| Simon Donaldson | |
 Simon Donaldson | |
| Born | August 20, 1957 Cambridge |
|---|---|
| Nationality | English |
| Known for | topology of smooth (differentiable) four-dimensional manifolds |
| Notable awards | 1986, Fields Medal |
Simon Kirwan Donaldson, born in Cambridge in 1957, is an English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. Events 636 - Battle of Yarmouk: Arab forces led by Khalid ibn al-Walid take control of Syria and Palestine Year 1957 ( MCMLVII) was a Common year starting on Tuesday (link displays the 1957 Gregorian calendar) The city of Cambridge (ˈkeɪmbrɪdʒ is a university town and the administrative centre of the county of Cambridgeshire, England England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Year 1986 ( MCMLXXXVI) was a Common year starting on Wednesday (link displays 1986 Gregorian calendar) The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical The city of Cambridge (ˈkeɪmbrɪdʒ is a university town and the administrative centre of the county of Cambridgeshire, England Year 1957 ( MCMLVII) was a Common year starting on Tuesday (link displays the 1957 Gregorian calendar) England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be
Donaldson gained a BA 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Pembroke College is a college of the University of Cambridge, home to over six hundred students and fellows, and is the third oldest of the colleges Year 1979 ( MCMLXXIX) was a Common year starting on Monday (link displays the 1979 Gregorian calendar) Year 1980 ( MCMLXXX) was a Leap year starting on Tuesday (link displays the 1980 Gregorian calendar) Nigel Hitchin (b 2 August 1946 in Holbrook Derbyshire) is a British Mathematician working in the fields of Differential Sir Michael Francis Atiyah, OM, FRS, FRSE (b April 22, 1929) is a British Mathematician, and one of the Still a graduate student, Donaldson soon 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. Year 1983 ( MCMLXXXIII) was a Common year starting on Saturday (link displays the 1983 Gregorian calendar) 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. Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, U An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In quantum field theory (QFT the forces between particles are mediated by other particles 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. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations 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 doctorate 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 at Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the Year 1983 ( MCMLXXXIII) was a Common year starting on Saturday (link displays the 1983 Gregorian calendar) All Souls College (in full The Warden and College of the Souls of all Faithful People deceased in the University of Oxford) is one of the constituent colleges The Institute for Advanced Study, located in Princeton New Jersey, United States is a center for theoretical research Princeton University is a private Coeducational research university located in Princeton, New Jersey. The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the Year 1985 ( MCMLXXXV) was a Common year starting on Tuesday (link displays 1985 Gregorian calendar) In 1999, he moved to Imperial College London. Year 1999 ( MCMXCIX) was a Common year starting on Friday (link will display full 1999 Gregorian calendar) Imperial College London (officially The Imperial College of Science Technology and Medicine as given in its Royal Charter It is one of only three universities to have reached
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. The Whitehead Prize is awarded yearly by the London Mathematical Society to a mathematician working in the United Kingdom who is at an early stage of their career The London Mathematical Society ( LMS) is the leading mathematical society in England. Year 1985 ( MCMLXXXV) was a Common year starting on Tuesday (link displays 1985 Gregorian calendar) The Royal Society of London for the Improvement of Natural Knowledge, known simply as The Royal Society, is a Learned society for science that was founded in 1660 The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical He was, however, turned down for fellowship of the Institute of Mathematics and its Applications on the grounds that he applied too soon after his doctorate. The Institute of Mathematics and its Applications (IMA is distinct from the Institute '''for''' Mathematics and its Applications in Minneapolis. He was awarded the 1994 Crafoord Prize. The annual Crafoord Prize is a science prize established in 1980 by Holger Crafoord, a Swedish industrialist and his wife Anna-Greta Crafoord
In February 2006 Dr 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. The Frederic Esser Nemmers Prize in Mathematics is awarded biennially from Northwestern University.
