Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge, and his mother earned a science degree there. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford, at first under Nigel Hitchin and later under Michael Atiyah's supervision. Still a postgraduate student, Donaldson proved in 1982 a result that would establish his fame. He published the result in a paper "Self-dual connections and the topology of smooth 4-manifolds" which appeared in 1983. In the words of Atiyah, the paper "stunned the mathematical world." Whereas Michael Freedman classified topological four-manifolds, Donaldson's work focused on four-manifolds admitting a differentiable structure, using instantons, a particular solution to the equations of Yang–Millsgauge theory which has its origin in quantum field theory. One of Donaldson's first results gave severe restrictions on the intersection form of a smooth four-manifold. As a consequence, a large class of the topological four-manifolds do not admit any smooth structure at all. Donaldson also derived polynomial invariants from gauge theory. These were new topological invariants sensitive to the underlying smooth structure of the four-manifold. They made it possible to deduce the existence of "exotic" smooth structures—certain topological four-manifolds could carry an infinite family of different smooth structures. After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford, he spent the academic year 1983–84 at the Institute for Advanced Study in Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998 as Professor of Pure Mathematics. In 2014, he joined the Simons Center for Geometry and Physics at Stony Brook University in New York, United States.
The diagonalizability theorem : If the intersection form of a smooth, closed, simply connected 4-manifold is positive- or negative-definite then it is diagonalizable over the integers. This result is sometimes called Donaldson's theorem.
A smooth h-cobordism between simply connected 4-manifolds need not be trivial. This contrasts with the situation in higher dimensions.
A non-singular, projective algebraic surface can be diffeomorphic to the connected sum of two oriented 4-manifolds only if one of them has negative-definite intersection form. This was an early application of the Donaldson invariant.
Donaldson's recent work centers on a problem in complex differential geometry concerning a conjectural relationship between algebro-geometric "stability" conditions for smooth projective varieties and the existence of "extremal" Kähler metrics, typically those with constant scalar curvature. Donaldson obtained results in the toric case of the problem. He then solved the Kähler–Einstein case of the problem in 2012, in collaboration with Chen and Sun. This latest spectacular achievement involved a number of difficult and technical papers. The first of these was the paper of on Gromov-Hausdorff limits. The summary of the existence proof for Kähler–Einstein metrics appears in. Full details of the proofs appear in.
In 2019, Donaldson was awarded the Oswald Veblen Prize in Geometry, together with Xiuxiong Chen and Song Sun, for proving a long-standing conjecture on Fano manifolds, which states "that a Fano manifold admits a Kähler–Einstein metricif and only if it is K-stable". It had been one of the most actively investigated topics in geometry since its proposal in the 1980s by Shing-Tung Yau after he proved the Calabi conjecture. It was later generalized by Gang Tian and Donaldson. The solution by Chen, Donaldson and Sun was published in the Journal of the American Mathematical Society in 2015 as a three-article series, "Kähler–Einstein metrics on Fano manifolds, I, II and III".
