# Math 607

## Connections and Characteristic Classes

## Winter 2021

- Lecture 1: Welcome.

notes,
worksheet.
- Lecture 2: Connections.

notes,
worksheet.
- Lecture 3: Connections, continued.

notes,
no worksheet.
- Lecture 4: Grassmannians.

notes,
no worksheet.
- Lecture 5: Associated and principal bundles.

notes,
worksheet.
- Lecture 6: Reduction of the structure group.

notes,
worksheet.
- Lecture 7: Integrable G-structures.

notes,
continue previous worksheet.
- Lecture 8: More on G-structures and connections.

notes,
no worksheet.
- Lecture 9: The Levi-Civita connection.

notes,
worksheet.
- Lecture 10: The Chern connection.

notes,
no worksheet.
- Lecture 11: Connections on principal bundles.

notes,
worksheet.
- Lecture 12: Curvature.

notes,
no worksheet.
- Lecture 13: Curvature continued.

notes,
no worksheet.
- Lecture 14: Curvature on principal bundles.

notes,
worksheet.
- Lecture 15: Curvature on principal bundles continued.

notes,
worksheet.
- Lecture 16: Odds and ends.

notes,
just talked about the worksheet in lecture.
- Lecture 17: Trace, determinant, etc. of cuvature.

notes,
worksheet.
- Lecture 18: Chern forms give well-defined cohomology classes.

notes,
worksheet.
- Lecture 19: Pullbacks, dual bundles, direct sums, tensor products.

notes,
worksheet postponed.
- Lecture 20: The Hopf fibration and the Fubini–Study metric.

notes,
no worksheet.
- Lecture 21: Chern–Gauss–Bonnet: reductions.

notes,
worksheet postponed again.
- Lecture 22: Chern–Gauss–Bonnet: main step.

notes,
worksheet.
- Lecture 23: Chern–Gauss–Bonnet: real case.

notes,
worksheet.
- Lecture 24: Chern–Gauss–Bonnet: cleaning up.

notes,
worksheet.
- Lecture 25: Lower Chern classes.

notes,
worksheet.
- Lecture 26: Lower Chern classes, continued.

notes,
continue previous worksheet. note on transversality.
- Lecture 27: Cohomology of classifying spaces.

notes,
worksheet.
- Lecture 28: Čech cohomology and principal bundles.

notes,
no worksheet.
- Lecture 20: Differential forms and electromagnetism.

notes,
no worksheet.