## 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.