# Andrew Wiles

Sir Andrew John Wiles (born 11 April 1953)[1] is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize[6] and the 2017 Copley Medal by the Royal Society.[3] He was appointed Knight Commander of the Order of the British Empire in 2000, and in 2018 was appointed as the first Regius Professor of Mathematics at Oxford.[7]

Sir Andrew Wiles

Wiles at the 61st birthday conference for Pierre Deligne at the Institute for Advanced Study in 2005
Born
Andrew John Wiles

11 April 1953 (age 65)[1]
Cambridge, England
NationalityBritish
EducationKing's College School, Cambridge
The Leys School[1]
Alma mater
Known forProving the Taniyama–Shimura Conjecture for semistable elliptic curves, thereby proving Fermat's Last Theorem
Proving the main conjecture of Iwasawa theory
Awards
Scientific career
FieldsMathematics
Institutions
ThesisReciprocity Laws and the Conjecture of Birch and Swinnerton-Dyer (1979)
Doctoral students

## Education and early life

Wiles was born on 11 April [8] 1953 in Cambridge, England, the son of Maurice Frank Wiles (1923–2005), the Regius Professor of Divinity at the University of Oxford,[1] and Patricia Wiles (née Mowll). His father worked as the chaplain at Ridley Hall, Cambridge, for the years 1952–55. Wiles attended King's College School, Cambridge, and The Leys School, Cambridge.[9]

Wiles states that he came across Fermat's Last Theorem on his way home from school when he was 10 years old. He stopped at his local library where he found a book about the theorem.[10] Fascinated by the existence of a theorem that was so easy to state that he, a ten year old, could understand it, but that no one had proven, he decided to be the first person to prove it. However, he soon realised that his knowledge was too limited, so he abandoned his childhood dream, until it was brought back to his attention at the age of 33 by Ken Ribet's 1986 proof of the epsilon conjecture, which Gerhard Frey had previously linked to Fermat's famous equation.[11]

## Career and research

Wiles earned his bachelor's degree in mathematics in 1974 at Merton College, Oxford, and a PhD in 1980 as a graduate student of Clare College, Cambridge.[5] After a stay at the Institute for Advanced Study in Princeton, New Jersey in 1981, Wiles became a Professor of Mathematics at Princeton University. In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford, and then he returned to Princeton. From 1994 - 2009, Wiles was a Eugene Higgins Professor at Princeton. He rejoined Oxford in 2011 as Royal Society Research Professor.[12] In May 2018 he was appointed Regius Professor of Mathematics at Oxford, the first in the university's history.[7]

Wiles's graduate research was guided by John Coates beginning in the summer of 1975. Together these colleagues 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 the rational numbers, and soon afterward, he generalised this result to totally real fields.[13]

His biographical page at Princeton University's website states that "Andrew has few equals in terms of his impact on modern number theory. Many of the world’s very best young number theorists received their Ph.D.'s under Andrew ... and many of these are today leaders and professors at top institutions around the world".[14]

### Proof of Fermat's Last Theorem

Starting in mid-1986, based on successive progress of the previous few years of Gerhard Frey, Jean-Pierre Serre and Ken Ribet, it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture"). The modularity theorem involved elliptic curves, which was also Wiles's own specialist area.[15]

The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove.[16]:203–205, 223, 226 For example, Wiles's ex-supervisor John Coates states that it seemed "impossible to actually prove",[16]:226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."[16]:223

Despite this, Wiles, with his from-childhood fascination with Fermat's Last Theorem, decided to undertake the challenge of proving the conjecture, at least to the extent needed for Frey's curve.[16]:226 He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.[16]:229–230

In June 1993, he presented his proof to the public for the first time at a conference in Cambridge.

He gave a lecture a day on Monday, Tuesday and Wednesday with the title 'Modular Forms, Elliptic Curves and Galois Representations.' There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said. ... Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat's last theorem was true. Q.E.D.[17]

In August 1993, it was discovered that the proof contained a flaw in one area. Wiles tried and failed for over a year to repair his proof. According to Wiles, the crucial idea for circumventing, rather than closing this area, came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a second paper which circumvented the problem and thus completed the proof. Both papers were published in May 1995 in a dedicated issue of the Annals of Mathematics.[18][19]

### Awards and honours

Andrew Wiles in front of the statue of Pierre de Fermat in Beaumont-de-Lomagne (October 1995)

Wiles's proof of Fermat's Last Theorem has stood up to the scrutiny of the world's other mathematical experts. Wiles was interviewed for an episode of the BBC documentary series Horizon[20] that focused on Fermat's Last Theorem. This was renamed "The Proof", and it was made an episode of the US Public Broadcasting Service's science television series Nova.[10] His work and life are also described in great detail in Simon Singh's popular book Fermat's Last Theorem.

Wiles has been awarded a number of major prizes in mathematics and science:

Wiles's 1987 certificate of election to the Royal Society reads:

Andrew Wiles is almost unique amongst number-theorists in his ability to bring to bear new tools and new ideas on some of the most intractable problems of number theory. His finest achievement to date has been his proof, in joint work with Mazur, of the "main conjecture" of Iwasawa theory for cyclotomic extensions of the rational field. This work settles many of the basic problems on cyclotomic fields which go back to Kummer, and is unquestionably one of the major advances in number theory in our times. Earlier he did deep work on the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication – one offshoot of this was his proof of an unexpected and beautiful generalisation of the classical explicit reciprocity laws of Artin–Hasse–Iwasawa. Most recently, he has made new progress on the construction of ℓ-adic representations attached to Hilbert modular forms, and has applied these to prove the "main conjecture" for cyclotomic extensions of totally real fields – again a remarkable result since none of the classical tools of cyclotomic fields applied to these problems.[22]

