Shing-Tung Yau

Shing-Tung Yau
File:ShingTungYau.jpg Shing-Tung Yau at Harvard Law School dining hall
Born	(1949-04-04) April 4, 1949 Shantou, Guangdong Province, China
Alma mater	Chinese University of Hong Kong (B.A. 1969) UC Berkeley (Ph. D 1971)
Awards	Fields Medal (1982) Veblen Prize (1981) Crafoord Prize (1994) National Medal of Science (1997) Wolf Prize (2010)
Scientific career
Fields	Mathematics
Institutions	Harvard University, Chinese University of Hong Kong Zhejiang University
Doctoral advisor	Shiing-Shen Chern
Doctoral students	Richard Schoen (Stanford, 1977) Huai-Dong Cao (Princeton, 1986) Gang Tian (Harvard, 1988) Lizhen Ji (Northeastern, 1991) Kefeng Liu (Harvard, 1993)

Shing-Tung Yau (Chinese: 丘成桐; born April 4, 1949) is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province. His work has led to the important Calabi-Yau manifolds, which are the basic building blocks of the universe, according to String Theory. He developed and put the field of differential equations on manifolds into the mainstream of mathematics, and his work on minimal manifolds has applications to topology and general relativity theory.

Yau has made lasting impacts on both physics and mathematics. Calabi-Yau manifolds are among the ‘standard toolkit’ for string theorists today. He has been very active in the exciting interface between geometry and theoretical physics. His proof of the positive energy theorem in general relativity finally demonstrated—sixty years after its discovery—that Einstein’s theory is consistent and stable. His proof of the Calabi conjecture allowed physicists—using Calabi-Yau compactification—to show that string theory is a viable candidate for a unified theory of nature.

Biography

Yau was born in Shantou, Guangdong Province, China with an ancestry in Jiaoling (also in Guangdong) in a family of eight children. When he was only a few months old, his family emigrated to Hongkong, where they lived first in Yuen Long and then 5 years later in Shatin. When Yau was fourteen, his father, a philosophy professor, died.

After graduating from Pui Ching Middle School, he studied mathematics at the Chinese University of Hong Kong from 1966 to 1969. Yau went to UC Berkeley in the fall of 1969. At the age of 22, Yau was awarded the Ph.D. degree under the supervision of Shiing-Shen Chern at Berkeley in two years. He spent a year as a member of the Institute for Advanced Study, Princeton, New Jersey, and two years at the State University of New York at Stony Brook. Then he went to Stanford University.

Since 1987, he has been at Harvard University , where he has had numerous Ph.D. students. He is also involved in the activities of research institutes in Hong Kong and China. He takes an interest in the state of K-12 mathematics education in China, and his criticisms of the Chinese education system, corruption in the academic world in China, and the quality of mathematical research and education, have been widely publicized.

Major contributions to mathematics

Duong Hong Phong of Columbia University has commented on the wide influence of Yau's research in geometric analysis.

Work on conjectures

Calabi Conjecture

Further information: Calabi conjecture

Yau's solution of the Calabi conjecture, concerning the existence of an Einstein-Kähler metric, has far-reaching consequences. The existence of such a canonical unique metric allows one to give explicit representatives of characteristic classes. Calabi-Yau manifolds are now fundamental in string theory, where the Calabi conjecture provides an essential piece in the model.

In algebraic geometry, the Calabi conjecture implies the Miyaoka-Yau inequality on Chern numbers of surfaces, a characterization of the complex projective plane and quotients of the two-dimensional complex unit ball, an important class of Shimura varieties.

Yau also made a contribution in the case that the first Chern number c₁ > 0, and conjectured its relation to the stability in the sense of geometric invariant theory in algebraic geometry. This has motivated the work of Simon Donaldson on scalar curvature and stability. Another important result of Donaldson-Uhlenbeck-Yau is that a holomorphic vector bundle is stable (in the sense of David Mumford) if and only if there exists an Hermitian-Yang-Mills metric on it. This has many consequences in algebraic geometry, for example, the characterization of certain symmetric spaces, Chern number inequalities for stable bundles, and the restriction of the fundamental groups of a Kähler manifold.

Positive mass conjecture and existence of black holes

Further information: Positive energy theorem

Yau pioneered the method of using minimal surfaces to study geometry and topology. By an analysis of how minimal surfaces behave in space-time, Yau and Richard Schoen proved the long standing conjecture that the total mass in general relativity is positive.

