丘成桐Shing-Tung Yau

丘成桐是父母陈国才 Ying Chiou 和 Yeuk-Lam Leung Chiou 八个孩子中的第五个。他有一个弟弟 Stephen Shing-Toung 丘成桐（生于1952年），也是一位杰出的数学家。丘成桐出生时，他的父亲是一位在中国南方工作的经济学家和哲学家。然而，到1949年底，共产党已控制了中国几乎全境，丘成桐一家逃往香港，他的父亲在那里获得了一所学院的教职。（该学院后来成为香港中文大学的一部分。）在访谈[76]中，丘成桐给出了更多细节：-

我父亲去日本学习经济学，但在1937年日本入侵之前，他回来帮助中国人自卫。战争结束时，他为联合国向穷人分发食物和衣服。1949年革命后，他担心与共产党发生麻烦，所以把全家带到了香港。我们非常贫穷——起初我们几乎饿死——但父亲有一大群学生经常在家里讨论哲学和文学。我10岁、11岁、12岁，逐渐习惯了抽象推理。父亲让我们背诵长篇散文和诗歌。当时我不明白它们的意思，但我记住了它们，后来派上了用场。

家庭生活极其艰难，所以丘成桐的母亲编织物品出售以补贴微薄的收入。丘成桐住在香港城外的一个村庄里，房子没有电也没有自来水，生活很艰苦，在这个阶段他经常逃学，更喜欢当街头帮派的头目。丘成桐的父亲对他影响很大，鼓励他对哲学和数学的兴趣。2003年，丘成桐评论道[43]：-

事实上，我觉得在我学了几何之后，我能更好地理解父亲的谈话。

丘成桐的父亲受到压力去为中国做间谍，但他拒绝了，并辞去了香港中文大学的教授职位。不幸的是，丘成桐的姐姐于1962年去世，一个哥哥得了脑病，当丘成桐十四岁时，他的父亲去世了。在2011年父亲百年诞辰时，丘成桐写了一篇致敬父亲的文章[80]：-

一个鼓舞人心的起伏人生，瞬间被征服。
尽管他的东西方智慧仍在我心中回响。
我从未充分享受他的爱，这让我沮丧。
青春的花朵已从我身边逝去，我的头发变灰。
然而我常常回首那个决定性的时刻，当我还是个粗心的少年。
当他那晚离开时，那是多么悲伤，那么久远而遥远。
我想知道，如果他能说，他会告诉我们什么？
尽管我永远不会听到那些话，但他的思想永远与我同在。

到十四岁时，丘成桐正在名为培正的学校愉快地接受中学教育，但他也不得不帮助养家。他通过做家教来帮助家里维持经济。尽管培正学校享有盛誉，但他后来觉得那里的物理教得不太好，他也没有获得足够的学科直觉。1965年离开学校后，他在香港的金芳蓉崇基学院继续学业。该学院成立于1951年，但在1963年并入香港中文大学，成为其三所创始学院之一。丘成桐于1968年未毕业便离开了该学院。幸运的是，他在该学院的一位讲师曾在加利福尼亚大学乔治·伯克利分校学习，看到丘成桐的巨大潜力后，建议他去那里攻读博士学位。

这位讲师是 Stephen Salaff（1938-2012），他出生于美国康涅狄格州斯坦福，曾就读于哥伦比亚大学，之后前往英国，在伦敦大学攻读数学学士学位。他在加利福尼亚大学伯克利分校获得数学博士学位，并在那里结识了学生 Janet Weitzner。他们结婚后，由于 Janet 的研究需要前往中国，夫妇俩移居香港，Stephen 在香港中文大学讲授数学。丘成桐一直对 Salaff 的支持极为感激。他在2012年说[60]：-

没有 Steve 的帮助，我不会成为现在的数学家。

丘成桐在采访[38]中解释了 Salaff 如何支持他：-

实际上 Stephen 试图让我提前获得本科学位。香港中文大学当时发生了很大的争执。当然，他们现在后悔了，因为我获得了约翰·查尔斯·菲尔兹奖章。如果我毕业于香港中文大学，那对他们来说会是一个加分项。

Salaff 联系了伯克利的副教授 Donald Sarason，并且很可能在陈省身的帮助下，他们设法让伯克利豁免了入学要求，并授予他两个国际商业机器公司奖学金之一，资助他在加利福尼亚大学伯克利分校进行研究。丘成桐说[38]：-

我申请时并不认识陈省身，但在我被录取后，陈省身被香港中文大学授予荣誉学位。我想他是在六月从美国来香港领取学位的，我在六月中旬见到了他。那是在我被录取之后。但我确信他一定在某种程度上帮了忙。

