Seminar on Characteristic Classes



Organizers

Overview

The point of this workgroup is to understand the Chern character. Along the way, we will investigate K-theory and de Rham cohomology. The idea is to study the (commutative) geometry (spaces, vector bundles, differential forms...), and the dual algebra (commutative algebras, projective modules, Kähler differentials...), in parallel.

References

  1. Vector bundles and K-theory
  2. De Rham cohomology
    • Greub, Halperin and Vanstone, Connections, Curvature, and Cohomology, Vol. I: De Rham cohomology of manifolds and vector bundles. Pure and Applied Mathematics, Vol. 47. Academic Press, New York-London, 1972.
    • Bott and Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82. Springer-Verlag New York-Berlin, 1982.
  3. Characteristic classes
    • Milnor and Stasheff, Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.

Schedule

Every second Monday, from 1:15 - 2:15, location King Edward B15. Starts September 25.

Tentative Outline

  1. (Sept. 25) Fibre bundles and vector bundles; constructions.
  2. (Oct. 16) Topological and algebraic K-theory; Swan's theorem.
  3. (Oct. 30) Smooth manifolds; tensor fields; differential forms and the exterior algebra.
  4. (Nov. 13) Exterior derivative and de Rham cohomology. Kähler differentials.
  5. (Nov. 27) Chern classes: axioms, applications, existence, the Chern character.


September 25

Speaker: Jonathan Scott

Overview. Fibre bundles, vector bundles and structure groups. Constructions on vector bundles (pullback, Whitney sum and tensor product); topological K-theory.

October 16

Speaker: Jonathan Scott

Constructions on vector bundles, topological K-theory, projective modules and algebraic K-theory, Serre-Swan theorem.

October 30

Speaker: Jonathan Scott

Projective modules and algebraic K-theory. For open subsets of Euclidean space: tangent and cotangent spaces, differential forms. A review of multilinear algebra. The de Rham complex of an open subset of Euclidean space.

Last updated: 29/10/2006