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.’
For Nick Katz, too, this was a tense period: ‘By October the only people who knew about the error were myself, Illusie, the other referees of other chapters and Andrew – in principle that was all. My attitude was that as a referee I was supposed to act in confidentiality. I certainly didn’t feel that it was my business to discuss this matter with anyone except Andrew, so I just didn’t say a word about it. I think externally he appeared normal but at this point he was keeping a secret from the world, and I think he must have been pretty uncomfortable about it. Andrew’s attitude was that with just another day he would solve it, but as the fall went on, and no manuscript was available, rumours started circulating that there was a problem.’
In particular, Ken Ribet, another of the referees, began to feel the pressure of keeping the secret: ‘For some completely accidental reason I became known as the “Fermat Information Service”. There was an initial article in the New York Times, where Andrew asked me to speak to the reporter in his place, and the article said, ‘Ribet who is acting as a spokesperson for Andrew Wiles …’, or something to that effect. After that I became a magnet for all kinds of interest in Fermat’s Last Theorem, both from inside and outside the mathematics community. People were calling from the press, from all around the world in fact, and also I gave a very large number of lectures over a period of two or three months. In these lectures I stressed what a magnificent achievement this was and I outlined the proof and I talked about the parts that I knew best, but after a while people started getting impatient and began asking awkward questions.
‘You see Wiles had made this very public announcement, but no one outside of the very small group of referees had seen a copy of the manuscript. So mathematicians were waiting for this manuscript that Andrew had promised a few weeks after the initial announcement in June. People said, “Okay, well this theorem has been announced – we’d like to see what’s going on. What’s he doing? Why don’t we hear anything?” People were a little upset that they were being held in ignorance and they simply wanted to know what was going on. Then things got even worse because slowly this cloud gathered over the proof and people kept telling me about these rumours, which claimed there was a gap in chapter 3. They’d ask me what I knew about it, and I just didn’t know what to say.’
With Wiles and the referees denying any knowledge of a gap, or at the very least refusing to comment, speculation began to run wild. In desperation mathematicians began sending e-mails to each other in the hope of getting to the bottom of the mystery.
In every tea-room of every mathematics department the gossip surrounding Wiles’s proof escalated every day. In response to the rumours and the speculative e-mails some mathematicians tried to return a sense of calm to the community.
Despite the calls for calm, the e-mails continued unabated. As well as discussing the putative error, mathematicians were now arguing over the ethics of pre-empting the referees’ announcement.
While the furore over his elusive proof was increasing, Wiles did his best to ignore the controversy and speculation. ‘I really shut myself off because I didn’t want to know what people were saying about me. I just went into seclusion but periodically my colleague Peter Sarnak would say to me, “You know that there’s a storm out there.” I’d listen, but, for myself, I really just wanted to cut myself off completely, just to focus completely on the problem.’
Peter Sarnak had joined the Princeton Mathematics Department at the same time as Wiles, and over the years they had become close friends. During this intense period of turmoil Sarnak was one of the few people in whom Wiles would confide. ‘Well, I never knew the exact details, but it was clear that he was trying to overcome this one serious issue. But every time he would fix this one part of the calculation, it would cause some other difficulty in another part of the proof. It was like he was trying to put a carpet in a room where the carpet might be bigger than the room. So Andrew could fit the carpet in any one corner, only to find that it would pop up in another corner. Whether you could or could not fit the carpet in the room was not something he was able to decide. Mind you, even with the error, Andrew had made a giant step. Before him there was no one who had any approach to the Taniyama–Shimura conjecture, but now everybody got really excited because he showed us so many new ideas. They were fundamental, new things that nobody had considered before. So even if it couldn’t be fixed this was a very major advance – but of course Fermat would still be unsolved.’
Eventually Wiles realised that he could not maintain his silence forever. The solution to the mistake was not just round the corner, and it was time to put an end to the speculation. After an autumn of dismal failure he sent the following e-mail to the mathematical bulletin board:
Few were convinced by Wiles’s optimism. Almost six months had passed without the error being corrected, and there was no reason to think anything would change in the next six months. In any case, if he really could ‘finish this in the near future’, then why bother issuing the e-mail? Why not just maintain the silence for a few more weeks and then release the finished manuscript? The February lecture course which he mentioned in his e-mail failed to give any of the promised detail, and the mathematical community suspected that Wiles was just trying to buy himself extra time.
