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Sir Andrew John Wiles (born April 11 , 1953 ) is a British Mathematician living in the United States . He was educated at The Leys School Cambridge and in 1974 he graduated from the University Of Oxford . He then completed his Ph.D. at Clare College of the University Of Cambridge in 1979 and is currently a Professor of Number Theory and the chair of the department of Mathematics at Princeton University in Fine Hall. Wiles is well known for his proof of Fermat's Last Theorem and before he proved it, he developed his reputation as a brilliant number theorist when he worked under the supervision of John Coates on Elliptic Curves . Working on elliptic curves he and John Coates made some of the first breakthroughs on the famous Birch And Swinnerton-Dyer Conjecture ( One Of The Millennium Prize Problems ), and he and Barry Mazur did important work on the main conjecture of Iwasawa Theory . Andrew Wiles is an international expert on the Birch And Swinnerton-Dyer Conjecture He still writes papers on elliptic curves. In 1999 Wiles opened the Wiles Centre for Technology at King's College School[http://www.kcs.cambs.sch.uk/Publisher/Article.aspx?id=18861 . INTEREST IN MATHEMATICS AT A YOUNG AGE Wiles loves mathematics and his love of mathematics dates from his early childhood days where he grew up in Cambridge in England. He loved working on the mathematical problems from school and he would make up his own problems that were new to him. The best problem he has ever comes across is Fermat's Last Theorem , because it is very simple to understand but the great mathematicians before him had not been able to solve it. FERMAT'S LAST THEOREM Fermat's Last Theorem (FLT) states that if n > 2, then the equation, xn + yn = zn has no solutions where x, y, and z are Integers . Pierre De Fermat did not discover his equation, but he did say that he had a proof of the fact that there is no solution to his equation. Fermat's proof has never been recovered. E.T. Bell wrote a book on the problem titled, ''The Last Problem''. When Wiles was 10 years old, he found the book in his local library and he tried to solve it himself, thinking that all the people in the world expect Fermat who had tried to prove it missed something that maybe he would not miss. He studied all the techniques that had been used and decided that none of them were really going to work. In graduate school Wiles stopped working on the problem, and began working with John Coates on elliptic curves. ELLIPTIC CURVES, MODULAR FORMS, AND FERMAT'S LAST THEOREM In the 1950s and 1960s a connection between elliptic curves and modular forms was conjectured by the two Japanese mathematicians Taniyama and Shimura . In the West it became well known through a paper Weil wrote. Weil gave conceptual evidence for it. The conjecture is now a theorem and was proved in 1999. This theorem is called the Taniyama-Shimura Theorem . It states that every rational elliptic curve's given j-invariant is modular. While it was not proved until 1999, papers were published saying what would be the results of the conjecture being correct. The problem with that is if it were wrong than anything that relied on the conjecture would be too. The following equation, , is a hypothetical Elliptic Curve . While it had been studied long before the connection between elliptic curves, the Fermat equation, and modular forms was made, Frey was the first to suggest that if it had any solutions it was not modular and it represented the Fermat equation as an elliptic curve. Serre created the precise mechanism that related modular forms with Fermat's equation. It was called the epsilon conjecture and was proved by Ribet . The work Ribet did told mathematicians not only that the epsilon conjecture was correct but that only a proof of the semistable elliptic curve case of Taniyama-Shimura would imply Fermat's Last Theorem. Just shortly after he had learnt that Ribet had proved the epsilon conjecture in 1986, Wiles made the decision that what he would work on was the Taniyama-Shimura conjecture. To many people the Taniyama-Shimura conjecture was just completely inaccessible, because no-one had been able to show that there is the same number of elliptic curves and modular forms. Ken Ribet (the man who proved that the semi-stable case of Taniyama-Shimura implied Fermat's Last Theorem) thinks that Andrew Wiles may have been one of a few people who had the audacity to dream that the conjecture could really be proved. Although Wiles was not trying to prove the complete Taniyama-Shimura conjecture the case he was trying to prove was still nearly as difficult to prove as the full conjecture. When Wiles first began studying Taniyama-Shimura, he would casually mention Fermat to people. He found that doing so created too much interest. He wanted to be able to work on his problem that demanded a lot of concentration. If people were talking to him a lot, then he would not have been able to focus on his problem. This was a problem that would take years to solve. Wiles did not do any research that was not related to Taniyama-Shimura. He did continue to work for Princeton university, and continued to attend seminars, lecture to undergraduates, and give tutorials. Simon Singh has suggested that he also did it so he would be the one to prove FLT. The problem was how do you count and it was the big question on the subject. Wiles introduced the correct counting technique. Wiles converted elliptic curves by their j-invariant to their corresponding Galois representation to study elliptic curves. He did studies on representations according to partations and said in his paper on this, "suppose for the moment that is irreducible". He tells us why 3 is important. He found a surprising link between Galois Representations and Modular Forms and '''the interpretation