nTo show that Q∗k ∗ δr (x) is continuous for k ≥ 2, we first require the following technical lemma.\nLemma 13 Let ν be a positive measure on a compact metric group G, and let f be any\nmeasurable, bounded function f with discontinuities on a set Df . Let xD\n−1\nf denote the set\n{xd−1 : d ∈ Df}. Given x, if ν(xD−1f ) = 0, then the convolution\nh(x) = ν ∗ f (x) =\n∫\nG\nf(z−1x) dν(z)\nis continuous at x.\nProof To show h(x) is continuous at x, consider any sequence xn ∈ G such that xn → x. It\nmust be shown that h(xn) → h(x).\nLet wn(z) = f(z\n−1xn) and w(z) = f(z−1x). Since f is bounded, all the wn and w, being\ntranslates of f , are uniformly bounded by some constant function. This constant function is in\nL1(G, ν), since G is compact.\nAlso, wn(z) → w(z) pointwise for all z 6∈ xD−1f , since f is continuous at those points. By the\nassumption on Df we have pointwise convergence almost everywhere. By Lebesgue’s dominated\nconvergence theorem,\n∫\nG wn(z) dν(z) →\n∫\nG w(z) dν(z), which is precisely the statement\nh(xn) → h(x). This completes the proof of Lemma 13. \u0003\nWe can now prove Proposition 5.\n17\nProof We apply Lemma 13 setting f(x) = δr(x) and ν = Q. By inspection, δr is bounded by\n1. The lemma implies Q ∗ δr(x) is continuous everywhere except possibly at x = id. This may\nbe seen by observing that the discontinuity set of δr is ∂B̃r, the boundary of B̃r. This is the\npre-image of a circle on S2. On the other hand, Q regarded as a measure on S2 is uniformly\nsupported on a circle at latitude θ from the north pole. Any two circles on S2 intersect in at\nmost two points, unless they are identical. Hence Q(∂B̃r) = 0 unless the support of Q intersects\n∂B̃r, which only occurs when B̃r = B̃n,θ. This corresponds to a discontinuity in Q ∗ δr(x) at\nx = id when r = θ.\nWe now apply Lemma 13 again to show that Q∗k ∗ δr(x) is continuous for k = 2. The preceding\nobservations show that Q∗δr(x) is continuous almost everywhere (except possibly at the identity\nwhich is not in the support of Q). It is bounded by 1. Applying the lemma for f(x) = Q ∗ δr(x)\nand ν = Q shows that Q∗2 ∗ δr(x) is continuous everywhere.\nNow proceed by induction on k. For k ≥ 3, let ν = Q and let f(x) = Q∗(k−1) ∗ δr(x), which is\ncontinuous. Then Lemma 13 shows that ν ∗ f(x) = Q∗k ∗ δr(x) is continuous. \u0003\nProof of Proposition 8\nThis proves a Legendre bound for small θ.\nProof Let x = cos θ. From [14, p.62],\n|Pn(x)| ≤ 2\nπ\n∫ π\/2\n0\ne−nz\n2w2\/2 dw\nfor z = 2π (1 − x2)1\/2. Substitute t = n1\/2zw, and set A = n1\/2zπ\/2 = n1\/2 sin θ. Obtain\n|Pn(x)| ≤ 1\nA\n∫ A\n0\ne−t\n2\/2 dt.\nTo estimate this integral, square both sides and consider the double integral over a square in\nfirst quadrant of the plane, and then change to polar coordinates:\n|Pn(x)|2 ≤ 1\nA2\n∫ A\n0\n∫ A\n0\ne(−t\n2\n1−t22)\/2 dt1 dt2\n≤ 2\nA2\n∫ π\/4\nφ=0\n∫ A\/ cos φ\nr=0\ne−r\n2\/2 r dr dφ\n≤ 2\nA2\n∫ π\/4\nφ=0\n(1 − e−A2\/2 cos2 φ) dφ .\nFor 0 ≤ φ ≤ π\/4, y = A22 sec2 φ ≤ A2 which by assumption is less than .9. Now for y < .9, the\ninequality 1 − e−y ≤ y − 3y28 holds, and yields\n|Pn(x)|2 ≤ 2\nA2\n∫ π\/4\nφ=0\n(\nA2\n2\nsec2 φ − 3A\n4\n32\nsec4 φ\n)\ndφ .\nIntegrating the right hand side gives\n|Pn(x)|2 ≤\n[\ntan φ − 3A\n2\n16\n(\ntan3 φ\n3\n+ tan φ\n)]∣∣∣∣\nπ\/4\n0\n≤ 1 − A\n2\n4\n. \u0003\n18\nReferences\n[1] Beck, J. and Chen, W. Irregularities of Distribution, Cambridge Univ. Press, 1987.\n[2] Bloom, W.R. and Heyer, H. Harmonic Analysis of Probability Measures on Hypergroups, Walter\nde Gruyter & Co., Berlin, 1995.\n[3] Brocker, T. and tom Dieck, T. Representations of Compact Lie Groups. Springer-Verlag, 1985.\n[4] Diaconis, P. Group Representations in Probability and Statistics. IMS Lecture Notes-Monograph\nSeries, vol. 11, 1988.\n[5] Diaconis, P. and Shahshahani, M. Time to reach stationarity in the Bernoulli-Laplace diffusion\nmodel. SIAM J. Math. Analysis 18(1987), 208-218.\n[6] Dieudonne, J. Treatise on Analysis, vol. VI, Academic Press, 1978.\n[7] Dieudonne, J. 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Spanier, eds., Springer-Verlag, 2000, 448-458.\n[23] Voit, M. A central limit theorem for isotropic random walks on n-spheres for n → ∞, J. Math.\nAnal. Appl. 189(1995), 215-224.\n19\n[24] Voit, M. Limit theorems for compact two-point homogeneous spaces of large dimensions, J.\nTheoret. Probab. 9(1996), 353-370.\n[25] Voit, M. Rate of convergence to Gaussian measures on n-spheres and Jacobi Hypergroups, Ann.\nProbab. 25 (1997), 457-477.\n20\n"}
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{"id":"91nH7p51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/advanced\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":43,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","advanced"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["advanced – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:18 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Fri Jul 12 00:00:00 UTC 2019","Sat Jun 29 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-1141067063548442680,"exact_signature_unique_b":true,"fuzzy_signature_l":4855755626405248550,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["advanced"],"h2_txt":["How to Use Fun Facts","Liouville Numbers","Does Order of Addition Matter?","Irrationality by Infinite Descent","Dedekind Cuts of Rational Numbers","Computability of Real Numbers","Fermat’s Last Theorem","Rationals Dense but Sparse","Ordinal Numbers","Risk-Free Betting on Different Beliefs"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Lawnmower.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Lawnmower 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":690,"text_t":"advanced – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! advanced. These fun facts are at the advanced level. How to Use Fun Facts Posted on July 12, 2019. by. Molly Reeves Fun Facts are designed for, and work best in, lower-division college courses where the typical student is not a math student. The... Continue reading... Liouville Numbers Posted on June 29, 2019. by. Samuel Nunoo Are there any real numbers that are NOT algebraic, i.e., expressible as the root of a non-zero polynomial with integer coefficients? In fact,... Continue reading... Does Order of Addition Matter? Posted on June 29, 2019. by. Samuel Nunoo From your earliest days of math you learned that the order in which you add two numbers doesn’t matter: 3+5... Continue reading... Irrationality by Infinite Descent Posted on June 29, 2019. by. Samuel Nunoo The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the... Continue reading... Dedekind Cuts of Rational Numbers Posted on June 29, 2019. by. Samuel Nunoo Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the... Continue reading... Computability of Real Numbers Posted on June 29, 2019. by. Samuel Nunoo We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We... Continue reading... Fermat’s Last Theorem Posted on June 29, 2019. by. Samuel Nunoo There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =... Continue reading... Rationals Dense but Sparse Posted on June 29, 2019. by. Samuel Nunoo Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the... Continue reading... Ordinal Numbers Posted on June 29, 2019. by. Samuel Nunoo One of the most useful properties of the whole numbers is that every non-empty subset has a least element; this... Continue reading... Risk-Free Betting on Different Beliefs Posted on June 29, 2019. by. Samuel Nunoo Alice believes that the 49ers will win the Super Bowl with probability 5\/8. Bob believes that Ravens will win the... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. Current Page Parent. About Math Fun Facts Toggle Child Menu. Current Page:. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! How to Use Fun Facts. Fun Facts are designed for, and work best in, lower-division college courses where the typical student is not a math student. The point of Fun Facts is to generate interest, arouse curiosity, and cause such students to look at mathematics differently! I do one per class, usually at the beginning of the hour. Some are shorter, some are longer, but try not to spend more than 5 minutes presenting one, since it can eat up valuable class time. The purpose of the fact to only to pique their interest in math; you can go into more detail after class for those interested. However, do not be afraid to give them a glimpse of some. advanced mathematics. For instance, they will enjoy hearing the word “fundamental group” even if they don’t understand what it means. If all students do is pick up the “buzzwords”, it still helps to popularize mathematics. For instance, how many of us know what “string theory” really is, but doesn’t its mystique give physics a lot of glamor? Sometimes I choose Fun Facts related to the theme of the current or upcoming class. However, it is also good to do several that are unrelated to the course material, for variety. If a Fun Fact works in well with the lesson, I’ll do it in the middle of the hour. If I want them to think about a Fun Fact, I may begin one at the end, give them a day to think about it, and then finish it up the following lecture. Sequences of Fun Facts that relate to each. other are fun to present over several days. Also, feel free to link the “Random Fun Fact” URL to your web page as a fun way for your students to learn some Fun Facts at their leisure. The Fun Facts in this database are ones which we have collected, but I’m sure you can think of many more—so feel free to be creative. And if you have good ideas, consider submitting them! —. Francis Su p.s. Please see the . disclaimer . Tags:. advanced ,. other Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! demonstration. Liouville Numbers Posted on June 29, 2019. by. Samuel Nunoo Are there any real numbers that are NOT algebraic, i.e., expressible as the root of a non-zero polynomial with integer coefficients? In fact,... Continue reading... Computability of Real Numbers Posted on June 29, 2019. by. Samuel Nunoo We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We... Continue reading... Fermat’s Last Theorem Posted on June 29, 2019. by. Samuel Nunoo There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =... Continue reading... Projective Planes Posted on June 29, 2019. by. Samuel Nunoo Most of you know how to make a Mobius band—take a strip of paper and glue the ends with a... Continue reading... Birthday Problem Posted on June 26, 2019. by. Samuel Nunoo How many people do you need in a group to ensure at least a 50 percent probability that 2 people... Continue reading... Suspended Rope Trick Posted on June 26, 2019. by. Samuel Nunoo What is the shape of a suspended rope? Is there some function that describes it? Answer: it’s the cosh curve!... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"xwTObW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2020\/06\/Writing-Math-Well-HMC-Math.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":90,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2020","06"],"url_file_name_s":"Writing-Math-Well-HMC-Math","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Writing-Math-Well-HMC-Math.pdf"],"author":"pdfTeX-1.40.10","author_sxt":["pdfTeX-1.40.10"],"last_modified":"Mon Aug 30 22:06:42 UTC 2010","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2537116829694031388,"exact_signature_unique_b":true,"fuzzy_signature_l":8605931833580834948,"fuzzy_signature_unique_b":true,"text_t":"HARVEY MUDD COLLEGE DEPARTMENT OF MATHEMATICS\nWRITING MATHEMATICS WELL\nCommunicating mathematics well is an important part of doing mathematics. As you write up your\nhomework solutions, keep these things in mind:\n• Write in sentences.\nComplete thoughts are sentences that end in periods. You may still highlight important\nequations by displaying them, but even displayed equations should have punctuation!\nUse paragraphs to separate important ideas.\n• Use helpful connective phrases.\n“If”, “then”, “so”, “therefore”, “we see that”, “recall that”, ...\n• Your audience is other students in the class who have not seen this problem before.\nRemind the reader of any relevant facts from class or the book. Your solution should give\nadequate detail so that the reader can follow your solution.\n• It is possible to write too much!\nIf you write out every triviality, the reader may get lost in the details. This is not good\nwriting, either. (In particular, really trivial calculations need not be shown.)\n• Avoid shorthand.\nDon’t use arrows, and write out ’for all’, ’there exists’.\n• You may wish to outline your problem-solving strategy at the beginning of the problem.\nExample. Here are two different solutions to the same problem. Which one is easier to understand?\n(0− 3)2 + (x− 2)2 = 25\n32 = 9 + (x2 − 4x+ 4) = 25\nx2 − 4x− 12\n(x− 6)(x+ 2) =⇒ x = −2, 6 x > 0 x = 6\nWHY THIS IS POORLY WRITTEN:\n• You don’t know what problem the writer was solving.\n• You can’t tell what’s an assumption and what’s a conclusion.\n• Where does one thought end and another begin? There are no sentences!\n• In the 2nd line: combining two thoughts can create untruths (32 is 9 but it isn’t 25).\n• The 3rd line dangles; what’s being asserted here? It’s not a sentence.\n• What’s the relationship between all these phrases? Connective phrases would help!\nProblem. Find a point in the plane on the positive x-axis that has distance 5 from the point (2, 3).\nSolution. The desired point is (6, 0).\nTo find this, we note if (x, 0) is a solution, then x must must satisfy the equation (x − 2)2 + (0 − 3)2 =\n25, which follows from the planar distance formula between the points (x, 0) and (2, 3). It follows that\nx2 − 4x+ 13 = 25. Then\nx2 − 4x− 12 = 0.\nFactoring, we obtain\n(x− 6)(x+ 2) = 0,\nsatisfied by either x = −2 or x = 6. Since we assumed x > 0 and y = 0, we see (6, 0) is the desired point.\nWHY THIS IS WELL-WRITTEN:\n• The writer described the problem, and strategy for solution.\n• Every thought is a complete sentence with subject and verb (the “equals” sign is a verb).\n• She answered the question right at the beginning. (Boxing answers is customary.)\n• Notice even the equations have punctuation (comma, periods) as they are part of sentences.\n• She highlighted important ingredients, displayed important equations, avoided trivial algebra.\nWriting well will benefit you, too! It helps you structure your own thinking, and you will thank yourself\nwhen you re-read your solutions later.\n","wordcount_i":488,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:00 UTC 2022","fresh_date_dt":"Tue Jun 06 17:23:06 UTC 2023","publisher_t":"pdfTeX-1.40.10","language_s":"en","size_i":76684,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088060554543104}
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Jun 08 17:22:03 UTC 2022","fresh_date_dt":"Wed Jun 08 17:22:03 UTC 2022","referrer_id_s":"J9NUmP51bhow","language_s":"en","size_i":47350,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":0,"references_internal_i":0,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088064336756736,"text_t":"High-Dimensional Spheres in Cubes – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! High-Dimensional Spheres in Cubes. How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle in each. Now, draw a circle whose center is at the center of the big square and whose radius is just big enough to touch the four circles you just drew. We can perform an equivalent operation in a cube, inscribing spheres in each of its eight octants and then placing a. sphere in the middle, just large enough to touch the. other spheres. It’s. easy to see that the central circle is much smaller than the square and that the central. sphere is much smaller than the cube. What happens if we keep doing this in higher dimensions? Will the central sphere grow or shrink in diameter, relative to the side-length of the cube, as you change the dimension? Our intuition suggests that the central. sphere doesn’t grow. After all, its boundary is determined by spheres which lie inside of the cube, hugging each of the corners. It turns out though, that in the ninth dimension the central. sphere is tangent to the cube, and in much higher dimensions the volume of the sphere is actually larger than the cube! Presentation Suggestions:. Draw the case for the square (and if you’re artistic enough, the cube as well) on the board. Ask the students what they think happens in higher dimensions. Maybe the diameter of the central. sphere is always less than some constant ‘a’ (a. The Math Behind the Fact:. We’ll make the numbers. easy by using a cube (in dimension N) with side length 2. Then when we cut the cube into orthants, each sub-cube has side length 1. Let’s just look at one of those sub-cubes. The. sphere S inscribed in that sub-cube has diameter 1. When we draw the central sphere, its center is on a corner of that subcube. Draw the diagonal from that corner to the opposite corner of the sub-cube. That diagonal has length Sqrt[N]. The part of the diagonal going through the sphere S has length 1 because that is the diameter of S. Of the part that is left over, half of it is in the central sphere, and in fact forms the radius of that central sphere. So the central sphere has radius (Sqrt[N]-1)\/2. If N=9, the radius of the central. sphere is 1, so it is just tangent to the cube. If N9, then part of the central sphere bulges outside the cube! And, eventually the volume of the central sphere is actually larger than the cube. For a related surprise, see Volume of a Ball in N Dimensions. How to Cite this Page:. Su, Francis E., et al. “High-Dimensional Spheres in Cubes.”. Math Fun Facts. . . Fun Fact suggested by:. Joel Miller. Did you like this Fun Fact? Click to rate it. Submit Rating. Average rating. 3.9. \/ 5. Vote count:. 15. No votes so far! Be the first to rate this Fun Fact. Share:. Tags:. easy ,. geometry ,. medium ,. other ,. sphere Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved."}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! geometry. Fermat’s Last Theorem Posted on June 29, 2019. by. Samuel Nunoo There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =... Continue reading... Slices of Hanging Cubes Posted on June 29, 2019. by. Samuel Nunoo Hang a cube from one of its vertices. Now, if you slice it horizontally through its center, you get a... Continue reading... Koch Tetrahedron Posted on June 29, 2019. by. Samuel Nunoo In Koch Snowflake we saw an interesting fractal snowflake-like object that is obtained when gluing smaller triangles iteratively to the sides of... Continue reading... Hyperbolic Geometry Posted on June 29, 2019. by. Samuel Nunoo In the Fun Fact on Spherical Geometry, we saw an example of a space which is curved in such a way... Continue reading... Volume of a Ball in N Dimensions Posted on June 29, 2019. by. Samuel Nunoo The unit ball in Rn is defined as the set of points (x1,…,xn) such that x12 + … + xn2 . Continue reading... Sphere Eversions Posted on June 29, 2019. by. Samuel Nunoo If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it... Continue reading... Sierpinski-Mazurkiewicz Paradox Posted on June 29, 2019. by. Samuel Nunoo If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces... Continue reading... Area of a Circle or Regular Polygon Posted on June 29, 2019. by. Samuel Nunoo There’s a nice way to see why the formula for the area of a circle of radius R is: Pi... Continue reading... Chords of an Ellipse Posted on June 29, 2019. by. Samuel Nunoo Consider N equally spaced on points on the unit circle, with the point P=(1,0) as one of these equally spaced... Continue reading... Sliding Chords Posted on June 29, 2019. by. Samuel Nunoo Take a circle C, and a chord in the circle. Now slide the chord around the circle. As you do this, the... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Fibonacci. Chords of an Ellipse Posted on June 29, 2019. by. Samuel Nunoo Consider N equally spaced on points on the unit circle, with the point P=(1,0) as one of these equally spaced... Continue reading... Fibonacci GCD’s, please Posted on June 27, 2019. by. Samuel Nunoo Fibonacci numbers exhibit striking patterns. Here’s one that may not be so obvious, but is striking when you see it.... Continue reading... Domino and Square Tilings Posted on June 26, 2019. by. Samuel Nunoo How many ways can you tile a (1 x n) board with (1 x 1) squares and (1 x 2)... Continue reading... Fibonacci Number Formula Posted on June 26, 2019. by. Samuel Nunoo The Fibonacci numbers are generated by setting F0 = 0, F1 = 1, and then using the recursive formulaFn = Fn-1 +... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Napoleon’s Theorem. Take any generic triangle, and construct equilateral. triangles on each side whose side lengths are the same as the length of each side of the original triangle. Surprise: the centers of the equilateral. triangles form an equilateral triangle! Presentation Suggestions:. Show the truth of the statement using some extreme cases for the initial triangle (a particularly instructive example is a triangle with one sidelength very close to zero). The Math Behind the Fact:. This theorem is credited to Napoleon, who was fond of mathematics, though many doubt that he knew enough math to discover it! How to Cite this Page:. Su, Francis E., et al. “Napoleon’s Theorem.”. Math Fun Facts. . . Fun Fact suggested by:. Michael Moody. Did you like this Fun Fact? Click to rate it. Submit Rating. Average rating. 4.7. \/ 5. Vote count:. 37. No votes so far! Be the first to rate this Fun Fact. Share:. Tags:. easy ,. geometry ,. triangles Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved. Loading Comments... Write a Comment... Email (Required). Name (Required). 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Skip to content Francis Su Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College. menu. New Book Bio CV Teaching Writing Math Well Real Analysis I (Math 131) Abstract Algebra II (Math 172) Real Analysis on YouTube Research Topology Text Author Webpage Math 188: Social Choice and Decision Making. Francis Su. Math 188: Social Choice and Decision Making. Term:. Summer 2021, May 24 – June 11. Course meets:. Virtually, Mon-Tue-Thu-Fri, 8:45-noon (Pacific time). The private Zoom link was sent by e-mail, and is also available on the course. Sakai page . Office Hours:. By appointment, usually at 1:30pm. People make decisions everyday — small ones such as what to eat or wear, to large ones such as which job to take or whether to buy a house. Groups of people also make decisions, such as which president to elect, or what policies to adopt, what business strategies to pursue. Mathematics has much to offer the decision-maker — both in the analysis of strategies and suggestions for a course of action. In this course, we will focus on the modeling of individual and group decisions using mathematical techniques from “game theory”, the area of mathematics that was pioneered in the 1950’s by John Nash and others, but now has applications to a wide variety of disciplines: economics, biology, computer science. Topics will include: basic concepts of game theory and social choice theory, representations of games, Nash equilibria, utility theory, non-cooperative games, cooperative games, voting games, paradoxes, Arrow’s impossibility theorem, Shapley value, power indices, “fair division” problems, and applications. Prerequisites. Math 55 (or some experience with discrete math) is recommended, but not required. Texts. Phil Straffin,. Game Theory and Strategy. . Alan Taylor and Allison Pacelli,. Mathematics and Politics. . E-books are available through the. Claremont Colleges Library at these links:. Straffin ,. Taylor-Pacelli . Coursework. As a 3-week summer course, there will be daily homework, which in the latter part of the course will include time to work on final projects. There will also be an exam at the beginning of week 3. Homework. Homeworks will be graded based on clarity of ideas and communication. You’ll note that Straffin has all answers at the back of the text, but these are rather minimal. (On the Honor Code, do NOT look at them until AFTER you have attempted each problem.) You’ll need to explain how you arrived at your answers to get full credit. Good communication is important in this class; be sure to read my. Guidelines for Good Mathematical Writing . All homework will be submitted through Gradescop. e. Access this through the. Sakai course page . On the left tab, you’ll see a link for Gradescope. You will need to upload by scanning your homework, and tagging which portion is associated with specific problems. Here is some guidance on how to do that:. https:\/\/www.gradescope.com\/get_started#student-submission . Exam. There will be one take-home exam due at the start of Week 3. Final Project. There will be a final project involving a critical analysis of the relationship of the mathematical methods of game theory to contemporary society. This will be a. group project. (2 to 3 students). You will create a video presentation of up to 10 minutes. Grading. Homework, Exam, Final Project are each worth 25%, with highest component worth 25%. The grader for the course is Allen Wu. HW #1 – due Tues 5\/25. READ. Straffin: Preface and Chapters 1, 2, and 7. Taylor-Pacelli: Preface and Sections 6.1-6.3. Note:. Problems marked with an R. means. “READ the problem and think about it, but do not hand it in.”. DO. Straffin Chapter 7 (3, 4, 5, 6abc, R6d) and. Problem A. Find a short news article involving some sort of interaction or. decision problem. Then, in just a few brief sentences:. (1) summarize the decision problem in the article,. (2) then identify each of the following:: (a) the players, (b) some possible strategies for each player, (c) some possible aspects of conflict or cooperation between players. (3) Include a scan of the article with your assignment. Remember to write clearly and in complete sentences and explain your reasoning carefully. Box answers where appropriate. Good communication is important in this class; be sure to read my. Guidelines for Good Mathematical Writing . HW #2 – due Thu 5\/27. READ. Taylor-Pacelli: Sections 4.1-4.3, AND. Straffin: Chapter 3. “R” means READ and THINK about, but do not do the problem. DO. Straffin Chap.2 ( 1, 2, 3, R6, 7 ) AND Taylor Chap. 4 ( 6 ). HW #3 – due Fri 5\/28. READ. Straffin: Chapters 4, 5, 6. Also, take a moment to reach out to someone else in the class, and have a chat with them about anything that strikes you as interesting from the class material. DO. Straffin Chap. 3 ( 1, 2, 3, 4, R5, 7 ). HW #4 – due Mon 5\/31. READ. Straffin Chapter 4, 5, 6, 9. Taylor-Pacelli Chapter 10. Also, take a moment to reach out to someone else in the class, and have a chat with them about anything that strikes you as interesting from the class material. DO. Straffin. Chap. 3 ( 5, R9 ). Chap.4 ( R2 ). Chap. 7 ( 6d ). Chap 9 ( R1, 2, 3, 5 ). HW #5 – due Tue 6\/1. READ. Taylor-Pacelli Chapter 4. Straffin Chapters 10, 11, 12. DO. Straffin Chapter 10 ( R3 ), Chap 11 ( 1 ), Chap 12 ( 4, 6 ) and. Taylor-Pacelli Chapter 4 ( 13, 15, R23 ). HW #6 – due Thu 6\/3. READ. Straffin Chapters 16-17. DO. On this. Handout ( R21, R22, R23, especially R24 ). and on this. Handout ( 7, R9, 11Y, 12Y, 15). and. Problem B. Meet with at least one other person in this class to discuss ideas that might be interesting to model using game theory. (See the last HW tab below under 6\/10 and 6\/11 for the final project guidelines.) On this homework, write down a few sentences about what you discussed. (Let me know if you need help connecting with someone else in the class.). HW #7 – due Fri 6\/4. READ. Straffin Chapter 19, 23, 25 and Taylor-Pacelli Chapter 2, 3. DO. Straffin Chap. 23 ( 2, 3abc, 3def ) 25 ( R1, 6 ) AND. Problem C. Identify a group for your final project. For 5 points on this homework ,state who you are working with (if anyone) and state any preliminary thoughts you have about a project topic, in a couple of sentences. HW #8 – due Mon 6\/7. REVIEW. Concepts from the course in the last 2 weeks. You may also look at the take-home exam (on Gradescope) and get started on that if you wish (due Tuesday). DO. Taylor-Pacelli Chapter 2 ( 2, R3, 10 ) AND. Problem D. Meet with your project group (if you have one). If you have not yet picked a topic,. discuss 3 ideas you have for a project (see HW for 6\/10 and 6\/11 below for guidelines on the final project). Summarize the pros and cons of pursuing each of those ideas for a final project (5 points each, 15 points total). If you have already settled on a topic,. describe: the topic idea, the way you expect game theory enters into analyzing that topic, and develop a plan for work on your project presentation (5 points each, 15 points total). TAKE-HOME EXAM – due Tue 6\/8. Look on Gradescope. The exam will be available Friday June 4, and is due Tuesday at 5pm on June 8. due Fri 6\/11 at 8am: Final Project. Guidelines here. Your. final project. will involve a critical analysis of the relationship of the mathematical methods of game theory to contemporary society. This will be a group project (2 to 3 students). You will create a video presentation of up to 10 minutes. You’ll model a real-life situation of your choice using the techniques of decision analysis and game theory, or choose an existing model in the literature to critique. Some examples could include:. I. Choose a past historical event or situation, model the agents and strategies involved, and use your model to explain why events transpired the way they did. Ex. Division of post-war Germany. Ex. Intervention of 3rd party candidates in nat’l elections. Ex. Predator-prey relationships in biological ecosystems. II. Choose a current situation or event, model the agents and strategies involved, and then use your model to (a) suggest a course of action or (b) make a prediction about what will happen. Ex. marketing strategies for a new business. Ex. the Palestinian conflict. Ex. strategies to contain the spread of AIDS. III. Choose a current social choice problem and understand it in terms of ideas from cooperative game theory. Ex. coalition formation in an organization. Ex. analysis of the power structure in a business or organization. Ex. resource allocation and “fair division”. Your presentation should also contain a critical analysis of the methods used. For instance, does mathematics provide a good model for decision making in the context you describe? What are the limitations of the model? As people often make decisions based on such models, what would be appropriate or inappropriate uses of this model? Your project video. will be judged on the basis of :. (i). clarity of explanation. : can the video be understood by another student in the course? (ii). game theoretic connection. : topic identifies where game theory is used, and is mathematically correct. (iii). insightfulness. : your project identifies clearly what you are bringing beyond a mere description of the topic. This can be achieved by demonstrating originality in choosing a model, offering your own critique of an existing model, or offering a new perspective or insight. You can submit the video to the Google Drive folder that was linked in an e-mail sent June 5. It’s also accessible through the Course Videos folder where my lectures have been posted (linked from the Sakai page). HOMEWORKS BELOW NOT FINAL UNTIL THEY APPEAR ABOVE THIS LINE. –. You can work ahead, but some assignments may change. My New Book! On . Amazon. , or. Bookshop. [support independent bookstores], or your favorite bookstore! Read a . review  or . two . See the. book webpage for more info. Contact Information. Mailing Address. Department of Mathematics Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711. Office Phone. 909.607.3616. Fax. 909.621.8366. Department Phone. 909.621.8023. Email. su@math.hmc.edu Social Media. Twitter Instagram LinkedIn Facebook [I accept follows]. Theme:. Vogue by Kaira."}
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{"id":"2yCdYp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/sphere\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":41,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","sphere"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["sphere – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:29 UTC 2022","dates_in_content_count_i":4,"dates_in_content_dts":["Sat Jun 29 00:00:00 UTC 2019","Fri Jun 28 00:00:00 UTC 2019","Thu Jun 27 00:00:00 UTC 2019","Wed Jun 26 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-8717328547691153668,"exact_signature_unique_b":true,"fuzzy_signature_l":5774736656875936714,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["sphere"],"h2_txt":["Klein Bottle","Sphere Eversions","High-Dimensional Spheres in Cubes","Spherical Pythagorean Theorem","Hairy Ball Theorem","Connected Sums","Surface Area of a Sphere","Spherical Geometry","Fundamental Group","Spherical Bands"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/BrokenMudd.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 BrokenMudd Broken Mudd 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":700,"text_t":"sphere – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! sphere. Klein Bottle Posted on June 29, 2019. by. Samuel Nunoo A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;... Continue reading... Sphere Eversions Posted on June 29, 2019. by. Samuel Nunoo If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it... Continue reading... High-Dimensional Spheres in Cubes Posted on June 28, 2019. by. Samuel Nunoo How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle... Continue reading... Spherical Pythagorean Theorem Posted on June 27, 2019. by. Samuel Nunoo Did you know there is a version of the Pythagorean Theorem for right triangles on spheres? First, let’s define precisely what we... Continue reading... Hairy Ball Theorem Posted on June 27, 2019. by. Samuel Nunoo Another fun theorem from topology is the Hairy Ball Theorem. It states that given a ball with hairs all over it, it... Continue reading... Connected Sums Posted on June 27, 2019. by. Samuel Nunoo A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,... Continue reading... Surface Area of a Sphere Posted on June 26, 2019. by. Samuel Nunoo The area of a disk enclosed by a circle of radius R is Pi*R2. The formula for the circumference of a... Continue reading... Spherical Geometry Posted on June 26, 2019. by. Samuel Nunoo Remember high school geometry? The sum of the angles of a planar triangle is always 180 degrees or Pi radians. However, triangles... Continue reading... Fundamental Group Posted on June 26, 2019. by. Samuel Nunoo You are abducted by space aliens and dumped, blindfolded, on a strange asteroid! Removing your blindfold, you decide to play... Continue reading... Spherical Bands Posted on June 26, 2019. by. Samuel Nunoo Take any sphere and slice it up by parallel knives 1 inch apart. The areas of the spherical bands between the cuts... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"2Ja5mp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/paradox\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":42,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","paradox"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["paradox – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:30 UTC 2022","dates_in_content_count_i":4,"dates_in_content_dts":["Sat Jun 29 00:00:00 UTC 2019","Fri Jun 28 00:00:00 UTC 2019","Thu Jun 27 00:00:00 UTC 2019","Wed Jun 26 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":6828986415325119906,"exact_signature_unique_b":true,"fuzzy_signature_l":-8719955770870380819,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["paradox"],"h2_txt":["St. Petersburg Paradox","Banach-Tarski Paradox","Rubber Bands Stuck on a Torus","Arrow’s Impossibility Theorem","All Numbers are Interesting","Two Envelopes Paradox","Gabriel’s Trumpet","Derivative Paradox","One Equals Zero: Integral Form","One Equals Zero!"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Faucet.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Faucet 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":648,"text_t":"paradox – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! paradox. St. Petersburg Paradox Posted on June 29, 2019. by. Samuel Nunoo A new casino offers the following game: you toss a coin until it comes up heads. If the first heads... Continue reading... Banach-Tarski Paradox Posted on June 29, 2019. by. Samuel Nunoo Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using... Continue reading... Rubber Bands Stuck on a Torus Posted on June 28, 2019. by. Samuel Nunoo Consider a rubber tire (a torus) with a hole in it. Suppose that there is a green rubber band stuck... Continue reading... Arrow’s Impossibility Theorem Posted on June 27, 2019. by. Samuel Nunoo Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences... Continue reading... All Numbers are Interesting Posted on June 27, 2019. by. Samuel Nunoo There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first... Continue reading... Two Envelopes Paradox Posted on June 26, 2019. by. Samuel Nunoo I have two envelopes, and inside each I have put some money. In fact, one envelope contains twice as much... Continue reading... Gabriel’s Trumpet Posted on June 26, 2019. by. Samuel Nunoo The surface of revolution of y = 1\/xfor x greater than 1, has finite volume but infinite surface area! Put another way, this... Continue reading... Derivative Paradox Posted on June 26, 2019. by. Samuel Nunoo Here’s a fun (but untrue) fact. You know from calculus that the derivative of x2 is 2x. But what’s wrong with the following... Continue reading... One Equals Zero: Integral Form Posted on June 26, 2019. by. Samuel Nunoo Consider the following integral: INTEGRAL (1\/x) dx Perform integration by parts: let u = 1\/x , dv = dx du = -1\/x2 dx... Continue reading... One Equals Zero! Posted on June 26, 2019. by. Samuel Nunoo The following is a “proof” that one equals zero. Consider two non-zero numbers x and y such that x =... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Thinned-Out Harmonic Series Posted on June 27, 2019. by. Samuel Nunoo You’re probably already aware that the harmonic series, which is the sum of the reciprocals of all natural numbers, diverges. In... Continue reading... Sums of Reciprocal Powers Posted on June 26, 2019. by. Samuel Nunoo You have seen that the harmonic series diverges. What about the sum of reciprocal squares? In fact, they converge, and to something... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! probabilities. Risk-Free Betting on Different Beliefs Posted on June 29, 2019. by. Samuel Nunoo Alice believes that the 49ers will win the Super Bowl with probability 5\/8. Bob believes that Ravens will win the... Continue reading... Medical Tests and Bayes’ Theorem Posted on June 29, 2019. by. Samuel Nunoo Suppose that you are worried that you might have a rare disease. You decide to get tested, and suppose that... Continue reading... Poker Probability Paradox Posted on June 26, 2019. by. Samuel Nunoo In poker, there is a hierarchy of desirable hands which relate to how unlikely the hand is. For instance, a... Continue reading... Monty Hall Problem Posted on June 26, 2019. by. Samuel Nunoo Here’s a problem that makes the round every few years, and each time, it is hotly debated. You are on... Continue reading... 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{"id":"Q5piip51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/does-order-of-addition-matter\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":60,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["funfacts","does-order-of-addition-matter"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Does Order of Addition Matter? – Math Fun Facts"],"last_modified":"Wed Jun 08 17:23:06 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":4752460808993508788,"exact_signature_unique_b":true,"fuzzy_signature_l":-910213266035540707,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["Does Order of Addition Matter?"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"bold_txt":["Fun Fact suggested by:","Principles of Mathematical Analysis","References:","Math Fun Facts","How to Cite this Page:","The Math Behind the Fact:","alternating harmonic series","Presentation Suggestions:","conditionally convergent","absolutely","series"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Lawnmower.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Lawnmower 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":687,"text_t":"Does Order of Addition Matter? – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Does Order of Addition Matter? From your earliest days of math you learned that the order in which you add two numbers doesn’t matter: 3+5 and 5+3 give the same result. The same is true for the addition of any finite set of numbers. But what if you are adding an infinite set of numbers? For instance, in. calculus you encounter. series. , which are infinite sums. Does the order of addition matter? Surprisingly, yes, but only for some series. A series is said to converge. absolutely. if the sum of the absolute values of the terms converges. It is a fact that a series which converges absolutely must converge, and no matter how you rearrange the terms the sum will be the same. Series that do not converge absolutely are wilder; they may or may not converge (so they are called. conditionally convergent. ). But, if such a sequence does converge, it is even stranger still— by rearranging the terms in the sequence, you can get the series to converge to any value you want! Presentation Suggestions:. Do an example with the. alternating. harmonic series :. 1 – (1\/2) + (1\/3) – (1\/4) + …. This series is conditionally convergent (because the. harmonic series diverges). But by the. alternating series test, it converges—in fact, to ln(2). As a demonstration, show students how to rearrange it to converge to any value (see Math Behind the Fact). The Math Behind the Fact:. This fact is due to Riemann, and the basic idea is simple. To obtain a rearrangement that sums to X, just add enough positive terms from the series (in their original order) until the partial sums first exceed X, then add enough negative terms from the series (in their original order) until the partial sums first dip below X, then repeat and continue with the successive positive terms, then the successive negative terms, etc. This will yield a rearrangement of the original series that sums to X. This fact shows how careful one has to be when switching the order of addition in a conditionally convergent series. How to Cite this Page:.  . Su, Francis E., et al. “Does Order of Addition Matter?.” . Math Fun Facts. . . References:. Walter Rudin,. Principles of Mathematical Analysis. , Chapter 3. Fun Fact suggested by:. Francis Su. Did you like this Fun Fact? Click to rate it. Submit Rating. Average rating. 4.6. \/ 5. Vote count:. 53. No votes so far! Be the first to rate this Fun Fact. Share:. Tags:. absolute convergence ,. advanced ,. alternating series ,. analysis ,. calculus ,. conditional convergence Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"QIc-Yp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/polynomial\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":45,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","polynomial"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["polynomial – Math Fun Facts"],"last_modified":"Wed Jun 08 17:23:07 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Sat Jun 29 00:00:00 UTC 2019","Wed Jun 26 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-4217974127838566550,"exact_signature_unique_b":true,"fuzzy_signature_l":-552202109046199062,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["polynomial"],"h2_txt":["Liouville Numbers","Lucas’ Theorem","Odd Numbers in Pascal’s Triangle","Sierpinski-Mazurkiewicz Paradox","Descartes’ Rule of Signs","Formula for Primes?","Successive Differences of Powers","Pascal’s Triangle"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Lawnmower.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Lawnmower 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":642,"text_t":"polynomial – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! polynomial. Liouville Numbers Posted on June 29, 2019. by. Samuel Nunoo Are there any real numbers that are NOT algebraic, i.e., expressible as the root of a non-zero polynomial with integer coefficients? In fact,... Continue reading... Lucas’ Theorem Posted on June 29, 2019. by. Samuel Nunoo Lucas’ Theorem: If p is a prime number, and N has base p representation (aj,…,a1,a0) and k has base p... Continue reading... Odd Numbers in Pascal’s Triangle Posted on June 29, 2019. by. Samuel Nunoo Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N... Continue reading... Sierpinski-Mazurkiewicz Paradox Posted on June 29, 2019. by. Samuel Nunoo If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces... Continue reading... Descartes’ Rule of Signs Posted on June 26, 2019. by. Samuel Nunoo Given a polynomial such as: x4 + 7×3 – 4×2 – x – 7 it is possible to say anything about how many positive real... Continue reading... Formula for Primes? Posted on June 26, 2019. by. Samuel Nunoo Is there a nice cozy formula that will always spit out primes? Try this one: f(n) = n2 + n +... Continue reading... Successive Differences of Powers Posted on June 26, 2019. by. Samuel Nunoo List the squares: 0, 1, 4, 9, 16, 25, 36, 49, … Then take their successive differences: 1, 3, 5,... Continue reading... Pascal’s Triangle Posted on June 26, 2019. by. Samuel Nunoo Consider the triangle in Figure 1, called Pascal’s triangle. It consists of numbers where each entry is the sum of the... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"QFHpJW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Four-Person-Envy-Free-Chore-Division.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":100,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Four-Person-Envy-Free-Chore-Division","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["notes.dvi"],"author":"Acrobat Distiller 4.0 for Windows","author_sxt":["Acrobat Distiller 4.0 for Windows"],"last_modified":"Tue Apr 23 14:43:43 UTC 2002","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Apr 02 00:00:00 UTC 2002"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":1024315328812785412,"exact_signature_unique_b":true,"fuzzy_signature_l":-5159282743967558520,"fuzzy_signature_unique_b":true,"wordcount_i":2931,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:07 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:30 UTC 2023","referrer_id_s":"K7fAEW51bhow","publisher_t":"Acrobat Distiller 4.0 for Windows","language_s":"en","size_i":96857,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088131694133248,"text_t":"N O T E S\nFour-Person Envy-Free Chore Division\nELISHA PETERSON\nMagdalen College, Oxford University\nOxford OX1 4AU, U.K.\nFRANCIS EDWARD SU\nHarvey Mudd College\nClaremont, CA 91711\nIn this article we explore the problem of chore division, which is closely related to a\nclassical question, due to Steinhaus [10], of how to cut a cake fairly. We focus on con-\nstructive solutions, that is, those obtained via a well-defined procedure or algorithm.\nAmong the many notions of fairness is envy-freeness: an envy-free cake division is a\nset of cuts and an allocation of the pieces that gives each person what she feels is the\nlargest piece. It is non-trivial to find such a division, since the cake may not be homo-\ngeneous and player valuations on subsets of cake will differ, in general. Much progress\nhas been made on finding constructive algorithms for achieving envy-free cake divi-\nsions; most recently, Brams and Taylor [3] produced the first general n-person pro-\ncedure. The recent books by Brams and Taylor [4] and Robertson and Webb [8] give\nsurveys on the cake-cutting literature.\nIn contrast to cakes, which are desirable, the dual problem of chore division is\nconcerned with dividing an object deemed undesirable. Here, each player would like\nto receive what he considers to be the smallest piece of, say, a set of chores. This\nproblem appears to have been first introduced by Martin Gardner [6].\nMuch less work has been done to develop algorithms for chore division than for\ncake-cutting. Of course, for 2 people, the familiar I-cut-you-choose cake-cutting pro-\ncedure also works for dividing chores: one cuts the chores and the other chooses what\nshe feels is the smallest piece. Oskui [8, p. 73] gave the first envy-free solutions for\nchore division among 3 people. Su [12] developed an envy-free chore-division algo-\nrithm for an arbitrary number of players; however, it does not yield an exact solution,\nbut only an \u0001-approximate one. There appear to be no exact envy-free chore-division\nalgorithms for more than three players in the literature; in unpublished manuscripts,\nBrams and Taylor [2] and Peterson and Su [7] offer n-person algorithms but these are\nnot bounded in the number of steps they require.\nIn this article, we develop a simple and bounded procedure for envy-free chore di-\nvision among 4 players. The reader will find that many of the ideas involved—moving\nknives, trimming and lumping, and a notion of “irrevocable advantage”—provide a\nnice introduction to similar techniques that arise in the literature on fair division prob-\nlems. As a warm-up to some of these ideas, we also present a 3-person solution that is\nsimpler and more symmetrical than the procedure of Oskui.\nWe assume throughout this paper that chores are infinitely divisible. This is not\nunreasonable, as a finite set of chores can be partitioned by dividing up each chore (for\ninstance, a lawn to be mowed could be divided just as if it were a cake), or dividing the\ntime spent on them. We also assume that player valuations over subsets of the chores\n117\n118 MATHEMATICS MAGAZINE\nare additive, that is, no value is destroyed or created by cutting or lumping pieces\ntogether. (The proper context for modeling player valuations is measure theory, but we\ncan avoid that for the purposes of this article.)\nA 3-person chore-division procedure We first describe a simpler 3-person chore-\ndivision procedure, which introduces some ideas that are important later.\nOur 3-person chore-division procedure relies on Austin’s procedure [1] for dividing\na cake into two pieces so that each of two players believes it is a 50-50 split. For\ncompleteness, we review it here. Let one player hold two knives over the cake, with one\nat the left edge, such that the portion of cake between them is what she believes to be\nexactly half. If the second player agrees that it is exactly half, we are done. Otherwise,\nlet the first player move the knives across the cake from left to right, keeping the\nportion between them exactly half (in her estimation), until the second player agrees it\nis exactly half. (There must be such a point because when the rightmost knife reaches\nthe right edge, the leftmost knife must be where the rightmost knife began, hence\nthe second player must by that point have changed preferences.) At this point cuts\nare made and the pieces of cake outside the knives are lumped together, yielding two\npieces that both players agree are exactly equal.\nAustin’s procedure is an example of what is sometimes called a “moving-knife”\nprocedure in the cake-cutting literature [5]. Our 3-person chore-division algorithm is\nalso a moving-knife procedure. The key idea is to divide the chores into six pieces and\nassign each player two of the pieces that he feels are at least as small as each pair of\npieces the other players receive.\nA THREE-PERSON ENVY-FREE CHORE-DIVISION PROCEDURE\nStep 1. Divide the chores into three portions using any 3-person envy-free cake-\ndivision procedure (that guarantees players a piece they think is largest), such\nas the Stromquist moving-knife procedure [11]. Now label each portion by\nthe name of the player to whom the cake-division procedure would assign that\nportion (this player believes that portion is largest).\nStep 2. Let player i divide portion i into 2 pieces (which she feels is exactly half) and\nassign those pieces to the other two players such that they each feel they have\nreceived no more than half of portion i . (This can be achieved via Austin’s\nprocedure: letting player i and one other player, say j , agree on a 50-50 split,\nlet the remaining player choose the half she thinks is smallest, and give the\nother half to j .)\nStep 3. Repeat Step 2 for each player, then end the procedure.\nWe now verify that each player has been assigned two out of six total pieces such\nthat each feels her share is smallest.\nCall the players i , j , and k. Player i will not envy player j because one piece of\neach of their pairs came from the portion labelled k, and i feels her half of that portion\nwas no larger than j’s. As for her other piece, player i feels it was no more than half\nof the portion it came from, and therefore cannot be as large as player j’s other piece,\nwhich i felt was exactly half of the largest portion. The same argument holds for any\npermutation of i , j , and k. See FIGURE 1.\nThis procedure requires at most 8 cuts (Step 1 uses 2 cuts, and Austin’s proce-\ndure uses at most 2 cuts each time it is applied; in FIGURE 1, some pieces may have\nbeen reassembled for simplicity). It is also less complicated than the discrete proce-\ndure of Oskui [8]. There are some 3-person moving-knife schemes that require fewer\ncuts [8, 9], but our approach is distinguished by being symmetric with respect to the\nplayers and being based on a cake-cutting procedure. The former property simplifies\nVOL. 75, NO. 2, APRIL 2002 119\nto k\nto j\nto j\ni j\nto i to i\nk\nto k\nFigure 1 An envy-free assignment of six pieces (of chores) to three people\nthe verification of envy-freeness, while the latter property may help in generalizing the\nscheme to more players via known cake-cutting algorithms.\nA 4-Person Chore-division Procedure We now describe our 4-person chore-\ndivision procedure, which is also a moving-knife procedure and requires at most 16\ncuts. It draws ideas from both the Brams-Taylor-Zwicker 4-person envy-free moving-\nknife scheme for cakes [5] and the Oskui 3-person envy-free discrete chore-division\nscheme [8]. We also show how the notion of irrevocable advantage, important in\ncake-cutting [4], can be applied in chore division.\nSuppose the players are named Alice, Betty, Carl, and Debbie. For convenience, we\nassume the chores are a rectangular block that may be divided by vertical cuts. Let\nAlice and Betty divide the chores into four pieces they both agree are all equal, by\nperforming three applications of Austin’s procedure (using at most 6 cuts).\nCall the pieces X1, X2, X3, and X4. Note that if Debbie and Carl disagree on which\npiece is the smallest, we can immediately allocate the pieces. Thus we may assume\nthey agree that one piece is strictly smaller than the others, say X4. Then each person\nthinks the following:\nAlice: X1 = X2 = X3 = X4\nBetty: X1 = X2 = X3 = X4\nCarl: X4 < X1, X2, X3\nDebbie: X4 < X1, X2, X3.\nNow, for each of X1, X2, and X3, let Debbie and Carl mark how they would trim them\nto make them the same size as X4. As each piece is rectangular, assume the trimmings\nare marked from the top edge, so that a person receives the piece below her mark.\nSee FIGURE 2. Hence, we can speak of one mark as being higher than another. The\nfollowing procedure will yield an envy-free chore division (we’ve already described\nStep 1):\nA FOUR-PERSON ENVY-FREE CHORE-DIVISION PROCEDURE\nStep 1. Let Alice and Betty use three applications of Austin’s procedure (6 cuts) to\nobtain 4 pieces (X1, X2, X3, X4) that Alice and Betty believe are exactly equal\nin size. If Carl and Debbie disagree on which piece is smallest, then allocate\nthe pieces accordingly and end the procedure. Otherwise, call X4 the piece\nthat Carl and Debbie agree is smallest.\nStep 2. Let Carl and Debbie mark X1, X2, X3 where they would cut them to create\nties for smallest with X4. Without loss of generality, suppose Debbie has more\nmarks higher than Carl’s. Trim the pieces at the higher marks (3 cuts), and set\naside the trimmings.\n120 MATHEMATICS MAGAZINE\nCarl’s marks\nDebbie’s marks\n4X X X X1 2 3\nFigure 2 This figure shows a possible set of markings made in Step 1 of our 4-person\nprocedure\nStep 3. Let Betty add back to one piece some of the corresponding trimming (1 cut)\nto create a two-way tie for smallest piece.\nStep 4. Let players choose from these pieces in the order Alice-Betty-Carl-Debbie,\nwith Betty required to take the added-back piece if Alice didn’t, and Carl\nrequired to choose a piece trimmed at his marking if it is available. This will\nallocate everything except the trimmings in an envy-free fashion.\nStep 5. Divide the trimmings (6 cuts), exploiting an irrevocable advantage of Betty\nover whomever receives X4. (The concept of an irrevocable advantage is de-\nfined and explained later.) This will allocate the trimmings in an envy-free\nfashion.\nWe now verify that this yields an envy-free solution. We assumed in Step 2 that\nDebbie has more higher marks than Carl (if not, just reverse the roles of Carl and\nDebbie in what follows). This produces two cases, the first in which Debbie has three\nhigher marks and the second in which Debbie has two.\nCase I: Debbie has three higher marks. Assume that Debbie’s marks are all at or\nabove Carl’s marks. Following Step 2, let Debbie trim X1, X2, and X3 at her marks\nto obtain a four-way tie for the smallest piece. Call the trimmed pieces X ′1, X\n′\n2, and\nX ′3, and the trimmings T1, T2, and T3, which are set aside for later. At this point, each\nperson thinks:\nAlice: X ′1, X\n′\n2, X\n′\n3 < X4\nBetty: X ′1, X\n′\n2, X\n′\n3 < X4\nCarl: X4 ≤ X ′1, X ′2, X ′3\nDebbie: X ′1 = X ′2 = X ′3 = X4.\nCarl must believe X4 is the smallest, or tied for the smallest, because his marks were\nall at or below Debbie’s (meaning he believes more should be trimmed to make them\nequal to X4).\nOf the remaining pieces X ′1, X\n′\n2, X\n′\n3, suppose without loss of generality that Betty\nbelieves X ′3 ≤ X ′2 ≤ X ′1. Following Step 3, let Betty return some of the trimmings T3\nto X ′3 to create a two-way tie for the smallest piece. (We still call the modified piece\nX ′3, and its trimmings T3.) Thus player valuations change:\nAlice: X ′1, X\n′\n2, X\n′\n3 < X4\nBetty: X ′2 = X ′3 ≤ X ′1 < X4\nCarl: X4 ≤ X ′1, X ′2, X ′3\nDebbie: X ′1 = X ′2 = X4 < X ′3.\nVOL. 75, NO. 2, APRIL 2002 121\nFollowing Step 4, Alice chooses first (hence is envy-free), then Betty, who is required\nto take X ′3 if it was not chosen by Alice. Since Betty has at least one of X\n′\n2, X\n′\n3 to choose\nfrom, she is envy-free. Then Carl chooses, and will clearly still have his smallest piece\nX4 available. Debbie will have one of her three smallest pieces available because Betty\ntook X ′3 if Alice did not. Thus all are non-envious of the portions they have recieved\nso far.\nDividing the Trimmings. The trimmings still need to be divided and assigned. With-\nout loss of generality, suppose that Betty chose X ′3 in the procedure above. Then Betty\nthinks\nBetty: T1, T2 ≤ T3.\nNote that because Betty believed X3 = X4, she could receive all of T3 and still not\nenvy Carl. In fact, by the above inequality she could receive 1\n3\n(T1 + T2 + T3) ≤ T3 and\nstill not envy him. We will say that Betty has an irrevocable advantage over Carl with\nrespect to the trimmings.\nSo, lump all the trimmings together (say, T = T1 + T2 + T3), and let Alice and\nDebbie use Austin’s procedure to divide T into four pieces that they both agree are\nall equal. Then let the players choose in the order Carl, Betty, and then (in any order)\nDebbie and Alice.\nWith respect to the trimmings, Carl will envy no one because he chooses first. Betty,\nchoosing the smallest of the remaining three pieces, will have a piece that she believes\nis at most 1\n3\nT and therefore will not envy Carl. Alice and Debbie will not envy Betty\nor Carl because they think all four pieces are equal. Thus the trimmings can be divided\nin an envy-free fashion.\nCase II: Debbie has two higher marks. Assume now that Debbie has two marks at\nor above Carl’s marks. Without loss of generality suppose that Carl has a higher mark\non X3 than Debbie, as in FIGURE 2. Following Step 2, let cuts be made at all three\nhighest marks. Then\nAlice: X ′1, X\n′\n2, X\n′\n3 < X4\nBetty: X ′1, X\n′\n2, X\n′\n3 < X4\nCarl: X ′3 = X4 ≤ X ′1, X ′2\nDebbie: X ′1 = X ′2 = X4 < X ′3.\nFollowing Step 3, let Betty create a two-way tie for the smallest piece (as before)\nby returning to the smallest piece some of the corresponding trimmings. She may add\neither to X ′1, X\n′\n2 or X\n′\n3. The X\n′\n1 and X\n′\n2 cases are equivalent, so we have two subcases.\nSuppose Betty adds to X ′3 until it is as large as, say, X\n′\n2. Then\nAlice: X ′1, X\n′\n2, X\n′\n3 < X4\nBetty: X ′2 = X ′3 ≤ X ′1 < X4\nCarl: X4 ≤ X ′1, X ′2, X ′3\nDebbie: X ′1 = X ′2 = X4 ≤ X ′3.\nThese inequalities are identical to those in Case I, and thus our procedure works in the\nsame way. Moreover, the trimmings can be handled just as before, since Betty has an\nirrevocable advantage over Carl (who receives X4).\nOtherwise, suppose Betty adds to X ′1 until it is as large as, say, X\n′\n2 (the X\n′\n3 case is\nsimilar). Then\n122 MATHEMATICS MAGAZINE\nAlice: X ′1, X\n′\n2, X\n′\n3 < X4\nBetty: X ′1 = X ′2 ≤ X ′3 < X4\nCarl: X ′3 = X4 ≤ X ′1, X ′2\nDebbie: X ′2 = X4 ≤ X ′1, X ′3.\nFollowing Step 4, let Alice choose first. When Betty chooses, she will have one of her\ntwo smallest pieces available (and will take X ′1 if available as the procedure requires).\nNext, the procedure requires Carl to take X ′3 if available (since it was the piece trimmed\nat his marking) and otherwise he recieves X4; either way he believes he has the smallest\npiece. As X ′1 and X\n′\n3 are allocated by this point, we know that Debbie will receive\neither X ′2 or X4, and hence is also envy-free. For the trimmings, note that Betty has an\nirrevocable advantage over whomever receives the X4 piece, so the trimmings can be\ndivided using the method discussed earlier.\nThis concludes the verification of envy-freeness for all cases. Note that we could\nalternately have presented tables for each case that list envy-free assignments for Betty,\nCarl, and Debbie given what Alice chose first. However, remembering those tables\nwould not be as easy as remembering the steps of our procedure.\nA bounded n-person procedure? Our procedure gives the first known bounded pro-\ncedure for 4-person envy-free chore division, requiring at most 16 cuts. Actually, this\ncan be reduced to 15 cuts with a modification much like Brams-Taylor-Zwicker’s 5-cut\nmodification [5] of the triple application of Austin’s procedure.\nAlthough the reader may be tempted to try to further reduce the number of cuts\nneeded for 4-person envy-free chore division, progress in this direction is not as impor-\ntant as the more compelling problem of finding any bounded procedure for more than\n4 players. While there do exist finite n-person envy-free chore-division procedures\n([2], [7]), these are not bounded in the number of steps or cuts, that is, depending on\nplayer preferences, they could take arbitrarily long to resolve. For cake-cutting as well\nas chore division, the existence of bounded n-person envy-free division procedures\nremains a major unsolved problem that will probably require new techniques.\nAcknowledgment. The authors gratefully acknowledge a Harvey Mudd College Beckman Research Award,\nwhich supports research by undergraduates with faculty.\nREFERENCES\n1. A. K. Austin, Sharing a cake, Mathematical Gazette 6:437 (1982), 212–215.\n2. S. J. Brams and A. D. Taylor, An envy-free cake division algorithm, Economic Research Reports, C. V. Starr\nCenter for Applied Economics, New York University, 1992.\n3. S. J. Brams and A. D. Taylor, An envy-free cake division protocol, Amer. Math. Monthly 102:1 (1995), 9–18.\n4. S. J. Brams and A. D. Taylor, Fair Division: From Cake-Cutting to Dispute Resolution, Cambridge University\nPress, 1996.\n5. S. J. Brams, A. D. Taylor, and W. S. Zwicker, A moving-knife solution to the four-person envy-free cake-\ndivision problem, Proc. Amer. Math. Soc. 125 (1997), 547–554.\n6. M. Gardner, aha! Insight, W. F. Freeman and Co., New York, 1978.\n7. E. Peterson and F. E. Su, Exact procedures for envy-free chore division, HMC Mathematics Department\ntechnical report, http:\/\/www.math.hmc.edu\/˜su\/papers.html, 1998.\n8. J. M. Robertson and W. A. Webb, Cake-Cutting Algorithms: Be Fair If You Can, A. K. Peters Ltd., 1998.\n9. F. W. Simmons, personal communication, 1998.\n10. H. Steinhaus, The problem of fair division, Econometrica 16:1 (1948), 101–104.\n11. W. Stromquist, How to cut a cake fairly, Amer. Math. Monthly, 87 (1980), 640–644.\n12. F. E. Su, Rental harmony: Sperner’s lemma in fair division, Amer. Math. Monthly, 106 (1999), 930–942.\n"}
