西蒙·唐纳森Simon Donaldson

西蒙·唐纳森的父亲是一名工程师，最初的职业生涯在海军。大约在他结婚时，他换了工作，到剑桥大学生理学实验室工作。在那里，他为旨在理解神经系统的实验制造仪器。唐纳森的母亲在剑桥长大，毕业于剑桥大学自然科学专业，但结婚后从未有过职业生涯。唐纳森是他父母的四个孩子之一，有一个哥哥、一个姐姐和一个弟弟。他的两个兄弟都成为了工程师。

唐纳森成长的家庭是一个人人都有项目的家庭。他的父亲制作模型飞机，修理家里的东西，通常忙于某个项目。唐纳森说[8]：-

我早年记得他津津有味地说：“……然后我就能回去做研究了”（大概是在完成他向我描述的一些家务之后）。我不知道“研究”可能是什么，但从那时起，这个词就带上了魅力和浪漫的色彩。

这是一个每个人都忙碌而活跃的家庭。他的外祖父对孩子们的教育也有很大影响。他是一位退休校长，对四个孩子的教育都非常关心，但也许唐纳森比其他三个孩子得到他的关注稍多一些。

唐纳森在剑桥的一所预备学校就读，他最喜欢的科目是历史。他原本以为自己的未来教育会进入一所独立的剑桥中学，但事与愿违，因为在他十二岁时，全家搬到了伦敦以南约30公里的七橡树附近的新家。这次搬家是因为他的父亲离开了剑桥大学的工作，与一些他在那里共事过的人一起去了伦敦一个由医学研究理事会资助的新研究单位。他是一个开发神经植入物团队的成员。这次搬家意味着唐纳森要换学校，因此他的中学教育是在肯特郡七橡树的七橡树学校完成的，他从1970年到1975年在那里就读。

七橡树学校是独立的，但由于七橡树没有公立文法学校，大约一半的学生由国家资助，一半是自费。这意味着它的社会构成比普通独立学校更加多元。起初唐纳森在学校不太开心，花了几年时间才适应。有帮助的一件事是他对设计游艇的热情。他不像家里其他人那样热爱实际项目，所以他对游艇设计的兴趣是理论性的。这种兴趣帮助他融入了学校，因为学校有帆船传统。他读书以帮助自己理解船舶设计背后的理论，这些书包含的数学比他学校里学到的更多。他的父亲教了他一些数学技巧，他在学校教授微积分之前就开始学习微积分了。

到十四岁时，他已经放弃了设计游艇，开始用外祖父为他买的书自学数学。他想成为一名数学家，但也有成为会计师的备用计划。在七橡树学校期间，他获得了奖学金，于1976年进入剑桥大学彭布罗克学院学习数学和物理。在那里，他学习了分析、拓扑学和数学物理，但他最喜欢的主题是几何学，尽管剑桥的教学大纲中这个主题不多。他于1979年获得学士学位，并继续在剑桥学习荣誉学位考试的第三部分直到1980年。他在剑桥的一位导师形容他是一个非常好的学生，但肯定不是他那一年的顶尖学生。然而，弗兰克·亚当斯对唐纳森在期末考试中的表现印象深刻，并写信祝贺他的解答。弗兰克·亚当斯是他申请在牛津攻读博士学位的推荐人之一。

1980年，唐纳森在牛津大学伍斯特学院开始研究生工作，最初由Nigel Hitchin指导。然而，他发现围绕迈克尔·阿蒂亚和罗杰·彭罗斯发展起来的研究活动和文化最具吸引力。Hitchin建议他研究他和迈克尔·阿蒂亚几年前提出的一个猜想。唐纳森利用偏微分方程和拓扑学的方法攻克了这个问题。一年后，Hitchin建议他转投迈克尔·阿蒂亚门下。他很快取得了非凡的突破。迈克尔·阿蒂亚在[3]中写道：-

1982年，当他还是二年级研究生时，唐纳森证明了一个震惊数学界的结果。

这个结果由唐纳森发表在论文Self-dual connections and the topology of smooth 4-manifolds中，该论文于1983年发表在美国数学会的Bulletin上。迈克尔·阿蒂亚继续描述唐纳森的工作[3]：-

