陶哲轩点评望月新一、张益唐和佩雷尔曼

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证明或者没有证明，这是一个问题！向来直言的陶哲轩尖锐点评望月新一，张益唐，和佩雷尔曼！

辞旧迎新之际，国际数学界最引人注目的话题，莫过于日本数学家望月新一关于ABC猜想的证明。据称，这一论文已经通过同行评议，即将发表于日本京都大学数理解析研究所出版的Publications of Research Institute for Mathematical Sciences 期刊上。然而该论文漫长且艰深晦涩，而望月又深居简出，令数学家们议论纷纷。这不，芝加哥大学数论专家Frank Calegari教授近日就发表题为《The ABC conjecture has (still) not been proved》的博客，认为ABC猜想尚未得到证明，并对望月的态度提出批评，引起围观。而吃瓜群众，就包括天才华人数学家陶哲轩教授和哈佛大学Benedict Gross教授。

陶哲轩

与通常出世的数学家不同，陶哲轩一向直言，并不掩饰自己的观点，包括争议极大的政治观点。2016年美国大选如火如荼之际，陶哲轩就发表川普的新衣与蓝眼睛谜团的博客，直言特拉普没有胜任美国总统的资格和能力，他所批着的皇帝新衣路人兼知，只是需要人大胆地说出来。不过陶的断言并未能影响美国大选，特拉普轻松当选，而这也没有阻止陶哲轩继续发表尖锐的观点。这不，围绕望月ABC猜想证明的讨论，陶哲轩又在Calegari的博客下发表评论，将望月的论文与张益唐和佩雷尔曼的工作进行比较。知社为您翻译并摘录如下，仅供参考。

张益唐

陶哲轩：

我没有足够的知识对望月的论文做专业的评价，但对您所提到的张益唐和佩雷尔曼的工作非常熟悉。它们之间一个显著的区别在于张益唐和佩雷尔曼在论文的开始就给出简洁的“概念验证”，而他们所发展的方法也能很快地用于相应领域，得到一些有意思的非平凡新结果，或是给出一些已有的非平凡结论的新证明。望月的论文缺乏这样的“概念验证”。
在佩雷尔曼的论文中，第5页就已经给出了Ricci流的全新解释：它将Ricci流视为梯度流，看起来非常有潜力。在第7页，他就用该解释建立了一个关于Ricci流的精彩定理。虽然这个定理相距最终证明庞加莱猜想甚远，但它本身就是一个新奇而有趣的结果，使得这个领域的专家迅速认定这篇论文有很多“好东西”。

张益唐的54页论文沿袭了解析数论的传统，将所要用到的引理放在论文的开头，因此有不少对专家而言是标准性的内容。但是这些引理陈列之后的第6页，张益唐就给出了一个非平凡的观察：只要能改进Bombieri-Vinogradov定理对光滑模的估计，就能证明素数间距离有限。这并非这篇论文最深刻的部分，但是它将原问题简化为更容易处理的问题。与此相反，无数试图攻克像黎曼猜想这样大问题的论文不断将原有问题复杂化，直到奇迹发生，而这样的奇迹通常只是一个错误。

从我了解的信息来看，望月工作的“概念验证”就要300多页，这样才能证明ABC猜想。在我看来，如果能有一个更简短的，比如少于100页的 “概念验证”，就有可能帮助人们消除对这一证明的疑问。如果需要300多页来建立一个全新的独立体系，而这个体系只能用来证明ABC猜想，却没有任何其他的外在意义，这将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.”

佩雷尔曼

而哈佛大学Benedict Gross教授的批评则更为直接，望月需要给大家讲授他的想法和论证，而不是贴一篇300多页的文稿就完事：

“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 猜想究竟何去何从？我们拭目以待！

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