# Symplectic Geometry Study Group

Fall 2021

Thursday, 3:00pm-4:00pm

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.

## Upcoming Seminars

Date: November 11, 2021

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.

## Past Seminars

Date: October 21, 2021

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: October 7, 2021

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: September 23, 2021

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: September 9, 2021

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.