This theorem implies that flat space-time is stable, a fundamental issue for the theory of general relativity. Briefly, the positive mass conjecture says that if a three-dimensional manifold has positive scalar curvature and is asymptotically flat, then a constant that appears in the asymptotic expansion of the metric is positive. A continuation of the above work gives another result in relativity proved by Yau, an existence theorem for black holes. Yau and Schoen continued their work on manifolds with positive scalar curvature, which led to Schoen's final solution of the Yamabe problem.

Smith conjecture

Further information: Smith conjecture

Yau and William H. Meeks solved the well-known question whether the Douglas solution of a minimal disk for an external Jordan curve, the Plateau problem, in three space, is always embedded if the boundary curve is a subset of a convex boundary. They then went on to prove that these embedded minimal surfaces are equivariant for finite group actions. Combining this work with a result of William Thurston, Cameron Gordon assembled a proof of the Smith conjecture: for any cyclic group acting on a sphere, the set of fixed points is not a knotted curve.

Hermitian Yang-Mills connection and stable vector bundles

Yau and Karen Uhlenbeck proved the existence and uniqueness of Hermitian-Einstein metrics (or equivalently Hermitian Yang-Mills connections) for stable bundles on any compact Kähler manifold, extending an earlier result of Donaldson for projective algebraic surfaces, and M. S. Narasimhan and C. S. Seshadri for algebraic curves. Both the results and methods of this paper have been influential in parts of both algebraic geometry and string theory. This result is now usually called the Donaldson-Uhlenbeck-Yau Theorem.

Frankel conjecture

Yau and Yum-Tong Siu proved the 1981 Frankel conjecture in complex geometry, stating that any compact positively-curved Kähler manifold is biholomorphic to complex projective space. An independent proof was given by Shigefumi Mori, using methods of algebraic geometry in positive characteristic.

Mirror conjecture

With Bong Lian and Kefeng Liu, Yau proved mirror formulas conjectured by string theorists. These formulas give the explicit numbers of rational curves of all degrees in a large class of Calabi-Yau manifolds, in terms of the Picard-Fuchs equations of the corresponding mirror manifolds.

New methods and concepts

Gradient estimates and Harnack inequalities

Yau developed the method of gradient estimates for Harnack's inequalities. This method has been used and refined by him and other people to attack for example, bounds on the heat kernel. Early in 1981, Yau suggested to Richard Hamilton that he use the Ricci flow to realize naturally the canonical decomposition of a three-dimensional manifold into pieces, each of which has a geometric structure, in the Thurston program. Hamilton amplified their results, to what is now called the Li-Yau-Hamilton inequality for the Ricci flow equations.

Gradient estimates were also used crucially in Yau's joint work with S. Y. Cheng to give a complete proof of the higher dimensional Hermann Minkowski problem and the Dirichlet problem for the real Monge–Ampère equation, and other results on the Kähler-Einstein metric of bounded pseudoconvex domains.

Uniformization of complex manifolds

When Yau was a graduate student, he started to generalize the uniformization theorem of Riemann surfaces to higher-dimensional complex Kähler manifolds. For a compact manifold with Positive bisectional curvature, the Frankel conjecture proved by Siu and Yau, and independently by Mori, shows that it is complex projective space. Yau proposed a series of conjectures when the manifold is non-compact, and made contributions towards their solutions. For example, when the bisectional curvature is positive, it must be biholomorphic to Cⁿ. Work on these conjectures were taken from the solution of the Poincaré conjecture by Perelman.

Harmonic maps and rigidity

When Yau was working on his thesis about manifolds with non-positive curvature and their fundamental groups, he realized that it is possible to use harmonic maps to give alternative proofs of some results there. He was aware of the Mostow rigidity theorem for locally symmetric spaces, which was used by him to prove the uniqueness of complex structure of quotients of complex balls. Naturally he proposed that harmonic maps be used to prove rigidity of the complex structure for Kähler manifolds with strongly negative curvature, a program that was successfully carried out by Yum-Tong Siu. This method, the so-called Siu-Yau method, has been extended to prove strong and super-rigidities of many locally symmetric spaces.

Minimal submanifolds

Minimal submanifolds have been used by Yau in the solutions of the Positive Mass Conjecture, the Smith conjecture, the Frankel conjecture, and else. Many people others have since applied minimal surfaces to other problems. Mikhail Gromov's introduction of pseudo-holomorphic curves in symplectic geometry has also had an important impact on that field.

Open problems

Yau has compiled an influential set of open problems in geometry.

Harmonic functions with controlled growth

One of Yau’s problems is about bounded harmonic functions, and harmonic functions on noncompact manifolds of polynomial growth. After proving non-existence of bounded harmonic functions on manifolds with positive curvatures, he turned around and proposed the Dirichlet problem at infinity for bounded harmonic functions on negatively curved manifolds, and then proceeded to harmonic functions of polynomial growth. Dennis Sullivan tells a story about Yau's geometric intuition, and how it led him to reject an analytical proof of Sullivan's. Michael Anderson independently found the same result about bounded harmonic function on simply connected negatively curved manifolds using a geometric convexity construction.

