Office: 383M

Phone: 723-2226

E-mail: andras "at" math.stanford.edu

Tentative office hours: TBA

Class location: MWF 10-10:50am, McCullough 126.

Textbook: `The Atiyah-Patodi-Singer Index Theorem' by R. B. Melrose and volume 2 of Michael Taylor's `Partial differential equations', both recommended. Some of the lectures will be typed up as lecture notes.

A knowledge of microlocal analysis is NOT a pre-requisite. I will provide a quick overview of the background. For a more thorough background on this, please see Richard Melrose's lecture notes and volume 2 of Michael Taylor's PDE book.

This is an advanced graduate PDE class, focusing on Melrose's so-called b-(pseudo)differential operators, but no PDE background is required. (Thus, 256A is not a prerequisite.) However, a thorough knowledge of functional analysis and Fourier analysis (as presented in the Math 205 sequence) is a must. In the Riemannian world b-analysis includes manifolds with cylindrical ends as well as asymptotically Euclidean spaces, and in the Lorentzian world such diverse spaces as Minkowski space, a neighborhood of the static patch of de Sitter space, and Kerr-de Sitter space, as well as various spaces which asymptotically have a similar (but not necessarily the same!) structure. In addition, b-analysis helps analyze the standard boundary value problems in wave propagation; time permitting this will be discussed as well. Students with a strong background can take 205B concurrently with 256B; this requires permission of the instructor.Rough outline of the course (probably too optimistic):

- Lecture 1: general overview. Lecture notes are available.
- Week 1-3: background in microlocal analysis (pseudodifferential operators, microlocalization). Lecture notes and the shorter preliminary lecture notes are available.
- Week 4-6: Real principal type propagation, radial points. Lecture notes are available.
- Week 7: Applications: Asymptotically Euclidean scattering theory, Klein-Gordon type equations on asymptotically Minkowski type spaces, analytic continuation of the resolvent on even asymptotically hyperbolic spaces. Lecture notes for the last part are available; the previous parts are in the scattering setting, and are essentially covered in the previous set of notes.
- Week 8: Asymptotically de Sitter and Kerr-de Sitter spaces
- Week 9: b-pseudodifferential operators (small calculus) and the Mellin transform. Lecture notes
- Week 10: Asymptotically Minkowski spaces

Grading policy: There may be a couple of (short) problem sets which must be handed in, but due to unavailability of graders, will not be graded carefully.

The homework will be due either in class or by 9pm in the instructor's mailbox on the designated day. You are allowed to discuss the homework with others in the class, but you must write up your homework solution by yourself. Thus, you should understand the solution, and be able to reproduce it yourself.