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{"id":"PKvlWW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Fair-Division-Procedures.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":88,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Fair-Division-Procedures","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Fair-Division-Procedures.pdf"],"author":"ESP Ghostscript 7.07","author_sxt":["ESP Ghostscript 7.07"],"last_modified":"Thu Jun 13 05:01:27 UTC 2019","dates_in_content_count_i":1,"dates_in_content_dts":["Thu Feb 10 00:00:00 UTC 2005"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2624765188301090265,"exact_signature_unique_b":true,"fuzzy_signature_l":185947410256438530,"fuzzy_signature_unique_b":true,"wordcount_i":6798,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:08 UTC 2022","fresh_date_dt":"Wed Dec 06 11:33:59 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"ESP Ghostscript 7.07","language_s":"en","size_i":224944,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088132656726016,"text_t":"Fair Division Procedures: Why use Mathematics?\nClaus-Jochen Haake\nInstitute of Mathematical Economics\nBielefeld University\nP.O. Box 100131\n33501 Bielefeld, Germany\nchaake@wiwi.uni-bielefeld.de\nFrancis Edward Su\nDepartment of Mathematics\nHarvey Mudd College\nClaremont, CA 91711, USA\nsu@math.hmc.edu\nFebruary 10, 2005\nContents\nIntroduction 2\n1 Mathematical Descriptions 4\n1.1 Modeling a Conflict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.2 Fairness Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n1.3 Solutions to the Bargaining Problem . . . . . . . . . . . . . . . . . . . . . . 10\n2 From Solutions to Procedures 14\n3 Why do we (have to) use Mathematics? 18\n4 Concluding Remarks 19\nGood sense is, of all things among men, the most equally distributed; for everyone\nthinks himself so abundantly provided with it, that those even who are the most\ndifficult to satisfy in everything else, do not usually desire a larger measure of\nthe quality than they already possess.\n—René Descartes, Discourse on Method, 1637\nIntroduction\nUnlike “good sense”, there are many things in the world that (by necessity or desire) must be\ndistributed among people or parties, but for which agreements on how to make that division\nare not easy to construct. This is a seed for conflict and a starting point of a negotiation\nproblem. As Descartes alluded to, the fundamental problem is how to “satisfy” the parties\naccording to a criterion such as “equal distribution” or some notion of fairness. Difficulties\narise because people often view the shares in any proposed distribution differently. And\n“good sense” is often not enough to help the parties resolve their conflict!\nMathematics may have something to offer in the quest to develop methods for conflict\nresolution. Problems of fair division —how to divide an object fairly— is one area in which\nmathematics has not only been fruitful in modeling the basic structure of the conflict and\nlocating solutions, but also has suggested useful methods and procedures for parties to reach\nagreement.\nOur goal in this paper is to indicate for a non-technical audience how mathematics can\nbe used to support the development of procedures for fair division problems. We wish to\nadvance an approach to conflict resolution, which is guided by the use of theoretical models,\nas we consider it to be complementary to “traditional” ones. To demonstrate this, consider\nthe Exercise below. Although it is a simple division problem, it provides a concrete context\nin which to illustrate our points (without being distracted by the technical mathematics that\na more complicated example would require).\nExercise: Tony and Chris wish to share a cake. They have differing opinions\nof the worth of various portions of cake; for instance, Tony prefers portions with\nchocolate and Chris prefers portions with cherries. They would like to determine\na “fair” division of the cake. As a mediator, can you suggest a method for helping\nthem solve their problem? 2\nUsing mathematics to model (and resolve) conflicts follows a general pattern. First, the\nstructure of the conflict is described in mathematical terms. Since a class of seemingly\n∼\/work\/zif\/contrib.tex 2 February 10, 2005\ndifferent division problems may have a similar structure, a mathematical model can be used\nto analyze a whole class of such conflicts. The model clarifies what the possible outcomes in\nthe conflict look like.\nSecond, one formulates desirable properties that a solution should satisfy. Here, the word\nsolution does not only refer to a particular outcome in a particular problem, but it means\na “rule” of which outcome to pick for any problem in the class of conflicts described by the\nmodel.\nThird, the work of the mathematician is to show that one can really find a solution that\nsatisfies the desired properties. The work does not end there, since knowing the existence of\na solution does not necessarily mean knowing how to reach it; there is often a gap between\ntheoretical existence results and a practical, applicable solution. A very important task,\nthen, is to convert theoretical results into a step-by-step manual to a solution. Each step\nshould be easy to comprehend, acceptable to the parties, and should bring the parties one\nstep further to a solution. We call such a manual a procedure. A procedural approach to\nresolve a conflict is a process that runs in stages to produce a solution, with a theoretical\nfoundation that shows why each step in the procedure works. The solution obtained fulfills\nall desired properties that were built into this process right at the beginning.\nIn this paper, we demonstrate by our running example (the Exercise) how to take a conflict\nand use mathematics to produce procedures for resolving the conflict. The reader may\ncompare her solution to the Exercise with our approaches, and judge whether he or she sees\nadvantages of using formal methods for developing procedures for a solution.\nThis paper is organized as follows: In Section 1 we start with the description of a conflict by\nmeans of a mathematical model. In the remainder of this section we discuss fairness concepts\n(envy-freeness and equitability) and their “formal” counterparts. In Section 2 we discuss the\nstep from theory to applicability. The question we address is: how can we achieve a solution\nby a procedure? Section 3 addresses the need, use and advantages of using mathematics in\nconflict resolution processes. Section 4 gives historical and technical as well as non-technical\nreferences for further reading.\n∼\/work\/zif\/contrib.tex 3 February 10, 2005\n1 Mathematical Descriptions\n1.1 Modeling a Conflict\nMathematics offers a wide range of tools for analyzing a fair division problem. Such an\nanalysis begins with a mathematical model, whose function is to describe the structure of\nthe problem and the set of possible outcomes unambiguously. Then, in order to determine\nsolutions for the problem, we have to know (and model) the persons’ preferences over the\npossible outcomes, which may vary from person to person (and indeed often do— it is\nembodied in the nature of a conflict that some of the possible outcomes are liked by some\nbut not by others).\nWe show how to do this in the context of our Exercise, which we now examine in a more\ndetailed fashion.\nExample 1.1 Suppose Tony and Chris wish to divide a cake among themselves. We use\nthe word “cake” only for convenience; the object to be divided could be anything (e.g.,\nland, money, time). Certainly, the cake might have a complicated form (in the Figure, for\ninstance, it consists of a number of geometric lumps), but as a first step in modeling we make\na simplifying assumption that does not change the fundamental nature of the problem: we\nassume that the cake has a rectangular form and is 10 cm long. The mathematical description\n\u0000 \u0001 \u0002\u0003\u0000\nFigure 1: Cake Division\nof the possible divisions of the cake clearly depends on the number and type of cuts that are\nallowed; hence the next step in modeling is to make assumptions regarding this.\nSuppose we assume that a knife cuts the cake into two pieces using just one vertical cut.\nThen each piece is specified by (a) a cut location, and (b) the side of the cut on which the\npiece lies, either “left” or “right”. For instance, (x, left) denotes the piece to the left of a cut\n∼\/work\/zif\/contrib.tex 4 February 10, 2005\n1.1 Modeling a Conflict\n\u0004 \u0005 \u0006\u0003\u0004\n\u0007\t\b\u000b\n\r\f\u000f\u000e\nFigure 2: Cake Division: One vertical cut\nat x. Hence each possible division is determined by (a) a cut location, and (b) an assignment\nof who gets which piece. (Mathematically, this is a pair (x, π), where x is a number between\n0 and 10, and π is a function that takes C or T to either “left” or “right”. We write πC\nfor the piece that Chris gets, and πT for the remaining piece, the one that Tony gets.) This\ndescribes the structure of the problem.\nNow we describe the preferences. It is likely that Chris and Tony have different preferences\nover such divisions of the cake, and the size of a piece of cake may not be their main criterion\nfor deciding which piece they want. For example, if Chris prefers cherries (which are more\nfrequent on the left side of the cake), and Tony likes chocolate (which is distributed evenly\nacross the cake), and neither care about the mass of the cake itself, then cutting the cake\nexactly in the middle and assigning Chris the left hand side might have the following effect:\nChris might think that he has more than 50% of what the total cake is worth to him. Tony\nmight think that the two pieces are equal (if both contain the same amount of chocolate) and\nmight be indifferent between getting the left or the right piece (which means he evaluates\nhis piece at 50%).\nTo elicit a complete description of a person’s preferences, we must know how that person\nwould evaluate each possible division of cake. As a “scale” for evaluation, we could let each\nperson assign a total of 100 points to the whole cake and distribute this number among the\ntwo pieces. Thus preferences could be described by the graphs in Figure 3.\nThe solid lines in Figure 3 correspond to those divisions of the cake in which Chris gets\nthe left piece and Tony gets the right piece, whereas the dashed lines represent the opposite\nassignment. Figure 3 tells us how Chris and Tony will evaluate their piece of cake as a\nfunction of the position of the cut. 2\n∼\/work\/zif\/contrib.tex 5 February 10, 2005\n1.1 Modeling a Conflict\n\u0010 \u0011\u0013\u0012 \u0012 \u0010 \u0011\u0013\u0012\n\u0014\u0016\u0015\u0018\u0017\u001a\u0019 \u001b\u001c\u0019\r\u0015\u001e\u001d\u0015 \u001f\u0016!#\"$\u001b\n\u0014\u0016\u0015\u0018\u0019%\u001d\u0018\u001b&\u0017\n\u0011\u0013\u0012\u0018\u0012\n\u0010\u001e\u0012\n\u0012\n'%( \u001f)\u001b+*,\u0019 ( ! (\n- \u0019%.\u001e\/0\u001b1*,\u0019 ( ! (\n2 \/ - \u0019)\u0017#3 (5476\u001e' \" 6 \u001b\u001c\u0019%\u0015\u001e\u001d 89\u0015\u0018\u001d0:\u00033 \u0017 (5476\u001e' \" 6 \u001b\u001c\u0019%\u0015\u001e\u001d\n\u0011;\u0012\u001e\u0012\n\u0010 \u0012\n\u0012\n'%( \u001f<\u001b+*,\u0019 ( ! (\n- \u0019%.\u001e\/0\u001b1*,\u0019 ( ! (\n\u0012 =\nFigure 3: Preferences over divisions of the cake\nThis example reveals a few issues concerning the mathematical description of the two-person\ncake-cutting problem.\n1. What is the appropriate method to cut the cake?\n2. What is the best way to express preferences?\n3. How practical is the process of specifying preferences?\nWe address each of these issues in turn:\n1) We assumed that the cake is divided by a single vertical cut, but in practice, one could\nthink of many other versions of cutting (e.g. rotating the knife, cutting more than once, or\nmaking no cuts). All these alternatives could just as well be formalized by slight modifica-\ntions. In doing so, one must bear in mind some external considerations: what is practicable\n(e.g., can we rotate the knife?), what will allow for the existence of a solution (e.g., making\nno cuts will not allow for an envy-free division), what is efficient (e.g., is more than one cut\nnecessary?), and what is procedural (e.g., can the solution be obtained by some practical\nmethod)?\n2) The scale used to express preferences should be chosen with respect to the desired out-\ncome. In the example, each person was to divide 100 points. With this kind of preference\nannouncement, it is not possible to compare how one person evaluates the cake with respect\nto how the other one does. If we were interested in dividing the cake such that each person\ngets the “same total benefit” (suppose this could be measured) then we would have to give a\nsmall piece to the person who loves cake and a large piece to the person who does not really\n∼\/work\/zif\/contrib.tex 6 February 10, 2005\n1.2 Fairness Concepts\ncare about cake, and the scale chosen above would not be the appropriate one. On the other\nhand, if we were interested in an envy-free division of the cake (i.e. neither person wants to\ntrade their assigned pieces), then the point assessment is helpful. In assigning 100 points,\nit is only important how each person evaluates the two pieces, but it is not relevant how he\nevaluates them in comparison to the other person. Thus the method by which preferences\nare described is strongly connected with the kind of solution of concern.\n3) In the example, each person must evaluate both pieces for a continuum of possible cuts.\nThis may not be practical. Thus there is a trade-off between practicability and accuracy\nwhen describing preferences. For practical reasons, it may be sufficient to ask Chris and\nTony how they assess the pieces for cuts at 0cm, 1cm, 2cm, ... , 10cm. Both would then\nonly have to announce 22 numbers (2 pieces x 11 cuts) in order to describe their preferences,\nwhich is much less demanding than before. In this scenario, one could either estimate Chris\nand Tony’s values of pieces at other cut locations, or simply limit the set of possible cuts\nto these 11 positions. Both versions have their drawbacks— estimation runs the risk that\nthe assumed preferences may not be the correct ones, and limitation runs the risk of not\nreaching an agreement at all. In the mathematical description of fair division problems one\nalways has to balance practicability with the accuracy of the data.\nAll these questions show that one should carefully decide how to describe the problem and\nhow to express personal preferences, before one can even speak about how to resolve the\nconflict.\n1.2 Fairness Concepts\nIn the previous subsection we were concerned with the descriptions of all possible outcomes\nof the conflict. Now we account for the fact that the desired outcome will be assessed by its\nfairness properties. For instance, in our Exercise, Chris and Tony may desire an envy-free\noutcome, i.e., a division of cake so that neither envies the other. Alternatively, they may\nwant an equitable division, i.e., one in which Chris’ assessment of his piece (in points) is the\nsame as Tony’s assessment of his piece. Thus the set of desirable outcomes will depend on\nthe notion of fairness chosen.\nEnvy-freeness\nIf Chris and Tony are interested in an envy-free division of the cake, then we might model\neach of their preferences by a utility function that assigns to each piece of cake a number that\ngives a measure of its “worth” to that person. Let uC and uT denote Chris and Tony’s utility\n∼\/work\/zif\/contrib.tex 7 February 10, 2005\n1.2 Fairness Concepts\nfunctions. Then, for instance, uT (x, left) is a number that describes Tony’s utility for the\npiece to the left of x. Then Chris does not envy Tony if the inequality uC(x, πC) ≥ uC(x, πT )\nholds true. Analogously, Tony is not envious if uT (x, πT ) ≥ uT (x, πC) is satisfied.\nExample 1.2 (Example 1.1 cont.)\nA closer look at the preferences in Figure 3 reveals the following: if the cut were made at\nx = 2, Chris would be indifferent between the left and the right piece. For x > 2 he prefers\nthe left piece and for x < 2 he would choose the right one. Hence, Chris’ condition\nuC(x, left) ≥ uC(x, right)\nis satisfied if and only if x ≥ 2. Analogously, Tony’s no-envy condition uT (x, left) ≥\nuT (x, right) is satisfied if and only if x ≥ 5. To summarize, we can say:\n• For x ≤ 2 both prefer the right piece over the left one.\n• For x ≥ 5 both prefer the left piece over the right one.\n• For 2 ≤ x ≤ 5 Chris prefers the left piece and Tony prefers the right one.\nThis means that the set of all solutions that do not create envy are those where πC is “left”,\nπT is “right” and x is between 2 and 5. This is a complete description of the envy-free\nsolutions for this problem. 2\nNote that in the example a cut at x = 2 and assigning the left piece to Chris, leaves Chris\nindifferent, i.e. he thinks he has got 50% of what the whole cake is worth to him. Tony,\nreceiving the right piece thinks he got more than 90% of what the cake is worth to him.\nAlthough this is an envy-free solution, it may be a division that is undesirable with respect\nto other fairness criteria, such as equitability.\nEquitability\nThe analysis in example 1.2 has to be modified if we change the fairness criterion. Let us\nassume now that Tony and Chris both consider a solution as “fair” if each one feels he got\njust as much in his evaluation as the other did in his. We say a solution is equitable if the\nequation uC(x, πC) = uT (x, πT ) is satisfied. That is, Chris assigns the same number of points\nto his piece (obtained by the cut at x) as Tony does to his piece.\n∼\/work\/zif\/contrib.tex 8 February 10, 2005\n1.2 Fairness Concepts\nExample 1.3 (Example 1.1 cont.)\nTo solve the above equation in our Exercise, Figure 4 rearranges the utility functions. In the\nleft diagram we have Chris’ and Tony’s utility functions for the case that Chris gets the left\nand Tony the right piece. The right diagram shows utilities for the opposite assignment.\n> ?;@ @ > ?;@\nACB\u0018D\u001cE F\u001cE%B\u0018GB\u001eH\u0016I5JKF\nA\u0016B\u0018E%G0F&D\n?;@\u001e@\n> @\n@\n?\u0013@\u0018@\n>\u001e@\n@\n@ L;M LL;M L\nN\u0003OQPSR9T\u001aU OWV\nN\u0003X1PSR9T\u001aU XYV\nN\u0003OWPSR9T\u001aU OWV\nN X PSR9T\u001aU XZV\nU X\\[^] E%_\u0018`0F T\u001aU Oa[cb\rd H<FU X\\[^b%d H)F T\u001cU Oa[^] E%_\u0018`0F\ne&f M g\nf#e M g\nFigure 4: Piecewise preferences over divisions of the cake\nThe two utility functions on the left diagram intersect for x = 4.4, i.e. uC(4.4, left) =\nuT (4.4, right) = 72.5. With a cut at position 4.4 and Chris obtaining the left piece, both\npersons assign 72.5 points to their shares. Switching the piece assignment results in the\nright diagram of Figure 4, where at the intersection of graphs we have uC(4.4, right) =\nuT (4.4, left) = 27.5 (since the assessments of left and right pieces for any person must add\nto 100). Thus there are two different outcomes that are equitable (although it is easy to see\nthat one of the two solutions is strictly preferred by both). 2\nIn this example, we see that the concepts of envy-freeness and equitability do not exclude\neach other. For example, the outcome (x = 4.4, πC = left, πT = right) also satisfies the\nenvy-freeness conditions. Also, observe the differences between the two concepts— whereas\neach envy-freeness condition involves only a single person’s preference, the equation that\ndetermines equitability uses both utility functions at the same time.\nNext, we consider a different way of presenting the persons’ preferences. We have described\nthe set of possible outcomes of the cake cutting problem. To each such outcome we can\nassign a pair of numbers consisting of Chris’ and Tony’s assessment (in points). These pairs\ncan be depicted in a two-dimensional diagram as it is done in Figure 5.\n∼\/work\/zif\/contrib.tex 9 February 10, 2005\n1.3 Solutions to the Bargaining Problem\nh\rikjml nmopiqjml n\u001cr\ns\ns\nt5s\nums\u0013s\nt5s ums\u0013s\nv\u0003w\u0013x y7zq{\u0016|\tw\u0013y }\nv\u0003w;x y;zq{Q~9\u001e€kx {\nFigure 5: assessments of possible outcomes\nThe solid line represents all possible pairs of points in which the left piece goes to Chris and\nthe right piece is for Tony, whereas the dashed line corresponds to the opposite assignment.\nHow can we identify equitable solutions from this picture? Since the diagonal (45 degree\ndotted line) contains all pairs with equal numbers of points, we simply locate the intersection\nof the diagonal with the two curves. Here we obtain the two pairs (27.5, 27.5) and (72.5, 72.5)\n(cf. Example 1.3).\nSince there are only two persons involved, each person does not envy the other if he thinks\nhis piece is at least 50 points worth (since then the value of the other piece is at most 50\npoints). So the pairs that correspond to envy-free outcomes are located on the solid line in\nthe shaded region of Figure 5.\n1.3 Solutions to the Bargaining Problem\nFigure 5 presents preferences in an aggregated form; from the set of possible allocations\n(pairs) of points, individual preferences cannot be extracted in general. In particular, the\nFigure does not exhibit the underlying outcomes but only the results in terms of utility (here\nmeasured in points).\nWhen one is interested in finding envy-free solutions, this “utility possibility set” is in general\nnot the appropriate description for analysis, because one has to compare the pieces and not\nthe points of the piece assigned. In the previous example we could display envy-free solutions\nonly because there are just two persons involved and utilities were normalized to 100 points.\nHence one’s valuation of one piece also reveals one’s valuation of the other piece. This is\nno longer possible as soon as there are more than two persons or without normalization\n∼\/work\/zif\/contrib.tex 10 February 10, 2005\n1.3 Solutions to the Bargaining Problem\nconstraints.\nHowever, utility possibility sets are a very useful tool to analyze fairness criteria that are\nbased on allocations of utility. For example, an equitable division corresponds to a point\nin the utility possibility set, in which each agent obtains the same fraction of his maximal\npossible utility. In the (axiomatic) theory of bargaining, utility possibility sets are used to\ndescribe the conflict. This form of representing a conflict was introduced in an early work\nof Noble-Prize winner John Nash (see Nash (1950)). A bargaining problem arises anytime\none has a utility possibility set U together with a “disagreement” allocation d in U that\nrepresents the utility allocation that prevails when no agreement can be reached. (The\nsource of utilities is a secondary question.) A bargaining solution is a “rule” that assigns to\neach bargaining problem a specific allocation of utilities (a point in U).\nThe axiomatic approach searches for bargaining solutions that satisfy certain desirable prop-\nerties (axioms), such as:\nPareto Optimality: The bargaining solution should always select a point for which no\nother point in U “dominates” it, i.e., there is no other utility allocation in U which\nleaves everyone with at least as much utility as before and does better for some of the\nplayers.\nIndividual Rationality: The solution should assign to each person at least as much utility\nas the disagreement point d does. That means everyone should have an incentive to\nparticipate in the bargaining process.\nSymmetry: The order of the persons does not matter for the solution. This means that\nthe solution only depends on players’ preferences and is not influenced by how a player\nis named.\nScale Invariance: The solution should not depend on the scale that is chosen to measure\nutilities.\nIndividual Monotonicity (IM): Suppose there are two bargaining problems (U, d) and\n(U ′, d), such that the set of outcomes U ′ of one contains all the outcomes U of the other,\nand such that player 1’s maximum outcome is the same in both, then the bargaining\nsolution should give player 2 at least as much in U ′ as in U .\nThe last two axioms call for an interpretation. The Scale Invariance property roughly says\nthat the solution should not depend on how many points (in total) are assigned to some\noutcome but on how many points are assigned relative to the maximal number of points.\n∼\/work\/zif\/contrib.tex 11 February 10, 2005\n1.3 Solutions to the Bargaining Problem\nSo, if Chris doubles all points that he assigns (resulting in a maximal number of 200 points),\nthen a solution that is scale invariant selects the same outcome as before.\nFigure 6 illustrates the basic idea behind the IM property. In (U ′, d) there are more possibil-\nities to allocate utility than in (U, d). Facing the “bigger” bargaining problem, Tony could\nargue that for each fixed utility level (number of points) for Chris, he can obtain more utility\nin U ′ than in U . Therefore the solution ϕ should give him more in U ′. In this spirit the\nsituation U ′ is more advantageous for Tony than U is. Chris is indifferent between U ′ and U ,\nbecause he could get the same maximal utility (90 points) in either case. For this reason the\nIM property only requires that the advantaged player should get more. Applied to the cake\n\u000b‚\u0018ƒp„Y…k†k‡ ˆ\n ‚ ƒp„C†q‡ ˆ\n‰\u0018Š,Š\n‹,Š ‰\u001eŠ,Š\nŒZ,Ž\u0003C ‘“’\u000b”#•<–<•)”m\n—™˜›š •S‘ \u000b’\u000b”#•<–<•)”m\nœ“\u001až Ÿk¡5¢¤£\n‡¦¥cƒS§\u0018†q§7ˆ\nœ“\u001až¨¡#¢¤£©#ª\n«#ª\n«mª\n¬\n¬®­\nFigure 6: The IM property\ncutting problem, an enlargement of the set U to U ′ could be achieved as follows: suppose\nTony loves nuts and Chris is indifferent to them. Now the cake is refined with additional\nnuts. Tony is delighted, because the cake (and every piece of it) is now worth more to him,\nwhile Chris does not realize a change. Comparing the solutions for the two versions of the\ncake, Tony should be at least as well off in the situation with nuts as without.\nIn the 1970’s, Ehud Kalai and Meir Smorodinsky introduced a bargaining solution1 for (a cer-\ntain class of) bargaining problems for two persons that satisfies all these five properties. The\nidea for this solution goes back to Raiffa (1953). Roughly speaking, the Kalai-Smorodinsky\n(KS) bargaining solution can be constructed as follows: One connects the disagreement point\nwith the point that represents each player obtaining his maximal utility. The latter point\nis (due to the nature of a conflict) in general not located in U . The KS solution is now\n1see Kalai & Smorodinsky (1975)\n∼\/work\/zif\/contrib.tex 12 February 10, 2005\n1.3 Solutions to the Bargaining Problem\nthe upper right most point on this connection line that is still in the utility possibility set\nFigure 7 illustrates the construction for a typical cake cutting problem from Section 1.\n¯\u001c°\u0013°\n¯\u001c°\u0013°\n±,²5³7´0µ ¶¸·7¹»º ¼ º ¹p´\n½›¾ ¿ º ¶\u001cµ7·7¹»º ¼ º ¹p´\nÀ\nÁ›ÂÄà Å\u001cÆSÅqÇ\nÈÊÉ\u0016Ë<Ì9Í»Î;Ï\nÐ\u000fÑ\nÐ\u000fÑ\nÒ ÅqÅ\u001cÆ Ò ÅqÅ\u001aÇ\nFigure 7: The The Kalai Smorodinsky solution\nIt can be shown that the above discussed properties are sufficient to uniquely characterize\nthe KS bargaining solution. This means that any bargaining solution with these properties\nmust be the KS bargaining solution. This way of determining solution concepts by their\nproperties is frequently referred to as axiomatization.\nThere is also a description of the fairness concept behind the KS solution that comes from\nits construction. In the KS solution point, Chris and Tony both get the same fraction of\ntheir maximal possible utility (In Figure 7 this fraction is 0.7). In the cake cutting scenario,\nthis means that both get an equal share of what they think the whole cake is worth. This\nis an equality in terms of utility. Note that this is exactly the same argumentation that we\nused in Section 1 for equitable solutions. Indeed, the KS solution is the bargaining theoretic\ncounterpart to equitability.2\nIn summary, there are three approaches to the KS solution: 1) the construction method (con-\nnection line), 2) the axiomatic approach (KS solution is characterized by certain properties)\nand 3) the fairness approach (equal fractions of maximal utility).\nBesides the KS solution there are many other bargaining solutions that can be approached the\nsame way, e.g., the Nash solution (cf. Nash (1950) and Nash (1953)) or the Perles-Maschler\nsolution (cf. Perles & Maschler (1981)).\n2Raith & Welzel (1998) showed this analogy by analyzing the Adjusted Winner procedure that was\nintroduced in Brams & Taylor (1996) to perform equitable divisions of a set of objects.\n∼\/work\/zif\/contrib.tex 13 February 10, 2005\n2 From Solutions to Procedures\nThe approaches that we have described above not only model the structure of a fair division\nproblem but also determine solutions. However, there is one decisive question left open: How\ndo we arrive at a solution?\nFor the players, the acceptance of a solution is strongly connected to the transparency of\nhow and why it is determined. Theoretical results can answer the “why”, but the “how” is\nwidely ignored. In Examples 1.2 and 1.3 envy-free and equitable solutions were constructed\nby modeling and analyzing the preferences. But the calculation of the final solution can be\ncarried out without involving Chris and Tony. Thus, there is a danger that the conflicting\nparties may not accept the solution because they do not understand the method by which\nit was determined.\nProcedural approaches are exactly concerned with these practical difficulties. Without leav-\ning the ground of a formal (mathematical) description, a procedure wants to give “step by\nstep instructions” to the solution. We believe that such a guide should exhibit the following\nfeatures:3 Each step should be\n1. intuitive, i.e. it should be easy to comprehend.\n2. plausible, i.e. should be simple to argue.\n3. manageable, i.e. must be straight forward to compute.\nMoreover, a procedure should lead to a solution after finitely many steps. In this section\nwe examine three procedures for our Exercise: the well-known Divide and Choose proce-\ndure, Austin’s Moving knives procedure, and a polling procedure. In these examples we\ndemonstrate how mathematics is used to reveal fairness properties of the outcomes.\nExample 2.1 (Divide and Choose)\nProbably the most famous procedure that most people know from the division of chocolate\nbars in their childhood is Divide and Choose.4 The rules are very simple to understand.\nChris (names could be interchanged) takes the knife and cuts the cake into two pieces. After\nthat, Tony may choose among the two pieces. Fine! But what property does the outcome\nsatisfy?\nLet us restate the rules combined with suggestions. Chris cuts the cake into what he considers\n3see also Haake, Raith & Su (2002)\n4see also Brams & Taylor (1996)\n∼\/work\/zif\/contrib.tex 14 February 10, 2005\nto be equal pieces. Then Tony chooses the piece that he likes most. Now it is clear that the\noutcome should be envy free. Chris does not envy Tony because he considers both pieces as\nequal, and Tony had the choice. With respect to envy-freeness the solution is fair. 2\nHowever, the procedure has its drawbacks with respect to other notions of fairness, such as\nequitability, since the outcome generally favors the chooser (since the cutter gets exactly 50%\nof what the cake is worth to him and the chooser usually gets more). So the final outcome\nis almost certainly not equitable. (And, if the cutter has information about the chooser’s\npreferences, the procedure can be exploited by the cutter to favor the cutter: he can achieve\nmore than 50% by cutting the cake so that the other player considers the pieces almost equal\nbut leaving the piece the cutter likes a little less attractive to the chooser).\nExample 2.2 (Moving knife)\nAustin’s procedure (Austin (1982)) is a method for dividing a cake into two pieces between\ntwo persons so that each person believes it is divided exactly in half. Let Chris hold two\nknives over the cake, with one at the left edge, such that the portion of cake between them\nis what he believes to be exactly half. If Tony agrees that it is exactly half, the procedure\nis finished. Otherwise, let Chris move the knives across the cake from left to right, keeping\nthe portion between them exactly half (in Chris’ estimation), until Tony agrees it is exactly\nhalf. (There must be such a point because when the rightmost knife reaches the right edge,\nthe leftmost knife must be where the rightmost knife began, hence Tony must by that point\nhave changed preferences.) At this point cuts are made and the cake outside the knives are\nlumped together, yielding two pieces which both agree are exactly equal. 2\nÓ Ô;ÕÕ\nÖ¤× Ö›Ø\nÙÛڙÜ\u0018ÝqÞ9ßáà¦Ýqâ›ã\u000bâ ä\nåWæZç ß\rÞàèÝqâ\u000bã\u000bâ\nä\nåWæYç ß\rÞ\\àèÝqâ\u000bã\u000bâ\nFigure 8: The Moving Knife Procedure\nThe basic difference between the two procedures is the following: the Divide and Choose\nprocedure generates envy-free outcomes, the Moving Knives provide an equitable division\n∼\/work\/zif\/contrib.tex 15 February 10, 2005\nof the cake. Clearly, Chris and Tony get exactly half of the cake, provided they follow the\nrules of the procedure and represent their preferences honestly. But there is a drawback with\nAustin’s procedure. As we have seen in Section 1, there are very well equitable divisions so\nthat both get more than 50% in their assessments. Austin’s procedure gives both exactly\n50%. So the property of Pareto efficiency (see Subsection 1.3) is not fulfilled. In general\nthere are divisions that are equitable but ensure both a higher utility.\nWe may check the fairness properties of both procedures by returning to the formal model\ndeveloped in Section 1. For this, consider once again Figure 3. Divide and Choose tells Chris\n(as cutter) to select the division of the cake so that he thinks both pieces are equally valuable\n(i.e. at x = 2). In the left diagram, the position of the cut is given by the intersection of\nChris’ utility functions (since the sum of his points for both pieces is always 100). With\na very mild formal condition on preferences (called continuity), one can always guarantee\nexistence of such an intersection point. Therefore the procedure can be carried out as stated,\nand as argued above, the outcome is envy-free. The mathematical description also reveals\nthat the procedure benefits the chooser.\né\u000bê7ë\u000fì í\u001aî7ï#ð1ñ\u001eò#ì ó5ô\u000fí\nõ ò#ó5ö\u0018î í\u0016ï#ð1ñ\u001eò#ì ó5ô\u000fí\n÷qø ôpô\u000fì ó7ùáñ\u001eò#ì ó5ô\nú ì ó7û\nú ì ó û\nü\nü\ný\u0003þ\ný›ÿ\n\u00007ü\n\u0000;ü\nFigure 9: The Moving Knife Preferences\nFor the Moving Knife procedure, consider Figures 8 and 9. Figure 8 simply clarifies who\ngets which piece. Figure 9 displays Chris and Tony’s preferences in a slightly subtle way.\nAny point in the x1-x2 diagram reflects the position of first and second knife. Because the\nx1-knife is always on the right of the x2-knife we only have to consider points below the\ndiagonal (here x1 ≥ x2 holds). The solid line now represents all sets of cuts for which Chris\nis indifferent between getting the interior or the outer piece. Hence, with all points on that\n∼\/work\/zif\/contrib.tex 16 February 10, 2005\nline, Chris obtains a 50% share of the cake. Tony’s 50% line is the dashed one. His line\nstarts to the right of Chris’ line, because at the beginning, Tony strictly prefers the right\n(outer) over the left (interior) piece. Tony’s line must (and does) terminate above Chris’\nline, because assignments just have changed compared to the starting situation. Then (this\nis what math tells us in general) there must be an intersection point of the two lines, i.e. a\nset of cuts in which both receive exactly a 50% share. Roughly speaking, the procedure finds\nthis point by “moving along” the solid line and stopping at the intersection point.\nExample 2.3 (A Polling Procedure)\nFinally, we describe a third procedure that illustrates some of the earlier considerations\nregarding practicability. Suppose, that because of practical reasons (e.g., time or limited\nresources) we can only ask for the preferences of the players at a finite set of cuts 0cm, 1cm,\n2cm, ... , 10cm. Can we find an envy-free division?\nThis polling procedure begins by successively moving the knife, from left to right, to the\npositions 0cm, 1cm, 2cm, ... , 10cm, pausing to ask each player at each cut “Which piece\nwould you prefer if the cake were cut here?” Players just answer either L or R at each cut\nposition (they do not need to assess their utilities of pieces). Based on these answers we can\nsuggest a division that is approximately envy-free.\nFirst note that if there were a cut for which Chris and Tony answered differently, then we\ncould cut there and assign them the pieces that they requested. But this may not always\nbe possible (e.g., if Chris and Tony were to answer the same way at all 11 cuts). However,\nwe can always find a 1cm segment of cake where at opposite endpoints Chris answered one\nway and Tony answered the other way. Then cut the cake in the middle of this segment and\nassign Chris and Tony the side of the cut they preferred at the endpoints. This solution may\nnot be envy-free, but it is envy-free to within the utility specified by one segment of cake.\nTake for instance the problem illustrated in Figure 3. If we ask Chris and Tony, which piece\nthey preferred at the 1cm mark then both respond “right piece”. At 2cm Chris is indifferent,\nwhereas Tony sticks to “right”. We can finally cut the cake at the 3cm mark, as Chris and\nTony prefer different pieces. 2\nHow do we know there is such a segment? Again, we appeal to mathematical argument—\nfor the cut at 0cm, both players will prefer piece R, and for the cut at 10cm, both prefer L.\nWe may assume that they answer the same way at all cut locations (else we would have an\n∼\/work\/zif\/contrib.tex 17 February 10, 2005\nenvy-free division). So as we proceed along the 10 segments from 0cm to 10cm, there must\nbe a segment on whose endpoints their preferences will switch (from L to R).\nAt first, a procedure to obtain an approximate envy-free division seems unnecessary— after\nall, Divide-and-Choose is a procedure for exact envy-free division and it is simple enough\nto implement. However, as the number of players increase for the cake-cutting problem, the\ndifficulty of obtaining a practical procedure for exact envy-free division rises quickly (Brams\n& Taylor 1995), making approximate procedures more attractive. Our two-person polling\nprocedure illustrates the basic idea of the approach advocated by Su (1999) for approximate\nenvy-free solutions of n-person fair division problems.\n3 Why do we (have to) use Mathematics?\nTo summarize, what lessons do we learn about the relationship between mathematics and\nprocedures for solving fair division problems?\nEven if we do not “see” the mathematics in the final description of a procedure, it is an\nindispensable analytic tool. As we observed, it is used to guarantee that the procedure\ndoes work and produces a well-defined and desirable outcome. Second, and this is probably\nmore important, mathematical methods reveal the fairness that the outcome of a procedure\nshould satisfy. If all agents follow the rules of the procedure, then the outcome is guaran-\nteed to satisfy some specific fairness criterion like, e.g.,envy-freeness or equitability. Third,\nmathematical insights also reveal the limitations of a given procedure (e.g., in our discussion\non Divide-and-Choose, the proposed solution is envy-free but still favors one party using a\ndifferent fairness criterion). Therefore, theoretical models not only help to characterize and\nfind fair outcomes, but they are also necessary for a critical assessment of conflict resolution\nprocedures.\nAlso, the mathematics of setting up a model for a division problem requires preferences to\nbe elicited and the structure of allowable agreements to be made precise. This process can\nbe beneficial to players, as it can help them to pin down what their preferences really are\nand equip them to assess their desires critically. (People who have not done this critical\nassessment often have a harder time making decisions.) Moreover, it will help them think\nbroadly about what divisions are possible.\nFinally, the axiomatic notion of a “solution” being a rule (that hopefully can also be sup-\nported by a procedure), which solves a whole class of conflicts, is a powerful idea. It frees us\n∼\/work\/zif\/contrib.tex 18 February 10, 2005\nfrom focusing only on just the one problem at hand, and allows one to pick fairness criteria\n(axioms) before getting caught up in a particular struggle. A solution based on axioms will\nseem less arbitrary to players than an ad-hoc, situation-specific method developed only for\nthe one conflict at hand.\n4 Concluding Remarks\nAlthough fair division problems have been around since antiquity, Steinhaus (1948) was\nperhaps the first to pose the question as a serious academic endeavor when he asked: how\nto cut a cake fairly? Since then, there has been a lot of work on this classical problem as\nwell as related problems: burden division (dividing an undesirable cake), dividing mixtures\nof goods and burdens, allocations of indivisible objects, division into unequal shares, and\nexplorations of all these types of divisions according to various notions of fairness. Two\nnotable surveys of the fair division literature are the books by Brams & Taylor (1996) and\nRobertson & Webb (1998), the latter being somewhat more mathematical in nature.\nAnd, just as traced in our paper, there is often a gap between the development of work that\nfocuses on existence of solutions, versus work that develops usable procedures. For instance,\nthe existence of cake-cutting solutions can be traced back to the 1940’s but the first general\nn-person procedure for cake-cutting due to Brams-Taylor was not developed until the 1990’s.\nEven that is not a usable procedure in practice, so there remains much work to be done.\nIn addition to the debate about the importance of various fairness criteria, there is also (and\nshould be) healthy debate about the aspects of procedures that make them acceptable to\nparticipants. For instance, how do various procedures compare on how intuitive, or plausible,\nor manageable they are? The examples we gave for 2-person cake-cutting are all very easy\nto follow and would rate well in all three aspects, but n-person counterparts for this and\nother fair division problems can be a bit more complex.\nEven if some of these procedures are lacking in the “manageability” arena, one exciting recent\ndevelopment is the possibility of using computers as “negotiation support tools” to help give\na mediator a hand in managing a complex procedure whose foundation is mathematical.\nThe Fair Division Calculator (Su 2000) is one example that was recently incorporated into\nARTUS, an online negotiation platform (Raith & Wilker 1998). This is a promising first\nstep to give automated procedural support to mediators in a live negotiation.\nIn this article, we hope that we have given the reader a sense of how mathematics can be\n∼\/work\/zif\/contrib.tex 19 February 10, 2005\nREFERENCES\nused to support the development of procedures for fair division problems, not just ones that\nare mathematically-supported but ones that can actually lead to practicable uses.\nReferences\nAlkan, A., G. Demange & D. Gale (1991): “Fair Allocation of Indivisible Goods and\nCriteria of Justice”, Econometrica, 59, 1023–1039.\nAragones, E. (1995): “A Derivation of the Money Rawlsian Solution”, Social Choice and\nWelfare, 12, 267–276.\nAustin, A. (1982): “Sharing a Cake”, Mathematical Gazette, 66(437), 213–215.\nBrams, S. J. & D. M. Kilgour (2001): “Competitive Fair Division”, Journal of Political\nEconomy, 109(2), 418–443.\nBrams, S. J. & A. D. Taylor (1995): “An envy-free cake division protocol”, American\nMathematical Monthly, 102, 9–18.\n(1996): Fair Division – from cake cutting to dispute resolution. Cambridge Univer-\nsity Press, Cambridge.\nFoley, D. (1967): “Resource Allocation and the Public Sector”, Yale Economic Essays, 7,\n45–98.\nHaake, C.-J., M. G. Raith & F. E. Su (2002): “Bidding for envy-freeness: A Procedural\nApproach to n-Player Fair Division Problems”, Social Choice and Welfare, 19(4), 723–749.\nKalai, E. & M. Smorodinsky (1975): “Other Solutions to Nash’s Bargaining Problem”,\nEconometrica, 43, 513–518.\nKlijn, F. (2000): “An Algorithm for Envy-free Allocations in an Economy with Indivisible\nObjects and Money”, Social Choice and Welfare, 17, 201–216.\nKnaster, B. (1946): “Sur le Problème du Partage Pragmatique de H. Steinhaus”, Annales\nde la Societé Polonaise de Mathematique, 19, 228–230.\nNash, J. F. (1950): “The Bargaining Problem”, Econometrica, 18, 155–162.\n(1953): “Two–Person Cooperative Games”, Econometrica, 21, 128–140.\n∼\/work\/zif\/contrib.tex 20 February 10, 2005\nREFERENCES\nPerles, M. & M. Maschler (1981): “The Present Status of the Superadditive Solu-\ntion”, in Essays in Game Theory and Mathematical Economics. Bibliographisches Institut,\nMannheim.\nRaiffa, H. (1953): “Arbitration Schemes for Generalized Two-Person Games”, in Contri-\nbutions to the Theory of Games, ed. by H. Kuhn & A. W. Tucker, vol. 1 of Annals of\nMathematical Studies 24, pp. 361–387. Princeton University Press, Princeton, NJ.\nRaith, M. G. & A. Welzel (1998): “Adjusted Winner: An Algorithm for Implementing\nBargaining Solutions in Multi-Issue Negotiations”, IMW Working Paper 295, University\nof Bielefeld.\nRaith, M. G. & H. Wilker (1998): “ARTUS: The Adaptable Round Table with a User-\nSpecific Surface”, IMW Working Paper 298, University of Bielefeld.\nRobertson, J. & W. Webb (1998): Cake-Cutting Algorithms. A K Peters, Natick, Mass.\nSteinhaus, H. (1948): “The Problem of Fair Division”, Econometrica, 16, 101–104.\nSu, F. E. (1999): “Rental Harmony: Sperner’s Lemma in Fair Division”, American Math-\nematical Monthly, 106, 930–942.\nSu, F. E. (2000): “The Fair Division Calculator”,\nhttp:\/\/www.math.hmc.edu\/∼su\/fairdivision\/.\nYoung, P. H. (1994): Equity: In Theory and Practice. Princeton University Press, Prince-\nton, New Jersey.\n∼\/work\/zif\/contrib.tex 21 February 10, 2005\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! combinatorics. Van der Waerden Theorem Posted on June 29, 2019. by. Samuel Nunoo Can you color the integers red and blue such that there are no monochromatic arithmetic progressions (AP’s) extending infintely in... Continue reading... Dinner Party Problem Posted on June 29, 2019. by. Samuel Nunoo How many people must you have at dinner to ensure that there are a subset of 3 people who all... Continue reading... Continuum Hypothesis Posted on June 29, 2019. by. Samuel Nunoo We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 correspondence with... Continue reading... Lucas’ Theorem Posted on June 29, 2019. by. Samuel Nunoo Lucas’ Theorem: If p is a prime number, and N has base p representation (aj,…,a1,a0) and k has base p... Continue reading... Cantor Diagonalization Posted on June 29, 2019. by. Samuel Nunoo We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as... Continue reading... Envy-free Cake Division Posted on June 29, 2019. by. Samuel Nunoo Say you and a friend wish to share a cake. What is a “fair” way to split it? Probably you... Continue reading... Farmers and Pesky Birds Posted on June 29, 2019. by. Molly Reeves Alice and Bob are two farmers each wanting to plant a (countably infinite) row of seeds, side by side in... Continue reading... Odd Numbers in Pascal’s Triangle Posted on June 29, 2019. by. Samuel Nunoo Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N... Continue reading... How many Rationals? Posted on June 29, 2019. by. Samuel Nunoo How many rational numbers are there? Yes, infinitely many, I hear you say. But how large is that infinity? Are... Continue reading... Dominoes on a Chessboard Posted on June 27, 2019. by. Samuel Nunoo A standard 8×8 chessboard can easily be covered (tiled) with non-overlapping dominoes (1×2 pieces): simply use 4 dominoes in each... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. Current Page Parent. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Current Page:. Contributors All Fun Facts Get a Random Fun Fact! Fun Facts Contributors. Project Director and Site Creator:. Francis Edward Su Primary Fun Fact Contributors:. Jorge Aarao, Art Benjamin, Michael Moody,. Francis Su, Ravi Vakil, Lesley Ward. Writing and Editing:. Francis Su, on all Fun Facts except the two suggested by Allen Stenger. Technical Design:. Christian Jones and Francis Su. (version 1.0, launched July 1999). Jonathan Lee, John Suarez, & Michelle Velea. (version 0.5, May 1999). with help from Gregory Rae. Counter by Dominic Mazzoni, and Perl help from Claire Connelly. Website redesign by Molly Reeves, Ashley Tung, Samuel Nunoo, and Shanaya Stephenson (. August 2019. ). Graphics and Animated Headers:. Ian Schempp, Francis Su. We would also like to thank the Mellon Foundation and Harvey Mudd College for generous support of this project. 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{"id":"OMa79W51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Cake-Cutting-Algorithms.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":87,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Cake-Cutting-Algorithms","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Cake-Cutting-Algorithms.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:12 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":6959063671892218792,"exact_signature_unique_b":true,"fuzzy_signature_l":-6755632078536411479,"fuzzy_signature_unique_b":true,"wordcount_i":1977,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:10 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:39 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":97682,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088134256852992,"text_t":"appeared in: Amer. Math. Monthly, 107(2000), 185-188.\nCake-Cutting Algorithms: Be Fair If You Can. By Jack Robertson and\nWilliam Webb. A K Peters, 1998, x + 181 pp., $38.\nReviewed by Francis Edward Su\nMany feel that mathematics will receive as much of its direction\nand vigor in the future from problems in the social sciences as it\nhas received in the past from the physical sciences. (p. 2)\nAs I grow older, I find myself yearning more and more for a connection\nbetween my work and the complex problems facing the world. And unlike\nmost people, I am in a profession where I have the flexibility to choose what\nI do. Should this yearning influence the kind of research that I pursue?\nShould it affect how and what I teach my students? At the very least I\nshould vigilantly ponder these questions.\nBecause of this yearning, I have taken up the challenge of a problem moti-\nvated by the social sciences—the problem of fair division. Mathematically, it\nis a very rich subject with many applications and a variety of open questions,\nranging from very deep problems to accessible ones that my undergraduates\ncan pursue. A very nice survey of the subject can be found in Cake-Cutting\nAlgorithms, by Jack Robertson and William Webb.\nThe basic question—how to cut a cake fairly?—is surely an ancient one.\nHowever, Robertson and Webb’s book demonstrates that the subject has\nevolved considerably in the fifty years since Steinhaus [10] first posed the\nquestion as a serious mathematical endeavor. Now much more than a collec-\ntion of ad hoc results, the subject has matured, exhibited fertile connections\nwith many areas of mathematics, and proved itself practical in social appli-\ncations [4], [8].\nThe question is loaded with terms that must be made precise. In today’s\ntheory, “cake” could mean any desirable set of goods (or burdens, or mixtures\nof goods and burdens), each with various properties (such as being divisible or\nindivisible) and restrictions (such as the number of goods a player may get).\nThe word “cut” refers to kinds of divisions that can be carried out in practice,\nsuch as discrete procedures or continuous moving-knife schemes for real cakes,\nand compensation procedures for the division of estates (using money as a\n1\ndivisible medium to negotiate the allocation of indivisible objects). The\nquestion “how?” suggests that we are not only interested in the existence\nof solutions, but desire constructive procedures for finding them. And there\nare many possible “fairness” notions. Two of the better known examples\nare proportional (or simple fair) division, in which each of n players feels\nshe received at least 1\/n of the cake, and the stronger notion of envy-free\ndivision, in which each person feels that no one else received a larger piece.\nThe main issues in the field are the following:\n1. (Existence) Does a division with the desired fairness property exist?\n2. (Construction) Is there a discrete or continuous procedure for locating\nsolutions? If not, can one locate an approximate solution?\n3. (Properties) Can one give bounds for the number of steps or cuts re-\nquired? What can be accomplished with a limited number of cuts?\nWhat kinds of procedures are provably impossible?\n4. (Implementation) Which procedures are useful in applications such as\ndivorce settlements and estate division? Are they convincingly fair to\neach player? Do they induce players to be truthful? Can negotiators,\nlawyers, lay people, or even countries use them to resolve conflicts?\nAs a mathematical survey, Cake-Cutting Algorithms focuses primarily on\nissues (1), (2), and (3). The extensive mathematical development of (3) and\nthe emphasis on cake division distinguish it from an earlier book [3] that takes\na complementary and wider view of the subject by additionally addressing\n(4) while scaling back discussion of mathematical issues.\nIdeas and techniques for solving fair division questions come from many\ndifferent arenas. The following are some of my favorite examples.\nMeasure Theory Let C denote the cake to be divided. A natural model\nfor a player’s valuation of the cake is a probability measure on subsets of C.\nThus µi(A) represents the value of subset A to player i as a fraction of the\nwhole cake. The cake-cutting problem becomes non-trivial and interesting\nonly when the player valuations differ; however, one lesson of the subject,\nnoted by Steinhaus and developed further by Robertson and Webb, is the\n“serendipity of disagreement”. Players can often get away with more if there\nare differences in player opinions.\n2\nAs an example, consider the following nice result of Barbanel [1]. A divi-\nsion of a cake is super envy-free if every player feels all other players received\nstrictly less than 1\/n of the total value of the cake. Clearly super envy-free\nimplies envy-free. Barbanel [1] proved that super envy-free divisions exist if\nand only if the player measures are linearly independent!\nThis result shows only existence. Robertson and Webb show how to\nconstruct a super envy-free division by using a set of pieces that “witness”\nthe linear independence of the measures.\nGraph Theory Hall’s famous marriage theorem can be used to develop a\ndiscrete proportional cake-cutting algorithm, as Robertson and Webb show.\nGiven pieces of cake X1, X2, . . . , Xn to be assigned to players P1, P2,\n. . . , Pn, we may construct a bipartite graph G with the players and objects\nas vertices; there is an edge between Pi and Xj if Pi thinks Xj is at least 1\/n\nof the cake. A perfect matching in G would yield an assignment of pieces to\nplayers that is proportionally fair. In this context, Hall’s theorem says that\nthere exists a perfect matching if and only if every subset S of pieces satisfies\nN(S) ≥ |S|, where N(S) denotes the number of players who find at least one\npiece in S acceptable and |S| denotes the size of S.\nWe can ensure that such subsets of size 1 exist by letting P1 cut the\ncake into n pieces equally desirable to her. Using those pieces, Robertson\nand Webb show that there exists a smallest subset of pieces T such that\nN(T ) = |T | ≥ 1 but all non-empty proper subsets S ⊂ T satisfy the strict\ninequality N(S) > |S|. So apply Hall’s theorem to assign pieces in T to the\ncorresponding N(T ) players. The remaining players (if there are any) do not\ndesire pieces in T and so are happy to give them away; they can now lump\nthe unassigned pieces together and divide the lump fairly among themselves\nby appealing to induction. This process can be converted to a procedure\nusing constructive matching algorithms.\nComplexity Robertson and Webb survey several algorithms for propor-\ntional and envy-free division and compare the number of required cuts.\nBrams and Taylor [2] and later Robertson and Webb [7] gave procedures\nfor envy-free division for n players. Though finite, both these procedures are\nunbounded, meaning they could take arbitrarily long to resolve, depending\non player preferences. Whether there exists a finite, bounded procedure for\nenvy-free cake division is a major open question.\n3\nFor proportional division, there do exist bounded procedures for n players;\nthe “divide-and-conquer” algorithm due to Even and Paz takes O(n log n)\ncuts and is the most efficient currently known for general n. Is this best\npossible? For specific values of n, other algorithms are more efficient than\nthe Even-Paz algorithm, and limited evidence suggests that O(n) cuts may\nbe sufficient. The question is interesting because we know from existence\ntheorems that n−1 cuts are necessary and sufficient; but what is the minimal\nnumber of cuts guaranteed by a procedure?\nRobertson and Webb tabulate known values for this and the dual question\nof how much cake can be guaranteed each player by a limited number of cuts.\nReaders may enjoy the challenge of improving these bounds for specific values\nof n.\nCombinatorial Topology Known n-person envy-free procedures involve\na large number of cuts and would, practically speaking, decimate the cake\neven for the case of 4 players.\nA new development [11], not discussed in the Robertson\/Webb book, is\nthe use of theorems from combinatorial topology to provide approximate solu-\ntions in which the number of cuts is minimal. For example, Sperner’s lemma\nconcerns triangulations of n-balls whose vertices are labelled by n+ 1 labels.\nIt states that under appropriate boundary conditions, at least one n-simplex\nin the triangulation possesses all labels. More importantly, a constructive\nproof of Sperner’s lemma surfaced in the 1960s and led to some startling\nconstructive methods for finding fixed points [12], since the lemma is equiv-\nalent to the Brouwer fixed point theorem. Sperner’s lemma relates to cake\ndivision via an extra ingredient not present in the fixed point connection—\nthe existence of an “owner-labelling” of a triangulated ball—which allows for\nthe accommodation of the preferences of n different players. Moreover, the\nconstructive proof of Sperner’s lemma can be converted into an algorithm\nfor approximate envy-free division.\nRelated combinatorial theorems, such as Tucker’s lemma, and their con-\nstructive proofs can be used to generate new algorithms for other fair division\nproblems. The exciting news is that the connection also runs the other way—\nfair division questions now motivate the development of new combinatorial\ntheorems; see [9].\nRobertson and Webb describe many more connections between fair divi-\nsion questions and other areas of mathematics.\n4\nCake-Cutting Algorithms is accessible to a wide mathematical audience.\nThe language is relaxed and informal, sometimes to the point of being chatty.\nHowever, the friendly style serves to engage the uninitiated reader, and stu-\ndents will find it a welcome difference from stiff textbook prose. The book\nis effectively divided into two parts: chapters 1-6, which assume a minimal\nmathematical background, and chapters 8-11, which are quite technical and\nrequire much more mathematical maturity. These are separated by chapter\n7, which summarizes the known cake-cutting algorithms.\nBy careful selection of topics, one could use material in the first seven\nchapters to supplement a math for liberal arts course or a topical freshman\nseminar, since the subject of fair division lends itself well to both mathemat-\nical excursions and philosophical discussions about “fairness”. At the other\nend of the spectrum, advanced readers will find some important theoretical\nresults in the final chapters that are primarily the work of the authors: impos-\nsibility theorems showing that certain kinds of divisions cannot be achieved\nby finite procedures, some very delicate arguments about what can be ac-\ncomplished with a limited number of cuts, and the use of Ramsey partitions\nto address the problem of division into unequal shares.\nOne of the best features of the Robertson-Webb book is the variety of\nquestions that it poses at all levels. There are exercises and projects at the\nend of each chapter as well as questions scattered throughout the text. Some\nare suitable for undergraduate projects, while others are tantalizing open\nproblems that may take years to resolve.\nFor the past two years, I have had my undergraduate research students\nfocus on fair division questions. I am happy to report that the Robertson-\nWebb text provided mathematical background as well as many launching\npoints for their research; they pursued problems in envy-free chore division\n(in which players desire the smallest, rather than largest, piece) [5], examined\ncake division into unequal ratios, and implemented fair division algorithms\non the Internet [6].\nIn summary, Cake-Cutting Algorithms will engage and challenge both\nveteran and novice mathematicians. Those who wish to develop their own\napproaches to the subject will find the book an indispensable guide to past\nwork on the problem of cake-cutting. Mathematicians in other fields may\nfind that the book stimulates them to think about the connections between\ntheir own work and contemporary problems in the social sciences.\n5\nReferences\n1. J. Barbanel, Super envy-free cake division and independence of measures,\nJ. Math. Anal. Appl. 197 (1996) 54-60.\n2. S. J. Brams and A. D. Taylor, An envy-free cake division protocol, this\nMonthly 102 (1995) 9-18.\n3. S. J. Brams and A. D. Taylor, Fair Division: from Cake-Cutting to Dis-\npute Resolution, Cambridge University Press, New York, 1996.\n4. S. J. Brams and A. D. Taylor, The Win-Win Solution, W. W. Norton,\nNew York, 1999.\n5. E. Peterson and F. E. Su, Exact procedures for envy-free chore division,\n1998, preprint.\n6. E. Peterson, F. E. Su, and P. Vinograd, The Fair Division Calculator\n(java applet), http:\/\/www.math.hmc.edu\/˜su\/fairdivision\/, version 2.0,\nMay 1999.\n7. J. Robertson and W. Webb, Near exact and envy-free cake division, Ars.\nComb. 45 (1997) 97-108.\n8. J. K. Sebenius, Negotiating the Law of the Sea, Harvard University Press,\nCambridge, 1984.\n9. F. W. Simmons and F. E. Su, Fair division via a generalized Tucker’s\nlemma, 1999, preprint.\n10. H. Steinhaus, The problem of fair division, Econometrica 16 (1948) 101-\n104.\n11. F. E. Su, Rental harmony: Sperner’s lemma in fair division, this\nMonthly 106 (1999) 930-942.\n12. M. J. Todd, The Computation of Fixed Points and Applications, Lecture\nNotes in Economics and Mathematical Systems, Springer-Verlag, New\nYork, 1976.\nHarvey Mudd College, Claremont, CA 91711\nsu@math.hmc.edu\n6\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Taylor series. Taylor-made Pi Posted on June 29, 2019. by. Samuel Nunoo After learning about the Taylor series for 1\/(1+x) in calculus, you can find an interesting expression for Pi very easily. Start with... Continue reading... Sum of Prime Reciprocals Posted on June 26, 2019. by. Samuel Nunoo It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges. But what about... Continue reading... Euler’s Formula Posted on June 26, 2019. by. Samuel Nunoo The five most important numbers in mathematics all appear in a single equation!ei*Pi + 1 = 0.In fact, this is a... Continue reading... Suspended Rope Trick Posted on June 26, 2019. by. Samuel Nunoo What is the shape of a suspended rope? Is there some function that describes it? Answer: it’s the cosh curve!... Continue reading... Search Fun Facts. 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An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! irrational. Irrationality by Infinite Descent Posted on June 29, 2019. by. Samuel Nunoo The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the... Continue reading... Dedekind Cuts of Rational Numbers Posted on June 29, 2019. by. Samuel Nunoo Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the... Continue reading... Computability of Real Numbers Posted on June 29, 2019. by. Samuel Nunoo We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We... Continue reading... Rationals Dense but Sparse Posted on June 29, 2019. by. Samuel Nunoo Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the... Continue reading... Rational Irrational Power Posted on June 29, 2019. by. Samuel Nunoo If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to... Continue reading... Square Root of Two is Irrational Posted on June 29, 2019. by. Samuel Nunoo An irrational number is a number that cannot be expressed as a fraction. But are there any irrational numbers? It was known to... Continue reading... Fourier Ears Only Posted on June 26, 2019. by. Samuel Nunoo Did you know that every sufficiently smooth function on an interval can be expressed as an infinite sum of sines... Continue reading... Pi Approximations Posted on June 26, 2019. by. Samuel Nunoo Pi is the ratio of the circumference of a circle to its diameter. It is known to be irrational and its decimal... Continue reading... Pick’s Theorem Posted on June 26, 2019. by. Samuel Nunoo A lattice point in the plane is any point that has integer coordinates. Let P be a polygon in the plane whose vertices have... Continue reading... Memorizing Pi Posted on June 26, 2019. by. Samuel Nunoo The digits of Pi are fascinating. As the ratio of the circumference of a circle to its diameter, Pi has... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"WroKeW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/A-Polytopal-Generalization.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":90,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"A-Polytopal-Generalization","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["A Polytopal Generalization of Sperner’s Lemma"],"author":"Loera, J. A. D. et al.","author_sxt":["Loera, J. A. D. et al."],"last_modified":"Tue Sep 24 07:57:40 UTC 2002","dates_in_content_count_i":1,"dates_in_content_dts":["Sun Oct 09 00:00:00 UTC 2011"],"keywords":"Sperner’s lemma; polytopes; path-following; simplicial algorithms.","http_unique_b":false,"www_unique_b":false,"exact_signature_l":5330978014879795113,"exact_signature_unique_b":true,"fuzzy_signature_l":-2368108997355916582,"fuzzy_signature_unique_b":true,"wordcount_i":9851,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:02 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:24 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"Acrobat Distiller 4.05 for Windows","language_s":"en","size_i":240381,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088126391484416,"text_t":"Journal of Combinatorial Theory, Series A 100, 1–26 (2002)\ndoi:10.1006\/jcta.2002.3274APolytopal Generalization of Sperner’s Lemma\nJesus A. De Loera1\nDepartment of Mathematics, University of California, Davis, California 95616\nE-mail: deloera@math:ucdavis:edu\nElisha Peterson2\nMagdalen College, Oxford University, Oxford OX1 4AU, United Kingdom\nE-mail: elisha:peterson@magdalen:oxford:ac:uk\nand\nFrancis Edward Su2,3\nDepartment of Mathematics, Harvey Mudd College, Claremont, California 91711\nE-mail: su@math:hmc:edu\nReceived July 7, 2001; published online July 8, 2002\nWe prove the following conjecture of Atanassov (Studia Sci. Math. Hungar. 32\n(1996), 71–74). Let T be a triangulation of a d-dimensional polytope P with n vertices\nv1; v2; . . . ; vn: Label the vertices of T by 1; 2; . . . ; n in such a way that a vertex of T\nbelonging to the interior of a face F of P can only be labelled by j if vj is on F : Then\nthere are at least n\u0001 d full dimensional simplices of T ; each labelled with d þ 1\ndifferent labels. We provide two proofs of this result: a non-constructive proof\nintroducing the notion of a pebble set of a polytope, and a constructive proof using a\npath-following argument. Our non-constructive proof has interesting relations to\nminimal simplicial covers of convex polyhedra and their chamber complexes, as in\nAlekseyevskaya (Discrete Math. 157 (1996), 15–37) and Billera et al. (J. Combin.\nTheory Ser. B 57 (1993), 258–268). # 2002 Elsevier Science (USA)\nKey Words: Sperner’s lemma; polytopes; path-following; simplicial algorithms.1. INTRODUCTION\nSperner’s lemma is a combinatorial statement about labellings of\ntriangulated simplices whose claim to fame is its equivalence with the1Research partially supported by NSF Grant DMS-0073815.\n2Research partially supported by a Beckman Research Grant at Harvey Mudd College.\n3To whom correspondence should be addressed.\n1\n0097-3165\/02 $35.00\n# 2002 Elsevier Science (USA)\nAll rights reserved.\nDE LOERA, PETERSON, AND SU2topological fixed-point theorem of Brouwer [12, 20]. In this paper, we prove\na generalization of Sperner’s lemma that settles a conjecture proposed by\nAtanassov [2].\nConsider a convex polytope P in Rd defined by n vertices v1; . . . ; vn 2 R\nd :\nFor brevity, we will call such polytope an ðn; dÞ-polytope. Throughout the\npaper we will follow the terminology of the book [24]. By a triangulation T\nof the polytope P we mean a finite collection of distinct simplices such that:\n(i) the union of the simplices of T is P; (ii) every face of a simplex in T is in\nT ; and (iii) any two simplices in T intersect in a face common to both. The\npoints v1; . . . ; vn are called vertices of P to distinguish them from vertices of\nT ; the triangulation. Similarly, a simplex spanned by vertices of P will be\ncalled a simplex of P to distinguish it from simplices involving other vertices\nof T : If S is a subset of P; then the carrier of S; denoted carrðSÞ; is the\nsmallest face F of P that contains S: In that case we say S is carried by F : A\ncover C of a convex polytope P is a collection of full dimensional simplices in\nP such that\nS\ns2C s ¼ P: The size of a cover is the number of simplices in the\ncover.\nLet T be a triangulation of P; and suppose that the vertices of T have a\nlabelling satisfying these conditions: each vertex of P is assigned a unique\nlabel from the set f1; 2; . . . ; ng; and each other vertex v of T is assigned a\nlabel of one of the vertices of P in carrðfvgÞ: Such a labelling is called a\nSperner labelling of T : We say that a d-simplex in the triangulation is a fully\nlabelled simplex or simply a full cell if all its labels are distinct. The following\nresult was proved by Sperner [20] in 1928:\nSperner’s Lemma. Any Sperner labelling of a triangulation of a\nd-simplex must contain an odd number of full cells; in particular, there is at\nleast one.\nConstructive proofs of Sperner’s lemma [8, 13, 17] emerged in the 1960s,\nand these were used to develop constructive methods for locating fixed\npoints [22, 23]. Sperner’s lemma and its variants continue to be useful in\napplications. For example, they have recently been used to solve fair division\nproblems in game theory [18, 21].\nThe main purpose of this paper is to present a solution of the following\nconjecture: any Sperner labelling of a triangulation of an ðn; dÞ-polytope\nmust contain at least n\u0001 d full cells. In 1996, Atanassov [2] stated this\nconjecture and gave a proof for the case where d ¼ 2: Note that Sperner’s\nlemma is exactly the case n ¼ d þ 1: In this paper we prove this conjecture\nfor all ðn; dÞ-polytopes:\nTheorem 1. Any Sperner labelling of a triangulation T of an ðn; dÞ-\npolytope P must contain at least n\u0001 d full cells.\nA POLYTOPAL SPERNER’S LEMMA 3We provide a non-constructive and a constructive proof of Theorem 1.\nThe non-constructive proof that we give in Section 2 relies on a\nknown result about the surjectivity of the piecewise linear map induced\nby a labelled triangulation of P (Proposition 3, cf. [5, 14]) and the\nnotion of a pebble set that we develop. Pebble sets are also used in\nSection 2 to prove the following result of independent interest in discrete\ngeometry.