# School of Mathematics

## Modularity of Galois Representations

In this expository talk, I will outline a plausible story of how the study of congruences between modular forms of Serre and Swinnerton-Dyer, which was inspired by Ramanujan's celebrated congruences for his tau-function, led to the formulation of Serre's modularity conjecture. I will give some hints of the ideas used in its proof given in joint work with J-P. Wintenberger. I will end by pointing out just one of the many interesting obstructions to generalising the strategy of the proof to get modularity results in more general situations.

## Combinatorial Theorems in Random Sets

## Implied Existence for 3-D Reeb Dynamics

## Perturbation Theory for Band Matrices

**ANALYSIS/MATHEMATICAL PHYSICS SEMINAR**

## Endoscopic Transfer of Depth-Zero Suprcuspidal L-Packets

## Potential Automorphy for Compatible Systems of l-Adic Galois Representations

I will describe a joint work with Barnet-Lamb, Gee and Taylor where we establish a potential automorphy result for compatible systems of Galois representations over totally real and CM fields. This is deduced from a potential automorphy result for single l-adic Galois representations satisfying a `diagonalizability' condition at the places dividing l.

## Lecture 6

## A New Approach to the Local Langlands Correspondence for $GL_n$ Over p-Adic Fields

## Planar Convexity, Infinite Perfect Graphs and Lipschitz Continuity

Infinite continuous graphs emerge naturally in the geometric analysis of closed planar sets which cannot be presented as countable union of convex sets. The classification of such graphs leads in turn to properties of large classes of real functions - e.g. the class of Lipschitz continuous functions - and to meta-mathematical properties of sub-ideals of the meager ideal (the sigma-ideal generated by nowhere dense sets over a Polish space) which reduce to finite Ramsey-type relations between random graphs and perfect graphs.