Wiles, Andrew John

▪ 1994

      In June 1993, at a small conference of mathematicians at the Isaac Newton Institute, Cambridge, Andrew Wiles dropped a historic bombshell. He had solved one of mathematics' oldest mysteries, Fermat's last theorem. The Princeton University professor's seven-year attack on the 350-year-old problem, one that many mathematicians had declared unsolvable, ended when on the third and final day of his lecture on elliptic curves, he announced his proof of the theorem. In December, during review of the 200-page proof for publication, a small snag was found, but the pervasive feeling among Wiles's peers was that he had indeed pulled off the seemingly impossible.

      Fermat's last theorem, so called because it was the only remaining theorem of the 17th-century French mathematician Pierre de Fermat that had been neither proved nor disproved, dates to 1637. While reading a copy of the Arithmetica by the ancient Greek mathematician Diophantus, Fermat scribbled the theorem in the book's margin along with the note that he had a "remarkable proof" but that "the margin is too small to contain it." So began a mystery that stymied the world's greatest mathematical minds for the next three centuries. Particularly vexing for would-be solvers was the theorem's simplicity. It states that there are no positive integer solutions to xⁿ + yⁿ = zⁿ when n is greater than two. Fermat himself eventually found a space large enough to write a proof for n = 4, and others elaborated proofs for specific cases of n. A universal solution, however, remained elusive until Wiles made his June announcement.

      The heart of Wiles's proof relies upon the Taniyama-Weil conjecture, a difficult problem in number theory dealing with the nature of elliptic curves (equations of the form y² = x³ + ax + b, in which a and b are constants). In 1986 Kenneth A. Ribet of the University of California at Berkeley showed that if the conjecture could be proved, a proof of Fermat's theorem would follow. Inspired by Ribet's work, Wiles laboured privately and secretly in the attic office of his Princeton home for seven years. By May 1993 he had succeeded in solving a special case of the Taniyama-Weil conjecture—enough to prove Fermat's last theorem—and he quickly signed on to speak at the Cambridge conference. After the first day's lecture, speculation swelled that Wiles had a solution to Fermat's problem. On the third day, when Wiles finally revealed the proof, he received a standing ovation from the audience.

      Wiles was born April 11, 1953, in Cambridge. He had dreamed of solving Fermat's last theorem from the age of 10, when he first encountered the problem in a book. In fact, he credited the problem with inspiring his interest in mathematics and leading him to his life's work as a number theorist. Wiles, who was also married to a mathematician (they used to recite the decimal places of pi to each other), studied mathematics at Clare College, Cambridge, earning an M.A. in 1977 and a Ph.D. in 1980. He joined the mathematics faculty of Princeton University in 1980.

      (JAMES HENNELLY)

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▪ English mathematician

born April 11, 1953, Cambridge, Eng.

      British mathematician who proved Fermat's last theorem; in recognition he was awarded a special silver plaque—he was beyond the traditional age limit of 40 years for receiving the gold Fields Medal—by the International Mathematical Union in 1998.

      Wiles was educated at Merton College, Oxford (B.A., 1974), and Clare College, Cambridge (Ph.D., 1980). Following a junior research fellowship at Cambridge (1977–80), Wiles held an appointment at Harvard University, Cambridge, Mass., U.S., and in 1982 moved to Princeton (N.J.) University. Wiles worked on a number of outstanding problems in number theory: the Birch and Swinnerton-Dyer conjectures, the principal conjecture of Iwasawa theory, and the Shimura-Taniyama-Weil conjecture. The last work provided resolution of the legendary Fermat's last theorem (not really a theorem but a long-standing conjecture)—i.e., that there do not exist positive integer solutions of xⁿ + yⁿ = zⁿ for n > 2. In the 17th century Fermat had claimed a solution to this problem, posed 14 centuries earlier by Diophantus, but he gave no proof, claiming insufficient room in the margin. Many mathematicians had tried to solve it over the intervening centuries, but with no success. Wiles had been fascinated by the problem from the age of 10, when he first saw the conjecture. In his paper in which the proof of the theorem appears, Wiles starts off with Fermat's quote (in Latin) about the margin being too narrow and then proceeds to give a recent history of the problem leading up to his solution.

      During the seven years Wiles devoted to developing his proof, he worked on little else. His solution involves elliptic curves and modular forms and builds on the work of Gerhard Frey, Barry Mazur, Kenneth Ribet, Karl Rubin, Jean-Pierre Serre, and many others. The results were first announced in a series of lectures at Cambridge in June 1993—lectures innocently titled “Modular Forms, Elliptic Curves, and Galois Representations.” When the implications of the lectures became clear, it created a sensation, but, as often happens in the case of complicated proofs of extremely difficult problems, there were some gaps in the argument that had to be filled in, and this process was not completed until 1995, with help from Richard Taylor. Following the proof of the Fermat problem, Wiles won the Wolf Prize (1995–96).

      His paper “Modular Elliptic Curves and Fermat's Last Theorem” was published in the Annals of Mathematics 141:3 (1995), pp. 443–551, accompanied by a necessary additional article, “Ring-Theoretic Properties of Certain Hecke Algebras,” coauthored with Taylor.

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