つづき （参考） 関連： 望月新一（数理研） www.kurims.kyoto-u.ac.jp/~motizuki/ News - Ivan Fesenko https://www.maths.nottingham.ac.uk/plp/pmzibf/nov.html Explicit estimates in inter-universal Teichmuller theory, by S. Mochizuki, I. Fesenko, Y. Hoshi, A. Minamide, W. Porowski, RIMS preprint in November 2020, updated in June 2021, accepted for publication in September 2021 https://ivanfesenko.org/wp-content/uploads/2021/11/Explicit-estimates-in-IUT.pdf NEW!! (2020-11-30) いわゆる南出論文 より P4 Theorem A. (Effective versions of ABC/Szpiro inequalities over mono-complex number fields) Theorem B. (Effective version of a conjecture of Szpiro) Corollary C. (Application to “Fermat’s Last Theorem”) P56 Corollary 5.9. (Application to a generalized version of “Fermat’s Last Theorem”) Let l, m, n be positive integers such that min{l, m, n} > max{2.453 ・ 10^30, log2 ||rst||C, 10 + 5 log2(rad(rst))}. Then there does not exist any triple (x, y, z) ∈ S of coprime [i.e., the set of prime numbers which divide x, y, and z is empty] integers that satisfies the equation
Theorem Bで、Effective ”Szpiro”も出る 但し、”effective versions of the Vojta”への言及がないので、Vojtaは 未だみたい ここ、一山当てる狙い目かもねｗ 他に、IUT関連 ・[R8] Higher adelic theory, talk at the Como School, September 2021 https://ivanfesenko.org/wp-content/uploads/2021/10/hat.pdf ・[R7] IUT and modern number theory, talk at the RIMS workshop on IUT Summit, September 2021 https://ivanfesenko.org/wp-content/uploads/2021/10/mntiut.pdf ? [R5] Class field theory, its three main generalisations, and applications, May 2021, EMS Surveys 8(2021) 107-133 https://ivanfesenko.org/wp-content/uploads/2021/10/232.pdf
Yourpedia 宇宙際タイヒミュラー理論 （URLが通らないので検索たのむ） https://ja.wikipedia.org/wiki/%E5%AE%87%E5%AE%99%E9%9A%9B%E3%82%BF%E3%82%A4%E3%83%92%E3%83%9F%E3%83%A5%E3%83%A9%E3%83%BC%E7%90%86%E8%AB%96 宇宙際タイヒミュラー理論 Wikipedia https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory 英Inter-universal Teichmuller theory 英 Wikipedia https://ja.wikipedia.org/wiki/ABC%E4%BA%88%E6%83%B3 ABC予想 https://en.wikipedia.org/wiki/Abc_conjecture 英abc conjecture
https://www.math.arizona.edu/~kirti/ から Recent Research へ入る Kirti Joshi Recent Research論文集 新論文（IUTに着想を得た新理論） https://arxiv.org/pdf/2106.11452.pdf Construction of Arithmetic Teichmuller Spaces and some applications Preliminary version for comments Kirti Joshi June 23, 2021
https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage] 論文集 なお、（メモ）TAYLOR DUPUYは、arxiv投稿で [SS17]を潰した（下記） https://arxiv.org/pdf/2004.13108.pdf PROBABILISTIC SZPIRO, BABY SZPIRO, AND EXPLICIT SZPIRO FROM MOCHIZUKI’S COROLLARY 3.12 TAYLOR DUPUY AND ANTON HILADO Date: April 30, 2020. P14 Remark 3.8.3. (1) The assertion of [SS17, pg 10] is that (3.3) is the only relation between the q-pilot and Θ-pilot degrees. The assertion of [Moc18, C14] is that [SS17, pg 10] is not what occurs in [Moc15a]. The reasoning of [SS17, pg 10] is something like what follows:
P15 (2) We would like to point out that the diagram on page 10 of [SS17] is very similar to the diagram on §8.4 part 7, page 76 of the unpublished manuscript [Tan18] which Scholze and Stix were reading while preparing [SS17]. References [SS17] Peter Scholze and Jakob Stix, Why abc is still a conjecture., 2017. 1, 1, 1e, 2, 7.5.3 ( https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf Date: July 16, 2018. https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf Date: August 23, 2018. ) [Tan18] Fucheng Tan, Note on IUT, 2018. 1, 2
