Alan Stapledon (MSRI)
Weighted Ehrhart theory and orbifold cohomology
Motivated by geometry, we consider a `less discrete' way of counting lattice points in polytopes, in which one assigns a certain `weight' to each lattice point. On the combinatorial side, this approach reveals some `hidden symmetry' which improves upon and makes transparent some classical results in Ehrhart theory. On the geometric side, the combinatorial invariants count orbifold betti numbers of toric stacks. If time permits, we will discuss a generalization involving motivic integration. .

Alex Küronya (BME, Budapest)
Okounkov bodies on low-dimensional varieties
We take a closer look at a construction of Okounkov and Lazarsfeld- Mustata that attaches a convex body to a divisor on a smooth projective variety over the complex numbers. After having recalled the basic properties of Okounkov bodies, we will mainly focus on studying varieties of low dimension. We will show that on a smooth surface, Okounkov bodies are finite polygons, and give higher dimensional examples that show that even Fano varieties have ample divisors with non-polyhedral Okounkov bodies. This is an account of joint work with Victor Lozovanu (University of Michigan), and Catriona Maclean (Institute Fourier, Université de Grenoble).

Jim Bryan (UBC)
Motivic degree zero Donaldson-Thomas invariants
The Hilbert scheme X[n] of n points on variety X parameterizes length n, zero dimensional subschemes of X. When X is a smooth surface, X[n] is also smooth and a beautiful formula for its motive was determined by Gottsche. When X is a threefold, X[n] is in general, singular, of the wrong dimension, and reducible. However if X is smooth Calabi-Yau threefold, X[n] has a canonical virtual motive --- a motivication of the degree zero Donaldson-Thomas invariants. We give a formula analogous to Gottsche's for the virtual motive of X[n]. The key computation gives a refinement of the classical formula of MacMahon which counts 3D partitions.