Tutte seminar - Kevin Purbhoo

Eigenvalues of Hermitian matrices and the Belkale-Kumar product

Abstract:

The story of this talk begins with the very classical Hermitian sum problem, which asks: if the eigenvalues of two Hermitian matrices are known, what can we say about the possible eigenvalues of their sum? 

The answer is, nowadays, quite a bit (though it took over a hundred years for us to get there). We know that the possible lists of eigenvalues form a convex polyhedal cone. The integer points in this cone have an interpretation in representation theory. The facets of this cone are described via the geometry of Schubert varieties, which in turn produces a combinatorial description. 
Recently, Belkale and Kumar introduced a ring structure, which they essentially defined by taking a complicated ring (the cohomology ring of a generalized flag variety), and changing the product so as to ignore the complicated parts. Amazingly, this simplification led to a number breakthroughts in generalizing the Hermitian sum picture, including descriptions of facets for related problems, and information about higher codimension faces. 
My goal in this talk is to give an overview of the mathematics surrounding the Hermitian sum problem, and to discuss, in particular, the combinatorics of the Belkale-Kumar product. No knowledge of representation theory, Schubert varieties, or cohomology will be assumed. This is joint work (in progress) with Allen Knutson.