Mathematics 790, Fall 2022

Mathematics 790 (790-90-04), Fall 2022

Minicourse: An introduction to contact geometry

Time: MW 10:15 - 11:30 am
Dates: September 21* - October 26
Location: Physics 205
Instructor: Lenny Ng

This minicourse will cover some of the basic ideas and techniques in contact geometry. Contact geometry, often described as the odd-dimensional analogue of symplectic geometry, is a subject whose origins date back to the 19th century in geometric optics; it's now grown into a large and beautiful subject that is closely tied to three-dimensional topology as well as symplectic and complex geometry. I'll try to lay out some of the motivating questions and foundational results of contact geometry, with special emphasis on ties to 3-manifold topology. 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 are some topics that I hope to cover (however briefly). This list is almost certainly too ambitious, but we can focus on specific topics depending on interest.

contact and symplectic structures; Hamiltonian and Liouville vector fields
Darboux's Theorem and Moser's method
tight versus overtwisted contact structures
Legendrian and transverse knots
Stein and Weinstein manifolds
open book decompositions; the Giroux correspondence.

* Please note the start date, which is one week earlier than originally scheduled. This is because I'll be at a workshop during the week of October 3 and will need to cancel classes on October 3 and 5; note also that we won't hold class on October 10 since that's during fall break.

Course lecture notes (will update as we go)