A WAVE of excitement is tearing through mathematics as experts set about deciphering two dense and difficult papers. They contain a possible solution to a hundred-year-old puzzle on which a million dollars of prize money rests.
The prize is on offer from the Clay Mathematics Institute in Cambridge, Massachusetts, for a proof of the notorious Poincar茅 conjecture. The money will be awarded if the proof withstands two years of scrutiny.
But that almost pales into insignificance compared with what else is being claimed 鈥� proof of a much broader statement called the geometrisation conjecture. That is a breakthrough which would revolutionise the branch of mathematics known as topology. 鈥淭his conjecture has been such a dominant force in the field, few people are untouched by it,鈥� says Mark Brittenham, a topologist at the University of Nebraska, Lincoln.
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Topology is concerned with shapes 鈥� but not as we normally think of them. Topologists study characteristics that do not change when an object is stretched or twisted. Working out the rules that different shapes must obey reveals which shapes can exist in different numbers of dimensions, including what form the Universe itself might take.
The Poincar茅 conjecture is based on an idea that seems simple enough. If you tie a piece of elastic around the two-dimensional surface of a sphere, it can always be pulled tight to a point. There is no other type of object with a 2D surface for which this is true. But what about 3D surfaces or spaces? Henri Poincar茅 claimed that the analogous statement holds 鈥� the 3D equivalent of a sphere is the only object that meets this condition.
The geometrisation conjecture is much more ambitious. Proposed by William Thurston in the 1970s, it characterises all 3D surfaces, saying they follow a set of simple rules. If you can prove Thurston right, Poincar茅鈥檚 statement is automatically true.
This is what Grigori Perelman from the Steklov Institute of Mathematics in St Petersburg, Russia, says he has achieved. News of the claim has been rumbling since Perelman published the first of two papers on the subject in November last year (see and ).
It will take months for the new proof to be thoroughly checked, and Perelman is giving no interviews to the press, but he is now touring the US to talk to colleagues about his work for the first time. They are impressed. 鈥淲e鈥檇 been studying the papers for a number of months and we had a list of questions we asked,鈥� says Tom Mrowka of MIT, who attended Perelman鈥檚 packed-out lectures last week. 鈥淗e鈥檚 really, really on top of his stuff.鈥�
The excitement is spreading. 鈥淥ver our beers we were speculating about what our field will look like if this paper holds up,鈥� says Brittenham. 鈥淟ife is going to be very different.鈥� That is because much of the work topologists do is aimed at proving the geometrisation conjecture for specific cases or conditions. A correct proof from Perelman would make all that redundant.
Thurston, who has been trying to prove the conjecture himself, is facing the fact that he may have been beaten to it. 鈥淭his has been a big goal for me and many others,鈥� he told New 杏吧原创. 鈥淚f Perelman鈥檚 proof holds up it is a great achievement.鈥