Luis Ferroni: Polytopes, Ehrhart theory, matroids, and beyond...

Luis Ferroni (IAS)

A recurring idea in combinatorics is to associate with a given combinatorial object (say, a graph, finite poset, matroid, etc.) a convex polytope. Often, the geometric properties of the convex polytope provide insights into the behavior of the object it is derived from. For example, starting from a finite poset P, one can associate the order polytope of P, whose normalized volume equals the number of linear extensions of P.

In this talk, we will focus on matroids and explain the natural polytopes associated with them. Rather than concentrating solely on the volume of these polytopes, we will study a more general invariant called the Ehrhart polynomial. While this polynomial is exceedingly subtle in the general setting of convex polytopes, for the special case of matroid polytopes, certain desirable properties have been established, while others (long conjectured to hold true) have been recently disproved.

We will explore some of the techniques used to tackle these problems on the Ehrhart polynomials of matroids. We will also highlight further generalizations that lead to a systematic method for constructing potential counterexamples to other conjectures in matroid theory.