迈克尔·阿蒂亚Michael Atiyah

迈克尔·阿蒂亚的父亲爱德华·塞利姆阿蒂亚（1903-1964）是黎巴嫩人，母亲简·利文斯是苏格兰人。爱德华的父亲是喀土穆的一名医生，爱德华本人曾在牛津大学布雷齐诺斯学院接受教育，后来成为喀土穆的一名公务员。他也是一位作家，并在第二次世界大战期间建立了一个无线电广播服务。他是巴勒斯坦事业的坚定支持者。阿蒂亚在[32]接受采访时谈到了他的父亲：-

我父亲的主要梦想是去牛津。他想把自己变成一个英国人。但这并没有完全实现。当他回到苏丹时，他发现自己不属于英国阶级结构的一部分，他被视为下层阶级之一，尽管他受过牛津教育，并在文化上认为自己是个英国人。这让他有些转变。他在某种程度上成了阿拉伯民族主义者。他的一生都在渴望成为英国人和同情大英帝国内阿拉伯政治立场之间分裂。

阿蒂亚的母亲简，尽管有苏格兰血统，却是约克郡一座教堂牧师的女儿。她住在牛津，并在那里的大学学习过。爱德华和简就是在牛津相遇的。他们有四个孩子，三个儿子阿蒂亚（长子，也是本传记的主人公）、帕特里克·塞利姆（生于1931年3月5日，后来成为英国律师和学者）和约瑟夫（被称为乔，四个孩子中最小的，在剑桥大学获得数学学位后，成为从事计算机软件和电信工作的计算机科学家），以及一个女儿塞尔玛（在美国一所大学学习英语，现居美国）。尽管阿蒂亚出生在伦敦，但他是在喀土穆长大的。然而，为了避开那里的夏季酷热，全家通常在那时返回英国。阿蒂亚的小学教育是在喀土穆的教区学校，他于1934年五岁时入学。他于1941年完成小学教育，全家照例返回英国。

黎巴嫩曾受法国控制，1940年法国沦陷后，它落入维希政府控制之下。在英国之行后，阿蒂亚一家于1941年经法国返回黎巴嫩，阿蒂亚回到一所法国学校。然而，就在这开始后不久，英国和自由法国开始争夺黎巴嫩的控制权。阿蒂亚被送到开罗的维多利亚学院。这是一所仿照英国寄宿学校制度建立的寄宿学校，也是Edward Atiyah曾就读的学校。阿蒂亚在自传[3]中写道：-

在维多利亚学院，我接受了良好的基础教育，但不得不适应比班上大多数同学小两岁的情况。我通过帮助大男孩做作业而生存下来，因此受到他们的保护，免受寄宿学校不可避免的欺凌。

阿蒂亚在[35]中谈到了他是如何选择数学的：-

我从小就对数学很感兴趣。……我的父母一直认为我从小就是当数学家的料，一直都是这样。……但在[开罗的维多利亚学院]有一个阶段，我对化学非常感兴趣，我觉得那会是很棒的事情；在学了大约一年高等化学之后，我认定那不是我想要做的事，于是又回到了数学。我从未认真考虑过做其他任何事情。

他在采访中对自己在化学与数学之间的抉择作了稍为详细的描述。他说，是无机化学让他对这门学科失去了兴趣[15]：——

就是如何制造硫酸之类的东西。一堆事实，纯粹是事实，你得记住大量的材料。有机化学更有意思，它有一点结构。但无机化学只是像这样的书里堆积如山的事实。确实，在数学里你并不真的需要惊人的记忆力。大多数东西你都可以自己推导出来，记住几条原理就行。如果你擅长这个，那它来得就容易。如果你想做别的事情，你就得下功夫去学大量的事实。我想，这是一个原因。但我喜欢思考，我擅长思考，并且会继续下去。

1945年战争结束后，Edward Atiyah回到英国永久定居。阿蒂亚进入了曼彻斯特文法学校，这是全国最好的数学学校之一。尽管他只有十六岁，但他已经在开罗的维多利亚学院比同龄人高出两个年级，并已经参加了A-level考试。他在曼彻斯特文法学校的两年时间都用来准备剑桥奖学金考试。然而，正是在这所学校，他开始热爱几何[3]：——

