**
TALK BY DIMITRI GIOEV
**

** "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.