Fermat's Last Theorem (36 page)

‘On 23 June Andrew began his third and final lecture,' recalls John Coates. ‘What was remarkable was that practically everyone who contributed to the ideas behind the proof was there in the room, Mazur, Ribet, Kolyvagin, and many, many others.'

By this point the rumour was so persistent that everyone from the Cambridge mathematics community turned up for the final lecture. The lucky ones were crammed into the auditorium, while the others had to wait in the corridor, where they stood on tip-toe and peered through the window. Ken Ribet had made sure that he would not miss out on the most important mathematical announcement of the century: ‘I came relatively early and I sat in the front row along with Barry Mazur. I had my camera with me just to record the event. There was a very charged atmosphere and people were very excited. We certainly had the sense that we were participating in a historical moment. People had grins on their faces before and during the lecture. The tension had built up over the course of several days. Then there was this marvellous moment when we were coming close to a proof of Fermat's Last Theorem.'

Barry Mazur had already been given a copy of the proof by Wiles, but even he was astonished by the performance. ‘I've never seen such a glorious lecture, full of such wonderful ideas, with such dramatic tension, and what a build-up. There was only one possible punch line.'

After seven years of intense effort Wiles was about to announce his proof to the world. Curiously Wiles cannot remember the final moments of the lecture in great detail, but does recall the atmosphere: ‘Although the press had already got wind of the lecture, fortunately they were not at the lecture. But there were plenty of people in the audience who were taking photographs towards the end and the Director of the Institute certainly had come well prepared with a bottle of champagne. There was a typical dignified
silence while I read out the proof and then I just wrote up the statement of Fermat's Last Theorem. I said, “I think I'll stop here”, and then there was sustained applause.'

Strangely, Wiles was ambivalent about the lecture: ‘It was obviously a great occasion, but I had mixed feelings. This had been part of me for seven years: it had been my whole working life. I got so wrapped up in the problem that I really felt I had it all to myself, but now I was letting go. There was a feeling that I was giving up a part of me.'

Wiles's colleague Ken Ribet had no such qualms: ‘It was a completely remarkable event. I mean, you go to a conference and there are some routine lectures, there are some good lectures and there are some very special lectures, but it's only once in a lifetime that you get a lecture where someone claims to solve a problem that has endured for 350 years. People were looking at each other and saying, “My God, you know we've just witnessed an historical event.” Then people asked a few questions about technicalities of the proof and possible applications to other equations, and then there was more silence and all of a sudden a second round of applause. The next talk was given by one Ken Ribet, yours truly. I gave the lecture, people took notes, people applauded, and no one present, including me, has any idea what I said in that lecture.'

While mathematicians were spreading the good news via e-mail, the rest of the world had to wait for the evening news, or the following day's newspapers. TV crews and science reporters descended upon the Newton Institute, all demanding interviews with the ‘greatest mathematician of the century'. The
Guardian
exclaimed, ‘The Number's Up for Maths' Last Riddle', and the front page of
Le Monde
read, ‘Le théorèm de Fermat enfin résolu'. Journalists everywhere asked mathematicians for their expert opinion on Wiles's work, and professors, still recovering from the shock, were expected to briefly explain the most complicated mathematical proof ever, or provide a soundbite which would clarify the Taniyama–Shimura conjecture.

The first time Professor Shimura heard about the proof of his own conjecture was when he read the front page of the
New York Times —
‘At Last, Shout of “Eureka!” In Age-Old Math Mystery'. Thirty-five years after his friend Yutaka Taniyama had committed suicide, the conjecture which they had created together had now been vindicated. For many professional mathematicians the proof of the Taniyama–Shimura conjecture was a far more important achievement than the solution of Fermat's Last Theorem, because it had immense consequences for many other mathematical theorems. The journalists covering the story tended to concentrate on Fermat and mentioned Taniyama–Shimura only in passing, if at all.

Shimura, a modest and gentle man, was not unduly bothered by the lack of attention given to his role in the proof of Fermat's Last Theorem, but he was concerned that he and Taniyama had been relegated from being nouns to adjectives. ‘It is very curious that people write about the Taniyama–Shimura conjecture, but nobody writes about Taniyama and Shimura.'

This was the first time that mathematics had hit the headlines since Yoichi Miyaoka announced his so-called proof in 1988: the only difference this time was that there was twice as much coverage and nobody expressed any doubt over the calculation. Overnight Wiles became the most famous, in fact the only famous, mathematician in the world, and
People
magazine even listed him among ‘The 25 most intriguing people of the year', along with Princess Diana and Oprah Winfrey. The ultimate accolade came from an international clothing chain who asked the mild-mannered genius to endorse their new range of menswear.

While the media circus continued and while mathematicians made the most of being in the spotlight, the serious work of checking the proof was under way. As with all scientific disciplines each new piece of work has to be thoroughly examined, before it could be accepted as accurate and correct. Wiles's proof had to be submitted to the ordeal of trial by referee. Although Wiles's lectures at the Isaac Newton Institute had provided the world with an outline of his calculation, this did not qualify as official peer review. Academic protocol demands that any mathematician submits a complete manuscript to a respected journal, the editor of which then sends it to a team of referees whose job it is to examine the proof line by line. Wiles had to spend the summer anxiously waiting for the referees' report, hoping that eventually he would get their blessing.

A problem worthy of attack
Proves its worth by fighting back.

Piet Hein

As soon as the Cambridge lecture was over, the Wolfskehl committee was informed of Wiles's proof. They could not award the prize immediately because the rules of the contest clearly demand verification by other mathematicians and official publication of the proof:

The
Königliche Gesellschaft der Wissenschaften
in Göttingen … will only take into consideration those mathematical memoirs which have appeared in the form of a monograph in the periodicals, or which are for sale in the bookshops … The award of the Prize by the Society will take place not earlier than two years after the publication of the memoir to be crowned. The interval of time is intended to allow German and foreign mathematicians to voice their opinion about the validity of the solution published.

