Inter-universal geometry と ABC予想 (応援スレ) 45

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706: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/05/14(木) 21:00:54 ID:xfdjzzyp

>>698
>これに関してアンチ望月にならないための反論ある？

それ 意味分かんないけど
S・Sレポ8月版 って、2018年でしょ？
そして、2020年4月3日に「査読終り。IUT成立。SSレポートはアウト」の記者会見
その後、Woitブログで、ショルツ先生 vs Dupuy先生のバトルが、下記

https://www.math.columbia.edu/~woit/wordpress/?p=11709 （woitブログ）
Not Even Wrong Latest on abc Posted on April 3, 2020 by woit

より、以下抜粋

１）
Peter Scholze says:
April 17, 2020 at 7:15 pm
PS: I just realized that maybe the following information is worth sharing. Namely, as an outsider one may wonder that the questions being discussed at length in these comments (e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments),
and feel almost philosophical, so one might wonder that one is not looking at the heart of the matter.

However, the discussions in Kyoto went along extremely similar lines, and these discussions were actually very much led, certainly initially, by Mochizuki.
He first wanted to carefully explain the need for distinct copies, by way of perfections of rings, and then of the log-link, leading to discussions rather close to the one I was having with UF here.
He agreed that one first has to understand these basic points before it makes sense to introduce all further layers of complexity. (I should add that we did also go through the substance of the papers, but kept getting back at how this reflects on the basic points, as we all agreed that this is the key of the matter.)

要するに、いまさら、「IUTは難しいので、基本的なポイントを理解する必要がある」などという

つづく

http://rio2016.5ch.net/test/read.cgi/math/1588552720/706

707: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/05/14(木) 21:02:03 ID:xfdjzzyp

>>706
つづき

２）
Peter Scholze says:
April 30, 2020 at 3:32 am
Dear Taylor,

I certainly understood your point there ? you might also take the ring Z[√(-1)].

There is of course a big difference between the ring Z and the “theory” it defines, i.e. roughly the class of all subsets of all finite powers Z^n  that are definable by polynomial equations.
The latter is indeed a highly nontrivial category (where morphisms are given by definable graphs); it is of course not equivalent to a category with one object and two morphisms.
If a category like this is in place in Mochizuki’s work, I’m happy to hear about it!

Reading the IUT papers, however, you are presented with some extremely difficult notion of a Hodge theater, together with a highly non-obvious notion of isomorphisms of such: Isomorphisms do not preserve nearly as much structure as you would expect them to, and this is by design as Mochizuki points out.
So I find it very hard to “guess” what something like a surrounding “theory” might be. For all I can see, Hodge theaters fit neither into the framework of “structures” as used in the wikipedia entry https://en.wikipedia.org/wiki/Interpretation_(model_theory) you linked to, nor the topos-theoretic framework of Caramello.
(Regarding the first one: A “structure” in the sense of model theory has first of all an underlying set. I find it hard to take a Hodge theater and produce some interesting set that is functorial in isomorphisms of Hodge theaters, the problem being the very lax notion of isomorphisms of Hodge theaters.)

However, these long discussions are all about interpretations. Regarding the mathematics proper: I stand by the claim made in our manuscript, and have indicated the proof above.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1588552720/707

724: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/05/14(木) 23:42:30 ID:xfdjzzyp

代数的整数論は、日本の伝統であり、お家芸とも言えるものです！
高木−志村五郎・谷山豊−（Shimura varieties）−岩澤理論−（ノイキルヒ・）内田−＜中村博明、玉川、望月（遠アーベル幾何学）＞
その伝統の上に IUTがあるので〜す！！（＾＾

（参考）
https://ja.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E7%9A%84%E6%95%B4%E6%95%B0%E8%AB%96
代数的整数論
目次
1 代数的整数論の歴史
1.1 ディオファントス
ヒルベルト
彼は類体論に関する一連の予想をたてた。構想は非常に影響的で、彼自身の貢献はヒルベルト類体と局所類体論（英語版）のヒルベルト記号（英語版）の名前に生き続けている。結果は高木貞治による研究の後1930年までにはほとんど証明された[注 1]。
現代理論
1955年頃、日本人数学者志村五郎と谷山豊は2つの一見全く異なる数学の分野、楕円曲線とモジュラー形式の間につながりがあるかもしれないことを観察した。結果のモジュラー性定理（志村・谷山予想）は、すべての楕円曲線はモジュラー（英語版）である、つまり一意的なモジュラー形式に付随できる、という主張である。
アンドリュー・ワイルズ
証明はまた、スキームの圏や岩澤理論や、フェルマーには利用可能でなかった他の20世紀の技術のような、現代的な代数幾何の標準的な構成を用いる。

https://en.wikipedia.org/wiki/Arithmetic_geometry
In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves.[20] Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.[21]

https://ja.wikipedia.org/wiki/%E3%83%8E%E3%82%A4%E3%82%AD%E3%83%AB%E3%83%92%E3%83%BB%E5%86%85%E7%94%B0%E3%81%AE%E5%AE%9A%E7%90%86
ノイキルヒ・内田の定理

＜中村博明、玉川、望月＞
https://ja.wikipedia.org/wiki/%E9%81%A0%E3%82%A2%E3%83%BC%E3%83%99%E3%83%AB%E5%B9%BE%E4%BD%95%E5%AD%A6
遠アーベル幾何学

http://rio2016.5ch.net/test/read.cgi/math/1588552720/724

728: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/05/15(金) 08:00:08 ID:Jy/2KfWb

下記のArithmetic geometry
Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond
のその先にIUTが位置するのだろうと 見ています（＾＾

