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Last Updated: Monday, 6 September, 2004, 19:07 GMT 20:07 UK
Big maths solution 'looks good'
By Paul Rincon
BBC News Online science staff, at the BA festival

Numbers on a page (BBC)
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One of the toughest problems in maths may have been solved by the Russian scientist Dr Grigori Perelman, of the Steklov Institute of Mathematics.

He put forward his solution to the Poincare Conjecture two years ago - and it is still being checked by peers.

But Stanford University's Dr Keith Devlin told a UK science festival on Monday that it looks to be correct.

The Poincare Conjecture is a complex problem concerned with the study of shapes, spaces and surfaces.

Millennium problems

Dr Devlin, speaking at the British Association's Festival of Science in Exeter, said the delay in confirming or rejecting the Perelman solution gave an indication of just how complicated it was.

"Many experts think that Grigori Perelman's proof of the Poincare Conjecture is correct, but it is likely to take many more months before the experts are sure whether it is right or wrong," said Dr Devlin.

"I believe that this proof may well turn out to be correct; and even if not, the new ideas Perelman has introduced will still have many important ramifications for the subject."

He added that it was impossible to anticipate where the Poincare Conjecture might have its most profound implications.

Four years ago, the Clay Mathematics Institute in the US announced seven Millennium Problems in maths, offering $1m to anyone who could solve one of them.

Higher levels

The problem, which is 100 years old this year, was devised by Henri Poincare, a French mathematician and physicist.

The Poincare Conjecture tries to understand the shapes of spaces. This includes three-dimensional and four-dimensional space-time.

"Once you go into four dimensions, you are talking about spaces you can't visualise. The easiest way to visualise this is by studying what happens one dimension down - with two-dimensional surfaces," said Dr Devlin, executive director of Stanford's Center for the Study of Language and Information.

But Poincare found that generalising this method from two-dimensional surfaces to three-dimensional ones is not as simple as it sounds, because the objects they cover exist in higher dimensions.

So he formulated the conjecture to see if what applied in two dimensions also applied for three.

"One of the odd things about this conjecture is that if you go even higher in dimensions - four, five, six manifolds, the Poincare Conjecture is true as it is for two manifolds (dimensions)," said Dr Devlin.

But it fails for three manifolds. The one case that is really of interest in physics is the one case in which it fails."

Sceptical community

Poincare made significant contributions to the fields of optics, electricity and fluid mechanics.

"He was a physicist as well as a mathematician and geometrist. The Poincare Conjecture grew out of his interest in physics. He almost got to relativity before Einstein, but didn't push it through to the end," said Dr Devlin.

Mathematician Louis de Branges of Purdue University, US, also claims to have solved the Riemann Hypothesis, another of Clay's Millennium Problems.

But Professor Marcus Du Sautoy, of the University of Oxford, said mathematicians were currently much more sceptical about de Branges' claim.

He accepted, however, that de Branges solution to another long-standing problem was also originally greeted with scepticism, but later proved to be correct.

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