Postgraduate Opportunities

I have supervised 19 students to PhD and 22 to MSc degrees. Currently I supervise 2 students in mathematics:  their projects range from model theory through representation theory to category, theory but all involve modules and share a fairly extensive conceptual and technical background.

Keywords

• Modules, representations of algebras, model theory, Ziegler spectrum, abelian and triangulated categories, geometric representation theory, model theory in categories, non-commutative geometry.

Research Group(s)

• Algebra
• Mathematical Logic

Professor of Pure Mathematics, School of Mathematics

My mathematical focus is the category of modules over a ring, and similar categories.  Usually these are far too complicated to understand in their entirety so we concentrate on the more interesting parts.  We also look for associated structures, which might be algebraic, logical, topological or geometric, that organise some of the information in the category and which reflect the complexity of the category.  So this is a mixture of algebra, model theory, category theory, ....  The interaction between these various parts of mathematics is, for me, an important part of the fun.

My interest in algebra and model theory formed while I was an undergraduate at the University of Aberdeen.  Given these interests, Leeds was a natural choice for where to go to work for a PhD.  After a couple of years lecturing at Leeds, a Science Research Council fellowship at Bedford College (London) and three years lecturing in the USA (2 years at Northern Illinois University, 1 year at Yale), I returned to spend three years as a University Research Fellow at the University of Liverpool and finally ended up here.

For my research and teaching see other parts of this, and my own webpage.  My main public engagement activity was involvement with the Greater Manchester Mathematics Challenge over the thirteen years of its existence.  My main "service" activities, apart from the usual refereeing, reviewing, writing of references, have been ten years as an editorial adviser for the journals of the London Mathematical Society (LMS) and being LMS regional organiser for the North of England.

My research lies within pure mathematics, specifically algebra, with more than a nod towards model theory and category theory.

For many mathematical structures there is a natural notion of action on other structures; if these other structures have an underlying addition then we have a module.  All the possible modules for the first-mentioned structure, together with the structure-preserving maps between these modules, form a "category".  This is itself a structure which we can try to understand.  Sometimes this is easy: usually it is impossible.  At least, it is impossible to gain complete understanding which, in any case, is probably more than we could possibly find use for.  So we focus on the interesting parts and on efficient ways of organising the most important information.

That's all very general, as is appropriate for a front page like this. For specifics see my research and publication webpages (there are some expository/survey as well as research papers there).