Simon Donaldson

Simon Donaldson
Donaldson in 2009
Simon Donaldson
Born	Simon Kirwan Donaldson 20 8, 1957
Birthplace	Cambridge, England
Nationality	British
Occupation	Mathematician
Title	Professor of Pure Mathematics
Employer	Imperial College London
Known for	Donaldson's theorem, Donaldson–Thomas theory, contributions to Kähler geometry
Education	DPhil, University of Oxford
Awards	Fields Medal (1986), Breakthrough Prize in Mathematics (2014), Wolf Prize in Mathematics (2020)
Website	[http://www.ma.ic.ac.uk/~skdona/ Official site]

Sir Simon Kirwan Donaldson (born 20 August 1957) is a British mathematician whose work on the topology of smooth four-dimensional manifolds, Donaldson–Thomas theory, and Kähler geometry has reshaped significant areas of modern mathematics. Born in Cambridge, England, Donaldson rose to international prominence in the early 1980s when, as a doctoral student at the University of Oxford, he proved striking results about four-manifolds using techniques drawn from gauge theory and mathematical physics. His doctoral thesis, supervised by Michael Atiyah and Nigel Hitchin, introduced the use of Yang–Mills equations on Kähler manifolds and yielded what became known as Donaldson's theorem — a result that overturned long-standing assumptions about the topology of four-dimensional spaces. For this work, he was awarded the Fields Medal in 1986 at the age of 29, becoming one of the youngest recipients of the prize. In the decades since, Donaldson has continued to produce foundational contributions across differential geometry and algebraic geometry. He currently holds a Chair in Pure Mathematics at Imperial College London and holds Emeritus Professor status at the Simons Center for Geometry and Physics (SCGP) and Stony Brook University. He was knighted in the 2012 New Year Honours for services to mathematics.

Early Life

Simon Kirwan Donaldson was born on 20 August 1957 in Cambridge, England. Growing up in a city long associated with academic and scientific achievement, Donaldson developed an early interest in mathematics. Cambridge's university environment, with its deep traditions in the mathematical sciences, provided an intellectually stimulating backdrop during his formative years.

Details of Donaldson's family background and childhood are not extensively documented in public sources. What is known is that he proceeded from his early education in England to study at the University of Cambridge, where he would begin his formal mathematical training as an undergraduate. His aptitude for abstract mathematical reasoning became apparent during his university studies, and he quickly distinguished himself among his peers in the Cambridge mathematical community.

Education

Donaldson pursued his undergraduate studies at Pembroke College, Cambridge, where he earned a Bachelor of Arts degree. He then moved to the University of Oxford for his doctoral work, enrolling at Worcester College. At Oxford, he had the exceptional fortune of being supervised jointly by two of the most influential mathematicians of the twentieth century: Michael Atiyah and Nigel Hitchin.

His doctoral thesis, completed in 1983, was titled "The Yang–Mills Equations on Kähler Manifolds." The thesis brought together ideas from mathematical physics — specifically the Yang–Mills gauge theory that had been developed in the context of theoretical particle physics — and applied them to problems in differential geometry and topology. The results contained in this doctoral thesis were already considered remarkable at the time of their completion, and they would go on to have profound consequences for the study of four-dimensional manifolds. Donaldson's DPhil from Oxford marked the beginning of one of the most celebrated research careers in contemporary mathematics.

Career

Early Academic Career and the Oxford Years

Following the completion of his doctorate in 1983, Donaldson began his academic career at the University of Oxford, where he held positions that allowed him to continue developing the ideas first articulated in his thesis. His early results demonstrated that the moduli spaces of solutions to the Yang–Mills equations — mathematical objects that parametrise families of gauge-theoretic solutions — carried rich topological information about the underlying four-dimensional manifolds on which the equations were defined. This insight was entirely novel and opened up a new area of research at the intersection of geometry, topology, and mathematical physics.

