# Mathematics: Frontiers and Perspectives

Share this page *Edited by *
*V. Arnold; M. Atiyah; P. Lax; B. Mazur*

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Society.

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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.

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.

#### Readership

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

#### Reviews & Endorsements

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

-- European Mathematical Society Newsletter

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.

-- The LMS Newsletter

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

#### Table of Contents

# Table of Contents

## Mathematics: Frontiers and Perspectives

- Cover Cover11
- Title page iii4
- Copyright iv5
- Contents v6
- Preface vii8
- Number theory, transcendence and Diophantine geometry in the next millennium 114
- Harmonic analysis and combinatorics: How much may they contribute to each other? 1326
- Back to Riemann 3346
- Noncommutative geometry and the Riemann zeta function 3548
- Polynomials, vanishing cycles and Floer homology 5568
- The two cultures of mathematics 6578
- Ten problems 7992
- An algebraic integration 93106
- Mathematics: The right choice? 117130
- On some challenging problems in nonlinear partial differential equations 121134
- Real world turbulence and modern applied mathematics 137150
- Mathematics as profession and vocation 153166
- Problems and conjectures in rigidity theory 161174
- A glimpse into symplectic geometry 175188
- Rational curves on algebraic varieties 189202
- The dawning of the age of stochasticity 197210
- Mathematical physics of the 20th and 21st centuries 219232
- Limitations to regularity 235248
- Conversations on mathematics with a visitor from outer space 251264
- Some problems in number theory, analysis and mathematical physics 261274
- Mathematical problems for the next century 271284
- Positivity problems and conjectures in algebraic combinatorics 295308
- On the future of mathematics/physics interaction 321334
- Twenty years of number theory 329342
- Magic, mystery, and matrix 343356
- Review of geometry and analysis 353366
- Polymathematics: Is mathematics a single science or a set of arts? 403416
- Mathematics and computing 417430
- The theme of 𝑝-adic variation 433446
- Back Cover Back Cover1473