Wiles was born in Cambridge to theologianMaurice Frank Wiles and Patricia Wiles. While spending much of his childhood in Nigeria, Wiles developed an interest in mathematics and in Fermat's Last Theorem in particular. After moving to Oxford and graduating from there in 1974, he worked on unifyingGalois representations,elliptic curves andmodular forms, starting withBarry Mazur's generalizations ofIwasawa theory. In the early 1980s, Wiles spent a few years at theUniversity of Cambridge before moving toPrinceton University, where he worked on expanding out and applyingHilbert modular forms. In 1986, upon readingKen Ribet's seminal work on Fermat's Last Theorem, Wiles set out to prove themodularity theorem for semistable elliptic curves, which implied Fermat's Last Theorem. By 1993, he had been able to convince a knowledgeable colleague that he had a proof of Fermat's Last Theorem, though a flaw was subsequently discovered. After an insight on 19 September 1994, Wiles and his studentRichard Taylor were able to circumvent the flaw, and published the results in 1995, to widespread acclaim.
In proving Fermat's Last Theorem, Wiles developed new tools for mathematicians to begin unifying disparate ideas and theorems. His former student Taylor along with three other mathematicians were able to prove the full modularity theorem by 2000, using Wiles' work. Upon receiving the Abel Prize in 2016, Wiles reflected on his legacy, expressing his belief that he did not just prove Fermat's Last Theorem, but pushed the whole of mathematics as a field towards theLanglands program of unifying number theory.[5]
Wiles began his formal schooling in Nigeria, while living there as a very young boy with his parents. However, according to letters written by his parents, for at least the first several months after he was supposed to be attending classes, he refused to go. From that fact, Wiles himself concluded that in his earliest years, he was not enthusiastic about spending time in academic institutions. In an interview withNadia Hasnaoui in 2021, he said he trusted the letters, yet he could not remember a time when he did not enjoy solving mathematical problems.[7]
Wiles attendedKing's College School, Cambridge,[8] andThe Leys School, Cambridge.[9] Wiles toldWGBH-TV in 1999 that he came across Fermat's Last Theorem on his way home from school when he was 10 years old. He stopped at his local library where he found a bookThe Last Problem, byEric Temple Bell, about the theorem.[10] Fascinated by the existence of a theorem that was so easy to state that he, a ten-year-old, could understand it, but that no one had proven, he decided to be the first person to prove it. However, he soon realised that his knowledge was too limited, so he abandoned his childhood dream until it was brought back to his attention at the age of 33 byKen Ribet's 1986 proof of theepsilon conjecture, whichGerhard Frey had previously linked to Fermat's equation.[11]
In 1989, Wiles was elected to theRoyal Society. At that point according to his election certificate, he had been working "on the construction of ℓ-adic representations attached toHilbert modular forms, and has applied these to prove the 'main conjecture' for cyclotomic extensions of totally real fields".[12]
From 1988 to 1990, Wiles was a Royal Society Research Professor at theUniversity of Oxford, and then he returned to Princeton.From 1994 to 2009, Wiles was aEugene Higgins Professor at Princeton.
Starting in mid-1986, based on successive progress of the previous few years ofGerhard Frey,Jean-Pierre Serre andKen Ribet, it became clear thatFermat's Last Theorem (the statement that no threepositiveintegersa,b, andc satisfy the equationan +bn =cn for any integer value ofn greater than2) could be proven as acorollary of a limited form of themodularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture").[15] The modularity theorem involved elliptic curves, which was also Wiles's own specialist area, and stated that all such curves have a modular form associated with them.[16][17] These curves can be thought of as mathematical objects resembling solutions for a torus' surface, and if Fermat's Last Theorem were false and solutions existed, "a peculiar curve would result". A proof of the theorem therefore would involve showing that such a curve would not exist.[18]
The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove.[19]: 203–205, 223, 226 For example, Wiles's ex-supervisorJohn Coates stated that it seemed "impossible to actually prove",[19]: 226 andKen Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."[19]: 223
Despite this, Wiles, with his from-childhood fascination with Fermat's Last Theorem, decided to undertake the challenge of proving the conjecture, at least to the extent needed forFrey's curve.[19]: 226 He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.[19]: 229–230
Wiles' research involved creating aproof by contradiction of Fermat's Last Theorem, which Ribet in his1986 work had found to have an elliptic curve and thus an associated modular form if true. Starting by assuming that the theorem was incorrect, Wiles then contradicted the Taniyama–Shimura–Weil conjecture as formulated under that assumption, with Ribet's theorem (which stated that ifn were aprime number, no such elliptic curve could have a modular form, so no odd prime counterexample to Fermat's equation could exist). Wiles also proved that the conjecture applied to the special case known as thesemistable elliptic curves to which Fermat's equation was tied. In other words, Wiles had found that the Taniyama–Shimura–Weil conjecture was true in the case of Fermat's equation, and Ribet's finding (that the conjecture holding for semistable elliptic curves could mean Fermat's Last Theorem is true) prevailed, thus proving Fermat's Last Theorem.[20][21][15]
In June 1993, he presented his proof to the public for the first time at a conference in Cambridge.Gina Kolata ofThe New York Times summed up the presentation as follows:
He gave a lecture a day on Monday, Tuesday and Wednesday with the title "Modular Forms, Elliptic Curves and Galois Representations". There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said. ... Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat's last theorem was true. Q.E.D.[18]
In August 1993, it was discovered that the proof contained a flaw in several areas, related to properties of theSelmer group and use of a tool called anEuler system.[22][23] Wiles tried and failed for over a year to repair his proof. According to Wiles, the crucial idea for circumventing—rather than closing—this area came to him on 19 September 1994, when he was on the verge of giving up. The circumvention usedGalois representations to replace elliptic curves, reduced the problem to aclass number formula and solved it, among other matters, all usingVictor Kolyvagin's ideas as a basis for fixingMatthias Flach's approach with Iwasawa theory.[23][22] Together with his former studentRichard Taylor, Wiles published a second paper which contained the circumvention and thus completed the proof. Both papers were published in May 1995 in a dedicated issue of theAnnals of Mathematics.[24][25]
In 2016, upon receiving theAbel Prize, Wiles said about his proof of Fermat's Last Theorem, "The methods that solved it opened up a new way of attacking one of the big webs of conjectures of contemporary mathematics called theLanglands Program, which as a grand vision tries to unify different branches of mathematics. It’s given us a new way to look at that".[5]
Wiles's proof of Fermat's Last Theorem has stood up to the scrutiny of the world's other mathematical experts. Wiles was interviewed for an episode of theBBC documentary seriesHorizon[27] about Fermat's Last Theorem. This was broadcast as an episode of the PBS science television seriesNova with the title "The Proof".[10] His work and life are also described in great detail inSimon Singh's popular bookFermat's Last Theorem.
