Andrew Wiles

For the French mathematician known for work in the area of elliptic curves, see André Weil.

Andrew Wiles
Sir Andrew John Wiles
Born	(1953-04-11) April 11, 1953 Cambridge, England
Nationality	British
Alma mater	Oxford University Cambridge University
Known for	Proving Fermat's Last Theorem
Awards	Wolf Prize (1995) Royal Medal (1996) Fermat Prize (1995) Shaw Prize (2005)
Scientific career
Fields	Mathematics
Institutions	Princeton University
Doctoral advisor	John Coates
Doctoral students	Manjul Bhargava Brian Conrad Karl Rubin Chris Skinner Richard Taylor

Sir Andrew John Wiles KBE FRS (born April 11 1953) is a British mathematician and a professor at Princeton University, specialising in number theory. He is most famous for proving Fermat's Last Theorem.

Early work

Andrew Wiles was born in Cambridge, England in 1953 and attended King's College School, Cambridge (where his maths teacher, David Higginbottom first introduced Fermat's Last Theorem to him) and The Leys School, Cambridge; and earned his BA degree in 1974 after study at Merton College, Oxford, and a Ph.D. in 1980 after research at Clare College, Cambridge. His graduate research was guided by John Coates beginning in the summer of 1975. Together they worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory. He further worked with Barry Mazur on the main conjecture of Iwasawa theory over Q and soon afterwards generalized this result to totally real fields. Taking approximately seven years to complete the work, Wiles was the first person to prove Fermat's Last Theorem, earning him a place in history.

Solution of Fermat's Last Theorem

Andrew Wiles' most famous mathematical result is that all rational semistable elliptic curves are modular which, in particular, implies Fermat's Last Theorem.

Wiles was introduced to Fermat's Last Theorem at the age of ten. He tried to prove the theorem using textbook methods and later studied the work of mathematicians who had tried to prove it. When he began his graduate studies he stopped trying to prove it and began studying elliptic curves under the supervision of John Coates.

In the 1950s and 1960s a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on some ideas that Yutaka Taniyama posed. In the West it became well known through a paper by André Weil. With Weil giving conceptual evidence for it, it is sometimes called the Shimura-Taniyama-Weil conjecture. It states that every rational elliptic curve is modular. The full conjecture was proven by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor in 1998 using many of the methods that Andrew Wiles used in his 1995 published papers.

Fermat's Last Theorem states that no nontrivial integer solutions exist for the equation: xn + yn = zn if n is an integer greater than two.

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The bridge between Fermat and Taniyama

If p is an odd prime and a, b, and c are positive integers such that ap+bp=cp, then a corresponding equation y² = x(x - ap)(x + bp) defines a hypothetical elliptic curve, called the Frey curve, which must exist if there is a counterexample to Fermat's Last Theorem. Following on work by Yves Hellegouarch who first considered this curve, Frey pointed out that if such a curve existed it had peculiar properties, and suggested in particular that it might not be modular.

A connection between Taniyama-Shimura and Fermat was made by Ken Ribet, following on work by Barry Mazur and Jean-Pierre Serre, with his proof of the epsilon conjecture showing that Frey's idea that the Frey curve could not be modular was correct. In particular, this showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Wiles made the decision that he would work exclusively on the Taniyama-Shimura conjecture shortly after he had learned that Ribet had proven the epsilon conjecture in 1986. While many mathematicians thought the Taniyama-Shimura conjecture was inaccessible, Wiles resolved to follow that approach.

When Wiles first began studying Taniyama-Shimura, he would casually mention Fermat to people, but he found that doing so created too much interest. He wanted to be able to work on his problem in a concentrated fashion, and if people were expressing too much interest then he would not have been able to focus on his problem. Consequently he let only Nicholas Katz know what he was working on. Wiles did not do any research that was not related to Taniyama-Shimura, though of course he did continue in his teaching duties at Princeton University; continuing to attend seminars, lecture undergraduates, and give tutorials.

