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Mathematics: Frontiers and Perspectives
Edited by: V. Arnold, University of Paris IX, France, and Steklov Mathematical Institute, Moscow, Russia, M. Atiyah, University of Edinburgh, Scotland, P. Lax, New York University-Courant Institute of Mathematical Sciences, NY, and B. Mazur, Harvard University, Cambridge, MA
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2000; 459 pp; softcover
ISBN-10: 0-8218-2697-2
ISBN-13: 978-0-8218-2697-3
List Price: US$45 Member Price: US$36
Order Code: MFP.S

This remarkable book is a celebration of the state of mathematics at the end of the millennium. Produced under the auspices of the International Mathematical Union (IMU), the volume was born as part of the activities observing the World Mathematical Year 2000.

The volume consists of 30 articles written by some of the most influential mathematicians of our time. Authors of 15 contributions were recognized in various years by the IMU as recipients of the Fields Medal, from K. F. Roth (Fields Medalist, 1958) to W. T. Gowers (Fields Medalist, 1998). The articles offer valuable reflections about the amazing mathematical progress we have witnessed in this century and insightful speculations about the possible development of mathematics over the next century.

Some articles formulate important problems, challenging future mathematicians. Others pay explicit homage to the famous set of Hilbert Problems posed one hundred years ago, giving enlightening commentary. Yet other papers offer a deeply personal perspective, allowing singular insight into the minds and hearts of people doing mathematics today.

Mathematics: Frontiers and Perspectives is a unique volume that pertains to a broad mathematical audience of various backgrounds and levels of interest. It offers readers true and unequaled insight into the wonderful world of mathematics at this important juncture: the turn of the millennium.

The work is one of those rare volumes that can be browsed, and if you do simply browse through it, you get a wonderful sense of mathematics today. Yet it also can be intensely studied on a detailed technical level for gaining insight into some of the great problems on which mathematicians are currently working.

Editors Michael Atiyah and Peter Lax were winners of the famous Abel Prize awarded by The Norwegian Academy of Science and Letters for outstanding work in mathematics.

Individual members of mathematical societies of the IMU member countries can purchase this volume at the AMS member price when buying directly from the AMS.

Graduate students and research mathematicians; general mathematical audience, including historians.

Reviews

"This book should be in the library of every working mathematician."

"Many papers are ... broad in that they address the history of their subject and also make predictions about future developments. Many readers will be drawn to the general interest material ... Readers in search of controversy will find plenty ... especially intrigued by Arnold's elaborate schema in which the triple tetrahedron-octahedron-icosahedron corresponds to the triple reals-complexes-quaternions ... an excellent book."

-- MAA Online

"This collection demonstrates well that mathematics is alive and vital."

-- American Scientist

"You can read this book as if listening to a succession of high-powered old school friends who are passing through ... most tell you about the mathematics that animates them ... you get a good sense of what they do, what's difficult about it, and why it matters ... provocative remarks ... vigorous account of much of the Russian school of mathematics ... thought-provoking reflections about mathematical life and language ... The most pleasing feature of this handsome book is the emphasis on the unity of mathematics ... The connections many people here want to make between mathematics and physics, von Neumann algebras and knot theory, number theory and analysis, are not only fresh and vivid, but oddly coherent. They give a sense not only of mathematics undergoing one of its characteristic contractions around a few organising principles, but how productive this reorganisation can be."

"One hundred years ago, David Hilbert's famous list of 26 mathematical problems began to catalyze the collective efforts of the world's mathematicians toward a century's worth of new research and achievement. The International Mathematical Union commissioned the current volume to do the same for the century just beginning. Fully half the contributors here own a Fields Medal, mathematics' highest honor (and that does not even count Andrew Wiles). Obviously, simply by dint of the prestige and caliber of the authors, this volume deserves reader attention and a place on every library's shelves. The essays themselves vary from the entirely technical to the purely personal. The sort of reading encounter they offer can set the direction of a whole career, so the undergraduate who picks up this volume now may expect to return here many times in the years to come."

-- CHOICE

"The mere names of the editors ensure that [the book] is fascinating and exciting ... Several of the 29 authors touch on the nature and/or future of mathematics ... the most interesting essays are those whose authors get out on a limb and dogmatically announce, as saving truth, propositions radically different from common opinion ... Among the authors there are many famous pure mathematicians whose contributions constitute a smorgasbord of delicacies sufficient to satisfy every taste. Do sample them!"

-- CMS Notes

"This collection of essays will reward and stimulate anyone who dips into it.It belongs on every mathematician's bookshelf and in the mathematics collection of every library."

-- MAA Monthly

• A. Baker and G. Wüstholz -- Number theory, transcendence and Diophantine geometry in the next millennium
• J. Bourgain -- Harmonic analysis and combinatorics: How much may they contribute to each other?
• S.-S. Chern -- Back to Riemann
• A. Connes -- Noncommutative geometry and the Riemann zeta function
• S. K. Donaldson -- Polynomials, vanishing cycles and Floer homology
• W. T. Gowers -- The two cultures of mathematics
• V. F. R. Jones -- Ten problems
• D. Kazhdan -- An algebraic integration
• F. Kirwan -- Mathematics: The right choice?
• P.-L. Lions -- On some challenging problems in nonlinear partial differential equations
• A. J. Majda -- Real world turbulence and modern applied mathematics
• Yu. I. Manin -- Mathematics as profession and vocation
• G. Margulis -- Problems and conjectures in rigidity theory
• D. McDuff -- A glimpse into symplectic geometry
• S. Mori -- Rational curves on algebraic varieties
• D. Mumford -- The dawning of the age of stochasticity
• R. Penrose -- Mathematical physics of the 20$$^{\text{th}}$$ and 21$$^{\text{st}}$$ centuries
• K. F. Roth -- Limitations to regularity
• D. Ruelle -- Conversations on mathematics with a visitor from outer space
• P. Sarnak -- Some problems in number theory, analysis and mathematical physics
• S. Smale -- Mathematical problems for the next century
• R. P. Stanley -- Positivity problems and conjectures in algebraic combinatorics
• C. Vafa -- On the future of mathematics/physics interaction
• A. Wiles -- Twenty years of number theory
• E. Witten -- Magic, mystery, and matrix
• S.-T. Yau -- Review of geometry and analysis
• V. I. Arnold -- Polymathematics: Is mathematics a single science or a set of arts?
• P. D. Lax -- Mathematics and computing
• B. Mazur -- The theme of $$p$$-adic variation