Ehrhart positivity and matroids

A fundamental invariant associated to a lattice polytope is its Ehrhart polynomial, which encodes the number of lattice points inside all the integer dilations of the polytope and much more arithmetic, algebraic and combinatorial information. I will discuss some results on Ehrhart polynomials with positive coefficients, with emphasizing the case of matroid polytopes, for which we will prove several Ehrhart positivity results, and outline a way of constructing a non-Ehrhart positive example.