The machinery of Euler systems (originating in the work of Kolyvagin and Thaine in the late 1980s) is an extremely powerful tool for studying the cohomology of Galois representations, and hence for attacking big conjectures such as Birch–Swinnerton-Dyer. However, current approaches to this theory require the Galois representation to satisfy some sort of "ordinarity" condition, which is a serious restriction in applications. I will discuss recent joint work with Sarah Zerbes in which we extend the Euler system machine to cover situations where this ordinary condition doesn't hold, using a surprising new ingredient (adapted from earlier work of Naomi Sweeting): non-principal ultrafilters, which serve to keep track of the sequences of auxiliary primes arising in Kolyvagin's argument. Applications of this theory, including new cases of the Iwasawa main conjecture, will be discussed in Sarah's talk later the same afternoon.

I will discuss the Brill-Noether theory of a general elliptic 𝐾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general 𝑘-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.