Research
Some of the questions that I have been thinking about:
- Toric mirror symmetry. For example, understanding derived equivalence between different crepant resolutions of affine toric Gorenstein singularities (categorification of GKZ systems), unification of different combinatorial mirror constructions (Batyrev-Borisov and Berglund-Hübsch-Krawitz), relations to tropical geometry, etc.
- Derived categories of algebraic varieties. For example, questions related to Kuznetsov’s Homological Projective Duality, categorical resolutions and noncommutative resolution of singularities.
- Bridgeland stability conditions and their relations to mirror symmetry.
- Grothendieck ring of varieties, D- and L- equivalences.
Summary of my research papers
Hanlon-Hicks-Lazarev resolution revisited, with Lev Borisov.
Hanlon, Hicks and Lazarev constructed resolutions of structure sheaves of toric substacks by certain line bundles on the ambient toric stacks. In this paper, we give a new and substantially simpler proof of their result.
Stringy Hodge numbers of Pfaffian double mirrors and Homological Projective Duality.
In this paper we study the so-called generalized Pfaffian double mirrors, which are predicted to share the same mirror family, hence are expected to have equivalent derived categories (or strictly speaking, certain categorical resolutions) and equal Hodge numbers (interpreted in terms of Batyrev’s stringy Hodge numbers). The odd-dimensional cases are well-behaved and both statements are established (c.f. Rennemo-Segal and Borisov-Libgober), however the even-dimensional cases are more subtle. More precisely, to obtain the desired equality of Hodge numbers, one has to modify the discrepancies of a log resolution of Pfaffian varieties. It would be interesting to understand the modification conceptually, and ideally to replace it with a less ad-hoc construction.
Central charges in local mirror symmetry via hypergeometric duality.
A byproduct of an (unsuccessful) attempt to construct a global integral structure for bbGKZ systems. We proved that certain A-brane integral structure defined by homology group of the complement of an affine hypersurface in algebraic torus is equivalent to the B-brane integral structure defined by the Grothendieck group of the corresponding toric CY.
Analytic continuation of better-behaved GKZ systems and Fourier-Mukai transforms.
On hypergeometric duality conjecture, with Lev Borisov.
We construct explicit non-degenerate pairing between solution spaces of bbGKZ systems in arbitrary dimensions, which can be seen as a globalization of the Euler pairing on derived categories of toric CYs.
On duality of certain GKZ hypergeometric systems, with Lev Borisov and Chengxi Wang.
An undergrad research project. We settled the hypergeometric duality conjecture in dim 2 by a direct computation.