Math
797SG, Symplectic Geometry and Floer Homology

Fall 2012

Fall 2012

Lectures: TuTh 2:30-3:45, LGRT 1234. If possible/necessary, we may schedule extra meetings for student presentations. TBD.

Instructor: Mike Sullivan
Office: LGRT 1544 Phone: 545-1909
email: my last name <at> math <at> umass
<at> edu

Office hours: By appointment Tues, Thurs, and frequent Weds,9AM-4PM.

Grade: One expository paper and one presentation (negotiable).

Reading material

(Free) Downloaded texts:

Lectures on symplectic geometry
,
by Ana Cannas da Silva

Lectures on Floer homology
,
by Dietmar Salamon

Lectures on Morse homology (with an eye towards Floer theory and holomorphic curves)
,
by Michael Hutchings

Introductory lectures on contact geometry
,
by John Etnyre

Recommended, but not required (non-free) books:

Introduction to symplectic geometry
, by
Dusa McDuff and Dietmar Salamon.

J-holomorphic Curves and Symplectic Topology
, by
Dusa McDuff and Dietmar Salamon.

An Introduction to Contact Topology
, by Hansjorg Geiges

Course Overview:
Symplectic geometry is a central topic in mathematics with connections to algebraic geometry, differential geometry, complex geometry and topology. A major tool which has generated much recent research interest and has many applications in a diverse set of fields, is Floer theory. This course will denote the beginning portion of the semester on a general introduction to symplectic geometry. Once the necessary background is complete, the course will introduce Floer theory, defined using holomorphic curves. The course may also discuss applications and computations of the theory in low-dimensional topology, and possibly symplectic and contact dynamics.

Prereqs:
Smooth manifolds and differential forms (Math 703-704),
basic topology (Math 671), Complex Analysis (Math 621).
Some exposure to homology and cohomology is very helpful
but not absolutely necessary (for example, Algebraic Topology
Math 782, 781 or 797AT and/or Homological Methods Math
797EG)

Syllabus:
Ideally, the course will be made of four parts.

Part 0 will be a one week overview without detail.

Part 1 will take 3-4 weeks and introduce basics of symplectic geometry:
symplectic and Lagrangian submanifolds, almost complex structures, Darboux theorem, ...
This will follow Chapters 1-3, 6-9, 12-14 of da Silva.
If you have a copy of McDuff-Salamon ``Intro" book,
you can also look at Chapters 2-4 and parts of Chapter 11.
There may also be a little bit of contact geometry, da Silva chapters 10-11,
Etnyre's notes, and Chapter 2 of Geiges book for more details.

Part 2 will take 4-6 weeks and introduce Floer homology.
**I expect this to be harder than Part 1.**
I will teach at one of three of levels of detailed
based on how much complex and functional analysis we want to prove versus assume.

Easier level: Following Hutchings' notes, we cover Morse theory in detail, which is a finite-dimensional toy-model of Floer theory, but useful in its
own right. We then make, without much proof, the corresponding Floer theory
constructions and proceed to applications. Some stand-alone Floer
topics may be covered by student projects.

Middle level: We cover Floer theory following Salamon's notes. This is still incomplete, but we will delve into more functional analysis. Again student projects can come in here, but not necessarily.

Harder level: We cover Floer theory following Oh's notes. This is almost at the level of McDuff-Salamon's ``J-holomorphic..." book, chapters 2,3,4,12 and
appendices A,B,C (an excellent reference for all the relevant complex and functional analysis).
Oh's notes also treats
the more complicated case of Lagrangian boundary conditions.

The original main application of Floer homology is Arnold's Conjecture in symplectic dynamics, which will be covered in all three cases.

Part 3 will take 4-5 weeks and focus on more applications/extensions of Floer homology.
There are two highly active research areas:
Heegaard Floer Homology, used in the study of low-dimensional topology; and Contact Homology, used in the study of contact dynamics and rigidity. If we really have time to either one of these topics, I will post relevant articles at that point.
Either direction provides many excellent topics for student projects.

Announcements:

09/23: Suggested exercises.

da Silva: HW1 # 7,9. HW2 # 1,2. HW 3 # 1,3. HW6 # 1,3. HW8 # 1,3.

Etnyre: 2.3, 2.9, 2.10, 2.14

Hutchings: Section 2 # 1. Section 4 # 1,2. Section 5 # 1,2,4.

Salamon: 1.14, 1.15, 1.17, 1.18, 1.20, 1.22, 2.1, 2.8, 3.8, 4.3

09/01:Welcome! This may be the only announcement. But check back, especially if you miss a class.