An introduction to Horn's problem

报告题目:An introduction to Horn's problem

报告人:Thierry Levy 教授 (巴黎六大)



摘要: Horn's problem is the following : given the eigenvalues of two Hermitian matrices of the same size, determine the possible eigenvalues of their sum. Except in trivial cases (that is, when one of the matrices is scalar), there are infinitely many possible eigenvalues for the sum, depending on the relative position of the eigenspaces of the matrices.

It took almost a century to bring a complete answer to this apparently elementary problem, which along the way turned out to be connected with many areas of mathematics : linear algebra of course, but also combinatorics, algebraic geometry, symplectic geometry, and even probability theory.

I will describe the problem and give an overview of the structure of its solution, which involves in a rather complicated way contributions of many mathematicians including (but this list could easily and rightfully be made much longer) Hermann Weyl, Ky Fan, Alfred Horn, Alexander Klyachko, Allen Knutson and Terence Tao.


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