Eigenfunctions of the Laplacian on a bounded domain represent the modes of vibration of a vibrating drum. The behavior of these eigenfunctions is closely related to the behavior of the underlying dynamical system of the billiard table. In this talk I first give a brief exposition on this relation and then I talk about the boundary traces of eigenfunctions and a recent joint work with Han, Hassell and Zelditch.

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.]