Nationality British Name Simon Donaldson | Role Mathematician Fields Mathematics | |

Institutions Stony Brook UniversityImperial College LondonInstitute for Advanced StudyStanford UniversityUniversity of Oxford Alma mater University of OxfordUniversity of Cambridge Thesis The Yang-Mills Equations on Kahler Manifolds (1983) Doctoral students Oscar Garcia-PradaDominic JoyceDieter KotschickGraham NelsonPaul SeidelVicente MunozRichard Thomas Books The Geometry of Four-manifolds Education Pembroke College, Cambridge, Worcester College, Oxford, University of Oxford, University of Cambridge Similar People |

## Simon donaldson 2015 breakthrough prize in mathematics symposium

**Sir Simon Kirwan Donaldson** FRS (born 20 August 1957), is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.

## Contents

- Simon donaldson 2015 breakthrough prize in mathematics symposium
- Some recent developments in Khler geometry and exceptional holonomy Simon Donaldson ICM2018
- Biography
- Awards and honours
- Donaldsons work
- Selected publications
- References

## Some recent developments in Kähler geometry and exceptional holonomy – Simon Donaldson – ICM2018

## Biography

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" (Atiyah 1986).

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 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.

In 2014, he joined the Simons Center for Geometry and Physics at Stony Brook University in New York, United States.

## Awards and honours

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 awarded the 1994 Crafoord Prize.

In February 2006, 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, a mathematics prize awarded by Northwestern University.

In 2009 he was awarded the Shaw Prize in Mathematics (jointly with Clifford Taubes) for their contributions to geometry in 3 and 4 dimensions.

In 2010, he was elected a foreign member of the Royal Swedish Academy of Sciences.

Donaldson was knighted in the 2012 New Year Honours for services to mathematics.

In 2012 he became a fellow of the American Mathematical Society.

In March 2014, he was awarded the degree "Docteur Honoris Causa" by Université Joseph Fourier, Grenoble.

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."

In January 2017, he was awarded the degree "DOCTOR HONORIS CAUSA" by the Universidad Complutense de Madrid, Spain.

## Donaldson's work

Donaldson's work is on the 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 have been mentioned:

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).

See also Donaldson theory.

## Selected publications

*J. Differential Geom.*

**18**(2): 279–315. MR 710056.

*Bull. Amer. Math. Soc.*

**8**(1): 81–83. MR 0682827. doi:10.1090/S0273-0979-1983-15090-5.

*J. Differential Geom.*

**26**(3): 397–428. MR 910015.

*J. Differential Geom*.

**26**(1): 141–168. MR 0892034.

*Duke Math. J.*

**54**(1): 231–247. MR 0885784. doi:10.1215/S0012-7094-87-05414-7.

*Topology*.

**29**(3): 257–315. MR 1066174. doi:10.1016/0040-9383(90)90001-Z.

*J. Differential Geom.*

**53**(2): 205–236. MR 1802722.

*J. Differential Geom.*

**59**(3): 479–522. MR 1916953.

*Riemann surfaces*. Oxford Graduate Texts in Mathematics.

**22**. Oxford: Oxford University Press. ISBN 978-0-19-960674-0. MR 2856237. doi:10.1093/acprof:oso/9780198526391.001.0001.

*The geometry of four-manifolds*. Oxford Mathematical Monographs. New York: Oxford University Press. ISBN 0-19-853553-8. MR 1079726.

*Acta Math.*

**213**(1): 63–106. MR 3261011. doi:10.1007/s11511-014-0116-3.

*Int. Math. Res. Notices*.

**8**: 2119–2125. MR 3194014. doi:10.1093/imrn/rns279.

*J. Amer. Math. Soc.*

**28**(1): 183–197. MR 3264766. doi:10.1090/S0894-0347-2014-00799-2.

*J. Amer. Math. Soc.*

**28**(1): 199–234. MR 3264767. doi:10.1090/S0894-0347-2014-00800-6.

*J. Amer. Math. Soc.*

**28**(1): 235–278. MR 3264768. doi:10.1090/S0894-0347-2014-00801-8.