我发现我必须非常努力才能跟上班级，竞争很激烈。我们有一位老派但鼓舞人心的老师，他1912年毕业于牛津，从他那里我获得了对射影几何的热爱，其优雅的综合证明，这种热爱从未离开过我。我成为了，并且一直是，一个几何学家，尽管这个词在不同的层次上有不同的解释。我还被引介了威廉·哈密顿关于四元数的工作，其优美令我着迷，至今依然如此。

1947年，他获得了剑桥大学三一学院的奖学金。然而，他没有直接去上大学——那是一个选项——而是决定服两年国民服役，这在当时是强制性的。他担任文书官员，并利用这个机会阅读数学书籍和文章。他在这时读了戈弗雷·哈罗德·哈代和爱德华·梅特兰·赖特的Number Theory，还读了关于群理论的文章。他获得特别许可，缩短了兵役的最后一年，在剑桥度过。在那里，他打了很多网球，并在图书馆里如饥似渴地自学数学。他于1949年秋季在三一学院注册入学。他的许多同学都决定推迟国民服役，所以阿蒂亚是他那一届学生中年龄较大的之一。凭借他非凡的才能、额外的成熟以及开课前所做的学习，尽管有许多非常有才华的同学，他仍然排名第一，这一点也不奇怪。还在本科阶段，他就写了他的第一篇论文A note on the tangents of a twisted cubic（1952年）。

1952年获得学士学位后，阿蒂亚继续在剑桥大学三一学院进行研究，于1955年凭借学位论文 Some Applications of Topological Methods in Algebraic Geometry获得博士学位。他的学位论文导师是威廉·瓦兰斯·道格拉斯·霍奇。谈到学位论文的工作，阿蒂亚说[35]：-

我来到剑桥时，几何学的重点还是老式的经典射影代数几何，我对此非常喜欢。我本会继续在那个领域工作，只是威廉·瓦兰斯·道格拉斯·霍奇代表了更现代的观点——与拓扑学相关的微分几何；我认识到了这一点。这对我来说是一个非常重大的决定。我本可以研究更传统的东西，但我认为这是一个明智的选择，而与他一起工作让我更多地接触到现代思想。他给了我很好的建议，有一个阶段我们还合作过。当时法国在层论方面有一些新近的工作。我对此产生了兴趣，他也产生了兴趣，我们一起工作并写了一篇联合论文，那是我学位论文的一部分。这对我非常有益。

阿蒂亚与他的学位论文导师威廉·瓦兰斯·道格拉斯·霍奇发表了两篇联合论文，Formes de seconde espèce sur une variété algébrique Ⓣ（代数簇上的第二类形式）（1954）和Integrals of the second kind on an algebraic variety（1955）。他还发表了独著论文Complex fibre bundles and ruled surfaces（1955）。1954年，他被选为剑桥大学三一学院会士。1955年7月30日，他与Lily Brown结婚；他们有三个儿子弗瑞兹·约翰、弗洛伦斯·南丁格尔·大卫和Robin。Lily 1928年生于爱丁堡，是罗塞斯海军船厂一名码头工人的女儿。她先是在爱丁堡大学学习数学，然后参加了剑桥的荣誉学位考试。她后来获得博士学位，师从玛丽·卡特赖特。Lily在剑桥结识了阿蒂亚，但到他们结婚时，她已是伦敦贝德福德学院的讲师。阿蒂亚获得英联邦奖学金，于1955-56学年在普林斯顿高等爱德华·斯图迪研究所学习。Lily不得不决定是保留她在贝德福德学院的工作，还是随丈夫去普林斯顿。她选择随丈夫去普林斯顿，并辞去了贝德福德学院的职位。对阿蒂亚来说，这是重要的一年，他结识了让-皮埃尔·塞尔、弗里德里希·希策布鲁赫、小平邦彦、唐纳德·斯宾塞、拉乌尔·博特和伊萨多·辛格等人。回到剑桥后，他于1957年成为学院讲师，1958年成为彭布罗克学院会士。他一直留在剑桥，直到1961年转到牛津大学担任高级讲师，并在那里成为圣凯瑟琳学院会士。

阿蒂亚很快于1963年填补了牛津大学极负盛名的Savilian几何学讲席，担任该讲席直到1969年，然后被任命为普林斯顿高等斯图迪研究所的数学教授。在普林斯顿三年后，阿蒂亚回到英国，成为牛津大学的皇家学会研究教授。他还被选为牛津大学圣凯瑟琳学院会士。牛津一直是阿蒂亚的基地，直到1990年他成为剑桥大学三一学院院长和新成立的剑桥艾萨克·牛顿数学科学研究所所长。

