Simon Donaldson

Infobox Scientist
name = Simon Kirwan Donaldson
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birth_date = birth date and age|1957|08|20
birth_place = Cambridge, England
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nationality = flagicon|UK British
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field = Mathematics
work_institutions = Imperial College London
Institute for Advanced Study
University of Oxford
alma_mater = University of Oxford
University of Cambridge
doctoral_advisor = Michael Atiyah
Nigel Hitchin
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known_for = Topology of smooth (differentiable) four-dimensional manifolds
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prizes = Junior Whitehead Prize (1985)
Fields Medal (1986)
Crafoord Prize (1994)
King Faisal International Prize (2006)
Nemmers Prize in Mathematics (2008)
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Simon Kirwan Donaldson (born August 20, 1957, in Cambridge, England), is an English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds.

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 graduate student, Donaldson soon 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-Mills gauge 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 at Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. In 1999, he moved to Imperial College London.

Donaldson received the Junior Whitehead Prize from the London Mathematical Society in 1985 and in the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal. He was, however, turned down for fellowship of the Institute of Mathematics and its Applications on the grounds that he applied too soon after his doctorate. He was awarded the 1994 Crafoord Prize.

In February 2006 Dr Donaldson was awarded the King Faisal International Prize for science for his work in pure mathematical theories linked to physics, which have helped in forming an understanding of the laws of matter at a subnuclear level.

In April 2008, he was awarded the Nemmers Prize in Mathematics.

Donaldson's work

A thread running through Donaldson's work is the creative application of mathematical analysis (especially the analysis of elliptic partial differential equations) to problems in geometry. The problems mainly concern 4-manifolds, complex differential geometry and symplectic geometry. The following theorems rank among his most striking achievements:

* The diagonalizability theorem (Donaldson 1983a, 1983b): if the intersection form of a smooth, closed, simply connected 4-manifold is positive- or negative-definite then it is diagonalizable over the integers. (The simple connectivity hypothesis has since been shown to be unnecessary using Seiberg-Witten theory.) This result is sometimes called Donaldson's theorem.

* A smooth h-cobordism between 4-manifolds need not be trivial (Donaldson 1987a). This contrasts with the situation in higher dimensions.

* A stable holomorphic vector bundle over a non-singular projective algebraic variety admits a Hermitian-Einstein metric (Donaldson 1987b). This was proved independently by Karen Uhlenbeck and Shing-Tung Yau (Uhlenbeck & Yau 1986).

* A non-singular, projective algebraic surface can only be diffeomorphic to the connected sum of two oriented 4-manifolds if one of them has negative-definite intersection form (Donaldson 1990). This was an early application of the Donaldson invariants (or instanton invariants).

* Any compact symplectic manifold admits a symplectic Lefschetz pencil (Donaldson 1999).

Donaldson's recent work centers on a difficult problem in complex differential geometry concerning a conjectural relationship between algebro-geometric "stability" conditions for smooth projective varieties and the existence of "optimal" Kähler metrics, typically those with constant scalar curvature. Definitive results have not yet been obtained, but substantial progress has been made (see for example Donaldson 2001).

References

* Donaldson, S. K. An application of gauge theory to four-dimensional topology. "J. Differential Geom.", 18, (1983), 279–315.
* Donaldson, S. K. Self-dual connections and the topology of smooth 4-manifolds. "Bull. Amer. Math. Soc.". 8, (1983), 81–83.
* Donaldson, S. K. Irrationality and the h-cobordism conjecture. "J. Differential Geom." 26 (1987), no. 1, 141–168.
* Donaldson, S. K. Infinite determinants, stable bundles and curvature. "Duke Math. J." 54 (1987), no. 1, 231–247.
* Donaldson, S. K. Polynomial invariants for smooth four-manifolds. "Topology" 29 (1990), no. 3, 257–315.
* Donaldson, S. K. Lefschetz pencils on symplectic manifolds. "J. Differential Geom." 53 (1999), no. 2, 205–236.
* Donaldson, S. K. Scalar curvature and projective embeddings. I. "J. Differential Geom." 59 (2001), no. 3, 479–522.
* Donaldson, S. K. and Kronheimer, P. B. "The geometry of four-manifolds". Oxford Mathematical Monographs, Oxford University Press, New York, (1990) ISBN 0-19-853553-8.
* Uhlenbeck, K.; Yau, S.-T. On the existence of Hermitian-Yang-Mills connections in stable vector bundles. "Comm. Pure Appl. Math." 39 (1986), no. S, suppl., S257–S293.

External links

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* [http://www.ma.ic.ac.uk/~skdona/ Home page at Imperial College]

Persondata
NAME= Donaldson, Simon Kirwan
ALTERNATIVE NAMES=
SHORT DESCRIPTION= English mathematician and Fields medalist
DATE OF BIRTH= August 20, 1957
PLACE OF BIRTH= Cambridge, England
DATE OF DEATH=
PLACE OF DEATH=