丘成桐向朋友借钱，以便能负担去美国的机票。他于1969年9月飞往旧金山，与另外三个寻找住处的人会合后，他们租到了一个地方。尽管他每月从奖学金中获得300美元，但他把一半寄回香港的家人，因此生活很艰难。他开始上课[29]：-

我报名了许多课程，因为我觉得自己对现代数学的许多不同领域都不了解，并且我开始对几何感兴趣。我从伯克利的不同院系学到了很多东西，在第一学期我学习了流形。我学习了黎曼几何，但不够，只是初等水平。

他参加了Charles Bradfield Morrey Jr.（1907-1984）讲授的非线性偏微分方程课程，他觉得这门课非常出色，由于无力回家，他利用圣诞假期在图书馆工作，阅读研究论文和专著。最重要的是，他喜欢阅读Journal of Differential Geometry。特别是，他读了一篇约翰·米尔诺的论文，描述了流形的基本群如何受曲率影响。他发现约翰·米尔诺的论文写得非常好，易于理解。它引用了早期的论文，丘成桐找到并阅读了这些论文。尽管作为研究生只过了几个月，他发现自己能够推广这些论文中的一些结果。

陈省身在丘成桐作为研究生的第一年并不在伯克利，但当他在丘成桐第二年开始时返回时，丘成桐问陈省身是否愿意做他的学位论文导师。他同意了，丘成桐继续在伯克利学习，由陈省身担任导师。在[21]中，Arthur Fischer描述了当时伯克利对数学研究生的情况：-

当时的数学系，在1969年和1970年，挤在Campbell Hall，即天文楼里，直到1971年才占据现在位于Evans Hall顶四层的所在地。在这个小体积、低熵状态下，教师、讲师、研究生和访问教师之间有相当多的互动。学术报告会常常挤满了活跃的人群，大约一百人在一个类似剧院的场所，这很多人，至少与普林斯顿学术报告会的小观众和宁静氛围相比。正是在这种低熵状态下，我在1969年秋天第一次遇到了丘成桐，当时他刚开始作为第一年研究生。... 我在1969年秋天在Campbell Hall的复印室... 丘成桐走了进来。我们模糊地知道彼此的存在，但仅此而已。丘成桐在那里复印他关于曲率与拓扑学之间联系的笔记。我很感兴趣。他告诉了我更多。我谈到了我正在做的工作，广义相对论哈密顿结构的全局表述。这很成功。这些可能相关。我们都兴奋起来。

一旦成为陈省身的学生，丘成桐大部分时间都在研究复流形。特别是，他阅读了Eugenio Calabi的论文，并对Calabi猜想着迷。仅仅两年后，他于1971年获得博士学位，学位论文为On the Fundamental Group of Compact Manifolds of Non-positive Curvature。到这时，他已经有三篇论文发表，On the fundamental group of manifolds of non-positive curvature （1970）、On the fundamental group of compact manifolds of non-positive curvature（1971）和Compact flat Riemannian manifolds（1971）。

在1971-72学年，丘成桐是普林斯顿高等爱德华·斯图迪研究所的成员。1972年，他被任命为纽约州立大学石溪分校的助理教授。1974年，他被任命为斯坦福大学的副教授。约翰·亨利·科茨此时遇到了丘成桐。他写道[12]：-

我第一次见到丘成桐是在1974年他来到斯坦福大学时，那是我到达斯坦福一年之后。当时，丘成桐的两个品质迅速打动了我。首先，他有非常广泛的数学兴趣（例如，令我惊讶的是，他耐心地听了我讲授的一门相当技术性的岩泽健吉理论研究生课程）。其次，他非常勤奋。事实上，我偶尔会在接近午夜时走进斯坦福数学系，每次我都记得发现丘成桐在那里，要么在做自己的工作，要么与他迅速聚集在身边的一群热切的研究生交谈。

1976年，丘成桐与物理学家郭玉云结婚；他们有两个儿子，Isaac Yau和Michael Yau。他在[80]中写道：-

我也很幸运地爱上并最终娶了一位与我观点一致的女性，她认为生活不仅仅是追求个人财富、物质财产和奢侈品——更大的回报可以来自学术努力。我很自豪地看到我们的儿子们也在学术道路上走得很远。

