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「Israel Gelfand」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

Israel Gelfand , who made profound and prolific contributions to many areas of mathematics, including functional analysis, representation theory, operator algebras, and partial differential equations, died on Monday , age 96.

Gelfand’s beautiful theory of -algebras and related spaces made quite an impact on me as a graduate student in Princeton, to the point where I was seriously considering working in this area; but there was not much activity in operator algebras at the time there, and I ended up working in harmonic analysis under Eli Stein instead. (Though I am currently involved in another operator algebras project, of which I hope to be able to discuss in the near future. The commutative vers

已知结果和反例

I met Gelfand only once, in one of the famous “Gelfand seminars” at the IHES in 2000. The speaker was Tim Gowers, on his new proof of Szemerédi’s theorem. (Endre Szemerédi, incidentally, was Gelfand’s student.) Gelfand’s introduction to the seminar, on the subject of Banach spaces which both mathematicians contributed so greatly to, was approximately as long as Gowers’ talk itself!

There are far too many contributions to mathematics by Gelfand to name here, so I will only mention two. The first are the Gelfand-Tsetlin patterns , induced by an Hermitian matrix . Such matrices have real eigenvalues . If one takes the top minor, this is another Hermitian matrix, whose eigenvalues intersperse the eigenvalues of the original matrix: for every . This interspersing can be easily seen from the minimax characterisation

证明或构造的主线

of the eigenvalues of , with the eigenvalues of the minor being similar but with now restricted to a subspace of rather than .

Similarly, the eigenvalues of the top minor of intersperse those of the previous minor. Repeating this procedure one eventually gets a pyramid of real numbers of height and width , with the numbers in each row interspersing the ones in the row below. Such a pattern is known as a Gelfand-Tsetlin pattern . The space of such patterns forms a convex cone, and (if one fixes the initial eigenvalues ) becomes a compact convex polytope. If one fixes the initial eigenvalues of but cho

阅读时建议盯住的点

The structure, symplectic geometry, and representation theory of Gelfand-Tsetlin patterns was enormously influential in my own work with Allen Knutson on honeycomb patterns, which control the sums of Hermitian matrices and also the structure constants of the tensor product operation for representations of ; indeed, Gelfand-Tsetlin patterns arise as the degenerate limit of honeycombs in three different ways, and we in fact discovered honeycombs by trying to glue three Gelfand-

The second contribution of Gelfand I want to discuss is the Gelfand-Levitan-Marchenko equation for solving the one-dimensional inverse scattering problem: given the scattering data of an unknown potential function , recover . This is already interesting in and of itself, but is also instrumental in solving integrable systems such as the Korteweg-de Vries equation, because the Lax pair formulation of such equations implies that they can be linearised (and solved explicitly) by

值得单独记下的条目

• Given the scattering data , one can recover .
• Given , one can recover .

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。

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