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Andrew Wiles

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Sir Andrew Wiles
KBEFRS
Wiles in 2005
Born
Andrew John Wiles

(1953-04-11)11 April 1953(age 71)
Cambridge,England
NationalityBritish
EducationKing's College School, Cambridge
The Leys School
Alma mater
Known forProving theTaniyama–Shimura conjecturefor semistable elliptic curves, thereby provingFermat's Last Theorem
Proving themain conjecture of Iwasawa theory
Awards
Scientific career
FieldsMathematics
Institutions
ThesisReciprocity Laws and the Conjecture of Birch and Swinnerton-Dyer(1979)
Doctoral advisorJohn Coates[2][3]
Doctoral students

Sir Andrew John WilesKBEFRS(born 11 April 1953) is an English mathematician and aRoyal SocietyResearch Professor at theUniversity of Oxford,specialising innumber theory.He is best known forprovingFermat's Last Theorem,for which he was awarded the 2016Abel Prizeand the 2017Copley Medaland for which he was appointed aKnight Commander of the Order of the British Empirein 2000.[1]In 2018, Wiles was appointed the firstRegius Professorof Mathematics at Oxford.[4]Wiles is also a1997 MacArthur Fellow.

Wiles was born in Cambridge to theologianMaurice Frank Wilesand Patricia Wiles. While spending much of his childhood in Nigeria, Wiles developed an interest in mathematics and in Fermat's Last Theorem in particular. After moving to Oxford and graduating from there in 1974, he worked on unifyingGalois representations,elliptic curvesandmodular forms,starting withBarry Mazur's generalizations ofIwasawa theory.In the early 1980s, Wiles spent a few years at theUniversity of Cambridgebefore moving toPrinceton University,where he worked on expanding out and applyingHilbert modular forms.In 1986, upon readingKen Ribet's seminal work on Fermat's Last Theorem, Wiles set out to prove themodularity theoremforsemistable elliptic curves,which implied Fermat's Last Theorem. By 1993, he had been able to prove Fermat's Last Theorem, though a flaw was discovered. After an insight on 19 September 1994, Wiles and his studentRichard Taylorwere able to circumvent the flaw, and published the results in 1995, to widespread acclaim.

In proving Fermat's Last Theorem, Wiles developed new tools for mathematicians to begin unifying disparate ideas and theorems. His former student Taylor along with three other mathematicians were able to prove the full modularity theorem by 2000, using Wiles' work. Upon receiving the Abel Prize in 2016, Wiles reflected on his legacy, expressing his belief that he did not just prove Fermat's Last Theorem, but pushed the whole of mathematics as a field towards theLanglands programof unifying number theory.[5]

Education and early life

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Wiles was born on 11 April 1953 inCambridge,England, the son ofMaurice Frank Wiles(1923–2005) and Patricia Wiles (née Mowll). From 1952 to 1955, his father worked as the chaplain atRidley Hall, Cambridge,and later became theRegius Professor of Divinity at the University of Oxford.[6]

Wiles began his formal schooling in Nigeria, while living there as a very young boy with his parents. However, according to letters written by his parents, for at least the first several months after he was supposed to be attending classes, he refused to go. From that fact, Wiles himself concluded that in his earliest years, he was not enthusiastic about spending time in academic institutions. In an interview withNadia Hasnaouiin 2021, he said he trusted the letters, yet he could not remember a time when he did not enjoy solving mathematical problems.[7]

Wiles attendedKing's College School, Cambridge,[8]andThe Leys School, Cambridge.[9]Wiles toldWGBH-TVin 1999 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 bookThe Last Problem,byEric Temple Bell,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 byKen Ribet's 1986 proof of theepsilon conjecture,whichGerhard Freyhad previously linked to Fermat's equation.[11]

Early career

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In 1974, Wiles earned hisbachelor's degreeinmathematicsatMerton College, Oxford.[6]Wiles's graduate research was guided byJohn Coates,beginning in the summer of 1975. Together they worked on the arithmetic ofelliptic curveswithcomplex multiplicationby the methods ofIwasawa theory.He further worked withBarry Mazuron themain conjecture of Iwasawa theoryover therational numbers,and soon afterward, he generalised this result tototally real fields.[12][13]

In 1980, Wiles earned a PhD while atClare College, Cambridge.[3]After a stay at theInstitute for Advanced StudyinPrinceton, New Jersey,in 1981, Wiles became aProfessor of MathematicsatPrinceton University.[14]

In 1985–86, Wiles was aGuggenheim Fellowat theInstitut des Hautes Études Scientifiquesnear Paris and at theÉcole Normale Supérieure.[14]

In 1989, Wiles was elected to theRoyal Society.At that point according to his election certificate, he had been working "on the construction of ℓ-adic representations attached toHilbert modular forms,and has applied these to prove the 'main conjecture' for cyclotomic extensions of totally real fields ".[12]

Proof of Fermat's Last Theorem

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Main article:Wiles's proof of Fermat's Last Theorem

From 1988 to 1990, Wiles was a Royal Society Research Professor at theUniversity of Oxford,and then he returned to Princeton. From 1994 to 2009, Wiles was aEugene Higgins Professorat Princeton.

