MATHEMATICAL PHYSICS SEMINAR
FALL 2009 Lectures
Regular meeting time:
Tuesdays 4:30--5:30
Place: Jadwin 343
| Date | Speaker | Title |
| Nov. 10 | Mathieu Lewin, Université de Cergy-Pontoise |
A variational model for crystals with defects The main idea is to describe at the same time the electrons bound by the defect and the (nonlinear) behavior of the infinite crystal. This leads to a bounded-below nonlinear functional whose variable is however an operator of infinite-rank. I will provide the correct functional setting for this functional, state the existence of global-in-time solutions to the associated time-dependent Schrödinger equation, and discuss the existence, the properties and the stability of bound states. In particular I will define the dielectric permittivity of the perfect crystal and relate this to some properties of ground states. This is a review of joint works with Eric Cancès and Amélie Deleurence (Ecoledes Ponts, Paris). |
| Nov. 17 | Jakob Yngvason, University of Vienna |
The emergence of a giant vortex in a fast rotating Bose gas. A Bose gas in fast rotation normally exhibits a growing number of vortices of unit strength if the angular velocity is increased. In an anharmonic trap at sufficiently high velocity, however, a phase transition is expected: Vortices in the bulk should disappear and all vorticity become concentrated in a region where the density is very low. Moreover, the critical velocity for the transition is expected to increase with on the interaction strength in a definite manner. In the lecture rigorous results on this behavior within two-dimensional Gross-Pitaevskii theory will be presented. This is joint work with Michele Correggi and Nicolas Rougerie. |
| Nov. 24 | Alexander Sodin, Tel Aviv Univ. |
The spectral edge of random band matrices We consider random periodic N X N band matrices of band width W. If the band is wide (W >> N^{5/6}), the spectral statistics at the edge behave similarly to those of GUE matrices; in particular, the largest eigenvalue converges in distribution to the Tracy -- Widom law. Otherwise, a different limit appears. The results are consistent with the Thouless criterion for localization, adapted to the band matrix setting by Fyodorov and Mirlin. |
| Dec. 1 | Vieri Mastropietro, Univ. Rome II |
Universal Relations for Non Solvable Statistical Models We present the first rigorous derivation of a number of universal relations for a class of models with continuously varying indices (among which are interacting quantum spin chains, 1D Fermi systems and interacting planar Ising models), for which an exact solution is not known, except in a few special cases. These formulas were conjectured by Luther and Peschel, Kadanoff, Haldane, but only checked in the special solvable models. The proof is based on the combination of exact Renormalization Group methods with Ward Identities based on approximate local gauge invariance; lattice effects are crucial and are rigorously taken into account. Applications of such methods to an effective model of graphene with long range interactions will be also described. Talk based on joint works with G.Benfatto and P. Falco. |
| Dec. 8 | H-T Yau, Harvard University |
Universality of Random Matrices and Dyson Brownian Motion The universality for eigenvalue spacing distributions is a central question in the random matrix theory. In this talk, we introduce a new general approach based on comparing the Dyson Brownian motion with a new related dynamics, the local relaxation flow. This method can be applied to prove the universality for the eigenvalue spacing distributions for the symmetric, hermitian, self-dual quaternion matrices and the real and complex Wishart matrices. A central tool in this approach is to estimate the entropy flow via the logarithmic Sobolev inequality. |