The Birch--Swinnerton-Dyer conjecture is now a theorem, under some mild hypotheses, for elliptic curves over Q with analytic rank ≤ 1. One of the main ingredients in the proof is Kolyvagin's theory of Euler systems: compatible families of cohomology classes which can be seen as an "arithmetic avatar'' of an L-function. The existence of Euler systems in other settings would have similarly strong arithmetical applications, but only a small number of examples are known.

In this talk, I'll introduce Euler systems and their uses, and I'll describe the construction of a new Euler system, which is attached to the Rankin--Selberg convolution of two modular forms; this is joint work with Antonio Lei and David Loeffler. I'll also explain recent work with Loeffler and Guido Kings where we prove an explicit reciprocity law for this Euler system, and use this to prove cases of the BSD conjecture and the finiteness of Tate--Shafarevich groups.

We will lay the groundwork needed to discuss some results that use homogeneous dynamics to bound the Hausdorff dimension of sets arising in number theory. Specifically, we will define mixing flows, Lie groups and algebras, homogeneous spaces, and expanding horospherical subgroups, and illustrate these concepts with a few basic examples.

Woodin's $P_{max}$ forcing when applied to a model of Determinacy produces a model which is maximal for sets of countable ordinals. We will briefly introduce $P_{max}$ and its applications and variations, and outline a proof of the maximality of $P_{max}$ extensions.