# Shing-Tung Yau

### Quick Info

Kwuntung (now Guangdong), China

**Yau**is a Chinese mathematician who won a Fields Medal for his work on differential geometry and partial differential equations.

### Biography

**Shing-Tung Yau**was the fifth of the eight children of his parents Chen Ying Chiou and Yeuk-Lam Leung Chiou. 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.) Times were difficult for the family, however, and 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 a recent talk Yau commented:-

In fact, I felt I could understand my father's conversations better after I learned geometry.Sadly, when he was fourteen years old, his father died but by this time Yau was enjoying his school education. He helped out with the family's finances by acting as a tutor. After leaving school in 1965, he continued his education at Chung Chi College in Hong Kong, leaving the College in 1968 before graduating. By good fortune, one of his lecturers at the College had studied at the University of California, and, seeing Yau's enormous potential, suggested that he go there to study for a doctorate. He obtained a fellowship from the International Business Machines Corporation and began his research at Berkeley.

Yau studied for his doctorate at the University of California at Berkeley under Chern's supervision. He received his Ph.D. in 1971 and, during session 1971-72, Yau was a member of the Institute for Advanced Study at Princeton.

Yau 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. He 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.

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 [5] 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 progess 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:-

... 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. 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:-

...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 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:-

... for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.G Tian [6] 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.Yau was elected to the National Academy of Sciences in 1993. He was awarded the National Medal of Science in 1997. 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. However, recently he has been 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 is one of these and he has been criticised by some for comments which people felt did not give Perelman full credit. However 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:-

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.

### References (show)

- Biography in
*Encyclopaedia Britannica.*http://www.britannica.com/biography/Shing-Tung-Yau - H Araki, Profiles of 1982 Fields Medal winners (Japanese),
*Sugaku***35**(1) (1983), 70-77. - S Kobayashi, The work of Shing Tung Yau (Japanese),
*Sugaku***35**(2) (1983), 121-127. - L Nirenberg, The work of Yau Shing-Tung,
*Proceedings of the International Congress of Mathematicians, Warsaw 1983***1**(Warsaw, 1984), 15-19. - L Nirenberg, The work of Shing-Tung Yau,
*Notices Amer. Math. Soc.***29**(1982), 501-502. - R Stern and G Tian, Donaldson and Yau receive Crafoord prize,
*Notices Amer. Math. Soc.***41**(7) (1994), 794-796.

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update September 2009

Last Update September 2009