Georgia Institute of Technology College of Sciences

Description: This course focus on the existence of symplectic structures on manifolds, their symplectic submanifolds, and the relation between symplectic geometry and topology. We will begin by surveying some of the basic results in symplectic geometry before moving on to Donaldson’s breakthrough in constructing symplectic submanifolds and what is currently known about, and approaches to, the existence questions for symplectic structures on manifolds. We will also discuss (1) representing symplectic 4-manifolds through Lefschetz fibrations and as branched covers; (2) the symplectic isotopy problem; (3) relations with Seiberg-Witten theory; and (4) analogous issues in contact topology. Along the way we will also develop useful tools from differential topology, Riemannian and complex geometry, and algebraic topology.