MA726
“I was not, and was conceived. I loved and did a little work. I am not and grieve not”.

William Kingdon Clifford.

This is a graduate class in differential geometry. This semester the subject is spin geometry. This is the syllabus. Time: T/Th 12:30 PM -- 1:45 PM.
Location: CGS 323

Course materials

There is not an official textbook for this class, but many topics will follow the presentation of Michelson and Lawson's Spin Geometry (the ``orange book''). I will post lecture notes below in addition to the following standalone summaries:
  • Notes on Clifford algebra. (Last updated: 2/1)
  • Notes on the construction of Atiyah, Bott, and Shapiro. (Last updated 2/22)
  • Notes on spin structures in geometry. (Last updated 2/24) The list of references below are excellent (they are not mine).
  • Notes on bordism by Dan Freed. Great source for background on bundles, principal bundles, and characteristic classes.
  • Class on spin geometry taught by José Figueroa-O'Farrill.
  • Linear field equations and self-dual spaces , Nigel Hitchin. This is a linear or "free" version of the Penrose-Ward correspondence between solutions of holomorphic PDE's on twistor space with conformally invariants PDE's on the base manifold of twistor space.
  • , the famous paper of Atiyah, Hitchin, and Singer.

    • Course log


  • January 21: Definition of the Clifford algebra. See Notes on Clifford algebra.
  • January 23: Definition of (S)pin group. Class notes.
  • January 28: Spin as a universal covering space. Class notes.
  • January 30: Classification of Clifford algebras. Class notes.
  • February 4: Clifford modules and spin representations. Class notes.
  • February 13: Classification of Clifford modules and K-theory. Class notes.
  • February 20: Complex K-theory, ring structure, Bott periodicity. Class notes.
  • February 25: The ABS construction. Begin spin structures.
  • February 27: Principal bundles, characteristic classes. Class notes.
  • March 4: Reduction of structure group, spin structure. Class notes.
  • March 6: Classifying spaces, the first Chern class. Class notes.
  • March 18: Spin structures on complex vector bundles. Class notes.
  • March 20: Sheaves, sheaf cohomology. Class notes.
  • March 25 Spin structures on Riemann surfaces. Moving towards twistors. Class notes.
  • March 27: Almost complex structures and isotropic subspaces.
  • April 8: Pure spinors and twistor space.
  • April 10: Twistor space and conformal structures. Integrability. Class notes.
  • April 15: The Penrose-Ward correspondence I; jets and linear fields. Class notes.
  • April 17: The Penrose-Ward correspondence II; more linear fields. Class notes .
  • April 22: The Penrose-Ward correspondence III; Penrose operators and line bundles. Class notes .

    • Problem sets

  • Problem set 1. Clifford algebras and spin groups (last updated: January 25).
  • Problem set 2. Some Clifford modules, mostly spin structures (Last updated: February 12).
  • Problem set 3. Pure spinors and complex structures.