As one of the deepest and most beautiful theorems in geometry, the
Hodge theorem builds a bridge between Riemannian metric and
topological invariants. It gives an isomorphism between the space of
harmonic p forms on a Riemannian manifold and the pde Rham cohomology group of a smooth structure. By the de Rham theorem, we see the
isomorphism between the space of harmonic p forms and p real singular cohomology group.

The Hodge theorem is a good example of how PDEs help us understand geometric structure and even topological structure. In this talk, we
will give an introduction to this theorem, explain the idea behind it, and give some applications in Riemannian geometry.

The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

Topologically all Cantor sets are the same. Nevertheless, thee are many ways to assign a quantitative characteristic to Cantor sets, and these notions play important role in applications to dynamical systems, number theory, spectral theory, and other areas of mathematics. We will describe some of the characteristics (e.g. fractal dimentions, thickness) of Cantor sets and the ways one can calculate and use them.

* Pizza and Soda to be served!

We will review the recent (and not so recent) results on dynamics of piecewise isometries (especially piecewise translations), both in one and in higher dimensional case. Some interesting results (by Suzuki, Goetz, Zhuravlev, Boshernitzan, Bruin, Troubetzkoy, Buzzi) are known, but most of natural questions are still open. The main goal of the talk is to expose these open questions to potential researchers.

We study discrete Schr ̈odinger operators with trigonomet-
ric potentials. In particular, we are interested in the connection be-
tween the absolutely continuous spectrum in the almost periodic case
and the spectra in the periodic case. We prove a weak form of a precise
conjecture relating the two.
We also bound the measure of the spectrum in the periodic case in
terms of the Lyapunov exponent in the almost periodic case.
In the proofs, we use a partial generalization of Chambers formula.
As an additional application of this generalization, we provide a new
proof of Hermans lower bound for the Lyapunov exponent.