University of Oxford

The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space.

I'll discuss work of Wise and Ollivier-Wise that gives cubulations of certain small cancellation and random groups, which in turn shows that they do not have property (T).

I will be looking at some conjectures and theorems closely related to the h-cobordism theorem and will try to show some connections between them and some group theoretic conjectures.

If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations, that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs in this setting and outline some novel applications of this approach.

I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. The proof is based on an extension theorem for probability measures that can be considered as a generalization of a Girsanov-type change of measure.

In the second part of my talk I will illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.

Let F be a free group, and N a normal subgroup of F with derived subgroup N'. The Magnus embedding gives a way of seeing F/N' as a subgroup of a wreath product of a free abelian group over over F/N. The aim is to show that the Magnus embedding is a quasi-isometric embedding (hence "Q.I." in the title). For this I will use an alternative geometric definition of the embedding (hence "picture"), which I will show is equivalent to the definition which uses Fox calculus. Please note that we will assume no prior knowledge of calculus.

I will present the basics of mathematical finance, and what probabilists do there. More specifically, I will present the basic concepts of replication of a derivative contract by trading, market completeness, arbitrage, and the link with Backward Stochastic Differential Equations (BSDEs).

I will explain a construction of a group acting on a rooted tree, related to the Grigorchuk group. Those groups have exponential growth, at least under certain circumstances. I will also show how it can be seen that any two elements fulfil a non-trivial relation, implying the absence of non-cyclic free subgroups.

The study of free groups and their automorphisms has a long pedigree, going back to the work of Nielsen and Dehn in the early 20th century, but in many ways the subject only truly reached maturity with the introduction of Outer Space by Culler and Vogtmann in the “Big Bang” of 1986. In this (non-expert) talk, I will walk us through the construction of Outer Space and some related complexes, and survey some group-theoretic applications.