Probability of a gap in the spectrum of a matrix from the
Gaussian Unitary Ensemble is given by a Fredholm determinant. Its asymptotics when the gap becomes large is an interesting problem related to Painleve equations, random permutations, etc. These asymptotics for the gap in the bulk of the spectrum were conjectured by Dyson. The proof was given over the years by Widom, Deift, Its , Zhou, and the speaker. In particular, the proof of the multiplicative constant in the asymptotics
was the last difficulty recently resolved. I will explain the method of determining this constant and the rest of the asymptotics (applicable also to other important Fredholm, and also Hankel, and Toeplitz determinants where the corresponding constant is not yet determined). The method uses the Riemann-Hilbert approach. This part of the talk will be based on the works of Deift, Its, Zhou, and the speaker.

I will describe our recent proof of localization at the bottom of the spectrum for Schrodinger operators with Poisson random potentials. Poisson random potentials are the most natural model for describing a material with impurities. This has been a longstanding open problem. I will give a very informal talk on work in progress.

In my talk I will discuss recent results obtained in collaboration with Th. Bodineau, D. Ioffe and R. Schonmann concerning the fine details of the geometry of the random macroscopic droplet of minus-phase, floating in the plus-phase of the 3D Ising model.