by Simon Singh
The first step was achieved when Wiles realised the power of Galois’s groups. A handful of solutions from every elliptic equation could be used to form a group. After months of analysis Wiles proved that the group led to one undeniable conclusion – the first element in every E-series could indeed be paired with the first one in an M-series. Thanks to Galois, Wiles had been able to topple the first domino. The next step of his inductive proof required him to find a way of showing that if any one element of the E-series matched the corresponding element in the M-series, then so must the next element match.
Getting this far had already taken two years, and there was no hint of how long it would take to find a way of extending the proof. Wiles was well aware of the task ahead: ‘You might ask how could I devote an unlimited amount of time to a problem that might simply not be soluble. The answer is that I just loved working on this problem and I was obsessed. I enjoyed pitting my wits against it. Furthermore, I always knew that the mathematics I was thinking about, even if it wasn’t strong enough to prove Taniyama–Shimura, and hence Fermat, would prove something. I wasn’t going up a back alley, it was certainly good mathematics and that was true all along. There was certainly a possibility that I would never get to Fermat, but there was no question that I was simply wasting my time.’
‘Fermat’s Theorem Solved?’
Although it was only the first step towards proving the Taniyama–Shimura conjecture, Wiles’s Galois strategy was a brilliant mathematical breakthrough, worthy of publication in its own right. As a result of his self-imposed seclusion he could not announce the result to the rest of the world, but similarly he had no idea who else might be making equally significant breakthroughs.
Wiles recalls his philosophical attitude towards any potential rivals: ‘Well, obviously no one wants to spend years trying to solve something and then find that someone else just solves it a few weeks before you do. But curiously, because I was trying a problem that’s considered impossible, I didn’t really have much fear of competition. I simply didn’t think I or anyone else had any real idea how to do it.’
On 8 March 1988 Wiles was shocked to read front-page headlines announcing that Fermat’s Last Theorem had been solved. The Washington Post and the New York Times claimed that thirty-eight-year-old Yoichi Miyaoka of the Tokyo Metropolitan University had discovered a solution to the world’s hardest problem. At this stage Miyaoka had not yet published his proof, but only described its outline at a seminar at the Max Planck Institute for Mathematics in Bonn. Don Zagier who was in the audience summarised the community’s optimism, ‘Miyaoka’s proof is very exciting and some people feel that there is a very good chance that it is going to work. It’s still not definite, but it looks fine so far.’
In Bonn, Miyaoka had described how he had approached the problem from a completely new angle, namely differential geometry. For decades differential geometers had developed a rich understanding of mathematical shapes and in particular the properties of their surfaces. Then in the 1970s a team of Russians led by Professor S. Arakelov attempted to draw parallels between problems in differential geometry and problems in number theory. This was one strand of the Langlands programme, and the hope was that unanswered problems in number theory could be solved by examining the corresponding question in differential geometry which had already been answered. This was known as the philosophy of parallelism.
Differential geometrists who tried to tackle problems in number theory became known as ‘arithmetic algebraic geometrists’, and in 1983 they claimed their first significant victory, when Gerd Faltings at the Institute for Advanced Study at Princeton made a major contribution towards understanding Fermat’s Last Theorem. Remember that Fermat claimed that there were no whole number solutions to the equation:
Faltings believed he could make some progress towards proving the Last Theorem by studying the geometric shapes associated with different values of n. The shapes corresponding to each of the equations are all different, but they do have one thing in common – they are all punctured with holes. The shapes are four-dimensional, rather like modular forms. All the shapes are like multi-dimensional doughnuts, with several holes rather than just one. The larger the value of n in the equation, the more holes there are in the corresponding shape.
Faltings was able to prove that, because these shapes always have more than one hole, the associated Fermat equation could only have a finite number of whole number solutions. A finite number of solutions could be anything from zero, which was Fermat’s own claim, to a million or a billion. So Faltings had not proved Fermat’s Last Theorem, but he had at least been able to discount the possibility of an infinity of solutions.
Five years later Miyaoka claimed he could go one step further. While still in his early twenties he had created a conjecture concerning the so-called Miyaoka inequality. It became clear that proof of his own geometrical conjecture would demonstrate that the number of solutions for Fermat’s equation was not only finite, but zero. Miyaoka’s approach was analogous to Wiles’s in that they were both trying to prove the Last Theorem by connecting it to a fundamental conjecture in a different field of mathematics. In Miyaoka’s case it was differential geometry; for Wiles the proof was via elliptic equations and modular forms. Unfortunately for Wiles he was still struggling to prove the Taniyama–Shimura conjecture when Miyaoka announced a full proof relating to his own conjecture, and therefore a proof of Fermat’s Last Theorem.
