The Riemann Hypothesis (RH) and Poincaré Conjecture are two of the seven Millennium Prize Problems, with significant implications for mathematics and beyond. A proof of RH would validate the work of mathematicians relying on its truth, potentially revolutionize calculations, and impact internet security due to its connection with prime numbers. The Poincaré Conjecture, which concerns the topology of three-dimensional spaces, has seen progress through Grigori Perelman's work, although his proof remains unverified. Both problems, if solved, promise profound insights into mathematics and the nature of spacetime.
PREREQUISITESMathematicians, theoretical physicists, and anyone interested in advanced mathematics, particularly those focused on topology, number theory, and the implications of solving major mathematical conjectures.
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 Poincaré 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 (£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 Poincaré 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 Poincaré 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 Poincaré 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 Poincaré 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 Poincaré 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 Poincaré 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."