Abstract
Stability is a multivariate generalization for real-rootedness in univariate polynomials. Within the past ten years, the theory of stable polynomials has contributed to breakthroughs in combinatorics, convex optimization, and operator theory. I will introduce a generalization of stability, called complete log-concavity, that satisfies many of the same desirable properties. These polynomials were inspired by recent work of Adiprasito, Huh, and Katz on combinatorial Hodge theory, but can be defined and understood in elementary terms. I will discuss the beautiful geometry underlying these polynomials and discuss some applications to approximate counting problems in matroid theory. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.