Given a convex polytope P, what is the number of integer points in P? This problem is of great interest in combinatorics and discrete geometry, with many important applications ranging from integer programming to statistics. From a computational point of view it is hopeless in any dimensions, as the knapsack problem is a special case. Perhaps surprisingly, in bounded dimension the problem becomes tractable. How far can one go? Can one count points in projections of P, finite intersections of such projections, etc?
We will survey both classical and recent results on the problem, emphasizing both algorithmic and complexity aspects. Some elegant hardness results will make an appearance in dimension as small as three. If time permits, we will discuss connections to Presburger Arithmetic and decidability problems for irrational polyhedra.
Joint work with Danny Nguyen.