# Symplectic Geometry Study Group

Spring 2022

Friday, 10:00am-11:30am

Co-organize with Bart Van Steirteghem and David Pham

Our seminar will be online in the Fall semester 2021. One of the goals is to understand Gromov’s nonsqueezing theorem.

Please email Fei Ye for the meeting link if you are interested in attending the seminar.

## Past Seminars

Date: Friday, May 13, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 8

Date: Friday, May 6, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 7

Date: Friday, April 29, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 6

Date: Friday, April 15, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 5

Date: Friday, April 8, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 4

Date: Friday, April 1, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 3

Date: Friday, March 25, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 2

Date: Friday, March 18, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Differential Topology 1

Date: Friday, March 11, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Equivariant Symplectic Geometry 3

Date: Friday, March 4, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Equivariant Symplectic Geometry 2

Date: Friday, February 25, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: TBA

Title: Equivariant Symplectic Geometry 1

Date: Friday, February 18, 2022

Time: 9:30am-11:00am

Room: Online (Email Fei for the Zoom link)

Speaker: David Pham (QCC)

Title: A Short Introduction to Moment Map

Date: Tuesday, February 8, 2022

Time: 10:00am-11:30am

Room: Online (Email Fei for the Zoom link)

Speaker: David Pham (QCC)

Title: Organizational Meeting

Abstract: We will discuss what to learn and plan the seminars for this semester.

Date: Thursday, November 11, 2021

Time: 3:00pm-4:00pm

Room: Online (Email Fei for the Zoom link)

Speaker: David Pham (QCC)

Title: The Lie group $\mathrm{Sp}(2n,\mathbb{R})$

Abstract: We review some results concerning the Lie group of symplectic matrices.

Date: Thursday, October 21, 2021

Time: 3:00pm-4:00pm

Room: Online (Email Fei for the Zoom link)

Speaker: David Pham (QCC)

Title: Neighborhood Theorem I

Abstract: I will prove a standard neighborhood theorem from symplectic geometry using Moser’s trick. Specifically, let $M$ be a manifold and let $Q$ be a submanifold of $M$. Suppose $\omega_0$ and $\omega_1$ are two symplectic forms on $M$ such that $\omega_0 = \omega_1$ on $TM|_Q$. I will show there exists an (open) neighborhood $W$ of $Q$ and a diffeomorphism $F: W \to W$ which is the identity on $Q$ and satisfies $F^*\omega_1=\omega_0$. In the special case where $Q$ is a point, we obtain Darboux’s theorem as a corollary. Following the talk there will be a free discussion.

Date: Thursday, October 7, 2021

Time: 3:00pm-4:00pm

Room: Online (Email Fei for the Zoom link)

Speaker: Fei Ye (QCC)

Title: Hamiltonian and Symplectic Vector Fields

Abstract: In this talk, we will discuss relations between Hamiltonian and symplectic fields as well as their properties and applications.

Date: Thursday, September 23, 2021

Time: 3:00pm-4:00pm

Room: Online (Email Fei for the Zoom link)

Speaker: David Pham (QCC)

Title: Moser's Trick & Darboux's Theorem

Abstract: In this talk, we will begin with a quick review of vector field flows and time-dependent vector fields which will serve as the necessary background for Moser’s trick. After covering Moser’s trick, we will use it to prove Darboux’s theorem. Darboux’s theorem is the statement that all symplectic manifolds of the same dimension are locally isomorphic. Hence, symplectic manifolds have no local invariants.

Date: Thursday, September 9, 2021

Time: 3:00pm-4:00pm

Room: Online (Email Fei for the Zoom link)

Speaker: David Pham (QCC)

Title: Organization Meeting

Abstract: I will explain Gromov’s nonsqueezing theorem. We will plan the seminar and discuss topics and questions that we would like to study. Participants are encouraged to bring in their own research interests in symplectic geometry for discussion.