Math 226B: Differential Geometry
Introduction to Symplectic Geometry
Winter 2011

Time and Place: MWF 1212:50 pm in MS 5148

Instructor: Ciprian Manolescu

Email: cm_at_math.ucla.edu

Office: MS 6921

Office hours: Wed 11am12pm and by appointment
Outline:
Symplectic manifolds are an intermediate case between
real and complex (Kaehler) manifolds. The original motivation for studying
them comes from physics: the phase space of a mechanical system,
describing both position and momentum, is in the most general case a
symplectic manifold. Symplectic manifolds still play an important role in
recent topics in physics, such as string theory. They have also proved
useful in understanding the structure of fourdimensional real manifolds.
Furthermore, Hamiltonian systems appear in the study of partial differential
equations.
This course will serve as an introduction to symplectic manifolds and their
properties. At the end I hope to sketch the proofs of two major results in the
field, Gromov's NonSqueezing Theorem and Arnold's Conjecture (in the monotone
case).
Prerequisites:
A solid knowledge of manifolds, differential
forms, and deRham cohomology, at the level of Math 225A and 225B. Math 226A is
not a prerequisite!
Topics to be covered:
 the standard
symplectic structure on Euclidean space, motivation from Hamiltonian mechanics;
 linear symplectic geometry: Lagrangian
and symplectic subspaces, the symplectic linear group, the Maslov index;
 symplectic manifolds in general, Hamiltonian vector fields,
Lagrangian submanifolds;
 Moser's trick, Darboux's theorem, other neighborhood theorems;
 complex and almost complex structures, Kaehler manifolds;
 moment maps, symplectic reduction;
 pseudoholomorphic curves, Gromov's nonsqueezing theorem;
 an introduction to GromovWitten invariants and Floer homology.
Textbook:
D. McDuff and D. Salamon, "Introduction to Symplectic
Topology," Oxford University Press, New York, 1998.
We will also use material from some additional sources, such as:
 B. Aebischer et al., "Symplectic geometry," Progress in Math. 124,
Birkhauser, Basel, 1994

A. Weinstein, "Lectures on symplectic manifolds," American Mathematical
Society, Providence, 1977
 D. McDuff and D. Salamon, "Jholomorphic curves and Symplectic Topology,"
American Mathematical Society, Providence, 2004
 A. Banyaga and D. Hurtubise, "Lectures on Morse Homology", Kluwer,
Dordrecht, 2004
 R. Berndt, "An introduction to symplectic geometry," Graduate Studies
in Math. vol. 26, AMS, Providence, 2001
Available online:
 A. Cannas da Silva, "Lectures on
Symplectic Geometry", Springer
Verlag, Berlin, Heidelberg, 2001
 A. Cannas da Silva, "Symplectic
geometry" (overview), in Handbook of
Differential Geometry, vol.2, North Holland, 2005
 D. McDuff and D. Salamon, "Jholomorphic
curves and Quantum Cohomology",
American Mathematical Society, Providence, 1994
 D. Salamon, "Lectures on
Floer homology", in "Symplectic Geometry and Topology", IAS/Park City
Mathematics Series, American Mathematical Society, Providence, 1999

Here is Gromov's symplectic camel.
Can you squeeze it through the eye of the needle using only symplectic
transformations?
