# 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

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

Last updated: February 21, 2019.