Prof Toby Stafford - research
My research interests are in Pure Mathematics. My research is primarily in noncommutative algebra although there are substantial overlaps between this and neighbouring subjects like algebraic geometry and Lie theory.
A particular interest is "noncommutative algebraic geometry.'' This has two complimentary components. The first is to use the techniques and intuition from algebraic geometry to understand the structure of important noncommutative algebras, while the second is to build up a theory of noncommutative geometry itself. Curiously, some of the most substantial applications of this principle occur with projective rather than affine geometry. This is perhaps due to the fact that the techniques of affine geometry tend to rely on techniques like localisation that are rarely availably in the noncommutative universe, whereas the more global and categorical approaches to projective geometry can and have been generalized. We have used this, for example, to classify all noncommutative graded rings of quadratic growth (sometimes called noncommutative curves) in terms of commutative curves. The next step, that of classifying noncommutative surfaces, or graded algebras with cubic growth. A solution of this problem is not in sight, but this has lead to the construction of some fascinating algebras and techniques that can be widely applied.
My other interests are in representation theory, broadly construed. In particular, I am interested in rings of differential operators, finite dimensional Lie algebras and related objects like Cherednik algebras and their spherical subalgebras. These three topics are closely related to each other, and to both noncommutative and commutative algebraic geometry.