\nTheorem 2. Let cðPÞ denote the covering number of an ðn; dÞ-polytope P;\nwhich is the size of the smallest cover of P: Then, cðPÞ5n\u0001 d: This\nresult is best possible as the equality is attained for stacked\npolytopes.\nThis result, which bounds the size of a polytope cover, is somewhat\nreminiscent of Barnette’s lower bound theorem [3] for bounding the\nnumber of facets of simplicial polytopes. Other results bounding the\nsize of polytope covers may be found in [6, 7, 16] and references\nwithin.\nWe emphasize that Theorem 2 holds for all covers, not just for those\ncovers needed in the proof of Theorem 1. To be specific, Theorem 1 uses the\nfact that the collection of full cells in T corresponds to a cover of P under the\npiecewise linear map that sends each vertex of T to the vertex of P that\nshares the same label. In such a cover, any pair of simplices is connected by a\nsequence of simplices that meet face-to-face, and it is easier to prove a result\nlike Theorem 2 for such covers.\nHowever, not all covers are necessarily of this type. For example, in the\nleft-hand side of Fig. 1 we specify a cover of an 8-vertex, three-dimensional\nconvex polytope that cannot be obtained from a piecewise linear map of P\nto P: Nevertheless, Theorem 2 still applies.\nIn principle, failing to satisfy this ‘‘face-to-face’’ property can lead to very\nsmall covers; in Fig. 1 we display a star-shaped 12-gon that can be covered\nwith just two triangles. Thus, the assertion of Theorem 2 is not true for\ncovers of non-convex polygons (even though it is true for triangulations of\nnon-convex polygons). The significance of Theorem 2 is that it holds for\nall covers of convex polytopes, not just triangulations or ‘‘face-to-face’’\ncovers.\nSection 3 develops background on path-following arguments in\npolytopes that is closely related to classical path-following arguments\nfor Sperner’s lemma [8, 13, 22]. This is applied to give a constructive\nproof of Theorem 1 for simplicial polytopes. In Section 4, we extend\nthe construction to prove the conjecture for arbitrary polytopes.\nSection 5 of the paper is devoted to applications, remarks, and open\nquestions.\n1\n2\n3\n4\n5\n6\n7\n8\n1346\n1467\n1457\n1578\n1358\nupper part\n1368\n1678\nof cover is \nlower part\nof cover is\n2345\n2347\n2367\n2478\n2458\n2356\n2568\n9\n10\n11 12 2\n4\n568\n7\n1\n3\nFIG. 1. Pathological covers.\nDE LOERA, PETERSON, AND SU42. A NON-CONSTRUCTIVE PROOF USING PEBBLE SETS\nOur non-constructive proof of Theorem 1 relies on the notion of a pebble\nset that we develop in this section. Constructing such a set will yield the\ndesired lower bound for the number of full cells.\nIn what follows, we will use the notion of chamber complex of a polytope\nP (see [1] and [4]): let S be the set of all d-simplices of P: Denote by bdryðsÞ\nthe boundary of simplex s: Consider the set of open polyhedra P\u0001S\ns2S bdryðsÞ: A chamber is the closure of one of these components. The\nchamber complex of P is the polyhedral complex given by all chambers and\ntheir faces.\nLet P be an ðn; dÞ-polytope with Sperner-labelled triangulation T :\nConsider the piecewise linear (PL) map f : P ! P that maps each vertex\nof T to the vertex of P that shares the same label, and is linear on each\nd-simplex of T : The next proposition will be very useful:\nProposition 3. The map f : P ! P defined as above is surjective, and\nthus the collection of full cells in T forms a cover of P under f :\nThe proof of the surjectivity of f that we show here is taken directly from\nthe forthcoming book [5]. The surjectivity of f can also be proved as a\nconsequence of the KKM-type result of [14, Theorem 10]. Similar\nA POLYTOPAL SPERNER’S LEMMA 5surjectivity results for maps arising from labellings by facets of P\n(rather than vertices of P) can be found in [10, Theorem 4; 23, Theorem\n14.5.3].\nProof. First, note that because of the Sperner labelling of the\ntriangulation T of P; the map f satisfies f ðFÞ \n F for any face F of P:\nSince this condition is hereditary for faces, to show that f is surjective,\nit suffices to show that each point y in the interior of P has a pre-image. For\ncontradiction, suppose that some y 2 int P is not in the image of f : For\nx 2 P; consider the ray emanating from f ðxÞ and passing through y; and let\ngðxÞ be the unique intersection of that ray with the boundary of P: This g is\na well defined and continuous map P ! P; and by Brouwer’s fixed point\ntheorem, there is an x0 2 P with gðx0Þ ¼ x0: The point x0 lies on the\nboundary of P; in some proper face F : But f ðx0Þ cannot lie in F (because the\nsegment from x0 ¼ gðx0Þ to f ðx0Þ passes through the point y outside F)\nwhich contradicts the fact that f ðFÞ \n F :\nThus f is surjective. Moreover, the collection of images of full cells under\nf suffices to cover P because interiors of chambers of P are only covered by\nimages of full cells, and boundaries of chambers are covered by any simplex\nthat covers an adjacent chamber. ]\nAs a consequence of Proposition 3, if we can find a set of points in P such\nthat any d-simplex spanned by ðd þ 1Þ vertices of P contains at most\none such point in its interior, then the pre-image of each such point\nwill correspond to a full cell in P (in fact, an odd number, because the\nnumber of pre-images, counted with sign, is 1). Thus finding full cells in\nTheorem 1 corresponds precisely to looking for the following kind of finite\npoint set:\nDefinition. A pebble set of a ðn; dÞ-polytope P is a finite set of points\n(pebbles) such that each d-simplex of P contains at most one pebble interior\nto chambers.\nIt is worth noting two facts about pebble sets. Firstly, the larger the\npebble set, the more full cells we can identify, i.e., the number of full cells is\nat least the cardinality of the largest size pebble set in P: Secondly, by the\ndefinition of a chamber, only one pebble can exist within a chamber and\nwhen choosing a pebble p we have the freedom to replace it by any point p0\nin the interior of the same chamber because p and p0 are contained by the\nsame set of d-simplices. We now show that a pebble set of size n\u0001 d exists\nfor any ðn; dÞ-polytope P by a ‘‘facet-pivoting’’ construction.\nIn the simplest situation, if one of the facets of P is a simplex, call this\nsimplex the base facet. Choose any point q0 (the basepoint) in the interior of\nDE LOERA, PETERSON, AND SU6this base facet. Now for each vertex vi not in the base facet, choose a point pi\nalong a line between q0 and vi but very close to vi: Exactly how close will be\nspecified in the proof. The collection of all such points fpig forms a pebble\nset; it is size n\u0001 d because the simplicial base facet has d vertices. See, for\nexample, Fig. 2 for the case of a pentagon; it is a ð5; 2Þ-polytope with pebble\nset fp1; p2; p3g:\nIf none of the facets are simplicial, then one must choose a non-simplicial\nfacet as base. In this case, choose a pebble set fqig for the base facet (an\ninductive hypothesis is used here) and then use any one of them for a\nbasepoint q0 to construct pi as above. The remaining pebbles are obtained\nfrom the other qi by perturbing them so they are interior to P: See\nFig. 3.\nTheorem 4. Any ðn; dÞ-polytope contains a pebble set of size n\u0001 d:\nProof. We induct on the dimension d: For dimension d ¼ 1; a polytope\nis just a line segment spanned by two vertices. Hence n\u0001 d ¼ 1 and clearly\nany point in the interior of the line segment forms a pebble set.\nFor any other dimension d; let V ¼ fv1; . . . ; vng 2 R\nd denote the vertices\nof the given ðn; dÞ-polytope P: Choose any facet F of P as a ‘‘base facet’’,\nand suppose without loss of generality that it is the convex hull of the last k\nvertices vn\u0001kþ1; . . . ; vn 2 R\nd ; k5d: Then F is a ðd \u0001 1Þ-dimensional polytope\nwith k vertices, and by the inductive hypothesis, F has a pebble set QF with\nðk\u0001 d þ 1Þ pebbles q0; q1; . . . ; qk\u0001d : (If F is a simplex, then k ¼ d and QF\nconsists of one point q0; which can be taken to be any point on the interior\nof F :)2\nq\n0\np1 p3\np\nFIG. 2. A pebble set with pebbles p1; p2; p3:\np\n1\n2p\n3\np\nq\n0\nq\n1\n2\nq\nFIG. 3. A pebble set with pebbles p1; p2; p3; q18; q28: The pebbles q18; q28 (not shown) lie just\nabove q1; q2 on the base of the polytope. Note how q0; q1; q2 arise from the pebble set\nconstruction in Fig. 2.\nA POLYTOPAL SPERNER’S LEMMA 7Let diamðPÞ denote the diameter of the polytope P; i.e., the maximum\npairwise distance between any two points in P: Let H be the minimum\ndistance between any vertex v 2 V and the convex hull of the vertices in\nV =fvg: Since there are finitely many such distances and the vertices are in\nconvex position, H exists and is positive. Set\ne ¼\nH\n2 diamðPÞ\n: ð1Þ\nUsing q0; let Q ¼ fp1; . . . ; pn\u0001kg denote the collection of n\u0001 k points\ndefined by\npi ¼ eq0 þ ð1 \u0001 eÞvi; ð2Þ\nfor 14i4n\u0001 k; where e is a small positive constant given by (1). Therefore\npoints in Q lie along straight lines extending from q0 and very close to the\nvertices of P not in F :\nBecause qi is in F ; it lies on the boundary of P and borders exactly one\nchamber of P (since by induction it is interior to a single chamber in the\nfacet F). Ignoring q0 momentarily, for 14i4k\u0001 d; let qi8 denote a point\nobtained by ‘‘pushing’’ qi into the interior of the unique chamber that it\nborders. Let QF 8 ¼ fq81; . . . ; q8k\u0001dg: We shall show that\nQ [QF8 ¼ fp1; . . . ; pn\u0001k; q81; . . . ; q8k\u0001dg\nis a pebble set for the polytope P: Note that if P has a simplicial facet F ;\nthen with this facet as base, the set Q suffices; for then QF 8 is empty and\nDE LOERA, PETERSON, AND SU8construction (2) yields the required number of pebbles by choosing any q0 in\nthe interior of a simplicial facet F : First, we prove some important facts\nabout the pi and qi8:\nLemma 5. Let S be a d-simplex spanned by vertices of P: If S contains pi;\nthen S must also contain vi as one of its vertices.\nProof. By construction, each pi has the property that pi is not in the\nconvex hull of V =fvig: This follows because\njjvi \u0001 pijj\njjvi \u0001 q0jj\n¼ e ¼\nH\n2 diamðPÞ\n4\nH\n2jjvi \u0001 q0jj\n;\nhence jjvi \u0001 pijj4H=2; implying that the distance of pi from the convex hull\nof V =fvig is greater than or equal to H=2:\nSince the convex hull of V =fvig does not contain pi; if S is to contain pi it\nmust contain vi as one of its vertices. ]\nLemma 6. Let S be a non-degenerate d-simplex spanned by vertices of P:\nThen qi8 is in S if and only if qi is in S \\ F :\nProof. Since qi8 is in the unique chamber of P that qi borders, any non-\ndegenerate simplex containing qi must contain qi8: Conversely, any simplex\nS containing qi8 must contain its chamber and therefore contains qi: Since qi\nis in F ; then qi is in S \\ F : ]\nThe next three lemmas will show that Q [QF 8 is a pebble set for P:\nLemma 7. Any d-simplex S spanned by vertices of P contains no more\nthan one pebble of Q:\nProof. If S is degenerate (i.e., the convex hull of those vertices is not full\ndimensional), then it clearly contains no pebbles of Q because the pi are by\nconstruction in the interior of a chamber. So we may assume that S is non-\ndegenerate.\nLet s1; . . . ; sdþ1 2 V denote the vertices of S: Suppose by way of\ncontradiction that S contained more than one point of Q: Then pi0 and pj0\nare contained in S for distinct i0; j0; where 14i0; j04n\u0001 k: Lemma 5 implies\nthat vi0 ; vj0 ; the vertices of P associated to pi0 and pj0 ; must both be vertices of\nS: Without loss of generality, let s1 ¼ vi0 and s2 ¼ vj0 : Let A be a matrix\nwhose columns consist of q0 and the vertices of S; adjoined with a\nA POLYTOPAL SPERNER’S LEMMA 9row of 1’s:\nA ¼\ns1 s2 \u000f \u000f \u000f sdþ1 q0\n1 1 \u000f \u000f \u000f 1 1\n2\n6664\n3\n7775:\nThis is a ðd þ 1Þ \u0010 ðd þ 2Þ matrix that has rank ðd þ 1Þ because the si are\naffinely independent (by the non-degeneracy of S). So the kernel of A; ker\nðAÞ is one dimensional. Note that pi0 2 S implies that it is a convex\ncombination of the first ðd þ 1Þ columns of A: On the other hand, by\nconstruction, it is also a convex combination of s1 and q0: Thus there exist\nconstants 04x1; x2; . . . ; xdþ141 satisfying\npi0\n1\n2\n6664\n3\n7775 ¼ A\n1 \u0001 e\n0\n..\n.\n0\ne\n2\n66666664\n3\n77777775\n¼ A\nx1\nx2\n..\n.\nxdþ1\n0\n2\n66666664\n3\n77777775\n; ð3Þ\nwhere the first equality follows from (2). Similarly, pj0 2 S implies\npj0\n1\n2\n66664\n3\n77775\n¼ A\n0\n1 \u0001 e\n..\n.\n0\ne\n2\n66666664\n3\n77777775\n¼ A\ny1\ny2\n..\n.\nydþ1\n0\n2\n66666664\n3\n77777775\n;\nfor some constants 04y1; y2; . . . ; ydþ141: The above equations show that\nðx1 þ e\u0001 1; x2; x3 . . . ; xdþ1;\u0001eÞ\nT\nand ðy1; y2 þ e\u0001 1; y3; . . . ; ydþ1;\u0001eÞ\nT\nare\nboth in kerðAÞ: But since kerðAÞ is one dimensional, and the last coordinates\nof these vectors are equal, all entries of these vectors are identical. In\nparticular, x1 þ e\u0001 1 ¼ y1:\nWe now claim that x1 þ e51 and hence y150; which would show that pj0\ncould not have been in S after all, a contradiction.\nTo establish the claim, use Eqs. (2) and (3) to express q0 as an affine\ncombination of the vertices of S:\nq0 ¼\n1\ne\nððx1 þ e\u0001 1Þs1 þ x2s2 þ \u000f \u000f \u000f þ xdþ1sdþ1Þ: ð4Þ\nDE LOERA, PETERSON, AND SU10Since x2; . . . ; xdþ150 and q0 is not in the interior of S; then either (i) one of\nthe coefficients of the si in (4) is equal to zero or (ii) x1 þ e51:\nIf case (i) holds, then q0 is on some facet of S; hence q0 is spanned\nby d vertices of S which lie on the facet F of P: Thus the vertices of S\ninclude those d vertices but by Lemma 5, vi0 and vj0 as well. Since vi0 ; vj0\nwere not on the facet F , we obtain a contradiction since S cannot contain\nmore than d þ 1 vertices.\nHence case (ii) holds, namely that x1 þ e51; which is the desired\ncontradiction. ]\nLemma 8. Any d-simplex S spanned by vertices of P contains no more\nthan one pebble in QF8:\nProof. Since S \\ F is a simplex in F that contains at most one point of\nQF ; then by Lemma 6, S can contain at most one point of QF 8: ]\nLemma 9. Any d-simplex S spanned by vertices of P cannot contain\npebbles of Q and QF 8 simultaneously.\nProof. Suppose S contained a point qi8 of QF 8: Then by Lemma 6, S \\ F\ncontains qi of QF : Since QF was a pebble set for the facet F ; S \\ F cannot\nalso contain q0:\nIf S also contained a pebble pi0 of Q; then by Lemma 5, S contains vi0\nas a vertex. Since S \\ F contains qi which is interior to a chamber of F ; S\nmust also contain d vertices of F : Since q0 is in F (but not in S \\ F), q0 is\nexpressible as a linear combination (but not convex combination) of\nthose d vertices. This linear combination, when substituted for q0 in (2),\nwould show that the pebble pi0 is not a convex combination of vi0 and\nthose d vertices. This contradicts the fact that pi0 was in S to begin\nwith. ]\nTogether, the three lemmas above show that S cannot contain more than\none point of Q [QF 8; which concludes the proof of Theorem 4. ]\nTheorem 1 now follows from Theorem 4, in light of the remarks following\nthe proof of Proposition 3.\nTheorem 2 also follows from our pebble set construction. Since\neach element of a cover can contain at most one pebble, Theorem 4\nshows that any cover must have at least n\u0001 d elements. This bound is\nbest possible, because stacked polytopes have triangulations of size n\u0001 d\n[16].\nA POLYTOPAL SPERNER’S LEMMA 113. GRAPHS FOR PATH-FOLLOWING AND SIMPLICIAL\nPOLYTOPES\nSperner’s lemma has a number of constructive proofs which rely on\n‘‘path-following’’ arguments (see, for example, the survey of Todd [22]).\nPath-following arguments work by using a labelling to determine a path\nthrough simplices in a triangulation, in which one endpoint is known and\nthe other endpoint is a full cell. In this section, we adapt these ideas for\nSperner-labelled polytopes, which are used in the next section to give a\nconstructive ‘‘path-following’’ proof of Theorem 1.\nLet P be an ðn; dÞ-polytope with triangulation T and a Sperner labelling\nusing the label set L ¼ f1; 2; . . . ; ng: We define some further terminology\nand notation that we will use from now on. Let LðsÞ; the label set\nof s; denote the set of distinct labels of vertices of s: Let LðFÞ denote\nthe label set of a face F of P: As defined earlier, a d-simplex s in T is\na full cell if the vertex labels of s are all distinct. Similarly, a ðd \u0001 1Þ-simplex\nt in T is a full facet if the vertex labels of t are all distinct. Note that\na full facet on the boundary of P can be regarded as a full cell in that\nfacet.\nDefinition. Given a Sperner-labelled triangulation T of a polytope P;\nwe define three useful graphs:\n1. The nerve graph G is an undirected graph with nodes that are\nsimplices of T whose label set is of size at least d: Two nodes in G are\nadjacent if (as simplices) one is a facet of the other.\n2. If K is a subset of the label set L ¼ f1; 2; . . . ; ng of size ðd \u0001 1Þ; the\nderived graph GK is the subgraph of the nerve graph G consisting of nodes in\nG whose label sets contain K :\n3. Let G0 denote the full cell graph, whose nodes are full cells in the\nnerve graph G: Two full cells s; t are adjacent in G0 if there exists a path\nfrom s to t in G that does not intersect any other full cell. If %G is a connected\ncomponent of G; construct the full cell graph %G\n0\nsimilarly.\nThus the nodes of G and GK are either full cells, full facets, or d-simplices\nwith exactly one repeated label. The full cell graph G0 only has full cells as\nnodes.\nExample. The pentagon in Fig. 4 has dimension d ¼ 2: Let K ¼ f1g; a\nset of cardinality ðd \u0001 1Þ: Then the derived graph GK consists of 1- and\n2-simplices that appear darkly shaded, and it is a subgraph of the nerve\ngraph G consisting of the dark- and light-shaded 1- and 2-simplices in Fig. 4.\n1\n1\n2\n3\n4\n4\n4\n4\n5\n5\n5\n3\n51\n2\n2\n1\n2 3\n3\n3\nC\nB\nA\n3\n2\n2\n3\n3\n2\n5\n5\n5\nFIG. 4. A triangulated ð5; 2Þ-polytope (a pentagon) with Sperner labelling. If K ¼ f1g;\nnodes of GK consist of the dark-shaded simplices, nodes of G consist of dark- and light-shaded\nsimplices, and nodes of G0 consist of the three full cells marked by A;B;C:\nDE LOERA, PETERSON, AND SU12In Fig. 4, G0 is a 3-node graph with nodes A;B;C; the full cells. In G0; A is\nadjacent to B; and B is adjacent to C; but A is not adjacent to C:\nAs the example illustrates, the nerve graph G branches in ðd þ 1Þ\ndirections at full cells, while the derived graph GK is the subgraph consisting\nof paths or loops that ‘‘follow’’ the labels of K along the boundary of the\nsimplices in G: We prove these assertions.\nLemma 10. The nodes of the derived graph GK are either of degree 1 or 2\nfor any K of size d \u0001 1: A node s is of degree 1 if and only if s is a full facet\non the boundary of P: Hence GK is a graph whose connected components\nare either loops or paths that connect pairs of full facets on the boundary\nof P:\nProof. Recall that each node s of GK has a label set containing K\nand is either a full cell, full facet, or d-simplex with exactly one repeated\nlabel.\nIf s is a full cell, since jK j ¼ d \u0001 1 we see that LðsÞ consists of labels in K\nand two other labels l1; l2: There are exactly two facets of s whose label sets\ncontain K ; these are the full facets with label sets K [ l1 and K [ l2;\nrespectively. Thus s has degree 2.\nA POLYTOPAL SPERNER’S LEMMA 13If s is a full facet with label set containing K ; then it is the face of exactly\ntwo d-simplices, unless s is on the boundary of P; in which case it is the face\nof exactly one d-simplex. Thus s is degree 1 or 2 in GK ; and degree 1 when s\nis a full facet on the boundary of P:\nIf s is a d-simplex with exactly one repeated label, then it must possess\nexactly two full facets. Since K \u0011 LðsÞ; these full facets must also have label\nsets that contain K : Hence these two full facets are the neighbors of s in GK ;\nso s has degree 2. ]\nLemma 11. The nodes of the nerve graph G are of degree 1, 2, or d þ 1: A\nnode s is of degree 1 if and only if s is a full facet on the boundary of P: A node\ns is of degree d þ 1 if and only if s is a full cell.\nProof. As noted before, each node s of G is either a full cell, full facet, or\nd-simplex with exactly one repeated label. The arguments for the latter two\ncases are identical to those in the proof of Lemma 10 by letting K be the\nempty set.\nIf s is a full cell, then every facet of s is a full facet, hence the degree of s is\nðd þ 1Þ in G: ]\nThe nerve graph G may have several components, as in Fig. 4. In\nTheorem 15, we will establish an interesting relation between the labels\noccurring in a component %G and the number of full cells it contains. First,\nwe show that all the labels in a component are carried by the full cells.\nLemma 12. If s is adjacent to t in G; then LðsÞ \n LðtÞ; unless s is a full\ncell, in which case LðtÞ \u0011 LðsÞ: Adjacent nodes in %G carry exactly the same\nlabels unless one of them is a full cell.\nProof. If s is a D-simplex with exactly one repeated label, then it is\nadjacent to two full facets with exactly the same label set, so the conclusion\nholds. If s is a full cell, any simplex t adjacent to s in G is contained in s as a\nfacet, so LðtÞ \u0011 LðsÞ in that case. Otherwise, if s is a full facet, then it is\nadjacent to two d-simplices that contain it as a facet, hence LðsÞ \n LðtÞ for\nt adjacent to s:\nThese observations combine to show that adjacent nodes in %G carry\nexactly the same labels unless one of them is a full cell. ]\nLemma 13. Suppose %G is connected component of G: If %G contains at least\none full cell as a node, then all the labels occurring in %G are carried by its full\ncells.\nFor example, in Fig. 4, G has two components. One of them has no full\ncells. In the other component, all of its labels f1; 2; 3; 4; 5g are carried by its\nfull cells A;B;C:\nDE LOERA, PETERSON, AND SU14Proof. Since %G is connected and contains at least one full cell, each\nsimplex s that is not a full cell is connected to a full cell t via a path in %G that\ndoes not intersect any other full cell in %G: Call this path fs ¼ s1; s2; . . . ;\nsp ¼ tg: By Lemma 12, Lðs1Þ ¼ Lðs2Þ ¼ \u000f \u000f \u000f ¼ Lðsp\u00011Þ \u0011 LðtÞ: Therefore\nlabels carried by the full cells contain all labels carried by any other node of\nthe graph. ]\nSince the label information in a nerve graph is found in its full cells, it\nsuffices to understand how the full cells connect to each other.\nLemma 14. Any two adjacent nodes in G0 are full cells in T whose label\nsets contain at least d labels in common.\nProof. Let s1 and s2 be adjacent nodes in G\n0: By construction they must\nbe simplices connected by a path in G; let t be any such node along this path.\nRepeated application of Lemma 12 yields LðtÞ \u0011 Lðs1Þ and LðtÞ \u0011 Lðs2Þ;\nso Lðs1Þ \\ Lðs2Þ contains at least the d labels in LðtÞ: ]\nWe will say the full cell graph G0 is a fully d-labelled graph because it\nclearly satisfies four properties:\n(a) all nodes in the graph are assigned ðd þ 1Þ labels (simply assign to a\nnode s of G0 the label set LðsÞ),\n(b) all edges are assigned d labels (assign an edge ðs1; s2Þ of G\n0 the d\nlabels specified in Lemma 14),\n(c) the label set of an edge ðs; tÞ (denoted by Lðs; tÞ) is contained in\nLðsÞ \\ LðtÞ; and\n(d) if t; t0 are nodes each adjacent to s; then Lðt; sÞ=Lðt0; sÞ (facets of\na full cell s must have different label sets).\nProposition 15. Suppose G0 is a connected fully d-labelled graph. Let\nLðG0Þ denote the set of all labels carried by simplices in G0 and jG0j the number\nof nodes in G0: Then\njG0j5jLðG0Þj \u0001 d:\nWe shall use this theorem for graphs G0 arising as a full cell graph of one\nconnected component of a nerve graph G: In Fig. 4, the full cell graph G0 has\njust one connected component, and LðG0Þ ¼ 5; jG0j ¼ 3 and d ¼ 2; and\nindeed 355 \u0001 2:\nProof. We induct on jG0j: If jG0j ¼ 1; the one full cell in G0 has d þ 1\nlabels. Hence jLðG0Þj \u0001 d ¼ ðd þ 1Þ \u0001 d ¼ 1; so the statement holds.\nA POLYTOPAL SPERNER’S LEMMA 15We now assume the statement holds for fully d-labelled graphs with less\nthan j nodes, and show it holds for fully d-labelled graphs G0 with jG0j ¼ j:\nAssume G0 has j full cells. We claim that it is possible to remove a vertex v\nfrom G0 and leave G0 connected. This is true because G0 contains a maximal\nspanning tree, and the removal of any leaf from this tree will leave the rest of\nthe nodes in G0 connected by a path in this tree.\nNow G0 with v and all its incident edges removed is a new graph (denoted\nby G0 \u0001 v) with j \u0001 1 nodes. Note that this new graph is still fully d-labelled,\nso by the inductive hypothesis, jG0 \u0001 vj5jLðG0 \u0001 vÞj \u0001 d:\nClearly jG0 \u0001 vj ¼ jG0j \u0001 1; and jLðG0 \u0001 vÞj5LðG0Þ \u0001 1 because v has\nat least d labels in common with some vertex in G0 \u0001 v; by Lemma 14.\nHence jG0j \u0001 15LðG0Þ \u0001 1 \u0001 d: Adding 1 to both sides gives the desired\nconclusion. ]\nThis will prove the following useful result.\nTheorem 16. Let T be a Sperner-labelled triangulation of an ðn; dÞ-\npolytope P: If the nerve graph G has a component %G that carries all the labels\nof G; then T contains at least n\u0001 d full cells.\nProof. Suppose that there is a component %G such that Lð %GÞ ¼ n: Use %G\nto construct the full cell graph %G\n0\nas above, which is a fully d-labelled graph.\nNote that if %G is connected then %G\n0\nis also connected. By Lemma 13,\nLð %GÞ ¼ Lð %G\n0\nÞ: Using Proposition 15, we have\nj %G\n0\nj5jLð %GÞj \u0001 d ¼ n\u0001 d;\nwhich shows there are at least n\u0001 d full cells in %G; and hence in G itself. ]\nThus to prove Atanassov’s conjecture for a given ðn; dÞ-polytope it suffices\nto find some component %G of the nerve graph G for which Lð %GÞ ¼ n: This is\nthe central idea of the proofs in the next sections.\nWe now use path-following ideas to outline a proof of Atanassov’s\nconjecture in the special case where the polytope is simplicial. This will\nmotivate the proof of Theorem 1 for arbitrary ðn; dÞ-polytopes in the\nsubsequent section.\nTheorem 17. If P is a simplicial polytope, there is some component %G\nof the nerve graph G which meets every facet of P; and hence carries all labels\nof G:\nProof. Let F be a simplicial facet of the polytope P: Let wðG;FÞ\ncount the number of nodes of G that are simplices in F : This may be\nDE LOERA, PETERSON, AND SU16thought of as the number of endpoints of paths in G that terminate on the\nfacet F :\nConsider two ‘‘adjacent’’ facets F1;F2 of P; whose intersection is a ridge\nof the polytope P; i.e., a co-dimension two face of P spanned by ðd \u0001 1Þ\nvertices of P: These vertices have distinct labels; let K be their label set. By\nLemma 10, the derived graph GK consists of loops or paths whose endpoints\nin GK must be full facets in F1 or F2; since the Sperner labelling guarantees\nthat no other facet of P has a label set containing K :\nSince every facet of P is simplicial, all the full facets in F1 and F2\ncontain K in their label set. Thus all the nodes of G that are full facets\nin F1 and F2 must also be nodes in the graph GK : Since GK is a subgraph\nof G and consists of paths that pair up full facets in F1 and F2; we see\nthat wðG;F1Þ \u0012 wðG;F2Þmod 2: In fact, since paths in GK are contained\nwithin a single connected component of G; this argument shows\nthat\nwð %G;F1Þ \u0012 wð %G;F2Þmod 2\nfor any connected component %G of G:\nSince F1 and F2 were arbitrary, the same argument holds for any two\nadjacent facets. This yields the somewhat surprising conclusion that the\nparity of wð %G;FÞ is independent of the facet F : We denote this parity by\nrð %GÞ: Since wðG;FÞ is also independent of facet, we can define rðGÞ\nsimilarly.\nSince wðG;FÞ is the sum of wð %G;FÞ over all connected components %G of G;\nit follows that rðGÞ \u0012\nP\nrð %GÞmod 2 over all connected components %G of G:\nMoreover, rðGÞ \u0012 1 mod 2 because the usual Sperner’s lemma applied to\n(any) simplicial facet F shows that there are an odd number of full facets of\nT in the facet F :\nHence there must be some %G such that rð %GÞ \u0012 1 mod 2; i.e., this %G meets\nevery facet of P: Because the facets of P are simplicial, %G carries every label,\ni.e., jLð %GÞj ¼ n: ]\nTheorem 18. Any Sperner-labelled triangulation of a simplicial ðn; dÞ-\npolytope must contain at least n\u0001 d full cells.\nProof. This follows immediately from Theorems 17 and 16. ]\nTo extend this proof for non-simplicial polytopes requires some new ideas\nbut follows the basic pattern: (1) find a function w that counts the number of\ntimes a component %G of G meets a certain facet in a certain way, and show\nthat this function only depends on %G; and (2) appeal to the usual Sperner’s\nlemma for simplices in a lower dimension to constructively show that the\nparity of w summed over all components %G must be odd. For the\nA POLYTOPAL SPERNER’S LEMMA 17non-simplicial case, we cannot guarantee that any faces of P except those in\ndimension 1 are simplicial. How to connect dimension 1 to dimension d is\ntackled in the next section, and the flag graph introduced there gives a\nconstructive procedure for finding certain full cells. Then we construct a\ncounting function w to show that there are at least n\u0001 d full cells for an\nðn; dÞ-polytope.\n4. THE FLAG GRAPH AND ARBITRARY POLYTOPES\nThroughout this section, let the symbol \u0012 denote equivalence mod 2:\nRecall that LðFÞ denotes the label set of a face F : We call an i-dimensional\nface of P an i-face of P: Let F denote a flag of the polytope P; i.e., a choice\nof faces F1 \u0011 F2 \u0011 \u000f \u000f \u000f \u0011 Fd where Fi is an i-face of P: When the choice of Fi\nis not understood by context, we refer to the i-face of a particular flag F by\nwriting FiðFÞ:\nGiven a flag F; it will be extremely useful to construct ‘‘super-paths’’\ncontaining simplices of P of various dimensions whose endpoints are either\non a one-dimensional edge or a d-dimensional full cell.\nDefinition. Let P be an ðn; dÞ-polytope with a Sperner-labelled\ntriangulation T : Let F ¼ fF1 \u0011 F2 \u0011 \u000f \u000f \u000f \u0011 Fdg be a flag of P: We define\nthe flag graph GF in the following way. For 14k4d; a k-simplex s 2 T is a\nnode in the graph GF if and only if s is one of four types:\n(I) the k-simplex s is carried by the k-face Fk and\njLðsÞ \\ LðFiÞj ¼ i þ 1 for all 14i4k;\n(II) the k-simplex s is carried by the ðkþ 1Þ-face Fkþ1 and\njLðsÞ \\ LðFiÞj ¼ i þ 1 for all 14i4k;\n(III) the k-simplex s is carried by the k-face Fk and\njLðsÞ \\ LðFkÞj ¼ k\nand\njLðsÞ \\ LðFiÞj ¼ i þ 1 for all 14i4k\u0001 1;\n(IV) the k-simplex s is carried by the k-face Fk and there is an I such that\njLðsÞ \\ LðFkÞj ¼ kþ 1;\njLðsÞ \\ LðFiÞj ¼ i þ 2 for all I4i4k\u0001 1\nDE LOERA, PETERSON, AND SU18and\njLðsÞ \\ LðFiÞj ¼ i þ 1 for all 14i5I :\nTwo nodes s and t carried by Fk are adjacent in GF if (as simplices) s is a\nfacet of t: Nodes s carried by Fk\u00011 and t carried by Fk are adjacent in GF if\ns is a facet of t and s is of type (I). There are no other adjacencies in GF:\nNote that if s is a type (I) simplex in GF; then it is a ‘‘non-degenerate’’ full\ncell of the k-face that it is carried in, i.e., in every i-face of the flag, the\nvertices of P corresponding to the labels in LðsÞ span an i-dimensional\nsimplex (rather than something of lower dimension). A type (II) simplex is a\nnon-degenerate full facet in the ðkþ 1Þ-face that it is carried in. A type (III)\nsimplex has just one repeated label and satisfies a certain kind of non-\ndegeneracy (that ensures its two full facets are non-degenerate). A type (IV)\nsimplex is one kind of degenerate full cell in the k-face that it is carried in\n(but such that it has exactly two facets which are non-degenerate).\nAdjacencies in the graph GF can only occur between two nodes s; t carried\nby the same k-face or carried by faces differing in dimension by 1. In the\nformer case, the adjacency conditions guarantee that the lower-dimensional\nsimplex s is of type (II). In the latter case, the simplex s carried by the lower-\ndimensional face is required to be of type (I).\nExample. Let P be a ð7; 3Þ-polytope P; i.e., a three-dimensional\npolytope with 7 vertices, and suppose T is a Sperner-labelled triangulation\nof P: Let F1 \u0011 F2 \u0011 F3 be a flag F of P with label sets f1; 2g \u0011 f1; 2; 3; 4;\n5g \u0011 f1; 2; . . . ; 7g; respectively.\nConsider the following collection of simplices shown in Fig. 5. Let s1;\n. . . ; s7 be simplices with label sets: Lðs1Þ ¼ f1; 2g; Lðs2Þ ¼ f1; 2g (with\nrepeated label 2), Lðs3Þ ¼ f1; 2g; Lðs4Þ ¼ f1; 2; 3g; Lðs5Þ ¼ f1; 2; 3; 4g; L\nðs6Þ ¼ f1; 2; 4g: Lðs7Þ ¼ f1; 2; 4; 6g such that in each pair fsi; siþ1g; one is a\nfacet of the other. The face F1 carries s1; the face F2 carries s2; s3; s4; and\nthe face F3 carries s5; s6; s7: Each of these simplices is a node in the graph\nGF: simplices s1; s4; s7 are type (I), s3; s6 are of type (II), s2 is of type (III),\nand s5 is of type (IV). Furthermore, each pair si and siþ1 are adjacent in\nGF: Except for s1 and s7; each of these simplices has exactly 2 neighbors in\nGF so the above sequence traces out a path.\nThe following result shows that GF does, in fact, consist of a collection of\nloops or paths whose endpoints are either one and d dimensional.\nLemma 19. Every node s of GF has degree 1 or 2; and has degree 1 only\nwhen s is a 1-simplex or a d-simplex in GF:\n6\n4\n3\n2\n1\n2\nadmits labels 1,2,3,4,5,6,7\nF2\n3\nadmits labels 1,2,3,4,5\nF\n122 12\n123\n1234\n1241246\n12\nF2\nF3\nFIG. 5. A path in the flag-graph of a ð7; 3Þ-polytope. The figure at left shows simplices along\na path in the triangulation. Simplices carried by F2 are shaded. The figure at right shows the\nlabel sets of the simplices along this path. Simplices s1; . . . ;s7 occur in counterclockwise order\nalong this path.\nA POLYTOPAL SPERNER’S LEMMA 19Proof.\nCase (I). Consider a k-simplex s of type (I). If k52; then s has a facet\ndetermined by the k labels in LðsÞ \\ LðFk\u00011Þ; and this facet is a ðk\u0001 1Þ-\nsimplex of type (I) or (II) so it is adjacent to s: No other facets of s are\ntypes (I)–(IV). If k4d \u0001 1; then s is a facet of exactly one ðkþ 1Þ-simplex\nt (carried by Fkþ1) that must be of type (I) or (III) or (IV). Thus a\ntype (I) simplex has degree 2 unless k ¼ 1 or d; in which case it has\ndegree 1.\nCase (II). A type (II) k-simplex s is the facet of exactly two ðkþ 1Þ-\nsimplices in Fkþ1; these are either of type (I) or (III) or (IV) and are thus\nneighbors of s in GF: According to the adjacency rules for GF; if there were\nany other neighbor of s; it would have to be a facet of s of type (I) in Fk or\ntype (II) in Fkþ1: But because s is a k-simplex, a facet of s carried in Fk\ncannot be of type (I), and a facet of s carried in Fkþ1 cannot be of type (II).\nThus type (II) vertices have degree 2.\nCase (III). A type (III) k-simplex s has exactly two facets determined by\nthe k labels in LðsÞ; each of these is a ðk\u0001 1Þ-simplex adjacent to s in GF\nbecause it is either of type (I) in Fk\u00011 or of type (II) in Fk: No other facets of\ns are of type (I) or (II). Thus type (III) vertices have degree 2.\nCase (IV). The labelling rules for a type (IV) simplex s show that the set\nLðsÞ \\ ðLðFIÞ=LðFI\u00011ÞÞ is of size two. Call these labels a and b: There is\nexactly one facet of s that omits the label a and one facet which omits the\nlabel b; each of these is a ðk\u0001 1Þ-simplex of type (I) in Fk\u00011 or of type (II) in\nFk; so is adjacent to s in GF: No other facets of s are non-degenerate, hence\nDE LOERA, PETERSON, AND SU20the other facets cannot be of type (I) or (II). Thus type (IV) vertices have\ndegree 2. ]\nThus the components of GF are paths that wind their way through nodes\ncarried by faces of the flag. The adjacency rules for GF require that a node s\nin a lower-dimensional face is adjacent to a node t in the next higher-\ndimensional face only when s is of type (I). This condition is important\nbecause otherwise some nodes in GF could have degree greater than 2. For\ninstance, in a Sperner-labelled, triangulated ð9; 4Þ-polytope, suppose F1 \u0011\nF2 \u0011 F3 \u0011 F4 is a flag F of P with label sets f1; 2g \u0011 f1; 2; 3; 4; 5g \u0011\nf1; 2; . . . ; 7g \u0011 f1; 2; . . . ; 9g; respectively. If t is a 4-simplex in F4 with label\nset f1; 2; 3; 4; 6g such that its face s with labels f1; 2; 3; 4g is carried in F3;\nthen both s and t are of type (IV). Each already has two facets that are\nnodes in GF of type (I) or (II), so we would not want to define s and t to be\nadjacent to each other, even though one is a facet of the other.\nTheorem 20. A Sperner-labelled triangulation of an ðn; dÞ-polytope\ncontains, for each edge F1 of P; a non-degenerate full cell whose labels\ncontain LðF1Þ:\nProof. For any flag F containing the edge F1; Lemma 19 shows that GF\nconsist of loops or paths whose endpoints are non-degenerate full cells in F1\nor Fd ; thus the total number of such endpoints (full cells in F1 and in Fd)\nmust be of the same parity. On the other hand, the one-dimensional\nSperner’s lemma shows that the number of full cells in F1 is odd. So the\nnumber of non-degenerate full cells in Fd in GF must be odd. In particular\nthere is at least one non-degenerate full cell in Fd whose label set contains\nLðF1Þ: ]\nNotice that the above proof is constructive; the graph GF yields a method\nfor locating a non-degenerate full cell for any choice of flag F; by starting at\none of the full cells on the edge F1 (an odd number of them are available)\nand following its path component in GF: At most an even number of edges\nof F1 are matched by paths in GF; so at least one of them is matched by a\npath to a non-degenerate full cell in Fd :\nHowever, as we show now, more can be said about the location of full\ncells. Rather than locating all of them at the endpoints of paths in a flag-\ngraph, we can show that there is some component %G of the nerve graph G\nthat contains at least n\u0001 d full cells. One can trace paths through this\ncomponent to find them. We find a component %G of G that carries all labels.\nTheorem 15 will imply that the component must have n\u0001 d full cells. As in\nthe case for simplicial polytopes, the key rests on defining a function w that\ncounts the number of times that %G meets a facet in a certain way, and then\nA POLYTOPAL SPERNER’S LEMMA 21showing that the parity of w exhibits a certain kind of invariance}it really\nonly depends on %G: Any component with non-zero parity will be the desired\ncomponent.\nDefinition. Suppose F is a facet of P and R is a ridge of P that is a\nfacet of F : Let wðG;F ;RÞ denote the number of nodes s of the nerve graph G\nin the facet F such that jLðsÞ \\ LðRÞj ¼ d \u0001 1:\nSimilarly if K is any ðd \u0001 1Þ-subset of LðFÞ; let wðG;F ;KÞ denote the\nnumber of nodes s of the graph G in the facet F such that jLðsÞ \\ K j ¼\nd \u0001 1:\nIf %G is a connected component of G; define wð %G;F ;RÞ and wð %G;F ;KÞ\nsimilarly using %G instead of G:\nThus wðG;F ;RÞ (resp. wð %G;F ;RÞ) counts non-degenerate full cells of type\n(I) from GF (resp. %GF) in the facet F ; for all flags F of P such that\nF ¼ Fd\u00011ðFÞ and R ¼ Fd\u00012ðFÞ: It is easy to show that the parity of wðG;\nF ;RÞ is independent of both F and R:\nTheorem 21. Given any flag F of P; suppose F ¼ Fd\u00011ðFÞ and R ¼\nFd\u00012ðFÞ: Then\nwðG;F ;RÞ \u0012 1:\nProof. Consider the subgraph of GF that contains simplices of\ndimension ðd \u0001 1Þ or lower. This subgraph must be a collection of loops\nor paths (since GF is) whose endpoints (an even number of them) are non-\ndegenerate full cells in either F1 or Fd\u00011: But Sperner’s lemma in one-\ndimension (or simple inspection) shows that the number of full cells in F1\nmust be odd. Hence the number of non-degenerate full cells of GF that meet\nFd\u00011 must be odd as well. ]\nThe next two theorems show that for connected components %G of G; the\nparity of wð %G;F ;RÞ is also independent of F and R: This fact does not follow\ndirectly from Theorem 21 since we do not know that endpoints of GF for\ndifferent flags are connected in %G: To establish this we need to trace\nconnected paths in the nerve graph G rather than the flag graph GF:\nLemma 22. Let %G be a connected component of G: Suppose that R;R0\nare ridges of P and F a facet of P such that R;R0 are both facets of F :\nThen\nwð %G;F ;RÞ \u0012 wð %G;F ;R0Þ:\nDE LOERA, PETERSON, AND SU22Proof. First, assume that R and R0 are ‘‘adjacent’’ ridges sharing a\ncommon facet C (this has dimension d \u0001 3). We claim that\nX\nA;x\nwð %G;F ;A [ xÞ \u0012 0;\nwhere x runs over all labels in LðFÞ that are not in LðCÞ and A runs over all\nðd \u0001 2Þ-subsets of LðCÞ: (Here we write A [ x instead of A [ fxg to reduce\nnotation.) The above sum holds because it only counts fully labelled\nsimplices s from %G in F that contain a ðd \u0001 2Þ-subset A of LðCÞ; and every\nsuch s appears exactly twice in this sum; if LðsÞ ¼ A [ a [ b; then s is\ncounted once each in wð %G;F ;A [ aÞ and in wð %G;F ;A [ bÞ:\nOn the other hand, if K ¼ A [ x is not the label set of any ridge of P; then\nany fully labelled simplex on the boundary of P that contains K must be\ncontained in the facet F : Since K is of size d \u0001 1; by Lemma 10, we see that\nwð %G;F ;KÞ \u0012 0 because there is an even number of endpoints of paths in GK ;\nand such paths are connected subgraphs of the connected graph %G:\nThus the only terms surviving the above sum correspond to label sets of\nthe two ridges R;R0 that are facets of F and share a common face C; i.e.,\nwð %G;F ;RÞ þ wð %G;F ;R0Þ \u0012 0;\nwhich yields the desired conclusion for neighboring ridges R;R0:\nSince any two ridges of a facet F are connected by a chain of adjacent\nridges, the general conclusion holds. ]\nLemma 23. Let %G be a connected component of G: Let F ;F 0 be adjacent\nfacets of the polytope P bordering on a common ridge R: Then\nwð %G;F ;RÞ \u0012 wð %G;F 0;RÞ:\nProof. Let R denote the ridge common to both F and F 0: Let K be any\nnon-degenerate subset of LðRÞ of size ðd \u0001 1Þ; i.e., K is not a subset of LðFiÞ\nfor i5ðd \u0001 2Þ: Consider the derived graph GK : By Lemma 10, this graph\nconsists of paths connecting full cells from %G on the boundary of P that\ncontain the label set K : Since these paths are connected subgraphs of %G;\nthere is an even number of endpoints of these paths in %G:\nOn the other hand, because of the Sperner labelling, all such endpoints\nmust lie in facets of P that contain R: There are exactly two such facets, F\nand F 0: Hence\nwð %G;F ;RÞ þ wð %G;F 0;RÞ \u0012 0;\nwhich produces the desired conclusion. ]\nA POLYTOPAL SPERNER’S LEMMA 23Theorem 24. Let %G be a connected component of G: The parity of\nwð %G;F ;RÞ is independent of F and R:\nProof. Since all facet-ridge pairs ðF ;RÞ are connected by a sequence\nof adjacent facets and ridges, the statement follows from Lemmas 22\nand 23. ]\nHence we may define the parity of %G to be rð %GÞ \u0012 wð %G;F ;RÞ for any\nfacet-ridge pair ðF ;RÞ: Similarly, define the parity of G to be rðGÞ \u0012\nwðG;F ;RÞ for any facet-ridge pair ðF ;RÞ; which is well defined and\ncongruent to 1 in light of Theorem 21. Now we may prove\nTheorem 25. If P is an ðn; dÞ-polytope, there is some component %G of G\nwhich carries all labels of P:\nProof. Fix some flag F of P; and let F ¼ Fd\u00011ðFÞ and R ¼ Fd\u00012ðFÞ:\nSince wðG;F ;RÞ is the sum of wð %G;F ;RÞ over all connected components %G of\nG; it follows that rðGÞ \u0012\nP\nrð %GÞ over all connected components %G of G:\nMoreover, Theorem 21 shows that rðGÞ \u0012 1:\nHence there must be some %G such that rð %GÞ \u0012 1; i.e., this %G carries the\nlabels in LðRÞ: Since the flag F was arbitrary, %G must carry all labels\nof P: ]\nThis concludes our alternate ‘‘path-following’’ proof of Theorem 1,\nbecause the n\u0001 d count follows immediately from Theorems 25 and 16,\nwhile the covering property follows (as before) from Proposition 3.\n5. CONCLUSION\nAs an application of Theorem 1, we establish a version of a KKM-type\nintersection result of [14, Theorem 10] which now includes more cardinality\ninformation.\nCorollary 26. Let P be an ðn; dÞ-polytope with vertices v1; . . . ; vn: Let\nfCj j j ¼ 1; . . . ; ng be a collection of closed sets covering the ðn; dÞ-polytope P;\nsuch that each face F is covered by [fCh j vh 2 Fg:\nThen, for each p 2 P; there exists a subset Jp \u0011 f1; 2; . . . ; ng such that ð1Þ p\nlies in the convex hull of the vertices vj with j 2 Jp; ð2Þ Jp has cardinality\nd þ 1; ð3Þ\nT\nj2Jp\nCj=|; and ð4Þ if p and q are interior points of the same\nchamber of P; then Jp ¼ Jq: There are at least cðPÞ; the covering number,\ndifferent such subsets, and the simplices of P indicated by the labels in these\nsubsets form a cover of P:\nC\nC\nC\nC\n1\n2\nC3 C4\n5\n6\n1\n2\n3\n5\n6 1\n2 5\n6\n(A) (B)\n4 3 4\nFIG. 6. Part (A) shows several closed sets covering a hexagon and their four intersection\npoints. The points of intersection correspond to a cover of the hexagon, in this case a\ntriangulation, illustrated in part (B).\nDE LOERA, PETERSON, AND SU24Fig. 6 illustrates with an example the content of the above corollary.\nProof of Corollary 26. Let Cj; j ¼ 1; . . . ; n be the closed sets in the\nstatement. Consider an infinite sequence of triangulations Tk of the polytope\nP with the property that the maximal diameter of their simplices tends\nto zero as k goes to infinity. For each triangulation, we label a vertex y\nof Tk with i ¼ minfj 2 f1; 2; . . . ; ng j y 2 C\njg: This is clearly a Sperner\nlabelling.\nBy Theorem 1, each triangulation Tk specifies a collection of simplices of\nP corresponding to full cells in Tk: There are only finitely many possible\ncollections (since they are subsets of the set of all simplices of P), and\nbecause there are infinitely many Tk; some collection C of simplices must be\nspecified infinitely many times by a subsequence Tki of Tk: By Proposition 3,\nthis collection C is a cover of P and therefore has at least cðPÞ elements.\nFor each simplex s in C; choose one full cell si in Tki that shares the same\nlabel set. The si form a sequence of simplices decreasing in size. By the\ncompactness of P; some subsequence of these triangles converges to a point,\nwhich (by the labelling rule) must be in the intersection of the closed sets Cj\nwith j 2 LðsÞ:\nThus given a point p 2 P; choose any simplex s of C that contains P\n(since C is a cover of P), and let Jp ¼ LðsÞ: Then the above remarks show\nthat Jp satisfies the conditions in the conclusion of the theorem. Moreover,\nthere are at least cðPÞ different such subsets, one for each s in C: ]\nWe close with a couple of questions. For a specific polytope P; define the\npebble number pðPÞ to be the size of its largest pebble set. The n\u0001 d lower\nbound of Theorem 1 is tight, achieved by stacked polytopes whose vertices\nA POLYTOPAL SPERNER’S LEMMA 25are assigned different labels. But for a specific polytope P; the arguments of\nSection 2 show that the lower bound n\u0001 d can be improved to pðPÞ: What\ncan be said about the value of pðPÞ?\nWe can provide at least two upper bounds for this number. On the one\nhand pðPÞ4cðPÞ; because for a maximal pebble set, at most one pebble lies\nin each simplex of a minimal size cover. On the other hand, consider the\nsimplex-chamber incidence 0=1 matrix M introduced in [1]. As the columns\ncorrespond to chambers a pebble selection is essentially a selection of a\n‘‘row-echelon’’ submatrix; therefore the rank of M is an upper bound on the\nsize of pebble sets. Our pebble construction gives an algorithm for selecting\nan explicit independent set of columns of M (although this may not always\nbe a basis).\nA related question is: for a specific polytope P; how can one determine the\nminimal cover size cðPÞ? Although Theorem 2 gives a general sharp lower\nbound for all polytopes, we know that sometimes minimal covers are much\nlarger for specific polytopes, such as for cubes (as the volume arguments in\n[19] show). Also note that the minimal cover may be strictly smaller than the\nminimal triangulation (an example is contained in [6]).\nFinding other explicit constructions of pebble sets (besides our ‘‘facet-\npivoting’’ construction of Section 2) that work for specific polytopes may\nshed some light on these questions.\nACKNOWLEDGMENTS\nThe authors thank Francisco Santos, G .unter Ziegler, and the anonymous referees for many\nhelpful suggestions. The material in Section 3 first appeared in the undergraduate thesis of\nPeterson [15], for which Su served as faculty advisor.\nREFERENCES\n1. T. V. Alekseyevskaya, Combinatorial bases in systems of simplices and chambers, Discrete\nMath. 157 (1996), 15–37.\n2. K. T. Atanassov, On Sperner’s lemma, Studia Sci. Math. Hungar. 32 (1996), 71–74.\n3. D. Barnette, A proof of the lower bound conjecture for convex polytopes, Pacific J. Math.\n46 (1973), 349–354.\n4. L. J Billera, I. M. Gel’fand, and B. Sturmfels, Duality and minors of secondary polyhedra,\nJ. 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E. Su, Rental harmony: Sperner’s lemma in fair division, Amer. Math. Monthly 106\n(1999), 930–942.\n22. M. J. Todd, ‘‘The Computation of Fixed Points and Applications,’’ Lecture Notes in\nEconomics and Mathematical Systems, Springer-Verlag, New York, 1976.\n23. Z. Yang, ‘‘Computing Equilibria and Fixed Points,’’ Kluwer Academic Publishers, Boston,\n1999.\n24. G. M. Ziegler, ‘‘Lectures on Polytopes,’’ Springer-Verlag, New York, 1995.\n"}
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{"id":"WbO35p51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/easy\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":39,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","easy"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["easy – Math Fun Facts"],"last_modified":"Wed Jun 08 17:23:02 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Sat Jun 29 00:00:00 UTC 2019","Fri Jun 28 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-6009490167971240289,"exact_signature_unique_b":true,"fuzzy_signature_l":4380539077062089178,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["easy"],"h2_txt":["Irrationality by Infinite Descent","Computability of Real Numbers","Van der Waerden Theorem","Rational Irrational Power","Sum of Cubes and Beyond","Odd Numbers in Pascal’s Triangle","Pretzel Unlinking","Impossible Integral?","Pythagorean Triples","High-Dimensional Spheres in Cubes"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/TheWave.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 TheWave The Wave 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":716,"text_t":"easy – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! easy. These fun facts are at the easy level. Irrationality by Infinite Descent Posted on June 29, 2019. by. Samuel Nunoo The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the... Continue reading... Computability of Real Numbers Posted on June 29, 2019. by. Samuel Nunoo We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We... Continue reading... Van der Waerden Theorem Posted on June 29, 2019. by. Samuel Nunoo Can you color the integers red and blue such that there are no monochromatic arithmetic progressions (AP’s) extending infintely in... Continue reading... Rational Irrational Power Posted on June 29, 2019. by. Samuel Nunoo If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to... Continue reading... Sum of Cubes and Beyond Posted on June 29, 2019. by. Samuel Nunoo We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the... Continue reading... Odd Numbers in Pascal’s Triangle Posted on June 29, 2019. by. Samuel Nunoo Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N... Continue reading... Pretzel Unlinking Posted on June 29, 2019. by. Samuel Nunoo Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and... Continue reading... Impossible Integral? Posted on June 29, 2019. by. Samuel Nunoo The following integral may be problematic for a freshman calculus student, even if armed with a list of antiderivatives: INTEGRAL0... Continue reading... Pythagorean Triples Posted on June 28, 2019. by. Samuel Nunoo Which triples of whole numbers {a, b, c} satisfy a2 + b2 = c2 ? Such triples are called Pythagorean triples because they are integer... Continue reading... High-Dimensional Spheres in Cubes Posted on June 28, 2019. by. Samuel Nunoo How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"V14lIW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Random-walks.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":76,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Random-walks","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Random-walks.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:38 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2966671589796514605,"exact_signature_unique_b":true,"fuzzy_signature_l":5065191805163394865,"fuzzy_signature_unique_b":true,"wordcount_i":2508,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:03 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:15 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":199114,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088126840274944,"text_t":"To appear in: Unusual Applications of Number Theory, M. Nathanson, ed.,\nDIMACS Series in Discrete Mathematics and Theoretical Computer Science, 2002.\nRandom walks with badly approximable numbers\nDoug Hensley and Francis Edward Su\nAbstract. Using the discrepancy metric, we analyze the rate of convergence\nof a random walk on the circle generated by d rotations, and establish sharp\nrates that show that badly approximable d-tuples in Rd give rise to walks with\nthe fastest convergence.\n1. Introduction\nFix ᾱ = (α1, α2, ..., αd), a d-tuple of real numbers, not all rational. Consider\na random walk on the circle S1 which proceeds as follows: at each step one of\nthe αi is chosen (with probability 1\/d) and the walk moves forwards or backwards\n(with probability 1\/2) on the circle by an angle 2παi. The distribution of this walk\nconverges to the uniform (Haar) measure on S1; in this paper, we investigate how\nquickly this convergence can take place.\nTo quantify this, identify S1 with the unit circle in R2, and let Q be the prob-\nability measure supported on {e±2πiαj}, j = 1, ..., d, with equal weights 1\/2d. If\nat time k = 0 the random walk starts at 1, then the convolution power Q∗k repre-\nsents the probability distribution of the random walk at time k. In this framework,\nwe study how quickly Q∗k approaches U , the uniform (Haar) measure on the unit\ncircle. Define the discrepancy D(P ) of a probability measure on S1 by\nD(P ) = sup\nI\n|P (I)− U(I)|(1)\nwhere I represents any connected interval on S1. The discrepancy D(P ) measures\nhow uniform P is, and metrizes weak-* convergence to the uniform distribution on\nS1. The usual definition of discrepancy found in the study of uniform distribution\nof sequences mod 1 (see [6]) is a special case of our definition, by letting P be an\nequally weighted measure on a sequence.\nWe seek bounds on how quickly D(Q∗k) diminishes as a function of the step\nsize k and the numbers αi. Clearly the convergence will not occur if all αi are\nrational.\nThe case d = 1 reduces to a random walk generated by one irrational rotation;\nthis problem was posed by Diaconis in [3, p. 34]. Su [12] obtained discrepancy\nbounds for this walk, showing that it converges most quickly when α is a quadratic\nirrational. In that case,\nC1√\nk\n≤ D(Q∗k) ≤ C2√\nk\nfor constants C1, C2 which can be determined given α. We investigate whether the\ninclusion of additional generators can speed up the convergence of this walk. In\n2000 Mathematics Subject Classification. Primary 60B15; Secondary 11J13, 11K38, 11K60.\nFrancis Su is grateful to the School of Operations Research at Cornell University for their\nhospitality during a sabbatical in which this was completed.\n1\n2 DOUG HENSLEY AND FRANCIS EDWARD SU\nparticular we show that if ᾱ is a badly approximable vector in Rd, then the random\nwalk generated by the ᾱ converges as fast as possible for a set of d generators. In\nTheorem 1 we obtain matching upper and lower bounds:\nC1\nkd\/2\n≤ D(Q∗k) ≤ C2\nkd\/2\n.\nWe also show how the constants depend on ᾱ.\nThis result is reminiscent of results in the theory of uniform distribution of\nsequences mod 1; however, there is a notable lack of any log terms in the upper\nand lower bounds. To compare for d = 1, it is known (e.g., see [6]) that for a fixed\nirrational α ∈ R the discrepancy of the sequence {α, 2α, 3α, . . . , kα} in R mod 1\nfalls as log k\/k up to constant factors. For our corresponding random walk, it is\nnot a surprise that the exponent gets halved, but it is quite surprising that the log\nterm disappears.\nWhile the total variation metric is more frequently used to study random walk\nconvergence (e.g., see [3]), it is not appropriate for this walk, because Q∗k is a\nfinitely supported measure and its total variation distance from U remains at 1 for\nall k. Moreover, we favor the use of discrepancy over other common metrics on\nprobabilities (e.g., Wasserstein, Prokhorov) because bounding techniques for such\nmetrics are not well developed, and the use of Fourier bounds for the discrepancy\nmetric admits the possibility of sharp analysis. An analogous discrepancy metric\nmay be used to study random walks on other spaces, e.g., [11, 13, 14]. For a\nsurvey of bounds relating various metrics, see [5].\nWe also remark that while the literature contains many results on rates of\nconvergence for random walks on finite groups (e.g., see [3]), and a few results for\nwalks on infinite compact groups ([8],[9]), very little has been done for discrete\nwalks on infinite compact groups, mainly due to lack of bounding techniques for\nappropriate metrics. Our analysis of the d-generator random walk on the circle\nreveals that bounds for the discrepancy metric are refined enough to yield sharp\nrates of convergence; this and [12] are the only sharp results we are aware of in this\ndirection.\n2. Random Walk Bounds\nThroughout this paper, let ‖x‖ denote the Euclidean distance of x ∈ Rd to\nthe nearest integer lattice point (the dimension inherent in the notation ‖ · ‖ is to\nbe understood by context). Given an arbitrary d-tuple ᾱ = (α1, α2, ..., αd), the\nDirichlet Approximation Theorem [4, p.68] implies that there exists a constant\nB = B(ᾱ) such that for any q, there exists a positive integer N ≤ qd such that\n‖Nᾱ‖ < B\nq\n≤ B\nN1\/d\n.(2)\nIn fact for any d-tuple ᾱ, the above bound holds for B =\n√\nd.\nThe d-tuple ᾱ is said to be badly approximable if there exists a constant β =\nβ(ᾱ) > 0 such that for all positive integers N ,\n‖Nᾱ‖ ≥ β\nN1\/d\n.(3)\nWe refer to B and β as approximation constants for ᾱ. The approximation\nconstant β is only defined for badly approximable ᾱ.\nOur main result is the following:\nRANDOM WALKS WITH BADLY APPROXIMABLE NUMBERS 3\nTheorem 1. Suppose ᾱ = (α1, α2, ..., αd) is badly approximable. Let Q denote\nthe generating measure of the random walk on S1 generated by the d generators αi.\nThen the discrepancy of the k-th step probability distribution of the walk satisfies\nC1\nkd\/2\n≤ D(Q∗k) ≤ C2\nkd\/2\n(4)\nwith constants that depend on d and ᾱ:\nC1 = 0.0947 (\n√\nd\/5B)d\nC2 = 19.857 (d\n√\nd\/β)d\nwhere B, β are approximation constants for ᾱ. The lower bound holds for walks\ngenerated by an arbitrary d-tuple.\nTherefore, among all d-tuples, walks generated by badly approximable d-tuples\nconverge the fastest. Note that the constants C1, C2 depend on the approximation\nconstants of ᾱ. However since one may choose B =\n√\nd, it follows that C1 ≥\n0.0947(1\/5)d.\nProof. We establish the lower bound using an inequality of Su [13] for the\ndiscrepancy of an arbitrary probability measure P on S1:\nD(P ) ≥\n(\n2\nπ2\n∞∑\nm=1\n|P̂ (m)|2\nm2\n)1\/2\n.(5)\nHere P̂ (m) denotes the m-th Fourier coefficient of P (viewed on R mod Z). Since\nevery term is positive, we can use a dominant term in the sum as a lower bound.\nThe Fourier coefficients for the d-generator walk are\nQ̂(m) =\nd∑\nl=1\n1\n2d\n(e2πimαl + e−2πimαl) =\n1\nd\nd∑\nl=1\ncos(2πmαl).(6)\nSince cos(2πx) = cos(2π‖x‖) ≥ 1− 2π2‖x‖2, for any m we have\nQ̂(m) ≥ 1− 2π\n2\nd\n‖mᾱ‖2(7)\nThe inequality (1− x)k ≥ 1− kx holds for k ≥ 1 and x ≤ 1. Then\n|Q̂(m)|k ≥ 1− 2π\n2k\nd\n‖mᾱ‖2\nas long as 2π2k‖mᾱ‖2\/d < 1. We ensure this by setting Z1 < 1 (to be specified\nshortly) and let q = (2π2B2k\/Z1d)\n1\/2. Then (2) implies there exists an integer\nm ≤ qd such that ‖mᾱ‖ < B\/q, which yields 2π2k‖mᾱ‖2\/d < Z1 < 1, as desired.\n(Note that q was chosen to ensure |Q̂(m)|k ≥ 1− Z1.)\nFor this m we use just the m-th term in (5), and since m < qd and Q̂∗k(m) =\nQ̂k(m), we obtain\nD(Q∗k) ≥\n√\n2|Q̂(m)|k\nπm\n≥ C1 k−d\/2(8)\nwith\nC1 =\n√\n2(1− Z1)\nπ\n(\nZ1d\n2π2B2\n)d\/2\n.\n4 DOUG HENSLEY AND FRANCIS EDWARD SU\nChoosing Z1 = 2π\n2\/25 < 1, we recover the lower bound (4) of the theorem.\nThe upper bound is trickier to compute. We start with the Erdös-Turán in-\nequality [7]: given a probability measure P on S1, for any integer M ,\nD(P ) ≤ 4\nM + 1\n+\n4\nπ\nM∑\nm=1\n|P̂ (m)|\nm\n(9)\nwhere P̂ represents the Fourier transform of P . Note that one may choose M in\nthe Erdös-Turán inequality so as to optimize the bound obtained.\nSince | cos(2πx)| ≤ 1− 4‖2x‖2 for all x ∈ R, it follows from (6) that\n|Q̂(m)| ≤ 1\nd\nd∑\nl=1\n| cos(2πmαl)|\n≤ 1− 4\nd\nd∑\nl=1\n‖2mαl‖2\n≤ exp(−4\nd\n‖2mᾱ‖2).\nIn light of the Erdös-Turán inequality and Q̂∗k(m) = Q̂k(m), we need to esti-\nmate a sum of the form\nM∑\nm=1\n|Q̂k(m)|\nm\n≤\nM∑\nm=1\n1\nm\nexp(−4k\nd\n‖2mᾱ‖2) =: S.\nSince M may be chosen freely, choose an integer M such that\nM ≤ 1\n2\n(β2k\/d3)d\/2 < M + 1.(10)\nRecall that β is an approximability constant for ᾱ from (3) and k is the number of\nsteps in the walk. The reason for this choice of M will be evident later. Choose an\ninteger J such that\n2J−1 ≤M ≤ 2J − 1.(11)\nThe sum in S may be grouped into J cohorts of integers m ∈ [2j−1, 2j − 1]\nfor j = 1, ..., J . Within each such cohort, the use of inequality (3) yields ‖2mᾱ‖ ≥\nβ\/(2m)1\/d ≥ β\/2(j+1)\/d. This says that the points of the sequence {2mᾱ} (mod\n1) in the unit cube in Rd are bounded away from the corners of the unit cube. In\nfact, they are bounded away from each other, since if m1,m2 ∈ [2j−1, 2j − 1], then\n‖2(m1 −m2)ᾱ‖ ≥ β\/2(j+1)\/d(12)\nas well. Therefore if s = β\/\n√\nd 2(j+1)\/d, then any box of side length s can contain\nat most one of the multiples from the sequence {2mᾱ} (mod 1), m ∈ [2j−1, 2j − 1],\nwith no such multiples in boxes containing the origin.\nSo divide up the unit d-cube into disjoint boxes of side length s and sides\nparallel to the axes. In the worst case, all M points are distributed in the boxes\nin the 2d corners of the unit cube nearest the origin. The nearest points in such\nboxes (under the L1 metric) are at integral multiples of s and extend out in layers\nto distance at most M1\/d. A crude upper bound for the number of boxes whose\nnearest point is at L1-distance ns from the origin is (n+1)\nd−1, and in the Euclidean\nRANDOM WALKS WITH BADLY APPROXIMABLE NUMBERS 5\nmetric the nearest point in such boxes is at least distance ns\/\n√\nd from the origin.\nHence we can bound S by grouping first by cohorts, then by corners and layers:\nS ≤\nJ∑\nj=1\n2j−1∑\nm=2j−1\n1\nm\nexp(−4k\nd\n‖2mᾱ‖2)\n≤\nJ∑\nj=1\n2d\nM1\/d∑\nn=1\n(n+ 1)d−1\n2j−1\nexp(−4k\nd\nβ2n2\nd · 22(j+1)\/d\n)\n≤\nJ∑\nj=1\n2d\n2j−1\n∞∑\nn=1\n(n+ 1)d−1 exp(−4n2d22(J−j−1)\/d).\nThe second inequality used the bound ‖2mᾱ‖ ≥ ns\/\n√\nd and the definition of s. The\nthird inequality follows by noting k ≥ d322J\/d\/β2 from the definitions of J and M\nin (10) and (11).\nUsing j ≤ J , the log derivative of the expression of the innermost sum with\nrespect to n can be bounded:\nd− 1\nn+ 1\n− 8nd22(J−j−1)\/d ≤ d− 1\nn+ 1\n− 8nd2−2\/d ≤ d− 1\n2\n− d ≤ −1\nfor all choices of n ≥ 1 and d ≥ 1. Hence the expression in the inner sum decreases\ngeometrically (by at least ratio e−1) as j increases, and so the inner sum is bounded\nby the first term (at n = 1) times the constant Z2 = 1\/(1− e−1) ≈ 1.5820. Thus\nS ≤\nJ∑\nj=1\nZ2 2\n2d−j exp(−4d22(J−j−1)\/d).\nThis sum may be bounded by noting that the largest term occurs when j = J . For\nj ≤ J , the log derivative of the terms with respect to j is ln 2(−1+8 ·22(J−j−1)\/d) ≥\nln 2 for d ≥ 1. Thus the sum decreases geometrically (with at least ratio 1\/2) as\nj ≤ J decreases, so the sum is therefore bounded by twice the final term at j = J :\nS ≤ 2Z2 22d−J exp(−4d2−2\/d)\n≤ Z22\n2d+1 exp(−4d2−2\/d)\nM + 1\n≤ 4.6559\nM + 1\n(13)\nwhere the second inequality follows from (11), and the final inequality from noting\nthat the numerator is greatest for d = 1. Using the Erdös-Turán inequality we\nobtain the upper bound\nD(Q∗k) ≤ 4\nM + 1\n+\n4\nπ\nS ≤ 4 + (4\/π)4.6559\nM + 1\n≤ 9.9281\nM + 1\nfor all d ≥ 1. An application of (10) produces\nD(Q∗k) ≤ 19.857 d3d\/2β−dk−d\/2\nwhich establishes the constant C2 in the statement of the theorem.\nWe remark that this argument simplifies the proof given in [12] for the case\nd = 1. For specific d, the constants C1, C2 can be improved by adjusting the\nderivations of the constants Z1, Z2 and the bound for the last inequality in (13).\n6 DOUG HENSLEY AND FRANCIS EDWARD SU\n3. Choosing Generators\nBadly approximable d-tuples are plentiful. Cassels [1] shows that there un-\ncountably many badly approximable d-tuples in Rd. Moreover, results of Schmidt\n[10, pp. 53-59] imply that the Hausdorff dimension of the set of badly approximable\nvectors is positive. For some concrete examples, it can be shown (see, e.g., [4, p.\n68]) that if 1, α1, ..., αd are linearly independent over Z, and if the degree of the\nextension [Q(α1, ..., αd) : Q] = d+ 1, then ᾱ is badly approximable.\nWe have shown that badly approximable d-tuples in Rd give rise to random\nwalks with the fastest convergence, established sharp rates of convergence, and\nexhibited how the constants depend on the approximation constants β,B of the\ngiven d-tuple. However, just among badly approximable d-tuples, can we say which\nrandom walks converge “the fastest”? By Theorem 1, this amounts to identifying\nd-tuples with the largest possible approximation constant β.\nFor d = 1, the number φ =\n√\n5−1\n2\nsatisfies (3) and is therefore badly approx-\nimable; in fact, it is known to be the “most” badly approximable number in the\nsense that its approximation constant β is larger than for any other number.\nFor d = 2, we conjecture that the “most” badly approximable 2-tuple is the\nvector v = (γ−2, γ−1), where γ ≈ 1.3247 is the unique real root of x3 − x− 1. For\nthis v, we believe (based on heuristic arguments and numerical evidence) that for\nany β < 0.54850..., there is a sufficiently large N such that n > N implies that\n‖nv‖ > β n−1\/2.(14)\nFurthermore, no other vector can have a much larger approximation constant β,\nbecause the work of Davenport and Mahler [2] implies that if β > (2\/\n√\n23)1\/2 ≈\n.64577..., then there is no vector v in R2 for which (14) holds for all n.\nThus the pair (γ−2, γ−1) yields a random walk on the circle with 2 generators\nwhose convergence rate appears close to fastest possible over all badly approximable\npairs. Is it fastest? For larger d the question is also open.\nReferences\n[1] J. W. S. Cassels, Simultaneous Diophantine Approximation II, Proc. London Math. Soc.\n(3) 5(1955), 435-448.\n[2] H. Davenport and K. Mahler, Simultaneous Diophantine Approximation, Duke Math J.\n13(1946), 105-111.\n[3] P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical\nStatistics Lecture Notes, Vol. 11, Hayward, CA, 1988.\n[4] M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in\nMath. 1651, Springer-Verlag, 1997.\n[5] A. Gibbs and F. E. Su, On choosing and bounding probability metrics. Preprint, 2000.\n[6] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974.\n[7] H. Niederreiter and W. Philipp, Berry-Esseen bounds and a theorem of Erdös and Turán\non uniform distribution mod 1, Duke Math. J. 40(1973), 633-649.\n[8] U. Porod, The cut-off phenomenon for random reflections, Ann. Probab. 24(1996), 74-96.\n[9] J. S. Rosenthal, Random rotations: characters and random walks on SO(n), Ann. Probab.\n22(1994), 398-423.\n[10] W. M. Schmidt, Diophantine Approximation, Lecture Notes in Math. 785, Springer-Verlag,\n1980.\n[11] F. E. Su Methods of Quantifying Rates of Convergence for Random Walks on Groups, Ph.D.\nThesis, Harvard University, 1995.\n[12] F. E. Su, Convergence of random walks on the circle generated by an irrational rotation,\nTrans. Amer. Math. Soc., 350(1998), 3717-3741.\nRANDOM WALKS WITH BADLY APPROXIMABLE NUMBERS 7\n[13] F. E. Su, A LeVeque-type lower bound for discrepancy, in Monte Carlo and Quasi-Monte\nCarlo Methods 1998, H. Niederreiter and J. Spanier (eds.), Springer-Verlag, 2000, pp. 448-\n458.