Two weeks after his announcement in Bonn, Miyaoka released the five pages of algebra which detailed his proof and then the scrutiny began. Number theorists and differential geometrists around the world examined the proof line by line, looking for the slightest gap in the logic or the merest hint of a false assumption. Within a few days several mathematicians highlighted what seemed to be a worrying contradiction within the proof. Part of Miyaoka’s work led to a particular conclusion in number theory, which when translated back to differential geometry conflicted with a result which had already been proved years earlier. Although this did not necessarily invalidate Miyaoka’s entire proof, it did clash with the philosophy of parallelism between number theory and differential geometry.
Another two weeks passed when Gerd Faltings, who had paved the way for Miyaoka, announced that he had pinpointed the exact reason for the apparent breakdown in parallelism – a gap in the logic. The Japanese mathematician was predominantly a geometrist and he had not been absolutely rigorous in translating his ideas into the less familiar territory of number theory. An army of number theorists attempted to help Miyaoka patch up the error but their efforts ended in failure. Two months after the initial announcement the consensus was that the original proof was destined to fail.
As with several other failed proofs in the past, Miyaoka had created new and interesting mathematics. Individual chunks of the proof stood on their own as ingenious applications of differential geometry to number theory, and in later years other mathematicians would build on them in order to prove other theorems, but never Fermat’s Last Theorem.
The fuss over Fermat soon died down and the newspapers ran short updates explaining that the 300-year-old puzzle remained unsolved. No doubt inspired by all the media attention a new piece of graffiti found its way on to New York’s Eighth Street subway station:
I have discovered a truly remarkable proof of this,
but I can’t write it now because my train is coming.
The Dark Mansion
Unknown to the world Wiles breathed a sigh of relief. Fermat’s Last Theorem remained unconquered and he could continue with his battle to prove it via the Taniyama–Shimura conjecture. ‘Much of the time I would sit writing at my desk, but sometimes I could reduce the problem to something very specific – there’s a clue, something that strikes me as strange, something just below the paper which I can’t quite put my finger on. If there was one particular thing buzzing in my mind then I didn’t need anything to write with or any desk to work at, so instead I would go for a walk down by the lake
. When I’m walking I find I can concentrate my mind on one very particular aspect of a problem, focusing on it completely. I’d always have a pencil and paper ready, so if I had an idea I could sit down at a bench and start scribbling away.’
After three years of non-stop effort Wiles had made a series of breakthroughs. He had applied Galois groups to elliptic equations, he had broken the elliptic equations into an infinite number of pieces, and then he had proved that the first piece of every elliptic equation had to be modular. He had toppled the first domino and now he was exploring techniques which might lead to the collapse of all the others. In hindsight this seemed like the natural route to a proof, but getting this far had required enormous determination to overcome the periods of self-doubt. Wiles describes his experience of doing mathematics in terms of a journey through a dark unexplored mansion. ‘One enters the first room of the mansion and it’s dark. Completely dark. One stumbles around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them.’
In 1990 Wiles found himself in what seemed to be the darkest room of all. He had been exploring it for almost two years. He still had no way of showing that if one piece of the elliptic equation was modular then so was the next piece. Having tried every tool and technique in the published literature, he had found that they were all inadequate. ‘I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal. It could be that the methods needed to solve this particular problem may simply be beyond present-day mathematics. Perhaps the methods I needed to complete the proof would not be invented for a hundred years. So even if I was on the right track, I could be living in the wrong century.’
Undaunted, Wiles persevered for another year. He began working on a technique called Iwasawa theory. Iwasawa theory was a method of analysing elliptic equations which he had learnt as a student in Cambridge under John Coates. Although the method as it stood was inadequate, he hoped that he could modify it and make it powerful enough to generate a domino effect.
Since making the initial breakthrough with Galois groups, Wiles had become increasingly frustrated. Whenever the pressure became too great, he would turn to his family. Since beginning work on Fermat’s Last Theorem in 1986, he had become a father twice over. ‘The only way I could relax was when I was with my children. Young children simply aren’t interested in Fermat, they just want to hear a story and they’re not going to let you do anything else.’
The Method of Kolyvagin and Flach
By the summer of 1991 Wiles felt he had lost the battle to adapt Iwasawa theory. He had to prove that every domino, if it itself had been toppled, would topple the next domino – that if one element in the elliptic equation E-series matched one element in the modular form M-series, then so would the next one. He also had to be sure that this would be the case for every elliptic equation and every modular form. Iwasawa theory could not give him the guarantee he required. He completed another exhaustive search of the literature and was still unable to find an alternative technique which would give him the breakthrough he needed. Having been a virtual recluse in Princeton for the last five years, he decided it was time to get back into circulation in order to find out the latest mathematical gossip. Perhaps somebody somewhere was working on an innovative new technique, and as yet, for whatever reason, had not published it. He headed north to Boston to attend a major conference on elliptic equations, where he would be sure of meeting the major players in the subject.
