Siyang Liu's Homepage
Welcome to my homepage! I'm a fourth year PhD student at University of Southern California. My advisor is Sheel Ganatra.
My research interest is symplectic geometry and geometric representation theory. For more information, see the "research" page.
Self-identified as cat, and use the name "Wildcat" in social accounts.
Contact me by email.Education
| Year | School | Degree | Description |
| 2020-Now | University of Southern California | PhD in Mathematics | 2020 - current |
| 2016-2020 | Sun Yat-Sen University | Bachelor in Science | In mathematics; Advisor: Prof. Jianxun Hu; Thesis: The Arnold Conjecture for Lagrangian Intersections. |
Seminars & Notes
Here is some of my unpublished notes(English) and links to seminars.
Notes:
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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. - 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.
- 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.
Translation:
- Classifications of Irregular Connections of One Variable by B. Malgrange in 1982.
Seminars:
- Spring 2022
- Summer 2022
- Fall 2022
- Spring 2023
- Summer 2023
- Fall 2023
- Spring 2024
- USC Student Topology Seminar
- Topics: Contact Topology and Convex Srufaces
- USC Student Topology Seminar
- 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.
Teaching
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