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.

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).


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.

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