SPIN PG course on Spin Geometry SPIN

This is the website for the PG course on Spin Geometry taught by José Figueroa-O'Farrill in the Spring Semester of 2010. For other uses of the word spin, please see this wikipedia page. The use given to this word here is closest to the fourth entry on that page.

The course will consist of 10 11 two-hour lectures. Unless otherwise noted, lectures will take place on Tuesdays at 2:00pm in JCMB 4312.

Schedule of lectures

The following is a tentative list of lectures with topics. Not having taught this course before, the list of topics may require modification, but any such changes will be reflected in this web page as soon as possible.

As the lectures progress I will link the lecture notes to the title of each of the lectures. A snapshot of the latest full set of lecture notes can be found here.

1. Tuesday, 19 January 2010
   Clifford algebras: basic notions
2. Tuesday, 26 January 2010
   Clifford algebras: classification
3. Tuesday, 2 February 2010
   Spin groups and spinorial representations
4. Tuesday, 9 February 2010
   Spin structures on manifolds
5. Tuesday, 16 February 2010
   Connections on fibre bundles
6. Tuesday, 23 February 2010
   The spin connection
7. Tuesday, 2 March 2010
   Holonomy
8. Tuesday, 9 March 2010
   Riemannian holonomy
9. Tuesday, 16 March 2010
   Parallel and Killing spinors
10. Tuesday, 13 April 2010
    Calibrated Geometry
11. Tuesday, 20 April 2010
    Generalized Geometry

Problem sheets

Problem sheets will appear here in due time. One learns by doing, so the problems are an essential part of the course. They will be used freely in the lectures.

1. Clifford algebras I
2. Clifford algebras II
3. Pinors and spinors
4. Spin structures
5. Connections on principal fibre bundles
6. Holonomy
7. Geometries associated to Killing spinors

References

• Applications of Grassmann's extensive algebra
  William Kingdon Clifford
  Of historical interest, perhaps, but this is the original paper on Clifford algebras.
• Clifford modules
  Michael Francis Atiyah, Raoul Bott and Arnold Shapiro
  The first modern paper on Clifford algebras.
• Spin Geometry
  H Blaine Lawson and Marie-Louise Michelsohn
• Spinors and Calibrations
  F Reese Harvey
• Twistor and Killing spinors on riemannian manifolds
  Helga Baum, Thomas Friedrich, Ralf Grunewald and Ines Kath
• Parallel spinors and parallel forms
  McKenzie Y Wang
• Real Killing spinors and holonomy
  Christian Bär
  This is the seminal paper on the classification of riemannian geometries admitting Killing spinors. It contains the very useful cone construction.
• Majorana spinors
  José M Figueroa-O'Farrill
  Some old notes of mine on spinors and Clifford algebras with Physics in mind. Despite the title, they are not just about Majorana spinors.