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