The Ehrhart Theory of Matroid Polytopes

In this thesis we investigate the Ehrhart polynomials of the basis polytope and the independence polytope of a matroid. We disprove conjectures by De Loera, Haws and Koppe (2007) and by Castillo and Liu (2015) asserting the positivity of the coefficients of these polynomials. Through the way we prove several positive results such as the Ehrhart positivity of hypersimplices and matroids of rank 2. Also, we provide an explicit formula for the Ehrhart polynomial of the basis polytope and the independence polytope of sparse paving matroids, a class that is conjecturally predominant.