The representation theory of the symmetric group, of Chevalley groups particularly in positive characteristic and of Lie algebraic systems, has undergone some remarkable developments in recent years. Many techniques are inspired by the great works of Issai Schur who passed away some 60 years ago. This volume is dedicated to his memory. This is a unified presentation consisting of an extended biography of Schur---written in collaboration with some of his former students---as well as survey articles on Schur's legacy (Schur theory, functions, etc). Additionally, there are articles covering the areas of orbits, crystals and representation theory, with special emphasis on canonical bases and their crystal limits, and on the geometric approach linking orbits to representations and Hecke algebra techniques. Extensions of representation theory to mathematical physics and geometry will also be presented. Contributors: Biography: W. Ledermann, B. Neumann, P.M. Neumann, H. Abelin- Schur; Review of work: H. Dym, V. Katznelson; Original papers: H. H. Andersen, A. Braverman, S. Donkin, V. Ivanov, D. Kazhdan, B. Kostant, A. Lascoux, N. Lauritzen, B. Leclerc, P. Littelmann, G. Luzstig, O.Mathieu, M. Nazarov, M. Reinek, J.-Y. Thibon, G. Olshanski, E. Opdam, A. Regev, C.S. Seshadri, M. Varagnolo, E. Vasserot, A. Vershik This volume will serve as a comprehensive reference as well as a good text for graduate seminars in representation theory, algebra, and mathematical physics.

The first European Congress of Mathematics was held in Paris from July 6 to July 10, 1992, at the Sorbonne and Pantheon-Sorbonne universities. It was hoped that the Congress would constitute a symbol of the development of the community of European nations. More than 1,300 persons attended the Congress. The purpose of the Congress was twofold. On the one hand, there was a scientific facet which consisted of forty-nine invited mathematical lectures that were intended to establish the state of the art in the various branches of pure and applied mathematics. This scientific facet also included poster sessions where participants had the opportunity of presenting their work. Furthermore, twenty-four specialized meetings were held before and after the Congress. The second facet of the Congress was more original. It consisted of six teen round tables whose aim was to review the prospects for the interactions of mathematics, not only with other sciences, but also with society and in particular with education, European policy and industry. In connection with this second goal, the Congress also succeeded in bring ing mathematics to a broader public. In addition to the round tables specific ally devoted to this question, there was a mini-festival of mathematical films and two mathematical exhibits. Moreover, a Junior Mathematical Congress was organized, in parallel with the Congress, which brought together two hundred high school students."

Table of contents: Plenary Lectures * V.I. Arnold: The Vassiliev Theory of Discriminants and Knots * L. Babai: Transparent Proofs and Limits to Approximation * C. De Concini: Poisson Algebraic Groups and Representations of Quantum Groups at Roots of 1 * S.K. Donaldson: Gauge Theory and Four-Manifold Topology * W. Muller: Spectral Theory and Geometry * D. Mumford: Pattern Theory: A Unifying Perspective * A.-S. Sznitman: Brownian Motion and Obstacles * M. Vergne: Geometric Quantization and Equivariant Cohomology * Parallel Lectures * Z. Adamowicz: The Power of Exponentiation in Arithmetic * A. Bjorner: Subspace Arrangements * B. Bojanov: Optimal Recovery of Functions and Integrals * J.-M. Bony: Existence globale et diffusion pour les modeles discrets * R.E. Borcherds: Sporadic Groups and String Theory * J. Bourgain: A Harmonic Analysis Approach to Problems in Nonlinear Partial Differatial Equations * F. Catanese: (Some) Old and New Results on Algebraic Surfaces * Ch. Deninger: Evidence for a Cohomological Approach to Analytic Number Theory * S. Dostoglou and D.A. Salamon: Cauchy-Riemann Operators, Self-Duality, and the Spectral Flow