丘成桐在斯坦福大学被提升为正教授，然后于1979年回到普林斯顿高等斯图迪研究所。1980年，他被任命为普林斯顿高等斯图迪研究所的教授，他一直担任该职位直到1984年，然后他转到加州大学圣地亚哥分校的讲席。1988年，他被任命为哈佛大学的教授。他是哈佛大学的威廉·卡斯帕·格劳斯坦数学教授，但在2022年4月，他从哈佛退休，被授予威廉·卡斯帕·格劳斯坦数学荣休教授，成为北京清华大学的数学教授。

丘成桐于1982年被授予约翰·查尔斯·菲尔兹奖章，以表彰他对偏微分方程、代数几何中的卡拉比猜想、广义相对论的正质量猜想以及实和复加斯帕尔·蒙日-安德烈-马里·安培方程的贡献。事实上，1982年的约翰·查尔斯·菲尔兹奖章是在1982年8月初在华沙举行的国际数学联盟大会全体会议上宣布的。它们直到在华沙举行的国际大会才颁发，该大会未能按计划于1982年举行，推迟到了次年。

路易·尼伦伯格在1983年华沙国际数学家大会上描述了丘成桐的工作。在1982年约翰·查尔斯·菲尔兹奖章宣布后，路易·尼伦伯格在[49]中写道：-

丘成桐在微分几何和偏微分方程方面做了极其深刻而强大的工作。他是一位分析学家的几何学家（或几何学家的分析学家），拥有巨大的技术能力和洞察力。他破解了多年来进展停滞的问题。

路易·尼伦伯格简要描述了丘成桐的工作领域。关于1954年提出的卡拉比猜想，他写道，这[49]：-

……来自代数几何，涉及证明在紧埃里希·凯勒流形上存在一个具有指定体积形式的埃里希·凯勒度量。分析问题在于证明一个高度非线性（复加斯帕尔·蒙日-安德烈-马里·安培）微分方程解的存在性。丘成桐的解在精神上是经典的，通过先验估计。他对估计的推导是一项绝技，在代数几何中的应用是优美的。

丘成桐于1976年解决了卡拉比猜想。1997年，在被授予国家科学奖章后，他谈到了他在试图证明或推翻该猜想时遇到的困难；见THIS LINK。

丘成桐解决的另一个猜想是正质量猜想，它来自黎曼几何。丘成桐在合作工作中构造了极小曲面，研究了它们的稳定性，并深入分析了它们在时空中的行为。他在这里的工作对黑洞的形成有应用。

约瑟夫·普拉托问题由约瑟夫·普拉托、卡尔·魏尔斯特拉斯、波恩哈德·黎曼和赫尔曼·阿曼杜斯·施瓦茨研究过，但最终由杰西·道格拉斯和蒂博尔·拉多解决。然而，仍有问题涉及杰西·道格拉斯的解，已知它是一个光滑浸入曲面，是否实际上是嵌入的。丘成桐与W H Meeks合作，于1980年解决了这个问题。

1981年，丘成桐被授予奥斯瓦尔德·维布伦几何学奖[89]：-

……因他在非线性偏微分方程方面的工作，他对可微流形拓扑学的贡献，以及他在紧复流形上复加斯帕尔·蒙日-安德烈-马里·安培方程方面的工作。

同年，他被授予国家科学院的弗瑞兹·约翰 J Carty 科学进步奖。该奖的颁发条件是：-

……以表彰在国家科学院章程规定的任何科学领域内取得的显著而杰出的成就。

在丘成桐与凯伦·乌伦贝克On the existence of Hermitian Yang-Mills connections in stable bundles（1986年）的合作工作中，他们解决了Hitchin-Kobayashi猜想的高维版本。他们的工作扩展了西蒙·唐纳森在1985年关于此主题的研究。

Royal Swedish Academy of Sciences的克拉福德奖于1994年授予丘成桐[75]：-

……因其在微分几何中发展非线性技巧，从而解决了若干突出问题。

G Tian[74]总结了丘成桐迄今为止的工作，这些工作使他获得了克拉福德奖：-

由于丘成桐过去二十年的工作，基础偏微分方程在几何中的角色和理解在数学领域内发生了巨大变化并大幅扩展。他的工作已经并将继续对数学和物理学的多个领域产生重大影响，这些领域包括拓扑学、代数几何、表示论、广义相对论以及微分几何和偏微分方程。

