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).

Non-Gaussian functional integrals arising in Quantum Field Theory are notoriously difficult to define and compute. This applies even to relatively simple models such as Chern–Simons gauge theory, where an exact solution was obtained without a direct evaluation of the functional integral. I will explain how to use supersymmetric localization to reduce in some cases a non-Gaussian functional integral to an ordinary integral. This technique can be used to evaluate ex-
actly some observables in Chern–Simons theory as well as in certain
supersymmetric gauge theories in three dimensions and to test various
duality conjectures concerning such theories.

We study the spectrum of discrete Schrodinger operators with potential given by a primitive invertible substitution sequence (and in fact our results hold for a larger class of potentials). We show this family of operators has a spectrum which is a dynamically defined Cantor set of zero Lebesgue measure. We also show that the Hausdorff dimension of this set depends analytically on the coping constant lambda and tends to 1 as lambda tends to 0. Finally, we show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly.

As, beginning with the famous Hofstadter's butterfly, all
numerical studies of spectral and dynamical quantities related to
quasiperiodic operators are actually performed for their rational
frequency approximants, the questions of continuity upon such
approximation are of fundamental importance. The fact that continuity
issues may be delicate is illustrated by the recently discovered
discontinuity of the Lyapunov exponent for non-analytic potentials.

I will review the subject and then focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences.

This is the first of (likely) two talks, where an almost entire proof will be presented. For understanding most of the talk knowledge of spectral theory should not be necessary and just knowing some basic harmonic analysis should suffice.

As, beginning with the famous Hofstadter's butterfly, all
numerical studies of spectral and dynamical quantities related to
quasiperiodic operators are actually performed for their rational
frequency approximants, the questions of continuity upon such
approximation are of fundamental importance. The fact that continuity
issues may be delicate is illustrated by the recently discovered
discontinuity of the Lyapunov exponent for non-analytic potentials.

I will review the subject and then focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences.

This is the first of (likely) two talks, where an almost entire proof will be presented. For understanding most of the talk knowledge of spectral theory should not be necessary and just knowing some basic harmonic analysis should suffice.