1. Biography

2. Awards and honours

3. Donaldson's work

4. Selected publications

5. Notes

6. References

7. External links

Sir Simon Kirwan Donaldson {{postnominal|country=GBR|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, 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. 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 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."^[7]

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).^[8]

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 {{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}}. (Another proof of a somewhat more general result was given by {{harvtxt|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 {{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 Donaldson theory.

Selected publications

• {{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 |url=http://projecteuclid.org/euclid.jdg/1214437665 | mr=710056 | ref=harv}}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=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 | ref=harv}}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=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 | ref=harv|bibcode=1984CMaPh..93..453D |doi=10.1007/BF01212289 }}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=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 |url=http://projecteuclid.org/euclid.jdg/1214441485 | mr=910015 | ref=harv}}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=3 |title=Irrationality and the h-cobordism conjecture |journal=J. Differential Geom. |volume=26 |issue=1 |year=1987b |pages=141–168 |mr=0892034 | url=http://projecteuclid.org/euclid.jdg/1214441179 | ref=harv}}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=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 | ref=harv}}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=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 | ref=harv}}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=3 |title=Lefschetz pencils on symplectic manifolds |journal=J. Differential Geom. |volume=53 |year=1999 |issue=2 |pages=205–236 |url=http://projecteuclid.org/euclid.jdg/1214425535 | mr=1802722 | ref=harv}}
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=3 |title=Scalar curvature and projective embeddings. I |journal=J. Differential Geom. |volume=59 |year=2001 |issue=3 |pages=479–522 |url=http://projecteuclid.org/euclid.jdg/1090349449 | mr=1916953 | ref=harv}}
• {{Cite book|last=Donaldson |first=Simon K. |authormask=3 |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}}^[9]
• {{Cite book |last=Donaldson |first=Simon K. |authormask=3 |lastauthoramp=yes |last2=Kronheimer |first2=Peter 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 }}^[10]
• {{Cite journal |last=Donaldson |first=Simon K. |authormask=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 | ref=harv|arxiv=1206.2609 }}
• {{Cite journal |last=Chen |first=Xiuxiong |last2=Donaldson |first2=Simon |last3=Sun |first3=Song |title=Kähler-Einstein metrics and stability |journal=Int. Math. Res. Notices |volume=8 |issue= |year=2014 |pages=2119–2125 |doi=10.1093/imrn/rns279 | mr=3194014 | ref=harv|arxiv=1210.7494 }}
• {{Cite journal |last=Chen |first=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 | ref=harv|arxiv=1211.4566 }}
• {{Cite journal |last=Chen |first=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 | ref=harv|arxiv=1212.4714 }}
• {{Cite journal |last=Chen |first=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 | ref=harv|arxiv=1302.0282 }}

Notes

References

• {{Cite journal |last=Atiyah |first=M. |authorlink=Michael Atiyah |lastauthoramp=yes |title=On the work of Simon Donaldson |journal=Proceedings of the International Congress of Mathematicians |year=1986}}
• {{Cite journal |last=Uhlenbeck |first=Karen |authorlink=Karen Uhlenbeck |lastauthoramp=yes |last2=Yau |first2=Shing-Tung |authorlink2=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 | ref=harv | mr=0861491}}

External links

• {{MacTutor Biography|id=Donaldson}}
• {{MathGenealogy|id=36909}}
• 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}} (Plenary Lecture 1)

• Brenton St. John
• Brenton, West Virginia
• Brenton Woods
• Brentor railway station
• Brent Palfreyman
• Brent Paris
• Brent Pease
• Brent Pelham Windmill
• Brent Peterson (b.1972)
• Brent Peterson (ice hockey, born 1972)
• Brent Peterson (left wing)
• Brent Peterson (left winger)
• Brent Petway
• Brent pipeline
• Brent Pope (Rugby Analyst)
• Brent Pope (rugby analyst)
• Brent R. Appel
• Brent Raymer
• Brent Redstone
• Brent Reiber
• Brent Renouf
• Brent reservoir
• Brents
• Brent Sadler
• Brent School