Morning Lectures

Tuesday, April 7 2009

9:15 - 10:15    Kirwan Lecture Theatre

Jürgen Berndt (University College Cork):
Foliations and cohomogeneity one actions

An isometric action on a connected complete Riemannian manifold is of cohomogeneity one if its orbit space is one-dimensional. Any such action induces a foliation on the manifold, possibly with one or two singular leaves. The aim of the talk is to provide a brief introduction to the general theory of such foliations, and to present an overview of the current status of the classification of such foliations on symmetric spaces.

9:15 - 10:15    Cairnes Lecture Theatre

Hinke Osinga (University of Bristol and Cornell University):
The Lorenz manifold: from mathematics to steel

The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. The organisation of the dynamics in the Lorenz system and also how the dynamics depends on the system parameters has long been an object of study. This talk addresses the role of the stable and unstable manifolds in organising the dynamics more globally. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold (the Lorenz manifold) and the other two equilibria have two-dimensional unstable manifolds. The intersections of these manifolds in three-dimensional space are heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent study of these manifolds reveals an intriguing combinatoric structure and how it ties in with related structures known for the periodic orbits and homoclinic bifurcations.
  The fascination of the Lorenz system goes far beyond mathematics and you will see how the Lorenz manifold was turned into a steel sculpture.

10:15 - 11:15    Cairnes Lecture Theatre

Eamonn O'Brien (University of Auckland):
Towards effective algorithms for linear groups

Despite their ubiquity, only limited algorithmic tools exist to study groups of matrices defined over finite fields. We will identify a few of the inherent difficulties, and outline some new and more promising strategies.

10:15 - 11:15    Kirwan Lecture Theatre

Tony Carbery (University of Edinburgh):
Tube-nullity and Fourier Analysis: results and open problems

We introduce the notion of tube-nullity of a set in Euclidean space. It is analogous to the usual notion of Lebesgue nullity but coverings by balls are replaced with coverings by tubes and the volume of balls is replaced by the cross-sectional area of tubes. We show how this notion arises very naturally in questions in Fourier analysis and go on to describe what is known and what is not known about such sets.

Wednesday, April 8 2009

9:15 - 10:15    Kirwan Lecture Theatre

Dominic Welsh (University of Oxford):
The Random Planar Graph

As late as 1995 very little was known about the two fundamental questions raised below:
  Problem 1. How does one generate a random simple planar graph uniformly at random from the set of simple labelled planar graphs on n vertices?
  Problem 2. What does this random planar graph look like?
  In this talk I shall survey the progress made since then and present some of the outstanding open problems.

9:15 - 10:15    Cairnes Lecture Theatre

Reidun Twarock (University of York):
Applications of Group Theory in Virology: Affine extensions of noncrystallographic groups predicting virus architecture

Viruses have protein containers that encapsulate, and hence provide protection for, their genomic material. For a significant number of viruses these containers are organised according to icosahedral symmetry, which allows us to model their structural organisation via symmetry techniques. We show here that a wide spectrum of distinct viral features can be predicted in striking detail via a classification of affine extensions of the icosahedral group. Examples discussed in thi s talk include the sizes and shapes of the protein building blocks of the containers, the double-shelled genomic RNA structure in MS2, the dodecahedral RNA cage in Pariacoto virus, and the heterogeneity in the genomic organisation of Picorn aviridae. Some of the implications of this fundamental geometric principle of virus architecture for virus assembly and evolution are also discussed.

Thursday, April 9 2009

9:15 - 10:15    Kirwan Lecture Theatre

Martin Mathieu (Queen's University Belfast):
On some non-commutative Banach-Stone theorems

The classical Banach-Stone theorem describes the surjective isometries between spaces of continuous functions on compact Hausdorff spaces. Over the years, many extensions of this important result have been obtained by a multitude of authors and in a variety of settings. I shall discuss several of the generalisations of the Banach-Stone theorem for non-commutative Banach algebras, in particular for operators preserving certain spectral properties on C*-algebras. It is surprising that the strongest results appear to be available for very non-commutative algebras.
  

9:15 - 10:15    Cairnes Lecture Theatre

Martin Kilian (University College Cork):
Flows of constant mean curvature surfaces

A surface has constant mean curvature if it extremizes the area subject to a volume constraint. Integrable systems techniques have recently made it possible to study the moduli space of such surfaces. In this talk I will outline some of the recent progress made in applying flow techniques to solve classification problems.

10:15 - 11:15    Kirwan Lecture Theatre

Lars Olsen (University of St Andrews):
Multifractals, Baire category and prevalence

Measures with widely varying intensities often occur naturally, e.g. the occupation measure on a strange attractor often has widely varying intensity. Such measures are called multifractals and have recently been the focus of much attention in the mathematics literature. In particular, during the past 15 years there has been an enormous interest in computing the so-called multifractal spectra and generalized dimensions of such measures.
  This talk will survey some of these results and discuss the multifractal structure of ``generic" measures. In particular, the talk will focus on two different notions of ``generic", namely:
  1) the classical topological notion of ``generic" from the 1920's based on Baire category;
  2) a very recently measure theoretical notion of ``generic", called prevalence, from the 1980's due to Christensen and, independently, Hunt, Sauer & Yorke.

10:15 - 11:15    Cairnes Lecture Theatre

Ian Leary (Ohio):
An infinite Smith group

In the 1930's, P A Smith proved a strong fixed-point theorem for actions of finite groups of prime power order on topological spaces. Recently, a rather large team has constructed an infinite group with a similar property, answering a question posed in the 1990's by P H Kropholler. I shall state the P A Smith theorem, outline subsequent developments, and explain some aspects of the construction of our group.