2010年10月25日～30日に開催されたJoint MSJ RIMS Conference:The 3rd Seasonal Institute of the Mathematical Society of Japan. “Development of Galois Teichmuller Theory and Anabelian Geometry””において、10月29日10時～11時の1時間の講演で、望月新一氏が宇宙際タイヒミュラー理論の概要を発表しました。
October 29 Friday 10:0011:00
Inter-universal Teichmüller Theory: A Progress Report
Abstract: The analogy between number fields and function fields of curves (e.g., hyperbolic curves) over finite fields is quite classical. In the present talk, we survey work in progress concerning a theory developed by the lecturer during the last decade — in the spirit of this analogy — whose goal is to construct an analogue for number fields “equipped with an elliptic curve” of the p-adic Teichmüller theory developed by the lecturer during the early 1990’s for hyperbolic curves over a finite field “equipped with a nilpotent ordinary indigenous bundle”. From an even more classical point of view, one may think of this theory as a sort of analogue for number fields of classical complex Teichmüller theory, in which canonical deformations of the holomorphic structure of a hyperbolic Riemann surface of finite type are constructed by dilating one of the two underlying real dimensions of the Riemann surface, while leaving the other dimension fixed (i.e., “undeformed”). In the case of number fields equipped with an elliptic curve, one thinks of the ring structure of the number field as a sort of “arithmetic holomorphic structure”. One then constructs canonical deformations of this arithmetic holomorphic structure — i.e., analogues of the canonical liftings of p-adic Teichmüller theory — by applying the general theory of Frobenioids, as well as the theory of the Frobenioid-theoretic theta function (developed in earlier papers by the lecturer). At a more concrete level, if one thinks of the ring structure (i.e., “arithmetic holomorphic structure”) of the given number field as consisting of “two underlying combinatorial dimensions” corresponding to addition and multiplication, then working with Frobenioids corresponds, roughly speaking, to performing operations with the multiplicative monoids involved (i.e., multiplicative portions of the rings involved) — in a fashion motivated by the theory of log structures; in particular, such operations are not necessarily compatible with the additive portions of the ring structures involved. Alternatively, if one thinks of the ring structure (i.e., “arithmetic holomorphic structure”) of the various local fields that arise as localizations of the given number field as consisting of “two underlying combinatorial dimensions” corresponding to the group of units and the value group, then one may think of these canonical deformations of the arithmetic holomorphic structure as deformations in which the value groups are (canonically!) dilated — by means of the theta function — while the units are left undeformed. Since such “arithmetic Teichmüller dilations” are manifestly incompatible with the ring structure of the given number field, it follows that they are not compatible, in general, with various classical scheme-theoretic constructions performed over the number field which depend on this ring structure. In particular, these arithmetic Teichmüller dilations fail to be compatible with the various basepoints of arithmetic fundamental groups involved (e.g., Galois groups) which are defined by considering scheme-theoretic geometric points. The resulting incompatibility of (conventional scheme-theoretic) basepoints on either side of the “arithmetic Teichmüller dilation” gives rise to numerous indeterminacies; these indeterminacies lead naturally to the introduction of tools from anabelian geometry. It is this fundamental aspect of the theory that is referred to by the term “inter-universal”. The (expected) main theorem of inter-universal Teichmüller theory consists of a fairly explicit computation, up to certain relatively mild indetermacies, of the “arithmetic Teichmüller deformations of a number field equipped with an elliptic curve” discussed above by applying various results obtained in previous papers by the lecturer concerning local and global absolute anabelian geometry, tempered anabelian geometry, and the étale theta function. This passage from the Frobenioid-theoretic definition of the arithmetic deformations involved to a more explicit Galois-theoretic description may be thought of as a sort of global arithmetic analogue of the classical computation of the Gaussian integral (i.e., R ∞ −∞ e −x 2 dx) by means of the passage from cartesian to polar coordinates. Inequalities of interest in diophantine geometry may then be obtained as (expected) corollaries of this (expected) main theorem.
After an eight-year struggle, Shinichi Mochizuki has finally gotten his 600-page proof of the abc conjecture accepted in a peer-reviewed journal. But some experts are still unconvinced.
Photo credit: Kyoto Universityhttps://t.co/vUQw5sfQu3pic.twitter.com/0zlHJgovae
Unfortunately, it has been brought to my attention that, despite the developments discussed in §1.1, fundamental misunderstandings concerning the mathematical content of inter-universal Teichmuller theory persist in certain sectors of the mathematical community. These misunderstandings center around a certain oversimplification — which is patently flawed, i.e., leads to an immediate contradiction — of inter-universal Teichmuller theory. This oversimplified version of interuniversal Teichmuller theory is based on assertions of redundancy concerning various multiple copies of certain mathematical objects that appear in interuniversal Teichmuller theory.
下のブログ(Peter Woit氏のNot Even Wrong)では、望月氏の論文に関してはPeter Scholze氏とJacob Stix氏が欠陥を指摘しており、それに対して適切な説明が望月氏からも論文を掲載したジャーナルエディターからもなされていないので、「ABC予想は証明されていない」というのが現在の数学界の認識であると説明しています。
In case the documentary doesn’t make this clear, the current consensus of experts in the field is that there is no proof. Peter Scholze and Jacob Stix identified a problem with Mochizuki’s proof in 2018 (discussed in detail by Scholze and othershere), and Mochizuki has not provided a convincing answer to their objections. No one else (including the journal editors who published the proof in PRIMS) has been able to provide a clear explanation of theproblematic part of the proof.
Taylor Dupuy is stillmaking implausible claimsthat Scholze’s criticism of the proof is invalid. To judge for yourself, seeherea long detailed discussion of the issue between them involving several other experts.
（引用元：ABC on NHK Posted on April 9, 2022 by woit）太字強調は当サイト
ゲノム編集技術CRISPR/Cas9の開発： Jennifer A. Doudna(ジェニファー・ダウドナ)、 Emmanuel Charpentier(エマニュエル・シャルパンティエ)、Feng Zhang（フェン・チャン）、Yoshizumi Ishino（石野 良純）、George M. Church (ジョージ・チャーチ)、Virginijus Šikšnys（ヴィルジニュス・シクシュニス）
“.. Yoshiki struck me as a happy scientist. He spoke softly and with a unique smile as he described Japanese traditions or revealed his astonishing findings. .. Sasai was a master at deciphering the code by which cells learn their place in a developing embryo. ..Yoshiki Sasai (1962–2014):Stem-cell biologist who decoded signals in embryos. Arturo Alvarez-Buylla. Nature 513,34 (04 September 2014) doi:10.1038/513034a
“.. Yoshiki had an unmatched ability to decipher the embryo—specifically, to uncover how this developmental marvel generates the extraordinary diversity of cell types that become organized into unique structures, like the pituitary gland, the brain, or the eye. .. “Obituary Yoshiki Sasai (1962–2014). Arnold R. Kriegsteinemail DOI: http://dx.doi.org/10.1016/j.stem.2014.08.007 Cell Stem Cell Volume 15, Issue 3, p265–266, 4 September 2014
“.. Yoshiki had a unique ability to see things clearly while others were left wandering in the dark. .. “OBITUARY Yoshiki Sasai: stem cell Sensei
Stefano Piccolo Development (2014) 141, 1-2 doi:10.1242/dev.116509