# School of Mathematics

## 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.

## Configuration Spaces of Hard Discs in a Box

The "hard discs" model of matter has been studied intensely in statistical mechanics and theoretical chemistry for decades. From computer simulations it appears that there is a solid--liquid phase transition once the relative area of the discs is about 0.71, but little seems known mathematically. Indeed, Gian-Carlo Rota suggested that if we knew the total measure of the underlying configuration space, "we would know, for example, why water boils at 100 degrees on the basis of purely atomic calculations."

## Fractional Perfect Matchings in Hypergraphs

A perfect matching in a k-uniform hypergraph H = (V, E) on n vertices

is a set of n/k disjoint edges of H, while a fractional perfect matching

in H is a function w : E → [0, 1] such that for each v ∈ V we have

e∋v w(e) = 1. Given n ≥ 3 and 3 ≤ k ≤ n, let m be the smallest

integer such that whenever the minimum vertex degree in H satisfies

δ(H) ≥ m then H contains a perfect matching, and let m∗ be defined

analogously with respect to fractional perfect matchings. Clearly, m∗ ≤

m.