Research
My research interest lies in low dimensional topology and symplectic topology. In particular I am interested in symplectic normal crossing divisors and its applications to symplectic topology, for instantce symplectic fillings, contact structures, Hamiltonian torus actions, etc. Recently, I have also been thinking about symplectic cohomology and birational geometry of affine varieties.
Papers and preprints: (arXiv, GoogleScholar)
[6] Almost complex geometry of symplectic log Calabi-Yau pairs with applications to almost toric fibrations (with Shengzhen Ning) Submitted. Preprint.
Given a symplectic log Calabi-Yau pair, we study the almost complex structures that are adapted to it. For these almost complex structures, we study the behavior of the curve cone, prove a Nakai-Moishezon criterion and describe the almost K\"{a}hler cone for the generic ones. As an application, we prove a decomposition theorem for Seiberg-Witten non-trivial classes. This leads to an alternative proof of the existence of almost toric fibrations on $c_1$-positive rational surfaces shown in [5].
[5] Almost toric presentations of symplectic log Calabi-Yau pairs (with Tian-Jun Li, Shengzhen Ning) Submitted. arXiv:2303.09964
We show that any symplectic log Calabi-Yau divisor can be realized as the boundary divisor of an almost toric fibration. This realization is canonical once we choose an extra data called the framing on the space of LCY. This is achieved by considering the symplectic analogue of the toric model used in the algebraic geometrical settings.Â
[4] Finite group actions on symplectic rational surfaces (with Weiwei Wu, Shuo Zhang) In preparation.
[3] Enumerative aspect of symplectic log Calabi-Yau divisors and almost toric fibrations (with Tian-Jun Li, Shengzhen Ning) To appear Israel Journal of Mathematics. arXiv:2203.08544
we are interested in the isotopy classes of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi-Yau divisors in a restrictive region of c1-nef cone. A detailed count in the case of 2- and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant polygons is reduced to the arrangement of homology classes. Then we study its relation with almost toric fibrations. We raise the problem of realizing all symplectic log Calabi-Yau divisors by some almost toric fibrations and verify it together with another conjecture of Symington in a special region.
[2] Circular spherical divisors and their contact topology (with Tian-Jun Li, Cheuk Yu Mak) To appear Communications in Analysis and Geometry. arXiv:2002.10504
This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors D that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such D. When D is anticanonical and convex, we give explicit betti number bounds for Stein fillings of its boundary contact torus bundle.
[1] Local geometry of symplectic divisors with applications to contact torus bundles (with Tian-Jun Li) To appear Proceedings of the International Congress of Chinese Mathematicians (Beijing 2019). arXiv:2101.05981 (arxiv version contains slightly more stuff than the published version)
In this note we study the contact geometry of symplectic divisors. We show the contact structure induced on the boundary of a divisor neighborhood is invariant under toric and interior blow-ups and blow-downs. We also construct an open book decomposition on the boundary of a concave divisor neighborhood and apply it to the study of universally tight contact structures of contact torus bundles.
This is my PhD thesis, which contains mostly preliminary results from papers [1], [2], [3] above. But it also contains a bit more stuff on achiral Lefschetz fibrations.
Work in progress:.
Maximal divisors and relative Kodaira dimension (with Tian-Jun Li).
Contact cut complex and Weinstein L-invariant (with Nick Castro, Gabe Islambouli, Sumeyra Sakalli, Laura Starkston and Angela Wu)
Expository writings:
Quantitative transversality and symplectic topology. MPhil thesis.
Symplectic caps and fillings.
Convex symplectic manifolds.
Research Talks:
Circular spherical divisors and contact topology, AMS sectional meeting, Cincinnatti, April 2023
Moduli space of symplectic log Calabi-Yau divisors and torus fibrations, Fields Institute, Jan 2023 video
Moduli space of symplectic log Calabi-Yau divisors and torus fibrations, 2nd Youth Forum, IGP USTC, Dec 2022
Moduli space of symplectic log Calabi-Yau divisors and torus fibrations, University of Massachusetts Amherst, Sep 2022
Symplectic log Calabi-Yau divisors and almost torus fibrations, University of Arkansas, Mar 2022
Symplectic divisors in dimension 4, Oberseminar, MPIM, Nov 2021
Moduli space of symplectic log Calabi-Yau divisors and torus fibrations, Freemath Seminar, Oct 2021 video
Moduli space of symplectic log Calabi-Yau divisors and torus fibrations, University of Minnesota, Oct 2021
Circular spherical divisors and contact topology, Shanghai Jiaotong University, July 2021
Symplectic log Calabi-Yau surfaces - contact aspects, AMS sectional meeting, Purdue, April 2020 slides
Graphs and symplectic capping, MAA North Central Sectional Meeting, Oct 2019 slides
Selected Expository Talks:
What is a symplectic cobordism, TWIGS, University of Massachusetts Amherst, Sep 2022
Introduction to Fukaya categories, MPIM topology seminar, Jan 2022
Symplectic birational geometry (short talk), MPIM topology seminar, Nov 2021
Symplectic fillings of ADC manifolds, UMN Student symplectic seminar, April 2021
Growth rate of symplectic cohomology, UMN Student symplectic seminar, Feb 2021
Immersed Lagrangian sphere and SYZ mirror symmetry for Grassmannians, Symplectic cut seminar, KCL, Nov 2020 (online) slides
Simple homotopy equivalence of nearby Lagrangians, Symplectic cut seminar, KCL, July 2020 (online) slides
Wall crossing in SYZ mirror symmetry, Symplectic zoom learning seminar, July 2020 (online)
Toric degeneration and potential function in $ S^2\times S^2 $, UMN Student symplectic seminar, June 2020
Symplectic fillings of simple singularities, UMN student symplectic serminar, Nov 2019
Symplectic fillings of Seifert fibered spaces, UMN student symplectic seminar, Sep 2018
Lecture Series on Fukaya category, UMN student symplectic seminar, Spring 2018
Symplectic Kodaira Dimension 0, Kylerec student workshop on symplectic geometry, Truckee, May 2017