Rank rigidity of nonpositively curved manifolds

Again otivated by Mostow's strong rigidity theorem, Yau called for a notion of rank for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics. Advances in this direction have been made by Ballmann, Brin and Eberlein in their work on non-positive curved manifolds, Gromov's and Eberlein's metric rigidity theorems for higher rank locally symmetric spaces and the classification of closed higher rank manifolds of non-positive curvature by Ballmann and Burns-Spatzier. This leaves rank 1 manifolds of non-positive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.

Kähler–Einstein metrics and stability of manifolds

It is known that if a complex manifold has a Kähler–Einstein metric, then its tangent bundle is stable. Yau realized early in 1980s that the existence of special metrics on Kähler manifolds is equivalent to the stability of the manifolds. Various people such as S.Donaldson, a Fields medalist, have made progress to understand such a relation.

Mirror symmetry

He has collaborated with string theorists including Strominger, Vafa and Witten, and as post-doctorals from theoretical physics with B. Greene, E. Zaslow and A. Klemm . The Strominger-Yau-Zaslow program is to construct explicitly mirror manifolds. David Gieseker wrote of the seminal role of the Calabi conjecture in relating string theory with algebraic geometry, in particular for the developments of the SYZ program, mirror conjecture and Yau-Zaslow conjecture.

Initiatives in mainland China and Taiwan

Yau was born in China but grew up in Hong Kong. After the door of China was opened to the west in the late 1970s, Yau revisited China in 1979 on the invitation of Hua Luogeng. To help develop Chinese mathematics, Yau started by educating students from China, then establishing mathematics research institutes and centers, organizing conferences at all levels, initiating out-reach programs, and raising private funds for these purposes. John Coates has commented on Yau's success as fundraiser. The first of Yau's initiatives is The Institute of Mathematical Sciences at The Chinese University of Hong Kong in 1993. The goal is to “organize activities related to a broad variety of fields including both pure and Applied mathematics, scientific computation, image processing, mathematical physics and statistics. The emphasis is on interaction and linkages with the physical sciences, engineering, industry and commerce.”

The second one is the Morningside Center of Mathematics in Beijing, established in 1996. Part of the money for the building and regular operations was raised by Yau from the Morningside Foundation in Hong Kong. Yau proposed organizing the International Congress of Chinese Mathematicians, now held every three years. The first congress was held at the Morningside Center from December 12 to 18, 1998. The third is the Center of Mathematical Sciences at Zhejiang University. It was established in 2002. Yau is the director of all these three math institutes and visits them on a regular basis.

Yau went came to Taiwan to attend a conference in 1985 thematics. In 1990, he was invited by Dr. C.-S. Liu, then the President of National Tsinghua University, to visit the university for a year. A few years later, he convinced Liu, by then the chairman of National Science Council, to create the National Center of Theoretical Sciences (NCTS), which was established at Hsinchu in 1998. He was the chairman of the Advisory Board of the NCTS until 2005 and was followed by H. T. Yau of Harvard University.

Outreach

His classmate at college Y.-C.Siu speaks of Yau as an ambassador of mathematics. In Hong Kong, with the support of Ronnie Chan, Yau set up the Hang Lung Award for high school students. He has also organized and participated in meetings for high school and college students, for example, the panel discussions Why Math? Ask Masters! in Hangzhou, July 2004, and The Wonder of Mathematics in Hong Kong, December 2004. Yau organized the JDG conference surveying developments in geometry and related fields, and the annual Current development of mathematics conference. Yau also co-initiated a series of books on popular mathematics, "Mathematics and Mathematical People".

Honors and awards

Yau has received numerous honors and awards in his life, including:

Prizes and awards

• 1979, California Scientist of the Year.
• 1981, Oswald Veblen Prize in Geometry.
• 1981, John J. Carty Award for the Advancement of Science, United States National Academy of Sciences.
• 1982, Fields Medal, 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".
• 1984, Science Digest, Americia’s 100 Brightest Scientists under 40.
• 1991, Humboldt Research Award, Alexander von Humboldt Foundation, Germany.
• 1994, Crafoord Prize.
• 1997, United States National Medal of Science.
• 2003, China International Scientific and Technological Cooperation Award, for “his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers”.
• 2010, Wolf Prize in Mathematics, for "his work in geometric analysis and mathematical physics".

