Shing-Tung Yau has made fundamental contributions to differential geometry which have influenced a wide range of scientific disciplines, including astronomy and theoretical physics. With Richard Schoen, Yau solved a longstanding question in Albert Einstein’s theory of relativity by proving that the sum of the energy in the universe is positive; their proof has provided an important tool for understanding how black holes form. In 1982 Yau was awarded the Fields Medal, the highest award in mathematics, and in 1994 he shared with Simon Donaldson of Oxford University the Crafoord Prize of the Royal Swedish Society, in recognition of his “development of nonlinear techniques in differential geometry leading to the solution of several outstanding problems.”
Yau was born in 1949 in Swatow, in southern China, the fifth of the eight children of Chen Ying Chiou and Yeuk-Lam Leung Chiou. Within the year, Communists had overthrown the government and the family fled to Hong Kong, where his father, a respected economist and philosopher, obtained a position at a college which would later be part of the Chinese University of Hong Kong. His mother knitted and created other goods by hand to help support the family, for professors were poorly paid. He credits his father, who died when Yau was fourteen, with encouraging him to study mathematics, and he has retained a passion for it: “It’s clean, clear-cut, beautiful, and has a lot of applications,” he told the Harvard Gazette. Yau entered Chung Chi College in Hong Kong, earning his undergraduate degree in 1968. One of his professors had attended the University of California at Berkeley and suggested that Yau study there. A fellowship from the International Business Machines Corp. (now IBM) made it possible; Yau studied with Shiing-Shen Chern, the legendary geometer (Yau would later edit a collection of papers honoring his teacher). Yau completed his doctorate in mathematics in 1971 at the age of twenty-two.
In the late 1970’s, Yau proved the Calabi conjecture in differential geometry. At first, Yau’s technically astounding proof provided pure mathematicians with a gigantic catalogue of new solutions to Einstein’s equations for the gravitational field in an even dimensional manifold. Then, in 1985, theoretical physicists realized that Yau's solutions were building blocks for the theory of superstrings, as explained in the book The Elegant Universe, by Brian Greene. Almost overnight, Yau’s work became the central focus in a huge and unprecedented collaboration between mathematicians and physicists.
Differential geometry, which is Yau’s field, was developed during the 1800’s, and it uses derivatives and integrals to describe geometric objects such as surfaces and curves. Differential geometry is particularly concerned with geometrical calculations across many dimensions. The simplest kind of geometry would be one- and two-dimensional, analyzing figures such as squares or circles; the geometry of a three-dimensional figure, such as a cube or a cylinder, is more complicated. Differential geometry is primarily concerned with calculations about geometrical figures in four or more dimensions. An example of a four- dimensional figure would be a three-dimensional one that changes over time—the stretching and snapping of a rubber band, for instance, or a drop of water splashing on a surface. One of the most important applications of differential geometry is Einstein’s theory of relativity: Einstein used differential geometry in his original calculations, and it was central to his theory of gravity. The general theory of relativity includes a conjecture—that is, an unproven postulate—which proposes that in an isolated physical system the total energy, including gravity and matter, would be positive. Called the positive mass conjecture, this was fundamental to the theory of relativity, but no one had been able to prove it.
Yau’s first major contribution to differential geometry was his proof of another conjecture, called the Calabi conjecture, which concerns how volume and distance can be measured not in four, but in five or more dimensions. In 1979 Yau and Richard Schoen proved Einstein’s positive mass conjecture by applying methods devised by Yau. The proof was based on their work with minimal surfaces. A minimal surface is one in which a small deformation creates a surface with a larger area—soap films are often used as an example of minimal surfaces. The mathematical equations that must be used to describe minimal surfaces differ from those used for most problems in differential geometry. The latter use differential equations to describe curves and surfaces, while mathematicians working with minimal surfaces use partial, nonlinear differential equations, which are far more difficult to work with. Shoen and Yau's proof analyzed how such surfaces behave in space and time and showed that Einstein had correctly defined mass. Their methods allowed for the development of a new theory of minimal surfaces in higher dimensions, and they have had an impact on topology, algebraic geometry, and general relativity.