Seminar - Cynthia Vinzant

Determinants, Hyperbolicity and Interlacing

Abstract: 

Writing a multivariate polynomial as the determinant of a matrix of linear forms is a classical problem in algebraic geometry and
complexity theory. Requiring that this matrix is Hermitian and positive definite at some point puts topological and algebraic restrictions on the polynomials that appear as the determinant and its minors. In particular the real zero sets of these polynomials are hyperbolic (or real stable) and interlace. I'll talk about the beautiful geometry behind these determinants
and its connection to semidefinite programming in optimization and matroid theory in combinatorics.