Wiles was welcomed by colleagues from around the world, who were delighted to see him after such a long absence from the conference circuit. They were still unaware of what he had been working on and Wiles was careful not to give away any clues. They did not suspect his ulterior motive when he asked them the latest news concerning elliptic equations. Initially the responses were of no relevance to Wiles’s plight but an encounter with his former supervisor John Coates was more fruitful: ‘Coates mentioned to me that a student of his named Matheus Flach was writing a beautiful paper in which he was analysing elliptic equations. He was building on a recent method devised by Kolyvagin and it looked as though his method was tailor-made for my problem. It seemed to be exactly what I needed, although I knew I would still have to further develop this so-called Kolyvagin–Flach method. I put aside completely the old approach I’d been trying and I devoted myself night and day to extending Kolyvagin–Flach.’
In theory this new method could extend Wiles’s argument from the first piece of the elliptic equation to all pieces of the elliptic equation, and potentially it could work for every elliptic equation. Professor Kolyvagin had devised an immensely powerful mathematical method, and Matheus Flach had refined it to make it even more potent. Neither of them realised that Wiles intended to incorporate their work into the world’s most important proof.
Wiles returned to Princeton, spent several months familiarising himself with his newly discovered technique, and then began the mammoth task of adapting it and implementing it. Soon for a particular elliptic equation he could make the inductive proof work – he could topple all the dominoes. Unfortunately the Kolyvagin–Flach method that worked for one particular elliptic equation did not necessarily work for another elliptic equation. He eventually realised that all the elliptic equations could be classified into various families. Once modified to work on one elliptic equation, the Kolyvagin–Flach method would work for all the other elliptic equations in that family. The challenge was to adapt the Kolyvagin–Flach method to work for each family. Although some families were harder to conquer than others, Wiles was confident that he could work his way through them one by one.
After six years of intense effort Wiles believed that the end was in sight. Week after week he was making progress, proving that newer and bigger families of elliptic curves must be modular. It seemed to be just a question of time before he would mop up the outstanding elliptic equations. During this final stage of the proof, Wiles began to appreciate that his whole proof relied on exploiting a technique which he had only discovered a few months earlier. He began to question whether he was using the Kolyvagin–Flach method in a fully rigorous manner.
‘During that year I worked extremely hard trying to make the Kolyvagin–Flach method work, but it involved a lot of sophisticated machinery that I wasn’t really familiar with. There was a lot of hard algebra which required me to learn a lot of new mathematics. Then around early January of 1993 I decided that I needed to confide in someone who was an expert in the kind of geometric techniques I was invoking for this. I wanted to choose very carefully who I told because they would have to keep it confidential. I chose to tell Nick Katz.’
Professor Nick Katz also worked in Princeton University’s Mathematics Department and had known Wiles for several years. Despite their closeness Katz was oblivious to what was going on literally just along the corridor. He recalls every detail of the moment Wiles revealed his secret: ‘One day Andrew came up to me at tea and asked me if I could come up to his office – there was something he wanted to talk to me about. I had no idea of what this could be. I went up to his office and he closed the door. He said he thought that he would be able to prove the Taniyama–Shimura conjecture. I was just amazed, flabbergasted – this was fantastic.
‘He explained that there was a big part of the proof that relied on his extension of the work of Flach and Kolyvagin but it was pretty technical. He really felt shaky on this highly technical part of the proof and he wanted to go through it with somebody because he wanted to be sure it was correct. He thought I was the right person to help him check it, but I
think there was another reason why he asked me in particular. He was sure that I would keep my mouth shut and not tell other people about the proof.’
After six years in isolation Wiles had let go of his secret. Now it was Katz’s job to get to grips with a mountain of spectacular calculations based on the Kolyvagin–Flach method. Virtually everything Wiles had done was revolutionary and Katz gave a great deal of thought as to the best way to examine it thoroughly: ‘What Andrew had to explain was so big and long that it wouldn’t have worked to try and just explain it in his office in informal conversations. For something this big we really needed to have the formal structure of weekly scheduled lectures, otherwise the thing would just degenerate. So, that’s why we decided to set up a lecture course.’
They decided that the best strategy would be to announce a series of lectures open to the department’s graduate students. Wiles would give the course and Katz would be in the audience. The course would effectively cover the part of the proof that needed checking but the graduate students would have no idea of this. The beauty of disguising the checking of the proof in this way was that it would force Wiles to explain everything step by step, and yet it would not arouse any suspicion within the department. As far as everyone else was concerned this was just another graduate course.
‘So Andrew announced this lecture course called “Calculations on Elliptic Curves”,’ recalls Katz with a sly smile, ‘which is a completely innocuous title – it could mean anything. He didn’t mention Fermat, he didn’t mention Taniyama–Shimura, he just started by diving right into doing technical calculations. There was no way in the world that anyone could have guessed what it was really about. It was done in such a way that unless you knew what this was for, then the calculations would just seem incredibly technical and tedious. And when you don’t know what the mathematics is for, it’s impossible to follow it. It’s pretty hard to follow it even when you do know what it’s for. Anyway, one by one the graduate students just drifted away and after a few weeks I was the only person left in the audience.’