安德鲁·怀尔斯Andrew Wiles

安德鲁·怀尔斯的父亲Maurice Frank Wiles（1923-2005）在剑桥大学学习道德科学和神学，然后在剑桥Ridley Hall接受按立培训。在斯托克波特担任副牧师两年后，他于1950年至1955年成为剑桥Ridley Hall的牧师。怀尔斯的母亲是Patricia Margaret Mowll（1927-2005，被称为Paddy）。她于1950年在英格兰苏塞克斯郡Cuckfield与Maurice Wiles结婚。Maurice和Patricia 怀尔斯有三个孩子：Patrick David Wiles（1951年5月2日出生）；怀尔斯 John Wiles，本传记的主题；以及Alison Ruth Wiles（1955年8月3日出生）。

1955年，怀尔斯两岁时，他的父亲被任命为尼日利亚伊巴丹新约研究的讲师。正是在尼日利亚，怀尔斯开始了他的学业，但像许多孩子一样，他起初不愿上学。1959年，Maurice Wiles回到剑桥，成为克莱尔学院的院长。全家乘坐Winneba从拉各斯航行到伦敦，于1959年7月3日抵达英国。在1959-60学年开始时，怀尔斯在剑桥国王学院学校开始学习，这是一所由国王亨利六世于1441年创立的预备学校。在这所学校，玛丽·布里格斯教他数学，她是校长弗洛伦斯·南丁格尔·大卫布里格斯的妻子。

怀尔斯对费马大定理的兴趣始于他在国王学院学校学习期间。他说[53]：-

我碰巧在当地公共图书馆里看书，发现了一本关于数学的书，它讲述了这个问题的历史，有人在300年前解决了这个问题，但没有人见过证明。没有人知道是否有证明。从那以后，人们一直在寻找证明。而这是一个我，一个十岁的孩子，能理解的问题，过去所有伟大的数学家都无法解决。从那一刻起，当然，我只是试图自己解决它。这是一个如此大的挑战，如此美丽的问题。这个问题就是皮埃尔·德·费马的最后定理。

怀尔斯正在看的书是埃里克·坦普尔·贝尔写的The Last Problem。这本书关于皮埃尔·德·费马的最后定理，于1961年出版，即埃里克·坦普尔·贝尔去世后的那一年。找到这本书后，怀尔斯寻找其他数论书籍，并开始研究其他皮埃尔·德·费马感兴趣的主题，如解同余式。

1966年，当他十二岁时，怀尔斯在剑桥的莱斯学校开始学习。这所私立学校最初于1875年作为卫理公会学校创立，有许多剑桥大学教职员工的儿子作为学生。怀尔斯是北A楼的成员，位于上四方院，对面是教堂。在这所学校，一位拥有数论博士学位的老师教他数学，并给了他一本戈弗雷·哈罗德·哈代和爱德华·梅特兰·赖特的An Introduction to the Theory of Numbers。他还发现了哈罗德·达文波特的The Higher Arithmetic，这两本书激发了他研究数论。

1971年，怀尔斯进入牛津大学默顿学院，尽管他乐于修读几何、逻辑和应用数学的课程，但他觉得这些课程分散了他对真正想要深入研究的领域的注意力，即代数和数论。虽然他的老师允许他选修额外的数论课程，但他希望能学到比牛津所能提供的更多关于该主题的知识。他尝试阅读皮埃尔·德·费马的拉丁文原著并不十分成功，因为皮埃尔·德·费马几乎不使用任何符号，他发现很难理解。他于1974年从牛津大学获得学士学位，然后进入剑桥大学克莱尔学院攻读博士学位。在开始研究[61]之前，他有一年的预备年：-

我有一年，一个预备年，在那一年里我只是学习了一系列科目，然后我可以做一个专题论文。约翰·亨利·科茨当时还不在剑桥，但我想他帮助了我，也许是在夏天。总之，那个夏天我遇见了他，并立即开始与他合作，那真是太棒了。从本科工作——你只是阅读和学习——到研究的转变，对我来说是真正的突破。那真是太棒了。

