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「Simons Lecture I: Structure and randomness in Fourier analysis and number theory」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

This week I am in Boston, giving this year’s Simons lectures at MIT together with David Donoho . (These lectures, incidentally, are endowed by Jim Simons , who was mentioned in some earlier discussion here .) While preparing these lectures, it occurred to me that I may as well post my lecture notes on this blog, since this medium is essentially just an asynchronous version of a traditional lecture series, and the hypertext capability is in some ways more convenient and inform

I am giving three lectures, each expounding on some aspects of the theme “the dichotomy between structure and randomness”, which I also spoke about (and wrote about ) for the ICM last August. This theme seems to pervade many of the areas of mathematics that I work in, and my lectures aim to explore how this theme manifests itself in several of these. In this, the first lecture, I describe the dichotomy as it appears in Fourier analysis and in number theory. (In the second , I

已知结果和反例

The “dichotomy between structure and randomness” seems to apply in circumstances in which one is considering a “high-dimensional” class of objects (e.g. sets of integers, functions on a space, dynamical systems, graphs, solutions to PDE, etc.). For sake of concreteness, let us focus today on sets of integers (later lectures will focus on other classes of objects). There are many different types of objects in these classes, however one can broadly classify them into three cate

A recurring question in many areas of analysis is the following: given a specific object (such as the prime numbers), can one determine precisely what the structured components are within the object, and how pseudorandom the remaining components of the object are? One reason for asking this question is that it often helps one compute various statistics (averages, sums, integrals, correlations, norms, etc.) of the object being studied. For instance, one can ask for how many tw

证明或构造的主线

The problem of determining exactly what the structured and pseudorandom components are of any given object is still largely intractable. However, what we have learnt in many cases is that we can at least show that an arbitrary object can be decomposed into some structured component and some pseudorandom component. Also there is often an orthogonality property (or dichotomy ): if an object is orthogonal (or has small correlation with) all structured objects, then it is necessa

To illustrate these general principles, let us focus now on a specific area in analytic number theory, namely that of finding additive patterns in the prime numbers {2, 3, 5, 7, …}. Despite centuries of progress on these problems, many questions are still unsolved, for instance:

阅读时建议盯住的点

On the other hand, we do have some positive results:

As a general rule, it appears that it is feasible (after non-trivial effort) to find patterns in the primes involving two or more degrees of freedom (as described by the parameters n, n’ in above examples), but we still do not have the proper technology for finding patterns in the primes involving only one degree of freedom n. (This is of course an oversimplification; for instance, the pattern n, n+2, n’, n’+2 has two degrees of freedom, but finding infinitely many of these p

值得单独记下的条目

• Twin prime conjecture : There are infinitely many positive integers n such that n, n+2 are both prime.
• Sophie Germain prime conjecture : There are infinitely many positive integers n such that n, 2n+1 are both prime.
• Even Goldbach conjecture : For every even number , there is a natural number n such that n, N-n are both prime.
• Vinogradov’s theorem : For every sufficiently large odd number N, there are positive integers n, n’ such that n, n’, N-n-n’ are all prime. (The best explicit bound currently known for “sufficiently large” is ; the result has also been verif
• van der Corput’s theorem : There are infinitely many positive integers n, n’ such that n, n+n’, n+2n’ are all prime.
• Green-Tao theorem : For any positive integer k, there are infinitely many positive integers n, n’ such that n, n+n’, …, n+(k-1)n’ are all prime.
• A polynomial generalisation : For any integer-valued polynomials with , there are infinitely many positive integers n, n’ such that are all prime.
• Question : does there there exist infinitely many r-tuples of positive integers such that are simulatenously prime?

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