The topics of every lecture appear here, possibly tentatively.
Chapter references refer to the course notes.
Jan 23: Lecture 1: Chapter 1-2.1
Introduction to vector bundles. Organizational matters.
Jan 30: Lecture 2: Chapter 2.2
Vector bundles as sheaves.
Feb 6: Presentation 1 (Eric), 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 13: Lecture 4: Chapter 3.1
Lie groups, Stiefel manifolds and Grassmann manifolds
Feb 20: Presentation 2 (Oliver), Lecture 5: Chapter 2.5-2.6
Smooth manifolds, tangent bundles, normal bundles, differential forms.
Feb 27: Lecture 6: Chapter 3.2-3.3
Simplicial spaces and simplicial categories, geometric realization.
Classifying spaces and the bar construction.
Mar 6: Presentation 3 (Nima), Lecture 7: Chapter 3.4-3.5
Schubert cells, paracompactness.
Mar 13: Exam week, no class
Mar 20: Lecture 8: Chapter 3.5-3.6
Homotopy invariance of pullback bundles, fiber bundles, universal bundles
Mar 27: Lecture 9: Chapter 4.1-4.3
Čech cohomology, the long exact sequence and Mayer-Vietoris sequence
Apr 3: Good Friday, no class
Apr 10: Presentation 4 (Rune), Chapter 6.1-6.3, 4.4-4.5
Riemannian manifolds, connections, and curvature; products in cohomology and de Rham cohomology
Apr 17: Presentation 5 (Stefano), Chapter 7.1-7.4
Schemes, cycles, Chow groups
Apr 24: Chapter 4.4-4.5
Fine sheaves, homotopy invariance and some cohomology computations
May 1: May Day, no class
May 8: Chapter 4.6-7, 5.1, Presentation (Kaj)
The cross and cup products in cohomology, cohomology of BU(n) and Chern classes
May 15: Chapter 5.1.1, 7.5-6, Presentation (Magnus)
Computations of Chern classes, Chern classes and Segre classes in algebraic geometry
May 22: Chapter 4.8, 6.4-5, Presentation (Eric)
De Rham cohomology and the de Rham theorem, Chern classes from curvature
May 29: Chapter 5.2-3, Presentation (Bashar)
Applications of characteristic classes, Stiefel-Whitney and Pontryagin classes