なお "[SS17] Peter Scholze and Jakob Stix, Why abc is still a conjecture., 2017."は、2018の気がする ”[Tan18] Fucheng Tan, Note on IUT, 2018. 1, 2”が見つからない。”the unpublished manuscript [Tan18]”とはあるのだが（＾＾ 代わりに、ヒットした下記でも、どぞ （2018の何月かが不明だが、2018.3のSS以降かも）
www.kurims.kyoto-u.ac.jp/~motizuki/Tan%20---%20Introduction%20to%20inter-universal%20Teichmuller%20theory%20(slides).pdf Introduction to Inter-universal Teichm¨uller theory Fucheng Tan RIMS, Kyoto University 2018 To my limited experiences, the following seem to be an option for people who wish to get to know IUT without spending too much time on all the details. ・ Regard the anabelian results and the general theory of Frobenioids as blackbox. ・ Proceed to read Sections 1, 2 of [EtTh], which is the basis of IUT. ・ Read [IUT-I] and [IUT-II] (briefly), so as to know the basic definitions. ・ Read [IUT-III] carefully. To make sense of the various definitions/constructions in the second half of [IUT-III], one needs all the previous definitions/results. ・ The results in [IUT-IV] were in fact discovered first. Section 1 of [IUT-IV] allows one to see the construction in [IUT-III] in a rather concrete way, hence can be read together with [IUT-III], or even before. S. Mochizuki, The ´etale theta function and its Frobenioid-theoretic manifestations. S. Mochizuki, Inter-universal Teichm¨uller Theory I, II, III, IV.
www.kurims.kyoto-u.ac.jp/daigakuin/Tan.pdf 教員名: 譚 福成（Tan, Fucheng） P-adic Hodge theory plays an essential role in Mochizuki's proof of Grothendieck's Anabelian Conjecture. Recently, I have been studying anabeian geometry and Mochizuki's Inter-universal Teichmuller theory, which is in certain sense a global simulation of p-adic comparison theorem.
https://www.maths.nottingham.ac.uk/plp/pmzibf/mp.html Ivan Fesenko - Research in texts https://www.maths.nottingham.ac.uk/plp/pmzibf/232.pdf [R5] Class field theory, its three main generalisations, and applications pdf, May 2021
P16の後半に面白い図がある
コピーペースト下記 Here are some relations between the three generalisations of CFT and their further developments:
2dLC?－－ 2dAAG－－－ IUT l ／ ｜ ｜ l ／ ｜ ｜ l／ ｜ ｜ LC 2dCFT anabelian geometry ＼ ｜ ／ ＼ ｜ ／ ＼ ｜ ／ CFT 注）記号： Class Field Theory (CFT), Langlands correspondences (LC), 2dAAG = 2d adelic analysis and geometry, two-dimensional (2d) (P8 "These generalisations use fundamental groups: the etale fundamental group in anabelian geometry, representations of the etale fundamental group (thus, forgetting something very essential about the full fundamental group) in Langlands correspondences and the (abelian) motivic A1 fundamental group (i.e. Milnor K2) in two-dimensional (2d) higher class field theory.") https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHorizIUT21/WS4/documents/Fesenko%20-%20IUT%20and%20modern%20number%20theory.pdf Fesenko IUT and modern number theory つづく
（IUTに対する批判的レビュー） https://zbmath.org/07317908 https://zbmath.org/pdf/07317908.pdf Mochizuki, Shinichi Inter-universal Teichmuller theory. I: Construction of Hodge theaters. (English) Zbl 07317908 Publ. Res. Inst. Math. Sci. 57, No. 1-2, 3-207 (2021). Reviewer: Peter Scholze (Bonn)
BuzzardのICM22講演原稿 Inter-universal geometry とABC 予想47 https://rio2016.5ch.net/test/read.cgi/math/1635332056/84 84 名前：38[] 投稿日：2021/12/23(木) 19:42:33.42 ID:iz9G4jw+ [1/2] Buzzardの原稿が出たヨ！ https://arxiv.org/abs/2112.11598 >A great example is Mochizuki’s claimed proof of the ABC conjecture [Moc21]. >This proof has now been published in a serious research journal, however >it is clear that it is not accepted by the mathematical community in general.
