Topic: Manifolds asymptotic to partially hyperbolic tori in Hamiltonian Systems January 27

Presenter: Misha Rudnev, University of Texas, Austin

Topic: Weak type interpolation and Sobolev embedding theorems January 31

Presenter: Michael Cwikel, Technion and Princeton University

Abstract: This is joint work with Evgeniy Pustylnik. We obtain a version of the Marcinkiewicz theorem for spaces very "near" the"endpoint"spaces. One application is a new proof of the Brezis- Wainger and Hansson version of the Sobolev embedding theorem in the limiting case. We generalize it and show it is optimal.

Topic: Percolation in a dependent random environment February 1

Presenter: Yuval Peres, Hebrew University

Topic: Physics and mathematics of oscillatory integrals February 4

Presenter: Slava Rychkov, Princeton University

Abstract: We will have fun with a number of things related to short-wave asymptotics of oscillatory integrals. Probably the easiest physical interpretation of the theory is the intensity of light near caustics. The method of calculating the asymptotics is based on resolving the singularities of the phase function by pulling it back to a toric variety. The construction is governed by the Newton polyhedron of the phase function.

Topic: Vorticity in the Ginzburg-Landau model of superconductivity February 7

Presenter: Sylvia Sefraty, L'Ecole Normale Superieure de Cachan

Abstract: The Ginzburg-Landau functional $$J(u,A)=\frac{1}{2}\int_{\Omega} |\nabla_A u|^2 + |h-h_{ex}|^2 + \frac{1}{2\epsilon^2} (1-|u|^2)^2,$$ is the energy of a superconductor submitted to a magnetic field $h_{ex}$. The main feature is the apparition of vortices for certain values of the applied field. After the work of Bethuel- Brezis- Helein on a simplified energy (without magnetic field), we (partly joint work with E. Sandier) have studied this full functional in the asymptotics of small $\,\epsilon$, and developed a similar analysis for it. We have particularly focused on describing the energy-minimizing configurations, their vortices, and determining a mean-vorticity measure.

Topic: L^2 harmonic forms on some Kaehler manifolds February 14

Presenter: Jeff McNeal, Ohio State University