阿蒂亚展示了空间上向量丛的研究如何可以被视为上同调理论的研究，称为K理论。亚历山大·格罗滕迪克也对K理论的发展做出了重大贡献。在[13]中，阿蒂亚的早期数学工作被描述如下：-

阿蒂亚对数学中围绕几何与分析相互作用的广泛主题做出了贡献。他的第一个重大贡献（与弗里德里希·希策布鲁赫合作）是在拓扑学中发展了一种新的强大技术（K理论），这导致了许多突出困难问题的解决。随后（与伊萨多·辛格合作），他建立了一个关于椭圆微分方程解的数目的重要定理。这个“指标定理”在代数几何中有先例，并导致了微分几何、拓扑学和分析之间的重要新联系。结合对称性的考虑，它（与拉乌尔·博特合作）导致了一个具有广泛适用性的新的、更精细的“不动点定理”。

由于这些早期成就，阿蒂亚于1966年在莫斯科国际数学家大会上被授予约翰·查尔斯·菲尔兹奖章。亨利·嘉当在大会上作了关于阿蒂亚贡献的报告，见[18]。K理论和指标定理在阿蒂亚的书K-theory（1967，1989年重印）以及他与G B 尔文·席格合作的The Index of Elliptic Operators I-V中得到了研究，后者发表在Annals of Mathematics第88和93卷（1968，1971）。阿蒂亚还在1973年作为美国数学会学术报告会讲座所作的The index of elliptic operators中描述了他关于指标定理的工作。

导致阿蒂亚被授予菲尔兹奖的那些思想，后来被认为与基本粒子的规范理论有关。我们再次引用[13]：-

指标定理可以用量子理论来解释，并且已被证明是理论物理学家的有用工具。除了这些线性问题之外，规范理论还涉及深刻而有趣的非线性微分方程。特别是，杨-米尔斯方程已被证明对数学家特别富有成果。阿蒂亚开创了该领域早期的大部分工作，他的学生西蒙·唐纳森继续在4维几何中 spectacularly 运用了这些思想。最近，阿蒂亚在强调拓扑学在量子场论中的作用以及将理论物理学家的工作，特别是爱德华·威滕的工作，引起数学界的注意方面具有影响力。

超空间和超引力理论以及基本粒子的弦理论，以新颖而出人意料的方式涉及波恩哈德·黎曼曲面的理论，都是理论物理学的领域，它们利用阿蒂亚所引入的思想发展起来。

阿蒂亚出版了许多极具影响力的书籍：K-theory（1967）；与I G Macdonald合著Introduction to commutative algebra（1969）；Vector fields on manifolds （1970）；Elliptic operators and compact groups（1974）；Geometry on Yang-Mills fields（1979）；与N J Hitchin合著The geometry and dynamics of magnetic monopoles（1988）；The geometry and physics of knots（1990）；（视频）The mysteries of space（1992）；Siamo tutti MatematiciⓉ（我们都是数学家）（2007）；以及Edinburgh Lectures on Geometry, Analysis and Physics（2010）。

我们给出这些书的一些评论摘录、一些序言摘录以及THIS LINK上的一些出版商描述。

阿蒂亚和约翰·泰特在2000年5月24日于巴黎的一次演讲中描述了克莱数学研究所千年奖问题。阿蒂亚的演讲涵盖了儒勒·昂利·庞加莱猜想、威廉·瓦兰斯·道格拉斯·霍奇猜想、量子杨-米尔斯理论和克洛德-路易·纳维-乔治·加布里埃尔·斯托克斯方程。他解释了这些问题并将它们置于历史背景中。他还讨论了如果找到这些问题的解决方案，对数学和物理各个领域的影响。该演讲的60分钟视频可用，标题为The millennium prize problems。

阿蒂亚的Collected Works已出版了六卷。这些卷包含阿蒂亚的评论，在序言中他评论了在作者生前出版“文集”的做法：-

在作者去世很久之前出版“文集”似乎越来越流行。这对各方都有几个明显的优势：后代省去了进行收集的麻烦，而作者可以通过评论添加一些个人色彩。也有缺点：评论会有偏见，作者可能会觉得自己被解雇了。

