Alexander Barvinok (Michigan), Christian Haase (Berlin)
I plan to discuss efficient algorithms for counting integer points in polyhedra. We will discuss both the case of (relatively) low dimensions where the exact counting is feasible and the case of higher dimensions, where asymptotic and approximate methods take over.
Adding two points with integer coordinates (aka lattice points) from a convex polytope P, one obtains a lattice point in the second dilate 2P of P. Conversely, take a lattice point in 2P, can we write it as a sum of two lattice points in P? I will discuss this innocent looking question, its applications and relatives starting in dimension two.