In Bass-Serre theory, one derives structural properties of groups from their actions on simplicial trees. In this talk, we introduce the theory of groups acting on $\mathbb{R}$-trees. In particular, we explain how the Rips machine is used to classify finitely generated groups which act freely on $\mathbb{R}$-trees.

I will look at some decidability questions for subgroups of Aut($F_n$) for general $n$. I will then discuss semisimple actions of Aut($F_n$) on complete CAT(0) spaces proving that the Nielsen moves will act elliptically. I will also look at proving Aut($F_3$) is large and if time permits discuss the fact that Aut($F_n$) is not Kähler

This talk will give an almost complete proof of the h-cobordism theorem, paying special attention to the sources of the dimensional restrictions in the theorem. If time allows, the alterations needed to prove its cousin, the s-cobordism theorem, will also be sketched.

If G is a semi-simple Lie group, it is known that all lattices
are arithmetic unless (up to finite index) G=SO(n,1) or SU(n,1).
Non-arithmetic lattices have been constructed in SO(n,1) for
all n and there are infinitely many non-arithmetic lattices in
SU(1,1). Mostow and Deligne-Mostow constructed 9 commensurability
classes of non-arithmetic lattices in SU(2,1) and a single
example in SU(3,1). The problem is open for n at least 4.
I will survey the history of this problem, and then describe
recent joint work with Martin Deraux and Julien Paupert, where
we construct 10 new commensurability classes of non-arithmetic
lattices in SU(2,1). These are the first examples to be constructed
since the work of Deligne and Mostow in 1986.

Building a suitable family of walls in the Cayley complex of a finitely
presented group G leads to a nontrivial action of G on a CAT(0) cube
complex, which shows that G does not have Kazhdan's property (T).  I
will discuss how this can be done for certain random groups in Gromov's
density model.  Ollivier and Wise (building on earlier work of Wise on
small-cancellation groups) have built suitable walls at densities <1/5,
but their method fails at higher densities.  In recent joint work with
Piotr Przytycki we give a new construction which finds walls at densites
<5/24.

By Thurston's geometrisation theorem, the complement of any knot admits a unique hyperbolic structure, provided that the knot is not the unknot, a torus knot or a satellite knot. However, this is purely an existence result, and does not give any information about important geometric quantities, such as volume, cusp volume or the length and location of short geodesics. In my talk, I will explain how some of this information may be computed easily, in the case of alternating knots. The arguments involve a detailed analysis of the geometry of certain subsurfaces.

The Nottingham Group of a finite field is an object of great interest in profinite group theory, owing to its extreme structural properties and the relative ease with which explicit computations can be made within it. In this talk I shall explore both of these themes, before describing some new work on efficient short-word approximation in the Nottingham Group, based on the profinite Solovay-Kitaev procedure. Time permitting, I shall give an application to the dynamics of compositions of random power series.

This talk concerns the numerical solution of elliptic partial differential equations posed on general smooth surfaces by the Closest Point Method. Based on the closest point representation of the surface, we formulate an embedding equation in a narrow band surrounding the surface, then discretize it using standard finite differences and interpolation schemes. Numerical convergence of the method will be discussed. In order to solve the resulting large sparse linear systems, we propose a specific geometric multigrid method which makes use of the closest point representation of the surface.