阿蒂亚贡献的另一个重要方面是他指导的博士生的非凡集合。

我们在THIS LINK上列出了他的学生及其学位论文的标题和日期，对于那些我们知道已经进入学术生涯的人，还列出了他们曾任教的一所大学。

我们还有一条链接提供给读者。阿蒂亚为普通读者撰写的八篇论文的简短摘录见THIS LINK。

阿蒂亚在其职业生涯中获得了许多荣誉，除上述约翰·查尔斯·菲尔兹奖章外，虽然我们无法一一列举，但将给出相当完整的叙述。他于1962年32岁时当选为伦敦皇家学会会士。他于1968年获得该学会的皇家奖章，并于1988年获得其科普利奖章。他于1975年作了皇家学会关于Global geometry的贝克里安讲座，并于1990年至1995年担任皇家学会主席。他于2005年至2008年担任爱丁堡皇家学会主席。

他获得的奖项包括1981年Accademia Nazionale dei Lincei的费尔特里内利奖、1987年的费萨尔国王国际科学奖、1990年爱丁堡皇家学会的冈宁维多利亚禧年奖、1993年的本杰明·富兰克林奖章、1993年的贾瓦哈拉尔·尼赫鲁纪念奖章、1997年委内瑞拉共和国颁发的安德烈斯·贝略勋章（一级）、2003年爱丁堡皇家学会的皇家奖章、2005年黎巴嫩颁发的功绩勋章（金质）以及2008年物理学会的主席奖章。2004年，阿蒂亚和伊萨多·辛格被挪威科学与文学院授予了48万英镑的尼尔斯·阿贝尔奖：-

……因他们发现并证明了指标定理，将拓扑学、几何学和分析学结合在一起，以及他们在数学与理论物理学之间搭建新桥梁方面所发挥的杰出作用。

挪威国王哈拉尔五世在奥斯陆的一个仪式上向他们颁发了该奖项。

阿蒂亚是1973年美国数学会学术讨论会讲师。他于1974-76年担任伦敦数学会主席，并于1980年获得其奥古斯塔斯·德摩根奖章。阿蒂亚于1983年被封为爵士，并于1992年成为功绩勋章成员。

他当选为许多国家科学院的外籍院士，包括：American Academy of Arts and Sciences（1969年）、Royal Swedish Academy of Sciences（1972年）、German Academy of Scientist Leopoldina（1977年）、Académie des Sciences, Paris（1978年）、United States National Academy of Sciences（1978年）、Royal Irish Academy（1979年）、第三世界科学院（1983年）、Australian Academy of Sciences（1992年）、Ukrainian Academy of Sciences（1992年）、Indian National Science Academy（1993年）、Russian Academy of Sciences（1994年）、Georgian Academy of Sciences（1996年）、委内瑞拉物理、数学和自然科学科学院（1997年）、美国哲学学会（1998年）、Accademia Nazionale dei Lincei, Rome（1999年）、Royal Norwegian Society of Sciences and Letters（2001年）、Czechoslovakia Union of Mathematics（2001年）、Moscow Mathematical Society（2001年）、Spanish Royal Academy of Sciences（2002年）、黎巴嫩科学院（2008年）、Norwegian Academy of Science and Letters（2009年）。他被授予以下机构的荣誉研究员或成员：剑桥大学三一学院（1976年）、剑桥大学彭布罗克学院（1983年）、皇家学会（1991年）、牛津大学圣凯瑟琳学院（1991年）、剑桥大学达尔文学院（1992年）、皇家工程院（1993年）、牛津大学新学院（1999年）、精算师学院（1999年）、医学科学院（2000年）。许多大学授予他荣誉学位，包括：波恩（1968年）、华威（1969年）、杜伦（1979年）、圣安德鲁斯（1981年）、都柏林三一学院（1983年）、芝加哥（1983年）、爱丁堡（1984年）、剑桥（1984年）、埃塞克斯（1985年）、伦敦（1985年）、苏塞克斯（1986年）、根特（1987年）、雷丁（1990年）、赫尔辛基（1990年）、莱斯特（1991年）、罗格斯（1992年）、萨拉曼卡（1992年）、蒙特利尔（1993年）、滑铁卢（1993年）、威尔士（1993年）、皇后-金斯顿（1994年）、基尔（1994年）、伯明翰（1994年）、开放大学（1995年）、曼彻斯特（1996年）、香港中文大学（1996年）、布朗大学（1997年）、牛津（1998年）、威尔士斯旺西大学（1998年）、布拉格查理大学（1998年）、赫瑞瓦特大学（1999年）、墨西哥大学（2001年）、贝鲁特美国大学（2004年）、约克（2005年）、哈佛大学（2006年）、比萨高等师范学校（2007年）、加泰罗尼亚理工大学（2008年）。

