Symplectic duality and Hikita conjecture
Symplectic duality is a facinating phenomena matching together (conical) symplectic singularities. One of the predictions of the symplectic duality tells that for symplectically dual varieties the algebra of cohomology of a variety is isomorphic to the ring of function on a schematic fixed points of the dual. We explore the limits of this conjecture together with Vasily Krylov.
1. Hikita-Nakajima conjecture for the Gieseker variety, with V.Krylov, Sel. Math. New Ser. 31, 37 (2025).
2. Hikita conjecture for the minimal nilpotent orbit, Proceedings of the American Mathematical Society, pp. 1 (2020)
Finite group actions on nilpotent orbits
We study actions of finite groups (mostly symmetric group) by inner automorphisms on the nilpotent orbits in Lie algebra of type A. In particular, we showed that minimal nilpotent orbits are distinguished among all the others because for them the corresponding quotient of the resolution will not instantly be terminal. We then proceed to construct a corresponding Q-factorial terminalization.
3. Finite group actions on nilpotent orbits in type A (in preparation), with Misha Feigin and Gwyn Bellamy
Quiver varieties
Together with Sam Lewis we classify all 4-dimensional quiver varieties using combinatorial method called "graph balancing". It so happens (proven by Crawley-Boevey) that all weighted graphs in question can be described using a special set of roots.
4. Nakajima quiver varieties in dimension 4, with S. Lewis, (coming soon)