MAT 562, Fall 2026

MAT 562: Symplectic Geometry and Topology

Stony Brook Fall 2026

General Course Information, A Proof and a Modern ``Proof"

References

All readings listed in the table below are from Foundations of Symplectic Geometry (work in progress, comments/corrections are encouraged; beginning of Appendix A.1 expanded 10/22 with existence of limits of flows), except as noted; please read the background material contained in Chapter 1 and Appendicies A,B as needed. MS refers to McDuff&Salamon's Introduction to Symplectic Topology, 3rd Ed. Warner refers to his Foundations of Differentiable Manifolds and Lie Groups. EG Notes refers to this.

Tentative Schedule

Date	Topic	Read	Problem Set
8/24, M	no lecture	MS Intro & Chapters 1,2 EG Notes Sections 1,4	ps1 solutions
8/26, W	Examples of symplectic manifolds	Sections 1.1/5
8/31, M	Connections and flows of vector fields	Sections 2.1/2, B.1/2, MS pp94-98 Warner pp36-41,69-72
9/2, W	Hamiltonian flows	Section 1.4, MS pp99-107
9/9	Moser's trick	Section 2.6; MS Section 3.2 Warner pp220-3	ps2
9/14	Symplectic Neighborhood Theorem	Section 2.3; MS Section 3.4
9/16	Lagrangian Neighborhood Theorem	Section 2.3; MS Section 3.4
9/21	Almost complex structures	Section 1.3 MS Sections 4.1,4.2	ps3 solutions
9/23	Almost complex structures	Section 1.3 MS Sections 4.1,4.2
9/28	Pseudoholomorphic maps	Section 1.6; MS Section 4.5
9/30	Kähler manifolds	MS Sections 4.3,4.4
10/5	Group actions on manifolds	Sections 1.2,2.1 Warner 3.1-3.10,3.29-3.33,3.44-3.47	ps4 solutions
10/7	Equivariant neighborhood theorems	Sections 2.2-2.4
10/14	Hamiltonian group actions	Sections 2.4,2.5; MS Sections 5.1-5.3
10/19	Local structure of moment maps	Section 2.5
10/21	Local-to-Global Principle	Section 3.5	ps5
10/26	Reeb quotients	Section 3.6
10/28	Path metrics
11/2	Wrapup of proof of L2G Principle
11/4	Convexity theorems	Section 3.1	ps6
11/9	Morse-Bott theory	Appendix A
11/11	Applications to Hamiltonian S¹-actions	Section 3.2
11/16	Classical approaches to convexity theorems	Section 3.3; Atiyah's paper §1,2 Guillemin-Sternberg's paper §4,5

	Kahler case
	Symplectic quotients
	Delzant theorem: surjectivity
	Hamiltonian symplectic cut
	Delzant theorem: injectivity
	Symplectic cut

This page is maintained by Aleksey Zinger.

Last modified: November 6, 2026.