让我们以这些悲伤的事实结束这篇传记：阿蒂亚的长子约翰于2002年6月24日在比利牛斯山与妻子徒步度假时去世，而阿蒂亚的兄弟帕特里克的小儿子杰里米于2006年4月12日在意大利徒步时去世。莉莉阿蒂亚于2018年3月13日去世，享年90岁。阿蒂亚本人于2019年1月11日去世。阿蒂亚和莉莉阿蒂亚都安葬在苏格兰泰宁汉的宾宁纪念林地。见https://www.binningwood.co.uk/。

Michael Atiyah's father, Edward Selim Atiyah (1903-1964), was Lebanese and his mother, Jean Levens, was Scottish. Edward, whose father was a medical doctor in Khartoum, had been educated at Brasenose College, Oxford, and became a civil servant in Khartoum. He was also an author and set up a radio broadcasting service during World War II. He was a strong supporter of the Palestinian cause. Michael Atiyah, when interviewed in [32], spoke about his father:-

My father's main dream was to go to Oxford. He wanted to convert himself into an Englishman. It didn't quite work out. When he came back to Sudan, he found he wasn't part of the English class structure, he was regarded as one of the lower classes, although he was Oxford-educated and regarded himself as culturally English. That turned him over a bit. He became an Arab nationalist to some extent. All his life was divided between wanting passionately to be English and yet sympathising with the Arab political position within the British empire.

Michael's mother Jean, although of Scottish descent, was the daughter of a minister of a church in Yorkshire. She lived in Oxford and had studied at the university there. It was in Oxford that Edward and Jean met. They had four children, three sons Michael (the eldest and subject of this biography), Patrick Selim (born 5 March 1931, who went on to become an English lawyer and academic) and Joseph (known as Joe, the youngest of the four children who after a mathematics degree from Cambridge University, became a computer scientist working in computer software and telecommunications), and a daughter Selma (who studied English at an American University and lives in America). Although he was born in London, Michael grew up in Khartoum. However, to avoid the summer heat there the family usually returned to England at that time. Michael's primary school education was at the Diocesan school in Khartoum which he entered in 1934 at the age of five. He completed his primary education in 1941 and the family, as usual, returned to England.

Lebanon had been controlled by the French and, after the fall of France in 1940, it came under the control of the Vichy government. After their trip to England, the Atiyah family returned to Lebanon via France in 1941 and Michael returned to a French school. However, just after this began, the British and Free French began fighting to gain control of the Lebanon. Michael was sent to Victoria College in Cairo. This was a boarding school modelled on the English boarding school system and it was a school that Edward Atiyah had attended. Atiyah writes in the autobiography [3]:-

At Victoria College I got a good basic education but had to adapt to being two years younger than most others in my class. I survived by helping bigger boys with their homework and so was protected by them from the inevitable bullying of a boarding school.

Atiyah talked in [35] about how he came to chose mathematics:-

I was always interested in mathematics from a very young age. ... My parents always thought that I was cut out to be a mathematician from a very young age, all the way through. ... But there was a stage [at Victoria College in Cairo] when I got very interested in chemistry, and I thought that would be a great thing; after about a year of advanced chemistry I decided that it wasn't what I wanted to do and I went back to mathematics. I never seriously considered doing anything else.

He gave a somewhat fuller description of his decision between chemistry and mathematics in the interview. He said that it was inorganic chemistry that put him off the subject [15]:-

It was how to make sulphuric acid and all that sort of stuff. Lists of facts, just facts, you had to memorize a vast amount of material. Organic chemistry was more interesting, there was a bit of structure to it. But inorganic chemistry was just a mountain of facts in books like this. It's true that in mathematics you don't really need an enormous memory. You can work most things out for yourself, remember a few principles. If you're good at that, then it comes easily. If you want to do other things, you've got to work hard to learn a lot of facts. There was one reason, I think. But I enjoyed thinking, I'm good at it, and will continue with it.

