Donaldson's father was an electrical engineer in the physiology department at theUniversity of Cambridge, and his mother earned a science degree there.[2] Donaldson gained aBA degree in mathematics fromPembroke College, Cambridge, in 1979, and in 1980 began postgraduate work atWorcester College, Oxford, at first underNigel Hitchin and later underMichael 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."[3]
WhereasMichael Freedman classified topological four-manifolds, Donaldson's work focused on four-manifolds admitting adifferentiable structure, usinginstantons, a particular solution to the equations ofYang–Millsgauge theory which has its origin inquantum field theory. One of Donaldson's first results gave severe restrictions on theintersection form of a smooth four-manifold. As a consequence, a large class of the topological four-manifolds do not admit anysmooth structure at all. Donaldson also derived polynomial invariants fromgauge 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.
In 2009, he was awarded theShaw Prize in Mathematics (jointly withClifford Taubes) for their contributions to geometry in 3 and 4 dimensions.[10]
In 2014, he was awarded theBreakthrough Prize in Mathematics "for the new revolutionary invariants of 4-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties."[11]
In March 2014, he was awarded the degree "Docteur Honoris Causa" byUniversité Joseph Fourier, Grenoble. In January 2017, he was awarded the degree "Doctor Honoris Causa" by the Universidad Complutense de Madrid, Spain.[18]
The diagonalizability theorem (Donaldson 1983a,1983b,1987a): If theintersection form of a smooth, closed, simply connected4-manifold is positive- or negative-definite then it is diagonalizable over the integers. This result is sometimes calledDonaldson's theorem.
A smoothh-cobordism between simply connected 4-manifolds need not be trivial (Donaldson 1987b). 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 (Donaldson 1990). This was an early application of theDonaldson invariant (orinstanton invariants).
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 constantscalar curvature (see for examplecscK metric). Donaldson obtained results in the toric case of the problem (see for exampleDonaldson (2001)). He then solved theKähler–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 ofDonaldson & Sun (2014) on Gromov–Hausdorff limits. The summary of the existence proof for Kähler–Einstein metrics appears inChen, Donaldson & Sun (2014). Full details of the proofs appear in Chen, Donaldson, and Sun (2015a,2015b,2015c).
Donaldson, S.K. (2002).Floer homology groups in Yang-Mills theory. Cambridge Tracts in Mathematics. Vol. 147. Cambridge: Cambridge University Press.ISBN0-521-80803-0.
^Donaldson, Simon K (1986). "The geometry of 4-manifolds". In AM Gleason (ed.).Proceedings of the International Congress of Mathematicians (Berkeley 1986). Vol. 1. pp. 43–54.CiteSeerX10.1.1.641.1867.
