麻豆传媒

MATHS used to be a thankless pastime, the preserve of people considered eccentric recluses. It was never a route to fame or riches. But all that is changing. Hugely successful Hollywood films such as A Beautiful Mind and Good Will Hunting had mathematics at the very core of their plots. A book describing the solution of Fermat鈥檚 last theorem has become a surprise bestseller. Something in the air of the 21st century is turning mathematical research into a glamorous way to spend your life.

And there鈥檚 money and fame on offer to prove it. Millions of dollars are available to mathematicians who can solve some of the subject鈥檚 most baffling problems (see 鈥淓yes on the prize鈥�). And from next year there will be a mathematical equivalent of the Nobel Prize, in the form of the Abel Prize funded by the Norwegian government.

But the world鈥檚 sharpest minds have been attacking some of these problems for centuries, to no avail. Will this promise of cash and glory bring on any answers?

Well, it might. For a start, mathematicians have always liked a bet. The Hungarian mathematician Paul Erd枚s set hundreds, if not thousands, of problems in letters to colleagues and in published papers. The prizes on offer start at 25 cents, but they add up to around $25,000. Erd枚s died in 1996, but Ronald Graham, a mathematician at the University of California, San Diego, has undertaken to honour any of these commitments. And Cambridge University mathematicians Alex Selby and Oliver Riordan can testify that, even in maths, money talks: they spent six months solving the Eternity Puzzle, a 209-piece jigsaw from hell, and earned a million pounds for their trouble. They would never have applied their skills to the problem without the cash incentive (see 鈥淎nd the winner is鈥︹��).

Southampton University mathematician Martin Dunwoody hopes to get his million for a proof of the Poincar茅 conjecture that he posted on the Net earlier this year. His proof has to be peer reviewed, published, and then remain unchallenged for two years. Then and only then will the Clay Mathematics Institute in Cambridge, Massachusetts, hand over the cash. The institute, which aims to further the cause of maths across the globe, first offered this money two years ago. It set up a $7 million fund for the solutions to seven of the most difficult problems in maths. The prizes aren鈥檛 really about generating solutions. They are about raising the profile of mathematics and luring young people into the subject.

Nonetheless, solutions would be great for mathematics. Many of the Clay questions have been studied for generations, without success. Mathematicians make the effort not because the answers necessarily tell us anything essential, but because they are bothered that they don鈥檛 know how to find them. In pure mathematics the solution of easily stated but hard to solve problems often opens up new analytical methods. And those methods are frequently valuable elsewhere in mathematics and science.

The Poincar茅 conjecture is a good example. Formulated in 1904 by the French mathematician Jules Henri Poincar茅, it lies at the heart of a branch of mathematics known as topology. Topology is often referred to as 鈥渞ubber sheet geometry鈥�, because it deals with those properties of geometric shapes that remain valid if the shape is stretched, twisted, bent or subjected to any other any other transformation that does not introduce discontinuities, like tearing or cutting. Topology has become a cornerstone of modern mathematics, including mathematical physics, because it is about understanding the properties of space.

The Poincar茅 conjecture is concerned with something called a 3-sphere. To describe this strange object, it鈥檚 best to start with the 2-sphere: an ordinary sphere. You can make a 2-sphere by taking a disc and squashing the entire circumference down to a single point, rather like pulling on the drawstring of a duffel bag. If you start with a solid ball and squash its entire surface down to a single point with a sort of hyper-drawstring, you get a 3-sphere.

Now imagine a piece of string forming a closed loop inside the 3-sphere. By making the loop contract, you can shrink it until it vanishes. Poincar茅鈥檚 conjecture asserts that this 3-sphere is the only 3-dimensional space in which every closed loop can be continuously shrunk to a point. It doesn鈥檛 sound like much, but no one鈥檚 ever been able to prove it鈥檚 true. And it鈥檚 important to mathematicians because it represents a major stumbling block to their understanding of such hypersurfaces. A big part of topology is stumped as long as the Poincar茅 conjecture remains unproved.

The Poincar茅 conjecture does relate to some real world issues, though. Our Universe may well be a 3-sphere, so cosmologists would be interested in an understanding of this idea. And some other Clay Prize questions are definitely of practical concern. The Navier-Stokes problem, for example, is about solving the equations for fluid flow. In 1821 the French engineer Claude Navier, who was also an accomplished mathematician, began to write down equations for the flow of a viscous fluid. The derivation of these equations was later improved by the Irishman George Gabriel Stokes, and they are known as the Navier-Stokes equations.

