In one of the great success stories in the history of mathematics, Andrew 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 a famous conjecture of Birch and Swinnerton-Dyer, and he also did important work on the main conjecture of Iwasawa theory. He was and is a professor at Princeton University.
Fermat's Last Theorem (FLT) asserts that there are no positive natural numbers x, y, and z such that
Wiles' odyssey began in 1985 when Ken Ribet, inspired by an idea of Gerhard Frey, proved that FLT would follow from another conjecture of Taniyama and Shimura, to the effect that every elliptic curve can be parametrized by modular forms. Though less familiar than Fermat's Last Theorem, the Taniyama-Shimura conjecture 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 conjecture, and of hence 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 Newton Institute 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.
In the coming 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, so the final version of the proof differed from his original one. This difficulty was overcome with the help of Richard Taylor.
The final version of Wiles' proof was published in the Annals of Mathematics 141 (1995), p. 443-551, together with another, supporting article by Wiles and Taylor titled "Ring-theoretic properties of certain Hecke algebras" (Annals of Mathematics 141 (1995), p. 553-572).
Wiles was awarded a Wolf Prize in mathematics, in 1996.
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