你可以在THIS LINK阅读丘成桐获得克拉福德奖的引文摘录。

丘成桐于1993年当选为国家科学院会士。他于1997年被授予国家科学奖章[17]：-

……因他在数学和物理学方面的基础性贡献。通过他的工作，对基础几何微分方程的理解已经改变，并且他在数学中极大地扩展了它们的角色。

你可以在THIS LINK阅读丘成桐获得国家科学奖章的引文摘录。

2010年，丘成桐被授予极具声望的沃尔夫奖[70]：-

……因他在几何分析方面的工作，对几何学和物理学的许多领域产生了深刻而巨大的影响

你可以在THIS LINK阅读授予丘成桐沃尔夫奖的颁奖词摘录。

2023年，丘成桐获得了另一项极具声望的奖项——邵逸夫奖，因其[69]：-

……与数学物理、算术几何、微分几何和凯勒几何相关的贡献。

你可以在THIS LINK阅读授予丘成桐邵逸夫奖的颁奖词摘录。

他投入了大量精力建设中国数学，在哈佛暑假期间访问中国，帮助顶尖中国学生赴美攻读博士学位，并为在香港、北京和杭州建立数学研究所而努力。2004年，他在位于北京天安门广场西侧的人民大会堂受到表彰，以表彰他对中国数学的贡献。

2015年3月19日丘成桐数学科学中心在清华大学揭牌时，他谈到了自己帮助中国建设数学的工作。丘成桐在开幕式上发表了讲话[79]：-

多年来，我在中国各地建立了几个数学中心。我的目标很简单，就是希望中国能够发展像数学这样一门基础学科。我非常感激大多数中国数学家与我有着共同的信念。杨乐教授是一位好朋友……在过去的几十年里，他给了我们巨大的支持。还有许多来自世界各地的人帮助了中国数学教育和研究的发展。从中国科学院开始，我在中国各地建立了研究中心和团队；我还在国外培养了数十名中国学生。其中一些人今天就在这里。事实上，我鼓励他们回到中国，以便我们能够合作改善中国的数学。我们也得到了中国许多数学家的帮助：清华大学、Chinese Academy of Sciences和浙江大学有许多志同道合的人为我们提供了强有力的支持。我们共同无私地投入时间和精力来组织和支持各种数学活动。

丘成桐写了许多书，大多是合著的。我们在THIS LINK给出了其中八本书的信息，包括序言摘录和评论。

遗憾的是，他卷入了一场关于儒勒·昂利·庞加莱猜想证明的不幸争议。俄罗斯数学家格里戈里·佩雷尔曼在2003年概述了该猜想的一个证明，几个团队开始致力于给出完整全面的证明。丘成桐的团队是其中之一，他因一些言论而受到批评，人们认为这些言论没有给予格里戈里·佩雷尔曼充分的认可。特别是 New Yorker的文章[47]对丘成桐提出了高度批评。哥伦比亚大学数学教授Richard S Hamilton写了公开信[91]为丘成桐辩护。信的开头是：-

我对丘成桐丘成桐在《纽约客》文章中被描绘的不公平方式感到非常不安。我在下面提供我的想法以澄清事实。如果对丘成桐有帮助，我授权你将这封信分享给《纽约客》和公众。

Richard Hamilton的信结尾如下：-

[丘成桐]所建立的是一个人才的集合，而不是一个权力的帝国，人们被他的精力、他的卓越想法以及他对一流数学的不懈支持所吸引，丘成桐将这些人聚集在一起研究最困难的问题。丘成桐和我多年来花了无数时间一起研究格雷戈里奥·里奇-库尔巴斯托罗流和其他问题，甚至常常工作到深夜。他总是慷慨地与我分享他的建议，从颈部收缩的观察到后来，从不要求功劳。事实上，就在去年冬天，当我终于设法证明了格雷戈里奥·里奇-库尔巴斯托罗流的卡尔·古斯塔夫·阿克塞尔·哈纳克不等式的局部版本时——这是我们多年来共同研究的问题——我说我应该把他的名字加到论文上，他谦虚地拒绝了。不幸的是，他的品格被如此严重地歪曲了。据我所知，他从未提出过任何功劳百分比，也从未提出格里戈里·佩雷尔曼应该与除我之外的任何人分享儒勒·昂利·庞加莱猜想的功劳；这是合理的，因为确实没有人比格里戈里·佩雷尔曼本人更慷慨地认可我的工作。他远没有窃取格里戈里·佩雷尔曼成就的功劳，反而赞扬了格里戈里·佩雷尔曼的工作，并和我一起支持他获得约翰·查尔斯·菲尔兹奖章。事实上，在创建格雷戈里奥·里奇-库尔巴斯托罗流项目方面，没有人比丘成桐贡献更大，而格里戈里·佩雷尔曼正是利用这个项目赢得了该奖项。

