In a celebrated paper, Dyson shows that the spectrum of a random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as non-colliding Brownian motions held together by a drift term. The universal edge, bulk and gap scalings for Hermitian random matrices, applied to the Dyson process, lead to novel stochastic processes, Markovian and non-Markovian; among them, the Airy, Sine and Pearcey processes. The integrable theory around the KdV and KP equations provides useful information on these new processes.

I will describe some recent progress on the regularity theory
for minimal hypersurfaces. Assuming stability of the hypersurfaces, the results to be presented establish a rather complete local regularity theory that is applicable near points of volume density less than 3. I will also present an existence result. The latter is joint work with
Leon Simon.

Special Lagrangian 3-folds are of interest in mirror symmetry, and in particular play an important role in the SYZ conjecture. One wishes to understand the singularities that can develop in families of these 3-folds; the relevant local model is provided by special Lagrangian cones in complex 3-space. When the link of the cone is a torus, there is a natural invariant g associated to the cone, namely the genus of its spectral curve. We show that for each g there are countably many real (g-2)-dimensional families of such special Lagrangian cones.

We consider collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature. Such sequences may arise from dilating about infinite time singularities. For finite time singularities no collapse can occur by a result Perelman. Using Fukaya theory, we study some geometric (not topological) aspects of such collapse. When the limit solution is 1-dimensional we construct a virtual 2-dimensional rotationally symmetric limit. This is joint work with David Glickenstein and Peng Lu.