## Lecture 6

Pierre Colmez
National Center for Scientific Research; Member, School of Mathematics
November 18, 2010

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

David Geraghty
Princeton University; Member, School of Mathematics
November 18, 2010

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.

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

Tasho Kaletha
Princeton University; Member, School of Mathematics
November 18, 2010

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

Peter Scholze
University of Bonn
November 17, 2010

## Honest Doubt

Paul Hodgson
November 17, 2010

Artist Paul Hodgson was a Director's Visitor at the Institute in 2010. In a Friends Forum, he discussed the "difficulties in making a judgement and dubtfulness in choosing one thing over another," that underlie his current practice and emerge "both in the way that I fabricate the work, and the images that I choose to present."

## Fractional Perfect Matchings in Hypergraphs

Andrzej Rucinski
Adam Mickiewicz University in Polznan, Poland; Emory University
November 15, 2010

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.

Matthew Kahle