The newspapers leapt on the story once again and mathematicians were reminded of Miyaoka’s failed proof in 1988. History was repeating itself. Number theorists were now waiting for the next e-mail which would explain why the proof was irretrievably flawed. A handful of mathematicians had expressed doubts over the proof back in the summer, and now their pessimism seemed to have been justified. One story claims that Professor Alan Baker at the University of Cambridge offered to bet one hundred bottles of wine against a single bottle that the proof would be shown to be invalid within a year. Baker denies the anecdote, but proudly admits to having expressed a ‘healthy scepticism’.
Less than six months after his lecture at the Newton Institute Wiles’s proof was in tatters. The pleasure, passion and hope that carried him through the years of secret calculations were replaced with embarrassment and despair. He recalls how his childhood dream had become a nightmare: ‘The first seven years that I worked on this problem I enjoyed the private combat. No matter how hard it had been, no matter how insurmountable things seemed, I was engaged in my favourite problem. It was my childhood passion, I just couldn’t put it down, I didn’t want to leave it for a moment. Then I’d spoken about it publicly, and in speaking about it there was actually a certain sense of loss. It was a very mixed emotion. It was wonderful to see other people reacting to the proof, to see how the arguments could completely change the whole direction of mathematics, but at the same time I’d lost that personal quest. It was now open to the world and I no longer had this private dream which I was fulfilling. And then, after there was a problem with it, there were dozens, hundreds, thousands of people who wanted to distract me. Doing maths in that kind of rather overexposed way is certainly not my style and I didn’t at all enjoy this very public way of doing it.’
Number theorists around the world empathised with Wiles’s position. Ken Ribet had himself been through the same nightmare eight years earlier when he tried to prove the link between the Taniyama–Shimura conjecture and Fermat’s Last Theorem. ‘I was giving a lecture about the proof at the Mathematical Sciences Research Institute in Berkeley and someone from the audience said, “Well, wait a minute, how do you know that such and such is true?” I responded immediately giving my reason and they said, “Well that doesn’t apply in this situation.” I had an immediate terror. I kind of broke out into a sweat and I was very upset about it. Then I realised that there was only one possibility for justifying this, which was to go back to the fundamental work on the subject and see exactly how it was done in a similar situation. I looked in the relevant paper and I saw that the method did indeed apply in my case, and within a day or two I had the thing all fixed up. In my next lecture I was able to give the justification. But you always live with this fear that if you announce something important, a fundamental mistake can be discovered.
‘When you find an error in a manuscript it can go two ways. Sometimes there’s an immediate confidence and the proof can be resurrected with little difficulty. And sometimes there’s the opposite. It’s very disquieting, there’s a sinking feeling when you realise that you’ve made a fundamental error and there’s no way to repair it. It’s possible that when a hole develops the theorem really just falls apart completely, because the more you try to patch it the more trouble you get into. But in Wiles’s case each chapter of the proof was a significant article in its own right. The manuscript was seven years’ work, it was basically several important papers pieced together and each one of the papers has a great deal of interest. The error occurred in one of the papers, in chapter 3, but even if you take out chapter 3 what remained was absolutely wonderful.’
But without chapter 3 there was no proof of the Taniyama–Shimura conjecture and therefore no proof of Fermat’s Last Theorem. There was a sense of frustration in the mathematical community that the proof behind two great problems was in jeopardy. Moreover, after six months of waiting still nobody, beyond Wiles and the referees, had access to the manuscript. There was a growing clamour for more openness, so everyone could see for themselves the details of the error. The hope was that somebody somewhere might see something that Wiles had missed, and conjure up a calculation to fix the gap in the proof. Some mathematicians claimed that the proof was too valuable to be left in the hands of just one man. Number theorists had become the butt of jibes from other mathematicians, who sarcastically questioned whether or not they understood the concept of proof. What should have been the proudest moment in the history of mathematics was turning into a joke.
Despite the pressure Wiles refused to release the manuscript. After seven years of devoted effort he was not ready to sit back and watch someone else complete the proof and steal the glory. The person who proves Fermat’s Last Theorem is not the person that puts in the most work, it’s the person who delivers the final and complete proof. Wiles knew that once the manuscript was published in its flawed state he would immediately be swamped by questions and demands for clarification from would-be gap-fixers, and these distractions would destroy his own hopes of mending the proof while giving others vital clues.
