#19 - JRL 7169
The Independent (UK)
May 6, 2003
Has an unknown Russian quietly solved a puzzle that has baffled the finest mathematicians for a century?
By Steve Connor, Science Editor

For almost a century Poincaré's Conjecture has tempted, taunted and ultimately vanquished some of the brightest minds in mathematics. Now, a little-known Russian scholar has astonished the rarefied world of topology – the science of surfaces – by coming up with what appears to be the first formal proof – probably.

For eight solitary years, Grigori Perelman, from the Steklov Institute of Mathematics in St Petersburg, has quietly toiled away on one of the hardest problems in mathematics, proposed in 1904 by the French-man Henri Poincaré.

It deals with the properties of surfaces in two, three or more dimensions, and, in essence, posits that all shapes can be reduced to spheres or doughnuts – but there is more to it than that, of course.

Most people would find the simplest explanation of the conjecture a puzzle, never mind its solution. But solving it could lead to understanding the shape of the universe, as well as earning Dr Perelman $1m (£620,000) in prize money.

For the past month, Dr Perelman has been touring universities in America explaining his proof to his peers for the first time. Many have come away convinced that he has done it, but nobody is prepared to say for sure.

"We're desperately trying to understand what he has done here," said Tomasz Mrowka, a topologist at the Massachusetts Institute of Technology, which hosted a series of lectures by Dr Perelman. "With a problem like this, that has had such a long history, including many failed attempts, people are quite cautious about making definitive statements." However, he added, the audiences had been "very appreciative".

Dr Perelman will not give interviews on his work, but some are comparing it to that of Andrew Wiles, the English mathematician who solved Fermat's Last Theorem in 1995 at Cambridge University. Two weeks ago, Professor Wiles was in the fifth row of the Taplin Auditorium at Princeton University, New Jersey, where he holds a chair in mathematics, to hear Dr Perelman. (Three rows behind him sat John Nash, the Nobel Laureate who inspired the film A Beautiful Mind.)

Professor Wiles stressed at the weekend that he was no expert in this field of maths and was in no position to assess the proof. "It all hinges on the detail," he said. "I think it is always very exciting when someone presents an argument that may provide a solution to one of these long-term problems."

The story of Dr Perelman's proposal began last November when he quietly posted a pre-print of his first research paper on the internet in which he stated enigmatically: "We give a sketch of an eclectic proof of this conjecture." Some thought he was proposing a method of attack rather than a solution, but he followed up with a second papercontaining more details of a putative answer in March. A third paper, details of which he included in his American lectures, was meant to finish the job.

The Clay Mathematics Institute in Cambridge, Massachusetts – the arbiters of the prize – will take at least two years to decide whether his work stands up to scrutiny. In that time, he can expect the world's sharpest mathematical minds to do everything they can to pick holes in his argument. If he survives the onslaught, he will receive Clay's top prize of $1m, one of seven that it is awarding for the greatest unsolved mathematical problems (see box).

It is an area littered with false dawns, as Martin Dunwoody, Professor of Mathematics at Southampton University, knows to his cost. Last year his explanation of Poincaré's Conjecturewas quickly torn apart by rival scholars who dismembered it equation by equation. "I thought I had a solution, but I was wrong," he said.

Even the simplest definition of the conjectureescapes an easy explanation. One way of looking at it is to imagine the surface of a football. Whether the ball is inflated or deflated its surface is the same and in this respect it is said the surface is two-dimensional.

A rubber band stretched over the ball's surface can be squeezed into a single point anywhere on the surface without breaking or tearing either the ball or the band. This is not the case if a rubber band can be made to stretch around a similar surface with a hole in it – a "torus", or doughnut shape.

The football is the only two-dimensional shape with this property, and Poincaré, recognising that there was a three-dimensional space, known as the three-sphere, for which it was also true, asked whether it was the only such three- dimensional space. Dr Perelman claims to have shown it is.

What is astonishing about Dr Perelman's proposal is that he was trying to achieve something far grander than merely solving Poincaré. He was trying to prove the Geometrisation Conjecture proposed by the American mathematician William Thurston in the 1970s – a far more ambitious set of rules that defines and characterises all three-dimensional surfaces.

Thurston's conjecture is an extension of Poincaré's. Therefore, if Dr Perelman solved Thurston's conjecture, it followed that he had solved Poincaré's, Graham Niblo, senior lecturer in mathematics at Southampton University, said. "It means that we would be able to produce a catalogue of all possible three-dimensional shapes in the universe. It means we could ultimately describe the actual shape of the universe itself," he said.

Sir Michael Atiyah, one of Britain's most distinguished mathematicians and a past president of the Royal Society, said the word was out that Dr Perelman was on to something.

"I think it's serious and most of the people I've spoken to have told me that parts of it have stood up to close examination, but lots of things have to be checked," he said. "This is probably one of the easiest unsolved problems in mathematics to explain, but proving it is difficult. There is a big gap between the conceptual side and the verification side."

A DOUGHNUT PLUS AN ORANGE PLUS A RUBBER BAND EQUALS ONE HARD NUT TO CRACK

If we stretch a rubber band around the surface of an orange, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the orange is "simply connected", but that the surface of the doughnut is not. Poincaré knew that a two-dimensional sphere was characterised by the property of simple connectivity, and asked the corresponding question for the three-dimensional sphere (the set of points in four-dimensional space at unit distance from the origin). The question turned out to be very difficult. Mathematicians have been struggling to resolve it ever since.

Source: Clay Mathematics Institute