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Determinants, polynomials, and matroids

Cynthia Vinzant, Dept. of Mathematics, North Carolina State University
Tuesday, October 8, 2019 - 4:00pm to 5:00pm
CSE2 G04

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 beautiful real and combinatorial geometry underlying these polynomials and applications of this theory to counting and sampling problems involving combinatorial objects called matroids. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.