Vorlesung: Symplectic Topology and Holomorphic Curves, Wintersemester 2018-19

Lecture summaries / reading suggestions / exercises (updated every week)


General information

Instructor: Prof. Chris Wendl (for contact information and office hours see my homepage)

Time and place: Lectures on Tuesdays 11:00-13:00 (c.t.) in room 1.115 (Rudower Chaussee 25), plus Problem Classes every other Friday 13:00-15:00 (c.t.) in room 1-1304 (Rudower Chaussee 26)

Language: The course will be taught in English.

Prerequisites: I will assume that all students are comfortable with the essentials of differential geometry (smooth manifolds, vector fields, differential forms and Stokes' theorem, de Rham cohomology), as well as some basic algebraic topology (fundamental group, singular homology and cohomology) and functional analysis (continuous linear operators on Banach spaces, the standard Lp-spaces). Some previous knowledge of additional topics from topology (homological intersection theory, the first Chern class) and functional analysis or PDE theory (Fourier transforms, distributions, Sobolev spaces) will occasionally be helpful, but I will give quick introductions to these topics as needed. Students who have not yet taken Topology 2 or Functional Analysis are strongly encouraged to take them concurrently with this course.

Symplectic manifolds were invented around the turn of the 20th century as the natural geometric setting in which to study Hamilton's equations of motion from classical mechanics. The subject of symplectic geometry has developed considerably since then, and it retains a close connection with theoretical physics despite being technically a branch of “pure” mathematics. In particular, the subfield known as symplectic topology, which deals with “global” rather than “local” properties of symplectic manifolds, has witnessed an explosion of activity since the introduction of techniques from elliptic PDE theory in the 1980s. The most spectacular advances came from Gromov's theory of pseudoholomorphic curves, which has led to a wide assortment of algebraic invariants of symplectic manifolds, some of them related to structures that physicists study in quantum field theory or string theory.

After a brief introduction to symplectic manifolds and the kinds of questions that symplectic topologists study, the goal of this course will be to develop enough of the basic theory of pseudoholomorphic curves to understand some of its classic applications, with emphasis on the 4-dimensional case. Pseudoholomorphic curves are a generalization of the notion of a holomorphic map from complex analysis, but in a geometric setting where most of the techniques of complex analysis do not easily apply. What applies instead is the theory of elliptic PDEs, thus holomorphic curve theory has a fairly analytical flavor and requires techniques from nonlinear functional analysis (also known as “infinite-dimensional differential geometry”). We will not have time to cover those techniques in full detail, but my objective will be to convey enough of the main ideas to make everything seem believable and at least somewhat intuitive, and to give you pointers for further reading if you want to see the complete details. A tentative outline of the topics to be covered is as follows:

  1. Introduction to Hamiltonian systems, symplectic manifolds and contact manifolds, symplectic fillings, Lagrangian submanifolds, examples of results in symplectic topology
  2. Darboux's theorem, Moser and Gray stability theorems, neighborhood theorems, almost complex structures and the first Chern class
  3. Holomorphic curve basics: the nonlinear Cauchy-Riemann equation, linearized Cauchy-Riemann operators, Sobolev inequalities and elliptic estimates, linear regularity results
  4. The similarity principle, nonlinear regularity results, local models for intersections, simple curves and multiple covers
  5. Fredholm theory and the Riemann-Roch formula
  6. The moduli space of unparametrized holomorphic curves, Fredholm regularity and the implicit function theorem
  7. The Sard-Smale theorem and transversality for simple holomorphic curves, transversality of the evaluation map
  8. Bubbling off analysis, removal of singularities, Gromov's compactness theorem
  9. Special properties in dimension four: automatic transversality, positivity of intersections, adjunction formula
  10. Symplectic and complex blowups, ruled surfaces and Lefschetz fibrations, statement of McDuff's theorems on rational or ruled symplectic 4-manifolds
  11. Compactness and index counting relations for McDuff's results
  12. Proofs of McDuff's theorems on holomorphic exceptional spheres and rational/ruled symplectic 4-manifolds
  13. Construction of the Gromov-Witten invariants via pseudocycles in dimension four, some basic computations, sketch of the proof that uniruled implies rational/ruled
  14. Contact manifolds, symplectic fillings and caps, pseudoconvexity, uniqueness of fillings for the 3-sphere and cotangent bundles of the 2-sphere and 2-torus, non-uniqueness for higher genus surfaces, nonexistence of semifillings with disconnected boundary, McDuff's examples with disconnected boundary
  15. Sketch of symplectic field theory (if time permits)

References: Much of the course will roughly follow my recently published book, Holomorphic Curves in Low Dimensions: From Symplectic Ruled Surfaces to Planar Contact Manifolds (Springer LNM, 2018). I have about 45 free copies of this book currently in a box in my office, so if you show up to the class, you can have one. (I'm also happy to send an electronic copy ahead of time to anyone who asks.) The course will go into slightly more detail than the book does on some of the analytical results, so for these, you may also want to consult my slightly older, denser and perpetually unfinished Lecture Notes on Holomorphic Curves in Symplectic and Contact Geometry. If at some point you decide to get really serious about symplectic topology and/or holomorphic curve theory, then you'll eventually also need to buy both of the books by Dusa McDuff and Dietmar Salamon:

Homework: I will assign exercises sometimes. Sometimes I will discuss them in the problem class. They will not be graded.

Grades: Since this is an advanced course, I have a fairly relaxed attitude about grades. If you stay with me for the whole semester, you can come to my office for a conversation (let's pretend that's the English translation of “mündliche Prüfung”). If you demonstrate that you learned something interesting from the course, you'll get a good grade.

Werbung: You might also be interested in Klaus Mohnke's seminar on Floer homology running this semester, which is closely related to the subject of this course and should serve as an ideal accompaniment. Klaus will also be teaching a followup lecture course in Sommersemester 2019 whose precise topic remains to be decided.

Chris Wendl's homepage