The Geometry and Combinatorics of Ehrhart

Ehrhart theory concerns the enumeration of lattice points in a lattice polytope. More specically, to any lattice polytope P, one can associate a polynomial $delta_P(t)$, called the Ehrhart $delta$-polynomial of P, which encodes the number of lattice points in all dilations of P. In this thesis, we introduce a refined version of Ehrhart theory, called weighted Ehrhart theory. In particular, we consider polynomials $delta^{lambda}(t)$, which record the number of lattice points with a certain `weight', depending on a parameter $lambda$, in dilations of P. On the one hand, these polynomials are interesting from a combinatorial viewpoint and lead to natural generalizations of some classical results in Ehrhart theory. On the other hand, we show that these polynomials have a geometric interpretation as motivic integrals on a toric stack XP associated to P. As an application, we obtain an interpretation of the coefficients of $delta_P(t)$ as sums of dimensions of orbifold cohomology groups of $X_P$.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/63786/1/astapldn_1.pd