与迈克尔·弗里德曼的重要工作一起，唐纳森的结果意味着存在“异种”4-空间，即拓扑上等价但微分上不等价于标准欧几里得4-空间$\mathbb{R}^{4}$的4维可微流形。这个结果之所以如此令人惊讶，是因为n = 4是唯一存在这种异种n-空间的值。这些异种4-空间具有显著的性质：与$\mathbb{R}^{4}$不同，它们包含不能包含在任何可微嵌入的3-球内的紧集！

1983年在牛津获得博士学位后，唐纳森被任命为牛津万灵学院的初级研究员。他在普林斯顿高等爱德华·斯图迪研究所度过了1983-84学年。在这一年里，他前往马里兰大学，在那里遇到了诺拉。她于1975年获得哥伦比亚山谷大学的文学学士学位，随后于1981年获得弗吉尼亚理工学院的理学硕士学位。当唐纳森在马里兰大学遇到她时，她正在攻读数理统计博士学位。他们结婚并育有三个孩子安德烈斯、简和尼古拉斯。他还有一个继女阿德里亚娜。诺拉于1988年获得马里兰大学博士学位，学位论文为A non-parametric estimation of the tumour onset time in a serial screening experiment。

回到牛津后，他继续在万灵学院担任博士后研究员，直到1985年。1985年，他成为万灵学院Quondam Fellow，并于同年被任命为牛津大学圣安妮学院约翰·沃利斯数学讲席教授。他在斯坦福度过了1997-8学年，然后于1999年转到伦敦帝国理工学院。他的妻子Nora于1999年被任命为国王学院医院研发部生物统计学部门负责人。他继续在帝国理工学院担任皇家学会研究教授，但自2014年以来，他还是美国石溪大学Simons几何与物理中心的永久成员。Nora Donaldson在国王学院医院工作到2005年，之后她加入牙科研究所担任生物统计学Reader。

唐纳森因其工作获得了许多荣誉。他于1985年获得伦敦数学会颁发的初级Whitehead奖。次年，他当选为皇家学会会士，同样在1986年，他在乔治·伯克利举行的国际数学家大会上获得了约翰·查尔斯·菲尔兹奖章。奖章由拉尔斯·阿尔福斯颁发给他。三年前，他在华沙举行的大会上作了报告Gauge theory and topology，在1986年伯克利大会上，他作了全会报告The geometry of 4-manifolds。我们还注意到，1998年他是国际数学联盟副主席，并在1998年柏林国际数学家大会上作了讲座Lefschetz Fibrations in Symplectic Geometry。他还在1990年国际数学物理大会和1992年欧洲数学家大会上担任全会报告人。

1991年，唐纳森获得剑桥哲学学会颁发的威廉·霍普金斯奖，次年获得皇家学会颁发的皇家奖章。他于1999年获得伦敦数学会的乔治·波利亚奖，几年后获得2006年费萨尔国王奖。他还于1994年获得瑞典皇家科学院颁发的克拉福德奖[5]：-

……因其通过应用瞬子对四维几何的根本性研究，特别是他发现的新微分不变量……

迈克尔·阿蒂亚描述了导致唐纳森在[3]中获得约翰·查尔斯·菲尔兹奖章的贡献。他总结了唐纳森的贡献：-

唐纳森在4维流形上做出了他最初的一些结果，这些思想对几何学家和拓扑学家来说如此新颖和陌生，以至于他们只能困惑地赞叹凝视。慢慢地，这一信息被传达开来，现在唐纳森的思想开始被其他人以各种方式使用。……唐纳森开辟了一个全新的领域；关于4维几何的意外而神秘的现象已被发现。此外，这些方法是新的且极其微妙，使用了困难的非线性偏微分方程。另一方面，这一理论牢固地处于数学的主流之中，与过去有着密切的联系，融合了理论物理学的思想，并与代数几何美妙地结合在一起。

文章[7]非常有趣，既提供了唐纳森关于他在牛津大学读研究生时如何做出重大发现的回忆集，也提供了他近年来所研究领域的综述。唐纳森在[7]中写道，他几乎所有的工作都归在以下标题下：-

