MATH 268

MATH 268: Topics in Differential Geometry: Symplectic Geometry

Description: (From the catalog) This course will cover the fundamentals of symplectic topology and geometry.

The first third of the course will be devoted to 'classical' symplectic geometry: Lagrangians, Legendre transformations, Hamiltonians, symplectic manifolds and the Darboux-Weinstein theorem, symmetries and conservation laws and the Arnold-Liouville theorem, momentum mappings, reduction, and convexity.

The second third of the course will be devoted to developing elliptic methods: pseudo-holomorphic curves, Gromov compactness and moduli, applications to packing and (non)-squeezing theorems, etc.

The final third will cover related topics and recent developments, perhaps relations with toric varieties, representation theory, or other topics that depend on the interests of the class.

Prerequisites: Mathematics 267 (Differential Geometry) and Math 262 (Algebraic Topology II). Specifically I will assume that the students are familiar with standard topics in Riemannian geometry: manifolds, metrics, connections, curvature, differential forms, de Rham cohomology, and some basic differential topology (a la, say, Bott and Tu).

Instructor: Robert L. Bryant

Texts:

• An introduction to Lie groups and symplectic geometry, by R. Bryant.
• Introduction to symplectic topology, by McDuff and Salamon.