Wiles attempted to return to the same state of isolation which had allowed him to create the original proof, and reverted to his habit of studying intensely in his attic. Occasionally he would wander down by the Princeton lake, as he had done in the past. The joggers, cyclists and rowers who had previously passed him by with a brief wave now stopped and asked him whether there was any progress with the gap. Wiles had appeared on front pages around the world, he had been featured in People magazine and he had even been interviewed on CNN. The previous summer Wiles had become the world’s first mathematical celebrity, and already his image was tarnished.
Meanwhile back in the mathematics department the gossip continued. Princeton mathematician Professor John H. Conway remembers the atmosphere in the department’s tea-room: ‘We’d gather for tea at 3 o’clock and make a rush for the cookies. Sometimes we’d discuss mathematical problems, sometimes we’d discuss the O.J. Simpson trial, and sometimes we’d discuss Andrew’s progress. Because nobody actually liked to come out and ask him how he’s getting on with the proof, we were behaving a little bit like Kremlinologists. So somebody would say: “I saw Andrew this morning” – “Did he smile?” – “Well, yes, but he didn’t look too happy.” We could only gauge his feelings by his face.’
As winter deepened, hopes of a breakthrough faded, and more mathematicians argued that it was Wiles’s duty to release the manuscript. The rumours continued and one newspaper article claimed that Wiles had given up and that the proof had irrevocably collapsed. Although this was an exaggeration, it was certainly true that Wiles had exhausted dozens of approaches which might have circumvented the error and he could see no other potential routes to a solution.
Wiles admitted to Peter Sarnak that the situation was getting desperate and that he was on the point of accepting defeat. Sarnak suggested that part of the difficulty was that Wiles had nobody he could trust on a day-to-day basis; there was nobody he could bounce ideas off or who could inspire him to explore more lateral approaches. He suggested that Wiles took somebody into his confidence and try once more to fill the gap. Wiles needed somebody who was an expert in manipulating the Kolyvagm—Flach method and who could also keep the details of the problem secret. After giving the matter prolonged thought, he decided to invite Richard Taylor, a Cambridge lecturer, to Princeton to work alongside him.
Taylor was one of the referees responsible for verifying the proof and a former student of Wiles, and as such he could be doubly trusted. The previous year he had been in the audience at the Isaac Newton Institute watching his former supervisor present the proof of the century. Now it was his job to help rescue the flawed proof.
By January Wiles, with the help of Taylor, was once again tirelessly exploring the Kolyvagin–Flach method, trying to find a way out of the problem. Occasionally after days of effort they would enter new territory, but inevitably they would find themselves back where they started. Having ventured further than ever before and failing over and over again, they both realised that they were in the heart of an unimaginably vast labyrinth. Their deepest fear was that the labyrinth was infinite and without exit, and that they would be doomed to wander aimlessly and endlessly.
Then in the spring of 1994, just when it looked as though things could not get any worse, the following e-mail hit computer screens around the world:
Noam Elkies was the Harvard professor who back in 1988 had found a counter-example to Euler’s conjecture, thereby proving that it was false:
Now he had apparently discovered a counter-example to Fermat’s Last Theorem, proving that it too was false. This was a tragic blow for Wiles – the reason he could not fix the proof was that the so-called error was a direct result of the falsity of the Last Theorem. It was an even greater blow for the mathematical community at large, because if Fermat’s Last Theorem was false, then Frey had already shown that this would lead to an elliptic equation which was not modular, a direct contradiction to the Taniyama-Shimura conjecture. Elkies had not only found a counter-example to Fermat, he had indirectly found a counter-example to Taniyama–Shimura.
The death of the Taniyama–Shimura conjecture would have devastating repercussions throughout number theory, because for two decades mathematicians had tacitly assumed its truth. In Chapter 5 it was explained that mathematicians had written dozens of proofs which began with ‘Assuming the Taniyama–Shimura conjecture’, but now Elkies had shown that this assumption was wrong and all those proofs had simultaneously collapsed. Mathematicians immediately began to demand more information and bombarded Elkies with questions, but there was no response and no explanation as to why he was remaining tight-lipped. Nobody could even find the exact details of the counter-example.
After one or two days of turmoil some mathematicians took a second look at the e-mail and began to realise that, although it was typically dated 2 April or 3 April, this was a result of having received it second or third hand. The original message was dated 1 April. The e-mail was a mischievous hoax perpetrated by the Canadian number theorist Henri Darmon. The rogue e-mail served as a suitable lesson for the Fermat rumour-mongers, and for a while the Last Theorem, Wiles, Taylor and the damaged proof were left in peace.