Table of Contents: D. Duffie: Martingales, Arbitrage, and Portfolio Choice * J. Frohlich: Mathematical Aspects of the Quantum Hall Effect * M. Giaquinta: Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings * U. Hamenstadt: Harmonic Measures for Leafwise Elliptic Operators Along Foliations * M. Kontsevich: Feynman Diagrams and Low-Dimensional Topology * S.B. Kuksin: KAM-Theory for Partial Differential Equations * M. Laczkovich: Paradoxical Decompositions: A Survey of Recent Results * J.-F. Le Gall: A Path-Valued Markov Process and its Connections with Partial Differential Equations * I. Madsen: The Cyclotomic Trace in Algebraic K-Theory * A.S. Merkurjev: Algebraic K-Theory and Galois Cohomology * J. Nekovar: Values of L-Functions and p-Adic Cohomology * Y.A. Neretin: Mantles, Trains and Representations of Infinite Dimensional Groups * M.A. Nowak: The Evolutionary Dynamics of HIV Infections * R. Piene: On the Enumeration of Algebraic Curves - from Circles to Instantons * A. Quarteroni: Mathematical Aspects of Domain Decomposition Methods * A. Schrijver: Paths in Graphs and Curves on Surfaces * B. Silverman: Function Estimation and Functional Data Analysis * V. Strassen: Algebra and Complexity * P. Tukia: Generalizations of Fuchsian and Kleinian Groups * C. Viterbo: Properties of Embedded Lagrange Manifolds * D. Voiculescu: Alternative Entropies in Operator Algebras * M. Wodzicki : Algebraic K-Theory and Functional Analysis * D. Zagier: Values of Zeta Functions and Their Applications

This volume is dedicated to the memory of Issai Schur. It opens with some biographical reminiscences of the famous school he established in Berlin, his brutal dismissal by the Nazi regime and his tragic end in Palestine. This is followed by an extensive review of the extraordinary impact of his lesser known analytic work. Finally, leading mathematicians in the representation theory of the symmetric groups, of semisimple and affine Lie algebras and of Chevalley groups have contributed original and outstanding articles. These concern many areas inspired by Schur's work as well as more recent developments involving crystal and canonical bases, Hecke algebras, and the geometric approach linking orbits to representations. Contributors: Biography: H. Abelin-Schur, W. Ledermann, Y. Ne'eman, B. Neumann, P.M. Neumann, M. Sonis. Review: H. Dym, V. Katsnelson. Original papers: H.H. Andersen, A. Braverman, S. Donkin, V. Ivanov, D. Kazhdan, B. Kostant, A. Lascoux, N. Lauritzen, B. Leclerc, P. Littelmann, G. Luzstig, O. Mathieu, M. Nazarov, G. Olshanski, E. Opdam, A. Regev, M. Reineke, C.S. Seshadri, J.-Y. Thibon, M. Varagnolo, E. Vasserot, A. Vershik. The first part of this book will appeal to a general audience. The second part will be of interest to graduate students especially those in analysis, while the third part is addressed to specialists in Lie algebras.

The first European Congress of Mathematics was held in Paris from July 6 to July 10, 1992, at the Sorbonne and Pantheon-Sorbonne universities. It was hoped that the Congress would constitute a symbol of the development of the community of European nations. More than 1,300 persons attended the Congress. The purpose of the Congress was twofold. On the one hand, there was a scientific facet which consisted of forty-nine invited mathematical lectures that were intended to establish the state of the art in the various branches of pure and applied mathematics. This scientific facet also included poster sessions where participants had the opportunity of presenting their work. Furthermore, twenty four specialized meetings were held before and after the Congress. The second facet of the Congress was more original. It consisted of sixteen round tables whose aim was to review the prospects for the interactions of mathe matics, not only with other sciences, but also with society and in particular with education, European policy and industry. In connection with this second goal, the Congress also succeeded in bringing mathematics to a broader public. In addition to the round tables specifically devoted to this question, there was a mini-festival of mathematical films and two mathematical exhibits. Moreover, a Junior Mathematical Congress was organized, in parallel with the Congress, which brought together two hundred high school students."

