Mathematics

by Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B.

• ISBN13: 9780821826973
• ISBN10: 0821826972
• Format: Paperback
• Copyright: 2000-12-01
• Publisher: Amer Mathematical Society

Summary

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. 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.

Table of Contents

Preface	vii
Michael Atiyah
Number theory, transcendence and Diophantine geometry in the next millennium	1	(12)
A. Baker
G. Wustholz
Harmonic analysis and combinatorics: How much may they contribute to each other?	13	(20)
J. Bourgain
Back to Riemann	33	(2)
Shing-Shen Chern
Noncommutative geometry and the Riemann zeta function	35	(20)
Alain Connes
Polynomials, vanishing cycles and Floer homology	55	(10)
S. K. Donaldson
The two cultures of mathematics	65	(14)
W. T. Gowers
Ten problems	79	(14)
V. F. R. Jones
An algebraic integration	93	(24)
David Kazhdan
Mathematics: The right choice?	117	(4)
Frances Kirwan
On some challenging problems in nonlinear partial differential equations	121	(16)
P.-L. Lions
Real world turbulence and modern applied mathematics	137	(16)
Andrew J. Majda
Mathematics as profession and vocation	153	(8)
Yu. I. Manin
Problems and conjectures in rigidity theory	161	(14)
Gregory Margulis
A glimpse into symplectic geometry	175	(14)
Dusa McDuff
Rational curves on algebraic varieties	189	(8)
Shigefumi Mori
The dawning of the age of stochasticity	197	(22)
David Mumford
Mathematical physics on the 20th and 21st century	219	(16)
Roger Penrose
Limitations to regularity	235	(16)
K. F. Roth
Conversations on mathematics with a visitor from outer space	251	(10)
David Ruelle
Some problems in number theory, analysis and mathematical physics	261	(10)
Peter Sarnak
Mathematical problems for the next century	271	(24)
Steve Smale
Positivity problems and conjectures in algebraic combinatorics	295	(26)
Richard P. Stanley
On the future of mathematics/physics interaction	321	(8)
Cumrun Vafa
Twenty years of number theory	329	(14)
Andrew Wiles
Magic, mystery, and matrix	343	(10)
Edward Witten
Review of geometry and analysis	353	(50)
S.-T. Yau
Polymathematics: Is mathematics a single science or a set of arts?	403	(14)
V. I. Arnold
Mathematics and computing	417	(16)
Peter D. Lax
The theme of p-adic variation	433
Barry Mazur