# Introduction to Microlocal Analysis: 18.157 (Spring 1998)

### General information

I will be following my lecture notes
(here is the current version of chapter 2, which is undergoing revision
chapter2.ps ) at least at the beginning. I hope to get far enough to cover
- Global pseudodifferential operators on Euclidean space
- Pseudodifferential operators on compact manifolds
- An example of pseudodifferential operators on a(nother) class of non-compact manifolds

and in each case to do some serious application -- determinants, homology, K-theory as time permits.
- Homework:
- Due February 19, 1998. Homework1 (Postscript file).

- Lectures (in 2-255 2:30 - 4:00 unless otherwise noted)
- Tuesday February 3: Schwartz test functions, tempered distributions.
- Thursday February 5: Symbols and approximation.
- Tuesday February 10: Pseudodifferential operators defined.
- Wednesday February 11: (2-3 in 2-102): Asymptotic summation, kernels of residual operators, reduction theorem (composition theorem?).
- Thursday February 19: [Problem set 1 due] Composition theorem. Principal symbol. Ellipticity. Parametrices.
- Tuesday February 24: Elliptic regularity, Laplacian. Schur's Lemma. Square root of a symbol.
- Wednesday February 25: Approximate square root. Boundedness on $L^2.$ Sobolev spaces.
- Thursday February 26: Weighted Sobolev spaces. Polyhomogeneity. Isotropic calculus.
- Tuesday March 3: Compact supports. Local coordinate invariance.
- Wednesday March 4: No lecture.
- Thursday March 5: Transformation of symbol. Pseudodifferential operators on a (compact) manifold.
- Thursday March 12: Ellipticity.
- Tuesday March 17: Fredholm condition.
- Wednesday March 18: Vector bundles, deRham complex.
- Thursday March 19: Metrics, inner products and Hodge theorem.
- April 7: Dirac operators and manifolds with boundary.
- April 9: Boundary problems and the Calderon projector.