18.157: Introduction to Microlocal Analysis

Instructor:

Richard Melrose
MIT, Department of Mathematics
Room 2-174
Email: rbm at math.mit.edu

Time and Place:

Tuesday and Thursday, 9:30-11:00am.
Room 2-151

Outline of the Course

I hope to run this at least partially as a seminar. Initially at least following the notes from earlier versions of the course:-

Changelog (15 Dec, 2007)
Front
Chapter1 Tempered distributions and the Fourier transform
Chapter2 Pseudodifferential operators on Euclidean space
Chapter3 Residual, or Schwartz, algebra
Chapter4 Isotropic calculus
Chapter5 Microlocalization
Chapter6 Pseudodifferential operators on manifolds
Chapter7 Scattering calculus
Chapter8 Elliptic boundary problems
Chapter9 The wave kernel
Chapter10 K-theory
Chapter11 Hochschild homology
Chapter12 (The index formula)

Material covered and projected

Lecture 1: 6 Sept:- Symbols. Notes, Chapter 2, section 2.1.Photos
Lecture 2: 11 Sept:- Chris Kottke, Section 2.2 + me on Section 2.3 and 2.4.Photos
Lecture 3: 13 Sept:- Jacob Bernstein, Section 2.5Photos
Lecture 4: 18 Sept:- Adjoints, composition, principal symbol, symbol sequence. Photos
Lecture 5: 20 Sept:- Chris or Jacob on ellipticity and the Laplacian. Photos
Lecture 6: 25 Sept:- Vedran Sohinger on L² boundedness. Photos
Lecture 7: 27 Sept:- Vedran on Sobolev boundedness. Other algebras.Photos
Lecture 8: 2 Oct:- Fang Wang on distributions and coordinate-invariance.Photos
Lecture 9: 4 Oct:- Lu Wang on compact manifolds.Photos
Lecture 10: 11 Oct:- Lu continues. Jacob on WF.
Lecture 11: 16 Oct:- Jacob on microlocality. Brief comments on microlocalization.
Lecture 12: 18 Oct:- Sheel Ganatra on elliptic operators on manifolds.
Lecture 13: 23 Oct:- Sheel concluding.
Lecture 14: 25 Oct:- Steven Sivek on Hamiltonian vector fields. Photos
Lecture 15: 30 Oct:- Isotropic calculus
Lecture 16: 1 Nov:- Peter Speh on propagation of singularities. Photos
Lecture 17: 6 Nov:- Peter concludes, Nikola Kamburov on Hörmander's propagation theorem.
Lecture 20: 15 Nov:- Vedran on products etc.Photos
Lecture 21: 20 Nov:- Chris on odd K-theory.
Lecture 22: 27 Nov:- Vedran on WF and operations
Lecture 23: 29 Nov:- Summary of families Atiyah-Singer theorem Photos
Lecture 24: 4 Dec:- Fang on isotropic index theorem Photos
Lecture 25: 6 Dec:- Fang continuing on Bott periodicity, Chris on semiclassical limit Photos
Lecture 26: 11 Dec:- Chris on semiclassical calculus and projections. Brief sketch of Atiyah-Singer theorem (again). Photos