\n[14] F. E. Su, Discrepancy convergence for the drunkard’s walk on the sphere. To appear, Elec-\ntron. J. Probab., 2001.\nDepartment of Mathematics, Texas A&M University, College Station, TX 77843\nE-mail address: Doug.Hensley@math.tamu.edu\nDepartment of Mathematics, Harvey Mudd College, Claremont, CA 91711\nE-mail address: su@math.hmc.edu\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! other. How to Use Fun Facts Posted on July 12, 2019. by. Molly Reeves Fun Facts are designed for, and work best in, lower-division college courses where the typical student is not a math student. The... Continue reading... About Math Fun Facts Posted on July 11, 2019. by. Francis Su I developed Math Fun Facts in 1994 as a warm-up activity for the calculus courses I taught as a graduate... Continue reading... Multiplication by 111 Posted on July 1, 2019. by. Ashley Tung If you liked the Fun Fact Multiplication by 11, you’ll enjoy seeing how to take that idea one step farther. Here’s... Continue reading... Liouville Numbers Posted on June 29, 2019. by. Samuel Nunoo Are there any real numbers that are NOT algebraic, i.e., expressible as the root of a non-zero polynomial with integer coefficients? In fact,... Continue reading... Dedekind Cuts of Rational Numbers Posted on June 29, 2019. by. Samuel Nunoo Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the... Continue reading... Computability of Real Numbers Posted on June 29, 2019. by. Samuel Nunoo We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We... Continue reading... Fermat’s Last Theorem Posted on June 29, 2019. by. Samuel Nunoo There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =... Continue reading... Rationals Dense but Sparse Posted on June 29, 2019. by. Samuel Nunoo Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the... Continue reading... Ordinal Numbers Posted on June 29, 2019. by. Samuel Nunoo One of the most useful properties of the whole numbers is that every non-empty subset has a least element; this... Continue reading... Risk-Free Betting on Different Beliefs Posted on June 29, 2019. by. Samuel Nunoo Alice believes that the 49ers will win the Super Bowl with probability 5\/8. Bob believes that Ravens will win the... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved. Loading Comments... Write a Comment... Email (Required). Name (Required). 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{"id":"S5D0WW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/11\/good-math-writing.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":81,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","11"],"url_file_name_s":"good-math-writing","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["good-math-writing.pdf"],"author":"Mac OS X 10.8.2 Quartz PDFContext","author_sxt":["Mac OS X 10.8.2 Quartz PDFContext"],"last_modified":"Mon Mar 18 15:11:02 UTC 2013","dates_in_content_count_i":1,"dates_in_content_dts":["Wed Jan 19 00:00:00 UTC 2011"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-8422294637848444932,"exact_signature_unique_b":true,"fuzzy_signature_l":8610320536060035182,"fuzzy_signature_unique_b":true,"wordcount_i":1657,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:05 UTC 2022","fresh_date_dt":"Tue Sep 19 17:05:02 UTC 2023","referrer_id_s":"p2v2CW51bhow","publisher_t":"Mac OS X 10.8.2 Quartz PDFContext","language_s":"en","size_i":102904,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":2,"references_internal_i":2,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088129352663040,"text_t":"GUIDELINES FOR GOOD MATHEMATICAL WRITING\nFRANCIS EDWARD SU\nCommunicating mathematics well is an important part of doing mathematics. Whether you are speak-\ning or writing, learning to communicate effectively is not just a service to your audience; it is also an\nexercise in clarifying and structuring your own thinking. Moreover, beyond developing a basic compe-\ntence, there is an art and elegance to good writing that every writer should strive for. And writing, as a\nwork of art, can bring great personal satisfaction.\nThese guidelines may serve as a starting point for good mathematical writing.\n1. BASICS\nKnow your audience. This is the most important consideration for writers. Put yourself in your reader’s\nshoes. What background can we assume of the reader? What terminology should we define? What kind\nof “voice” do we want to project: casual or professional, serious or inviting, terse or loquacious?\nIf you are a student writing solutions for a homework set and your professor has not specified your\naudience, a good rule of thumb is to assume you are writing to another student in the course who has\nnot yet done the assignment. Though you may assume that she has attended all the same lectures and\nhas read the same textbook, it is standard courtesy to remind your reader of any relevant items that she\nhas recently learned in class or from the textbook, or things she should know but might have forgotten.\nFor instance, if the concept of a rational number was only recently learned in class, you might insert\n“Recall that a rational number can be expressed as a fraction;” before saying: “since x is rational, we can\nset x = m\/n where m and n are integers.”\nSet an invitational tone. It is traditional to create an inviting atmosphere in one’s mathematical writing.\nIn effect, we invite readers to join us in our reasoning process by writing in the present tense, using the\npronoun “we” instead of “I” (e.g., “we construct a tangent plane...”), and directing the reader with gentle\ncommands (e.g., “let n be...”, “recall that...”, or “consider the set of...”).\nUse complete sentences. All mathematics should be written in complete sentences. Open any mathe-\nmatics text and you’ll see that this is true. Equations, even displayed ones, have punctuation that help\nyou see where it fits in the context of a larger sentence. Consider this piece of writing:\n(x\u0000 2)2 + (x\u0000 1)2 = 52 52 = 25\n(x\u0000 2)2 = x2 \u0000 4x + 4 + x2 \u0000 2x + 1 = 25\n2x2 \u0000 6x\u0000 20\n2(x + 2)(x\u0000 5) x = \u00002, 5 x > 0 x = 5\nCan you figure out what the writer is doing? What’s being assumed? What’s being proved? Where\ndoes one thought end and another begin? What’s the relationship between these phrases? Some phrases\nare dangling, and others, as statements, are not even true. The reader should not have to figure out what\nthe writer was thinking!\nNow consider the work of another writer who has attempted the same problem:\nDate: January 19, 2011.\n1\nProblem. Find a point on the line y = x which is distance 5 from the point (2, 1) and such\nthat x > 0.\nSolution. We wish to solve (x\u00002)2 +(x\u00001)2 = 52, an equation obtained from the distance\nformula in the plane. Some algebra turns this equation into:\n2x2 \u0000 6x\u0000 20 = 0.\nFactoring the left side, we obtain\n2(x + 2)(x\u0000 5) = 0,\nwhose solutions are evidently x = \u00002 and x = 5. Since we assumed x > 0, we have (5, 5)\nas the desired point on the line y = x.\nHere, the writer has clearly stated the problem and described her path to a solution. She has set an\ninvitational tone, and every thought is expressed in a complete sentence. Now it is clear that x > 0\nis a condition, not a result. Notice the punctuation in equations: one ended with a period because her\nthought was complete, the other ended with a comma because she wanted to continue the thought. Since\nshe assumed her audience could do algebra, she didn’t bore them with trivial algebraic manipulation,\nwhich would only obscure the thread of her arguments. But she did show the most interesting parts: the\nresulting polynomial and its factoring. And she made sure she answered the original question.\nAvoid shorthand in formal writing. The many types of mathematical writing can be loosely grouped\ninto formal and informal writing. Informal writing includes writing on a blackboard during lecture, or\nexplaining something to a friend on a piece of scratch paper. Formal writing includes the kind of writing\nexpected on a homework assignment, or in a paper. There are differences in what is acceptable.\nFor instance, in informal writing, it is common to use shorthand for quantifiers and logical connec-\ntives: symbols such as 8,9, =) , () , or abbreviations like “iff” and “s.t.”.\nHowever, in formal writing such shorthand should generally be avoided. You should write out “for\nall”, “there exists”, “implies”, “if and only if”, and “such that”. Most other symbols are acceptable in\nformal writing, after defining them where needed. The membership symbol 2 (“is an element of”) is\ntraditionally acceptable in formal writing, as are relations (e.g., <, >,⇥,+,[,\\, etc.), variable names\n(e.g., x, y, z, etc.), and symbols for sets (Q, R, Z, etc.). Here is an acceptable use of symbols in formal\nmathematical writing.\nLet A, B be two subsets of R. We say A dominates B if for every x 2 A there exists y 2 B\nsuch that y > x.\nLearn the etiquette. The above example also illustrates two common conventions of mathematical eti-\nquette. It is customary to avoid beginning sentences or phrases with a number or symbol; 41 people\ntold me not to do it because it looks bad. It is also customary to emphasize unfamiliar words that we are\nabout to define, either by italicizing or underlining them. Other rules of etiquette can be learned through\nobserving the norms of the mathematical people or textbooks in your area of study.\n2. TOWARDS ELEGANCE\nDecide what’s important to say. Writing well does not necessarily mean writing more. If your solution\nis too wordy, it can sometimes obscure the points you are making.\nA well-written solution will present just enough details and highlight the most interesting or unex-\npected parts of the argument. What theorems or axioms were crucial in getting your solution, and where\nwere they used? Your role as a writer is not primarily to communicate details (though that can be quite\nimportant); your primary role is to give insight.\n2\nHighlight structure. If your argument is going to be a long one, with lots of technical details, then try\nto help the reader by summarize the outline of your argument at the beginning. Then, throughout your\nwriting, help your reader see how you are progressing through your outline.\nUse paragraphs to emphasize blocks of ideas that are related. The role of the first sentences of para-\ngraphs are crucial: imagine a reader skimming your writing and only reading the first sentences. Will\nthey see the flow of your argument? Similarly, you might want to display only the most important\nequations. Replacing an oft-repeated argument by a good lemma can help streamline the flow of your\nargument, as well as highlight a key idea.\nChoose good examples. A difficult idea may be easier to digest if accompanied by an example. Choose\none simple enough to follow, but interesting enough to retain the salient features. A proof of a very\ngeneral idea could be preceded by an example in a specific context. A long exposition might benefit\nfrom a running example— one in which the same example is used multiple times in different contexts.\nAvoid red herrings! Omit details that have no bearing on the solution of the problem, as they may\nthrow the reader off. For example, if you say “we express the rational r as m\/n where m and n have no\ncommon factors,” you are leaving a clue that later, you will use the “no common factors” idea. So if you\nnever use that fact, you should omit saying it. It’s extraneous.\nStep back and simplify! After writing a proof, step back and ask: how can I simplify this argument?\nDid I use every tool I pulled out to solve this problem? Can I streamline this argument? For example,\nconsider this apparent proof by contradiction:\nProblem. Show that if 4 divides an integer n, then n is even.\nProof (by contradiction). Suppose n were not even.\nSince 4 divides n, we have n = 4k for some integer k. Thus n = 2(2k), which is even.\nThis contradicts our hypothesis that n is not even. ⇤\nDo you see why this is not really a proof by contradiction? The contradictory hypothesis in the first\nsentence was never used! Strip away the first and last sentence, and you have an elegant, direct proof.\nRefine, refine, refine. Good writing is a process of successive approximations. You should not expect\nyour first draft to be perfect. You will find that when you review your writing, you will see ways to\nshorten an argument, or say something in a better way. This is the part of the writing process that will\nhelp clarify your own thinking as well.\nOften, after completing a draft, a writer may notice that a particular choice of notation or a choice of\ndefinition was not optimal. A lazy writer would leave things as they are, but a thoughtful writer will\ntake the time to go back and make changes.\n3. FINALLY...\nObserve the culture. Good communication is inseparable from the culture in which it takes place. This\nrealization may unsettle budding mathematicians who are attracted to the logical absolutes of mathe-\nmatics. But even these absolutes are expressed in different ways by mathematicians of different eras, as\ncan be seen by comparing Newton’s writings to any of today’s calculus texts. The rules of mathematical\netiquette have evolved over time.\nWhile these guidelines attempt to draw up some common principles for formal writing, there will\nalways be exceptions— because some subset of mathematicians may have a slightly different norm. The\nbest rule of thumb to get a sense of what is acceptable or conventional in your cultural context is to\nbrowse several highly regarded textbooks or papers related to the document you are writing.\nAcknowledgements. It is good to acknowledge any support you received. My revisions of this piece of\nwriting benefited greatly from the helpful feedback of Jon Jacobsen and Lesley Ward.\n3\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! universe. Klein Bottle Posted on June 29, 2019. by. Samuel Nunoo A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;... Continue reading... Projective Planes Posted on June 29, 2019. by. Samuel Nunoo Most of you know how to make a Mobius band—take a strip of paper and glue the ends with a... Continue reading... Spherical Geometry Posted on June 26, 2019. by. Samuel Nunoo Remember high school geometry? The sum of the angles of a planar triangle is always 180 degrees or Pi radians. However, triangles... Continue reading... Making History by Card Shuffling Posted on June 26, 2019. by. Samuel Nunoo Did you know that whenever you shuffle a deck of cards, it is quite likely that you are making history?... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! hard. Slices of Hanging Cubes Posted on June 29, 2019. by. Samuel Nunoo Hang a cube from one of its vertices. Now, if you slice it horizontally through its center, you get a... Continue reading... Continuum Hypothesis Posted on June 29, 2019. by. Samuel Nunoo We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 correspondence with... Continue reading... Banach-Tarski Paradox Posted on June 29, 2019. by. Samuel Nunoo Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using... Continue reading... Face Derivatives and Computer Vision Posted on June 29, 2019. by. Samuel Nunoo One challenge in robotics is the problem of computer vision: how do you program a computer to interpret and “understand”... Continue reading... Pretzel Unlinking Posted on June 29, 2019. by. Samuel Nunoo Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and... Continue reading... Unbelievable Unlinking Posted on June 28, 2019. by. Samuel Nunoo Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber.... Continue reading... Red-Black Pairs Card Trick Posted on June 26, 2019. by. Samuel Nunoo Here’s a terrific mathematical card trick that will impress your friends. When you do this trick, the effect of the... Continue reading... Prime Number Theorem Posted on June 26, 2019. by. Samuel Nunoo Fix some number N. What fraction of the integers less than or equal to N are prime? Thinking about it,... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! medium. These fun facts are at the medium level. Multiplication by 111 Posted on July 1, 2019. by. Ashley Tung If you liked the Fun Fact Multiplication by 11, you’ll enjoy seeing how to take that idea one step farther. Here’s... Continue reading... i to the i is a Real Number Posted on June 29, 2019. by. Samuel Nunoo If you are familiar with complex numbers, the “imaginary” number i has the property that the square of i is -1.... Continue reading... Face Derivatives and Computer Vision Posted on June 29, 2019. by. Samuel Nunoo One challenge in robotics is the problem of computer vision: how do you program a computer to interpret and “understand”... Continue reading... Square Root of Two is Irrational Posted on June 29, 2019. by. Samuel Nunoo An irrational number is a number that cannot be expressed as a fraction. But are there any irrational numbers? It was known to... Continue reading... Finding the N-th digit of Pi Posted on June 29, 2019. by. Samuel Nunoo Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:Pi =... Continue reading... Ellipsoidal Paths Posted on June 29, 2019. by. Samuel Nunoo Given an ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a... Continue reading... Large Counterexample Posted on June 29, 2019. by. Samuel Nunoo A positive integer is said to be of even type if its factorization into primes has an even number of primes. Otherwise it is... Continue reading... Whiffle Ball Posted on June 29, 2019. by. Samuel Nunoo Does a ball take longer to come down than go up, or does it take the same amount of time... Continue reading... Area of a Circle or Regular Polygon Posted on June 29, 2019. by. Samuel Nunoo There’s a nice way to see why the formula for the area of a circle of radius R is: Pi... Continue reading... Pretzel Unlinking Posted on June 29, 2019. by. Samuel Nunoo Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"JZUbfW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/11\/hw3.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":67,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","11"],"url_file_name_s":"hw3","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["hw3.pdf"],"author":"pdfTeX-1.40.16","author_sxt":["pdfTeX-1.40.16"],"last_modified":"Fri Sep 25 20:48:38 UTC 2015","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-2198088499180449498,"exact_signature_unique_b":true,"fuzzy_signature_l":-7327670550758105489,"fuzzy_signature_unique_b":true,"text_t":"Math 131 — Homework 3\nREAD Problem D. Let u be an upper bound of non-empty set A in R.\nProve that u is the supremum of A if and only if for all \u000f > 0 there is an\na ∈ A such that u− \u000f < a.\nNote that to show that “S if and only if T” you must show that S implies\nT , and T implies S.)\nREAD Problem E. LetA,B be nonempty subsets of R that are bounded\nabove, and let A+B = {a+ b : a ∈ A, b ∈ B}. Show that\nsup(A+B) = supA+ supB.\nProblem F. Let A,B be nonempty subsets of positive real numbers that\nare bounded above, and let A ·B = {ab : a ∈ A, b ∈ B}. Show that\nsup(A ·B) = supA supB.\nProblem G. (a) Let A be a nonempty subset of R and suppose that\ns = supA belongs to A. If b is not in A, show that sup(A ∪ {b}) is equal to\nthe larger of the two numbers s and b.\n(b) Use this to show that a nonempty finite set A contains its supremum.\n[Hint— use induction: show it is true first for a one-element set, then show\nthat if it is true for an n-element set then it must be true for an (n + 1)-\nelement set.]\nDo also Chapter 1 ( 6ab, 6cd, 12, R13, 15 ).\nComment: When you are asked a question, e.g., problem 1.15, you should\nalways give justification.\n1\n","wordcount_i":236,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:13 UTC 2022","fresh_date_dt":"Tue Sep 19 16:32:45 UTC 2023","referrer_id_s":"p2v2CW51bhow","publisher_t":"pdfTeX-1.40.16","language_s":"en","size_i":83884,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088137267314688}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! geometric series. Rationals Dense but Sparse Posted on June 29, 2019. by. Samuel Nunoo Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the... Continue reading... Taylor-made Pi Posted on June 29, 2019. by. Samuel Nunoo After learning about the Taylor series for 1\/(1+x) in calculus, you can find an interesting expression for Pi very easily. Start with... Continue reading... Cantor Set Posted on June 26, 2019. by. Samuel Nunoo Start with the interval [0,1]. Remove the (open) middle third of it, i.e. get (1\/3, 2\/3). Now remove the middle... Continue reading... Repunit Fun Posted on June 26, 2019. by. Samuel Nunoo A repunit is a whole number consisting of only 1’s, such as 1, 111, or 111111111. (It’s like a repeating unit.) These numbers have... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights 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{"id":"IDEmtp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/calculus\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":43,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","calculus"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["calculus – Math Fun Facts"],"last_modified":"Wed Jun 08 17:23:13 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Thu Jul 11 00:00:00 UTC 2019","Sat Jun 29 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":4706832964531940469,"exact_signature_unique_b":true,"fuzzy_signature_l":4714532641511046594,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["calculus"],"h2_txt":["About Math Fun Facts","Does Order of Addition Matter?","Irrationality by Infinite Descent","Dedekind Cuts of Rational Numbers","Rationals Dense but Sparse","Devil’s Staircase","Space-filling Curves","Rational Irrational Power","Multidimensional Newton’s Method","Continuous but Nowhere Differentiable"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Marching.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Marching 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":696,"text_t":"calculus – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! calculus. About Math Fun Facts Posted on July 11, 2019. by. Francis Su I developed Math Fun Facts in 1994 as a warm-up activity for the calculus courses I taught as a graduate... Continue reading... Does Order of Addition Matter? Posted on June 29, 2019. by. Samuel Nunoo From your earliest days of math you learned that the order in which you add two numbers doesn’t matter: 3+5... Continue reading... Irrationality by Infinite Descent Posted on June 29, 2019. by. Samuel Nunoo The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the... Continue reading... Dedekind Cuts of Rational Numbers Posted on June 29, 2019. by. Samuel Nunoo Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the... Continue reading... Rationals Dense but Sparse Posted on June 29, 2019. by. Samuel Nunoo Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the... Continue reading... Devil’s Staircase Posted on June 29, 2019. by. Samuel Nunoo Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically... Continue reading... Space-filling Curves Posted on June 29, 2019. by. Samuel Nunoo Consider a square in the plane. Is it possible to draw a curve in the square that touches every point inside the... Continue reading... Rational Irrational Power Posted on June 29, 2019. by. Samuel Nunoo If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to... Continue reading... Multidimensional Newton’s Method Posted on June 29, 2019. by. Samuel Nunoo You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did... Continue reading... Continuous but Nowhere Differentiable Posted on June 29, 2019. by. Samuel Nunoo You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved. Loading Comments... Write a Comment... Email (Required). Name (Required). 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{"id":"IBcn2W51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/ON-CHOOSING-AND-BOUNDING-PROBABILITY.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":100,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"ON-CHOOSING-AND-BOUNDING-PROBABILITY","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["ON-CHOOSING-AND-BOUNDING-PROBABILITY.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:25 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-5394378780356018964,"exact_signature_unique_b":true,"fuzzy_signature_l":6748020896415938994,"fuzzy_signature_unique_b":true,"wordcount_i":6949,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:23:15 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:40 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":264483,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088139177820160,"text_t":"ON CHOOSING AND BOUNDING PROBABILITY\nMETRICS\nALISON L. GIBBS AND FRANCIS EDWARD SU\nManuscript version January 2002\nAbstract. When studying convergence of measures, an important is-\nsue is the choice of probability metric. We provide a summary and some\nnew results concerning bounds among some important probability met-\nrics\/distances that are used by statisticians and probabilists. Knowledge\nof other metrics can provide a means of deriving bounds for another one\nin an applied problem. Considering other metrics can also provide alter-\nnate insights. We also give examples that show that rates of convergence\ncan strongly depend on the metric chosen. Careful consideration is nec-\nessary when choosing a metric.\nAbrégé. Le choix de métrique de probabilité est une décision très\nimportante lorsqu’on étudie la convergence des mesures. Nous vous\nfournissons avec un sommaire de plusieurs métriques\/distances de prob-\nabilité couramment utilisées par des statisticiens(nes) at par des proba-\nbilistes, ainsi que certains nouveaux résultats qui se rapportent à leurs\nbornes. Avoir connaissance d’autres métriques peut vous fournir avec un\nmoyen de dériver des bornes pour une autre métrique dans un problème\nappliqué. Le fait de prendre en considération plusieurs métriques vous\npermettra d’approcher des problèmes d’une manière différente. Ainsi,\nnous vous démontrons que les taux de convergence peuvent dépendre de\nfaçon importante sur votre choix de métrique. Il est donc important de\ntout considérer lorsqu’on doit choisir une métrique.\n1. Introduction\nDetermining whether a sequence of probability measures converges is a\ncommon task for a statistician or probabilist. In many applications it is\nimportant also to quantify that convergence in terms of some probability\nmetric; hard numbers can then be interpreted by the metric’s meaning,\nand one can proceed to ask qualitative questions about the nature of that\nconvergence.\nKey words and phrases. discrepancy, Hellinger distance, probability metrics, Prokhorov\nmetric, relative entropy, rates of convergence, Wasserstein distance.\nFirst author supported in part by an NSERC postdoctoral fellowship. Second author\nacknowledges the hospitality of the Cornell School of Operations Research during a sab-\nbatical in which this was completed. The authors thank Jeff Rosenthal and Persi Diaconis\nfor their encouragement of this project and Neal Madras for helpful discussions.\n1\n2 ALISON L. GIBBS AND FRANCIS EDWARD SU\nAbbreviation Metric\nD Discrepancy\nH Hellinger distance\nI Relative entropy (or Kullback-Leibler divergence)\nK Kolmogorov (or Uniform) metric\nL Lévy metric\nP Prokhorov metric\nS Separation distance\nTV Total variation distance\nW Wasserstein (or Kantorovich) metric\nχ2 χ2 distance\nTable 1. Abbreviations for metrics used in Figure 1.\nThere are a host of metrics available to quantify the distance between\nprobability measures; some are not even metrics in the strict sense of the\nword, but are simply notions of “distance” that have proven useful to con-\nsider. How does one choose among all these metrics? Issues that can affect\na metric’s desirability include whether it has an interpretation applicable to\nthe problem at hand, important theoretical properties, or useful bounding\ntechniques.\nMoreover, even after a metric is chosen, it can still be useful to familiarize\noneself with other metrics, especially if one also considers the relationships\namong them. One reason is that bounding techniques for one metric can be\nexploited to yield bounds for the desired metric. Alternatively, analysis of a\nproblem using several different metrics can provide complementary insights.\nThe purpose of this paper is to review some of the most important met-\nrics on probability measures and the relationships among them. This project\narose out of the authors’ frustrations in discovering that, while encyclopedic\naccounts of probability metrics are available (e.g., Rachev (1991)), rela-\ntionships among such metrics are mostly scattered through the literature\nor unavailable. Hence in this review we collect in one place descriptions\nof, and bounds between, ten important probability metrics. We focus on\nthese metrics because they are either well-known, commonly used, or admit\npractical bounding techniques. We limit ourselves to metrics between prob-\nability measures (simple metrics) rather than the broader context of metrics\nbetween random variables (compound metrics).\nWe summarize these relationships in a handy reference diagram (Figure\n1), and provide new bounds between several metrics. We also give examples\nto illustrate that, depending on the choice of metric, rates of convergence\ncan differ both quantitatively and qualitatively.\nThis paper is organized as follows. Section 2 reviews properties of our\nten chosen metrics. Section 3 contains references or proofs of the bounds in\nFigure 1. Some examples of their applications are described in Section 4. In\nON CHOOSING AND BOUNDING PROBABILITY METRICS 3\nS\n@\n@\n@\n@\n@\n@I x\nTV\n-x\n\u001b√\n2x\nH\nW\non\nmetric\nspaces\n@\n@\n@\n@\n@\n@R\nx\/dmin @\n@\n@\n@\n@\n@I diamΩ · x\n-\n√\nx\n\u001b\n(diamΩ + 1)x\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0012√\nx\/2\nν dom µ\n@\n@\n@\n@\n@\n@I\n√\nx6√\nx\/2\nI -\nlog(1 + x)\nχ2\nnon-metric\ndistances\n6√\nx\nν dom µ\n6\nx\nP\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0012x\nD\n?\n2x\n6x\nK L on IR\n6x\n\u001b\nx\n-(1 + sup |G\n′|)x\n-x+ φ(x)\nFigure 1. Relationships among probability metrics. A di-\nrected arrow from A toB annotated by a function h(x) means\nthat dA ≤ h(dB). The symbol diam Ω denotes the diameter\nof the probability space Ω; bounds involving it are only use-\nful if Ω is bounded. For Ω finite, dmin = infx,y∈Ω d(x, y).\nThe probability metrics take arguments µ, ν; “ν dom µ” in-\ndicates that the given bound only holds if ν dominates µ.\nOther notation and restrictions on applicability are discussed\nin Section 3.\n4 ALISON L. GIBBS AND FRANCIS EDWARD SU\nSection 5 we give examples to show that the choice of metric can strongly\naffect both the rate and nature of the convergence.\n2. Ten metrics on probability measures\nThroughout this paper, let Ω denote a measurable space with σ-algebra\nB. Let M be the space of all probability measures on (Ω,B). We consider\nconvergence in M under various notions of distance, whose definitions are\nreviewed in this section. Some of these are not strictly metrics, but are non-\nnegative notions of “distance” between probability distributions on Ω that\nhave proven useful in practice. These distances are reviewed in the order\ngiven by Table 1.\nIn what follows, let µ, ν denote two probability measures on Ω. Let f\nand g denote their corresponding density functions with respect to a σ-\nfinite dominating measure λ (for example, λ can be taken to be (µ+ ν)\/2).\nIf Ω = IR, let F , G denote their corresponding distribution functions. When\nneeded, X,Y will denote random variables on Ω such that L(X) = µ and\nL(Y ) = ν. If Ω is a metric space, it will be understood to be a measurable\nspace with the Borel σ-algebra. If Ω is a bounded metric space with metric\nd, let diam(Ω) = sup{d(x, y) : x, y ∈ Ω} denote the diameter of Ω.\nDiscrepancy metric.\n1. State space: Ω any metric space.\n2. Definition:\ndD(µ, ν) := sup\nall closed balls B\n|µ(B)− ν(B)|.\nIt assumes values in [0, 1].\n3. The discrepancy metric recognizes the metric topology of the underly-\ning space Ω. However, the discrepancy is scale-invariant: multiplying\nthe metric of Ω by a positive constant does not affect the discrepancy.\n4. The discrepancy metric admits Fourier bounds, which makes it useful\nto study convergence of random walks on groups (Diaconis 1988, p. 34).\nHellinger distance.\n1. State space: Ω any measurable space.\n2. Definition: if f , g are densities of the measures µ, ν with respect to a\ndominating measure λ,\ndH(µ, ν) :=\n[∫\nΩ\n(\n√\nf −√g)2 dλ\n]1\/2\n=\n[\n2\n(\n1−\n∫\nΩ\n√\nfg dλ\n)]1\/2\n.\nThis definition is independent of the choice of dominating measure λ.\nFor a countable state space Ω,\ndH(µ, ν) :=\n[∑\nω∈Ω\n(√\nµ(ω)−\n√\nν(ω)\n)2]1\/2\n(Diaconis and Zabell 1982).\nON CHOOSING AND BOUNDING PROBABILITY METRICS 5\n3. It assumes values in [0,\n√\n2]. Some texts, e.g., LeCam (1986), intro-\nduce a factor of a square root of two in the definition of the Hellinger\ndistance to normalize its range of possible values to [0, 1]. We follow\nZolotarev (1983). Other sources, e.g., Borovkov (1998), Diaconis and\nZabell (1982), define the Hellinger distance to be the square of dH .\n(While dH is a metric, d\n2\nH is not.) An important property is that for\nproduct measures µ = µ1×µ2, ν = ν1×ν2 on a product space Ω1×Ω2,\n1− 1\n2\nd2H(µ, ν) =\n(\n1− 1\n2\nd2H(µ1, ν1)\n)(\n1− 1\n2\nd2H(µ2, ν2)\n)\n(Zolotarev 1983, p. 279). Thus one can express the distance between\ndistributions of vectors with independent components in terms of the\ncomponent-wise distances. A consequence (Reiss 1989, p. 100) of the\nabove formula is d2H(µ, ν) ≤ d2H(µ1, ν1) + d2H(µ2, ν2).\n4. The quantity (1 − 1\n2\nd2H) is called the Hellinger affinity. Apparently\nHellinger (1907) used a similar quantity in operator theory, but Kaku-\ntani (1948, p. 216) appears responsible for popularizing the Hellinger\naffinity and the form dH in his investigation of infinite products of mea-\nsures. Le Cam and Yang (1990) and Liese and Vajda (1987) contain\nfurther historical references.\nRelative entropy (or Kullback-Leibler divergence).\n1. State space: Ω any measurable space.\n2. Definition: if f , g are densities of the measures µ, ν with respect to a\ndominating measure λ,\ndI(µ, ν) :=\n∫\nS(µ)\nf log(f\/g) dλ,\nwhere S(µ) is the support of µ on Ω. The definition is independent of\nthe choice of dominating measure λ. For Ω a countable space,\ndI(µ, ν) :=\n∑\nω∈Ω\nµ(ω) log\nµ(ω)\nν(ω)\n.\nThe usual convention, based on continuity arguments, is to take 0 log 0\nq\n=\n0 for all real q and p log\np\n0\n=∞ for all real non-zero p. Hence the rela-\ntive entropy assumes values in [0,∞].\n3. Relative entropy is not a metric, since it is not symmetric and does not\nsatisfy the triangle inequality. However, it has many useful properties,\nincluding additivity over marginals of product measures: if µ = µ1×µ2,\nν = ν1 × ν2 on a product space Ω1 × Ω2,\ndI(µ, ν) = dI(µ1, ν1) + dI(µ2, ν2)\n(Cover and Thomas 1991, Reiss 1989, p. 100).\n4. Relative entropy was first defined by Kullback and Leibler (1951) as\na generalization of the entropy notion of Shannon (1948). A standard\nreference on its properties is Cover and Thomas (1991).\n6 ALISON L. GIBBS AND FRANCIS EDWARD SU\nKolmogorov (or Uniform) metric.\n1. State space: Ω = IR.\n2. Definition:\ndK(F,G) := sup\nx\n|F (x)−G(x)|, x ∈ IR.\n(Since µ, ν are measures on IR, it is customary to express the Kol-\nmogorov metric as a distance between their distribution functions F,G.)\n3. It assumes values in [0, 1], and is invariant under all increasing one-to-\none transformations of the line.\n4. This metric, due to Kolmogorov (1933), is also called the uniform met-\nric (Zolotarev 1983).\nLévy metric.\n1. State space: Ω = IR.\n2. Definition:\ndL(F,G) := inf{\u000f > 0 : G(x− \u000f)− \u000f ≤ F (x) ≤ G(x+ \u000f) + \u000f,∀x ∈ IR}.\n(Since µ, ν are measures on IR, it is customary to express the Lévy\nmetric as a distance between their distribution functions F,G.)\n3. It assumes values in [0, 1]. While not easy to compute, the Lévy metric\ndoes metrize weak convergence of measures on IR (Lukacs 1975, p. 71).\nIt is shift invariant, but not scale invariant.\n4. This metric was introduced by Lévy (1925, p. 199-200).\nProkhorov (or Lévy-Prokhorov) metric.\n1. State space: Ω any metric space.\n2. Definition:\ndP (µ, ν) := inf{\u000f > 0 : µ(B) ≤ ν(B\u000f) + \u000f for all Borel sets B}\nwhere B\u000f = {x : infy∈B d(x, y) ≤ \u000f}. It assumes values in [0, 1].\n3. It is possible to show that this metric is symmetric in µ, ν. See (Huber\n1981, p. 27).\n4. This metric was defined by Prokhorov (1956) as the analogue of the\nLévy metric for more general spaces. While not easy to compute, this\nmetric is theoretically important because it metrizes weak convergence\non any separable metric space (Huber 1981, p. 28). Moreover, dP (µ, ν)\nis precisely the minimum distance “in probability” between random\nvariables distributed according to µ, ν. This was shown by Strassen\n(1965) for complete separable metric spaces and extended by Dudley\n(1968) to arbitrary separable metric spaces.\nSeparation distance.\n1. State space: Ω a countable space.\n2. Definition:\ndS(µ, ν) := max\ni\n(\n1− µ(i)\nν(i)\n)\n.\nON CHOOSING AND BOUNDING PROBABILITY METRICS 7\n3. It assumes values in [0, 1]. However, it not a metric.\n4. The separation distance was advocated by Aldous and Diaconis (1987)\nto study Markov chains because it admits a useful characterization in\nterms of strong uniform times.\nTotal variation distance.\n1. State space: Ω any measurable space.\n2. Definition:\ndTV (µ, ν) := sup\nA⊂Ω\n| µ(A)− ν(A) |(1)\n=\n1\n2\nmax\n|h|≤1\n∣∣∣∣\n∫\nh dµ−\n∫\nh dν\n∣∣∣∣(2)\nwhere h : Ω → IR satisfies |h(x)| ≤ 1. This metric assumes values in\n[0, 1].\nFor a countable state space Ω, the definition above becomes\ndTV :=\n1\n2\n∑\nx∈Ω\n|µ(x)− ν(x) |\nwhich is half the L1-norm between the two measures. Some authors\n(for example, Tierney (1996)) define total variation distance as twice\nthis definition.\n3. Total variation distance has a coupling characterization:\ndTV (µ, ν) = inf{Pr(X 6= Y ) : r.v. X, Y s.t. L(X) = µ, L(Y ) = ν}\n(Lindvall 1992, p. 19).\nWasserstein (or Kantorovich) metric.\n1. State space: IR or any metric space.\n2. Definition: For Ω = IR, if F,G are the distribution functions of µ, ν\nrespectively, the Kantorovich metric is defined by\ndW (µ, ν) :=\n∫ ∞\n−∞\n|F (x)−G(x)| dx\n=\n∫ 1\n0\n|F−1(t)−G−1(t)| dt.\nHere F−1, G−1 are the inverse functions of the distribution functions\nF,G. For any separable metric space, this is equivalent to\ndW (µ, ν) := sup\n{∣∣∣∣\n∫\nh dµ−\n∫\nh dν\n∣∣∣∣ : ‖h‖L ≤ 1\n}\n,(3)\nthe supremum being taken over all h satisfying the Lipschitz condition\n|h(x)− h(y)| ≤ d(x, y), where d is the metric on Ω.\n8 ALISON L. GIBBS AND FRANCIS EDWARD SU\n3. The Wasserstein metric assumes values in [0,diam(Ω)], where diam(Ω)\nis the diameter of the metric space (Ω, d). This metric metrizes weak\nconvergence on spaces of bounded diameter, as is evident from Theo-\nrem 2 below.\n4. By the Kantorovich-Rubinstein theorem, the Kantorovich metric is\nequal to the Wasserstein metric:\ndW (µ, ν) = inf\nJ\n{E[d(X,Y )] : L(X) = µ,L(Y ) = ν},\nwhere the infimum is taken over all joint distributions J with marginals\nµ, ν. See Szulga (1982, Theorem 2). Dudley (1989, p. 342) traces some\nof the history of these metrics.\nχ2-distance.\n1. State space: Ω any measurable space.\n2. Definition: if f , g are densities of the measures µ, ν with respect to a\ndominating measure λ, and S(µ), S(ν) are their supports on Ω,\ndχ2(µ, ν) :=\n∫\nS(µ)∪S(ν)\n(f − g)2\ng\ndλ.\nThis definition is independent of the choice of dominating measure λ.\nThis metric assumes values in [0,∞]. For a countable space Ω this\nreduces to:\ndχ2(µ, ν) :=\n∑\nω∈S(µ)∪S(ν)\n(µ(ω)− ν(ω))2\nν(ω)\n.\nThis distance is not symmetric in µ and ν; beware that the order of\nthe arguments varies from author to author. We follow Csiszar (1967)\nand Liese and Vajda (1987) because of the remarks following equation\n(4) below and the natural order of arguments suggested by inequalities\n(10) and (11). Reiss (1989, p. 98) takes the opposite convention as well\nas defining the χ2-distance as the square root of the above expression.\n3. The χ2-distance is not symmetric, and therefore not a metric. How-\never, like the Hellinger distance and relative entropy, the χ2-distance\nbetween product measures can be bounded in terms of the distances\nbetween their marginals. See Reiss (1989, p. 100).\n4. The χ2-distance has origins in mathematical statistics dating back to\nPearson. See Liese and Vajda (1987, p. 51) for some history.\nWe remark that several distance notions in this section are instances of a\nfamily of distances known as f-divergences (Csiszar 1967). For any convex\nfunction f , one may define\ndf (µ, ν) =\n∑\nω\nν(ω)f(\nµ(ω)\nν(ω)\n).(4)\nON CHOOSING AND BOUNDING PROBABILITY METRICS 9\nThen choosing f(x) = (x − 1)2 yields dχ2 , f(x) = x log x yields dI , f(x) =\n|x − 1|\/2 yields dTV , and f(x) = (\n√\nx − 1)2 yields d2H . The family of f -\ndivergences are studied in detail in Liese and Vajda (1987).\n3. Some Relationships Among Probability Metrics\nIn this section we describe in detail the relationships illustrated in Fig-\nure 1. We give references for relationships known to appear elsewhere, and\nprove several new bounds which we state as theorems. In choosing the order\nof presentation, we loosely follow the diagram from bottom to top.\nAt the end of this section, we summarize in Theorem 6 what is known\nabout how these metrics relate to weak convergence of measures.\nThe Kolmogorov and Lévy metrics on IR. For probability measures\nµ, ν on IR with distribution functions F,G,\ndL(F,G) ≤ dK(F,G).\nSee Huber (1981, p. 34). Petrov (1995, p. 43) notes that if G(x) (i.e., ν) is\nabsolutely continuous (with respect to Lebesgue measure), then\ndK(F,G) ≤\n(\n1 + sup\nx\n|G′(x)|\n)\ndL(F,G).\nThe Discrepancy and Kolmogorov metrics on IR. It is evident that\nfor probability measures on IR,\ndK ≤ dD ≤ 2 dK .(5)\nThis follows from the regularity of Borel sets in IR and expressing closed\nintervals in IR as difference of rays.\nThe Prokhorov and Lévy metrics on IR. For probability measures on\nIR,\ndL ≤ dP .\nSee Huber (1981, p. 34).\nThe Prokhorov and Discrepancy metrics. The following theorem shows\nhow discrepancy may be bounded by the Prokhorov metric by finding a suit-\nable right-continuous function φ. For bounded Ω, φ(\u000f) gives an upper bound\non the additional ν-measure of the extended ball B\u000f over the ball B, where\nB\u000f = {x : infy∈B d(x, y) ≤ \u000f}. Note that this theorem also gives an upper\nbound for dK through (5) above.\nTheorem 1. Let Ω be any metric space, and let ν be any probability measure\nsatisfying\nν(B\u000f) ≤ ν(B) + φ(\u000f)\nfor all balls B and complements of balls B and some right-continuous func-\ntion φ. Then for any other probability measure µ, if dP (µ, ν) = x, then\ndD(µ, ν) ≤ x+ φ(x).\n10 ALISON L. GIBBS AND FRANCIS EDWARD SU\nAs an example, on the circle or line, if ν = U is the uniform distribution,\nthen φ(x) = 2x and hence\ndD(µ,U) ≤ 3 dP (µ,U).\nProof. For µ, ν as above,\nµ(B)− ν(Bx) ≥ µ(B)− ν(B)− φ(x).\nAnd if dP (µ, ν) = x, then µ(B)− ν(Bx̃) ≤ x̃ for all x̃ > x and all Borel sets\nB. Combining with the above inequality, we see that\nµ(B)− ν(B)− φ(x̃) ≤ x̃.\nBy taking the supremum over B which are balls or complements of balls,\nobtain\nsup\nB\n(µ(B)− ν(B)) ≤ x̃+ φ(x̃).\nThe same result may be obtained for ν(B) − µ(B) by noting that ν(B) −\nµ(B) = µ(Bc)− ν(Bc) which, after taking the supremum over B which are\nballs or complements of balls, obtain\nsup\nB\n(ν(B)− µ(B)) = sup\nBc\n(µ(Bc)− ν(Bc)) ≤ x̃+ φ(x̃)\nas before. Since the supremum over balls and complements of balls will\nbe larger than the supremum over balls, if dP (µ, ν) = x, then dD(µ, ν) ≤\nx̃+φ(x̃) for all x̃ > x. For right-continuous φ, the theorem follows by taking\nthe limit as x̃ decreases to x.\nThe Prokhorov and Wasserstein metrics. Huber (1981, p. 33) shows\nthat\n(dP )\n2 ≤ dW ≤ 2 dP\nfor probability measures on a complete separable metric space whose metric\nd is bounded by 1. More generally, we show the following:\nTheorem 2. The Wasserstein and Prokhorov metrics satisfy\n(dP )\n2 ≤ dW ≤ (diam(Ω) + 1) dP\nwhere diam(Ω) = sup{d(x, y) : x, y ∈ Ω}.\nProof. For any joint distribution J on random variables X,Y ,\nEJ [d(X,Y )] ≤ \u000f · Pr(d(X,Y ) ≤ \u000f) + diam(Ω) · Pr(d(X,Y ) > \u000f)\n= \u000f+ (diam(Ω)− \u000f) · Pr(d(X,Y ) > \u000f)\nIf dP (µ, ν) ≤ \u000f, we can choose a coupling so that Pr(d(X,Y ) > \u000f) is bounded\nby \u000f (Huber 1981, p. 27). Thus\nEJ [d(X,Y )] ≤ \u000f+ (diam(Ω)− \u000f)\u000f ≤ (diam(Ω) + 1)\u000f.\nTaking the infimum of both sides over all couplings, we obtain\ndW ≤ (diam(Ω) + 1)dP .\nON CHOOSING AND BOUNDING PROBABILITY METRICS 11\nTo bound Prokhorov by Wasserstein, use Markov’s inequality and choose\n\u000f such that dW (µ, ν) = \u000f\n2. Then\nPr(d(X,Y ) > \u000f) ≤ 1\n\u000f\nEJ [d(X,Y )] ≤ \u000f\nwhere J is any joint distribution on X,Y . By Strassen’s theorem (Huber\n1981, Theorem 3.7), Pr(d(X,Y ) > \u000f) ≤ \u000f is equivalent to µ(B) ≤ ν(B\u000f) + \u000f\nfor all Borel sets B, giving d2P ≤ dW .\nNo such upper bound on dW holds if Ω is not bounded. Dudley (1989,\np. 330) cites the following example on IR. Let δx denote the delta measure\nat x. The measures Pn := ((n− 1)δ0 + δn)\/n converge to P := δ0 under the\nProkhorov metric, but dW (Pn, P ) = 1 for all n. Thus Wasserstein metric\nmetrizes weak convergence only on state spaces of bounded diameter.\nThe Wasserstein and Discrepancy metrics. The following bound can\nbe recovered using the bounds through total variation (and is therefore not\nincluded on Figure 1), but we include this direct proof for completeness.\nTheorem 3. If Ω is finite,\ndmin · dD ≤ dW\nwhere dmin = minx 6=y d(x, y) over distinct pairs of points in Ω.\nProof. In the equivalent form of the Wasserstein metric, Equation (3), take\nh(x) =\n{\ndmin for x in B\n0 otherwise\nfor B any closed ball. h(x) satisfies the Lipschitz condition. Then\ndmin · |µ(B)− ν(B)| =\n∣∣∣∣\n∫\nΩ\nh dµ−\n∫\nΩ\nh dν\n∣∣∣∣\n≤ dW (µ, ν)\nand takingB to be the ball that maximizes |µ(B)−ν(B)| gives the result.\nOn continuous spaces, it is possible for dW to converge to 0 while dD\nremains at 1. For example, take delta measures δ\u000f converging on δ0.\nThe Total Variation and Discrepancy metrics. It is clear that\ndD ≤ dTV(6)\nsince total variation is the supremum over a larger class of sets than discrep-\nancy.\nNo expression of the reverse type can hold since the total variation dis-\ntance between a discrete and continuous distribution is 1 while the discrep-\nancy may be very small. Further examples are discussed in Section 5.\nThe Total Variation and Prokhorov metrics. Huber (1981, p. 34)\nproves the following bound for probabilities on metric spaces:\ndP ≤ dTV .\n12 ALISON L. GIBBS AND FRANCIS EDWARD SU\nThe Wasserstein and Total Variation metrics.\nTheorem 4. The Wasserstein metric and the total variation distance sat-\nisfy the following relation:\ndW ≤ diam(Ω) · dTV\nwhere diam(Ω) = sup{d(x, y) : x, y ∈ Ω}. If Ω is a finite set, there is a\nbound the other way. If dmin = minx6=y d(x, y) over distinct pairs of points\nin Ω, then\ndmin · dTV ≤ dW .(7)\nNote that on an infinite set no such relation of the second type can occur\nbecause dW may converge to 0 while dTV remains fixed at 1. (minx 6=y d(x, y)\ncould be 0 on an infinite set.)\nProof. The first inequality follows from the coupling characterizations of\nWasserstein and total variation by taking the infimum of the expected value\nover all possible joint distributions of both sides of:\nd(X,Y ) ≤ 1X 6=Y · diam(Ω).\nThe reverse inequality follows similarly from:\nd(X,Y ) ≥ 1X 6=Y ·min\na 6=b\nd(a, b).\nThe Hellinger and Total Variation metrics.\n(dH)\n2\n2\n≤ dTV ≤ dH .(8)\nSee LeCam (1969, p. 35).\nThe Separation distance and Total Variation. It is easy to show (see,\ne.g., Aldous and Diaconis (1987, p. 71)) that\ndTV ≤ dS .(9)\nAs Aldous and Diaconis note, there is no general reverse inequality, since\nif µ is uniform on {1, 2, ..., n} and ν is uniform on {1, 2, ..., n − 1} then\ndTV (µ, ν) = 1\/n but dS(µ, ν) = 1.\nRelative Entropy and Total Variation. For countable state spaces Ω,\n2 (dTV )\n2 ≤ dI .\nThis inequality is due to Kullback (1967). Some small refinements are pos-\nsible where the left side of the inequality is replaced with a polynomial in\ndTV with more terms; see Mathai and Rathie (1975, p. 110-112).\nRelative Entropy and the Hellinger distance.\n(dH)\n2 ≤ dI .\nSee Reiss (1989, p. 99).\nON CHOOSING AND BOUNDING PROBABILITY METRICS 13\nThe χ2-distance and Hellinger distance.\ndH(µ, ν) ≤\n√\n2( dχ2(µ, ν))\n1\/4.\nSee Reiss (1989, p. 99), who also shows that if the measure µ is dominated\nby ν, then the above inequality can be strengthened:\ndH(µ, ν) ≤ (dχ2(µ, ν))1\/2.\nThe χ2-distance and Total Variation. For a countable state space Ω,\ndTV (µ, ν) =\n1\n2\n∑\nω∈Ω\n|µ(ω)− ν(ω)|√\nν(ω)\n√\nν(ω) ≤ 1\n2\n√\ndχ2(µ, ν)\nwhere the inequality follows from Cauchy-Schwarz. On a continuous state\nspace, if µ is dominated by ν the same relationship holds; see Reiss (1989,\np. 99).\nThe χ2-distance and Relative Entropy.\nTheorem 5. The relative entropy dI and the χ\n2-distance dχ2 satisfy\ndI(µ, ν) ≤ log\n[\n1 + dχ2(µ, ν)\n]\n.(10)\nIn particular, dI(µ, ν) ≤ dχ2(µ, ν).\nProof. Since log is a concave function, Jensen’s inequality yields\ndI(µ, ν) ≤ log\n(∫\nΩ\n(f\/g) f dλ\n)\n≤ log\n(\n1 + dχ2(µ, ν)\n)\n≤ dχ2(µ, ν),\nwhere the second inequality is obtained by noting that\n∫\nΩ\n(f − g)2\ng\ndλ =\n∫\nΩ\n(\nf2\ng\n− 2f + g\n)\ndλ =\n∫\nΩ\nf2\ng\ndλ− 1.\nDiaconis and Saloff-Coste (1996, p. 710) derive the following alternate upper\nbound for the relative entropy in terms of both the χ2 and total variation\ndistances.\ndI(µ, ν) ≤ dTV (µ, ν) +\n1\n2\ndχ2(µ, ν).(11)\nWeak convergence. In addition to using Figure 1 to recall specific bounds,\nour diagram there can also be used to discern relationships between topolo-\ngies on the space of measures. For instance, we can see from the mutual\narrows between the total variation and Hellinger metrics that they generate\nequivalent topologies. Other mutual arrows on the diagram indicate similar\nrelationships, subject to the restrictions given on those bounds.\nMoreover, since we know that the Prokhorov and Lévy metrics both\nmetrize weak convergence, we can also tell which other metrics metrize weak\nconvergence on which spaces, which we summarize in the following theorem:\n14 ALISON L. GIBBS AND FRANCIS EDWARD SU\nTheorem 6. For measures on IR, the Lévy metric metrizes weak conver-\ngence. Convergence under the discrepancy and Kolmogorov metrics imply\nweak convergence (via the Lévy metric). Furthermore, these metrics metrize\nweak convergence µn → ν if the limiting metric ν is absolutely continuous\nwith respect to Lebesgue measure on IR.\nFor measures on a measurable space Ω, the Prokhorov metric metrizes\nweak convergence. Convergence under the Wasserstein metric implies weak\nconvergence.\nFurthermore, if Ω is bounded, the Wasserstein metric metrizes weak con-\nvergence (via the Prokhorov metric), and convergence under any of the fol-\nlowing metrics implies weak convergence: total variation, Hellinger, separa-\ntion, relative entropy, and the χ2-distance.\nIf Ω is both bounded and finite, the total variation and Hellinger metrics\nboth metrize weak convergence.\nThis follows from chasing the diagram in Figure 1, noting the existence\nof mutual bounds of the Lévy and Prokhorov metrics with other metrics\n(using the results surveyed in this section) and reviewing conditions under\nwhich they apply.\n4. Some Applications of Metrics and Metric Relationships\nWe describe some of the applications of these metrics in order to give\nthe reader a feel for how they have been used, and describe how some au-\nthors have exploited metric relationships to obtain bounds for one metric\nvia another.\nThe notion of weak convergence of measures is an important concept in\nboth statistics and probability. For instance, when considering a statistic T\nthat is a functional of an empirical distribution F , the “robustness” of the\nstatistic under small deviations of F corresponds to the continuity of T with\nrespect to the weak topology on the space of measures. See Huber (1981).\nThe Lévy and Prokhorov metrics (and Wasserstein metric on a bounded\nstate space) provide quantitative ways of metrizing this topology.\nHowever, other distances that do not metrize this topology can still be\nuseful for other reasons. The total variation distance is one of the most com-\nmonly used probability metrics, because it admits natural interpretations as\nwell as useful bounding techniques. For instance, in (1), if A is any event,\nthen total variation can be interpreted as an upper bound on the difference\nof probabilities that the event occurs under two measures. In Bayesian sta-\ntistics, the error in an expected loss function due to the approximation of\none measure by another is given (for bounded loss functions) by the total\nvariation distance through its representation in equation (2).\nIn extending theorems on the ergodic behavior of Markov chains on dis-\ncrete state spaces to general measurable spaces, the total variation norm is\nON CHOOSING AND BOUNDING PROBABILITY METRICS 15\nused in a number of results (Orey 1971, Nummelin 1984). More recently, to-\ntal variation has found applications in bounding rates of convergence of ran-\ndom walks on groups (e.g., Diaconis (1988), Rosenthal (1995)) and Markov\nchain Monte Carlo algorithms (e.g., Tierney (1994), Gilks, Richardson and\nSpiegelhalter (1996)). Much of the success in obtaining rates of convergence\nin these settings is a result of the coupling characterization of the total\nvariation distance, as well as Fourier bounds.\nGibbs (2000) considers a Markov chain Monte Carlo algorithm which\nconverges in total variation distance, but for which coupling bounds are\ndifficult to apply since the state space is continuous and one must wait for\nrandom variables to couple exactly. The Wasserstein metric has a coupling\ncharacterization that depends on the distance between two random variables,\nso one may instead consider only the time required for the random variables\nto couple to within \u000f, a fact exploited by Gibbs. For a related example with\na discrete state space, Gibbs uses the bound (7) to obtain total variation\nbounds.\nLike the total variation property (2), the Wasserstein metric also repre-\nsents the error in the expected value of a certain class of functions due to\nthe approximation of one measure by another, as in (3), which is of interest\nin applications in Bayesian statistics. The fact that the Wasserstein metric\nis a minimal distance of two random variables with fixed distributions has\nalso led to its use in the study of distributions with fixed marginals (e.g.,\nRüschendorf, Schweizer and Taylor (1996)).\nBecause the separation distance has a characterization in terms of strong\nuniform times (like the coupling relationship for total variation), conver-\ngence of a Markov chain under the separation distance may be studied by\nconstructing a strong uniform time for the chain and estimating the proba-\nbility in the tail of its distribution. See Aldous and Diaconis (1987) for such\nexamples; they also exploit inequality (9) to obtain upper bounds on the\ntotal variation.\nSimilarly, total variation lower bounds may be obtained via (6) and lower\nbounds on the discrepancy metric. A version of this metric is popular among\nnumber theorists to study uniform distribution of sequences (Kuipers and\nNiederreiter 1974); Diaconis (1988) suggested its use to study random walks\non groups. Su (1998) uses the discrepancy metric to bound the convergence\ntime of a random walk on the circle generated by a single irrational rotation.\nThis walk converges weakly, but not in total variation distance because its\nn-th step probability distribution is finitely supported but its limiting mea-\nsure is continuous (in fact, uniform). While the Prokhorov metric metrizes\nweak convergence, it is not easy to bound. On the other hand, for this\nwalk, discrepancy convergence implies weak convergence when the limiting\nmeasure is uniform; and Fourier techniques for discrepancy allow the calcu-\nlation of quantitative bounds. The discrepancy metric can be used similarly\nto study random walks on other homogeneous spaces, e.g., Su (2001).\n16 ALISON L. GIBBS AND FRANCIS EDWARD SU\nOther metrics are useful because of their special properties. For instance,\nthe Hellinger distance is convenient when working with convergence of prod-\nuct measures because it factors nicely in terms of the convergence of the com-\nponents. Reiss (1989) uses this fact and the relationships (8) between the\nHellinger and total variation distances to obtain total variation bounds. The\nHellinger distance is also used in the theory of asymptotic efficiency (e.g.,\nsee LeCam (1986)) and minimum Hellinger distance estimation (e.g., see\nLindsay (1994)). It is used throughout Ibragmiov and Has’minskii (1981) to\nquantify the rate of convergence of sequences of consistent estimators to their\nparameter. Kakutani (1948) gives a criterion (now known as the Kakutani\nalternative) using the Hellinger affinity to determine when infinite products\nof equivalent measures are equivalent; this has applications to stochastic\nprocesses and can be used to show the consistency of the likelihood-ratio\ntest. See Jacod and Shiryaev (1987), Williams (1991) for applications.\nDiaconis and Saloff-Coste (1996) use log-Sobolev techniques to bound the\nχ2 convergence of Markov chains to their limiting distributions, noting that\nthese also give total variation and entropy bounds. The χ2-distance bears\nits name because in the discrete case it is the well-known χ2 statistic used,\nfor example, in the classic goodness-of-fit test, e.g., see Borovkov (1998,\np. 184). Similarly, the Kolmogorov metric between a distribution function\nand its empirical estimate is used as the test statistic in the Kolmogorov-\nSmirnov goodness-of-fit test, e.g., see Lehmann (1994, p. 336).\nRelative entropy is widely used because it is a quantity that arises natu-\nrally, especially in information theory (Cover and Thomas 1991). Statistical\napplications include proving central limit theorems (Linnik 1959, Barron\n1986) and evaluating the loss when using a maximum likelihood versus a\nBayes density estimate (Hartigan 1998). In the testing of an empirical dis-\ntribution against an alternative, Borovkov (1998, p. 256) gives the relation-\nship of the asymptotic behaviour of the Type II error to the relative entropy\nbetween the empirical and alternative distributions. In Bayesian statistics,\nBernardo and Smith (1994, p. 75) suggest that relative entropy is the nat-\nural measure for the lack of fit of an approximation of a distribution when\npreferences are described by a logarithmic score function.\nUp to a constant, the asymptotic behaviour of relative entropy, the Hellinger,\nand the χ2-distance are identical when the ratio of the density functions is\nnear 1 (Borovkov 1998, p. 178). These three distances are used extensively\nin parametric families of distributions to quantify the distance between mea-\nsures from the same family indexed by different parameters. Borovkov (1998,\npp. 180-181) shows how these distances are related to the Fisher information\nin the limit as the difference in the parameters goes to zero.\n5. Rates of Convergence that Depend on the Metric\nWe now illustrate the ways in which the choice of metric can affect rates of\nconvergence in one context: the convergence of a random walk to its limiting\ndistribution. Such examples point to the need for practitioners to choose\nON CHOOSING AND BOUNDING PROBABILITY METRICS 17\na metric carefully when measuring convergence, paying attention to that\nmetric’s qualitative and quantitative features. We give several examples of\nrandom walks whose qualitative convergence behavior depend strongly on\nthe metric chosen, and suggest reasons for this phenomenon.\nAs a first basic fact, it is possible for convergence to occur in one metric\nbut not another. An elementary example is the convergence of a stan-\ndardized Binomial (n, p) random variable with distribution µn which con-\nverges to the standard normal distribution, ν, as n → ∞. For all n < ∞,\ndTV (µn, ν) = 1, while dD(µn, ν) → 0 as n → ∞. In the random walk con-\ntext, Su (1998) shows that a random walk on the circle generated by an\nirrational rotation converges in discrepancy, but not total variation. The\nlatter fact follows because the n-th step probability distribution is finitely\nsupported, and remains total variation distance 1 away from its continuous\nlimiting distribution.\nHowever, more interesting behavior can arise. Below, we cite a family\nof random walks on a product space, indexed by some parameter, which\nnot only converges under each of total variation, relative entropy, and the\nχ2-distance, but exhibits different rates of convergence as a function of the\nparameter.\nWe then cite another example of a family of walks that not only has\ndifferent convergence rates under two different metrics, but also exhibits\nqualitatively different convergence behavior. This family exhibits a cutoff\nphenomenon under the first metric but only exponential decay under the\nsecond.\nExample: convergence rates that depend on the metric. The follow-\ning family of random walks show that convergence rates in total variation,\nrelative entropy, and χ2-distance may differ. Recall that a family of random\nwalks, indexed by some parameter n, is said to converge in f(n) steps us-\ning some metric\/distance if that metric\/distance can be uniformly bounded\nfrom above after f(n) steps and is uniformly bounded away from zero before\nf(n) steps.\nLet G = Z mod g, a finite group with g elements. Then Gn is the set\nof all n-tuples of elements from G. Consider the following continuous-time\nrandom walk on Gn: start at (0, 0, ..., 0) and according to a Poisson process\nrunning at rate 1, pick a coordinate uniformly at random and replace that\ncoordinate by a uniformly chosen element from G. (Thus each coordinate is\nan independent Poisson process running at rate 1\nn\n.)\nSu (1995) proves that if g grows with n exponentially according to g = 2n,\nthen the random walk on Gn described above converges in 2n log n steps\nunder the relative entropy distance, and n2 log 2 steps under the χ2-distance.\nHowever, it converges in at most n log n steps under total variation.\nIn this example, the relative entropy and χ2-distance are calculated by\nusing properties of these metrics, while the total variation upper bound in\n18 ALISON L. GIBBS AND FRANCIS EDWARD SU\nthis example is, in fact, calculated via its metric relationship (9) with the\nseparation distance.\nWhy does the difference in rates of convergence occur? This is due to the\nfact that total variation distance, unlike relative entropy or the χ2-distance,\nis insensitive to big or small elementwise differences when the total of those\ndifferences remains the same. For instance, consider the follow measures\nµ, ν on Z10. Let\nµ(i) =\n\n\n0.6 if i = 0\n0.1 if i = 1, 2, 3, 4\n0 else\nν(i) =\n{\n0.2 if i = 0, 1, 2, 3, 4\n0 else\n.\nLet U be the uniform distribution on Z10. In this example dI(µ,U) ≈\n1.075 and dI(ν, U) = 0.693, but the total variation distances are the same:\ndTV (µ,U) = dTV (ν, U) = 0.5. This is true because we could redistribute\nmass among the points on which µ exceeded ν without affecting the total\nvariation distance.\nThus it is possible for an unbalanced measure (with large elementwise\ndifferences from uniform) to have the same total variation distance as a\nmore balanced measure. The random walk on the product space has a\nnatural feature which rewards total variation distance but hinders relative\nentropy— the randomization in each coordinate drops the variation distance\nquickly, but balancing the measure might take longer.\nExample: qualitatively different convergence behavior. Chung, Di-\naconis and Graham (1987) study a family of walks that exhibits a cutoff\nphenomenon under total variation; the total variation distance stays near 1\nbut drops off sharply and steeply to zero after a certain cutoff point. The\nfamily of random walks is on the integers mod p where p is an odd num-\nber, with a randomness multiplier: the process is given by X0 = 0 and\nXn = 2Xn−1 + \u000fn (mod p) where the \u000fi are i.i.d. taking values 0, ±1 each\nwith probability 1\/3. The stationary distribution for this process is uni-\nform. Using Fourier analysis, Chung et al. (1987) show for the family of\nwalks when p = 2t − 1 that O(log2 p log log2 p) steps are sufficient and nec-\nessary for total variation convergence, for t a positive integer. There is, in\nfact, a cutoff phenomenon.\nHowever, as proven in Su (1995, pp. 29-31), O(log p) steps are sufficient for\nconvergence in discrepancy. Moreover, the convergence is qualitatively dif-\nferent from total variation, because there is no cutoff in discrepancy. Rather,\nthere is only exponential decay.\nAgain, analysis under these two metrics sheds some light on what the\nwalk is doing. For p = 2t − 1, the “doubling” nature of the process keeps\nthe walk supported very near powers of 2. The discrepancy metric, which\naccounts for the topology of the space, falls very quickly because the walk\nhas spread itself out even though it is supported on a small set of values.\nHowever, the total variation does not “see” this spreading; it only falls when\nthe support of the walk is large enough.\nON CHOOSING AND BOUNDING PROBABILITY METRICS 19\nThus, the difference in rates of convergence sheds some light on the nature\nof the random walk as well as the metrics themselves.\nReferences\nAldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks, Adv.\nin Appl. Math. 8: 69–97.\nBarron, A. (1986). Entropy and central limit theorem, Ann. Probab. 14: 336–42.\nBernardo, J. M. and Smith, A. F. M. (1994). Bayesian Theory, John Wiley & Sons,\nChichester, England.\nBorovkov, A. A. (1998). Mathematical Statistics, Gordon and Breach Science Publishers,\nAmsterdam. Translated from the Russian by A. Moullagaliev and revised by the\nauthor.\nChung, F., Diaconis, P. and Graham, R. L. (1987). Random walks arising in random\nnumber generation, Ann. Probab. 15: 1148–1165.\nCover, T. M. and Thomas, J. A. (1991). Elements of Information Theory, John Wiley &\nSons Inc., New York. A Wiley-Interscience Publication.\nCsiszar, I. (1967). Information type measures of difference of probability distributions and\nindirect observations, Studia Sci. Math. Hungar. 2: 299–318.\nDiaconis, P. (1988). Group Representations in Probability and Statistics, Institute of Math-\nematical Statistics, Hayward, CA.\nDiaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov\nchains, Ann. Appl. Probab. 6(3): 695–750.\nDiaconis, P. and Zabell, S. L. (1982). Updating subjective probability, J. Amer. Statist.\nAssoc. 77: 822–830.\nDudley, R. M. (1968). Distances of probability measures and random variables, Ann. Math.\nStatist. 39: 1563–1572.\nDudley, R. M. (1989). Real Analysis and Probability, Wadsworth & Brooks\/Cole, Belmont,\nCA.\nGibbs, A. L. (2000). Convergence of Markov Chain Monte Carlo Algorithms with Appli-\ncations to Image Restoration, PhD thesis, University of Toronto.\nGilks, W. R., Richardson, S. and Spiegelhalter, D. J. (eds) (1996). Markov Chain Monte\nCarlo in Practice, Chapman and Hall, London.\nHartigan, J. A. (1998). The maximum likelihood prior, Ann. Statist. 26: 2083–103.\nHellinger, E. (1907). Die Orthogonalinvarianten quadratischer Formen von unendlich vie-\nlen Variablen, Dissertation, Göttingen.\nHuber, P. J. (1981). Robust Statistics, John Wiley & Sons, New York.\nIbragmiov, I. A. and Has’minskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory,\nSpringer-Verlag, New York.\nJacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes, Springer-\nVerlag, Berlin.\nKakutani, S. (1948). On equivalence of infinite product measures, Ann. of Math. (2)\n49: 214–224.\nKolmogorov, A. N. (1933). Sulla determinazione empirica di una legge di distribuzione,\nGiornale dell’Istituto Italiano degli Attuari 4: 83–91.\nKuipers, L. and Niederreiter, H. (1974). Uniform Distribution of Sequences, Wiley-\nInterscience [John Wiley & Sons], New York. Pure and Applied Mathematics.\nKullback, S. (1967). A lower bound for discrimination in terms of variation, IEEE Trans.\nInform. Theory 4: 126–127.\nKullback, S. and Leibler, R. A. (1951). On information and sufficiency, Ann. Math. Sta-\ntistics 22: 79–86.\nLe Cam, L. and Yang, G. L. (1990). Asymptotics in Statistics: Some Basic Concepts,\nSpringer-Verlag, New York.