Simon Donaldson

Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge, and his mother earned a science degree there.[http://www.shawprize.org/en/shaw.php?tmp=3&twoid=12&threeid=41&fourid=22&fiveid=11 Simon Donaldson Autobiography, The Shaw Prize, 2009] 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."{{cite journal |last=Atiyah |first=M. |author-link = Michael Atiyah |name-list-style=amp |title = On the work of Simon Donaldson |journal =Proceedings of the International Congress of Mathematicians |year=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,[http://www.debretts.com/people/biographies/browse/d/1854/Simon+Kirwan.aspx Biography at DeBretts] {{webarchive|url=https://web.archive.org/web/20130620191352/http://www.debretts.com/people/biographies/browse/d/1854/Simon+Kirwan.aspx |date=20 June 2013 }} he moved to Imperial College London in 1998 as Professor of Pure Mathematics.[https://www.ukwhoswho.com/view/10.1093/ww/9780199540884.001.0001/ww-9780199540884-e-13910 "Donaldson, Sir Simon (Kirwan)"], Who's Who (online ed., Oxford University Press, December 2018). Retrieved 2 June 2019.

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

Donaldson was an invited speaker of the International Congress of Mathematicians (ICM) in 1983,{{cite web | title=ICM Plenary and Invited Speakers | website=International Mathematical Union (IMU) | url=https://www.mathunion.org/icm-plenary-and-invited-speakers?combine=Donaldson | access-date=2022-09-03}} and a plenary speaker at the ICM in 1986,{{cite book|author=Donaldson, Simon K|chapter=The geometry of 4-manifolds|title=Proceedings of the International Congress of Mathematicians (Berkeley 1986)|editor=AM Gleason|volume=1|pages=43–54|year=1986|citeseerx=10.1.1.641.1867}} 1998,{{cite book|author=Donaldson, S. K.|chapter=Lefschetz fibrations in symplectic geometry|title=Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II|year=1998|pages=309–314|chapter-url=https://www.elibm.org/ft/10011708000}} and 2018.{{cite web|url=https://www.mathunion.org/icm-plenary-and-invited-speakers?combine=Donaldson|title=ICM Plenary and Invited Speakers, International Mathematical Union (IMU)|website=mathunion.org}}

In 1985, Donaldson received the Junior Whitehead Prize from the London Mathematical Society. In 1994, he was awarded the Crafoord Prize in Mathematics. 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.{{cite web |title=The Shaw Prize|website=shawprize.org |date=16 June 2009 |url=http://www.shawprize.org/laureates/mathematical-sciences/2009 }}

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."{{cite web |url=https://breakthroughprize.org/News/18 |title= Five Winners Receive Inaugural Breakthrough Prize in Mathematics|website=breakthroughprize.org |date=23 June 2014 |access-date=21 May 2022}}

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).[https://wolffund.org.il/the-wolf-prize/#Laureates Wolf Prize 2020], wolffund.org.il. Accessed 8 January 2023.

In 1986, he was elected a Fellow of the Royal Society and received a Fields Medal at the International Congress of Mathematicians (ICM) in Berkeley. In 2010, Donaldson was elected a foreign member of the Royal Swedish Academy of Sciences.[http://www.kva.se/en/News/2010/New-foreign-members-elected-to-the-academy/ New foreign members elected to the academy], press announcement from the Royal Swedish Academy of Sciences, 26 May 2010.

He was knighted in the 2012 New Year Honours for services to mathematics.{{London Gazette |issue=60009 |date=31 December 2011 |page=1 |supp=y}} In 2012, he became a fellow of the American Mathematical Society.[http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society]. Retrieved 10 November 2012.

In March 2014, he was awarded the degree "Docteur Honoris Causa" by Université Joseph Fourier, Grenoble. In January 2017, he was awarded the degree "Doctor Honoris Causa" by the Universidad Complutense de Madrid, Spain.{{Cite web |title=Doctores "Honoris Causa" {{!}} Facultad de Ciencias Matemáticas |url=https://matematicas.ucm.es/doctores-honoris-causa |access-date=2023-10-27 |website=matematicas.ucm.es}}

{{Further|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:{{by whom|date=March 2021}}