The first European Congress of Mathematics was held in Paris from July 6 to July 10, 1992, at the Sorbonne and Pantheon-Sorbonne universities. It was hoped that the Congress would constitute a symbol of the development of the community of European nations. More than 1,300 persons attended the Congress. The purpose of the Congress was twofold. On the one hand, there was a scientific facet which consisted of forty-nine invited mathematical lectures that were intended to establish the state of the art in the various branches of pure and applied mathematics. This scientific facet also included poster sessions where participants had the opportunity of presenting their work. Furthermore, twenty four specialized meetings were held before and after the Congress. The second facet of the Congress was more original. It consisted of sixteen round tables whose aim was to review the prospects for the interactions of mathe matics, not only with other sciences, but also with society and in particular with education, European policy and industry. In connection with this second goal, the Congress also succeeded in bringing mathematics to a broader public. In addition to the round tables specifically devoted to this question, there was a mini-festival of mathematical films and two mathematical exhibits. Moreover, a Junior Mathematical Congress was organized, in parallel with the Congress, which brought together two hundred high school students."

Table of Contents: D. Duffie: Martingales, Arbitrage, and Portfolio Choice * J. Frohlich: Mathematical Aspects of the Quantum Hall Effect * M. Giaquinta: Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings * U. Hamenstadt: Harmonic Measures for Leafwise Elliptic Operators Along Foliations * M. Kontsevich: Feynman Diagrams and Low-Dimensional Topology * S.B. Kuksin: KAM-Theory for Partial Differential Equations * M. Laczkovich: Paradoxical Decompositions: A Survey of Recent Results * J.-F. Le Gall: A Path-Valued Markov Process and its Connections with Partial Differential Equations * I. Madsen: The Cyclotomic Trace in Algebraic K-Theory * A.S. Merkurjev: Algebraic K-Theory and Galois Cohomology * J. Nekovar: Values of L-Functions and p-Adic Cohomology * Y.A. Neretin: Mantles, Trains and Representations of Infinite Dimensional Groups * M.A. Nowak: The Evolutionary Dynamics of HIV Infections * R. Piene: On the Enumeration of Algebraic Curves - from Circles to Instantons * A. Quarteroni: Mathematical Aspects of Domain Decomposition Methods * A. Schrijver: Paths in Graphs and Curves on Surfaces * B. Silverman: Function Estimation and Functional Data Analysis * V. Strassen: Algebra and Complexity * P. Tukia: Generalizations of Fuchsian and Kleinian Groups * C. Viterbo: Properties of Embedded Lagrange Manifolds * D. Voiculescu: Alternative Entropies in Operator Algebras * M. Wodzicki : Algebraic K-Theory and Functional Analysis * D. Zagier: Values of Zeta Functions and Their Applications

Table of contents: Plenary Lectures * V.I. Arnold: The Vassiliev Theory of Discriminants and Knots * L. Babai: Transparent Proofs and Limits to Approximation * C. De Concini: Poisson Algebraic Groups and Representations of Quantum Groups at Roots of 1 * S.K. Donaldson: Gauge Theory and Four-Manifold Topology * W. Muller: Spectral Theory and Geometry * D. Mumford: Pattern Theory: A Unifying Perspective * A.-S. Sznitman: Brownian Motion and Obstacles * M. Vergne: Geometric Quantization and Equivariant Cohomology * Parallel Lectures * Z. Adamowicz: The Power of Exponentiation in Arithmetic * A. Bjorner: Subspace Arrangements * B. Bojanov: Optimal Recovery of Functions and Integrals * J.-M. Bony: Existence globale et diffusion pour les modeles discrets * R.E. Borcherds: Sporadic Groups and String Theory * J. Bourgain: A Harmonic Analysis Approach to Problems in Nonlinear Partial Differatial Equations * F. Catanese: (Some) Old and New Results on Algebraic Surfaces * Ch. Deninger: Evidence for a Cohomological Approach to Analytic Number Theory * S. Dostoglou and D.A. Salamon: Cauchy-Riemann Operators, Self-Duality, and the Spectral Flow