## References

1. Anon (2017) WILES, Sir Andrew (John). ukwhoswho.com. Who's Who (online Oxford University Press ed.). A & C Black, an imprint of Bloomsbury Publishing plc. doi:10.1093/ww/9780199540884.013.39819. (subscription required)
2. ^ a b Castelvecchi, Davide (2016). "Fermat's last theorem earns Andrew Wiles the Abel Prize". Nature. 531 (7594): 287–287. Bibcode:2016Natur.531..287C. doi:10.1038/nature.2016.19552. PMID 26983518.
3. ^ a b c "Mathematician Sir Andrew Wiles FRS wins the Royal Society's prestigious Copley Medal". The Royal Society. Retrieved 27 May 2017.
4. ^ a b
5. ^ a b Wiles, Andrew John (1978). Reciprocity laws and the conjecture of birch and swinnerton-dyer. lib.cam.ac.uk (PhD thesis). University of Cambridge. OCLC 500589130. EThOS uk.bl.ethos.477263.
6. ^ "The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2016 to Sir Andrew J. Wiles". Abelprisen.no. Retrieved 23 August 2018.
7. ^ a b "Sir Andrew Wiles appointed first Regius Professor of Mathematics at Oxford". News & Events. University of Oxford. 2018-05-31. Retrieved 2018-06-01.
8. ^ "Andrew Wiles". famous-mathematicians.com.
9. ^ "Cambridge-born mathematician awarded top prize for solving centuries-old numerical problem". Cambridge News. Retrieved 16 March 2016.
10. ^ a b "Andrew Wiles on Solving Fermat". WGBH. Retrieved 16 March 2016.
11. ^ Chang, Sooyoung (2011). Academic Genealogy of Mathematicians. p. 207. ISBN 9789814282291.
12. ^ a b c d O'Connor, John J.; Robertson, Edmund F. (September 2009). "Wiles Biography". MacTutor History of Mathematics archive. Retrieved 16 March 2016.
13. ^ "Andrew Wiles". National Academy of Sciences. Retrieved 16 March 2016.
14. ^ "Andrew John Wiles - Dean of the Faculty". dof.princeton.edu.
15. ^ Brown, Peter (28 May 2015). "How Math's Most Famous Proof Nearly Broke". Nautilus. Retrieved 16 March 2016.
16. Simon Singh (1997). Fermat's Last Theorem. ISBN 1-85702-521-0
17. ^ Kolata, Gina (24 June 1993). "At Last, Shout of 'Eureka!' In Age-Old Math Mystery". The New York Times. Retrieved 21 January 2013.
18. ^ Wiles, Andrew (May 1995). "Issue 3". Annals of Mathematics. 141: 1–551 – via JSTOR.
19. ^ "Are mathematicians finally satisfied with Andrew Wiles's proof of Fermat's Last Theorem? Why has this theorem been so difficult to prove?". Scientific American. 21 October 1999. Retrieved 16 March 2016.
20. ^ "BBC TWO, Horizon Fermat's Last Theorem". BBC. 16 December 2010. Retrieved 12 June 2014.
21. ^ "Sir Andrew Wiles KBE FRS". London: Royal Society. Archived from the original on 17 November 2015. One or more of the preceding sentences incorporates text from the royalsociety.org website where:

All text published under the heading 'Biography' on Fellow profile pages is available under Creative Commons Attribution 4.0 International License." --"Royal Society Terms, conditions and policies". Archived from the original on 25 September 2015. Retrieved 9 March 2016.CS1 maint: BOT: original-url status unknown (link)

22. ^ a b "EC/1989/39: Wiles, Sir Andrew John". The Royal Society. Retrieved 16 March 2016.
23. ^ a b c Wiles Receives 2005 Shaw Prize. American Mathematical Society. Retrieved 16 March 2016.
24. ^ "NAS Award in Mathematics". National Academy of Sciences. Archived from the original on 29 December 2010. Retrieved 13 February 2011.
25. ^ Wiles Receives Ostrowski Prize. American Mathematical Society. Retrieved 16 March 2016.
26. ^ "1997 Cole Prize, Notices of the AMS" (PDF). American Mathematical Society. Retrieved 13 April 2008.
27. ^ Paul Wolfskehl and the Wolfskehl Prize. American Mathematical Society. Retrieved 16 March 2016.
28. ^ "Andrew J. Wiles Awarded the "IMU Silver Plaque"". American Mathematical Society. 11 April 1953. Retrieved 12 June 2014.
29. ^ "Andrew Wiles Receives Faisal Prize" (PDF). American Mathematical Society. Retrieved 12 June 2014.
30. ^ "Premio Pitagora" (in Italian). University of Calabria. Archived from the original on 15 January 2014. Retrieved 16 March 2016.
31. ^ "JPL Small-Body Database Browser". NASA. Retrieved 11 May 2009.
32. ^ "No. 55710". The London Gazette (Supplement). 31 December 1999. p. 34.
33. ^ "Mathematical Institute". University of Oxford. Retrieved 16 March 2016.
34. ^ "British mathematician Sir Andrew Wiles gets Abel math prize". The Washington Post. Associated Press. 15 March 2016. Archived from the original on 15 March 2016.CS1 maint: BOT: original-url status unknown (link)
35. ^ Sheena McKenzie, CNN (16 March 2016). "300-year-old math question solved, professor wins \$700k - CNN". CNN.
36. ^ "A British mathematician just won a \$700,000 prize for solving this fascinating centuries-old math problem 22 years ago". Business Insider. Retrieved 19 March 2016.
37. ^ Iyengar, Rishi. "Andrew Wiles Wins 2016 Abel Prize for Fermat's Last Theorem". Time. Retrieved 19 March 2016.