**Seminar "Characteristic Classes"**

Dr. Christian Lange, Dr. Jonas Stelzig

Wintersemester 2020/21

organizational meeting: first week of lectures, date to be fixed

How to give a nice seminar talk?
(by Prof. Manfred Lehn in German)

Characteristic classes are cohomological invariants of vector/principal bundles over topological spaces.
More precisely, to such a bundle one associates a cohomology class of the base in a natural way. Characteristic classes are important
in many branches of geometry.

In the seminar we will first discuss several constructions of vector bundles. In particular, we will see that under mild assumptions each vector bundle can be obtained
from a universal bundle over a so-called classifying space. Then we will introduce Stiefel-Whitney classes, the Euler class, and Chern classes and
study some of their properties and applications. We may then discuss Chern-Weil theory, which describes how characteristic classes can be computed in terms of
the curvature forms.

Students should ideally have some background in algebraic topology. Knowledge about differential geometry is also helpful. We encourage everyone who is interested to participate in the organizational meeting, regardless of the exact previous knowledge.

A tentative list of possible seminar topics can be found here. It might be modified depending on the knowledge and number of participants.

If you are interested in attending the seminar please write an email to clange@math.uni-koeln.de or jonas.stelzig@math.lmu.de, mentioning your prior knowledge.

It is planned to hold the seminar in one block of several consecutive days, after the end of the lectures. Details will be discussed at an **organizational meeting at the beginning of the semester (you'll receive all details via email).**

**Talks**

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**Literatur:**

[MS] | J. W. Milnor and J. D. Stasheff,
Characteristic Classes. | ||||||

[H] | A. Hatcher,
Vector bundles and K-Theory. | ||||||

[K] | M. Kreck,
Differential Algebraic Topology. |