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.
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.
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.
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
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
- M. Abouzaid, Framed
bordism and Lagrangian embeddings proves that some exotic
(4k+1)-spheres are detected by the symplectomorphism type of their
- T. Ekholm, I. Smith, Exact Lagrangian immersions
with a single double point gives strong restrictions on the
objects described by the title.
- 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,
- Started to discuss the Arnold conjecture for fixed points of Hamiltonian symplectomorphisms, will explain more next time.
(1/31) Definition of contact manifolds, statement of the Weinstein conjecture and motivation. See section 1 of the above survey article.
- 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.
More about contact manifolds, Legendrian submanifolds.
- Review of Hodge theory.
- Started on Darboux's theorem etc.
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.
- 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.
(2/16) 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
(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
(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).
- 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.
(3/20) Hamiltonian G-actions and moment maps. See little McDuff-Salamon section 5.2 or Cannas da Silva sections 22 and 26.
Symplectomorphisms of S^2 x S^2, see big McDuff-Salamon section 9.5.
- Symplectic quotients by Hamiltonian S^1 actions.