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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. 

  • 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 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

  • 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

  • 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.

  • ·         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.