Hodge Theory of Matroids

Introduction Logarithmic concavity is a property of a sequence of real numbers, occurring throughout algebraic geometry, convex geometry, and combinatorics. A sequence of positive numbers a0,... ,ad is log-concave if a2 i ≥ ai−1ai+1 for all i. This means that the logarithms, log(ai), form a concave sequence. The condition implies unimodality of the sequence (ai), a property easier to visualize: the sequence is unimodal if there is an index i such that a0 ≤ ⋯ ≤ ai−1 ≤ ai ≥ ai+1 ≥ ⋯ ≥ ad. We will discuss our work on establishing log-concavity of various combinatorial sequences, such as the coefficients of the chromatic polynomial of graphs and the face numbers of matroid complexes. Our method is motivated by complex algebraic geometry, in particular Hodge theory. From a given combinatorial object M (a matroid), we construct a graded commutative algebra over the real numbers