Shing-Tung Yau
Encyclopedia
Shing-Tung Yau 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.
Yau's contributions have had an influence on both physics
and mathematics
. Calabi–Yau manifolds are among the ‘standard toolkit’ for string theorists today. He has been active at the interface between geometry
and theoretical physics
. His proof of the positive energy theorem
in general relativity
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.
, 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 Hong Kong
, where they lived first in Yuen Long
and then 5 years later in Shatin. When Yau was fourteen, his father Qiuzhen Ying, 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 the University of California, 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.
has commented on the wide influence of Yau's research in geometric analysis
.
, 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
.
s 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
.
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 point
s is not a knotted curve.
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.
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.
, 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.
. 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.
of Riemann surfaces to higher-dimensional complex Kähler manifold
s. 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.
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. 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.
, 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.
s, and harmonic functions on noncompact manifolds of polynomial
growth. After proving non-existence of bounded harmonic functions on manifolds with positive curvatures
, he 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.
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.
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 including Simon Donaldson
have made progress to understand such a relation.
, 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.
.
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 to Taiwan
to attend a conference in 1985. 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.
, 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".
article on the Poincaré conjecture
, "Manifold Destiny
", discussed Yau's relationship to that famous problem. The printed edition had a cartoon showing Yau as trying to steal Grigori Perelman
's Fields Medal. Perelman is quoted saying that he is disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to the efforts of Fields medalist Shing-Tung Yau to downplay Perelman's role in the proof and play up the work of Cao
and Zhu
.
Yau stated: “Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle.” Perelman stated of Yau, "I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest." Yau claimed that this article was defamatory, and in September 2006 he established a public relations website, managed by the public relations firm Spector & Associates, to dispute points in it and demand an apology. Some mathematicians, including two quoted in the New Yorker article, posted letters of support.
On October 17, 2006, a more sympathetic profile of Yau appeared in the New York Times
. It devoted about half its length to the Perelman affair. The article stated that Yau had alienated some colleagues, but represented Yau's position as that Perelman's proof was not generally understood and he "had a duty to dig out the truth of the proof."
Chinese American
Chinese Americans represent Americans of Chinese descent. Chinese Americans constitute one group of overseas Chinese and also a subgroup of East Asian Americans, which is further a subgroup of Asian Americans...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
working in differential geometry. He was born in Shantou
Shantou
Shantou , historically known as Swatow or Suátao, is a prefecture-level city on the eastern coast of Guangdong province, People's Republic of China, with a total population of 5,391,028 as of 2010 and an administrative area of...
, Guangdong Province, China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
into a family of scholars from Jiaoling
Jiaoling
Jiaoling County is a northeastern county of Guangdong province....
, Guangdong Province.
Yau's contributions have had an influence on both physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. Calabi–Yau manifolds are among the ‘standard toolkit’ for string theorists today. He has been active at the interface between geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
. His proof of the positive energy theorem
Positive energy theorem
In general relativity, the positive energy theorem states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non-negative; furthermore, the mass is zero only for Minkowski spacetime...
in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
demonstrated—sixty years after its discovery—that Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theory is consistent and stable. His proof of the Calabi conjecture
Calabi conjecture
In mathematics, the Calabi conjecture was a conjecture about the existence of good Riemannian metrics on complex manifolds, made by and proved by ....
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 ShantouShantou
Shantou , historically known as Swatow or Suátao, is a prefecture-level city on the eastern coast of Guangdong province, People's Republic of China, with a total population of 5,391,028 as of 2010 and an administrative area of...
, Guangdong Province, China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
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 Hong Kong
Hong Kong
Hong Kong is one of two Special Administrative Regions of the People's Republic of China , the other being Macau. A city-state situated on China's south coast and enclosed by the Pearl River Delta and South China Sea, it is renowned for its expansive skyline and deep natural harbour...
, where they lived first in Yuen Long
Yuen Long
Yuen Long , formerly Un Long, is an area and town located in the northwest of Hong Kong, on the Yuen Long Plain. To its west lie Hung Shui Kiu and Ha Tsuen, to the south Shap Pat Heung and Tai Tong, to the east Au Tau and Kam Tin, and to the north Nam Sang Wai.-Name:The Cantonese name Yuen Long 元朗...
and then 5 years later in Shatin. When Yau was fourteen, his father Qiuzhen Ying, a philosophy professor, died.
