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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.

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.

Ice streams are narrow bands of rapidly sliding ice within an otherwise

slowly flowing continental ice sheet. Unlike the rest of the ice sheet,

which flows as a typical viscous gravity current, ice streams experience

weak friction at their base and behave more like viscous 'free films' or

membranes. The reason for the weak friction is the presence of liquid

water at high pressure at the base of the ice; the water is in turn

generated as a result of dissipation of heat by the flow of the ice

stream. I will explain briefly how this positive feedback can explain the

observed (or inferred, as the time scales are rather long) oscillatory

behaviour of ice streams as a relaxation oscillation. A key parameter in

simple models for such ice stream 'surges' is the width of an ice stream.

Relatively little is understood about what controls how the width of an

ice stream evolves in time. I will focus on this problem for most of the

talk, showing how intense heat dissipation in the margins of an ice stream

combined with large heat fluxes associated with a switch in thermal

boundary conditions may control the rate at which the margin of an ice

stream migrates. The relevant mathematics involves a somewhat non-standard

contact problem, in which a scalar parameter must be chosen to control the

location of the contact region. I will demonstrate how the problem can be

solved using the Wiener-Hopf method, and show recent extensions of this

work to more realistic physics using a finite element discretization.

A form of PDE-constrained inversion is today used as an engineering tool for seismic imaging. Today there are some successful studies and good workflows are available. However, mathematicians will find some important unanswered questions: (1) robustness of inversion with highly nonconvex objective functions; (2) scalable solution highly oscillatory problem; and (3) handling of uncertainties. We shall briefly illustrate these challenges, and mention some possible solutions.

One of the main obstacles to forecasting sea level rise over the coming centuries is the problem of predicting changes in the flow of ice sheets, and in particular their fast-flowing outlet glaciers. While numerical models of ice sheet flow exist, they are often hampered by a lack of input data, particularly concerning the bedrock topography beneath the ice. Measurements of this topography are relatively scarce, expensive to obtain, and often error-prone. In contrast, observations of surface elevations and velocities are widespread and accurate.

In an ideal world, we could combine surface observations with our understanding of ice flow to invert for the bed topography. However, this problem is ill-posed, and solutions are both unstable and non-unique. Conventionally, this problem is circumvented by the use of regularization terms in the inversion, but these are often arbitrary and the numerical methods are still somewhat unstable.

One philosophically appealing option is to apply a fully Bayesian framework to the problem. Although some success has been had in this area, the resulting distributions are extremely difficult to work with, both from an interpretive standpoint and a numerical one. In particular, certain forms of prior information, such as constraints on the bedrock slope and roughness, are extremely difficult to represent in this framework.

A more profitable avenue for exploration is a semi-Bayesian approach, whereby a classical inverse method is regularized using terms derived from a Bayesian model of the problem. This allows for the inclusion of quite sophisticated forms of prior information, while retaining the tractability of the classical inverse problem. In particular, we can account for the severely non-Gaussian error distribution of many of our measurements, which was previously impossible.

Deciding whether or not two elements of a group are conjugate might seem like a trivial problem. However, there exist finitely presented groups where this problem is undecidable: there is no algorithm to output yes or no for any two elements chosen. In this talk Houghton groups (a family of groups all having solvable conjugacy problem) will be introduced as will the idea of twisted conjugacy: a generalisation of the conjugacy problem where an automorphism is also given. This will be our main tool in answering whether finite extensions and finite index subgroups of any Houghton group have solvable conjugacy problem.