(1) 全纯向量丛的微分几何。
(2) 规范理论在4维流形拓扑学中的应用。

并且他将自己的贡献与该领域许多其他人的贡献联系起来。

唐纳森的工作由R 莫里茨·亚伯拉罕·斯特恩在[36]中总结如下：-

1982年，唐纳森开始了一段丰富的几何之旅，这段旅程正引领我们走向本世纪激动人心的结论。他创造了一个全新且激动人心的研究领域，数学的许多部分都经过这个领域，并且它继续产生关于光滑4维流形拓扑学和几何学的神秘而意外的现象。

唐纳森于2000年当选国家科学院会士。2006年2月，他因[43]被授予费萨尔国王国际科学奖：-

……对理论的奠基性贡献，这些理论加强了数学与物理学之间的联系，并帮助为物理理论提供了严格的基础，从而对亚核层次物质定律给出了非常好的描述。

2008年4月，他被西北大学授予弗雷德里克·埃瑟·内默斯数学奖。该奖因他的[44]而授予他：-

……在四维拓扑学、辛几何和规范理论方面的开创性工作，以及他卓越地运用物理学思想推进纯粹数学。

西北大学数学讲席教授弗瑞兹·约翰弗兰克斯提供了更多关于唐纳森导致获奖的贡献的细节[44]：-

唐纳森的突破性工作发展了四维流形几何及其光滑结构研究的新技术。他的方法被描述为极其精细，使用了困难的非线性偏微分方程。利用瞬子，即杨-米尔斯规范理论方程的解，他对闭四维流形的结构获得了重要洞见。规范理论技术还使他能够证明存在没有光滑结构的四维流形，以及另一些具有无穷多个光滑结构的四维流形。他的工作为其他人在四维流形研究中的工作提供了奠基性步骤。

2009年，唐纳森与Clifford H Taubes共同获得邵逸夫数学科学奖。颁奖委员会写道，唐纳森和Taubes [45]：-

……是两位几何学家，他们通过开创性的技术和源自理论物理（包括量子理论）的思想，改变了整个学科，并彻底改变了我们对空间和时间的几何理解。

关于该奖项的更多信息，包括唐纳森获奖引文的更多摘录，请参见THIS LINK。

唐纳森于2012年被封为爵士，同年他当选为美国数学会会士。他的下一个重要奖项是2015年获得的数学突破奖[26]：-

……因四维流形的新革命性不变量，以及因研究代数几何与整体微分几何中稳定性的关系，包括对丛和基诺·法诺簇的研究。

突破奖于2012年设立，旨在表彰基础物理学（2012年首次颁奖）、生命科学（2013年首次颁奖）和数学（2015年首次颁奖）领域重要的、主要是近期的成就：-

“万物皆数，”毕达哥拉斯教导说。尽管现代数学所涵盖的远不止数字本身，这一原则依然成立。数学是自然的通用语言。数学也是知识增长的基础，因为它是支撑所有科学的脚手架。它与物理学的关系尤为密切。从虚数到大卫·希尔伯特空间，曾经看似纯粹抽象的东西，结果却构成了真实物理过程的基础。此外，当今生命科学的所有领域都利用统计和计算方法的力量进行研究。数学奖项奖励该学科众多分支中的重大发现。它们由Yuri Milner创立，并由Yuri和茹利亚 Milner所设立的基金会提供资助。

2015年突破奖颁奖典礼于2014年11月9日在加利福尼亚州山景城的NASA一号机库举行；这里是约翰内斯·开普勒任务的总部，该任务旨在寻找能够支持生命的行星。为启动数学奖项，公布了五位获奖者，每人获得300万美元。该奖项的全部五位获奖者都同意担任遴选委员会成员，负责从数学界提名的竞争者中选出该奖项的后续获奖者。2015年的颁奖典礼有许多演员和其他名人出席，并在多个电视频道播出。

关于完整的引文和唐纳森获奖后的回应，见THIS LINK。

2019年，唐纳森是获得奥斯瓦尔德·维布伦奖的三位合作者之一。2019年奥斯瓦尔德·维布伦几何学奖[1]：-

……授予Xiuxiong 陈国才、唐纳森——两人均来自石溪大学——以及加利福尼亚大学伯克利分校的Song Sun，以表彰他们分三部分组成的系列论文“法诺流形上的Kähler-阿尔伯特·爱因斯坦度量，I、II和III”，该系列于2015年发表在Journal of the American Mathematical Society上，他们在其中证明了微分几何中一个长期存在的猜想。

