T*******x
发帖数: 8565 | 1 【 以下文字转载自 Military 讨论区 】
发信人: Chromo (CHROMO), 信区: Military
标 题: 陶哲轩对望月新一的证明很不爽
发信站: BBS 未名空间站 (Thu Dec 28 14:26:50 2017, 美东)
陶哲轩：我没有足够的专业知识去对望月新一的论文做一个直接的评价，但是对于你文
章中提到张益唐及佩雷尔曼的工作却更加熟悉。它们之间一个显著的区别在于张益唐与
佩雷尔曼的工作里有着较短的“方法验证”，即用他们的方法能很快得到相应领域里一
些很有意思的不平凡的结论（或是发展出一些已有不平凡结论的新证明）。这些事情发
生在论文发表之后，而不是之前。
在佩雷尔曼的工作中，第五页就已经给出了Ricci流的一个全新解释：它将Ricci流看成
了梯度流，而这是一个看起来非常有前途的方法。在第七页，他就用该解释建立了一个
关于Ricci流的非常精彩的定理。虽然这个定理离最后证明庞加莱猜想还有千里之遥，
但是它本身就是一个新奇且有趣的结果。这也是为什么这个领域的专家迅速认定这篇文
章中有很多“好东西”。
在张益唐54页的论文中，有很多对专家来说是标准性的内容（特别地，这篇文章沿袭了
解析数论界的传统，将所有要用到的引理放在了文章的开头）。但是仅仅6页的引理陈
列之后，张益唐就从中做出了不平凡的观测：只要能改进Bombieri-Vinogradov定理对
光滑模的估计，我们就能证明素数间距离有限。（其实这个观测也被Motohashi和Pintz
独立地导出，但其形式却无法被得心应手地运用在张益唐后面30多页的证明中）。这并
不是张益唐论文中最深奥的部分，但是它却将原问题化为了一个看起来更容易处理的问
题。与无数试图攻克像黎曼猜想等大问题的论文相比，它没有将原问题不断转化为看起
来更复杂的问题（只有奇迹发生，才能将这些复杂的问题转化回一个简单的问题。）
从我了解的信息来看，对于望月新一所用的方法，最短的“方法验证”就是300多页ABC
猜想的证明。如果能有一个更短的 “方法验证”（比如少于100页）就可以帮助人们消
除对于这篇论文的怀疑。 如果说一个300多页的独立体系只能用来证明ABC猜想，且这
个体系里不能衍生出任何证明其它结论的子方法，这将是一件非常邪门的事情。 | T*******x
发帖数: 8565 | 2 我觉得陶哲轩的批评有一定道理。
【在 T*******x 的大作中提到】
: 【 以下文字转载自 Military 讨论区 】
: 发信人: Chromo (CHROMO), 信区: Military
: 标 题: 陶哲轩对望月新一的证明很不爽
: 发信站: BBS 未名空间站 (Thu Dec 28 14:26:50 2017, 美东)
: 陶哲轩：我没有足够的专业知识去对望月新一的论文做一个直接的评价，但是对于你文
: 章中提到张益唐及佩雷尔曼的工作却更加熟悉。它们之间一个显著的区别在于张益唐与
: 佩雷尔曼的工作里有着较短的“方法验证”，即用他们的方法能很快得到相应领域里一
: 些很有意思的不平凡的结论（或是发展出一些已有不平凡结论的新证明）。这些事情发
: 生在论文发表之后，而不是之前。
:
| A***e
发帖数: 130 | 3 Tao's original post https://galoisrepresentations.wordpress.com/2017/12/17/
the-abc-conjecture-has-still-not-been-proved/):
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. 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.
【在 T*******x 的大作中提到】
: 我觉得陶哲轩的批评有一定道理。
| d********m
发帖数: 3662 | |
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