Research

My research interest is symplectic geometry, in particular computations of symplectic invariants, e.g. symplectic cohomology, (partially) wrapped Fukaya category, in good cases, with the hope of getting some new symplectic topological results. Here're descriptions of my research projects.

No.	Title
	Description
1	(with Sukjoo Lee, Yin Li and Cheuk-Yu Mak) Hypertoric Convolution Algebras as Fukaya Categories II. in Preparation.
	To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (or equivalently, a Liouville sector) $M(\mathbb{V})$, whose underlying manifold is the complement of the union of the complexified hyperplanes in the arrangement, endowed with a Liouville structure induced by a very affine embedding, and the stop is determined by the polarization. In this article, we study the symplectic topology of these stopped Liouville manifolds. In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers in the hyperplane arrangement. A computation of the Fukaya $A_\infty$-algebra of the generating objects then enables us to identity these wrapped Fukaya categories with $\mathbb{G}_m^d$-equivariant convolution algebras $\widetilde{B}(\mathbb{V})$ associated to $\mathbb{V}$. This confirms a conjecture of Lauda-Licata-Manion [LLM] and Lekili-Segal [LS].
2	(with Sheel Ganatra and Wenyuan Li) Degeneration as Localization for Wrapped Fukaya Categories. In Preparation.
	When a variety is obtained from a family of "smoother" varieties, we can subtract out information of this variety from its nearby "general fibers", together with the action of the topology of the underlying parameter spaces. In this paper, we explore this philosophy in the symplectic category. More precisely, we showed that when a Weinstein sector is a degeneration of better Weinstein sectors, then its (wrapped) Fukaya category is obtained from the Fukaya category of the nearby sector localizing the induced monodromy action. As an application, we derive an analogue of Seidel's long exact sequence in some special cases and finish the conjecture of Lauda-Licata-Manion [LLM].
3	(with Peng Zhou) Fukaya-Seidel Category of Noncritical Landau-Ginzburg Models. In Progress.

4	(with Junxiao Wang) Gamma Conjecture for Very Affine Hypersurfaces. In Progress.

5	Local-to-Global Mirror Symmetry for Complements of Very Affine Hypersurfaces. In Progress.

References

[A] Auroux, D. (2010). Fukaya categories of symmetric products and bordered Heegaard-Floer homology. arXiv preprint arXiv:1001.4323.
[BLPW] Braden, T., Licata, A., Proudfoot, N., & Webster, B. (2010). Gale duality and Koszul duality. Advances in Mathematics, 225(4), 2002-2049.
[GPS2] Ganatra, S., Pardon, J., & Shende, V. (2018). Sectorial descent for wrapped Fukaya categories. arXiv preprint arXiv:1809.03427.
[LLM] Lauda, A. D., Licata, A. M., & Manion, A. (2020). From hypertoric geometry to bordered Floer homology via the m= 1 amplituhedron. arXiv preprint arXiv:2009.03981.
[McL] McLean, M. (2012). The growth rate of symplectic homology and affine varieties. Geometric And Functional Analysis, 22, 369-442.
[Sei06] Seidel, P. (2006). A biased view of symplectic cohomology. In Current developments in mathematics, 2006 (Vol. 2006, pp. 211-254). International Press of Boston.