Title: Quasiperiodic operators with analytic potential at low coupling:
sharp non-perturbative results.

Abstract: We will discuss recent results on analytic Schrodinger cocycles
at small couplings, with applications including the dry version of Ten
Martini problem and 1/2-Holder continuity of the integrated density of
states for Diophantine frequencies.  Bloch structure of solutions is
equivalent to the analytic reducibility of cocycle and is linked to the
existence of localized eigenfunctions for the dual model. However, there
is a generic set of energies in the spectrum for which no localized
eigenfunctions exist. For such energies the solutions are still linked
through a possibly divergent Fourier series.  We show that for all
energies there are solutions (for the dual model) that are localized on a
large set, between a sparse sequence of resonances.  This allows to give
sharp estimates on the dynamics (and therefore solutions) for all
energies. This is joint work with Artur Avila.