### Born: 11 April 1953 in Cambridge, England

**Andrew Wiles**'s interest in Fermat's Last Theorem began at a young age. He said:-

*... I was a ten year old and one day 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 and I, a ten year old, could understand it. From that moment I tried to solve it myself, it was such a challenge, such a beautiful problem, this problem was Fermat's Last Theorem.*

In 1971, Wiles entered Merton College, Oxford, graduating with a B.A. in 1974. He then entered Clare College, Cambridge to study for his doctorate. His Ph.D. supervisor at Cambridge was John Coates who said:-

*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 1980 he was awarded his doctorate, then spent a while at the Sonderforschungsbereich Theoretische Mathematik in Bonn. He returned to the United States near the end of 1981 to take up a post at the Institute for Advanced Study in Princeton. He was appointed a professor at Princeton the following year and, also during 1982, he spent a while as a visiting professor in Paris.

Wiles was awarded a Guggenheim Fellowship which enabled him to visit the Institut des Hautes Études Scientifique in Paris and also the École Normale Supérieure in Paris during 1985-86. In [1] 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*

*a*^{n} + *b*^{n} = *c*^{n}

*is a counterexample to Fermat's Last Theorem, then the elliptic curve*

*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.*

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...*

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 [1] 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 R 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 R Taylor 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 included the Wolf Prize and was elected as a foreign member to the National Academy of Sciences of the United States, receiving its mathematics prize.

Wiles said:-

*... 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 [1] his worked is summed up:-

*Wiles's work is highly original, a technical tour de force, and a monument to individual perseverance.*

**Article by:** *J J O'Connor* and *E F Robertson*

**April 1997**

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