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「At the Fefferman conference」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 am currently at Princeton for the conference “ The power of Analysis ” honouring Charlie Fefferman ‘s 60th birthday. I myself gave a talk at this conference entitled “Recent progress on the Kakeya conjecture”; I plan to post a version of this talk on this blog shortly.

But one nice thing about attending these sorts of conferences is that one can also learn some neat mathematical facts, and I wanted to show two such small gems here; neither is particularly deep, but I found both of them cute. The first one, which I learned from my former student Soonsik Kwon , is a unified way to view the mean , median , and mode of a probability distribution on the real line. If one assumes that this is a continuous distribution for some smooth, rapidly dec

已知结果和反例

the median is the value of that minimises the first moment

and the mode is the value of that maximises the “pseudo-negative first moment”

证明或构造的主线

(Note that the Dirac delta function has the same scaling as , hence my terminology “pseudo-negative first moment”.)

The other fact, which I learned from my former classmate Diego Córdoba (and used in a joint paper of Diego with Antonio Córdoba ), is a pointwise inequality

阅读时建议盯住的点

for the fractional differentiation operators applied to a sufficiently nice real-valued function (e.g. Schwartz class will do), in any dimension and for any ; this should be compared with the product rule .

The proof is as follows. By a limiting argument we may assume that . In this case, there is a formula

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