Math 242: Symplectic Geometry
UC Berkeley, Spring 2019
Office: 923 Evans.
Tentative office hours: Thursday
The main references for this course are the following two books. We will also be discussing some topics which are not in either of these books, particularly in regard to holomorphic curve techniques, and I will give references for those as we go along.
- 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.
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.
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 either write a short (5-10 page) expository article about it, or give a short (30 minute) presentation.
Lecture summaries and references
After each lecture, brief summaries and references will be posted here.
- (Tuesday 1/22)
- Definition of symplectic manifold, basic examples. See Cannas da Silva, chapters 1 and 2.
- Hamiltonian vector fields.
- Introduction to the problem of finding closed characteristics on a hypersurface in a symplectic manifold; for some related discussion see McDuff-Salamon section 1.2, and section 1 of this survey article.
- (Thursday 1/24)
- Symplectic isotopies, Hamiltonian isoptopies, and Hamiltonian symplectomorphisms. See McDuff-Salamon section 3.1. (There is a deeper discussion in chapter 10.)
- Introduction to the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms. See McDuff-Salamon section 11.1.
- (Tuesday 1/29)
- Introduction to Gromov nonsqueezing and symplectic embedding problems. See McDuff-Salamon section 12.1, this survey article I wrote a few years ago, or this recent survey by Schlenk.
- Introduction to Lagrangian submanifolds. See Cannas da Silva chapter 3.
- (Thursday 1/31)
- More about Lagrangians, and an introduction to the notion of displaceability. For deeper discussion see McDuff-Salamon sections 9.4 and 11.3 or this introduction to Fukaya categories by Auroux (hopefully we will have time to learn more about this later in the course).
- Started to explain the Moser trick. See Cannas da Silva chapter 7.
- (Tuesday 2/5)
- Relative Moser trick. Proof of Darboux's theorem. Sketch of proof of Weinstein's Lagrangian neighborhood theorem. For more details see Cannas da Silva chapters 7 and 8.
- (Thursday 2/7)
- Introduction to contact manifolds. The book by Geiges, "An introduction to contact topology", is a very nice reference for this. For more about contact type hypersurfaces (which I discussed at the end without saying that word yet), see also the beginning of my survey article above (1/22). For more about Legendrian contact homology (which I just briefly mentioned), see this paper by Etnyre-Ng-Sabloff.
- (Tuesday 2/12)
- Contact type hypersurfaces.
- Other examples of contact manifolds.
- Started reviewing connections on principal circle bundles, among other reasons to prepare to discuss Boothby-Wang manifolds, also known as prequantization spaces.
- (Thursday 2/14)
- Review of the Euler class of an oriented S^1-bundle. For more details see e.g. section 11 of these lecture notes.
- Curvature of a connection on a principal S^1-bundle, and relation with the Euler class. (For more about principal G-bundles in general, see e.g. Spivak's comprehensive introduction to differential geometry, volume 2.)
- Back to contact geometry: prequantization spaces.
- Darboux's theorem for contact forms. For more about this see Geiges's book.
- (Tuesday 2/19)
- Gray's stability theorem. See Geiges section 2.2.
- Introduction to tight and overtwisted contact structures in three dimensions. For more, see this survey article by Etnyre.
- Started to explain the Legendre transform. See McDuff-Salamon section 1.1.
- (Thursday 2/21)
- Finished discussing the Legendre transform. For more about this see Cannas da Silva chapters 19 and 20.
- Started discussing Hamiltonian S^1 actions and their symplectic quotients. See McDuff-Salamon section 5.1.
- (Tuesday 2/26)
- Fun with toric symplectic manifolds. See McDuff-Salamon section 5.5 or Cannas da Silva chapters 28-30.
- Started talking about omega-compatible almost complex structures. (Note that I stated the last lemma incorrectly because I forgot to write the word "symplectic" on the right hand side. I'll finish this next time.)
- (Thursday 2/28)
- Finished discussing the space of omega-compatible almost complex structures. For more about this see McDuff-Salamon, section 2.5.
- Definition of J-holomorphic maps.
- Proof that for an omega-tame almost complex structure J, the area of a J-holomorphic map agrees with the integral of omega.
- Proof that for an omega-compatible almost complex structure, a J-holomorphic map minimizes area in its homology class.