After the war ended in 1945, Edward Atiyah returned to live permanently in England. Michael Atiyah attended Manchester Grammar School, one of the best schools for mathematics in the country. Although he was only sixteen years old, he had already taken his A-level examinations having been two years ahead of his age groups in Victoria College, Cairo. His two years at Manchester Grammar School were spent training to take the Cambridge scholarship examinations. However, it was at this school that he came to love geometry [3]:-

I found that I had to work very hard to keep up with the class and the competition was stiff. We had an old-fashioned but inspiring teacher who had graduated from Oxford in 1912 and from him I acquired a love of projective geometry, with its elegant synthetic proofs, which has never left me. I became, and remained, primarily a geometer though that word has been reinterpreted in different ways at different levels. I was also introduced to Hamilton's work on quaternions, whose beauty fascinated me, and still does.

He won a scholarship to Trinity College, Cambridge in 1947. However, rather than go straight to university, which was an option, he decided to do his two-years National Service, which was compulsory at the time. He served as a clerical officer and took the opportunity to read mathematics books and articles. He read Hardy and Wright's Number Theory at this time and also read articles on group theory. He was granted special permission to cut short the final year of his military service and spend it at Cambridge. There he played a lot of tennis and avidly studied mathematics on his own in the library. He matriculated at Trinity College in the autumn of 1949. Many of his fellow students had decided to postpone their National Service, so Atiyah was one of the older of the students in his year. With his exceptional talent, his extra maturity, and the studying he had done before starting his course, it is not at all surprising that he came out ranked first despite having many very talented fellow students. While still an undergraduate, he wrote his first paper A note on the tangents of a twisted cubic (1952).

After graduating with his BA in 1952, Atiyah continued to undertake research at Trinity College, Cambridge obtaining his doctorate in 1955 with his thesis Some Applications of Topological Methods in Algebraic Geometry. His thesis advisor was William V D Hodge. Speaking of the work for his thesis, Atiyah said [35]:-

I'd come up to Cambridge at a time when the emphasis in geometry was on classical projective algebraic geometry of the old-fashioned type, which I thoroughly enjoyed. I would have gone on working in that area except that Hodge represented a more modern point of view - differential geometry in relation to topology; I recognized that. It was a very important decision for me. I could have worked in more traditional things, but I think that it was a wise choice, and by working with him I got much more involved with modern ideas. He gave me good advice and at one stage we collaborated together. There was some recent work in France at the time on sheaf theory. I got interested in it, he got interested in it, and we worked together and wrote a joint paper which was part of my thesis. That was very beneficial for me.

Atiyah published two joint papers with his thesis advisor William Hodge, Formes de seconde espèce sur une variété algébrique Ⓣ (1954) and Integrals of the second kind on an algebraic variety (1955). He also published the single author papers Complex fibre bundles and ruled surfaces (1955). He was made a fellow of Trinity College, Cambridge in 1954. He married Lily Brown on 30 July 1955; they had three sons John, David and Robin. Lily, born in Edinburgh in 1928, was the daughter of a dock worker at the Rosyth naval yard. She had studied mathematics first at the University of Edinburgh and then took the Cambridge Tripos. She went on to obtain a doctorate, working under Mary Cartwright. Lily had met Michael Atiyah at Cambridge but, by the time they married, she was a lecturer at Bedford College, London. Atiyah was awarded a Commonwealth Fellow to study at the Institute for Advanced Study in Princeton during session 1955-56. Lily had to decide whether to keep her job at Bedford College or go to Princeton with her husband. She chose to go to Princeton with her husband and resigned her position at Bedford College. This was an important year for Atiyah who met, among others, Jean-Pierre Serre, Friedrich Hirzebruch, Kunihiko Kodaira, Donald Spencer, Raoul Bott and Isadore Singer. Returning to Cambridge, he was a college lecturer from 1957 and a Fellow of Pembroke College from 1958. He remained at Cambridge until 1961 when he moved to a readership at the University of Oxford where he became a Fellow of St Catherine's College.

