The Ehrhart polynomial of a polytope with integral vertices counts the integer points in the polytope as it gets dilated by an integer factor. One reason for being interested in this counting function is that its leading term gives the volume of the polytope. We will present a way of computing Ehrhart polynomials for polytopes using complex-analytic methods.
En route to their application to the Birkhoff polytope, we will introduce ideas and concepts from discrete geometry, number theory, and combinatorics. The application of our methods to the Birkhoff polytope provides an example showing that pure mathematics and computationally efficient algorithms are not mutually exclusive.
This is joint work with Dennis Pixton (Binghamton University, SUNY).