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.?

Given a set X, the notion of VC-dimension provides a way to measure randomness in collections of subsets of X. Specifically, the VC-dimension of a collection S of subsets of X is the largest integer d (if it exists) such that some d-element subset Y of X is ""shattered"" by S, meaning that every subset of Y can be obtained as the intersection of Y with some element of S. In this talk, we will focus on the case that X is a group G, and S is the collection of left translates of some fixed subset A of G.

The Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi (1982) and Leighton (1983 )states that if a graph of n vertices and e>4n edges is drawn in the plane, then the number of crossings between its edges must be at least constant times e^3/n^2. This statement, which is asymptotically tight, has found many applications in combinatorial geometry and in additive combinatorics. However, most results that were obtained using the Crossing Lemma do not appear to be optimal, and there is a quest for improved versions of the lemma for graphs satisfying certain special properties.

The study of the representations theoretic properties of the group of diffeormorphisms of locally compact non compact Riemmanian manifolds which equal to the identity outside a compact set is is linked to a natural quasi invariant action of the group which moves all points of a Poisson point process according to the diffeomorphism (Gelfand-Graev-Vershik and Goldin et al.). Neretin noticed that the local diffeomorphism group is a subgroup of a larger group which he called GMS and that GMS also acts in a similar manner on the Poisson point process.