Andrew John Wiles

Quick Info

11 April 1953
Cambridge, England

Andrew Wiles is an English mathematician famous for having proved Fermat's Last Theorem in 1995. He has received numerous honours including being made a Knight Commander of the Order of the British Empire by the Queen in 2000, and being awarded the Wolf Prize, the Shaw Prize, the Clay Research Award and the Abel Prize.


Andrew Wiles' father, Maurice Frank Wiles (1923-2005), studied moral sciences and theology at the University of Cambridge, then trained for ordination at Ridley Hall, Cambridge. After two years as a curate in Stockport, he became the chaplain at Ridley Hall, Cambridge, from 1950 to 1955. Andrew's mother was Patricia Margaret Mowll (1927-2005, known as Paddy). She married Maurice Wiles in 1950, at Cuckfield, Sussex, England. Maurice and Patricia Wiles had three children: Patrick David Wiles (born 2 May 1951); Andrew John Wiles, the subject of this biography; and Alison Ruth Wiles (born 3 August 1955).

In 1955, when Andrew was two years old, his father was appointed as a lecturer in New Testament Studies at Ibadan, Nigeria. It was in Nigeria that Andrew began his schooling but, like many children, he was at first reluctant to attend school. In 1959 Maurice Wiles returned to Cambridge, to become dean of Clare College. The family sailed from Lagos to London, on the Winneba, arriving in England on 3 July 1959. At the start of the 1959-60 school year, Andrew began his studies at King's College School, Cambridge, a preparatory school founded by King Henry VI in 1441. At this school he was taught mathematics by Mary Briggs, the wife of the headmaster David Briggs.

Wiles' interest in Fermat's Last Theorem began while he was studying at King's College School. He said [53]:-
I happened to be looking in my local public library, and I found a book on maths and it told a bit about the history of this problem, that someone had resolved this problem 300 years ago, but no one had ever seen the proof. No one knew if there was a proof. And people ever since had looked for the proof. And here was a problem that I, a ten-year-old, could understand, that none of the great mathematicians in the past had been able to resolve. And from that moment, of course, I just tried to solve it myself. It was such a challenge, such a beautiful problem. This problem was Fermat's last theorem.
The book that Wiles was looking at was The Last Problem by Eric Temple Bell. This book, about Fermat's Last Theorem, was published in 1961 in the year following Bell's death. After finding this book, Andrew looked for other books on number theory and began to study other topics which had interested Fermat like solving congruences.

In 1966, when he was twelve years old, Andrew began his studies at The Leys School in Cambridge. This private school, originally founded as a Methodist school in 1875, had many sons of University of Cambridge staff as pupils. Andrew was a member of North A House, located on the upper quad, opposite the chapel. At this school he was taught mathematics by a teacher with a Ph.D. in number theory who gave him a copy of Hardy and Wright's An Introduction to the Theory of Numbers. He also found Davenport's The Higher Arithmetic and these two books inspired him to study number theory.

In 1971, Wiles entered Merton College, Oxford, and, although he was happy to take courses on geometry, logic and applied mathematics, he felt these were distracting him from the things he really wanted to study in depth, namely algebra and number theory. Although his teachers allowed him to take additional number theory courses, he would have liked to have been able to learn more on that topic than Oxford could offer. His attempts to read Fermat in the original Latin were not really successful since Fermat wrote with hardly any symbols and he found it hard to understand. He graduated from the University of Oxford with a B.A. in 1974 and then entered Clare College, Cambridge to study for his doctorate. He had a preliminary year before starting research [61]:-
I had a year, a preliminary year, in which I just studied a range of subjects and then I could do a special paper. John Coates was not yet at Cambridge, but I think he helped me, maybe over the summer. Anyway, that summer I met him and started working with him right away, and that was just wonderful. The transition from undergraduate work, where you were just reading and studying, to research - that was the real break for me. It was just wonderful.
His Ph.D. supervisor, John Coates, had arrived in Cambridge in the summer of 1975 and began to supervise Wiles' research. Coates said [52]:-
I have been very fortunate to have had Andrew as a student. Even as a research student he was a wonderful person to work with, he had very deep ideas then and it was always clear he was a mathematician who would do great things.
Wiles did not work on Fermat's Last Theorem for his doctorate. He said:-
... the problem with working on Fermat is that you could spend years getting nothing so when I went to Cambridge my advisor John Coates was working on Iwasawa theory of elliptic curves and I started working with him ...
From 1977 until 1980 Wiles was a Junior Research fellow at Clare College, Cambridge and also a Benjamin Peirce Assistant Professor at Harvard University in Cambridge, Massachusetts, USA. At Harvard he worked with Barry Mazur on modular forms. Their collaboration led to the jointly authored paper Class fields of abelian extensions of Q\mathbb{Q} (1984) in which they proved the Main Conjecture of Iwasawa theory for real abelian extensions of Q\mathbb{Q} and odd primes pp. In 1980 Wiles was awarded his doctorate for his thesis Reciprocity laws and the conjecture of Birch and Swinnerton-Dyer in which he proved certain special cases of the Birch and Swinnerton-Dyer conjecture.

