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.

We saw earlier that a subquadratic isoperimetric inequality implies a linear one. I will give examples of groups, due to Brady and Bridson, which prove that this is the only gap in the isoperimetric spectrum. 

We are pleased to announce that Andrew Wiles will present the inaugural Oxford Mathematics Christmas Public Lecture. Please register by emailing @email

In this talk, I will introduce an internal, structural 
characterisation of certain convergence properties (Fréchet-Urysohn, or 
more generally, radiality) and apply this structure to understand when 
Stone spaces have these properties. This work can be generalised to 
certain Zariski topologies and perhaps to larger classes of spaces 
obtained from other structures.

It is well known that piecewise smooth signals are approximately sparse in a wavelet basis. However, other sparse representations are possible, such as the discrete gradient basis. It turns out that signals drawn from a random piecewise constant model have sparser representations in the discrete gradient basis than in Haar wavelets (with high probability). I will talk about this result and its implications, and also show some numerical experiments in which the use of the gradient basis improves compressive signal reconstruction.

The Banach-Tarski paradox is a celebrated result showing that, using the axiom of choice, it is possible to deconstruct a ball into finitely many pieces that may be rearranged to build two copies of that ball. In this seminar we will sketch the proof of the paradox trying to emphasize the key ideas.