Course Syllabus
The topics of every lecture appear here, possibly tentatively. Chapter references refer to the course notes.
| Sep 2, 10–12 | Introduction to vector bundles. Organizational matters. | Chapter 1–2.1 |
| Sep 9, 14–16 | Vector bundles as sheaves | Chapter 2.2 |
| Sep 20, 13-15 | Vector bundles as cocycles; Presentations groups 1+2 | Chapter 2.3.1 |
| Sep 27, 13-15 | Cocycles as functors, presentation group 3; operations on bundles. Slides by Lukas G. | Chapter 2.3.2-2.4 |
| Oct 4, 13-15 | Operations on vector bundles, presentation group 4; smooth manifolds via sheaves | Chapter 2.5-2.6 |
| Oct 11 | class cancelled | |
| Oct 18, 13-15 | Presentation group 5; Tangent bundles, bundles with additional structure. Stiefel and Grassmann manifolds, presentation group 2. Slides by Vahid. | Chapter 2.6,3.2 |
| Oct 21, 10-12 | Simplicial spaces, presentation group 1, geometric realization, presentation group 5 | Chapter 3.3 |
| Oct 28, 3-5 | Overview of universal bundles and characteristic classes. Schubert cells, presentation group 4. Slides by Stepan Maximov | Chapter 3.4 |
| Nov 4, 3-5 (remote) | Proof of existence of universal bundles. Cech cohomology and functoriality (lecture slides by Gregory), presentation group 3 | Chapters 3.5 -3.6, 4.1-4.2 |
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Nov 11, 3-5 (remote) |
Homotopy invariance of cohomology, Meyer-Vietoris sequence, product in cohomology. Lecture slides by Gregory | |
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Nov 18, 3-5 (remote) |
Cohomology ring, the Kunneth formula. Lecture slides by Gregory . Presentation group 1, Leray Theory and the Gysin sequence (sections 5.1.1 and 5.1.2) | |
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Nov 25, 3-5 (hybrid) |
Presentation group 2, Chern classes (5.1.3). Stiefel-Whitney classes. Some examples. | |
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Dec 2, 3-5 (hybrid) |
Applications to immersion. Presentation by Robin Stoll on de Rham cohomology. | |
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Dec 7, 3-5 (remote) |
Here's the division into groups. Please let me know asap if somebody drops out.
| # | Participants | Talk 1 | Talk 2 | Talk 3 |
| 1 | Klara, Axel, Scott, Errol | 2.3.1 up to Prop 2.3.8 | 3.3 | 5.1.1-5.1.2 |
| 2 | Nausica, Christian, Wen-Kui, Björn | 2.3.1, starting after the proof of Prop 2.3.8 (as far as you get) | 3.2 | 5.1.3 |
| 3 | Stefan, Louis, Kirthana, Felix | 2.3.2 | 4.1-4.2 | |
| 4 | Isaac, Zi, Stepan, Petter, Lukas | 2.4.2,4,5 | 3.4 | |
| 5 | Vahid, Andrés, Tobias, Lukas, Sven | 2.5.2 | 3.3.2 |
Course Summary:
| Date | Details | Due |
|---|---|---|