Welcome to my homepage! I received my PhD from University of Southern California. My advisor was Sheel Ganatra. I'm currently working in Rhenish Friedrich-Wilhelms-Universität Bonn.

欢迎来到我的个人主页！我已经逃离南加州大学了，在南加州大学时博士导师是Sheel Ganatra。我现在在德国波恩大学避难。

I'll be a Chapman Fellow at Imperial College of London starting September.

My research interest is symplectic geometry, algebraic geometry and geometric representation theory. For more information, see the "research" page.

我的研究兴趣是辛几何、代数几何和几何表示论。更多信息请见“研究”页面。

Self-identified as cat, and use the name "Wildcat" in social accounts.

在互联网上想要找到我可以寻找名为“野猫"的账号，欢迎加关注一键三连喵！

Education & Career

教育经历

Seminars & Notes

研讨会与笔记

Here is some of my unpublished notes(English) and links to seminars.

以下是我写的一些未发表的笔记（英文）和研讨会链接。

Notes:

笔记：

• ▷▽

  Floer Theory for Lagrangian Intersections

  Abstract:

  In this thesis we give an exposition of Arnold's conjecture on Lagrangian intersections, which was one of the main topics in late 1980s. We will mainly focus on Floer's proof under the topological condition $\pi_2 (P,L)=0$, given in his series of papers, with some slight generalizations made by precessors in most of the parts of the proof.
     In the first section we give a brief introduction to the history of Arnold conjecture, including its origin: the Poincar\'e's last geometric theorem, how it was proposed by Arnold in his 1965 paper \cite{Arnold1965}, and a summary of his first exposition on this problem.
     Then we present Floer's approach to this Arnold conjecture. In the second section we apply variation to an action functional and derive the Cauchy-Riemann equation on a pseudo-holomorphic strip, which indicates the trajectories associated to this action functional. We prove that this trajectories tend to critical points as expected.
     The third section is the main part of this paper. We construct the Floer chain complex and prove that this is exactly a chain complex so that we could take the cohomology. This step requires much techniques from non-linear analysis and some study of non-linear elliptic partial differential equations. Some of the technical part is presented in the appendix.
     Finally in the last section, we show that the given cohomology is independent of the choice of some generic structures along the process we construct the chain complex, so is actually a topological invariant. We could then reduce to the easiest case of a ``classical'' phase space and prove that the Floer cohomology group is isomorphic to the Morse cohomology group, hence proving the Arnold conjecture.
• ▷▽

  拉格朗日相交的Floer理论

  摘要：

  在这篇论文中，我们介绍了Arnold关于拉格朗日相交的猜想，这是20世纪80年代末的一个重要课题。我们主要关注Floer在拓扑条件$\pi_2 (P,L)=0$下的证明，内容基于他的一系列论文，并在大部分证明过程中结合了前人的一些推广结果。
     第一部分简要介绍了Arnold猜想的历史背景，包括其起源——Poincaré的最后几何定理，Arnold在1965年论文中的提出过程，以及他对该问题的首次阐述。
     接下来，我们介绍Floer对Arnold猜想的研究方法。第二部分通过对作用泛函的变分，推导出伪全纯带上的Cauchy-Riemann方程，揭示了与该作用泛函相关的轨道，并证明这些轨道如预期般收敛到临界点。
     第三部分是本文的核心。我们构造了Floer链复形，并证明其确实构成一个链复形，从而可以进行上同调运算。这一步需要大量非线性分析技术以及对非线性椭圆偏微分方程的研究，部分技术细节在附录中给出。
     最后一部分，我们证明所得到的上同调与构造链复形过程中所选取的某些一般结构无关，因此实际上是一个拓扑不变量。最终可以归约到“经典”相空间的最简单情形，并证明Floer上同调群同构于Morse上同调群，从而证明了Arnold猜想。
• From Lie Groups to Group Algebras
  This one is written for student seminar on representation theory of Lie groups. We discuss in this note the relation between representations of Lie groups and of group algebras, and some of the applications of this point of view. We also discuss briefly how to go back from group algebra to Lie groups, leading also to complexification of Lie groups.
• 从李群到群代数
  这篇文章是为李群表示理论的学生研讨会写的。我们在这篇文章中讨论了李群的表示与群代数的表示之间的关系，以及这一观点的一些应用。我们还简要讨论了如何从群代数回到李群，进而得到李群的复化。
• Combinatorics and Topology of Hyperplane Arrangements
  This is a note written for the Graduate Colloquium at USC. The aim of this note is to introduce the concept of hyperplane arrangement, describe its combinatorial structures, and show how it's related to the topology of some good spaces.
• 超平面排列的组合与拓扑
  这是一篇为南加州大学研究生座谈会写的笔记。本文旨在介绍超平面排列的概念，描述其组合结构，并展示其与某些良好空间的拓扑之间的关系。

Translation:

翻译：

• Classifications of Irregular Connections of One Variable by B. Malgrange in 1982.
• 单变量非正规联络的分类，B. Malgrange，1982年。

Seminars:

• Spring 2022
  • MATH 740: Stable Homotopy Theory
• Summer 2022
  • Microseminar 1: Topology of Milnor Fibrations
  • Microseminar 2: Holomorphic Curves
• Fall 2022
• MATH 710: Compact Lie Groups
• Spring 2023
• USC Student Topology Seminar(complementary to USC topology seminar)
• Summer 2023
• Linear Algebraic Groups
• Symplectic Cohomology
• Fall 2023
  • USC Student Topology Seminar(independent)
• Spring 2024
  • USC Student Topology Seminar
    • Topics: Contact Topology and Convex Srufaces
• Summer 2024
  • Hodge Reading Group: in person at Caltech
• Fall 2024
  • GHHPS seminar: online. In order to understand the recent paper by Ganatra-Hanlon-Hicks-Pomerleano-Sheridan.
  • Calabi-Yau structure on wrapped Fukaya category: an online seminar on Symplectic Cohomology and Duality for the Wrapped Fukaya Category.
• Winter 2025
  • Seminar on SYZ mirror symmetry.
  • Co-organizing Symplectic Geometry Seminar at University of Bonn.