After graduating from Pui Ching Middle School
Pui Ching Middle School (Hong Kong)
Pui Ching Middle School is a Baptist secondary school in Ho Man Tin, Kowloon, Hong Kong. Founded in 1889, it currently has sister schools in Macau and Guangzhou.-History:The Hong Kong branch was established in 1933 in Ho Man Tin...
, he studied mathematics at the Chinese University of Hong Kong
Chinese University of Hong Kong
The Chinese University of Hong Kong is a research-led university in Hong Kong.CUHK is the only tertiary education institution in Hong Kong with Nobel Prize winners on its faculty, including Chen Ning Yang, James Mirrlees, Robert Alexander Mundell and Charles K. Kao...
from 1966 to 1969. Yau went to the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
in the fall of 1969. At the age of 22, Yau was awarded the Ph.D. degree under the supervision of Shiing-Shen Chern
Shiing-Shen Chern
Shiing-Shen Chern was a Chinese American mathematician, one of the leaders in differential geometry of the twentieth century.-Early years in China:...
at Berkeley in two years. He spent a year as a member of the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
, Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, and two years at the State University of New York at Stony Brook
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....
. Then he went to Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
.
Since 1987, he has been at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, 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
K-12
K–12 is a designation for the sum of primary and secondary education. It is used in the United States, Canada, Australia, and New Zealand where P–12 is also commonly used...
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.
Contributions to mathematics
Duong Hong Phong of Columbia UniversityColumbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
has commented on the wide influence of Yau's research in geometric analysis
Geometric analysis
Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations. It includes both the use of geometrical methods in the study of partial differential equations , and the application of the theory of partial differential equations to geometry...
.
Calabi conjecture
Yau's solution of the Calabi conjectureCalabi conjecture
In mathematics, the Calabi conjecture was a conjecture about the existence of good Riemannian metrics on complex manifolds, made by and proved by ....
, 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
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, where the Calabi conjecture provides an essential piece in the model.
In algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, 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
Unit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
, 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
Geometric invariant theory
In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces...
in algebraic geometry. This has motivated the work of Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
on scalar curvature
Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
and stability. Another important result of Donaldson-Uhlenbeck-Yau is that a holomorphic vector bundle
Holomorphic vector bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map \pi:E\to X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the...
is stable (in the sense of David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
) 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
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
.
Positive mass conjecture and existence of black holes
Yau pioneered the method of using minimal surfaceMinimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s to study geometry and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. By an analysis of how minimal surfaces behave in space-time, Yau and Richard Schoen
Richard Schoen
Richard Melvin Schoen is an American mathematician. Born in Fort Recovery, Ohio, he received his PhD in 1977 from Stanford University where he is currently the Anne T. and Robert M. Bass Professor of Humanities and Sciences...
proved the long-standing conjecture that the total mass in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
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
Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
and is asymptotically flat, then a constant that appears in the asymptotic expansion
Asymptotic expansion
In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular,...
of the metric is positive. A continuation of the above work gives another result in relativity proved by Yau, an existence theorem
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...
for black holes. Yau and Schoen continued their work on manifolds with positive scalar curvature
Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
, which led to Schoen's final solution of the Yamabe problem
Yamabe problem
The Yamabe problem in differential geometry concerns the existence of Riemannian metrics with constant scalar curvature, and takes its name from the mathematician Hidehiko Yamabe. Although claimed to have a solution in 1960, a critical error...
.
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 convexConvex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
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
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
, Cameron Gordon
Cameron Gordon (mathematician)
Cameron Gordon is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem...
assembled a proof of the Smith conjecture
Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere, of finite order then the fixed point set of f cannot be a nontrivial knot....
: for any cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
acting on a sphere, the set of fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
s is not a knotted curve.
Hermitian Yang–Mills connection and stable vector bundles
Yau and Karen UhlenbeckKaren Uhlenbeck
Karen Keskulla Uhlenbeck is a professor and Sid W. Richardson Regents Chairholder in the Department of Mathematics at The University of Texas in Austin. In 1998 she was selected to be a Noether Lecturer. In 2000, she became a recipient of the National Medal of Science...