Starting in mid-1986, based on successive progress of the previous few years ofGerhard Frey,Jean-Pierre SerreandKen Ribet,it became clear thatFermat's Last Theorem(the statement that no threepositiveintegersa,b,andcsatisfy the equationan+bn=cnfor any integer value ofngreater than2) could be proven as acorollaryof a limited form of themodularity theorem(unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture" ).[15]The modularity theorem involved elliptic curves, which was also Wiles's own specialist area, and stated that all such curves have a modular form associated with them.[16][17]These curves can be thought of as mathematical objects resembling solutions for a torus' surface, and if Fermat's Last Theorem were false and solutions existed, "a peculiar curve would result". A proof of the theorem therefore would involve showing that such a curve would not exist.[18]

The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove.[19]: 203–205, 223, 226 For example, Wiles's ex-supervisorJohn Coatesstated that it seemed "impossible to actually prove",[19]: 226 andKen Ribetconsidered 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]."[19]: 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 forFrey's curve.[19]: 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.[19]: 229–230 

Wiles' research involved creating aproof by contradictionof Fermat's Last Theorem, which Ribet in his1986 workhad found to have an elliptic curve and thus an associated modular form if true. Starting by assuming that the theorem was incorrect, Wiles then contradicted the Taniyama–Shimura–Weil conjecture as formulated under that assumption, with Ribet's theorem (which stated that ifnwere aprime number,no such elliptic curve could have a modular form, so no odd prime counterexample to Fermat's equation could exist). Wiles also proved that the conjecture applied to the special case known as thesemistable elliptic curvesto which Fermat's equation was tied. In other words, Wiles had found that the Taniyama–Shimura–Weil conjecture was true in the case of Fermat's equation, and Ribet's finding (that the conjecture holding for semistable elliptic curves could mean Fermat's Last Theorem is true) prevailed, thus proving Fermat's Last Theorem.[20][21][15]

In June 1993, he presented his proof to the public for the first time at a conference in Cambridge.Gina KolataofThe New York Timessummed up the presentation as follows:

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.[18]

In August 1993, it was discovered that the proof contained a flaw in several areas, related to properties of theSelmer groupand use of a tool called anEuler system.[22][23]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. The circumvention usedGalois representationsto replace elliptic curves, reduced the problem to aclass number formulaand solved it, among other matters, all usingVictor Kolyvagin's ideas as a basis for fixingMatthias Flach's approach with Iwasawa theory.[23][22]Together with his former studentRichard Taylor,Wiles published a second paper which contained the circumvention and thus completed the proof. Both papers were published in May 1995 in a dedicated issue of theAnnals of Mathematics.[24][25]

Later career

[edit]

In 2011, Wiles rejoined the University of Oxford as Royal Society Research Professor.[14]

In May 2018, Wiles was appointedRegius Professorof Mathematics at Oxford, the first in the university's history.[4]

Legacy

[edit]

Wiles' work has been used in many fields of mathematics. Notably, in 1999, three of his former students,Richard Taylor,Brian Conrad,andFred Diamond,working withChristophe Breuil,built upon Wiles' proof to prove the full modularity theorem.[26][15]Wiles's doctoral students have also includedManjul Bhargava(2014 winner of theFields Medal),Ehud de Shalit,Ritabrata Munshi(winner of theSSB PrizeandICTP Ramanujan Prize),Karl Rubin(son ofVera Rubin),Christopher Skinner,andVinayak Vatsal(2007 winner of theCoxeter–James Prize).

In 2016, upon receiving theAbel Prize,Wiles said about his proof of Fermat's Last Theorem, "The methods that solved it opened up a new way of attacking one of the big webs of conjectures of contemporary mathematics called theLanglands Program,which as a grand vision tries to unify different branches of mathematics. It’s given us a new way to look at that ".[5]

Awards and honours

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Andrew Wiles in front of the statue ofPierre de FermatinBeaumont-de-Lomagnein 1995, Fermat's birthplace in southern France

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 theBBCdocumentary seriesHorizon[27]about Fermat's Last Theorem. This was broadcast as an episode of the PBS science television seriesNovawith the title "The Proof".[10]His work and life are also described in great detail inSimon Singh's popular bookFermat's Last Theorem.

