# Introduction to Micro-local Analysis

## Spring, 2004

### Instructor: Lennie Friedlander

### Office: Room 716, Mathematics building, tel: 6212742

### Office hours: T 11:00-11:50, Th 2:00-2:50

The main topics of the course are:

1. Review of the elementary theory of distributions.

2. Local solvability of PDE with constant coefficients.

3. The wave front set of a distribution.

4. Pseudo-differential operators.

5. The index of an elliptic operator.

6. The zeta function of an elliptic operator.

7. The non-commutative residue and analytic continuation of the zeta function.

8. Elements of the theory of Fourier Integral Operators (time permitting.)

There will be no textbook for the course. A good book on the distribution
theory is *"Introduction to the theory of distributions"*,
second edition, by G. Friedlander and M. Joshi. I can recommend
*"Pseudodifferential Operators and Spectral Theory"*, by M. Shubin,
and *"Pseudodifferential Operators"*, by M. Taylor.
The non-commutative residue has not been covered in any textbook that I
am aware of; I will mostly follow the original approach of V. Guillemin.

There will be no formal exams. From time to time, I will formulate problems
during lectures. The list of these problems will be kept at a
separate web page. I expect everybody
to work on the problems. In the end, everybody will make a 30 minute
presentation. The topics for the presentations will be distributed right
after the spring break.

Below are the links to my notes.

1. Analytic continuation of the
distribution |x|^{λ}

2. The wave front set of a
distribution

3. Propagation of singularities for
the wave equation