The Poincaré conjecture: in search of the shape of the universe

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.

“o'shea Tells The Fascinating Story Of This Mathematical Mystery And Its Solution By The Eccentric Mr. Perelman.”—wall Street Journal

in 1904, Henri Poincaré, A Giant Among Mathematicians Who Transformed The Fledging Area Of Topology Into A Powerful Field Essential To All Mathematics And Physics, Posed The Poincaré Conjecture, A Tantalizing Puzzle That Speaks To The Possible Shape Of The Universe. For More Than A Century, The Conjecture Resisted Attempts To Prove Or Disprove It. As Donal O’shea Reveals In His Elegant Narrative, Poincaré’s Conjecture Opens A Door To The History Of Geometry, From The Pythagoreans Of Ancient Greece To The Celebrated Geniuses Of The Nineteenth-century German Academy And, Ultimately, To A Fascinating Array Of Personalities—poincaré And Bernhard Riemann, William Thurston And Richard Hamilton, And The Eccentric Genius Who Appears To Have Solved It, Grigory Perelman. The Solution Seems Certain To Open Up New Corners Of The Mathematical Universe.

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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.)

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Conceived In 1904, The Poincaré Conjecture, A Puzzle That Speaks To The Possible Shape Of The Universe And Lies At The Heart Of Modern Topology And Geometry, Has Resisted Attempts By Generations Of Mathematicians To Prove Or To Disprove It. Despite A Million-dollar Prize For A Solution, Russian Mathematician Grigory Perelman, Posted His Solution On The Internet Instead Of Publishing It In A Peer-reviewed Journal. This Book Tells The Story Of The Fascinating Personalities, Institutions, And Scholarship Behind The Centuries Of Mathematics That Have Led To Perelman's Dramatic Proof. The Author Also Chronicles Dramatic Events At The 2006 International Congress Of Mathematicians In Madrid, Where Perelman Was Awarded A Fields Medal For His Solution, Which He Declined. Cambridge, April 2003 -- The Shape Of The Earth -- Possible Worlds -- The Shape Of The Universe -- Euclid's Geometry -- The Non-euclideans -- Bernhard Riemann's Probationary Lecture -- Riemann's Legacy -- Klein And Poincaré -- Poincaré's Topological Papers -- The Great Savants -- The Conjecture Takes Hold -- Higher Dimensions -- A Solution In The New Millennium -- Madrid, August 2006. Donal O'shea. Includes Bibliographical References (p. 259-272) And Index. Henri PoincarE 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 PoincarE 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. PoincarE'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 PoincarE 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 A history of one of mathematics' most baffling conundrums looks at the PoincarGe Conjecture, a principle of modern geometry, topology, and the possible shape of the universe, as well as efforts by mathematicians to find a solution to the problem.