This is a joint work with P.Deift.
We give a rigorous proof of the Universality Conjecture
in Random Matrix Theory for orthogonal (beta=1) and
symplectic (beta=4) ensembles in the scaling limit
for a class of polynomial potentials
whose equilibrium measure is supported on
a single interval.
Our starting point is Widom's representation
of the correlation kernels for the beta=1,4 cases
in terms of the unitary (beta=2) correlation kernel
plus a correction.
In the asymptotic analysis of the correction terms
we use amongst other things differential equations for the derivatives
of orthogonal polynomials (OP's) due to Tracy-Widom,
and uniform Plancherel-Rotach type asymptotics for OP's
due to Deift-Kriecherbauer-McLaughlin-Venakides-Zhou.
The problem reduces to a small norm problem
for a certain matrix of a fixed size
that is equal to the degree of the polynomial potential.