https://www.math.columbia.edu/~woit/wordpress/?p=10560 Not Even Wrong Scholze and Stix on the Mochizuki Proof Posted on September 20, 2018 by woit 0245現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 20:56:18.94ID:id6ENHqe>>244 さて、
＜IUTの現状分析＞ １．2012年のIUT論文4つが完成以来、いまだ成否定まらず ・特記として、2018年9月のScholze and Stixの誤りだという指摘と、それへの反論があった ・1)IUT成立派（RIMS以外にも）と、2)IUT不成立派（国際的には、Scholze and Stix以外に何人か） ・3)中間派：この中でも、IUTに好意的な人達が何人かいる。来年のIUTワークショップの1本目に参加表明している人達 ２．来年IUTのシンポジュームを打って、４本のワークショップが企画されている ・多分、IUT成立派は、これを最大限利用して、IUT成立の国際的合意を得たいだろう （果たして）
https://en.wikipedia.org/wiki/Outline_of_category_theory Outline of category theory (抜粋) The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
Contents 1 Essence of category theory 2 Branches of category theory 3 Specific categories 4 Objects 5 Morphisms 6 Functors 7 Limits 8 Additive structure 9 Dagger categories 10 Monoidal categories 11 Cartesian closed category 12 Structure 13 Topoi, toposes 14 History of category theory 15 Persons influential in the field of category theory
Essence of category theory Category ? Functor ? Natural transformation ?
Branches of category theory Homological algebra ? Diagram chasing ? Topos theory ? Enriched category theory ? Higher category theory ?
組織委員長：望月新一（京都大学数理解析研究所） 組織委員：星裕一郎（京都大学数理解析研究所） Ivan Fesenko （英・ノッティンガム大学） 田口雄一郎（東京工業大学） 加藤文元（東京工業大学） 栗原将人（慶応義塾大学） 志甫淳（東京大学）
Foundations and Perspectives of Anabelian Geometry 部屋：420号室 期間：2020-05-18?2020-05-22 組織委員：Ivan Fesenko （英・ノッティンガム大学） 南出新（英・ノッティンガム大学） 譚福成（京都大学数理解析研究所）
組合せ論的遠アーベル幾何とその周辺 部屋：420号室 期間：2020-06-29?2020-07-03 組織委員：星裕一郎（京都大学数理解析研究所） 望月新一（京都大学数理解析研究所） Ivan Fesenko （英・ノッティンガム大学） 南出新 （英・ノッティンガム大学）
宇宙際タイヒミューラー理論への誘い（いざない） 部屋：420号室 期間：2020-09-01?2020-09-04 組織委員：星裕一郎（京都大学数理解析研究所） 望月新一（京都大学数理解析研究所） Ivan Fesenko （英・ノッティンガム大学） 田口雄一郎 （東京工業大学）
宇宙際タイヒミューラー理論サミット２０２０ 部屋：420号室 期間：2020-09-08?2020-09-11 組織委員：星裕一郎（京都大学数理解析研究所） 望月新一（京都大学数理解析研究所） Ivan Fesenko （英・ノッティンガム大学） 田口雄一郎 （東京工業大学） 0303現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/06(金) 11:28:36.04ID:mXy02Ftq 青天の霹靂というけれど ”In March 2018, Peter Scholze and Jakob Stix visited Kyoto University for five days of discussions” は、事前に分かっていたはずだし、内容も事前に大体の情報は得ていたろう
”five days of discussions”で、合意には至らなかった だが、IUTが事実上論破されたのに、（論破されたのを分かりつつ予算の都合上）屁理屈こね回して、反論しているというような 例えば”IUTで、多くの人が参加して、数学で巧妙な振り込め詐欺をやっている”みたいな見方は、如何なものか（Peter Scholzeでさえ理解できないのかというのは、驚きだったかも知れないが）
http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV: ¨ LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki October 2019 Abstract. (抜粋) The present paper forms the fourth and final paper in a seriesof papers concerning “inter-universal Teichm¨uller theory”. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”. (引用終り) 0307現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 09:16:44.55ID:H2e5WMAT>>306
(抜粋) In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves. (引用終り)
政治的に場合分けすると
１．もし、望月IUTが正しいとすると、 ”the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.” の３つは、今後、「おれが解決したぞ」という人が出てきても、二番煎じの扱いにしかならない
(引用開始) １．もし、望月IUTが正しいとすると、 ”the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.” の３つは、今後、「おれが解決したぞ」という人が出てきても、二番煎じの扱いにしかならない (引用終り)
もし、IUTを経由しない、従来の数学手法の改良で、 ”the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.” を証明できて、それがずっと簡明ならば、かなり高評価なのでしょうね 政治的には
などの円周率 π や自然対数の底 e の大抵の和、積、べき乗は、有理数であるのか無理数であるのか超越的であるのか否かは証明されていない[注 4]。
とあるけど？ 証明できたなら論文になるよ。 でもこの人の数学内容は全然信用できないけどね。 0322１３２人目の素数さん2019/12/07(土) 16:23:51.68ID:qvPzzpXn>>317 以前、IUT とかの数論幾何は標数0の実数体Rには無力ではないか、 という旨の内容のレスを IUT スレに書いたことがある。 そうしたら、IUT スレでその通りという返事が返って来た。 0323現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 16:24:59.86ID:H2e5WMAT>>306 補足 http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV:LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki October 2019 (抜粋) P3 Theorem A. (Diophantine Inequalities)
