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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.

"Wildcat" is my name for most of my social accounts, and I'm happy with any nicknames generated from this(if it sounds good).

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Here is some of my unpublished notes(English) and links to seminars.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