Simon Donaldson

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 in New York,[1] and a Professor in Pure Mathematics at Imperial College London.

Sir

Simon Donaldson

FRS
Born
Simon Kirwan Donaldson

(1957-08-20) 20 August 1957
Cambridge, England
NationalityBritish
Alma materWorcester College, Oxford
Pembroke College, Cambridge
Known forTopology of smooth (differentiable) four-dimensional manifolds
Donaldson theory
Donaldson theorem
AwardsJunior 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)
Oswald Veblen Prize (2019)
Wolf Prize in Mathematics (2020)
Scientific career
FieldsMathematics
InstitutionsImperial College London
Stony Brook University
Institute for Advanced Study
Stanford University
All Souls College, Oxford
ThesisThe Yang–Mills Equations on Kähler Manifolds (1983)
Doctoral advisorMichael Atiyah
Nigel Hitchin
Doctoral studentsDominic Joyce
Dieter Kotschick
Graham Nelson
Paul Seidel
Ivan Smith
Gábor Székelyhidi
Richard Thomas

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."[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–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,[4] he moved to Imperial College London in 1998 as Professor of Pure Mathematics.[5]

In 2014, he joined the Simons Center for Geometry and Physics at Stony Brook University in New York, United States.[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 at the International Congress of Mathematicians (ICM) in Berkeley. In addition to being a plenary speaker of the ICM in 1986,[6] he was an invited speaker of the ICM in 1983 in Warsaw and in 1998 in Berlin,[7] as well as a plenary speaker of the ICM in 2018 in Rio de Janeiro.[8] 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.[9]

Donaldson was knighted in the 2012 New Year Honours for services to mathematics.[10]

In 2012 he became a fellow of the American Mathematical Society.[11]

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."[12]

In January 2017, he was awarded the degree "Doctor Honoris Causa" by the Universidad Complutense de Madrid, Spain.

In January 2019, he was awarded the Oswald Veblen Prize in Geometry (jointly with Xiuxiong Chen and Song Sun).[13]

In 2020 he received the Wolf Prize in Mathematics (jointly with Yakov Eliashberg).[14]

Contributions
Further information: Donaldson theory

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 gauge theory, 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).

Conjecture on Fano manifolds and Veblen Prize

In 2019, Donaldson was awarded the Oswald Veblen Prize in Geometry, together with Xiuxiong Chen and Song Sun, for proving a long-standing conjecture on Fano manifolds, which states "that a Fano manifold admits a Kähler–Einstein metric if and only if it is K-stable". It had been one of the most actively investigated topics in geometry since its proposal in the 1980s by Shing-Tung Yau after he proved the Calabi conjecture. It was later generalized by Gang Tian and Donaldson. The solution by Chen, Donaldson and Sun was published in the Journal of the American Mathematical Society in 2015 as a three-article series, "Kähler–Einstein metrics on Fano manifolds, I, II and III".[13]

Selected publications
  • Donaldson, Simon K. (1983a). "An application of gauge theory to four-dimensional topology". J. Differential Geom. 18 (2): 279–315. doi:10.4310/jdg/1214437665. MR 0710056.CS1 maint: ref=harv (link)
  • ——— (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.CS1 maint: ref=harv (link)
  • ——— (1984b). "Instantons and geometric invariant theory". Comm. Math. Phys. 93 (4): 453–460. Bibcode:1984CMaPh..93..453D. doi:10.1007/BF01212289. MR 0892034.CS1 maint: ref=harv (link)
  • ——— (1987a). "The orientation of Yang-Mills moduli spaces and 4-manifold topology". J. Differential Geom. 26 (3): 397–428. doi:10.4310/jdg/1214441485. MR 0910015.CS1 maint: ref=harv (link)
  • ——— (1987b). "Irrationality and the h-cobordism conjecture". J. Differential Geom. 26 (1): 141–168. doi:10.4310/jdg/1214441179. MR 0892034.CS1 maint: ref=harv (link)
  • ——— (1987c). "Infinite determinants, stable bundles and curvature". Duke Math. J. 54 (1): 231–247. doi:10.1215/S0012-7094-87-05414-7. MR 0885784.CS1 maint: ref=harv (link)
  • ——— (1990). "Polynomial invariants for smooth four-manifolds". Topology. 29 (3): 257–315. doi:10.1016/0040-9383(90)90001-Z. MR 1066174.CS1 maint: ref=harv (link)
  • ——— (1999). "Lefschetz pencils on symplectic manifolds". J. Differential Geom. 53 (2): 205–236. doi:10.4310/jdg/1214425535. MR 1802722.CS1 maint: ref=harv (link)
  • ——— (2001). "Scalar curvature and projective embeddings. I". J. Differential Geom. 59 (3): 479–522. doi:10.4310/jdg/1090349449. MR 1916953.CS1 maint: ref=harv (link)
  • ———; Sun, Song (2014). "Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry". Acta Math. 213 (1): 63–106. arXiv:1206.2609. doi:10.1007/s11511-014-0116-3. MR 3261011.CS1 maint: ref=harv (link)
  • Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2014). "Kähler-Einstein metrics and stability". Int. Math. Res. Notices. 2014 (8): 2119–2125. arXiv:1210.7494. doi:10.1093/imrn/rns279. MR 3194014.CS1 maint: ref=harv (link)
  • 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. arXiv:1211.4566. doi:10.1090/S0894-0347-2014-00799-2. MR 3264766.CS1 maint: ref=harv (link)
  • 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. arXiv:1212.4714. doi:10.1090/S0894-0347-2014-00800-6. MR 3264767.CS1 maint: ref=harv (link)
  • 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. arXiv:1302.0282. doi:10.1090/S0894-0347-2014-00801-8. MR 3264768.CS1 maint: ref=harv (link)

Books

gollark: Unless it's on some stupidly long cycle.
gollark: I guess we'll be able to see if it's day-based tomorrow?
gollark: What was debacleish about them?
gollark: Where's the moon phases page?
gollark: Simple.

References
  1. "Simon Donaldson, Simons Center for Geometry and Physics".
  2. Simon Donaldson Autobiography, The Shaw Prize, 2009
  3. Atiyah, M. (1986). "On the work of Simon Donaldson". Proceedings of the International Congress of Mathematicians.
  4. Biography at DeBretts Archived 20 June 2013 at the Wayback Machine
  5. "Donaldson, Sir Simon (Kirwan)", Who's Who (online ed., Oxford University Press, December 2018). Retrieved 2 June 2019.
  6. 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. CiteSeerX 10.1.1.641.1867.
  7. Donaldson, S. K. (1998). "Lefschetz vibrations in symplectic geometry". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 309–314.
  8. "ICM Plenary and Invited Speakers, International Mathematical Union (IMU)". www.mathunion.org.
  9. New foreign members elected to the academy, press announcement from the Royal Swedish Academy of Sciences 2010-05-26
  10. "No. 60009". The London Gazette (Supplement). 31 December 2011. p. 1.
  11. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
  12. , retrieved 2014-06-26.
  13. "2019 Oswald Veblen Prize in Geometry to Xiuxiong Chen, Simon Donaldson, and Song Sun". American Mathematical Society. 19 November 2018. Retrieved 9 April 2019.
  14. Wolf Prize 2020
  15. Another proof of a somewhat more general result was given by 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.CS1 maint: ref=harv (link)
  16. 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.
  17. 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.
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