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The many diverse articles presented in these three volumes, collected on the occasion of Alexander Grothendieck's sixtieth birthday and originally published in 1990, were offered as a tribute to one of the world's greatest living mathematicians. Grothendieck changed the very way we think about many branches of mathematics. Many of his ideas, revolutionary when introduced, now seem so natural as to have been inevitable. Indeed, it is difficult to fully grasp the influence his vast contributions to modern mathematics have subsequently had on new generations of mathematicians.Many of the groundbreaking contributions in these volumes contain material that is now considered foundational to the subject. Topics addressed by these top-notch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, K-theory, category theory, and homological algebra.CONTRIBUTORS to Volume II: P. Cartier; C. Contou-Carrère; P. Deligne; T. Ekedahl; G. Faltings; J.-M. Fontaine; H. Hamm; Y. Ihara; L. Illusie; M. Kashiwara; V.A. Kolyvagin; R. Langlands; Lé D.T.; D. Shelstad; and A. Voros.

The many diverse articles presented in these three volumes, collected on the occasion of Alexander Grothendieck's sixtieth birthday and originally published in 1990, were offered as a tribute to one of the world's greatest living mathematicians. Grothendieck changed the very way we think about many branches of mathematics. Many of his ideas, revolutionary when introduced, now seem so natural as to have been inevitable. Indeed, it is difficult to fully grasp the influence his vast contributions to modern mathematics have subsequently had on new generations of mathematicians.Many of the groundbreaking contributions in these volumes contain material that is now considered foundational to the subject. Topics addressed by these top-notch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, K-theory, category theory, and homological algebra.CONTRIBUTORS to Volume I: A. Altman; M. Artin; V. Balaji; A. Beauville; A.A. Beilinson; P. Berthelot; J.-M. Bismut; S. Bloch; L. Breen; J.-L. Brylinski; J. Dieudonné; H. Gillet; A.B. Goncharov; K. Kato; S. Kleiman; W. Messing; V.V. Schechtman; C.S. Seshadri; C. Soulé; J. Tate; M. van den Bergh; and A.N. Varchenko.

The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed, leading up to Trépreau's theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category.At the beginning of the book, no prerequisites are assumed beyond calculus and linear algebra. Later on, basic facts from distribution theory and functional analysis are needed. In a few places, a more extensive background in differential geometry or pseudodifferential calculus is required, but these sections can be bypassed with no loss of continuity. The major part of the book should therefore be accessible to graduate students so that it can serve as an introduction to complex analysis in one and several variables. The last sections, however, are written mainly for readers familiar with microlocal analysis.

Developed from the authors' introductory combinatorics course, this book focuses on a branch of mathematics which plays a crucial role in computer science. Combinatorial methods provide many analytical tools used for determining the expected performance of computer algorithms. Elementary subjects such as combinations and permutations, and mathematical tools such as generating functions and Pólya's Theory of Counting, are covered, as are analyses of specific problems such as Ramsey Theory, matchings, and Hamiltonian and Eulerian paths. This introduction will provide students with a solid foundation in the subject. ---- "This is a delightful little paperback which presents a day-by-day transcription of a course taught jointly by Pólya and Tarjan at Stanford University. Woods, the teaching assistant for the class, did a very good job of merging class notes into an interesting mini-textbook; he also included the exercises, homework, and tests assigned in the class (a very helpful addition for other instructors in the field). The notes are very well illustrated throughout and Woods and the Birkhäuser publishers produced a very pleasant text. One can count on [Pólya and Tarjan] for new insights and a fresh outlook. Both instructors taught by presenting a succession of examples rather than by presenting a body of theory...[The book] is very well suited as supplementary material for any introductory class on combinatorics; as such, it is very highly recommended. Finally, for all of us who like the topic and delight in observing skilled professionals at work, this book is entertaining and, yes, instructive, reading."-Mathematical Reviews (Review of the original hardcover edition) "The mathematical community welcomes this book as a final contribution to honour the teacher G. Pólya."-ZentralblattMATH (Review of the original hardcover edition)

