In this talk, I’ll sketch a way of unifying a wide variety of set theoretic approaches for generating new models from old models. The underlying methodology will draw from techniques in Sheaf Theory and the theory of Boolean Ultrapowers.

By Faltings' theorem, any curve over Q of genus at least two has only finitely many rational points—but the bounds coming from known proofs of Faltings' theorem are often far from optimal. Chabauty's method gives much sharper bounds for curves whose Jacobian has low rank, and can even be refined to give uniform bounds on the number of rational points. I'll discuss Kim's non-abelian analogue of Chabauty's method, which uses the unipotent fundamental group of the curve to replace the restriction on the rank with a weaker technical condition that is conjectured to hold for all hyperbolic curves. I will give an overview of this method and discuss my recent work with Ellenberg where we prove the necessary condition for any curve that dominates a CM curve, from which we deduce finiteness of rational points on any superelliptic curve.

When different issues come up in teaching, there are many different people who can potentially help...  we'll play a game related to deciding whom to ask for assistance in different circumstances (as well as when something can probably be handled on your own).  

We will discuss several open problems in dynamical systems (related to random dynamical systems with parameters, sums of Cantor sets, etc.) that can potentially turn into a research project suitable for graduate students.