Course Description:
We will investigate the geometry and topology of symplectic manifolds. The study of symplectic structures first arose in the study of Hamiltonian mechanics. However, in recent years they have developed deep connections with 3- and 4-dimensional topology. Symplectic structures can be thought of as generalizations of complex (or Kahler) structures on even dimensional manifolds. However, symplectic geometry is much more flexible than complex geometry.

We aim to cover the following topics:

• Linear Symplectic Geometry
• Symplectic Manifolds, symplectomorphisms, Lagrangian submanifolds
• Darboux and Moser Theorems
• Weinstein Tubular Neighborhood Theorem
• Contact Manifolds
• Almost complex structures; compatibility and integrability; J-holomorphic curves
• Smooth 4-manifolds; existence and uniqueness of symplectic structures, fiber connected sums

• Introduction to Symplectic Topology by Dusa McDuff and Dietmar Salamon
• 4-Manifolds and Kirby Calculus by Robert Gompf and Andras Stipsicz
• J-holomorphic Curves and Symplectic Topology by Dusa McDuff and Dietmar Salamon