\n20 ALISON L. GIBBS AND FRANCIS EDWARD SU\nLeCam, L. M. (1969). Théorie Asymptotique de la Décision Statistique, Les Presses de\nl’Université de Montréal, Montréal.\nLeCam, L. M. (1986). Asymptotic Methods in Statistical Decision Theory, Springer-Verlag,\nNew York.\nLehmann, E. L. (1994). Testing Statistical Hypotheses, 2nd edn, Chapman & Hall, New\nYork.\nLévy, P. (1925). Calcul des Probabilitiés, Gauthiers-Villars, Paris.\nLiese, F. and Vajda, I. (1987). Convex Statistical Distances, B.G. Teubner Verlagsge-\nsellschaft, Leipzig.\nLindsay, B. G. (1994). Efficiency versus robustness: The case for minimum Hellinger\ndistance and related methods, Ann. Statist. 22: 1081–1114.\nLindvall, T. (1992). Lectures on the Coupling Method, John Wiley & Sons, New York.\nLinnik, Y. V. (1959). An information-theoretic proof of the central limit theorem with the\nLindeberg condition, Theory Probab. Appl. 4: 288–99.\nLukacs, E. (1975). Stochastic convergence, second edn, Academic Press [Harcourt Brace\nJovanovich Publishers], New York. Probability and Mathematical Statistics, Vol. 30.\nMathai, A. M. and Rathie, P. N. (1975). Basic Concepts in Information Theory, Wiley\nEastern, New Delhi.\nNummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators,\nCambridge University Press, Cambridge.\nOrey, S. (1971). Lecture Notes on Limit Theorems for Markov Chain Transition Proba-\nbilities, Van Nostrand Reinhold Co., London. Van Nostrand Reinhold Mathematical\nStudies, No. 34.\nPetrov, V. V. (1995). Limit theorems of probability theory, The Clarendon Press Oxford\nUniversity Press, New York. Sequences of independent random variables, Oxford\nScience Publications.\nProkhorov, Y. V. (1956). Convergence of random processes and limit theorems in proba-\nbility theory, Teor. Veroyatnost. i Primenen. 1: 177–238. English translation: Theory\nProbab. Appl. 1 (1956), 157-214.\nRachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models, John\nWiley & Sons, Chichester, New York.\nReiss, R.-D. (1989). Approximate Distributions of Order Statistics, Springer-Verlag, New\nYork.\nRosenthal, J. S. (1995). Convergence rates of Markov chains, SIAM Rev. 37: 387–405.\nRüschendorf, L., Schweizer, B. and Taylor, M. D. (eds) (1996). Distributions with Fixed\nMarginals and Related Topics, Vol. 28 of Lecture Notes – Monograph Series, Institute\nof Mathematical Statistics, Hayward, California.\nShannon, C. E. (1948). A mathematical theory of communication, Bell System Tech. J.\n27: 379–423, 623–656.\nStrassen, V. (1965). The existence of probability measures with given marginals, Ann.\nMath. Statist 36: 423–439.\nSu, F. E. (1995). Methods for Quantifying Rates of Convergence for Random Walks on\nGroups, PhD thesis, Harvard University.\nSu, F. E. (1998). Convergence of random walks on the circle generated by an irrational\nrotation, Trans. Amer. Math. Soc. 350: 3717–3741.\nSu, F. E. (2001). Discrepancy convergence for the drunkard’s walk on the sphere, Electron.\nJ. Probab. 6: no. 2, 20 pp. (electronic).\nSzulga, A. (1982). On minimal metrics in the space of random variables, Theory Probab.\nAppl. 27: 424–430.\nTierney, L. (1994). Markov chains for exploring posterior distributions, with discussion,\nAnn. Statist. 22: 1701–1762.\nON CHOOSING AND BOUNDING PROBABILITY METRICS 21\nTierney, L. (1996). Introduction to general state-space Markov chain theory, in W. R.\nGilks, S. Richardson and D. J. Spiegelhalter (eds), Markov Chain Monte Carlo in\nPractice, Chapman and Hall, London, pp. 59–74.\nWilliams, D. (1991). Probability with Martingales, Cambridge University Press, Cam-\nbridge.\nZolotarev, V. M. (1983). Probability metrics, Theory Probab. Appl. 28: 278–302.\nDepartment of Mathematics and Statistics, York University, 4700 Keele\nStreet, Toronto, Ontario, Canada, M3J 1P3\nE-mail address: agibbs@mathstat.yorku.ca\nDepartment of Mathematics, Harvey Mudd College, Claremont, CA, U.S.A.\n91711\nE-mail address: su@math.hmc.edu\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! number theory. Liouville Numbers Posted on June 29, 2019. by. Samuel Nunoo Are there any real numbers that are NOT algebraic, i.e., expressible as the root of a non-zero polynomial with integer coefficients? In fact,... Continue reading... Irrationality by Infinite Descent Posted on June 29, 2019. by. Samuel Nunoo The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the... Continue reading... Fermat’s Last Theorem Posted on June 29, 2019. by. Samuel Nunoo There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =... Continue reading... Greedy to Avoid Progressions Posted on June 29, 2019. by. Samuel Nunoo An arithmetic progression is a sequence of 3 or more integers whose terms differ by a constant, e.g., 20, 23, 26, 29... Continue reading... Riemann Hypothesis Posted on June 29, 2019. by. Samuel Nunoo If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our... Continue reading... Lucas’ Theorem Posted on June 29, 2019. by. Samuel Nunoo Lucas’ Theorem: If p is a prime number, and N has base p representation (aj,…,a1,a0) and k has base p... Continue reading... Rational Irrational Power Posted on June 29, 2019. by. Samuel Nunoo If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to... Continue reading... Odd Numbers in Pascal’s Triangle Posted on June 29, 2019. by. Samuel Nunoo Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N... Continue reading... Euler’s Product Formula Posted on June 29, 2019. by. Samuel Nunoo Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it... Continue reading... Square Root of Two is Irrational Posted on June 29, 2019. by. Samuel Nunoo An irrational number is a number that cannot be expressed as a fraction. But are there any irrational numbers? It was known to... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"10_jPW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Bidding-for-envy-freeness.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":89,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Bidding-for-envy-freeness","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["149_p 723..749"],"author":"Acrobat Distiller 4.05 for Windows","author_sxt":["Acrobat Distiller 4.05 for Windows"],"last_modified":"Fri Oct 04 11:42:42 UTC 2002","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-125049098361021596,"exact_signature_unique_b":true,"fuzzy_signature_l":-7476523065272444983,"fuzzy_signature_unique_b":true,"wordcount_i":10894,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:32 UTC 2022","fresh_date_dt":"Wed Dec 06 11:46:38 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"Acrobat Distiller 4.05 for Windows","language_s":"en","size_i":199847,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088094783209472,"text_t":"Soc Choice Welfare (2002) 19: 723–749\n99992002\nBidding for envy-freeness: A procedural approach to\nn-player fair-division problems\nClaus-Jochen Haake1, Matthias G. Raith2, Francis Edward Su3\n1 Institute of Mathematical Economics, University of Bielefeld, P.O. Box\n100131, 33501 Bielefeld, Germany (e-mail: chaake@wiwi.uni-bielefeld.de,\nmraith@wiwi.uni-bielefeld.de)\n2 Department of Economics, University of Magdeburg, P.O. Box 4120,\n39016 Magdeburg, Germany (e-mail: raith@ww.uni-magdeburg.de)\n3 Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA\n(e-mail: su@math.hmc.edu)\nReceived: 6 March 2000\/Accepted: 21 May 2001\nAbstract. We develop a procedure for implementing an e‰cient and envy-free\nallocation of m objects among n individuals with the possibility of monetary\nside-payments, assuming that players have quasi–linear utility functions. The\nprocedure eliminates envy by compensating envious players. It is fully descrip-\ntive and says explicitly which compensations should be made, and in what\norder. Moreover, it is simple enough to be carried out without computer sup-\nport. We formally characterize the properties of the procedure, show how it\nestablishes envy-freeness with minimal resources, and demonstrate its appli-\ncation to a wide class of fair-division problems.\n1 Introduction\nIn this paper we consider problems of fair division, in which a group of\nindividuals must decide how to allocate several objects (goods or burdens)\n‘‘fairly’’ among the group’s members, given the possibility of (monetary) side-\npayments. A variety of situations fit into this setting: heirs inheriting an estate,\nemployees splitting a list of duties, developers staking claims in a new frontier,\nor students renting a house together. In many cases, such problems can also\ninvolve additional costs or compensations for the group as a whole.\nAs a basic notion of fairness, we focus on envy-freeness, which means that\nWe would like to thank Steven Brams and Marc Kilgour for stimulating discussions,\nand we are grateful to an anonymous referee for helpful comments. Financial support\nby the ‘Ministerium für Wissenschaft und Forschung, NRW’ (Raith) and the ‘Gra-\nduiertenkolleg’ Mathematical Economics, University of Bielefeld’ (Su) is gratefully\nacknowledged.\nno individual will wish to trade shares with anyone else.1 In proving existence\nof envy-free solutions, particular attention has been given to the development\nof constructive proofs that yield implementable algorithms to find the desired\nallocation. This aspect is emphasized by Alkan et al. (1991) and Su (1999).2\nBecause both these approaches assume non-linear utility in money, the algo-\nrithms are not finite; instead they converge on an envy-free solution. Exact\nsolutions can be obtained using the algorithms of Aragones (1995) and Klijn\n(2000) who assume that players’ preferences for money are characterized by\nlinear utility functions. This is a strong restriction, but one that seems appro-\npriate for the present context, nevertheless.\nOur approach in this paper is not only constructive, but we consider it to\nbe procedural as well. In the context of fair division, we view a procedure as\nfeaturing the following characteristics: it is intuitive, meaning that each step\nmust be easy to understand; it is plausible, meaning that each step must be\nsimple to argue; and it is manageable, meaning that each step must be straight-\nforward to compute. We see these subjective criteria as relevant for the prac-\ntical implementation of a fair-division outcome, in particular when parties in\nreal life prefer to establish fairness by themselves, rather than trust the ‘‘magic’’\nof a computer algorithm.\nKnaster and Steinhaus (1948) proposed a fair-division method that is pro-\ncedural in this sense, as it is simple to apply and largely intuitive. However, as\nBrams and Taylor argue as they revisit this approach, the procedure does not\nensure envy-freeness for more than two players. In a more recent approach,\nBrams and Kilgour (2001) characterize a procedure that assigns a given num-\nber of objects to the same number of players e‰ciently, establishing fairness\nthrough internal ‘‘market prices’’. The procedure is intuitive, plausible, and\nsimple to manage. But for more than three players it also does not guarantee\nenvy-freeness.\nIn the following sections we develop a compensation procedure that estab-\nlishes envy-freeness for any number of players and a possibly di¤erent number\nof objects. We thus extend the Knaster-Steinhaus procedure, while preserving\nenvy-freeness for any number of players. The procedure eliminates envy in\nrounds by compensating envious players. It is fully descriptive and says explic-\nitly which compensations should be made, and in what order. The procedure\nmimics and thus supports a natural mediation process with the objective of\nimplementing an envy-free outcome. We formally characterize the properties\nof the procedure and illustrate how it works in practice.\nWe assume that players can articulate their preferences over bundles of\nobjects through monetary bids and that their utility of money is linear. In\naddition we impose a qualification condition for each person taking part in\nthe fair division, requiring that her valuations of all bundles sum to at least\n1 The concept of envy-freeness was first used in an economic context by Foley (1967),\nalthough under a di¤erent label.\n2 The algorithm of Su (1999) is even interactive in the sense that it sequentially gives\neach player the opportunity to choose her favorite alternative at evolving prices.\n724 C.-J. Haake et al.\nthe total cost for the group as a whole. This is not needed for an envy-free\nsolution, but it guarantees that a player will never have to pay more than her\nbid.3\nWe consider two alternative methods for financing the necessary monetary\nside-payments. In the method of ex-ante payments, players first pay the amount\nthey bid for the objects they receive. Everyone thus begins with the same ad-\nvantage, viz. none, and there is an aggregate amount of money left to divide\namong the players. Under the method of ex-post payments, players submit\npayments only after the monetary compensations needed to establish envy-\nfreeness are determined via the procedure. A distinctive feature of both meth-\nods is that the resources for compensation are generated by the group itself.\nBecause we find ex-ante payments more intuitive for players, we focus mainly\non this method, but we do provide a comparison of the outcomes.\nIn Sect. 2, we formulate the first step of our procedure: an assignment of\nobjects to players that maximizes the sum of players’ utilities. We call this the\nutilitarian assignment.4 When the bundling of objects is restricted, the utili-\ntarian assignment will generally cause envy among the players.\nIn Sect. 3, we describe non-technically the individual steps of the compen-\nsation procedure which will eliminate envy by using the surplus from players’\ninitial payments. We demonstrate its application with a numerical example\nthat we use throughout the paper.\nThe formal characterization of our compensation procedure and thus a\nconstructive existence proof for an envy-free allocation is given in Sect. 4.\nHere we prove that there is always at least one player who is non-envious at\nthe start and then show how our procedure successively eliminates the envy\nof players who are envious of non-envious players. In contrast to the algo-\nrithms of Aragones (1995) and Klijn (2000), our procedure does not need to\nkeep track of all envy relations, since it uses only maximum envy.5 A further\nfeature distinguishing our compensation procedure from Klijn’s algorithm\nis that it requires only minimal financial resources to establish envy-freeness.\nThe necessary amount is automatically determined through the compensa-\ntions.\nIn Sect. 5, we study alternative methods of dividing the surplus that gen-\nerally remains after envy-freeness is established.6 The simplest way to obtain\n3 Brams and Kilgour (2001) use a similar, but stricter constraint, basically for the same\npurpose.\n4 The same starting point is chosen by Steinhaus (1948), Aragones (1995), or Brams\nand Kilgour (2001). A notable exception is Klijn (2000).\n5 Of course all envy relations are assessed by players within the procedure in order to\ndetermine maximal envy; but the non-maximal assessments are not retained for the\nprocedure.\n6 There will generally be an infinite number of envy-free outcomes to choose from.\nAnd, as Tadenuma and Thomson (1991) verify, there exists no proper sub-solution\nsatisfying the notion of consistency. A sub-solution is consistent in their sense if it\nallocates the objects and money received by a subset of players in the same way as the\nsolution assigns total resources to all players.\nBidding for envy-freeness 725\na unique outcome is to divide the surplus equally. However, this always leads\nto an outcome on the boundary of the set of envy-free prices that favors a\nparticular player. We therefore propose an alternative method that implements\na unique outcome, generally in the interior of the envy-free set, thus treating\nplayers more symmetrically.\nIn Sect. 6, we allow the initial assignment of exogenously given bundles\nto be ine‰cient (i.e., non-utilitarian with respect to those bundles). We find\nthis aspect crucial, because an e‰cient assignment may be di‰cult to find\nin practice if the fair-division problem involves many players and objects.\nWe show how our compensation procedure is easily adapted to non-\nutilitarian initial assignments. We use the fact that envy-freeness requires an\ne‰cient assignment of bundles (cf. Svensson 1983). If the assignment is in-\ne‰cient, our envy-reducing procedure will create an envy cycle indicating a\ncyclical trade of bundles that increases the sum of players’ utilities; this is\nanalogous to the permutation procedure of Klijn (2000). Strict application\nof the compensation procedure thus produces cyclical trades of bundles lead-\ning to an e‰cient assignment, and then establishes envy-freeness as described\nabove.\nSection 7 concludes with some practical considerations. The Appendix\ncontains numerical examples that illustrate the analysis.\n2 Characterization of a utilitarian assignment\nWe consider a group of players I ¼ f1; . . . ; ng who wish to assign a set of\nobjects K ¼ f1; . . . ;mg among themselves in an envy-free fashion.\nAssumption 1. Players value bundles of objects in a common divisible unit of\naccount, e.g., money.\nEach player i A I can express her valuations of bundles Bi JK of objects\nthrough (monetary) bids, which we characterize by functions bi : 2\nK ! R.\nNote that Assumption 1 does not say anything specific about the relationship\nbetween a player’s valuation of a bundle and her valuations of the individual\nobjects contained within. In specific cases, however, players may have addi-\ntively separable preferences over the objects in K, such that the value of a\nbundle simply equals the sum of values of the individual objects, i.e., biðBjÞ ¼P\nk ABj\n~biðkÞ ði; j A IÞ, where ~bi : K ! R denotes a player’s bid for specific\nobjects. Moreover, as we will see below, when players have linear preferences\nover sub-divisions of objects that are divisible, the outcome of our fair-division\nprocedure is the same whether or not K contains divisible objects.\nWe assume no specific relationship between the number of objects and the\nnumber of players – this can be included through additional restrictions\n(addl. restr.) on the objects. The group’s assignment determines the bundle\nBi JK of objects that each player i receives. Possible assignments are char-\nacterized by\n726 C.-J. Haake et al.\nB ¼\n\u0002\nðB1; . . . ;BnÞ jBi JK ;Bi XBj ¼ q;\n6\ni A I\nBi ¼ K ; ðþaddl: restr: on BiÞ\n\u0003\n:\nAn assignment B A B thus groups the m objects of K into n separate bundles\nBi without dividing them; if there are fewer objects than players, B will nec-\nessarily also include empty bundles.\nThe set of assignments B may be further restricted by specific requirements\nfor the individual bundles. In the simplest case, the additional restriction may\njust be an exogenously given bundling of objects. For an endogenous bundl-\ning of objects, one could specify that all players receive the same number of\nobjects ðjBij ¼ m=nÞ, or that each player receives a minimum number of\nobjects (jBijbm , where mam=n). Or, in a di¤erent context, assume that\nthe objects are distinct territories in a geographical region. One may then\nwish to have the territories in each bundle be connected in some specific form,\ne.g., lying within a single sub-region. Generally, the set of assignments B can\ninclude any restriction on the objects of K that is player-anonymous; in par-\nticular, there are no restrictions of the form ‘‘Player i must (or must not) receive\nobject k.’’ Indeed, with an exogenously imposed restriction of this type, no\nprocedure can guarantee envy-freeness. However, if the players themselves\nwish to give a particular player special attention, they can express this directly\nvia their preferences over the objects.\nWe view the assignment of the m objects to the n players as a joint venture,\nfor which there is a total cost C (measured in the common unit of account)\nthat must also be divided among the players. Denoting by ci the contribution\nthat is to be paid by the player receiving bundle Bi, this implies\nP\ni A I ci ¼ C.\nAssumption 2. Players have linear preferences over values measured in the com-\nmon unit of account.\nUnder Assumptions 1 and 2, we can characterize players’ preferences\nthrough quasi-linear utility functions ui : 2\nK \t R ! R, with\nuiðBj ; cjÞ ¼ biðBjÞ \n cj; i; j A I :\nIn order to implement an e‰cient outcome, our fair-division procedure is\nbased on an assignment that maximizes the (unweighted) sum of players’ util-\nities. We characterize such a utilitarian assignment B A B by\nB A arg max\nB AB\nX\ni A I\nuiðBi; ciÞ\n¼ arg max\nB AB\nX\ni A I\nbiðBiÞ \n C\n¼ arg max\nB AB\nX\ni A I\nbiðBiÞ:\nThe utilitarian assignment B yields the maximum sum of players’ bids,\nBidding for envy-freeness 727\nwhich we denote by M. Let Bi denote the bundle assigned to player i; thenP\ni A I biðBiÞ ¼ M.\nRegardless of the total cost C of the joint venture, the utilitarian assign-\nment B endogenously bundles the objects of K, while acknowledging the ad-\nditional restrictions, and assigns these bundles to the individual players.7 If\nthere are no additional restrictions on the bundles, the utilitarian assignment\nis easy to implement when preferences are additively separable: simply assign\neach object to the player who values it most (if there are several, choose one\nplayer arbitrarily). However, in general assignments will be complicated by the\nadditional restrictions specifying how bundles are to be created. When bundles\nare given exogenously, we call an assignment e‰cient if no re-assignment of\nthe bundles yields a larger sum of bids. The utilitarian assignment thus allo-\ncates given bundles e‰ciently among players.\nWe characterize a utilitarian allocation as envy-free if no player values the\nbundle of any other player (net of its cost) higher than her own bundle (net\nof its cost):\nuiðBi; ciÞb uiðBj; cjÞ; i; j A I :\nWe wish to determine an envy-free pricing of utilitarian bundles, with prices\nthat sum to the total cost C of the joint venture, such that no player pays more\nthan she thinks her bundle is worth. The procedure described in the next sec-\ntion will accomplish this, if we impose the following additional requirement.\nAssumption 3. The sum of each player’s bids for all the bundles of a utilitarian\nassignment is at least equal to the total cost, i.e.,\nPn\nj¼1 biðBjÞbC, Ei A I .\nAssumption 3 can be seen as an individual qualification constraint for each\ngroup member. If the objects to be distributed are assigned across several\nplayers, then player i is qualified if, by teaming with other players of identical\npreferences, this group of players would be able to a¤ord the joint venture. As\nthe procedure will show, the qualification constraint is not required to produce\nenvy-freeness, but it guarantees that no player will pay more than her bid.\n3 The compensation procedure\nOur procedure with ex-ante payments begins by having each player contribute\nthe amount that they bid for their assigned bundle, yielding M dollars from\nwhich the cost is paid. The remaining surplus M \n C will be returned to the\nplayers in the form of discounts in a way which will guarantee envy-freeness.\nIn each round of the compensation procedure, discounts are determined on\nthe basis of players’ assessments, and then assessments are revised taking dis-\ncount changes into account.\n7 Note that, when players have linear preferences over divisible objects, the utilitarian\nassignment would only divide an object if the value added by its inclusion in a player’s\nbundle is the same for two or more players. In this case we may just as well assume that\nthe object is fully assigned to just one player.\n728 C.-J. Haake et al.\nLet aij denote Player i ’s assessment of the value of Player j’s bundle minus\nits cost:\naij ¼ biðBjÞ \n cj ¼ biðBjÞ \n bjðBjÞ þ dj; i; j A I ; ð1Þ\nwhere dj is the discount that Player j has received during the procedure (at the\nstart dj ¼ 0). We call A ¼ ðaijÞ the assessment matrix. Note that if aii < aij\nthen Player i will experience envy for Player j. Without additional restrictions\non bundles, the utilitarian assignment simply assigns each object to the player\nwho values it most. The assessment matrix will thus be envy-free from the\nstart, since aii b aij . With additional restrictions, however, this will generally\nnot be the case, and envious players will need to be compensated. The com-\nplete compensation procedure is described as follows.\nThe compensation procedure for a utilitarian assignment\n1. Assign bundles to players using the utilitarian assignment. Each player ini-\ntially contributes her bid on her assigned bundle, yielding a pool of size M\nfrom which the cost C is paid.\n2. Calculate the assessment matrix. Note that there will always be at least\none player who experiences no envy (see Theorem 1). If all players are non-\nenvious, skip to Step 5.\n3. Now perform a round of compensations: use the assessment matrix to iden-\ntify all players whose maximum envy is directed towards a non-envious\nplayer, and compensate these individuals from the surplus by their maxi-\nmum envy di¤erence.8 Then recalculate the assessment matrix (but only\nafter all the compensations have been made in this round).\n4. Perform additional compensation rounds until all envy is eliminated. (Theo-\nrem 2 shows that at most ðn\n 1Þ compensation rounds will be needed.)\n5. The sum of the compensations made in Steps 3 and 4 is minimal (see The-\norem 3), and it will never exceed the surplus M \n C (see Theorem 4).\nTherefore distribute any remaining surplus in a way that maintains envy-\nfreeness; e.g., one could simply divide it equally among all players. (Sect. 5\ndiscusses an alternative method for post-envy allocation of the remaining\nsurplus.)\nTo illustrate we give an example. Suppose there are four players (denoted\nPi) who submit bids for a joint venture that has a total cost of C ¼ 100. The\nutilitarian assignment determines four bundles (denoted Bi) for which players\nhave the valuations given in Table 1.\nThe bids in the utilitarian assignment (the framed entries along the diago-\nnal of Table 1) are collected as initial payments. Since they sum to 145, after\npaying the cost of 100, there is a surplus of 45 left to return to the players in\nthe form of discounts. The assessment matrix can be computed by subtracting\n8 Alternatively, one could compensate all envious players. However, this would require\nmore compensations in each round, but not fewer rounds.\nBidding for envy-freeness 729\nthe diagonal entry from each column. In Table 2, row i then shows Player\ni ’s assessment of Player j’s bundle. We keep track of discounts in a separate\nrow.\nThe assessment matrix in Table 2 shows (by comparing entries in each\nrow) that Player 2 envies Player 1 (a22 < a21) and Player 3 envies Player 4\n(a33 < a34). Therefore we must compensate Player 2 by giving her a discount\nof 10, and Player 3 a discount of 5. To recalculate the assessment matrix, we\nmay add 10 to column 2 and add 5 to column 3. The new assessment matrix\nis given in Table 3.\nNow both Player 3 and Player 4 envy Player 2, and Player 2 feels tied with\nPlayer 1 (who remains non-envious). We must compensate Player 3 and Player\n4 by giving them both additional discounts of 5. Adding 5 to both columns 3\nand 4, we obtain Table 4.\nTable 1. Players’ bids for bundles\nB1 B2 B3 B4\nP1 50 20 10 20\nP2 60 40 15 10\nP3 0 40 25 35\nP4 50 35 10 30\nInitial payment 50 40 25 30\nTable 2. The initial assessment matrix\nP1 P2 P3 P4\nP1 0 \n20 \n15 \n10\nP2 10 0 \n10 \n20\nP3 \n50 0 0 5\nP4 0 \n5 \n15 0\nDiscounts 0 0 0 0\nTable 3. The modified assessment matrix\nP1 P2 P3 P4\nP1 0 \n10 \n10 \n10\nP2 10 10 \n5 \n20\nP3 \n50 10 5 5\nP4 0 5 \n10 0\nDiscounts 0 10 5 0\n730 C.-J. Haake et al.\nNow all envy has been eliminated, since each diagonal element is the largest\nentry in its row. The discounts used 25 units of the surplus, and the remain-\ning surplus of 20 can be equally divided among the four players, yielding\ntotal discounts given by d ¼ ð5; 15; 15; 10Þ. This gives final envy-free costs of\nc ¼ ð45; 25; 10; 20Þ.\nIt is important to note that, in formulating the compensation procedure,\nwe do not make any assumptions concerning the signs of players’ bids bi or\ntheir contributions ci. Therefore, our procedure can also be applied to sit-\nuations where the objects are burdens for which the group as a whole receives\na compensation (C < 0). Players’ negative bids are then requested payments\nthat express their disutility of accepting these burdens, and Assumption 3 states\nthat a player is qualified if her demands for bearing all burdens do not exceed\nthe total compensation, i.e., \nbiðKÞa\nC. We provide an example for the\ndivision of burdens in the Appendix.\nMore generally, our procedure can be applied to fair-division problems that\ninvolve both goods and burdens. For example, a group of individuals that\ndecides to share a house cannot only derive a fair allocation of rooms and\nrents, but they can also include all the (group’s) chores that come with the\nhouse (e.g., lawn mowing, cleaning, cooking, etc.). Of course, if there is no\nextra compensation, the qualification constraint (Assumption 3) becomes more\nbinding as chores are added. But this is only plausible – in order to qualify, a\nhousemate must not only be willing to pay the necessary rent, she must also\nbe willing to perform the necessary chores.\n4 Properties of the compensation procedure\nWe now show that our procedure works as indicated. One property of the as-\nsessment matrix A is crucial in all that follows. We define a permutation sum\nof an n\t n matrix A to be any sum of the form\nP\ni aipðiÞ, where p : I ! I is a\npermutation of n elements. Thus a permutation sum picks one element of each\ncolumn and row and forms their sum.\nLemma 1. At any step of the procedure, the largest permutation sum of A occurs\nalong the diagonal.\nTable 4. The envy-free assessment matrix\nP1 P2 P3 P4\nP1 0 \n10 \n5 \n5\nP2 10 10 0 \n15\nP3 \n50 10 10 10\nP4 0 5 \n5 5\nDiscounts 0 10 10 5\nBidding for envy-freeness 731\nProof. We check:X\ni\naipðiÞ ¼\nX\ni\nðbiðBpðiÞÞ \n bpðiÞðBpðiÞÞ þ dpðiÞÞ\n¼\nX\ni\nbiðBpðiÞÞ \nM þ\nX\ni\ndi a\nX\ni\ndi;\nwhich follows from the definition of M, the maximum sum of bids. The in-\nequality is clearly an equality when p is the diagonal assignment. r\nWe keep track of ‘‘chains’’ of envy with the following notation. We write\ni ! j if j is the player that i envies the most (if there are several, pick one ar-\nbitrarily). In this case aii < aij and aij is the largest entry in row i. The single\narrows thus only indicate maximum envy relations. We use a double arrow\ni ) j if i envies no one but feels tied with j and this tie was the result of an\nearlier compensation. Thus aii ¼ aij, and these are the largest entries in row i.\nNote that double arrows only keep track of created ties, and are not used for\nties occurring ‘‘by coincidence’’ (e.g., where aii ¼ aij but this was not the result\nof a previous compensation). In the sequel when we refer to ‘‘arrow’’ we shall\nmean either a single or double arrow unless explicitly specified.\nWe form a directed graph G in which the vertices represent players and the\nedges are given by the arrow relations between players.9 (We shall speak of\n‘‘players’’ and ‘‘vertices’’ interchangeably in all that follows.) Throughout the\nprocedure, the directed graph G will change. Keeping track of how G evolves\nis the key to showing that the procedure terminates.\nLemma 2. At any step in the procedure, the directed graph G contains no cycles.\nProof. If there were a cycle of single arrows, say i1 ! i2 ! \f \f \f ! ik ! i1, then\nai1i1 < ai1i2\nai2i2 < ai2i3\n..\n.\naikik < aiki1 ;\nð2Þ\nand by adding these relations one would find that\nai1i1 þ ai2i2 þ \f \f \f þ aikik < ai1i2 þ ai2i3 þ \f \f \f þ aiki1 ; ð3Þ\nwhich augmented by the other diagonal terms would contradict Lemma 1.\nA nearly identical argument can be used for cycles in which some (but\nnot all) of the arrows are double; some of the envy inequalities in (2) would\nbecome equalities, but the inequality (3) would remain strict as long as there\nwere a single arrow in the cycle.\nThe only other possibility is a cycle consisting entirely of double arrows.\nBut this cannot arise, because double arrows only originate from single arrows\n9 It is important to note that our directed graph G is simpler than the graphs con-\nstructed by Aragones (1995) and Klijn (2000), since our procedure only keeps track of\nmaximum envy.\n732 C.-J. Haake et al.\n(as compensations are made to envious players). Thus if there were a cycle of\ndouble arrows, by stepping backwards through the procedure one would find\na prior round in which that cycle contained a single arrow – a possibility that\nwas ruled out above. r\nThe outdegree of a vertex represents the number of arrows that originate\nat a vertex. Because of the way we defined arrows, every vertex in G has an\noutdegree of at most 1. Hence there is a uniquely defined path that ‘‘flows’’\nfrom any given vertex. Since G cannot have any cycles at any step in the\nprocedure, we deduce that G must always be a disjoint set of directed trees,\neach of which has a unique root, a vertex of outdegree 0 to which all other\nvertices flow. Roots of trees correspond to non-envious players who have not\nyet experienced envy.\nTheorem 1. At the start there is at least one player who will remain non-envious\nthroughout the entire procedure.\nProof. A vertex with no arrow is a root, and a vertex with a single or double\narrow corresponds to a player who experiences or has experienced envy. It\nfollows that the outdegree of a vertex can never decrease – once a root earns\nan arrow it can never become a root again.\nIf by the end of the procedure there were no roots, then any path following\nthe arrows in G would eventually cycle, contradicting Lemma 2. Thus there is\na root which must have been a root throughout the entire procedure, corre-\nsponding to a person who remains non-envious. r\nBecause every vertex in a tree has a unique path to its root, we may classify\nvertices in G by ‘‘levels’’ – a level 0 vertex is a root, and a level k vertex is one\nthat is k arrows away from the root of its tree. A vertex i is said to be an an-\ncestor of vertex j in the tree if there is a chain of arrows flowing from j to i.\nThus the root of a tree is an ancestor of every other vertex in the tree. The\nancestral path of vertex i is the set of all vertices on the path from i to its root\n– including the root, but not including i. (Note that the ancestral path does\nnot depend on the type of arrows along the path.) Vertices may change levels\nas G evolves throughout the procedure.\nLemma 3. During a compensation round, if a player P changes her ancestral\npath, the new path must contain a newly compensated player Q between P and\nthe root. Thus in the new graph, P will have a higher level number than Q.\nProof. During a compensation round, the only new envy that can be intro-\nduced is directed at players receiving compensations during that round. Thus\nif player P or any of her ancestors changes her maximum envy (hence the direc-\ntion of her arrow) to a newly compensated player Q, then after the compen-\nsation round, the path flowing from P must pass through Q. From the defini-\ntion of level this means that P will have a higher level number than Q. r\nLemma 4. After k compensation rounds, there are no envious players on levels\n0 through k.\nBidding for envy-freeness 733\nProof. We prove this by induction on k. For k ¼ 0 (before any compensa-\ntions) the statement trivially holds, since there are by definition no envious\nroots.\nNow assume that the lemma holds for k. We show that it also holds for\nk þ 1. By the inductive hypothesis, after k compensation rounds all envious\nplayers must be on levels ðk þ 1Þ or higher. A subset of these will be com-\npensated during the ðk þ 1Þ-st round; in particular this must include all the\nenvious players on level ðk þ 1Þ, since their maximum envy is for non-envious\nplayers on level k.\nSo consider a player P anywhere in G. If P’s ancestral path did not change\nas a result of the ðk þ 1Þ-st compensation round, then she remains on the same\nlevel. Moreover, if she is on level ðk þ 1Þ, then she must also now not be en-\nvious (because she was either compensated or was not envious to begin with).\nIf P’s ancestral path did change as a result of the ðk þ 1Þ-st compensation\nround, then we show her new level number must be ðk þ 2Þ or greater. Con-\nsider the new path that flows from P; by Lemma 3, it contains some newly\ncompensated player. Thus it makes sense to speak of the newly compensated\nplayer on P’s path who is closest to the root; call this player Q. Q’s ancestral\npath cannot have been changed by the last compensation round (otherwise\nLemma 3 would have produced some other newly compensated player closer\nto the root than Q). So Q’s level was unchanged, and being a newly compen-\nsated player, Q must have had level number ðk þ 1Þ or greater. Since P has a\nhigher level number than Q, P must have a level number of ðk þ 2Þ or greater\nin the new graph.\nThus if any player changed levels as a result of the ðk þ 1Þ-st compensation\nround, they must now be at level number ðk þ 2Þ or greater. The players who\nremain on levels 0 through k were non-envious before the round and must still\nbe non-envious, while the ones remaining on level ðk þ 1Þ have been compen-\nsated and are also non-envious. r\nTheorem 2. The procedure requires no more than n\n 1 compensation rounds to\neliminate envy.\nProof. Because G has only n vertices, and every tree in G must have a root, no\nvertex can be on level n or greater. Hence, by Lemma 4, all players will be\nnon-envious after at most ðn\n 1Þ compensation rounds. r\nIt is not yet clear what e¤ect the initial assignment has on the resulting\nenvy-free discounts. Moreover, when the utilitarian assignment for n given\nbundles is not unique, players have multiple choices for the starting point\nof the compensation procedure. Remarkably, the outcome is not a¤ected by\nthe choice of assignment. We first prove two lemmas needed to establish this\nresult.\nLemma 5. If there are two utilitarian assignments involving the same bundles,\nthen a vector of discounts which yields envy-free assessments under one assign-\nment will also be envy-free under the other assignment.\n734 C.-J. Haake et al.\nProof. We have\naij :¼ biðBjÞ \n bjðBjÞ þ dj; i; j A I :\nLet p : I ! I be the permutation that transforms the original utilitarian\nassignment into the other one; i.e., in the second assignment Player i receives\nthe bundle that Player pðiÞ would receive in the original assignment. Envy-\nfreeness of the discounts di in particular yields aii b aipðiÞ.\nTaking the sum over di¤erences yieldsX\ni\nðaii \n aipðiÞÞ ¼\nX\ni\nðdi \n dpðiÞÞ þ\nX\ni\nðbpðiÞðBpðiÞÞ \n biðBpðiÞÞÞ\n¼ 0 þM \nM ¼ 0:\nSince each addend on the left-hand side is non-negative, and their sum is zero,\naii ¼ aipðiÞ; Ei A I : r\nSaid another way, Lemma 5 shows that the compact convex set of all pos-\nsible envy-free discount vectors is independent of the utilitarian assignment.10\nNote, however, that Lemma 5 says nothing specific about the discount vector\ninduced by the compensation procedure.\nLemma 6. The compensation procedure yields a unique minimal vector of non-\nnegative discounts that make a given utilitarian assignment envy-free.\nProof. Given a utilitarian assignment, the compensation procedure yields a\nvector of player discounts ðdiÞ which makes every player envy-free. If there\nwere some other vector of (non-negative) envy-free player discounts ðmiÞ which\nwas smaller for some player, then we obtain a contradiction.\nSuppose mk0 < dk0 , for player k0. Thus dk0 is strictly positive (because\nmk0 b 0), i.e., player k0 was compensated sometime during the compensa-\ntion procedure. This implies that in the final envy graph at the conclusion of\nthe compensation procedure, Player k0 is not at the root of her tree. Let\nk0; k1; k2; . . . ; kl ¼ r be the path from k0 to the root r of her tree. Since each\nof the arrows in the final envy-graph are double arrows, akiki ¼ akikiþ1 for all\n0 a ia l \n 1. Using (1) to express this in terms of the bids and discounts, we\nhave\ndki ¼ bkiðBkiþ1Þ \n bkiþ1ðBkiþ1Þ þ dkiþ1 ;\nfor all 0 a ia l \n 1. But the ðmiÞ, being an envy-free vector of discounts, must\nsatisfy\nmki b bkiðBkiþ1Þ \n bkiþ1ðBkiþ1Þ þmkiþ1 ;\nsince the left side is what Player ki would receive under these discounts and the\nright side is what Player ki believes Player kiþ1 would receive. Subtracting\nthese two equations we obtain\n10 Lemma 5 corresponds to Aragones’ (1995) Lemma 4.\nBidding for envy-freeness 735\ndki \n dkiþ1 amki \nmkiþ1 :\nBy summing this over all 0 a ia l \n 1, we have\ndk0 \n dkl amk0 \nmkl :\nBut dkl ¼ 0 because kl was the root, i.e., a player not compensated throughout\nthe entire procedure. And since the discount mkl is non-negative, we have\ndk0 ¼ dk0 \n dkl amk0 \nmkl amk0 :\nThis contradicts the fact that mk0 < dk0 . r\nTogether Lemmas 5 and 6 allow us to establish the following practical re-\nsult.\nTheorem 3. For given bundles, the outcome of the compensation procedure yields\na unique minimal vector of player discounts which does not depend on the utili-\ntarian assignment chosen.\nProof. Suppose there were two utilitarian assignments – the non-prime and\nthe prime assignment. The compensation procedure applied to each assign-\nment yields corresponding vectors of player discounts ðdiÞ and ðd\n0\ni Þ.\nNote that the discounts ðd 0i Þ are also envy-free discounts for the non-prime\nassignment (Lemma 5), and because the ðdiÞ are minimal for the non-prime\nassignment (Lemma 6), we must have di a d\n0\ni for all i. Similarly, the ðdiÞ are\nalso envy-free discounts for the prime assignment, and minimality of the ðd 0i Þ\nfor the prime assignment implies that d 0i a di for all i. Therefore di ¼ d\n0\ni for\nall i. r\nThe sum of discounts thus yields the minimal amount of money required\nfor envy-freeness. With n bundles to allocate among the individual players,\nmultiple utilitarian assignments do not cause a coordination problem, because\nthey all lead to the same discounts. Hence there is no problem of choosing an\nappropriate starting point for the compensation procedure as long as the as-\nsignment maximizes the sum of players’ bids.\nWe have shown that this procedure will terminate with envy-free costs\nwithout having used the individual qualification condition (Assumption 3).\nThe only need for this condition is to ensure that the surplus is never exceeded\nat the end of the compensation rounds. (If the surplus were exceeded at the\nend, one could still obtain envy-free prices by charging all players equally for\nthe overdraft, but then some players might end up paying more than what\nthey bid on their bundle.)\nTheorem 4. If each person meets the qualification condition (Assumption 3),\nthen by the end of the procedure, the compensations will not have exceeded the\nsurplus.\nProof. Since the prices obtained at the end of the procedure are envy-free,\naii b aij for all i; j. Using (1) this implies\ndj \n di a bjðBjÞ \n biðBjÞ; ð4Þ\n736 C.-J. Haake et al.\nfor all i; j. Summing the above equation over all j, one obtains that for all i,X\nj\nðdj \n diÞaM \n\nX\nj\nbiðBjÞ: ð5Þ\nChoose any player i who was not compensated throughout the entire proce-\ndure (there must be at least one, by Theorem 1). Since di ¼ 0, and since the\nsum of the bids of any player is C or greater, we must have thatX\nj\ndj aM \n C;\nwhich shows that the sum of the compensations does not exceed the sur-\nplus. r\n5 Dividing the remaining surplus\nAccording to Theorem 4, after the compensation procedure has established\nenvy-freeness, the remaining surplus S is given by\nS ¼ M \n C \n\nX\nj A I\ndj b 0:\nWith S > 0, there is surplus left to distribute among the players. The distri-\nbution schemes that are of interest here are those that maintain envy-freeness.\nThere is a convex and compact set of envy-free discounts to choose from.11\nWe consider two alternative methods for implementing a unique solution.\nEqual distribution of the surplus or ‘‘ex-post equal payments’’\nWith an envy-free assessment matrix at the end of the compensation proce-\ndure, no envy will be created if all entries in the matrix are increased by the\nsame amount. This is easily achieved through an equal distribution of the\nremaining surplus among all players. Denoting players’ final discounts under\nthis equal distribution scheme by d ei , this gives\nd ei ¼ di þ\n1\nn\nS: ð6Þ\nThe equal distribution of the remaining surplus was demonstrated in our exam-\nple in Sect. 3. Despite the simplicity of this distribution scheme, some parties\nmay dislike the procedural asymmetry because they pay di¤erent amounts for\ntheir bundles but receive identical shares of the remaining surplus.\nTherefore, consider the following modification of the compensation pro-\ncedure: parties are assigned bundles, but they do not pay in advance. Instead,\na (hypothetical) mediator finances the compensation procedure and charges\nthe group afterwards for total compensations and the cost C. The mediator\n11 The set of envy-free prices is given by a convex polyhedron characterized by n2\ninequalities that can be derived from the envy-free assessment matrix.\nBidding for envy-freeness 737\nlets each party pay an equal share of the total costs. We call this the com-\npensation procedure with ex-post equal payments.\nWith this modification, Player i ’s assessment of Player j’s bundle becomes\naij ¼ biðBjÞ þ ~dj, where ~dj denotes the compensation under the modified pro-\ncedure. Thus, in this case the initial assessment matrix is the bid matrix, so it\nfollows that Theorems 1–3 also hold for the method of ex-post equal pay-\nments. And, similar to the proof of Theorem 4, it can be shown using the quali-\nfication condition that\n1\nn\nð\nP\nj\n~dj þ CÞa biðBiÞ þ ~di, Ei A I . Hence no player pays\nmore than what she thinks her share is worth.\nGenerally, the directed graphs for ex-ante and ex-post payments will fea-\nture di¤erent envy relations. Nevertheless, the seemingly di¤erent procedures\nexhibit an interesting equivalence: if the final directed graphs turn out to be\nthe same and exactly one and the same player remains non-envious in both\nprocedures, then the ex-post equal payments procedure and the ex-ante pay-\nments procedure with an equal distribution of the remaining surplus will yield\nthe same outcome.12\nTheorem 5. Consider the two procedures: (1) ex-post equal payments and (2)\nex-ante payments with equal distribution of the remaining surplus. Suppose that\nthe final envy graphs of both procedures coincide and have exactly one root.\nThen the outcome of both procedures will be the same.\nProof. With ex-post payments, each player i A I begins with an initial value\nbiðBiÞ. Player i envies Player j if Player i thinks that Player j receives more\nthan Player i. In the (final) directed graph, which we denote by ~G, i ) j then\nimplies\n~di ¼ biðBjÞ \n biðBiÞ þ ~dj; ð7Þ\nwhere ~di denotes the compensation under the modified procedure.\nIn contrast, the compensation procedure with ex-ante payments begins by\nhaving each player i A I pay biðBiÞ for her assigned bundle. Player i envies\nPlayer j if Player j pays less for her bundle than what Player i thinks it is\nworth. In the (final) directed graph G, i ) j implies\ndi ¼ biðBjÞ \n bjðBjÞ þ dj: ð8Þ\nCoincidence of the final graphs enables us to subtract (8) from (7) and to ob-\ntain:\nbiðBiÞ þ ~di \n di ¼ bjðBjÞ þ ~dj \n dj:\nAssume that Player r is a root of the directed graphs G and ~G. Since Player\nr does not receive a compensation under either procedure, dr ¼ ~dr ¼ 0, hence\nbiðBiÞ þ ~di \n di ¼ brðBrÞ ð9Þ\n12 This is the case for our example given in Table 1. The ex-post payments procedure\nyields final discounts ~d ¼ ð0; 20; 35; 25Þ. With C ¼ 100, there is a total of 180 units to\nbe shared equally by all four players. The final cost for each player is thus the same as\nunder the ex-ante method.\n738 C.-J. Haake et al.\nfor every player i in the directed graph leading (with double arrows) to Player\nr. Summing up both sides in (9) we obtain in particular\n1\nn\nX\nj\n½bjðBjÞ þ ~dj \n dj \u000f ¼ brðBrÞ: ð10Þ\nNote that the final value that Player i receives under the compensation\nprocedure with ex-ante payments and an equal distribution of the remaining sur-\nplus is\nbiðBiÞ \n biðBiÞ þ di þ\n1\nn\nX\nj\nbjðBjÞ \n C \n\nX\nj\ndj\n\" #\n¼ biðBiÞ \n ½biðBiÞ \n di\u000f þ\n1\nn\nX\nj\n½bjðBjÞ \n dj\u000f \n\n1\nn\nC:\nThis is the value of Player i ’s bundle, minus its (discounted) cost, plus an equal\nshare of the total payments, minus an equal share of the total costs.\nConsider now the final value that Player i receives under the compensation\nprocedure with ex-post payments and an equal distribution of total compensa-\ntions and total costs C:\nbiðBiÞ þ ~di \n\n1\nn\nX\nj\n~dj \n\n1\nn\nC:\nThe final value is the value of the bundle, plus additional compensation, minus\nan equal share of all costs.\nBy comparing their final values, one can see that both procedures lead to\nthe same outcome if and only if\nbiðBiÞ þ ~di \n di ¼\n1\nn\nX\nj\n½bjðBjÞ þ ~dj \n dj\u000f:\nEquations (9) and (10) show that this condition is satisfied if the final envy-\ngraphs of both procedures are the same and possess a single root. r\nBut even when there is a di¤erence in the outcomes (due to di¤erent envy\nrelations), the compensation procedure with ex-post equal payments estab-\nlishes envy-freeness with minimal resources and thus minimal side-payments\nbetween players. Afterwards each player is charged an equal amount, just\nenough to cover the cost of envy-freeness plus the cost of the joint venture, so\nthat there is no remaining surplus to be distributed. Hence, if the sole objec-\ntive is to implement a unique envy-free outcome, the method with ex-post\npayments has a practical advantage. We demonstrate the application of this\nprocedure in Sect. 6.\nThe average discount method or ‘‘The average biased mediator’’\nAn equal distribution of the remaining surplus may not be the most plau-\nsible approach if the set of envy-free discounts is asymmetric in the sense that\nBidding for envy-freeness 739\nplayers’ maximum possible discounts under envy-freeness di¤er. The asym-\nmetry becomes apparent if the remaining surplus is used to maximize the dis-\ncount of a specific player i, while raising the discounts of the other players just\nenough to maintain envy-freeness. If this is done with each of the n players,\none obtains n extreme discount distributions, each favoring a specific player.\nThe average discount method takes the average of these n extreme surplus\ndistributions, yielding a unique outcome, generally in the interior of the envy-\nfree set.13\nFocusing on a specific player i A I , we denote by diji Player i ’s own maxi-\nmum discount and the minimum corresponding discounts of all other players\nj A I by djji. At the start, all players’ discounts are as given by the compensa-\ntion procedure, i.e., djji :¼ dj, j A I .14 The extreme discounts djji are then\nupdated according to the following four-step algorithm.\nThe average discount method\n(i) Begin by placing Player i in the set D of players whose discounts are to\nbe increased. Formally: D :¼ fig.\n(ii) Add to the set D all the players who are not in D but feel tied with some\nplayer in D. Note that these are the players who would become envious if\nthe discounts of the players in D were increased. Continue looking for\nadditional players to be included in D until only those players are left\n(outside of D) who value the discounted bundle of every player in D\nless than their own. Formally: Dþ :¼ DW fh A InD j ahj ¼ ahh; j A Dg. If\nDþ 0D, then D :¼ Dþ, and repeat Step (ii).\n(iii) If the set D contains all the players of I, then distribute the remaining\nsurplus equally among the players. Otherwise, determine how far the dis-\ncounts of the players in D can be raised without creating envy for any\nplayer who is not in D. The discounts of the players in D can then be\nincreased up to this maximum amount as long as there is enough sur-\nplus left. Update the discounts of the players in D, recalculate the corre-\nsponding columns of the assessment matrix, and recalculate the remain-\ning surplus. Formally: if D ¼ I , then increase the discounts of all players\nby dþ :¼ S\njDj\n; else determine dþ :¼ min minh A InDfahh \n ahj ; j A Dg;\nS\njDj\n\u0007 \b\n.\nUpdate: djji :¼ djji þ d\nþ; aij :¼ aij þ d\nþ, Ej A D; S :¼ M \n C \n\nP\nj A I djji.\n(iv) If there is no surplus left (which happens if dþ ¼ S=jDj), quit. Otherwise,\nif some surplus remains, we need to update D by returning to Step (ii).\n(The updated discounts from the last step will have created players, not\nin D, who feel tied with players belonging to D.) Formally: if S ¼ 0, quit.\nIf S > 0, return to Step (ii).\n13 More generally, one could also maximize the discounts of every subset of players,\nthus tracing out all the corners of the polyhedron of envy-free prices, and then take the\nmidpoint of all outcomes.\n14 We use ‘‘:¼’’ as an assignment operator to update variables in the following way:\nthe value of the term on the right-hand side is assigned to the variable on the left-hand\nside.\n740 C.-J. Haake et al.\nAfter the algorithm has been applied for each player i A I , determine the\naverage extreme discount of Player i:\nd ai ¼\n1\nn\nX\nj A I\ndij j: ð11Þ\nWe demonstrate the calculation of average discounts by using our example\nof Sect. 3. The final outcome of the compensation procedure with ex-ante\npayments is shown in Table 4, with a remaining surplus of S ¼ 20 to be dis-\ntributed among the four players. Beginning with Player 1, the final discounts\nare ð5; 15; 15; 10Þ; beginning with Player 2, the final discounts are ð1:25; 16:25;\n16:25; 11:25Þ; beginning with Player 3, the final discounts are ð3:75; 13:75;\n18:75; 8:75Þ; and beginning with Player 4, the final discounts are ð2:5; 12:5;\n17:5; 12:5Þ. A detailed derivation of these discounts is given in the Appendix.\nTaking all four extreme discount distributions into account, the average dis-\ncounts, given by Eq. (11), are d a ¼ ð3:125; 14:375; 16:875; 10:625Þ, yielding\nfinal costs of c ¼ ð46:875; 25:625; 8:125; 19:375Þ.\nIn all four cases of individual discount maximization, as soon as Player 1 is\nadded to D, all other players are included as well, and the remaining surplus is\ndivided equally among the whole group. This is because Player 1 is the root of\na unique tree of double arrows at the end of the compensation procedure. So\nif her discount is increased, all other players must benefit equally in order to\navoid new envy.\nMore generally, when there are several roots, as soon as the last uncom-\npensated player i with di ¼ 0 joins D, the whole group must belong to D,\ni.e., D ¼ I . At this point, however, the aggregate discounts cannot yet have\nexceeded the surplus if all players met the qualification constraint (Assump-\ntion 3) – this follows from the proof of Theorem 4.\nThe maximization of a specific player i ’s discount according to the method\nabove could also be interpreted as the outcome of a modified compensa-\ntion procedure implemented by a biased mediator who favors Player i. This\nonly requires modifying Step 3 of the compensation procedure: whenever the\nfavored player is compensated for her maximum envy and then, if noone\nenvies her, increase her discount as much as possible without raising the envy\nof any other player.15 Compensate each other player the same as before.\nDistribute the remaining surplus according to the average discount method,\nfavoring the same player. This biased procedure can be applied for each player.\nThe average over all biased compensations thus yields the outcome of the av-\nerage biased mediator.\nRecall that under an equal distribution of the remaining surplus, given by\nEq. (6), each player received an increment of 5 to her discount. In the pre-\n15 In the directed graphs used in the proofs of Lemmas 2–4, a double arrow i ) j\nthen means that i envies no one, but feels a (weak) advantage over j, and this lead was\nthe result of an earlier compensation. Thus aii b aij , and these are the largest entries in\nrow i. As before, double arrows only keep track of created leads. Note that the biased\ncompensations during the compensation procedure do not change the envy graph.\nBidding for envy-freeness 741\nceding analysis of extreme envy-free prices, we found that 5 is the maximum\ndiscount that Player 1 (the root player) can receive. In contrast, for Player 3,\nan increase of 5 is the minimum increment that preserves envy-freeness. The\nequal distribution of the remaining surplus thus implements the most favor-\nable outcome for the initially non-envious (and therefore uncompensated)\nplayer on the boundary of the set of envy-free prices. This may be di‰cult to\njustify in practice.\nBy contrast, the average discount method acknowledges an additional no-\ntion of fairness: Player 1, who did not experience envy throughout the entire\ncompensation procedure and who is always at the verge of being envied by\nPlayer 2, receives only a small share of the remaining surplus. Player 3, on the\nother hand, who quickly becomes envious when other players are compen-\nsated, but who is relatively far from being envied by anyone else, receives\na larger share. From a practical viewpoint, the average discounts enhance\nthe stability of the outcome: by choosing discounts in the interior of the set\nof envy-free prices, each player strictly prefers her own bundle to any other\n(unless, of course, some players have identical preferences).\n6 Cycling to e‰ciency\nIn problems of fair division that involve only a few parties and a few objects, a\nutilitarian assignment will usually be easy to identify in practice. When many\nplayers are involved, the complexity of this initial step quickly rises. Clearly, if\nplayers must rely on computational assistance to perform the necessary cal-\nculations before compensations can be made, this will diminish the attractive-\nness of a procedure that is supposed to work without computer support.\nIn our previous analysis, we used a utilitarian assignment as the starting\npoint for our procedure, thus ensuring that the envy-free allocation is also ef-\nficient. However, e‰ciency is not just a further desirable property of the out-\ncome. Indeed, envy-freeness can only be achieved if the assignment of n given\nbundles is e‰cient.\nLemma 7. Let d be a vector of envy-free discounts for a given assignment B A B.\nThen B must be an e‰cient assignment of the n associated bundles.\nProof. Consider some assignment B A B and an assignment Bp A B, which is\nobtained through a permutation p of the n bundles of B. Let d be the vector\nof envy-free discounts associated with the assignment B. Envy-freeness then\nimplies aii b aipðiÞ, or, using definition (1),\ndi b biðBpðiÞÞ \n bpðiÞðBpðiÞÞ þ dpðiÞ: ð12Þ\nSince\nP\ni di ¼\nP\ni dpðiÞ and\nP\ni bpðiÞðBpðiÞÞ ¼\nP\ni biðBiÞ, we can sum over both\nsides of (12) to obtainX\ni\nbiðBiÞb\nX\ni\nbiðBpðiÞÞ;\n742 C.-J. Haake et al.\nfor any permutation p. Hence B is an e‰cient assignment of the n bun-\ndles. r\nSo, what if the initial assignment B A B is not e‰cient? With exogenously\ngiven bundles, the utilitarian assignment B can be obtained from B through\na permutation of bundles. This is equivalent to the setting studied by Klijn\n(2000) who assumes that the number of objects equals the number of players,\nwith the additional restriction that each player must receive one object. If this\nis the case, Lemma 7 implies that envy-freeness can only be established for a\nutilitarian assignment.16\nConsider an arbitrary assignment B A B. If B results in an e‰cient alloca-\ntion of bundles, the compensation procedure will lead to envy-free discounts.\nIf the assignment is not e‰cient, then Lemma 7 implies that the compensation\nprocedure cannot establish envy-freeness, so it must lead to an envy cycle, i.e.,\nduring the compensation procedure the envy graph exhibits a cycle with at\nleast one single arrow.\nTheorem 6. Let B A B be an ine‰cient assignment of n given bundles. Applying\nthe compensation procedure will then create an envy cycle (in the directed graph\nG) after at most n\n 1 compensation rounds. Cycling bundles in the opposite\ndirection of the arrows (re-assigning to each player in the cycle the bundle of the\nplayer she envies or previously envied) will increase the sum of players’ utilities.\nProof. We begin by proving the first claim. If the directed graph G of the inef-\nficient assignment B contains cycles in single arrows (i.e., before compensa-\ntions are made), the claim trivially holds. Assume therefore that G contains\nno envy cycles in single arrows. In that case there is at least one player who is\nnon-envious.\nThe compensation procedure eliminates envy by compensating envious\nplayers. Throughout the procedure the directed graph G evolves, with single\narrows being converted into double arrows and new single arrows emerging\nthrough the creation of new envy. In Lemmas 3 and 4 the notion of a level\nonly applies to a non-cyclical graph, but otherwise both lemmas are valid re-\ngardless of whether or not the assignment is e‰cient. Assume by way of con-\ntradiction that an envy cycle is never created. Then the procedure must ter-\nminate in at most n\n 1 rounds in an envy-free solution. This contradicts\nLemma 7. Thus after at most n\n 1 rounds the procedure must create a cycle\n(consisting of at least one single arrow and the rest double arrows) in the\ndirected graph G.\nIn order to establish the second claim, consider the situation where a cycle\nis created, in which a non-envious player k0 becomes envious of a newly com-\npensated player kl . (Note that k0 does not have to be a root in G.) The cycle in\nG has the following structure:\nkl ) \f \f \f ) k1 ) k0 ! kl ; 1 a l a n\n 1;\n16 This is equivalent to a result established by Svensson (1983).\nBidding for envy-freeness 743\nwhere the cycle may contain more than one single arrow. Due to the com-\npensations of (not necessarily all) ancestors of kl in the graph G,\ndkl ¼ bkl ðBkl Þ \n bkl\n1ðBkl\n1Þ þ dkl\n1\na\nXl\nh¼1\n½bkhðBkh\n1Þ \n bkh\n1ðBkh\n1Þ\u000f þ dk0 :\nHowever, since k0 is now envious of kl ,\ndk0 < bk0ðBkl Þ \n bkl ðBkl Þ þ dkl ;\nand therefore,\nXl\nh¼0\nbkhðBkhÞ < bk0ðBkl Þ þ\nXl\nh¼1\nbkhðBkh\n1Þ;\nwhich shows that a reassignment of bundles in the opposite direction of the\narrows strictly increases the sum of players’ utilities. r\nApplying the compensation procedure to any assignment B A B, for which\nthe directed graph G contains no envy cycles, will either lead to envy-freeness\nor to a cycle in G that allows a utility-increasing reassignment of bundles.\nRepeated application of the compensation procedure (restarting with zero dis-\ncounts after each permutation of bundles), leads to an envy-free e‰cient out-\ncome with respect to the given bundles, since the number of assignments is\nfinite. Note that, unlike the permutation procedure of Klijn (2000), our pro-\ncedure does not need to keep track of all envy relations. This is because the\nsingle arrows in the directed graph identify only maximum envy, i.e., possible\ncycles in non-maximum envy are simply ignored.\nSince the computation of assessment matrices requires additional calcu-\nlations when the assignment changes during the procedure, we use the com-\npensation procedure with ex-post equal payments, where initial assessments\nare made directly on the basis of players’ bids. From the previous section, we\nknow that this procedure also satisfies Theorems 1–3. The modified compen-\nsation procedure consists of the following steps.\nThe compensation procedure for an arbitrary assignment\n1. Consider all players to be non-envious at the start. For n exogenously given\nbundles, assign to each player an initial bundle. (For practical reasons, one\ncould let each player state her ‘‘best guess’’ of a utilitarian assignment and\nthen choose the assignment with the highest sum of bids as the initial start-\ning point.) Use the bid matrix as the initial assessment matrix.\n2. Use the assessment matrix to identify each player’s maximum envy. If no\nplayer is envious, skip to Step 5.\n3. If a previously non-envious player has become envious, check for an envy\ncycle. If a cycle exists, re-assign bundles by giving each player in the cycle\nthe bundle of the player she envies (single arrow) or previously envied\n744 C.-J. Haake et al.\n(double arrow). Have all players return any compensations they have re-\nceived, and repeat Step 2 to restart the compensation procedure.\n4. If there are envious players, perform a round of compensations: identify\nthose players whose maximum envy is directed towards a non-envious\nplayer. Compensate them by their maximum envy di¤erence, thus making\nthem non-envious. Then recalculate the assessment matrix (but only after\nall the compensations have been made in this round), and repeat Step 2.\n5. Have each player pay an equal share of the total compensations and the\ntotal cost C.\nWe provide an example of the modified compensation procedure in the Ap-\npendix.\nIt is important to note that the procedure as formulated above cycles to a\nutilitarian assignment and establishes envy-freeness only for n exogenously\ngiven bundles. In particular, when each bundle contains only a single object,\nthe final assignment will be utilitarian. In our more general framework,\nthough, an assignment does not require the same number of objects as players.\nConsequently, an assignment is non-utilitarian not only when the n bundles of\nthe assignment are allocated ine‰ciently, but also if the m objects are bundled\nine‰ciently. Since cyclical trades under the compensation procedure involve\ncomplete bundles of objects, players may thus still wish to trade individual\nobjects when the procedure is finished, even though they will not want to trade\ncomplete bundles. However, this type of envy will be di‰cult to identify under\nthe compensation procedure, because the discounts that players receive apply\nonly to their complete bundles; they cannot be broken down into partial dis-\ncounts and associated with individual objects. The compensation procedure\nthus establishes envy-freeness for the highest aggregate utility that any player\ncan identify. In order to obtain an e‰cient bundling of objects, players may\nstill have to resort to computational support in order to improve their best\ninitial guess.\n7 Conclusions\nOur objective in this paper was to formally develop a practical procedure for\nmultilateral problems of fair-division. Our procedural approach eliminates\nenvy in a ‘‘natural’’ way: it first identifies the players who are non-envious (we\nshowed that there will always be at least one if the assignment of bundles\nis e‰cient), and then it compensates those players whose maximum envy is\ndirected towards non-envious players. By first beginning with an e‰cient as-\nsignment, we formulated the procedure for establishing envy-freeness using\nintuitive, plausible, and manageably simple steps. We then showed that the\nprocedure is also capable of guiding parties to an e‰cient assignment when\nthe bundles are fixed. Moreover, the outcome of the compensation procedure\nis the same no matter which e‰cient assignment is eventually reached.\nIn our analysis, we placed no bound on the level of individual compensa-\nBidding for envy-freeness 745\ntions. Consequently, we do not preclude the possibility that an individual may\nend up being paid by the others to take a bundle of goods. In the context of fair\ndivision we do not find this problematic at all. Indeed, if a group does not wish\nto exclude any of its members, then there is no reason why the group should\nnot subsidize a member for receiving an undesired bundle. Moreover, the qual-\nification constraint guarantees that subsidization is never a consequence of a\nplayer’s insu‰cient valuation of the complete set of objects to be distributed.\nAs with all cooperative procedures, our compensation scheme is theoreti-\ncally vulnerable to strategic manipulation. However, practically speaking, this\nrequires detailed information on the other players’ preferences. If players’ bids\nare disclosed simultaneously, strategic bids under incomplete information can\neasily backfire: an untruthful player may have to pay more than what she\nthinks her bundle is worth or may envy some other players for their bundles.