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Poincare Conjecture: In Search of the Shape of the Universe

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  • 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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