Some of the remarkable phenomena that emerge when a trapped, ultracold Bose gas is set in rapid rotational motion will be reviewed. In anharmonic traps, where the rotational velocity can in principle be arbitrarily large, one can distinguish three critical velocities at which the flow pattern changes radically. The first is the velocity at which vorticity sets in, eventually leading to a lattice of vortices, at the second a 'hole' is created and the condensate becomes concentrated in an annulus while the vortex lattice persists in the bulk, and at the third a transition to a 'giant vortex' state takes place in which all vorticity disappears from the bulk but a macroscopic circulation around the hole remains.

The mathematical model used for analysis of these phenomena has similarities with Ginzburg-Landau (GL) Theory in superconductivity with the critical velocities in rotating gases playing an analogous role to the critical magnetic fields in GL theory, and techniques originally developed in the context of GL theory have, indeed, been important for understanding rotating gases. There are, however, also important differences, and in particular the theory of the giant vortex transition requires significant modifications of the GL setting. Theses similarities and differences will be discussed. (Joint work with Michele Correggi and Nicoals Rougerie.)

We will describe how the "special value theory" in function field arithmetic is an interesting mixture of very strong theorems determining all algebraic relations in some cases, emerging partial conjectural pictures in some cases, and quite wild phenomena often.

The addition of quenched disorder has a rounding effect on 1st order phase trans ition in systems of sufficiently low dimension (d=2, and up to � 4 in case of co ntinuous symmetry). The talk will focus on the recent extension to quantum pha se transition of a result that was previously proven for classical systems, and on a currently studied question concerning the nature of the disorder-dominated state in d=2 dimensions. (Work done in different collaborations.)

Percolation is perhaps the simplest non-trivial model in statistical mechanics, but has remained under active study for more than forty years. In 2-D it exhibits a second-order phase transition, at which a number of interesting and little-understood symmetries manifest themselves. We discuss three of these: (a) the horizontal crossing probability, which reveals a triangular symmetry, (b) an exact factorization of certain correlation functions, and (c) a generalization of this factorization that shows a mysterious independence of one coordinate. We demonstrate (c) via the explicit calculation of a certain six-point correlation function. Both (b) and (c) generalize to a variety of other two-dimensional critical points. The main tool employed is conformal field theory.

In 1995, G. Courtois, S. Gallot and myself constructed a family of maps with very good properties regarding volume elements between certain manifolds. We used it to give an alternative proof of Mostow's rigidity for rank one closed symmetric spaces as well as a rigidity result for their geodesic flow, conjectured by A. Katok. Various modifications of the original construction have been made since yielding new results in different settings. We shall describe the basic construction, the modifications, some applications and open questions.

Review of rigorous results on thermostats. Families of exact formal solutions of the BBGKY hierarchy for hard sphere systems with free boundary conditions at collisions and Fourier equation emergence, to first order in the temperature difference, after boundary conditions are imposed. Formal means that the solutions are given by series with well defined terms but whose convergence is not discussed.