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       1 ‘I Think I’ll Stop Here’

      Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

      G.H. Hardy

      23 June 1993, Cambridge

      It was the most important mathematics lecture of the century. Two hundred mathematicians were transfixed. Only a quarter of them fully understood the dense mixture of Greek symbols and algebra that covered the blackboard. The rest were there merely to witness what they hoped would be a truly historic occasion.

      The rumours had started the previous day. Electronic mail over the Internet had hinted that the lecture would culminate in a solution to Fermat’s Last Theorem, the world’s most famous mathematical problem. Such gossip was not uncommon. The subject of Fermat’s Last Theorem would often crop up over tea, and mathematicians would speculate as to who might be doing what. Sometimes mathematical mutterings in the senior common room would turn the speculation into rumours of a breakthrough, but nothing had ever materialised.

      This time the rumour was different. One Cambridge research student was so convinced that it was true that he dashed to the bookies to bet £10 that Fermat’s Last Theorem would be solved within the week. However, the bookie smelt a rat and refused to accept his wager. This was the fifth student to have approached him that day, all of them asking to place the identical bet. Fermat’s Last Theorem had baffled the greatest minds on the planet for over three centuries, but now even bookmakers were beginning to suspect that it was on the verge of being proved.

      The three blackboards became filled with calculations and the lecturer paused. The first board was erased and the algebra continued. Each line of mathematics appeared to be one tiny step closer to the solution, but after thirty minutes the lecturer had still not announced the proof. The professors crammed into the front rows waited eagerly for the conclusion. The students standing at the back looked to their seniors for hints of what the conclusion might be. Were they watching a complete proof to Fermat’s Last Theorem, or was the lecturer merely outlining an incomplete and anticlimactic argument?

      The lecturer was Andrew Wiles, a reserved Englishman who had emigrated to America in the 1980s and taken up a professorship at Princeton University where he had earned a reputation as one of the most talented mathematicians of his generation. However, in recent years he had almost vanished from the annual round of conferences and seminars, and colleagues had begun to assume that Wiles was finished. It is not unusual for brilliant young minds to burn out, a point noted by the mathematician Alfred Adler: ‘The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished.’

      ‘Young men should prove theorems, old men should write books,’ observed G.H. Hardy in his book A Mathematician’s Apology. ‘No mathematician should ever forget that mathematics, more than any other art or science, is a young man’s game. To take a simple illustration, the average age of election to the Royal Society is lowest in mathematics.’ His own most brilliant student Srinivasa Ramanujan was elected a Fellow of the Royal Society at the age of just thirty-one, having made a series of outstanding breakthroughs during his youth. Despite having received very little formal education in his home village of Kumbakonam in South India, Ramanujan was able to create theorems and solutions which had evaded mathematicians in the West. In mathematics the experience that comes with age seems less important than the intuition and daring of youth. When he posted his results to Hardy, the Cambridge professor was so impressed that he invited him to abandon his job as a lowly clerk in South India and attend Trinity College, where he could interact with some of the world’s foremost number theorists. Sadly the harsh East Anglian winters were too much for Ramanujan who contracted tuberculosis and died at the age of thirty-three.

      Other mathematicians have had equally brilliant but short careers. The nineteenth-century Norwegian Niels Henrik Abel made his greatest contribution to mathematics at the age of nineteen and died in poverty, just eight years later, also of tuberculosis. Charles Hermite said of him, ‘He has left mathematicians something to keep them busy for five hundred years’, and it is certainly true that Abel’s discoveries still have a profound influence on today’s number theorists. Abel’s equally gifted contemporary Evariste Galois also made his breakthroughs while still a teenager and then died aged just twenty-one.

      These examples are not intended to show that mathematicians die prematurely and tragically but rather that their most profound ideas are generally conceived while they are young, and as Hardy once said, ‘I do not know an instance of a major mathematical advance initiated by a man past fifty.’ Middle-aged mathematicians often fade into the background and occupy their remaining years teaching or administrating rather than researching. In the case of Andrew Wiles nothing could be further from the truth. Although he had reached the grand old age of forty he had spent the last seven years working in complete secrecy, attempting to solve the single greatest problem in mathematics. While others suspected he had dried up, Wiles was making fantastic progress, inventing new techniques and tools which he was now ready to reveal. His decision to work in absolute isolation was a high-risk strategy and one which was unheard of in the world of mathematics.

      Without inventions to patent, the mathematics department of any university is the least secretive of all. The community prides itself in an open and free exchange of ideas and tea-time breaks have evolved into daily rituals during which concepts are shared and explored over biscuits and Earl Grey. As a result it is increasingly common to find papers being published by co-authors or teams of mathematicians and consequently the glory is shared out equally. However, if Professor Wiles had genuinely discovered a complete and accurate proof of Fermat’s Last Theorem, then the most wanted prize in mathematics was his and his alone. The price he had to pay for his secrecy was that he had not previously discussed or tested any of his ideas with the mathematics community and therefore there was a significant chance that he had made some fundamental error.

      Ideally Wiles had wanted to spend more time going over his work to allow him to check fully his final manuscript. Then the unique opportunity arose to announce his discovery at the Isaac Newton Institute in Cambridge and he abandoned caution. The sole aim of the institute’s existence is to bring together the world’s greatest intellects for a few weeks in order to hold seminars on a cutting-edge research topic of their choice. Situated on the outskirts of the university, away from students and other distractions, the building is especially designed to encourage the academics to concentrate on collaboration and brainstorming. There are no dead-end corridors in which to hide and every office faces a central forum. The mathematicians are supposed to spend time in this open area, and are discouraged from keeping their office doors closed. Collaboration while moving around the institute is also encouraged – even the elevator, which only travels three floors, contains a blackboard. In fact every room in the building has at least one blackboard, including the bathrooms. On this occasion the seminars at the Newton Institute came under the heading of ‘L-functions and Arithmetic’. All the world’s top number theorists had been gathered together in order to discuss problems relating to this highly specialised area of pure mathematics, but only Wiles realised that L-functions might hold the key to solving Fermat’s Last

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