# School of Mathematics

## Topological and arithmetic intersection numbers attached to real quadratic cycles

Abstract: I will discuss a recent conjecture formulated in an ongoing project with Jan Vonk relating the intersection numbers of one-dimensional topological cycles on certain Shimura curves to the arithmetic intersections of associated real multiplication points on the Drinfeld p-adic upper half-plane. Numerical experiments carried out with Vonk and James Rickards supporting the conjecture will be described.

## An Euler system for genus 2 Siegel modular forms

Abstract: Euler systems are compatible families of cohomology classes for a global Galois represenation, which plan an important role in studying Selmer groups. I will outline the construction of a new Euler system, for the Galois representation associated to a cohomological cuspidal automorphic representation on the symplectic group GSp(4). This is joint work with Chris Skinner and Sarah Zerbes.

## A derived Hecke algebra in the context of the mod $p$ Langlands program

Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of smooth representations of G and the category of H-modules. When the field of coefficients has characteristic zero this link is well understood by work of Bernstein and Borel.

## Computations in the topology of locally symmetric spaces

## Time quasi-periodic gravity water waves in finite depth

## Higher Hida theory

## Modularity lifting theorems for non-regular symplectic representations

Abstract: We prove an ordinary modularity lifting theorem for certain non-regular 4-dimensional symplectic representations over totally real fields. The argument uses both higher Hida theory and the Calegari-Geraghty version of the Taylor-Wiles method. We also present some applications of these theorems to abelian surfaces. (Joint work with F. Calegari, T. Gee, and V. Pilloni.)

## Potential automorphy of some compatible systems over CM fields

## Automorphy of mod 3 representations over CM fields

Abstract: Wiles' work on modularity of elliptic curves over the rationals, used as a starting point that odd, irreducible represenations $G_Q \rightarrow GL_2 (F_3)$ arise from cohomological cusp forms (i.e. new forms of weight $K \geq 2$).