\nBy contrast, since a player’s compensation is based on her own subjective\nassessments, truthful behavior will always guarantee her envy-freeness, regard-\nless of the others’ behavior. In practice, this insurance creates a disincentive\nfor attempting to distort one’s preferences.\nImplemented as a computer algorithm, our compensation procedure is\npolynomially bounded, thus making it comparable to alternative algorithms.\nBut this is not our point: as a procedure, it is not meant to be run on a machine;\ninstead it is designed to be used live in a mediation process.\nAppendix\nEnvy-free division of burdens\nConsider the following example of four players that have to divide a number of\nchores among each other. The total compensation for all chores is: \nC ¼ 300.\nThe utilitarian assignment bundles the chores and thereby determines the min-\nimum total requested payment \nM > 0. Table 5 shows players (negative) bids\nfor the four bundles of burdens.17\n17 The entries in Table 5 are equal to the entries in Table 1 minus 100.\nTable 5. Players’ requested payments\nB1 B2 B3 B4\nP1 \n50 \n80 \n90 \n80\nP2 \n40 \n60 \n85 \n90\nP3 \n100 \n60 \n75 \n65\nP4 \n50 \n65 \n90 \n70\nInitial reward 50 60 75 70\n746 C.-J. Haake et al.\nPlayers’ demands under the utilitarian assignment (along the diagonal) are\nsatisfied through initial payments, which sum to 255. The total compensation\nof 300 thus leaves an extra 45 to be granted to the players in the form of\nbonuses. The assessment matrix can be computed by subtracting the diagonal\nentry from each column.\nThe resulting initial assessment matrix is identical to the one in Table 2\nwhere row i now shows Player i ’s assessment of Player j’s bundle of chores.\nConsequently, the remaining surplus is also the same as in the example of\nSect. 3. Hence, the compensation procedure together with the distribution\nof the remaining surplus yields the same outcome, whether for the distribution\nof goods with a total contribution, or for the distribution of chores with a\ntotal compensation, or for a combination of goods and chores with a total\ncost or compensation.\nNumerical calculations for the average discount method\nIn what follows, (i)–(iv) denote the steps of the average discount method\nalgorithm.\nBeginning with Player 1, the procedure places this player in set D (i). Since\nPlayer 2 sees her bundle tied with that of Player 1 (see Row 2 of Table 4),\nPlayer 2 is added to set D (ii). In further comparisons, one now finds that\nPlayers 3 and 4 see their bundles tied with that of Player 2, so they must be\nadded to D as well (ii). With all players included in D, the remaining surplus is\ndivided equally among them (iii). The updated discounts are ð5; 15; 15; 10Þ,\nand the procedure for Player 1 ends because the surplus is used up (iv).\nBeginning with Player 2 (i), Players 3 and 4 must be added to set D (ii).\nWith only these players included in D, each one’s discount can be raised by 5\nbefore Player 1 experiences envy (see Row 1 of Table 4). This leaves a remain-\ning surplus of 20 \n 15 ¼ 5 (iii), so the procedure continues (iv). Player 1 now\nsees her bundle tied with those of Players 3 and 4, so she must be added to D\n(ii). The remaining surplus is divided equally among all players giving each an\nadditional 1.25 (iii). The discounts are ð1:25; 16:25; 16:25; 11:25Þ, and the pro-\ncedure for Player 2 ends (iv).\nBeginning with Player 3 in set D (i), there is no player who feels tied with\nthis player (ii). Player 3’s discount can thus be raised by 5 without making any\nother player envious (iii). With a remaining surplus of 20 \n 5 ¼ 15, the pro-\ncedure continues (iv). Now Player 1 is tied with Player 3, Player 2 is tied with\nPlayer 1 and Player 4 is tied with Player 2, so they all must be added to D (ii).\nThe remaining surplus is divided equally, giving each player an additional 3.75\n(iii). The discounts are now ð3:75; 13:75; 18:75; 8:75Þ, and the procedure for\nPlayer 3 ends (iv).\nBeginning with Player 4 in set D (i), Player 3 must be added because of\nthe tie with Player 4 (ii). The discounts of these two players can be raised\nby 5 before Player 1 becomes envious (iii). With a remaining surplus of\n20 \n 10 ¼ 10, the procedure continues (iv). Now Player 1 is added to D and,\ntherefore, also Player 2, because of the tie with Player 1 (ii). The remaining\nBidding for envy-freeness 747\nsurplus is divided equally among all players, giving each an additional 2.5 (iii).\nThe discounts are ð2:5; 12:5; 17:5; 12:5Þ, and the procedure for Player 4 ends\n(iv).\nEliminating envy-cycles with the compensation procedure\nThe following example shows how our modified compensation procedure\nwith ex-post equal payments works for a non-utilitarian assignment. Consider\nthe bid matrix shown in Table 6, which reproduces Table 1 with columns\nexchanged. The framed bids indicate the non-utilitarian assignment. The cor-\nresponding envy graph, constructed on the basis of bids, is denoted by ~G.\nStrictly following the compensation procedure, Players 1 and 4 are now\ncompensated by 30 and 15. (Player 3 is not yet compensated, because she\nenvies an envious player.) Accordingly, the bid matrix changes to Table 7.\nNow Players 3 and 4 envy Player 1. In order to eliminate their envy, they\nreceive compensations of 40 and 10, respectively. The result is shown in Table\n8. However, this second round of compensations involving the last envious\nplayers (3 and 4) creates new envy: Player 4 feels tied with 1 and Player 1 feels\ntied with 2, but Player 2 now envies 4, thus creating a cycle in the directed\ngraph ~G.\nThe modified procedure now calls for a trade in the opposite direction of\nthe arrows in ~G. Hence, Player 4 gives her bundle to Player 2, Player 2 gives\nher bundle to Player 1, and Player 1 gives her bundle to Player 2. In addi-\nTable 6. Players’ bids before compensations are made\nB4 B1 B3 B2\nP1 20 50 10 20\nP2 10 60 15 40\nGraph ~G\nP3 35 0 25 40\n1 !\nP4 30 50 10 35\n3 ! 4 !\n2\n0 0 0 0\nTable 7. Players’ valuations of bundles after the first round of compensations\nB4 B1 B3 B2\nP1 50 50 10 35\nP2 40 60 15 55\nGraph ~G\nP3 65 0 25 55\n3 !\nP4 60 50 10 50\n4 !\n1 ) 2\n30 0 0 15\n748 C.-J. Haake et al.\ntion all compensations must be returned, and the compensation procedure is\nrestarted. Note that the bid matrix after the required permutation of columns\nis now given by Table 1. As we know this new assignment is utilitarian and the\ncompensation procedure with ex-post payments will lead to an envy-free out-\ncome.\nReferences\nAlkan A, Demange G, Gale D (1991) Fair allocation of indivisible goods and criteria\nof justice. Econometrica 59: 1023–1039\nAragones E (1995) A derivation of the money Rawlsian solution. Soc Choice Welfare\n12: 267–276\nBrams SJ, Kilgour DM (2001) Competitive fair division. J Political Econ (forthcoming)\nBrams SJ, Taylor AD (1996) Fair division – from cake-cutting to dispute resolution.\nCambridge University Press, Cambridge, MA\nFoley DK (1967) Resource allocation and the public sector. Yale Econ Essays 7: 45–98\nKlijn F (2000) An algorithm for envy-free allocations in an economy with indivisible\nobjects and money. Soc Choice Welfare 17: 201–216\nKnaster B (1946) Sur le problème du partage pragmatique de H Steinhaus. Ann Soc\nPolonaise Math 19: 228–230\nSteinhaus H (1948) The problem of fair division. Econometrica 16: 101–104\nSu FE (1999) Rental harmony: Sperner’s lemma in fair division. Am Math Monthly\n106: 930–942\nSvensson LG (1983) Large indivisibles: An analysis with respect to price equilibrium\nand fairness. Econometrica 51: 939–954\nTadenuma K, Thomson W (1991) No-envy and consistency in economies with indi-\nvisible goods. Econometrica 59: 1755–1767\nTable 8. Players’ valuations of bundles after the second round of compensations\nB4 B1 B3 B2\nP1 50 50 50 45\nP2 40 60 55 65\nGraph ~G\nP3 65 0 65 65\n3 )\nP4 60 50 50 60\n4 )\n1 ) 2 ! 4\n30 0 40 25\nBidding for envy-freeness 749\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! binomial coefficients. Misleading Sequence Posted on June 26, 2019. by. Samuel Nunoo Place n points along a unit circle, in such a way that when you draw all lines connecting every pair of... Continue reading... Zero to the Zero Power Posted on June 26, 2019. by. Samuel Nunoo It is commonly taught that any number to the zero power is 1, and zero to any power is 0.... Continue reading... Pascal’s Triangle Posted on June 26, 2019. by. Samuel Nunoo Consider the triangle in Figure 1, called Pascal’s triangle. It consists of numbers where each entry is the sum of the... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! functions. Devil’s Staircase Posted on June 29, 2019. by. Samuel Nunoo Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically... Continue reading... Multidimensional Newton’s Method Posted on June 29, 2019. by. Samuel Nunoo You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did... Continue reading... Sum of Cubes and Beyond Posted on June 29, 2019. by. Samuel Nunoo We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the... Continue reading... Continuous but Nowhere Differentiable Posted on June 29, 2019. by. Samuel Nunoo You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have... Continue reading... Euler’s Product Formula Posted on June 29, 2019. by. Samuel Nunoo Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it... Continue reading... Connected Sums Posted on June 27, 2019. by. Samuel Nunoo A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,... Continue reading... Fourier Ears Only Posted on June 26, 2019. by. Samuel Nunoo Did you know that every sufficiently smooth function on an interval can be expressed as an infinite sum of sines... Continue reading... Quick Square Roots Posted on June 26, 2019. by. Samuel Nunoo Here’s a very quick way to generate the square root of N. Let A0=N. Then generate a sequence of numbers A1, A2,... Continue reading... Sine of (1\/55555….) Posted on June 26, 2019. by. Samuel Nunoo Take your calculator, and enter a number consisting of several 5’s, such as 5555555. Now take the reciprocal of this number. Now take the... Continue reading... Area of a Hyperbolic Wedge Posted on June 26, 2019. by. Samuel Nunoo The unit circle can be parametrized by (cos w, sin w). Given a point on it, the region cut out... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! ellipse. Ellipsoidal Paths Posted on June 29, 2019. by. Samuel Nunoo Given an ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a... Continue reading... Chords of an Ellipse Posted on June 29, 2019. by. Samuel Nunoo Consider N equally spaced on points on the unit circle, with the point P=(1,0) as one of these equally spaced... Continue reading... Eccentricity of Conics Posted on June 26, 2019. by. Samuel Nunoo To each conic section (ellipse, parabola, hyperbola) there is a number called the eccentricity that uniquely characterizes the shape of the curve. A circle... Continue reading... Area of an Ellipse Posted on June 26, 2019. by. Samuel Nunoo You know the formula for the area of a circle of radius R. It is Pi * R2. But what... Continue reading... Search Fun Facts. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! arithmetic. Greedy to Avoid Progressions Posted on June 29, 2019. by. Samuel Nunoo An arithmetic progression is a sequence of 3 or more integers whose terms differ by a constant, e.g., 20, 23, 26, 29... Continue reading... Van der Waerden Theorem Posted on June 29, 2019. by. Samuel Nunoo Can you color the integers red and blue such that there are no monochromatic arithmetic progressions (AP’s) extending infintely in... Continue reading... Four Fours Problem Posted on June 26, 2019. by. Samuel Nunoo Here’s a challenge that you may wish to try: can you express all the numbers from 1 to 100 using... Continue reading... Proof with Words Posted on June 26, 2019. by. Samuel Nunoo 2 + 11 – 1 = 12. Figure 1 demonstrates a “proof with words”! Presentation Suggestions:Write the letters for two and... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Jun 08 17:22:54 UTC 2022","fresh_date_dt":"Wed Jun 08 17:22:55 UTC 2022","referrer_id_s":"J9NUmP51bhow","language_s":"en","size_i":46724,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":28,"references_internal_i":28,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088118167502848,"text_t":"Birthday Problem – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Birthday Problem. How many people do you need in a group to ensure at least a 50 percent. probability that 2 people in the group share a birthday? Let’s take a show of hands. How many people think 30 people is enough? 60? 90? 180? 360? Surprisingly, the answer is only 23 people to have at least a 50 percent chance of a match. This goes up to 70 percent for 30 people, 90 percent for 41 people, 95 percent for 47 people. With 57 people there is better than a 99 percent chance of a birthday match! Presentation Suggestions:. If you have a large class, it is fun to try to take a poll of birthdays: have people call out their birthdays. But of course, whether or not you have a match proves nothing…. The Math Behind the Fact:. Most people find this result surprising because they are tempted to calculate the. probability of a birthday match with one particular person. But the calculation should be done over all pairs of people. Here is a trick that makes the calculation easier. To calculate the. probability of a match, calculate the probability of no match and subtract from 1. But the probability of no match among n people is just (365\/365)(364\/365)(363\/365)(362\/365)…((366-n)\/365), where the k-th term in the product arises from considering the probability that the k-th person in the group doesn’t have a birthday match with the (k-1) people before her. If you want to do this calculation quickly, you can use an approximation: note that for. i. much smaller than. 365. , the term (1-i\/365) can be approximated by EXP(-i\/365). Hence, for n much smaller than 365, the. probability of no match is close to. EXP( – SUM. i=1 to (n-1). i\/365) = EXP( – n(n-1)\/(2*365)). When n=23, this evaluates to 0.499998 for the. probability of no match. The probability of at least one match is thus 1 minus this quantity. For still more fun, if you know some probability: to find the. probability that in a given set of n people there are exactly M matches, you can use a Poisson approximation. The Poisson distribution is usually used to model a random variable that counts a number of “rare events”, each independent and identically distributed and with average frequency lambda. Here, the. probability of a match in a given pair is 1\/365. The matches can be considered to be approximately independent. The frequency lambda is the product of the number of pairs times the probability of a match in a pair: (n choose 2)\/365. Then the approximate probability that there are exactly M matches is: (lambda). M. * EXP(-lambda) \/ M! which gives the same formula as above when M=0 and n=-365. How to Cite this Page:.  . Su, Francis E., et al. “Birthday Problem.” . Math Fun Facts. . . Fun Fact suggested by:. Francis Su. Did you like this Fun Fact? Click to rate it. Submit Rating. Average rating. 4.7. \/ 5. Vote count:. 15. No votes so far! Be the first to rate this Fun Fact. Share:. Tags:. demonstration ,. easy ,. probability Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved."}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! multivariable calculus. Volume of a Ball in N Dimensions Posted on June 29, 2019. by. Samuel Nunoo The unit ball in Rn is defined as the set of points (x1,…,xn) such that x12 + … + xn2 . Continue reading... Impossible Integral? Posted on June 29, 2019. by. Samuel Nunoo The following integral may be problematic for a freshman calculus student, even if armed with a list of antiderivatives: INTEGRAL0... Continue reading... Polyhedral Face Vectors Posted on June 26, 2019. by. Samuel Nunoo A polyhedron is any closed region of 3-space cut out by a finite set of planes. Take any polyhedron and do the... Continue reading... Spherical Bands Posted on June 26, 2019. by. Samuel Nunoo Take any sphere and slice it up by parallel knives 1 inch apart. The areas of the spherical bands between the cuts... Continue reading... Right Double Cone Posted on June 26, 2019. by. Samuel Nunoo Take a solid cylinder of radius R and height 2R. Remove the right “double cone” which passes through the center... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! analysis. Does Order of Addition Matter? Posted on June 29, 2019. by. Samuel Nunoo From your earliest days of math you learned that the order in which you add two numbers doesn’t matter: 3+5... Continue reading... Dedekind Cuts of Rational Numbers Posted on June 29, 2019. by. Samuel Nunoo Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the... Continue reading... Rationals Dense but Sparse Posted on June 29, 2019. by. Samuel Nunoo Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the... Continue reading... Devil’s Staircase Posted on June 29, 2019. by. Samuel Nunoo Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically... Continue reading... Space-filling Curves Posted on June 29, 2019. by. Samuel Nunoo Consider a square in the plane. Is it possible to draw a curve in the square that touches every point inside the... Continue reading... Rational Irrational Power Posted on June 29, 2019. by. Samuel Nunoo If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to... Continue reading... Multidimensional Newton’s Method Posted on June 29, 2019. by. Samuel Nunoo You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did... Continue reading... Cantor Diagonalization Posted on June 29, 2019. by. Samuel Nunoo We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as... Continue reading... Continuous but Nowhere Differentiable Posted on June 29, 2019. by. Samuel Nunoo You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have... Continue reading... Euler’s Product Formula Posted on June 29, 2019. by. Samuel Nunoo Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"erIiXp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/about\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":36,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["funfacts","about"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["About Math Fun Facts – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:55 UTC 2022","dates_in_content_count_i":1,"dates_in_content_dts":["Tue Jul 20 00:00:00 UTC 1999"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-4282837363532988858,"exact_signature_unique_b":true,"fuzzy_signature_l":-4861958382450615634,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["About Math Fun Facts"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"bold_txt":["Disclaimer:"],"imagescount_i":5,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Faucet.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2021\/02\/su-lake-louise-tight-square-1024x1024.jpeg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,1024,294,-1],"images_width_val":[-1,-1,1024,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Faucet 2021 02 su lake louise tight square 1024x1024 jpeg 1024x 1024 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":658,"text_t":"About Math Fun Facts – Math Fun Facts. Skip to content Expand Menu. Current Page:. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! About Math Fun Facts. I developed Math Fun Facts in 1994 as a warm-up activity for the. calculus courses I taught as a graduate student at Harvard. Calculus was the final math course that most of my students would take, and I worried that they would emerge from college with the mistaken notion that all of mathematics was “just more calculus”. Most college students never see the interesting stuff that motivates mathematicians to study the subject. So, I began to tell them “Fun Facts”—daily mathematical tidbits from all areas of mathematics (not just calculus), meant to arouse their curiosity and fascination with the subject. Fun Facts give students a glimpse that mathematics is full of interesting ideas, patterns, and new modes of thinking. The student reaction to Fun Facts was highly positive. Once I started, I couldn’t stop—students would clamor for them if I forgot to give one. They stopped me after class to discuss them, began to create their own, and were motivated to take. other math courses because of the Fun Facts they learned. I got the idea from something that one of my college professors (. John Jones ) did; he once wrote an interesting. infinite series on the board and called it a “fun fact”. I thought to myself: why not do something like this everyday and include brief fun facts from all areas of mathematics? I told myself that if I became a professor someday, I would try it…. So here, on this site, is an archive of many of the Fun Facts that my colleagues and I have collected. I launched this site on July 20, 1999 so that others might use it as a resource and repository of ideas, and it continues to grow. I hope you enjoy this site… do feel free to visit often! Best wishes,. Francis Su. July 1999. Disclaimer:. I make no representation, neither express nor implied, as to the accuracy, completeness, timeliness, or reliability of the materials or information or external links provided on this website. By using this site, you agree that your access to, and usage of, the content of this site is on an “as-is” basis, and in no event shall the author, designers, hosts, institution, affiliates, or any. other persons associated with the creation or management of this site be liable for any direct or indirect damages arising out of or related to the use, inability to use, performance or non-performance of this site. Tags:. calculus ,. infinite series ,. other Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved. Loading Comments... Write a Comment... Email (Required). Name (Required). Website.","wordcount_i":633,"linkscount_i":83,"linksnofollowcount_i":2,"inboundlinkscount_i":75,"outboundlinkscount_i":8,"inboundlinks_protocol_sxt":["000-https","001-https","002-https","003-https","004-https","005-https","006-https","007-https","008-https","009-https","010-https","011-https","012-https","013-https","014-https","015-https","016-https","017-https","018-https","019-https","020-https","021-https","022-https","023-https","024-https","025-https","026-https","027-https","028-https","029-https","030-https","031-https","032-https","033-https","034-https","035-https","036-https","037-https","038-https","039-https","040-https","041-https","042-https","043-https","044-https","045-https","046-https","047-https","048-https","049-https","050-https","051-https","052-https","053-https","054-https","055-https","056-https","057-https","058-https","059-https","060-https","061-https","062-https","063-https","064-https","065-https","066-https","067-https","068-https","069-https","070-https","071-https","072-https","073-https","074-https"],"inboundlinks_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/plugins\/rate-my-post\/public\/css\/fonts\/ratemypost.ttf","math.hmc.edu\/funfacts\/wp-json\/","math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/852","math.hmc.edu\/funfacts\/xmlrpc.php?rsd=","math.hmc.edu\/funfacts\/wp-includes\/wlwmanifest.xml","math.hmc.edu\/funfacts\/about\/","math.hmc.edu\/funfacts\/?p=852","math.hmc.edu\/funfacts\/wp-json\/oembed\/1.0\/embed?url=https%3A%2F%2Fmath.hmc.edu%2Ffunfacts%2Fabout%2F","math.hmc.edu\/funfacts\/wp-json\/oembed\/1.0\/embed?url=https%3A%2F%2Fmath.hmc.edu%2Ffunfacts%2Fabout%2F&","math.hmc.edu\/funfacts\/","math.hmc.edu\/funfacts\/howto\/","math.hmc.edu\/funfacts\/fun-facts-contributors\/","math.hmc.edu\/funfacts\/all\/","www.math.hmc.edu\/funfacts\/random","math.hmc.edu\/funfacts\/tag\/calculus\/","math.hmc.edu\/funfacts\/tag\/other\/","math.hmc.edu\/funfacts\/tag\/infinite-series\/","math.hmc.edu\/funfacts\/tag\/easy","math.hmc.edu\/funfacts\/tag\/medium","math.hmc.edu\/funfacts\/tag\/advanced","math.hmc.edu\/funfacts\/tag\/algebra","math.hmc.edu\/funfacts\/tag\/calculus","math.hmc.edu\/funfacts\/tag\/combinatorics","math.hmc.edu\/funfacts\/tag\/geometry","math.hmc.edu\/funfacts\/tag\/numtheory","math.hmc.edu\/funfacts\/tag\/probability","math.hmc.edu\/funfacts\/tag\/topology","math.hmc.edu\/funfacts\/tag\/other","math.hmc.edu\/funfacts\/fibonacci-gcds-please\/","math.hmc.edu\/funfacts\/birthday-problem\/","math.hmc.edu\/funfacts\/brouwer-fixed-point-theorem\/","math.hmc.edu\/funfacts\/napoleons-theorem\/","math.hmc.edu\/funfacts\/does-order-of-addition-matter\/","math.hmc.edu\/funfacts\/tag\/advanced\/","math.hmc.edu\/funfacts\/tag\/algebra\/","math.hmc.edu\/funfacts\/tag\/analysis\/","math.hmc.edu\/funfacts\/tag\/approximations\/","math.hmc.edu\/funfacts\/tag\/arithmetic\/","math.hmc.edu\/funfacts\/tag\/bicycle\/","math.hmc.edu\/funfacts\/tag\/binomialcoefficients\/","math.hmc.edu\/funfacts\/tag\/combinatorics\/","math.hmc.edu\/funfacts\/tag\/complex-numbers\/","math.hmc.edu\/funfacts\/tag\/demonstration\/","math.hmc.edu\/funfacts\/tag\/easy\/","math.hmc.edu\/funfacts\/tag\/ellipse\/","math.hmc.edu\/funfacts\/tag\/fibonacci\/","math.hmc.edu\/funfacts\/tag\/functions\/","math.hmc.edu\/funfacts\/tag\/game-theory\/","math.hmc.edu\/funfacts\/tag\/geometric-series\/","math.hmc.edu\/funfacts\/tag\/geometry\/","math.hmc.edu\/funfacts\/tag\/hard\/","math.hmc.edu\/funfacts\/tag\/irrational\/","math.hmc.edu\/funfacts\/tag\/lightning-arithmetic\/","math.hmc.edu\/funfacts\/tag\/logic\/","math.hmc.edu\/funfacts\/tag\/matrix\/","math.hmc.edu\/funfacts\/tag\/medium\/","math.hmc.edu\/funfacts\/tag\/multivariable-calculus\/","math.hmc.edu\/funfacts\/tag\/numtheory\/","math.hmc.edu\/funfacts\/tag\/paradox\/","math.hmc.edu\/funfacts\/tag\/pi\/","math.hmc.edu\/funfacts\/tag\/pi-formula\/","math.hmc.edu\/funfacts\/tag\/polynomial\/","math.hmc.edu\/funfacts\/tag\/prime\/","math.hmc.edu\/funfacts\/tag\/probabilities\/","math.hmc.edu\/funfacts\/tag\/probability\/","math.hmc.edu\/funfacts\/tag\/puzzle\/","math.hmc.edu\/funfacts\/tag\/real-analysis\/","math.hmc.edu\/funfacts\/tag\/riemann-zeta-function\/","math.hmc.edu\/funfacts\/tag\/sphere\/","math.hmc.edu\/funfacts\/tag\/taylor-series\/","math.hmc.edu\/funfacts\/tag\/topology\/","math.hmc.edu\/funfacts\/tag\/triangles\/","math.hmc.edu\/funfacts\/tag\/unit-circle\/","math.hmc.edu\/funfacts\/tag\/universe\/","math.hmc.edu\/funfacts\/tag\/unsolved-problem\/"],"inboundlinks_anchortext_txt":["","","","","","","","","","","How 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{"id":"dQSK9W51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/LINEAR-RECURRENCES.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":82,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"LINEAR-RECURRENCES","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["LINEAR-RECURRENCES.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:30 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":1803249181320454024,"exact_signature_unique_b":true,"fuzzy_signature_l":4659696498839949730,"fuzzy_signature_unique_b":true,"wordcount_i":4947,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:57 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:10 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":217976,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088120303452160,"text_t":"LINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS\nCHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nAbstract. We present a tiling interpretation for kth order linear recurrences, which yields new\ncombinatorial proofs for recurrence identities. Moreover, viewing the tiling process as a Markov\nchain also yields closed form Binet-like expressions for these recurrences.\n1. Introduction\nThe theme of this paper is the use of tilings and a random tiling process as a general method\nfor understanding and proving identities involving k-th order linear recurrences. Typically, such\nidentities are proved by algebraic means (such as induction or generating functions) which generally\ngives very little insight into their nature; by contrast, our combinatorial approach enables visual\ninterpretations of such identities— facilitating a clearer understanding of them, unifying them, and\nmaking them (and their proofs) easy to remember.\nA simple example of this approach is the well-known interpretation [4] of Fibonacci numbers\n(generated by the recurrence fn = fn−1 + fn−2, f0 = 1, f1 = 1) as the number of ways to tile\nan n × 1 board using squares and dominoes. Less well-known is an analogous interpretation for\na recurrence with different initial conditions; these involve phased tilings and are treated in [2],\nwhere they are used as a unifying method for proving a host of identities in [11]. In this paper,\nwe develop a tiling interpretation for higher-order recurrences with non-negative integer coefficients\nand arbitrary initial conditions. We show how this interpretation can be used to prove associated\nrecurrence identities as well as a new closed form expression for the n-th term of such recurrences,\nextending formulae of [5, 8, 9].\nThe paper is organized as follows. In Section 2, we analyze 2nd-order recurrences for motivation,\nand demonstrate the usefulness of this combinatorial model in deriving several identities. This is\nfollowed in Section 3 by a general tiling interpretation for k-th order linear recurrences. Section 4\n1\n2 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\n3 4 5 8 910 2 6 7\nblue redyellowgreenred\"phased\"\nFigure 1. A 9-board tiled with colored squares and dominoes. The first tile is\ngiven a phase instead of a color.\nbuilds on this interpretation with a Markov chain model to derive a Binet-like formula for 3-bonacci\nsequences, and this is generalized in Section 5 to handle k-th order linear recurrences with special\ninitial conditions, and further generalized in Section 6 to handle arbitrary initial conditions.\n2. 2nd Order Linear Recurrences\nLet s, t, H0, and H1 be real numbers, and for n ≥ 2, define\nHn = sHn−1 + tHn−2.(1)\nWhen s, t, H0 andH1 are non-negative integers, thenHn can be given a combinatorial interpretation.\nWe define an n-board to be an array of n cells, numbered 1 through n. See Figure 1 for a typical\nn-board, covered with colored square and domino tiles.\nTheorem 1. For n ≥ 1, Hn counts the number of ways to cover an n-board with (length one)\nsquares and (length two) dominoes, where all tiles, except for the initial one, are given a color.\nThere are s colors for squares and t colors for dominoes. The initial tile is given a phase and there\nare H1 phases for an initial square and tH0 phases for an initial domino.\nProof. The number of ways to tile a 1-board is H1. The number of ways to tile a 2-board (with two\nsquares or a single domino) is sH1 + tH0 = H2. For n > 2, by conditioning on whether the last tile\nis a square or domino, we have Hn = sHn−1 + tHn−2.\nThis interpretation allows us to combinatorially explain many identities for sequences generated\nby second order recurrences. We illustrate with several examples. In the identities that follow we\nassume that all quantities are non-negative. We shall relax this assumption in the next section.\nIdentity 1. t\n∑n\nk=0 s\nn−kHk = Hn+2 − sn+1H1.\nLINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS 3\nProof. The right side of this identity counts the number ways to tile an (n+ 2)-board, excluding the\ntilings consisting of all squares. It remains to show that the left side counts the same quantity.\nSpecifically, we show that the left side counts the number of (n+ 2)-tilings where the last domino\noccupies cells k + 1 and k + 2 for some 0 ≤ k ≤ n. For 1 ≤ k ≤ n, there are Hk ways to tile cells 1\nthrough k, t ways to color the domino on cells k + 1 and k + 2, and sn−k ways to color the squares\non cells k+ 3 to n+ 2; consequently there are tHks\nn−k such tilings. When k = 0 there are tH0 ways\nto choose the initial domino, and sn ways to color the subsequent squares, resulting in tH0s\nn tilings.\nAltogether, we have t\n∑n\nk=0 s\nn−kHk tilings of an (n+ 2)-board with at least one domino.\nWhen tiling a board of even length 2n, the last square, if it exists, must cover an even cell 2k for\nsome 1 ≤ k ≤ n. The preceding squares are tiled H2k−1 ways and the last square and subsequent\nn− k dominoes can be colored stn−k ways. Consequently, we have\nIdentity 2. H2n = H0t\nn + s\n∑n\nk=1 t\nn−kH2k−1,\nwhere the first term on the right side enumerates the all-domino tilings.\nSimilarly, when tiling a (2n + 1)-board, a last square must exist at some cell 2k + 1 for some\n0 ≤ k ≤ n. Separating the k = 0 case from the rest leads to\nIdentity 3. H2n+1 = H1t\nn + s\n∑n\nk=1 t\nn−kH2k.\nThe next identity invites a more intricate, but natural interpretation on pairs of tilings.\nIdentity 4. H22n = t\n2nH20 + s\n∑2n\nk=1 t\n2n−kHk−1Hk.\nProof. The quantity on the left side counts the number of ordered pairs (A,B), where A and B\nare (2n)-tilings. The first term on the right side counts those (A,B) where A and B consist only\nof dominoes. For any (2n)-tiling X, let kX to be the last cell of tiling X covered by a square.\nIf X is all dominoes, set kX to infinity. For (A,B) to have at least one square, the minimum\nof kA and kB must be finite and even. Let k = max{kA, kB − 1}. When k is even, A and B\nhave dominoes covering cells k + 1 through 2n and A has a square at cell k. In this way, the\nnumber of tilings (A,B) with even k is the number of ways to tile A times the number of ways\n4 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nto tile B, i.e. Hk−1st\n(2n−k)\/2 · Hkt(2n−k)\/2 = st2n−kHk−1Hk. When k is odd, A has dominoes\ncovering cells k through 2n and B has dominoes covering cells k+ 2 through 2n and a square at cell\nk + 1, so the number of tiling pairs is Hk−1t\n(2n−k+1)\/2 ·Hkst(2n−k−1)\/2 = sHk−1Hkt2n−k, the same\nexpression as for even k. Altogether the number of tiling pairs (A,B) with at least one square is\ns\n∑2n\nk=1 t\n2n−kHk−1Hk.\nA similar approach easily leads to\nIdentity 5. H22n+1 = t\n2nH21 + s\n∑2n+1\ni=2 t\n2n+1−iHi−1Hi.\nThe above formulae are just a few examples of identities for 2nd-order linear recurrences that can\nbe easily assimilated, explained, and remembered by our combinatorial interpretation.\n3. k-th Order Linear Recurrences\nIn this section we present a combinatorial interpretation of sequences generated by k-th order lin-\near recurrences with non-negative integer coefficients. Specifically, by the reasoning and terminology\nof the last section (i.e., by conditioning on the last tile), we obtain the following theorem.\nTheorem 2. Given non-negative integers c1, c2, . . . , ck, a0, a1, . . . , ak−1, consider for n ≥ k, the\nlinear recurrence\nan = c1an−1 + c2an−2 + · · ·+ ckan−k.(2)\nThen for n ≥ 1, an counts the number of ways to tile an n-board using colored tiles of various lengths\nwhere each tile, except the initial one, has a color. Specifically, for 1 ≤ i ≤ k, each tile of length i\nmay be assigned any of ci different colors. The initial tile is assinged a phase and if it has length\n1 ≤ i ≤ k, the number of phases for that tile is\npi = ai −\ni−1∑\nj=1\ncjai−j .(3)\nIn particular, an initial square has p1 = a1 phases, and by (2) and (3), it follows that pk = cka0.\nFor k-th order recurrences where k > 2, this combinatorial interpretation is only valid when the\ninitial conditions a0, a1, . . . , ak−1 are sufficiently “spread out” so that for 1 ≤ i < k, the number of\nphases pi is non-negative. This restriction will be removed later in this section.\nLINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS 5\nTo demonstrate the utility of our combinatorial interpretation, we give new proofs of the following\nthree identities for “generalized Tribonacci” sequences that were originally proved using matrix\nmethods in [12].\nIdentity 6. Consider the sequence generated by integers a0, a1, a2 and for n ≥ 3, an = c1an−1 +\nc2an−2 + c3an−3. Then for n ≥ 0,\ncn1 (c3a0 + c1a2) + (c3 + c1c2)\nn∑\ni=1\ncn−i1 ai = c1an+2 + c3an.\nProof. In this identity, we tile an (n+ 3) board with squares, dominoes and 3-ominoes. Each tile of\nlength i, unless it is the first tile of our tiling, has ci color choices. If the first tile has length i, then\nit can be phased pi ways, where p1 = a1, p2 = a2 − c1a1, and p3 = c3a0. For n ≥ 1, (when n = 0,\nthe statement is obvious), the right side of our identity counts the number of ways to tile a board\nof length n+ 3 such that the last tile is either a colored square (which can be preceded an+2 ways)\nor a colored 3-omino (which can be preceded an ways).\nTo combinatorially interpret the left side, we first count, for 1 ≤ i ≤ n, tilings whose last domino\nor 3-omino begins at cell i+ 1. There are (c3 + c1c2)c\nn−i\n1 ai such tilings since cells i+ 1, i+ 2, i+ 3\nconsists of either a 3-omino or a domino followed by a square (c3 + c1c2 choices), the tiles in front of\ncell i+3 must all be squares (cn−i1 choices) and cells 1 through i may be tiled arbitrarily (ai choices).\nNote that by our ending condition, a last domino may not begin at cell (n+ 2). The only uncounted\ntilings are those with only squares on cells 4 through n+3 (cn1 choices) and cells 1 through 3 contains\neither a phased 3-omino (p3 = c3a0 choices) or has a square at cell 3 (c1a2 choices). Altogether, our\ntilings can be constructed cn1 (c3a0 + c1a2) + (c3 + c1c2)\n∑n\ni=1 c\nn−i\n1 ai ways.\nIn similar fashion, we can also prove for sequences {an} generated by the same recurrence and\ninitial conditions,\nIdentity 7. For n ≥ 1,\ncn2 (a2 − c1a1) + (c3 + c1c2)\nn∑\ni=1\ncn−i2 a2i−1 = c2a2n + c3a2n−1.\n6 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nProof. Here the right side of the identity counts (2n + 2)-tilings that are restricted to end with\na domino or 3-omino. For the left side, we first count, for 1 ≤ i ≤ n, tilings whose last square\nor 3-omino begins at cell 2i. Such a tiling has cells 2i, 2i + 1, and 2i + 2 consisting of either a\n3-omino or a square followed by a domino (c3 + c1c2 choices) preceded by an arbitrary (2i − 1)-\ntiling (a2i−1 choices) and followed by all dominoes (c\nn−i\n2 choices). The only uncounted tiles are\nthe all-domino tilings (p2c\nn\n2 = (a2 − c1a1)cn2 choices). Altogether our tilings can be constructed\ncn2 (a2 − c1a1) + (c3 + c1c2)\n∑n\ni=1 c\nn−i\n2 a2i−1 ways.\nBy a similar argument, this time with boards of odd length, we obtain:\nIdentity 8. For n ≥ 1,\ncn−12 (c3a0 + c2a1) + (c3 + c1c2)\nn−1∑\ni=1\ncn−1−i2 a2i = c2a2n−1 + c3a2n−2.\nFinally, we illustrate the power of the combinatorial approach by establishing the next identity,\nproved by more sophisticated methods in [7] and [8]:\nIdentity 9. Let gn be the k-th order Fibonacci sequence defined by g0 = 1, and for 1 ≤ n < k,\ngn = gn−1 + gn−2 + · · ·+ g0. For n ≥ k, gn = gn−1 + gn−2 + · · ·+ gn−k. Then for n ≥ 0,\ngn =\n∑\nn1\n∑\nn2\n· · ·\n∑\nnk\n(\nn1 + n2 + · · ·nk\nn1, n2, . . . , nk\n)\n,\nwhere the summation is over all non-negative integers n1, n2, . . . , nk such that n1 +2n2 + · · ·+knk =\nn.\nProof. Here gn counts the number of ways to tile an n-board with (colorless and phaseless) tiles of\nlength at most k. (This may seen directly by conditioning on the last tile or one can derive from\nequation (3) that for 1 ≤ j ≤ k, cj = 1 and pj = 2j−1 − (2j−1 − 1) = 1.) The right side of this\nidentity conditions on how many such tilings use exactly ni tiles of length i for 1 ≤ i ≤ k. To\nbe non-zero, the sum of the lengths of the tiles must be n. The number of ways to arrange these\nn1 + n2 + · · ·nk tiles is given by the multinomial coefficient.\nLINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS 7\nWe proved Identities 1 through 5 under the assumption that the initial conditions were non-\nnegative, and Identities 6 through 8 made the stronger assumption that the initial conditions were\nsufficiently spread out so that pi ≥ 0 for 1 ≤ i ≤ k − 1. We conclude this section by demonstrating\nthat, by exploiting linearity, these identities remain true for arbitrary real (or complex) initial\nconditions.\nFor any given numbers a and b, let H denote the set of all sequences (H0,H1, . . . ) that satisfy the\nrecurrence of equation (1), where the initial conditions H0 and H1 are arbitrary real (or complex)\nnumbers. Then H is a two-dimensional real vector space, with basis sequences H(1, 0) and H(0, 1)\nwhere H(x, y) is the sequence in H with initial conditions H0 = x and H1 = y. The function\nL : R2 → H is linear, where L(x, y) = H(x, y). Many identities can be viewed as a linear function\nI : H → R. For example, Identity 1 can be viewed as I : H → R defined for H ∈ H by I(H) =\nt\n∑n\nk=0 s\nn−kHk −Hn+2− sn+1H1. Identity 1 asserts that I(H) = 0 for all H ∈ H where H is of the\nform H(x, y) with x and y non-negative integers. Since the composed linear function I ◦L : R2 → R\nis equal to 0 for basis vectors (1, 0) and (0, 1), then the identity is true for all initial conditions. The\nsame argument applies to Identities 2 and 3.\nTo extend this reasoning to linear k-th order recurrences like Identites 6, 7, and 8, we sim-\nply to need to find a basis of non-negative integer vectors that are sufficiently “spread out”,\nwhich can always be done. For instance, if k = 3 in recurrence (2), a suitable basis would be\n{(0, 0, 1), (0, 1, c1), (1, c1, c21 + c2)}. Thus Identities 6, 7, and 8 are valid for any initial conditions.\nFinally, identities such as 4 and 5 can be viewed as quadratic functions on H ×H, that is they\nare of the form Q(H ′,H ′′) = 0 where H ′,H ′′ ∈ H, and are linear in both H ′ and H ′′. Thus if the\nidentity holds for any pair of basis vectors in H, (e.g., when H ′ and H ′′ begin with (0, 1) or (1, 0))\nthen the identity holds for all initial conditions. A similar argument can be made for quadratic\nidentities for linear k-th order recurrences when k > 2.\n4. Binet’s Formula for 3-bonacci Tilings\nSo far the tiling interpretation of k-th order recurrences has yielded natural combinatorial proofs of\nseveral identities for such recurrences. We now show how our combinatorial interpretation, together\nwith a stochastic element, even allows us to prove recurrence identities involving irrational numbers.\n8 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nA closed-form expression for the n-th Fibonacci number (where f0 = f1 = 1) is given by Binet’s\nformula:\nfn =\n1√\n5\n[\nφn+1 −\n(\n−1\nφ\n)n+1]\n,(4)\nwhere φ is the golden mean (1 +\n√\n5)\/2. A novel proof can be obtained through a combinatorial\ninterpretation of fn as the number of ways to tile an n-board using squares and dominoes. See [1].\nThe above formula, which can be generalized to handle Fibonacci recurrences with arbitrary initial\nconditions, then arises by interpreting the tiling as a process in which a square or domino is laid\nsequentially on a board that is infinitely long.\nWe define Fk,n to be the number of ways of tiling an n-board with two types of tiles: squares\nand k-ominoes (a tile that covers k cells). Naturally, for 0 ≤ n ≤ k − 1, Fk,n = 1 and for n ≥ k\nFk,n = Fk,n−1 + Fk,n−k. Now we show how a random tiling process yields a formula analogous\nto Equation (4) for F3,n. This will motivate the analysis in the following sections, in which more\ngeneral recurrences are handled.\nSuppose we are given an infinitely long board with cells numbered 1, 2, 3, . . . , which we shall cover\nwith squares and 3-ominoes in a random manner. Specifically, starting at cell 1, we place a square\nwith probability 1\/τ1 and place a 3-omino with probability 1\/τ\n3\n1 , where τ1 is the (unique) real root\nof\n1\nτ\n+\n1\nτ3\n= 1.\nThis ensures that the probability that our tiling begins with a specific length n tiling is τ−n1 , re-\ngardless of how many squares or 3-ominoes are used. We see that τ1 satisfies the characteristic\nequation\nτ3 − τ2 − 1 = 0.(5)\nDescartes’s rule of signs shows that a positive real root τ1 of this equation exists and is unique (the\nnumber of positive real roots is bounded by the number of sign changes in the coefficients and equal\nin parity). We denote the other two (complex) roots of this equation by τ2 and τ3.\nWe say that a tiling is breakable at cell n if a new tile begins at cell (n + 1). For example, the\ntiling in Figure 1 is breakable at cells 0, 2, 3, 5, 7, 8. Let qn denote the probability that the tiling\nLINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS 9\n...\n...\nn\n......\n...\n...\nn+3n+2\nB\nB\nB 1\n2\n0\nn+1n-1\nFigure 2. Examples of the three Markov Chain states at cell n.\nis breakable at cell n. Since there are F3,n ways to tile the first n cells, and each such tiling has\nprobability 1\/τn1 of occurring, we have\nqn =\nF3,n\nτn1\n.(6)\nWe determine qn (and hence F3,n) using a stochastic model. The process of randomly placing tiles\nas we advance one unit along the board can be described by a Markov chain that moves between\nthree states: B0 (breakable at the current cell), B1 (a 3-omino ends one cell later), and B2 (a\n3-omino ends two cells later). (See Figure 2.)\nThe matrix of transition probabilities is:\nP =\n\n\nB2 B1 B0\nB2 0 1 0\nB1 0 0 1\nB0 1\nτ31\n0 1\nτ1\n\n\nwhere pij is the probability of going from state i to state j. Note the presence of 1’s in the first\ntwo rows. These occur because once a 3-omino is placed, the next two states in the process are\ndetermined; the next choice occurs at the break where the 3-omino is completed. At time (cell) 0,\nthe chain begins in the breakable state. So qn, the probability that this tiling is breakable at cell n,\nis the (3, 3) entry of Pn. By diagonalizing P , we obtain:\n10 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nPn =\n\n\n1 1 1\nτ2\nτ1\nτ3\nτ1\n1\nτ22\nτ21\nτ23\nτ21\n1\n\n\n\n\n(\nτ2\nτ1\n)n\n0 0\n0\n(\nτ3\nτ1\n)n\n0\n0 0\n(\nτ1\nτ1\n)n\n\n\n\n\nτ1τ3\n(τ2−τ1)(τ2−τ3)\nτ1(τ1+τ3)\n(τ2−τ1)(τ2−τ3)\nτ21\n(τ2−τ1)(τ2−τ3)\nτ1τ2\n(τ3−τ1)(τ3−τ2)\nτ1(τ1+τ2)\n(τ3−τ1)(τ3−τ2)\nτ21\n(τ3−τ1)(τ3−τ2)\nτ2τ3\n(τ1−τ2)(τ1−τ3)\nτ1(τ2+τ3)\n(τ1−τ2)(τ1−τ3)\nτ21\n(τ1−τ2)(τ1−τ3)\n\n .\nThe (3, 3) entry of Pn simplifies to\nqn =\n1\nτ1n\n[\nτn+11\n3τ1 − 2\n+\nτn+12\n3τ2 − 2\n+\nτn+13\n3τ3 − 2\n]\n.\nIt follows directly from (6) that\nF3,n =\nτn+11\n3τ1 − 2\n+\nτn+12\n3τ2 − 2\n+\nτn+13\n3τ3 − 2\n,(7)\ngiving a closed form expression for F3,n in terms of the roots of Equation (5). This formula was\nderived in [10] by algebraic means.\n5. A Binet-like Formula for kth Order Linear Recurrences\nWe next observe how our Markov chain model changes when we consider the generalized tiling\ninterpretation of k-th order linear recurrences described in Section 3. We first describe these for\n“ideal” initial conditions, then extend our results to arbitrary conditions. The ideal initial conditions\narise from setting pi = ci for 1 ≤ i ≤ k in the tiling interpretation for such recurrences. Equivalently,\nfor a given k-th order linear recurrence, we define aj = 0 for j < 0, a0 = 1, and for n ≥ 1\nan = c1an−1 + c2an−2 + · · ·+ ckan−k.(8)\nThis recurrence has characteristic polynomial\nf(x) = xk − c1xk−1 − c2xn−2 − · · · − ck−1x− ck.(9)\nLet µ = µ1 denote the unique positive real root of Equation (9) (which exists by Descartes’s rule of\nsigns), and let µ2, µ3, . . . , µk denote the other roots. We consider only the case where the roots are\nall distinct.\nAs in the last section, we now create a random tiling of an infinitely long board. We begin by\nplacing a random colored tile beginning at cell 1. For 1 ≤ i ≤ k, such a tile will have length i with\nprobability ci\/µ\ni, and the color will be chosen at random (uniformly) from the ci available colors.\n(Thus any colored tile of length i has probability 1\/µi of being selected.) All subsequent tiles will\nLINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS 11\nbe chosen randomly and independently with these probabilities. Notice that these probabilities sum\nto 1 since\n∑k\ni=1\nci\nµi\n= 1 follows from Equation (9).\nAs before, let qn denote the probability that the tiling is breakable at cell n. Since there are an\nways to tile cells 1 through n, each with probability 1\/µn, we have\nqn =\nan\nµn\n.(10)\nMouline and Rachidi [6] study the asymptotic behavior of this expression; in contrast, we are con-\ncerned with obtaining an exact Binet-like expression for qn.\nThe Markov chain represented by our tiling consists of k states: B0, B1, B2, . . . , Bk−1, where B0\nis the state in which the tiling is breakable at the current cell, and Bi is the state in which the\ncurrent tile ends after i more cells. The matrix P of transition probabilities is:\nP =\n\n\nBk−1 Bk−2 Bk−3 Bk−4 · · · B0\nBk−1 0 1 0 0 · · · 0\nBk−2 0 0 1 0 0\nBk−3 0 0 0 1\n...\n...\n...\n. . .\n. . . 0\nB1 0 0 0 0 1\nB0 ck\nµk\nck−1\nµk−1\nck−2\nµk−2\n· · · c2\nµ2\nc1\nµ\n\n\nwhere pij is the probability of going from state i to state j.\nA similar matrix appears in Kalman [5], where it is derived by the matrix representation of a k-th\norder linear recurrence. We follow a similar analysis to derive an expression for qn. At time (cell)\n0, the chain begins in the breakable state. Hence qn, the probability that the tiling is breakable at\ncell n, is the (k, k) entry of Pn:\nqn = [0, 0, 0, . . . , 1]P\nn[0, 0, 0, . . . , 1]T.(11)\nThe eigenvalues λi of the matrix P are determined by taking the determinant of λI − P , which\nyields:\n(λµ)k − c1(λµ)k−1 − c2(λµ)k−2 − · · · − ck−1(λµ)− ck = 0.(12)\n12 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nThis expression for λ shows that (λµ) satisfies the characteristic equation (9), so the k eigenvalues\nof P are related to the k roots of (9) by:\nλi = µi\/µ,\nfor 1 ≤ i ≤ k. By equation (12),the vector [1, λi, λ2i , ..., λ\nk−1\ni ]\nT is an eigenvector corresponding to λi.\nUsing the Vandermonde array\nS =\n\n\n1 1 1 · · · 1\nλ1 λ2 λ3 · · · λk\nλ1\n2\nλ2\n2\nλ3\n2 · · · λk2\n...\n...\nλ1\nk−1\nλ2\nk−1\nλ3\nk−1 · · · λkk−1\n\n\n,\nand the diagonal matrix D (with dii = λi), we can diagonalize P = SDS\n−1. Using this diagonal-\nization and (11) we have\nqn = [0, 0, 0, . . . , 1]SD\nnS−1[0, 0, 0, . . . , 1]T.\nThe product of the first three matrices is [λn+k−11 , λ\nn+k−1\n2 , . . . , λ\nn+k−1\nk ]. Letting the remaining two\nmatrices be represented by [y1, y2, . . . , yk]\nT, we have\nqn =\nk∑\ni=1\nyiλ\nn+k−1\ni .(13)\nWe can find the yi by solving S[y1, y2, y3, . . . , yk]\nT = [0, 0, 0, . . . , 1]T. Cramer’s Rule gives\nyi =\nµk−1\nΠj 6=i(µi − µj)\n.(14)\nNote that the denominator can be expressed as f ′(µi), where f is the characteristic polynomial in\nequation (9). Combining (10), (13), and (14),\nan =\nk∑\ni=1\nµn+k−1i\nf ′(µi)\n.(15)\nOur Markov chain method has thus yielded a closed form Binet-like expression for the recurrence\nan in terms of the roots of the characteristic equation. Note that f\n′(µi) = 0 if and only if µi is a\nrepeated root, in which case (15) is undefined. This equation was derived in [5, 9] using a recurrence\nmatrix rather than a Markov chain approach.\nLINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS 13\nAs a specific case of this formula, consider a tiling of a board using (colorless and phaseless)\nsquares and k-ominoes. This yields the characteristic equation\nxk − xk−1 − 1 = 0\nwhose roots µi are manifested in the Binet-like formula\nFk,n =\nk∑\ni=1\nµn+1i\nkµi − k + 1\n.\nThis formula recovers (7) in the case k = 3 and the original Binet formula (4) when k = 2.\n6. Extension to Arbitrary Initial Conditions\nLet ci, 1 ≤ i ≤ k, be non-negative integers, and let A0, A1, . . . , Ak−1 be integers. Consider the\nseries αi = Ai for 0 ≤ i ≤ k, and for n ≥ k, let\nαn = c1αn−1 + c2αn−2 + · · ·+ ck−1αn−k+1 + ckαn−k.(16)\nWe now prove a Binet-like formula for this very general recurrence. Theorem 3 generalizes Equation\n(15) and extends the formulae found in [5, 8, 9].\nTheorem 3. Given a recurrence of the form (16), with initial conditions Am, 0 ≤ m ≤ k − 1,\nαn =\nk∑\ni=1\nk∑\nj=1\nj∑\nm=1\ncj Ak−m µ\nn+m−j−1\ni\nf ′(µi)\n(17)\nholds whenever the characteristic polynomial f(x) = xk − c1xk−1 − c2xn−2 − · · · − ck−1x − ck has\ndistinct roots µi, 1 ≤ i ≤ k.\nThe denominator f ′(µi) vanishes if and only if µi is a repeated root, so the expression (17) is\nalways valid when it is defined.\nNote that for the Fibonacci sequence where k = 2, c1 = c2 = 1, A0 = A1 = 1, the inner two sums\nin (17) become\n1\n2µi − 1\n[\nµn−1i + (µ\nn−2\ni + µ\nn−1\ni )\n]\n=\n1\n2µi − 1\n[\nµn−1i + µ\nn\ni\n]\n=\nµn+1i\n2µi − 1\n.\nNoting for i = 1, 2 that 2µi − 1 = ±\n√\n5, and summing over i, then recovers Equation (4).\nThis theorem will be proved using a set of “basis series” e0n, e\n1\nn, . . . , e\nk−1\nn satisfying (16). Then\nevery series satisfying (16) can be represented as a linear combination of these basis series.\n14 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nProof. The set of all sequences that satisfy the recurrence (16) forms a k-dimensional vector space.\nEach sequence is completely determined by its first k terms.\nLet ein be the n-th term of the sequence determined by the initial conditions e\ni\ni = 1 and e\ni\nj = 0\nfor all other j in 0, 1, . . . , k − 1. (Thus the vector of initial conditions has a 1 in the i-th position\nand zeroes elsewhere.) Any k-th order linear recurrence with initial conditions A0, A1, . . . , Ak−1\ncan then be represented as\nαn = A0e\n0\nn +A1e\n1\nn +A2e\n2\nn + · · ·+Ak−1ek−1n .(18)\nThe basis series ein can be expressed as a linear combination of “shifts” of the specific recurrence\nan studied in the last section, arising from setting pi = ci in the tiling interpretation. We list the\nfirst few terms below:\nTerm an\n0 1\n1 c1\n2 c1a1 + c2\n3 c1a2 + c2a1 + c3\n...\n...\nk − 2 c1ak−3 + c2ak−4 + · · ·+ ck−2\nk − 1 c1ak−2 + c2ak−3 + · · ·+ ck−2a1 + ck−1\nNote that this series can also be obtained by setting a0 = 1 and a−1 = a−2 = · · · = a1−k = 0 and\nusing the recurrence (8) to generate the later terms. Using these negative-indexed terms and the\ntable above, observe that for all n ≥ 1,\ne0n = an − c1an−1 − c2an−2 − · · · − ck−1an−k+1 = ckan−k.\nLINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS 15\nIn general, we can express the basis series ein, for 0 ≤ i ≤ k − 1, in terms of (k − i) “shifts” of the\nsequence an:\nein = an−i − c1an−i−1 − · · · − ck−i−1an−k+1\n= ck−ian−k + · · ·+ ckan−k−i.(19)\nHence by equations (16), (19), and (15),\nαn =\nk−1∑\n`=0\nA` e\n`\nn\n=\nk−1∑\n`=0\nk∑\nj=k−`\nA` cj an−`−j\n=\nk−1∑\n`=0\nk∑\nj=k−`\nk∑\ni=1\nA` cj\nµ\nn−`−j+k−1\ni\nf ′(µi)\n=\nk∑\ni=1\nk∑\nj=1\nk−1∑\n`=k−j\nA` cj µ\nn−`−j+k−1\ni\nf ′(µi)\n=\nk∑\ni=1\nk∑\nj=1\nj∑\nm=1\nAk−m cj µ\nn+m−j−1\ni\nf ′(µi)\nas desired.\n7. Discussion\nOur combinatorial interpretation of linear recurrences as solutions to tiling problems gives a pow-\nerful method for understanding recurrence identities. This approach allows one to quickly assimilate\nand visually interpret recurrence identities as well as their proofs. Moreover, an associated Markov\nchain on tilings even allows one to recover identities that at first glance to not appear to be combi-\nnatorial, such as the “Binet-like” formula of Theorem 3. Our tiling and random tiling intepretations\nare a unifying approach to understanding k-th order linear recurrences.\nAcknowledgments. We wish to thank Henry A. Krieger and Jennifer J. Quinn for valuable\ninput on this paper.\n16 CHRISTOPHER R. H. HANUSA, ARTHUR T. BENJAMIN, AND FRANCIS EDWARD SU\nReferences\n[1] Benjamin, A.T., Levin, G.M., Mahlburg, K., and Quinn, J.J. (2000). Random approaches to Fibonacci identities,\nAmer. Math. Monthly 107(6), 511-516.\n[2] Benjamin, A.T., Quinn, J.J., and Su, F.E. (2000). Phased tilings and generalized Fibonacci identities, Fibonacci\nQuart. 38(3), 282-288.\n[3] Benjamin, A.T., Quinn, J.J., and Su, F.E. (2000). Counting on continued fractions, Math. Mag. 73, 98-104.\n[4] Brigham, R.C., Caron, R.M., Chinn, P.Z., and Grimaldi, R.P. (1996). A tiling scheme for the Fibonacci numbers,\nJ. Recreational Math., 28(1), 10-16.\n[5] Kalman, D. (1982). Generalized Fibonacci numbers by matrix methods, Fibonacci Quart. 20(1), 73-76.\n[6] Mouline, M. and Rachidi, M. (1995). Suites de Fibonacci Généralisées et châınes de Markov, Rev. Real Acad.\nCienc. Exact. F́ıs. Natur. Madrid 89(1-2), 61-77.\n[7] Philippou, A.N. (1983). A note on the Fibonacci sequence of order K and the multinomial coefficients, Fibonacci\nQuart. 21(2), 82-86.\n[8] Philippou, A.N. and Muwafi, A.A. (1982). Waiting for the K-th consecutive success and the Fibonacci sequence\nof order K, Fibonacci Quart. 20(1), 28-32.\n[9] Shannon, A.G. (1974). Some properties of a fundamental recursion sequence of arbitrary order, Fibonacci Quart.\n12(4), 327-335.\n[10] Spickerman, W.R. (1982). Binet’s formula for the tribonacci sequence, Fibonacci Quart. 20(2), 118-120.\n[11] Vajda, S. (1989). Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications, John Wiley\nand Sons, New York.\n[12] Waddill, M.E. (1991). Using matrix techniques to establish properties of a generalized Tribonacci sequence,\nApplications of Fibonacci numbers, Vol. 4, Kluwer Academic Publishers, 299-308.\nDepartment of Mathematics, University of Washington, Seattle, WA 98195\nE-mail address: chanusa@washington.edu\nDepartment of Mathematics, Harvey Mudd College, Claremont, CA 91711\nE-mail address: benjamin@hmc.edu\nDepartment of Mathematics, Harvey Mudd College, Claremont, CA 91711\nE-mail address: su@math.hmc.edu\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! triangles. Koch Tetrahedron Posted on June 29, 2019. by. Samuel Nunoo In Koch Snowflake we saw an interesting fractal snowflake-like object that is obtained when gluing smaller triangles iteratively to the sides of... Continue reading... Spherical Pythagorean Theorem Posted on June 27, 2019. by. Samuel Nunoo Did you know there is a version of the Pythagorean Theorem for right triangles on spheres? First, let’s define precisely what we... Continue reading... Sperner’s Lemma Posted on June 26, 2019. by. Samuel Nunoo Divide a triangle T into lots of baby triangles, so that baby triangles only meet at a common edge or a common... Continue reading... Spherical Geometry Posted on June 26, 2019. by. Samuel Nunoo Remember high school geometry? The sum of the angles of a planar triangle is always 180 degrees or Pi radians. However, triangles... Continue reading... Behold! the Pythagorean Theorem Posted on June 26, 2019. by. Samuel Nunoo Figure 1 shows one of the simplest proofs of the Pythagorean Theorem. It is also perhaps the earliest recorded proof, known... Continue reading... Napoleon’s Theorem Posted on June 26, 2019. by. Samuel Nunoo Take any generic triangle, and construct equilateral triangles on each side whose side lengths are the same as the length... Continue reading... Shortest Pythagorean Proof? Posted on June 26, 2019. by. Samuel Nunoo Here is one of the shortest proofs of the Pythagorean Theorem. Suppose we are given any right triangle with sides of... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! probability. Risk-Free Betting on Different Beliefs Posted on June 29, 2019. by. Samuel Nunoo Alice believes that the 49ers will win the Super Bowl with probability 5\/8. Bob believes that Ravens will win the... Continue reading... Medical Tests and Bayes’ Theorem Posted on June 29, 2019. by. Samuel Nunoo Suppose that you are worried that you might have a rare disease. You decide to get tested, and suppose that... Continue reading... St. Petersburg Paradox Posted on June 29, 2019. by. Samuel Nunoo A new casino offers the following game: you toss a coin until it comes up heads. If the first heads... Continue reading... Farmers and Pesky Birds Posted on June 29, 2019. by. Molly Reeves Alice and Bob are two farmers each wanting to plant a (countably infinite) row of seeds, side by side in... Continue reading... Volume of a Ball in N Dimensions Posted on June 29, 2019. by. Samuel Nunoo The unit ball in Rn is defined as the set of points (x1,…,xn) such that x12 + … + xn2 . Continue reading... Poker Probability Paradox Posted on June 26, 2019. by. Samuel Nunoo In poker, there is a hierarchy of desirable hands which relate to how unlikely the hand is. For instance, a... Continue reading... Monty Hall Problem Posted on June 26, 2019. by. Samuel Nunoo Here’s a problem that makes the round every few years, and each time, it is hotly debated. You are on... Continue reading... Deal or No Deal Posted on June 26, 2019. by. Samuel Nunoo On the TV show “Deal or No Deal”, a contestant is faced with a number of briefcases (26 in the... Continue reading... Seven Shuffles Posted on June 26, 2019. by. Samuel Nunoo How many shuffles does it take to randomize a deck of cards? The answer, of course, depends on what kind... Continue reading... Pass the Beer Posted on June 26, 2019. by. Samuel Nunoo Suppose you and a bunch of friends are sitting around a table. There are N of you. You have a... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"n7vrUp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/unsolved-problem\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":51,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","unsolved-problem"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["unsolved problem – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:45 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Sat Jun 29 00:00:00 UTC 2019","Wed Jun 26 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":5597198204422086786,"exact_signature_unique_b":true,"fuzzy_signature_l":6945420496085639130,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["unsolved problem"],"h2_txt":["Fermat’s Last Theorem","Riemann Hypothesis","Hailstone Numbers","Goldbach’s Conjecture"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Lawnmower.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Lawnmower 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":633,"text_t":"unsolved problem – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! unsolved problem. Fermat’s Last Theorem Posted on June 29, 2019. by. Samuel Nunoo There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =... Continue reading... Riemann Hypothesis Posted on June 29, 2019. by. Samuel Nunoo If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our... Continue reading... Hailstone Numbers Posted on June 26, 2019. by. Samuel Nunoo Start with any positive integer (an initial seed) and obtain a sequence of numbers by following these rules. 1. If the... Continue reading... Goldbach’s Conjecture Posted on June 26, 2019. by. Samuel Nunoo Here’s a famous unsolved problem: is every even number greater than 2 the sum of 2 primes? The Goldbach conjecture, dating from... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"nSRbEp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/fibonacci-gcds-please\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":52,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["funfacts","fibonacci-gcds-please"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Fibonacci GCD’s, please – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:46 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-5929111894052181551,"exact_signature_unique_b":true,"fuzzy_signature_l":-9059484091754344563,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["Fibonacci GCD’s, please"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"bold_txt":["Fun Fact suggested by:","Math Fun Facts","How to Cite this Page:","The Math Behind the Fact:","Presentation Suggestions:"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/BrokenMudd.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 BrokenMudd Broken Mudd 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":626,"text_t":"Fibonacci GCD’s, please – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Fibonacci GCD’s, please. Fibonacci numbers exhibit striking patterns. Here’s one that may not be so obvious, but is striking when you see it. Recall the Fibonacci numbers:.  n: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,  12,  13,  14, … . f. n. : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …. Now let’s look at some of their greatest common divisors. (gcd’s): gcd(f. 10. , f. 7. ) = gcd(55, 13) = 1 = f. 1. gcd(f. 6. , f. 9. ) = gcd(8, 34) = 2 = f. 3. gcd(f. 6. , f. 12. ) = gcd(8, 144) = 8 = f. 6. gcd(f. 7. , f. 14. ) = gcd(13, 377) = 13 = f. 7. gcd(f. 10. , f. 12. ) = gcd(55, 144) = 1 = f. 2. Do you see the pattern? The. greatest common divisor of any two. Fibonacci numbers is also a Fibonacci number! Which one? If you look even closer, you’ll see the amazing general result:. gcd(f. m. , f. n. ) = f. gcd(m, n). . Presentation Suggestions:. After presenting the general result, go back to the examples to verify that it holds. You may wish to prepare a transparency beforehand with a table of. Fibonacci numbers on it, so you can refer to it throughout the presentation. The Math Behind the Fact:. The proof is based on the following lemmas which are interesting in their own right. All can be proved by induction. . a) gcd(f. n. , f. n-1. ) = 1, for all n . b) f. m+n.  = f. m+1.  f. n.  + f. m.  f. n-1.  . c) if m divides n, then f. m.  divides f. n.  . and the ever important Euclidean Algorithm which states: if n= qm + r, then gcd(n, m) = gcd(m, r). For such n,m we have. gcd(f. m. , f. n. ) = gcd(f. m. , f. qm+r. ) = gcd(f. m. , f. qm+1. f. r. + f. qm. f. r-1. ) = gcd(f. m. , f. qm+1. f. r. ) = gcd(f. m. , f. r. ). where the 2nd equality follows from (b), the 3rd equality from (c) noting that m divides qm, and the 4th equality from noting that f. m.  divides f. qm.  which is relatively. prime to f. qm+1. . Thus. gcd(f. n. , f. m. ) = gcd(f. m. , f. r. ). which looks a lot like the Euclidean algorithm but with f’s on top! For example since. gcd(100, 80) = gcd(80, 20) = gcd(20, 0) = 20, then. gcd(f. 100. , f. 80. ) = gcd(f. 80. , f. 20. ) = gcd(f. 20. , f. 0. = 0) = f. 20. . How to Cite this Page:. Su, Francis E., et al. “. Fibonacci GCD’s, please.”. Math Fun Facts. . . Fun Fact suggested by:. Arthur Benjamin. Did you like this Fun Fact? Click to rate it. Submit Rating. Average rating. 4.7. \/ 5. Vote count:. 26. No votes so far! Be the first to rate this Fun Fact. Share:. Tags:. Fibonacci ,. greatest common divisor ,. medium ,. number theory ,. prime Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"lofk7W51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/FAIR-DIVISION-AND-REDISTRICTING.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":95,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"FAIR-DIVISION-AND-REDISTRICTING","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["FAIR-DIVISION-AND-REDISTRICTING.pdf"],"author":"dvips + GPL Ghostscript GIT PRERELEASE 9.22","author_sxt":["dvips + GPL Ghostscript GIT PRERELEASE 9.22"],"last_modified":"Mon Sep 10 18:19:53 UTC 2018","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-3099915406908269609,"exact_signature_unique_b":true,"fuzzy_signature_l":2309640327574378796,"fuzzy_signature_unique_b":true,"wordcount_i":6883,"linkscount_i":4,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":4,"outboundlinks_urlstub_sxt":["www.senate.leg.state.mn.us\/departments\/scr\/redist\/","www.nytimes.com\/2005\/11\/11\/opinion\/11mann.html","clerk.house.gov\/members\/electionInfo\/elections.html","arxiv.org\/abs\/1402.0862v1"],"outboundlinks_anchortext_txt":["","","",""],"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:48 UTC 2022","fresh_date_dt":"Wed Dec 06 11:33:29 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"dvips + GPL Ghostscript GIT PRERELEASE 9.22","language_s":"en","size_i":222318,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088110893531136,"text_t":"ar\nX\niv\n:1\n4\n0\n2\n.0\n8\n6\n2\nv\n1\n  \n[c\ns.\nG\nT\n] \n 4\n F\neb\n 2\n0\n1\n4\nFAIR DIVISION AND REDISTRICTING\nZEPH LANDAU AND FRANCIS EDWARD SU\n1. Introduction\nRedistricting is the political practice of dividing states into electoral districts\nof equal population. It is mandated to occur every ten years, after the census,\nto ensure equal representation in the legislative body. Where the boundaries are\ndrawn can dramatically alter the number of districts a given political party can\nwin. As a result, a political party which has control over the legislature, can\n(and does) manipulate the boundaries to win a larger number of districts, thus\naffecting the balance of power in the U.S. House of Representatives. This kind of\nboundary manipulation occurs even with certain legal and legislative constraints\nthat restrict some aspect of how districts can be drawn and mandate that, where\nappropriate, districts should be created to have a majority of voters consisting of\na racial minority. (See [11] for a detailed summary of these constraints.)