University of Oxford

We will investigate what one can detect about a discrete group from its profinite completion, with an emphasis on considering geometric properties.

I am going to introduce Thompson's groups F, T and V. They can be seen in two ways: as functions on [0,1] or as isomorphisms acting on trees.

Given a block, b, of a finite group, Alperin's weight conjecture predicts a miraculous equality between the number of isomorphism classes of simple b-modules and the number of G-orbits of b-weights. Radha Kessar showed that the latter can be written in terms of the fusion system of the block and Markus Linckelmann has computed it as an Euler characteristic of a certain space (provided certain conditions hold). We discuss these reformulations and give some examples.

This talk will be an introduction to property (T). It was originally introduced by Kazhdan as a method of showing that certain discrete subgroups of Lie groups are finitely generated, but has expanded to become a widely used tool in group theory. We will take a short tour of some of its uses.

We show the existence of global-in-time weak solutions to a general class of bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.

The talk is based on joint work with John W. Barrett [Imperial College London].

We develop a theory based on relative entropy to show stabilityand uniqueness of extremal entropic Rankine-Hugoniot discontinuities forsystems of conservation laws (typically 1-shocks, n-shocks, 1-contactdiscontinuities and n-contact discontinuities of big amplitude), amongbounded entropic weak solutions having an additional strong traceproperty. The existence of a convex entropy is needed. No BV estimateis needed on the weak solutions considered. The theory holds withoutsmallness condition. The assumptions are quite general. For instance, thestrict hyperbolicity is not needed globally. For fluid mechanics, thetheory handles solutions with vacuum.

Not only does the definition of an (abstract) saturated fusion system provide us with an interesting way to think about finite groups, it also permits the construction of exotic examples, i.e. objects that are non-realisable by any finite group. After recalling the relevant definitions of fusion systems and saturation, we construct an exotic fusion system at the prime 3 as the fusion system of the completion of a tree of finite groups. We then sketch a proof that it is indeed exotic by appealing to The Classification of Finite Simple Groups.

We will introduce both the classical Hanna Neumann Conjecture and its strengthened version, discuss Stallings' reformulation in terms of immersions of graphs, and look at some partial results. If time allows we shall also look at the new approach of Joel Friedmann.