# 2010 NZMRI Summer Workshop/Abstracts

### From MathsDept

**Martin Bridson**: Groups that want to be free

This series will explore residually free groups, emphasising the connections between geometry, group theory and logic.

**Michel Broué**: Local representation theory of finite groups and cyclotomic algebras

Taking its origin from the efforts to classify the finite simple groups, the local theory of finite groups has grown up into local representation theory, and has suggested some challenging conjectures about homological behaviour of finite group algebras over finite fields. When applied to finite reductive groups, these conjectures lead to considerations involving complex reflections groups, their associated braid groups, and cyclotomic Hecke algebras. We shall present an overview of that panorama.

**Persi Diaconis**: Probability, Combinatorics and Group Extensions

The classical subject of group extensions (how to stick two groups together) has so far escaped the community's enumerative eye. These introductory talks will explain the basics, give many examples and point to much work still to be done.

Lecture 1: On Adding Up a List of Numbers

The basic process of adding a list of numbers gives "carries" along the way. How many carries are typical (about 1/2?). How are the carries distributed? It turns out that these questions have neat answers in terms of "one-dependent, determinental processes" (explanations provided). More or less, the whole story goes through for general central extensions. This is joint work with Jason Fulman and Alexei Borodin.

Lecture 2: Carries, Shuffling, and an "Amazing Matrix"

When several long integers are added, there are carries "along the top." These turn out to be distributed as a Markov chain whose transition matrix has "amazing properties" discovered by Holte. This same matrix comes up in the usual method of shuffling cards and in the combinatorics of the Veronese imbedding. These various occurrences interact in useful ways. This is joint work with Jason Fulman.

Lecture 3: Carries, Extensions and Koszul Algebras

Koszul algebras are a particularly nice species of (perhaps non-commutative) algebra. It turns out (Polishchuk-Positselski) that any Koszul algebra generates a one dependent, determinental process. These include many (all?) carries processes. The connections are tantalizing and there is much to understand. This is joint work with Jason Fulman, Bob Guralnick, and David Eisenbud.

**Roger Howe**: Representations of the general linear group, an algebraic perspective

The general linear group plays a central role in representation theory and in its connections with other areas, especially combinatorics and algebraic geometry. In the 1940s, two superficially different descriptions of the structure of representations of GL_n were given, one by Hodge, with motivations and methods from algebraic geometry, and one by Gelfand and Tsetlin, based on ideas internal to representation theory. Recently, ideas from the theory of Groebner bases have been used to unify and simplify these descriptions. These talks will outline this development and point to further applications.

**Gus Lehrer**: Knot invariants, Hecke algebras and cellular algebras

Many important concepts in algebra are motivated by geometry and topology, and vice versa; the Jones polynomial for knots is a celebrated example. A key idea, which forms a bridge between geometry and algebra, is to construct algebras whose operations (like multiplication) are described using diagrams. Easily understood in its elementary forms, this idea is gaining importance in current research. The notion of a "cellular algebra" is an abstraction of this notion, which is finding applications in several areas of mathematics.

The three lectures will give an elementary introduction to ideas which bear on the topics of the title. Subjects touched on will be roughly as indicated below.

Lecture 1. The space Mn of configurations of distinct points in C; its quotient Xn by the symmetric group. The n-string braid group as the fundamental group of Xn. The algebra of braids. From braids to oriented links (and knots); link invariants. Alexander's and Markov's theorems.

Lecture 2. Skein relations for link invariants lead to Hecke algebras. The Hecke algebra Hn(q) of type A. Trace function on Hn(q) and the Temperley-Lieb algebra as a quotient. Some Frobenius theory for the symmetric group: partitions and representations. Temperley-Lieb diagrams, descent of trace function. The Jones and Homfly polynomials.

Lecture 3. The Temperley-Lieb algebra as a cellular algebra. Examples of semisimple and non-semisimple algebras; radicals. Cellularity and degeneration of semisimplicity. Small examples. Canonical bases of Hecke algebras, cellularity.

**Marcus du Sautoy**: Through the looking glass: groups from a number theoretic perspective

Ever since Riemann’s seminal paper on the primes, the zeta function has proved a powerful weapon in the mathematician’s arsenal. In recent years, group theorists have discovered that non-commutative analogues of classical zeta functions in number theory provide an interesting new perspective on the theory of infinite groups. These zeta functions encode in a Dirichlet series arithmetic information about the lattice of subgroups of an infinite group. These lecture will explain how these zeta functions are providing a new bridge between the theory of nilpotent groups and classical arithmetic geometry.