Optimal Control brings together many of the important advances in 'nonsmooth' optimal control over the last several decades concerning necessary conditions, minimizer regularity, and global optimality conditions associated with the Hamilton–Jacobi equation. The book is largely self-contained and incorporates numerous simplifications and unifying features for the subject’s key concepts and foundations.Features and Topics:* a comprehensive overview is provided for specialists and nonspecialists* authoritative, coherent, and accessible coverage of the role of nonsmooth analysis in investigating minimizing curves for optimal control* chapter coverage of dynamic programming and the regularity of minimizers* explains the necessary conditions for nonconvex problemsThis book is an excellent presentation of the foundations and applications of nonsmooth optimal control for postgraduates, researchers, and professionals in systems, control, optimization, and applied mathematics.-----Each chapter contains a well-written introduction and notes. They include the author's deep insights on the subject matter and provide historical comments and guidance to related literature. This book may well become an important milestone in the literature of optimal control.—Mathematical ReviewsThis remarkable book presents Optimal Control seen as a natural development of Calculus of Variations so as to deal with the control of engineering devices. ... Thanks to a great effort to be self-contained, it renders accessibly the subject to a wide audience. Therefore, it is recommended to all researchers and professionals interested in Optimal Control and its engineering and economic applications. It can serve as an excellent textbook for graduate courses in Optimal Control (with special emphasis on Nonsmooth Analysis). —AutomaticaThe book may be an essential resource for potential readers, experts in control and optimization, as well as postgraduates and applied mathematicians, and it will be valued for its accessibility and clear exposition.—Applications of Mathematics

A highly readable introduction to stochastic integration and stochastic differential equations, this book combines developments of the basic theory with applications. It is written in a style suitable for the text of a graduate course in stochastic calculus, following a course in probability.Using the modern approach, the stochastic integral is defined for predictable integrands and local martingales; then It¿s change of variable formula is developed for continuous martingales. Applications include a characterization of Brownian motion, Hermite polynomials of martingales, the Feynman¿Kac functional and the Schrödinger equation. For Brownian motion, the topics of local time, reflected Brownian motion, and time change are discussed.New to the second edition are a discussion of the Cameron¿Martin¿Girsanov transformation and a final chapter which provides an introduction to stochastic differential equations, as well as many exercises for classroom use.This book willbe a valuable resource to all mathematicians, statisticians, economists, and engineers employing the modern tools of stochastic analysis.The text also proves that stochastic integration has made an important impact on mathematical progress over the last decades and that stochastic calculus has become one of the most powerful tools in modern probability theory. ¿Journal of the American Statistical Association An attractive text¿written in [a] lean and precise style¿eminently readable. Especially pleasant are the care and attention devoted to details¿ A very fine book.¿Mathematical Reviews

Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. It is no coincidence that the fundamental science of numbers has come to be known as the "e;Queen of Mathematics."e; Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory.Andr Weil, one of the outstanding contributors to number theory, has written an historical exposition of this subject; his study examines texts that span roughly thirty-six centuries of arithmetical work - from an Old Babylonian tablet, datable to the time of Hammurapi to Legendre's Essai sur la Thorie des Nombres (1798). Motivated by a desire to present the substance of his field to the educated reader, Weil employs an historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. In the course of his study Weil accompanies the reader into the workshops of four major authors of modern number theory (Fermat, Euler, Lagrange and Legendre) and there he conducts a detailed and critical examination of their work. Enriched by a broad coverage of intellectual history, Number Theory represents a major contribution to the understanding of our cultural heritage.

This volume contains the first two out of four chapters which are intended to survey a large part of the theory of theta functions. These notes grew out of a series of lectures given at the Tata Institute of Fundamental Research in the period October, 1978, to March, 1979, on which notes were taken and excellently written up by C. Musili and M. Nori. I subsequently lectured at greater length on the contents of Chapter III at Harvard in the fall of 1979 and at a Summer School in Montreal in August, 1980, and again notes were very capably put together by E. Previato and M. Stillman, respectively. Both the Tata Institute and the University of Montreal publish lecture note series in which I had promised to place write-ups of my lectures there. However, as the project grew, it became clear that it was better to tie all these results together, rearranging and consolidating the material, and to make them available from one place. I am very grateful to the Tata Institute and the University of Montreal for permission to do this, and to Birkhauser-Boston for publishing the final result. The first 2 chapters study theta functions strictly from the viewpoint of classical analysis. In particular, in Chapter I, my goal was to explain in the simplest cases why the theta functions attracted attention.

