Date, time, location
The first meeting of this course is on Tuesday, January 22, 10:15 in room 3418 at KTH.
There is no class on January 29. From Tuesday, February 5, we meet 10:15-12:00 every week in the same room.
The course stretches over periods 3 and 4.
Lecturer
Course content
- Introduction to vector bundles. Bundles as parametrized vector spaces, as sheaves, and as cocycles. Operations on bundles. Algebraic bundles. Tangent and normal bundles. Bundles with additional structure
- Lie groups, Grassmannians, universal bundles, and classifying spaces. Simplicial spaces and paracompactness.
- Čech cohmology, the cup product, de Rham cohomology
- The definition and computation of characteristic classes: Stiefel-Whitney classes, Chern classes, and Pontryagin classes
- Introduction to differential geometry: connections, curvature
- Chern-Weil theory and generalized Gauss-Bonnet theorems
- Characteristic classes in algebraic geometry, Chow groups, Segre classes
- An advanced topic such as cobordism, characteristic numbers, genera, the Hirzebruch signature theorem, or the Hirzebruch-Riemann-Roch theorem.
Prerequisites
Required: Familiarity with basic algebraic structures such as groups, rings, fields, modules. Familiarity with basic topological notions: topo- logical space, compactness.
Desirable: One or more of: homological algebra, homology of topological spaces, varieties and sheaves, Riemannian manifolds.
Literature
Lecture notes will be provided for the students. They will contain a bibliography but no textbook will be used for the course.