丘成桐已经明确表示，他的本意只是说他的团队使格里戈里·佩雷尔曼的证明能为更广泛的数学家所理解。

让我们引用加州大学河滨分校数学教授Bun Wong和Yat Sun Poon的话来结束这篇传记[84]：-

丘成桐在数学上的成就在数学界广为人知。同样广为人知的是，他成功培养了近50名数学博士研究生，并在全球有许多合作者。也许，鲜为人知的是，他捐赠个人资金为数学学生设立奖学金，向教育机构捐赠了数万册书籍，帮助筹集了数千万美元以促进数学教育和研究，并筹集资金以促进跨学科和跨国界的科学家之间的交流。

最后，这里是丘成桐在2002年写的一首诗：-

空间/时间

Time, time
why does it vanish?
All manner of things
what infinite variety.
Three thousand rivers
all from one source.
Time, space
mind, matter, reciprocal.
Time, time
it never returns.
Space, space
how much can it hold?
In constant motion
always in flux.
Black holes lurking
mysteries afoot.
Space and time
one without bounds.
Infinite, infinite
the secrets of the universe.
Inexhaustible, lovely
in every detail.
Measure time, measure space
no one can do it.
Watched through a straw
what's to be learned has no end.

丘成桐
北京，2002年

Shing-Tung Yau was the fifth of the eight children of his parents Chen Ying Chiou and Yeuk-Lam Leung Chiou. He has a younger brother Stephen Shing-Toung Yau (born 1952) who is also an exceptional mathematician. His father was an economist and philosopher working in southern China when Yau was born. However, by late 1949 the Communists were in control of almost all of China and Yau's family fled to Hong Kong where his father obtained a position teaching at a College. (The College later became a part of the Chinese University of Hong Kong.) In the interview [76] Yau gives more details:-

My father went to Japan to study economics, but he came back to help the Chinese defend themselves before the Japanese invaded in 1937. By the end of the war he was distributing food and clothes to the poor for the U.N. After the revolution in 1949, he worried about getting in trouble with the Communists, so he brought the whole family to Hong Kong. We were very poor - at first we were almost starving - but my father had a large group of students constantly at home to talk about philosophy and literature. I was 10, 11, 12 years old, and I grew accustomed to abstract reasoning. My father made us memorise long essays and poems. At the time I didn't understand what they meant, but I remembered them and later made use of it.

Times were extremely difficult for the family so Yau's mother knitted goods to sell in order to supplement their low income. Life was tough for Yau living in a village outside Hong Kong city in a house which had no electricity or running water and at this stage of his life he often played truant from school preferring his role as leader of a street gang. Yau's father was a major influence on him, encouraging his interest in philosophy and mathematics. In 2003 Yau commented [43]:-

In fact, I felt I could understand my father's conversations better after I learned geometry.

Yau's father was pressured to spy on China but refused and resigned his professorship at the Chinese University of Hong Kong. Sadly, Yau's older sister died in 1962, an elder brother got a brain disease, and when Yau was fourteen years old, his father died. In a tribute to his father, written on his centennial birthday in 2011, Yau writes [80]:-

An inspiring life of ups and down, vanquished in a moment.
Though his wisdom of East and West still echoes in my heart.
I never enjoyed his love enough, which has left me in dismay.
The bloom of youth has passed me by, my hair turned to grey.
Yet I oft look back to that fateful time when I was just a careless teen.
How sad it was when he left that night, so long ago and faraway.
What might he have told us, I wonder, if only he could have said?
Though I'll never hear those words, his thoughts are with me, always.

By the time he was fourteen, Yau was enjoying his High School education at the school named Pui Ching but he also had to help support the family. He helped out with the family's finances by acting as a tutor. Despite the fact that Pui Ching school had a great reputation, he later felt that physics had not been taught very well there and he had not been given enough intuition for the subject. After leaving school in 1965, he continued his education at Chung Chi College in Hong Kong. This College had been founded in 1951 but, in 1963, was incorporated as one of the three foundation colleges of the Chinese University of Hong Kong. Yau left the College in 1968 before graduating. By good fortune, one of his lecturers at the College had studied at the University of California, Berkeley, and, seeing Yau's enormous potential, suggested that he go there to study for a doctorate.

This lecturer was Stephen Salaff (1938-2012) who was born in Stamford, Connecticut, USA, and had studied at Columbia University before going to England to study for a bachelor's degree in mathematics at the University of London. He had been awarded a Ph.D. in mathematics from the University of California at Berkeley where he met the student Janet Weitzner. They married and since Janet's research required going to China, the couple moved to Hong Kong and Stephen lectured in mathematics at the Chinese University there. Yau has always been extremely grateful for Salaff's support. He said in 2012 [60]:-

Without Steve's help, I would not be the mathematician I am now.

