In 1983 Dan Voiculescu used a family of unitary matrices, now
known as "Voiculescu's Unitaries," to provide the first counter-example to
an old conjecture of Halmos regarding "almost commuting" matrices. Later,
Ruy Exel and Terrry Loring used "Voiculescu's Unitaries" in an elementary
and elegant proof to provide another counter-example on "almost commuting"
matrices. In this talk, we present two new counter-examples using
"Voiculescu's Unitaries." The talk should be accessible to anyone with
knowledge of basic real analysis and linear algebra.

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.