Vector bundles can be thought of as vector spaces parametrized by a base space, where “space” can mean a topological space, an algebraic variety, or a manifold. They occurence is abundant in topology, geometry, and algebra. Characteristic classes are cohomological invariants of vector bundles and the most important and powerful tools to study them.

This course is a PhD level course on characteristic classes in topology, algebra, and geometry, including an introduction to vector bundles, cohomology, and differential geometry.

Course goals

The course goal is to understand and be able to apply the concept of characteristic classes in a range of mathematical disciplines. At the end of the course, the student will be able to follow current research literature and, if desired, pursue own research projects in this area.

Date, time, location

The first lecture will be Thursday, September 2, 10:15-12:00 via zoom:

https://kth-se.zoom.us/j/68846843780

The second lecture will be Thursday, September 9, 14:15-16:00.

The third lecture will be on Monday, September 20, 13:15-15:00, in person at KTH (room U61) and via zoom. All following classes will be on Mondays 13-15.

We will then decide, based on participants' schedules, whether we keep that time slot or move it to another time on Thursdays, or whether we will meet physically.

During the second half the course will be taught on Thursdays, 3-5. The first lecture of the second half will be delivered on Thursday, October 28, in Kräftriket, House 5, room 31.

The Zoom link for the second half of the course is https://stockholmuniversity.zoom.us/j/64437666741

Lecturer

T. Bauer, G. Arone

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.

Examination

The participants are divided up into groups. Each group prepares around three 30-minute talks. Each participant should talk at some point, but it is up the group to decide on the format of the presentations. Each group is collectively responsible for each talk they give.

The course is graded on a pass/fail scale.

Literature

Lecture notes