Proof of Poincaré conjecture is tipped for Fields medal
Some speculation always precedes the announcement of the Fields medals, the most illustrious awards in mathematics. But this year rumours have an extra dimension: one of the prizes to be awarded at the International Congress of Mathematicians in Madrid on 22 August may be given for the solution of a century-old problem.
Favourite to win a medal is Russian mathematician Grigory 'Grisha' Perelman, for his work on the Poincaré conjecture. Three papers now suggest that his proof of the conjecture is right. If it stands up to two years of scrutiny, Perelman will also be eligible for at least a share of a million-dollar mathematics prize (see 'Clay feats: the million dollar quastions').
“I am completely convinced that Perelman has proved the Poincaré conjecture,” says John Morgan, a topologist at Columbia University in New York and co-author of the most recent paper analysing Perelman's work.
The Poincaré conjecture is a seemingly simple statement about the nature of three-dimensional surfaces, but it had resisted proof since French mathematician Henri Poincaré put it forward in 1904.
In 2002, Perelman posted to an online preprint server a paper1 that claimed to “give a sketch of an eclectic proof of this conjecture”. By this, Perelman meant not just the Poincaré conjecture, but the geometrization conjecture, a much broader theorem of which Poincaré's statement is a special case.
The geometrization conjecture classifies all the possible forms that three-dimensional surfaces, called three-manifolds, can take. The Poincaré conjecture describes the properties of a subset known as 'simply connected' manifolds — in particular, those for which every loop drawn on the surface can be shrunk to a point.
To tackle the conjecture, Perelman used a mathematical tool known as Ricci flow, developed by Richard Hamilton of Columbia University. Perelman, formerly of the Steklov Institute in Moscow, advanced Hamilton's work by finding a way to deal with certain problems encountered in using Ricci flow to study the deformation of surfaces.
For people studying topology, the whole landscape has changed.
The advent of a possible proof shook a small contingent of mathematicians. “For people studying topology, the whole landscape has changed. It's like waking up one morning after an earthquake,” says Bruce Kleiner of Yale University in New Haven, Connecticut.
Perelman followed his first article with two further papers, and in the spring of 2003 he toured the United States to lecture about his work2. He has since retreated from the public eye, leaving other mathematicians to comb through his papers line by line, filling in details and searching for holes in his logic.
“How do you decide whether anything is correct?” asks John Ball, president of the International Mathematical Union and chair of the Fields medal committee. “All that can happen is that clever people, experts in the area, read it and come to an opinion.”
Three pairs of respected mathematicians have produced written accounts filling in the details of Perelman's work.
Kleiner and John Lott of the University of Michigan, Ann Arbor, posted their most recent paper to the arXiv preprint server on 25 May3. “There was a moment when I really thought there was a problem with the argument,” says Kleiner. “But that state of mind persisted for only a week or two before the issue was resolved.”
A 473-page paper4 by Morgan and Gang Tian of the Massachusetts Institute of Technology appeared on 25 July.
The third paper, from Huai-Dong Cao of Lehigh University in Bethlehem, Pennsylvania, and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, was published in June5. It claims to complete the proof of both conjectures, rather than simply fleshing out Perelman's work. This is justified, argues Shing-Tung Yau, an editor-in-chief on the journal, because Cao and Zhu follow a somewhat different argument to Perelman.
The appearance of these three papers has fuelled rumours that Perelman will receive a Fields prize. Up to four medals are awarded every four years to mathematicians no older than 40, making Perelman just eligible for this year's crop. Not that he will attend the Madrid meeting: the organizing committee received no reply when they invited him to give a plenary lecture.
Even so, many mathematicians think Perelman will be honoured with a medal. “I hope that they give him one. This is such a major achievement in mathematics, it would be funny if they didn't,” says Morgan. There is also a precedent. Two previous Fields medals have been awarded for proofs of an equivalent of the Poincaré conjecture in higher dimensions — to Stephen Smale in 1966 and Michael Freedman in 1986.
And in 2006? Morgan, along with 4,000 guests, will find out in a few weeks — when the medals are announced at the opening ceremony of the congress in Madrid.
Perelman, G. Preprint at http://arxiv.org/abs/math.DG/0211159 (2002).
Singer, E. Nature 427, 388–389 (2004).
Kleiner, B. & Lott, J. Preprint at http://arxiv.org/abs/math.DG/0605667 (2006).
Morgan, J. W. & Tian, G. Preprint at http://arxiv.org/abs/math.DG/0607607 (2006).
Cao, H. -D. & Zhu, X. -P. Asian J. Math. 10, 165–492 (2006).
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Hogan, J. Maths 'Nobel' rumoured for Russian recluse. Nature 442, 490 (2006). https://doi.org/10.1038/422490a