Consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrodinger operators. [This is joint work with Pedro Duarte.]

An interesting feature of extended Harper's model (EHM), a generalization of the
almost Mathieu operator popularized by DJ Thouless, is the appearance of a large
regime of coupling parameters invariant under Aubry duality (``self-dual regime'').
In this regime, extensive numerical analysis in physics literature conjecture a
``strange collapse'' from purely singular continuous to purely absolutely continuous
spectrum, determined by the symmetries of the model.

Based on earlier work on the model [2], we have recently proven this conjecture [1]
by excluding eigenvalues in the self-dual regime for a full measure set of phases
and frequencies. The work is joint with S. Jitomirskaya.

[1] S. Jitomirskaya, C. A. Marx, On the spectral theory of Extended Harper's Model,
preprint (2013).

[2] S. Jitomirskaya, C. A. Marx, Analytic quasi-periodic cocycles with singularities
and the Lyapunov Exponent of Extended Harper's Model, Commun. Math. Phys. 316,
237-267 (2012).}

We show that the set of limit-periodic Schroedinger operators with
purely absolutely continuous spectrum is dense in the space of
limit-periodic
Schroedinger operators in arbitrary dimensions. This result was previously
known only in dimension one.
The proof proceeds through the non-perturbative construction of
limit-periodic
extended states. The proof relies on a new estimate of the probability (in
quasi-momentum) that the Floquet Bloch operators have only simple
eigenvalues.

We consider analytic cocycles of d \times d matrices. Such cocycles
appear for instance for the transfer matrices of a quasi periodic
Schrödinger operator on a strip.
We prove joint continuity (depending on frequency and the analytic
function of d \times d matrices) of all Lyapunov exponents at irrational
frequencies. Moreover, the so called accelerations (previously defined
for SL(2,C) cocycles by A. Avila) are also quantized at irrational
frequencies.
As a consequence, we obtain that the set of dominated cocycles is dense
within the set of cocycles where one has at least 2 different Lyapunov
exponents.

joint work with A. Avila and S. Jitomirskaya

We study unitary random matrix ensembles using the theory of
orthogonal polynomials on the unit circle. In particular we explicitly
compute the joint eigenvalue statistics of their rank-one truncations.
We prove that this eigenvalue point process is universal under the
natural scaling limit for a class of subunitary operators. Putting it
differently, we compute the limiting density of zeros of orthogonal
polynomials on the unit circle with random Verblunsky coefficients.
Joint work with Rowan Killip (UCLA).