唐纳森与Yakov Eliashberg共同获得了2020年沃尔夫数学奖[28]：-

……以表彰他们对微分几何和拓扑学的贡献。

唐纳森的引文结束于[28]：-

唐纳森因其在过去35年中在几何学领域的领导地位而被授予沃尔夫奖。他的工作独特地结合了整体非线性分析、拓扑学、代数几何和理论物理中的新思想，这源于他在4-流形和规范理论方面的基础性工作。尤其引人注目的是他最近在辛几何和埃里希·凯勒几何方面的工作。

完整引文见THIS LINK。

让我们以唐纳森在其国家科学院网页[30]上对自己贡献的描述来结束这篇传记：-

我的研究兴趣在于数学中与几何、拓扑和分析相邻，并与数学物理有实质性联系的领域。我早期的大部分工作依赖于将杨-米尔斯方程的瞬子解——最初在粒子物理中引入——作为工具，来解决关于四维流形拓扑的纯数学问题。这导致了新颖且广泛的结果，这些结果无法通过其他方法获得，它们让我们得以一窥四维拓扑和几何的特殊性质。最近，我在这个方向上的工作集中在辛流形这一特殊类别上。我已经表明，复代数几何中的某些经典技术可以适应这一设定，目前我正在研究这对辛流形分类的意义。贯穿我研究的另一个主题是研究复微分几何中出现的某些偏微分方程。在20世纪80年代，我研究了与全纯向量丛相关的方程，最近一直在研究类似思想在埃里希·凯勒度量上的应用。

Simon Donaldson's father was an engineer whose first career was in the navy. At about the time he married, he changed jobs to work in the Physiology Laboratory in the University of Cambridge. There he constructed apparatus for experiments which were aimed at understanding the nervous system. Simon's mother was brought up in Cambridge and graduated in Natural Sciences from the University of Cambridge but after marrying never had a career. Simon was one of his parents' four children, having an older brother, an older sister and a younger brother. Both his brothers became engineers.

The home in which Simon grew up was one in which people had projects. His father built model airplanes, saw about repairing things in the home and generally was busy working on some project. Simon said [8]:-

I have an early memory of him saying with relish: "... and then I shall be able to get back to research" (presumably, after completing some chores which he had described to me). I had no idea what "research" might be, but from that time the word was tinged with glamour and romance.

It was a home in which everyone was busy and active. His maternal grandfather also had a big influence on the children's education. He was a retired schoolmaster and took a great interest in the education of all four children, but perhaps Simon got a little more of his attention than the other three children.

Simon attended a Preparatory School in Cambridge where his favourite subject was history. He saw his future education as progressing to an independent Cambridge Secondary School but this was not to be since when he was twelve year old the family moved to a new home near Sevenoaks, around 30 km south of London. The move came about because his father left his job at the University of Cambridge, going with some others whom he had worked with there to a new research unit in London funded by the Medical Research Council. He was part of a team developing neurological implants. The move meant a change of school for Simon and so his secondary school education was at Sevenoaks School in Sevenoaks, Kent which he attended from 1970 to 1975.

The Sevenoaks School was independent but, because there was no state grammar school in Sevenoaks, about half the pupils were funded by the state and half were private. This meant it had a much greater social mix than a normal independent school. At first Simon was not very happy at the school and it took him a few years to settle in. One thing that helped was his passion he had for designing yachts. He did not have a love of practical projects like the rest of his family, so his interest in yacht design was theoretical. This interest helped him fit into the school, since it had a sailing tradition. He read books to help him understand the theory behind the design of ships and these contained more mathematics than he had learnt at school. His father taught him some mathematical techniques and he began to study calculus well before it was taught at school.

By the age of fourteen he had given up designing yachts and was studying mathematics on his own from books that his maternal grandfather bought for him. He wanted to be a mathematician but had a backup plan to become an accountant. While at Sevenoaks School he won a scholarship to study mathematics and physics at Pembroke College, Cambridge which he entered in 1976. There he studied analysis, topology and mathematical physics but the topic he liked most was geometry, although there was not much of that topic in the Cambridge syllabus. He was awarded his B.A. in 1979 and continued to study Part III of the tripos at Cambridge until 1980. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Frank Adams, however, was very impressed with the work Donaldson did in his final examinations and wrote to him to congratulate him on his solutions. Frank Adams was one of his referees for his application to study for a doctorate at Oxford.

