Abstract
Complete log-concavity is a generalization of hyperbolicity for multivariate polynomials that is closed related to notions of discrete convexity, including matroids, submodular functions, and generalized permutohedra. Many of the nice properties of hyperbolic polynomials extend to this more general class. While inspired by work of Adiprasito, Huh, and Katz on combinatorial Hodge theory, these polynomials can be defined and understood in elementary terms. I will introduce this class of polynomials, the underlying real algebraic geometry, and some important examples coming from combinatorics and convex optimization. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.