Math 277, Topics in Differential Geometry:
"Applied Holomorphic Curve Theory"

Instructor

Syllabus

• Using holomorphic curves to prove results that do not include holomorphic curves in their statement, either explicitly or implicitly.
• Gaining a working knowledge of the use of holomorphic curves, and being aware of technical pitfalls, but using the underlying analytical results as "black boxes" when possible.

Lecture summaries and references

• (Tuesday 1/19) Introduction to some of the applications of holomorphic curves that we will be talking about in this class, including the original applications in Gromov's classic paper, symplectic embedding problems, the Weinstein conjecture and generalizations, restrictions on Lagrangian submanifolds, etc. For Gromov's results and some of their generalizations, see McDuff-Salamon, J-holomorphic curves in symplectic topology, Chapter 9.
• (Thursday 1/21) Basic definitions: tame and compatible almost complex structures, holomorphic curves. Energy and area. First examples of holomorphic curves. In addition to McDuff-Salamon, another reference for foundational material about holomorphic curves is the set of lecture notes by Chris Wendl.
• (Tuesday 1/26) Monotonicity lemma for minimal surfaces. Proof of Gromov nonsqueezing, assuming the existence of a certain holomorphic curve. Muliply covered curves versus somewhere injective curves. Started talking about transversality for holomorphic curves, to be continued next time.
• (Thursday 1/28) Discussed the linearized d-bar operator for a holomorphic curve (with fixed domain), and its index. Outlined the proof that if J is generic then all somewhere injective holomorphic curves are regular. For details see McDuff-Salamon chapter 3, or Chris Wendl's notes.
• (Tuesday 2/2) More (but certainly not all) details of the proof of transversality for generic J. The Carleman similarity principle and automatic transversality. Discussion of transversality for curves with varying conformal structure and marked points. Examples related to Gromov non-squeezing.
• (Thursday 2/4) Introduction to Gromov compactness in the simplest case. Completion of the proof of Gromov nonsqueezing (modulo the many details that we skipped). Discussion of intersection positivity in four dimensions. (The latter is proved in detail in an appendix to McDuff-Salamon.)
• (Tuesday 2/9) The adjunction formula in four dimensions. Holomorphic foliations of S^2 x S^2.
• (Thursday 2/11) The key idea in the proofs of Gromov's theorems on (1) the recognition of R^4, and (2) the diffeomorphism group of S^2 x S^2. For more general versions of these see McDuff-Salamon sections 9.4 and 9.5. Started discussing Hamiltonian Floer homology. For what is to come, everyone should know how to define the Morse complex and continuation maps. For a brief review, see e.g. section 5 of this survey. The first four chapters of these old notes might also be helpful, but beware of mistakes.
• (Tuesday 2/16) Basics of Hamiltonian Floer homology. A good reference for this is Salamon's lecture notes on Floer homology.
• (Thursday 2/18) Introduction to index and spectral flow. See this paper by Robbin and Salamon for the technical details.
• (Tuesday 2/23) The Conley-Zehnder index. See Salamon's lecture notes for an overview. Full details are given in various papers by Salamon and coauthors.
• (Thursday 2/25 and Tuesday 3/1) Definition of Hamiltonian Floer homology for monotone symplectic manifolds. Computation for small autonomous Hamiltonians; see the 1992 paper by Salamon-Zehnder.
• (Thursday 3/3) Explained as much as time permitted of the proof that Hamiltonian Floer homology (for monotone symplectic manifolds) does not depend on the time-dependent Hamiltonian or almost complex structure. These are standard arguments due to Floer, which we will be extending to other situations later in the course.
• (Tuesday 3/8) Introduction to contact manifolds, the Weinstein conjecture, etc. See e.g. the first couple of sections of my survey article about the Weinstein conjecture.
• (Thursday 3/10) Started talking about holomorphic curves in the symplectization of a contact manifold.
• (Tuesday 3/15) More about holomorphic curves in symplectizations: the index formula and BEHWZ compactness.
• (Thursday 3/17) Definition of cylindrical contact homology (fantasizing that transversality works; we will deal with transversality problems later).
• (Tuesday 3/29) Using cylindrical contact homology to define symplectic capacities for (dynamically) convex domains.
• (Thursday 3/31) Invariance of cylindrical contact homology, and maps induced by exact symplectic cobordisms.
• (Tuesday 4/5) Introduction to the contact homology algebra and linearized contact homology (again modulo transversality, which we will deal with soon).
• (Thursday 4/7, Tuesday 4/12) Symplectic homology. I am following the treatment in this paper by Jean Gutt.
• (Thursday 4/14) I attempted to explain how to compute S^1-equivariant homology via Morse theory, as a warmup to defining S^1-equivariant symplectic homology (using the approach of Bourgeois-Oancea). The version of Floer homology for families that enters into this is defined in section 6 of this paper (see also the erratum).
• (Tuesday 4/19) Introduction to S^1-equivariant symplectic homology.
• (Thursday 4/21) All about Morse-Bott theory in the finite dimensional case: the older approach using chains on the critical submanifolds, and the newer approach using cascades.
• (Tuesday 4/26) Using Morse-Bott Hamiltonian Floer theory to partially describe (positive S^1-equivariant) symplectic homology in terms of Reeb orbits. This story is developed in several papers of Bourgeois-Oancea, and the above paper by Gutt.
• (Thursday 4/28) Student presentations, day 1 of 3.
• (Tuesday 5/3) Student presentations, day 2 of 3.
• (Thursday 5/5) Student presentations, day 3 of 3.