High level people は自分達で勝手に立てたスレ28へどうぞ!sage進行推奨(^^; また、スレ43は、私が立てたスレではないので、私は行きません。そこでは、私はスレ主では無くなりますからね。このスレに不満な人は、そちらへ。 http://rio2016.2ch.net/test/read.cgi/math/1506152332/ 旧スレが512KBオーバー(又は間近)で、新スレ立てる (スレ主の趣味で上記以外にも脱線しています。ネタにスレ主も理解できていないページのURLも貼ります。関連のアーカイブの役も期待して。)
いやはや、(文系) High level people たち( ID:jEMrGWmk さん含め)の、数学ディベートもどきは面白いですね(^^; ”手強い?”とは・・、まさに、ディベートですね
私ら、理系の出典(URL)とコピペベース、ロジック(論証)&証明重視のスタンスと、ディベートもどきスタイル(2CHスタイル?)とは、明白に違いますね 私ら、(文系) High level people たちとの議論は、時間とスペースの無駄。レベルが高すぎてついていけませんね。典拠もなしによく議論しますね。よく分かりましたよ(^^;
4 Answers Update: The lecture notes of the CAGA lecture series on perfectoid spaces at the IHES can now be found online, cf. http://www.ihes.fr/~abbes/CAGA/scholze.html.
It seems that it's my job to answer this question, so let me just briefly explain everything. A more detailed account will be online soon.
We start with a complete non-archimedean field K of mixed characteristic (0,p) (i.e., K has characteristic 0, but its residue field has characteristic p), equipped with a non-discrete valuation of rank 1, such that (and this is the crucial condition) Frobenius is surjective on K+/p, where K+⊂K is the subring of elements of norm ?1.
https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ Quanta magazine Titans of Mathematics Clash Over Epic Proof of ABC Conjecture Erica Klarreich September 20, 2018 (抜粋) In a report posted online today, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a “serious, unfixable gap” within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. (引用終り)
http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf Why abc is still a conjecture PETER SCHOLZE AND JAKOB STIX Date: August 23, 2018. (抜粋) In March 2018, the authors spent a week in Kyoto at RIMS of intense and constructive discussions with Prof. Mochizuki and Prof. Hoshi about the suggested proof of the abc conjecture. We thank our hosts for their hospitality and generosity which made this week very special. We, the authors of this note, came to the conclusion that there is no proof. We are going to explain where, in our opinion, the suggested proof has a problem, a problem so severe that in our opinion small modifications will not rescue the proof strategy. We supplement our report by mentioning dissenting views from Prof. Mochizuki and Prof. Hoshi about the issues we raise with the proof and whether it constitutes a gap at all, cf. the report by Mochizuki. (引用終り) 0043現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/23(日) 09:45:33.79ID:A1h249rM>>42 補足
1.”http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf Why abc is still a conjecture PETER SCHOLZE AND JAKOB STIX Date: August 23, 2018.” が、「In a report posted online today」と書かれているが、 URLは、http://www.kurims.kyoto-u.ac.jp/~motizuki/なので、望月のサイトにアップされたんだが 望月のホームページには、この投稿に触れた文は掲載されていない 2.Erica Klarreichが、September 20に、”posted online today”と書けたのは、だれかからのたれ込み、おそらくは二人の筆者の一人からか、それを聞きつけた数学関係者から連絡を貰ったのだろう 3.望月がおそらく、この論文を掲載したと連絡したのだろうね 4.この論文は、Date: August 23, 2018.だから、September 20に”posted online today”と書くことは、偶然この論文を見つけただけでは断言できないから 5.不思議なのは、望月側がこの論文について、いまだ何も語っていないこと。なんか言えよという感じが個人的にはするがね
http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html March 2018 Discussions on IUTeich (抜粋) In March 2018, discussions concerning inter-universal Teichmuller theory (IUTeich) were held at RIMS, Kyoto University. Participation in these discussions was restricted to four mathematicians. This web page was created, with the consent of the four participants in these March 2018 discussions, in order to make various files related to these discussions available to the mathematical public. It is hoped that the material posted on this web page will help to stimulate further constructive mathematical discussions concerning this material. In this context, we wish to emphasize that the material posted on the present web page is by no means in final form and will be subject to further updates as discussions progress.
