Introduction to Symplectic Topology Winter 2022

Instructor: Danny Calegari

Eckhart 203 MWF 2:30-3:20

Description of course:

This class is an introduction to symplectic topology. We shall introduce definitions, study some key examples, and develop enough of the technical machinery to prove some foundational results. In principle, this class aims to provide some `background' for the minicourse that Leonid Polterovich will teach in the first 3 weeks of the quarter.

Warning: We hope to give a topologist's view of the subject, not an analyst's.

Cancellations:

None yet.

Notices:

The first two weeks of classes will be online. The link to these classes will be via the Canvas site for the class.

Syllabus:

• Symplectic linear algebra
• Examples of symplectic manifolds; local structure
• Hamiltonian flows, Poisson brackets, principle of least action
• Symplectic reduction (applications eg to gauge theory)
• Pseudo-holomorphic curves: Gromov compactness, formal computation of dimension
• Some applications: non-squeezing; characterization of standard symplectic R^4
• Capacities and C^0 symplectic topology
• Floer homology

Lectures

Notes from class:

Notes for this class will be posted online here and updated as we go along.

References:

The main references are:

• Arnold and Givental, Symplectic Geometry in Encyc. Math. Sci. Dyn. Systems IV
• Mc Duff and Salamon, Introduction to Symplectic Topology Oxford Math. Monog.