Andy Berget

Western Washington University
, Online - Zoom

Abstract

Log-concavity of matroid \(h\)-vectors

A sequence of numbers \((h_0,h_1,\dots,h_m)\) is said to be log-concave if for all \(1 < i < m\), 
\[h_{i-1}h_{i+1} \leq h_i^2.\]

Many sequences of integers in algebra, combinatorics and geometry are known to be log-concave. The presence of this property is often a shadow of deep underlaying results, such as the Alexander-Fenchel inequality for mixed volumes of convex bodies, or the Hodge index theorem. Proving log-concavity of a given (interesting) sequence is often difficult. In this talk I will describe joint work with Hunter Spink (Stanford) and Dennis Tseng (MIT) on our improvement, and resolution of, of Dawson's 1984 conjecture on the log-concavity of matroid \(h\)-vectors.  In its original form, this conjecture was resolved concurrently by Federico Ardila, Graham Dehnam and June Huh (who give their own strengthening in a different direction). I will give an account of this interesting story that is accessible to the non-expert. The actors range from elementary linear algebra and graph theory, to the deep connections Huh and his collaborators have unearthed between matroids, convex geometry and Hodge theory.