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

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.
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).

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.