I will present a new framework for determining effectively the spectrum and stability of traveling waves on networks with symmetries, such as rings and lattices, by computing master stability curves (MSCs). Unlike traditional methods, MSCs are independent of system size and can be readily used to assess wave destabilization and multi-stability in small and large networks.

I will briefly introduce the Bloch-Kato conjecture, a very general conjecture relating special values of L-functions to arithmetic, and explain how it generalises many more familiar theorems and conjectures such as the BSD conjecture for elliptic curves. I will then introduce the concept of an "Euler system", which is a powerful tool in proving cases of these conjectures, and survey some recent constructions of Euler systems using the geometry of Shimura varieties.

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.