Diff Geo & Morse Theory

Prof. Jo Nelson
Math 402/500
Spring 2021

Email: jo [dot] nelson [at] rice [dot] edu
Discussion/Lectures: TTh 11.20am-12.40pm
Location: HBH 227 + Zoom
Office Hours:
TBA
Leo Problem Session: Th 9pm CST

Assessment, % of Course Grade

Homework, 40%; Discussion, 26%; Final presentation, 34%
The cutoff for a B- in this course is 55%.

Discussion Assessment

Students will be assessed on their participation during the regularly scheduled TTh discussion time, and each week of participation is worth 2% of the course grade. Successful participation includes asking clarifying questions about the lecture material and providing solutions or hints to the practice homework problems, at least once per week. Uncollegial behavior will result in a 0% discussion assessment for the week.

Resources

We will cover the necessary background from smooth manifolds from Lee (free download from Rice ezproxy):
J. Lee, Introduction to Smooth Manifolds, Second Edition, Springer Graduate Texts in Mathematics.

We will need the notion of a connection and the Hessian, which is developed in more detail than we need in Lee's book (free download from Rice ezproxy):
J. Lee, Introduction to Riemannian Manifolds, Second Edition, Springer Graduate Texts in Mathematics.

Our primary resource for Morse theory are these free, unpublished lecture notes by Prof. Michael Hutchings:
Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), Fall 2002

Our secondary resource for Morse theory is this textbook by M. Audin and M. Damian (free download from Rice ezproxy):
Morse Theory and Floer Homology

Teaching Assistant

The teaching assistant for this course is Leo Digiosia. Leo will hold a weekly discussion/example session. Leo will grade homework and I will grade the presentations.

Homework

In lieu of weekly homework sets graduate students who have passed or are preparing for Advanced Exams and have formal coursework on manifolds at the graduate level may LaTeX 4-6 proofs or detailed examples which are missing or could be better explained from the Hutchings notes. There will be 8 homework sets and homework will count for 40% of your final grade. You must upload your homework to gradescope by 5pm on alternating Fridays. Clearly print your first and last name on your assignment and indicate those students that you worked with. Late homework will not be accepted. Your lowest homework score will be dropped.

Final presentation

You will be expected to give one final presentation, worth 34% of your course grade, which we will reserve the last two weeks of classes. This presentation should be 20-30 minutes, detailing anything tangentially related to the world of manifolds (including but not limited to examples, theory, applications, or generalizations) at the level of an RTG or CMS talk. You should submit a one page outline of your talk to me ahead of your presentation. I am happy to provide suggestions and resources to help you prepare for this talk. Undergrads are welcome to have their grade solely consist of weekly homework and attendance.

In the event of illness or family emergency I must be notified ideally at least 24 hours in advance and documentation from the dean and doctor must be provided to me.

Help

If you find yourself confused, please seek help sooner rather than later. I will be available to answer questions during my office hours and Leo will hold a problem session and office hour Thursdays at 9pm CST. You should use Canvas to post questions about the course, including questions about topics covered in class or regarding the homework.

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.

Outline

The basic plan is to spend the first few weeks getting us caught up on differential geometry background from Lee. Then we will work our way through Hutchings' notes, covering Morse theory with an eye towards Floer theory and pseudoholomorphic curves. If time permits, we will also cover a little bit of Hamiltonian Floer theory at the end of the semester.

I will try to teach the basic ideas 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. My goal is that you'll be comfortable with Morse theory when you encounter it in the wild after taking my course. 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 Morse theory
Background from Lee
Hutchings 2: Morse chain complex
Hutchings 3: Isomorphism between singular homology and Morse homology
Hutchings 4: A priori invariance of Morse homology
Hutchings 5: Genericity and transversality
Salamon: Hamiltonian Floer theory

Schedule & Assignments

Date	Material Covered	Homework (Fridays)
1/26	Introductions. Outline of Morse theory
1/28	Lee 3: Differential (Pushforwards)
2/2	Lee 8-9: Vector fields and flow	HW 1 LaTeX Due 2/5
2/4	Lee 11: Cotangent bundle and covectors
2/9	Lee 11: Cotangent bundle and covectors	HW 2 LaTeX Due 2/12
2/11	Lee 13: TM = T*M via musical Isomorphisms
2/16	SNOW DAY ☃️	HW 3 LaTeX Due 2/26
2/18	SNOW DAY ☃️
2/23	Lee Riem. 11: Connections + Hessians	Break from Homework
2/25	Prof Jo: Transversality
3/2	1: History and Overview	HW 4 LaTeX Due 3/12
3/4	2.1: Morse functions and gradient flow
3/9	2.2: Gradient flow, Morse-Smale condition, a few words on background	HW 5 LaTeX Due 3/19
3/11	Cult Classic: Sphere Eversion 2.3: Compactification by broken flow lines
3/16	Sprinkle day
3/18	2.4: The Morse chain complex and Examples	HW 6 LaTeX Due Monday 3/29
3/23	3.1: Outline of Morse = Singular 3.2: Chain map
3/25	3.3: Left inverse chain map 3.4: Chain homotopy
3/30	4.1: Continuation maps
4/1	4.2: Chain homotopies	HW 7 LaTeX Due Monday 4/12
4/6	5.1: The Sard-Smale theorem
4/8	SPRINKLE Day
4/13	5.2: Generic functions are Morse
4/15	5.3: Spectral flow
4/20	5.4: Generic Morse-Smale transversality
4/22	Student presentations
4/27	Hamiltonian Floer theory