An expert problem solver must be endowed with two incompatible qualities – a restless imagination and a patient pertinacity.
Howard W. Eves
‘It was one evening at the end of the summer of 1986 when I was sipping iced tea at the house of a friend. Casually in the middle of a conversation he told me that Ken Ribet had proved the link between Taniyama–Shimura and Fermat’s Last Theorem. I was electrified. I knew that moment that the course of my life was changing because this meant that to prove Fermat’s Last Theorem all I had to do was to prove the Taniyama–Shimura conjecture. It meant that my childhood dream was now a respectable thing to work on. I just knew that I could never let that go. I just knew that I would go home and work on the Taniyama–Shimura conjecture.’
Over two decades had passed since Andrew Wiles had discovered the library book that inspired him to take up Fermat’s challenge, but now, for the first time, he could see a path towards achieving his childhood dream. Wiles recalls how his attitude to Taniyama–Shimura changed overnight: ‘I remembered one mathematician who’d written about the Taniyama–Shimura conjecture and cheekily suggested it as an exercise for the interested reader. Well, I guess now I was interested!’
Since completing his Ph.D. with Professor John Coates at Cambridge, Wiles had moved across the Atlantic to Princeton University where he himself was now a professor. Thanks to Coates’s guidance Wiles probably knew more about elliptic equations than anybody else in the world, but he was well aware that even with his enormous background knowledge and mathematical skills the task ahead was immense.
Most other mathematicians, including John Coates, believed that embarking on the proof was a futile exercise: ‘I myself was very sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime.’
Wiles was aware that the odds were against him, but even if he ultimately failed in proving Fermat’s Last Theorem he felt his efforts would not be wasted: ‘Of course the Taniyama–Shimura conjecture had been open for many years. No one had had any idea how to approach it but at least it was mainstream mathematics. I could try and prove results, which, even if they didn’t get the whole thing, would be worthwhile mathematics. I didn’t feel I’d be wasting my time. So the romance of Fermat which had held me all my life was now combined with a problem that was professionally acceptable.’
At the turn of the century the great logician David Hilbert was asked why he never attempted a proof of Fermat’s Last Theorem. He replied, ‘Before beginning I should have to put in three years of intensive study, and I haven’t that much time to squander on a probable failure.’ Wiles realised that to have any hope of finding a proof he would first have to completely immerse himself in the problem, but unlike Hilbert he was prepared to take the risk. He read all the most recent journals and then played with the latest techniques over and over again until they became second nature to him. Gathering the necessary weapons for the battle ahead would require Wiles to spend the next eighteen months familiarising himself with every bit of mathematics which had ever been applied to, or had been derived from, elliptic equations or modular forms. This was a comparatively minor investment, bearing in mind that he fully expected that any serious attempt on the proof could easily require ten years of single-minded effort.
Wiles abandoned any work which was not directly relevant to proving Fermat’s Last Theorem and stopped attending the never-ending round of conferences and colloquia. Because he still had responsibilities in the Princeton Mathematics Department, Wiles continued to attend seminars, lecture to undergraduates and give tutorials. Whenever possible he would avoid the distractions of being a faculty member by working at home where he could retreat into his attic study. Here he would attempt to expand and extend the power of the established techniques, hoping to develop a strategy for his attack on the Taniyama–Shimura conjecture.
‘I used to come up to my study, and start trying to find patterns. I tried doing calculations which explain some little piece of mathematics. I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about. Sometimes that would involve going and looking it up in a book to see how it’s done there. Sometimes it was a question of modifying things a bit, doing a little extra calculation. And sometimes I realised that nothing that had ever been done before was any use at all. Then I just had to find something completely new – it’s a mystery where that comes from.
‘Basically it’s just a matter of thinking. Often you write something down to clarify your thoughts, but not necessarily. In particular when you’ve reached a real impasse, when there’s a real problem that you want to overcome, then the routine kind of mathematical thinking is of no use to you. Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to really think about nothing but that problem – just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over and it’s during that time that some new insight comes.’
From the moment he embarked on the proof, Wiles made the remarkable decision to work in complete isolation and secrecy. Modern mathematics has developed a culture of cooperation and collaboration, and so Wiles’s decision appeared to hark back to a previous era. It was as if he was imitating the approach of Fermat himself, the most famous of mathematical hermits. Wiles explained that part of the reason for his decision to work in secrecy was his desire to work without being distracted: ‘I realised that anything to do with Fermat’s Last Theorem generates too much interest. You can’t really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed.’
Another motivation for Wiles’s secrecy must have been his craving for glory. He feared the situation arising whereby he had completed the bulk of the proof but was still missing the final element of the calculation. At this point, if news of his breakthroughs were to leak out, there would be nothing stopping a rival mathematician building on Wiles’s work, completing the proof and stealing the prize.