他的博士导师约翰·亨利·科茨于1975年夏天抵达剑桥，并开始指导怀尔斯的研究。约翰·亨利·科茨说[52]：-

我非常幸运能有怀尔斯作为学生。即使作为一名研究生，他也是一个很好共事的人，他当时就有非常深刻的想法，而且始终清楚他是一位会做出伟大成就的数学家。

怀尔斯的博士论文并非研究皮埃尔·德·费马大定理。他说：-

……研究皮埃尔·德·费马的问题在于你可能花费数年却一无所获，所以当我来到剑桥时，我的导师约翰·亨利·科茨正在研究椭圆曲线的岩泽健吉理论，我就开始与他合作……

从1977年到1980年，怀尔斯是剑桥大学克莱尔学院的研究员，同时也是美国马萨诸塞州剑桥市哈佛大学的本杰明·皮尔斯助理教授。在哈佛，他与巴里·马聚尔合作研究模形式。他们的合作促成了共同署名的论文Class fields of abelian extensions of $\mathbb{Q}$（1984年），在其中他们证明了岩泽健吉理论对于$\mathbb{Q}$的实阿贝尔扩张和奇素数$p$的主猜想。1980年，怀尔斯因其学位论文Reciprocity laws and the conjecture of Birch and Swinnerton-Dyer获得博士学位，在论文中他证明了Birch和Swinnerton-Dyer猜想的某些特殊情形。

然后他作为波恩理论数学特别研究领域的访问教授度过了一段时间，于1981年春季就职。他于1981年秋季返回美国，在普林斯顿高等爱德华·斯图迪研究所任职。次年，他被任命为普林斯顿大学教授，同样在1982年，他在法国奥赛的巴黎大学担任春季访问教授。

怀尔斯获得了古根海姆奖学金，这使他得以在1985-86年间访问巴黎的高等科学研究所和巴黎高等师范学院。在[24]中描述了这一时期之后改变怀尔斯研究方向的重要事件：-

...大约十年前，G Frey提出并由K Ribet证明（基于巴里·马聚尔和让-皮埃尔·塞尔的想法），皮埃尔·德·费马的最后定理可由志村五郎-谷山丰猜想推出，即定义在有理数上的每条椭圆曲线都是模的。确切地说，如果

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

是皮埃尔·德·费马最后定理的一个反例，那么椭圆曲线

$y^{2} = x(x - a^{n})(x + b^{n})$

不可能是模的，从而违反志村五郎-谷山丰猜想。这一结果为怀尔斯的工作奠定了基础。

怀尔斯说[52]：-

傍晚时分，我在朋友家喝冰茶，他在谈话中随口提到：“顺便问一下，你听说Ken Ribet证明了epsilon猜想了吗？”我顿时感到一阵激动。那一刻我知道我的人生轨迹正在改变，因为这意味着要证明皮埃尔·德·费马大定理，我只需证明谷山丰-志村五郎猜想。从那一刻起，这就是我一直在研究的东西。我就知道我会回家研究谷山丰-志村五郎猜想。

事实上，怀尔斯在听说已被证明的内容后，放弃了他所有其他研究，七年来，他全神贯注于试图证明志村五郎-谷山丰猜想，因为他知道这样一来就能证明皮埃尔·德·费马大定理。怀尔斯说：-

……几年后我意识到，随便与人谈论皮埃尔·德·费马是不可能的，因为它引起了太多兴趣，除非你有这种不被分心的专注，否则你无法多年专注于自己，而太多旁观者会破坏这种专注……

怀尔斯于1988年与Nada Canaan结婚。她曾是普林斯顿大学的学生，1983年获得第一个学位，1988年夏天，她在普林斯顿大学获得分子生物学博士学位。怀尔斯和Nada 怀尔斯有女儿：Clare，1990年8月出生；Kate，1991年11月出生；Olivia，1994年5月出生。

