Math Labs

Simon Donaldson

fl. 1957, Cambridge, England

TopologyGeometryDifferential equations and dynamical systemsFields Medal 1986Crafoord Prize in Mathematics 1994Shaw Prize in Mathematical Sciences 2009Breakthrough Prize in Mathematics 2015Wolf Prize in Mathematics 2020

British mathematician who founded smooth four-manifold theory through gauge-theoretic Donaldson invariants, and later resolved the existence of constant scalar curvature Kähler metrics on toric Fano manifolds.

Simon Kirwan Donaldson was born on 20 August 1957 in Cambridge, England. He studied at Pembroke College, Cambridge, then took a second undergraduate year at Worcester College, Oxford, before beginning doctoral work under Michael Atiyah and Nigel Hitchin. His 1983 thesis on anti-self-dual connections instantly changed four-dimensional topology; he became a fellow of All Souls College and professor at Oxford, moved to Imperial College London in 1998, and was knighted in 2012.

Donaldson's instanton theory showed that smooth structure in dimension four carries information invisible to topology: his theorem on diagonalizable intersection forms, together with Freedman's classification, implied that R4\mathbb{R}^4 admits uncountably many exotic smooth structures. With Uhlenbeck he proved that a stable holomorphic bundle over a compact Kähler manifold admits a Hermitian Yang–Mills connection — the Donaldson–Uhlenbeck–Yau theorem — linking algebraic stability to differential geometry. From the late 1990s he developed the conjectural equivalence between K-stability of Fano manifolds and existence of Kähler–Einstein metrics, proved for smooth toric Fano surfaces and manifolds; the general case was resolved by Chen–Donaldson–Sun in 2012–2014, settling the Yau–Tian–Donaldson conjecture.

Since 2014 he has divided his time between Imperial College London and the Simons Center for Geometry and Physics at Stony Brook University. His honors include the Fields Medal (1986), the Crafoord Prize (1994, shared with Yau), the Shaw Prize (2009, with Clifford Taubes), the inaugural Breakthrough Prize in Mathematics (2014 award for 2015) and the Wolf Prize (2020, with Yakov Eliashberg).

Workplaces: Imperial College London, Simons Center for Geometry and Physics, Stony Brook University, University of Oxford

United Kingdom

Award-winning work

Fields Medal 1986For using gauge theory to study the topology of four-dimensional manifolds. Donaldson analyzed the moduli space of anti-self-dual Yang–Mills connections (instantons) on a smooth 4-manifold and extracted from it the Donaldson invariants, proving that a definite intersection form of a smooth simply-connected 4-manifold must be diagonalizable over the integers — a constraint with no counterpart in the topological category. Combined with Michael Freedman's topological classification, this implied the existence of exotic smooth structures on R4\mathbb{R}^4 and opened the field of smooth four-manifold theory, later revolutionized again by the Seiberg–Witten equations that his framework anticipated.

Contributions, linked to the library

Library topicDifferential geometryGauge theory (instantons, Hermitian Yang–Mills connections) as a tool of differential geometry and topology.Great problemSmooth four-dimensional Poincaré conjectureDonaldson invariants detect exotic smooth structures in dimension four; the smooth 4-dimensional Poincaré conjecture remains the central open question of the field he founded.