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