We study the extended states regime of the discrete Anderson model. The perturbative approach requires precise estimates on the free propagator,
$(a- e(p)+i\eta)^{-1}$,$\eta>0$, $\alpha\in \bR$,
where $e(p)= \sum_{i=1}^3 [1- \cos (p_i)]$, $p=(p_1, p_2, p_3)$,
is the dispersion relation of the three dimensional cubic lattice. The level surfaces of the function $e(p)$ have vanishing curvature. We will present new bounds on the Fourier transform of such surfaces. This will yield estimates on the probability that
a quantum particle travelling in a weak random environment
recollides with obstacles visited earlier.

I will consider inverse scattering theory for Jacobi operators which
are
short range perturbation of a quasi-periodic finite-gap background
operator.
In particular I want to investigate the algebraic constraints on the
scattering
data in this situation.

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.