He then spent a while as a Visiting Professor at the Sonderforschungsbereich Theoretische Mathematik in Bonn, taking up the appointment in the spring of 1981. He returned to the United States in the autumn of 1981 to take up a position at the Institute for Advanced Study in Princeton. He was appointed a professor at Princeton the following year and, also during 1982, he spent the spring as a visiting professor in the University of Paris, Orsay, France.

Wiles was awarded a Guggenheim Fellowship which enabled him to visit the Institut des Hautes Études Scientifiques in Paris and also the École Normale Supérieure in Paris during 1985-86. In [24] the important events which changed the direction of Wiles's research after this period are described:-
... about ten years ago, G Frey suggested and K Ribet proved (building on ideas of B Mazur and J-P Serre) that Fermat's Last Theorem follows from the Shimura-Taniyama conjecture that every elliptic curve defined over the rational numbers is modular. Precisely, if
an+bn=cna^{n} + b^{n} = c^{n}
is a counterexample to Fermat's Last Theorem, then the elliptic curve
y2=x(xan)(x+bn)y^{2} = x(x - a^{n})(x + b^{n})
cannot be modular, thus violating the Shimura-Taniyama conjecture. This result set the stage for Wiles's work.
Wiles said [52]:-
I was at a friend's house sipping iced tea early in the evening, and he just mentioned casually in the middle of a conversation, "By the way, did you hear that Ken Ribet has proved the epsilon conjecture?" And I was just electrified. I knew that moment the course of my life was changing, because this meant that to prove Fermat's Last Theorem, I just had to prove the Taniyama-Shimura conjecture. From that moment, that was what I was working on. I just knew I would go home and work on the Taniyama-Shimura conjecture.
In fact Wiles abandoned all his other research when he heard what had been proved and, for seven years, he concentrated solely on attempting to prove the Shimura-Taniyama conjecture, knowing that a proof of Fermat's Last Theorem then followed. Wiles said:-
... after a few years I realised that talking to people casually about Fermat was impossible because it generated too much interest and you cannot focus yourself for years unless you have this kind of undivided concentration which too many spectators would destroy ...
Andrew Wiles married Nada Canaan in 1988. She was a student at Princeton University, obtaining her first degree in 1983 and, in the summer of 1988, she was awarded a Ph.D. in molecular biology from Princeton. Andrew and Nada Wiles have there daughters: Clare, born in August 1990; Kate, born in November 1991; and Olivia, born May 1994.

In fact married life was a rather restricted affair for Wiles who said:-
... my wife has only known me while I have been working on Fermat. I told her a few days after I got married. I decided that I really only had time for my problem and my family and while I was concentrating very hard then I found with young children that it was the best possible way to relax. When you're talking to young children they're simply not interested in Fermat ...
In 1988 Wiles went to Oxford University where he spent two years as a Royal Society Research Professor. While in Oxford he was elected, in 1989, a Fellow of the Royal Society. In [24] the course of his research is described:-
Using Mazur's deformation theory of Galois representations, recent results on Serre's conjecture on the modularity of Galois representations, and deep arithmetical properties of Hecke algebras, Wiles (with one key step due jointly to Wiles and Richard Taylor) succeeded in proving that all semistable elliptic curves defined over the rational numbers are modular. Although less than the full Shimura-Taniyama conjecture, this result does imply that the elliptic curve given above is modular, thereby proving Fermat's Last Theorem.
In fact the path to the proof was not as smooth as suggested by this description. In 1993 Wiles told two other mathematicians that he was close to a proof of Fermat's Last Theorem. He filled what he thought were the remaining few gaps and gave a series of lectures at the Isaac Newton Institute in Cambridge ending on 23 June 1993. At the end of the final lecture he announced he had a proof of Fermat's Last Theorem. When the results were written up for publication, however, a subtle error was discovered. Wiles said:-
... the first seven years I had worked on this problem I loved every minute of it however hard it had been. There had been setbacks, things which had seemed insurmountable but it was a kind of private and very personal battle I was engaged in and then after there was a problem with it, doing mathematics in that kind of rather overexposed way is certainly not my style, I certainly have no wish to repeat it ...
Wiles worked hard for about a year, helped in particular by Richard Taylor whom we referred to above, and by 19 September 1994, having almost given up, he decided to have one last try:-
... suddenly, totally unexpectedly, I had this incredible revelation. It was the most important moment of my working life. Nothing I ever do again ... it was so indescribably beautiful, it was so simple and so elegant, and I just stared in disbelief for twenty minutes, then during the day I walked round the department. I'd keep coming back to my desk to see it was still there - it was still there.
In 1994 Wiles was appointed Eugene Higgins Professor of Mathematics at Princeton. His paper which proves Fermat's Last Theorem is Modular elliptic curves and Fermat's Last Theorem which appeared in the Annals of Mathematics in 1995. From 1995 Wiles began to receive many honours for this outstanding piece of work. He was awarded the Schock Prize in Mathematics from the Royal Swedish Academy of Sciences and the Prix Fermat from the Université Paul Sabatier. In 1996 he received further awards including the Wolf Prize which he shared with Robert Langlands. The citation for his award states it is given to Wiles [89]:-
... for spectacular contributions to number theory and related fields, major advances on fundamental conjectures, and for settling Fermat's last theorem.
Also in 1996, he was elected as a foreign member to the National Academy of Sciences of the United States, receiving its mathematics prize.

