Spaces of sections of tensor powers of the theta line bundle on moduli spaces of semistable arbitrary rank bundles on a compact Riemann surface are subject to a level-rank duality: each space of sections is geometrically isomorphic to the dual of the space of sections obtained by interchanging the tensor power (level) of the theta bundle on the moduli space and the rank of the bundles that make up the moduli space.
This corresponds in representation theory to an isomorphism of conformal blocks of representations of affine Lie algebras, when the rank of the algebra and the level of the representation are interchanged.
Dr. Marian will sketch a proof of the geometric statement, which is the result of joint work with Dragos Oprea, and draws inspiration from work by Prakash Belkale who established the isomorphism for a generic Riemann surface.

Dr. Krieger will discuss a recent result, joint with W. Schlag and D. Tataru, which establishes reguarity breakdown for wave maps with suitable initial data and target S^{2} in the energy critical dimension. The breakdown occurs via the bubbling off of a ground state harmonic map.

Many remarkable classical questions about prime numbers have natural analogues in the context of elliptic curves. In this talk I will give a brief introduction to the theory of elliptic curves, and discuss how "higher dimensional" analogues of open questions about prime numbers appear naturally in the study of the reductions of an elliptic curve. In particular, I will discuss progress made towards the resolution of variations of a Lang-Trotter conjecture from 1976.

In this talk I will review some recent developments in the area of reaction-diffusion-advection equations. I will concentrate on the phenomenon of quenching (extinction) of flames by a strong flow. These questions naturally lead to the related problem of estimating the relaxation speed for the solution of a corresponding passive scalar equation, which will also be discussed.

The main focus of my talk shall be on the well-posedness for the interface problem between a viscous fluid and an elastic solid. This is a two-phases problem, where each phase satisfies its own natural equation of evolution, and where the interaction between the two phases comes from the natural continuity of velocity field and normal stress across the unknown moving
interface. The methods known in fluid moving boundary problems (viscous or inviscid) cannot handle the apparent incompatibility between the regularity of the two phases, which has led previous authors to consider the case where the solid satisfies a simplified law where the difficulties are not present. I shall present the new methods that where required in order to allow the treatment of classical elasticity laws in this moving
interface problem.

I shall then briefly explain how some of these ideas and some new tools preserving the transport structure of the Euler equations can provide the well-posedness for the free surface Euler equations with (or without) surface tension, without any restriction on the curl of the initial velocity.