Remember the Navier-Stokes equations next time you get on a aircraft, because they describe just the kind of airflow that keeps planes flying. Engineers designing jumbo jets and the space shuttle have to make do with approximate solutions to the equations, worked out by computers, because no one has ever found a formula that solves them exactly. And to get your hands on the Clay Institute鈥檚 cash, you don鈥檛 actually have to solve them either. The prize will go to the first person to prove that for fluids in 3-dimensional space, a solution to the Navier-Stokes equations exists that will never break down. Such 鈥渆xistence theorems鈥� are important to mathematics, because many methods for finding or approximating solutions start by assuming that a lasting solution exists. If it doesn鈥檛, the method might in principle give something that you thought was a solution, but actually is not.

There is a particularly compelling reason for being interested in the existence question. If the answer is no, then some fluid flows must develop singularities: that is, they must behave in some very strange way that invalidates the equations. It鈥檚 a sobering thought that, for all our prowess in mathematics and engineering, we don鈥檛 yet know whether fluids will always flow over aircraft wings in a way that we can predict.

Another problem that has some practical significance in the real world is the 鈥淧 = NP?鈥� problem. This is about the efficiency of computer algorithms, the mathematical skeletons from which computer programs are made. The crucial concept here is how the running time of an algorithm depends on the size of the input data.

A problem is class P if an algorithm to solve it has polynomial running time. In other words, its running time is proportional to some fixed power (such as the square or cube) of the size of the input data. Sorting a list of words into alphabetical order, for instance, is a class P problem.

If a problem is in class NP (short for nondeterministic polynomial) any guess at an answer can be checked in a time proportional to some fixed power of the size of input data. Take a jigsaw puzzle with lots of pieces, for example. Solving it is hard, but checking that someone has indeed solved it takes little more than a quick glance. Another, more rigorous, example of an NP problem is the fact that a guess at a prime factor of a large number can be quickly checked by doing a single division sum. Finding the prime factor may be very hard; checking it once found is easy.

Any P problem is automatically NP, but it doesn鈥檛 work the other way round. Many important problems are known to be NP but have no known P algorithm. An example is the travelling salesman problem. Given n cities, find the shortest route that visits each once only. This is NP because checking any 鈥渟olution鈥� is relatively easy, but the fastest known algorithm for solving this problem has exponential running time: the equations involve some fixed constant raised to the power of n. Even so, there鈥檚 no proof that, with sufficient ingenuity, you can鈥檛 design a P algorithm for it.

So the million-dollar P = NP? problem asks whether P and NP are actually the same. If an answer to a question can always be checked in polynomial time, can an answer always be found in polynomial time? If the answer is yes, then P does equal NP and we know something very important: there are no hard problems other than those that are 鈥渢rivially鈥� hard because of the quantity of data needed to output the answer. If P = NP then it would be possible to find fast, efficient algorithms for scheduling airline flights, or optimising factory output, for instance. If the answer is no, we鈥檒l have a cast-iron guarantee that all the apparently hard problems really are hard, and we will be able to stop wasting time trying to find fast algorithms for them. We win either way. What鈥檚 a nuisance is not knowing which way it goes. And that鈥檚 why there鈥檚 a million dollars for the answer.

Other Clay Prize problems are perhaps not so immediately practical, but they still manage to keep mathematicians awake at night. And sometimes physicists, too. Edward Witten, for instance, is a physicist at Caltech and the Princeton Institute for Advanced Study. He also sits on the Clay Institute鈥檚 scientific advisory board. Perhaps that鈥檚 why the Yang-Mills mass-gap hypothesis is among the open questions that could win a million bucks. This hypothesis asserts that, if a particle has mass, there is a lower limit to the mass it can have. It鈥檚 an idea that partly derives from Witten鈥檚 work in the 1970s.

An awful lot of physicists would like to see this problem solved. The mass-gap hypothesis is central to theoretical quantum physics because it would explain why mass is quantised. It鈥檚 also a mathematical issue because explaining the mass gap involves finding the equations that unite all the forces of nature. Once physicists have those they will essentially have their long-awaited 鈥渢heory of everything鈥�. If anyone wins the million dollars, the Nobel Prize for Physics will probably not be far behind.

