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Yau was appointed assistant professor at the State University of New York at Stony Brook in 1972. In 1974 he was appointed an associate professor at Stanford University. He was promoted to full professor at Stanford before returning to the Institute for Advanced Study at Princeton in 1979. In 1980 he was made a professor at the Institute for Advanced Study at Princeton, a position he held until 1984 when he moved to a chair at the University of California at San Diego. In 1988 he was appointed professor at Harvard University.

Yau was awarded a Fields Medal in 1982 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. In fact the 1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year.

Nirenberg described Yau's work at the International Congress in Warsaw in 1983. Writing in after the Fields Medal awards were announced in 1982, Nirenberg wrote:

S-T Yau has done extremely deep and powerful work in differential geometry and partial differential equations. He is an analyst's geometer (or geometer's analyst) with enormous technical power and insight. He has cracked problems on which progess has been stopped for years.

Nirenberg describes briefly the areas of Yau's work. On the Calabi conjecture, which was made in 1954, he writes that this:

... comes from algebraic geometry and involves proving the existence of a Kähler metric, on a compact Kähler manifold , having a prescribed volume form. The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge - Ampère ) differential equation . Yau's solution is classical in spirit, via a priori estimates. His derivation of the estimates is a tour de force and the applications in algebraic geometry are beautiful.

Yau solved the Calabi conjecture in 1976. Another conjecture solved by Yau was the positive mass conjecture, which comes from Riemannian geometry. Yau, in joint work, constructed minimal surfaces, studied their stability and made a deep analysis of how they behave in space-time. His work here has applications to the formation of black holes.

The Plateau problem was studied by Plateau , Weierstrass , Riemann and Schwarz but it was finally solved by Douglas and Radó . However, there were still questions relating to whether Douglas 's solution, which was known to be a smooth immersed surface, is actually embedded. Yau, working with W H Meeks solved this problem in 1980.

In 1981 Yau was awarded The Oswald Veblen Prize in Geometry:

...for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge - Ampère equation on compact complex manifolds.

In joint work of Yau with Karen Uhlenbeck On the existence of Hermitian Yang-Mills connections in stable bundles (1986), they solved higher dimensional versions of the Hitchin-Kobayashi conjecture. Their work extended that of Donaldson on this topic in 1985.

The Crafoord Prize of the Royal Swedish Academy of Sciences was awarded to Yau in 1994:

... for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.

G Tian sums up Yau's work to date which led to his being awarded the Crafoord Prize:

As a result of Yau's work over the past twenty years, the role and understanding of basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics. His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory , and general relativity as well as differential geometry and partial differential equations.

Yau was elected to the National Academy of Sciences in 1993.

Source:School of Mathematics and Statistics University of St Andrews, Scotland