Yau explained in the interview [38] how Salaff supported him:-

Actually Stephen tried to get me an early undergraduate degree. There was a big fight at the Chinese University of Hong Kong. Of course, they regret it now because I got a Fields Medal. If I was a graduate from the Chinese University it would count as a good point for them.

Salaff contacted Donald Sarason, an Associate Professor at Berkeley and, probably with Shiing-shen Chern's help, they managed to get Berkeley to waive their entrance requirements and award him one of two International Business Machines Corporation Fellowships to fund him to undertake research at the University of California at Berkeley. Yau said [38]:-

When I applied, I didn't know Chern, but after I was admitted Chern was offered an honorary degree by the Chinese University of Hong Kong. I think in June he came to Hong Kong from America to pick up the degree and I met him in the middle of June. That was after I was admitted. But I'm sure he must have helped somewhat.

Yau asked his friends to lend him some money so he could afford the air fare to America. He flew to San Francisco in September 1969 and, meeting up with three others looking for accommodation, they were able to rent a place. Although he received $300 a month from his scholarship, he sent half back to his family in Hong Kong, so life was hard. He began by attending courses [29]:-

I enrolled in many classes because I felt I didn't know many different areas of modern mathematics and I started to be interested in geometry. I learned quite a lot of things from different faculties in Berkeley and during the first semester I learned about manifolds. I learned Riemannian geometry, but not enough, it was just elementary.

He attended a course on nonlinear partial differential equations given by Charles Bradfield Morrey Jr. (1907-1984) which he felt was excellent and, unable to afford to return home, spent the Christmas vacation working in the library reading research papers and monographs. Most of all, he enjoyed reading the Journal of Differential Geometry. In particular, he read a paper by John Milnor describing how the fundamental group of a manifold is influenced by curvature. He found Milnor's paper very well written and easy to understand. It referred to earlier papers which Yau found and read. Although only a few months into his time as a graduate student, he found himself able to generalise some of the results in these papers.

Chern had not been in Berkeley during Yau's first year as a graduate student but when he returned at the beginning of Yau's second year Yau asked Chern if he would be his thesis advisor. He agreed and Yau continued studying at Berkeley with Chern as his supervisor. In [21] Arthur Fischer describes what Berkeley was like at that time for mathematics graduate students:-

The mathematics department at that time, in 1969 and 1970, was crammed into Campbell Hall, the Astronomy building, and was not to occupy its current home on the top four floors of Evans Hall until 1971. In this small volume, low entropy state, there was considerable interaction between faculty, lecturers, graduate students, and visiting faculty. The colloquia were often packed with a lively crowd, a hundred people or so in a theatre-like setting, which was a lot of people, at least as compared with the small audiences and tranquil demeanour of the Princeton colloquia. It was in this low entropy state that I first met Yau in the Fall of 1969, when he was just starting out as a first year graduate student. ... I was in the photocopy room of Campbell Hall in the Fall of 1969 ... and Yau walked in. We vaguely had known of each other's existence, but not much more. Yau was there to photocopy his notes on links between curvature and topology. I was interested. He told me more. I talked about what I was working on, a global formulation of the Hamiltonian structure of general relativity. It was a hit. These might be related. We both got excited.

Once he was Chern's student, Yau spent most of his time studying complex manifolds. In particular, he read Eugenio Calabi's papers and became fascinated by the Calabi conjecture. After only two years, he received his Ph.D. in 1971 for his thesis On the Fundamental Group of Compact Manifolds of Non-positive Curvature. By this time he had three papers in print, On the fundamental group of manifolds of non-positive curvature (1970), On the fundamental group of compact manifolds of non-positive curvature (1971), and Compact flat Riemannian manifolds (1971).

During session 1971-72, Yau was a member of the Institute for Advanced Study at Princeton. He was appointed assistant professor at the State University of New York at Stony Brook in 1972. In 1974 he was appointed an associate professor at Stanford University. John Coates met Yau at this time. He writes [12]:-

I first met Yau when he came to Stanford University in 1974, a year after I had arrived at Stanford. I was rapidly struck by two qualities of Yau at this time. Firstly, he had very broad mathematical interests (for example, I was amazed that he sat patiently though a rather technical graduate course which I gave on Iwasawa theory). Secondly, he was remarkably hard working. Indeed, I would occasionally come into the Stanford Mathematics Department towards midnight, and on every occasion I remember finding Yau there either doing his own work or talking to the keen group of graduate students which he quickly gathered around him.