Wiles spoke about how much Fermat's Last Theorem meant to him:-
... there's no other problem that will mean the same to me. I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream. I know its a rare privilege but I know if one can do this it's more rewarding than anything one can imagine.
In [24] his worked is summed up:-
Wiles's work is highly original, a technical tour de force, and a monument to individual perseverance.
In addition to the prizes and awards mentioned above, Wiles has continued to receive many honours for his outstanding work. In 1998, not being eligible for the award of a Fields medal on the grounds that he was over forty years of age, the International Mathematical Union presented him with a silver plaque at the International Congress of Mathematicians held that year in Berlin. He gave an evening lecture at the Congress which attracted an audience of over 2300. In the same year of 1998 he was awarded the King Faisal Prize. He [44]:-
... received the prize at a special ceremony in Riyadh, Saudi Arabia, on January 6, 1998. The prize consists of a $200,000 cash award and a commemorative gold medal. The prize citation noted that Wiles's contribution not only was a major addition to mathematical knowledge but also has had a positive influence on public perceptions of mathematics.
In the following year, he received the Clay Research Award from the Clay Mathematics Institute on 10 May 1999 in Cambridge, Massachusetts [17]:-
...... for his role in the development of number theory.
In the same year he was honoured by having "asteroid 9999 Wiles" named after him.

In 2000, Andrew Wiles became "Sir Andrew Wiles" when he was made a Knight Commander of the Order of the British Empire by the Queen. He received the Pythagoras Award in Crotone, Italy, in 2004 and, in the following year, Wiles received the Shaw Prize. Sir Run Run Shaw, a leader of the Hong Kong media industry, established this Prize in 2002. The first award was made in 2004 to:-
... individuals, regardless of race, nationality and religious belief, who have achieved significant breakthrough in academic and scientific research or application, and whose work has resulted in a positive and profound impact on mankind.
The 2005 Shaw Prize for Mathematical Sciences was made to Wiles:-
... for his proof of Fermat's last theorem.
Wiles was at Princeton University between 1982 and 2010, except for short periods of leave. In 2010 he returned to England when appointed as a Royal Society Research Professor in Oxford. The University of Oxford formally opened the new Andrew Wiles Building on 3 October 2013 [88]:-
The building brought together the department's members from three buildings and, as importantly, is a venue where we can welcome our collaborators, our friends and the researchers of the future to share the beauty and power of mathematics.
You can read the history behind this building at [87].

