Last week an eccentric Russian was tipped to win a 'Maths Nobel' for solving the fiendishly difficult Poincare Conjecture, one of seven major 'millennium' mathematical mysteries that have been earmarked for urgent solution. But these problems are just the tip of a scientific iceberg. Other fields are also beset by frustrating theoretical failures.
Here Robin McKie highlights the most baffling of those questions and suggests how science will one day provide those elusive answers.
Q Is there a God particle?
Next year, in a vast cavern deep below the Jura mountains on the French-Swiss border, Europe's nuclear research facility Cern will start operations at its £4bn Large Hadron Collider. Streams of sub-atomic particles will be fired at immense velocities in different directions round the collider's 17-mile circumference tunnel. Then these beams will be crossed over and smashed into each other. The purpose will be to re-create the colossal energies that existed during the universe's explosive birth, and from the nuclear debris thrown up in these nuclear collisions, to find proof of the existence of the Higgs boson, a strange, elusive entity that is sometimes known as the God particle.
The Higgs boson is the Holy Grail of particle physics. According to the man who dreamed up its existence, Edinburgh University theorist Peter Higgs, it latches on to other particles and gives them mass. It is bit like walking through mud. A particle flies through a field of Higgs bosons and some of them stick, giving that particle mass.
If it was not for the Higgs, nothing would have mass and the universe would be a decidedly odder place than it is now. Hence scientists urgent desire to pinpoint one from the nuclear debris they will create in their collider's tunnels next year.
Is there life on any other planet?
Fifty years ago, life was expected to be ubiquitous throughout our solar system and across our galaxy. Since then, Russian and American space probes, in combination with radio surveys of the sky, have uncovered not a single sign of life 'out there'. As a result, some scientists now say they suspect complex life forms may be extremely rare if not non-existent elsewhere in the cosmos. Earth may be unique and very special. Others point to other possible causes for our failure to find aliens. Technology, once unleashed, may invariably doom its creators, just as global warming threatens to trigger devastating climate changes on Earth, they argue.
Finding an extraterrestrial life form has therefore become an urgent philosophical and scientific quest and has involved scientists setting up three major projects. The first involves sending probes to nearby planets, such as Mars, in order to find simple micro-organisms that may exist there. Over the next 10 years, a series of increasingly complex missions aimed at finding water and life on the Red Planet will be launched by Europe and the US. A different technique, being pursued both by Nasa, and the European Space Agency, would involve sending flotillas of telescopes to pinpoint planets in orbit round distant stars. By exploiting advanced optical technologies, these telescopes could analyse the make-up of their atmospheres and pinpoint chemicals that would indicate the presence of living organisms.
And finally, scientists based at Seti - the Search for Extraterrestrial Intelligence - in America are building increasingly complex arrays of radio telescopes, linked to powerful computers, to pinpoint any faint alien radio beacon that might be beaming signals across space and which would reveal that we are not alone.
Is the number up for primes?
Some numbers have a special property. They cannot be expressed as the product of two smaller numbers. These are called prime numbers and include 3, 7, and 17. Prime numbers are used in codes that protect financial transactions between mainframe computers. However, mathematicians had great difficulty in finding a pattern to prime number distribution until, in the 19th century, GFB Riemann worked out an elaborate equation which suggested the existence of such a pattern. Proving that Riemann was right is now regarded as the greatest challenge in mathematics. Several academics claim to have succeeded but none have convinced their peers that they have succeeded. Most mathematicians expect a proof to be worked out in the near future, nevertheless.
Will the cosmos close with a crunch?
The universe is flying apart with its galaxies hurtling away from each other at colossal speeds. But will the cosmos expand forever with the stars slowly flickering out one by one leaving a cold, dead universe for eternity? Or will it slow down, halt and possibly collapse on itself? In other words, will it be a Heat Death or a Big Crunch? The answer, say scientists, depends on how much matter the universe possesses. If there were a lot, it would create enough gravity to pull things back together again, causing a collapse; not enough and everything would continue to fly apart. At present, scientists cannot agree about how much matter the universe possesses. Telescope studies suggest there is not enough and that a heat death awaits. But many astronomers believe there is an invisible form of 'dark matter', made up of as yet unknown class of sub-atomic particles which permeates the cosmos, unseen and undetected, and which could halt the universe's expansion. Searches for these particles are now planned at several research centres round the world.