In the years to come Wiles was to make a series of extraordinary discoveries, none of which would be discussed or published until his proof was complete. Even close colleagues were oblivious to his research. John Coates can recall exchanges with Wiles during which he was given no clues as to what was going on: ‘I remember saying to him on a number of occasions, “It’s all very well this link to Fermat’s Last Theorem but it’s still hopeless to try and prove Taniyama–Shimura.” I think he just smiled.’
Ken Ribet, who completed the link between Fermat and Taniyama–Shimura, was also completely unaware of Wiles’s clandestine activities. ‘This is probably the only case I know where someone worked for such a long time without divulging what he was doing, without talking about the progress he was making. It’s just unprecedented in my experience. In our community people have always shared their ideas. Mathematicians come together at conferences, they visit each other to give seminars, they send e-mail to each other, they talk on the telephone, they ask for insights, they ask for feedback – mathematicians are always in communication. When you talk to other people you get a pat on the back; people tell you that what you’ve done is important, they give you ideas. It’s sort of nourishing and if you cut yourself off from this, then you are doing something that’s probably psychologically very odd.’
In order not to arouse suspicion Wiles devised a cunning ploy which would throw his colleagues off the scent. During the early 1980s he had been working on a major piece of research on a particular type of elliptic equation, which he was about to publish in its entirety, until the discoveries of Ribet and Frey made him change his mind. Wiles decided to publish his research bit by bit, releasing another minor paper every six months or so. This apparent productivity would convince his colleagues that he was still continuing with his usual research. For as long as he could maintain this charade, Wiles could continue working on his true obsession without revealing any of his breakthroughs.
The only person who was aware of Wiles’s secret was his wife, Nada. They married soon after Wiles began working on the proof, and as the calculation progressed he confided in her and her alone. In the years that followed, his family would be his only distraction. ‘My wife’s only known me while I’ve been working on Fermat. I told her on our honeymoon, just a few days after we got married. My wife had heard of Fermat’s Last Theorem, but at that time she had no idea of the romantic significance it had for mathematicians, that it had been such a thorn in our flesh for so many years.’
In order to prove Fermat’s Last Theorem Wiles had to prove the Taniyama–Shimura conjecture: every single elliptic equation can be correlated with a modular form. Even before the link to Fermat’s Last Theorem mathematicians had tried desperately to prove the conjecture, but every attempt had ended in failure. Wiles was acquainted with the failures of the past: ‘Ultimately what one would naïvely have tried to do, and what people certainly did try to do, was to count elliptic equations and count modular forms, and show that there are the same number of each. But nobody has ever found any simple way of doing that. The first problem is that there are an infinite number of each and you can’t count an infinite number. One simply doesn’t have a way of doing it.’
In order to find a solution, Wiles adopted his usual approach to solving difficult problems. ‘I sometimes write scribbles or doodles. They’re not important doodles, just subconscious doodles. I never use a computer.’ In this case, as with many problems in number theory, computers would be of no use whatsoever. The Taniyama–Shimura conjecture applied to an infinite number of equations and, although a computer could check an individual case in a few seconds, it could never check all cases. Instead what was required was a logical step-by-step argument which would effectively give a reason and explain why every elliptic equation had to be modular. To find the proof Wiles relied solely on a piece of paper, a pencil and his mind. ‘I carried this thought around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.’
After a year of contemplation Wiles decided to adopt a general strategy known as induction as the basis for his proof. Induction is an immensely powerful form of proof, because it can allow a mathematician to prove that a statement is true for an infinite number of cases by only proving it for just one case. For example, imagine that a mathematician wants to prove that a statement is true for every counting number up to infinity. The first step is to prove that the statement is true for the number 1, which presumably is a fairly straightforward task. The next step is to show that if the statment is true for the number 1 then it must be true for the number 2, and if it is true for the number 2 then it must be true for the number 3, and if it is true for the number 3 then it must be true for the number 4, and so on. More generally, the mathematician has to show that if the statement is true for any number n, then it must be true for the next number n + 1.
Proof by induction is essentially a two step process:
(1) Prove that the statement is true for the first case.
(2) Prove that if the statement is true for any one case, then it must be true for the next case.
Another way to think of proof by induction is to imagine the infinite number of cases as an infinite line of dominoes. In order to prove every case it is necessary to find a way of knocking down every one of the dominoes. Knocking them down one by one would take an infinite amount of time and effort, but proof by induction allows mathematicians to knock them all down by just knocking down the first one. If the dominoes are carefully arranged, then knocking down the first domino will knock down the second domino, which will in turn knock down the third domino, and so on to infinity. Proof by induction invokes the domino effect. This form of mathematical domino-toppling allows an infinite number of cases to be proved by just proving the first one. Appendix 10 shows how proof by induction can be used to prove a relatively simple mathematical statement about all numbers.
The challenge for Wiles was to construct an inductive argument which showed that each of the infinity of elliptic equations could be matched to each of the infinity of modular forms. Somehow he had to break the proof down into an infinite number of individual cases and then prove the first case. Next, he had to demonstrate that, having proved the first case, all the others would topple. Eventually he discovered the first step to his inductive proof hidden in the work of a tragic genius from nineteenth-century France.
