The Poincare Conjecture In Search of the Shape of the Universe

Donal O'Shea

Walker Books




304 Pages



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Henri Poincaré, a giant among nineteenth- and twentieth-century mathematicians, transformed the fledgling area of topology—which studies properties of geometric shapes, such as the Möbius strip, that are unchanged by stretching or twisting—into a powerful field essential to all of modern mathematics and physics. He also posed a puzzle that speaks to the possible shape of the universe and lies at the heart of modern topology and geometry.

Conceived in 1904, the Poincaré conjecture has resisted attempts by generations of mathematicians to prove or disprove it. Donal O'Shea charts the trajectory of geometric knowledge from the scribes of vanished Babylon to the great trailblazers of mathematics in our times that led to Poincaré's famed proposition, and recounts the competitive attempts to resolve it over the past century.

Signaling its significance, in 2000 the Clay Mathematics Institute identified the conjecture as one of the seven great and essential unsolved conundrums of the new millennium, and offered a one-million-dollar prize for its solution. Seven years later, the conjecture appears to have been proved. Russian mathematician Grigory Perelman, who shuns the limelight, shocked the mathematics world by posting his solution on the Internet instead of publishing it in a peer-reviewed journal. It has withstood years of intense vetting by several teams of mathematicians, and now seems certain to open up new corners of the mathematical universe.

The Poincaré Conjecture tells of the personalities, institutions, and scholarship behind centuries of mathematics that have led to Perelman's dramatic proof, and which have, in the process, broadened the understanding of how the universe works. Donal O'Shea brings alive the achievements of Poincaré, Bernhard Riemann, William Thurston, Richard Hamilton, and others whose genius has transformed the field in the last century. O'Shea chronicles the events at the 2006 International Congress of Mathematicians in Madrid, where the eccentric Perelman was awarded a Field Medal only to turn it down. The Poincaré Conjecture offers a glimpse into our collective search for knowledge about the universe.


Praise for The Poincare Conjecture

"Donal O'Shea's beautifully written new book, The Poincaré Conjecture, begins with a quick tour of the history of ideas of geometry and topology that set the stage for Poincaré . . . O'Shea goes on to trace the history of modern geometry and the mathematical giants who founded it in the second half of the 19th century . . . Poincaré comes on the scene as the natural heir to [Bernhard] Riemann and as the father of topology. O'Shea explains that . . . Poincaré had made a long study of two-dimensional spaces (surfaces) . . . In his 1895-1904 papers he established the basics of these [higher-dimensional] spaces, proved some important results and then formulated his famous conjecture, which concerns how one can characterize the three-dimensional analogue of the sphere among all three-dimensional spaces . . . O'Shea briefly gives the story of some of the leading 20th-century topologists and their attacks on the Poincaré conjecture and its generalizations to higher dimensions. Then O'Shea turns to the developments that eventually led to the positive resolution of the conjecture . . . The depth and beauty of mathematics, the intellectual sweep and power of the ideas, are notoriously difficult, if not downright impossible, to explain to those who have no mathematical training and even to natural scientists, who do have extensive mathematical training . . . To deal with the abstraction and to keep the subject on the right track, mathematicians have developed a very precise language that is foreign to the nonspecialist. Recently, there have been a few books that have managed to present some of the most spectacular mathematical advances in a way that makes them accessible to a wider audience and at the same time does justice to the mathematics. O'Shea's volume is one of these. He has a gift for metaphor and often is able to capture the essence of a mathematical development or idea without falling into the jargon of the discipline or impenetrable prose. The Poincaré Conjecture is a fine example of mathematical writing for a general audience. Those with an interest in knowing what mathematics is about, in understanding mathematical ideas, and in tracing the history of famous mathematical problems should read it. No matter what their mathematical background, readers will find much to further stimulate their interest and will learn about both the history of mathematics and its most recent spectacular advances. The book should also be required reading for mathematicians, who will derive great pleasure from seeing their subject so well presented. It should serve as a model to inspire more good general writing by mathematicians, whose field has so much worth explaining and so few good expositors."—John Morgan, American Scientist

"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

"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 Math Gene and The Millennium Problems

"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 traces the footsteps of mathematicians like Euclid, Gauss, Riemann, and their contemporaries to prove one of mathematics' greatest puzzles, which eventually culminated with Grigory 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."—Ian D. Gordon, Library Journal

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  • Donal O'Shea

  • 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.