Math 277: Contact homology

UC Berkeley, Fall 2012


Michael Hutchings. Tentative office hours: Tuesday 2:00-3:30, 923 Evans. More office hours can be scheduled as needed, and you can always ask me questions by email at [My last name with the last letter removed]

Course outline

The goal of this course is to introduce some invariants of contact manifolds (and related invariants of symplectic manifolds) which are defined by counting holomorphic curves, and to describe some applications of these invariants. The invariants we may discuss include cylindrical contact homology, linearized contact homology, the contact homology algebra, symplectic homology, symplectic field theory, Legendrian contact homology, and embedded contact homology. Applications of these invariants include distinguishing contact manifolds, the Weinstein conjecture and generalizations, symplectic embedding problems, and calculating Gromov-Witten type invariants of closed symplectic manifolds by cutting them along contact-type hypersurfaces. The analytical foundations of this subject are difficult, and not all of these theories have been rigorously defined (yet); I will give an introduction to the foundational issues, but will not go into too much detail about them. This is a large and active area of research, and my hope is that by the end of the course, you will be prepared to read recent papers in the subject and think about unsolved problems.

If you want to learn about a particular topic in more depth, you can give a presentation on it at the end of the course. This is optional.

The following is the rough plan for the course. After each lecture, I will write a sentence or two about what was actually covered in the lecture, and give references as appropriate. (There is no textbook for the course, but the two books by McDuff-Salamon, while mostly concerned with closed symplectic manifolds, contain a lot of useful background and foundational material.)

Lecture summaries and references