Introduction to microlocal analysis
18.157 Spring 2000
In the first half of the course I will define pseudodifferential operators
and discuss their basic properties. To do this I will start with a
description of the Fourier transform on tempered distributions on R^n
and use this to develop the standard oscillatory integral representation of
the kernels of pseudodifferential operators. After describing composition
and invariance properties I will discuss the standard algebra of
pseudodifferential operators on a compact manifold and the fundamental
`symbol sequence'.
In the second part of the semester I will discuss some at least of the
following related global results:
- Spectrum of elliptic operators.
- Hodge (and deRham) theorem.
- Traces, star products and homology.
- Determinants and the determinant bundle.
- Eta invariants.
- Adiabatic limits.
- Relations to K-theory.