For singular operators of the form (H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n, we prove such operators have purely singular spectrum on the set {E: \delta{(\alpha,\theta)}>L(E)\}, where f and g are both analytic functions.

We prove that Schrodinger operators with meromorphic potentials have purely singular continuous spectrum on the set {E: \delta{(\alpha,\theta)}>L(E)\} where \alpha is the frequency, \theta is the phase, delta is an explicit function, and L is the Lyapunov exponent. This extends a result of Jitomirskaya and Liu for the Maryland model to the general class of meromorphic potentials.

The exact solutions of the Korteweg-de Vries (KdV) equation obtained by travelling wave and similarity reductions may be expressed in terms of elliptic functions and Painleve transcendents respectively. Discrete versions of the KdV equation may be obtained from chains of commuting Backlund transformations of the KdV equation. These systems are considered integrable in their own right. This introductory talk will demonstrate how solutions obtained as reductions of the discrete KdV equation give us discrete analogues of elliptic equations and discrete Painleve equations, mimicking the case for the KdV equation.

  For the second  phase transition line $\lambda=e^{\beta}$ of  the almost Mathieu operator, we prove that for dense $\alpha$, the operator has purely singular continuous spectrum for every phases, and  for dense $\alpha$, the operator has pure point spectrum for almost every phases. This is joint work with Artur Avila and Svetlana Jitomirskaya. 

Consider the generalized Anderson Model
$H^\omega=\Delta+\sum_{n\in\mathcal{N}}\omega_n P_n$, where $\mathcal{N}$ is a countable set, $\{\omega_n\}_{n\in\mathcal{N}}$ are iid randomvariables and $P_n$ are rank $N<\infty$ projections. For these models one can prove theorems analogous to that of Jak\v{s}i\'{c}-Last on the
equivalence of measures.

We show that if the projection $Q_m^\omega P_n$ (where $Q^\omega_m$ is
cannonical projection on the subspace generated by $H^\omega$ and range of
$P_m$) has same rank as $P_n$, then the trace measure
$\sigma_i(\cdot)=tr(P_iE_{H^\omega}(\cdot)P_i)$ and absolute continuous
part of the measure $P_iE_{H^\omega}(\cdot)P_i$ are equivalent for $i=n,m$.