Donaldson's work
A thread running through Donaldson's work is the creative application of mathematical analysis (especially the analysis of elliptic partial differential equations) to problems in geometry. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i The problems mainly concern 4-manifolds, complex differential geometry and symplectic geometry. In Mathematics, 4-manifold is a 4-dimensional Topological manifold. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a The following theorems rank among his most striking achievements:
- The diagonalizability theorem (Donaldson 1983a, 1983b): if the intersection form of a smooth, closed, simply connected 4-manifold is positive- or negative-definite then it is diagonalizable over the integers. In Mathematics, 4-manifold is a 4-dimensional Topological manifold. (The simple connectivity hypothesis has since been shown to be unnecessary using Seiberg-Witten theory. In Theoretical physics, Seiberg-Witten gauge theory is a set of calculations that determine the low-energy physics &mdash namely the Moduli space and the masses of ) This result is sometimes called Donaldson's theorem. In Mathematics, Donaldson's theorem states that a Positive definite Intersection form of a Simply connected Smooth manifold of
- A smooth h-cobordism between 4-manifolds need not be trivial (Donaldson 1987a). A Cobordism W between M and N is an h -cobordism if the inclusion maps M \hookrightarrow W \quad\mbox{and}\quad 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 1987b). In Mathematics, a holomorphic vector bundle is a Complex vector bundle over a Complex manifold X such that the total space E is complex This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety This was proved independently by Karen Uhlenbeck and Shing-Tung Yau (Uhlenbeck & Yau 1986). Karen Keskulla Uhlenbeck (born August 24, 1942 in Cleveland Ohio) is a Professor and Sid W Shing-Tung Yau ( born April 4, 1949) is a Chinese American Mathematician working in Differential geometry, and involved
- A non-singular, projective algebraic surface can only be diffeomorphic to the connected sum of two oriented 4-manifolds if one of them has negative-definite intersection form (Donaldson 1990). This was an early application of the Donaldson invariants (or instanton invariants). An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics.
- Any compact symplectic manifold admits a symplectic Lefschetz pencil (Donaldson 1999). In Mathematics, a Lefschetz pencil is a construction in Algebraic geometry considered by Solomon Lefschetz, in order to analyse the Algebraic topology
Donaldson's recent work centers on a difficult problem in complex differential geometry concerning a conjectural relationship between algebro-geometric "stability" conditions for smooth projective varieties and the existence of "optimal" Kähler metrics, typically those with constant scalar curvature. In Mathematics, a Kähler manifold is a Manifold with unitary structure (a ''U''(''n''-structure) satisfying an Integrability condition In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest Curvature invariant of a Riemannian manifold. Definitive results have not yet been obtained, but substantial progress has been made (see for example Donaldson 2001).
External links
- O'Connor, John J. & Robertson, Edmund F. , “Simon Donaldson”, MacTutor History of Mathematics archive
- Simon Donaldson at the Mathematics Genealogy Project
- Home page at Imperial College
References
- Donaldson, S. The MacTutor History of Mathematics archive is an award-winning website maintained by John J The Mathematics Genealogy Project is a web-based Database that gives an Academic genealogy based on Dissertation supervision relations K. An application of gauge theory to four-dimensional topology. J. Differential Geom. , 18, (1983), 279–315.
- Donaldson, S. K. Self-dual connections and the topology of smooth 4-manifolds. Bull. Amer. Math. Soc. . 8, (1983), 81–83.
- Donaldson, S. K. Irrationality and the h-cobordism conjecture. J. Differential Geom. 26 (1987), no. 1, 141–168.
- Donaldson, S. K. Infinite determinants, stable bundles and curvature. Duke Math. J. 54 (1987), no. 1, 231–247.
- Donaldson, S. K. Polynomial invariants for smooth four-manifolds. Topology 29 (1990), no. 3, 257–315.
- Donaldson, S. K. Lefschetz pencils on symplectic manifolds. J. Differential Geom. 53 (1999), no. 2, 205–236.
- Donaldson, S. K. Scalar curvature and projective embeddings. I. J. Differential Geom. 59 (2001), no. 3, 479–522.
- Donaldson, S. K. and Kronheimer, P. B. The geometry of four-manifolds. Oxford Mathematical Monographs, Oxford University Press, New York, (1990) ISBN 0-19-853553-8.
- Uhlenbeck, K. ; Yau, S. -T. On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Comm. Pure Appl. Math. 39 (1986), no. S, suppl. , S257–S293.
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