（参考）
https://en.wikipedia.org/wiki/Arithmetic_geometry
Arithmetic geometry
(抜粋)
Overview
The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers.
Rational points can be directly characterized by height functions which measure their arithmetic complexity.[5]
The structure of algebraic varieties defined over non-algebraically-closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry.
Over finite fields, etale cohomology provides topological invariants associated to algebraic varieties.[6] p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.[7]

History
Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond

In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell?Weil theorem only demonstrates finite generation of the set of rational points as opposed to finiteness).[25][26]

In 2001, the proof of the local Langlands conjectures for GLn was based on the geometry of certain Shimura varieties.[27]

In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application to Galois representations and certain cases of the weight-monodromy conjecture.[28][29]

https://ja.wikipedia.org/wiki/%E6%95%B0%E8%AB%96%E5%B9%BE%E4%BD%95%E5%AD%A6
数論幾何学

http://rio2016.5ch.net/test/read.cgi/math/1588552720/728

736: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [sage] 2020/05/15(金) 17:41:14 ID:zHVN/jcs

>>735

つづき

上記 PDF NEW !! (2011-12-22)と 下記(Revised November 11, 2010)で、微妙に違う（＾＾；
http://www.kurims.kyoto-u.ac.jp/~yuichiro/papers.html
星裕一の論文
組み合わせ論的遠アーベル幾何学関連
https://projecteuclid.org/download/pdf_1/euclid.hmj/1323700038
Hiroshima Math. J.
41 (2011), 275?342
On the combinatorial anabelian geometry of nodally nondegenerate outer representations
Yuichiro Hoshi and Shinichi Mochizuki
(Received August 19, 2009)
(Revised November 11, 2010)
Abstract. Let S be a nonempty set of prime numbers. In the present paper, we continue the study, initiated in a previous paper by the second author, of the combinatorial
anabelian geometry of semi-graphs of anabelioids of pro-S PSC-type, i.e., roughly
speaking, semi-graphs of anabelioids associated to pointed stable curves. Our first
main result is a partial generalization of one of the main combinatorial anabelian results
of this previous paper to the case of nodally nondegenerate outer representations, i.e.,
roughly speaking, a sort of abstract combinatorial group-theoretic generalization of the
scheme-theoretic notion of a family of pointed stable curves over the spectrum of a
discrete valuation ring. We then apply this result to obtain a generalization, to the case
of proper hyperbolic curves, of a certain injectivity result, obtained in another paper by
the second author, concerning outer automorphisms of the pro-S fundamental group of a
configuration space associated to a hyperbolic curve, as the dimension of this configuration
space is lowered from two to one. This injectivity allows one to generalize a certain
well-known injectivity theorem of Matsumoto to the case of proper hyperbolic curves.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1588552720/736

738: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [sage] 2020/05/15(金) 18:16:34 ID:zHVN/jcs

>>726 追加
> １．IUTとは何か？　
>　文元本（算数（たし算かけ算））より上の大学レベルの数学的説明がほしい

（参考）
下記は、山下先生が書いた手書き講演のOHPを彷彿とさせるな
q^(j^2)の話とかABC予想の話が見えて面白いね。ここらを、ちゃんと 手を入れてタイプアップと 説明文書を入れて
発行してほしいと思う
今日このこの頃 （＾＾

http://www.kurims.kyoto-u.ac.jp/~motizuki/travel-japanese.html
望月 出張・講演

（この2004-01 P1に、通常の集合論の拡張とか、ラベルの使用とか、∈- ループ っぽい話が書いてあるね（＾＾；　）
http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Brief%20Introduction%20to%20Inter-universal%20Geometry%20(Tokyo%202004-01).pdf
[7] A Brief Introduction to Inter-universal Geometry (東京大学 2004年1月). PDF

http://www.kurims.kyoto-u.ac.jp/~motizuki/Categorical%20Representation%20of%20Arithmetic%20Log%20Schemes%20-%20with%20Applications%20to%20the%20Arithmetic%20of%20Elliptic%20Curves%20(Tokyo%202004-02).pdf
[8] Categorical Representation of Arithmetic Log Schemes ? with Applications to the Arithmetic of Elliptic Curves (東京大学 2004年2月). PDF

http://www.kurims.kyoto-u.ac.jp/~motizuki/Anabelian%20Geometry%20from%20an%20Inter-universal%20Point%20of%20View%20(RIMS%20Kyoto%202004-09).pdf
[9] Anabelian Geometry from an Inter-universal Point of View (京都大学数理解析研究所 2004年9月).　PDF

http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Hodge-Arakelov%20Theory%20(RIMS%20Kyoto%202005-12).pdf
[13] Inter-universal Hodge-Arakelov Theory (京都大学数理解析研究所 2005年12月). PDF

http://rio2016.5ch.net/test/read.cgi/math/1588552720/738

746: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/05/15(金) 23:04:47 ID:Jy/2KfWb

>>744
どうも
情報ありがとう
立派な修論ですね〜（＾＾；

https://arxiv.org/abs/1902.03767
[Submitted on 11 Feb 2019 (v1), last revised 24 Aug 2019 (this version, v3)]
A topological approach to indices of geometric operators on manifolds with fibered boundaries
Mayuko Yamashita
In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries.
We define K-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete Φ or edge metrics, can be regarded as the index pairing over these K-groups.
We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.
Comments: 69 pages; the final version
Subjects: K-Theory and Homology (math.KT); Differential Geometry (math.DG)

http://rio2016.5ch.net/test/read.cgi/math/1588552720/746

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