Contact Geometry and Topology

Prof. Jo Nelson
Math 541
Spring 2022

Email: jo [dot] nelson [at] rice [dot] edu
Lectures: Tues/Th 10:50am-12:05pm
Location: HBH 453 + Zoom
Office Hours:
TBA HBH 402

Resources

McDuff and Salamon Introduction to Symplectic Topology
Geiges An Introduction to Contact Topology
Wendl Lecture notes on Symplectic Field Theory
Hutchings Lecture notes on ECH

My favorite resources for background on smooth manifolds are Lee's textbooks: (free download from Rice ezproxy)
J. Lee, Introduction to Smooth Manifolds, Second Edition, Springer Graduate Texts in Mathematics.
J. Lee, Introduction to Riemannian Manifolds, Second Edition, Springer Graduate Texts in Mathematics.

Outline

Symplectic and contact structures first arose in the study of classical mechanical systems, such as planetary motion and thermodynamics. Solutions to the equations of motion can be recast in terms of the flow lines of Hamiltonian or Reeb vector fields on a symplectic or contact manifold, respectively. Understanding the dynamics of these vector fields led to the development of global Floer theoretic homological invariants, which originated from Floer's breakthrough to synthesize variational methods, pseudoholomorphic curves, and the Morse-Smale-Witten complex. See my survey article, From Dynamics to Contact and Symplectic Topology and Back.

This course will start with an introduction to some of basic results and constructions in symplectic and contact geometry and then we will delve into the foundations and applications of embedded contact homology (ECH).

The prerequisites for the course are differential topology and algebraic topology. Specifically, I will be assuming you are familiar with smooth manifolds, differential forms, distributions, Lie derivatives, transversality, the fundamental group, homology, and cohomology. It should be do-able to familiarize yourself with these topics via self-study from Lee's Smooth Manifolds textbook; please reach out for guidance as needed.

I will motivate many of the basic ideas and constructions which will help prepare you to read many sources depending on your research, some establishing foundations of the material discussed in class, others going further with it. In class I will introduce the main ideas, explain where they come from, and demonstrate how to use them. I will leave most proofs and technical lemmas for you to read (or not).

Overview of symplectic geometry and topology (McDuff-Salamon)
Overview of contact geometry and topology (Geiges)
Overview of pseudoholomorphic curves (Wendl)
Definition and background of embedded contact homology (Hutchings)
Applications of embedded contact homology (Hutchings)

Grading

Your grade will be based on class participation and some class assignments OR seminar presentations. For the class assignments, in each of the months of February and March you need to either turn in 4 problems mentioned in class or in one of the text books (relating to material being covered in class). In lieu of one or both written assignments, you may give one or two presentations in any student seminar at some point during the spring term (tangentially related to the course material).

If you give a seminar talk, please email me the date/time, topic, and notes from your talk. The written assignments must be turned in by the second week of the month and will be graded on a pass/fail basis. A reasonable attempt will receive a pass. By reasonable attempt, I mean that the solutions to 3 out of the 4 problems must essentially be correct even if there might be some minor errors. If you give a presentation, that will count as a completed assignment for that month. For solving the written problems you are welcome to consult with me or work with other students. But the problems must be written up by you, in your own words.

To receive passing grade you must complete and pass the assignments and foster an atmosphere of collegiality.

For an A you must also attend at least 90% of the classes (or have an excused absence), for a B you must attend at least 80% of the classes, for a C, 70%, and for a D, 60%.

Schedule & Assignments

Date	Material Covered	Exercises (in lieu of presentations)
1/11	Symplectic forms	Symplectic exercises LaTeX
1/13	Symplectic linear transformations
1/18	The symplectic linear group
1/20	Symplectic and complex vector bundles
1/25	From symplectic forms to Hamiltonian flows
1/27	Contact structures and Reeb vector fields	Contact exercices LaTeX
2/1	Examples of contact manifolds
2/3	The (unit) cotangent bundle
2/8	Relationship between contact and symplectic
2/10	Spring Recess: No class
2/15	Open book decompositions
2/17	Overview of Morse theory
2/22	Overview of Hamiltonian Floer theory
2/24	Pseudoholomorphic curves in symplectizations I
3/1	Pseudoholomorphic curves in symplectizations II
3/3	Pseudoholomorphic curves in symplectizations III
3/8	Sketch of Hofer's proof of the Weinstein Conjecture
3/10	Overflow
3/15	Spring Break: No class
3/17	Spring Break: No class
3/22	Overview of ECH
3/24	ECH index I
3/29	ECH index II
3/31	Computations of ECH
4/5	Applications of ECH to symplectic embeddings
4/7	Obstruction bundle gluing I
4/12	Obstruction bundle gluing II
4/14	Obstruction bundle gluing III
4/19	Taubes' proof of the Weinstein Conjecture
4/21	Overflow

Additional Course Policies

Comportment Expectations. The Department of Mathematics supports an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. We expect all participants in mathematics courses (students and faculty alike) to treat each other with courtesy and respect, and to adhere to the Mathematics Department Standards of Collegiality, Respect, and Sensitivity as well as the Rice Student Code of Conduct. If you think you have experienced or witnessed unprofessional or antagonistic behavior, then the matter should be brought to the attention of the instructor and/or department chair. The Ombudsperson is also available as an intermediate, informal option, and contacting them will not necessarily trigger a formal inquiry.

Title IX Responsible Employee Notification. Rice University cares about your wellbeing and safety. Rice encourages any student who has experienced an incident of harassment, pregnancy discrimination or gender discrimination or relationship, sexual, or other forms interpersonal violence to seek support through The SAFE Office. Students should be aware when seeking support on campus that most employees, including myself, as the instructor/TA, are required by Title IX to disclose all incidents of non-consensual interpersonal behaviors to Title IX professionals on campus who can act to support that student and meet their needs. For more information, please visit safe.rice.edu or email titleixsupport@rice.edu.

Disability-related Academic Accommodations. In order to receive disability-related academic accommodations, students must first be registered with the Disability Resource Center (DRC). Students who may need accommodations in this course should give me a written letter from the DRC within the first two weeks. More information on the DRC registration process is available online at https://drc.rice.edu/. Registered students must present an accommodation letter to the professor before exams or other accommodations can be provided. Students who have, or think they may have, a disability are invited to contact DRC for a confidential discussion.