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「Mikhail Gromov wins 2009 Abel prize」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

The 2009 Abel prize has been awarded to Mikhail Gromov , for his contributions to numerous areas of geometry, including Riemannian geometry, symplectic geometry, and geometric group theory.

The prize is, of course, richly deserved. I have mentioned some of Gromov’s work here on this blog, including the Bishop-Gromov inequality in Riemannian geometry (which (together with its parabolic counterpart, the monotonicity of Perelman reduced volume) plays an important role in Perelman’s proof of the Poincaré conjecture), the concept of Gromov-Hausdorff convergence (a version of which is also key in the proof of the Poincaré conjecture), and Gromov’s celebrated theorem o

已知结果和反例

Another well-known result of Gromov that I am quite fond of is his nonsqueezing theorem in symplectic geometry (or Hamiltonian mechanics ). In its original form, the theorem states that a ball of radius R in a symplectic vector space (with the usual symplectic structure ) cannot be mapped by a symplectomorphism into any cylinder which is narrower than the ball (i.e. ). This result, which was one of the foundational results in the modern theory of symplectic invariants, is som

I can sketch Gromov’s original proof of the non-squeezing theorem here. The symplectic space can be identified with the complex space , and in particular gives an almost complex structure J on the ball (roughly speaking, J allows one to multiply tangent vectors v by complex numbers, and in particular Jv can be viewed as v multiplied by the unit imaginary i). This almost complex structure J is compatible with the symplectic form ; in particular J is tamed by , which basically

证明或构造的主线

Now suppose for contradiction that there is a symplectic embedding from the ball to a smaller cylinder. Then we can push forward the almost complex structure J on the ball to give an almost complex structure on the image . This new structure is still tamed by the symplectic form on this image.

Just as complex structures can be used to define holomorphic functions, almost complex structures can be used to define pseudo-holomorphic or J-holomorphic curves. These are curves of one complex dimension (i.e. two real dimensions, that is to say a surface) which obey the analogue of the Cauchy-Riemann equations in the almost complex setting (i.e. the tangent space of the curve is preserved by J). The theory of such curves was pioneered by Gromov in the paper where the nonsq

阅读时建议盯住的点

Now, the point lies in the cylinder and in particular lies in a disk of symplectic area spanning this cylinder. This disk will not be pseudo-holomorphic in general, but it turns out that it can be deformed to obtain a pseudo-holomorphic disk spanning passing through of symplectic area at most . Pulling this back by , we obtain a minimal surface spanning passing through the origin that has surface area at most . However, any minimal surface spanning and passing through the ori

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