"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas.

The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest.

This is the first part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas.

The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest.

This is the second part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrodinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations.

Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art.""

This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers.

The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrodinger operators, nonlinear Schrodinger and wave equations, and the Euler and Navier-Stokes equations."