\nThe ability of one political party to gain political advantage by carefully choosing\nthe boundaries during redistricting has been recognized as a serious problem with\nthe redistricting process in the United States; we shall refer to this as the problem\nof partisan unfairness. Attempts to fix and\/or mitigate the problem of partisan\nunfairness (beyond the legal restrictions) have taken one of two approaches: try-\ning to constrain the process to reduce the amount of political gain achievable, and\ntrying to remove politics from the redistricting process. Examples of the first ap-\nproach include attempting to limit the power of the drawing party by more strictly\nprescribing the allowable shapes of districts, and banning the use of registration\nand voting data within the redistricting process. Examples of the second approach\ninclude assigning the task of redistricting to bipartisan or non-partisan panels1,\nand using computer programs to generate redistricting maps that optimize certain\ncarefully chosen criteria.\nLandau-Reid-Yershov [6] took a different approach to provide a novel solution\nto the problem of partisan unfairness: rather than trying to fix the problem by\nrestricting the shape of the possible maps or by assigning the power to draw the\nmap to nonbiased entities, their solution ensures fairness by balancing competing\ninterests against each other. This kind of solution is an example of what are known\nas “fair division” solutions— such solutions account explicitly for the preferences of\nall parties, are determined by a procedure in which all parties are actively involved,\nand are accompanied by rigorous guarantees of a specified notion of fairness.\n2000 Mathematics Subject Classification. Primary 91F10; Secondary 91B32, 91B12.\nResearch partially supported by NSF Grants DMS-0701308 and DMS-1002938 (Su).\n1Eleven states redistrict using bipartisan or independent commissions.\n1\n2 ZEPH LANDAU AND FRANCIS EDWARD SU\nThe goal of this article is to provide an exposition of this redistricting method\nin the context of a detailed sample “map”, and make a stronger connection to the\nideas of fair division than is provided in [6]. In particular, we propose a specific\nnotion of fairness that was used but not made explicit in [6]. This notion of fairness\ncan be used in concert with (not substitute for) other necessary or desired criteria\nfor a good redistricting solution. We clarify how fair division ideas can play an\nimportant role in a realistic redistricting solution by introducing an interactive step\nthat involves multilateral evaluation, procedural fairness, and fairness guarantees.\nAnd by making the bridge between fair division ideas and redistricting solutions\nmore explicit, we hope to encourage the flow of ideas between the two areas.\nWe begin with an introduction to the ideas of fair division in Section 2. We\ndescribe the problem of partisan unfairness in more detail in Section 3. Despite the\nability to easily recognize the “unfairness” inherent in the redistricting process, it\nhas been hard to give a reasonable definition of what would be fair. In Section 4\nwe give an explicit definition of fairness—the geometric target—that incorporates\ngeometric considerations such as how constituent voters are distributed and how\ndistricts are shaped. Section 6 examines the protocol of [6] by analyzing its behavior\nin a specific example that demonstrates most aspects of the solution. Within this\nexample, the fairness of the protocol is discussed in detail.\nWe stress that the ideas discussed here are well suited to be combined with\nother necessary or desired criteria for a good redistricting solution. Our definition\nof fairness involving geometric targets in Section 4 can incorporate independently\ndesired requirements for district shape or competitiveness. The protocol of [6] can\nbe easily adjusted to approximate fairness under these additional requirements. If a\nsolution involving an independent commission is desired, these definitions of fairness\ncan be used as a target for the commission. Similarly if a computer assisted solution\nthat consists of optimizing some function is sought, this measure of fairness can be\nused as a component of the function to be optimized. Separately, the simple fair\ndivision ranking protocol given in Section 5 can be used to incorporate some degree\nof legislative preference within any proposed redistricting solution that otherwise\ndoes not include legislative influence.\n2. Fair Division\nThe problem of fair division, as Steinhaus put it [14] is essentially a question of\nhow to divide some object fairly. Usually, this object is affectionately referred to\nas cake [12], but in general it could be desirable or undesirable (e.g., the division\nof chores [9]) or a mixture of desirable and undesirable goods [15]. The cake may\nbe infinitely divisible (as we usually regard real cake) or only divisible into discrete\npieces (such as a pieces of an estate). Applications of fair division ideas include\nmethods for resolving international disputes and divorce settlements [3].\nThere are several notions of fairness that one might consider, but an important\naspect of fair division problems is that this fairness notion is evaluated by the parties\ninvolved in the negotation, rather than an outside arbiter. Thus, the outcome of\na fair division procedure will give each party what it considers to be a fair share,\naccording to its own evaluation.\nThe simplest example of a fair division procedure is the familiar “I-cut-you-\nchoose” method for dividing a cake among two people. One might consider a fair\nFAIR DIVISION AND REDISTRICTING 3\npiece in which each party does not envy the other; we call such a solution an envy-\nfree solution. Again, note that envy is measured by each party according to its\nown evaluation. If one person cuts the cake (into two pieces that she is indifferent\nbetween) and the other person is allowed to choose first (picking the piece that he\nmost desires), then both people will end up with a piece for which they experience no\nenvy. This simplest of all fair division procedures already highlights some interesting\nfeatures common to all fair division procedures:\n(1) Multilateral evaluation. Fairness is evaluated according to each party’s own\npreferences. Therefore, parties don’t have to agree on what is valuable; each\nwill obtain a share they would consider fair in their own estimation (and\nthey do not need to know the other party’s preferences).\n(2) Procedural fairness. There is a process by which preferences are elicited,\nall parties are involved in the process, and they understand the criteria\nby which fairness is measured. Because of this, parties are more likely\nto feel that the process is fair, more so than a decision imposed by an\noutside arbiter (see e.g., [13]). The procedure guides parties to a mutually\nacceptable division.\n(3) Fairness guarantee. By following the procedure, as long as you tell the\ntruth about your preferences, you will obtain what you feel is fair (even if\neveryone else lies). Thus there is an incentive to be truthful (and if you lie\nabout your preferences, it can backfire).\nA reader might object to the above particular cake-division solution, because the\ncutter only gets what he perceives to be 50 percent of the cake in his valuation, while\nthe other person might end up with more. The solution is envy-free (neither person\nenvies the other person’s share) but it is not equitable, meaning the perceived share\nof cake each player gets (in his own valuation) is different. This is not a deficit of the\nprocedure (which only guaranteed envy-freeness and not equitability) as much as it\nis a deficit of the procedure chosen. An active area of research in mathematics [12],\neconomics [7], and political science [3] is the development of fair division procedures\nin various settings and with various fairness criteria.\nA more interesting fair division solution that has found application by practi-\ntioners is the Adjusted Winner procedure of Brams and Taylor [4]. It is procedure\nfor dividing a set of goods between two parties in such a way that the division\nis: envy-free, equitable, and efficient (or Pareto-optimal). The last criterion means\nthat there is no division that dominates the given one, i.e., there is no other division\nthat is just as good for both parties and strictly better for one party. Thus the\nAdjusted Winner solution gives each party a share for which they do not wish to\ntrade shares, and in which they feel they got just as good a portion as the other\nparty feels it got, and there is no other solution that dominates. At most one of\nthe goods may have to be divided in the procedure (though one cannot predict\nbeforehand which procedure it is). We note that if there are more than two parties,\nit may not always be possible to satisfy these three properties in cake division, as\ndiscussed in [2].\nThe Adjusted Winner procedure has found application in divorce settlements [4]\nbecause of its fairness guarantees as well as its ease of use. In the procedure, both\nparties are given 100 points to divide by assigning over the objects. This is the part\nof the procedure where preferences are elicited. Then objects are initially given to\nthe party that valued them most; such a division is efficient, but it may not be\n4 ZEPH LANDAU AND FRANCIS EDWARD SU\nenvy-free or equitable. Call the party who ends up with the largest fractional share\n(in its own evaluation) the winner, and the other, the loser. In the next phase of\nthe procedure, the assignment of goods is “adjusted” by transferring goods from\nthe winner to the other party in a particular order until both fractional shares are\nequalized.\nThe Adjusted Winner procedure has the 3 features described above for a fair\ndivision procedure. It has a fairness guarantee: what results is an outcome that\nis provably envy-free and efficient, in addition to being equitable. It relies on\nmutilateral evaluation: the preferences of both parties are taken into account, and\nthe resulting division achieves the fairness guarantee for both parties using their own\nestimation. And it is procedurally fair: parties using the Adjusted Winner procedure\ncan understand and verify the fairness guarantees for a particular solution (without\nhaving to understand the proofs); because they participated in the procedure by\nstating their preferences, they are more likely to feel that the outcome is fair.\nAs we shall see, these three fair division ideas can offer some helpful ideas to\ncurrent thinking about redistricting which can be combined with other desired\nideas for a good redistricting solution. They underlie the redistricting procedure\nof [6] that we will now explain. First, we will explain the problem of partisan\nunfairness that [6] attempts to address.\n3. Redistricting: the problem of partisan unfairness\nIn most of the 50 states in the U.S., the districting protocol is to have one party\ndraw all the boundaries. If the drawing party’s goal is to win as many districts as\nit can, the strategy is clear: draw boundaries so that each district either a) has a\nsmall majority of its voters, or b) has a large majority of the other party’s voters.\nIn other words, for any district, the drawing party should strive to either win it by\na small margin or lose it by a large margin (See [1] for a detailed discussion).\nIn general, with such a strategy, a drawing party with X% of support of the\nvoters can win just under min(2X%, 100%) of the districts if there are no geomet-\nric constraints (e.g., requiring districts be compact or contiguous, etc.). In reality,\ngeometric constraints usually mean that this ideal outcome cannot be achieved;\nhowever, in most cases, the drawing party can still win a significantly larger per-\ncentage than X% of the districts, even with only partial knowledge of voting trends.\nThis is not just a theoretical issue, as has often been demonstrated when the\nparty in control changes. We cite two examples:\n• When Republicans took control of the Texas legislature in 2002, they re-\ndrew state districts mid-decade, and the Texas delegation changed from 15\nRepublicans and 17 Democrats to 22 Republicans and 10 Democrats. [8]\n• In Michigan, the 2000 election produced 7 Republican representatives and 9\nDemocratic representatives. After the census, a new district map was drawn\nresulting in 9 Republican representatives and 6 Democratic representatives\nin the 2002 election (Michigan lost 1 seat due to the census).[8]\nThis ability of one party to draw districts in such a way as to gain political\nadvantage is viewed as one of the major problems with redistricting in the United\nStates; we shall refer to this as the problem of partisan unfairness. The districting\nprotocol proposed in [6] avoids this inherent unfairness by ensuring that either party\ncan win a percentage of districts that is very close to their fair share.\nFAIR DIVISION AND REDISTRICTING 5\n4. What is a party’s fair share?\nDefining a “fair” share is not as straightforward as it may seem. A reasonable\nfirst attempt would be to say that the percentage of districts won by a party should\nbe close to the percentage of constituent voters in the party. However, geometric\nconstraints can make this impossible. For instance, if one party enjoys a statewide\n60% - 40% advantage in the electorate and if the parties are mixed homogeneously\nthroughout the state, then any reasonable district would have a similar 60% -\n40% advantage for the same party. In such a case, if we believe that districts\nshould contain people who live in contiguous and relatively compact regions, then\nall districts would go to the majority party and we would consider this outcome\nfair even though it differed dramatically from the percentage of constituent voters.\nThis example demonstrates the need to incorporate geometric constraints into any\nreasonable notion of fair shares.\nIt may come as a surprise that such a fairness notion can be defined, despite the\ncomplexity of possible geometric constraints. In this section, we define a fairness\nnotion that incorporates geometric considerations such as how constituent voters\nare distributed and how districts are shaped. Before doing so, we make some\npreliminary definitions.\nAny division of a state into districts that satisfies all desirable constraints (in-\ncluding legal constraints that district maps are subject to) will be called a viable\ndivision. The voting outcome Vout is a description of how every voter actually votes\nin a given election. Any party with a role in redistricting attempts to predict as-\npects of Vout to guide their choices of district boundaries. A voting model V shall\nbe such a prediction of how every voter will vote.\nA party’s rating system R is an assignment of a number to any proposed viable\ndivision of a state that gives a measure of how desirable that viable division is. A\nsimple example of a rating system is to rate a viable division by the number of\ndistricts the party thinks it can win; we will denote this rating system by Rwin.\nImplicit in calculating Rwin is a voting model V that the party is using to predict\nwhich districts it will win.\nIn reality, a party’s interests may be much more complicated then just the num-\nber of districts that it wins. A more general rating system is one where a party\ncould rate the desirability of each district in a division (assigning it a number), then\nsum these numbers over all the districts to give a rating for the division. As in [6],\nwe shall refer to such a rating sytem as an additive rating system, and denote any\nparticular instance of it as Rsum. The rating system Rwin is a special case of Rsum\nin which a party assigns a 1 to districts it expects to win and a 0 to those it expects\nto lose.\nThe more general rating system Rsum allows a party to take other considera-\ntions into account. Politically, these can be important, as the following examples\ndemonstrate:\n• perhaps some district has an incumbent who is on an important congres-\nsional committee, so winning that district is more valuable to the party\n(hence rated higher than other winnable districts),\n• perhaps some district has an important landmark (a stadium or a construc-\ntion project) worth more to a party than some other district,\n6 ZEPH LANDAU AND FRANCIS EDWARD SU\n• perhaps some district encompasses the supporters of two incumbents from\nthe opposition party, so that even though the district will be lost, the\nelimination of one strong incumbent from the other party is valuable.\nEquipped with the notions of viable division, voting model, and rating system,\nwe can now define our fairness notion:\nDefinition (Geometric Target). Given a voting model V and a rating system R,\na party’s geometric target with respect to V and R is the average of its highest and\nlowest ratings among the set of viable divisions. A party’s geometric target with\nrespect to Vout and Rwin will be called the absolute geometric target.\nThe absolute geometric target is a definition of fairness that takes into account\ngeometric constraints. There are several compelling reasons why this is a good\ndefinition. First, the definition seems conceptually fair as it lies exactly between\nthe best and worst outcome (in terms of number of districts won) for each party.\nSecond, when there are no geometric constraints, the absolute geometric target\ncoincides with the percentage of constituent voters—as already mentioned, if the\nminority party has X% of the vote, its best outcome is to win about 2X% of the\ndistricts, while its worst outcome is to lose all the districts 0% and so the absolute\ngeometric target in this case would be approximately 2X+0\n2\n= X% of the districts.\nThird, because this definition uses only viable district maps, it incorporates geo-\nmetric constraints by restricting attention to realizable outcomes. For instance, in\nthe example at the beginning of this section (with the homogeneous 60% - 40%\nelectorate split), both the best and worst rating for the minority party would be to\nwin 0 districts which is the only possible outcome (and thus fair) in that case. We\nremark that the absolute geometric target could be used as a target for fairness for\nindependent commissions.\nThe protocol for districting proposed in [6] (and described subsequently), allows\neach party the opportunity to achieve an outcome that is close to their own absolute\ngeometric target. Moreover, it allows each party the opportunity to achieve an\noutcome that is close to a geometric target with respect to any voting model V and\nany additive rating system R.\nNote that the geometric target with respect to a voting model V and a rating\nsystem R is a notion that captures the fair division principle of multilateral evalua-\ntion: that party preferences should be taken into account. Each party has its own\ngeometric target, based on its own voting model V and rating system R(derived\nfrom its preferences). This is to be distinguished from any absolute notions of fair-\nness that might be imposed by an external arbiter (including the absolute geometric\ntarget).\nWe shall soon see that districting protocol of [6] will, in addition, possess the\nother features of a fair division procedure— procedural fairness, and a fairness\nguarantee (see Section 2).\n5. The ranking protocol\nBefore presenting the redistricting protocol of [6], we present a simple but useful\nprotocol, which we call the ranking protocol, for how two parties can decide on one\nof a bunch of outcomes. For our purposes, the setting will be that of two political\nparties, A and B, where the choice of outcomes are different proposed divisions of\na state. The protocol is then simple: both parties rank the proposed divisions from\nFAIR DIVISION AND REDISTRICTING 7\nbest to worst from their perspective. Each proposed division then has two rankings.\nSelect the proposed division whose worse ranking is best (this reflects the Rawls’\nmaximin criterion, as discussed in [10]). If there are 2 such proposals (there can be\nat most 2), randomly choose one.\nNotice that if there are n proposed divisions, the division chosen is guaranteed to\nbe in the top n\n2\n+ 1 of both lists. Said another way, the ranking protocol provides\nan outcome that, for either party, is no worse than one less than their median\noutcome among the choices (and can be much better if the two parties desires are\nnot diametrically opposed). As we shall see in the next section, the ranking protocol\nis used as an augmentation step for the core redistricting protocol of [6].\nWe point out more generally that the ranking protocol could be used with any\nproposed method for generating divisions of a state, be it divisions created by\nindependent panels, computers, or based on any kind of optimization scheme.\n6. The fair division redistricting protocol\nWe now describe the core protocol for redistricting that was presented in [6]. It\nwill involve three parties: two parties with vested interest in the division (e.g. the\ndemocratic and republican parties, or the majority and minority party in a state)\ncalled parties A and B, and an independent agent which we’ll refer to as I.\nThe formal precise description of the protocol can be found in [6]; here we will\ndescribe the protocol while working through an example that will illustrate many\nof the aspects of both the protocol and the resulting solution.\n6.1. The State. Consider the map in Figure 1, which represents a state consisting\nof 25 parcels, which we are thinking of as indivisible units (here, they are rectangles\nor squares). Each parcel contains the same number of people; thus the smaller the\nparcel, the denser the population. Loosely, we can think of this state as having a city\nlocated at the small rectangles (T,U,V,W,X,Y), with suburban areas surrounding\nthe city, and the remaining areas rural.\nSuppose our goal is to divide this state into 5 districts, each containing exactly\n5 parcels.\nY\nB C D\nE F G H\nI J K L M\nA\nN\nO P Q\nR S\nT U V\nW X\nFigure 1. Sample map of districts.\nSuppose that the voting outcome (Vout) of the ensuing election is given by Figure\n2, which shows in each parcel the percentage of votes that party A receives.\n8 ZEPH LANDAU AND FRANCIS EDWARD SU\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\nFigure 2. Vote totals by district.\nThe ability to use the redistricting process to gain political advantage relies on\nthe ability to predict some features of Vout. In general, of course, Vout is not known\nprecisely at the time districts are being re-drawn. However, the combination of\ndata from previous elections and opinion polling increasingly gives a more and\nmore accurate model of how votes will be distributed. For the purposes of this\nexample we will assume that the working voting model V of both parties coincides\nwith Vout in Figure 2. In this example, party A has a slim statewide majority,\nreceiving 50.12% of the total vote.\nFor this particular example, we will assume that the only thing the two parties\ncare about is maximizing the number of districts they can win; thus their preferences\nare diametrically opposed with each having rating system Rwin. We emphasize that\nthis is an assumption we make for this example but that the protocol is designed\nto work under much more general preferences—the additive ratings system Rsum,\ndiscussed earlier.\nNotice that even though parties A and B each have approximately half the\nvoters over the state, if either party is given complete control of the district-drawing\nprocess, they can draw districts so that they are the majority in 4 of the 5 districts.\nSee Figure 3. In this example, the absolute geometric target for either party is\n1+4\n2\n= 2.5 districts.\nFAIR DIVISION AND REDISTRICTING 9\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 3. The left diagram shows a division in which A can win\n4 districts. The right diagram shows a division in which B can win\n4 districts. Districts that A wins are shaded.\n6.2. The Protocol. There are three core steps to the redistricting protocol along\nwith an augmenting fourth step. After outlining them in general, we will work\nthrough each step in the above example.\n• Split Sequence Generation. This step is performed by the independent\nagent I. The agent generates a sequence of so-called k-splits: a k-split is\na division of the state into two pieces (piece 1 and piece 2) such that the\npopulation within piece 1 totals the number of people in k districts. The\nindependent agent I generates a split sequence: a 1-split, a 2-split, a 3-split,\netc. with each split building on the previous so that piece 1 of the j-split\ncontains piece 1 of the (j − 1)-split for all j.\n• Preference. For each of the k-splits, the two parties are each asked which\nof the following options they would prefer:\n(1) to have party A divide piece 1 of the split into k districts and have\nparty B divide piece 2 into n−k districts (where n is the total number\nof districts).\n(2) to have party B divide piece 1 of the split into k districts and have\nparty A divide piece 2 into n− k districts.\nEach party has the option of saying that they are indifferent to the two\nchoices2.\n• Resolution. If there exists an i-split such that parties A and B both prefer\nthe same option in the preference step above then create a map using that\noption. If there exists an i-split such that one party is indifferent, then\ncreate a map using the option selected by the party that was not indifferent.\nIf there exists an i-split such that both parties are indifferent, then create\na map by randomly choosing one of the options for that i-split.\nIf none of the above scenarios occur it means that the parties have oppo-\nsite preferences for each i. Find the first i0, 1 ≤ i0 ≤ n− 2 for which party\nA prefers option (2) for i = i0 and switches preferences to option (1) when\ni = i0 + 1. (This scenario is guaranteed to occur at least once since party\n2This additional option is a modification of the original protocol in [6].\n10 ZEPH LANDAU AND FRANCIS EDWARD SU\nA prefers option (2) when i = 1 and prefers option (1) when i = n − 1.)\nRandomly choose to divide the state from the following four prescriptions:\ni. use option (1) for the i0-split,\nii. use option (2) for i0-split,\niii. use option (1) for (i0 + 1)-split,\niv. use option (2) for (i0 + 1)-split.\n• Augmentation. We perform the above 3 steps for a number of different\nsplit sequences to produce a number of divisions of the state. We then\nuse the ranking protocol of Section 5 to choose among these maps, i.e.\neach party ranks the divisions (from best to worst) according to their own\npreferences, and the division whose worst ranking (among both parties) is\nhighest is the one that is chosen. If there are two such splits, select one of\nthem at random.\n6.3. The protocol in action. We now show how the protocol works for the ex-\nample described in Figure 2.\nSplit Sequence Generation step. Suppose the Split Generation step yields the\nsplit sequence in Figure 4. In each diagram piece 1 will be the left piece and piece\n2 will be the right piece.\nFigure 4. Split sequence generation. From left to right, a 1-split,\n2-split, 3-split, 4-split.\nPreference step. For the 1-split, we exhibit in Figure 5 a possible way each party\ncan optimize its own interests in each of the two options. In option (1), A divides\npiece 1 and B divides piece 2. In option (2), B divides piece 1 and A divides piece\n2.\nFAIR DIVISION AND REDISTRICTING 11\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 5. Two options for the 1-split. The first diagram is option\n(1): in which A divides the left piece and B divides the right piece.\nThe second diagram is option (2) in which B divides the left piece\nand A divides the right piece.\nThus party A would prefer option (2) (since it would win 4 out of 5 districts)\nand party B would prefer option (1) (since it would win 4 out of 5 districts). This\nis not surprising since there is no opportunity to gerrymander the left piece.\nWe then consider the same question for the 2-split. Figure 6 shows one possible\nway each party can optimize its own interests in each of the two options. Here,\nparty A will still prefer option (2) and party B would still prefer option (1).\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 6. Two options for the 2-split. The first diagram is option\n(1): in which A divides the left piece and B divides the right piece.\nThe second diagram is option (2) in which B divides the left piece\nand A divides the right piece.\nFor the 3-split, Figure 7 shows one possible way each party can optimize its own\ninterests in each of the two options. Notice that now the parties preferences have\nchanged: party A now prefers (1) and party B prefers (2).\n12 ZEPH LANDAU AND FRANCIS EDWARD SU\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 7. Two options for the 3-split. The first diagram is option\n(1): in which A divides the left piece and B divides the right piece.\nThe second diagram is option (2) in which B divides the left piece\nand A divides the right piece.\nFinally, for the 4-split, Figure 8 shows one possible way each party can optimize\nits own interests in each of the two options. Again, party A prefers (1) and party\nB prefers (2).\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 8. Two options for the 4-split. The first diagram is option\n(1): in which A divides the left piece and B divides the right piece.\nThe second diagram is option (2) in which B divides the left piece\nand A divides the right piece.\nWe summarize the results from party A’s point of view in the following table:\nFAIR DIVISION AND REDISTRICTING 13\nTable 1. The number of districts A wins on the left\/right sides\nof split and total.\noption (1): left\/right=total option (2): left\/right = total\n1-split 1\/1=2 1\/3=4\n2-split 1\/1=2 1\/2=3\n3-split 3\/0=3 1\/1=2\n4-split 4\/0=4 1\/0=1\nWith all the preferences now stated, we are ready to move to the final step of\nthe protocol.\nResolution Step. Since the two parties prefer different options in each of the\nfour splits, we find the point at which party A’s preference switches; here it occurs\nbetween the 2-split and the 3-split. Thus i0 = 2 and we randomly choose between\nthe four prescriptions corresponding to the options listed in the second and third\nrow of Table 1.\ni. option (1) for the 2-split with the result: party A wins 2 districts, party B\nwins 3.\nii. option (2) for the 2-split with the result: party A wins 3 districts, party B\nwins 2.\niii. option (1) for the 3-split with the result: party A wins 3 districts, party B\nwins 2.\niv. option (2) for the 3-split with the result: party A wins 2 districts, party B\nwins 3.\nNotice that these results have party A winning either 40% or 60% of the districts,\nthe two closest achievable percentages to both the percentage of votes for party A\n(50.12%) and the percentage of districts given by the absolute geometric target\n(2.5\/5 = 50%). This is the result that the protocol is designed to produce; it\nis argued in [6] that a rigorous result establishing a good choice property of the\nprotocol combined with the way i0 is chosen will result in this kind of behavior for\nmost choice of split sequences. We discuss this in the next section.\nAugmentation Step. For our example, we run the same protocol for the following\nfour additional split sequences:\nFigure 9. Vertical Split Sequence.\n14 ZEPH LANDAU AND FRANCIS EDWARD SU\nFigure 10. Horizontal Split Sequence.\nFigure 11. Diagonal Split Sequence 1.\nFigure 12. Diagonal Split Sequence 2.\nThese have the following outcomes from party A’s perspective, listed in Table 2.\nFAIR DIVISION AND REDISTRICTING 15\nTable 2. The number of districts A wins in piece 1, piece 2, and total.\noption (1): pc.1\/pc.2 = total option (2): pc.1\/pc.2 = total\nVertical Split\n1-split 1\/1=2 1\/3=4\n2-split 1\/1=2 1\/2=3\n3-split 2\/0=2 1\/1=2\n4-split 4\/0=4 1\/0=1\nHorizontal Split\n1-split 0\/1=1 0\/4=4\n2-split 0\/2=2 0\/3=3\n3-split 2\/1=3 1\/2=3\n4-split 3\/0=3 1\/0=1\nDiagonal Split 1\n1-split 0\/1=1 0\/3=3\n2-split 1\/1=2 0\/3=3\n3-split 2\/1=3 1\/2=3\n4-split 3\/0=3 1\/0=1\nDiagonal Split 2\n1-split 1\/1=2 1\/3=4\n2-split 2\/0=2 2\/2=4\n3-split 3\/0=3 2\/1=3\n4-split 4\/0=4 1\/0=1\nUnlike the first split sequence from Figure 4 that we explored in detail, the\nresult of each of these split sequences is that the parties will be indifferent to one\nsplit. Here are possible maps from the results of the protocol for each of these split\nsequences:\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 13. Outcomes for Vertical Split Sequence.\n16 ZEPH LANDAU AND FRANCIS EDWARD SU\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 14. Outcomes for Horizontal Split Sequence.\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 15. Outcomes for Diagonal Split Sequence 1.\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\n.65 .65.59\n.59 .59 .65\n.41\n.41\n.41\n.41\n.41\n.53 .53 .53\n.53\n.53\n.53.53\n.35\n.47\n.47\n.47\n.35\n.47\n.47\nFigure 16. Outcomes for Diagonal Split Sequence 2.\nIn the ranking protocol, both parties would rank the 5 different outcomes from\nbest to worst. Depending on the outcome of the random choice of prescription\nfor the first split sequence (Figure 4), the outcomes would be party A winning 3\ndistricts 3 or 4 times (Horizontal Split sequence, both Diagonal split sequences,\nFAIR DIVISION AND REDISTRICTING 17\nand possibly the first split sequence), and winning 2 districts 1 or 2 times (Vertical\nsplit sequence, and possibly the first split sequence). Since the result of the ranking\nprotocol will result in one of the top three outcomes for both parties, in this case,\nthe final outcome will be party A winning 3 districts and party B winning 2. We\nremark that this resolution is as close as one can get to both the absolute geometric\ntarget (2.5) and the proportion of constituent voters (50.12 %)\n6.4. Fairness qualities of the protocol. Having explained the protocol we turn\nto a discussion of why the protocol is fair. We will analyze the protocol from the\npoint of view of party A; the identical analysis can be made for party B. We address\nthe following two questions:\n• If the map is created from a choice party A preferred or was indifferent to,\nwill it be fair for party A?\n• What if the randomness in the algorithm results in a choice party A did not\nprefer?\nThe first question is answered in the affirmative by establishing that the protocol\nhas the good choice property [6]. Approximately, the good choice property says that\nif party A is using a voting model V and has an additive rating system R, there\nwill be a choice for party A that achieves an outcome that is at least as good as a\nnumber close to the geometric target for V and R (see Section 4 for definitions).\nPrecisely, for a given k-split define the party’s k-split geometric target for V and R\nto be the average rating of the best and worst outcomes over all viable divisions\nthat include the dividing line of the k-split. Then the good choice property is:\nTheorem 6.1 (Good Choice Property [6]). For any voting model V and rating\nsystem R, one of the choices given in the protocol achieves an outcome that is at\nleast as good as the party’s k-split geometric target for V and R.\nFor our example above party A has been acting according to its interests with\nV = Vout and R = Rwin. The good choice property follows from the following\nobservation that is perhaps best seen pictorially (see Figure 17): given a particular\nk-split, the average of the number of districts won by party A under options (1)\nand (2) is equal to the average of the number of districts that party A wins if it\nhad complete control (which would result in the best outcome for party A) and if\nit had no control (which would result in an outcome no worse than the worst for\nparty A).\nThus at least one of the two options is better than the average outcome between\nthe best and worst scenario for party A, which is precisely the definition of the\nk-split geometric target when V = Vout and R = Rwin. It should be clear that the\nsame argument holds regardless of choice of V and additive rating system R.\nThe astute reader will notice that the k-split geometric target for V and R can\ndiffer from the geometric target for V and R; in other words, insisting that the\ndivision includes the boundary given by the k-split can penalize one party. Two\nobservation suggest that should this happen, the penalty will not be large. First,\nthe choice of split is made by an independent (neutral) third party and thus should\nbe no more biased against one party than a random choice. Second, in the case\nwhere there are no geometric constraints (see [6]), the absolute geometric target\nand the k-split geometric target for Vout and Rwin can differ by at most\n1\n2\n. In\nour example, we see this difference between the k-split geometric target and the\nabsolute geometric target: for party A, in the 3-split in Vertical Split sequence, and\n18 ZEPH LANDAU AND FRANCIS EDWARD SU\nBest for A\nWorst for A\nWorst for A\nBest for A\nBest for A Worst for A\nWorst for A Best for A\nFigure 17. The good choice property.\nfor party B, in the 3-splits on the final three split sequences of the protocol. In\neach of these cases, however, the difference from the absolute geometric target is\nas small as it could be: 1\n2\n. It is reasonable to assume that most splits will either\nnot particularly favor either party, or favor a party by a small amount. It is then\nthe augmentation step of the protocol that ensures that a rare “bad” split for a\nparticular party will not come into play (since the affected party would put such a\nsplit towards the bottom of their rankings).\nWe see therefore, that the good choice property, when coupled with the aug-\nmented protocol, implies that party A should be satisfied if the division is created\nby an option that it chose.\nWe now turn to the second question—how party A will fare if the randomness\nin the algorithm results in a choice they did not prefer. The randomness is im-\nplemented only if for each i-split, the two parties have opposite preferences (for\ninstance in the first split sequence described in Figure 4). We suppose the random\nprescription in the Resolution Step (see Section 6.2) is one not preferred by party\nA, for instance prescription (i.) in the Resolution step (i.e. option (1) for i = i0).\nIn our example, this would correspond to i0 = 2 in the first split sequence. Party\nA, however, prefered option (2): to divide piece 2 and have party B divide piece\n1. Notice, however, that party A would prefer to divide up piece 1 in the i0 + 1\nsplit, and this piece 1 only differs from the piece 1 of the i0 split by a small region\nwith a population equal to the size of a single district. (Similarly piece 2 in the i0\nand i0 + 1 splits only differ by this same small region). Because party A prefers\noption (2) for the i0 split and option (1) for the i0 +1 split (and because piece 1 of\nthese two splits do not differ by very much), it is reasonable to expect that party\nA’s preference for option (2) over option (1) for the i0 split is mild. (In our exam-\nple, this is indeed the case as party A achieves an outcome of winning 2 districts\nwhich is of minimal negative deviation from the absolute geometric target of 2.5).\nFAIR DIVISION AND REDISTRICTING 19\nIf indeed this is the case, then party A’s discontent with the division would only\nbe mild as we have shown (by the good choice property) that party A would have\nbeen satisfied with the slightly better option of (2).\nEven though the first pieces of the two splits differ by a small amount, one can\nconstruct scenarios where that small amount makes a big difference. However, recall\nthat the creation of the splits was done by an independent party and therefore one\nwould expect this type of scenario to be rare. Again, choosing to use the augmented\nprotocol would ensure that this rare scenario would not occur in the division chosen.\n7. Conclusion\nReplacing current redistricting procedures with the protocol presented here surely\npresents substantial political obstacles. It has been observed numerous times (e.g.,\nsee [5]) that any proposed change should be structured to take effect far enough in\nthe future so that it could not be interpreted as a power grab by one party. However,\nas noted in the last paragraph of the introduction, some of the ideas presented here\ncould be incorporated into current processes without requiring a complete overhaul\nof the redistricting process.\nIn this article, we have used a detailed example to explore the redistricting pro-\ntocol of [6]. We have shown how this procedure retains the usual constraints that\nmay be desirable to impose on a redistricting solution, while incorporating some of\nthe best features of a fair division procedure: multilateral evaluation, procedural\nfairness, and fairness guarantees. Procedural fairness is apparent in the protocol,\nthe geometric targets incorporate multilateral evaluation, and the ability to ensure\noutcomes near geometric targets provides the fairness guarantee. The result is a so-\nlution that accounts for both parties having different interests, involves a resolution\nprocess and an interactive protocol to elicit preferences, and provides mathematical\nconfidence that the outcome will be fair.\nReferences\n[1] N. Apollonio, R. I. Becker, I. Lari, F. Ricca, and B. Simeone, The Sunfish against the\nOctopus: opposing compactness to gerrymandering, in Mathematics and Democracy. Recent\nadvances in Voting Systems and Collective Choice, Studies in Choice and Welfare (2006),\n19–41.\n[2] S. J. Brams, M. A. Jones and C. Klamler, N-Person Cake-Cutting: There May Be No\nPerfect Division, Amer. Math. Monthly 120 (2013), 35–47.\n[3] S. J. Brams and A. D. Taylor, Fair Division: from Cake-Cutting to Dispute Resolution,\nCambridge University Press, 1996 .\n[4] S. J. Brams and A. D. Taylor, The Win-Win Solution: Guaranteeing Fair Shares to Every-\nbody, New York: W. W. Norton, Inc., 1999.\n[5] S. Hirsch and T. E. Mann, “For Election Reform, a Heartening Defeat”, New York Times,\nNov. 11, 2005, also at http:\/\/www.nytimes.com\/2005\/11\/11\/opinion\/11mann.html.\n[6] Z. Landau, O. Reid and I. Yershov, A Fair Division Solution to the Problem of Redistricting,\nSocial Choice and Welfare 32, Issue 3 (2009), 479–492.\n[7] H. Moulin, Fair Division and Collective Welfare, MIT Press, 2003.\n[8] Office of the Clerk, U.S. House of Representatives website:\nhttp:\/\/clerk.house.gov\/members\/electionInfo\/elections.html\n[9] E. Peterson and F. E. Su, Four-person envy-free chore division, Math. Mag. 75 (2002),\n117—122.\n[10] J. Rawls, A Theory of Justice, Harvard University Press, 1971.\n[11] Redistricting Law 2000. Denver, CO: National Conference of State Legislatures, 1999. Avail-\nable at\n20 ZEPH LANDAU AND FRANCIS EDWARD SU\nhttp:\/\/www.senate.leg.state.mn.us\/departments\/scr\/redist\/\nred2000\/red-tc.htm\n[12] J. M. Robertson and W. A. Webb, Cake-Cutting Algorithms: Be Fair If You Can, A. K.\nPeters Ltd., 1998.\n[13] B. I. Spector, Analytical Support to Negotiations: An Empirical Assessment, Group Deci-\nsion and Negotiation 6 (1997), 421–436.\n[14] H. Steinhaus, The problem of fair division, Econometrica 16 (1948), 101–104.\n[15] F. E. Su, Rental harmony: Sperner’s lemma in fair division, Amer. Math. Monthly 106\n(1999), 930–942.\nDepartment of Engineering and Computer Science, University of California at Berke-\nley, Berkeley CA 94720\nE-mail address: zeph.landau@gmail.com\nDepartment of Mathematics, Harvey Mudd College, Claremont, CA 91711\nE-mail address: su@math.hmc.edu\n"}
{"index":{}}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! logic. Computability of Real Numbers Posted on June 29, 2019. by. Samuel Nunoo We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We... Continue reading... All Horses are the Same Color Posted on June 29, 2019. by. Samuel Nunoo If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the... Continue reading... Continuum Hypothesis Posted on June 29, 2019. by. Samuel Nunoo We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 correspondence with... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). 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{"id":"lKhCDW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Procedural-support-for-cooperative-negotiations.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":111,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Procedural-support-for-cooperative-negotiations","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Procedural-support-for-cooperative-negotiations.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:15 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":2083995334385482475,"exact_signature_unique_b":true,"fuzzy_signature_l":2798223087140017866,"fuzzy_signature_unique_b":true,"wordcount_i":2190,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:48 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:05 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":79471,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088111621242880,"text_t":"appeared in: Advances in Decision Technology and Intelligent Information Systems, Volume I (eds. Kurt J. Engemann and George E. Lasker),\nThe International Institute for Advanced Studies in Systems Research and Cybernetics, Windsor, Canada, 2000, pp. 31-36.\nProcedural Support for Cooperative Negotiations\nTheoretical Design and Practical Implementation\nMatthias G. Raith\nUniversity of Bielefeld, Germany\nFrancis Edward Su\nHarvey Mudd College and Cornell University, USA\nAbstract: We discuss the theoretical design of algorithms for solving distributional con-\nflicts within groups. We consider an algorithm to be procedural if the implementation of\nthe outcome requires the participation of the players, or if it can even be conducted by\nthe players themselves without computational assistance. We compare two procedures for\nmultilateral problems of fair division; both establish envy-freeness, given the possibility of\nmonetary compensations between players.\n1 Introduction\nNegotiation is always related to distribution — either of benefits or of costs. Since this\ndistributive dimension is present in every bargaining situation, negotiating parties must\ninevitably deal with conflict in one way or another. Negotiation analysis is a practically\noriented field of research that seeks to offer negotiators prescriptive advice on how to establish\na desirable group decision in practice. A major goal is the development of procedures for\ndealing with conflicts in a constructive manner. Since negotiation always involves the division\nof some “pie,” a promising approach seems to be the analysis of problems of fair division.\nThe theoretical development of fair-division procedures has become quite popular over\nthe past decade. Of particular interest for group decisions are procedures dealing with mul-\ntilateral distribution problems. New theoretical results show conditions for the existence of\nenvy-free solutions, where an outcome is considered to be envy free if each party believes\nthat its share is at least as desirable as that of any other party. The mere proof of existence,\nhowever, does not yet provide the desired outcome. More helpful in this respect are con-\nstructive proofs of envy-freeness, yielding solutions via algorithms that are implementable,\ne.g., as a computer algorithm.\nIn this paper we extend the notion of a constructive proof: we consider an algorithm to\nbe procedural if the implementation of the group decision involves the active participation of\nthe players, or if it can even be conducted by the players themselves without computational\nassistance. This aspect seems to be essential for the acceptance of analytical support, be-\ncause, as Spector (1997) observes, practitioners are often reluctant to use support tools that\noffer solutions simply at the push of a button. What is required in his view is (procedural)\nassistance that guides negotiators to a mutually acceptable agreement.\n1\n2 A Multilateral Fair-Division Problem\nConsider the following situation: Four individuals wish to rent a house with four bedrooms\nfor a total rent of, say $100. Is there a possibility of allocating rooms to players and dividing\nthe total rent between rooms in such a way that no individual will wish to trade rooms\nwith another? Under relatively mild assumptions concerning players’ preferences such an\nenvy-free allocation exists — the question is how to find it.\nSuppose that each player Pi, i=1, . . . , 4, is able to express her valuations of the individual\nrooms Rj, j=1, . . . , 4, via monetary bids, as given in Table 1 (cf. Haake, Raith, and Su, 1999).\nR1 R2 R3 R4\nP1 50 20 10 20\nP2 60 40 15 10\nP3 0 40 25 35\nP4 50 35 10 30\nTable 1: Players’ monetary valuations (bids) of rooms\nIn addition, if players’ preferences over money are linear, then each player Pi will prefer\nthe room Rj which offers her the maximal discount, given by the difference between her\nindividual bid and the actual price of the room.\n3 An Interactive Choice Procedure\nA possible procedure for allocating rooms and splitting the rent is to assign prices to the\nindividual rooms that sum to the total rent, and then let each player choose the room she\nlikes best. For example, if all rooms are priced equally at $25, then P1, P2, and P4 will all\nwish to have R1, while P3 would choose R2. With only one player allowed in each room,\nthis price distribution is sure to create envy. However, if a pricing scheme can be found such\nthat each room is desired by a different player, then the allocation will be envy-free.\nThe interactive fair-division procedure developed by Su (1999) is based on a constructive\nproof of the combinatorial lemma given by Sperner (1928), reformulated to find an envy-\nfree pricing scheme. The algorithm sequentially offers each player a specific pricing scheme\nand then lets this player choose her favorite alternative. For example if P1 is offered the\npricing scheme [0,0,0,100], her best choice will be R1, because this maximizes her discount.\nUnder the Sperner approach, each player’s choice of an alternative determines which pricing\nscheme is offered next and to which player. The interactive procedure consistently works\nits way towards envy-freeness, and it provably converges on an envy-free pricing scheme\nfor any given level of precision with which players can determine the value of rooms. For\nour example above, the pricing scheme [45,25,10,20] induces P1, P2, P3, and P4 to choose\nR1, R2, R3, and R4, respectively. Subtracting room prices from players’ valuations, given\n2\nin Table 1, shows that these choices give players maximum discounts for the given pricing\nscheme.\nImplemented on a computer, the Sperner choice procedure operates in the form of\nan artificial mediator. The mediation procedure can be employed using the web-based\n“Fair-Division Calculator” (http:\/\/www.math.hmc.edu\/̃ su\/fairdivision\/calc\/). Par-\nties can actively take part in the process and are allowed to make choices to their individual\nbenefit. The Fair Division Calculator plays the part of the mediator by offering pricing\nschemes in a manner that will lead the group to an envy-free allocation, provided that par-\nticipants’ choices are consistent and truthful. If a player behaves inconsistently, either by\naccident or due to a change of preferences, the mediator can react by pointing to the inconsis-\ntency or by accepting the modified preferences, thus ignoring previous (inconsistent) choices.\nStrategic choices, on the other hand, cannot be unmasked as long as they are consistent.\nHowever, players will find it quite difficult to manipulate the procedure, because the other\nplayers’ preferences are never fully revealed. Indeed, if the choices of the other players are\nkept confidential up to the final round, then a player, when offered a pricing scheme, will\nnever know whether the next choice will be her last.\nViewed as a mediation process, the main difficulty with the Sperner choice procedure is\nthat the operations of the mediator are largely incomprehensible to the group. Although each\nplayer has a free choice of alternatives and thereby determines the direction of the process,\nshe will generally have a problem understanding the logic behind the evolution of pricing\nschemes. Players must completely rely on the artificial mediator to arrive at an envy-free\noutcome. Without this trust, the choice procedure may not be acceptable for the group.\n4 A Mediating Compensation Procedure\nAcceptability of a live mediation procedure typically requires additional features that relate\nthe process to the players involved. To be acceptable, a mediation procedure should be in-\ntuitive, meaning that the implementation of an outcome must accord to the players’ way of\nreasoning. In addition, the procedure should be plausible, meaning that the individual steps\nmust match the players’ way of arguing. And finally, the procedure should be manageable,\nmeaning that the required calculations must correspond to the players’ computational abil-\nities. With the compensation procedure developed by Haake, Raith, and Su (1999), players\nare able to see how the outcome evolves through the interaction of their preferences. They\ncan, therefore, understand the mediation process, and even conduct it themselves.\nThe compensation procedure is initialized by having players individually and simultane-\nously submit monetary bids for the individual rooms. All bids, disclosed together, then form\nthe basis of the mediation process leading to envy-freeness.\nThe first step of the mediation procedure is to assign rooms to players, such that the\nsum of players’ bids is maximized. In our example, this utilitarian allocation is achieved\nby assigning R1, R2, R3, and R4 to players P1, P2,P3, and P4, respectively (in Table 1,\nthe diagonal assignment). The utilitarian assignment ensures that no sub-group of players\ncan mutually gain by trading rooms. Consequently, there can be no two-way or cyclical\n3\nenvy, since this could be eliminated via a mutually beneficial trade. However, a utilitarian\nassignment may still leave individual players envious. As Table 1 shows, P1 is satisfied with\nR1, but P2 and P4 prefer R1 as well, while P3 would rather have R2. As any envious player\ncan verify, though, no player is willing to enter a trade with her.\nThe second step of the procedure is to compensate all envious players by an amount\nsufficient to eliminate their envy. The most plausible starting point is to begin with those\nplayers whose maximum envy is directed towards a non-envious player. A non-envious\nplayer always exists, because if all players were envious, there would be efficient trading\nopportunities. As Table 1 shows, P1 is the non-envious player, and P2 and P4 need to be\ncompensated for their envy; P3 envies an envious player and can, therefore, be taken care\nof in a later round. By adding $20 to the value of R2 and R4, P2 and P4 are not envious\nof P1 anymore. Indeed, P2 is now envy-free, but, as the left-hand panel of Table 2 shows,\nP4 now envies P2, and P3 is even more envious of P2 than before. In the second round P3\nand P4 are, therefore, compensated by adding $35 to R3 and an additional $5 to R4. As the\nright-hand panel of Table 2 shows, all players are now satisfied with their assigned rooms.\nR1 R2 R3 R4\nP1 50 40 10 40\nP2 60 60 15 30\nP3 0 60 25 50\nP4 50 55 10 50\ncomp. 0 20 0 20\nR1 R2 R3 R4\nP1 50 40 45 45\nP2 60 60 50 35\nP3 0 60 60 55\nP4 50 55 45 55\ncomp. 0 20 35 25\nTable 2: Players’ most preferred rooms after one and two rounds of compensations\nAs Haake, Raith, and Su (1999) formally show, after k rounds of compensation at least k+\n1 players will be envy-free. Envy-freeness for 4 players thus requires at most 3 compensation\nrounds; in our example only two are necessary. Note that, even if players do not understand\nthe formal proof of this claim, they can verify its validity at every step of the procedure.\nAfter envy-freeness is achieved, the group must bear the cost of compensations, $80, plus\nthe rent for the house, $100. If the total cost is shared equally, each player pays $45. Net of\ncompensations, the final envy-free prices for the rooms are [45,25,10,20].\nAlthough the compensation procedure can generally be conducted without computa-\ntional assistance, larger groups may wish to have precise guidance as the mediation process\nbecomes longer. Moreover, the utilitarian assignment becomes increasingly difficult to find\nwithout analytical support. Implemented on a computer, the compensation procedure pro-\nvides the complete mediation process, based only on players’ initial valuations of rooms —\nthe compensation procedure can be employed using the Fair Division Calculator (FDC).\nHowever, in contrast to the Sperner choice procedure in which the FDC works its way to\nenvy-freeness by interacting with the players, the FDC does not operate as a mediator under\nthe compensation procedure. Instead, it supplies the mediation process that can be used by\na mediator or the group itself as a map to envy-freeness.\n4\n5 Decentralized Procedural Support\nAdapting group interaction to the support environment is a common feature of many com-\nputerized support systems. However, if the system imposes too many restrictions on the\ngroup decision process, users may find it too artificial and, therefore, be reluctant to use it\nin practice. Practitioners often find their flexibility constrained by the way they are forced to\ninteract with the system. The challenge in designing a practicable support system, therefore,\nlies in the development of analytical techniques that complement the group’s interaction in\na “natural” way.\nAs a web-based application the Fair Division Calculator can be employed from any remote\nlocation. Equipped with a variety of different mediation procedures, the FDC provides full\nanalytical mediation support. Depending on the amount of information that parties are\nwilling to reveal, the FDC can assist by offering the group a workable mediation process, or\nit can be part of the process and operate as an artificial mediator.\nSince the FDC offers group support via a single user interface, decentralized group inter-\naction requires additional support facilities, allowing parties to interact with each other from\nremote locations. This has been accomplished by integrating the FDC into a negotiation\nplatform under the the internet system ARTUS (http:\/\/artus.wiwi.uni-bielefeld.de\/).\nARTUS provides an infrastructure for group decision making over the Internet in the form\nof virtual tables. The system includes options for defining the method and structure of\nmultilateral interaction, and it records the complete interaction for security, documentation,\nand evaluation (for a detailed documentation see Raith and Wilker, 1998).\nDue to its accessibility via the internet and its variable user interfaces, ARTUS can be\nadapted to many different forms of decentralized group interaction, with or without analytical\nsupport. A mediation process via ARTUS allows the participation of a natural mediator,\nequipped with all the psychological skills necessary for guiding parties in a conflict situation,\nand assisted by the analytical skills of a support system such as the Fair Division Calulator.\nGroup support can thus be integrated naturally in an artificial environment.\nReferences\nHaake, C.J., M.G. Raith, F.E. Su (1999): “Bidding for Envy-Freeness,” University of Biele-\nfeld, Institute of Mathematical Economics, WP 311\nRaith, M.G., H. Wilker (1998): “ARTUS: The Adaptable Round Table with a User-Specific\nSurface,” University of Bielefeld, Institute of Mathematical Economics, WP 298\nSpector, B.I. (1997): “Analytical Support to Negotiations: An Empirical Assessment,”\nGroup Decision and Negotiation, 6, 421-436\nSperner, E (1928): “Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes,”\nAbh. Math. Sem., Univ. Hamburg, 6, 265-272\nSu, F.E. (1999): “Rental Harmony: Sperner’s Lemma in Fair Division,” American Mathe-\nmatical Monthly, 106, 922-934\n5\n"}
