Math 242: Symplectic geometry

UC Berkeley, Spring 2012

Instructor

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

Textbooks

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.

A. Cannas da Silva, Lectures on Symplectic Geometry, 2006. These online lecture notes clearly explain the basic structures of symplectic geometry.
D. McDuff and D. Salamon, Introduction to symplectic topology, 2nd edition, 1998. A very nice introduction to some of the more topological aspects of symplectic geometry.
D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, 2004. Detailed explanation of holomorphic curves and some of their applications.
H. Geiges, An introduction to contact topology, 2008. A nice introduction to contact geometry, the odd-dimensional counterpart of symplectic geometry, which we will also discuss in this course.

Syllabus

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)

Definition of symplectic manifold and Lagrangian submanifold and basic examples.
Statements of some deep results that can be proved using holomorphic curves. I plan to discuss the results of Gromov's 1985 paper later in the course. The others I just mentioned for motivation. Here are some references. (I don't expect you to be able to read these yet, but hopefully this course will help prepare you to understand them.)
- M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. This is the seminal paper that, as its title suggests, introduced the use of holomorphic curves in symplectic geometry. Among other things it proves the nonsqueezing theorem and the nonexistence of a compact exact Lagrangian in C^n.
- Here is a survey article I wrote which discusses McDuff's theorem on ellipsoid embeddings.
- M. Abouzaid, Framed bordism and Lagrangian embeddings proves that some exotic (4k+1)-spheres are detected by the symplectomorphism type of their cotangent bundle.
- T. Ekholm, I. Smith, Exact Lagrangian immersions with a single double point gives strong restrictions on the objects described by the title.

(1/19)

Hamiltonian vector fields.
How Hamiltonian vector fields arise from classical mechanics via the Legendre transform. (This was mainly for motivation, although we did use it to explain why geodesic flow on the cotangent bundle is the Hamiltonian flow of the norm squared of cotangent vectors.) For a brief treatment of this see the beginning of McDuff-Salamon, Intro to Symp Top. For much more, see the beautiful book
- V. I. Arnold, Mathematical methods of classical mechanics, second edition.
Started to discuss the Arnold conjecture for fixed points of Hamiltonian symplectomorphisms, will explain more next time.

(1/24)

Defined Hamiltonian symplectomorphisms.
Defined the flux of a symplectic isotopy. (See McDuff-Salamon, IST.)
Stated the two basic versions of the Arnold conjecture for fixed points of Hamiltonian symplectomorphisms (original version and nondegenerate version). Later in the course we will explain how the existence of Floer homology implies the nondegenerate version.
Started to discuss the problem of periodic orbits of time-independent Hamiltonian vector fields on regular hypersurfaces, and introduced the characteristic foliation. Will say more about this next time. See section 1 of this survey article.

(1/31) Definition of contact manifolds, statement of the Weinstein conjecture and motivation. See section 1 of the above survey article.

(2/2)

More about contact manifolds, Legendrian submanifolds.
Review of Hodge theory.
Started on Darboux's theorem etc.

(2/7)

Using the Moser method to prove stability of symplectic forms, Darboux's theorem, local structure of symplectic and Lagrangian submanifolds. The details are nicely explained in Cannas da Silva and McDuff-Salamon.
Gray's stability theorem; for the details of this and Darboux's theorem for contact forms see Geiges.

(2/9)

Review of the first Chern class. For the obstruction theory definition, see chapter 11 of these notes. (The notes discuss the Euler class of an oriented S^k bundle; when k=1 this is the same as the first Chern class.) I don't know a reference for the curvature definition the way I presented it, but this is very standard.
Contact structures on circle bundles.

(2/14)

Brief review of the Euler class. (These notes explain more details, but this is beyond the scope of this course.)
Examples of contact structures in three dimensions, definition of overtwisted and tight contact structures, statements of key theorems about these.

(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)

Symplectomorphisms of S^2 x S^2, see big McDuff-Salamon section 9.5.
Symplectic quotients by Hamiltonian S^1 actions.

(3/20) Hamiltonian G-actions and moment maps. See little McDuff-Salamon section 5.2 or Cannas da Silva sections 22 and 26.