Atiyah was soon to fill the highly prestigious Savilian Chair of Geometry at Oxford from 1963, holding this chair until 1969 when he was appointed professor of mathematics at the Institute for Advanced Study in Princeton. After three years in Princeton, Atiyah returned to England, becoming a Royal Society Research Professor at Oxford. He was also elected a Fellow of St Catherine's College, Oxford. Oxford was to remain Atiyah's base until 1990 when he became Master of Trinity College, Cambridge and Director of the newly opened Isaac Newton Institute for Mathematical Sciences in Cambridge.

Atiyah showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory, called K-theory. Grothendieck also contributed substantially to the development of K-theory. In [13] Atiyah's early mathematical work is described as follows:-

Michael Atiyah has contributed to a wide range of topics in mathematics centring around the interaction between geometry and analysis. His first major contribution (in collaboration with F Hirzebruch) was the development of a new and powerful technique in topology (K-theory) which led to the solution of many outstanding difficult problems. Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations. This 'index theorem' had antecedents in algebraic geometry and led to important new links between differential geometry, topology and analysis. Combined with considerations of symmetry it led (jointly with Raoul Bott) to a new and refined 'fixed point theorem' with wide applicability.

For these early achievements Atiyah was awarded a Fields Medal at the International Congress at Moscow in 1966. An address concerning Atiyah's contributions was given at the Congress by Henri Cartan, see [18]. The K-theory and the index theorem are studied in Atiyah's book K-theory (1967, reprinted 1989) and his joint work with G B Segal, The Index of Elliptic Operators I-V, in the Annals of Mathematics, volumes 88 and 93 (1968, 1971). Atiyah also described his work on the index theorem in The index of elliptic operators given as an American Mathematical Society Colloquium Lecture in 1973.

The ideas which led to Atiyah being awarded a Fields Medal were later seen to be relevant to gauge theories of elementary particles. Again we quote [13]:-

The index theorem could be interpreted in terms of quantum theory and has proved a useful tool for theoretical physicists. Beyond these linear problems, gauge theories involved deep and interesting nonlinear differential equations. In particular, the Yang-Mills equations have turned out to be particularly fruitful for mathematicians. Atiyah initiated much of the early work in this field and his student Simon Donaldson went on to make spectacular use of these ideas in 4-dimensional geometry. More recently Atiyah has been influential in stressing the role of topology in quantum field theory and in bringing the work of theoretical physicists, notably E Witten, to the attention of the mathematical community.

The theories of superspace and supergravity and the string theory of fundamental particles, which involves the theory of Riemann surfaces in novel and unexpected ways, were all areas of theoretical physics which developed using the ideas which Atiyah was introducing.

Atiyah has published a number of highly influential books: K-theory (1967); (with I G Macdonald) Introduction to commutative algebra (1969); Vector fields on manifolds (1970); Elliptic operators and compact groups (1974); Geometry on Yang-Mills fields (1979); (with N J Hitchin) The geometry and dynamics of magnetic monopoles (1988); The geometry and physics of knots (1990); (Video) The mysteries of space (1992); Siamo tutti Matematici Ⓣ (2007); and Edinburgh Lectures on Geometry, Analysis and Physics (2010).

We give extracts from some reviews of these books, some extracts from Prefaces and some Publisher's descriptions at THIS LINK.

Atiyah and John Tate described the Clay Mathematics Institute Millennium Prize Problems in a lecture in Paris on 24 May 2000. Atiyah's lecture covered the Poincaré conjecture, the Hodge conjecture, quantum Yang-Mills theory and the Navier-Stokes equation. He explained the problems and placed them in their historical context. He also discussed the implications for various fields of mathematics and physics if solutions to these problems were found. A 60-minute video of the lecture is available entitled The millennium prize problems.

Six volumes of Atiyah's Collected Works have been published. These contain a commentary by Atiyah and in the Preface he comments on the practice of publishing 'collected works' during the lifetime of their author:-

It appears to be increasingly fashionable to publish 'collected works' long before the author's demise. There are several clear advantages to all parties: posterity is saved the trouble of undertaking the collection, while the author can add some personal touches by way of a commentary. There are also disadvantages: the commentary will be biased, and the author may feel that he is being pensioned off.

Another important aspect of Atiyah's contribution is the remarkable collection of doctoral students he supervised.

We have listed his students with the title and date of their thesis and, for those who we know have gone on to an academic career, a university at which they have taught, at THIS LINK.

