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News from the world of maths: Perelman awarded Millennium Prize for proof of Poincaré

Wednesday, March 24, 2010

Last week the Clay Mathematics Institute announced that Grigoriy Perelman has won the Millennium Prize for his proof of the century old Poincaré Conjecture. And almost as soon as it was announced the speculation began as to whether Perelman would accept the prize, and the $US 1,000,000 of prize money.

The Poincaré Conjecture is a question essentially about the nature of shapes in space. Mathematicians have long understood the nature of every possible 2D surface in 3D space. For example the surface of a sphere, such as the outside of a ball, is completely characterised by being simply connected — it has no edge, and any loop on the surface can be slid off without being cut or torn. And these two properties are true not matter how much a sphere is squashed or stretched out of shape. However they aren't true for any other kind of 2D surface, for example the surface of a doughnut: a loop through the centre hole of a donut can't be removed without being cut. That is because a doughnut is not the same, topologically speaking, as a sphere.

Poincaré proposed that all 3D spheres can be characterised by the same two properties. However for over a century the result remained unproven despite the efforts of some of the best mathematical minds. The problem was seen as so important that it was included in the list of seven Millennium Problems chosen by the Clay Institute in 2000. The solution to any of these Millennium Problems would be a monumental advancement in mathematics, and the Clay Institute offered a prize of $US 1,000,000 for the solution for each.

In 2003 Perelman surpised the mathematical world by posting a proof of a much wider conjecture online. He claimed to have proved Thurston's Geometrisation Conjecture, that characterised every 3D surface. The Poincaré Conjecture would be proven true as a consequence of this wider result.

After much examination, discussion and exposition, the mathematical community accepted that Perelman had proved the Poincaré conjecture and he was awarded the Fields Medal in 2006, the highest prize in mathematics. Controversially Perelman declined to accept the prize, the first person to ever do so. He withdrew from mathematics and now lives a reclusive life in the outskirts of St Petersburg.

Now that Perelman's work has survived several years of critical review and has been accepted by the mathematical world the Clay Institute has awarded him the Millennium Prize for his proof of the Poincaré Conjecture, the first of the Millennium Problems to be solved. However most people in the mathematical community expect that, like the Fields medal, Perelman will not accept this prize or the prize money. Despite some reports in the media, the Clay Institute told Plus that they had been in contact with Perelman and that "he said he would think about it".

Whether or not Perelman's decides to accept the Millennium Prize at a ceremony in June, his enormous contributions to mathematics will be celebrated for many years, and we hope that he is able to live his life happily in whatever way he chooses. It might seem hard for most of us to understand how someone could refuse such wealth and fame but, as Marcus du Sautoy explained to BBC Radio 4's The World Tonight, for some people other things are more important:

"I think there is something noble in that he values solving a mathematical problem above the glory of being in the limelight and winning prizes and getting vast sums of money. There is something rather nice about Perelman's choice to just enjoy the mathematics."

You can read more about the Poincaré Conjecture and the Clay Millennium Problems on Plus, and you can read more about his award, including his original papers at the Clay Mathematics Institute.

posted by Plus @ 9:26 AM 

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