My research lies in the areas of noncommutative geometry, algebraic geometry, symplectic geometry and knot theory.

I have developed a system of noncommutative calculus, which puts in a unified framework various notions, such as the noncommutative de Rham complex, the necklace Lie algebra, double Poisson algebras, Calabi-Yau completions, etc, and explain their close analogy (and indeed direct relations) with their commutative counterpart.

In another project, I have also studied the problem of the change in derived categories under flips and flops. In particular, I have developed the notion of homological flips and homological flops. This allows one to prove that under a certain class of 3-fold flips of type A, the derived categories are related by a fully faithful functor. Hopefully this result can be extended to more general cases.

In another project, I have also studied a knot invariant called knot contact homology. More specifically, in collaboration with Yu. Berest and A. Eshmatov, we developed an algebraic framework called homotopy braid closure that produces knot invariants out of braid group actions of a certain form. When a certain braid group action constructed by S. Gelfand, R. MacPherson and K. Vilonen is put into this framework, the result is an extension of knot contact homology, and is expected to be derived Morita equivalent to a certain partially wrapped Fukaya category. Knot contact homology is also conjectured to be closely related to the colored HOMFLY polynomials. I wish to explore some of these relations in the future.