Charles Rex Arbogast/AP
British number theorist Andrew Wiles has received the 2016 Abel Prize for his solution to Fermat’s last theorem — a problem that stumped some of the world’s greatest minds for three and a half centuries. The Norwegian Academy of Science and Letters announced the award — considered by some to be the 'Nobel of mathematics' — on 15 March.
Wiles, who is 62 and now at the University of Oxford, UK, will receive 6 million kroner (US$700,000) for his 1994 proof of the theorem, which states that there cannot be any positive whole numbers x, y and z such that xn + yn = zn, if n is greater than 2.
Soon after receiving the news on the morning of 15 March, Wiles told Nature that the award came to him as a “total surprise”.
That he solved a problem considered too hard by so many — and yet a problem relatively simple to state — has made Wiles arguably “the most celebrated mathematician of the twentieth century”, says Martin Bridson, director of Oxford's Mathematical Institute — which is housed in a building named after Wiles. Although his achievement is now two decades old, he continues to inspire young minds, something that is apparent when school children show up at his public lectures. “They treat him like a rock star,” Bridson says. “They line up to have their photos taken with him.”
Wiles's story has become a classic tale of tenacity and resilience. While a faculty member at Princeton University in New Jersey in the 1980s, he embarked on a solitary, seven-year quest to solve the problem, working in his attic without telling anyone except for his wife. He went on to make a historic announcement at a conference in his hometown of Cambridge, UK, in June 1993, only to hear from a colleague two months later that his proof contained a serious mistake. But after another frantic year of work — and with the help of one of his former students, Richard Taylor, who is now at the Institute for Advanced Study in Princeton — he was able to patch up the proof. When the resulting two papers were published in 1995, they made up an entire issue of the Annals of Mathematics1, 2.
But after Wiles's original claim had already made front-page news around the world, the pressure on the shy mathematician to save his work almost crippled him. “Doing mathematics in that kind of overexposed way is certainly not my style, and I have no wish to repeat it,” he said in a BBC documentary in 1996, still visibly shaken by the experience. “It’s almost unbelievable that he was able to get something done” at that point, says John Rognes, a mathematician at the University of Oslo and chair of the Abel Committee.
“It was very, very intense,” says Wiles. “Unfortunately as human beings we succeed by trial and error. It’s the people who overcome the setbacks who succeed.”
Wiles first learnt about French mathematician Pierre de Fermat as a child growing up in Cambridge. As he was told, Fermat formulated his eponymous theorem in a handwritten note in the margins of a book in 1637: “I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain,” he wrote (in Latin).
“I think it has a very romantic story,” Wiles says of Fermat's idea. “The kind of story that catches people’s imagination when they’re young and thinking of entering mathematics.”
But although he may have thought he had a proof at the time, only a proof for one special case has survived him, for exponent n = 4. A century later, Leonhard Euler proved it for n = 3, and Sophie Germain's work led to a proof for infinitely many exponents, but still not for all. Experts now tend to concur that the most general form of the statement would have been impossible to crack without mathematical tools that became available only in the twentieth century.
In 1983, German mathematician Gerd Faltings, now at the Max Planck Institute for Mathematics in Bonn, took a huge leap forward by proving that Fermat's statement had, at most, a finite number of solutions, although he could not show that the number should be zero. (In fact, he proved a result viewed by specialists as deeper and more interesting than Fermat's last theorem itself; it demonstrated that a broader class of equations has, at most, a finite number of solutions.)
The winning number
To narrow it to zero, Wiles took a different approach: he proved the Shimura-Taniyama conjecture, a 1950s proposal that describes how two very different branches of mathematics, called elliptic curves and modular forms, are conceptually equivalent. Others had shown that proof of this equivalence would imply proof of Fermat — and, like Faltings' result, most mathematicians regard this as much more profound than Fermat’s last theorem itself. (The full citation for the Abel Prize states that it was awarded to Wiles “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.”)
The link between the Shimura–Taniyama conjecture and Fermat's theorum was first proposed in 1984 by number theorist Gerhard Frey, now at the University of Duisburg-Essen in Germany. He claimed that any counterexample to Fermat's last theorem would also lead to a counterexample to the Shimura–Taniyama conjecture.
Kenneth Ribet, a mathematician at the University of California, Berkeley, soon proved that Frey was right, and therefore that anyone who proved the more recent conjecture would also bag Fermat's. Still, that did not seem to make the task any easier. “Andrew Wiles is probably one of the few people on Earth who had the audacity to dream that he can actually go and prove this conjecture,” Ribet told the BBC in the 1996 documentary.
Fermat's last theorem is also connected to another deep question in number theory called the abc conjecture, Rognes points out. Mathematician Shinichi Mochizuki of Kyoto University's Research Institute for Mathematical Sciences in Japan claimed to have proved that conjecture in 2012, although his roughly 500-page proof is still being vetted by his peers. Some mathematicians say that Mochizuki's work could provide, as an extra perk, an alternative way of proving Fermat, although Wiles says that sees those hopes with scepticism.
Wiles helped to arrange an Oxford workshop on Mochizuki's work last December, although his research interests are somewhat different. Lately, he has focused his efforts on another major, unsolved conjecture in number theory, which has been listed as one of seven Millennium Prize problems posed by the Clay Mathematics Institute in Oxford, UK. He still works very hard and thinks about mathematics for most of his waking hours, including as he walks to the office in the morning. “He doesn’t want to cycle,” Bridson says. “He thinks it would be a bit dangerous for him to do it while thinking about mathematics.”
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