The central result from this period, now known as Donaldson's theorem, showed that the intersection forms of smooth, compact, simply connected four-manifolds are subject to stringent constraints that do not apply in the purely topological category. Specifically, Donaldson proved that if such a manifold has a definite intersection form, then that form must be diagonalisable over the integers. This result was surprising because Michael Freedman's contemporaneous work on topological four-manifolds had shown that every unimodular symmetric bilinear form could be realised as the intersection form of some topological four-manifold. The discrepancy between the topological and smooth categories revealed by Donaldson's theorem demonstrated that four-dimensional differential topology is fundamentally different from the topology of manifolds in other dimensions — a phenomenon that continues to drive research in the field.

Donaldson further developed his gauge-theoretic techniques to define new invariants of smooth four-manifolds, which became known as Donaldson invariants or Donaldson polynomial invariants. These invariants, constructed from the topology of moduli spaces of anti-self-dual connections (instantons) on principal bundles over four-manifolds, provided powerful tools for distinguishing between smooth structures on four-manifolds that are homeomorphic but not diffeomorphic. The construction of these invariants represented a major advance in the application of ideas from physics to pure mathematics.

Stanford and the Institute for Advanced Study

Over the course of his career, Donaldson held visiting and permanent positions at several leading institutions. He was affiliated with Stanford University and the Institute for Advanced Study in Princeton, New Jersey, in addition to his longstanding base at Oxford. These affiliations allowed him to collaborate with a broad network of mathematicians and physicists working on related problems in geometry, topology, and mathematical physics.

Imperial College London

Donaldson joined the faculty of Imperial College London, where he has held a Chair in Pure Mathematics. At Imperial, he continued his research programme in Kähler geometry, focusing in particular on problems related to the existence of special metrics on algebraic varieties. A central theme of this later work has been the relationship between the existence of constant scalar curvature Kähler metrics (cscK metrics) on a projective variety and algebraic-geometric notions of stability — a circle of ideas closely related to the Yau–Tian–Donaldson conjecture. This conjecture, which connects differential geometry with algebraic geometry in a deep way, proposes that a smooth polarised variety admits a cscK metric if and only if it is K-polystable. Donaldson made substantial contributions toward the resolution of this conjecture, and the programme of research he initiated in this area has been continued by numerous mathematicians, including several of his former doctoral students.

His official academic homepage has been maintained through Imperial College London.

Simons Center for Geometry and Physics and Stony Brook University

In addition to his position at Imperial College London, Donaldson became a permanent member of the Simons Center for Geometry and Physics (SCGP) at Stony Brook University in New York, and also held a professorship in Stony Brook's Department of Mathematics. The SCGP, founded in 2010 with support from the Simons Foundation, is dedicated to research at the interface of mathematics and theoretical physics — a natural intellectual home for Donaldson given the gauge-theoretic roots of his most influential work. After serving as a member of the SCGP for approximately a decade, Donaldson holds Emeritus Professor status at the SCGP and Stony Brook University.

Donaldson–Thomas Theory

Beyond his foundational work on four-manifold topology and Kähler geometry, Donaldson is also known for Donaldson–Thomas theory, developed in collaboration with his former doctoral student Richard Thomas. Donaldson–Thomas invariants are algebro-geometric invariants that count sheaves (or ideal sheaves) on Calabi–Yau threefolds. These invariants have deep connections to string theory, enumerative geometry, and the theory of moduli spaces, and they have become a central subject of study in modern algebraic geometry and mathematical physics. The significance of this theory was further highlighted in 2024 when the American Mathematical Society announced that Soheyla Feyzbakhsh and Richard Thomas, both of Imperial College London, would receive the 2025 AMS Oswald Veblen Prize in Geometry for work related to this area.

Doctoral Students

Donaldson has supervised numerous doctoral students who have themselves gone on to distinguished careers in mathematics. His doctoral students include Dominic Joyce, Paul Seidel, Ivan Smith, Richard Thomas, Michael Thaddeus, Gábor Székelyhidi, Oscar Garcia-Prada, Dieter Kotschick, and Graham Nelson. Several of these former students have received major international prizes in their own right, reflecting the broad influence of Donaldson's research programme and mentoring.