Teaching

I was TA for:

• MATH 125 in Fall 2021
• MATH 126 in Spring 2022
• MATH 255 in Spring 2023
• MATH 229 in Fall 2023
• MATH 445 in Spring 2024
• MATH 245 in Summer 2024

I was a grader for:

• MATH 425b in Spring 2024

I'll be Instructor for:

• Introduction to Symplectic Topology in Summer 2026

Travels & Talks

• Sep. 23rd, 2024: Speak at University of Southern California
  • Title: Symplectic Geometry of Hyperplane Arrangements
  • Abstract: Motivated from Heegaard-Floer theory, Lauda-Licata-Manion proposed a conjecture relating symplectic geometry of hyperplane arrangements to representation theory of hypertoric varieties. In this talk, I'll explain our recent work on resolving this conjecture first for hyperplanes with transversal intersection and finally for all possible arrangements. This is based on the joint work with Sukjoo Lee, Yin Li and Cheuk Yu Mak and recent joint work in preparation with Sheel Ganatra, Wenyuan Li and Peng Zhou.
• Sep. 30th, 2024: Speak at Stanford University
  • Title: Symplectic Geometry of Hyperplane Arranngements
  • Abstract: Motivated from Heegaard-Floer theory, Lauda-Licata-Manion conjectured a relation between symplectic geometry of hyperplane arrangements and representation theory of hypertoric varieties. In this talk, I'll explain our recent work on proving this conjecture firstly for hyperplanes with transversal intersections and finally for all possible configurations. This is based on the recent joint work with Sukjoo Lee, Yin Li and Cheuk Yu Mak and work in preparation with Sheel Ganatra, Wenyuan Li and Peng Zhou.
• Oct. 17th, 2024: Speak at University of Minnesota, Minneapolis (online)
  • Title: Symplectic Topology of Hyperplane Arrangements
  • Abstract: In this talk, I’ll explain our recent work studying the symplectic topology of complements of complexified hyperplane arrangements. When hyperplanes intersect transversely, we proved that symplectic handles can be found essentially from the intersection points of the hyperplanes. If time permits, I’ll also discuss the strategy of extending the result to general hyperplane arrangements. This is based on the recent joint work with Sukjoo Lee, Yin Li and Cheuk Yu Mak and work in preparation with Sheel Ganatra, Wenyuan Li and Peng Zhou.
• Oct. 26th, 2024: Speak in AMS Special Session on the Topology and Geometry of Contact and Symplectic Manifolds
  • Title: Degenerations as Localizations for Weinstein manifolds
  • Abstract: In this talk, I'll advertise our recent work in progress on studying degenerations of Weinstein manifolds. We will show that under degeneration, Weinstein manifolds undergo handle detachments where the detaching data is given by cocores that are not fixed by the monodromy action. The corresponding symplectic invariant, e.g. the wrapped Fukaya category, will undergo a localization, or quotient by the monodromy action. As a byproduct, an inductive handle structure of a family of degenerations will also be shown. This is joint work in preparation with Sheel Ganatra, Wenyuan Li and Peng Zhou.
• Dec. 18th, 2024: Speak in Tsinghua-BIMSA symplectic geometry seminar
  • Title: Symplectic Aspects of Representation Theory of Hypertoric Varieties
  • Abstract: In this talk we will discuss a symplectic model for T-equivariant hypertoric category \mathscr{O} as constructed by Braden-Licata-Proudfoot-Webster, which is the Fukaya-Seidel category of complexified complement of hyperplane arrangements associated to the corresponding hypertoric variety. This would provide a categorification of K-theoretic stable envelopes for hypertoric varieties. This is partly based on the recent work collaborated with S. Lee, Y. Li and C. Y. Mak and the work in preparation with S. Ganatra, W. Li and P. Zhou.
• Feb. 12th, 2025: Speak in Purdue Algebraic Geometry Seminar
  • Title: Wonderful Compactifications (pretalk)
  • Abstract: I’ll talk about a wonderful model for resolutions of certain good singularities.
  • Title: Hyperplane Arrangements, Hypertoric Variety, and Representation Theory
  • Abstract: Complements of complex hyperplane arrangements are affine varieties with interesting combinatorial structures and deep relations to geometric representation theory. They encode necessary data for hypertoric varieties coming from algebraic symplectic reduction of quaternion vector spaces by algebraic tori. Braden-Licata-Proudfoot-Webster constructed "hypertoric enveloping algebra" associated to a given toric hypertoric variety and studied the BGG-type category O of this algebra. In the recent joint work with Sukjoo Lee, Yin Li and Cheuk Yu Mak, we showed that universal deformations of the category O can be in fact recovered from the geometry of complexified hyperplane arrangements, in particular simple and standard modules can be realized as certain submanifolds, and endomorphisms provided by intersection points. This also verifies a conjecture of Lauda-Licata-Manion and a proposal by Lekili-Segal.