Cultural references

• Wiles's work on Fermat's Last Theorem was commemorated (in fictional form) in the musical Fermat's Last Tango, written by Joanne Sydney Lessner and Joshua Rosenblum.[1]

• Wiles and his work on Fermat's last theorem were mentioned in the Star Trek: Deep Space Nine episode "Facets". This also served as a correction for Fermat's last theorem being said to be unsolved in the earlier Star Trek: The Next Generation episode "The Royale". Details may be found at Fermat's Last Theorem in fiction.

• Wiles is mentioned in Tom Lehrer's song "That's Mathematics"

• Information about Wiles' proof of Fermat's Last Theorem appears in The Parrot's Theorem

• Wiles was interviewed for an episode of BBC's documentary series Horizon that focused on Fermat's Last Theorem

• One of two multiple choice questions, on a fake television quiz show watched in episode 15 of the anime series Lucky Star, concerns "Princeton's Sir Wiles", and asks which theorem he proved in 1994.

• In the 2007 mystery film The Oxford Murders, "Professor Wilkes" (Martin Nigel Davey) presents his proof of "Bormat"'s Last Theorem at Cambridge.

• Wiles was the subject of the documentary "The Proof", an episode of the PBS television science series Nova.

• The AOL Instant Messenger "Trout Bots" occasionally ask the question, "Mathematician Andrew Wiles is hideously ugly, don't you think?"

Awards

Wiles has been awarded several major prizes in mathematics and science

• Fellow of the Royal Society (1989)
• Junior Whitehead Prize of the LMS, (1988)
• Schock Prize (1995)
• Cole Prize (1997)
• National Academy of Sciences Award in Mathematics from the American Mathematical Society (1996) [2]
• Ostrowski Prize (1996) [3][4]
• Royal Medal (1996)
• Wolf Prize (1996)
• Wolfskehl Prize (1997) [5] - see Paul Wolfskehl
• A silver plaque from the International Mathematical Union (1998) recognizing his achievements, in place of the Fields Medal, which is restricted to those under 40 (Wiles was born in 1953 and proved the theorem in 1994). [6]
• King Faisal Prize (1998) [7]
• Clay Research Award (1999)
• Named Knight Commander of the Order of the British Empire (2000).
• Shaw Prize (2005) [8]

Other Biographical details

Wiles' father is Rev. Prof. Maurice Frank Wiles (1923-2005), Regius Professor of Divinity at the University of Oxford and his mother Patricia Wiles (née Mowll). His father worked as Chaplain at Ridley Hall Cambridge for the period 1952-55. Wiles is married to Nada Canaan Wiles, who has a PhD in microbiology from Princeton, and they have two daughters Since 1994 he has been Eugene Higgins Professor at Princeton. He is a Foreign member of the US National Academy of Sciences since 1996 (as he remains a British citizen).

He is currently Chair of the Mathematics Department at Princeton.

References

• Andrew Wiles (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. doi:10.2307/2118559. {{cite journal}}: Unknown parameter |month= ignored (help)
• Richard Taylor and Andrew Wiles (1995). "Ring-theoretic properties of certain Hecke algebras" (PDF). Annals of Mathematics. 141 (3): 553–572. doi:10.2307/2118560. {{cite journal}}: Unknown parameter |month= ignored (help)
• Gerd Faltings (1995). "The Proof of Fermat's last theorem by R. Taylor and A. Wiles" (PDF). Notices of the AMS. 42 (7): 743–746.
• John Coates (1996). "Wiles Receives NAS Award in Mathematics" (PDF). Notices of the AMS. 43 (7): 760–763. {{cite journal}}: Unknown parameter |month= ignored (help)
• Singh, Simon (1997). Fermat's Enigma. ISBN 0-8027-1331-9.
• Mozzochi, Charles (2000). The Fermat Diary. ISBN 0-8218-2670-0.
• "[[NOVA (TV series)|NOVA Online]]: The Proof". WGBH. 1997. Retrieved 2006-05-03. {{cite web}}: URL–wikilink conflict (help)
• "Bluff your way in Fermat's Last Theorem". Retrieved 2006-05-03.
• "Fermat's Last Tango". Retrieved 2006-05-22.

External links

• Andrew Wiles at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Andrew Wiles", MacTutor History of Mathematics Archive, University of St Andrews
• Princeton home page
• Andrew Wiles' bibliography