86 名前：１３２人目の素数さん[] 投稿日：2021/12/23(木) 20:46:56.21 ID:a0F2ZqKI >>84 ホントに出ていたね。その引用部分の少し後に次のことが書かれている。 Furthermore, the key sticking point right now is that the unbelievers argue that more details are needed in the proof of Corollary 3.12 in the main paper, and the state of the art right now is simply that one cannot begin to formalise this corollary without access to these details in some form (for example a paper proof containing far more information about the argument) (引用終り)
”Comments: 28 pages, companion paper to ICM 2022 talk”と明記もあるね 思うに、その意図は、「反論あるなら言ってきてね。反論の機会を与える。反論なき場合はこのまま総会発表とする」ってことか （西洋流で、「黙っていたから 認めたってことじゃん」みたいなｗ） 普通は、こんな形でプレプリ出さない気がするな さあ、面白くなってきたかも ドンパチ派手にやってほしい
>>18 2020年の解明論文 Kumar, A., Bechhoefer, J. Exponentially faster cooling in a colloidal system. Nature 584, 64–68 (2020). https://doi.org/10.1038/s41586-020-2560-x https://arxiv.org/pdf/2008.02373.pdf
>>13 補足 https://ivanfesenko.org/wp-content/uploads/2021/11/232.pdf CLASS FIELD THEORY, ITS THREE MAIN GENERALISATIONS, AND APPLICATIONS IVAN FESENKO This text was published in EMS Surveys, 8(2021) 107?133.
Here are some relations between the three generalisations of CFT and their further developments: 2dLC?－－ 2dAAG－－－ IUT l ／ ｜ ｜ l ／ ｜ ｜ l／ ｜ ｜ LC 2dCFT anabelian geometry ＼ ｜ ／ ＼ ｜ ／ ＼ ｜ ／ CFT (引用終り)
下記のProblemがいいね (FESENKO) Problem 1. Develop fuller LC analogues of Parts II-IV of CFT. Problem 2. Find a version of enhanced arithmetic LC parallel to GCFT. Problem 3. Find a version of enhanced LC parallel to SCFT but which works over all number fields. Problem 4. Find a version of enhanced LC parallel to some of post-cohomological CFT, thus circumventing the problem of using non-abelian cohomology. Problem 5. Develop a general ramification theory for surfaces compatible with 2dCFT and taking into careful account ramification theory at the one-dimensional residue level. Problem 6. Develop a special higher CFT which uses torsion structures, to provide new insights into 2dCFT. Problem 7. Find more direct relations between the generalisations of CFT. Use them to produce a single unified generalisation of CFT.
>>18>>22 2017年論文は2つ ・Lu, Zhiyue; Raz, Oren (16 May 2017). "Nonequilibrium thermodynamics of the Markovian Mpemba effect and its inverse". Proceedings of the National Academy of Sciences. 114 (20): 5083–5088. arXiv:1609.05271. Bibcode:2017PNAS..114.5083L. doi:10.1073/pnas.1701264114. ISSN 0027-8424. PMC 5441807. PMID 28461467.
https://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 April 2020
P5 If, moreover, one thinks of Z as being constructed, in the usual way, via axiomatic set theory, then one may interpret the “absolute” - i.e., “tautologically unrelativizable” - nature of conventional scheme theory over Z at a purely settheoretic level. Indeed, from the point of view of the “∈-structure” of axiomatic set theory, there is no way to treat sets constructed at distinct levels of this ∈-structure as being on a par with one another. On the other hand, if one focuses not on the level of the ∈-structure to which a set belongs, but rather on species, then the notion of a species allows one to relate - i.e., to treat on a par with one another - objects belonging to the species that arise from sets constructed at distinct levels of the ∈-structure. That is to say, 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).
P68 On the other hand, by the axiom of foundation, there do not exist infinite descending chains of universes V0 V1 V2 V3 ... Vn ... - where n ranges over the natural numbers.
