Andrew Wiles

Last updated

Sir

Andrew Wiles

KBE FRS
Andrew wiles1-3.jpg
Wiles in 2005
Born
Andrew John Wiles

(1953-04-11) 11 April 1953 (age 70)
Cambridge, England
NationalityBritish
Education King's College School, Cambridge
The Leys School
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
Fields Mathematics
Institutions
Thesis Reciprocity Laws and the Conjecture of Birch and Swinnerton-Dyer  (1979)
Doctoral advisor John Coates [2] [3]
Doctoral students

Sir Andrew John Wiles KBE FRS (born 11 April 1953) is an English 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 and the 2017 Copley Medal and for which he was appointed a Knight Commander of the Order of the British Empire in 2000. [1] In 2018, Wiles was appointed the first Regius Professor of Mathematics at Oxford. [4] Wiles is also a 1997 MacArthur Fellow.

Contents

Wiles was born in Cambridge to theologian Maurice Frank Wiles and his wife Patricia. 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 unifying Galois representations, elliptic curves and modular forms, starting with Barry Mazur’s generalizations of Iwasawa theory. In the early 1980s, Wiles moved to Princeton University from Cambridge and worked on expanding out and applying Hilbert modular forms. In 1986, upon reading Ken Ribet’s seminal work on Fermat’s Last Theorem, Wiles set out to prove the modularity theorem for semistable 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 student Richard Taylor were 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 the Langlands program of unifying number theory. [5]

Education and early life

Wiles was born on 11 April 1953 in Cambridge, England, the son of Maurice Frank Wiles (1923–2005) and Patricia Wiles (née Mowll). From 1952 to 1955, his father worked as the chaplain at Ridley Hall, Cambridge, and later became the Regius 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. He trusts the letters, though he could not remember a time when he did not enjoy solving mathematical problems. [7]

Wiles attended King's College School, Cambridge, [8] 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 The Last Problem, by Eric 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 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

In 1974, Wiles earned his bachelor's degree in mathematics at Merton College, Oxford. [6] Wiles's 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 the rational numbers, and soon afterward, he generalised this result to totally real fields. [12] [13]

In 1980, Wiles earned a PhD while at Clare College, Cambridge. [3] After a stay at the Institute for Advanced Study in Princeton, New Jersey, in 1981, Wiles became a Professor of Mathematics at Princeton University. [14]

In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure .

In 1987, Wiles was elected to the Royal Society. At that point according to his election certificate, he had been working "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". [12]

From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford, and then he returned to Princeton. From 1994 to 2009, Wiles was a Eugene Higgins Professor at Princeton. He rejoined Oxford in 2011 as Royal Society Research Professor. [14]

In May 2018, Wiles was appointed Regius Professor of Mathematics at Oxford, the first in the university's history. [4]

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 (the statement that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2) 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, and stated that all such curves have a modular form associated with them. [15] [16] 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. [17]

The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove. [18] : 203–205, 223, 226  For example, Wiles's ex-supervisor John Coates stated that it seemed "impossible to actually prove", [18] : 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]." [18] : 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. [18] : 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. [18] : 229–230 

Wiles’ research involved creating a proof by contradiction of Fermat’s Last Theorem, which Ribet in his 1986 work had 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 if n were a prime number, no such elliptic curve could have a modular form, so no odd prime counterexample to Fermat's equation could exist), and Wiles also proved that the conjecture applied to the special case known as the semistable elliptic curves to 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. [19] [20] [21]

In June 1993, he presented his proof to the public for the first time at a conference in Cambridge. Gina Kolata of The New York Times summed 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. [17]

In August 1993, it was discovered that the proof contained a flaw in several areas, related to properties of the Selmer group and use of a tool called an Euler system. [22] 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. According to Eric W. Weisstein, the circumvention involved "replacing elliptic curves with Galois representations, reducing the problem to a class number formula, solving that problem, and tying up loose ends", all using Iwasawa theory to fix "results from Matthias Flach based on ideas from Victor Kolyvagin", and letting Iwasawa's and Flach's approaches strengthen each other. [21] [22] [23] Together with his former student Richard Taylor, he 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 the Annals of Mathematics. [24] [25]

Legacy

Wiles’ work has been used in many fields of mathematics. Notably, in 1999, his former student Richard Taylor and three other mathematicians built upon Wiles’ proof to prove the full modularity theorem. [26]

In 2016, upon receiving the Abel 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 the Langlands 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

Andrew Wiles in front of the statue of Pierre de Fermat in Beaumont-de-Lomagne in 1995, Fermat's birthplace in southern France Wiles vor Sockel.JPG
Andrew Wiles in front of the statue of Pierre de Fermat in Beaumont-de-Lomagne in 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 the BBC documentary series Horizon [27] about Fermat's Last Theorem. This was broadcast as an episode of the PBS science television series Nova with the title "The Proof". [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:

Related Research Articles

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

Gorō Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently, Ralph Greenberg has proposed an Iwasawa theory for motives.

