I am going to cover the following topics:
1. Review of the theory of distributions;
2. Oscillatory integrals;
3. Pseudo-differential Operators and Fourier Integral Operators;
4. Non-commutative residue;
5. Zeta function for elliptic operators;
6. Propagation of singularities for hyperbolic equations.
There is no textbook for the course. My favorite reference for the Theory Of Distributions is "Introduction to the Theory of Distributions", by G. Friedlander and M. Joshi, second edition. The book "Pseudo-differential operators and spectral theory", by M. Shubin contains a good introduction to pseudo-differential operators and Fourier integral operators. For Non-commutative residue and the theory of Zeta function, I will follow the paper "A New Proof of Weyl's Formula on the Asymptotic Distribution of Eigenvalues", by V. Guillemin, Advances in Mathematicas, Vol 55, 131--160 (1985).
I will not formally assign problems. However, from time to time, I will formulate problems during the lectures, and I will post them. I will always be happy to discuss them. In lieu of the final exam, everybody is expected to give a 30 minute presentation in the end of semester. The topics will be distributed in late October.
Notes
1. Analytic continuation of |x|λ