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「“Math Doesn’t Suck”, and the Chayes-McKellar-Winn theorem」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

As you may already know, Danica McKellar , the actress and UCLA mathematics alumnus, has recently launched her book “ Math Doesn’t Suck “, which is aimed at pre-teenage girls and is a friendly introduction to middle-school mathematics, such as the arithmetic of fractions. The book has received quite a bit of publicity , most of it rather favourable, and is selling quite well; at one point, it even made the Amazon top 20 bestseller list , which is a remarkable achievement for

I’m very happy that the book is successful for a number of reasons. Firstly, I got to know Danica for a few months (she took my Introduction to Topology class way back in 1997, and in fact was the second-best student there; the class web page has long since disappeared, but you can at least see the midterm and final ), and it is always very heartening to see a former student put her or his mathematical knowledge to good use 🙂 . Secondly, Danica is a wonderful role model and i

已知结果和反例

While I am not exactly in the target audience for this book, I can relate to its pedagogical approach. When I was a kid myself, one of my favourite maths books was a very obscure (and now completely out of print) book called “ Creating Calculus “, which introduced the basics of single-variable calculus via concocting a number of slightly silly and rather contrived stories which always involved one or more ants. For instance, to illustrate the concept of a derivative, in one o

Anyway, Danica’s book has already been reviewed in several places , and there’s not much more I can add to what has been said elsewhere. 一个常见想法是 however that I could talk about another of Danica’s contributions to mathematics, namely her paper “ Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on ” (PDF available here ), joint with Brandy Winn and my colleague Lincoln Chayes . (Brandy, incidentally, was the only student in my topology class who

证明或构造的主线

[ Update , Aug 23: I added a non-technical “executive summary” of what the Chayes-McKellar-Winn theorem is at the very end of this post.]

To begin the story, I would like to quickly review the theory of statistical mechanics . This is the theory which bridges the gap between the microscopic (particle physics) description of many-particle systems, and the macroscopic ( thermodynamic ) description, giving a semi-rigorous explanation of the empirical laws of the latter in terms of the fundamental laws of the former.

阅读时建议盯住的点

Statistical mechanics is a remarkably general theory for describing many-particle systems – for instance it treats classical and quantum systems in almost exactly the same way! But to simplify things I will just discuss a toy model of the microscopic dynamics of a many-particle system S – namely a finite Markov chain model . In this model, time is discrete, though the interval between discrete times should be thought of as extremely short. The state space is also discrete; at

If the graph of microstates was connected (i.e. one can get from any microstate to any other by some path along the graph), then after a sufficiently long period of time, the probability distribution of the microstates will converge towards normalised counting measure, as can be seen by basic Markov chain theory. However, if the system S is isolated (i.e. not interacting with the outside world), conservation laws intervene to disconnect the graph. In particular, if each micro

值得单独记下的条目

• In the four-state Potts model , we have the same four magnetisation states, but now the energy of a bond between two particles is -1 if they are in the same state and 0 otherwise.

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