Math 242: Symplectic geometry

UC Berkeley, Spring 2012


Michael Hutchings. [My last name with the last letter removed] Tentative office hours: Wednesday 2:00-4:00, 923 Evans.


There is no official textbook for this class, but the following are some useful ones. References to research papers will be provided as we go along.


The course has two basic parts (which will not be completely separate from each other). The first part introduces the basic structures of symplectic geometry (roughly corresponding to the official course description), as well as some basic notions of contact geometry. The second part is about holomorphic curves and some of their applications. I expect about 70 percent of this course to overlap with the 242 course I taught three years ago; the list of topics that I covered then can be found here.

Final project

Each student is expected to research a topic of interest and either write a 5-10 page expository article about it, or give a short presentation to the class. The articles will be posted here (if I am given permission). The books by McDuff-Salamon and Geiges suggest a number of good starting points with many references. I can give more references. (Since McDuff-Salamon is a few years old, it does not cite the latest works.)

Lecture summaries and references

  • (1/17)
  • (1/19)
  • (1/24)
  • (1/31) Definition of contact manifolds, statement of the Weinstein conjecture and motivation. See section 1 of the above survey article.
  • (2/2)
  • (2/7)
  • (2/9)
  • (2/14)
  • (2/16) More about three-dimensional contact geometry. See these lecture notes by John Etnyre.
  • (2/21-2/23) Almost complex structures (see little McDuff-Salamon for the linear algebra details), introduction to holomorphic curves. Monotonicity lemma for minimal surfaces, and explanation of why the Gromov nonsqueezing theorem follows from the existence of a certain holomorphic sphere.
  • (2/28-3/1) Review of Banach manifolds. Proof that the moduli space of somewhere injective J-holomorphic maps is a manifold for generic J, modulo some technical facts. For details see big McDuff-Salamon.
  • (3/6) Simplest case of Gromov compactness (for moduli spaces of holomorphic spheres with the smallest possible symplectic area).
  • (3/8) Automatic transversality for certain holomorphic spheres (completing the proof of Gromov nonsqueezing). Intersection positivity and adjunction formula.
  • (3/13) Proof of Gromov's theorem on the recognition of R^4, modulo one point which I got stuck on, which is explained in big McDuff-Salamon, Theorem 9.4.7(ii).
  • (3/15)
  • (3/20) Hamiltonian G-actions and moment maps. See little McDuff-Salamon section 5.2 or Cannas da Silva sections 22 and 26.