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:

- Proof of the index theorem using the heat equation method, following the book of John Roe.\smallskip
- Calculations of indices for various geometrically interesting operators, including operators of interest in Yang-Mills and Seiberg-Witten gauge theory.\smallskip
- Index theory on manifolds with boundary, $\eta$-invariants. \smallskip
- 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.