Math 211a: Fall 1998

Topics in Differential Geometry and Analysis

Schedule: Tuesday \& Friday, 10:30-12 am, Room 209 (conference room).

The class will cover various topics in index theory. A rough outline of the course is:

  1. Proof of the index theorem using the heat equation method, following the book of John Roe.\smallskip
  2. Calculations of indices for various geometrically interesting operators, including operators of interest in Yang-Mills and Seiberg-Witten gauge theory.\smallskip
  3. Index theory on manifolds with boundary, $\eta$-invariants. \smallskip
  4. Further topics related to 3 : spectral flow, relationship to gauge theory on manifolds with boundary, applications in topology.

A few references:

M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry: I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43--69.

M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry: II, Math. Proc. Camb. Phil. Soc. 78 (1975), 405--432.

M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry: III, Math. Proc. Camb. Phil. Soc. 79 (1976),71-99.

All three papers can be found in Atiyah's Collected Works, volume 4.

N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag (1992).

H.B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton (1989).

J. Roe, Elliptic operators, topology and asymptotic methods, Pitman (1988).

Prerequisites: Math 110ab or equivalent preparation.

Course requirements: I expect students to participate actively, and give (at least) one lecture on a topic related to the subject of the course.