In 1988, Wiles was awarded the JuniorWhitehead Prizeof theLondon Mathematical Society(1988).[6]In 1989, he was elected aFellow of the Royal Society (FRS)[28][12]

In 1994, Wiles was elected member of theAmerican Academy of Arts and Sciences.[29]Upon completing his proof of Fermat's Last Theorem in 1995, he was awarded theSchock Prize,[14]Fermat Prize,[30]andWolf Prize in Mathematicsthat year.[14]Wiles was elected aForeign Associate of the National Academy of Sciences[13]and won anNAS Award in Mathematicsfrom the National Academy of Sciences,[31]theRoyal Medal,and theOstrowski Prizein 1996.[32]He won theAmerican Mathematical Society'sCole Prize,[33]aMacArthur Fellowship,and theWolfskehl Prizein 1997,[34]and was elected member of theAmerican Philosophical Societythat year.[35]

In 1998, Wiles was awarded a silver plaque from theInternational Mathematical Unionrecognising his achievements, in place of theFields Medal,which is restricted to those under the age of 40 (Wiles was 41 when he proved the theorem in 1994).[36]That same year, he was awarded theKing Faisal Prize[37]along with theClay Research Awardin 1999,[14]the year theasteroid9999 Wileswas named after him.[38]

In 2000, he was awardedKnight Commander of the Order of the British Empire(2000)[39]In 2004 Wiles won the Premio Pitagora.[40]In 2005, he won theShaw Prize.[30]

The building at theUniversity of Oxfordhousing theMathematical Institutewas named after Wiles in 2016.[41]Later that year he won theAbel Prize.[42][43][44][45][46]In 2017, Wiles won theCopley Medal.[1]In 2019, he won theDe Morgan Medal.[47]

