Seminar on microlocal analysis

Content of the Seminar

A main tool for microlocal analysis is the algebra of pseudodifferential operators. This algebra includes differential operators. It also includes operators of negative degrees, e.g. the inverse of an invertible elliptic differential operator is again a pseudodifferential operator. It will be important to understand their symbols.

In the seminar we follow a new book on microlocal analysis by Peter Hintz. One objective of this book is to apply the powerful machinery of microlocal analysis for studying hyperbolic evolution equations, both for hyperbolic systems on ℝⁿ and for natural hyperbolic partial differential equations on Lorentzian manifolds.

It is worth learning microlocal analysis because it ...

• reveals both where singularities occur and the directions in which they oscillate,
• explains how singularities propagate through solutions of differential equations,
• provides a powerful framework for pseudodifferential operators and elliptic regularity,
• helps to understand the solutions of hyperbolic equation, and the singularities of its solutions are described by the wave front set,
• connects classical mechanics to quantum theory: the wave front set of the quantum solution moves with the geodesic flow,
• makes many difficult PDE arguments more systematic and geometrically transparent.

Literature

• Peter Hintz, An Introduction to Microlocal Analysis, Springer

Requirements

• Riemannian and Lorentzian geometry
• Distribution and Fourier transform
• Some facts about partial differential operators and the associated equations

Program

• The program (public version) of the seminar is available here.
• The intranet version of the program for the seminar is available here (use your NDS account!)

Time and Place

Registration

Preliminary meeting and distribution of the first talks: Monday, Sept 28, 10:15 in the "Sitzungszimmer" M201

Verbundene Webseiten

• Notes in the Commented List of Courses (KVV)

Formal requirements