The fundamental misunderstandings concerning IUTeich documented in these files --- as well as the situation surrounding such misunderstandings that has arisen in the mathematical community --- are most regrettable, not only for those directly involved in research and dissemination activities concerning IUTeich, but also for the mathematical community as a whole. On the other hand, at the time of writing, it appears that the only way in which meaningful progress in remedying this situation can be made is to further efforts to render the mathematical content that is the subject of these misunderstandings more explicit and more easily accessible, through further mathematical discussions and more detailed manuscripts. From this point of view, the files [SS2018-05], [SS2018-08] are a significant first step, but are still relatively short and do not contain detailed, rigorous arguments concerning numerous (often very strong) assertions. Moreover, it does not necessarily appear realistic to expect that further substantial efforts of the sort just described will be made by the authors of these files [SS2018-05], [SS2018-08] in the immediate future. In particular, further constructive mathematical involvement on the part of mathematicians who did not participate in the March 2018 discussions has the potential to yield substantial, meaningful progress in remedying this situation.
つづく 0056現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/24(月) 11:08:36.40ID:b5cCvILu>>55 つづき Throughout the six years (2012-2018) since the release of the four preprints on IUTeich, I, together with a number of colleagues, have addressed literally hundreds (perhaps thousands) of technical questions from quite a number of mathematicians concerning IUTeich via e-mail, skype, and in face-to-face discussions. Many of these colleagues have written, or are in the process of writing, detailed expositions of IUTeich. As is discussed in the final section of [Rpt2018], the contents of the files posted on the present web page have been discussed thoroughly with quite a number of these colleagues. Finally, we recall that, during these six years, RIMS, Kyoto University, has contributed quite substantial financial, administrative, and infrastractural resources to hosting two large-scale workshops on IUTeich, as well as quite a number of visitors to RIMS, for visits devoted (at least partially) to serious mathematical discussions concerning IUTeich.
Any professional mathematician (or graduate student) interested in engaging in serious, constructive mathematical discussions concerning this material --- especially, any such mathematician who feels that he/she has a solid mathematical understanding of the mathematical assertions made in the files [SS2018-05], [SS2018-08], and, moreover, is interested in engaging in constructive mathematical discussions concerning these assertions (where we note that it is by no means clear, at the time of writing, that the set of such mathematicians is nonempty!) --- is encouraged to contact Shinichi Mochizuki at the e-mail address given on the top page of this web site.
http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf [Rpt2018] Report by Shinichi Mochizuki (with the cooperation of Yuichiro Hoshi) on the March 2018 discussions (updated on 2018-09-21) つづく 0057現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/24(月) 11:09:22.80ID:b5cCvILu>>56 つづき
http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-05.pdf [SS2018-05] May 2018 Report by the other participants in the March 2018 discussions
2) [SS2018-08] August 2018 Report by the other participants in the March 2018 discussions ↓ http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf [SS2018-08] August 2018 Report by the other participants in the March 2018 discussions 0059現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/24(月) 11:21:41.16ID:b5cCvILu>>58 (追加情報) https://twitter.com/math_jin ?@math_jin ツィッター 9月22日
http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf [Rpt2018] Report by Shinichi Mochizuki (with the cooperation of Yuichiro Hoshi) on the March 2018 discussions (updated on 2018-09-21) (抜粋) From an organizational point of view, the discussions took the form of “negotations” between two “teams”: one team (HM), consisting of Hoshi and Mochizuki, played the role of explaining various aspects of IUTch; the other team (SS), consisting of Scholze and Stix, played the role of challenging various aspects of the explanations of HM. Most of the sessions were conducted in the following format: Mochizuki would stand and explain various aspects of IUTch, often supplementing oral explanations by writing on whiteboards using markers in various colors; the other participants remained seated, for the most part, but would, at times, make questions or comments or briefly stand to write on the whiteboards.