Thus, Theorem A asserts an inequality concerning the canonical height [i.e.,“htωX(D)”], the logarithmic different [i.e., “log-diffX”], and the logarithmic conductor [i.e., “log-condD”] of points of the curve UX valued in number fields whose extension degree over Q is ? d . In particular, the so-called Vojta Conjecture forhyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves all follow as special cases of Theorem A.
P54 Corollary 2.3. (Diophantine Inequalities)
P57 Remark 2.3.3. Corollary 2.3 may be thought of as an effective version of the Mordell Conjecture. From this point of view, it is perhaps of interest to compare the “essential ingredients” that are applied in the proof of Corollary 2.3 [i.e., in effect, that are applied in the present series of papers!] with the “essential ingredients” applied in [Falt]. The following discussion benefited substantially from numerous e-mail and skype exchanges with Ivan Fesenko during the summer of 2015. (引用終り)
Vojta Conjecture forhyperbolic curves、ABC Conjecture、Szpiro Conjecture for elliptic curves、an effective version of the Mordell Conjecture 全部IUTの射程内だという 本当なら、面白いじゃない？ｗ（＾＾ 0324１３２人目の素数さん2019/12/07(土) 16:31:09.21ID:qvPzzpXn>>320 あの問いは、元々私が証明したと主張した代数的な命題から派生してスレ主に出題した問いだ。
https://en.wikipedia.org/wiki/Weil_conjectures Weil conjectures (抜粋) Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps:
Use of Lefschetz pencils ・The theory of monodromy of Lefschetz pencils, introduced for complex varieties (and ordinary cohomology) by Lefschetz (1924), and extended by Grothendieck (1972) and Deligne & Katz (1973) to l-adic cohomology, relates the cohomology of V to that of its fibers. The relation depends on the space Ex of vanishing cycles, the subspace of the cohomology Hd?1(Vx) of a non-singular fiber Vx, spanned by classes that vanish on singular fibers. ・The Leray spectral sequence relates the middle cohomology group of V to the cohomology of the fiber and base. c(U,E), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces Ex of vanishing cycles.
The key estimate The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks. This is done by studying the zeta functions of the even powers Ek of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups. The crucial idea of considering even k powers of E was inspired by the paper Rankin (1939), who used a similar idea with k=2 for bounding the Ramanujan tau function. Langlands (1970, section 8) pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization. つづく 0337現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 19:39:38.79ID:H2e5WMAT>>336
つづき
Completion of the proof The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.
Deligne's second proof Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem. Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallee Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.
Inspired by the work of Witten (1982) on Morse theory, Laumon (1987) found another proof, using Deligne's l-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallee Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Kiehl & Weissauer (2001) used Laumon's proof as the basis for their exposition of Deligne's theorem. Katz (2001) gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Kedlaya (2006) gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology.
https://en.wikipedia.org/wiki/Lefschetz_pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.
It has been shown that Lefschetz pencils exist in characteristic zero. They apply in ways similar to, but more complicated than, Morse functions on smooth manifolds. It has also been shown that Lefschetz pencils exist in characteristic p for the etale topology.
Simon Donaldson has found a role for Lefschetz pencils in symplectic topology, leading to more recent research interest in them. (引用終り) 以上 0339現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 19:40:56.78ID:H2e5WMAT>>336
