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 R^{n}—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). |
S^{3} and SDS: | Click here for information. |
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 |