Spring 2021

Time: Tuesday and Thursday, 1-2:30PM ET, January 28 - May 6. No class Thursday, March 4.

Location: Zoom (please write me for the link)

Professor: Josh Greene

Course description: This course is an introduction to symplectic geometry, with a view towards J-holomorphic curves, Floer homology, and phenomena specific to dimension four. Roughly, I envision spending the first third of the semester on introductory topics, as covered in the recommended texts (2) and (3) below; the second third on properties and applications of J-holomorphic curves, as covered in the texts (4) and (5); the final third on Floer homology, as covered in (1) and Salamon's notes below; and the fourth third on four-dimensional phenomena, from various sources. Plans will adapt as we go. We will go light on the analytical aspects, but I hope to give some flavor of them, and to give some motivation for its need from Morse homology.

Post-mortem: Ultimately the course was half an introduction to symplectic geometry and half an introduction to pseudoholomorphic curves, with a little overview of Morse homology and Hamiltonian Floer homology at the end. Lecture notes are below. They are "as is", they certainly contain some errors, and they draw quite closely at parts from the sources below. I may clean them up if and when I reteach this course, and I welcome any comments that may improve them.

Recommended texts:

(1) Audin and Damian, Morse theory and Floer homology: a beautiful and thoughtful text, expanded from an actual graduate course (for actual graduate students).

(2) Cannas da Silva, Lectures on Symplectic Geometry: a beautiful and accessible text that covers all of the basics, also based on a graduate course the author gave.

(3) McDuff and Salamon, Introduction to Symplectic Topology: a broader and more advanced introduction.

(4) McDuff and Salamon, J-holomorphic Curves and Quantum Cohomology: this, and to an extent, (1), will guide our treatment of J-holomorphic curves.

(5) McDuff and Salamon, J-holomorphic Curves and Symplectic Topology: this is a vast outgrowth of the previous book and contains some of the applications of J-holomorphic curves we will cover.

I would highly recommend you get your hands on copies of (1) and (3). (I have the copy from the library of (3) on loan -- sorry.)

Lecture notes:

Lecture 1: Overview (notes incomplete)

Lecture 2: Physical origins

Lecture 3: Symplectic manifolds and Darboux's theorem

Lecture 4: Darboux's theorem, Moser's method, and action

Lecture 5: Action and Moser's theorem

Lecture 6: Relative Moser, a little symplectic linear algebra, and Liouville’s theorem

Lecture 7: Hamiltonian isotopies, the symplectic quotient, and CP^n

Lecture 8: The symplectic quotient, CP^n, and cotangent bundles

Lecture 9: Cotangent bundles and the nearby Lagrangian conjecture

Lecture 10: Almost complex structures and the Lagrangian neighborhood theorem

Lecture 11: Almost complex structures: existence and contractibility

Lecture 12: Sp(2n) and the Maslov class

Lecture 13: Dolbeaut cohomology and Kähler manifolds

Lecture 14: Kähler manifolds

Lecture 15: Pseudoholomorphic curves: first steps

Lecture 16: Energy and the d-bar operator

Lecture 17: Energy and the dimension theorem

Lecture 18: Gromov compactness

Lecture 19: The S^2 x S^2 theorem

Lecture 20: The moduli space of psh curves

Lecture 21: More about the moduli space of psh curves, and the R^4 recognition theorem

Lecture 22: Adjunction and the nonsqueezing theorem

Lecture 23: Lagrangian surfaces: tori in R^4

Lecture 24: Lagrangian surfaces: tori in R^4, part two

Lecture 25: Lagrangian surfaces and peg problems

Lecture 26: Morse homology

Lecture 27: Hamiltonian Floer homology and the Arnold conjecture

Other resources:

• Notes by Jackson van Dyke to a course on symplectic geometry that Michael Hutchings taught at UC Berkeley in Spring 2019.

• Notes to a course on symplectic manifolds and Lagrangian submanifolds that Denis Auroux taught at Harvard in Fall 2018.

• Notes to a course on symplectic geometry that John Etnyre taught at Georgia Tech in Fall 2018.

• A list of topics for a course on sympectic geometry that Steven Sivek taught in Bonn in Winter 2016-17.

• Notes to a course on Lagrangian Floer homology that James Pascaleff taught at UT Austin in Spring 2014.

• Notes to a course on symplectic topology that Jonny Evans taught at ETH in 2010.

• Notes to a course on symplectic geometry that Denis Auroux taught at MIT in Spring 2007.

• Notes to lectures on Floer homology that Dietmar Salamon gave at PCMI in 1997.

• Chris Wendl's book-in-progress Lectures on Holomorphic Curves in Contact and Symplectic Geometry.

• Musings on symplectic geometry by a couple of non-specialists: Henry Cohn and Rich Schwartz.