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.

The CMV matrix is a unitary operator on $\ell^2(\mathbb N)$ that is a central tool in the study of orthogonal polynomials on the unit circle. One may view it as a unitary analogue of the Jacobi matrix. We may extend the CMV matrix to be a unitary operator on $\ell^2(\mathbb Z)$. It is more natural to consider the extended CMV matrix in certain contexts: for example, if we wish to generate CMV matrices dynamically. The extended CMV matrix also plays an important role in the study of quantum random walks.

In this talk, we will discuss a Gordon lemma for the CMV matrix (The Gordon lemma is an important tool in the study of Jacobi matrices, used to rule out the possibility pure point spectrum). We will also discuss some results pertaining to the H\"older-continuity of the spectrum of the extended CMV matrix.

Quasi one-dimensional particle systems have domains which are infinite
in one direction and bounded in all other directions, e.g. an infinite
cylinder. We will show that for such particle systems with Coulomb
interactions and a neutralizing background, the so-called jellium,
there is translation symmetry breaking in the Gibbs measures at any
temperature. This extends a previous result on Laughlin states in
thin, two-dimensional strips by Jansen, Lieb and Seiler (2009). The
structural argument is akin to that employed by Aizenman and Martin
(1980) for a similar statement concerning symmetry breaking at all
temperatures in strictly one-dimensional Coulomb systems. The
extension is enabled through bounds which establish tightness of
finite-volume charge fluctuations. We will also discuss an
application to quantum one-dimensional jellium which extends an old
result of Brascamp and Lieb (1975).

A classical result by Pál Turán, estimates the global behavior
of an exponential polynomial on an interval by its supremum on any
arbitrary subinterval. In this talk we discuss Nazarov's extension of this
``global to local reduction'' to arbitrary Borel sets of positive Lebesgue
measure.

Using the idea of "acceleration", which is first introduced by Avila, I will construct a
cerntain class of analytic quasiperiodic Szego cocycles with uniformly positive LE.