Gauß—Manin Determinant Connections and Periods for Irregular Connections -- Problems in Hamiltonian PDE’S -- On a Classical Limit of Quantum Theory and the Non-Linear Hartree Equation -- Rough Structure and Classification -- Spaces and Questions -- Classification of Infinite-Dimensional Simple Groups of Supersymmetries and Quantum Field Theory -- Geometrization in Representation Theory -- Geometric and Unipotent Crystals -- ?-Functions of Representations and Lifting -- PDE as a Unified Subject -- Lessons for Turbulence -- The Mathematics of the Second Law of Thermodynamics -- Discrete and Continuous: Two Sides of the Same? -- Pythagorean and Platonic Conceptions in XXth Century Physics -- 1. Classical and Modern Topology 2. Topological Phenomena in Real World Physics -- Some Problems in the Theory of Dynamical Systems and Mathematical Physics -- Some Geometrical Concepts Arising in Harmonic Analysis.

"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.