This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kahler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac's quantization of the electrical charge.

The purpose of the present book is to offer an up-to-date account of the theory of viscosity solutions of first order partial differential equations of Hamilton-Jacobi type and its applications to optimal deterministic control and differential games. The theory of viscosity solutions, initiated in the early 80's by the papers of M.G. Crandall and P.L. Lions [CL81, CL83], M.G. Crandall, L.C. Evans and P.L. Lions [CEL84] and P.L. Lions' influential monograph [L82], provides an - tremely convenient PDE framework for dealing with the lack of smoothness of the value functions arising in dynamic optimization problems. The leading theme of this book is a description of the implementation of the viscosity solutions approach to a number of significant model problems in op- real deterministic control and differential games. We have tried to emphasize the advantages offered by this approach in establishing the well-posedness of the c- responding Hamilton-Jacobi equations and to point out its role (when combined with various techniques from optimal control theory and nonsmooth analysis) in the important issue of feedback synthesis.

All modem introductions to complex analysis follow, more or less explicitly, the pattern laid down in Whittaker and Watson [75]. In "e;part I'' we find the foundational material, the basic definitions and theorems. In "e;part II"e; we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part II in fact predates part I, so clearly it can be taught first. Why should the student have to wade through hundreds of pages before finding out what the subject is good for? In teaching complex analysis this way, we risk more than just boredom. Beginning with a series of unmotivated definitions gives a misleading impression of complex analy- sis in particular and of mathematics in general. The classical theory of analytic functions did not arise from the idle speculation of bored mathematicians on the possible conse- quences of an arbitrary set of definitions; it was the natural, even inevitable, consequence of the practical need to answer questions about specific examples. In standard texts, after hundreds of pages of theorems about generic analytic functions with only the rational and trigonometric functions as examples, students inevitably begin to believe that the purpose of complex analysis is to produce more such theorems. We require introductory com- plex analysis courses of our undergraduates and graduates because it is useful both within mathematics and beyond.

The soliton, a solitary wave impulse preserving its shape and strikingly similar to a particle, is one of the most fascinating and beautiful phenomena in the physics of nonlinear waves.  In this classic book, the concept of the soliton is traced from the beginning of the last century to modern times, with recent applications in biology, oceanography, solid state physics, electronics, elementary particle physics, and cosmology.The Versatile Soliton is an appropriate title indeed. There is much new historical information in the book...The book is written in a lively language and the physics presented in a clear, pedagogical style. Most of the chapters require only knowledge of fairly elementary mathematics and the main ideas of soliton physics are well explained without mathematics at all...Yet it contains valuable information and offers a historical review of soliton physics that cannot be found elsewhere.  -CentaurusIn summary, this book is a good elementary treatment of solitons and the related history of physics and mathematics, even for readers with little knowledge of advanced mathematics. For readers with the latter knowledge, it is still a good introduction to the physical ideas required for the understanding of solitons prior to the study of more mathematical treatments from other sources.  -Mathematical ReviewsThis engaging book is an excellent introduction into the wonderful world of soliton mechanics.   -Zentralblatt MathNo doubt, everyone can get new information from the book. First, the book is strongly recommended to young researchers. In a certain sense, the book is unique and definitely will find a niche among numerous textbooks on solitons.   -Physicala

More than twenty years ago I gave a course on Fourier Integral Op­ erators at the Catholic University of Nijmegen (1970-71) from which a set of lecture notes were written up; the Courant Institute of Mathematical Sciences in New York distributed these notes for many years, but they be­ came increasingly difficult to obtain. The current text is essentially a nicely TeXed version of those notes with some minor additions (e.g., figures) and corrections. Apparently an attractive aspect of our approach to Fourier Integral Operators was its introduction to symplectic differential geometry, the basic facts of which are needed for making the step from the local definitions to the global calculus. A first example of the latter is the definition of the wave front set of a distribution in terms of testing with oscillatory functions. This is obviously coordinate-invariant and automatically realizes the wave front set as a subset of the cotangent bundle, the symplectic manifold in which the global calculus takes place.