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  • My research is on various aspects of the interaction between representation theory, algebraic geometry and topology. In a very rough sense, I'm interested in both diagrammatic and geometric approaches to representation theory. We tend to think of representation theory as based around classical objects like Lie algebras or the symmetric group (and indeed it is), but we can gain insight on these categories by understanding them in different ways.

    One approach I'm very interested in is using the geometry of symplectic singularities. These are special affine varieties, one example of which is the variety of nilpotent square matrices. Using techniques combining algebraic geometry and non-commutative ring theory, one can assign an interesting non-commutative algebra to such a variety. In the case of nilpotent matrices, you'll end up with the universal enveloping algebra of a Lie algebra.

    However, there are lot of other interesting examples, such as rational Cherednik algebras, and other things that seem to be new to science. Some of these come out of toric geometry and the theory of quiver varieties. The general theory of these algebras is still being worked out, and much remains to be understood in special cases as well.

    Another approach that might seem diametrically opposite to this one, but which is actually closely related, is the diagrammatic perspective on representation theory. It's recently been realized that lots of interesting algebras or categories, including some with no previously known presentation, have unexpected presentations which can be prescribed with planar diagrams. This has created a large bestiary of interesting algebras based on ideas of Khovanov, Lauda and Rouquier and have connections to knot invariants, MV polytopes, canonical bases and Cherednik algebras.