Oxford

Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

Our work (which is joint with Simon Smith) began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations.
 

One is the generalisation in which point stabilisers are merely assumed to satisfy min-{\sc N}, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on the socle of~$M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

One way of understanding groups is by investigating their actions on special spaces, such as Hilbert and Banach spaces, non-positively curved spaces etc. Classical properties like Kazhdan property (T) and the Haagerup property are formulated in terms of such actions and turn out to be relevant in a wide range of areas, from the conjectures of Baum-Connes and Novikov to constructions of expanders. In this talk I shall overview various generalisations of property (T) and Haagerup to Banach spaces, especially in connection with classes of groups acting on non-positively curved spaces.

Recent developments in 3-dimensional topological quantum field theory allow us to understand the vector spaces assigned to surfaces as spaces of string diagrams. In the Reshetikhin-Turaev model, these string diagrams live inside a handlebody bounding the surface, while in the Turaev-Viro model, they live on the surface itself. There is a "lifting map" from the former to the latter, which sheds new light on a number of constructions. Joint with Gerrit Goosen.

Oxford Mathematics Public Lectures

MC Escher - Artist, Mathematician, Man 

Roger Penrose and Jon Chapman

This lecture has now sold out

The symbiosis between mathematics and art is personified by the relationship between Roger Penrose and the great Dutch graphic artist MC Escher. In this lecture Roger will give a personal perspective on Escher's work and his own relationship with the artist while Jon Chapman will demonstrate the mathematical imagination inherent in the work. 

The lecture will be preceded by a showing of the BBC 4 documentary on Escher presented by Sir Roger Penrose. Private Escher prints and artefacts will be on display outside the lecture theatre.

5pm

Lecture Theatre 1

Mathematical Institute

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

OX2 6GG

Roger Penrose is Emeritus Rouse Ball Professor at the Mathematical Institute in Oxford

Jon Chapman is Statutory Professor of Mathematics and Its Applications at the Mathematical Institute in Oxford

We give a classification of open Klein topological conformal field theories in terms of Calabi-Yau $A_\infty$-categories endowed with an involution. Given an open Klein topological conformal field theory, there is a universal open-closed extension whose closed part is the involutive version of the Hochschild chains associated to the open part.

This'll be a nice and slow paced introduction to topological quantum field theory in general, and 1-2-3 dimensional theories in particular. If time permits I will explain the spin version of these and their connection to physics. There will be lots of pictures.