of special values of L-functions '''. The proof is based on that link and is used to prove a hypothesis that is semistable at 3 by linking some of Commutative Algebra with a well-know type for a Class Number problem. ANNOUNCEMENT OF THE PROOF Andrew Wiles gave a lecture series called Modular Forms, Elliptic Curves, and Galois Representations at the Isaac Newton Institute for the programme L-functions and Arithmetic for three days in June of 1993. He was originally only given two days, but John Coates gave up his slot, because he thought Wiles had something very important to say. REFEREEING OF WILES' MANUSCRIPT AND PUBLICATION Wiles' manuscript was submitted to '' Inventiones Mathematicae '' and refereed by a team of six. The team was organized by Barry Mazur, editor of Inventiones Mathematicae, and included Ken Ribet , Nick Katz , and Richard Taylor . The first version of the proof depended on the construction of an object called an Euler system. This aspect proved problematic, as a flaw emerged during peer review. For almost a year Wiles thought that the flaw might be fatal, and that although he had made many important discoveries, the ultimate goal had eluded him. He was on the verge of giving up, when he decided to make one last attempt to solve the last remaining problem in his proof in collaboration with Richard Taylor , one of his former PhD students in 1994. 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. THE MATHEMATICAL COMMUNITY'S REACTION Barry Mazur said that he had never seen a lecture series in mathematics like the one Wiles gave. He said that the reason the series was unique was that there were many new glorious ideas and that it was suspenseful until the end. COMPLETE PROOF OF TANIYAMA SHIMURA The proof of the Taniyama-Shimura conjecture is a huge achivement. Before Andrew Wiles' work on the problem it was thought to be inaccessible and many people had tried to prove it. Its importance is based on the fact that it is at the centre of the arithmetic of elliptic curves. Richard Taylor formulated a novel strategy using Andrew Wiles' method, and the strategy was used for proving the Artin conjecture in the remaining (most interesting) case where the image of ½ in PGL2(C) is isomorphic to A5—the socalled icosahedral case. Enough of Taylor’s program has now been carried out in joint work with Kevin Buzzard, Mark Dickinson, and Nicholas Shepherd Barron to establish the truth of the Artin Conjecture for infinitely many icosahedral Galois representations.
WORK AFTER HIS WORK ON HIS MANUSCRIPTS Andrew Wiles has written papers on modular forms. He has given lectures at the Fermat meeting. He is on the Scientific Advisory Board of the Clay Mathematics Institute which says that it is dedicated to increasing and disseminating mathematical knowledge. He has written an article for the institute on the Birch And Swinnerton-Dyer Conjecture to formulate the problem, which is one of the seven Millennium Prize Problems. THE PUBLIC'S AWARENESS OF WILES' STORY A film directed by Simon Singh focuses on Wiles' story and interviews with Wiles and his colleagues. He wrote a book that focused more on the history of Fermat's Last Theorem and gave a deeper look at the mathematics involved than the film. AWARDS Wiles has been awarded several major prizes in mathematics: Schock Prize ( 1995 ), Royal Medal (1996), Cole Prize (1996), Wolf Prize ( 1996 ), a silver plate from the International Mathematical Union ( 1998 ), King Faisal Prize ( 1998 ), Clay Research Award ( 1999 ) and Shaw Prize ( 2005 ). He became a Knight Of The British Empire in 2000. Wiles cannot receive the Fields Medal as the award can only be given to those below 40 years of age (Wiles was born in 1953 and proved the theorem in 1994), a rule strictly adhered to. REFERENCES Papers by Andrew wiles
# (with J. Coates ) Kummer’s criterion for Hurwitz numbers, Algebraic Number Theory, Kyoto, (1977), 9-23. # (with J. Coates) Explicit reciprocity laws, Soci´et´e Math. France, Ast´erisque, 41-42, (1977), 7-17. # (with J. Coates) On the conjecture of Birch and Swinnerton-Dyer , Inventiones Mathematicae , 39 (1977), 223-251. # Higher explicit reciprocity laws, Annals of Mathematics, 107 (1978), 235-254. # (with J. Coates) On p-adic L-functions and elliptic units, Jour. Aus. Math. Soc., 26 (1978), 1-26. # On modular curves and the class group of Q(�p), Inventiones Mathematicae, 58 (1980), 1-35. # (with K. Rubin ) Mordell-Weil groups of elliptic curves over cyclotomic fields, Proceedings of a Conference on Number Theory related to Fermat’s Last Theorem, Boston: Birkhauser, (1982), 237-254. # (with B. Mazur ) Analogies between function fields and number fields, American Journal of Mathematics, June (1983), 507-521. # (with B. Mazur) Class fields of abelian extensions of Q, Inventiones Mathematicae, 76 (1984), 179-330. # On p-adic representations of totally real fields, Annals of Mathematics, 123 (1986), 407-456. # (with B. Mazur) On p-adic analytic families of Galois representations, Compositio Mathematicae, 59 (1986), 231-261. # On ordinary �-adic representations associated to modular forms, Inventiones Mathematicae, 94 (1988), 529-573. # The Iwasawa conjecture for totally real fields, Annals of Mathematics, 131 (1990), 493-540. # On a conjecture of Brumer, Annals of Mathematics, 131 (1990), 555-565. Andrew J. Wiles 2 # Modular Elliptic Curves and Fermat’s Last Theorem , Annals Of Mathematics , 141, (1995), 443-551. # (with R. Taylor ) Ring-theoretic properties of Certain Hecke Algebras , Annals of Mathematics, 141, (1995), 553-572. # (with C. Skinner ) Buse Change and a problem of Serre, Duke Math. J., 107, No. 1, (2001), 15-25. # (with C. Skinner), Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math (6), 10, No. I, (2001), 185-215. Books, television transcripts, and articles
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