"Ehrhart Polynomials, h*-vectors and triangulations of matroid polytopes"

David Haws
University of Kentucky

Monday, October 19, 2009
4:00 pm, 845 Patterson Office Tower

Abstract:

Matroids naturally encapsulate the combinatorial notion of independence. They are found in matrices, graphs, transversals, point configurations, hyperplane arrangements, greedy optimization, and pseudosphere arrangements to name a few. One of the reasons matroids have become fundamental objects in pure and applied combinatorics are their many equivalent axiomatizations. In this talk I will define matroids and matroid polytopes and present a new result that the Ehrhart polynomial of matroid polytopes can be computed in polynomial time when the rank is fixed. Second, I will discuss two conjectures about the h*-vector and coefficients of Ehrhart polynomials of matroid polytopes and provide theoretical and computational evidence for their validity. Time permitting I will also present a variant of White's conjecture which states that every matroid polytope has a regular unimodular triangulation. I will show computational evidence supporting this new conjecture and propose a combinatorial condition on simplices sufficient for unimodularity.