Reclusive mathematician Grigory Perelman proved the Poincar茅 conjecture. But he refused a $1聽million prize for solving this famous puzzle 鈥� in part because he believed he wasn鈥檛 the only one who deserved credit. Now the mathematician who Perelman wanted to see recognised has won a lucrative prize.
On Tuesday, the Shaw Prize Foundation in Hong Kong that it would split its annual $1聽million prize in the mathematical sciences, with half going to at Columbia University in New York, who devised a geometrical process that underpins Perelman鈥檚 proof of the Poincar茅 conjecture. The other half goes to at the Swiss Federal Institute of Technology Zurich, who works on black hole physics and general relativity.
Henri Poincar茅鈥檚 famous statement, posed in 1904, is one of the oldest and most fundamental conjectures in the field of topology, sometimes called 鈥渞ubber sheet geometry鈥� 鈥� the mathematical study of surfaces and shapes with all possible twists, turns and dimensions. It suggests a way to recognise hard-to-imagine geometrical objects that have more than three dimensions. More specifically, it teaches mathematicians how to recognise higher-dimensional spheres, even if they鈥檝e been smashed or distorted like Play-Doh.
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Beautiful idea
Though Poincar茅 suspected his test worked, he couldn鈥檛 prove it 鈥� and for a long time, nor could anyone else. In the 1980s, Hamilton devised Ricci flow, a powerful mathematical tool that could be applied to abstract shapes and smooth them out.
鈥淗e developed this theory of Ricci flow pretty much from scratch,鈥� says Jim Carlson, president of the Clay Mathematics Institute in Cambridge, Massachusetts. 鈥淰ery early on he had this beautiful idea of what kind of equation could govern the change of a shape. He proceeded to prove a whole series of fantastic results.鈥�
But, despite his efforts, these did not include the Poincar茅 conjecture. Hamilton鈥檚 idea of Ricci flow didn鈥檛 work on every different possible case 鈥� until Perelman came along and found a way around the roadblocks.
Hamilton鈥檚 dream
Perelman posted his proof on the arXiv.org site in three parts in 2002 and 2003. In 2006, after other mathematicians had verified the accuracy of his proof, he was awarded the Fields medal, one of the highest honours in mathematics. He refused it 鈥� and remained mostly mum about his reasons.
Last July, Perelman also turned down a $1聽million prize that the Clay institute awarded him for proving the Poincar茅 conjecture. On the eve of the Clay celebration, he that he thought the organised mathematical community was 鈥渦njust鈥� and that he did not like their decisions. Richard Hamilton, he said, deserved as much credit for the proof as he did.
鈥淭he solution to the Poincar茅 conjecture is an enormous step forward in topology and geometry that we now know is made possible by Hamilton鈥檚 ideas,鈥� adds Carlson. 鈥淧erelman should be viewed as achieving the dream that Hamilton had.鈥�
