Thanks to one of our students, we have some running lecture notes, which I will update periodically, here. Warning: These are just notes, they have only been lightly double-checked for correctness. Please let me know if you have any corrections!

Here are short summaries of the material covered in lecture.

Part 1: The basics

1/25: Syllabus and course policies. Origins of symplectic geometry in Hamiltonian mechanics. Examples of Hamiltonian systems: the harmonic oscillator; the planar pendulum; the Kepler problem. Statement of Liouville’s theorem. N-body problem choreographies. Loose reference: McDuff-Salamon, 1.1.

1/27: Preliminaries about differential forms. Visualizing the symplectic area. The Hamiltonian vector field as a symplectic gradient. Hamiltonian flows preserve the standard symplectic structure. Proof of Liouville’s theorem. Statement of Gromov’s non-squeezing theorem. Loose reference: McDuff-Salamon, 1.1-1.2.

2/1: The symplectic camel. Symplectic embeddings of ellipsoids and the Fibonacci staircase. Conservation of energy. Definition of a symplectic manifold and motivation. Loose reference: McDuff-Salamon, 1.1-1.2, see also the note by Henry Cohn in our supplements section.

2/3: Examples of symplectic manifolds; the cotangent bundle. Symplectic linear algebra: Lagrangian, symplectic, isotropic and coisotropic subspaces; symplectic bases; every finite dimensional symplectic vector space is standard. Reference: McDuff-Salamon, 2.1, 3.1.

2/8: Symplectic forms and volume forms. Examples of manifolds that do not admit symplectic forms. What obstructions are there for a closed manifold to admit a symplectic form? Survey of what is known in dimension \ge 6, and dim = 4. More symplectic linear algebra: the linear symplectic group, and its relations with the orthogonal, complex linear, and Hermitian groups. References: McDuff-Salamon, 2.2, 3.1, 14.1, see also the survey by Salamon in our supplements section.

2/10: Almost complex structures and compatible triples. Symplectic manifolds have no local invariants: statement of Darboux’s theorem. Moser’s trick: any two symplectic forms on a closed manifold that can be connected by a path of symplectic forms in the same cohomology class are symplectomorphic. References: McDuff-Salamon, 2.5, 3.2, see also our supplement on Cartan’s magic formula in the time-varying case.

2/15: Proof of Darboux’s Theorem. Moser’s relative isotopy trick. Submanifolds of symplectic manifolds. Weinstein’s Lagrangian neighborhood theorem. Weinstein’s “Lagrangian manifold creed”. Examples of Lagrangians: cotangent bundles, graphs. Reference: McDuff-Salamon, 3.2, 3.4

2/17: More about normal neighborhoods; comparison of the smooth, symplectic and Lagrangian case. Examples of symplectic spheres in four-manifolds. Arnold’s Conjecture. Intersections of Lagrangians. The Fubini-Study form. Reference: McDuff-Salamon, 3.4, 1.2, 4.3.

2/22: Almost complex structures. Compatible triples. The space of compatible almost complex structures is non-empty and contractible. Pseudoholomorphic curves. References: McDuff-Salamon, 4.1, 4.5; Canas da Silva, Chapters 12, 13.

Part 2: Pseudoholomorphic curves in closed manifolds.

2/24: Gromov’s non-squeezing theorem, part 1. Idea of the proof. The moduli space of J-holomorphic spheres. The action by Mobius transformations. References: McDuff-Salamon, 4.5; this blog post. See also: these notes (especially Chapter 7) for relevant facts about minimal surfaces

3/1: Bubbling; Gromov compactness (statement in a special case); Fredholm theory for J-holomorphic curves, part 1. References: Here are some excellent notes

3/3: Fredholm theory for J-holomorphic curves, part 2. References: Chapter 3 of https://people.math.ethz.ch/~salamon/PREPRINTS/jholsm.pdf

3/8: The index formula. Fredholm theory of J-holomorphic curves, part 3. Somewhere injective curves versus multiple covers. Some recollections of homology. References: Great lecture notes, and for background on homology see Chapter 2 of Hatcher.

