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, algebraic 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 lectures will be on Thursdays, 10:15-12:00
The first half of the course starts on September 4 (a Wednesday) and will be taught by Tilman in room E32 at KTH.
Last class (October 17,moved to October 25 instead) takes place in room D37.
The second half is taught by Greg at SU (location tbc).

In-person attendance is strongly encouraged, but if you are unable to come, you may follow a live stream at

https://kth-se.zoom.us/j/61857992732

The lectures will not be recorded. The online experience is provided as-is without any guarantee on its quality.

Lecturers

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: topological space, compactness.

Desirable: One or more of: homological algebra, homology of topological spaces, schemes, varieties and sheaves, Riemannian manifolds.

Examination

The examination is based on biweekly homework sets and short presentations. The course is graded on a pass/fail scale.

The rule for the homework is "2-2-2": Every two weeks there will be a homework set. You should work on these problems in groups of two (and turn in together). You solve all of the homework problems, but turn in only two of them from each set. (You may decide which ones. Maybe it's the ones you find easiest. Or maybe the ones you'd most like to get feedback on!)

There will be a total of 7 homework sets and every problem gives 0-4 points. Of the maximal possible score of 2*4*7=56 points, you need 28 points to pass the course.

There will likely also be a requirement that you present (part of) a solution of a problem in class. I will specify the details as soon as I know the size of the class!

Homework

Problem set 1, due September 16. Sample solutions
Problem set 2, due October 1. Sample solutions
Problem set 3, due October 15
Problem set 4, due November 5

Literature

Lecture notes

If you feel these lecture notes need to be completed or corrected, you are more than welcome to contribute (fork the github repository and make a pull request when you are done. Talk to Tilman if you are unsure what this means.) A significant contribution can replace a homework set in that topic area; discuss this with the lecturer before working on it.

Registration

Anyone is welcome to sit in and listen without registration. If you want to participate, have homework graded, and earn credit for the course, you have to register either by going through the KTH course registration system or by sending me an email with the data: Full name, personal number, home university, status (PhD student/postdoc/master student...). Unless you are a PhD student in math at KTH or SU, please also write a paragraph on why you want to read the course and what relevant prerequisites you have. We reserve the right to decline registration for external students.

Registered Participants

In case you want to get in touch with one another. Be aware that the officially registered participants under the "People" tab are only the KTH crowd.

Benedetta Andina, SU PhD student, andina.benedetta@gmail.com
Rolf Andreasson, Chalmers PhD student, rolfan@chalmers.se
Nami Arabyarmohammadi, SU Bachelor student, nami@math.su.se
Igor Caetano, SU Bachelor student, igor.caetano@math.su.se
Eric Dannetun, SU PhD student, eric.dannetun@gmail.com
Ask Ellingen, Uppsala PhD student, ask.ellingsen@math.uu.se
Victor Groth, KTH Master student, victor.groth73@gmail.com
Jianhao Guo, KTH PhD student, jianhaog@kth.se
Jon Pål Hamre, KTH PhD student, jphamre@kth.se
Georg Huppertz. Chalmers PhD student, huppertz@chalmers.se
Axel Janson, KTH PhD student, axejan@kth.se
Christina Kapatsori, KTH PhD student, christinakapa900@gmail.com
Kim Lukas Kiehn, KTH PhD student, kiehn@kth.se
Jonathan Krook, KTH PhD student, jkroo@kth.se
Lukas König, SU PhD student (physics), lukas.konig@fysik.su.se
Kilian Liebe, SU pre-PhD student, kilian.liebe@math.su.se
Oliver Lindström, SU PhD student, oliver.lindstrom@math.su.se
Tianqi Liu, MDU PhD student, tianqi.liu@mdu.se
Giacomo Maletto, KTH PhD student, gmaletto@kth.se
Edoardo Mason, SU PhD student, edo.mason@gmail.com
Elias Nyholm, Chalmers PhD student, eliasny@chalmers.se
Jon-Magnus Rosenblad, KTH PhD student, jmros@kth.se
Gabriel Saadia, SU PhD student, gabriel.saadia@math.su.se
Ludvig Svensson, Chalmers PhD student, ludsven@chalmers.se
Sjoerd de Vries, SU PhD student, sjoerd.devries@math.su.se
Björn Wehlin, KTH PhD student, bwehlin@kth.se
Carl Westerlund, Umeå PhD student, carl.westerlund@umu.se