## ANNOUNCEMENTS

Registration for Spring term opens Thursday, February 23, 2016.

This page was last updated: .

## COURSE DESCRIPTION

This course presents the heat equation proof of the Atiyah-Singer index theorem. The following topics will be covered: spin geometry, Dirac operator, pseudodifferential calculus, Fredholm operators, the heat kernel method for computing the Seeley-de Witt coefficients and the index of an elliptic operator on a closed manifold, the McKean-Singer index formula, the Hodge decomposition theorem, characteristic classes, the harmonic oscillator, Mehler's formula, and the Atiyah-Singer index theorem.

## PREREQUISITES

Acquaintance with manifolds, exterior derivative, Riemannian metrics, and the spectral theorem for compact self-adjoint operators on a Hilbert space, will be very useful.

## INSTRUCTORS

Farzad Fathizadeh

OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS

Sloan 358

626-395-4355

farzadf@caltech.edu

## POLICIES

### Grades

Based on final presentation and class attendance. Suggested references for choosing the topic of the final presentation will be posted at the bottom of this page.

### Homework Policy

There is no homework for this course.

## TEXTBOOKS

- John Roe,
*Elliptic operators, topology and asymptotic methods,*second edition, Longman, 1998. - P. B. Gilkey,
*Invariance theory, the heat equation, and the Atiyah-Singer index theorem,*Studies in Advanced Mathematics, CRC Press, 1995.