The Primitive Equations are a fundamental model describing large scale oceanic and atmospheric processes. They are derived from the fully compressible Navier-Stokes equations on a combined basis of scale analysis and meteorological data. While an extensive body of mathematical literature exists in the study of these systems, very little is known in the stochastic setting. In this talk we discuss recent joint work with M. Ziane concerning existence and uniqueness of solutions for the 2-D equations in the presence of multiplicative noise terms.

In this talk, I review the mathematical results of the
dynamcis of Bose-Einstein condensate (BEC) and present
some efficient and stable numerical methods to compute ground states
and dynamics of BEC. As preparatory steps,
we take the 3D Gross-Pitaevskii equation (GPE) with an angular
momentum rotation, scale it to obtain a four-parameter model
and show how to reduce it to 2D GPE in certain limiting regimes.
Then we study numerically and asymptotically the
ground states, excited states and quantized vortex
states as well as their energy and chemical potential diagram
in rotating BEC. Some very interesting numerical results are
observed. Finally, we study numerically stability and interaction
of quantized vortices in rotating BEC. Some interesting interaction
patterns will be reported.

Abstract: (Thanks to work of Abel Klein and others) it is understood how
to represent the quantum Ising model in terms of a certain classical model
of stochastic geometry called the `continuum random-cluster model'. In the
regime of large external field, this geometrical model is subcritical. By
developing bounds for its `ratio weak' mixing rate, one obtains estimates
involving the reduced density matrix of the quantum Ising model. The
implications of these estimates for entanglement do not appear to be best
possible, but they are at least robust for disordered systems. [Joint work
with Tobias Osborne and Petra Scudo.]

The T-adic L-function is a unversal L-function which
interpolates classical L-functions of all p-power order
exponential sums associated to a polynomial f(x) defined
over a finite field. We study its T-adic analytic properties
(analytic continuation and its T-adic Newton polygon).
The T-adic Newton polygon provides a uniform improvement
to previous results on p-adic Newton polygon of exponential sums
in the non-ordinary case. This is joint work with Chunlei Liu.