In 1976 Yau married the physicist Yu-yun Kuo; they have two sons, Isaac Yau and Michael Yau. He wrote in [80]:-

I was also lucky to have fallen in love with, and eventually married, a woman who shared my view that there is more to life than seeking personal wealth, material possessions, and luxuries - that greater rewards can come from scholarly endeavours. I'm proud to see that our sons have also ventured far along academic paths.

Yau was promoted to full professor at Stanford before returning to the Institute for Advanced Study at Princeton in 1979. In 1980 he was made a professor at the Institute for Advanced Study at Princeton, a position he held until 1984 when he moved to a chair at the University of California at San Diego. In 1988 he was appointed professor at Harvard University. He was William Caspar Graustein Professor of Mathematics at Harvard but, in April 2022, he retired from Harvard, where he was named William Caspar Graustein Professor of Mathematics Emeritus, to become a professor of mathematics at Tsinghua University in Beijing, China.

Yau was awarded a Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampère equations. In fact the 1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year.

Nirenberg described Yau's work at the International Congress in Warsaw in 1983. Writing in [49] after the Fields Medal awards were announced in 1982, Nirenberg wrote:-

S-T Yau has done extremely deep and powerful work in differential geometry and partial differential equations. He is an analyst's geometer (or geometer's analyst) with enormous technical power and insight. He has cracked problems on which progress has been stopped for years.

Nirenberg describes briefly the areas of Yau's work. On the Calabi conjecture, which was made in 1954, he writes that this [49]:-

... comes from algebraic geometry and involves proving the existence of a Kähler metric, on a compact Kähler manifold, having a prescribed volume form. The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge-Ampère ) differential equation. Yau's solution is classical in spirit, via a priori estimates. His derivation of the estimates is a tour de force and the applications in algebraic geometry are beautiful.

Yau solved the Calabi conjecture in 1976. In 1997, after being awarded the National Medal of Science, he spoke about the difficulties he encountered trying to either prove or disprove the conjecture; see THIS LINK.

Another conjecture solved by Yau was the positive mass conjecture, which comes from Riemannian geometry. Yau, in joint work, constructed minimal surfaces, studied their stability and made a deep analysis of how they behave in space-time. His work here has applications to the formation of black holes.

The Plateau problem was studied by Plateau, Weierstrass, Riemann and Schwarz but it was finally solved by Douglas and Radó. However, there were still questions relating to whether Douglas's solution, which was known to be a smooth immersed surface, is actually embedded. Yau, working with W H Meeks solved this problem in 1980.

In 1981 Yau was awarded The Oswald Veblen Prize in Geometry [89]:-

...for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampère equation on compact complex manifolds.

In the same year be was awarded the John J Carty Award for the Advancement of Science of the National Academy of Sciences. This is given:-

... to recognise noteworthy and distinguished accomplishments in any field of science within the National Academy of Science's charter.

In joint work of Yau with Karen Uhlenbeck On the existence of Hermitian Yang-Mills connections in stable bundles (1986), they solved higher dimensional versions of the Hitchin-Kobayashi conjecture. Their work extended that of Donaldson on this topic in 1985.

The Crafoord Prize of the Royal Swedish Academy of Sciences was awarded to Yau in 1994 [75]:-

... for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.

G Tian [74] sums up Yau's work to date which led to his being awarded the Crafoord Prize:-

As a result of Yau's work over the past twenty years, the role and understanding of basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics. His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations.

You can read extracts from the citation for the award of the Crafoord Prize to Yau at THIS LINK.

Yau was elected to the National Academy of Sciences in 1993. He was awarded the National Medal of Science in 1997 [17]:-

... for his fundamental contributions in mathematics and physics. Through his work, the understanding of basic geometric differential equations has been changed and he has expanded their role enormously within mathematics.

You can read extracts from the citation for the award of the National Medal of Science to Yau at THIS LINK.

In 2010, Yau was awarded the highly prestigious Wolf Prize [70]:-

... for his work in geometric analysis that has had a profound and dramatic impact on many areas of geometry and physics

You can read extracts from the citation for the award of the Wolf Prize to Yau at THIS LINK.

In 2023, Yau was awarded another very prestigious prize, the Shaw Prize, for his [69]:-

... contributions related to mathematical physics, to arithmetic geometry, to differential geometry and to Kähler geometry.

You can read extracts from the citation for the award of the Shaw Prize to Yau at THIS LINK.

