Tim Radford, science editorTuesday September 7, 2004The Guardian

“Mathematicians could be on the verge of solving two separate million dollar problems. If they are right – still a big if – and somebody really has cracked the so-called Riemann hypothesis, financial disaster might follow. Suddenly all cryptic codes could be breakable. No internet transaction would be safe.

On the other hand, if somebody has already sorted out the so-called PoincarÃ© conjecture, then scientists will understand something profound about the nature of spacetime, experts told the British Association science festival in Exeter yesterday.

Both problems have stood for a century or more. Each is almost dizzyingly arcane: the problems themselves are beyond simple explanation, and the candidate answers published on the internet are so intractable that they could baffle the biggest brains in the business for many months.”

They are two of the seven “millennium problems” and four years ago the Clay Mathematics Institute in the US offered $1m (Â£563,000) to anyone who could solve even one of these seven. The hypothesis formulated by Georg Friedrich Bernhard Riemann in 1859, according to Marcus du Sautoy of Oxford University, is the holy grail of mathematics. “Most mathematicians would trade their soul with Mephistopheles for a proof,” he said.

The Riemann hypothesis would explain the apparently random pattern of prime numbers – numbers such as 3, 17 and 31, for instance, are all prime numbers: they are divisible only by themselves and one. Prime numbers are the atoms of arithmetic. They are also the key to internet cryptography: in effect they keep banks safe and credit cards secure.

This year Louis de Branges, a French-born mathematician now at Purdue University in the US, claimed a proof of the Riemann hypothesis. So far, his colleagues are not convinced. They were not convinced, years ago, when de Branges produced an answer to another famous mathematical challenge, but in time they accepted his reasoning. This time, the mathematical community remains even more sceptical.

“The proof he has announced is rather incomprehensible. Now mathematicians are less sure that the million has been won,” Prof du Sautoy said.

“The whole of e-commerce depends on prime numbers. I have described the primes as atoms: what mathematicians are missing is a kind of mathematical prime spectrometer. Chemists have a machine that, if you give it a molecule, will tell you the atoms that it is built from.

Mathematicians haven’t invented a mathematical version of this. That is what we are after. If the Riemann hypothesis is true, it won’t produce a prime number spectrometer. But the proof should give us more understanding of how the primes work, and therefore the proof might be translated into something that might produce this prime spectrometer. If it does, it will bring the whole of e-commerce to its knees, overnight. So there are very big implications.”

The PoincarÃ© conjecture depends on the almost mind-numbing problem of understanding the shapes of spaces: mathematicians call it topology. Bernhard Riemann and other 19th century scholars wrapped up the mathematical problems of two-dimensional surfaces of three dimensional objects – the leather around a football, for instance, or the distortions of a rubber sheet.

But Henri PoincarÃ© raised the awkward question of objects with three dimensions, existing in the fourth dimension of time. He had already done groundbreaking work in optics, thermodynamics, celestial mechanics, quantum theory and even special relativity and he almost anticipated Einstein. And then in 1904 he asked the most fundamental question of all: what is the shape of the space in which we live? It turned out to be possible to prove the PoincarÃ© conjecture in unimaginable worlds, where objects have four or five or more dimensions, but not with three.

“The one case that is really of interest because it connects with physics, is the one case where the PoincarÃ© conjecture hasn’t been solved,” said Keith Devlin, of Stanford University in California.

In 2002 a Russian mathematician called Grigori Perelman posted the first of a series of internet papers. He had worked in the US, and was known to American mathematicians before he returned to St Petersburg. His proof – he called it only a sketch of a proof – was very similar in some ways to that of Fermat’s last theorem, cracked by the Briton Andrew Wiles in the last decade.

Like Wiles, Perelman is claiming to have proved a much more complicated general problem and in the course of it may have solved a special one that has tantalised mathematicians for a century. But his papers made not a single reference to PoincarÃ© or his conjecture. Even so, mathematicians the world over understood that he tackled the essential challenge. Once again the jury is still out: this time, however, his fellow mathematicians believe he may be onto something big.

The plus: the multidimensional topology of space in three dimensions will seem simple at last and a million dollar reward will be there for the asking. The minus: the solver does not claim to have found a solution, he doesn’t want the reward, and he certainly doesn’t want to talk to the media.

“There is good reason to think the kind of approach Perelman is taking is correct. However there are some problems. He is very reclusive, won’t talk to the press, has shown no indication of publishing this as a paper, which you would have to do if you wanted to get the prize from the Clay Institute, and has shown no interest in the prize whatsoever,” Dr Devlin said.

“Has it been proved? We don’t know. We have good reason to assume it has been and within the next 12 months, in the inner core of experts in differential geometry, which is the field we are speaking about, people will start to say, yes, OK, this looks right. But there is not going to be a golden moment.”

The implications of a proof of the PoincarÃ© conjecture would be enormous, but like the problem itself, very difficult to explain, he said. “It can’t fail to have huge ramifications: not only the result, but the methods as well. At that level of abstraction, that level of connection, so much can follow. Differential geometry is the subject that is really underneath understanding everything about space and spacetime.”

Seven baffling pillars of wisdom

1 Birch and Swinnerton-Dyer conjecture Euclid geometry for the 21st century, involving things called abelian points and zeta functions and both finite and infinite answers to algebraic equations

2 PoincarÃ© conjecture The surface of an apple is simply connected. But the surface of a doughnut is not. How do you start from the idea of simple connectivity and then characterise space in three dimensions?

3 Navier-Stokes equation The answers to wave and breeze turbulence lie somewhere in the solutions to these equations

4 P vs NP problem Some problems are just too big: you can quickly check if an answer is right, but it might take the lifetime of a universe to solve it from scratch. Can you prove which questions are truly hard, which not?

5 Riemann hypothesis Involving zeta functions, and an assertion that all “interesting” solutions to an equation lie on a straight line. It seems to be true for the first 1,500 million solutions, but does that mean it is true for them all?

6 Hodge conjecture At the frontier of algebra and geometry, involving the technical problems of building shapes by “gluing” geometric blocks together

7 Yang-Mills and Mass gap A problem that involves quantum mechanics and elementary particles. Physicists know it, computers have simulated it but nobody has found a theory to explain it

*Call me naive, but I am not real worried about the end of the net, there will always be new developments and improvements in cryptography. As one develops one answer there will be a response, it will go back and forth.*

Stacey

Can’t wait to see how if these proofs are verified. How exciting, but now it means I only have 5 more chances for the $1,000,000.