20200403の記者会見により、望月Inter-universal Teichmuller theory (abbreviated as IUT) （下記）は、新しい局面に入りました。 査読が終り、IUTが正しいことは、99％確定です。 このスレは、IUT応援スレとします。番号は前スレ43を継いでNo.44とします。 （なお、このスレは本体IUTスレの43からの分裂スレです）
https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory Inter-universal Teichmuller theory (abbreviated as IUT) (抜粋) Contents 1 History 2 Mathematical significance (引用終り)
https://en.wikipedia.org/wiki/Morita_equivalence Morita equivalence (抜粋) Significance in K-theory If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of (roughly) the classifying space of the nerve of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K-groups. 以上
IUT IV（>>624） ”the notion of a species allows one to “simulate ∈-loops” without violating the axiom of foundation of axiomatic set theory - cf. the discussion of Remark 3.3.1, (i).” の関係というか、繋がりを探しているが
あんまり関係ないが、メモ貼る https://en.wikipedia.org/wiki/Algebraic_combinatorics Algebraic combinatorics (抜粋) Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
Contents 1 History 2 Scope 3 Important topics 3.1 Symmetric functions 3.2 Association schemes 3.3 Strongly regular graphs 3.4 Young tableaux 3.5 Matroids 3.6 Finite geometries
History The term "algebraic combinatorics" was introduced in the late 1970s.[1] Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS Mathematics Subject Classification, introduced in 1991.
Scope Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common.
下記の Lawvere Ph.D. thesis は、一読の価値あるな ”The authors comments are F. William Lawvere, 2004.”の部分
710 名前：だけでも、読んでおく価値がある！（＾＾ http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories F. William Lawvere Originally published as: Ph.D. thesis, Columbia University, 1963 and in Reports of the Midwest Category Seminar II, 1968, 41-61, The authors comments are F. William Lawvere, 2004. http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf
https://en.wikipedia.org/wiki/Universal_algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study. History Starting with William Lawvere's thesis in 1963, techniques from category theory have become important in universal algebra.[9] Footnotes 9 Lawvere, William F. (1964), Functorial Semantics of Algebraic Theories (PhD Thesis)
https://en.wikipedia.org/wiki/Category_theory Category theory Historical notes Main article: Timeline of category theory and related mathematics https://en.wikipedia.org/wiki/Timeline_of_category_theory_and_related_mathematics Category theory may be viewed as an extension of universal algebra, as the latter studies algebraic structures, and the former applies to any kind of mathematical structure and studies also the relationships between structures of different nature. []
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.)
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.
１．ショルツ先生も、望月IUT難しすぎで、分からないことがある ２．でも、自分が正しいと思う（数学的には証明になってないみたいｗ（＾＾； ） ３．Dupuy先生納得せず。”I’m happy to continue any further discussions by e-mail.”（あとは、メールで）と ショルツ先生
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]
（参考） https://www.mathunion.org/icm/icm-2022 ICM 2022 The International Congress of Mathematicians 2022 (ICM 2022) will be held in St Petersburg, Russia, 6 - 14 July 2022.
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]
>>708 (抜粋) ３） Peter Scholze says: May 1, 2020 at 4:42 pm Dear Taylor, I’m happy to continue any further discussions by e-mail. (引用終り)
Taylor Dupuyの勝利に終わったのか？ それは、ともかく Taylor Dupuyのカベは、ショルツ先生には破れないだろうな（＾＾；
（参考） math jin：（IUTT情報サイト） https://twitter.com/math_jin (抜粋) math_jinさんがリツイート Taylor Dupuy 5月3日 返信先: @Rz6rX さん, @math_jin さん 他2人 The manuscript with Stix has problems. The worst being the diagram in section 2.2. That does not appear in Mochizuki’s manuscript. I think people should stop citing it.
math_jinさんがリツイート 太郎 @Rz6rX 5月3日 返信先: @DupuyTaylor さん, @math_jin さん 他2人 Excuse me, It may have been asked many times, but is Professor Scholze's point true? (deleted an unsolicited ad)
（下記みたいだな） www.kurims.kyoto-u.ac.jp/~motizuki/papers-japanese.html 望月 論文 遠アーベル幾何、圏の幾何 [25] On the Combinatorial Anabelian Geometry of Nodally Nondegenerate Outer Representations. PDF NEW !! (2011-12-22) Comments NEW !! (2019-07-20)
798 名前：(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 []
