"Universality Questions in Random Matrix Theory"
One of the main features of Random Matrix Theory (RMT) is that it provides accurate models for correlated quantities that describe various complex systems arising in a broad variety of problems in physics, pure and applied mathematics, and in other branches of knowledge. We will start by presenting various examples of such quantities. A mathematical reason for a wide applicability of RMT is that there should be certain versions of the Central Limit Theorem but now for certain classes of correlated random variables. In particular, the limiting behavior of the eigenvalues of a large random matrix should be independent of the details of the distribution of the matrix elements. This loose statement is known as the Universality Conjecture. We then introduce the three classical types of invariant random matrix ensembles. For the technically simplest case of the so-called unitary ensembles, we explain how all the statistical quantities of interest can be expressed in terms of orthogonal polynomials on the real line. The Universality Conjecture in the bulk of the spectrum for the unitary ensembles was established by Deift-Kriecherbauer-McLaughlin-Venakides-Zhou in 1999. The other two classes of invariant ensembles, orthogonal and symplectic, are more difficult to analyze. In the end of the talk, we will describe our recent work with Deift where we prove the Universality Conjecture both in the bulk and at the edge of the spectrum for the remaining two classes of invariant ensembles in great generality.