The Poincare Conjecture: In Search of the Shape of the Universe [NOOK Book]

Overview


Henri Poincaré was one of the greatest mathematicians of the late nineteenth and early twentieth century. He revolutionized the field of topology, which studies properties of geometric configurations that are unchanged by stretching or twisting. The Poincaré conjecture lies at the heart of modern geometry and topology, and even pertains to the possible shape of the universe. The conjecture states that there is only one shape possible for a ...
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The Poincare Conjecture: In Search of the Shape of the Universe

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Overview


Henri Poincaré was one of the greatest mathematicians of the late nineteenth and early twentieth century. He revolutionized the field of topology, which studies properties of geometric configurations that are unchanged by stretching or twisting. The Poincaré conjecture lies at the heart of modern geometry and topology, and even pertains to the possible shape of the universe. The conjecture states that there is only one shape possible for a finite universe in which every loop can be contracted to a single point.

Poincaré's conjecture is one of the seven "millennium problems" that bring a one-million-dollar award for a solution. Grigory Perelman, a Russian mathematician, has offered a proof that is likely to win the Fields Medal, the mathematical equivalent of a Nobel prize, in August 2006. He also will almost certainly share a Clay Institute millennium award.

In telling the vibrant story of The Poincaré Conjecture, Donal O'Shea makes accessible to general readers for the first time the meaning of the conjecture, and brings alive the field of mathematics and the achievements of generations of mathematicians whose work have led to Perelman's proof of this famous conjecture.
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Editorial Reviews

From Barnes & Noble
In everyday life, conjectures are cheap; in mathematics, they sometimes stimulate years of vigorous theorizing. In the case of Henri Poincaré (1854-1912), the supposition was simply stated. "Consider a compact 3-dimensional manifold V without boundary," he wrote. "Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?" At the time of his death, this pregnant thought was still in gestation. It would take almost a century for a mathematician to offer a persuasive proof of this major conundrum. Written for nonprofessional readers, The Poincaré Conjecture explains why the issue has loomed so large in scientific circles and also provides stimulating capsule biographies of the thinkers who prepared the way for its solution.
Publishers Weekly

The reclusive Russian mathematician Grigory Perelman became a minor media celebrity last summer when he refused the prestigious Fields medal, awarded every four years to a mathematician under the age of 40. Perelman had succeeded in solving the Poincaré conjecture, named for 19th-century French mathematician Henri Poincaré, and which contemporary cosmologists believe has implications for our understanding of the shape of the universe. O'Shea, a professor of mathematics at Mount Holyoke College, begins his account of the long and contentious search for a solution to the puzzle by looking at how we came to understand the shape of the Earth, beginning with the Greeks, in particular Pythagoras and Plato. Writing for generalist science buffs, O'Shea gives a brief course in geometry and in topology and the topological structures called manifolds that are the basis of Poincaré's puzzle. Inexplicably, however, O'Shea doesn't give readers a formal statement of the conjecture itself until well into the book. O'Shea describes mind-bending structures in topology as clearly as most of us can describe a cube, but readers will need to do a little Wikipedia-ing first to find out just what it is they're reading about. Illus. (Mar.)

Copyright 2006 Reed Business Information.
Library Journal
In 1904, Henri Poincaré (1854–1912) challenged the world to solve one of the 20th century's most famous mathematical problems, collectively named the Poincaré Conjecture and involving configuring multidimensional space using algebraic topology. Poincaré asked a basic question of what it means for mathematical space to be curved. O'Shea (mathematics, Mount Holyoke Coll.) traces the footsteps of mathematicians like Euclid, Gauss, Riemann, and their contemporaries to prove one of mathematics' greatest puzzles, which eventually culminated with Grigori Perelman's eclectic, brilliant solution in 2003. O'Shea inspires readers to note the beauty, application, and humanity involved with this mathematical journey. Writing for readers with limited mathematical background, O'Shea successfully weaves mathematical proofs with curious insights to tell a great story, along with reams of valuable endnotes and figures. For all mathematicians and academic and larger public libraries.
—Ian D. Gordon
From the Publisher

Praise for The Poincaré Conjecture:

“O'Shea inspires readers to note the beauty, application, and humanity involved with this mathematical journey.”—Library Journal

O'Shea describes mind-bending structures in topology as clearly as most of us can “describe a cube…”—Publishers Weekly

“Accessible…. valiant nonnumerical clarity…”—Booklist

"Fascinating….[O'Shea] does a good job of explaining the mathematics involved in solving the conjecture…"—Wall St Journal

“A layman’s guide to this mathematical odyssey is long overdue, and this one will appeal to math whizzes and interested novices alike.”—Discover magazine

“O’Shea shows that, just like chasing ‘sensual passions,’ the single-minded, relentless pursuit of proof can be a creative process.”—Chicago Tribune

“O’Shea tells the whole story in this book, neatly interweaving his main theme with the history of ideas about our planet and universe. There is good coverage of all the main personalities involved, each one set in the social and academic context of his time.”—New York Sun

