Oxford

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.

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.

In sieve theory, one is concerned with estimating the size of a sifted set, which avoids certain residue classes modulo many primes. For example, the problem of counting primes corresponds to the situation when the residue class 0 is removed for each prime in a suitable range. This talk will be concerned about what happens when a positive proportion of residue classes is removed for each prime, and especially when this proporition is more than a half. In doing so we will come across an algebraic question: given a polynomial f(x) in Z[x], what is the average size of the value set of f reduced modulo primes?