Essentially, a module is a mathematical object acting linearly. We can take the object acting to be a ring and the object acted upon to be an abelian group; each element of the ring should act as a group-structure-preserving map and the actions of the various elements of the ring should reflect the addition and multiplication in the ring.
If the ring is acutally a field of scalars then we have the notion of a vector space. Allowing a ring R in place of the field gives us the more general notion of an R-module.
Collections of linear actions, hence modules, arise all over mathematics and a central theme in my research has been the effort to understand modules and how they fit together.
The most basic way in which they fit together is given by fixing a ring R, taking all the R-modules and all the linear maps between these and regarding this as a single (huge) mathematical structure - the category of R-modules.
If R is a field then this category can be understood rather well because the structure of a vector space is determined by its vector-space dimension and the maps between vector spaces are uncomplicated. This is misleadingly simple since, in general, it is impossible to describe explicitly all actions of a ring. In general, therefore, we need ways of understanding the structure of the category of R-modules. This might involve applying measures of complexity or defining associated mathematical structures (algebraic, geometric, topological) from which we can obtain information about individual modules or the whole category of modules.
The most important modules, those which are "natural", which arise in connections with other parts of mathematics, are usually identifiable in terms of the category structure. That is, forming a category from all R-modules gives us a way of organising a situation which is too complex to understand in complete detail, while keeping most significant structure.
In practice we often want to vary the ring, perhaps in a discrete way but often also in some way which reflects some kind of underlying mathematical object such as a parameter space. As we vary the ring, the associated category of modules will vary and we can try to understand the effect of this.
As well as algebraic methods, techniques from model theory and from category theory have proved to be effective in understanding categories of modules.
There are three main directions to my current research.
- Understanding the category of modules, as understood through the Ziegler spectrum and associated structures, over specific rings, especially over finite-dimensional algebras. The Ziegler spectrum arose in the model theory of modules and the techniques used here combine methods of algebra (ring and module theory, representations of quivers, homological algebra) and the model theory of modules.
- Extending what has been achieved for module categories, using functor-category methods and the model theory of modules, to other additive categories (Grothendieck categories, definable categories, triangulated categories). There is a kind of non-commutative geometry underlying this.
- Developing a model theory which combines "classical" model theory and accessible category theory/topos theory. It turned out that classical model theory and category-theoretic model theory link together very well when applied to additive structures; the additive case seems to provide a good guide to linking these in much more general situations.