Math 226B: Differential Geometry

Introduction to Symplectic Geometry

Winter 2011



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:

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:

Available online:


Here is Gromov's symplectic camel.

Can you squeeze it through the eye of the needle
using only symplectic transformations?