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.

In this talk, we will introduce some advances on non-analytic quasi-periodic cocycles, including Schrodinger and non-Schrodinger situations. Moreover, we will discuss some conjectures in this area.

The talk is split into two parts. In the first half we present a strategy
to prove absence of point spectrum, on the example of the self-dual regime
of extended Harper's model for all but countably many phases and almost
all frequencies. The starting point is a dynamical formulation of Aubry
duality via rotation reducibility, previously used by Avila and
Jitomirskaya for the almost Mathieu operator.

The second half of the seminar is devoted to some on-going work on the
Lyapunov exponent (LE) of a quasi-periodic Schr\"odinger cocycle whose
potential is a trigonometric polynomial. Based on the strategy of ``almost
constant cocycles,'' we obtain upper bounds for the phase-complexified LE.
This allows to give an estimate on the regime of sub-critical behavior,
therefore complementing the classical results of Herman's on positivity of
the LE. Within the framework of Avila's global theory, sub-critical
behavior implies purely absolutely continuous spectrum for all phases.