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.

I'll begin with the Battelle Rencontre "Lectures in Mathematics and Physics" (Seattle, 1967) and end with S-duality as reflected in Mirror Symmetry and in the Electric-Magnetic Duality connection with the Geometric Langlands Program. In between will be a guided tour of special moments in the interaction of mathematicians and high energy theorists: gauge theory and fibre bundles, instantons and index theory, string theory and Calabi-Yau manifolds.

We discuss some recent developments in the problem of Khler metrics of constant scalar curvature and stability in geometric invariant theory. In particular, we discuss various notions of stability, both finite and infinite-dimensional, and various analytic methods for the problem. These include estimates for energy functionals, density of states and Tian-Yau-Zelditch and Lu asymptotic expansions, geometric heat flows, and both a priori estimates and pluripotential theory for the complex Monge-Ampre equation.