Yau Shing-Tung

Yau Shing-Tung the Chinese-American mathematician (born 1949) who won the Fields Medal in 1982 for his proof of the Calabi conjecture in differential geometry and who has been one of the most influential mathematicians of his generation.

Early Life and Education

Yau Shing-Tung was born on 4 April 1949 in Shantou, Guangdong province, and grew up in Hong Kong, to which his family moved when he was an infant. The family was poor and came from the countryside, and his father, a teacher of philosophy and literature, encouraged the boy to think independently and to question received authority.

He showed early mathematical talent and studied at Pui Ching Middle School in Hong Kong before entering the Chinese University of Hong Kong, where he graduated in 1969 after only three years. He then went to the United States and entered the University of California, Berkeley, completing his doctorate in 1971 under the supervision of Shiing-Shen Chern.

His doctoral dissertation concerned problems in differential geometry, the field in which he would spend his career. The meeting with Chern was decisive, and Yau later spoke of the importance of the older mathematician as a teacher, a collaborator and a model of dedication to the subject.

The Calabi Conjecture

In the 1950s the Italian-American geometer Eugenio Calabi had conjectured that a certain class of complex manifolds, spaces that support a notion of distance satisfying particular curvature conditions, always admit a canonical metric with prescribed Ricci curvature. The problem had remained open for decades and was regarded as a central question of geometry.

Yau proved the Calabi conjecture in 1976 and published the detailed proof in 1978. The proof required the invention of powerful new techniques for solving nonlinear partial differential equations on complex manifolds, including difficult a priori estimates for the solutions of Monge-Ampere equations.

The result, often called the Calabi-Yau theorem, guaranteed the existence of the metrics whose study had been conjectured and gave mathematicians a large new supply of examples. These spaces came to be known as Calabi-Yau manifolds, and they turned out to have an importance far beyond the question that had prompted their study.

Calabi-Yau Manifolds and Physics

Calabi-Yau manifolds became a central object of theoretical physics when it was realised that they provide the kind of compact internal space required by string theory. In these theories the extra dimensions of space are curled up into a small six-dimensional shape, and the geometry of that shape determines the physical properties of the particles that are observed.

The study of these manifolds drew mathematicians and physicists into a long and fertile exchange, in which insights from one field were used to attack problems in the other. Yau was a leading figure in this dialogue and wrote extensively on the mutual influence of geometry and physics.

He also developed the theory of mirror symmetry, which relates pairs of Calabi-Yau manifolds whose geometry appears very different but whose physical predictions agree, a subtlety that revealed deep connections between algebraic geometry, symplectic geometry and string theory.

Other Mathematical Work

Yau made many other fundamental contributions. He proved the positive mass theorem in general relativity, which states that the total mass of an isolated gravitational system cannot be negative, and he worked on the theory of minimal surfaces, on the structure of manifolds of positive scalar curvature and on eigenvalue problems in geometry.

With Richard Schoen he proved the existence of solutions of the Yamabe problem and made basic contributions to the understanding of spaces of positive scalar curvature. He solved the Frankel conjecture on the classification of complex manifolds and developed the theory of harmonic maps and their applications to geometry.

He also proved important results in the geometry of manifolds of positive curvature and in the theory of locally symmetric spaces, and he made contributions to the study of the Ricci flow, the evolution equation later used by Perelman in the proof of the Poincare conjecture. His range was extraordinarily broad.

Influence and Legacy

Yau was awarded the Fields Medal in 1982, the highest honour of the mathematical community, in recognition of his work on the Calabi conjecture and the positive mass theorem. He has received the Wolf Prize, the Crafoord Prize, the Shaw Prize and the National Medal of Science, and he is a member of many academies.

He has trained many outstanding students at the Institute for Advanced Study, at Harvard University and at the Chinese University of Hong Kong, where he led research centres and founded prizes and conferences. He has been a tireless promoter of mathematics in China and of the integration of geometry with physics and with applied subjects.

He has also been a prominent figure in public debates about the direction of mathematics and about methods of mathematical education. Whatever the controversies, his scientific achievements are secure, and Calabi-Yau manifolds and the theorems he proved remain at the centre of geometry and of theoretical physics today.

See also

• Calabi-Yau manifold
• Fields Medal
• Shiing-Shen Chern
• Differential geometry