陶哲軒點評望月新一、張益唐和佩雷爾曼

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引 言

證明或者沒有證明，這是一個問題！向來直言的陶哲軒尖銳點評望月新一，張益唐，和佩雷爾曼！

辭舊迎新之際，國際數學界最引人注目的話題，莫過於日本數學家望月新一關於ABC猜想的證明。據稱，這一論文已經通過同行評議，即將發表於日本京都大學數理解析研究所出版的Publications of Research Institute for Mathematical Sciences 期刊上。然而該論文漫長且艱深晦澀，而望月又深居簡出，令數學家們議論紛紛。這不，芝加哥大學數論專家Frank Calegari教授近日就發表題為《The ABC conjecture has (still) not been proved》的博客，認為ABC猜想尚未得到證明，並對望月的態度提出批評，引起圍觀。而吃瓜群眾，就包括天才華人數學家陶哲軒教授和哈佛大學Benedict Gross教授。

陶哲軒

與通常出世的數學家不同，陶哲軒一向直言，並不掩飾自己的觀點，包括爭議極大的政治觀點。2016年美國大選如火如荼之際，陶哲軒就發表川普的新衣與藍眼睛謎團的博客，直言特拉普沒有勝任美國總統的資格和能力，他所批著的皇帝新衣路人兼知，只是需要人大膽地說出來。不過陶的斷言並未能影響美國大選，特拉普輕鬆當選，而這也沒有阻止陶哲軒繼續發表尖銳的觀點。這不，圍繞望月ABC猜想證明的討論，陶哲軒又在Calegari的博客下發表評論，將望月的論文與張益唐和佩雷爾曼的工作進行比較。知社為您翻譯並摘錄如下，僅供參考。

張益唐

陶哲軒：

我沒有足夠的知識對望月的論文做專業的評價，但對您所提到的張益唐和佩雷爾曼的工作非常熟悉。它們之間一個顯著的區別在於張益唐和佩雷爾曼在論文的開始就給出簡潔的「概念驗證」，而他們所發展的方法也能很快地用於相應領域，得到一些有意思的非平凡新結果，或是給出一些已有的非平凡結論的新證明。望月的論文缺乏這樣的「概念驗證」。在佩雷爾曼的論文中，第5頁就已經給出了Ricci流的全新解釋：它將Ricci流視為梯度流，看起來非常有潛力。在第7頁，他就用該解釋建立了一個關於Ricci流的精彩定理。雖然這個定理相距最終證明龐加萊猜想甚遠，但它本身就是一個新奇而有趣的結果，使得這個領域的專家迅速認定這篇論文有很多「好東西」。

張益唐的54頁論文沿襲了解析數論的傳統，將所要用到的引理放在論文的開頭，因此有不少對專家而言是標準性的內容。但是這些引理陳列之後的第6頁，張益唐就給出了一個非平凡的觀察：只要能改進Bombieri-Vinogradov定理對光滑模的估計，就能證明素數間距離有限。這並非這篇論文最深刻的部分，但是它將原問題簡化為更容易處理的問題。與此相反，無數試圖攻克像黎曼猜想這樣大問題的論文不斷將原有問題複雜化，直到奇蹟發生，而這樣的奇蹟通常只是一個錯誤。

從我了解的信息來看，望月工作的「概念驗證」就要300多頁，這樣才能證明ABC猜想。在我看來，如果能有一個更簡短的，比如少於100頁的 「概念驗證」，就有可能幫助人們消除對這一證明的疑問。如果需要300多頁來建立一個全新的獨立體系，而這個體系只能用來證明ABC猜想，卻沒有任何其他的外在意義，這將是一件非常奇怪的事情。

Terrence Tao:

「 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. 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 straight forward 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 suchas 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.<100pages) would help dispel skepticism 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.」

佩雷爾曼

而哈佛大學Benedict Gross教授的批評則更為直接，望月需要給大家講授他的想法和論證，而不是貼一篇300多頁的文稿就完事：

「Terry’s comment (from the outside of number theory) is particularly telling. For those of us inside of it, the situation is infuriating. Shortly after Faltings announced his proof of Tate’s isogeny conjecture and the Mordell conjecture, he lectured on it at the Arbeitstagung, explaining the new tools he had introduced. Everyone in the audience who had thought about the problem was immediately convinced. Instead of producing 300+ pages of manuscript, Mochizuki needs to give one or two lectures (in Bonn, or Paris, or Boston, or…) clearly explaining the new ideas in his argument and showing how they lead to a proof of ABC. This shouldn’t be difficult — I have no idea why he refuses to do so.」

ABC 猜想究竟何去何從？我們拭目以待！

擴展閱讀

望月新一: 小李飛刀重現數學江湖

陶哲軒: 川普的新衣和藍眼睛謎團

張益唐：我的數學人生

陶哲軒: 數學少年的奇幻之旅

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