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Simmons , Francis Edward Su\na\nDepartment of Mathematics, Portland Community College, Portland, OR 97229,USA\nb\nDepartment of Mathematics, Harvey Mudd College, Claremont CA 91711,USA\nReceived 1 February 2001; received in revised form 1 March 2002; accepted 1 August 2002\nAbstract\nIn this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a\nconsensus-halving: a division of an object into two portions so that each ofn people believes the\nportions are equal. Moreover, the division takes at mostn cuts, which is best possible. This\nextends prior work using methods from combinatorial topology to solve fair division problems.\nSeveral applications of consensus-halving are discussed.\n   2002 Elsevier Science B.V. All rights reserved.\nKeywords: Tucker’s lemma; Borsuk-Ulam theorem; Fair division\nJEL classification: C61; D63; D74\nAMS classification: Primary 55M20; Secondary 91B32, 91A06\n1 . Introduction\nThe study of fair division problems is concerned with finding ways to divide an object\namong several parties according to some notion of fairness. The cake-cutting problem of\nSteinhaus (1948) is perhaps the best known example. Aside from the division of goods,\nother fair-division problems address the division of burdens (e.g., the chore-division\nproblem (Gardner, 1978; Peterson and Su, 2002)) and the division of mixtures of goods\nand burdens (e.g., ther nt-partitioning problem (Brams and Kilgour, 2001; Haake et al.,\n2002; Su, 1999): how to split rent so that housemates are satisfied by different rooms).\nRecently, ideas from combinatorial topology have provided new and constructive\n*Corresponding author. Tel.:11-909-607-3616; fax:11-909-621-8366.\nE-mail address: su@math.hmc.edu(F.E. Su).\n1\nResearch partially supported by a Beckman Research Grant at Harvey Mudd College.\n0165-4896\/02\/$ – see front matter   2002 Elsevier Science B.V. All rights reserved.\ndoi:10.1016\/S0165-4896(02)00087-2\n16 F.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25\nmethods for obtaining solutions to fair-division problems. Su (1999) discusses a\ncake-cutting procedure of Simmons that can be extended to obtain envy-free solutions\nfor chore division and rent-partitioning using variants of a result known as Sperner’s\nlemma, which is the combinatorial equivalent of the Brouwer fixed point theorem of\ntopology.\nIn this paper, we demonstrate how a result known as Tucker’s lemma, which is the\ncombinatorial equivalent of the Borsuk-Ulam theorem of topology, can be used to solve\na different kind of fair-division problem: is it possible to divide a mixture into two\nportions so that each ofn people believes both portions are the same size (a consensus-\nhalving)? Moreover, a constructive proof of Tucker’s lemma yields an effective\nalgorithm for locating an approximate solution that uses a minimal number of cuts.\nAs an application, an approximate consensus-halving procedure could allow two\nfamilies to split a piece of land into two regions such that every member of both families\nbelieves the land is nearly equally divided. Using a minimal number of cuts would also\nbe desirable in this setting. We discuss potential applications to the Law of the Sea\nTreaty (Brams and Taylor, 1996) and then cklace-splitting problem of Alon (1987).\nAnother application solves ateam-splitting problem: given a territory and a pair each of\nzoologists, botanists, and archaeologists, is it possible to divide the territory into two\nportions in such a way that members of any given pair will prefer to explore different\nportions? Thus the group could be split in anenvy-free fashion—into two teams with\none member of each specialty among them. We explain how a consensus-halving method\ncan be adapted for this purpose near the end of this article.\n2 . Remarks on procedures and algorithms\nAs we shall be interested in procedures of various kinds, we make a few associated\ndefinitions. We use the termalgorithm to refer to a procedure that proceeds in discrete\ntime steps and terminates after a finite number of steps. (Thus the so-called ‘moving-\nknife’ procedures (Brams et al., 1997) are not algorithms because they require\ncontinuous evaluation of the cake.) For problems that have a finite number of solution\ncandidates, one algorithm that can be used to find a solution is a brute-forceexhaustive\nsearch, which involves sequentially testing each of the finite number of possibilities.\nAny algorithm that locates a solution more systematically than an exhaustive search will\nbe calledeffective. We shall be concerned with effective algorithms in this paper.\n3 . Consensus-halving\nTo be precise, we assume that anyobject A to be divided is a measurable bounded set\n2 3\nin R or R , infinitely divisible, and that each playeri has a bounded continuous\nmeasurem on (measurable) subsets ofA which describes the (positive or negative) valuei\nthat she assigns to that subset. Thus the object we consider can be a mixture of desirable\nand undesirable parts (the players may in fact disagree on which parts are desirable and\nundesirable). By acontinuous measure, we mean absolutely continuous with respect to\nF.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25 17\nLebesgue measure, so that continuity of the measures (with respect to Lebesgue\nmeasure) forbids the existence of ‘point masses’—zero volume subsets with non-zero\nworth.\nAlthough we model player preferences with measures, we remark that none of our\nproofs will require them to be additive over subsets. (Them may as well be continuousi i\nset functions defined on the Borels-algebra and satisfying all the properties of measures\nexcept countable additivity.) Thus wedo not need to require that player valuations over\nsubsets ofA be additively separable.\nWe establish the following theorem.\nTheorem 1. (Consensus-Halving)Consider an object A, and n people whose prefer-\nences on A are modeled by continuous measures hm , . . . ,m j. Using at most n cuts by1 n\nparallel planes, A may be divided into two portions A and A such that each of n1 2\npeople thinks that A and A are exactly equal, i.e., m (A )5m (A ) for all1 2 i 1 i 2\ni [ h1,2,. . . ,nj. This number of cuts by parallel planes is best possible; there is a\ncollection of player measures which requires n cuts. Moreover, an effective algorithm\nexists for locating an ´-approximate solution, i.e., two portions A ,A such that1 2\num (A )2m (A )u#´ for all i [ h1,2,. . . ,nj.i 1 i 2\nNon-constructive versions of Theorem 1 have already been obtained; for instance, see\nGoldberg and West (1985) and Alon and West (1986). The latter uses the Borsuk-Ulam\ntheorem but in a fashion that requires additivity of the measuresm . Alon (1987) provesi\na generalization that producesk equal portions according ton probability measures; it\nyields our result whenk 5 2, but it is also non-constructive and based on a topological\n´ ´result of Barany et al. (1981). Another approach to produce the existence of the setsA1\nand A is to use Lyapunov’s theorem (see Barbanel (1996)); however, it is even less2\nconstructive because it does not even say how many cuts are required or what the sets\nA and A might look like.1 2\nBy contrast our result is constructive in the sense that the proof yields a method for\nconverging on a solution. If only an approximate solution is desired, then there is an\neffective algorithm for locating it, using a combinatorial result known as Tucker’s\nlemma. Our proof does not require additivity nor positivity of the measures, nor must the\nmeasures satisfym (A)5 1 (but they should be bounded for the conclusion to makei\nsense). A constructive proof of Tucker’s lemma yields a simplicial algorithm that\nguarantees ań-approximate solution, so that each person believes the two portions are\nwithin size ´ of each other.\nIn some sense approximate solutions are the best one can hope for; Robertson and\nWebb (1988, p. 104) have shown that there is no finite algorithm that will produce an\nexact equal division. If one allows continuous procedures such as ‘moving-knife’\nsolutions (Brams et al., 1997), then a procedure due to Austin (1982) will achieve an\nexact consensus-halving forn 52. However, there is no known generalization of\nAustin’s procedure forn $ 3.\nWe make a couple of remarks about our´-approximate solution. First, note that a\nbrute-force search algorithm is always possible for locating an´- pproximate solution.\nBecause we know (by this theorem or Theorem 1.2 of Alon (1987)) that a solution\n18 F.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25\nexists, and since the set of possible divisions byn parallel planes is a compact setD,\nonly a finite number of divisions need be checked to find one that is with´ of a\nconsensus-halving. However, while our approach would use those same divisions,\nregarded as vertices in an associated triangulation ofD (see Section 4), our approach is\nmore systematic than a brute-force search because it only checks the vertices near a\n‘path’ in the triangulation, which for for reasonable player preferences is generally very\nstraight and does not wind throughout all ofD.\nSecond, we emphasize that the number of cuts involved is minimal. Brams and Taylor\n(1996, pp. 131–132) give an inductive procedure for an approximate consensus-halving,\nand Robertson and Webb (1988, p. 128) propose a procedure for approximate division\ninto ratios; however, both these procedures involve a large number of cuts which grows\nas ´ decreases. By contrast, our approach never uses more thann cuts.\n4 . Tucker’s combinatorial lemma and the Borsuk-Ulam theorem\nm\nRecall that ann-simplex in R is the convex hull ofn 11 affinely independent points\nm\n(vertices) in R . A k-face of an n-simplex is thek-simplex spanned by any subset of\nk 1 1 vertices. Atriangulation of a setX is a collection of (distinct)n-simplices whose\nunion isX, with the property that any two of them intersect in a face common to both, or\nnot at all.\nn n\nRepresent then-ball B by the set of all pointsx5 (x )[R such thatux u1 ? ? ? 1i 1\nn\nux u#1. The boundary of this ‘octahedral’ ball is thes t of all points inR satisfyingn\nn21\nux u1 ? ? ? 1 ux u5 1 and may be thought of as an (21)-sphereS . A centrally1 n\nn21\nsymmetric triangulation of S is one such that ifs is any simplex of the triangulation,\nthen 2s also is.\nThe following combinatorial theorem of Tucker (1945) was proved for the casen 5 2.\nThe proof for generaln may be found in Lefschetz (1949).\nn\nTucker’s Lemma. Let T be a centrally symmetric triangulation of S whose vertices are\nassigned labels from h61,62, . . . ,6nj such that labels of antipodal vertices sum to\nzero, i.e., the labelling function l satisfies l(2x)5 2 l(x) for any vertex x. Then there\nexist adjacent vertices in the triangulation whose labels sum to zero.\nThis result is often stated for a triangulation of a ball, but for our purposes later, we\nhave cast it for a triangulation of a sphere (obtained by gluing twon-balls along their\nboundaries) (see Fig. 1).\nTucker’s lemma is equivalent (see Freund and Todd, 1981) to the following famous\ntheorem from topology:\nn n\nThe Borsuk-Ulam Theorem. For any continuous function f :S →R , there exist\nn\nantipodal points x,2 x[ S such that f(x)5 f(2x).\nThe equivalence is valuable because Tucker’s lemma has a constructive proof, while\nthe Borsuk-Ulam theorem can be used to prove fair division theorems. For instance, the\nF.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25 19\n2\nFig. 1. An ‘octahedral’ sphereS (with triangulation not shown). The conditions of Tucker’s lemma (e.g., see\nthe antipodal12,22 labels) implies the existence of an edge in the triangulation whose labels sum to zero\n(e.g., the11,21 labels).\nHam Sandwich Theorem, which says that that there exists a hyperplane that perfectly\nn\nbisectsn sets of positive measure inR , is well-known to be a consequence of the\nBorsuk-Ulam theorem (Rotman, 1988, p. 413) and therefore Tucker’s lemma can be\nused to find such a hyperplane. However, as a fair-division theorem, the Ham Sandwich\nTheorem is of little practical value when the dimension of the sets is greater than 3, and\neven in dimension 3, it is unneeded if one allows several cuts.\nWe seek somewhat more practical applications of the Borsuk-Ulam theorem and\nTucker’s lemma, such as the consensus-halving result of Theorem 1.\nProof of Theorem 1. For ease of expression, we refer to the objectA to be divided as\n‘cake’ even though players may find certain subsets undesirable.\nPlaceA in a coordinate system aligned with the cardinal directions of the compass.\nAssume without loss of generality that the (east \/west) width ofA is one unit. Suppose\nfurther thatA is to be divided by vertical, parallel north–south planes.\nn\nEach point (x , . . . , x ) of S corresponds to a set of cuts of the cake (called a1 n11\ncut-set) obtained by making north \/south cuts so that (from west to east) the pieces have\nwidths of ux u, ux u, . . . ,ux u. Use the respective signs ofx ,x , . . .x to determine1 2 n11 1 2 n11\nwhich portion of the division gets the corresponding piece: collect all the pieces for\nwhich x is positive, lump them together, and call this the portionA . The other piecesi 1\nwill be lumped together to create portionA (see Fig. 2).2\nThe existence of a division such thatA and A are deemed exactly equal by all1 2\nn n\nplayers follows easily from the Borsuk-Ulam theorem; consider the functionf :S →R\nsuch that thei-th coordinate functionf (x)5m (A ), playeri’s measure of the ‘value’ ofi i 1\nA . This is a continuous function ofx (because of the continuity assumption on the1\nmeasures), hence by the Borsuk-Ulam theorem there exists a pointx such that\nn21\nf(x)5 f(2x). But since antipodal points onS correspond to the same division with the\nroles of A and A interchanged, the Borsuk-Ulam pointx corresponds to a set of (at1 2\n20 F.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25\nFig. 2. A cut-set represented by (20.2,1 0.1,1 0.2,2 0.3,1 0.2). The portionA is the union of the white1\npieces;A is the union of the shaded pieces.2\nmost)n cuts (and fewer if the components ofA are adjacent) such thatm (A )5m (A )i i 1 i 2\nfor all i, i.e., the pieces are deemed equal by all players. One may see thatn cuts are also\nnecessary in the case in whichA is a line segment and the player measures have support\nin n disjoint subintervals ofA. This shows the existence of a solution to the consensus\nhalving problem.\nTo construct an approximate solution (to any pre-specified error tolerance´), use\nn\nTucker’s lemma. Recall that every point inS corresponds to a cut-set. Giveń. 0,\nn\nchoose a triangulation ofS with mesh size so small that, in the cut-sets corresponding\nto any two adjacent vertices, the portionsA (and alsoA ) differ by no more thań in1 2\nany of the player measures.\nWe now assign to every vertex a label in the seth1 1,2 1, . . . ,1 n,2 nj which\nconsists of a number and a sign. Thenumber assigned to a vertex will be the number of\nthe player who believes the difference betweenA and A is greatest for the cut-set1 2\ncorresponding to that vertex. (If there are players equally distressed about the difference,\nchoose the smallest-numbered player.) Thesign assigned to a vertex will signify the\nportion that the ‘most distressed’ playerp efers in the cut-set corresponding to that\nvertex: if pieceA (resp. A ) is preferred, the sign assigned is1 (resp.2). (In case1 2\nthat player prefers both pieces equally, choose the portion containing the west edge of\nthe cake.)\nNote that this gives an anti-symmetric labellingl in which l(2x)5 2 l(x) at every\nvertex, because when the roles ofA and A are reversed, the same player is most1 2\ndistressed but her preference is reversed. (Thus moving to the antipodal vertex leaves the\nlabel number the same but flips the sign.)\nApplying Tucker’s lemma, there exists a pair of adjacent vertices in the triangulation\nwith the same label number but opposite signs. Either of these vertices corresponds to a\ncut-set that is anapproximate consensus-halving, since at these two nearby cut-sets, the\nmaximally distressed player prefers different portions. For this player, both portions are\nwithin ´ of each other in value, and since this player’s distress was maximal, no other\nplayer will dispute this assessment by more than´.\nThese adjacent vertices may be found effectively by using the algorithm of Freund\nand Todd (1981), or more recent methods found in Yang (1999). These aresimplicial\nalgorithms that follow paths of simplices in the triangulation; we do not review them\nhere for lack of space. However, we do emphasize the important feature of such\nalgorithms is the fact that they use vertex labels to determine a path that finds the desired\nF.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25 21\nadjacent vertices. This is far more efficient than a brute-force search of the vertices. In\nour setting, each vertex corresponds to a cut-set, so the vertex labels can be determined\non the fly by moving from vertex to vertex and interactively polling the players for their\npreferences at the cut-sets along the path. (T. Prescott has noted that one can reduce the\nlength of the path produced by the Freund-Todd algorithm by modifying the above\nlabelling rule: assign a vertex the number of the most-distressed player (as before), but\nlet the sign of the vertex signify the portion which that playerl ast prefers.) h\nWe remark that the proof of Theorem 1 can be modified to address preference\nmeasures on a measurable set of any dimension as long as it can be mapped onto a\nbounded real interval such that the image measures (of the players’ measures) are\nabsolutely continuous. In this case, the inverse images of the cut sets of the interval yield\nthe cut sets of the object.\n5 . Remarks on implementation and efficiency\nIn an actual implementation, the algorithm for consensus-halving can be coded so that\na computer could proceed through the algorithm and interactively ask players at each\nstep which portion they would prefer and their perceived difference in size between the\nportions. (See Su (1999) for a similar fair division procedure based on Sperner’s\nlemma.) Convergence to a solution can be enhanced by existent homotopy algorithms in\nwhich ´ need not be specified in advance and generally decreases with the run time. See\nTodd (1976) or Yang (1999) for a survey of such methods applied to fixed point\nproblems.\nAn additional feature of our approach is that since the algorithm is interactive, players\ndo not have to reveal their a priori preferences (which may in general be very hard to\ndescribe). Moreover, during the procedure they do not need to reveal their preferences\nover all possible cut-sets, but only for cut-sets near a path followed by the simplicial\nalgorithm. On the other hand, if players can fully describe their preferences beforehand,\nthe algorithm can be run from the initial data alone. This may be possible with\nsufficiently nice preferences, or if the measures are describing some objective data, such\nas in the ‘necklace-splitting’ problem below.\nWe remark that issues such as strategic manipulability of the algorithm and Pareto-\nefficiency of the outcome are not meaningful in the consensus-halving problem if\nplayers are not assigned to one of the portions that result. For instance, a team ofn\npeople might wish to divide a project into two portions so that they can equally divide\ntheir work over 2 days. In this case all players desire the same goal: to get agreement by\nall n people that two portions of the project are nearly equal in size.\nOn the other hand, efficiency and manipulability are important issues when each\nplayer seeksdifferent goals, such as maximizing different pieces. For instance, if feuding\nfamilies wish to divide some territory between the two families, then a consensus-\nhalving among all involved may not be as desirable as a division in which each person\nbelieves that the portion his family received wasbigger than the other portion. In that\ncase, a consensus-halving would not be Pareto-efficient.\n22 F.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25\nWhat can be said about the efficiency of a consensus-halving? Some heuristic\nevidence suggests that for large numbers of players, any division is likely be efficient.\nAn interesting model of O’Neill, discussed by Roth (1985), shows that in a negotiation\nof n players amongm settlements, if the rank-orderings of the settlements by players are\nassumed equally likely, the likelihood of a Pareto-optimal settlement approaches 1 as the\nnumber of players grows. In fact, for as few as 10 players and 1000 settlements, the\nlikelihood of a Pareto-optimal settlement is more than 76%; for 15 players \/1000\nsettlements, more than 98%. Roth notes that ‘‘the larger the number of negotiators with\nindependent interests, the harder it is to propose a change from one settlement to another\nthat all regard as an improvement’’. In the consensus-halving problem, any cake division\nis a settlement. Thus if we consider divisions of the cake into 10 strips of width 1\/10,\n10\nthere are 2 51024 possible settlements. The O’Neill data suggests that for 10 players,\nabout 3\/4 of these settlements are efficient with respect to all 10-strip divisions.\nAs for strategic play in applications where players are assigned to one of the portions,\nwe can discourage such actions by specifying that the assignment be made only after the\nhalving has already been decided. Thus, no player would have any assurance that she\nwould be assigned to a piece that she tried to fatten up, and strategic play might backfire.\nOn the other hand, if she states her true intentions, she will be guaranteed an\napproximate consensus-halving. This provides a strong incentive to be truthful.\n6 . The law of the sea treaty and necklace splitting\nWe remarked earlier that a consensus-halving procedure could help families to split a\npiece of land into two regions in such a way that every member of both families believes\nthe land is nearly equally divided.\nAn analogous situation arises in the 1994 Convention of the Law of the Sea (Brams\nand Taylor, 1996, p. 10), which uses a divide-and-choose procedure to protect the\ninterests of developing countries when an industrialized nation wants to mine a portion\nof the seabed in international waters. An agency representing the developing countries\nchooses one of the two halves to reserve it for future mining by less-developed nations.\nIf a consensus-halving procedure were used instead of divide-and-choose, it would yield\na division into two portions such that every nation agreed both portions were almost\nequally valuable.\nOur algorithm also provides an effective solution to the discrete problem of ‘splitting\nnecklaces’ (Alon, 1987). Imagine a necklace of jewels ofn different colors but an even\nnumber of identical jewels of each color. (The position of each jewel is fixed relative to\nthe other jewels.) Using a minimal number of cuts, we desire a division of the necklace\ninto two portions such that for any color, both portions have the same number of jewels\nof that color. (We imagine the necklace laid out along a straight line, with cuts made\nperpendicular to this line.) Theorem 1 can be applied in this context by assigning each\nplayer a jewel color, and replacing each player’s subjective measure by a precise count\nof the number of jewels of her assigned color in each portion. (These measures are\nabsolutely continuous with respect to Lebesgue measure, as required by Theorem 1.) In\nthis case, since the measures are completely known from initial data, a simplicial\nF.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25 23\nalgorithm can be adapted to compute an exact solution: if´ is chosen to be one jewel,\nthen the simplicial algorithm will find adjacent vertices possessing opposite labels, and a\nlittle thought reveals that one of these vertices must represent cuts that divide all the\njewel colors in half. (Otherwise the same player could not be ‘most distressed’ at both\nvertices and still change preferences.)\n7 . Team-splitting\nEach consensus-halving result corresponds to a relatedenvy-free division problem for\ntwice the number of people, by averaging measures. For instance, our consensus-halving\nresult can be used to address the following ‘team-splitting’ problem.\nSuppose among the 2n explorers on an expedition there are two of each specialty: two\nzoologists, two botanists, two archaeologists etc. They want to know the fairest way to\nsplit both their team and their territory. In other words, they want to split into two teams\nin such a way that each specialty is represented on each team, and such that each team\nmember is satisfied that she is on the team with the best half of the territory to explore.\nTheorem 2. (Team-Splitting)Given a territory and such a collection of 2n explorers,\nthere exists a way to divide the territory and the people into two teams of n explorers\n(one of each type) such that each explorer is satisfied with his \/her territory.\nThis result assumes there are no coalitions (sets of people who desire to be on the\nsame team) and that the players have continuous (though not necessarily additive)\nmeasures over the territory.\nProof. The territory is the objectA that will be divided by consensus-halving. Consider\n9 9the i-th pair of scientists by specialty, with measuresl andl . Let m 5l 1l . Thesei i i i i\nform a collection of n measures with which to apply consensus-halving, obtaining two\nportions A and A for which1 2\n9 9l (A )1l (A )5l (A )1l (A )i 1 i 1 i 2 i 2\n9 9for all i. If l (A )5l (A ) andl (A )5l (A ) then both scientists of thei-th pairi 1 i 2 i 1 i 2\nare indifferent between the portions; we can flip a coin to make assignments. Otherwise\nwe conclude that one member of thei-th pair believesA is more valuable thanA , and1 2\nthe other believes the opposite. In this case assign each scientist of thei-th pair the\nportion of the territory that she prefers.h\nFrom the consensus-halving theorem, we see that the team-splitting solution is even\nreasonably practical—it would never involve more thann straight cuts through the\nterritory.\n24 F.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25\n8 . Open problems\nWe close with some open problems.\n(1) Consensus-splitting in an arbitrary ratio. Suppose we desired a division of cake into\ntwo portions so that each ofn people agreed the split was some other ratio, say\ntwo-to-one? Under what conditions can this be achieved effectively using a minimal\nnumber of cuts?\n(2) Consensus-1 \/k-division. Is there an effective algorithm for obtaining a division into\nk portions such that each ofn people believes allk portions are equal in size? Such\na method could be used, for instance, to divide an estate amongk children such that\neach ofn people (parents, children, and others) agreed that all children received\nequal portions.\n(3) A generalized Tucker’ s lemma. It seems quite likely that the above problem could\nbe addressed by proving some generalization of Tucker’s lemma. What is the\nappropriate combinatorial generalization, and is there a constructive proof?\nWe pose the following conjecture, a generalization of Tucker’s lemma which is\nmotivated by our cake-cutting interpretation. It would solve the consensus-1\/k-division\nquestion if true. Let\nn k k\nS 5 (z , . . . ,z ): z [C,Ouz u51, z 5 uz uh jk 0 n i i i i\nn11\nThis is a kind of a ‘branched sphere’ embedded inC ; it is a manifold that branches in\nn n\nk directions at every seam, and whenk 52 it is just the usual sphereS . Each point inSk\ncan be associated to a division of cake. Think ofuz u as widths of pieces of cake; thei\nk k\nz 5 uz u condition guarantees that thez are real scalar multiples of ak-th root of unity,i i i\nwhich is the ‘sign’ of the piece. These signs associate pieces of the cake intok portions.\n(The signs are1 1,21 whenk 5 2, i.e., consensus-halving). There is a natural action of\nn\nthe symmetric groupS on thek signs; it induces an action ofS which permutes thek k\nassignment of the pieces to portions, i.e., permuting the signs will permute the signs of\nthe portions accordingly.\nn(k21)\nConjecture. Suppose that S is triangulated symmetrically, i.e., if s is a face of thek\ntriangulation and s [ S , then s(s) is also a face of the triangulation. Let V be thek\nvertices of the triangulation, and let l:V → L be a labelling function such that the labels\nj 2p i \/k j\nin L 5 hv m:v 5 e , 1# j # k, 1#m # nj consist of a number m and a sign v .\nSuppose that l satisfies a ‘consistency condition’, such that if s [ S then l(s(v))5 s(l(v)).k\nThen there must exist k adjacent vertices in the triangulation with the same label\nnumber but different signs.\nA constructive proof of this conjecture would give an effective algorithm for\napproximate consensus-1\/k-division as well as associated discrete versions (Alon, 1987).\nThis conjecture appears to be related to recent work of G. Ziegler on colorings of Kneser\nhypergraphs (Ziegler, 2002).\nF.W. Simmons, F.E. Su \/ Mathematical Social Sciences 45 (2003) 15–25 25\nA cknowledgements\nWe thank anonymous referees and Adam Bliss, Steven Brams, Claus-Jochen Haake,\nMatthias Raith, and Timothy Prescott for suggestions that improved this paper.\nR eferences\nAlon, N., 1987. Splitting necklaces. Advances in Mathematics 63, 247–253.\nAlon, N., West, D.B., 1986. The Borsuk-Ulam theorem and the bisection of necklaces. Proceedings of the\nAmerican Mathematical Society 98, 623–628.\nAustin, A.K., 1982. Sharing a cake. Mathematical Gazette 6 (437), 212–215.\n´ ´ ¨Barany, I., Shlosman, S.B., Szucs, A., 1981. On a topological generalization of a theorem of Tverberg. Journal\nof the London Mathematical Society 23 (2), 158–164.\nBarbanel, J.B., 1996. Super envy-free cake division and independence of measures. Journal of Mathematical\nAnalysis and its Applications 197, 54–60.\nBrams, S.J., Taylor, A.D., 1996. Fair Division: from Cake-Cutting to Dispute Resolution. Cambridge\nUniversity Press, Cambridge.\nBrams, S.J., Kilgour, D.M., 2001. Competitive fair division. Journal of Political Economy 109, 418–443.\nBrams, S.J., Taylor, A.D., Zwicker, W.S., 1997. A moving-knife solution to the four-person envy-free\ncake-division problem. Proceedings of the American Mathematical Society 125, 547–554.\nFreund, R.M., Todd, M.J., 1981. A constructive proof of Tucker’s combinatorial lemma. Journal of\nCombinatorial Theory Series A 30, 321–325.\nGardner, M., 1978. Aha! Insight. W.F. Freeman, New York.\nGoldberg, C.H., West, D.B., 1985. Bisection of circle colorings. SIAM Journal on Algebraic and Discrete\nMethods 6, 93–106.\nHaake, C.J., Raith, M.G., Su, F.E., 2002. Bidding for envy-freeness: a procedural approach ton-player fair\ndivision problems. Social Choice and Welfare 19, 723–749.\nLefschetz, S., 1949. Introduction to Topology. Princeton University Press, Princeton, NJ.\nPeterson, E., Su, F.E., 2002. Four-person envy-free chore division. Mathematics Magazine 75, 117–122.\nRobertson, J.M., Webb, W.A., 1998. Cake-cutting Algorithms: Be Fair If You Can. A.K. Peters, Natick, MA.\nRoth, A.E., 1985. Some additional thoughts on post-settlement settlements. Negotiation Journal 1, 245–247.\nRotman, J.J., 1988. An Introduction to Algebraic Topology. Springer, New York.\nSteinhaus, H., 1948. The problem of fair division. Econometrica 16, 101–104.\nSu, F.E., 1999. Rental harmony: Sperner’s lemma in fair division. American Mathematical Monthly 106,\n930–942.\nTodd, M.J., 1976. The Computation of Fixed Points and Applications. Lecture Notes in Economics and\nMathematical Systems. Springer, New York.\nTucker, A.W., 1945. Some topological properties of the disk and sphere. In: Proceedings of the First Canadian\nMath. Congress, Montreal, pp. 285–309.\nYang, Z., 1999. Computing Equilibria and Fixed Points. Kluwer Academic Publishers, Boston, MA.\nZiegler, G.M., 2002. Generalized Kneser coloring theorems with combinatorial proofs. Inventiones Math. 147,\n671–691.\n"}
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{"id":"vSlE2p51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/unit-circle\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":46,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","unit-circle"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["unit circle – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:39 UTC 2022","dates_in_content_count_i":2,"dates_in_content_dts":["Sat Jun 29 00:00:00 UTC 2019","Wed Jun 26 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":4114765456923611309,"exact_signature_unique_b":true,"fuzzy_signature_l":6446804436819004652,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["unit circle"],"h2_txt":["Equidecomposability","Chords of an Ellipse","Arclength Surprise","Chords of a Unit Circle","Area of a Hyperbolic Wedge","Area of an Ellipse"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Anagram.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Anagram 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":631,"text_t":"unit circle – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! unit circle. Equidecomposability Posted on June 29, 2019. by. Samuel Nunoo Two sets A and B are said to be equidecomposable if you can partition set A into a finite number of subsets... Continue reading... Chords of an Ellipse Posted on June 29, 2019. by. Samuel Nunoo Consider N equally spaced on points on the unit circle, with the point P=(1,0) as one of these equally spaced... Continue reading... Arclength Surprise Posted on June 26, 2019. by. Samuel Nunoo Consider a unit circle, and any arc S on the unit circle in the first quadrant. No matter where S is... Continue reading... Chords of a Unit Circle Posted on June 26, 2019. by. Samuel Nunoo Take N equidistant points on the unit circle. Pick one of those points, then draw chords from it to all... Continue reading... Area of a Hyperbolic Wedge Posted on June 26, 2019. by. Samuel Nunoo The unit circle can be parametrized by (cos w, sin w). Given a point on it, the region cut out... Continue reading... Area of an Ellipse Posted on June 26, 2019. by. Samuel Nunoo You know the formula for the area of a circle of radius R. It is Pi * R2. But what... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. 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{"id":"uPSMAW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Borsuk-Ulam-implies-Brouwer.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":91,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Borsuk-Ulam-implies-Brouwer","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Borsuk-Ulam-implies-Brouwer.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:40 UTC 2019","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-5316532360697337239,"exact_signature_unique_b":true,"fuzzy_signature_l":4094838281748793764,"fuzzy_signature_unique_b":true,"wordcount_i":1979,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:40 UTC 2022","fresh_date_dt":"Wed Dec 06 11:46:41 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":128469,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088103390969856,"text_t":"appeared in: Amer. Math. Monthly 104 (1997), 855-859.\nBORSUK-ULAM IMPLIES BROUWER:\nA DIRECT CONSTRUCTION\nFRANCIS EDWARD SU\n1. Introduction\nThe Borsuk-Ulam theorem and the Brouwer fixed point theorem are well-known\ntheorems of topology with a very similar flavor. Both are non-constructive existence\nresults with somewhat surprising conclusions. Most topology textbooks that cover\nthese theorems (e.g., [4], [5], [6]) do not mention the two are related—although, in\nfact, the Borsuk-Ulam theorem implies the Brouwer Fixed Point Theorem.\nThe theorems themselves are often proved using the machinery of algebraic topol-\nogy or the concept of degree of a map. That one theorem implies the other can\ntherefore be established once one understands this machinery, but this requires\nbackground. Moreover, such proofs tend to be indirect, relying on the equivalence\nof these existence theorems with corresponding non-existence theorems. For in-\nstance, Dugundji and Granas [3] show that the Borsuk-Ulam theorem is equivalent\nto the statement that no antipode-preserving, continuous map f : Sn → Sn can be\nhomotopic to a constant map. From this one can see that the Brouwer fixed point\ntheorem is a special case, because it can be shown equivalent to the statement that\nthe identity map id : Sn → Sn (which is antipode-preserving) is not homotopic to\na constant map.\nHowever, such an indirect approach is not really necessary, and perhaps a more\ndirect proof would give insight as to how the two theorems are related. The purpose\nof this note is to provide a completely elementary proof that the Borsuk-Ulam the-\norem implies the Brouwer theorem by a direct construction, in which the existence\nof antipodal points in one theorem yields the asserted fixed point in the other.\n2. The Theorems\nLet Sn denote the unit n-sphere in Rn+1, i.e., all points at distance one from the\norigin. Two points are antipodal if they lie opposite each other on the sphere—i.e.,\n{x,−x} for some x.\nThe Borsuk-Ulam Theorem. Let f : Sn → Rn be a continuous map. There\nexists a pair of antipodal points on Sn that are mapped by f to the same point in\nR\nn.\nThis theorem was conjectured by S. Ulam and proved by K. Borsuk [1] in 1933.\nIn particular, it says that if f = (f1, f2, ..., fn) is a set of n continuous real-valued\nfunctions on the sphere, then there must be antipodal points on which all the\nfunctions agree. For instance, one interpretation for the case n = 2 is that there is\nalways a pair of antipodal points on the earth’s surface with the same temperature\nand barometric pressure (assuming, of course, that temperature and pressure vary\ncontinuously).\n1\n2 FRANCIS EDWARD SU\nLet Bn denote the unit n-ball in Rn. A fixed point for a map f from a space\ninto itself is a point y such that f(y) = y. The following theorem, due to L.E.J.\nBrouwer, is one of the most celebrated theorems in topology:\nThe Brouwer Fixed Point Theorem. Every continuous map f : Bn → Bn\npossesses a fixed point.\nBrouwer proved the case n = 3 in 1909, and Hadamard followed in 1910 with\na proof for all dimensions. Brouwer gave a different proof in 1912 [2]. See [3] for\nmore historical notes and a survey of fixed point theory.\nIn dimension three, the Brouwer theorem is often interpreted as follows: no\nmatter how you slosh around the coffee in a coffee cup (as long as you do it contin-\nuously), some point is always in the same position it was before the sloshing took\nplace (although it might have moved around in the meantime). Moreover, should\nyou try to move this point out of its original position, you will unavoidably move\nsome other point back into its original position.\n3. The Idea\nAs motivation we first briefly sketch a construction that shows how the Borsuk-\nUlam theorem implies the Brouwer fixed point theorem.\nWe choose to think of Bn as [−1, 1]n, the “n-cube” in Rn. Similarly, we choose\nto think of Sn as the boundary of the (n+ 1)-cube [−1, 1]n+1 in Rn+1:\nSn = {x | x = (x1, x2, ..., xn+1), |xi| ≤ 1 and max |xi| = 1}.\nThe “cubical” n-sphere is homeomorphic to the usual n-sphere via the rays from the\norigin. In fact, this is an antipode-preserving homeomorphism, so the Borsuk-Ulam\ntheorem holds for maps on cubical n-spheres. We choose to work with cubical n-\nspheres and n-balls because constructing and describing functions on such objects\nis easier in rectangular coordinates.\nGiven f : Bn → Bn, we would like to construct a map g : Sn → Rn that encodes\nf in such a manner that the existence of Borsuk-Ulam antipodal points for g implies\nthe existence of a Brouwer fixed point for f .\nThe idea is as follows: on the cubical n-sphere, the “top” and “bottom” faces of\nthe cube are homeomorphic copies of Bn. These are separated by an “equatorial”\nband, consisting of the other faces. Our task is to define a continuous function g\non these three regions of Sn. On the top face, we define g in such a way that a zero\nof g implies a fixed point for f . We then define g on the bottom face so that the\nimage of each point there is the negative of the image of its antipode on the top\nface. Such map is called antipode-preserving—meaning g(−x) = −g(x). If we can\npatch-in the equatorial region with a map that is also antipode-preserving but never\nzero, then the Borsuk-Ulam Theorem guarantees the existence of antipodal points\nthat get mapped by g to the same point. Because g is antipode-preserving, these\nantipodal points must get mapped to zero, which by construction cannot occur in\nthe equatorial band. A zero for g on the top or bottom face then implies a fixed\npoint for f .\n4. The Construction\nWe seek to construct a map g = (g1, g2, ..., gn) : S\nn → Rn that is continuous and\nantipode-preserving, i.e., g(−x) = −g(x).\nBORSUK-ULAM IMPLIES BROUWER 3\nWe first construct g on the “top” and “bottom” faces. Note that each face, on\nwhich some coordinate xk = ±1, is an n-cube. When xn+1 = ±1 we obtain:\nSntop = {x | x = (x1, x2, ..., xn, 1)} and Snbot = {x | x = (x1, x2, ..., xn,−1)},\nwhich denote the top and bottom faces of the cubical Sn. See Figure 1. Let\np : Rn+1 → Rn be defined by p(x) = (x1, ..., xn), i.e., p ignores the last coordinate.\nS\nn\nX\nn\nS\nS\nn\ntop\nbot\neq\nFigure 1.  Top, bottom, and equator of the cubical n-sphere (here n=2).\n                   An example of antipodal points is indicated.\nantipode -X\nFor all x in Sntop, define g(x) = p(x) − f(p(x)). For all x in Snbot, define g(x) =\np(x) + f(−p(x)).\nSince p(−x) = −p(x), one may check that g(−x) = −g(x). Thus g is, so far,\nantipode-preserving. It is continuous, since f and p are. If g(x) = 0 then p(x) is a\nfixed point for f .\nNow we want to define g on the “side” faces of the cubical n-sphere so that it\nmatches up continuously with g on Sntop and S\nn\nbot and is still antipode-preserving,\nbut is never zero on the sides. The latter is the tricky part.\nOne might try to extend the values of g linearly from top to bottom, but this\ndoes not guarantee that g 6= 0 on the sides. However, the following lemmas show\nthat if we define g suitably on the equator, we can linearly extend the values from\nthe equator to top and bottom without creating a new zero for g.\nLemma 1. Let F be a “side” face of the cubical Sn. That is, there exists some k,\n1 ≤ k ≤ n, such that for all x in F , xk is constant and equal to +1 or −1. Then\nfor all x in F ∩ (Snbot ∪ Sntop), the coordinate function gk(x) is either 0 or has the\nsame sign as xk.\nProof. By the definition of g, on F ∩ Sntop, gk(x) = xk − fk(p(x)) and on F ∩ Snbot,\ngk(x) = xk + fk(−p(x)). Since f is a map to an n-cube, |fk| ≤ 1. Hence, if xk = 1,\nthen gk(x) ≥ 0 for all x ∈ F ∩ (Snbot ∪ Sntop). If xk = −1, then gk(x) ≤ 0 for all\nx ∈ F ∩ (Snbot ∪ Sntop).\nNow let Sneq denote the “equator” of S\nn, i.e., {x ∈ Sn | x = (x1, ..., xn, 0)}. For\nall x ∈ Sneq, define g on Sneq by\ng(x) = p(x) +\ng(x1, ..., xn,−1) + g(x1, ..., xn, 1)\n2\n,(1)\n4 FRANCIS EDWARD SU\nand observe that g is now antipode-preserving on the equator.\nLemma 2. For all x ∈ Sneq, if |xk| = 1, then the coordinate function gk(x) is not\n0 and has the same sign as xk.\nProof. Lemma 1 shows that if |xk| = 1, gk has the same sign as xk on the top and\nbottom faces if it is not zero. Therefore, using (1) and pk(x) = xk = ±1, we see\nthat gk(x) is non-zero on the equator and has the same sign as xk.\nTo define g on the equator we have “averaged” the values of the corresponding\npoints on Sntop and S\nn\nbot and then “lifted” that average by p(x) (which equals xk in\nthe k-th coordinate) to pull it away from possibly being zero. See Figure 2.\nif  x    is negativek\n+\n                 Graphs show a cross-section of  g    values along on \"longitude\" of the\n                          cubical n-sphere on the faces determined by  x   = +1,-1.\n+\nif  x    is positivek\nk\nk\nk\nn\nbot S\nn\n_\neq S\nn\ng  (x)k g  (x)k\nS S\nS\nSbot eq\ntop\ntop\nn n n\nx xn+1 n+1\nxk\nFigure 2.  Extending the values of   g    linearly from the equator to top and bottom.\nkx\n_\nWe now define g continuously on the rest of Sn by extending it linearly from the\nequator to the values on Snbot and S\nn\ntop. That is, for 0 ≤ xn+1 ≤ 1, let\ng(x) = xn+1 g(x1, ..., xn, 1) + (1− xn+1) g(x1, ..., xn, 0).(2)\nFor −1 ≤ xn+1 ≤ 0, let\ng(x) = −xn+1 g(x1, ..., xn,−1) + (1 + xn+1) g(x1, ..., xn, 0).(3)\nRefer to Figure 2 again. Note that g is continuous and antipode-preserving. Fur-\nthermore, it can achieve 0 only on Sntop or S\nn\nbot, because of\nLemma 3. If |xn+1| < 1, then g(x) 6= 0.\nProof. Since |xn+1| < 1, we are on a side face and therefore there exists some k,\n1 ≤ k ≤ n, for which xk = ±1. We shall show that g(x) cannot be zero by showing\nthat the coordinate function gk(x) is non-zero.\nConsider (2) and (3). By Lemmas 1 and 2, gk(x1, x2, ..., xn,±1) and gk(x1, x2, ..., xn, 0)\nhave the same sign as xk, and the latter is non-zero. Moreover, (1 − xn+1) and\n(1 + xn+1) are strictly positive because |xn+1| < 1. Equations (2) and (3) now\nimply that gk(x) is non-zero and, in fact, has the same sign as xk.\nBORSUK-ULAM IMPLIES BROUWER 5\nNow that g is defined everywhere on Sn, the Borsuk-Ulam Theorem implies that\nthere exists a pair {x,−x} such that g(x) = g(−x). But g(x) = −g(−x), since g\nis antipode-preserving. Therefore g(x) = g(−x) = 0 which, by Lemma 3, implies\nthat one of the pair {x,−x} is in Sntop. Without loss of generality, suppose it is x.\nThen g(x) = p(x) − f(p(x)) = 0 on Sntop implies that for y = p(x) ∈ Bn, we have\nf(y) = y, which proves the Brouwer Fixed Point Theorem.\nAcknowledgements. I thank Michael Starbird for many lively discussions and\nCourtney Coleman for helpful stylistic suggestions.\nReferences\n[1] K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math., 20 (1933),\n177-190.\n[2] L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann., 71 (1912), 97-115.\n[3] J. Dugundji and A. Granas, Fixed Point Theory, Polish Scientific Publishers, Warsaw, 1982.\n[4] W.S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, New York, 1991.\n[5] J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, New York, 1984.\n[6] E.H. Spanier, Algebraic Topology, Springer-Verlag, New York, 1966.\nDepartment of Mathematics\nHarvey Mudd College\nClaremont, CA 91711\nsu@math.hmc.edu\n"}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! complex numbers. Really Complex Matrices Posted on June 29, 2019. by. Samuel Nunoo If you know how to multiply 2×2 matrices, and know about complex numbers, then you’ll enjoy this connection. Any complex number... Continue reading... Products of Sums of Two Squares Posted on June 27, 2019. by. Samuel Nunoo Here’s a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the... Continue reading... Chords of a Unit Circle Posted on June 26, 2019. by. Samuel Nunoo Take N equidistant points on the unit circle. Pick one of those points, then draw chords from it to all... Continue reading... Complex Roots Made Visible Posted on June 26, 2019. by. Samuel Nunoo Everyone learns that the roots of a polynomial have a graphical interpretation: they’re the places where the function crosses the x-axis. But... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! game theory. St. Petersburg Paradox Posted on June 29, 2019. by. Samuel Nunoo A new casino offers the following game: you toss a coin until it comes up heads. If the first heads... Continue reading... Envy-free Cake Division Posted on June 29, 2019. by. Samuel Nunoo Say you and a friend wish to share a cake. What is a “fair” way to split it? Probably you... Continue reading... Arrow’s Impossibility Theorem Posted on June 27, 2019. by. Samuel Nunoo Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences... Continue reading... Vickrey Auction Posted on June 26, 2019. by. Samuel Nunoo Suppose you are hosting a silent auction to sell your antique car. The rules are: (1) prospective buyers bid for... Continue reading... Deal or No Deal Posted on June 26, 2019. by. Samuel Nunoo On the TV show “Deal or No Deal”, a contestant is faced with a number of briefcases (26 in the... Continue reading... Two Envelopes Paradox Posted on June 26, 2019. by. Samuel Nunoo I have two envelopes, and inside each I have put some money. In fact, one envelope contains twice as much... Continue reading... Social Choice and the Condorcet Paradox Posted on June 26, 2019. by. Samuel Nunoo How should one select the winner of an election? If there are only two candidates, the answer is clear— choose the... Continue reading... Rental Harmony Posted on June 26, 2019. by. Samuel Nunoo You and your college friends decide to rent a house together, and the N of you have found a house... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"k3ZW9p51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/brouwer-fixed-point-theorem\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":58,"url_protocol_s":"https","url_paths_count_i":2,"url_paths_sxt":["funfacts","brouwer-fixed-point-theorem"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Brouwer Fixed Point Theorem – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:49 UTC 2022","dates_in_content_count_i":0,"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-2983127961258698319,"exact_signature_unique_b":true,"fuzzy_signature_l":4529238272118584538,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["Brouwer Fixed Point Theorem"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"bold_txt":["Fun Fact suggested by:","References:","Math Fun Facts","How to Cite this Page:","The Math Behind the Fact:","Presentation Suggestions:"],"imagescount_i":5,"images_protocol_sxt":["000-https","001-https","002-https","003-https","004-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/BrokenMudd.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/11\/20002.7.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,117,294,-1],"images_width_val":[-1,-1,250,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 BrokenMudd Broken Mudd 11 7 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":651,"text_t":"Brouwer Fixed Point Theorem – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! Brouwer Fixed Point Theorem. One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem. Take two sheets of paper, one lying directly above the other. If you crumple the top sheet, and place it on top of the. other sheet, then Brouwer’s theorem says that there must be at least one point on the top sheet that is directly above the corresponding point on the bottom sheet! Do you believe that? In dimension three, Brouwer’s theorem says that if you take a cup of coffee, and slosh it around, then after the sloshing there must be some point in the coffee which is in the exact spot that it was before you did the sloshing (though it might have moved around in between). Moreover, if you tried to slosh that point out of its original position, you can’t help but slosh another point back into its original position! More formally the theorem says that a continuous function from an N-ball into an N-ball must have a fixed point. Continuity of the function is essential (if you rip the paper or if you slosh discontinuously, then there may not be fixed point). Presentation Suggestions:. Bring a coffee cup and 2 sheets of paper with you and demonstrate as you present the fun fact. Draw a grid on the paper, number the gridboxes, then xerox that sheet of paper. After you crumple the paper, you can say that at least one number is on top of the corresponding number on the lower sheet of paper. Alternatively, bring a map of Claremont (or whatever city you are in) to class and drop it on the floor—then there must be some point in the map lying directly over the point that it represents! A good follow-up Fun Fact is the Borsuk-Ulam Theorem. The Math Behind the Fact:. Fixed point theorems are some of the most important theorems in all of mathematics. Among. other applications, they are used to show the existence of solutions to differential equations, as well as the existence of equilibria in game theory. There are many proofs of the Brouwer fixed point theorem. The. advanced student may wish to see if she can show the equivalence of this theorem with Sperner’s lemma, which yields a rather nice elementary proof. How to Cite this Page:. Su, Francis E., et al. “Brouwer Fixed Point Theorem.”. Math Fun Facts. . . References:. Any undergraduate text on topology. Fun Fact suggested by:. Francis Su. Did you like this Fun Fact? Click to rate it. Submit Rating. Average rating. 4.7. \/ 5. Vote count:. 54. No votes so far! Be the first to rate this Fun Fact. Share:. Tags:. advanced ,. analysis ,. medium ,. other ,. topology Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. All Rights Reserved. Loading Comments... Write a Comment... Email (Required). Name (Required). 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! lightning arithmetic. Squaring Quickly Posted on June 29, 2019. by. Samuel Nunoo You may have seen the Fun Fact on squares ending in 5; Here’s a trick that can help you square ANY... Continue reading... Squaring Numbers between 40 and 60 Posted on June 26, 2019. by. Samuel Nunoo Here’s a quick way to square numbers between 40 and 60. You know the squares of 40, 50, and 60,... Continue reading... Squares Ending in 5 Posted on June 26, 2019. by. Samuel Nunoo Give me any 2 digit number that ends in 5, and I’ll square it in my head! 452 = 2025 852 = 7225, etc.... Continue reading... Difference of Squares Posted on June 26, 2019. by. Samuel Nunoo You have all learned that a2 – b2 = (a + b)(a – b) But perhaps you haven’t thought about how to... Continue reading... Multiplication by 11 Posted on June 26, 2019. by. Samuel Nunoo Multiplication by 11 is easy! To multiply by a 2-digit number add the two digits and place the sum in... Continue reading... Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. 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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! real analysis. Dedekind Cuts of Rational Numbers Posted on June 29, 2019. by. Samuel Nunoo Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the... Continue reading... Rationals Dense but Sparse Posted on June 29, 2019. by. Samuel Nunoo Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the... Continue reading... Devil’s Staircase Posted on June 29, 2019. by. Samuel Nunoo Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically... Continue reading... Space-filling Curves Posted on June 29, 2019. by. Samuel Nunoo Consider a square in the plane. Is it possible to draw a curve in the square that touches every point inside the... Continue reading... Rational Irrational Power Posted on June 29, 2019. by. Samuel Nunoo If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to... Continue reading... Multidimensional Newton’s Method Posted on June 29, 2019. by. Samuel Nunoo You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did... Continue reading... Cantor Diagonalization Posted on June 29, 2019. by. Samuel Nunoo We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as... Continue reading... Continuous but Nowhere Differentiable Posted on June 29, 2019. by. Samuel Nunoo You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have... Continue reading... Banach-Tarski Paradox Posted on June 29, 2019. by. Samuel Nunoo Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using... Continue reading... Sierpinski-Mazurkiewicz Paradox Posted on June 29, 2019. by. Samuel Nunoo If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"iqeqbp51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/funfacts\/tag\/prime\/","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":40,"url_protocol_s":"https","url_paths_count_i":3,"url_paths_sxt":["funfacts","tag","prime"],"content_type":["text\/html"],"crawldepth_i":2,"collection_sxt":["user"],"title":["prime – Math Fun Facts"],"last_modified":"Wed Jun 08 17:22:52 UTC 2022","dates_in_content_count_i":3,"dates_in_content_dts":["Sat Jun 29 00:00:00 UTC 2019","Thu Jun 27 00:00:00 UTC 2019","Wed Jun 26 00:00:00 UTC 2019"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-8387472117781788532,"exact_signature_unique_b":true,"fuzzy_signature_l":-1321054975738485658,"fuzzy_signature_unique_b":true,"icons_protocol_sxt":["000-https","001-https"],"icons_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header-150x70.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"icons_rel_sxt":["icon","icon apple-touch-icon"],"icons_sizes_sxt":["32x32","192x192"],"h1_txt":["prime"],"h2_txt":["Riemann Hypothesis","Lucas’ Theorem","Sum of Cubes and Beyond","Sums of Two Squares","All Numbers are Interesting","Fibonacci GCD’s, please","Sum of Prime Reciprocals","How many Primes?","Gaps in Primes","Fermat’s Little Theorem"],"h3_txt":["Search Fun Facts","Search by Difficulty","Search by Subject","Most Popular Fun Facts","From the Creator of Math Fun Facts:","View Our Most Used Tags!"],"imagescount_i":4,"images_protocol_sxt":["000-https","001-https","002-https","003-https"],"images_urlstub_sxt":["math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/05\/20002.2.1.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/Lawnmower.gif","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/12\/Flourishing20Cover_edited-e1577727491927-202x300.jpg","math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2019\/06\/header.gif"],"images_alt_sxt":["Math Fun Facts","Math Fun Facts","Book Cover: Mathematics for Human Flourishing","Math Fun Facts"],"images_height_val":[-1,-1,294,-1],"images_width_val":[-1,-1,198,-1],"images_text_t":"math hmc edu funfacts wp content uploads sites 4 2019 05 20002 2 1 gif Math Fun Facts 06 Lawnmower 12 Flourishing20Cover edited e1577727491927 202x300 jpg Flourishing 20 Cover e 1577727491927 202x 300 Book Cover: Mathematics for Human Flourishing header","responsetime_i":660,"text_t":"prime – Math Fun Facts. Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! prime. Riemann Hypothesis Posted on June 29, 2019. by. Samuel Nunoo If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our... Continue reading... Lucas’ Theorem Posted on June 29, 2019. by. Samuel Nunoo Lucas’ Theorem: If p is a prime number, and N has base p representation (aj,…,a1,a0) and k has base p... Continue reading... Sum of Cubes and Beyond Posted on June 29, 2019. by. Samuel Nunoo We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the... Continue reading... Sums of Two Squares Posted on June 29, 2019. by. Samuel Nunoo Which whole numbers are expressible as sums of two (integer) squares? Here’s a theorem that completely answers the question, due... Continue reading... All Numbers are Interesting Posted on June 27, 2019. by. Samuel Nunoo There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first... Continue reading... Fibonacci GCD’s, please Posted on June 27, 2019. by. Samuel Nunoo Fibonacci numbers exhibit striking patterns. Here’s one that may not be so obvious, but is striking when you see it.... Continue reading... Sum of Prime Reciprocals Posted on June 26, 2019. by. Samuel Nunoo It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges. But what about... Continue reading... How many Primes? Posted on June 26, 2019. by. Samuel Nunoo Are there infinitely many primes? We’ll give a proof, due to Euclid, to show that there must be infinitely many primes.... Continue reading... Gaps in Primes Posted on June 26, 2019. by. Samuel Nunoo We know there are infinitely many primes, so are many interesting questions you can ask about the distribution of primes, i.e., how... Continue reading... Fermat’s Little Theorem Posted on June 26, 2019. by. Samuel Nunoo Fermat’s little theorem gives a condition that a prime must satisfy: Theorem. If P is a prime, then for any... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. An inclusive vision of mathematics: what it is, who it’s for, why anyone should learn it View Our Most Used Tags! advanced algebra analysis approximations arithmetic bicycle binomial coefficients calculus combinatorics complex numbers demonstration easy ellipse Fibonacci functions game theory geometric series geometry hard infinite series irrational lightning arithmetic logic matrix medium multivariable calculus number theory other paradox pi pi formula polynomial prime probabilities probability puzzle real analysis Riemann zeta function sphere Taylor series topology triangles unit circle universe unsolved problem 1999-2021 by Francis Su. 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{"id":"qawvUW51bhow","host_id_s":"51bhow","sku":"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2019\/06\/Counting-on-Continued-Fractions.pdf","host_s":"math.hmc.edu","host_organization_s":"hmc","url_chars_i":95,"url_protocol_s":"https","url_paths_count_i":7,"url_paths_sxt":["su","wp-content","uploads","sites","10","2019","06"],"url_file_name_s":"Counting-on-Continued-Fractions","url_file_ext_s":"pdf","content_type":["application\/pdf"],"crawldepth_i":2,"collection_sxt":["user"],"title":["Counting-on-Continued-Fractions.pdf"],"author":"pdfTeX-0.13d","author_sxt":["pdfTeX-0.13d"],"last_modified":"Thu Jun 13 04:34:08 UTC 2019","dates_in_content_count_i":1,"dates_in_content_dts":["Wed Sep 08 00:00:00 UTC 2010"],"http_unique_b":false,"www_unique_b":false,"exact_signature_l":-3864910911135739684,"exact_signature_unique_b":true,"fuzzy_signature_l":-6544282795464925463,"fuzzy_signature_unique_b":true,"wordcount_i":2339,"linkscount_i":0,"linksnofollowcount_i":0,"inboundlinkscount_i":0,"outboundlinkscount_i":0,"charset_s":"UTF-8","httpstatus_i":200,"load_date_dt":"Wed Jun 08 17:22:44 UTC 2022","fresh_date_dt":"Wed Dec 06 11:47:02 UTC 2023","referrer_id_s":"28_JZW51bhow","publisher_t":"pdfTeX-0.13d","language_s":"en","size_i":154251,"audiolinkscount_i":0,"videolinkscount_i":0,"applinkscount_i":0,"references_i":1,"references_internal_i":1,"references_external_i":0,"references_exthosts_i":0,"host_extent_i":-1,"_version_":1735088106976051200,"text_t":"appeared in: Mathematics Magazine 73(2000), pp. 98-104.\nCopyright the Mathematical Association of America 1999. All rights reserved.\nCounting on Continued Fractions\nArthur T. Benjamin\nFrancis Edward Su\nHarvey Mudd College\nClaremont, CA 91711\nJennifer J. Quinn\nOccidental College\nLos Angeles, CA 90041\nIntroduction\nYou might be surprised to learn that the finite continued fraction\n3 +\n1\n7 +\n1\n15 +\n1\n1 +\n1\n292\nand its reversal 292 +\n1\n1 +\n1\n15 +\n1\n7 +\n1\n3\nhave the same numerator. These fractions simplify to 103993\n33102\nand 103993\n355\nrespectively.\nIn this paper, we provide a combinatorial interpretation for the numerators and\ndenominators of continued fractions which makes this reversal phenomenon easy\nto see. Through the use of counting arguments, we illustrate how this and other\nimportant identities involving continued fractions can be easily visualized, derived,\nand remembered.\nWe begin by defining some basic terminology. Given an infinite sequence of\nintegers a0 ≥ 0, a1 ≥ 1, a2 ≥ 1, . . . , let [a0, a1, . . . , an] denote the finite continued\nBenjamin, Quinn, Su 2\nfraction\na0 +\n1\na1 +\n1\na2 +\n1\n. . . +\n1\nan\n.\nThe infinite continued fraction [a0, a1, a2, . . . ] is the limit of [a0, a1, . . . , an] as n→\n∞. This limit always exists and is some irrational number α [3]. The rational\nnumber rn := [a0, a1, . . . , an] is a fraction pn\/qn in lowest terms, called the n-th\nconvergent of α. It is well-known that pn and qn satisfy the recurrences\npn = anpn−1 + pn−2\nqn = anqn−1 + qn−2\nfor n ≥ 2, with initial conditions p0 = a0, p1 = a1a0 + 1, q0 = 1, q1 = a1.\nNow let’s do some combinatorics. For a given continued fraction [a0, a1, a2, . . . ],\nconsider the following tiling problem. Let Pn count the number of ways to tile a\n1 × (n + 1) board with dominoes and stackable square tiles. All cells (numbered\n0, 1, . . . , n) must be covered by a tile. Nothing can be stacked on top of a domino, but\ncell number i may be covered by a stack of as many as ai square tiles, i = 0, . . . , n.\nFigure 1 shows an empty board with the height conditions a0, a1, . . . , an indicated.\nFigure 2 gives an example of a valid tiling for a 1× 12 board with height conditions\n5, 10, 3, 1, 4, 8, 2, 7, 7, 4, 2, 3.\n@@\n@@ @@ @@ @@ @@ @@ @@ @@ @@Board\ncells: 0 1 2 3 4 5 6 n− 1 n\nheights:a0\na1\na2\na3\na4\na5\na6 an−1\nan\n@@\n· · ·\n· · ·\n· · ·\nFigure 1: An empty 1× (n+ 1) board.\nFor n ≥ 2, we show\nPn = anPn−1 + Pn−2.\nThis follows from the observation that a tiling either ends with a stack of square\ntiles or a single domino. In the first case, there are an choices for the stack size and\nBenjamin, Quinn, Su 3\n@@Board\ncells: 0 1 2 3 4 5 6 7 8 9 10 11\nheights: 5\n10\n3\n1\n4\n8\n2\n7 7\n4\n2\n3\n@@ @@\n@@\n@@\n@@\n@@\n@@\n@@\n@@\n@@ @@\n@@\n@@ @@\n@@\n@@\n@@\nFigure 2: A tiling satisfying the height conditions 5, 10, 3, 1, 4, 8, 2, 7, 7, 4, 2, 3.\n@@\na0 a0\na1\n@@ @@\n@@\n@@\n@@\n@@\n@@\n@@ @@\n@@\n@@\n- or -\na0\na1\n@@\n@@\n@@\nCounting\na0 a0 · a1 + 1\nP0: P1:\ntilings:\nFigure 3: Verifying the initial conditions for the recurrence relation Pn = anPn−1 +\nPn−2.\nPn−1 ways to tile cells 0 through n− 1. In the second case, there is only one choice\nfor the last domino, and there are Pn−2 ways to tile cells 0 through n − 2. Using\nFigure 3 one can check that P0 = a0 and P1 = a0a1 + 1. Since Pn and pn satisfy the\nsame recurrence and initial conditions, we have Pn = pn.\nRemoving cell 0 from the previous board, let Qn count the number of ways to\ntile the 1 × n board with dominoes and stackable square tiles, where the ith cell\nmay be covered by a stack of as many as ai square tiles, i = 1, . . . , n. By the same\nreasoning as before, (and letting Q0 = 1 denote the “empty” tiling) we see that\nQn = qn.\nTo illustrate, consider the continued fraction representation for π, which begins\n[3, 7, 15, 1, 292, . . . ]. See Figure 4. If we count the number of ways to tile cells 0, 1,\nand 2, we get p2 = 333. Counting the number of ways to tile only cells 1 and 2\neasily gives us q2 = 106. This produces the π approximation r2 = 333\/106. The\nreader should verify that tiling cells 0 through 3 produces r3 = 355\/113.\nBenjamin, Quinn, Su 4\n@@\n@@ @@\n0 1\n@@\n2\n3\n7\n15\n@@ @@\n1\n...\n292\n· · ·\n· · ·\n3 4 · · ·\nFigure 4: The beginning of the π board.\nWhen ai = 1 for all i ≥ 0, it is well-known that the nth convergent pn\/qn is the\nratio of two consecutive Fibonacci numbers. Specifically, if we define f0 = 1, f1 = 1,\nand for n ≥ 2, fn = fn−1 + fn−2, then pn = fn+1 and qn = fn. You may recall that\nthe Fibonacci number fn counts the number of ways to tile a 1×n board with 1×1\nsquares and 1 × 2 dominoes. So the continued fraction tiling problem generalizes\nthe tiling interpretation of Fibonacci numbers [1, 2].\nIdentities\nArmed with our tiling interpretation, many well-known continued fraction identities\ncan be explained combinatorially. We begin with the reversal identity.\nTheorem 1 If [a0, a1, . . . , an−1, an] = pn\/qn, then [an, an−1, . . . , a1, a0] = pn\/pn−1.\nProof. Although one can easily prove this by induction, the theorem is nearly\nobvious when viewed combinatorially. To understand the common numerator, we see\nthat the number of ways to tile the board with height conditions an, an−1, . . . , a1, a0\nis the same as the number of ways to tile the board with height conditions a0, a1, . . . ,\nan−1, an. The denominator of [an, an−1, . . . , a1, a0] is the number of ways to tile the\nboard with height conditions [an−1, . . . , a1, a0], which by reversal is pn−1.\nThe next few identities are useful for measuring the rate of convergence of con-\nvergents.\nTheorem 2 The difference between consecutive convergents of [a0, a1, a2, . . . ] is:\nBenjamin, Quinn, Su 5\nrn − rn−1 = (−1)n−1\/qnqn−1. Equivalently, after multiplying both sides by qnqn−1,\nwe have\npnqn−1 − pn−1qn = (−1)n−1.\nProof. Given height conditions a0, a1, . . . , an, let Pn denote the set of all tilings\non cells 0, . . . , n and let Qn denote the set of all tilings on cells 1, . . . , n. Note that\nthese sets have sizes |Pn| = pn and |Qn| = qn.\nWe will construct an almost one-to-one correspondence between the sets Pn ×\nQn−1 and Pn−1 ×Qn. Consider (S, T ) ∈ Pn ×Qn−1. For i ≥ 1, we say (S, T ) has a\nfault at cell i if both S and T have tiles that end at i. We say (S, T ) has a fault at\ncell 0 if S has a square at cell 0. For instance, in Figure 5, there are faults at cells\n0, 3, 5, and 6.\nIf (S, T ) has a fault, construct (S′, T ′) by swapping the “tails” of S and T\nafter the rightmost fault. See Figures 5 and 6. Note that (S′, T ′) ∈ Pn−1 × Qn.\nSince (S′, T ′) has the same rightmost fault as (S, T ), this procedure is completely\nreversible.\nS ∈ P11\nT ∈ Q10\n0 1 2 3 4 5 6 7 8 9 10 11\na0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11heights:\n\u001b\ntail\n\u001b -\ntail\n-\n6\n?\n6\n?\n6\n?\n6\n?\nFigure 5: A pair of tilings with faults and tails indicated.\nNotice when either S or T contains a square, (S, T ) must have a fault. Thus\nthe only fault-free pairs occur when S and T consist of all dominoes in staggered\nformation as illustrated in Figure 7. When n is odd (i.e., S and T both cover an\neven number of cells), there is precisely one fault-free element of Pn×Qn−1 and no\nfault-free elements of Pn−1 ×Qn. Therefore when n is odd, we have |Pn ×Qn−1| −\n|Pn−1 ×Qn| = 1.\nSimilarly when n is even, there are no fault-free elements of Pn × Qn−1 and\nexactly one fault-free element of Pn−1 ×Qn. Hence when n is even, |Pn ×Qn−1| −\nBenjamin, Quinn, Su 6\nS′ ∈ P10\nT ′ ∈ Q11\n0 1 2 3 4 5 6 7 8 9 10 11\na0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11heights:\nFigure 6: Result of swapping tails in Figure 5.\n0 1 2 3 4 5 6 7 8 9 10 11\na0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11\nFigure 7: The fault-free pair consists of staggered dominoes.\n|Pn−1 ×Qn| = −1. Treating the odd and even cases together, we obtain\npnqn−1 − pn−1qn = (−1)n−1.\nThe next identity shows that the even convergents are increasing, while the odd\nconvergents are decreasing.\nTheorem 3 rn−rn−2 = (−1)nan\/qnqn−2. Equivalently, after multiplying both sides\nby qnqn−2, we have\npnqn−2 − pn−2qn = (−1)nan.\nProof. As in the last proof, we use tail swapping after the last fault to create a\none-to-one correspondence between the “faulty” elements of Pn×Qn−2 and Pn−2×\nQn. The proof is essentially given in Figures 8, 9, and 10.\nThe only unmatched elements are those that are fault-free. When n is odd,\nthere are no fault-free elements of Pn ×Qn−2, but there are precisely an fault-free\nelements of Pn−2×Qn, consisting of a stack of squares on the nth cell, and dominoes\neverywhere else (Figure 10). Likewise when n is even, there are no fault-free elements\nof Pn−2×Qn, but there are an fault-free elements of Pn×Qn−2, consisting of a stack\nBenjamin, Quinn, Su 7\nS ∈ P11\nT ∈ Q9\n0 1 2 3 4 5 6 7 8 9 10 11\na0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11heights:\ntail\n6\n?\n-\ntail\n\u001b\n-\u001b\nFigure 8: An element of P11 ×Q9 with rightmost fault indicated.\nS′ ∈ P9\nT ′ ∈ Q11\n0 1 2 3 4 5 6 7 8 9 10 11\na0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11heights:\nFigure 9: The result of swapping tails in Figure 8.\nof squares on the nth cell, and dominoes everywhere else. Thus we have established\n|Pn ×Qn−2| − |Pn−2 ×Qn| = (−1)nan, as desired.\nUsing the combinatorially clear fact that qn →∞ as n→∞, the last two iden-\ntities demonstrate that (r0, r1), (r2, r3), (r4, r5), . . . is a sequence of nested intervals\nwhose lengths are going to zero. Hence, limn→∞ rn exists.\nExtensions\nNext we examine the quantity K(i, j), for i ≤ j, that counts the number of tilings\nof the sub-board with cells i, i+ 1, . . . , j with height conditions ai, ai+1, . . . , aj . For\nconvenience we define K(i, i − 1) = 1. We see that K(i, j) is the numerator of the\ncontinued fraction [ai, ai+1, . . . , aj ] and the denominator of the continued fraction\n[ai−1, ai, . . . , aj ]. Thus the K(i, j) are identical to the classical continuants of Euler\n[4].\nThe following theorem, due to Euler, can also be proved by the same tail-\nswapping technique.\nBenjamin, Quinn, Su 8\n0 1 2 3 4 5 6 7 8 9 10\na0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10heights: a11\n11\nFigure 10: Problem pairings are fault-free.\nam−1am aj aj+1\n· · ·\n· · ·\nan\n· · ·\nai\nK(i,m− 2)\n\u001b\nK(j + 2, n)fault-free region\n· · · · · ·· · ·heights:\n\u001b -\n· · ·\n-\u001b -\nFigure 11: When j −m is even, there are K(i,m − 2)K(j + 2, n) fault-free tilings\n(S, T ).\nTheorem 4 For i < m < j < n,\nK(i, j)K(m,n)−K(i, n)K(m, j) = (−1)j−mK(i,m− 2)K(j + 2, n)\nThis result follows by considering tilings of sub-boards S from cells i to j and T\nfrom m to n. Every faulty pair (S, T ) corresponds to another faulty pair (S′, T ′)\nobtained by swapping the tails after the last fault. The term on the right side\nof Theorems 4 counts the number of fault-free tilings that only occur when the\noverlapping regions (of S and T , or of S′ and T ′, depending on the parity of j −m)\nconsist entirely of dominoes in staggered formation. See Figures 11 and 12. Setting\ni = 0 and m = 1, Theorem 4 generalizes Theorems 2 and 3 by allowing us to compare\narbitrary convergents rj and rn.\nFinally, we generalize in a different direction. Suppose we allow dominoes to\nbe stacked as well as squares. Specifically, suppose we impose height conditions\nb1, b2, . . . so that we may stack as many as bi dominoes on cells i − 1 and i. We\nlet P̂n count the number of ways to tile the board with cells 0, 1, . . . , n and height\nconditions a0, . . . , an and b1 . . . , bn for the squares and dominoes respectively. We\nlet Q̂n count the same problem with cell 0 removed. As before, we see that P̂n and\nBenjamin, Quinn, Su 9\nam−1am aj aj+1\n· · ·\n· · ·\nan\n· · ·\nai\nK(i,m− 2) K(j + 2, n)fault-free region\n· · · · · ·· · ·heights:\n\u001b - \u001b\n· · ·\n-\u001b-\nFigure 12: When j −m is odd, there are K(i,m − 2)K(j + 2, n) fault-free tilings\n(S′, T ′).\nQ̂n satisfy\nP̂n = anP̂n−1 + bnP̂n−2\nQ̂n = anQ̂n−1 + bnQ̂n−2\nfor n ≥ 2, with initial conditions P̂0 = a0, P̂1 = a1a0 + b1, Q̂0 = 1, Q̂1 = a1. But\nthese are precisely the conditions that define the convergents of the expansion\na0 +\nb1\na1 +\nb2\na2 +\nb3\n. . . +\nbn\nan +\n.. .\nIn other words, when the above continued fraction is truncated at the bn\/an\nterm, it simplifies to the rational number P̂n\/Q̂n. All of the preceding theorems\nhave generalizations along these lines with similar combinatorial interpretations.\nWe invite the reader to continue these investigations.\nAcknowledgment We thank Chris Hanusa for asking an inspiring question.\nReferences\n[1] A. T. Benjamin and J. J. Quinn, Recounting Fibonacci and Lucas Identities,\nCollege Math. J. 30 (1999), 359-366.\n[2] R. C. Brigham, R. M. Caron, P. Z. Chinn, and R. P. Grimaldi, A tiling scheme\nfor the Fibonacci numbers, J. Recreational Math., Vol 28, No. 1 (1996-97) 10-16.\nBenjamin, Quinn, Su 10\n[3] I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the\nTheory of Numbers, John Wiley & Sons, Inc., New York, 1991.\n[4] O. Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Co., 1929.\n[5] S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section, John Wiley\nand Sons, New York, 1989.\n"}
{"index":{}}
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Skip to content Expand Menu. About Math Fun Facts Toggle Child Menu. How to Use Fun Facts Contributors All Fun Facts Get a Random Fun Fact! topology. Klein Bottle Posted on June 29, 2019. by. Samuel Nunoo A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;... Continue reading... Projective Planes Posted on June 29, 2019. by. Samuel Nunoo Most of you know how to make a Mobius band—take a strip of paper and glue the ends with a... Continue reading... Sphere Eversions Posted on June 29, 2019. by. Samuel Nunoo If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it... Continue reading... Pretzel Unlinking Posted on June 29, 2019. by. Samuel Nunoo Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and... Continue reading... Unbelievable Unlinking Posted on June 28, 2019. by. Samuel Nunoo Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber.... Continue reading... Rubber Bands Stuck on a Torus Posted on June 28, 2019. by. Samuel Nunoo Consider a rubber tire (a torus) with a hole in it. Suppose that there is a green rubber band stuck... Continue reading... Connected Sums Posted on June 27, 2019. by. Samuel Nunoo A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,... Continue reading... Borsuk-Ulam Theorem Posted on June 26, 2019. by. Samuel Nunoo The Borsuk-Ulam theorem is another amazing theorem from topology. An informal version of the theorem says that at any given... Continue reading... Brouwer Fixed Point Theorem Posted on June 26, 2019. by. Samuel Nunoo One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem.... Continue reading... Ham Sandwich Theorem Posted on June 26, 2019. by. Samuel Nunoo Here is one of my favorite theorems from topology, called the Ham Sandwich Theorem. It says: given globs of ham, bread,... Continue reading... Next Page » Search Fun Facts. Search by Difficulty. • Easy • Medium • Advanced Search by Subject. • Algebra • Calculus • Combinatorics • Geometry • Number Theory • Probability • Topology • Other Most Popular Fun Facts. 4.7. (26). Fibonacci GCD’s, please 4.7. (15). Birthday Problem 4.7. (54). Brouwer Fixed Point Theorem 4.7. (37). Napoleon’s Theorem 4.6. (53). Does Order of Addition Matter? From the Creator of Math Fun Facts:. Winner of the 2021 Euler Book Prize. from the Mathematical Association of America. 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