References

[edit]
  1. ^abc"Mathematician Sir Andrew Wiles FRS wins the Royal Society's prestigious Copley Medal".The Royal Society.Retrieved27 May2017.
  2. ^abAndrew Wilesat theMathematics Genealogy ProjectEdit this at Wikidata
  3. ^abWiles, Andrew John (1978).Reciprocity laws and the conjecture of Birch and Swinnerton-Dyer(PhD thesis). University of Cambridge.OCLC500589130.EThOSuk.bl.ethos.477263– via Cambridge University Library.
  4. ^ab"Sir Andrew Wiles appointed first Regius Professor of Mathematics at Oxford".News & Events.University of Oxford.31 May 2018.Retrieved1 June2018.
  5. ^abSample, Ian (15 March 2016)."Abel prize won by Oxford professor for Fermat's Last Theorem proof".The Guardian.Retrieved20 November2023.
  6. ^abcAnon (2017)."Wiles, Sir Andrew (John )".Who's Who(onlineOxford University Pressed.). Oxford: A & C Black.doi:10.1093/ww/9780199540884.013.39819.(Subscription orUK public library membershiprequired.)
  7. ^"Interview with Andrew Wiles".The Abel Prize. 10 March 2021.Retrieved15 November2023– via YouTube.
  8. ^"Alumni".King's College School, Cambridge.Retrieved1 February2022.
  9. ^"Old Leysian Prof Sir Andrew Wiles wins the Copley Medal".The Leys & St Faith's Schools Foundation. 2 November 2017.Retrieved1 February2022.
  10. ^ab"Andrew Wiles on Solving Fermat".WGBH.November 2000.Retrieved16 March2016.
  11. ^Chang, Sooyoung (2011).Academic Genealogy of Mathematicians.World Scientific. p. 207.ISBN9789814282291.
  12. ^abc"EC/1989/39: Wiles, Sir Andrew John".The Royal Society.Archived fromthe originalon 13 July 2015.Retrieved16 March2016.
  13. ^ab"Andrew Wiles".National Academy of Sciences.Retrieved16 March2016.
  14. ^abcdefO'Connor, John J.;Robertson, Edmund F.(September 2009)."Andrew John Wiles Biography".MacTutor History of Mathematics archive.Retrieved1 February2022.
  15. ^abcDarmon, Henri(December 1999)."A Proof of the Full Shimura- Taniyama-Weil Conjecture Is Announced"(PDF).Notices of the AMS.46(11).American Mathematical Society:1397-1401.Retrieved1 August2024.
  16. ^Brown, Peter (28 May 2015)."How Math's Most Famous Proof Nearly Broke".Nautilus. Archived fromthe originalon 15 March 2016.Retrieved16 March2016.
  17. ^Broad, William J.(31 January 2022)."Profiles in Science – The Texas Oil Heir Who Took on Math's Impossible Dare – James M. Vaughn Jr., wielding a fortune, argues that he brought about the Fermat breakthrough after the best and brightest had failed for centuries to solve the puzzle".The New York Times.Retrieved2 February2022.
  18. ^abKolata, Gina(24 June 1993)."At Last, Shout of 'Eureka!' In Age-Old Math Mystery".The New York Times.Archived fromthe originalon 20 November 2023.Retrieved21 January2013.
  19. ^abcdeSimon Singh(1997).Fermat’s Last Theorem.ISBN1-85702-521-0
  20. ^Stevens, Glenn H.(n.d.),An Overview of the Proof of Fermat's Last Theorem(PDF),Boston University
  21. ^Boston, Nick (Spring 2003),Proof of Fermat's Last Theorem(PDF),University of Wisconsin–Madison
  22. ^abFaltings, Gerd(July 1995)."The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles"(PDF).Notices of the AMS.42(7).American Mathematical Society:743-746.Retrieved1 August2024.
  23. ^abCipra, Barry Arthur(1995). "Princeton Mathematician Looks Back on Fermat Proof".Science.268(5214): 1133-1134.Bibcode:1995Sci...268.1133C.doi:10.1126/science.268.5214.1133.
  24. ^Wiles, Andrew (May 1995). "Issue 3".Annals of Mathematics.141:1–551.JSTORi310703.
  25. ^"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.Retrieved16 March2016.
  26. ^Devlin, Keith(21 July 1999)."Beyond Fermat's last theorem".The Guardian.Retrieved20 November2023.
  27. ^"BBC TWO, Horizon Fermat's Last Theorem".BBC. 16 December 2010.Retrieved12 June2014.
  28. ^"Sir Andrew Wiles KBE FRS".London:Royal Society.Archived fromthe originalon 17 November 2015.Retrieved1 February2022.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 underCreative Commons Attribution 4.0 International License.
  29. ^"Andrew J. Wiles".American Academy of Arts & Sciences.Retrieved10 December2021.
  30. ^abWiles Receives 2005 Shaw Prize.American Mathematical Society.Retrieved 16 March 2016.
  31. ^"NAS Award in Mathematics".National Academy of Sciences.Archived fromthe originalon 29 December 2010.Retrieved13 February2011.
  32. ^Wiles Receives Ostrowski Prize.American Mathematical Society.Retrieved 16 March 2016.
  33. ^"1997 Cole Prize, Notices of the AMS"(PDF).American Mathematical Society.Archived(PDF)from the original on 9 October 2022.Retrieved13 April2008.
  34. ^Paul Wolfskehl and the Wolfskehl Prize.American Mathematical Society.Retrieved 16 March 2016.
  35. ^"APS Member History".search.amphilsoc.org.Retrieved10 December2021.
  36. ^"Andrew J. Wiles Awarded the" IMU Silver Plaque "".American Mathematical Society.11 April 1953.Retrieved12 June2014.
  37. ^"Andrew Wiles Receives Faisal Prize"(PDF).American Mathematical Society.Archived(PDF)from the original on 9 October 2022.Retrieved12 June2014.
  38. ^"JPL Small-Body Database Browser".NASA.Retrieved11 May2009.
  39. ^"No. 55710".The London Gazette(Supplement). 31 December 1999. p. 34.
  40. ^"Premio Pitagora"(in Italian).University of Calabria.Archived fromthe originalon 15 January 2014.Retrieved16 March2016.
  41. ^"Mathematical Institute".University of Oxford.Archived fromthe originalon 13 January 2016.Retrieved16 March2016.
  42. ^Castelvecchi, Davide (2016)."Fermat's last theorem earns Andrew Wiles the Abel Prize".Nature.531(7594): 287.Bibcode:2016Natur.531..287C.doi:10.1038/nature.2016.19552.PMID26983518.
  43. ^"British mathematician Sir Andrew Wiles gets Abel math prize".The Washington Post.Associated Press. 15 March 2016. Archived fromthe originalon 15 March 2016.
  44. ^McKenzie, Sheena (16 March 2016)."300-year-old math question solved, professor wins $700k – CNN".CNN.
  45. ^"A British mathematician just won a $700,000 prize for solving this fascinating centuries-old math problem 22 years ago".Business Insider.Retrieved19 March2016.
  46. ^Iyengar, Rishi."Andrew Wiles Wins 2016 Abel Prize for Fermat's Last Theorem".Time.Retrieved19 March2016.
  47. ^"Winners of the De Morgan Medal of the LMS".MacTutor History of Mathematics Archive.St Andrews College.Retrieved29 January2024.
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