Research fellowships

• 1975-1976, Sloan Fellow.
• 1982, Guggenheim Fellowship.
• 1984-1985, MacArthur Fellow.

Honorary professorships

• Honorary Professor, Hunan Normal University (appointed on Dec 22, 2009).
• Honorary Professor, Northwest University (appointed on Jul 15, 2009).
• Honorary Professor, North University of China (appointed on Jun 18, 2009).
• Honorary Professor, Huazhong University of Science and Technology (appointed on Jan 15, 2006).
• Honorary Professor, University of Science and Technology of China (appointed in 1999).
• Honorary Professor, Peking University (appointed in 1998).
• Honorary Professor, Zhejiang University (appointed in 1998).
• Honorary Professor, Nankai University (appointed in 1993).
• Honorary Professor, Tsinghua University (appointed in 1987).
• Honorary Professor, Hangzhou University (appointed in 1987).
• Honorary Professor, Fudan University (appointed in 1983).

Honorary degrees

• Honorary Degree of Doctor of Science, The Chinese University of Hong Kong (1981)
• Honorary Master of Arts, Harvard University (1987)
• Honorary Degree of Doctor of Science, National Central University (Jul 9, 1993)
• Honorary Degree of Doctor of Science, National Chiao Tung University, Taiwan (1997)
• Honorary Degree of Doctor of Science, National Tsing Hua University, Taiwan (2000)
• Doctor of Science honoris causa, The University of Macau (2002)
• Honorary Doctorate, Zhejiang University (Mar 25, 2003).
• Doctor of Science honoris causa, The Hong Kong University of Science and Technology (Aug 26, 2004)
• Doctor of Science, Polytechnic University in Brooklyn, New York (2005)
• Doctor of Science, National Taiwan University (2005)

Academic memberships

• Foreign Member, National Academy of Lincei of Italy (elected in 2005)
• Foreign Member, Russian Academy of Sciences (elected in 2003).
• Foreign member, Chinese Academy of Sciences (elected in 1995).
• Member, United States National Academy of Science (elected in 1993).
• Academician, Academic Sinica (elected in 1984).
• Member, American Academy of Arts and Sciences (elected in 1982).
• Honorary Member, Academic Committee of the Institute of Mathematics, Chinese Academy of Sciences (elected in 1980).
• Fellow, American Association for the Advancement of Science.
• Fellow, Society for Industrial and Applied Mathematics.
• Fellow, American Physical Society.
• Member, Boston Academy of Arts and Sciences.
• Member, New York Academy of Science.
• Honorary Fellow, Shaw College of The Chinese University of Hong Kong.
• Fellow, American Mathematical Society (elected in 1971)

Poincaré conjecture debates

In August 2006, a New Yorker article on the Poincaré conjecture, "Manifold Destiny", discussed Yau's relationship to that famous problem. Yau claims this article is defamatory, and in September 2006 he established a public relations website, managed by the PR firm Spector and Associates, to dispute points in it and demand an apology. By April 5, 2007, fifteen mathematics professors, including two quoted in the New Yorker article, had posted letters of support on Yau's website. The New Yorker reportedly stands by its article.

On October 17, 2006, a more sympathetic profile of Yau appeared, along with photographs from different stages of his life, in the New York Times. After recounting Yau's humble beginnings and rise to academic stardom, it devotes about half its length to the Perelman affair. The article acknowledges that Yau's egotism and high-profile activities, including criticism of Chinese academia, have alienated some of his colleagues and that Yau's promotion of the Cao-Zhu paper "annoyed many mathematicians, who felt that Dr. Yau had slighted Dr. Perelman." In contrast to the "Manifold Destiny" article, it paints Yau as ultimately more concerned with the development of mathematics than with his reputation. In regards to the Perelman affair, the article focuses on Yau's position, which is that Perelman's proof was not understood by all, and he "had a duty to dig out the truth of the proof."

External links

• Interview (11 pages long in Traditional Chinese)
• Yau's autobiographical account (mostly English, some Chinese)
• O'Connor, John J.; Robertson, Edmund F., "Shing-Tung Yau", MacTutor History of Mathematics Archive, University of St Andrews
• Shing-Tung Yau at the Mathematics Genealogy Project
• Banquet photos
• Yau's website, with information on his legal action and letter to The New Yorker
• Yau's Public Relations agent, Spector and Associates
• Richard S Hamilton' Letter to Yau Shing-Tung' Attorney on September 25, 2006
• Plugging A Math Gap
• UC Irvine courting Yau with a $2.5 million professorship
• International Conference Celebrating Shing Tung Yau's Birthday 8/27/2008-9/1/2008 Harvard University