Simon Donaldson
Encyclopedia
Simon Kirwan Donaldson FRS (born 20 August 1957, in Cambridge
Cambridge
The city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...

, England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

), is an English
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

 mathematician known for his work on the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 of smooth (differentiable) four-dimensional manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London
Imperial College London
Imperial College London is a public research university located in London, United Kingdom, specialising in science, engineering, business and medicine...

. In 2010, he was elected a foreign member of the Royal Swedish Academy of Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

.

Biography

Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...

. Donaldson gained a BA
Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...

 degree in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 from Pembroke College, Cambridge
Pembroke College, Cambridge
Pembroke College is a constituent college of the University of Cambridge, England.The college has over seven hundred students and fellows, and is the third oldest college of the university. Physically, it is one of the university's larger colleges, with buildings from almost every century since its...

 in 1979, and in 1980 began postgraduate work at Worcester College, Oxford
Worcester College, Oxford
Worcester College is one of the constituent colleges of the University of Oxford in England. The college was founded in the eighteenth century, but its predecessor on the same site had been an institution of learning since the late thirteenth century...

, at first under Nigel Hitchin
Nigel Hitchin
Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...

 and later under Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

's supervision. Still a graduate 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
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...

 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
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

 which has its origin in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

. One of Donaldson's first results gave severe restrictions on the intersection form
Intersection form
Intersection form may refer to:*Intersection theory *intersection form...

 of a smooth four-manifold. As a consequence, a large class of the topological four-manifolds do not admit any smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....

 at all. Donaldson also derived polynomial invariants from gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a 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
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...

 degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford
All Souls College, Oxford
The Warden and the College of the Souls of all Faithful People deceased in the University of Oxford or All Souls College is one of the constituent colleges of the University of Oxford in England....

, he spent the academic year 1983–84 at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

 in Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...

, and returned to Oxford as Wallis Professor of Mathematics in 1985. In 1999, he moved to Imperial College London
Imperial College London
Imperial College London is a public research university located in London, United Kingdom, specialising in science, engineering, business and medicine...

.

Donaldson received the Junior Whitehead Prize from the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...

 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
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...

. He was, however, turned down for fellowship of the Institute of Mathematics and its Applications
Institute of Mathematics and its Applications
The Institute of Mathematics and its Applications is the UK's chartered professional body for mathematicians and one of the UK's learned societies for mathematics ....

 on the grounds that he applied too soon after his doctorate. He was awarded the 1994 Crafoord Prize
Crafoord Prize
The Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord...

.

In February 2006 Professor 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
Nemmers Prize in Mathematics
The Frederic Esser Nemmers Prize in Mathematics is awarded biennially from Northwestern University. It was initially endowed along with a companion prize, the Erwin Plein Nemmers Prize in Economics, as part of a $14 million donation from the Nemmers brothers. They envisioned creating an award that...

, a mathematics prize awarded by Northwestern University
Northwestern University
Northwestern University is a private research university in Evanston and Chicago, Illinois, USA. Northwestern has eleven undergraduate, graduate, and professional schools offering 124 undergraduate degrees and 145 graduate and professional degrees....

.

In 2009 he was awarded the Shaw Prize
Shaw Prize
The Shaw Prize is an annual award first presented by the Shaw Prize Foundation in 2004. Established in 2002 in Hong Kong, it honours living "individuals, regardless of race, nationality and religious belief, who have achieved significant breakthrough in academic and scientific research or...

 in Mathematics (jointly with Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...

) 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
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

.

Donaldson's work

A thread running through Donaldson's work is the application of mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 (especially the analysis of elliptic partial differential equations) to problems in geometry. The problems mainly concern 4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...

s, complex differential geometry and symplectic geometry. The following theorems rank among his most striking achievements:
  • The diagonalizability theorem (Donaldson 1983a, 1983b): if the intersection form
    Intersection theory
    In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...

     of a smooth, closed, simply connected 4-manifold
    4-manifold
    In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...

     is positive- or negative-definite then it is diagonalizable over the integers. (The simple connectivity hypothesis has since been shown to be unnecessary using Seiberg-Witten theory.) This result is sometimes called Donaldson's theorem
    Donaldson's theorem
    In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...

    .

  • A smooth h-cobordism
    H-cobordism
    A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...

     between 4-manifolds need not be trivial (Donaldson 1987a). This contrasts with the situation in higher dimensions.

  • A stable holomorphic vector bundle
    Holomorphic vector bundle
    In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map \pi:E\to X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the...

     over a non-singular projective algebraic variety
    Algebraic variety
    In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

      admits a Hermitian-Einstein metric (Donaldson 1987b). This was proved independently by Karen Uhlenbeck
    Karen Uhlenbeck
    Karen Keskulla Uhlenbeck is a professor and Sid W. Richardson Regents Chairholder in the Department of Mathematics at The University of Texas in Austin. In 1998 she was selected to be a Noether Lecturer. In 2000, she became a recipient of the National Medal of Science...

     and Shing-Tung Yau
    Shing-Tung Yau
    Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....

     (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 (Donaldson 1990). This was an early application of the Donaldson invariant (or instanton
    Instanton
    An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...

     invariants).

  • Any compact symplectic manifold admits a symplectic Lefschetz pencil
    Lefschetz pencil
    In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, in order to analyse the algebraic topology of an algebraic variety V. A pencil here is a particular kind of linear system of divisors on V, namely a one-parameter family, parametrised by the...

     (Donaldson 1999).


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
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

, typically those with constant scalar curvature
Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

. Definitive results have not yet been obtained, but substantial progress has been made (see for example Donaldson 2001).

See also Donaldson theory
Donaldson theory
Donaldson theory is the study of smooth 4-manifolds using gauge theory. It was started by Simon Donaldson who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold....

.

External links

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