I am very interested in taking PhD students (British or otherwise) and would welcome informal approaches by email from anybody who would like more information. Most of my students since 1995 have been studying the spectral theory of non-self-adjoint Schrodinger operators or the properties of resonances, and some have done bits of numerical computation to support their theoretical work. See Analysis for further information about the research group.
It is very hard to understand what might be involved in research when you are still an undergraduate, but you will probably spend a good part of your first year reading the book below. This grew out of a graduate course which I have given for many years, and Chapter 2 describes my own research. Before starting this you will need a good understanding of what is in first courses on functional analysis/operator theory.
It is the policy of the Department that students must pass a formal test of some type at the end of their first year, in order to be transferred to Ph D status. In Analysis this normally takes the form of three or so written examinations taken in May. I normally see my graduate students every week or so to discuss their progress on an individual basis.