• The diagonalizability theorem {{harvs|last=Donaldson|year1=1983a|year2=1983b|year3=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 {{harv|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 {{harv|Donaldson|1987c}}, proven using an inductive proof and the theory of determinant bundles and Quillen metrics.Another proof of a somewhat more general result was given by {{cite journal |last1=Uhlenbeck |first1=Karen |author1-link = Karen Uhlenbeck |name-list-style=amp |last2=Yau |first2=Shing-Tung |author2-link = Shing-Tung Yau |title = On the existence of Hermitian-Yang-Mills connections in stable vector bundles |journal=Comm. Pure Appl. Math. |volume=39 |year=1986 |issue = S, suppl |pages = S257–S293 |doi = 10.1002/cpa.3160390714 |mr=0861491 }}
• 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 {{harv|Donaldson|1990}}. This was an early application of the Donaldson invariant (or instanton invariants).
• Any compact symplectic manifold admits a symplectic Lefschetz pencil {{harv|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 {{harvtxt|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 {{harvtxt|Donaldson|Sun|2014}} on Gromov–Hausdorff limits. The summary of the existence proof for Kähler–Einstein metrics appears in {{harvtxt|Chen|Donaldson|Sun|2014}}. Full details of the proofs appear in {{harvs|txt|last1=Chen|last2=Donaldson|last3=Sun|year1=2015a|year2=2015b|year3=2015c}}.

{{See also|K-stability|K-stability of Fano varieties}}

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".{{cite web |url = http://www.ams.org/news?news_id=4705 |title=2019 Oswald Veblen Prize in Geometry to Xiuxiong Chen, Simon Donaldson, and Song Sun |date=2018-11-19 |website=American Mathematical Society |access-date=2019-04-09 }}

• {{cite journal |last=Donaldson |first=Simon K. |title=An application of gauge theory to four-dimensional topology |journal=J. Differential Geom. |volume=18 |issue=2 |year=1983a |pages=279–315 |mr=710056 |doi=10.4310/jdg/1214437665 |doi-access=free }}
• {{cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=Self-dual connections and the topology of smooth 4-manifolds |journal=Bull. Amer. Math. Soc. |volume=8 |issue=1 |year=1983b |pages=81–83 |doi=10.1090/S0273-0979-1983-15090-5 |mr=0682827 |doi-access=free }}
• {{Cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=Instantons and geometric invariant theory|journal=Comm. Math. Phys. |volume=93 |issue=4 |year=1984b |pages=453–460 |mr=0892034 |url= https://projecteuclid.org/euclid.cmp/1103941177 |bibcode=1984CMaPh..93..453D |doi=10.1007/BF01212289 |s2cid=120209762 }}
• {{Cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=The orientation of Yang-Mills moduli spaces and 4-manifold topology |journal=J. Differential Geom. |volume=26 |year=1987a |issue=3 |pages=397–428 |mr=910015 |doi=10.4310/jdg/1214441485 |doi-access=free }}
• {{Cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=Irrationality and the h-cobordism conjecture |journal=J. Differential Geom. |volume=26 |issue=1 |year=1987b |pages=141–168 |mr=0892034 |doi=10.4310/jdg/1214441179 |doi-access=free }}
• {{Cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=Infinite determinants, stable bundles and curvature |journal=Duke Math. J. |volume=54 |year=1987c |issue=1 |pages=231–247 |doi=10.1215/S0012-7094-87-05414-7 | mr=0885784 }}
• {{Cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=Polynomial invariants for smooth four-manifolds |journal=Topology |volume=29 |year=1990 |issue=3 |pages=257–315 |doi=10.1016/0040-9383(90)90001-Z | mr=1066174 |doi-access= }}
• {{Cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=Lefschetz pencils on symplectic manifolds |journal=J. Differential Geom. |volume=53 |year=1999 |issue=2 |pages=205–236 |mr=1802722 |doi=10.4310/jdg/1214425535 |doi-access=free }}
• {{Cite journal |last=Donaldson |first=Simon K. |author-mask=3 |title=Scalar curvature and projective embeddings. I |journal=J. Differential Geom. |volume=59 |year=2001 |issue=3 |pages=479–522 |mr=1916953 |doi=10.4310/jdg/1090349449 |doi-access=free }}
• {{Cite journal |last1=Donaldson |first1=Simon K. |author-mask=3 | last2=Sun |first2=Song |title= Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry |journal=Acta Math. |volume=213 |issue=1 |year=2014 |pages=63–106 |doi=10.1007/s11511-014-0116-3 | mr=3261011 |arxiv=1206.2609 |s2cid=120450769 }}
• {{Cite journal |last1=Chen |first1=Xiuxiong |last2=Donaldson |first2=Simon |last3=Sun |first3=Song |title=Kähler-Einstein metrics and stability |journal=Int. Math. Res. Notices |volume=2014 |issue= 8|year=2014 |pages=2119–2125 |doi=10.1093/imrn/rns279 | mr=3194014 |arxiv=1210.7494 |s2cid=119165036 }}
• {{Cite journal |last1=Chen |first1=Xiuxiong |last2=Donaldson |first2=Simon |last3=Sun |first3=Song |title=Kähler-Einstein metrics on Fano manifolds I: Approximation of metrics with cone singularities |journal=J. Amer. Math. Soc. |volume=28 |issue=1 |year=2015a |pages=183–197 |doi=10.1090/S0894-0347-2014-00799-2 |mr=3264766 |arxiv=1211.4566 |s2cid=119641827 }}
• {{Cite journal |last1=Chen |first1=Xiuxiong |last2=Donaldson |first2=Simon |last3=Sun |first3=Song |title=Kähler-Einstein metrics on Fano manifolds II: Limits with cone angle less than 2π |journal=J. Amer. Math. Soc. |volume=28 |issue=1 |year=2015b |pages=199–234 |doi=10.1090/S0894-0347-2014-00800-6 | mr=3264767 |arxiv=1212.4714 |s2cid=119140033 }}
• {{Cite journal |last1=Chen |first1=Xiuxiong |last2=Donaldson |first2=Simon |last3=Sun |first3=Song |title=Kähler-Einstein metrics on Fano manifolds III: Limits as cone angle approaches 2π and completion of the main proof |journal=J. Amer. Math. Soc. |volume=28 |issue=1 |year=2015c |pages=235–278 |doi=10.1090/S0894-0347-2014-00801-8 |mr=3264768 |arxiv=1302.0282 |s2cid=119575364 }}