That summer Wiles and Taylor made no progress. After eight years of unbroken effort and a lifetime’s obsession Wiles was prepared to admit defeat. He told Taylor that he could see no point in continuing with their attempts to fix the proof. Taylor had already planned to spend September in Princeton before returning to Cambridge, and so despite Wiles’s despondency, he suggested they persevere for one more month. If there was no sign of a fix by the end of September, then they would give up, publicly acknowledge their failure and publish the flawed proof to allow others an opportunity to examine it.
Although Wiles’s battle with the world’s hardest mathematical problem seemed doomed to end in failure, he could look back at the last seven years and be reassured by the knowledge that the bulk of his work was still valid. To begin with Wiles’s use of Galois groups had given everybody a new insight into the problem. He had shown that the first element of every elliptic equation could be paired with the first element of a modular form. Then the challenge was to show that if one element of the elliptic equation was modular, then so must the next piece be modular, and so must they all be modular.
During the middle years Wiles wrestled with the concept of extending the proof. He was trying to complete an inductive approach and had wrestled with Iwasawa theory in the hope that this would demonstrate that if one domino fell then they all would. Initially Iwasawa theory seemed powerful enough to cause the required domino effect but in the end it could not quite live up to his expectation. He had devoted two years of effort to a mathematical dead end.
In the summer of 1991, after a year in the doldrums, Wiles encountered the method of Kolyvagin and Flach and he abandoned Iwasawa theory in favour of this new technique. The following year the proof was announced in Cambridge and he was proclaimed a hero. Within two months the Kolyvagin–Flach method was shown to be flawed, and ever since the situation had only worsened. Every attempt to fix Kolyvagin–Flach had failed.
All of Wiles’s work apart from the final stage involving the Kolyvagin–Flach method was still worthwhile. The Taniyama–Shimura conjecture and Fermat’s Last Theorem might not have been solved; nevertheless he had provided mathematicians with a whole series of new techniques and strategies which they could exploit to prove other theorems. There was no shame in Wiles’s failure and he was beginning to come to terms with the prospect of being beaten.
As a consolation he at least wanted to understand why he had failed. While Taylor re-explored and re-examined alternative methods, Wiles decided to spend September looking one last time at the structure of the Kolyvagin–Flach method to try and pinpoint exactly why it was not working. He vividly remembers those final fateful days: ‘I was sitting at my desk one Monday morning, 19 September, examining the Kolyvagin–Flach method. It wasn’t that I believed I could make it work, but I thought that at least I could explain why it didn’t work. I thought I was clutching at straws, but I wanted to reassure myself. Suddenly, totally unexpectedly, I had this incredible revelation. I realised that, although the Kolyvagin–Flach method wasn’t working completely, it was all I needed to make my original Iwasawa theory work. I realised that I had enough from the Kolyvagin–Flach method to make my original approach to the problem from three years earlier work. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem.’
Iwasawa theory on its own had been inadequate. The Kolyvagin–Flach method on its own was also inadequate. Together they complemented each other perfectly. It was a moment of inspiration that Wiles will never forget. As he recounted these moments, the memory was so powerful that he was moved to tears: ‘It was so indescribably beautiful; it was so simple and so elegant. I couldn’t understand how I’d missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I’d keep coming back to my desk looking to see if it was still there. It was still there. I couldn’t contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.’
This was not only the fulfilment of a childhood dream and the culmination of eight years of concerted effort, but having been pushed to the brink of submission Wiles had fought back to prove his genius to the world. The last fourteen months had been the most painful, humiliating and depressing period of his mathematical career. Now one brilliant insight had brought an end to his suffering.
‘So the first night I went back home and slept on it. I checked through it again the next morning and by 11 o’clock I was satisfied, and I went down and told my wife, “I’ve got it! I think I’ve found it.” And it was so unexpected that she thought I was talking about a children’s toy or something, and she said, “Got what?” I said, “I’ve fixed my proof. I’ve got it.’”
The following month Wiles was able to make up for the promise he had failed to keep the previous year. ‘It was coming up to Nada’s birthday again and I remembered that last time I could not give her the present she wanted. This time, half a minute late for our dinner on the night of her birthday, I was able to give her the complete manuscript. I think she liked that present better than any other I had ever given her.’