Evariste Galois was born in Bourg-la-Reine, a small village just south of Paris, on 25 October 1811, just twenty-two years after the French Revolution. Napoleon Bonaparte was at the height of his powers, but the following year saw the disastrous Russian campaign, and in 1814 he was driven into exile and replaced by King Louis XVIII. In 1815 Napoleon escaped from Elba, entered Paris and reclaimed power but within a hundred days he was defeated at Waterloo and forced to abdicate once again in favour of Louis XVIII. Galois, like Sophie Germain, grew up during a period of immense upheaval, but whereas Germain shut herself away from the turmoils of the French Revolution and concentrated on mathematics, Galois repeatedly found himself at the centre of political controversy, which not only distracted him from a brilliant academic career, but also led to his untimely death.
In addition to the general unrest which impinged on everybody’s life, Galois’s interest in politics was inspired by his father, Nicolas-Gabriel Galois. When Evariste was just four years old his father was elected mayor of Bourg-la-Reine. This was during Napoleon’s triumphant return to power, a period when his father’s strong liberal values were in keeping with the mood of the nation. Nicolas-Gabriel Galois was a cultured and gracious man and during his early years as mayor he gained respect throughout the community, so even when Louis XVIII returned to the throne he retained his elected position. Outside of politics, his main interest seems to have been the composition of witty rhymes, which he would read at town meetings to the delight of his constituents. Many years later this charming talent for epigrams would lead to his downfall.
At the age of twelve Evariste Galois attended his first school, the Lycée of Louis-le-Grand, a prestigious but authoritarian institution. To begin with he did not encounter any courses in mathematics and his academic record was respectable but not outstanding. However, one event occurred during his first term which would influence the course of his life. The Lycée had previously been a Jesuit school and rumours began to circulate suggesting that it was about to be returned to the authority of the priests. During this period there was a continual struggle between republicans and monarchists to sway the balance of power between Louis XVIII and the people’s representatives, and the increasing influence of the priests was seen as an indication of a shift away from the people and towards the King. The students of the Lycée, who in the main had republican sympathies, planned a rebellion but the director of the school, Monsieur Berthod, uncovered the plot and immediately expelled the dozen or so ringleaders. The following day when Berthod demanded a demonstration of allegiance from the remaining senior scholars, they refused to drink a toast to Louis XVIII, whereupon another hundred students were expelled. Galois was too young to be involved in the failed rebellion and so remained at the Lycée. Nevertheless, watching his fellow students being humiliated in this way only served to inflame his republican tendencies.
It was not until the age of sixteen that Galois enrolled in his first mathematics class, a course which would, in the eyes of his teachers, transform him from a conscientious pupil into an unruly student. His school reports show that he neglected all his other subjects and concentrated solely on his new found passion:
This student works only in the highest realms of mathematics. The mathematical madness dominates this boy. I think it would be best for him if his parents would allow him to study nothing but this. Otherwise he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments.
Galois’s desire for mathematics soon outstripped the capacity of his teacher, and so he learnt directly from the very latest books written by the masters of the age. He readily absorbed the most complex of concepts, and by the time he was seventeen he published his first paper in the Annales de Gergonne. The path ahead seemed clear for the prodigy, except that his own sheer brilliance was to provide the greatest obstacle to his progress. Although he obviously knew more than enough mathematics to pass the Lycée’s examinations, Galois’s solutions were often so innovative and sophisticated that his examiners failed to appreciate them. To make matters worse Galois would perform so many calculations in his head that he would not bother to outline clearly his argument on paper, leaving the inadequate examiners even more perplexed and frustrated.
The young genius did not help the situation by having a quick temper and a rashness which did not endear him to his tutors or anybody else who crossed his path. When Galois applied to the Ecole Polytechnique, the most prestigious college in the land, his abruptness and lack of explanation in the oral examination meant that he was refused admission. Galois was desperate to attend the Polytechnique, not just because of its academic excellence but also because of its reputation for being a centre for republican activism. One year later he reapplied and once again his logical leaps in the oral examination only served to confuse his examiner, Monsieur Dinet. Sensing that he was about to be failed for a second time and frustrated that his brilliance was not being recognised, Galois lost his temper and threw a blackboard rubber at Dinet, scoring a direct hit. Galois was never to return to the hallowed halls of the Polytechnique.
Undaunted by the rejections, Galois remained confident of his mathematical talent and continued his own private researches. His main interest concerned finding solutions to equations, such as quadratic equations. Quadratic equations have the form
The challenge is to find the values of x for which the quadratic equation holds true. Rather than relying on trial and error mathematicians would prefer to have a recipe for finding solutions, and fortunately such a recipe exists:
Simply by substituting the values for a, b and c into the above recipe one can calculate the correct values for x. For instance, we can apply the recipe to solve the following equation:
By putting the values of a, b and c into the recipe, the solution turns out to be x = 1 or x = 2.
