Office: 923 Evans.

Tentative office hours: Thursday 9:00-11:00

- 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.

- (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.

- Introduction to spectral flow; for some precise statements and rigorous proofs see e.g. J. Robbin and D. Salamon,
- (Tuesday 4/23)
- (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.

- Computation of Hamiltonian Floer homology in the symplectically monotone case. See Salamon-Zehnder,
- (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)