# Research in Algebraic Model Theory

My research is based around categories of modules (modules are also referred to as representations: for instance, representations of a group are essentially the same as modules over the group algebra). So, primarily, this is algebra. But my view of modules (and, consequently, the questions that I and my students investigate) is definitely influenced by model theory, category theory and, to an increasing extent, algebraic geometry.

Even if you have not come across the term "module" you surely have come across some examples. Vector spaces are (rather simple) examples, as are abelian groups. Representations of algebras, of quivers and of groups all are modules, as are representations of Lie algebras. Modules arise in and are used in many areas of mathematics but my prime interest is in understanding modules per se. Modules are, however, far too varied, numerous and complicated to understand completely in any but the simplest cases. In practice it is only certain kinds of modules, the relations between them, and how they are organised into other structures that are of interest and which are investigated. It is particularly the last feature (the fact that (some) modules can be organised into categorical, topological or geometric structures) which serves as a focus for my research.

Much of my research has centred around the Ziegler spectrum: this is a topological space whose points are certain modules and which carries a great deal of information about the category of modules. It arose from the model-theoretic investigation of modules but turns out to be interesting and useful in a purely algebraic context. There are many open questions and avenues for exploration in connection with this spectrum. If you're interested in this you will need to have/acquire a strong background in algebra and be interested in using ideas from other parts of mathematics as well. Certainly there are many well-defined open problems here (as well as some less-well defined ones which are connected with "non-commutative geometry").

My other programmes of research are less developed. There are two main topics. One is a programme to develop a more functorial approach to model theory. There has been some initial success here in relating categories of imaginaries (these come from model theory) to categories of finitely presented and coherent functors, but there remains much to be done. The other direction, still rather speculative, is connected with varieties and schemes (in the sense of algebraic geometry), both varieties of modules (more precisely, module structures on some vector space) and the application of the scheme approach to the study of module categories and associated structures.

## Members of staff involved

NAME |
Title |
EMAIL @manchester.ac.uk |
PHONE |
LOCATION |
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Prest | Mike | Prof. | MPrest | 0161 27 55875 | 1.120 |