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 currentlya permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.

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."

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 as Professor of Pure Mathematics.

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

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, he was an invited speaker of the ICM in 1983 in Warsaw and in 1998 in Berlin, as well as a plenary speaker of the ICM in 2018 in Rio de Janeiro. 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.

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

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

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 varietyadmits a Hermitian–Einstein metric (Donaldson 1987c).
• 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 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".

• 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.
• ——— (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.
• ——— (1984b). "Instantons and geometric invariant theory". Comm. Math. Phys. 93 (4): 453–460. Bibcode:1984CMaPh..93..453D. doi:10.1007/BF01212289. MR 0892034.
• ——— (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.
• ——— (1987b). "Irrationality and the h-cobordism conjecture". J. Differential Geom. 26 (1): 141–168. doi:10.4310/jdg/1214441179. 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. doi:10.4310/jdg/1214425535. MR 1802722.
• ——— (2001). "Scalar curvature and projective embeddings. I". J. Differential Geom. 59 (3): 479–522. doi:10.4310/jdg/1090349449. 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.
• ——— & Kronheimer, Peter B. (1990). The geometry of four-manifolds. Oxford Mathematical Monographs. New York: Oxford University Press. ISBN 0-19-853553-8. MR 1079726.
• ———; 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.
• 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.
• 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.
• 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.
• 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.