Math 226B: Differential Geometry

Introduction to Symplectic Geometry

Winter 2011

Time and Place: MWF 12-12:50 pm in MS 5148
Instructor: Ciprian Manolescu
E-mail: cm_at_math.ucla.edu
Office: MS 6921
Office hours: Wed 11am-12pm 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 four-dimensional 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 Non-Squeezing 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;
pseudo-holomorphic curves, Gromov's non-squeezing theorem;
an introduction to Gromov-Witten 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, "J-holomorphic 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, "J-holomorphic 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?