The quadratic is a type of equation within a much larger class of equations known as polynomials. A more complicated type of polynomial is the cubic equation:
The extra complication comes from the additional term x3. By adding one more term x4, we get the next level of polynomial equation, known as the quartic:
By the nineteenth century, mathematicians also had recipes which could be used to find solutions to the cubic and the quartic equations, but there was no known method for finding solutions to the quintic equation:
Galois became obsessed with finding a recipe for solving quintic equations, one of the great challenges of the era, and by the age of seventeen he had made sufficient progress to submit two research papers to the Academy of Sciences. The referee appointed to judge the papers was Augustin-Louis Cauchy, who many years later would argue with Lamé over an ultimately flawed proof of Fermat’s Last Theorem. Cauchy was highly impressed by the young man’s work and judged it worthy of being entered for the Academy’s Grand Prize in Mathematics. In order to qualify for the competition the two papers would have to be re-submitted in the form of a single memoir, so Cauchy returned them to Galois and awaited his entry.
Having survived the criticisms of his teachers and rejection by the Ecole Polytechnique Galois’s genius was on the verge of being recognised, but over the course of the next three years a series of personal and professional tragedies would destroy his ambitions. In July of 1829 a new Jesuit priest arrived in the village of Bourgla-Reine, where Galois’s father was still mayor. The priest took exception to the mayor’s republican sympathies and began a campaign to oust him from office by spreading rumours aimed at discrediting him. In particular the scheming priest exploited Nicolas-Gabriel Galois’s reputation for composing clever rhymes. He wrote a series of vulgar verses ridiculing members of the community and signed them with the mayor’s name. The elder Galois could not survive the shame and the embarrassment which resulted and decided that the only honourable option was to commit suicide.
Evariste Galois returned to attend his father’s funeral and saw for himself the divisions that the priest had created in the village. As the coffin was being lowered into the grave, a scuffle broke out between the Jesuit priest, who was conducting the service, and supporters of the mayor, who realised that there had been a plot to undermine him. The priest suffered a gash to the head, the scuffle turned into a riot, and the coffin was left to drop unceremoniously into its grave. Watching the French establishment humiliate and destroy his father served only to consolidate Galois’s fervent support for the republican cause.
Upon returning to Paris, Galois combined his research papers well ahead of the competition deadline and submitted the memoir to the secrerary of the Academy, Joseph Fourier, who was supposed to pass it on to the judging committee. Galois’s paper did not offer a solution to the quintic problem but it did offer a brilliant insight and many mathematicians, including Cauchy, considered that it was a likely winner. To the shock of Galois and his friends, not only did he fail to win the prize, but he had not even been officially entered. Fourier had died a few weeks prior to the judging and, although a stack of competition entries was passed on to the committee, Galois’s memoir was not among them. The memoir was never found and the injustice was recorded by a French journalist.
Last year before March 1st, Monsieur Galois gave to the secretary of the Institute a memoir on the solution of numerical equations. This memoir should have been entered in the competition for the Grand Prize in Mathematics. It deserved the prize, for it could resolve some difficulties that Lagrange had failed to do. Monsieur Cauchy had conferred the highest praise on the author about this subject. And what happened? The memoir is lost and the prize is given without the participation of the young savant.
Le Globe, 1831
Galois felt that his memoir had been deliberately lost by a politically biased Academy, a belief that was reinforced a year later when the Academy rejected his next manuscript, claiming that ‘his argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigour’. He decided that there was a conspiracy to exclude him from the mathematical community, and as a result he neglected his research in favour of fighting for the republican cause. By this time he was a student at the Ecole Normale Supérieure, a slightly less prestigious college than the Ecole Polytechnique. At the Ecole Normale Galois’s notoriety as a trouble-maker was overtaking his reputation as a mathematician. This culminated during the July revolution of 1830 when Charles X fled France and the political factions fought for control in the streets of Paris. The Ecole’s director Monsieur Guigniault, a monarchist, was aware that the majority of his students were radical republicans and so confined them to their dormitories and locked the gates of the college. Galois was being prevented from fighting alongside his brothers, and his frustration and anger were compounded when the republicans were eventually defeated. When the opportunity arose he published a scathing attack on the college director, accusing him of cowardice. Not surprisingly, Guigniault expelled the insubordinate student and Galois’s formal mathematical career was at an end.