In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchin's supervision. It was, however, the activity and culture that developed around Michael Atiyah and Roger Penrose that he found most attractive. Hitchin suggested that he look at a conjecture that he and Atiyah had proposed a couple of years earlier. Donaldson attacked the problem using both methods from partial differential equations and from topology. After a year Hitchin suggested that he changed to become Atiyah's student. He soon made a remarkable breakthrough. Atiyah writes in [3]:-

In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.

This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [3]:-

Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space $\mathbb{R}^{4}$. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike $\mathbb{R}^{4}$) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere!

After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced Study at Princeton. During this year he made a trip to the University of Maryland where he met Nora. She had been awarded a Bachelor of Arts degree from the Universidad del Valle in Colombia in 1975, followed by a Master in Science degree from Virginia Tech in 1981. When Donaldson met her at the University of Maryland she was studying for a Ph.D. in Mathematical Statistics. They married and have three children Andres, Jane and Nicholas. He also has a step-daughter Adriana. Nora was awarded a Ph.D. by the University of Maryland in 1988 for the thesis A non-parametric estimation of the tumour onset time in a serial screening experiment.

After returning to Oxford he continued to hold his Post-Doctoral Research Fellowship at All Souls College until 1985. He became a Quondam Fellow of All Souls College in 1985 and, in that year, was appointed Wallis Professor of Mathematics at St Anne's College Oxford. He spent the year 1997-8 in Stanford before moving to Imperial College, London in 1999. His wife Nora was appointed head of the Biostatistics Unit in the Research and Development Department at King's College Hospital in 1999. He continues to work at Imperial College as Royal Society Research Professor, but since 2014 he is also a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in the United States. Nora Donaldson worked at King's College Hospital until 2005 when she joined the Dental Institute as a Reader in Biostatistics.

Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress of Mathematicians at Berkeley. He was presented with the medal by Lars Ahlfors. Three years earlier he had delivered the talk Gauge theory and topology at the Congress in Warsaw, and at the Berkeley Congress in 1986 he gave the plenary address The geometry of 4-manifolds. We note also that in 1998 he was the Vice-President of the International Mathematical Union and delivered the lecture Lefschetz Fibrations in Symplectic Geometry to the 1998 International Congress of Mathematicians in Berlin. He was also a plenary lecturer at the International Congress of Mathematical Physics in 1990 and at the European Congress of Mathematicians in 1992.

In 1991 Donaldson received the William Hopkins Prize from the Cambridge Philosophical Society and, in the following year, the Royal Medal from the Royal Society. He was awarded the London Mathematical Society's Polya prize in 1999 and some years later, the 2006 King Faisal Prize. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994 [5]:-

... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...

Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [3]. He sums up Donaldson's contribution:-

When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration. Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.

The article [7] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [7] that nearly all his work has come under the headings:-

(1) Differential geometry of holomorphic vector bundles.
(2) Applications of gauge theory to 4-manifold topology.

and he relates his contribution to that of many others in the field.

Donaldson's work in summed up by R Stern in [36]:-

In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds.

Donaldson was elected to the National Academy of Sciences in 2000. In February 2006 he was awarded the King Faisal International Prize for science for [43]:-

... seminal contributions to theories which have strengthened the links between mathematics and physics, and helped provide a rigorous foundation for physical theories giving a very good description of the laws of matter at the sub-nuclear level.

In April 2008, he was awarded the Frederic Esser Nemmers Prize in Mathematics from Northwestern University. The Prize was given for his [44]:-

... ground-breaking work in four-dimensional topology, symplectic geometry and gauge theory, and for his remarkable use of ideas from physics to advance pure mathematics.

John Franks, the Chair of Mathematics at Northwestern University, gave more details of Donaldson's contributions which led to the award [44]:-

Donaldson's breakthrough work developed new techniques in the geometry of four-manifolds and the study of their smooth structures. His methods have been described as extremely subtle, using difficult nonlinear partial differential equations. Using instantons, solutions to the equations of Yang-Mills gauge theory, he gained important insight into the structure of closed four-manifolds. Gauge theory techniques also enabled him to show the existence of four-manifolds with no smooth structure and others with infinitely many. His work has provided the seminal steps for the work of others in study of four-manifolds.