The Abel Prize is recognised as the highest possible award to a mathematician. It was presented to Andrew Wiles in 2016 [10]:-
... for his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.
The full citation ended by emphasising how the ideas introduced by Wiles in solving Fermat's Last Theorem have led to further advances [10]:-
The new ideas introduced by Wiles were crucial to many subsequent developments, including the proof in 2001 of the general case of the modularity conjecture by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. As recently as 2015, Nuno Freitas, Bao V Le Hung, and Samir Siksek proved the analogous modularity statement over real quadratic number fields. Few results have as rich a mathematical history and as dramatic a proof as Fermat's Last Theorem.
In 2017 Wiles was awarded the Royal Society's Copley Medal, the world's oldest scientific prize. Previous winners of this award include Carl Friedrich Gauss, Charles Darwin, Humphrey Davy and Albert Einstein. Venki Ramakrishnan, President of the Royal Society, presented the award saying [55]:-
Sir Andrew is a well-deserved recipient of the Copley Medal, the Royal Society's most prestigious prize. In proving Fermat's Last Theorem - a problem that had remained unsolved for hundreds of years - he not only made a major mathematical breakthrough, but also captured the imagination of the public. This is an inspirational story of a highly creative intellectual pursuit and the satisfaction of solving a deep fundamental problem in mathematics. He is a hero to an entire generation of mathematicians. The Royal Society is delighted to recognise this achievement.
In May 2018 Wiles was appointed by Her Majesty the Queen to be Oxford's first Regius Professor of Mathematics. Martin Bridson, Head of Oxford's Mathematical Institute, said [69]:-
It is entirely fitting that the first holder of this Professorship should be Sir Andrew Wiles. Nobody exemplifies the relentless pursuit of mathematical understanding in the service of mankind better than him. His dedication to solving problems that have defied mankind for centuries, and the stunning beauty of his solutions to these problems, provide a beacon to inspire and sustain everyone who wrestles with the fundamental challenges of mathematics and the world around us. We are immensely proud to have Andrew as a colleague at the Mathematical Institute in Oxford.
On Friday 13 July 2018, Wiles was awarded the degree of Doctor of Science honoris causa by the University of Bristol. Jon Keating, presenting him for the honorary degree, said [45]:-
Andrew's story exemplifies the academic values we all aspire to: intellectual courage and ambition, passion, intense hard work, relentless determination, and a focus on the most profound ideas and problems. His quest truly deserves to be called heroic, it is a testament to the human spirit. In honouring him today, we celebrate his achievements and those values.
In 2019 Wiles was awarded the London Mathematical Society's De Morgan Medal. The citation reads [85]:-
A De Morgan Medal is awarded to Professor Sir Andrew Wiles FRS of the University of Oxford for his seminal contributions to number theory and for his resolution of 'Fermat's Last Theorem' in particular, as well as for his numerous activities promoting mathematics in general.

Wiles did his PhD with John Coates and immediately attracted attention through their joint work on the conjecture of Birch and Swinnerton-Dyer. This was followed by a proof of the 'Main Conjecture' of Iwasawa Theory for cyclotomic fields, in collaboration with Mazur.

However it is his work on the Shimura-Taniyama-Weil conjecture, with its consequences for Fermat's Theorem, for which he will be forever known. The Shimura-Taniyama-Weil conjecture is a first step in the Langlands Program, a grand vision which seeks to unify and explain a vast panorama of individual arithmetic phenomena, of which only the smallest individual cases are currently understood. In Wiles' work a technique was developed to resolve the semistable case of the Shimura-Taniyama-Weil conjecture. This case was enough to handle Fermat's Last Theorem, but within mathematics the opening into the Shimura-Taniyama-Weil conjecture was even more important. Later workers developed Wiles' methods to establish the conjecture in full, but investigations into further extensions of these ideas within the Langlands programme are still a major industry within the subject.

The publicity associated with his resolution of Fermat's Last Theorem has put Wiles in a unique position to communicate the excitement of mathematics to students of all ages, and to society in general. He has given a large number of public talks, which are invariably 'sold out', and has reached an enormous audience across the globe.
On 23 June 2023 the Isaac Newton Institute for Mathematical Sciences in Cambridge, England, celebrated 30 years since Andrew Wiles announced his first proof of Fermat's Last Theorem. They produced a video interview, a podcast documentary and a written article [86]:-
... all aimed at delving deeper into why this proof of a more than 350-year-old problem had, and continues to have, such a profound effect upon not just the field of number theory, but also on mathematics as a subject, and even on the public perception of the science.
You can see the video and article by following the link to [86]. The text ends as follows:-
That moment 30 years ago was clearly a turning point in Wiles' career. He is one of the few mathematicians who is well-known outside of mathematics, and was recognised with a knighthood in 2000. Within mathematics he has received a wealth of honours and awards, including the prestigious Abel Prize in 2016. It's been such a pleasure to revisit this moment with all of these mathematicians, to hear the human story, as well as the mathematical one. Wiles told us, back in 2016, about some of the personal qualities a mathematician has to have - they have to be creative, and they have to be able to enjoy being stuck. And perseverance again appeared as a key thread in the story when we spoke to him for this article. Our final question was whether he would have kept on working on Fermat's Last Theorem even if he hadn't found a solution back in the early 90s. His answer was characteristic of his approach to mathematics: "I am not a person who gives up on a problem."

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Written by J J O'Connor and E F Robertson
Last Update December 2023