On 4 December the thwarted genius attempted to become a professional rebel by joining the Artillery of the National Guard, a republican branch of the militia otherwise known as the ‘Friends of the People’. Before the end of the month the new king Louis-Phillipe, anxious to avoid a further rebellion, abolished the Artillery of the National Guard, and Galois was left destitute and homeless. The most brilliant young talent in all of Paris was being persecuted at every turn and some of his former mathematical colleagues were becoming increasingly worried about his plight. Sophie Germain, who was by this time the shy elder stateswoman of French mathematics, expressed her concerns to friend of the family Count Libri-Carrucci:
Decidedly there is a misfortune concerning all that touches upon mathematics. The death of Monsieur Fourier has been the final blow for this student Galois who, in spite of his impertinence, showed signs of a clever disposition. He has been expelled from the Ecole Normale, he is without money, his mother has very little also and he continues his habit of insult. They say he will go completely mad. I fear this is true.
As long as Galois’s passion for politics continued it was inevitable that his fortunes would deteriorate further, a fact documented by the great French writer Alexandre Dumas. Dumas was at the restaurant Vendanges de Bourgogne when he happened upon a celebration banquet in honour of nineteen republicans aquitted of conspiracy charges:
Suddenly, in the midst of a private conversation which I was carrying on with the person on my left, the name Louis-Phillipe, followed by five or six whistles, caught my ear. I turned around. One of the most animated scenes was taking place fifteen or twenty seats from me. It would be difficult to find in all Paris two hundred persons more hostile to the government than those to be found reunited at five o’clock in the afternoon in the long hall on the ground floor above the garden.
A young man who had raised his glass and held an open dagger in the same hand was trying to make himself heard – Evariste Galois was one of the most ardent republicans. The noise was such that the very reason for this noise had become incomprehensible. All that I could perceive was that there was a threat and that the name of Louis-Phillipe had been mentioned: the intention was made clear by the open knife.
This went way beyond my own republican opinions. I yielded to the pressure from my neighbour on the left who, as one of the King’s comedians, didn’t care to be compromised, and we jumped from the window sill into the garden. I went home somewhat worried. It was clear this episode would have its consequences. Indeed, two or three days later, Evariste Galois was arrested.
After being detained at Sainte-Pélagie prison for a month Galois was charged with threatening the King’s life and brought to trial. Although there was little doubt from his actions that Galois was guilty, the raucous nature of the banquet meant that nobody could actually confirm that they had heard him make any direct threats. A sympathetic jury and the rebel’s tender age – he was still only twenty – led to his acquittal. The following month he was arrested again.
On Bastille Day, 14 July 1831, Galois marched through Paris dressed in the uniform of the outlawed Artillery Guard. Although this was merely a gesture of defiance, he was sentenced to six months in prison and returned to Sainte-Pélagie. During the following months the teetotal youth was driven to drink by the rogues who surrounded him. The botanist and ardent republican François Raspail, who was imprisoned for refusing to accept the Cross of the Legion of Honour from Louis-Phillipe, wrote an account of Galois’s first drinking bout:
He grasps the little glass like Socrates courageously taking the hemlock; he swallows it as one gulp, not without blinking and making a wry face. A second glass is not harder to empty than the first, and then the third. The beginner loses his equilibrium. Triumph! Homage to the Bacchus of the jail! You have intoxicated an ingenuous soul, who holds wine in horror.
A week later a sniper in a garret opposite the prison fired a shot into a cell wounding the man next to Galois. Galois was convinced that the bullet was intended for himself and that there was a government plot to assassinate him. The fear of political persecution terrorised him, and the isolation from his friends and family and rejection of his mathematical ideas plunged him into a state of depression. In a bout of drunken delirium he tried to stab himself to death, but Raspail and others managed to restrain and disarm him. Raspail recalls Galois’s words immediately prior to the suicide attempt:
Do you know what I lack my friend? I confide it only to you: it is someone I can love and love only in spirit. I have lost my father and no one has ever replaced him, do you hear me …?
In March 1832, a month before Galois’s sentence was due to finish, a cholera epidemic broke out in Paris and the prisoners of Sainte-Pélagie were released. What happened to Galois over the next few weeks has been the subject of intense speculation, but what is certain is that the events of this period were largely the consequence of a romance with a mysterious woman by the name of Stéphanie-Félicie Poterine du Motel, the daughter of a respected Parisian physician. Although there are no clues as to how the affair started, the details of its tragic end are well documented.
Stéphanie was already engaged to a gentleman by the name of Pescheux d’Herbinville, who uncovered his fiancée’s infidelity. D’Herbinville was furious and, being one the finest shots in France, he had no hesitation in immediately challenging Galois to a duel at dawn. Galois was well aware of his challenger’s reputation. During the evening prior to the confrontation, which he believed would be his last opportunity to commit his thoughts to paper, he wrote letters to his friends explaining his circumstances:
I beg my patriots, my friends, not to reproach me for dying otherwise than for my country. I died the victim of an infamous coquette and her two dupes. It is in a miserable piece of slander that I end my life. Oh! Why die for something so little, so contemptible? I call on heaven to witness that only under compulsion and force have I yielded to a provocation which I have tried to avert by every means.
