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Reading courses

Lectures on Characteristic Classes

 Reading course by Prof. Dr. Ilka Agricola (max. 8 participants)

 Characteristic classes are global topological invariants that serve to distinguish isomorphism classes of principal fibre bundles or vector bundles, and to study interesting properties like orientability, existence of a spin structure, bordisms, or curvature. 

  • Audience

    This course is intended for students at the master's level in mathematics or theoretical physics who already have a good knowledge of differential geometry and algebraic topology and who are familiar with the geometry of vector bundles and principal fiber bundles. For how to participate, see below.

  • Topics

    Topics to be covered are:

    Motivation, first examples, and the notion of a classifying space

    ·         Z_2-principal fibre bundles

    ·         Classifying spaces and universal bundles

    ·         Z-principal fibre bundles
    Stiefel-Whitney classes

    ·         The cohomology ring of real Grassmannians

    ·         Properties of SW classes, splitting principle, total SW class, homotopy invariance

    ·         Calculation of SW classes, relation to symmetric polynomials

    ·         Applications of SW classes: orientation, spin structures, immersions

    ·         SW numbers, theorems of Thom and Pontrjagin
    Chern classes

    ·         The Borel-Hirzebruch formalism

    ·         Properties of Chern classes, Todd class, Chern character

    ·         Euler class and Pontrjagin classes
    Signature theorem and Novikov conjecture

    ·         The signature of a 4k-dimensional manifold

    ·     L genus and  "A hat" genus

    ·         Hirzebruch’s signature theorem

  • Course format

    The lecture videos are recorded and can be studied any time by the student, one per week (approx. 90 minutes, classical blackboard lecture). Weekly assignments (homework).  

    Every week, there is a 1 h online meeting with me, serving a multiple purpose:

    ·         Possibility for the students to ask questions

    ·         Presentation of the solutions of the assignments by the students, discussion with the other participants

    ·        Discussion of the material, general feedback, additional reading if needed

  • How to sign up

    There are two ways to take this course:

    1.      You are already enrolled as a student of Philipps-Universität Marburg, presumably at Fachbereich 12 (Mathematics and Computer Science) or 13 (Physics): Please contact me personally for further information.

    2.      You are not a student in Marburg: Then you may be eligible for the “MarburgOnline: VirtualExchange - MO:VE” program. Please contact the staff of this virtual exchange program.

  • Literature

    ·         John Milnor, James Stasheff, Characteristic Classes. Annals of Mathematics Studies, vol. 76, Princeton University Press, 1974.

    ·         Allen Hatcher, Notes on vector bundles and K-theory, available online at https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html

     Additional course material will be provided by the lecturer.