We study the self-adjoint Schr\"odinger operator on the axis
\[
H_v = -\frac{d^2}{dx^2} + v(x),\ \ \ -\infty < x < \infty,
\]
with an almost periodic real-valued potential $v(x)$.
Let $\Lambda$ be a dense subgroup of the group $(\R,+)$. Denote by
$AP_{\Lambda}(\mathbf{R})$ the Banach space of all real-valued almost periodic
functions on $\R$ whose all frequencies belong to $\Lambda$, with the supremum norm.
\bigskip

\textbf{Theorem}
\ There exists a dense $G_{\delta}$ subset $X\subseteq AP_{\Lambda}(\mathbf{R})$,
such that for all $v\in X$ the operator $H_v$ has a nowhere dense spectrum.

The talk will discuss the stability of spectral properties of
Schroedinger operators under the decaying perturbation potentials. The
primary focus will be on one-dimensional operators and preservation of
absolutely continuous spectrum. The talk will review some known results,
present new results, and discuss some open problems and conjectures.

We distinguish a class of random point processes which we call
Giambelli compatible point processes. Our definition was partly
inspired by determinantal identities for averages of products and
ratios of characteristic polynomials for random matrices.
It is closely related to the classical Giambelli formula for Schur symmetric functions.

We show that orthogonal polynomial ensembles, z-measures on
partitions, and spectral measures of characters of generalized
regular representations of the infinite symmetric group generate
Giambelli compatible point processes. In particular, we prove
determinantal identities for averages of analogs of characteristic
polynomials for partitions.

Our approach provides a direct derivation of determinantal
formulas for correlation functions

Phase transitions in classical spin systems are well understood,
however phase transitions in quantum spin systems are not ... at least
that is what most mathematicians would say. (If anything, they might
question how well we even understand classical spin systems.) Physicists,
on the other hand, say that if the classical model has a phase transition,
then the quantum model does as well. We will prove that, for some special
models. The main tools are reflection positivity, coherent states, and a
new result which generalizes the Berezin-Lieb inequality to the level of
matrix elements.