Course Description: This course is an introduction to symplectic geometry and topology. We will start with some standard background material and then in the second half of the course focus on holomorphic curves techniques and Hamiltonian Floer theory. The main topics we plan to cover include:

• Symplectic manifolds, symplectomorphisms; Lagrangian submanifolds.
• Moser theorem, Darboux theorem, Lagrangian tubular neighborhoods theorem.
• Hamiltonian diffeomorphisms, flux, Hamiltonian flows, isotopy.
• (Almost) Complex Structures, symplectic and complex vector bundles, Chern classes, Maslov index
• Hamiltonian group actions, moment maps and symplectic reductions
• Pseudoholomorphic curves and Hamiltonian Floer theory (roughly 2nd part of the course)

This is meant to be mostly an introductory graduate level course, and there are no formal prerequisites for this class, although a good background in topology, geometry and analysis/PDEs is strongly recommended.

Recommended reading: We will not follow any references too closely but here are some suggestions:

• Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, (second edition), Oxford, 1998.
• Anna Cannas da Silva, Lectures on Symplectic Geometry, Springer (2001).
• Dietmar Salamon, Lectures on Floer Homology, available online.
• Michèle Audin and Mihai Damian, Morse Theory and Floer Homology, Springer (2014).