Andrew Wiles

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Andrew John Wiles (born April 11, 1953) is a British mathematician living in the United States. In 1974, he received his bachelor's degree from the University of Oxford. He then completed his Ph.D. at the University of Cambridge in 1979 and is currently a Professor at Princeton University, and chair of the department of mathematics, Princeton University. In one of the great success stories in the history of mathematics, Wiles (with help from Richard Taylor) proved Fermat's Last Theorem in 1994.

Before this result, Andrew Wiles had done outstanding work in number theory. In work with John Coates he obtained some of the first results on the famous Birch and Swinnerton-Dyer conjecture, and he also did important work on the main conjecture of Iwasawa theory.

Fermat's Last Theorem (FLT) asserts that there are no positive integers x, y, and z such that

in which n is a natural number greater than 2.

Wiles had been inspired by the problem as a child when he encountered it in E.T. Bell's book, The Last Problem. His odyssey towards the final proof began in 1985 when Ken Ribet, inspired by ideas of Jean-Pierre Serre and Gerhard Frey, proved that FLT would follow from another conjecture of Taniyama, Shimura and Weil, to the effect that every elliptic curve can be parametrized by modular forms. Though less familiar than Fermat's Last Theorem, the Taniyama-Shimura theorem is the more significant of the two, because it touches on truly deep currents in number theory. No one had any idea how to prove it. Working in absolute secrecy, and sharing his ideas and progress only with Nicholas Katz, another professor of mathematics at Princeton, Wiles eventually developed a proof of the Taniyama-Shimura-Weil conjecture, and hence of FLT. The proof is a tour de force introducing many new ideas.

Wiles was uncharacteristically dramatic in revealing the proof. He arranged to give three lectures at the Isaac Newton Institute, Cambridge, England, in June of 1993. He did not announce the topic of the lectures in advance, and as the audience and the world became aware of where the lectures were headed, the audience swelled so that the third lecture was to an overpacked room. At the end of the third lecture, he announced "(...) this proves Fermat's Last Theorem. I'll stop here", and received a standing ovation.

In the following months, the manuscript of the proof was circulated only to a small number of mathematicians while the world awaited. The first version of the proof depended on the construction of an object called an Euler system, and this aspect proved problematical, a flaw emerged during peer review of the subtle and complex mathematics involved. For almost a year it began to seem that Wiles' proof was destined like so many others to be fatally flawed, and that although he had made many important discoveries, the ultimate goal had eluded him. Wiles was on the point of giving up finally, when he decided to have one last try at solving the last remaining problem in his proof in collaboration with Richard Taylor, one of his former PhD students in 1994. He commented:

"... suddenly, totally unexpectedly, I had this incredible revelation. It was the most important moment of my working life. Nothing I ever do again will mean as much ... it was so indescribably beautiful, it was so simple and so elegant, and I just stared in disbelief for twenty minutes, then during the day I walked round the department. I'd keep coming back to my desk to see it was still there – it was still there."

The final version of Wiles' proof, which therefore differs from his original one, was published in the Annals of Mathematics 141 (1995), pp. 443–551, together with another, supporting article by Wiles and Taylor titled "Ring-theoretic properties of certain Hecke algebras" (Annals of Mathematics 141 (1995), pp. 553–572) relating to the final step of discovery.

Wiles was awarded several prizes in mathematics: Schock Prize (1995), Royal Medal (1996), Cole Prize (1996), Wolf Prize (1996), King Faisal Prize (1998), Clay Research Award (1999) and Shaw Prize (2005).

Andrew Wiles should not be confused with André Weil, another famous mathematician who, like Wiles, has done important work in elliptic curves.

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