Expanders are highly-connected sparse graphs widely used in computer science. The optimal expanders (Ramanujan graphs) were constructed in 1988 by Margulis, Lubotzky, Phillips and Sarnak using deep results from the theory of automorphic forms. In recent joint work with Bourgain and Sarnak tools from additive combinatorics were used to prove that a wide class of "congruence graphs" are expanders; this expansion property plays a crucial role in establishing novel sieving results.

The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of sheaves over a Riemann surface or a higher dimensional base. These moduli spaces also carry theta divisors, described as zero loci of "generalized" theta functions. I will discuss recent progress in the study of generalized theta functions. In particular, I will emphasize an unexpected geometric duality between spaces of generalized theta functions, as well as its geometric consequences for the study of the moduli spaces of sheaves.

Even though the regularity problem for the 3D Navier-Stokes equations is far from been solved, numerous regularity criteria have been proved since the work of Leray. We will discuss some classical results as well as their extensions in Besov spaces.

The geometric theory of Banach spaces underwent a tremendous development in the decade 1990-2000 with the solution of several outstanding conjectures by Gowers, Maurey, Odell and Schlumprecht.

Their discoveries both hinted at a previously unknown richness of the class of separable Banach spaces and also laid the beginnings of a classification program for separable Banach spaces due to Gowers.

However, since the initial steps done by Gowers, little progress was made on the classification program. We shall discuss some recent advances due to V. Ferenczi and myself on this by means of Ramsey theory and dichotomy theorems for the structure of Banach spaces. This simultaneously allows us to answer some related questions of Gowers concerning the quasiorder of subspaces of a Banach space under the relation of isomorphic embeddability.

It is well known that the square root of any integer can be written as a linear combination of roots of unity. A generalization of this fact is the "Kronecker-Weber Theorem", which states that in fact any element which generates an abelian Galois extension of the field of rational numbers Q can also be written as such a linear combination. The roots of unity may by viewed as the special values of the analytic function e(x) = exp(2*pi*i*x) where x is taken to be a rational number. Broadly speaking, Hilbert's 12th problem is to find an analogous result when Q is replaced by a general algebraic number field F, and in particular to find the analytic functions which play the role of e(x) in this general setting.

Hilbert's 12th problem has been solved in the case where F is an imaginary quadratic field, with the role of e(x) being played by certain modular forms. All other cases are, generally speaking, unresolved. In this talk I will discuss the case where F is a real quadratic field, and more generally, a totally real field. I will describe relevant conjectures of Stark and Gross, as well as current work using a p-adic approach and methods of Shintani. A proof of these conjectures would arguably provide a positive resolution of Hilbert's 12th problem in these cases.