18.157: Introduction to Microlocal Analysis
MIT, Department of Mathematics
Email: rbm at math.mit.edu
Time and Place:
Tuesday and Thursday, 9:30-11:00am.
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:-
(15 Dec, 2007)
Tempered distributions and the Fourier transform
Pseudodifferential operators on Euclidean space
Residual, or Schwartz, algebra
Pseudodifferential operators on manifolds
Elliptic boundary problems
The wave kernel
(The index formula)
Material covered and projected
Lecture 1: 6 Sept:- Symbols. Notes, Chapter 2, section 2.1.
Lecture 2: 11 Sept:- Chris Kottke, Section 2.2 + me on Section 2.3 and 2.4.
Lecture 3: 13 Sept:- Jacob Bernstein, Section 2.5
Lecture 4: 18 Sept:- Adjoints, composition, principal symbol, symbol sequence.
Lecture 5: 20 Sept:- Chris or Jacob on ellipticity and the Laplacian.
Lecture 6: 25 Sept:- Vedran Sohinger on L
Lecture 7: 27 Sept:- Vedran on Sobolev boundedness. Other algebras.
Lecture 8: 2 Oct:- Fang Wang on distributions and coordinate-invariance.
Lecture 9: 4 Oct:- Lu Wang on compact manifolds.
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.
Lecture 15: 30 Oct:- Isotropic calculus
Lecture 16: 1 Nov:- Peter Speh on propagation of singularities.
Lecture 17: 6 Nov:- Peter concludes, Nikola Kamburov on Hörmander's propagation theorem.
Lecture 20: 15 Nov:- Vedran on products etc.
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
Lecture 24: 4 Dec:- Fang on isotropic index theorem
Lecture 25: 6 Dec:- Fang continuing on Bott periodicity, Chris on semiclassical limit
Lecture 26: 11 Dec:- Chris on semiclassical calculus and projections. Brief sketch of Atiyah-Singer theorem (again).