- (Tuesday 3/5) MIDTERM! (Just kidding, there was no class. :-) )
- (Thursday 3/7)
- Holomorphic curves and simple examples.
- Proof of Gromov nonsqueezing, modulo the monotonicity lemma for minimal surfaces and the existence of a certain holomorphic curve. (Coming up we will quickly prove the former, and then take a bunch of time to explain some of the latter.)
- (Tuesday 3/12)
- Explained the monotonicity lemma for minimal surfaces (modulo a couple of technicalities.)
- Stated various facts about holomorphic curves which together imply Gromov nonsqueezing. In the next few lectures we will explain some of why these facts are true.
- (Thursday 3/14)
- Explained what it means for a holomorphic curve (to simplify the discussion we are just talking about holomorphic maps for now) to be cut out transversely, and how the moduli space is a manifold in this case with a nice dimension formula. In particular we needed a bit (and will need some more) introduction to calculus on Banach manifolds and elliptic differential operators.
- Most of this analysis can be found in big McDuff-Salamon chapter 3 and appendix A (and some of appendices B and C). For more about elliptic operators, the book Spin Geometry by Lawson and Michelson has a nice treatment.
- Next time we will outline how to prove transversality of simple curves for generic J.
- (Tuesday 3/19)
- Example: transversality of the product curves in the proof of Gromov nonsqueezing.
- Discussion of the Carleman similarity principle and its corollaries. For more details see big McDuff-Salamon section 2.3.
- Started explaining the proof of transversality of simple holomorphic maps for generic J. For more details see big McDuff-Salamon section 3.2.
- (Thursday 3/21)
- Finished the sketch of proof of transversality of simple holomorphic maps for generic J.
- A bit about the simplest case of Gromov compactness. For some inspiration from the history of Uhlenbeck's work on bubbling, see this article by Donaldson.
- (Spring break) Please think about what topic you want to do your final project on, and send me an email so that I can give feedback and references.
- (Tuesday 4/2)
- Intersection positivity and adjunction formula for holomorphic curves in four dimensions. (Only proved trivial cases; the general case is in an appendix to big McDuff-Salamon.)
- Holomorphic curves in S^2 x S^2, when the two factors have equal area. (This is discussed in a more general context in chapter 9 of big McDuff-Salamon.)
- Next time we will use this to prove Gromov's theorem on the recognition of R^4.
- (Thursday 4/4)
- Gromov's theorems on the symplectomorphism group of S^2 x S^2 and the recognition of R^4. For more details and more general results, see big McDuff-Salamon chapter 9.
- (Tuesday 4/9)
- Review of Morse homology, in preparation for Floer homology of (Hamiltonian) symplectomorphisms. See e.g. section 5 of my survey article on the Weinstein conjecture above.
- (Thursday 4/11)
- Outline of how to prove that Morse homology depends only on the smooth manifold (without comparing it to singular homology), as a model for proving that Floer homology is an invariant.
- (Tuesday 4/16)
- Started to explain the definition of Hamiltonian Floer homology in the symplectically aspherical case. See e.g. big McDuff-Salamon section 12.1 (note that they use some different sign conventions).
- (Thursday 4/18)
- Introduction to spectral flow; for some precise statements and rigorous proofs see e.g. J. Robbin and D. Salamon, The spectral flow and the Maslov index.
- Started talking about the Conley-Zehnder index of a path of symplectic matrices.
- (Tuesday 4/23)
- Expected dimension of moduli spaces of solutions to Floer's equation, in terms of the Conley-Zehnder index. For more details, see D. Salamon, Lectures on Floer homology, section 2.
- (Thursday 4/25)
- Finished sketching how to define Hamiltonian Floer homology in the symplectically aspherical case.
- (Tuesday 4/30)
- Computation of Hamiltonian Floer homology in the symplectically monotone case. See Salamon-Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index.
- Discussion of what is involved in defining Hamiltonian Floer homology in more general cases.
- (Thursday 5/2)
- Brief introduction to Lagrangian Floer homology. For more about this, see e.g. this expository article by Auroux.
- Brief introduction to cylindrical contact homology. The results of Gutt and myself that I mentioned are here.
- (Tuesday 5/7) (student presentations)
- (Thursday 5/9) (student presentations)