Spring 2017 Mathematics GR6403 section 001
Columbia University, Department of Mathematics
Day/Time: MW 10:10am-11:25am
Location: 307 Mathematics Building

Instructor: Anton Zeitlin

Recommended books:

  1. Manfredo do Carmo, Riemannian Geometry, chapters 1-9
  2. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry
  3. J. Milnor, J. Stasheff, Characteristic classes, Appendix C
  4. J. Dupont, Fibre bundles and Chern-Weil theory (pdf)
  5. L. I. Nicolaescu, Notes on Atiyah-Singer Index Theorem (pdf)

Topics covered:

  1. Riemannian geometry: Hopf-Rinow theorem, curvature and second fundamental form, Jacobi fields, the theorems of Gauss-Bonnet, Cartan-Hadamard, and Bonnet-Myers, spaces of constant sectional curvature...
  2. Connections and curvature on vector bundles, Spin structures, Spaces of flat connections, Characteristic classes via Chern-Weil appoach
  3. Atiyah-Singer index theorem

Projects for students:

    1. V. Mathai, D. Quillen,
      Superconnections, thom classes, and equivariant differential forms
    2. M. Atiyah, L. Jeffrey,
      Topological Lagrangians and cohomology
    1. A. Schwarz,
      Geometry of Batalin-Vilkovisky quantization
    2. A. Schwarz, O. Zaboronsky,
      Supersymmetry and localization
    1. J.J. Duistermaat, G.J. Heckman,
      On the variation in the cohomology of the symplectic form of the reduced phase space
    2. M. Atiyah, R. Bott,
      The moment map and equivariant cohomology
  1.     (for undergraduates only)