Personal Life

Donaldson was knighted in the 2012 New Year Honours for services to mathematics, and has since been formally styled as Sir Simon Donaldson. He maintains professional affiliations in both the United Kingdom and the United States. Beyond his knighthood and extensive list of academic honours, Donaldson has maintained a relatively private personal life, with limited publicly documented details about his family.

Recognition

Donaldson's contributions to mathematics have been recognised with numerous major international prizes and honours, placing him among the most decorated mathematicians of his generation.

Fields Medal (1986)

In 1986, Donaldson was awarded the Fields Medal, often described as the highest honour in mathematics, at the International Congress of Mathematicians in Berkeley, California. He received the medal at the age of 29 for his work on the topology of four-manifolds, particularly his use of gauge theory to derive new results about smooth structures on four-dimensional spaces. The Fields Medal is awarded every four years to mathematicians under the age of 40.

Breakthrough Prize in Mathematics (2014)

In June 2014, Donaldson was named as one of the five inaugural recipients of the Breakthrough Prize in Mathematics, receiving a prize of $3 million. The prize was financed by the Russian-born investor Yuri Milner. Donaldson was recognised for his contributions to differential geometry and topology.

Wolf Prize in Mathematics (2020)

In January 2020, Donaldson was named a laureate of the Wolf Prize in Mathematics, one of the most prestigious international awards in the field. He was recognised "for his contribution to differential geometry and topology." The Wolf Prize ceremony took place in 2021, delayed due to the COVID-19 pandemic, at which Donaldson received the joint 2020 Wolf Prize for Mathematics.

Oswald Veblen Prize in Geometry (2019)

In November 2018, the American Mathematical Society awarded the 2019 Oswald Veblen Prize in Geometry to Stony Brook University faculty members, with Donaldson among the honourees.

Shaw Prize

Donaldson has also been awarded the Shaw Prize in Mathematical Sciences, an international prize established in 2002 that honours individuals for achieving distinguished and significant advances in their respective fields.

Other Honours

Donaldson was elected as a Fellow of the American Mathematical Society. He was also elected as a Foreign Member of the Royal Swedish Academy of Sciences in 2010. His profile has been recorded in Debrett's, the authoritative guide to the British establishment.

Legacy

Simon Donaldson's work has had a transformative impact on several areas of mathematics. His introduction of gauge-theoretic methods into the study of four-dimensional topology in the early 1980s created an entirely new field of research and demonstrated, in a striking and unexpected way, that the smooth topology of four-manifolds is far richer and more complex than the topology of manifolds in any other dimension. The Donaldson invariants he constructed remain fundamental objects of study in four-manifold topology, even after the development of Seiberg–Witten invariants in the mid-1990s provided alternative (and in some respects technically simpler) tools for addressing many of the same questions.

His work on Kähler geometry, and in particular his contributions to the Yau–Tian–Donaldson conjecture, has shaped the direction of research in complex differential geometry and algebraic geometry for decades. The programme he initiated — connecting the existence of canonical metrics in Kähler geometry to algebraic notions of stability — has become one of the central themes in modern geometry, attracting contributions from mathematicians around the world.

Through Donaldson–Thomas theory, his influence extends into algebraic geometry, enumerative geometry, and mathematical physics, where Donaldson–Thomas invariants have become a standard tool. The fact that several of his former doctoral students — including Dominic Joyce, Richard Thomas, Paul Seidel, and Gábor Székelyhidi — have become leading figures in their own right is further evidence of the breadth and depth of his intellectual influence.

Donaldson's receipt of the Fields Medal (1986), the Breakthrough Prize in Mathematics (2014), and the Wolf Prize in Mathematics (2020) — three of the most significant international awards in mathematics — places him in a small group of mathematicians who have been honoured at the very highest levels throughout their careers.