We have one further link to give the reader. Brief extracts from eight papers written by Michael Atiyah for a general audience are at THIS LINK.

Atiyah has received many honours during his career, in addition to the Fields Medal referred to above, and although we cannot list them all we will give a fairly full account. He was elected a Fellow of the Royal Society of London in 1962 at the age of 32. He received the Royal Medal of the Society in 1968 and its Copley Medal in 1988. He gave the Royal Society's Bakerian Lecture on Global geometry in 1975 and was President of the Royal Society from 1990 to 1995. He was President of the Royal Society of Edinburgh from 2005 to 2008.

Among the prizes that he has received are the Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Gunning Victoria Jubilee Prize from the Royal Society of Edinburgh in 1990, the Benjamin Franklin Medal in 1993, the Jawaharlal Nehru Memorial Medal in 1993, the Order of Andres Bello (1st Class) from the Republic of Venezuela in 1997, the Royal Medal from the Royal Society of Edinburgh in 2003, the Order of Merit (Gold) from the Lebanon in 2005, and the President's Medal from the Institute of Physics in 2008. In 2004 Atiyah and Isadore Singer were awarded the Neils Abel prize of £480 000 by the Norwegian Academy of Science and Letters:-

... for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.

They were presented with the prize by King Harald V of Norway at a ceremony in Oslo.

Atiyah was the American Mathematical Society Colloquium Lecturer in 1973. He was President of the London Mathematical Society in 1974-76 receiving its De Morgan Medal in 1980. Atiyah was knighted in 1983 and made a member of the Order of Merit in 1992.

He has been elected a foreign member of many national academies including: the American Academy of Arts and Sciences (1969), Royal Swedish Academy of Sciences (1972), German Academy of Scientist Leopoldina (1977), Académie des Sciences, Paris (1978), United States National Academy of Sciences (1978), Royal Irish Academy (1979), Third World Academy of Science (1983), Australian Academy of Sciences (1992), Ukrainian Academy of Sciences (1992), Indian National Science Academy (1993), Russian Academy of Sciences (1994), Georgian Academy of Sciences (1996), Academy of Physical, Mathematical and Natural Sciences of Venezuela (1997), American Philosophical Society (1998), Accademia Nazionale dei Lincei, Rome (1999), Royal Norwegian Society of Sciences and Letters (2001), Czechoslovakia Union of Mathematics (2001), Moscow Mathematical Society (2001), Spanish Royal Academy of Sciences (2002), Lebanese Academy of Sciences (2008), Norwegian Academy of Science and Letters (2009). He has been made an Honorary Fellow or Member of: Trinity College, University of Cambridge (1976), Pembroke College, University of Cambridge (1983), Royal Institution (1991), St Catherine's College, University of Oxford, (1991), Darwin College, University of Cambridge (1992), Royal Academy of Engineering (1993), New College, University of Oxford (1999), Faculty of Actuaries (1999), Academy of Medical Sciences (2000). Many universities have awarded him an honorary degree including: Bonn (1968), Warwick (1969), Durham (1979), St Andrews (1981), Trinity College Dublin (1983), Chicago (1983), Edinburgh (1984), Cambridge (1984), Essex (1985), London (1985), Sussex (1986), Ghent (1987), Reading (1990), Helsinki (1990), Leicester (1991), Rutgers (1992), Salamanca (1992), Montreal (1993), Waterloo (1993), Wales (1993), Queen's-Kingston (1994), Keele (1994), Birmingham (1994), Open University (1995), Manchester (1996), Chinese University of Hong Kong (1996), Brown University (1997), Oxford (1998), University of Wales Swansea (1998), Charles University Prague (1998), Heriot-Watt University (1999), University of Mexico (2001), American University of Beirut (2004), York (2005), Harvard University (2006), Scuola Normale Pisa (2007), Universitat Politècnica de Catalunya (2008).

Let us end this biography by recording the sad facts that Atiyah's eldest son John died on 24 June 2002 while on a walking holiday in the Pyrenees with his wife, while Jeremy, the youngest son of Atiyah's brother Patrick, died on 12 April 2006 while walking in Italy. Lily Atiyah died on 13 March 2018 at the age of 90. Atiyah himself died on 11th January 2019. Both Michael and Lily Atiyah are buried in Binning Memorial Wood, Tyninghame, Scotland. See https://www.binningwood.co.uk/.