# Pure Mathematics Seminar

## Pure mathematics seminar 2022, Semester 1

**Organisers**: Jesse Gell-Redman, Ting Xue

**Schedule**

1 Jun **Andrea D’Agnolo** (Università di Padova) **On the irregular Riemann-Hilbert correspondence**

**15:15-16:15** Peter Hall 213

Abstract: The classical Riemann-Hilbert correspondence establishes an equivalence between regular holonomic D-modules and perverse sheaves.

A few years ago, jointly with Masaki Kashiwara, we proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular.

On the topological side of this correspondence, besides monodromy, the Stokes phenomenon comes into play.

In this talk I will present an overview of our construction, sweeping the technical points under the carpet.

27 May **Bryden Cais** (University of Arizona) **Iwasawa theory of class group schemes**

**11:30-13** Peter Hall 107

**Abstract:** Iwasawa theory is the study of the growth of arithmetic invariants in Galois extensions of global fields with Galois group a p-adic Lie group. Beginning with Iwasawa’s seminal work in which he proved that the p-primary part of the class group in Z_p-extensions of number fields grows with striking and unexpected regularity, Iwasawa theory has become a central strand of modern number theory and arithmetic geometry. While the theory has traditionally focused on towers of number fields, the function field setting has been studied extensively, and has important applications to the theory of p-adic modular forms. This talk will introduce an exciting new kind of p-adic Iwasawa theory for towers of function fields over finite fields of characteristic p, and discuss some applications to problems and open conjectures in the field.

21 March **Stefano Morra ** (Université Paris 8) **Moduli of Fontaine–Laffaille modules and local-global compatibility mod p**

14:15-15:15 Peter Hall 213

**Abstract: **The mod p-local Langlands program has developed from the proof of the Shimura–Taiyama–Weil conjecture performed by Breuil–Conrad–Diamond–Taylor, by the observation that certain invariants on local Galois deformation rings can be predicted by the modular representation theory of finite groups of Lie type. In particular, one would hope that a local Langlands correspondence for with mod coefficients will be realized in Hecke eigenspaces of the cohomology with infinite level at of compact unitary groups. In this talk we prove one direction of this expectation, namely that the smooth action on the Hecke eigenspaces with infinite level at determines the Galois parameter at -adic places, when the latter are Fontaine–Laffaille.