Features: Faculty Insights

## It's thirty years since Andrew Wiles announced his proof of Fermat's Last Theorem, a problem that had haunted mathematicians for centuries. Today researchers at the Department of Pure Mathematics and Mathematical Statistics lead the field that Wiles' work has opened up.

"I think I'll stop here." This is how, on 23rd June 1993, Andrew Wiles ended his series of lectures at the Isaac Newton Institute, our neighbour here at the Centre for Mathematical Sciences. The applause, so witnesses report, was thunderous. Wiles had just delivered a proof that had eluded mathematicians for over 350 years: Fermat's Last Theorem.

### An infamous scribble

The theorem concerns equations of the form

xn+yn=zn

where n is a natural number. The question is whether there are triples of non-zero natural numbers x,y,z, that satisfy such an equation. For n=2 the answer is yes. There are in fact infinitely many such triples, known as Pythagorean triples, because the numbers involved also satisfy Pythagoras' theorem for right-angled triangles.

"This was the first time that I had seen a human story attached to a mathematical problem. Not just the story of one person, but people talking to each other over a period of centuries." Jack Thorne

The 17th century mathematician Pierre de Fermat convinced himself that when the exponent n is greater than 2, however, there are no integer solutions to the equation. In 1637 he wrote into the margin of his maths textbook that he had found a "marvellous proof" for this fact, which the margin was too narrow to contain. That tantalising scribble was to taunt mathematicians for a long time. Andrew Wiles was one of them.

"I first found out about Fermat's Last theorem from the cover of a book by E.T. Bell when I was about ten years old," says Wiles, who earned his PhD here at Cambridge in 1980, and is now Regius Professor in Mathematics at the University of Oxford. "I was captured by the romantic history of [the problem], so I spent some of my teenage years and even [some time] in college trying to solve it. But then when I became a professional mathematician I realised that this was not something you should be working on because it probably wouldn't generate any results."

In the mid-1980s, however, work by the mathematicians Gerhard Frey, Jean-Pierre Serre and Ken Ribet provided a new way of attacking Fermat's Last Theorem. It showed that if you could prove another result, known as the modularity conjecture (also known as the Taniyama-Shimura-Weil conjecture) then you'd have automatically proved Fermat's Last Theorem too.

"I was sceptical when the first announcement came out, but when Ribet proved that connection I was completely hooked and I dropped everything and started working on Fermat straight away," says Wiles. Unusually for a mathematician, he decided to work on the problem alone and in secret, for a period of seven years.  "Very few people want to work on a problem for that long. To really commit yourself to a problem takes a certain kind of personality. I  did initially [talk about it] a tiny bit, but then I realised that it got so much unwanted attention when you said you were working on it, you wouldn't be left in peace. So I felt it was wiser to do it in private."

### An electric atmosphere

The result which Wiles finally proved was the modularity conjecture, in a setting that was general enough to imply that Fermat's Last Theorem was also true. He announced his proof at the Isaac Newton Institute on June 23, 1993. The announcement came at the end of a series of three lectures and nobody really knew that this was what Wiles had had in store.

"Rumours started to get around," says Professor Tom Körner of the Department of Pure Mathematics and Mathematical Statistics at Cambridge, who had the privilege of witnessing the lecture. "I do not know if people knew or just speculated, so I asked one of Andrew's students whether I  would regret missing the lecture, and he said yes. The atmosphere was electric."

"At the end of [the lecture] Andrew wrote up the statement of Fermat's Last Theorem, and indicated that what he had done, he felt, had proved it. There was tremendous applause and then the experts got up and asked questions which indicated that, although details of the proof remained to be thoroughly checked, it was a very plausible way of attacking the problem. It was also a new way of attacking the problem, so that whether it succeeded or not, it had added a substantial amount to mathematics."

