# 18.157 Homepage, Spring, 2004

### Introduction to Microlocal Analysis.

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.