He put a great deal of effort into building Chinese mathematics, visiting China during the Harvard summer vacation, helping top Chinese students go to the United States for doctoral studies, and working hard for the founding of mathematical institutes in Hong Kong, Beijing and Hangzhou. In 2004 he was honoured in the Great Hall of the People, located on the western side of Tiananmen Square in Beijing, for his contributions to Chinese mathematics.

He spoke about his work to help build mathematics in China when the Shing-Tung Yau Mathematical Science Centre was Inaugurated at Tsinghua University on 19 March 2015. Yau made a speech at the Opening Ceremony [79]:-

Over the years, I have established several mathematics centres in various places in China. My goal is simply to see that China could develop a subject as fundamental as mathematics. I am very grateful that most Chinese mathematicians share my convictions. Professor Lo Yang is a good friend ... and for the past several decades, he has given us tremendous support. There are many others from around the world who have helped the development of Chinese mathematics education and research. Starting in the Chinese Academy of Sciences, I have set up research centres and groups at various places in China; I have also trained dozens of Chinese students abroad. Some of them are here today. In fact, I encourage them to return to China so that we might collaborate in improving mathematics in China. We have also received help from many mathematicians within China: there are many like-minded people at Tsinghua University, the Chinese Academy of Sciences, and Zhejiang University providing us with strong support. Together we have selflessly devoted our time and energy to organise and support various mathematics activities.

Yau has written many books, mostly co-authored. We give information about eight of these books, including extracts from Prefaces and reviews, at THIS LINK.

Sadly, he was involved in an unfortunate dispute regarding the proof of the Poincaré conjecture. The Russian mathematician Grigory Perelman sketched a proof of the conjecture in 2003 and several teams began work on giving a full comprehensive proof. Yau's team was one of these and he has been criticised by some for comments which people felt did not give Perelman full credit. In particular the New Yorker article [47] is highly critical of Yau. Richard S Hamilton, Professor of Mathematics, Columbia University, wrote the open letter [91] defending Yau. It begins:-

I am very disturbed by the unfair manner in which Yau Shing-Tung has been portrayed in the New Yorker article. I am providing my thoughts below to set the record straight. I authorise you to share this letter with the New Yorker and the public if that will be helpful to Yau.

Richard Hamilton ends his letter as follows:-

[What] Yau has built is an assembly of talent, not an empire of power, people attracted by his energy, his brilliant ideas, and his unflagging support for first rate mathematics, people whom Yau has brought together to work on the hardest problems. Yau and I have spent innumerable hours over many years working together on the Ricci Flow and other problems, often even late at night. He has always generously shared his suggestions with me, starting with the observation of neck pinches, never asking for credit. In fact just last winter when I finally managed to prove a local version of the Harnack inequality for the Ricci Flow, a problem we had worked on together for many years, and I said I ought to add his name to the paper, he modestly declined. It is unfortunate that his character has been so badly misrepresented. He has never to my knowledge proposed any percentages of credit, nor that Perelman should share credit for the Poincaré conjecture with anyone but me; which is reasonable, as indeed no one has been more generous in crediting my work than Perelman himself. Far from stealing credit for Perelman's accomplishment, he has praised Perelman's work and joined me in supporting him for the Fields Medal. And indeed no one is more responsible than Yau for creating the program on Ricci Flow which Perelman used to win this prize.

Yau has clearly stated that it had only been his intention to say that his team made Perelman's proof understandable to a much wider range of mathematicians.

Let us end this biography by quoting Bun Wong and Yat Sun Poon, Professors of Mathematics at the University of California at Riverside [84]:-

Yau's achievement in Mathematics is well known within mathematics community. It is equally well known that he has successfully produced nearly 50 PhD students in mathematics and has many collaborators across the globe. Perhaps, it is less well known that he has donated personal fund to establish scholarships for mathematics students, has donated tens of thousands of books to educational institutions, has helped raise tens of millions of dollars to promote mathematics education and research, and has raised fund to promote interaction among scientists across subject boundaries and national borders.

Finally, here is a poem written by Yau in 2002:-

SPACE/TIME

Time, time
why does it vanish?
All manner of things
what infinite variety.
Three thousand rivers
all from one source.
Time, space
mind, matter, reciprocal.
Time, time
it never returns.
Space, space
how much can it hold?
In constant motion
always in flux.
Black holes lurking
mysteries afoot.
Space and time
one without bounds.
Infinite, infinite
the secrets of the universe.
Inexhaustible, lovely
in every detail.
Measure time, measure space
no one can do it.
Watched through a straw
what's to be learned has no end.

Shing-Tung Yau
Beijing, 2002