George G. Szpiro - Poincares Prize: The Hundred-Year Quest to Solve One of Maths Greatest Puzzles

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Table of Contents

A PLUME BOOK POINCARS PRIZE

GEORGE G. SZPIRO (GeorgeSzpiro.com) was born in 1950 in Vienna, Austria, and later moved to Zurich, Switzerland. He studied mathematics at the Eidgenssische Technische Hochschule (ETH) in Zurich, obtained an M.B.A. at Stanford University, and did his Ph.D. in mathematical economics at the Hebrew University in Jerusalem. Szpiro has been assistant professor of finance and decision sciences at the Wharton School of the University of Pennsylvania, and then at the Hebrew University. He has written thirty scientific papers in mathematics, physics, economics, and finance.

He became an Israel correspondent and science writer for the Swiss daily Neue Zrcher Zeitung and the weekly NZZ am Sonntag. He has been honored by the Swiss Academy of Sciences with the Prix Mdia, was a finalist for the Descartes Prize in Science Communication (awarded by the European Commission), and was a recipient of the Media Prize from the German Association of Mathematicians (DMV).

His previous books are Keplers Conjecture and The Secret Life of Numbers.

George lives with his family in Jerusalem.

PLUME
Published by the Penguin Group
Penguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A. Penguin Group
(Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3 (a division of
Pearson Penguin Canada Inc.) Penguin Books Ltd., 80 Strand, London WC2R 0RL, England Penguin
Ireland, 25 St. Stephens Green, Dublin 2, Ireland (a division of Penguin Books Ltd.) Penguin Group
(Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia
Group Pty. Ltd.) Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi - 110
017, India Penguin Group (NZ), 67 Apollo Drive, Rosedale, North Shore 0632, New Zealand (a division
of Pearson New Zealand Ltd.) Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank,
Johannesburg 2196, South Africa
Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England
Published by Plume, a member of Penguin Group (USA) Inc. Previously published in a Dutton edition.
First Plume Printing, August 2008

Copyright George Szpiro, 2007

All rights reserved
Mathematical editing: Christina Sormani, Lehman College and CUNY Graduate Center

REGISTERED TRADEMARKMARCA REGISTRADA
The Library of Congress has catalogued the Dutton edition as follows:
Szpiro, George, 1950
Poincars prize : the hundred-year quest to solve one of maths greatest puzzles / George Szpiro.
p. cm.
Includes bibliographical references and index.

eISBN : 978-0-452-28964-2

Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book.

PUBLISHERS NOTE

The scanning, uploading, and distribution of this book via the Internet or via any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions, and do not participate in or encourage electronic piracy of copyrighted materials. Your support of the authors rights is appreciated.

BOOKS ARE AVAILABLE AT QUANTITY DISCOUNTS WHEN USED TO PROMOTE PRODUCTS OR SERVICES. FOR INFORMATION PLEASE WRITE TO PREMIUM MARKETING DIVISION, PENGUIN GROUP (USA) INC., 375 HUDSON STREET, NEW YORK, NEW YORK 10014.

http://us.penguingroup.com

Dedicated to Grisha Perelman, a man of rare genius and modesty.
His devotion to mathematics and his refusal to let glory get
the better of him are admirable.

Chapter 1

Fit for a King

August is not the best time of year to visit Spain. Soaring temperatures during the late summer make even short sightseeing trips and brief walks through medieval cities a plight. But tourism was not the reason that several thousand men and women from all over the world flocked to Madrid on August 22, 2006. Their motivation for traveling to Spain from 143 countries and all continents except Antarctica was the 25th International Congress of Mathematicians. Commonly known as the ICM, this gathering is a grand affair that takes place only once every four years. The previous ICM had been held in Beijing, the one before in Berlin. Four thousand mathematicians from around the world, among them the most distinguished professors from the most prestigious universities and research institutes, had gathered in Spain this time to attend twenty plenary sessions and more than a thousand invited lectures, short talks, and poster sessions. Over ten days, scientific meetings alternated with concerts, exhibitions, and cultural activities. The ICMs main event was the opening ceremony, at which Juan Carlos I, the king of Spain, awarded the prestigious Fields Medals to four mathematicians.

Already two hours before the official opening at ten A.M., a long queue of people wound around the modern Palacio Municipal de Congresos in the northeastern part of the Spanish capital. The expected presence of the king necessitated strict security measures, and each entrant was searched before being given access to one of the conference halls. While the thousands of participants whiled away the time in a long human snake outside the conference center, the air was abuzz with speculations about who would be honored with a Fields Medal. Considered the equivalent of the Nobel Prize, these medals are the highest honor the mathematics profession has to offer. Even though the monetary reward is only a tiny fraction of the Nobel Prizes, the Fields Medals are arguably even more select than the Nobel Prizes, with at most four winners chosen every four years. The identities of the medalists are a closely kept secret up until the very last moment. To encourage talent, the rules specify that recipients must not be older than forty years at the start of the year in which the ICM takes place.

Over many months preceding the ICM, a secret committee consisting of nine of the worlds top mathematicians had screened the work of dozens of brilliant candidates. Finally, at the end of May, they had chosen the four best from among them. Those who knew the results were under strict orders not to divulge them. The awardees themselves had been informed during the summer, but they were not to tell anybody of their impending honor, except possibly their spouses. Not even they knew the identities of the other medalists, however.

To the mathematical world, a mysterious Russian from St. Petersburg was the most deserving of the impending honor. Grigori Perelman, the hands-down favorite in the queue outside the conference center, had all the right qualifications: He had proven his worth with sensational papers in the preceding years, some of them unpublished. He was just two months into his fortieth year, his birthday having fallen on the previous June 13. But most important, he had solved one of the oldest unsolved problems of mathematics: the 102-year-old Poincars Conjecture. This problem was considered of such fundamental importance by the community of mathematicians that when the former Fields Medalist Steve Smale was asked in 1998 to name the most important mathematical problems for the coming century, he put Poincars Conjecture near the top of his list. And the Clay Mathematics Institute in Boston listed it as one of the seven most mind-boggling unsolved math problems anyone on earth could imagine.

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