Despite his devotion to the republican cause and his romantic involvement, Galois had always maintained his passion for mathematics and one of his greatest fears was that his research, which had already been rejected by the Academy, would be lost forever. In a desperate attempt to gain recognition he worked through the night writing out the theorems which he believed fully explained the riddle of quintic equations. The pages were largely a transcription of the ideas he had already submitted to Cauchy and Fourier, but hidden within the complex algebra were occasional references to ‘Stéphanie’ or ‘une femme’ and exclamations of despair – ‘I have not time, I have not time!’ At the end of the night, when his calculations were complete, he wrote a covering letter to his friend Auguste Chevalier, requesting that, should he die, the papers be distributed to the greatest mathematicians in Europe:
My Dear Friend,
I have made some new discoveries in analysis. The first concern the theory of quintic equations, and others integral functions.
In the theory of equations I have researched the conditions for the solvability of equations by radicals; this has given me the occasion to deepen this theory and describe all the transformations possible on an equation even though it is not solvable by radicals. All this will be found here in three memoirs …
In my life I have often dared to advance propositions about which I was not sure. But all I have written down here has been clear in my head for over a year, and it would not be in my interest to leave myself open to the suspicion that I announce theorems of which I do not have a complete proof.
Make a public request of Jacobi or Gauss to give their opinions, not as to the truth, but as to the importance of these theorems. After that, I hope some men will find it profitable to sort out this mess.
I embrace you with effusion,
E. Galois
The following morning, Wednesday 30 May 1832, in an isolated field Galois and d’Herbinville faced each other at twenty-five paces armed with pistols. D’Herbinville was accompanied by seconds; Galois stood alone. He had told nobody of his plight: a messenger he had sent to his brother Alfred would not deliver the news of the duel until it was over and the letters he had written the previous night would not reach his friends for several days.
The pistols were raised and fired. D’Herbinville still stood, Galois was hit in the stomach. He lay helpless on the ground. There was no surgeon to hand and the victor calmly walked away leaving his wounded opponent to die. Some hours later Alfred arrived on the scene and carried his brother to Cochin hospital. It was too late, peritonitis had set in, and the following day Galois died.
His funeral was almost as farcical as his father’s. The police believed that it would be the focus of a political rally and arrested thirty comrades the previous night. Nonetheless two thousand republicans gathered for the service and inevitably scuffles broke out between Galois’s colleagues and the government officials who had arrived to monitor events.
The mourners were angry because of a growing belief that d’Herbinville was not a cuckolded fiancé but rather a government agent, and that Stéphanie was not just a lover but a scheming seductress. Events such as the shot which was fired at Galois while he was in Sainte-Pélagie prison already hinted at a conspiracy to assassinate the young trouble-maker, and therefore his friends concluded that he had been duped into a romance which was part of a political plot contrived to kill him. Historians have argued about whether the duel was the result of a tragic love affair or politically motivated, but either way one of the world’s greatest mathematicians was killed at the age of twenty, having studied mathematics for only five years.
Before distributing Galois’s papers his brother and Auguste Chevalier rewrote them in order to clarify and expand the explanations. Galois’s habit of explaining his ideas hastily and inadequately was no doubt exacerbated by the fact that he had only a single night to outline years of research. Although they dutifully sent copies of the manuscript to Carl Gauss, Carl Jacobi and others, there was no acknowledgment of Galois’s work for over a decade, until a copy reached Joseph Liouville in 1846. Liouville recognised the spark of genius in the calculation and spent months trying to interpret its meaning. Eventually he edited the papers and published them in his prestigious Journal de Mathématiques pures et appliquées. The response from other mathematicians was immediate and impressive because Galois had indeed formulated a complete understanding of how one could go about finding solutions to quintic equations. First Galois had classified all quintics into two types: those that were soluble and those that were not. Then, for those that were soluble, he devised a recipe for finding the solutions to the equations. Moreover, Galois examined equations of higher order than the quintic, those containing x6, x7, and so on, and could identify which of these were soluble. It was one of the masterpieces of nineteenth-century mathematics created by one of its most tragic heroes.
In his introduction to the paper Liouville reflected on why the young mathematician had been rejected by his seniors and how his own efforts had resurrected Galois:
An exaggerated desire for conciseness was the cause of this defect which one should strive above all else to avoid when treating the abstract and mysterious matters of pure Algebra. Clarity is, indeed, all the more necessary when one essays to lead the reader farther from the beaten path and into wilder territory. As Descartes said, ‘When transcendental questions are under discussion be transcendentally clear.’ Too often Galois neglected this precept; and we can understand how illustrious mathematicians may have judged it proper to try, by the harshness of their sage advice, to turn a beginner, full of genius but inexperienced, back on the right road. The author they censured was before them ardent, active; he could profit by their advice.
But now everything is changed. Galois is no more! Let us not indulge in useless criticisms; let us leave the defects there and look at the merits …
My zeal was well rewarded, and I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem.
