Cynthia Vinzant: Determinants, hyperbolicity, and log-concavity

Abstract:  The determinant of a symmetric matrix is a fundamental object in mathematics, whose discrete and functional properties have applications across the scientific disciplines. The determinant of a matrix is also a real polynomial in its entries.  Hyperbolic polynomials and, more generally, log-concave polynomials are real polynomials that share many useful functional properties of determinants. Like real-rooted univariate polynomials, they also have interesting combinatorial applications.  I will discuss the real, topological, and algebraic geometry underlying these polynomials and applications of this theory in operator theory, optimization, and combinatorics.