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

I will survey some applications of a special kind of stratification of an algebraic stack called a theta-stratification. The goal is to eventually be able to study semistability and wall-crossing 
in a large array of moduli problems beyond the well-known examples. The most general application is to studying the derived category of coherent sheaves on the stack, but one can use this to understand the topology (K-theory, Hodge-structures, etc.) of the semistable locus and how it changes as one varies the stability condition. I will also describe a ``virtual non-abelian localization theorem'' which computes the virtual index of certain classes in the K-theory of a stack with perfect obstruction theory. This generalizes the virtual localization theorem of Pandharipande-Graber and the K-theoretic localization formulas of Teleman and Woodward.

Igusa's p-adic zeta function $Z(s)$ attached to a polynomial $f$ in $N$ variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation $f=0$ modulo powers of a prime $p$. It is expressed as a $p$-adic integral, and Igusa proved that it is rational in $p^{-s}$ using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of $Z(s)$ is at most $N$, the number of variables in $f$. In 1999, Wim Veys conjectured that the only possible pole of order $N$ of the so-called topological zeta function of $f$ is minus the log canonical threshold of $f$. I will explain a proof of this conjecture, which also applies to the $p$-adic and motivic zeta functions. The proof is inspired by non-archimedean geometry and Mirror Symmetry, but the main technique that is used is the Minimal Model program in birational geometry. This talk is based on joint work with Chenyang Xu.