In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2024, only special cases of the conjecture have been proven.

<span class="mw-page-title-main">Richard Taylor (mathematician)</span> British mathematician

Richard Lawrence Taylor is a British mathematician working in the field of number theory. He is currently the Barbara Kimball Browning Professor in Humanities and Sciences at Stanford University.

Ribet's theorem is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture and the epsilon conjecture together imply that FLT is true.

<span class="mw-page-title-main">Gerhard Frey</span> German mathematician

Gerhard Frey is a German mathematician, known for his work in number theory. Following an original idea of Hellegouarch, he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a purported solution to the Fermat equation, that is central to Wiles's proof of Fermat's Last Theorem.

<span class="mw-page-title-main">Yutaka Taniyama</span> Japanese mathematician

Yutaka Taniyama was a Japanese mathematician known for the Taniyama–Shimura conjecture.

<span class="mw-page-title-main">Arithmetic geometry</span> Branch of algebraic geometry focused on problems in number theory

In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.

In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre, states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.

Brian Conrad is an American mathematician and number theorist, working at Stanford University. Previously, he taught at the University of Michigan and at Columbia University.

<span class="mw-page-title-main">Modular elliptic curve</span>

A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.

In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve

<span class="mw-page-title-main">Fermat's Last Theorem</span> 17th-century conjecture proved by Andrew Wiles in 1994

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

<span class="mw-page-title-main">Wiles's proof of Fermat's Last Theorem</span> 1995 publication in mathematics

Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using current knowledge by almost all contemporary mathematicians.

In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.

<span class="mw-page-title-main">Barry Mazur</span> American mathematician

Barry Charles Mazur is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.

<span class="mw-page-title-main">Ken Ribet</span> American mathematician

Kenneth Alan Ribet is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fermat's Last Theorem, as well as for his service as President of the American Mathematical Society from 2017 to 2019. He is currently a professor of mathematics at the University of California, Berkeley.

Christopher McLean Skinner is an American mathematician and professor at Princeton University. He works in algebraic number theory and arithmetic aspects of the Langlands program.

Vinayak Vatsal is a Canadian mathematician working in number theory and arithmetic geometry.

