The topics of every lecture appear here, possibly tentatively.
Chapter references refer to the course notes.
Jan 22: Lecture 1: Chapter 1-2.1
Introduction to vector bundles. Organizational matters.
Jan 29: no class.
Feb 5: Lecture 2: Chapter 2.2
Vector bundles as sheaves.
Feb 12: Presentation 1, Lecture 3: Chapter 2.3-2.4
Vector bundles as cocycles.
Operations on vector bundles, tensor, symmetric, and exterior products
(the last three as reading assignment).
Feb 19: Lecture 4: Chapter 3.1
Lie groups, Stiefel manifolds and Grassmann manifolds
Feb 26: Presentation 2, Lecture 5: Chapter 2.5-2.6
Smooth manifolds, tangent bundles, normal bundles, differential forms.
Mar 5: Lecture 6: Chapter 3.2-3.3
Simplicial spaces and simplicial categories, geometric realization.
Classifying spaces and the bar construction.
Mar 12: Lecture 7: Chapter 3.4-3.5, Presentation Nasrin
Schubert cells, paracompactness.
Mar 19: Lecture 8: Chapter 3.5-3.6
Homotopy invariance of pullback bundles, fiber bundles, universal bundles
Mar 26: Lecture 9: Chapter 4.1-4.3
Čech cohomology, the long exact sequence and Mayer-Vietoris sequence
Apr 2: Class cancelled. Reading assignment: Chapter 4.6-4.8
The cup product in cohomology; sample cohomology rings; de Rham cohomology
Apr 9: Chapter 6.1-3, 5.1.1, Presentation Jevgenija
Riemannian manifolds, connections, and curvature; the Leray-Hirsch theorem and Thom classes
April 16: 5.1, Presentations Frida, Lukas
Thom isomorphism, Euler class, Gysin sequence, cohomology of BU(n) and Chern classes
Apr 23: Presentation 5, Chapter 7, Presentation Erik
Schemes, cycles, Chow groups; Chern classes and Segre classes in algebraic geometry
April 30: Chapter 5.2-4, Presentations Axel, Menno, Jeroen
Stiefel-Whitney classes, Pontryagin classes and applications; the Hirzebruch-Riemann-Roch theorem
May 7: Presentations Alvin, Thomas
The Hirzebruch signature theorem; introduction to K-theory
May 14: Chapter 6.4-6, Presentations Jacob, Bernardo
Chern-Weil theory and the generalized Gauss-Bonnet theorem
