Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory.

He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal and for which he was appointed a Knight Commander of the Order of the British Empire in 2000.

After moving to Oxford and graduating from there in 1974, he worked on unifying Galois representations, elliptic curves and modular forms, starting with Barry Mazur's generalizations of Iwasawa theory.

After an insight on 19 September 1994, Wiles and his student Richard Taylor were able to circumvent the flaw, and published the results in 1995, to widespread acclaim.

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 the Langlands program of unifying number theory.

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 by Ken Ribet's 1986 proof of the epsilon conjecture, which Gerhard Frey had previously linked to Fermat's equation.

He further worked with Barry Mazur on the main conjecture of Iwasawa theory over the rational numbers, and soon afterward, he generalised this result to totally real fields.

[14] In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure.

At that point according to his election certificate, he had been working "on the construction of ℓ-adic representations attached to Hilbert modular forms, and has applied these to prove the 'main conjecture' for cyclotomic extensions of totally real fields".

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 a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2) 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").

"[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 for Frey'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 a proof by contradiction of Fermat's Last Theorem, which Ribet in his 1986 work had found to have an elliptic curve and thus an associated modular form if true.

Gina Kolata of The 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".

[18]In August 1993, it was discovered that the proof contained a flaw in several areas, related to properties of the Selmer group and use of a tool called an Euler system.

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.

Notably, in 1999, three of his former students, Richard Taylor, Brian Conrad, and Fred Diamond, working with Christophe Breuil, built upon Wiles' proof to prove the full modularity theorem.