3/10: Completion of the proof of Gromov’s non-squeezing theorem, assuming bubbling can not occur. Reference: Great lecture notes.

3/15: Ruling out bubbling. References: Great lecture notes and for background on \pi_2, see Chapter 4 of Hatcher.

Part 3: Contact homology

3/17: An introduction to contact geometry. Contact type hypersurfaces and Hamiltonian dynamics. Examples of hypersurfaces of contact type. Fillable contact manifolds. Reference: The first three chapters of this survey article.

Spring Break

3/29: The tight versus overtwisted dichotomy. Pseudoholomorphic curves in symplectizations. The contact homology setup. Reference: The first three chapters of this survey article.

3/31: Cylindrical contact homology: a first attempt, some potential problems. The SFT compactification; pseudoholomorphic buildings. Reference: Chapter 6 of this survey article; for the material on broken holomorphic curves (which we called holomorphic buildings), the discussion in Section 5.3 of here is excellent.

4/5: What kind of pseudoholomorphic buildings can arise from index 1 cylinders? Topological and index considerations. The transversality problem: failure of transversality to hold for multiple covers. Possible approaches to get around this. Reference: The first two chapters of these notes.

4/7: Cylindrical contact homology for convex domains. Reference: The third chapter of these notes.

Contact homology vacation

4/12: Guest lecture on Mirror Symmetry by Dan Pomerleano. References: Slides: Part 1, Part 2. Kontsevich’s ICM address:
Kontsevich, Homological Algebra of Mirror Symmetry, ICM address 1994, https://arxiv.org/abs/alg-geom/9411018; HMS for elliptic curves: Polishchuk and Zaslow, Categorical Mirror Symmetry: the elliptic curve, https://arxiv.org/abs/math/9801119; Ganatra, Perutz, Sheridan: GPS, Mirror Symmetry: from categories to curve counts, https://arxiv.org/abs/1510.03839; Calabi-Yau hypersurfaces:
Sheridan, Homological Mirror symmetry for Calabi-Yau hypersurfaces in projective space, Inventiones, 2015.
(https://arxiv.org/abs/1111.0632). Auroux, A beginner’s introduction to Fukaya categories, https://arxiv.org/abs/1301.7056.

Back to contact homology

4/14: More details about cylindrical contact homology for convex domains. Why is d^2 = 0? Gluing. Why is convexity helpful? The Conley-Zehnder index. Reference: The third chapter of these notes.

4/17: More details about cylindrical contact homology. Contact homology algebra. References: The third chapter of these notes. This paper. This video and its sequels.

4/19: More details about contact homology algebra: why do we expect d^2 = 0? An impressionistic sketch of virtual fundamental cycles. This paper. This video and its sequels.

4/24: Good versus bad Reeb orbits and connecting to gluing.
Computations and applications: the CCH of an ellipsoid. Reeb orbits on convex domains. References: Section 3.6 in these notes. Section 4 of these notes.

4/27 Student presentation on symplectic capacities and ECH capacities. More computations and applications: CHA of overtwisted contact structures vanishes; contact structures on tori spaces. References: The fourth chapter of these notes. This paper. Section 1 of these notes.

5/3 Student presentations on symplectic toric manifolds. More computations and applications: Morse-Bott techniques. References: The fourth chapter of these notes. The first three chapters of these notes.

5/5 Student presentations: more about symplectic toric manifolds; virtual fundamental cycles. References: This Book. This paper.

5/10 Student presentations: Mahler’s conjecture and Viterbo’s conjecture; bubbling is the only obstruction to compactness, and there are only a finite number of bubbles; how to tell if a manifold is almost contact. References: arXiv:1303.4197; see the chapter on compactness. For figuring out if a manifold is almost contact, etc., see these great notes by Keith Mills.