"On the one hand I was very excited to present [the result], but on the other there is always a tension the first time [you share the work]," says Wiles when recalling the announcement. "You have been thinking about this [for a long time], a lot of it on your own, so you [hope that you] haven't done anything stupid. I think people wanted to see the details, but they could see that this was a completely new approach and that it was going to prove something - whether it had all the details of the final claim remained to be seen."

The desire to see the details proved justified: it turned out the proof as it stood contained a hole, which it took Wiles, together with the mathematician Richard Taylor (also a Cambridge alumnus who had been Wiles' former PhD student at Princeton), nearly a year to fix. But then finally, in 1994, the centuries old problem that was so tantalisingly inspired by a note scribbled in the margin of a book was finally solved.

### The future of mathematics

You might think that, when an old problem is finally solved, a door closes on the area of mathematics involved. But this is rarely the case, as a solved problem usually opens up a range of unsolved ones. Wiles says that Fermat's Last Theorem has sparked two periods of intense progress in the past: one in the 19th century when the foundations for Wiles' areas of mathematics were laid in attempts to prove Fermat's Last Theorem, and one in the 1980s, which finally lead to the proof.

The proof itself, Wiles says, has helped to ring in a new era. "It opened another door, this time on problems of modularity. And these problems of modularity are themselves just one more door opening on this great perspective of what is called the Langlands programme — that's the future of mathematics."

It is difficult to explain the Langlands programme even to an expert, suffice to say that it consists of a web of far reaching conjectures made by Robert Langlands in the 1960s that draws extremely surprising connections between different fields of mathematics. Proving all these conjectures is seen by many as the single biggest project of modern mathematics.

The Langlands programme attracts some of the brightest minds in mathematics. Among them is Professor Jack Thorne of the Department of Pure Mathematics and Mathematical Statistics here at Cambridge. Thorne was six years old when Wiles announced the proof of Fermat's Last Theorem, and became interested in the result while doing his mathematics A levels.

"I found it quite exciting because doing A Level maths you learn how to do certain kinds of calculations, for example how to balance two balls on a rod and things like that," he says. "But this was the first time that I had seen a human story attached to a mathematical problem. Not just the story of one person, but people talking to each other over a period of centuries."

Despite his young age, Thorne is already a leading expert in his field. He has won a number of prizes, including the prestigious New Horizons in Mathematics Prize, and became the youngest living fellow of the Royal Society when he was elected in 2020. Thorne works on the Langlands programme, in particular on the connection it provides between number theory on the one hand and an area of maths that comes from generalisations of objects called modular forms on the other. "They are two worlds [for] which, a priori, it is not clear they should be connected, but [which] talk to each other in ways that are very mysterious and very striking," he explains. "It's really like there's a hidden telephone line."

The Langlands programme provides new tools for attacking problems in number theory. Thorne has used these tools to consider equations similar to the one of Fermat, but slightly more general: rather than requiring all the coefficients in the equation to be integers, you can ask yourself what happens if the coefficients come from larger number fields, for example fields containing more awkward irrational numbers such as the square root of 2. For some such classes of equations the theory generalised beautifully, says Thorne, but much work is still needed to push the field further. Wiles agrees that extending our theory of arithmetic to encompass more general number fields, using the tools provided by the Langlands programme, is one of the most important challenges of the future.

So while Wiles' proof settled a problem that is so easy to state that even a high school student can understand it, it has opened the door on a deep new area of mathematics which will see exciting developments in the next decade or so, in which mathematicians like Thorne are likely to play a leading role.

Fermat's Last Theorem has certainly defined Andrew Wiles' career. He is one of the few mathematicians who is well-known outside of mathematics, and was recognised with a knighthood in 2000. Within mathematics he has received a wealth of honours and awards, including the prestigious Abel Prize in 2016.

When asked whether he would have kept on working on Fermat's Last Theorem even if he hadn't found a solution back in the early 90s, his answer was characteristic of his approach to mathematics. "I am not a person who gives up on a problem."