At the heart of Galois’s calculations was a concept known as group theory, an idea which he had developed into a powerful tool capable of cracking previously insoluble problems. Mathematically, a group is a set of elements which can be combined together using some operation, such as addition or multiplication, and which satisfy certain conditions. An important defining property of a group is that, when any two of its elements are combined using the operation, the result is another element in the group. The group is said to be closed under that operation.
For example, positive and negative whole numbers form a group under the operation of ‘addition’. Combining one whole number with another under the operation of addition leads to a third whole number, e.g.
Mathematicians state that ‘positive and negative whole numbers are closed under addition and form a group’. On the other hand the whole numbers do not form a group under the operation of ‘division’, because dividing one whole number by another does not necessarily lead to another whole number, e.g.
The fraction 1⁄3 is not a whole number and is outside the original group. However, by considering a larger group which does include fractions, the so-called rational numbers, closure can be re-established: ‘the rational numbers are closed under division’. Having said this, one stills needs to be careful because division by the element zero results in infinity, which leads to various mathematical nightmares. For this reason it is more accurate to state that ‘the rational numbers (excluding zero) are closed under division’. In many ways closure is similar to the concept of completeness described in earlier chapters.
The whole numbers and the fractions form infinitely large groups, and one might assume that, the larger the group, the more interesting the mathematics it will generate. However, Galois had a ‘less is more’ philosophy, and showed that small carefully constructed groups could exhibit their own special richness. Instead of using the infinite groups, Galois began with a particular equation and constructed his group from the handful of solutions to that equation. It was groups formed from the solutions to quintic equations which allowed Galois to derive his results about these equations. A century and a half later Wiles would use Galois’s work as the foundation for his proof of the Taniyama–Shimura conjecture.
To prove the Taniyama–Shimura conjecture, mathematicians had to show that every one of the infinite number of elliptic equations could be paired with a modular form. Originally they had attempted to show that the whole DNA for one elliptic equation (the E-series) could be matched with the whole DNA for one modular form (the M-series), and then they would move on to the next elliptic equation. Although this is a perfectly sensible approach, nobody had found a way to repeat this process over and over again for the infinite number of elliptic equations and modular forms.
Wiles tackled the problem in a radically different way. Instead of trying to match all elements of one E-series and M-series and then moving on to the next E-series and M-series, he tried to match one element of all E-series and M-series and then move on to the next element. In other words each E-series has an infinite list of elements, individual genes which make up the DNA, and Wiles wanted to show that the first gene in every E-series could be matched with the first gene in every M-series. He would then go on to show that the second gene in every E-series could be matched with the second gene in every M-series, and so on.
In the traditional approach one had an infinite problem, which was that even if you could prove that all of one E-series matched all of one M-series, there were still infinitely many other E-series and M-series to be matched. Wiles’s approach still involved tackling infinity because even if he could prove that the first gene of every E-series was identical to the first gene of every M-series there were still infinitely many other genes to be matched. However, Wiles’s approach had one major advantage over the traditional approach.
In the old method, once you had proved that the whole of one E-series matched the whole of one M-series, you then had to ask, Which E-series and M-series do I try and match up next? The infinity of E-series and M-series have no natural order and so whichever one is tackled next is a largely arbitrary choice. Crucially, in Wiles’s method, the genes in the E-series do have a natural order, and so having proved that all the first genes match (E1 = M1), the next step is obviously to prove that all the second genes match (E2 = M2), and so on.
This natural order is exactly what Wiles needed in order to develop an inductive proof. Initially Wiles would have to show that the first element of every E-series could be paired with the first element of every M-series. Then he would have to show that if the first elements could be paired then so could the second elements, and if the second elements could be paired then so could the third elements, and so on. He had to topple the first domino, and then he had to prove that any falling domino would also topple the next one.
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.’
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.
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.’
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.’
Katz sat in the lecture theatre and listened carefully to every step of Wiles’s calculation. By the end of it his assessment was that the Kolyvagin–Flach method seemed to be working perfectly. Nobody else in the department realised what had been going on. Nobody suspected that Wiles was on the verge of claiming the most important prize in mathematics. Their plan had been a success.
Once the lecture series was over Wiles devoted all his efforts to completing the proof. He had successfully applied the Kolyvagin–Flach method to family after family of elliptic equations, and by this stage only one family refused to submit to the technique. Wiles describes how he attempted to complete the last element of the proof: ‘One morning in late May, Nada was out with the children and I was sitting at my desk thinking about the remaining family of elliptic equations. I was casually looking at a paper of Barry Mazur’s and there was one sentence there that just caught my attention. It mentioned a nineteenth-century construction, and I suddenly realised that I should be able to use that to make the Kolyvagin–Flach method work on the final family of elliptic equations. I went on into the afternoon and I forgot to go down for lunch, and by about three or four o’clock I was really convinced that this would solve the last remaining problem. It got to about tea-time and I went downstairs and Nada was very surprised that I’d arrived so late. Then I told her – I’d solved Fermat’s Last Theorem.’