Books

• {{cite book |last1=Donaldson |first1=S.K. |last2=Kronheimer |first2=P.B. |authorlink2=Peter B. Kronheimer |title=The geometry of four-manifolds |series=Oxford Mathematical Monographs |publisher=Oxford University Press |location=New York |year=1990 |isbn=0-19-853553-8 |mr=1079726 |url-access=registration |url=https://archive.org/details/geometryoffourma0000dona }}{{cite journal |last = Hitchin |first = Nigel |author-link=Nigel Hitchin |title = Review: The geometry of four-manifolds, by S. K. Donaldson and P. B. Kronheimer|journal=Bull. Amer. Math. Soc. (N.S.) |year=1993|volume=28|issue=2|pages=415–418 |url = http://www.ams.org/journals/bull/1993-28-02/S0273-0979-1993-00377-X/|doi=10.1090/s0273-0979-1993-00377-x |doi-access=free }}
• {{cite book|last1=Donaldson|first1=S.K.|title=Floer homology groups in Yang-Mills theory|series=Cambridge Tracts in Mathematics|publisher=Cambridge University Press|year=2002|location=Cambridge|isbn=0-521-80803-0|volume=147}}
• {{Cite book|last=Donaldson |first=Simon |title=Riemann surfaces |series=Oxford Graduate Texts in Mathematics |volume=22 |publisher=Oxford University Press |location = Oxford |year=2011 |isbn=978-0-19-960674-0 |mr=2856237 |doi=10.1093/acprof:oso/9780198526391.001.0001 }}{{cite journal |last=Kra |first = Irwin |author-link=Irwin Kra |title=Review: Riemann surfaces, by S. K. Donaldson |journal=Bull. Amer. Math. Soc. (N.S.) |year=2012 |volume=49 |issue=3 |pages=455–463 |url = http://www.ams.org/journals/bull/2012-49-03/S0273-0979-2012-01375-7/ |doi=10.1090/s0273-0979-2012-01375-7 |doi-access=free }}

{{Reflist}}

{{Wikiquote}}

• {{MacTutor Biography |id = Donaldson}}
• {{MathGenealogy |id = 36909 }}
• [http://www.ma.ic.ac.uk/~skdona/ Home page at Imperial College]
• {{cite web |url = https://www.youtube.com/watch?v=P8-MjtCmyzg |title=Some recent developments in Kähler geometry and exceptional holonomy – Simon Donaldson – ICM2018 |website = YouTube |date=19 September 2018 }} (Plenary Lecture 1)

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