From 1988 to 1990, Wiles was a Royal Society Research Professor at the
University of Oxford, and then he returned to Princeton. From 1994 to 2009, Wiles was a
Eugene Higgins Professor at Princeton. Starting in mid-1986, based on successive progress of the previous few years of
Gerhard Frey,
Jean-Pierre Serre and
Ken Ribet, it became clear that
Fermat's Last Theorem (the statement that no three
positive integers , , and satisfy the equation for any integer value of greater than ) could be proven as a
corollary of a limited form of the
modularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture"). These curves can be thought of as mathematical objects resembling solutions for a torus's surface, and if Fermat's Last Theorem were false and solutions existed, "a peculiar curve would result". A proof of the modularity theorem therefore would have as a consequence that such a curve would not exist. For example, Wiles's ex-supervisor
John Coates stated that it seemed "impossible to actually prove", 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 used
Galois representations to replace elliptic curves, reduced the problem to a
class number formula and solved it, among other matters, all using
Victor Kolyvagin's ideas as a basis for fixing
Matthias Flach's approach with Iwasawa theory. Together with his former student
Richard 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 the
Annals of Mathematics. ==Later career==