Math 242: Symplectic Geometry

UC Berkeley, Fall 2024

Instructor

Michael Hutchings
hutching@berkeley.edu.
Office: 923 Evans.
Tentative office hours: Thursday 10:00 am - noon

Textbooks

The main references for this course are the following two books.

A. Cannas da Silva, Lectures on Symplectic Geometry, authorized free download here. This book gives a nice explanation of the basic geometric constructions and lemmas in symplectic geometry.
D. McDuff and D. Salamon, Introduction to Symplectic Topology, third edition, Oxford University Press. This book is a useful reference for many of the topics that we will be discussing, as well as a good introduction to a wide range of topics of current research interest.

The following additional books are recommended:

V. Arnold, Mathematical Methods of Classical Mechanics. This book is kind of a "prequel" to the course and discusses the use of symplectic geometry in classical mechanics. It contains a lot of rich content beyond the scope of this course.
H. Geiges, An Introduction to Contact Topology. This is a good reference for contact geometry (the odd dimensional counterpart of symplectic geometry), which we will discuss a bit in the course.
D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology. This book explains the technical details of holomorphic curves in great detail. We will discuss some applications of holomorphic curves at the end of the course, but without a lot of technical details. Chapter 9 of this book explains some of the "classical" applications which we will be discussing.

Syllabus

In this course we will introduce the basic structures of symplectic geometry (corresponding to most of the course catalog description) as well as contact geometry (the odd dimensional counterpart of symplectic geometry). We will also introduce holomorphic curve techniques and applications (omitting hard analysis proofs). The precise topics covered will be similar to but not exactly the same as the last time I taught this course; you can see the summary here.

Final project

The only requirement for this course is to complete a final project, in which you will learn about a topic of interest in or related to symplectic geometry, and then write a short (around 5 pages, possibly a little longer but absolutely no more than 10 pages) expository paper about it. Your paper will then be submitted to the Journal of Math 242 Final Projects. It will be sent to an anonymous fellow student to referee, and you will be asked to anonymously referee a fellow student's paper. You will then revise your paper in response to the referee's comments. If your paper is accepted for publication in the Journal of Math 242 Final Projects, then you can (optionally) elect to have it published on the course webpage.

Homework exercises will occasionally be suggested, but these are optional and will not be graded.

Ed Discussion

You are welcome to discuss mathematical questions related to the course in the Ed Discussion here. I will contribute to these discussions when I can.

Lecture summaries and references

After each lecture, brief summaries and references will be posted here.

