My research has mainly been on the representation theory of finite-dimensional associative algebras (see fdlist), and related questions in linear algebra, ring and module theory, and algebraic geometry.
In recent years I have concentrated on representations of quivers and preprojective algebras. A quiver is essentially the same thing as a directed graph, and a representation associates a vector space to each vertex and a linear map to each arrow. The subject was started by P. Gabriel in 1972, when he discovered that the quivers with only finitely many indecomposable representations are exactly the ADE Dynkin diagrams which occur in Lie theory (for example a quiver of type E6 is illustrated on the left). Quivers and their representations now appear in all sorts of areas of mathematics and physics, including representation theory, cluster algebras, geometry (algebraic, differential, symplectic), noncommutative geometry, quantum groups, string theory, and more.
The preprojective algebra associated to a quiver was invented by I. M. Gelfand and V. A. Ponomarev. Its modules are intimately related to representations of the quiver, but it is often the modules for the preprojective algebra which are of relevance in other parts of mathematics. There is beautiful geometry linked to the preprojective algebra, including Kleinian singularities and H. Nakajima's quiver varieties. (The illustration on the right shows the real-valued points of varieties associated to a quiver of extended Dynkin, that is, affine, type D4.)
There are also links between the preprojective algebra and the classification of differential equations on the Riemann sphere. They are used in work on the Deligne-Simpson problem, which concerns the existence of matrices in prescribed conjugacy classes whose product is the identity matrix, or whose sum is the zero matrix. (The picture on the left shows loops on the punctured Riemann sphere which generate its fundamental group. Consideration of the monodromy around such loops links the classification of differential equations on the Riemann sphere to the Deligne-Simpson problem.)
In earlier times I was interested in tame algebras, matrix problems, and infinite-dimensional modules. Finite dimensional associative algebras naturally divide into three classes: algebras finite representation type with only finitely many indecomposable modules, wild algebras for which the indecomposable modules are unclassifiable (in a suitable sense), and those on the boundary between these classes, the tame algebras. There are many interesting classes of tame algebras, and it is often a major problem to actually give the classification of the indecomposable modules.
One way to study tame algebras is to convert the problem of classifying their modules into a matrix problem: putting a partitioned matrix into canonical form using not all elementary operations, but a subset defined by the partition. (The illustration on the right shows what an arbitrary matrix can be reduced to if you allow all row and column operations; it also shows an example of a partition of a matrix.) Using advanced methods based on this idea, Yu. A. Drozd proved his wonderful Tame and Wild Theorem showing that there is a wide gulf between the behaviour of tame and wild algebras. The same methods can be used to show that tame algebras are characterized by the behaviour of their infinite-dimensional modules. In fact, the behaviour of infinite-dimensional modules for tame algebras is extremely interesting, and not at all understood.