Physicists would also love you to solve the Riemann hypothesis. Georg Bernhard Riemann was a German mathematician who lived in the mid 19th century. He wanted to be able to calculate, given a number, x, roughly how many primes there are with value equal to or less than x. Earlier mathematicians had found a good approximation: divide x by its natural logarithm. Riemann wanted a better understanding of this, and the search led him to what is now called the Riemann zeta function.

Sometimes the function鈥檚 value is zero. There are infinitely many such zeros of the zeta function, but Riemann noticed what appears to be a pattern. The Riemann hypothesis states that every one of these zeros, aside from a few trivial ones, lie on the 鈥渃ritical line鈥� of complex numbers with real part 1/2. Recently it has been proved that the Riemann hypothesis is true for the first 1.5 billion of the zeros, but there are infinitely many more, any one of which might just fall off the critical line. To pick up your prize, you have to find a way to show that they don鈥檛. In so doing, you may shed new light on fundamental aspects of physics. The spacing of zeros of the Riemann zeta function appears to correspond to the spacing of energy levels in complex quantum mechanical systems such as atomic nuclei. Understanding the maths behind this could help us understand why the quantum world acts so strangely.

If a zeta function sounds a bit too abstract for your taste, don鈥檛 even think about another two Clay problems, the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture. They belong to a field called algebraic geometry, and just describing them in a way that鈥檚 intelligible to anyone but hardcore mathematicians is a headache.

The Hodge Conjecture states, roughly speaking, that on any smooth multidimensional surface defined by algebraic equations, every solution of the Laplacian partial differential equation鈥攁n important equation in mathematical physics鈥攊s equivalent to one that can be obtained by a purely algebraic procedure (I told you it was hard). The Birch and Swinnerton-Dyer Conjecture is about integer solutions of a type of equation called an elliptic curve. Fiendishly difficult as these problems certainly are, mathematicians consider them worth solving because solutions would enable them to make big progress in other areas.

Maybe the Clay money will get mathematicians the answers they want, but there鈥檚 no guarantee. In 2000 the publisher Faber & Faber offered a million-dollar prize as an incentive to solve a very simple-looking mathematical problem that some of the brightest minds in mathematics have been trying to prove or disprove for more than 250 years. It began in 1742, when Christian Goldbach noticed that every even integer larger than 2 appears to be the sum of two primes. A prime number is one that has no exact integer divisors, except itself and 1. As a convention, mathematicians exclude 1 from the list, so it begins 2, 3, 5, 7, 11, 13, 17, 19, 23. Goldbach noticed that if you add pairs of primes together, you seem to be able to make all the even numbers. For instance, 2 + 2 = 4, 3 + 3 = 6, 3 + 5 = 8, 3 + 7 = 10, 5 + 7 = 12, and so on.

We still have no idea whether the Goldbach conjecture holds for all even numbers. The best result so far states that every even integer (apart from 2) is either the sum of two primes, or it is the sum of a prime and an 鈥渁lmost prime鈥�: a number that can be obtained by multiplying two primes together. This was proved by Jing-Run Chen in 1973. Anyone who had been able to get rid of that 鈥渁lmost鈥� by the end of March could have had Faber鈥檚 million dollars. But no one did, and perhaps that shouldn鈥檛 be a surprise: a two-year window of opportunity is hardly a fair chance when 250 years have failed to yield a solution.

But while the prize hasn鈥檛 elicited the answer it was looking for, it does seem to have revealed something about mathematicians. For many of them money seems to be of far less interest than the problems themselves. Andrew Wiles famously proved Fermat鈥檚 last theorem in 1995 because he wanted to. He shut himself away and dedicated his life to the problem for seven years with absolutely no financial incentive. And if you want any more insight into the mind of most mathematicians, look no further than the fate of the Erd枚s prizes awarded so far. Ron Graham has made out a number of cheques, but few of them have ever been cashed. In most cases, the triumphant mathematicians frame the cheques and stick them on the wall.

You can show mathematicians the money, you can even give it to them. But you can鈥檛 make them spend it. Maths and mammon seldom mix.