陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「An integration approach to the Toeplitz square peg problem」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

I’ve just uploaded to the arXiv my paper “ An integration approach to the Toeplitz square peg problem “, submitted to Forum of Mathematics, Sigma . This paper resulted from my attempts recently to solve the Toeplitz square peg problem (also known as the inscribed square problem ):

Conjecture 1 (Toeplitz square peg problem) Let be a simple closed curve in the plane. Is it necessarily the case that contains four vertices of a square?

已知结果和反例

See this recent survey of Matschke in the Notices of the AMS for the latest results on this problem.

The route I took to the results in this paper was somewhat convoluted. I was motivated to look at this problem after lecturing recently on the Jordan curve theorem in my class. The problem is superficially similar to the Jordan curve theorem in that the result is known (and rather easy to prove) if is sufficiently regular (e.g. if it is a polygonal path), but seems to be significantly more difficult when the curve is merely assumed to be continuous. Roughly speaking, all the

证明或构造的主线

Inspired by my previous work on finite time blowup for various PDEs, I first tried looking for a counterexample in the category of (locally) self-similar curves that are smooth (or piecewise linear) away from a single origin where it can oscillate infinitely often; this is basically the smoothest type of curve that was not already covered by previous results. By a rescaling and compactness argument, it is not difficult to see that such a counterexample would exist if there wa

Conjecture 2 (Periodic square peg problem) Let be two disjoint simple closed piecewise linear curves in the cylinder which have a winding number of one, that is to say they are homologous to the loop from to . Then the union of and contains the four vertices of a square.

阅读时建议盯住的点

In contrast to Conjecture 1 , which is known for polygonal paths, Conjecture 2 is still open even under the hypothesis of polygonal paths; the homological arguments alluded to previously now show that the number of inscribed squares in the periodic setting is even rather than odd , which is not enough to conclude the conjecture. (This flipping of parity from odd to even due to an infinite amount of oscillation is reminiscent of the “ Eilenberg-Mazur swindle “, discussed in th

I therefore tried to construct counterexamples to Conjecture 2 . I began perturbatively, looking at curves that were small perturbations of constant functions. After some initial Taylor expansion, I was blocked from forming such a counterexample because an inspection of the leading Taylor coefficients required one to construct a continuous periodic function of mean zero that never vanished, which of course was impossible by the intermediate value theorem. I kept expanding to

值得单独记下的条目

• (i) Conjecture 1 holds when is the union of the graphs of two Lipschitz functions of Lipschitz constant less than one that agree at the endpoints.
• (ii) Conjecture 2 holds when are graphs of Lipschitz functions of Lipschitz constant less than one.
• (i) For any , the sums are non-zero.
• (ii) (Non-crossing) For any and with the same parity, the pairs and are non-crossing in the sense that
• (iii) (Non-crossing sums) For any , , of the same parity, one has

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源