Syllabus for Math 519, spring 2023

Syllabus for
Math 519, Section C1, Spring 2023
Differentiable Manifolds II

Professor: Eugene Lerman
Office: 37 Computer Applications Building (CAB)

Phone:

email: lerman at illinois edu
Home page: https://faculty.math.illinois.edu/~lerman
Course page: https://faculty.math.illinois.edu/~lerman/519/s23/519s23.html
homework assignments are on Moodle (Moodle login help is here )
Office Hours: TBA. Check my calendar here
Class meets:

12:00 pm - 12:50 pm MWF Altgeld Hall 447

Prerequisites: Math 518 or equivalent.

Textbooks: No textbook is required. I plan to write up my notes. Some of the material is not in textbooks.

Homework: Homework problems are to be assigned once a week. They are due the following week in class on Wednesdays. Late homework will be penalized. I will drop the lowest homework score.
There are several reasons why I want the homework to be turned in on time. First of all, if you are not keeping up with the homework, you are not keeping up with the class, and then you may well get lost in the lectures. Secondly, it is hard to grade fairly the homework that is turned in late. The grader would have to keep careful track of the grading rubrics and so on. In short: late homework is bad for you and means more work for the grader.
If you have questions about your homework score please take it up with the grader first. If you and the grader can't resolve the issue I will step in.
If you have questions about the homework itself, please use the forum on Moodle.

Exams: no exams.
Grade: The final grade is based on the homework.

Planned content of the course:

Basics of Lie groups up to Adjoint and adjoint representations (review?).
Categories and functors.
Operations on vector bundles
Fiber bundles, principal bundles, associated bundles.
Frame bundle of a vector bundle.
Connections on principal bundles as distributions and as Lie algebra-valued 1-forms. Induced covariant derivatives on the associated bundles.
Curvature. Chern classes. Chern-Weil homomorphism.
Diffeological spaces, sheaves of sets on a site, thin fundamental groupoid and parallel transport.
Sheaves of groupoids, descent, 1-stacks. Lie groupoids and geometric stacks over the site of manifolds (if time permits).

Attendance

Academic Integrity

Campus police safety information. Plus a one page handout

The Pandemic : Following University policy, all students are required to engage in appropriate behavior to protect the health and safety of the community. Students are also required to follow the campus COVID-19 protocols. Please refer to the University of Illinois Urbana-Champaign's COVID-19 website for further information.

Return to Eugene Lerman's home page

Last modified: Thu Jan 5 14:36:51 CST 2023

Syllabus for Math 519, Section C1, Spring 2023 Differentiable Manifolds II

Syllabus for
Math 519, Section C1, Spring 2023
Differentiable Manifolds II