Complete log-concavity is a functional property of real multivariate polynomials that translates to strong and useful conditions on its coefficients. I will introduce the class of completely log-concave polynomials in elementary terms, discuss the beautiful real and combinatorial geometry underlying these polynomials, and describe applications to random walks on simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. 

Cynthia Vinzant is an assistant professor in the department of mathematics at North Carolina State University. Her research involves real algebraic geometry, combinatorics, and convexity with applications in optimization and theoretical computer science.