The research topic for this IAS/PCMI Summer Session was nonlinear wave phenomena. Mathematicians from the more theoretical areas of PDEs were brought together with those involved in applications. The goal was to share ideas, knowledge, and perspectives. How waves, or "frequencies", interact in nonlinear phenomena has been a central issue in many of the recent developments in pure and applied analysis. It is believed that wavelet theorywith its simultaneous localization in both physical and frequency space and its lacunarityis and will be a fundamental new tool in the treatment of the phenomena. Included in this volume are writeups of the "general methods and tools" courses held by Jeff Rauch and Ingrid Daubechies. Rauch's article discusses geometric optics as an asymptotic limit of highfrequency phenomena. He shows how nonlinear effects are reflected in the asymptotic theory. In the article "Harmonic Analysis, Wavelets and Applications" by Daubechies and Gilbert the main structure of the wavelet theory is presented. Also included are articles on the more "specialized" courses that were presented, such as "Nonlinear Schrödinger Equations" by Jean Bourgain and "Waves and Transport" by George Papanicolaou and Leonid Ryzhik. Susan Friedlander provides a written version of her lecture series "Stability and Instability of an Ideal Fluid", given at the Mentoring Program for Women in Mathematics, a preliminary program to the Summer Session. This Summer Session brought together students, fellows, and established mathematicians from all over the globe to share ideas in a vibrant and exciting atmosphere. This book presents the compelling results. Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price. Readership Graduate students and research mathematicians working in partial differential equations. Table of Contents Nonlinear Schrödinger equations  J. Bourgain  Introduction
 J. Bourgain  Generalities and initial value problems
 J. Bourgain  The initial value problem (continued)
 J. Bourgain  A digressioin: The initial value problem for the KdV equation
 J. Bourgain  1D invariant Gibbs measures
 J. Bourgain  Invariant measures (2D)
 J. Bourgain  Quasiperiodic solutions of Hamiltonian PDE
 J. Bourgain  Time periodic solutions
 J. Bourgain  Time quasiperiodic solutions
 J. Bourgain  Normal forms
 J. Bourgain  Applications of symplectic capacities to Hamiltonian PDE
 J. Bourgain  Remarks on longtime behaviour of the flow of Hamiltonian PDE
Harmonic analysis, wavelets and applications  I. C. Daubechies and A. C. Gilbert  Introduction
 I. C. Daubechies and A. C. Gilbert  Constructing orthonormal wavelet bases: Multiresolution analysis
 I. C. Daubechies and A. C. Gilbert  Wavelet bases: Construction and algorithms
 I. C. Daubechies and A. C. Gilbert  More wavelet bases
 I. C. Daubechies and A. C. Gilbert  Wavelets in other functional spaces
 I. C. Daubechies and A. C. Gilbert  Pointwise convergence for wavelet expansions
 I. C. Daubechies and A. C. Gilbert  Twodimensional wavelets and operators
 I. C. Daubechies and A. C. Gilbert  Wavelets and differential equations
 I. C. Daubechies and A. C. Gilbert  References
Lectures on stability and instability of an ideal fluid  S. Friedlander  Introduction
 S. Friedlander  Equations of motion
 S. Friedlander  Initialboundary value problem
 S. Friedlander  The type of the Euler equations
 S. Friedlander  Vorticity
 S. Friedlander  Steady flows
 S. Friedlander  Stability/instability of an equilibrium state
 S. Friedlander  Twodimensional spectral problem
 S. Friedlander  "Arnold" criterion for nonlinear stability
 S. Friedlander  Plane parallel shear flow
 S. Friedlander  Instability in a vorticity norm
 S. Friedlander  Sufficient condition for instability
 S. Friedlander  Exponential stretching
 S. Friedlander  Integrable flows
 S. Friedlander  Baroclinic instability
 S. Friedlander  Nonlinear instability
 S. Friedlander  References
Waves and transport  G. Papanicolaou and L. Ryzhik  Introduction
 G. Papanicolaou and L. Ryzhik  The Schrödinger equation
 G. Papanicolaou and L. Ryzhik  Symmetric hyperbolic systems
 G. Papanicolaou and L. Ryzhik  Waves in random media
 G. Papanicolaou and L. Ryzhik  The diffusion approximation
 G. Papanicolaou and L. Ryzhik  The geophysical applications
 G. Papanicolaou and L. Ryzhik  References
Lectures on geometric optics  J. Rauch and M. Keel  Introduction
 J. Rauch and M. Keel  Basic linear existence theorems
 J. Rauch and M. Keel  Examples of propagation of singularities and of energy
 J. Rauch and M. Keel  Elliptic geometric optics
 J. Rauch and M. Keel  Linear hyperbolic geometric optics
 J. Rauch and M. Keel  Basic nonlinear existence theorems
 J. Rauch and M. Keel  One phase nonlinear geometric optics
 J. Rauch and M. Keel  Justification of one phase nonlinear geometric optics
 J. Rauch and M. Keel  References
