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.