## Laszlo Erdös (Ludwig-Maximillians Universität Müchen)

Cette conférence s'adresse à un large auditoire / Suitable for a general audience

### Lundi 19 mars 2012, 16h00 / Monday, March 19, 2012, 4:00 pm

Salle / Room 1360

Centre de recherches mathématiques

Pavillon André-Aisenstadt

Université de Montréal

2920, chemin de la Tour

**Universality of Spectral Statistics for Random Matrices**

Random matrices appear in many different branches of mathematics, with applications ranging from statistics, combinatorics and communication networks to quantum mechanics and even number theory. Despite many facets of random matrices, their spectral statistics exhibit a remarkably universal behavior. The celebrated Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics depend only on the symmetry class of the matrix and is independent of the detailed structure of the matrix ensemble. We have recently proved this conjecture by embedding the random matrix into a stochastic flow of matrices and analyzing the relaxation mechanism of the coupled stochastic differential equation for the eigenvalues. This approach has revealed the intrinsic underlying mechanism behind matrix universality. The talk will be an overview of these developments for non-experts, summarizing our recent joint works with P. Bourgade, A. Knowles, B. Schlein, H.-T. Yau and J. Yin.

*Une réception suivra la conférence au Salon Maurice-L'Abbé, Pavillon André-Aisenstadt (Salle 6245).*

*A reception will follow at the Salon Maurice-L'Abbé, Pavillon André-Aisenstadt (Room 6245).*

### Mardi 20 mars 2012, 15h00 / Tuesday, March 20, 2012, 3:00 pm

Salle / Room 6214

Centre de recherches mathématiques

Pavillon André-Aisenstadt

Université de Montréal

2920, chemin de la Tour

**The Local Version of Wigner's Semicircle Law and Dyson's Brownian Motion**

I will give some details of the proof of our recent result on the spectral universality of Wigner random matrices. The core of the argument is a hydrodynamical approach that is concise and can be presented in some details. To apply this theory, several preparatory results will be presented, most importantly I will discuss the local version of Wigner's semicircle law.

### Jeudi 22 mars 2012, 16h00 / Thursday, March 22, 2012, 4:00 pm

Salle / Room 6214

Centre de recherches mathématiques

Pavillon André-Aisenstadt

Université de Montréal

2920, chemin de la Tour

**Quantum Diffusion and Random Band Matrices**

Random band matrices can be viewed as intermediate models between random Schrödinger operators and mean field Wigner matrices. It is expected that an Anderson-type metal-insulator transition takes place as the key parameter, the bandwidth $W$, increases. The fundamental conjecture is that the localization length is of order $W^2$. We show that the system is diffusive on a certain time scale and we deduce a nontrivial lower bound on the localization length.