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
C. S. Seshadri
C.S.Seshadri FRS is an eminent Indian mathematician. He is currently Director-Emeritus of the Chennai Mathematical Institute, and is known for his work in algebraic geometry. The Seshadri constant is named after him. He is also a recipient of the prestigious Padma Bhushan in 2009, the third highest...
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 SiuYum-Tong Siu
Yum-Tong Siu is the William Elwood Byerly Professor of Mathematics at Harvard University.Dr. Siu has been a prominent figure in the mathematics of several complex variables for a quarter-century. He has mastered techniques at the interfaces between complex variables, differential geometry, and...
proved the 1981 Frankel conjecture in complex geometry
Complex geometry
In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
, stating that any compact positively-curved Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
is biholomorphic to complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
. An independent proof was given by Shigefumi Mori
Shigefumi Mori
-References:*Heisuke Hironaka, Fields Medallists Lectures, Michael F. Atiyah , Daniel Iagolnitzer ; World Scientific Publishing, 2007. ISBN 9810231172...
, using methods of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
in positive characteristic.
Mirror conjecture
With Bong Lian and Kefeng LiuKefeng Liu
Kefeng Liu , is an Chinese-American mathematician mainly working in topology. Liu is current head of the Department of Mathematics at Zhejiang University in Hangzhou. He is also the current Executive Director of the Center of Mathematical Sciences of ZJU.-Biography:Liu was born February 1965 in...
, Yau proved mirror formulas conjectured by string theorists. These formulas give the explicit numbers of rational curves of all degrees
Degree of an algebraic variety
The degree of an algebraic variety in mathematics is defined, for a projective variety V, by an elementary use of intersection theory.For V embedded in a projective space Pn and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined...
in a large class of Calabi–Yau manifolds, in terms of the Picard–Fuchs equations of the corresponding mirror manifolds.
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 kernelHeat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a particular domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some...
. Early in 1981, Yau suggested to Richard Hamilton that he use the Ricci flow
Ricci flow
In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
to realize naturally the canonical decomposition of a three-dimensional manifold into pieces, each of which has a geometric structure, in the Thurston program
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
. 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
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
problem and the Dirichlet problem
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
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 theoremUniformization theorem
In mathematics, the uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature...
of Riemann surfaces to higher-dimensional complex Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
s. 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
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
. 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.
Harmonic maps and rigidity
When Yau was working on his thesis about manifolds with non-positive curvatureNon-positive curvature
In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero...
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
Mostow rigidity theorem
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique...
for locally symmetric spaces, which was used by him to prove the uniqueness of complex structure of quotients of complex balls. 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
Yum-Tong Siu
Yum-Tong Siu is the William Elwood Byerly Professor of Mathematics at Harvard University.Dr. Siu has been a prominent figure in the mathematics of several complex variables for a quarter-century. He has mastered techniques at the interfaces between complex variables, differential geometry, and...
. 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 conjectureSmith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere, of finite order then the fixed point set of f cannot be a nontrivial knot....
, 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.
Harmonic functions with controlled growth
One of Yau’s problems is about bounded harmonic functionHarmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
s, and harmonic functions on noncompact manifolds of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
growth. After proving non-existence of bounded harmonic functions on manifolds with positive curvatures
Constant curvature
In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...
, he proposed the Dirichlet problem
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
at infinity for bounded harmonic functions on negatively curved manifolds, and then proceeded to harmonic functions of polynomial growth. Dennis Sullivan
Dennis Sullivan
Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...
tells a story about Yau's geometric intuition, and how it led him to reject an analytical proof of Sullivan's. Michael Anderson
Michael Anderson (differential geometer)
Michael Anderson is a Professor of Mathematics inState University of New York at Stony Brook. He is a differential geometer working on geometrization of 3-manifolds, general relativity, Einstein metrics and AdS/CFT correspondence et cetera.Anderson got his BA from UC Santa Barbara his MA from UC...
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 motivated by Mostow's strong rigidity theorem, Yau called for a notion of rankRank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
. Advances in this direction have been made by Ballmann
Hans Werner Ballmann
Hans Werner Ballmann is a German mathematician. His area of research is differential geometry with focus on geodesic flows, spaces of negative curvature as well as spectral theory of Dirac operators...
