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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
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Lie groups, Grassmannians, universal bundles, and classifying spaces. Sim-plicial spaces and paracompactness.
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Čech cohmology, the cup product, de Rham cohomology
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The definition and computation of characteristic classes: Stiefel-Whitney classes, Chern classes, and Pontryagin classes
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Introduction to differential geometry: connections, curvature
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Chern-Weil theory and generalized Gauss-Bonnet theorems
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Characteristic classes in algebraic geometry, Chow groups, Segre classes
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An advanced topic such as cobordism, characteristic numbers, genera, the Hirzebruch signature theorem, or the Hirzebruch-Riemann-Roch theorem.
FSF3709 Characteristic Classes 7.5 credits
![](https://kursinfostorageprod.blob.core.windows.net/kursinfo-image-container/Picture_by_MainFieldOfStudy_26_Default_picture.jpg)
Information for research students about course offerings
Spring 2019
About course offering
For course offering
Autumn 2024 Start 26 Aug 2024 programme students
Target group
No information insertedPart of programme
No information insertedPeriods
P1 (4.0 hp), P2 (3.5 hp)Duration
Pace of study
25%
Form of study
Normal Daytime
Language of instruction
English
Course location
KTH Campus
Number of places
Places are not limited
Planned modular schedule
Course memo
Course memo is not publishedSchedule
Schedule is not publishedApplication
For course offering
Autumn 2024 Start 26 Aug 2024 programme students
Application code
51070
Contact
For course offering
Autumn 2024 Start 26 Aug 2024 programme students
Contact
Tilman Bauer (tilmanb@kth.se)
Examiner
No information insertedCourse coordinator
No information insertedTeachers
No information insertedContent and learning outcomes
Course contents
Intended learning outcomes
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.
Literature and preparations
Specific prerequisites
Familiarity with basic algebraic structures such as groups, rings, fields, modules. Familiarity with basic topological notions: topo-logical space, compactness.
Recommended prerequisites
One or more of: homological algebra, homology of topological spaces, varieties and sheaves, Riemannian manifolds.
Equipment
Literature
Lecture notes will be provided for the students. They will contain a bibliography but no textbook will be used for the course.
Examination and completion
If the course is discontinued, students may request to be examined during the following two academic years.
Grading scale
Examination
- INL1 - Assignment, 7.5 credits, grading scale: P, F
Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.
The examiner may apply another examination format when re-examining individual students.
Homework and presentation.
Opportunity to complete the requirements via supplementary examination
Opportunity to raise an approved grade via renewed examination
Examiner
Ethical approach
- All members of a group are responsible for the group's work.
- In any assessment, every student shall honestly disclose any help received and sources used.
- In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.