I will discuss a problem that lies in the intersection of symplectic geometry and combinatorics. Given a compact symplectic manifold equipped with a Hamiltonian torus action, we can define a convex polytope called the moment polytope. This polytope has internal structure, and some interesting combinatorial questions include determining the number of chambers inside the moment polytope, and finding a way to distinguish the chambers. I will discuss some of the symplectic geometry and lots of the combinatorics involved in the answers to these questions. This is based in part on joint work with R. Goldin and L. Jeffrey.