Wiles submitted his manuscript to the journal
Inventiones Mathematicae
, whereupon its editor Barry Mazur began the process of selecting the referees. Wiles's paper involved such a variety of mathematical techniques, both ancient and modern, that Mazur made the exceptional decision to appoint not just two or three
referees, as is usual, but six. Each year thirty thousand papers are published in journals around the world, but the sheer size and importance of Wiles's manuscript meant that it would undergo a unique level of scrutiny. To simplify matters the 200-page proof was divided into six sections and each of the referees took responsibility for one of these chapters.

Chapter 3
was the responsibility of Nick Katz, who had already examined that part of Wiles's proof earlier in the year: ‘I happened to be in Paris for the summer to work at the Institut des Hautes Etudes Scientifiques, and I took with me the complete 200-page proof – my particular chapter was seventy pages long. When I got there I decided I wanted to have serious technical help, and so I insisted that Luc Illusie, who was also in Paris, become a joint referee on this chapter. We would meet a few times a week throughout that summer, basically lecturing to each other to try and understand this chapter. Literally we did nothing but look through this manuscript line by line to try and make sure that there were no mistakes. Sometimes we got confused by things and so every day, sometimes twice a day, I would e-mail Andrew with a question – I don't understand what you say on this page or it seems to be wrong on this line. Typically I would get a response that day or the next day which clarified the matter and then we'd go on to the next problem.'

The proof was a gigantic argument, intricately constructed from hundreds of mathematical calculations glued together by thousands of logical links. If just one of the calculations was flawed or if one of the links became unstuck then the entire proof was potentially worthless. Wiles, who was now back in Princeton, anxiously waited for the referees to complete their task. ‘I don't like to celebrate full out until I have the paper completely off my hands. In the meantime I had my work cut out dealing with the questions I was
getting via e-mail from the referees. I was still pretty confident that none of these questions would cause me much trouble.' He had already checked and double-checked the proof before releasing it to the referees, so he was expecting little more than the mathematical equivalent of grammatical or typographic errors, trivial mistakes which he could fix immediately.

‘These questions continued relatively uneventfully through till August,' recalls Katz, ‘until I got to what seemed like just one more little problem. Sometime around 23 August I e-mail Andrew, but it's a little bit complicated so he sends me back a fax. But the fax doesn't seem to answer the question so I e-mail him again and I get another fax which I'm still not satisfied with.'

Wiles had assumed that this error was as shallow as all the others, but Katz's persistence forced him to take it seriously: ‘I couldn't immediately resolve this one very innocent looking question. For a little while it seemed to be of the same order as the other problems, but then sometime in September I began to realise that this wasn't just a minor difficulty but a fundamental flaw. It was an error in a crucial part of the argument involving the Kolyvagin–Flach method, but it was something so subtle that I'd missed it completely until that point. The error is so abstract that it can't really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail.'

In essence the problem was that there was no guarantee that the Kolyvagin–Flach method worked as Wiles had intended. It was supposed to extend the proof from the first element of all elliptic equations and modular forms to cover all the elements, providing the toppling mechanism from one domino to the next. Originally the Kolyvagin–Flach method only worked under particularly
constrained circumstances, but Wiles believed he had adapted and strengthened it sufficiently to work for all his needs. According to Katz this was not necessarily the case, and the effects were dramatic and devastating.

The error did not necessarily mean that Wiles's work was beyond salvation, but it did mean that he would have to strengthen his proof. The absolutism of mathematics demanded that Wiles demonstrate beyond all doubt that his method worked for every element of every
E
-series and
M
-series.

When Katz realised the significance of the error which he had spotted, he began to ask himself how he had missed it in the spring when Wiles had lectured to him with the sole purpose of identifying any mistakes. ‘I think the answer is that there's a real tension when you're listening to a lecture between understanding everything and letting the lecturer get on with it. If you interrupt every second – I don't understand this or I don't understand that – then the guy never gets to explain anything and you don't get anywhere. On the other hand if you never interrupt you just sort of get lost and you're nodding your head politely, but you're not really checking anything. There's this real tension between asking too many questions and asking too few, and obviously by the end of those lectures, which is where this problem slipped through, I had erred on the side of too few questions.'

Only a few weeks earlier, newspapers around the globe had dubbed Wiles the most brilliant mathematician in the world, and after 350 years of frustration number theorists believed that they had at last got the better of Pierre de Fermat. Now Wiles was faced
with the humiliation of having to admit that he had made a mistake. Before confessing to the error he decided to try and make a concerted effort to fill in the gap. ‘I couldn't give up. I was obsessed by this problem and I still believed that the Kolyvagin–Flach method just needed a little tinkering. I just needed to modify it in some small way and then it would work just fine. I decided to go straight back into my old mode and completely shut myself off from the outside world. I had to focus again but this time under much more difficult circumstances. For a long time I would think that the fix was just round the corner, that I was just missing something simple and it would all just fit into place the next day. Of course it could have happened that way, but as time went by it seemed that the problem just became more intransigent.'

The hope was that he could fix the mistake before the mathematical community was aware that a mistake even existed. Wiles's wife, who had already witnessed the seven years of effort that had gone into the original proof, now had to watch her husband's agonising struggle with an error that could destroy everything. Wiles remembers her optimism: ‘In September Nada said to me that the only thing she wanted for her birthday was a correct proof. Her birthday is on 6 October. I had only two weeks to deliver the proof, and I failed.'