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Joint Mathematics Meetings

AMS Short Course on Random Matrices

The 2013 Short Course on Random Matrices is organized by Van Vu, Yale University.

Overview

The theory of random matrices is a rich topic in mathematics. Beside being interesting in its own right, random matrices play fundamental roles in various areas such as statistics, mathematical physics, combinatorics, theoretical computer science, number theory, and numerical analysis, to mention a few. A famous example here is the study of physicist E. Wigner, who used the spectrum of random matrices as a model in nuclear physics, and consequently discovered the fundamental semicircle law which describes the limiting distribution of the eigenvalues of a random hermitian matrix.

Special random matrices models where the entries are iid complex or real Gaussian random variables (GUE, GOE or Wishart) have been studied in detail. However, much less was known about general models, as the above-mentioned study relies very heavily on properties of the Gaussian distribution. In the last ten years or so we have witnessed considerable progress on several fundamental problems concerning general models, such as the Circular law conjecture or the Wigner-Dyson-Mehta conjecture. More importantly, these new results are proved using novel approaches which seem to be applicable to many other problems. The main goal of this Short Course is to introduce to a general audience these new results and methods, along with the several beautiful and surprising connections between the theory of random matrices with other areas of mathematics. We hope this will provide the audience a broad picture about this fascinating and rapidly developing field and encourage young researchers to participate in its study.

Each talk will be accessible to a general audience and will contain several open questions and/or suggestions for new directions of research. The methods presented in the first two talks, combined with earlier results by many researchers, have led to a complete solution of one of the most important conjectures in the field, the Wigner-Dyson-Mehta conjecture on the universality of the k-correlation functions of the eigenvalues. Both speakers will discuss this conjecture from different points of view.

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Monday’s Lectures

(1) Terence Tao, University of California Los Angeles, Random matrices: The universality phenomenon for the Wigner ensemble. We survey recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment theorem and its applications, including the universality of the sine-kernel and Central Limit theorem for various parameters. Lecture notes can be found here.

(2) Laszlo Erdös, Ludwig-Maximilians Universität, Universality of random matrices and Dyson Brownian motion. By proving an old conjecture of Dyson, we demonstrate that the strong local ergodicity of the Dyson Brownian motion is the the intrinsic mechanism behind universality for both invariant and noninvariant ensembles.

(3) Alice Guionnet, Massachusetts Institute of Technology, Free probability and random matrices. We will introduce the theory of free probability and give some applications of this theory to random matrix theory and operator algebra. Lecture notes can be found here.

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Tuesday’s Lectures

(4) Alan Edelman, Massachusetts Institute of Technology, Random matrices, numerical computation, and remarkable applications. In this talk requiring no special prerequisites, we will introduce the interplay of numerical computation with Random Matrix Theory. Two overarching themes will emerge: 1) computation is not a side show to the mathematics, but rather the math and computation go together and 2) The applications of Random Matrix Theory continue to surprise all of us. Lecture notes can be found here.

(5) Mark Rudelson, University of Michigan, Nonasymptotic theory of random matrices. We will discuss recent developments in the study of the spectral properties of random matrices of a large fixed size, concentrating on the extreme singular values. Bounds for the extreme singular values were crucial in establishing several limit laws of random matrix theory. Besides the random matrix theory itself, these bounds have applications in geometric functional analysis and computer science. Lecture notes can be found here.

(6) Djalil Chafai, Université Paris-Est Marne-la-Valleé, Around the circular law. We will present in an accessible way various stochastic models connected to the circular law phenomenon, including models of random matrices, random graphs, and random polynomials, investigated in recent years. We will also discuss some open problems. Lecture notes can be found here.

(7) Van Vu, Random Matrices: The universality phenomenon for nonhermitian random matrices. We discuss a very recent universality result for local statistics of eigenvalues of nonhermitian random matrices. As an application, we determine the number of real eigenvalues of a random matrix with iid (non-gaussian) real entries.

Registration

Advance registation fees for members of the AMS or MAA, US$104; nonmembers are US$150; and students/unemployed or emeritus members are US$52. These fees are in effect until December 17. If you choose to register on site, the fees for members of the AMS or MAA are US$138; nonmembers are US$180; and students/unemployed or emeritus members are US$73. Advance registration starts on September 1, 2012. Onsite registration will take place on Monday, January 7, outside Room 4, San Diego Convention Center upper level.

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