Simon Donaldson

Simon Kirwan Donaldson
Born	20 August 1957 Cambridge, England
Nationality	British
Fields	Mathematics
Institutions	Stony Brook University Imperial College London Institute for Advanced Study Stanford University University of Oxford
Alma mater	University of Oxford University of Cambridge
Thesis	The Yang-Mills Equations on Kähler Manifolds (1983)
Doctoral advisor	Michael Atiyah Nigel Hitchin
Doctoral students	Oscar Garcia-Prada Dominic Joyce Dieter Kotschick Graham Nelson Paul Seidel Vicente Muñoz Richard Thomas
Known for	Topology of smooth (differentiable) four-dimensional manifolds
Notable awards	Junior Whitehead Prize (1985) Fields Medal (1986) Royal Medal (1992) Crafoord Prize (1994) Pólya Prize (1999) King Faisal International Prize (2006) Nemmers Prize in Mathematics (2008) Shaw Prize in Mathematics (2009) Breakthrough Prize in Mathematics (2014)

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^[1] and a Professor in Pure Mathematics at Imperial College London.

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.^[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" (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,^[3] 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, USA.^[1]

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.^[4]

Donaldson was knighted in the 2012 New Year Honours for services to mathematics.^[5]

In 2012 he became a fellow of the American Mathematical Society.^[6]

In March 2014, he was awarded an honorary 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."^[7]

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:

• 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 stable holomorphic vector bundle over a non-singular projective algebraic variety admits a Hermitian–Einstein metric (Donaldson 1987c). (Another proof of a somewhat more general result was given by Uhlenbeck & Yau (1986).)
• 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).
• Any compact symplectic manifold admits a symplectic Lefschetz pencil (Donaldson 1999).

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 last 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

• Donaldson, Simon K. (1983a). "An application of gauge theory to four-dimensional topology". J. Differential Geom. 18 (2): 279–315. MR 710056.
• ——— (1983b). "Self-dual connections and the topology of smooth 4-manifolds". Bull. Amer. Math. Soc. 8 (1): 81–83. doi:10.1090/S0273-0979-1983-15090-5. MR 0682827.
• ——— (1987a). "The orientation of Yang-Mills moduli spaces and 4-manifold topology". J. Differential Geom. 26 (3): 397–428. MR 910015.
• ——— (1987b). "Irrationality and the h-cobordism conjecture". J. Differential Geom. 26 (1): 141–168. MR 0892034.
• ——— (1987c). "Infinite determinants, stable bundles and curvature". Duke Math. J. 54 (1): 231–247. doi:10.1215/S0012-7094-87-05414-7. MR 0885784.
• ——— (1990). "Polynomial invariants for smooth four-manifolds". Topology 29 (3): 257–315. doi:10.1016/0040-9383(90)90001-Z. MR 1066174.
• ——— (1999). "Lefschetz pencils on symplectic manifolds". J. Differential Geom. 53 (2): 205–236. MR 1802722.
• ——— (2001). "Scalar curvature and projective embeddings. I". J. Differential Geom. 59 (3): 479–522. MR 1916953.
• ——— (2011). Riemann surfaces. Oxford Graduate Texts in Mathematics 22. Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780198526391.001.0001. ISBN 978-0-19-960674-0. MR 2856237.^[8]
• ——— & Kronheimer, Peter B. (1990). The geometry of four-manifolds. Oxford Mathematical Monographs. New York: Oxford University Press. ISBN 0-19-853553-8. MR 1079726.^[9]
• ———; Sun, Song (2014). "Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry". Acta Math. 213 (1): 63–106. doi:10.1007/s11511-014-0116-3. MR 3261011.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2014). "Kähler-Einstein metrics and stability". Int. Math. Res. Notices 8: 2119–2125. doi:10.1093/imrn/rns279. MR 3194014.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2015a). "Kähler-Einstein metrics on Fano manifolds I: Approximation of metrics with cone singularities". J. Amer. Math. Soc. 28 (1): 183–197. doi:10.1090/S0894-0347-2014-00799-2. MR 3264766.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2015b). "Kähler-Einstein metrics on Fano manifolds II: Limits with cone angle less than 2π". J. Amer. Math. Soc. 28 (1): 199–234. doi:10.1090/S0894-0347-2014-00800-6. MR 3264767.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2015c). "Kähler-Einstein metrics on Fano manifolds III: Limits as cone angle approaches 2π and completion of the main proof". J. Amer. Math. Soc. 28 (1): 235–278. doi:10.1090/S0894-0347-2014-00801-8. MR 3264768.

Notes

1. "Simon Donaldson, Simons Center for Geometry and Physics".
2. Simon Donaldson Autobiography, The Shaw Prize, 2009
3. Biography at DeBretts
4. New foreign members elected to the academy, press announcement from the Royal Swedish Academy of Sciences 2010-05-26
5. The London Gazette: (Supplement) no. 60009. p. 1. 31 December 2011.
6. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
7. , retrieved 2014-06-26.
8. Kra, Irwin (2012). "Review: Riemann surfaces, by S. K. Donaldson". Bull. Amer. Math. Soc. (N.S.) 49 (3): 455–463. doi:10.1090/s0273-0979-2012-01375-7.
9. Hitchin, Nigel (1993). "Review: The geometry of four-manifolds, by S. K. Donaldson and P. B. Kronheimer". Bull. Amer. Math. Soc. (N.S.) 28 (2): 415–418. doi:10.1090/s0273-0979-1993-00377-x.

References

• Atiyah, M. (1986). "On the work of Simon Donaldson". Proceedings of the International Congress of Mathematicians.
• Uhlenbeck, Karen & Yau, Shing-Tung (1986). "On the existence of Hermitian-Yang-Mills connections in stable vector bundles". Comm. Pure Appl. Math. 39 (S, suppl.): S257–S293. doi:10.1002/cpa.3160390714. MR 0861491.

External links

• O'Connor, John J.; Robertson, Edmund F., "Simon Donaldson", MacTutor History of Mathematics archive, University of St Andrews.
• Simon Donaldson at the Mathematics Genealogy Project
• Home page at Imperial College