Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

of smooth (differentiable) four-dimensional

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

Donaldson–Thomas theory In mathematics, specifically algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of ...

, and his contributions to Kähler geometry. He is currently a permanent member of the Simons Center for Geometry and Physics at

Stony Brook University Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public university, public research university in Stony Brook, New York, United States, on Long Island. Along with the University at Buffalo, it is on ...

in New York, and a Professor in Pure Mathematics at

Imperial College London Imperial College London, also known as Imperial, is a Public university, public research university in London, England. Its history began with Prince Albert of Saxe-Coburg and Gotha, Prince Albert, husband of Queen Victoria, who envisioned a Al ...

Biography

Donaldson's father was an electrical engineer in the physiology department at the

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

, and his mother earned a science degree there. Donaldson gained a BA degree in mathematics from

Pembroke College, Cambridge Pembroke College is a constituent college of the University of Cambridge, England. The college is the third-oldest college of the university and has over 700 students and fellows. It is one of the university's larger colleges, with buildings from ...

, in 1979, and in 1980 began postgraduate work at

Worcester College, Oxford Worcester College ( ) is a constituent college of the University of Oxford in England. The college was founded in 1714 by the benefaction of Sir Thomas Cookes, 2nd Baronet (1648–1701) of Norgrove, Worcestershire, whose coat of arms was ad ...

, at first under Nigel Hitchin and later under

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...

'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 Michael Hartley Freedman (born April 21, 1951) is an American 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 4-dimensional gen ...

classified topological four-manifolds, Donaldson's work focused on four-manifolds admitting a

differentiable structure In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for d ...

, using instantons, a particular solution to the equations of Yang–Mills

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

which has its origin in

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

. 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 In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold. Definition A smooth structure on a manifold M ...

at all. Donaldson also derived polynomial invariants from

. 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 A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

degree from

Oxford University The University of Oxford is a collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the second-oldest continuously operating u ...

in 1983, Donaldson was appointed a Junior Research Fellow at

All Souls College, Oxford All Souls College (official name: The College of All Souls of the Faithful Departed, of Oxford) is a constituent college of the University of Oxford in England. Unique to All Souls, all of its members automatically become fellows (i.e., full me ...

. He spent the academic year 1983–84 at the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

in Princeton, and returned to

Oxford Oxford () is a City status in the United Kingdom, cathedral city and non-metropolitan district in Oxfordshire, England, of which it is the county town. The city is home to the University of Oxford, the List of oldest universities in continuou ...

as Wallis Professor of Mathematics in 1985. After spending one year visiting

Stanford University Leland Stanford Junior University, commonly referred to as Stanford University, is a Private university, private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth ...

, he moved to

in 1998 as Professor of Pure Mathematics. In 2014, he joined the Simons Center for Geometry and Physics at

in New York, United States.

Awards

Donaldson was an invited speaker of the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

(ICM) in 1983, and a plenary speaker at the ICM in 1986, 1998, and 2018. In 1985, Donaldson received the Junior Whitehead Prize from the

London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...

. In 1994, he was awarded the

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 following a donation to the Royal Swedish Academy of Sciences. It is awarded jointly by the Acade ...

in Mathematics. In February 2006, Donaldson was awarded the

King Faisal International Prize The King Faisal Prize (, formerly King Faisal International Prize), is an annual award sponsored by King Faisal Foundation presented to "dedicated men and women whose contributions make a positive difference". The foundation awards prizes in fiv ...

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 Northwestern University (NU) is a Private university, private research university in Evanston, Illinois, United States. Established in 1851 to serve the historic Northwest Territory, it is the oldest University charter, chartered university in ...

. 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 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 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 The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...

(jointly with Yakov Eliashberg). In 1986, he was elected a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

and received a

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

at the

(ICM) in Berkeley. In 2010, Donaldson was elected a foreign member of the

Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences () is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting nat ...

.New foreign members elected to the academy
press announcement from the Royal Swedish Academy of Sciences, 26 May 2010. He was

knighted A knight is a person granted an honorary title of a knighthood by a head of state (including the pope) or representative for service to the monarch, the church, or the country, especially in a military capacity. The concept of a knighthood ...

in the 2012

New Year Honours The New Year Honours is a part of the British honours system, with New Year's Day, 1 January, being marked by naming new members of orders of chivalry and recipients of other official honours. A number of other Commonwealth realms also mark this ...

for services to mathematics. In 2012, he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. 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.

Research

Donaldson's work is on the application of

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

(especially the analysis of elliptic

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...

) to problems in geometry. The problems mainly concern

4-manifold In mathematics, a 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. T ...

s, complex

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

. The following theorems have been mentioned: * The diagonalizability theorem : If the intersection form of a smooth, closed, simply connected

is positive- or negative-definite then it is diagonalizable over the integers. This result is sometimes called Donaldson's theorem. * A smooth

h-cobordism In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : ...

between simply connected 4-manifolds need not be trivial . This contrasts with the situation in higher dimensions. * A stable

holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...

over a non-singular projective

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...

admits a Hermitian–Einstein metric , proven using an

inductive proof Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a simple case, then ...

and the theory of determinant bundles and Quillen metrics. * 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 . This was an early application of the Donaldson invariant (or

instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...

invariants). * Any compact symplectic manifold admits a symplectic Lefschetz pencil . 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 In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...

(see for example cscK metric). Donaldson obtained results in the toric case of the problem (see for example ). 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 on Gromov–Hausdorff limits. The summary of the existence proof for Kähler–Einstein metrics appears in . Full details of the proofs appear in .

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 In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The ...

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 Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

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 The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abs ...

'' in 2015 as a three-article series, "Kähler–Einstein metrics on Fano manifolds, I, II and III".

Selected publications

* * * * * * * * * * * * * * Books * * *

References

External links

* *
Home page at Imperial College
* (Plenary Lecture 1) {{DEFAULTSORT:Donaldson, Simon 1957 births Living people 20th-century English mathematicians 21st-century English mathematicians Differential geometers Algebraic geometers Fellows of the Royal Society Foreign associates of the National Academy of Sciences Foreign members of the Russian Academy of Sciences Members of the Royal Swedish Academy of Sciences Members of the French Academy of Sciences Institute for Advanced Study visiting scholars Fields Medalists Wallis Professors of Mathematics Fellows of All Souls College, Oxford Academics of Imperial College London Alumni of Pembroke College, Cambridge Alumni of Worcester College, Oxford Royal Medal winners Whitehead Prize winners Knights Bachelor Fellows of the American Mathematical Society Wolf Prize in Mathematics laureates