18.157 Homepage, Spring, 2004

Introduction to Microlocal Analysis.

Instructor: András Vasy

Office: 2-277

Phone: 253-4386

E-mail: andras@math.mit.edu

Time and location of class: TR 2:30-4, Room 4-145

Tentative office hours: TR 1:30-2:30

Prerequisite: 18.155

Textbook: Grigis and Sjöstrand: Microlocal Analysis for Differential Operators, and Hörmander: The Analysis of Linear Partial Differential Operators, I (as reference).

Grading: there will be no graded homeworks or exams.

Course outline: First I quickly go through basic distribution theory, mostly intended as a reminder. Then I define pseudo-differential operators on Euclidean space and analyze their composition and invariance properties. In particular, this allows us to define ps.d.o's on manifolds and to introduce the notion of principal symbol there.

After thus developing the technical background, our main goal will be to study PDE's. For elliptic PDE's on compact manifolds, I go through the basic spectral theory and functional calculus, resulting in a rough form of Weyl's law for the counting function of eigenvalues. Then I introduce the notion of wave front set to study singularities of distributions, and describe the propagation of singularities for hyperbolic PDE's, such as the wave equation. This will also allow us to obtain an optimal version of Weyl's law.

The preliminary syllabus is available in postscript format.