Math 394 Homepage, Winter 2011-2012

Classics in Analysis

Instructor: András Vasy

Office: 383M

Phone: 723-2226

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

Tentative office hours: MT 11am-12pm, Th1-2pm.

Class location: MWF 2:15-3:30pm, Room 381U. The class will meet twice per week on average; three days are listed to help with scheduling conflicts.

Actual meeting days:

Week of Jan 9: Mon, Wed: Overview, Chernoff's paper
Week of Jan 16: Wed, Fri: Chernoff's paper, Hörmander's paper (Chris)
Week of Jan 23: Mon, Wed: Hörmander's paper (Chris)
Week of Jan 30: Fri only: Mourre's paper (David)
Week of Feb 6: Mon, Wed, Fri: Mourre's paper (David), Sylvester-Uhlmann paper (Otis)
Week of Feb 13: Mon, Wed: Sylvester-Uhlmann paper (Otis)
Week of Feb 20: Wed, Fri: Dencker's paper (Vitaly)
Week of Feb 27: Wed, Fri:

Textbook: Original papers, backed up, as needed, by texts.

Microlocal analysis, at the level of pseudodifferential operators on manifolds without boundary, will be useful, but is not necessary for many of the papers we should be going through (and the students should have sufficient flexibility to present on one of these non-microlocal papers). For a more thorough background on microlocal analysis, 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 topics that PDE courses often do not cover. Some possible topics:

Essential self-adjointness of the Laplacian, as in Chernoff's `Essential self-adjointness of powers of generators of hyperbolic equations'
Non-solvability of PDEs, as in Hörmander's `Differential equations without solutions'
Unique continuation theorems, as in Aronszajn's `A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order'
A study of the continuous spectrum of operators, as in Mourre's `Absence of singular continuous spectrum for certain self-adjoint operators'
Inverse problems, as in Sylvester and Uhlmann's `A global uniqueness theorem for an inverse boundary value problem'
Propagation for hyperbolic systems, as in Dencker's `On the propagation of polarization sets for systems of real principal type'
Boundary behavior of hyperbolic systems as in Taylor's `Reflection of singularities of solutions to systems of differential equations'
Index theory, as in Getzler's `A short proof of the local Atiyah-Singer index theorem'

Since some background is required for some of these, it is not yet clear how much ground we can cover, but this would be a reasonable indication of the planned scope.

Grading policy: The participants will be expected to present parts of the papers we will study.