References

  1. 1 2 3 "Mathematician Sir Andrew Wiles FRS wins the Royal Society's prestigious Copley Medal". The Royal Society. Retrieved 27 May 2017.
  2. 1 2 Andrew Wiles at the Mathematics Genealogy Project OOjs UI icon edit-ltr-progressive.svg
  3. 1 2 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.
  4. 1 2 "Sir Andrew Wiles appointed first Regius Professor of Mathematics at Oxford". News & Events. University of Oxford. 31 May 2018. Retrieved 1 June 2018.
  5. 1 2 Sample, Ian (15 March 2016). "Abel prize won by Oxford professor for Fermat's Last Theorem proof". the Guardian. Retrieved 20 November 2023.
  6. 1 2 3 Anon (2017). "Wiles, Sir Andrew (John )" . Who's Who (online Oxford University Press  ed.). Oxford: A & C Black. doi:10.1093/ww/9780199540884.013.39819.(Subscription or UK public library membership required.)
  7. ↑ "Interview with Andrew Wiles". YouTube. The Abel Prize. 10 March 2023. Retrieved 15 November 2023.
  8. ↑ "Alumni". King's College School, Cambridge. Retrieved 1 February 2022.
  9. ↑ "Old Leysian Prof Sir Andrew Wiles wins the Copley Medal". The Leys & St Faith's Schools Foundation. 2 November 2017. Retrieved 1 February 2022.
  10. 1 2 "Andrew Wiles on Solving Fermat". WGBH. November 2000. Retrieved 16 March 2016.
  11. ↑ Chang, Sooyoung (2011). Academic Genealogy of Mathematicians. World Scientific. p. 207. ISBN   9789814282291.
  12. 1 2 3 "EC/1989/39: Wiles, Sir Andrew John". The Royal Society. Archived from the original on 13 July 2015. Retrieved 16 March 2016.
  13. 1 2 "Andrew Wiles". National Academy of Sciences . Retrieved 16 March 2016.
  14. 1 2 3 4 5 O'Connor, John J.; Robertson, Edmund F. (September 2009). "Andrew John Wiles Biography". MacTutor History of Mathematics archive . Retrieved 1 February 2022.
  15. ↑ Brown, Peter (28 May 2015). "How Math's Most Famous Proof Nearly Broke". Nautilus. Archived from the original on 15 March 2016. Retrieved 16 March 2016.
  16. ↑ 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 . Retrieved 2 February 2022.
  17. 1 2 Kolata, Gina (24 June 1993). "At Last, Shout of 'Eureka!' In Age-Old Math Mystery". The New York Times . Archived from the original on 20 November 2023. Retrieved 21 January 2013.
  18. 1 2 3 4 5 Simon Singh (1997). Fermat's Last Theorem. ISBN   1-85702-521-0
  19. ↑ Stevens, Glenn (n.d.), An Overview of the Proof of Fermat's Last Theorem (PDF), Boston University
  20. ↑ Boston, Nick (Spring 2003), Proof of Fermat's Last Theorem (PDF), University of Wisconsin–Madison
  21. 1 2 "Fermat's Last Theorem / Useful Notes". TV Tropes. 2023. Retrieved 20 November 2023.
  22. 1 2 Weisstein, Eric W. (14 May 2004). "Fermat's Last Theorem -- from Wolfram MathWorld". Wolfram Research, Inc. Retrieved 20 November 2023.
  23. ↑ Weisstein, Eric W. (26 September 2009). "Taniyama-Shimura Conjecture -- from Wolfram MathWorld". Wolfram Research, Inc. Retrieved 20 November 2023.
  24. ↑ Wiles, Andrew (May 1995). "Issue 3". Annals of Mathematics. 141: 1–551. JSTOR   i310703.
  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. Retrieved 16 March 2016.
  26. ↑ Devlin, Keith (21 July 1999). "Beyond Fermat's last theorem". the Guardian. Retrieved 20 November 2023.
  27. ↑ "BBC TWO, Horizon Fermat's Last Theorem". BBC. 16 December 2010. Retrieved 12 June 2014.
  28. ↑ "Sir Andrew Wiles KBE FRS". London: Royal Society. Archived from the original on 17 November 2015. Retrieved 1 February 2022. 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.
  29. ↑ "Andrew J. Wiles". American Academy of Arts & Sciences. Retrieved 10 December 2021.
  30. 1 2 3 Wiles Receives 2005 Shaw Prize. American Mathematical Society. Retrieved 16 March 2016.
  31. ↑ "NAS Award in Mathematics". National Academy of Sciences. Archived from the original on 29 December 2010. Retrieved 13 February 2011.
  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. Retrieved 13 April 2008.
  34. ↑ Paul Wolfskehl and the Wolfskehl Prize. American Mathematical Society. Retrieved 16 March 2016.
  35. ↑ "APS Member History". search.amphilsoc.org. Retrieved 10 December 2021.
  36. ↑ "Andrew J. Wiles Awarded the "IMU Silver Plaque"". American Mathematical Society. 11 April 1953. Retrieved 12 June 2014.
  37. ↑ "Andrew Wiles Receives Faisal Prize" (PDF). American Mathematical Society. Archived (PDF) from the original on 9 October 2022. Retrieved 12 June 2014.
  38. ↑ "Premio Pitagora" (in Italian). University of Calabria. Archived from the original on 15 January 2014. Retrieved 16 March 2016.
  39. ↑ "JPL Small-Body Database Browser". NASA . Retrieved 11 May 2009.
  40. ↑ "No. 55710". The London Gazette (Supplement). 31 December 1999. p. 34.
  41. ↑ "Mathematical Institute". University of Oxford. Archived from the original on 13 January 2016. Retrieved 16 March 2016.
  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 . PMID   26983518.
  43. ↑ "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.
  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. Retrieved 19 March 2016.
  46. ↑ Iyengar, Rishi. "Andrew Wiles Wins 2016 Abel Prize for Fermat's Last Theorem". Time. Retrieved 19 March 2016.
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