事实上，婚姻生活对怀尔斯来说是一件相当受限的事情，他说：-

……我的妻子只在我研究皮埃尔·德·费马的时候认识我。结婚几天后我告诉了她。我决定我真的只有时间处理我的问题和我的家庭，而当我非常专注时，我发现与年幼的孩子相处是最好的放松方式。当你和年幼的孩子交谈时，他们根本不关心皮埃尔·德·费马……

1988年，怀尔斯前往牛津大学，在那里作为皇家学会研究教授度过了两年。在牛津期间，他于1989年当选为皇家学会会士。在[24]中描述了他的研究过程：-

利用巴里·马聚尔的埃瓦里斯特·伽罗瓦表示形变理论、关于让-皮埃尔·塞尔的埃瓦里斯特·伽罗瓦表示模性猜想的最新结果，以及埃里希·赫克代数的深刻算术性质，怀尔斯（其中一个关键步骤由怀尔斯和理查德.泰勒共同完成）成功证明了定义在有理数上的所有半稳定椭圆曲线都是模性的。尽管这还不及完整的志村五郎-谷山丰猜想，但这一结果确实意味着上述椭圆曲线是模性的，从而证明了皮埃尔·德·费马大定理。

事实上，证明之路并不像这段描述所暗示的那样顺利。1993年，怀尔斯告诉另外两位数学家，他接近证明皮埃尔·德·费马大定理。他填补了他认为剩余的几个缺口，并在剑桥的艾萨克·牛顿研究所做了一系列讲座，于1993年6月23日结束。在最后一讲结束时，他宣布他证明了皮埃尔·德·费马大定理。然而，当结果写成论文准备发表时，发现了一个微妙的错误。怀尔斯说：-

……最初七年我研究这个问题时，无论多么艰难，我都热爱其中的每一分钟。有过挫折，有些事情似乎无法逾越，但这是我参与的一种私人的、非常个人的战斗，然后当它出现问题后，以那种相当过度曝光的方式做数学肯定不是我的风格，我当然不想重复它……

怀尔斯努力工作了大约一年，特别得到了我们上面提到的理查德.泰勒的帮助，到1994年9月19日，几乎已经放弃的他决定做最后一次尝试：-

……突然，完全出乎意料地，我有了这个不可思议的启示。这是我工作生涯中最重要的时刻。我以后再也做不出……它是如此难以形容的美丽，如此简单而优雅，我难以置信地盯着看了二十分钟，然后那天我在系里走来走去。我会不断回到我的办公桌前看看它还在不在——它还在那里。

1994年，怀尔斯被任命为普林斯顿大学Eugene Higgins数学讲席教授。他证明皮埃尔·德·费马最后定理的论文是Modular elliptic curves and Fermat's Last Theorem，于1995年发表在Annals of Mathematics上。从1995年起，怀尔斯因这一杰出工作开始获得许多荣誉。他获得了瑞典皇家科学院颁发的Schock数学奖和Université Paul Sabatier颁发的Prix 皮埃尔·德·费马。1996年，他又获得了更多奖项，包括与罗伯特·朗兰兹共享的沃尔夫奖。他的获奖评语写道，该奖授予怀尔斯 [89]：-

……对数论及相关领域做出了引人注目的贡献，在基本猜想上取得了重大进展，并解决了皮埃尔·德·费马最后定理。

同样在1996年，他当选为国家科学院 of the United States的外籍成员，并获得了该院的数学奖。

怀尔斯谈到了皮埃尔·德·费马最后定理对他意味着什么：-

……没有其他问题会对我有同样的意义。我拥有这种非常罕见的特权，能够在成年后追求我童年的梦想。我知道这是一种罕见的特权，但我知道如果一个人能做到这一点，那比任何能想象到的事情都更有回报。

在[24]中，他的工作被总结为：-

怀尔斯的工作极具原创性，是技术上的杰作，也是个人毅力的丰碑。

除上述奖项和荣誉外，怀尔斯因其杰出工作继续获得许多荣誉。1998年，由于年龄超过四十岁而无资格获得约翰·查尔斯·菲尔兹奖章，International Mathematical Union在当年于柏林举行的国际数学家大会上向他颁发了一块银牌。他在大会上作了晚间演讲，吸引了超过2300名听众。同年1998年，他被授予费萨尔国王奖。他[44]：-

