The intersection cohomology of a matroid

In 1974, Dowling and Wilson conjectured that for a matroid of rank d, the number of flats of rank k is at most the number of flats of rank d-k for k ≤ d/2. For matroids which are realized by a configuration of vectors, the conjecture was proved by Huh and Wang using the intersection cohomology of an algebraic variety associated to the configuration. The general conjecture was then proved in joint work with Huh, Matherne, Proudfoot and Wang, by constructing an "intersection cohomology" module IH(M) for an arbitrary matroid which satisfies the properties needed for Huh and Wang's argument. The construction of IH(M) was complicated and difficult to work with, and it left open the question of giving a simpler characterization that would describe it uniquely. In this talk, after giving some background on matroids and the Dowling-Wilson conjecture, I will give such a characterization.