Donaldson's father was an electrical engineer in the physiology department at the
University of Cambridge, and his mother earned a science degree there.[2] 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."[3]
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.
In 2009, he was awarded the
Shaw Prize in Mathematics (jointly with
Clifford Taubes) for their contributions to geometry in 3 and 4 dimensions.[10]
In 2014, he was awarded the
Breakthrough 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" by
Université Joseph Fourier, Grenoble. In January 2017, he was awarded the degree "Doctor Honoris Causa" by the Universidad Complutense de Madrid, Spain.[17]
The diagonalizability theorem (Donaldson
1983a,
1983b,
1987a): 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 (
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 the
Donaldson invariant (or
instanton 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 constant
scalar curvature (see for example
cscK metric). Donaldson obtained results in the toric case of the problem (see for example
Donaldson (2001)). 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
Donaldson & Sun (2014) on Gromov–Hausdorff limits. The summary of the existence proof for Kähler–Einstein metrics appears in
Chen, 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.
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