%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ワークショップ「代数多様体のホモトピー理論とミルナー予想」 2002年7月5日(金) -- 8 日(月) 八ヶ岳高原 泉郷 (山梨県北巨摩郡大泉村谷戸字並木8741, 0551-38-2336, http://www.izumigo.co.jp) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <趣旨> 数論幾何の深い結果のいくつかは代数的トポロジーの方法の類似を追うことに よって証明されてきました. 例えば, グロタンディークとドリーニュによるヴェイ ユ予想の証明は代数的トポロジーにおける結果から影響を受けています. 近年, モレルとヴォエヴォドスキーはトポロジーの安定ホモトピー理論の類似 をたどりました. この方法によって系統的に様々なコホモロジー理論を構成す ることができます. 例えばモチヴィック・コホモロジー(エタール・コホモロ ジーを含む)やK理論が再構成されますし, 新しいコホモロジー理論を構成 することもできます. 代数多様体のホモトピー理論の最も驚くべき応用はヴォエヴォドスキーによる ミルナー予想(つまりミルナー$K$群からガロア・コホモロジーへのシンボル 写像 K^M_i(k)/2 -> H^i(k, Z/2Z) が同型であるという予想)の解決です. <文献> 代数多様体のホモトピー理論に関する概説としては Voevodsky's Seattle Lectures : K-Theory and Motivic Cohomology, by Charles Weibel (Algebraic K-theory, Seattle conference, Proc. Symp. Pure Math. 67) が参考になります. 主要な文献は A^1-homotopy theory, by Vladimir Voevodsky, Documenta Mathematica, Extra Volume ICM I (1998), 579-604. On 2-torsion in motivic cohomology, by Vladimir Voevodsky, preprint. です. また A^1-homotopy theory of schemes, by Fabian Morel and Vladimir Voevodsky, Publ. Math. IHES 90 (1999) Th\'eorie homotopique des S-sch\'emas, by Jo\"el Riou, D.E.A, 2002 (availabe at http://www.math.jussieu.fr/~riou/). La conjecture de Milnor, by Bruno Kahn, Seminar Bourbaki, Expos\'e 834. もご参照下さい. 以上の文献のいくつかは http://www.math.ias.edu/%7Evladimir/seminar.html 経由で入手できます. <プログラム> 7月5日(金) 15:00 集合 15:30 - 16:30, 16:45 - 17:45 Lars Hesselholt (MIT), Homotopy theory in algebraic topology I, II 7月6日(土) A^1-homotopy theory I - V 9:30 - 10:30 古庄 英和 (京大数理研), I : Spaces 10:45 - 11:45 南 範彦 (名工大), II : Unstable homotopy category 13:30 - 14:30 望月 哲史 (東大), III : Stable homotopy category 14:45 - 15:45 朝倉 政典 (九州大), IV : Spectra 16:30 - 17:30 大坪 紀之 (千葉大), V : Cohomology theories from spectra 7月7日(日) 9:30 - 10:30 未定 10:45 - 11:45 Thomas Geisser (USC・東大), 未定 13:30 - 14:30 竹田 雄一郎 (九州大), Proof of the Milnor conjecture I 14:45 - 15:45 花村 昌紀 (九州大), Proof of the Milnor conjecture II 16:30 - 17:30 加藤 和也 (京大理学部), 浦島太郎の驚き ---             ミルナー予想の解決への初老数学者の驚き 7月8日(月) 解散 ------------------------------------------------------------------------ Workshop on Homotopy Theory of Algebraic Varieties and the Milnor Conjecture Yatsugatake Izumigo, July 5-8, 2002 ------------------------------------------------------------------------ Several deep results in arithmetic geometry have been proven by immitating methods from algebraic topology. For example Deligne's proof of the Weil conjecture was inspired by results from algbraic topology. Recently, Morel and Voevodsky immitated the conststruction of stable homotopy theory in topology. This method gives a systematic way of constructing cohomology theories. For instance, old cohomology theories like motivic cohomology (including \'etale cohomology) or K-theory have been recovered, and new cohomology theories have been constructed this way. The most striking application of the homotopy theory of algebraic varieties is Voevodsky's proof of the Milnor conjecture, stating that the symbol map from Milnor K-theory to Galois cohomology $K_i^M(k)/2 \to H^i(k,\Z/2)$ is an isomorphism. We plan to hold a three day workshop on the homotopy theory of algebraic varieties and the proof of the Milnor conjecture. As an outline, we plan to give two talks on the topological background (friday), five talks on the homotopy theory of algebraic varieties (saturday), and five talks on the proof of the Milnor conjecture and applications (sunday). There is no talk on monday. A nice overview of the homotopy theory of algebraic varieties is given in "Voevodsky's Seattle Lectures K-theory and Motivic Cohomology", by Charles A. Weibel (K-theory, Seattle conference, Pro. Symp. AMS). The other main references will be "A1-homotopy theory", by Vladimir Voevodsky, and "On 2-torsion in motivic cohomology", by Vladimir Voevodsky. See also "A1-homotopy theory of schemes", by Fabien Morel Th\'eorie homotopique des S-sch\'emas, by Jo\"el Riou, D.E.A, 2002 (availabe at http://www.math.jussieu.fr/~riou/), and Vladimir Voevodsky and "La conjecture de Milnor", by Bruno Kahn (seminar Bourbaki). Some of these articles can be found via the web page: http://www.math.ias.edu/%7Evladimir/seminar.html Here is a web page of the place of the workshop: http://www.izumigo.co.jp/tab/yatsu_05_frame.html The cost of the hotel and the registration fee is 10,000 yen per night. The deadline of application is June 16th. Thomas Geisser (Univ. of Southern California / Univ. of Tokyo) geisser@ms.u-tokyo.ac.jp Noriyuki Otsubo (Chiba Univ.) otsubo@math.s.chiba-u.ac.jp