Log-concavity is an important feature of many functions and discrete sequences appearing across mathematics, including combinatorics, algebraic geometry, convex analysis, and optimization. In this talk I will describe recent developments in our understanding of discrete and functional log-concavity and a connection to combinatorial objects called matroids. At the heart of this connection is a classification of multivariate log-concave polynomials, which takes inspiration from the breakthroughs of Adiprasito, Huh and Katz in combinatorial Hodge theory. These polynomials give discrete probability distributions that can be approximately sampled efficiently using Markov chains. I will discuss the discrete and convex geometry of log-concave polynomials and consequences for the expansion of the basis-exchange graph of a matroid and approximate sampling. This is based on joint works with Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Thuy Duong Vuong. Speaker(s): Cynthia Vinzant (University of Washington)