, Brin and Eberlein in their work on non-positive curved
Non-positive curvature
In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero...
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 manifoldComplex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
has a Kähler–Einstein metric, then its tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
is stable. Yau realized early in 1980s that the existence of special metrics
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
on Kähler manifolds is equivalent to the stability of the manifolds. Various people including Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
have made progress to understand such a relation.
Mirror symmetry
He has collaborated with string theorists including StromingerAndrew Strominger
Andrew Eben Strominger is an American theoretical physicist who works on string theory and son of Jack L. Strominger. He is currently a professor at Harvard University and a senior fellow at the Society of Fellows...
, Vafa
Cumrun Vafa
Cumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
and Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
, and as post-doctorals from theoretical physics with B. Greene
Brian Greene
Brian Greene is an American theoretical physicist and string theorist. He has been a professor at Columbia University since 1996. Greene has worked on mirror symmetry, relating two different Calabi-Yau manifolds...
, 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 LuogengHua Luogeng
Hua Luogeng was a Chinese mathematician born in Jintan, Jiangsu. He was the founder and pioneer in many fields in mathematical research. He wrote more than 200 papers and monographs, many of which became classics. Since his sudden death while delivering a lecture at the University of Tokyo, Japan,...
.
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
Chinese University of Hong Kong
The Chinese University of Hong Kong is a research-led university in Hong Kong.CUHK is the only tertiary education institution in Hong Kong with Nobel Prize winners on its faculty, including Chen Ning Yang, James Mirrlees, Robert Alexander Mundell and Charles K. Kao...
in 1993. The goal is to “organize activities related to a broad variety of fields including both pure and Applied mathematics, scientific computation
Computational science
Computational science is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems...
, image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
, mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
. The emphasis is on interaction and linkages with the physical sciences, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, industry
Industry
Industry refers to the production of an economic good or service within an economy.-Industrial sectors:There are four key industrial economic sectors: the primary sector, largely raw material extraction industries such as mining and farming; the secondary sector, involving refining, construction,...
and commerce
Commerce
While business refers to the value-creating activities of an organization for profit, commerce means the whole system of an economy that constitutes an environment for business. The system includes legal, economic, political, social, cultural, and technological systems that are in operation in any...
.”
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
Zhejiang University
Zhejiang University , sometimes referred to as Zheda, is a national university in China. Founded in 1897, Zhejiang University is one of China's oldest institutions of higher education...
. It was established in 2002. Yau is the director of all these three math institutes and visits them on a regular basis.
Yau went to Taiwan
Taiwan
Taiwan , also known, especially in the past, as Formosa , is the largest island of the same-named island group of East Asia in the western Pacific Ocean and located off the southeastern coast of mainland China. The island forms over 99% of the current territory of the Republic of China following...
to attend a conference in 1985. In 1990, he was invited by Dr. C.-S. Liu, then the President of National Tsinghua University
National Tsing Hua University
National Tsing Hua University is one of the most prestigious universities in Taiwan. The university has a strong reputation in the studies of science and engineering. Times Higher Education - World University Rankings is107in the world. Engineering and Science are the best in Taiwan...
, to visit the university for a year. A few years later, he convinced Liu, by then the chairman of National Science Council
National Science Council
The National Science Council is the main governmental promotion and funding body for science research in Taiwan, Republic of China. It is a governmental body under the Executive Yuan.-External links:* *...
, to create the National Center of Theoretical Sciences (NCTS), which was established at Hsinchu
Hsinchu
Hsinchu City is a city in northern Taiwan. Hsinchu is popularly nicknamed "The Windy City" for its windy climate.Hsinchu City is administered as a special municipality within Taiwan . The city is bordered by Hsinchu County to the north and east, Miaoli County to the south, and the Taiwan Strait...
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 ChanRonnie Chan
Ronnie Chan Chi-chung is a Hong Kong entrepreneur. He graduated with an MBA from the University of Southern California....
, 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
Hangzhou
Hangzhou , formerly transliterated as Hangchow, is the capital and largest city of Zhejiang Province in Eastern China. Governed as a sub-provincial city, and as of 2010, its entire administrative division or prefecture had a registered population of 8.7 million people...
, 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".
Prizes and awards
- 1979, California Scientist of the Year.