……于1998年1月6日在沙特阿拉伯利雅得的一个特别仪式上领取了该奖。该奖包括20万美元现金奖励和一枚纪念金牌。奖项引文指出，怀尔斯的贡献不仅是对数学知识的重大补充，而且对公众对数学的认知产生了积极影响。

次年，他于1999年5月10日在马萨诸塞州剑桥从克莱数学研究所获得了克莱研究奖[17]：-

……因他在数论发展中的作用。

同年，他因“小行星9999 怀尔斯”以他命名而获得荣誉。

2000年，怀尔斯被女王封为“Sir 怀尔斯”，成为大英帝国爵级司令。2004年，他在意大利克罗托内获得毕达哥拉斯奖，次年，怀尔斯获得邵逸夫奖。香港传媒业领袖邵逸夫爵士于2002年设立此奖。首次颁奖于2004年，授予：-

……不论种族、国籍和宗教信仰，在学术和科学研究或应用方面取得重大突破，且其工作对人类产生积极而深远影响的个人。

2005年邵逸夫数学科学奖授予怀尔斯：-

……因其证明了皮埃尔·德·费马大定理。

1982年至2010年间，怀尔斯一直在普林斯顿大学，仅有短暂休假。2010年，他回到英国，被任命为牛津的皇家学会研究教授。牛津大学于2013年10月3日正式启用新的怀尔斯大楼[88]：-

这座大楼将该系成员从三座建筑中汇聚到一起，同样重要的是，它是一个场所，让我们能够欢迎我们的合作者、朋友和未来的研究者来分享数学之美与力量。

你可以在[87]读到这座建筑背后的历史。

尼尔斯·阿贝尔奖被公认为授予数学家的最高奖项。它于2016年颁发给怀尔斯[10]：-

……因为他通过半稳定椭圆曲线的模性猜想令人惊叹地证明了皮埃尔·德·费马大定理，开辟了数论的新时代。

完整的颁奖词在结尾强调，怀尔斯在解决皮埃尔·德·费马大定理时引入的思想如何带来了进一步的进展[10]：-

怀尔斯引入的新思想对许多后续发展至关重要，包括2001年Christophe Breuil、Brian Conrad、Fred Diamond和理查德.泰勒对模性猜想一般情形的证明。就在2015年，Nuno Freitas、Bao V Le Hung和Samir Siksek证明了实二次数域上的类似模性陈述。很少有结果像皮埃尔·德·费马大定理那样拥有如此丰富的数学历史和如此戏剧性的证明。

2017年，怀尔斯被授予皇家学会的科普利奖章，这是世界上最古老的科学奖项。该奖项此前的获奖者包括卡尔·弗里德里希·高斯、查尔斯·高尔顿·达尔文、Humphrey Davy和阿尔伯特·爱因斯坦。Royal Society主席Venki Ramakrishnan颁发了该奖项，他说[55]：-

怀尔斯爵士是科普利奖章的当之无愧的获得者，这是皇家学会最负盛名的奖项。在证明皮埃尔·德·费马最后定理——一个数百年来悬而未决的问题——的过程中，他不仅取得了重大的数学突破，还激发了公众的想象力。这是一个关于高度创造性的智力追求以及解决数学中一个深刻基本问题所带来的满足感的鼓舞人心的故事。他是整整一代数学家的英雄。Royal Society很高兴认可这一成就。

2018年5月，怀尔斯被女王陛下任命为牛津大学首位皇家数学讲席教授。牛津大学数学研究所所长Martin Bridson表示[69]：-

这一讲席的首位持有者由怀尔斯爵士担任，完全恰当。没有人比他更能体现为人类服务而对数学理解的不懈追求。他致力于解决困扰人类数百年的问题，以及他对这些问题解答的惊人美感，为所有与数学和我们周围世界的根本挑战搏斗的人提供了激励和支撑的灯塔。我们极为自豪能有怀尔斯作为牛津大学数学研究所的同事。