(Wednesday 8/28)
- Definition of symplectic manifold, basic examples. See Cannas da Silva, chapters 1 and 2.
(Friday 8/30)
- Hamiltonian vector fields. For practice, here is some optional homework.
(Wednesday 9/4)
- Hamiltonian and symplectic isotopies.
- Introduction to different versions of the Arnold conjecture on the minimum number of fixed points of a Hamiltonian symplectomorphism.
(Friday 9/6)
- The flux of a symplectic isotopy.
- Introduction to Lagrangian submanifolds.
(Monday 9/9)
- More about Lagrangians; introduction to Lagrangian intersection problems.
- For practice, the second optional homework is here.
(Wednesday 9/11 and Friday 9/13)
- The Moser method, Darboux's theorem, and Weinstein's Lagrangian neighborhood theorem. For more details see Cannas da Silva, Part III.
(Monday 9/16)
- Introduction to contact structures and Legendrian knots. For more about this see the book by Geiges or these lecture notes by Etnyre.
(Wednesday 9/18)
- Connections on principal S^1 bundles, and the resulting examples of contact forms. For connections on principal bundles with more general gauge groups, see e.g. Spivak's Comprehensive Introduction to Differential Geometry, Volume II, Chapter 8.
(Friday 9/20)
- Reeb vector fields and contact type hypersurfaces. See e.g. the first few pages of this survey article.
(Monday 9/23)
- The Legendre transform. (Didn't quite finish.)
(Wednesday 9/25)
- Finished the Legendre transform. See these notes.
- Started Gray stability.
(Friday 9/27)
- Hamiltonian circle actions. See McDuff-Salamon, section 5.1. Next time we will discuss some of the more general story in Cannas da Silva sections 27-29.
(Monday 9/30)
- Introduction to toric symplectic manifolds. See Cannas da Silva sections 27-29 and McDuff-Salamon section 5.5.
- Here is the third homework.
(Wednesday 10/2)
- More about toric symplectic manifolds, basic examples. For an extensive discussion of blowups, see McDuff-Salamon section 7.1.
(Friday 10/4)
- Statement of Gromov nonsqueezing.
- Compatible and tame almost complex structures. For details of the linear algebra see McDuff-Salamon section 2.5.
(Monday 10/7)
- Brief review of Riemann surfaces. Definition of a J-holomorphic curve. For a brief overview see "little" McDuff-Salamon (the symplectic topology book) section 4.5. For lots of details see "big" McDuff-Salamon (the J-holomorphic curves book).
(Wednesday 10/9)
- Area of J-holomorphic curves when J is tame or compatible.
(Friday 10/11)
- Monotonicity lemma for minimal surfaces.
- Proof of Gromov nonsqueezing, assuming the existence of a certain holomorphic sphere.
(Monday 10/14 - Friday 10/18)
- Tools for proving "generic transversality" results, including a brief review of calculus on Banach manifolds, and as an example a proof that generic functions are Morse. For more details about all of this see "big" McDuff-Salamon. There is also some discussion of this in chapter 5 of my Morse theory lecture notes, although beware that these notes have lots of mistakes and I need to revise them sometime.
(Monday 10/21)
- The linearized Cauchy-Riemann operator associated to a holomorphic map. For more about this see "big" McDuff-Salamon, section 3.1.
- For more about the nice properties of elliptic operators in general see e.g. the book by Lawson and Michelsohn, Spin Geometry, Chapter III.
(10/23-10/25)
- Proof of transversality of moduli spaces of simple holomorphic curves for generic J (modulo some technical details). See big McDuff-Salamon chapter 3 for details.
- Introduction to Gromov compactness in the simplest case. See big McDuff-Salamon chapter 4 for more details.
(10/30-11/6)
- Completion of the proof of Gromov non-squeezing.
- Automatic transversality, adjunction formula, and intersection positivity in dimension 4. See big McDuff-Salamon, Appendix E for more details.
- Gromov's theorems on the symplectomorphism group of S^2 x S^2 and the recognition of R^4. For more details see big McDuff-Salamon, sections 9.4 and 9.5.
(11/8-11/13)
- Review of Morse homology, as a model for Floer homology.
(Friday 11/15)
- Started discussing Floer homology for nondegenerate Hamiltonian symplectomorphisms in the symplectically aspherical case. For more details see the Lectures on Floer homology by Dietmar Salamon. (Beware of differing sign conventions.)
(Monday 11/18)
- Continued developing Floer theory and discussed the relation between index and spectral flow. For more details see this article by Robbin-Salamon. (Again, beware of differing sign conventions.)
(Wednesday 11/20)
- The Conley-Zehnder index of a periodic orbit, and the computation of the index of the deformation operator for Floer's equation. For more details see Salamon's lectures on Floer homology, chapter 2.
(Friday 11/22)
- Outline of the definition of Hamiltonian Floer homology in the symplectically aspherical case.
(Monday 11/25)
- Computation of Floer homology in the symplectically aspherical case. We used the argument in this paper by Salamon-Zehnder, which explains the details.
(Monday 12/2 and Friday 12/6)
- Generalization of Hamiltonian Floer homology to the monotone case, and to the "Calabi-Yau" case using Novikov rings.
- Brief introduction to some other kinds of Floer homology.