I study algebraic and geometric objects connected to mathematical physics, using tools from deformation theory and cohomology, algebraic differential operators, representation theory, and algebraic geometry. I am particularly interested in symplectic resolutions and singularities, which give rise to a vast new generalisation of the representation theory of semisimple Lie algebras. I also like log symplectic manifolds, which are mild generalisations of symplectic manifolds, which arise naturally in moduli spaces. These are all Poisson varieties, meaning they carry the mathematical structure coming from Hamiltonian mechanics. In my work I have invented new homology theories (eg Poisson-de Rham) and approaches to them, defined new nondegeneracy conditions ("holonomicity") on Poisson varieties, and applied them and other algebraic and geometric techniques to classify resolutions and deformations. I have also studied Hochschild (co)homology of algebras and orbifolds, algebraic differential operators, A-infinity and L-infinity algebras, preprojective algebras and their generalisations, and various constructions in noncommutative algebraic geometry.

From 2008--2013, I was a five-year fellow of the American Institute of Mathematics, and I received NSF standard grants.