Random matrices

December 13 to December 17, 2010

at the

American Institute of Mathematics, Palo Alto, California

organized by

Terence Tao and Van Vu

This workshop, sponsored by AIM and the NSF, will focus on recent developments on limiting distributions concerning spectrum of a random matrix. We will focus on the two main types of limiting distributions:

  1. Global: One would like to understand the limiting law of the counting measure generated by all eigenvalues. The most famous example here is the semi-circle law regarding the eigenvalues of random Hermitian matrices, discovered by Wigner in the 1950's.
  2. Local: One would like to understand the limiting law of fluctuation of individual eigenvalues (say the largest or smallest eigenvalues, or in general, the kth eigenvalues for any k), or local interaction among eigenvalues in a small neighborhood. Typical examples here are the Tracy-Widom law (for the extremal eigenvalues) and Dyson laws (for the distribution of gaps between consecutive eigenvalues and for correlation functions).
Both global and local laws are well understood in few special cases (such as GUE). The general belief is that these laws should hold for much larger classes of random matrices. This is known as the universality phenomenon (or invariance principle). This phenomenon has been supported by overwhelming numerical data and various conjectures, many of which (such as the Circular Law conjecture and Mehta conjectures) are among the most well known problems in the field.

In this workshop, we aim to first provide an overview about recent developments that establish (both global and local) universality in many important cases and in addition, we would like to discuss the techniques and directions for future research.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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