Math 531B/532B: Introduction to Microlocal Analysis

Gunther Uhlmann

Autumn/Winter 2000-2001, Monday/Wednesday/Friday 9:30-10:20


Microlocal analysis, which is, roughly speaking, local analysis in phase space, arose as a natural development of the methods of geometrical optics pioneered by Hamilton and others.  Hörmander introduced in the early 1970s the concept of wave front set of a distribution and developed a calculus of Fourier integral operators (FIO).  These were very important developments that led to many results in the study of singularities of solutions of partial differential equations (PDE) and have led to many important applications in PDE and other fields.

During the first quarter and a half we will develop the theory of microlocal analysis: wave front set and operations on distributions, the Lax parametrix construction and its global version, FIO.  In order to do this we will develop some of the basic concepts in symplectic geometry that we will use.  In particular, we will study Lagrangian manifolds in some detail.  We will then proceed to develop the calculus of FIO.

In the second half of the second quarter we will give applications of this calculus.  The applications will be chosen depending on the interests of the audience.  Possible topics are inversion of generalized Radon transforms, Weyl's formula for the counting function of eigenvalues, the Poisson summation formula for general Riemannian manifolds, and the inverse seismic migration problem.

I will pretty much follow the book by A. Grigis and J. Sjösstrand, Microlocal Analysis for Differential Operators.

Prerequisites:  The third quarter of Linear Analysis and Topology and Geometry of Manifolds.  Please contact the instructor if you have any doubts about the prerequisites.