§2. Scholze has, for some time, taken a somewhat negative position concerning IUTch, and his position, and indeed the position of SS, remained negative even after the March discussions. My own conclusion, and indeed the conclusion of HM, after engaging in the March discussions, is as follows:
The negative position of SS is a consequence of certain fundamental misunderstandings (to be explained in more detail in the remainder of the present report ? cf. §17 for a brief summary) on the part of SS concerning IUTch, and, in particular, does not imply the existence of any flaws whatsoever in IUTch. The essential gist of these misunderstandings ? many of which center around erroneous attempts to “simplify” IUTeich ? may be summarized very roughly as follows:
(Smm) Suppose that A and B are positive real numbers, which are defined so as to satisfy the relation ?2B = ?A (which corresponds to the Θ-link). One then proves a theorem ?2B <= ?2A + 1 (which corresponds to the multiradial representation of [IUTchIII], Theorem 3.11). This theorem, together with the above defining relation, implies a bound on A ?A <= ?2A + 1, i.e., A <= 1 (which corresponds to [IUTchIII], Corollary 3.12). From the point of view of this (very rough!) summary of IUTch, the misunderstandings of SS amount to the assertion that the theory remains essentially unaffected even if one takes A = B, which implies (in light of the above defining relation) that A = B = 0, in contradiction to the initial assumption that A and B are positive real numbers. In fact, however, the essential content (i.e., main results) of IUTch fail(s) to hold under the assumption “A = B”; moreover, the “contradiction” A = B = 0 is nothing more than a superficial consequence of the extraneous assumption “A = B” and, in particular, does not imply the existence of any flaws whatsoever in IUTch. (We refer to (SSIdEx), (ModEll), (HstMod) below for a “slightly less rough” explanation of the essential logical structure of an issue that is closely related to the extraneous assumption “A = B” in terms of ・ complex structures on real vector spaces or, alternatively (and essentially equivalently), in terms of the well-known classical theory of ・ moduli of complex elliptic curves. Additional comparisons with well-known classical topics such as ・ the invariance of heights of elliptic curves over number fields with respect to isogeny, ・ Grothendieck’s definition of the notion of a connection, and ・ the differential geometry surrounding SL2(R) may be found in §16.)
Indeed, in the present context, it is perhaps useful to recall the following well-known generalities concerning logical reasoning: (GLR1) Given any mathematical argument, it is always easy to derive a contradiction by arbitrarily identifying mathematical objects that must be regarded as distinct in the situation discussed in the argument. On the other hand, this does not, by any means, imply the existence of any logical flaws in the original mathematical argument! (GLR2) Put another way, the correct interpretation of the contradiction obtained in (GLR1) is ? not the conclusion that the original argument, in which the arbitrary identifications of (GLR1) were not in force, has logical flaws (!), but rather ? the conclusion that the contradiction obtained in (GLR1) implies that the distinct mathematical objects that were arbitrarily identified are indeed distinct, i.e., must be treated (in order, for instance, to arrive at an accurate understanding of the original argument!) as distinct mathematical objects! It is most unfortunate indeed that the March discussions were insufficient from the point of view of overcoming these misunderstandings. On the other hand, my own experience over the past six years with regard to exposing IUTch to other mathematicians is that this sort of short period (roughly a week) is never sufficient,
つづく 0063現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/24(月) 13:14:03.75ID:b5cCvILu>>62 つづき i.e., that substantial progress in understanding IUTch always requires discussions over an extended period of time, typically on the order of months. Indeed, the issue of lack of time became especially conspicuous during the afternoon of the final day of discussions. Typically, short periods of interaction center around reacting in real time and do not leave participants the time to reflect deeply on various aspects of the mathematics under discussion. This sort of deep reflection, which is absolutely necessary to achieve fundamental progress in understanding, can only occur in situations where the participants are afforded the opportunity to think at their leisure and forget about any time or deadline factors. (In this context, it is perhaps of interest to note that Scholze contacted me in May 2015 by e-mail concerning a question he had regarding the non-commutativity of the logtheta- lattice in IUTch (i.e., in effect, “(Ind3)”). This contact resulted in a short series of e-mail exchanges in May 2015, in which I addressed his (somewhat vaguely worded) question as best I could, but this did not satisfy him at the time. On the other hand, the March 2018 discussions centered around quite different issues, such as (Ind1, 2), as will be described in detail below.)