"Donal O’Shea has written a truly marvelous book. Not only does he explain the long-unsolved, beautiful Poincaré conjecture, he also makes clear how the Russian mathematician Grigory Perelman finally solved it. Around this drama O’Shea weaves a tapestry of elementary topology and astonishing concepts, such as the Ricci flow, that have contributed to Perelman’s brilliant achievement. One can’t read The Poincaré Conjecture without an overwhelming awe at the infinite depths and richness of a mathematical realm not made by us."—Martin Gardner, author of The Annotated Alice and Aha! Insight

"The history of the Poincaré conjecture is the story of one of the most important areas of modern mathematics. Donal O’Shea tells that story in a delightful and informative way—the concepts, the issues, and the people who made everything happen. I recommend it highly."—Keith Devlin, Stanford University, author of The Millennium Problems

"In The Poincaré Conjecture, Mr. O'Shea tells the fascinating story of this mathematical mystery and its solution by the eccentric Mr. Perelman . . . Mr. O'Shea does a good job of explaining the mathematics involved in solving the conjecture . . . [He] avoids cliché (we're spared the usual reference to coffee cups turning into doughnuts as an explanation of how surfaces might stretch without closing holes), and he tries to keep things lively."—Amir D. Aczel, The Wall Street Journal

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Product Details

  • ISBN-13: 9780802718945
  • Publisher: Bloomsbury USA
  • Publication date: 5/26/2009
  • Sold by: Barnes & Noble
  • Format: eBook
  • Edition number: 1
  • Pages: 304
  • Sales rank: 1,194,542
  • Product dimensions: 5.50 (w) x 8.25 (h) x 0.75 (d)
  • File size: 8 MB

Meet the Author

Donal O'Shea is professor of mathematics and dean of faculty at Mount Holyoke College. He has written scholarly books and monographs, and his research articles have appeared in numerous journals and collections. He lives in South Hadley, Massachusetts.
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Read an Excerpt

The Poincare Conjecture

In Search of the Shape of the Universe
By O'Shea, Donal

Walker & Company

Copyright © 2007 O'Shea, Donal
All right reserved.

ISBN: 080271532X

"Mathematical knowledge builds on the work of those who have gone before us. Any one of us with an elementary school education can solve arithmetic and algebraic problems that would have defeated the most learned Babylonian scribes. Any one of us with a few courses of calculus and linear algebra can solve problems that Pythagoras, Archimedes, or even Newton could not have touched. A mathematics graduate student today can handle topological calculations that Riemann and Poincaré could not have begun. We are not smarter than they. Rather, we are their beneficiaries."

Continues...

Excerpted from The Poincare Conjecture by O'Shea, Donal Copyright © 2007 by O'Shea, Donal. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents


Preface     ix
Cambridge, April 2003     1
The Shape of the Earth     6
Possible Worlds     21
The Shape of the Universe     32
Euclid's Geometry     46
The Non-Euclideans     57
Bernhard Riemann's Probationary Lecture     75
Riemann's Legacy     88
Klein and Poincare     106
Poincare's Topological Papers     122
The Great Savants     137
The Conjecture Takes Hold     151
Higher Dimensions     163
A Solution in the New Millennium     182
Madrid, August 2006     195
Notes     201
Glossary of Terms     241
Glossary of Names     247
Timeline     253
Bibliography     259
Further Reading     271
Art Credits     273
Acknowledgments     275
Index     279
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Sort by: Showing all of 2 Customer Reviews
  • Posted May 2, 2010

    Poincare's Conjecture

    Poincare's Conjecture is a mathematical guess about the ultimate shape of the physical universe. "Poincare's Conjecture", the book, is a fondly told story as much about the world of mathematicians and the role of mathematics, as it is the story of the Conjecture and its proof. I found it engrossing, but at the same time I have to say it is not light reading. The foot notes need to be read along with the text as they add much entry level information. I read it three times and found new insight with each reading.
    Henri Poincare lived in the last half of the 19th Century and the first half of the 20th. This was a time when European Universities were competing for prestige and the best mathematicians in a spirit of nationalism. It was a time of seminal mathematical progress. Carl Gauss laid down the laws of electromagnetism, the tools which make generators and electric engines as well as satellite TV possible. Bernhard Riemann developed the concept of space and geometry which revolutionized what 3 and higher dimensioned geometry might be like. Albert Einstein, a young scientist with some useful thoughts of his own about space curvature and the Universe, worked on his theories of relativity during this period. Poincare's conjecture in 1904 asked whether complicated convoluted 3-spaces might ultimately resolve to a simple 3-dimensioned sphere. In Poincare's words, "Is it possible that the fundamental group of a manifold could be the identity, but that the manifold might not be homeomorphic to the 3-dimensional sphere?" Manifolds and Geometries having constant curvature are the only spaces which allow motion of rigid bodies whose lengths and angles do not change. The search for a proof also produced some notable people. John Milnor as an undergraduate at Princeton mistook a long standing problem on closed curves as a homework problem and solved it, reminiscent of Matt Damon's character in the movie, "Good Will Hunting". John Nash, the central character in "A Beautiful Mind", solved another longstanding problem, making him famous prior to his bouts with depression.
    Donal O'Shea wrote this book "for the curious individual who remembers a little high school geometry, but not much more". He also reveals his love of the topic. I enjoyed the tour. I would like to take one his classes some day.

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    Posted April 4, 2011

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