### University of Toronto

## Introduction to Symplectic Geometry, Winter 2018-19

(MAT1344HS)

Student presentations summaries

Lecture notes
by Jesse Frohlich.
If you see anything confusing or wrong it's most likely Yael's fault.
Make sure to ask.

Partial lecture notes, by Yael.
Updated March 18, 2019.

**Lectures:** Mondays 12-2 in **HU1018**;
Wednesdays 10-11 in **BA B026**.

**Instructor:** Yael Karshon

**Email:** karshon@math.toronto.edu

**Office hours:** Wednesday after class, or by appointment.

**Office:** BA 6119

**Syllabus:** This is an introductory course in symplectic
geometry and topology. We will discuss a variety of concepts,
examples, and theorems, which may include, but are not restricted to,
these topics:
Moser's method and Darboux's theorem;
Hamiltonian group actions and momentum maps;
almost complex structures and holomorphic curves;
Gromov's nonsqueezing theorem.

**Prerequisites:** Manifolds and differential forms; homology.

**Requirements for credit:**
Participation, assignments, presentations and written summaries.

** Non-inclusive list of relevant books:**
Anna Cannas da Silva,
Lectures on Symplectic Geometry,
corrected 2nd printing, 2008.
Eckhard Meinrenken, lecture notes on Symplectic Geometry, on his website.
Dusa McDuff and Dietmar Salamon, "Introduction to Symplectic Topology",
3rd Edition, 2017.
Helmut Hofer and Eduard Zehnder,
"Symplectic invariants and Hamiltonian dynamics", 1994.
Victor Guillemin and Shlomo Sternberg,
"Symplectic Techniques in Physics", 1984.
B. Aebischer, M. Borer, M. Kalin, C. Leuenberger, and H.M.Bach,
"Symplectic Geometry, an introduction based on the seminar in Bern, 1992".
Alan Weinstein, Lectures on Symplectic Manifolds, 1977.
John Lee, "Introduction to Smooth Manifolds", 2nd edition, 2013,
Chapter 22 ("Symplectic Manifolds").

(A Crash course on manifolds
and a Crash course on flows
have been incorporated into Chapter 0 of Yael's partial lecture notes.

A crash course on homology has been incorporated into Chapter 6,
Section 16, of Yael's partial lecture notes.)