§3. On the other hand, it seems that the March discussions may in fact be regarded as constituting substantial progress in the following sense. Prior to the March discussions, (at least to my knowledge) negative positions concerning IUTch were always discussed in highly nonmathematical terms, i.e., by focusing on various aspects of the situation that were quite far removed from any sort of detailed, well-defined, mathematically substantive content. (引用終り) 0064現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/24(月) 13:24:34.76ID:b5cCvILu>>63
(私的な要約) ・ between two “teams”: one team (HM), consisting of Hoshi and Mochizuki 、team (SS) consisting of Scholze and Stix で討議した ・ (SS) は、IUTchに否定的 ・”hese misunderstandings ? many of which center around erroneous attempts to “simplify” IUTeich ? may be summarized very roughly as follows:(略)”
https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ Persiflage Galois Representations and more! The ABC conjecture has (still) not been proved Posted on December 17, 2017 (抜粋) 34 Responses to The ABC conjecture has (still) not been proved Terence Tao says: December 18, 2017 at 2:46 pm Thanks for this. I do not have the expertise to have an informed first-hand opinion on Mochizuki’s work, but on comparing this story with the work of Perelman and Yitang Zhang you mentioned that I am much more familiar with, one striking difference to me has been the presence of short “proof of concept” statements in the latter but not in the former, by which I mean ways in which the methods in the papers in question can be used relatively quickly to obtain new non-trivial results of interest (or even a new proof of an existing non-trivial result) in an existing field. In the case of Perelman’s work, already by the fifth page of the first paper Perelman had a novel interpretation of Ricci flow as a gradient flow which looked very promising, and by the seventh page he had used this interpretation to establish a “no breathers” theorem for the Ricci flow that, while being far short of what was needed to finish off the Poincare conjecture, was already a new and interesting result, and I think was one of the reasons why experts in the field were immediately convinced that there was lots of good stuff in these papers. つづく 0070現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/27(木) 06:23:32.93ID:xNwcF5GI>>69 つづき Yitang Zhang’s 54 page paper spends more time on material that is standard to the experts (in particular following the tradition common in analytic number theory to put all the routine lemmas needed later in the paper in a rather lengthy but straightforward early section), but about six pages after all the lemmas are presented, Yitang has made a non-trivial observation, which is that bounded gaps between primes would follow if one could make any improvement to the Bombieri-Vinogradov theorem for smooth moduli. (This particular observation was also previously made independently by Motohashi and Pintz, though not quite in a form that was amenable to Yitang’s arguments in the remaining 30 pages of the paper.) This is not the deepest part of Yitang’s paper, but it definitely reduces the problem to a more tractable-looking one, in contrast to the countless papers attacking some major problem such as the Riemann hypothesis in which one keeps on transforming the problem to one that becomes more and more difficult looking, until a miracle (i.e. error) occurs to dramatically simplify the problem.
From what I have read and heard, I gather that currently, the shortest “proof of concept” of a non-trivial result in an existing (i.e. non-IUTT) field in Mochizuki’s work is the 300+ page argument needed to establish the abc conjecture. It seems to me that having a shorter proof of concept (e.g. <100 pages) would help dispel scepticism about the argument. It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever. (引用終り) (参考 Yitang Zhang’s 54 page paperは、ガセか? そも、この論文に日付無し) http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.308.998&rep=rep1&type=pdf Bounded gaps between primes - CiteSeerX Yitang Zhang 0071現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/27(木) 07:13:38.66ID:xNwcF5GI>>70 >参考 Yitang Zhang’s 54 page paperは、ガセか?
なるほど、 http://annals.math.princeton.edu/2014/179-3/p07 Bounded gaps between primes Pages 1121-1174 from Volume 179 (2014), Issue 3 by Yitang Zhang Abstract It is proved that lim?inf n→∞(pn+1?pn)<7×1^07, where pn is the n-th prime.
Definition for sequences The limit inferior of a sequence (xn) is defined by lim inf_n→∞ x_n:= lim _n→∞ (inf m >x_m ) or lim inf _n→∞x_n:= sup _n>= 0 inf m>= n x_m = sup{inf{x_m : m>= n } : n>= 0 }
https://mathtrain.jp/supmax 高校数学の美しい物語 〜定期試験から数学オリンピックまで800記事〜 sup(上限)とinfの意味,maxとの違い 最終更新:2016/05/18 要素が実数である集合 A に対して maxA:A の最大値,maximum(英語),マックス(読み方の例) minA:A の最小値,minimum,ミン supA:A の上限,supremum,スープ infA:A の下限,infimum,インフ 大学の解析のしょっぱなで学ぶ sup の意味について解説します。 min は max の反対側,inf は sup の反対側なので,ここでは max,sup についてのみ解説します。 集合の上限 sup の定義です。 supA=c ←→ ・任意の x∈A に対して x?c かつ (c は A の上界) ・ c より小さい任意の実数 r に対して,r<x なる x∈A が存在する (少しでも小さくすると上界でなくなる) 日本語で言うと「上界の最小値」です。 0088現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/09/29(土) 18:34:19.44ID:f3v/fPif>>86 「時枝がガセだと思う理由」は、散々書いたので過去ログをどうぞ で、私の主張は、数学セミナーに書く程度の内容は、専門の論文かテキスト(教科書)で裏付けられるべき