I will discuss the proofs of some conjectural formulas
about Hodge integrals on moduli spaces of curves.
The generating series for all genera and all marked
points of such integrals are expressed in terms of
finite closed formulas from Chern-Simons knot invariants.
Such conjectures were made by string theorists based
on large N duality in string theory. I will explain
our proofs from localization techniques. Their relation
to toric Calabi-Yau manifolds and equivariant index
theory in gauge theory will also be discussed.
These are joint works with C.-C. Liu, J. Zhou and J. Li.

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.