# Group Theory

## On the local Langlands conjectures

### Abstract

The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and the representation theory of groups. More precisely, the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa.

In this talk we sketch a few aspects of the local Langlands correspondence using elementary examples. We then comment on some questions raised by the emerging "mod p" Langlands program.

### Biography

Professor Ollivier works in the Langlands Programme, a central theme in pure mathematics which predicts deep connections between number theory and representation theory. She has made profound contributions in the new branches of the "p-adic" and "mod-p" Langlands correspondence that emerged from Fontaine's work on studying the p-adic Galois representation, and is one of the pioneers shaping this new field. The first results on the mod-p Langlands correspondence were limited to the group GL2(Qp); but Dr. Ollivier has proved that this is the only group for which this holds, a surprising result which has motivated much subsequent research.

She has also made important and technically challenging contributions in the area of representation theory of p-adic groups, in particular, in the study of the Iwahori-Hecke algebra. In joint work with P. Schneider, Professor Ollivier used methods of Bruhat-Tits theory to make substantial progress in understanding these algebras. She has obtained deep results of algebraic nature, recently defining a new invariant that may shed light on the special properties of the group GL2(Qp).

Rachel Ollivier received her PhD from University Paris Diderot (Paris 7), and then held a research position at ENS Paris. She subsequently was an assistant professor at the University of Versailles and then Columbia University, before joining the UBC Department of Mathematics in 2013.

Rachel is the recepient of the 2015 UBC Mathematics and Pacific Institute for the Mathematical Sciences Faculty Award.

More information on this event is available on the event webpage

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## Finite Simple Groups and Applications

The classification of finite simple groups is of fundamental importance in mathematics. It is also one of the longest and most complicated proofs in mathematics.

We will very briefly discuss the result and a bit of history and then explain how it can and has been used to solve problems in many areas. We will end with mentioning some specific and perhaps surprising consequences in various fields.

## Iwahori-Hecke algebras are Gorenstein (part II)

n the local Langlands program the (smooth) representation theoryof p-adic reductive groups G in characteristic zero plays a key role. For any compact open subgroup K of G there is a so called Hecke algebra H(G,K). The representation theory of G is equivalent to the module theories over all these algebras H(G,K). Very important examples of such subgroups K are the Iwahori subgroup and the pro-p Iwahori subgroup. By a theorem of Bernstein the Heckealgebras of these subgroups (and many others) have finite global dimension.

In recent years the same representation theory of G but over an algebraically closed field of characteristic p has become more and more important. But little is known yet. Again one can define analogous Hecke algebras. Their relation to the representation theory of G is still very mysterious. Moreover they are no longer of finite global dimension. In joint work with R. Ollivier we prove that over any field the algebra H(G,K), for K the (pro-p) Iwahori subgroup, is Gorenstein.

## Iwahori-Hecke algebras are Gorenstein (part I)

**N.B. Due to problems with our camera, there is some audio distortion on this file and a small portion of the video has been removed.**

This lecture is the first of a two part series (part II).

In the local Langlands program the (smooth) representation theory

of p-adic reductive groups G in characteristic zero plays a key role. For any

compact open subgroup K of G there is a so called Hecke algebra H(G,K). The

representation theory of G is equivalent to the module theories over all these

algebras H(G,K). Very important examples of such subgroups K are the Iwahori

subgroup and the pro-p Iwahori subgroup. By a theorem of Bernstein the Hecke

algebras of these subgroups (and many others) have finite global dimension.

In recent years the same representation theory of G but over

an algebraically closed field of characteristic p has become more and more important. But little is known yet. Again one can define analogous Hecke algebras. Their relation to the representation theory of G is still very mysterious. Moreover

they are no longer of finite global dimension. In joint work with R. Ollivier

we prove that over any field the algebra H(G,K), for K the (pro-p) Iwahori

subgroup, is Gorenstein.

## Quadratic forms and finite groups

The study of quadratic forms is a classical and important topic of algebra and number theory. A natural example is the trace form of a finite Galois extension. This form has the additional property of being invariant under the Galois group,leading to the notion of "self-dual nornal basis", introduced by Lenstra. The aim of this talk is to give a survey of this area, and to present some recent joint results with Parimala and Serre.

## Asymptotic dimension

Most of the talk will be an introduction to (second) bounded cohomology of a discrete group. I will explain classical constructions of bounded cocycles and recent results (joint with Bromberg and Fujiwara) regarding mapping class groups and a construction of bounded cocycles with coefficients in an arbitrary unitary representation.

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## Second Bounded Cohomology

Most of the talk will be an introduction to (second) bounded cohomology of a discrete group. I will explain classical constructions of bounded cocycles and recent results (joint with Bromberg and Fujiwara) regarding mapping class groups and a construction of bounded cocycles with coefficients in an arbitrary unitary representation.

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## Expanders, Group Theory, Arithmetic Geometry, Cryptography and Much More

This is a lecture given on the occasion of the launch of the PIMS CRG in "L-functions and Number Theory".

The theory of expander graphs is undergoing intensive development. It finds more and more applications to diverse areas of mathematics. In this talk, aimed at a general audience, I will introduce the concept of expander graphs and discuss some interesting connections to arithmetic geometry, group theory and cryptography, including some very recent breakthroughs.

## Regular Permutation Groups and Cayley Graphs

Regular permutation groups are the `smallest' transitive groups of permutations, and have been studied for more than a century. They occur, in particular, as subgroups of automorphisms of Cayley graphs, and their applications range from obvious graph theoretic ones through to studying word growth in groups and modeling random selection for group computation. Recent work, using the finite simple group classification, has focused on the problem of classifying the finite primitive permutation groups that contain regular permutation groups as subgroups, and classifying various classes of vertex-primitive Cayley graphs. Both old and very recent work on regular permutation groups will be discussed.

## Algebraic Z^d-actions

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