Many polynomials in combinatorics (and in other areas of mathematics) have nice properties such as having all of their roots being real numbers, or having all of their coefficients being nonnegative.  By surveying recent advances in the Hodge theory of matroids (namely, the nonnegativity of Kazhdan-Lusztig polynomials of matroids and Dowling and Wilson’s top-heavy conjecture for the lattice of flats of a matroid), I will give several examples of well-behaved polynomials, and I will indicate some connections of these properties to geometry and representation theory.  The talk should be understandable to everyone, and should appeal to those with interests in at least one of the following topics: hyperplane arrangements, matroids, log-concavity, real-rooted polynomials, lattice theory, Coxeter groups, and the representation theory of Lie algebras.  It will contain results that are joint work with Tom Braden, Luis Ferroni, June Huh, Nicholas Proudfoot, Matthew Stevens, Lorenzo Vecchi, and Botong Wang.