This is simultaneously a thesis defense.

Committee: S. Jitomirskaya (chair),
A. Klein
A. Figotin

In this talk we will consider the spectrum of the Almost Mathieu operator, \[ \left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n-1} + b \cos\left(2 \pi n\omega + \phi\right)x_n, \] on $l^2(\mathbb{Z})$. We will show that for $b \ne 0,\pm 2$ and $\omega$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'', for these values of $b$ and $\omega$.

The proof uses a combination of results on reducibility, localization and duality for the Almost Mathieu operator, and its associated eigenvalue equation, sometimes called the Harper equation.

Finally, we will also show that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.

We discuss the trace map associated with the Fibonacci
quasicrystal. While the associated dynamical system has been studied heavily as a real dynamical system, it may also be regarded as a complex dynamical system. We study the stable set and give explicit bounds for the complex approximants. Quantum dynamical consequences of these results will be
explained. This is joint work with Serguei Tcheremchantsev.

We consider paraorthogonal polynomials P_n on the unit circle defined by
random recurrence (Verblunsky) coefficients. Their zeros are exactly
the eigenvalues of a special class of random unitary matrices (random CMV
matrices). We prove that the local statistical distribution of these zeros
converges to a Poisson distribution. This means that, for large n, there
is no local correlation between the zeros of the random polynomials P_n.

We define and study spaces of random operators which satisfy Hilbert-Schmidt and trace class - like conditions with respect to the trace per unit volume.

Such spaces appear in the derivation of the Kubo formula in linear response theory for random Schrodinger operators in a constant magnetic field.