Why a $1m prize doesn't add up to a hill of beans for an award-winning mathematician

Mathematicians are a notorious bunch for toiling in self-imposed obscurity: Grigori Perelman is just the latest example.

John Nash, the mathematician and Nobel laureate for his work in game theory, suffered with schizophrenia for about 40 years.

Most people would snap up US$1 million (Dh3.67m) for doing something that has already brought them worldwide acclaim. Not Grigori Perelman; last week he declared that he has no interest in the money or the fame that accompanied his achievement.

But then, Dr Perelman is not most people. He is a 43-year old recluse, lives in a tiny flat in St Petersburg, and is credited with being the first to solve one of the notorious Millennium Prize problems, mind-bendingly difficult mathematical conundrums whose resolution each carries a million-dollar reward. Dr Perelman has been in line for the prize since 2006, when mathematicians decided they were convinced by his proof of the Poincaré Conjecture. Put (very) simply, this states that there is an infallible way of proving that any given surface is spherical.

More than a century ago, the French mathematician Henri Poincaré believed that he had found a way, but could not prove it. So it remained a conjecture - maths-speak for an educated guess - until 2002, when Dr Perelman posted the first of a set of papers on the web outlining a full-blown proof. It took four years to convince other mathematicians, and another four for the Clay Institute, which administers the Millennium Prize, to offer the prize to Dr Perelman. Few who knew him were surprised by his announcement last week: he had previously turned down the Fields Medal, the "Nobel Prize" of mathematics.

There is nothing particularly new about a brilliant person rejecting wordly baubles. Jean-Paul Sartre refused to accept the Nobel Prize for Literature, while Marlon Brando rejected his Best Actor Oscar in 1973, sending a Native American activist along to the ceremony to explain his loathing of Hollywood. But brilliant mathematicians do have a reputation for being especially other-worldly. Take the case of Paul Erdos, widely regarded as one of the greatest mathematicians of the 20th century. He spent most of his life homeless, carrying his worldly possessions in two battered hold-alls. Mathematicians anywhere in the world could find a short, wizened figure standing on their doorstep, declaring, "My brain is open" - meaning he was looking to collaborate for a while. Once installed, Dr Erdos would take some Benzedrine, and then expect his collaborator to work with him for days on end, until the problem was cracked or he got bored - in which case he would pack his bags and head off again, muttering his life-long motto: "Another roof, another proof."

At the other extreme was (or, possibly, is - it is not entirely clear) the French mathematician Alexander Grothendieck. Born in Berlin in 1928, Dr Grothendieck was already carrying out world-class work in algebraic geometry in his early 20s, and created whole new fields of research that unified different branches of mathematics. In 1966 he won the Fields Medal, after which his life started to take an increasingly bizarre track.

Appalled by the Vietnam War, in 1970 he quit a prestigious professorship having learned that his institute was partly funded by military sources. He eventually returned to academia, but gained a reputation for being keener to lecture students on ecology and pacificism than maths. Even so, Dr Grothendieck's achievements led in 1988 to his being awarded the Crafoord Prize, which is even more valuable than the Nobel. He rejected it and not long afterwards burned thousands of pages of his manuscripts and then vanished. His whereabouts remain unknown; according to some reports he still lives as a hermit somewhere in southern France or the Spanish Pyrenées.

Many suspect Dr Grothendieck may have lost his mind. If true, he will have joined such illustrious figures as Georg Cantor, the 19th-century German mathematician driven to insanity by his pioneering work on infinity, and John Nash, the Nobel Prize-winning pioneer of game theory whose battle with schizophrenia featured in the film A Beautiful Mind. Why are so many brilliant mathematicians so close to madness? Even a glance at the six remaining Millennium Prize problems gives some inkling. Who in their right mind would want to devote themselves to tackling, say, the Hodge Conjecture, which requires a proof that "every harmonic differential form on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles".

One person who might once have been able to explain it is Dr Grothendieck. In 1969 he published a paper titled "Hodge's general conjecture is false for trivial reasons" - though you can be sure they are anything but trivial. Perhaps he resolved the Millennium Prize version of the conjecture long ago, but like Dr Perelman simply is not interested in either the fame or the money offered by the Clay Institute.

In truth, the world is unlikely to change much if anyone does come up with a proof of the Hodge Conjecture. It belongs to the stratospheric realms of advanced mathematics where even tiny steps require monumental effort. It takes a very unusual type of person to spend long periods of time in such places. But there is one Millennium Prize problem which is much more down to earth: the enigmatically named P versus NP Conjecture. This centres on a question of everyday importance: are we missing nifty ways of tackling a host of tough tasks, from airline scheduling to designing microchips?

In the 1960s mathematicians found a surprising link between such tasks, known technically as non-deterministic polynomial (NP) time complete problems. Put (as ever) very simply, such problems are like jigsaw puzzles: they are time-consuming to solve, but once done, very easily checked. They also have something else in common: if a quick way to solve just one NP complete problem can be found, there must also be quick ways of solving all the myriad of other such problems.

So the $1 million question is: does any such quick (in maths-speak, "polynomial time") solution exist for any NP problems? Or, in the terminology of the Millennium Prize, does P = NP? Most mathematicians think not: they think that problems like designing timetables or finding the shortest route connecting different towns will always be time-consuming. Of course, they could be wrong. But if history is any guide, anyone who tries to settle the matter should first do a careful calculation: is it worth risking one's sanity for $1 million ?

Robert Matthews is Visiting Reader in Science at Aston University, Birmingham, England