- 1981, Oswald Veblen Prize in GeometryOswald Veblen Prize in GeometryThe Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen...
. - 1981, John J. Carty Award for the Advancement of Science, United States National Academy of SciencesUnited States National Academy of SciencesThe National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
. - 1982, Fields MedalFields MedalThe Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
, for"his contributions to partial differential equations, to the Calabi conjecture in algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, 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 FoundationAlexander von Humboldt FoundationThe Alexander von Humboldt Foundation is a foundation set-up by the government of the Federal Republic and funded by the German Foreign Office, the Ministry of Education and Research, the Ministry of Economic Cooperation and Development and others for the promotion of international co-operation...
, Germany. - 1994, Crafoord PrizeCrafoord PrizeThe Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord...
. - 1997, United StatesUnited StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...
National Medal of ScienceNational Medal of ScienceThe National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and...
. - 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 MathematicsWolf Prize in MathematicsThe Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
, for "his work in geometric analysis and mathematical physics".
Research fellowships
- 1975-1976, Sloan Fellow.
- 1982, Guggenheim FellowshipGuggenheim FellowshipGuggenheim Fellowships are American grants that have been awarded annually since 1925 by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts." Each year, the foundation makes...
. - 1984-1985, MacArthur Fellow.
Honorary professorships
- Honorary Professor, Hunan Normal UniversityHunan Normal UniversityHunan Normal University , founded in 1938, is a higher education institution located in Changsha, Hunan Province, People's Republic of China. It has existed for 72 years...
(appointed on Dec 22, 2009). - Honorary Professor, Northwest University (appointed on Jul 15, 2009).
- Honorary Professor, North University of ChinaNorth University of ChinaNorth University of China is a university based in Taiyuan, Shanxi province, China，which was once called North China Institute of Technology.It was formerly known as Taihang Industrial School,which was founded in Sep. 8th,1941, and renamed Taiyuan Institute of Machinery in 1958...
(appointed on Jun 18, 2009). - Honorary Professor, Huazhong University of Science and TechnologyHuazhong University of Science and TechnologyThe Huazhong University of Science and Technology is a public, coeducational research university located in Wuhan, Hubei province, China. As a national key university, HUST is directly affiliated to the Ministry of Education of China. HUST has been referred as the flagship of China's higher...
(appointed on Jan 15, 2006). - Honorary Professor, University of Science and Technology of ChinaUniversity of Science and Technology of ChinaThe University of Science and Technology of China is a national research university in Hefei, People's Republic of China. It is a member of the C9 League formed by nine top universities in China...
(appointed in 1999). - Honorary Professor, Peking UniversityPeking UniversityPeking University , colloquially known in Chinese as Beida , is a major research university located in Beijing, China, and a member of the C9 League. It is the first established modern national university of China. It was founded as Imperial University of Peking in 1898 as a replacement of the...
(appointed in 1998). - Honorary Professor, Zhejiang UniversityZhejiang UniversityZhejiang University , sometimes referred to as Zheda, is a national university in China. Founded in 1897, Zhejiang University is one of China's oldest institutions of higher education...
(appointed in 1998). - Honorary Professor, Nankai UniversityNankai UniversityNankai University , commonly known as Nankai, is a public research university based in Tianjin on mainland China. Founded in 1919 by educators Zhang Boling and Yan Fansun , Nankai University is a member of the Nankai serial schools. It is the alma mater of former Chinese Premier and key historical...
(appointed in 1993). - Honorary Professor, Tsinghua UniversityTsinghua UniversityTsinghua University , colloquially known in Chinese as Qinghua, is a university in Beijing, China. The school is one of the nine universities of the C9 League. It was established in 1911 under the name "Tsinghua Xuetang" or "Tsinghua College" and was renamed the "Tsinghua School" one year later...
(appointed in 1987). - Honorary Professor, Hangzhou UniversityHangzhou UniversitySince 1998, Hangzhou University has been a part of Zhejiang University....
(appointed in 1987). - Honorary Professor, Fudan UniversityFudan UniversityFudan University , located in Shanghai, is one of the oldest and most selective universities in China, and is a member of the C9 League. Its institutional predecessor was founded in 1905, shortly before the end of China's imperial Qing dynasty...
(appointed in 1983).
