Let n be a large prime. A set A of residues modulo n is zero-sum-free if no subsetsum of A is divisible by n. Zero-sum-free sets have been studied for a long time but little was know about the following fundamental question: How many zero-sum-free sets are there ?In this talk, we shall present a sharp answer to this question, using new results about long arithmetic progressions in sumsets. In fact, we are able to characterize zero-sum-free sets: the main (and natural) reason for a set to be zero-sum-free is that the sum of its elements is less than n. (joint work with E. Szemeredi)

We describe two mass-like quantities arising from the Green's function for the Laplacian operator on surfaces. The Robin's mass is obtained by regularizing the logarithmic singularity of the Green's function. We show that the Robin's mass is connected to a spectral invariant. On spheres, we introduce a "geometrical mass", which is, a priori, a smooth function on the sphere. The goemetrical mass is shown to be independent of the point on the sphere, and it is also a spectral invariant. Moreover, a connection to a Sobolev-type inequality reveals that it is minimized at the standard round metric. The definition of the geometrical mass is inspired by the roles played by the Green's function for the conformal Laplacian and the Positive Mass Theorem in the solution to the Yamabe Problem.

A family of polynomials ${P_n}$ such that $P_n$ has degree n is a basis for the polynomial ring. A product $P_{n_1}$ $P_{n_2}$ ... $P_{n_k}$ can be expanded in this basis, and the coefficients in this expansion are called linearization coefficients. If the basis consists of orthogonal polynomials, these coefficients are generalizations of the moments of the measure of orthogonality. Just like moments, these coefficients have combinatorial significance for many classical families. For instance, for the Hermite polynomials they are the numbers of inhomogeneous matchings. I will describe the linearization coefficients for a number of classical families. The proofs are based on the relation between the polynomials and certain stochastic processes. They involve the machinery of combinatorial stochastic measures, introduced by Rota and Wallstrom. The number of examples treated by this method is increased significantly by using non-commutative stochastic processes, consisting of operators on a q-deformed full Fock space.