# Shing-Tung Yau Extras

We give information relating to Shing-Tung Yau being awarded (1) the Crafoord Prize 1994 (2) the National Medal of Science 1997 (3) the Wolf Prize in Mathematics 2010 and (4) the Shaw Prize 2023. Mostly this information is extracted (in slightly modified form) from the citations and the Press Releases for these prizes.

**1. The Royal Swedish Academy of Sciences awards the 1994 Crafoord Prize to differential geometry to Simon Donaldson and Shing-Tung Yau.**

The Prize is awarded to Simon Donaldson, University of Oxford, England, for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants: and Shing-Tung Yau, Harvard University, Cambridge, MA, USA, for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.

The Crafoord Prize amounts to c. USD 300 000, and is divided equally between the two prize winners. It was awarded in September 1994.

Modern geometry is divided in two main directions, differential geometry which takes distances into account, and topology, which only cares about shape. Gradually it has become clear that there is in topology a clear division between geometry in five dimensions or more on the one hand, and geometry in lower dimensions on the other. In the fifties and sixties topology in five and higher dimensions was investigated and today could be said to be well understood. As for low dimensions, from one to four, they have all turned out to have their own peculiarities. One and two dimensions are, because of their perspicuity, relatively simple. Three-dimensional geometry, through considerably more complicated, behaves much like that of lower, particularly two, dimensions. Geometry of four dimensions seems to be a boundary case. It differs fundamentally from lower dimensions while the methods used for higher dimensions do not work.

The study of five or more dimensions shows that in topology there is reason to take into account how strongly the form of geometrical objects are permitted to be changed. Continuity is the notion that is used when one allows the sharpest possible changes, differentiability the one used when only smooth changes are accepted. The understanding of this distinction in these dimensions is part of the achievements obtained in the fifties and sixties while this distinction may not be seen in one, two and three dimensions.

Thanks to Shing-Tung Yau's work over the past twenty years, the role and understanding of the basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics. His work has had an 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 is a student of the legendary Chinese mathematician Shiing-Shen Chern, with whom he studied at Berkeley. As a teacher he is very generous with his ideas and he has had many students and also collaborated with many mathematicians.

Shing-Tung Yau was born in Guandong in southern China in 1949. He received his Ph. D. from the University of California at Berkeley in 1971, and his D.Sc. from the Chinese University of Hong Kong in 1980. During the years 1971-72 and 1979-80 he was a member of the Institute for Advanced Study in Princeton, NJ, USA, and was a professor there 1980-84. After that Yau was a professor at the University of California at San Diego, La Jolla, for some years, and is now (since the end of the 1980's) at Harvard University.

**2. S-T Yau Receives National Medal of Science in 1997.**

Shing-Tung Yau has received the National Medal of Science, the nation's highest scientific honour. On 16 December 1997, President Clinton presented medals to Yau and eight other laureates in a ceremony at the Old Executive Office Building in Washington, DC.

Established by Congress in 1959, the National Medal of Science is bestowed annually by the president on a select group of individuals "deserving of special recognition by reason of their outstanding contributions to knowledge in the physical, biological, mathematical, or engineering sciences." Congress expanded this definition in 1980 to recognise outstanding work in the social and behavioural sciences. In 1962 President John F Kennedy awarded the first Medal of Science to the late Theodore Von Karman, president emeritus of aeronautical engineering at the California Institute of Technology. Including the 1997 winners, 353 individuals have been awarded the Medal of Science. In the past five years (1992-96) the National Medal of Science has been awarded to four who work in the mathematical sciences: Richard Karp and Steven Smale (1996), Martin Kruskal (1994), and Alberto Calderón (1992).

Yau was honoured "for profound contributions to mathematics that have had a great impact on fields as diverse as topology, algebraic geometry, general relativity and string theory. His work insightfully combines two different mathematical approaches and has resulted in the solution of several long-standing and important problems in mathematics."

Shing-Tung Yau was born on 4 April 1949, in Kwuntung, China. He received his Ph.D. from the University of California, Berkeley, in 1971, where his advisor was S S Chern, who received the National Medal of Science in 1975. In 1971 Yau went to the Institute for Advanced Study (IAS) and the following year became an assistant professor at the State University of New York, Stony Brook. After that came appointments as professor at Stanford University (1974-79), professor at the IAS (1979-84), chair and professor at the University of California, San Diego (1984-87), and professor at Harvard University (1987-present). Currently he is also an adjunct professor at the Chinese University of Hong Kong. Yau held a Visiting Professorship and Sid Richardson Centennial Chair in Mathematics at the University of Texas at Austin in 1986 and was a Fairchild Distinguished Scholar at Caltech in 1990. During 1991-92, he was Wilson T S Wang Distinguished Visiting Professor at the Chinese University of Hong Kong and held a Special Chair at the National Tsing Hua University in Hsinchu, Taiwan.

Yau received a Fields Medal at the International Congress of Mathematicians in Warsaw in 1983. His other awards and honours include the AMS Veblen Prize (1981), the Carty Prize of the National Academy of Sciences (1981), a MacArthur Fellowship (1985), and the Crafoord Prize of the Royal Swedish Academy of Sciences (1994) (for an account of Yau's research, see the announcement of the Crafoord Prize, Notices, September 1994, page 794). He is a member of the National Academy of Sciences, the New York Academy of Sciences, and the American Academy of Arts and Sciences. He is a foreign member of the Chinese Academy of Sciences and a foreign academician of the Academia Sinica.

