Andrew Wiles

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

(1953-04-11) 11 April 1953 (age 71)
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 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 unifying Galois representations, elliptic curves and modular forms, starting with Barry Mazur's generalizations of Iwasawa theory. In the early 1980s, Wiles spent a few years at the University of Cambridge before moving to Princeton University, where he 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. In an interview with Nadia Hasnaoui in 2021, he said he trusted the letters, yet 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 told WGBH-TV in 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 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 equation. [11]

Early career

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 . [14]

In 1989, 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]

Proof of Fermat's Last Theorem

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.

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"). [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-supervisor John Coates stated that it seemed "impossible to actually prove", [19] : 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]." [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 for Frey'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 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). 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. [20] [21] [15]

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. [18]

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] [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 used Galois representations to replace elliptic curves, reduced the problem to a class number formula and solved it, among other matters, all using Victor Kolyvagin's ideas as a basis for fixing Matthias Flach's approach with Iwasawa theory. [23] [22] Together with his former student Richard 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 the Annals of Mathematics. [24] [25]

Later career

In 2011, Wiles rejoined the University of Oxford 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]

Legacy

Wiles' work has been used in many fields of mathematics. Notably, in 1999, three of his former students, Richard Taylor, Brian Conrad, and Fred Diamond, working with Christophe Breuil, built upon Wiles' proof to prove the full modularity theorem. [26] [15] Wiles's doctoral students have also included Manjul Bhargava (2014 winner of the Fields Medal), Ehud de Shalit, Ritabrata Munshi (winner of the SSB Prize and ICTP Ramanujan Prize), Karl Rubin (son of Vera Rubin), Christopher Skinner, and Vinayak Vatsal (2007 winner of the Coxeter–James Prize).

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 .

In 1988, Wiles was awarded the Junior Whitehead Prize of the London Mathematical Society (1988). [6] In 1989, he was elected a Fellow of the Royal Society (FRS) [28] [12]

In 1994, Wiles was elected member of the American Academy of Arts and Sciences. [29] Upon completing his proof of Fermat's Last Theorem in 1995, he was awarded the Schock Prize, [14] Fermat Prize, [30] and Wolf Prize in Mathematics that year. [14] Wiles was elected a Foreign Associate of the National Academy of Sciences [13] and won an NAS Award in Mathematics from the National Academy of Sciences, [31] the Royal Medal, and the Ostrowski Prize in 1996. [32] He won the American Mathematical Society's Cole Prize, [33] a MacArthur Fellowship, and the Wolfskehl Prize in 1997, [34] and was elected member of the American Philosophical Society that year. [35]

In 1998, Wiles was awarded a silver plaque from the International Mathematical Union recognising his achievements, in place of the Fields 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 the King Faisal Prize [37] along with the Clay Research Award in 1999, [14] the year the asteroid 9999 Wiles was named after him. [38]

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

The building at the University of Oxford housing the Mathematical Institute was named after Wiles in 2016. [41] Later that year he won the Abel Prize. [42] [43] [44] [45] [46] In 2017, Wiles won the Copley Medal. [1] In 2019, he won the De Morgan Medal. [47]

See also

Related Research Articles

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. 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. 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 (born 1944)

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.

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> Mathematical concept

A modular elliptic curve is an elliptic curve E that admits a parametrization 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 associated with an ABC triple . This relates properties of solutions of equations to elliptic curves. This curve was popularized in its application to Fermat’s Last Theorem where one investigates a (hypothetical) solution of Fermat's equation

<span class="mw-page-title-main">Karl Rubin</span> American mathematician

Karl Cooper Rubin is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. Between 1997 and 2006, he was a professor at Stanford, and before that worked at Ohio State University between 1987 and 1999. His research interest is in elliptic curves. He was the first mathematician (1986) to show that some elliptic curves over the rationals have finite Tate–Shafarevich groups. It is widely believed that these groups are always finite.

<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 Sir 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 previous knowledge by almost all living mathematicians at the time.

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 (born 1937)

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

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  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 (PhD thesis). University of Cambridge. OCLC   500589130. EThOS   uk.bl.ethos.477263 – via Cambridge University Library.
  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.
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  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 . Retrieved 2 February 2022.
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  21. ↑ Boston, Nick (Spring 2003), Proof of Fermat's Last Theorem (PDF), University of Wisconsin–Madison
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  23. 1 2 Cipra, 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.
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