2018年7月13日星期五，怀尔斯被布里斯托大学授予荣誉理学博士学位。Jon Keating在为他颁发荣誉学位时表示[45]：-

怀尔斯的故事体现了我们所有人都向往的学术价值观：智力勇气和雄心、激情、刻苦努力、不懈决心，以及对最深刻思想和问题的专注。他的探索确实配得上被称为英雄般的，这是对人类精神的见证。今天向他致敬，我们庆祝他的成就和那些价值观。

2019年，怀尔斯被授予伦敦数学会的奥古斯塔斯·德摩根奖章。颁奖词写道[85]：-

奥古斯塔斯·德摩根奖章授予牛津大学的怀尔斯教授爵士FRS，以表彰他对数论的奠基性贡献，特别是他对“皮埃尔·德·费马大定理”的解决，以及他为推广数学所做的大量活动。

怀尔斯与约翰·亨利·科茨一起完成了他的博士学位，并立即通过他们关于Birch和Swinnerton-Dyer猜想的合作工作引起了关注。随后，他与巴里·马聚尔合作，证明了分圆域的岩泽健吉理论的‘主猜想’。

然而，正是他在志村五郎-谷山丰-安德烈·韦伊猜想上的工作，及其对皮埃尔·德·费马定理的影响，使他永远为人所知。志村五郎-谷山丰-安德烈·韦伊猜想是罗伯特·朗兰兹纲领的第一步，这是一个宏大的愿景，旨在统一和解释大量个别算术现象的全景，而目前仅理解了其中最小的个别情形。在怀尔斯的工作中，发展了一种技术来解决志村五郎-谷山丰-安德烈·韦伊猜想的半稳定情形。这一情形足以处理皮埃尔·德·费马最后定理，但在数学内部，通向志村五郎-谷山丰-安德烈·韦伊猜想的开端甚至更为重要。后来的工作者发展了怀尔斯的方法来完全建立该猜想，但在罗伯特·朗兰兹纲领内对这些思想的进一步扩展的研究仍然是该学科内的一个重要领域。

他解决皮埃尔·德·费马大定理所带来的公众关注，使怀尔斯处于一个独特的位置，能够向各年龄段的学生以及整个社会传达数学的激动人心之处。他做了大量公开演讲，这些演讲无一例外都“座无虚席”，并触及了全球大量听众。

2023年6月23日，位于英格兰剑桥的艾萨克·牛顿数学科学研究所庆祝了怀尔斯宣布其首次证明皮埃尔·德·费马大定理30周年。他们制作了一段视频采访、一部播客纪录片和一篇书面文章[86]：-

……所有这些都旨在更深入地探究，为什么这个对一个已有350多年历史的问题的证明曾经、并且继续对不仅数论领域，而且对数学这门学科，甚至对公众对科学的看法产生如此深远的影响。

你可以通过以下链接观看视频和文章[86]。文本结尾如下：-

30年前的那一刻显然是怀尔斯职业生涯的转折点。他是少数在数学之外也广为人知的数学家之一，并于2000年被授予爵士头衔。在数学界，他获得了大量荣誉和奖项，包括2016年享有盛誉的尼尔斯·阿贝尔奖。能与所有这些数学家一起重温这一刻，聆听其中的人性故事以及数学故事，真是非常愉快。怀尔斯在2016年告诉我们数学家必须具备的一些个人品质——他们必须有创造力，并且必须能够享受陷入困境。当我们为这篇文章采访他时，坚持不懈再次成为故事中的一条关键线索。我们最后的问题是，即使他在90年代初没有找到解决方案，他是否还会继续研究皮埃尔·德·费马大定理。他的回答体现了他对待数学的典型态度：“我不是一个会放弃问题的人。”

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 $\mathbb{Q}$ (1984) in which they proved the Main Conjecture of Iwasawa theory for real abelian extensions of $\mathbb{Q}$ and odd primes $p$. 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

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

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.

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 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."