After defining the spectral theory of orthogonal polynomials on the unit circle (OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szego's theorem as a sum rule for OPUC and the Killip--Simon sum rule for OPRL and their spectral consequences. Next I'll explain the original proof of Killip--Simon using representation theorems for meromorphic Herglotz functions. Finally I'll focus on recent work of Gambo, Nagel and Rouault who obtain the sum rules using large deviations for random matrices.

Kumura showed that there are no eigenvalues embedded in the essential
spectrum of the Laplacian on $n$-dimensional noncompact
complete Riemannian manifold $(M_n, g)$, if the radial curvature $K_{\rm
rad}+1=o(r^{-1})$ as $r$ goes to infinity.

Given any finite/countable set of positive energies $\{\lambda_n\}$, we
can
construct a Riemannian manifold with the decay order
$K_{\rm rad}+1=O(r^{-1})$/$K_{\rm rad}+1=\frac{C(r)}{r}$, where $C(r)\geq
0$ and $C(r) $ goes to infinity arbitrarily slowly, such that the
eigenvalues $\{\frac{(n-1)^2}{4}+\lambda_n\}$ are embedded in the
essential
spectrum $\sigma_{{\rm ess}}(-\Delta_g)=\left[\frac{(n-1)^2}{4},\infty
\right)$.

One of the classical questions about the evolution of a one
dimensional quantum system is the asymptotic rate of propagation of
the wave packet. It is usually captured through the notion of
transport exponents. Several methods were developed to estimate these
quantities in various models. However many authors only treated the
case of a state initially localized at a single site (in the discrete
setting). We show that some of these results can be extended to a
broad class of initial states with compact or even infinite support,
and explain what are the methods and obstacles to further
generalizations.

We study random permutations of the vertices of a hypercube  given by products of (uniform, independent) random transpositions on edges.  We establish the existence of a phase transition accompanied by emergence of cycles of diverging lengths. The problem is motivated by phase transitions in quantum spin models. (Joint work with Piotr Miłoś and Daniel Ueltschi.)