Mathematics 378, Fall 2010

Topics in Symplectic Geometry

Time: Mondays and Wednesdays, 1:15 - 2:30
Dates: November 8 through December 8
Location: Physics 227

This minicourse is intended to be a quick introduction to symplectic geometry and relevant techniques. Symplectic geometry is a vast subject, with relations to dynamical systems, algebraic geometry, gauge theory, topology, and string theory, among other fields; we'll content ourselves with hitting some highlights. The minicourse should be accessible to anyone familiar with basic differential geometry and algebraic topology.

We'll start with some basics: symplectic structures, almost complex structures and compatibility, Kähler manifolds, Lagrangian submanifolds, Hamiltonian vector fields, contact structures. Along the way, we'll prove Darboux's Theorem and discuss Gromov's ideas of flexibility versus rigidity.

Time permitting, we'll use the rest of the minicourse to explore some more specialized and recent topic(s) in the field. This could be something like: J-holomorphic curves and Gromov compactness; group actions and moment maps; Lagrangian intersection Floer homology; contact homology; Heegaard Floer homology. (Note that these topics are each rather large in scope, and we can only do a cursory treatment of any of these.) I might lean towards talking about J-holomorphic curves, but class input is encouraged and solicited.