Honorary degrees
- Honorary Degree of Doctor of Science, The Chinese University of Hong KongChinese University of Hong KongThe Chinese University of Hong Kong is a research-led university in Hong Kong.CUHK is the only tertiary education institution in Hong Kong with Nobel Prize winners on its faculty, including Chen Ning Yang, James Mirrlees, Robert Alexander Mundell and Charles K. Kao...
(1981) - Honorary Master of Arts, Harvard UniversityHarvard UniversityHarvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
(1987) - Honorary Degree of Doctor of Science, National Central UniversityNational Central UniversityNational Central University is a national comprehensive university in Taiwan .National Central University was founded in 1915 and originated in 258 CE at Nanjing, China. After NCU in Nanjing was renamed Nanjing University in 1949, NCU was re-established in Taiwan in 1962...
(Jul 9, 1993) - Honorary Degree of Doctor of Science, National Chiao Tung UniversityNational Chiao Tung UniversityNational Chiao Tung University is a public university located in Hsinchu, Taiwan. It is recognized as one of the most prestigious and selective universities in Taiwan and is renowned for its research and teaching excellence in electrical engineering, computer science, engineering, management, and...
, TaiwanTaiwanTaiwan , also known, especially in the past, as Formosa , is the largest island of the same-named island group of East Asia in the western Pacific Ocean and located off the southeastern coast of mainland China. The island forms over 99% of the current territory of the Republic of China following...
(1997) - Honorary Degree of Doctor of Science, National Tsing Hua UniversityNational Tsing Hua UniversityNational Tsing Hua University is one of the most prestigious universities in Taiwan. The university has a strong reputation in the studies of science and engineering. Times Higher Education - World University Rankings is107in the world. Engineering and Science are the best in Taiwan...
, Taiwan (2000) - Doctor of Science honoris causa, The University of MacauUniversity of MacauThe University of Macau, ;, established in 1981, was the first and currently the largest university in Macau, a former Portuguese colony. It was formerly known as University of East Asia , and was renamed the University of Macau in 1991. The university offers about 100 Doctoral, Master's and...
(2002) - Honorary Doctorate, Zhejiang UniversityZhejiang UniversityZhejiang University , sometimes referred to as Zheda, is a national university in China. Founded in 1897, Zhejiang University is one of China's oldest institutions of higher education...
(Mar 25, 2003). - Doctor of Science honoris causa, The Hong Kong University of Science and TechnologyHong Kong University of Science and TechnologyThe Hong Kong University of Science and Technology is a public university located in Hong Kong. Established in 1991 under Hong Kong Law Chapter 1141 , it is one of the nine universities in Hong Kong.Professor Tony F. Chan is the president of HKUST...
(Aug 26, 2004) - Doctor of Science, Polytechnic University in Brooklyn, New York (2005)
- Doctor of Science, National Taiwan UniversityNational Taiwan UniversityNational Taiwan University is a national co-educational university located in Taipei, Republic of China . In Taiwan, it is colloquially known as "Táidà" . Its main campus is set upon 1,086,167 square meters in Taipei's Da'an District. In addition, the university has 6 other campuses in Taiwan,...
(2005) - Doctor of Science, Lehigh UniversityLehigh UniversityLehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. It was established in 1865 by Asa Packer as a four-year technical school, but has grown to include studies in a wide variety of disciplines...
(2009) - Doctor of Science, National Cheng Kung UniversityNational Cheng Kung UniversityNational Cheng Kung University is a national university in Tainan City, Taiwan. Its abbreviation is NCKU. In Chinese, its name is shortened to 成大...
(2010) - Doctor of Mathematics, University of WaterlooUniversity of WaterlooThe University of Waterloo is a comprehensive public university in the city of Waterloo, Ontario, Canada. The school was founded in 1957 by Drs. Gerry Hagey and Ira G. Needles, and has since grown to an institution of more than 30,000 students, faculty, and staff...
(Jun 17, 2011)
Academic memberships
- Foreign Member, National Academy of LinceiAccademia dei LinceiThe Accademia dei Lincei, , is an Italian science academy, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy....
of Italy (elected in 2005) - Foreign Member, Russian Academy of SciencesRussian Academy of SciencesThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....
