C2

We know that the existence of a period three point for an interval map implies much about the dynamics of the map, but the restriction of the map to the periodic orbit itself is trivial. Countable invariant subsets arise naturally in many dynamical systems, for example as $\omega$-limit sets, but many of the usual notions of dynamics degenerate when restricted to countable sets. In this talk we look at what we can say about dynamics on countable compact spaces.  In particular, the theory of countable dynamical systems is the theory of the induced dynamics on countable invariant subsets of the interval and the theory of homeomorphic countable dynamics is the theory of compact countable invariant subsets of homeomorphisms of the plane.

Joint work with Columba Perez

The study of rational points on K3 surfaces has recently seen a lot of activity. We discuss how to compute the Picard rank of a K3 surface over a number field, and the implications for the Brauer-Manin obstruction.

I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.

In this talk I will define algebraic automorphic forms, first defined by Gross, which are objects that are conjectured to have Galois representations attached to them. I will explain how this fits into the general picture of the Langlands program and, giving some examples, briefly describe one method of proving certain cases of the conjecture.