# Shing-Tung Yau

### Quick Info

Kwuntung (now Guangdong), China

**Shing-Tung Yau**is a Chinese mathematician who won a Fields Medal for his work on differential geometry and partial differential equations. He has also received other prestigious mathematical prizes such as the Wolf Prize and the Shaw Prize.

### Biography

**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.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.

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.

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 physicsYou 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

Shing-Tung Yau
*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.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.

Beijing, 2002

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*Columbia University New York*.

https://secretary.columbia.edu/directory/shing-tung-yau - Shing-Tung Yau,
*Institute for Advanced Study*.

https://www.ias.edu/scholars/shing-tung-yau - Shing-Tung Yau, The Shaw Prize 2023,
*The Shaw Foundation*.

https://www.shawprize.org/laureates/2023-mathematical-sciences/?type=Contribution - Shing-Tung Yau: Wolf Prize in Mathematics 2010,
*Wolf Foundation*.

https://wolffund.org.il/2018/12/11/shing-tung-yau/ - Shing-Tung Yau Ph.D.,
*Simons Foundation*.

https://www.simonsfoundation.org/people/shing-tung-yau/ - Shing-Tung Yau Ph.D. Biography,
*Ganga Library*(2010).

https://www.gangalib.org/yau.php#complete - Shing-Tung Yau meets with Dr Choi Yeuk-Lin, Secretary for Education of the HKSAR in Tsinghua,
*Yau Mathematical Sciences Center, Tsinghus University*(30 September 2023).

https://ymsc.tsinghua.edu.cn/en/info/1024/2732.htm - R Stern and G Tian, Donaldson and Yau receive Crafoord prize,
*Notices of the American Mathematical Society***41**(7) (1994), 794-796. - The Royal Swedish Academy of Sciences awards The 1994 Crafoord Prize to differential geometry,
*The Crafoord Prize*(12 January 1994).

https://www.crafoordprize.se/news/the-royal-swedish-academy-of-sciences-awards-the-1994-crafoord-prize-to-differential-geometry/ - P Weintraub, Discover Interview: The Math Behind the Physics Behind the Universe,
*Discover Magazine*(27 April 2010).

https://www.discovermagazine.com/the-sciences/discover-interview-the-math-behind-the-physics-behind-the-universe - J C Wood, Review: Lectures on harmonic maps, by R Schoen and S T Yau,
*Mathematical Reviews*MR1474501**(98i:58072)**. - H H Wu, Review: Nonlinear analysis in geometry, by Shing-Tung Yau,
*Mathematical Reviews*MR0865650**(88e:53001)**. - S-T Yau, Inauguration of the Shing-Tung Yau Center at Tsinghua University,
*Notices of the International Congress of Chinese Mathematicians***3**(1) (2015), 86-88. - S-T Yau and S Nadis,
*The shape of a life. One mathematician's search for the universe's hidden geometry*(Yale University Press, New Haven, CT, 2019). - S-T Yau and S Nadis,
*The shape of inner space. String theory and the geometry of the universe's hidden dimensions*(Basic Books, New York, 2010). - S-T Yau College Student Mathematics Contest,
*Yau Mathematical Science Center, Tsinghua University*.

http://yau-contest.com/en/page-Introduction.html - L Xin, Chinese-born maths genius leaves Harvard to help China become a powerhouse on subject,
*South China Morning Post*(21 April 2022). - B Wong and Y S Poon, Visibility of Asian Americans in Mathematics,
*Notices of the American Mathematical Society***54**(1) (2007), 6. - N Hitchin, Review: The shape of inner space. String theory and the geometry of the universe's hidden dimensions, by S-T Yau and S Nadis,
*Notices of the American Mathematical Society***58**(2) (2011), 311-312. - Shing-Tung Yau,
*Prabook*.

https://prabook.com/web/shing-tung.yau/3577045 - G B Folland, Review: A history in sum. 150 years of mathematics at Harvard (1825-1975), by S Nadis and S-T Yau,
*The American Mathematical Monthly***122**(5) (2015), 508-510. - C Sherman, Review: The shape of a life. One mathematician's search for the universe's hidden geometry, by S-T Yau and S Nadis,
*Math Horizons***27**(2) (2019), 29. - E Pitcher, Veblen Prize, in
*A history of the second fifty years, American Mathematical Society 1939-88*(American Mathematical Society, 1988), 54-55. - P Hond, Holes in the Argument. Who solved the Poincaré Conjecture, one of the world's most difficult math problems?,
*Columbia Magazine*(Winter 2006-07).

https://magazine.columbia.edu/article/holes-argument - Letter to Howard M Cooper from Richard Hamilton (25 September 2006).

https://web.archive.org/web/20061012053048/https://doctoryau.com/hamiltonletter.pdf

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Written by J J O'Connor and E F Robertson

Last Update December 2023

Last Update December 2023