**Shing-Tung Yau's interview after the award of the National Medal of Science.**

The beauty of the structure really charms my own feeling; I cannot resist feeling in a very nice way. When I was a young boy I was fascinated by geometry. I found it very elegant and beautiful so I pursued it vigorously when I was a graduate student and I was very excited by how space could have some curvature; that space curves and how to capture them. So I was fascinated by trying to understand this concept until I saw a paper by some famous mathematician by the name of Calabi and he points to some lessons that I think are fundamental for us to understand what the curvature of space means.

So I pursued it, did a lot of work to understand it. At that point it was considered to be impossible and in fact people thought that the problem cannot be solved in that way. But I felt strongly that even though it cannot be solved, I still have to understand why. I worked on it trying to disprove what Mr Calabi said and I encountered great difficulty. Then I thought I had found how to disprove it. It turned out to be wrong. Then I spent the next three years trying to prove it, to demonstrate that it actually works. That was hard work but finally I understood it and I finished the proof in 1976.

For mathematics I can keep on thinking and solve a problem by myself or work together with other people. I see the beauty of it created by myself or created together with my friends. I always found that fascinating. The future is bright because the subject had brought in a unification among many different branches of mathematics. The most important for a scientist, as I see it, is your passion - a very strong passion for the subject you are interested in. In the process of enduring the difficulties in research, there is much fun in the end when you think about it because at the very last minute you found the answer you wanted. I think nothing can compare with such excitement. Being an immigrant to this country, to receive the National Medal of Science is certainly a very important moment. But the most exciting moment for me was the following. I overheard my two children, who are about fifteen years old, saying their father, that is me, was always saying around how good he is. Maybe he is actually good! So I was pleased to learn that statement, that my children feel I'm not that bad.

**3. Shing-Tung Yau awarded the Wolf Prize in Mathematics 2010.**

Shing-Tung Yau's affiliation at the time of the award is Harvard University, USA.

The award is made to Shing-Tung Yau "for his work in geometric analysis that has had a profound and dramatic impact on many areas of geometry and physics."

Shing-Tung Yau (born in 1949, China) has linked partial differential equations, geometry, and mathematical physics in a fundamentally new way, decisively shaping the field of geometric analysis. He has developed new analytical tools to solve several difficult nonlinear partial differential equations, particularly those of the Monge-Ampere type, critical to progress in Riemannian, Kahler and algebraic geometry and in algebraic topology, that radically transformed these fields. The Calabi-Yau manifolds, as these are known, a particular class of Kahler manifolds, have become a cornerstone of string theory aimed at understanding how the action of physical forces in a high-dimensional space might ultimately lead to our four-dimensional world of space and time. Prof Yau's work on T-duality is an important ingredient for mirror symmetry, a fundamental problem at the interface of string theory and algebraic and symplectic geometry. While settling the positive mass and energy conjectures in general relativity, he also created powerful analytical tools, which have broad applications in the investigation of the global geometry of space-time.

Prof Yau's eigenvalue and heat kernel estimates on Riemannian manifolds, count among the most profound achievements of analysis on manifolds . He studied minimal surfaces, solving several classical problems, and then used his results, to create a novel approach to geometric topology. Prof Yau has been exceptionally productive over several decades, with results radiating onto many areas of pure and applied mathematics and theoretical physics. In addition to his diverse and fundamental mathematical achievements, which have inspired generations of mathematicians, Prof Yau has also had an enormous impact, worldwide, on mathematical research, through training an extraordinary number of graduate students and establishing several active mathematical research centres.

**4. Shing-Tung Yau awarded the Shaw Prize in Mathematical Sciences 2023.**

The Shaw Prize in Mathematical Sciences 2023 is awarded in equal shares to Vladimir Drinfeld, Harry Pratt Judson Distinguished Service Professor of Mathematics at the University of Chicago, USA and Shing-Tung Yau, Chair Professor at Tsinghua University, PRC, for their contributions related to mathematical physics, to arithmetic geometry, to differential geometry and to Kähler geometry.

They share an interest in mathematical physics. Drinfeld launched with Beilinson the geometric Langlands program, which, to quote Witten, has some common features with aspects of quantum field theory, and yet stems from number theory. Yau worked on mathematical problems arising from general relativity and string theory.

Yau developed systematically partial differential equation methods in differential geometry. With these, he solved the Calabi conjecture, for which he was awarded the Fields medal in 1982, the existence of Hermitian Yang-Mills connections (with Uhlenbeck), and the positive mass conjecture (with Schoen) for which they used the theory of minimal surfaces. He introduced geometric methods to important problems in general relativity, which led for example to Schoen-Yau's black-hole existence theorem and to an intrinsic definition of quasi-local mass in general relativity.

Yau's work on the existence of a Kähler-Einstein metric led to the solution to the Calabi conjecture, and to the concept of Calabi-Yau manifolds, which are cornerstones both in string theory and in complex geometry. The Strominger-Yau-Zaslow construction has had a major impact on mirror symmetry.

His work (with P Li) on heat kernel estimates and differential Harnack inequalities has changed the analysis of geometric equations on manifolds. It has influenced the development of optimal transportation and Hamilton's work on Ricci flow.

Yau contributed to the fusion of geometry and analysis, now known as geometric analysis. His work has had a deep and lasting impact on both mathematics and theoretical physics.

Mathematical Sciences Selection Committee

The Shaw Prize

30 May 2023 Hong Kong

Last Updated December 2023