(elected in 2003). - Foreign member, Chinese Academy of SciencesChinese Academy of SciencesThe Chinese Academy of Sciences , formerly known as Academia Sinica, is the national academy for the natural sciences of the People's Republic of China. It is an institution of the State Council of China. It is headquartered in Beijing, with institutes all over the People's Republic of China...
(elected in 1995). - Member, United StatesUnited StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...
National Academy of Science (elected in 1993). - Academician, Academic Sinica (elected in 1984).
- Member, American Academy of Arts and SciencesAmerican Academy of Arts and SciencesThe American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
(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 ScienceAmerican Association for the Advancement of ScienceThe American Association for the Advancement of Science is an international non-profit organization with the stated goals of promoting cooperation among scientists, defending scientific freedom, encouraging scientific responsibility, and supporting scientific education and science outreach for the...
. - Fellow, Society for Industrial and Applied MathematicsSociety for Industrial and Applied MathematicsThe Society for Industrial and Applied Mathematics was founded by a small group of mathematicians from academia and industry who met in Philadelphia in 1951 to start an organization whose members would meet periodically to exchange ideas about the uses of mathematics in industry. This meeting led...
. - Fellow, American Physical SocietyAmerican Physical SocietyThe American Physical Society is the world's second largest organization of physicists, behind the Deutsche Physikalische Gesellschaft. The Society publishes more than a dozen scientific journals, including the world renowned Physical Review and Physical Review Letters, and organizes more than 20...
. - Member, Boston Academy of Arts and Sciences.
- Member, New York Academy of Science.
- Honorary Fellow, Shaw CollegeShaw CollegeThe Shaw College is the fourth college of The Chinese University of Hong Kong. It is named after its patron, Sir Run Run Shaw who donated five hundred million Hong Kong dollars for the establishment of the college...
of The Chinese University of Hong Kong. - Fellow, American Mathematical SocietyAmerican Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
(elected in 1971)
Poincaré conjecture controversy
In August 2006, a New YorkerThe New Yorker
The New Yorker is an American magazine of reportage, commentary, criticism, essays, fiction, satire, cartoons and poetry published by Condé Nast...
article on the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, "Manifold Destiny
Manifold Destiny
"Manifold Destiny" is an article in The New Yorker written by Sylvia Nasar and David Gruber and published in the August 28, 2006 issue of the magazine...
", discussed Yau's relationship to that famous problem. The printed edition had a cartoon showing Yau as trying to steal Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
's Fields Medal. Perelman is quoted saying that he is disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to the efforts of Fields medalist Shing-Tung Yau to downplay Perelman's role in the proof and play up the work of Cao
Huai-Dong Cao
Huai-Dong Cao is A. Everett Pitcher Professor of Mathematics at Lehigh University. He collaborated with Xi-Ping Zhu of Zhongshan University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team is one of three teams formed for this purpose...
and Zhu
Xi-Ping Zhu
Zhu Xiping is a Professor of Mathematics at Sun Yat-sen University. He collaborated with Cao Huaidong of Lehigh University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team was one of three teams formed for this purpose...
.
Yau stated: “Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle.” Perelman stated of Yau, "I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest." Yau claimed that this article was defamatory, and in September 2006 he established a public relations website, managed by the public relations firm Spector & Associates, to dispute points in it and demand an apology. Some mathematicians, including two quoted in the New Yorker article, posted letters of support.
On October 17, 2006, a more sympathetic profile of Yau appeared in the New York Times
The New York Times
The New York Times is an American daily newspaper founded and continuously published in New York City since 1851. The New York Times has won 106 Pulitzer Prizes, the most of any news organization...
. It devoted about half its length to the Perelman affair. The article stated that Yau had alienated some colleagues, but represented Yau's position as that Perelman's proof was not generally understood and he "had a duty to dig out the truth of the proof."
Publications
- 2010. (with Steve Nadis) The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic BooksBasic BooksBasic Books is a book publisher founded in 1952 and located in New York. It publishes books in the fields of psychology, philosophy, economics, science, politics, sociology, current affairs, and history.-History:...
. ISBN 978-0465020232.
External links
- Discover Magazine Interview, June 2010 issue
- Interview (11 pages long in Traditional Chinese)
- Yau's autobiographical account (mostly English, some Chinese)
- 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