In 2009 Donaldson, together with Clifford H Taubes, was awarded the Shaw Prize in Mathematical Sciences. The Committee who made the award wrote that Donaldson and Taubes [45]:-

... are the two geometers who have transformed the whole subject by pioneering techniques and ideas originating in theoretical physics, including quantum theory [and] have totally changed our geometrical understanding of space and time.

For more information about this award, including further extracts from the citation for Donaldson, see THIS LINK.

Donaldson was knighted in 2012, the same year that he was elected as a fellow of the American Mathematical Society. His next major award was the Breakthrough Prize in Mathematics which he received in 2015 [26]:-

... for the new revolutionary invariants of four-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties.

The Breakthrough Prize was launched in 2012 to honour important, primarily recent, achievements in Fundamental Physics (first awards 2012), Life Sciences (first awards 2013) and Mathematics (first awards 2015):-

"All is number," taught Pythagoras. Though modern mathematics encompasses far more than numbers alone, the principle remains true. Mathematics is the universal language of nature. Mathematics is also fundamental to the growth of knowledge, as it is the scaffolding that supports all the sciences. Its relationship to physics is particularly intimate. From imaginary numbers to Hilbert spaces, what once seemed pure abstractions have turned out to underlie real physical processes. In addition, all fields in the life sciences today utilise the power of statistical and computational approaches to research. The mathematics prizes reward significant discoveries across the many branches of the subject. They were founded by Yuri Milner and are funded by grants from the foundations established by Yuri and Julia Milner.

The 2015 Breakthrough Prize ceremony was held on 9 November 2014 at NASA's Hangar 1 in Mountain View, California; this is the home of the Kepler Mission to find planets capable of supporting life. To launch the mathematics award, five winners were named each receiving $3 million. All five recipients of the Prize agreed to serve on the Selection Committee, responsible for choosing subsequent winners of the prize from the pool of contenders nominated by the mathematics community. The 2015 ceremony, with many actors and other celebrities present, was broadcast on several television channels.

For the full citation and Donaldson's response on receiving the award, see THIS LINK.

In 2019 Donaldson was one of three collaborators who won the Oswald Veblen Prize. The 2019 Oswald Veblen Prize in Geometry was [1]:-

... awarded to Xiuxiong Chen, Simon Donaldson - both of Stony Brook University - and Song Sun, University of California, Berkeley, for their three-part series, "Kähler-Einstein metrics on Fano manifolds, I, II and III," published in 2015 in the Journal of the American Mathematical Society, in which they proved a long-standing conjecture in differential geometry.

Donaldson was a joint winner of the Wolf Prize in Mathematics in 2020 along with Yakov Eliashberg [28]:-

... for their contributions to differential geometry and topology.

The citation for Donaldson ends [28]:-

Professor Simon Donaldson is awarded the Wolf Prize for his leadership in geometry in the last 35 years. His work has been a unique combination of novel ideas in global non-linear analysis, topology, algebraic geometry, and theoretical physics, following his fundamental work on 4-manifolds and gauge theory. Especially remarkable is his recent work on symplectic and Kähler geometry.

For the full citation, see THIS LINK.

Let us end this biography by quoting Donaldson's own description of his contributions from his National Academy of Sciences web page [30]:-

My research interests lie in the area of mathematics bordering geometry, topology, and analysis and having substantial connections with mathematical physics. Much of my early work hinged on the application of the instanton solutions of the Yang-Mills equations - first introduced in particle physics - as tools to solve purely mathematical problems about the topology of four-dimensional manifolds. This has led to novel and wide-ranging results, not obtainable by other methods, that give a glimpse of the special nature of four-dimensional topology and geometry. More recently my work in this direction has focused on the special class of symplectic manifolds. I have shown that certain classical techniques from complex algebraic geometry can be adapted to this setting and am currently pursuing the implications of this for the classification of symplectic manifolds. Another theme running through my research is the study of certain partial differential equations arising in complex differential geometry. In the 1980s I worked on equations related to holomorphic vector bundles and have recently been studying the application of similar ideas to Kähler metrics.