## A Rigorous Renormalization Group Study of a p-Adic Quantum Field Theory

ANALYSIS/MATHEMATICAL PHYSICS SEMINAR

Abdelmalek Abdesselam

University of Virginia

November 12, 2010

ANALYSIS/MATHEMATICAL PHYSICS SEMINAR

Moderator: Harold Shapiro, President Emeritus and Professor of Economics and Public Affairs, Princeton University

Trustee, Institute for Advanced Study

November 13, 2010

Panelists:

Didier Fassin, James D. Wolfensohn Professor, School of Social ScienceJonathan Israel, Professor, School of Historical Studies

Avishai Margalit, George F. Kennan Professor, School of Historical Studies

Joan Wallach Scott, Harold F. Linder Professor, School of Social Science

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

Institute for Advanced Study

November 15, 2010

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."

Menachem Kojman

Ben Gurion University of the Negev; Member, School of Mathematics

November 16, 2010

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.

Peter Scholze

University of Bonn

November 17, 2010

Paul Hodgson

Institute for Advanced Study

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."

Pierre Colmez

National Center for Scientific Research; Member, School of Mathematics

November 18, 2010

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.

Tasho Kaletha

Princeton University; Member, School of Mathematics

November 18, 2010