This volume is a collection of notes from lectures given at the 2008 Clay Mathematics Institute Summer School, held in Zurich, Switzerland. The lectures were designed for graduate students and mathematicians within five years of the Ph.D., and the main focus of the programme was on recent progress in the theory of evolution equations. Such equations lie at the heart of many areas of mathematical physics and arise not only in situations with a manifest time evolution (such as linear and nonlinear wave and Schrodinger equations) but also in the high energy or semi-classical limits of elliptic problems.
The three main courses presented focused mainly on microlocal analysis and spectral and scattering theory, the theory of the nonlinear Schrodinger and wave equations, and evolution problems in general relativity. These major topics were supplemented by several mini-courses on the derivation of effective evolution equations from microscopic quantum dynamics, on wave maps with and without symmetries, on quantum N-body scattering, diffraction of waves, and symmetric spaces, and on nonlinear Schrodinger equations at critical regularity.
Although highly detailed treatments of some of these topics are now available in the published literature, in this collection the reader can learn the fundamental ideas and tools with a minimum of technical machinery. Moreover, the treatment in this volume emphasises common themes and techniques in the field, including exact and approximate conservation laws, energy methods, and positive commutator arguments.
Table of Contents:
Microlocal analysis and evolution equations: Lecture notes from the 2008
CMI/ETH Summer School by J. Wunsch
Some global aspects of linear wave equations by D. Baskin and R. Mazzeo
Lectures on black holes and linear waves by M. Dafermos and I. Rodnianski
The theory of nonlinear Schrodinger equations by G. Staffilani
On the singularity formation for the nonlinear Schrodinger equation by P.
Raphael
Nonlinear Schrodinger equations at critical regularity by R. Killip and M.
Visan
Geometry and analysis in many-body scattering by A. Vasy
Wave maps with and without symmetries by M. Struwe
Derivation of effective evolution equations from microscopic quantum dynamics
by B. Schlein
David Ellwood, Harvard University, Cambridge, MA, USA.
Igor Rodnianski, Massachusetts Institute of Technology, Cambridge, MA, USA.
Gigliola Staffilani, Massachusetts Institute of Technology, Cambridge, MA, USA.
Jared Wunsch, Northwestern University, Evanston, IL, USA.
