18.157, Introduction to Microlocal Analysis, Spring 2019

Instructor:Peter Hintz
Class hours:TR 11–12:30 in 2–151
Office hours:Tue 1:30–3:30 PM in 2–265
Grading:There will be no exam. The grade will be based on problem sets, which I will assign every other week; late solutions will not be accepted. Solutions can be submitted either in person or as a pdf file by email (phintz@mit.edu). No points will be given, but I will read your submissions and provide feedback. Collaboration on homework is encouraged, but individually written solutions are required; name all collaborators or other sources of information.
Outline:

Microlocal analysis provides powerful tools for the study of (linear) partial differential equations. In the first part of the course, I will define pseudodifferential operators on Rn—which are generalizations of differential operators—and discuss their composition, mapping, and invariance properties. A simple application is the construction of an approximate inverse of an elliptic operator, and the Fredholm property of elliptic PDE on closed manifolds.

The second part revolves around the key notion of the wave front set of a distribution, which captures its singularities in position and frequency. The main theorem concerns the propagation of singularities (or, in the reverse direction: regularity) for solutions of non-elliptic PDE such as the wave equation.

Prerequisites:I will assume basic familiarity with distributions, functional analysis, and the theory of manifolds.
Textbooks:I will draw material from various references, including Richard Melrose's lecture notes and Grigis and Sjöstrand, Microlocal Analysis for Differential Operators. Further references are given in the lecture notes.
Lecture notes:Available here (in progress).
S3 and SDS:Click here for information.

Schedule

TueFeb 5 Overview. Schwartz functions and tempered distributions. Fourier transform and inversion. The Schwartz representation theorem and the Schwartz kernel theorem.
ThuFeb 7 Differential operators. Symbols, ellipticity.
Homework 1, due Thursday, Feb 21.
TueFeb 12 Classical symbols, asymptotic summation. Quantization of symbols.
Office hours moved to Wednesday, Feb 13, 2–4 pm.
ThuFeb 14 Boundedness on Schwartz functions and tempered distributions. Left/right reduction.
TueFeb 19 No class (Monday schedule!)
Office hours moved to Thursday, Feb 21, 1–2 pm and 3–4 pm.
ThuFeb 21 Left/right reduction (continued). Composition. Pseudo-locality of ps.d.o.s. Principal symbol.
Homework 1 due at 5 pm.
Homework 2, due Thursday, Mar 7.
TueFeb 26 Principal symbol (continued). Classical operators. Elliptic parametrix. Boundedness on Sobolev spaces.
ThuFeb 28 Boundedness on Sobolev spaces (continued). Local coordinate invariance.
TueMar 5 Local coordinate invariance (continued). Manifolds, vector bundles. Density bundles, invariant integration on manifolds.
ThuMar 7 Differential and pseudodifferential operators on manifolds.
Homework 2 due at 5 pm.
Homework 3, due Thursday, Mar 21.
TueMar 12 Ps.d.o.s on manifold (continued), principal symbol. Operators on vector bundles.
ThuMar 14 Fredholm theory for elliptic operators on compact manifolds. Sobolev spaces.
TueMar 19 Elliptic theory with Sobolev spaces. Hamilton vector fields and commutators.
ThuMar 21 Operator wave front set.
Homework 3 due at 5 pm.
TueMar 26 No class (spring break)
ThuMar 28 No class (spring break)
TueApr 2 Wave front set. Microlocal elliptic regularity. Relative wave front set.
Homework 4, due Thursday, Apr 18.
ThuApr 4 Pairings and products of distributions. Symmetric hyperbolic systems.
TueApr 9 Scalar symmetric hyperbolic equations, Egorov's theorem, propagation of singularities.
ThuApr 11 Propagation of singularities: positive commutator argument.
TueApr 16 No class (Patriots' Day)
ThuApr 18 Propagation of singularities: conclusion of the positive commutator argument.
Homework 4 due at 5 pm.
TueApr 23
ThuApr 25
TueApr 30
ThuMay 2
TueMay 7 TBD (I will be out of town.)
ThuMay 9 TBD (I will be out of town.)
TueMay 14
ThuMay 16


Last updated: February 21, 2019.