After seven years of single-minded effort Wiles had completed a proof of the Taniyama–Shimura conjecture. As a consequence, and after thirty years of dreaming about it, he had also proved Fermat’s Last Theorem. It was now time to tell the rest of the world.
‘So by May 1993, I was convinced that I had the whole of Fermat’s Last Theorem in my hands,’ recalls Wiles. ‘I still wanted to check the proof some more but there was a conference which was coming up at the end of June in Cambridge, and I thought that would be a wonderful place to announce the proof – it’s my old home town, and I’d been a graduate student there.’
The conference was being held at the Isaac Newton Institute. This time the institute had planned a workshop on number theory with the obscure title ‘L-functions and Arithmetic’. One of the organisers was Wiles’s Ph.D. supervisor John Coates: ‘We brought people from all around the world who were working on this general circle of problems and, of course, Andrew was one of the people that we invited. We’d planned one week of concentrated lectures and originally, because there was a lot of demand for lecture slots, I only gave Andrew two lecture slots. But then I gathered he needed a third slot, and so in fact I arranged to give up my own slot for his third lecture. I knew that he had some big result to announce but I had no idea what.’
When Wiles arrived in Cambridge he had two and a half weeks before his lectures began and wanted to make the most of the opportunity: ‘I decided I would check the proof with one or two experts, in particular the Kolyvagin–Flach part. The first person I gave it to was Barry Mazur. I think I said to him, “I have a manuscript here with a proof to a certain theorem.” He looked very baffled for a while, and then I said, “Well, have a look at it.” I think it then took him some time to register. He appeared stunned. Anyway I told him that I was hoping to speak about it at the conference, and that I’d really like him to try and check it.’
One by one the most eminent figures in number theory began to arrive at the Newton Institute, including Ken Ribet whose calculation in 1986 had inspired Wiles’s seven-year ordeal. ‘I arrived at this conference on L-functions and elliptic curves and it didn’t seem to be anything out of the ordinary until people started telling me that they had been hearing weird rumours about Andrew Wiles’s proposed series of lectures. The rumour was that he had proved Fermat’s Last Theorem, and I just thought this was completely nuts. I thought it couldn’t possibly be true. There are lots of cases when rumours start circulating in mathematics, especially through electronic mail, and experience shows that you shouldn’t put too much stock in them. But the rumours were very persistent and Andrew was refusing to answer questions about it and he was behaving very very queerly. John Coates said to him, “Andrew, what have you proved? Shall we call the press?” Andrew just kind of shook his head and sort of kept his lips sealed. He was really going for high drama.
‘Then one afternoon Andrew came up to me and started asking me about what I’d done in 1986 and some of the history of Frey’s ideas. I thought to myself, this is incredible, he must have proved the Taniyama–Shimura conjecture and Fermat’s Last Theorem, otherwise he wouldn’t be asking me this. I didn’t ask him directly if this was true, because I saw that he was behaving very coyly and I knew I wouldn’t get a straight answer. So I just kind of said, “Well Andrew, if you have occasion to speak about this work, here’s what happened.” I sort of looked at him as though I knew something, but I didn’t really know what was going on. I was still just guessing.’
Wiles’s reaction to the rumours and the mounting pressure was simple: ‘People would ask me, leading up to my lectures, what exactly I was going to say. So I said, well, come to my lectures and see.’
Back in 1920 David Hilbert, then aged fifty-eight, gave a public lecture in Göttingen on the subject of Fermat’s Last Theorem. When asked if the problem would ever be solved, he replied that he would not live to see it, but perhaps younger members of the audience might witness the solution. Hilbert’s estimate for the date of the solution was proving to be fairly accurate. Wiles’s lecture was also well timed in relation to the Wolfskehl Prize. In his will Paul Wolfskehl had set a deadline of 13 September 2007.
The title of Wiles’s lecture series was ‘Modular Forms, Elliptic Curves and Galois Representations’. Once again, as with the graduate lectures he had given earlier in the year for the benefit of Nick Katz, the title of the lectures was so vague that it gave no hint of his ultimate aim. Wiles’s first lecture was apparently mundane, laying the foundations for his attack on the Taniyama–Shimura conjecture in the second and third. The majority of his audience were completely unaware of the gossip, did not appreciate the point of the lectures, and paid little attention to the details. Those in the know were looking for the slightest clue which might give credence to the rumours.
Immediately after the lecture ended the rumour mill started again with renewed vigour, and electronic mail flew around the world. Professor Karl Rubin, a former student of Wiles, reported back to his colleagues in America:
By the following day more people had heard the gossip, and so the audience for the second lecture was significantly larger. Wiles teased them with an intermediate calculation which showed that he was clearly trying to tackle the Taniyama–Shimura conjecture, but the audience was still left wondering if he had done enough to prove it and, as a consequence, conquer Fermat’s Last Theorem. A new batch of e-mails bounced off the satellites.
‘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.