https://ivanfesenko.org/?page_id=146 IVAN FESENKO Selected coorganised workshops since 2014
・Como School “Unifying themes in Geometry”, September 27-30 2021 https://utge.lakecomoschool.org/ Unifying Themes In Geometry Lake Como School of Advanced Studies, 27 - 30 September 2021
Organizers and school lecturers Olivia Caramello (University of Insubria and IHES) Ivan Fesenko (University of Nottingham) Laurent Lafforgue (Huawei)
Supporting lecturers Wojciech Porowski (University of Nottingham)
Invited speakers Alain Connes (IHES) Misha Gromov (IHES and Courant Institute, N.Y.) Maxim Kontsevich (IHES) Barry Mazur (Harvard University)
Sponsors We gratefully acknowledge the support of the Lake Como School of Advanced Studies, the University of Insubria and the EPSRC Programme Grant Symmetries and Correspondences.
Wojciech Porowski: “Introduction to anabelian geometry” [slides available here; video available here]
Abstract:
After recalling the notion of the étale fundamental group of a scheme, we will discuss how various properties of a hyperbolic curve can be reconstructed from its fundamental group.
https://utge.lakecomoschool.org/programme/ Ivan Fesenko: “Higher adelic theory” [slides available here; video available here] Abstract: This talk starts a series of lectures on higher adelic theory (HAT) in the case of arithmetic surfaces and its applications. 2D objects associated to the surfaces and two different adelic structures on the surfaces will be introduced. The use of analytic adelic structures in higher zeta integrals and applications will be presented. The talk will start with the origin of several key developments in modern number theory: class field theory and its generalisations. https://www.dropbox.com/s/u4n8070t6s526ad/FesenkoSlides.pdf?dl=0 slides available here Ivan Fesenko: “Higher adelic theory”20210927
>>148 補足 >https://www.dropbox.com/s/u4n8070t6s526ad/FesenkoSlides.pdf?dl=0 >slides available here Ivan Fesenko: “Higher adelic theory”20210927
このスライドの最後に、下記がある
https://ivanfesenko.org/wp-content/uploads/2021/10/prad-1.html Ivan Fesenko Higher adelic theory Como School, September 27 2021 36 / 37 List of open problems in HAT Selected open problems in 2d adelic analysis and geometry (updated December 2018) (D stands for ‘done’, P stands for ‘in progress’) 40 problems, 6 done, 9 partially done
Porowski氏は、上記は https://www.dropbox.com/s/kuiz1n7sqdggj9x/PorowskiSlides.pdf?dl=0 で、 （ここのP19のBelyi cuspidalization (2)の図は、まさにIUTのアニメの図でしょ（ https://www.kurims.kyoto-u.ac.jp/~motizuki/project-2021-japanese.html 宇宙際タイヒミューラー理論の拡がり ） それから、P30 References (2) で more detailed presentation として、Oxford IUT workshop, December 2015, slides by J. Stix, Kyoto IUT workshop, July 2016, slides by K. Higashiyama の二つを挙げている ） Porowski氏の下記 Overview of IUT theory 今読むと、絶品ですね。分かり易いわ。Porowski氏は優秀だね
（>>4より） https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHorizIUT21/WS4/ExpHorizIUT21-IUTSummit-notes.html Inter-universal Teichmuller Theory (IUT) Summit 2021, RIMS workshop, September 7 - September 10 2021 Notes and recordings of the workshop
https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHorizIUT21/WS4/documents/Fesenko%20-%20IUT%20and%20modern%20number%20theory.pdf van Fesenko IUT and modern number theory.
https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHorizIUT21/WS4/documents/Porowski%20-%20Overview%20of%20IUT.pdf Wojciech Porowski Overview of IUT theory
因みに、南出明示公式は下記 https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHorizIUT21/WS4/documents/Minamide%20-%20Explicit%20Estimate.pdf Arata Minamide Explicit Estimates in Inter-universal Teichmuller Theory I and II, Reference: [ExpEst]. []
(参考) https://en.wikipedia.org/wiki/Class_field_theory#Generalizations_of_class_field_theory Class field theory Generalizations of class field theory There are three main generalizations, each of great interest. They are: the Langlands program, anabelian geometry, and higher class field theory. Often, the Langlands correspondence is viewed as a nonabelian class field theory. If and when it is fully established, it would contain a certain theory of nonabelian Galois extensions of global fields. However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case. It also does not include an analog of the existence theorem in class field theory: the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence point of view.
Another generalization of class field theory is anabelian geometry, which studies algorithms to restore the original object (e.g. a number field or a hyperbolic curve over it) from the knowledge of its full absolute Galois group or algebraic fundamental group. Another natural generalization is higher class field theory, divided into higher local class field theory and higher global class field theory. It describes abelian extensions of higher local fields and higher global fields. The latter come as function fields of schemes of finite type over integers and their appropriate localizations and completions. It uses algebraic K-theory, and appropriate Milnor K-groups generalize the {\displaystyle K_{1}}K_{1} used in one-dimensional class field theory
https://en.wikipedia.org/wiki/Anabelian_geometry Anabelian geometry More recently, Mochizuki introduced and developed a so called mono-anabelian geometry which restores, for a certain class of hyperbolic curves over number fields or some other fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute Anabelian Geometry." Anabelian geometry can be viewed as one of generalizations of class field theory. Unlike two other generalizations ? abelian higher class field theory and representation theoretic Langlands program ? anabelian geometry is highly non-linear and non-abelian.
https://en.wikipedia.org/wiki/Langlands_program Langlands program 4 Current status 4.1 Local Langlands conjectures 4.2 Fundamental lemma 4.3 Implications
https://en.wikipedia.org/wiki/Higher_local_field Higher local class field theory Higher local class field theory is compatible with class field theory at the residue field level, using the border map of Milnor K-theory to create a commutative diagram involving the reciprocity map on the level of the field and the residue field.[7] General higher local class field theory was developed by Kazuya Kato[8] and by Ivan Fesenko.[9][10] (引用終り)
Higher local class field theory ”General higher local class field theory was developed by Kazuya Kato[8] and by Ivan Fesenko.[9][10]”ね Ivan Fesenko.出てくるね
Como Schoolの Porowskiの講演は、IUTの基礎の遠アーベルの部分で、講演自身ではIUTに触れてない>>148 だが、”P30 References (2) で more detailed presentation として、Oxford IUT workshop, December 2015, slides by J. Stix, Kyoto IUT workshop, July 2016, slides by K. Higashiyama の二つを挙げている”>>170 ことからすると
話の流れで、読む順は Como Schoolの Porowskiの講演 → IUT Summit Overview of IUT theory >>170 かも
>>201 >出してくる文献が古過ぎるやん 仕方ないですよ 昭和時代末期の話ですからね >圏論プログラミング Lambek, J. and Scott, P. J., 1986. Introduction to Higher Order Categorical Logic, Cambridge University Press. という本は大学の図書館で見ましたよ まあ読まなかったんですけど
学部の時読んだのは、これですか David Gries The Science of Programming
