Algebraic Geometry Seminar: "Tropical fans, mixed volumes, and log-concavity"

Abstract: Log-concave sequence are ubiquitous in algebra, geometry, and combinatorics. My favorite type of log-concave sequence is the sequence of mixed volumes of two convex bodies. On the other hand, the most famous type of log-concave sequence is the sequence of coefficients of the characteristic polynomial of a matroid. In the latter setting, log-concavity was conjecture almost half a century ago, but was verified only very recently by Adiprasito, Huh, and Katz, who proved it as a consequence of their “combinatorial Hodge theory.” In this talk, I’ll discuss recent and ongoing work with Anastasia Nathanson, Lauren Nowak, and Patrick O’Melveny that leads to a new mixed-volume-theoretic proof of log-concavity of characteristic polynomials of matroids.

Host: Ben Wormleighton