This is an indication of the area in which my current work is focussed, hence the area in which I would expect to supervise a student.
First, it's algebra with input from model theory and category theory.
The area of algebra is module (= representation) theory, especially representations of algebras.
In extremely general terms, the aim is to understand the structure of the category of modules. This might mean getting a description of some of the most interesting modules and the maps between them or it might mean finding some structure (topological, geometric, algebraic, ...) on a set of these, and investigating that 'larger-scale' structure on (part of) the category of modules.
The input of model theory (part of mathematical logic) in the specific context of the representation theory of finite-dimensional algebras, where interest is typically focussed on finite-dimensional representations, leads us to extend our interest to at least some of the infinite-dimensional representations, even if our eventual applications are back in the context of the finite-dimensional ones. The same general pattern, of looking at (somewhat) 'large' representations, can be seen over algebras which are not finite-dimensional.
Another input of model theory is the concept of interpretation which, in this context, can be seen as a certain kind of functor between categories of modules. Understanding how these link categories of modules is another rather general aim.
My website (www.maths.manchester.ac.uk/~mprest/publications.html) gives more (too much) information but some flavour of the area can be got by browsing around there.
Any offer of a place will include a description of a broad research problem but a specific project will be determined taking account of a variety of factors, in particular, the current state of knowledge and activity in the area and the interests and development of the student. It can also be that the direction of the project changes as it develops, in the light of what is discovered.