# Seminars & Events for Mathematical Physics Seminar

##### A variational model for crystals with defects

This talk will be devoted to the reduced Hartree-Fock 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).

##### 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.

##### The spectral edge of random band matrices

We consider random periodic $N\times N$ 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.

##### 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.

##### Disconnection and Random Interlacements

The general theme of the talk pertains to the question of understanding how paths of random walks can create large separating interfaces. This question has in particular been investigated in the context of random walk on a discrete cylinder with a large connected base, and when discussing the presence or absence of a giant component in the complement of the trajectory of a random walk on a large discrete torus. We will present an overview of some of the results, and paradigms, which have now emerged and explain how the above problems are related to questions of ercolation and to the model of random interlacements.

##### Superconcentration

We introduce the term 'superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a 'multiple valley picture' emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology.

##### On the Boltzmann limit of a homogeneous Fermi gas

##### On the formation of black holes

I will discuss some recent results obtained in collaboration with I. Rodnianski on the dynamic formation of black holes for the Vacuum Einstein equations. These results simplify and extend considerably the recent well known result of D. Christodoulou.

##### Quasi-adiabatic continuation and the Topology of Many-body Quantum Systems

Topological arguments play a key role in understanding quantum systems. For example, recently it has been shown that K-theory provides a tool for classifying different phases of non-interacting, or single-particle, systems. However, topological arguments have also been applied to interacting systems. I will explain the technique of quasi-adiabatic continuation, which provides a way to rigourously formulate many of the topological arguments made by physicists for these systems. In particular, I will discuss its application to a higher dimensional Lieb-Schultz-Mattis theorem (a statement about degeneracy of ground states, which can arise for topological reasons), where this technique was introduced in 2004, and its more recent application to proving quantum Hall conductance quantization for interacting systems.