Differentiable Manifolds
Prof. Jo Nelson
Math 451/551
Fall 2022
Email:
jo [dot] nelson [at] rice [dot] edu
Discussion/Lectures: TTh 9.25 - 10.40 am
Location: HBH 453
Office Hours: TBA
References (free download from Rice ezproxy):
J. Lee, Intro to Smooth Manifolds, 2nd. Ed., Springer GTM.
J. Lee, Intro to Riemannian Manifolds, 2nd. Ed., Springer GTM.
Outline
The basic plan is to cover most of the material in chapters 1-19 of Lee's book (adding a few interesting things which are not in the book). My goal is for you to understand the basic concepts listed below and to feel confident about encountering manifolds in the wild (e.g. outside of the classroom). This material is all essential background for graduate level geometry and topology research. In class I will try to introduce the main ideas, explain where they come from, and demonstrate how to use them, with an emphasis on examples. I will tend to leave most proofs and technical lemmas for you to read in Lee's book (or not).
Assessment, % of Course Grade
Undergraduates enrolled in Math 451 will have their grade based on homework (72%) and participation (28%). There will be approximately 11 weekly homework assignments; you may drop your two lowest (or nonexistent) homeworks. To receive a passing grade you must complete and pass the assignments, attend at least 70% of the classes, 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%.
Grad students enrolled in Math 551 will have their grade based on one in class presentation (36%), 2 seminar talks (36%), and participation (28%). Graduate students may elect to do the weekly homeworks in lieu of presentations; please inform Prof. Jo. To receive a passing grade you must give the in class presentation and one seminar talk, attend at least 70% of the classes, 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%. Please email me the date/time, topic, and notes from your seminar talk.
Participation
Students will be assessed on their participation during the regularly scheduled TTh class meetings, and each week of participation is worth 2% of the course grade. Successful participation includes attending class and asking clarifying questions about the lecture material and/or participating in class discussions. Uncollegial behavior will result in a 0% discussion assessment for the week.
Teaching Assistant
Tam Cheetham-West is the teaching assistant for this course. Tam will hold
a weekly discussion/example session. Tam will
grade homework and I will grade the presentations. I will review your homeworks and read your weekly homework reflections.
Homework (Undergraduates)
There will be 12 homework sets and homework will count for 72% of the final grade for undergraduates enrolled in Math 451. You must upload your homework to gradescope by 5pm on Fridays. Collaboration is encouraged but the write up of the solutions should be in your own words. Late homework will not be accepted without prior authorization from me. Your lowest two homework scores will be dropped. In the event of illness or family emergency I must be notified ideally at least 24 hours in advance and documentation from your magister or doctor may be requested.
In class presentation (Graduate students)
Graduate students enrolled in Math 551 will be expected to give one in class presentation, worth 36% of your
course grade. These presentations will be done the last two weeks of the semester. Your presentation should be 30-40 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, CMS, or learning seminar 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.
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 Tam will hold a problem session and office hour TBA.
Schedule & Assignments
Date
Material Covered
Homework (Fridays)
8/23
Definition of topological and smooth manifold, examples.
8/25
Diffeomorphisms. Tangent vectors.
8/30
Tangent space.
Derivative of a smooth map between smooth manifolds.
Homework 1 LaTeX
Due 9/2
9/1
Local coordinates.
9/6
Vector fields and the tangent bundle.
Homework 2 LaTeX
Due 9/9
9/8
Immersions, embeddings, and submersions.
Short movie ``Outside In".
9/13
Embeddings and submanifolds I
Homework 3 LaTeX
Due 9/16
9/15
Embeddings and submanifolds II
9/20
Transversality
Homework 4 LaTeX
Due 9/23 9/22
"Generic" transversality results
9/27
The Lie bracket of two vector fields. The flow of a vector field.
Hopf Fibration and Video Homework 5 LaTeX
Due 9/30
9/29
Lie Algebras
10/4
Lie Groups
Homework 6 LaTeX
Due 10/7
10/6
Multilinear algebra and tensors
10/11
Fall Recess
No classes Fall Break, No HW!
10/13
The cotangent bundle
Differentials revisited
10/18
Symmetric Tensors. Riemannian Metrics.
TM = T*M via musical isomorphisms
Homework 7 LaTeX
Due 10/21 10/20
Alternating tensors in gory detail
10/25
Differential forms in general.
Wedge product, pullback, and exterior derivative. Homework 8 LaTeX
Due 10/28 10/27
Lie derivatives
Commutator two VFs = 0 iff their flows commute.
11/1
Distributions of the integrable, involutive, and contact persuasion.
Contact Slides sketches of topology Homework 9 LaTeX
Due 11/4 11/3
Differential ideals, the Frobenius theorem.
Foliations.
11/8
Voting Day (Political Geometry)
Dr. Moon Duchin (Tufts) on The Mathematics of Redistricting Homework 10 LaTeX
Due 11/10
11/10
Orientations revisited.
Volume form on a Riemannian manifold
11/15
A one-form is exact iff its integral over every loop is 0.
Integration of differential forms. Homework 11 LaTeX
Due 11/18
11/17
Stokes' theorem.
11/22
Overview of de Rham cohomology.
(No more homework!)
11/24
Thanksgiving
No classes
11/29
Isomorphism between singular and de Rham cohomology
12/1
Student Presentations
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.
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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.