Mathematics 790, Fall 2021

Minicourse: An introduction to symplectic and contact geometry

Time: MW 10:15 - 11:30 am
Dates: August 23 to September 20
Location: Physics 205
Instructor: Lenny Ng

This minicourse will cover some of the basic ideas and techniques in symplectic and contact geometry. Symplectic and contact geometry is now a vast subject, with relations to dynamical systems, algebraic geometry, gauge theory, topology, and string theory, among other fields; I'll just try to hit a few highlights, with a bias toward the topological side of the subject. The minicourse should be accessible to anyone who is reasonably comfortable with smooth manifolds (along the lines of Math 620) and, not as crucially, algebraic topology (Math 611).

Here's a rough outline of topics that I plan to cover:

• symplectic structures; Hamiltonian vector fields; Lagrangian submanifolds
• Moser's method; Darboux's Theorem; Lagrangian neighborhood theorem
• almost complex structures; Kähler manifolds
• contact structures; Stein and Weinstein manifolds; constructing symplectic manifolds through handle attachment
• (if time permits) Arnold conjecture and Floer homology.

Course lecture notes (will update as we go)