Math 225A: Differentiable Manifolds
Fall 2021
Syllabus
This is the first quarter of a year-long sequence in geometry
and topology.
Instructor: Ko Honda
Office: MS 7919
Office Hours: M 3-4pm or by appointment
E-mail: honda at math dot ucla dot edu
URL: http://www.math.ucla.edu/~honda
TA: Eilon Reisin-Tzur; office hours TBA;
ereisint at math dot ucla dot
Class Meetings:
- Lectures: MWF 2pm -
2:50pm in MS 5117
- Discussion: Tu 2pm - 2:50pm in Franz Hall 2288
COVID-19 Policy:
Keeping our community safe depends on each of us
following the latest UCLA health and safety guidelines. Things
might change, but for the moment you:
- Are responsible, regardless of vaccination
status, for wearing an approved mask that fully covers our nose and mouth for the duration of class,
office hours, or other course-related activity.
Disposable masks are available at the Wooden Center for anyone
unable to obtain a mask or who has
forgotten to bring one to campus.
Appropriate masks include two-ply woven fabric masks, surgical
masks, non-woven KN95 masks, and N95
respirators. Please note that scarves,
balaclavas/ski masks, single-layer fabric masks and neck gaiters, bandanas, and turtleneck collars are not
adequate. For those that have a medical
reason not to wear a mask, you can contact the Center for Accessible Education (CAE) to have this exception
approved and sent to instructors.
- Must be fully vaccinated or have submitted an
exception request. Unvaccinated
students with pending or approved exceptions must comply with
twice-weekly testing.
- Are required to complete daily symptom checks
prior to coming to campus, regardless of vaccination status,
and must stay home if you are not cleared by the symptom
survey and/or are advised by the Exposure Management Team to
quarantine or isolate.
- Will refrain from eating or drinking in the
classroom. If you need to take a sip of
water or eat something quickly for medical reasons, please
step outside the room to do so.
Please stay home if sick or potentially
exposed. Email me if you need to stay home, and we will
arrange for you to access class recordings/notes.
The flip side of this requirement is that I
also cannot come in even if I have mild cold symptoms such as
a headache, runny nose, or sore throat. I think it's
likely this will happen at some point in the quarter because I
catch colds pretty often. In that case I'll email
everyone in the class and we'll
have to switch to Zoom for a lecture or two.
Be advised that refusal to comply with current
campus directives related to COVID-19 mitigation will result in
dismissal from the classroom and referral to the Office of
Student Conduct. If you have any questions or concerns about
UCLA’s COVID-19 protocol, go to https://covid-19.ucla.edu/information-for-students/.
Topics
- Review of advanced calculus (calculus on
R^n); inverse and implicit function theorems.
- Differentiable manifolds and their maps.
- Tangent and cotangent bundles, vector bundles.
- Differential forms: tensor and exterior
algebra, exterior differentiation, and Lie derivatives.
- Integration: Stokes' theorem, de Rham
cohomology, and computations using Meyer-Vietoris sequences.
- Vector fields, distributions, Frobenius'
theorem.
Prerequisites
- Knowledge of calculus on R^n, as presented
in the first three chapters of Spivak's Calculus
on Manifolds book.
- This course requires more mathematical
maturity than the average first-year graduate course in the
mathematics department.
Homework
There will be weekly problem sets; see the class
schedule. Homework is due on Fridays, although there
may be some exceptional weeks. The problem sets count for
a large percentage of your total grade (approximately
70%). You may work with others or consult other textbooks,
but the homework you turn in must be
written by you, in your own words, and you must cite your
sources used and your collaborators!
Final examination
There will be a take-home
final. This will be approximately 30% of your final grade.
References
I will follow my Differential Geometry Course Notes. The
main reference is Lee's book, where you can find more details and
examples.
- Differential
Geometry Course Notes
- Lee, Introduction to Smooth Manifolds
- Spivak, A Comprehensive
Introduction to Differential Geometry
- Tu, An Introduction
to Manifolds
- Warner, Foundations of Differentiable Manifolds and Lie
Groups
- Peter
Petersen's notes
WARNING: The course syllabus provides a general plan for
the course; deviations may become necessary.
Last modified: September 15, 2021. |