| Ehrhart theory concerns the enumeration of lattice points in a lattice polytope. More specifically, to any lattice polytope P, one can associate a polynomial deltaP(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 sonic 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 deltaP( t) as sums of dimensions of orbifold cohomology groups of XP . |
