TetsuoShima Posted December 29, 2006 Share Posted December 29, 2006 Solving the Poincare Conjecture wins Science's Breakthrough of the Year This year, Science (open access) named a breakthrough that has no connections to politics or religion: the proof of the PoincarÃƒÆ’Ã‚Â© Conjecture by Russian mathematician Grisha Perelman. The PoincarÃƒÆ’Ã‚Â© Conjecture was originally proposed by Henri PoincarÃƒÆ’Ã‚Â© in 1904 and deals with the topology of everyday objects, namely what, in topological terms, defines a sphere. The Conjecture remained unsolved for almost 100 years, although not for lack of trying, and in the year 2000 the Clay Mathematics Institute (CMI) named the PoincarÃƒÆ’Ã‚Â© Conjecture as one of its six Millennium Problems. These problems have solutions that have eluded mathematicians for years and carry a US $1,000,000 prize to anyone who solves them (either in a positive or negative manner). As stated in the CMI's official problem declaration, the PoincarÃƒÆ’Ã‚Â© Conjecture asks: "If a compact three-dimensional manifold M^{3} has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M^{3} is homeomorphic to the sphere S^{3}?" Over the years, special solutions for specific dimensions were developed. For one and two dimensions, the proof was trivial; seven or more dimensions were handled by a proof from Stephen Smale, developed in 1960. Smale then extended his proof to all dimensions greater than or equal to five, for which he was awarded the Fields Medal in 1966. More than 20 years later, Micheal Friedman proved the Conjecture for the four dimensional case in 1980 and was awarded the Fields Medal in 1986. PoincarÃƒÆ’Ã‚Â© had now been proven for ALL dimensions EXCEPT three, the original dimension for which Henri PoincarÃƒÆ’Ã‚Â© first proposed the problem. The final unproven dimension fell in a series of papers published to the web by Grisha Perelman, all available to the curious for free from arXiv.org. Dr. Perelman extended previous work done by Richard Hamilton, a mathematician who proposed that an arbitrarily lumpy space could "flow" towards a smooth spaceÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Âimagine a Koosh ball morphing into a soccer ballÃƒÂ¢Ã¢â€šÂ¬Ã¢â‚¬Âthrough equations akin to the heat equation, and named the process "Ricci flow." This series of papers was published in 2002 and 2003, yet it took the mathematics community at large three more years to accept this solution as a true proof to the PoincarÃƒÆ’Ã‚Â© Conjecture. However, the story does not end there. In 2006 the International Mathematical Union announced that it had awarded the Fields Medal to Grisha Perelman, but he declined it. In a rare interview with The New Yorker, Dr. Perelman announced that he was retiring from mathematics, stating he was disheartened by what he viewed as ethical lapses by some of his colleagues. This New Yorker article has caused quite a commotion in the mathematical world, with people claiming their words were distorted and threats of lawsuits traded. This has caused a black cloud to hang over what is the greatest mathematical breakthrough of the millennium (so far). Fortunately mathematical proofs are not affected by the feelings of those who create them. The proof of the PoincarÃƒÆ’Ã‚Â© Conjecture remains atop the list of the greatest scientific breakthroughs of the year. For those who wish to read more of this (though they probably allready have, if they're interested in these things), a list of Accompanying Hyperlinks: http://www.sciencemag.org/cgi/content/full/314/5807/1848 http://mathworld.wolfram.com/PoincareConjecture.html http://www.claymath.org/millennium/ http://mathworld.wolfram.com/Topology.html http://www.tti-c.org/smale.html http://research.microsoft.com/theory/freedman/ http://arxiv.org/abs/math.DG/0211159 | http://arxiv.org/abs/math.DG/0303109 | - > These 3 are the published papers!! http://arxiv.org/abs/math.DG/0307245 | http://www.newyorker.com/fact/content/articles/060828fa_fact2 http://arstechnica.com/news.ars/post/20061227-8509.html The runners-up in Science's "Breakthroughs of 2006" One of the worst aspects of the various "Best of" lists that proliferate at the end of the year is that they rarely encompass all the good material that's appeared within the proceeding year. So, although Science magazine names a "Breakthrough of the Year" (which we'll cover separately), they also recognize the other nine discoveries that round out their top-10 list of breakthroughs. The important work of the past year covers a broad range of science, and includes some work with important implications for humans and the planet we occupy. The remaining 9 of their top 10 were: Ancient DNA and the Neanderthal genome The Vanishing Ice Tiktaalik is First with a Neck The Vanishing Objects Maintaining Sight in Spite of Macular Degeneration Species do the splits Diffract This Memories go Long-term Piwi RNAs are Big That's what Science thinks was big. The Nobel Intent staff is in the process of making our own list, but feel encouraged to suggest what you think the biggest science stories of 2006 were in the discussion. For more info on the seperate runners-up, visit arstechnica (link below). Accompanying Hyperlinks: http://www.sciencemag.org/cgi/content/full/314/5807/1850a http://arstechnica.com/news.ars/post/20061227-8508.html Link to comment Share on other sites More sharing options...

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