01 · A life in mathematics
The person behind the work
Simon Donaldson was born in Cambridge, England, and worked at Oxford when he received the 1986 Fields Medal. He brought gauge theory, originally developed in physics, into the study of smooth four-dimensional manifolds. His analysis of solutions to the anti-self-dual Yang–Mills equations imposed unexpected restrictions on their topology. In particular, his diagonalisation theorem sharply constrained definite intersection forms of smooth closed simply connected four-manifolds. Compared with Freedman’s topological classification, these restrictions exposed a dramatic gap between continuous and smooth geometry. Their interaction also revealed exotic smooth structures on four-dimensional Euclidean space. Donaldson’s work concerns smooth structure, not a classification that ignores differentiability.
The contribution, explained
Surfaces inside an oriented four-manifold intersect in isolated points, and their signed intersections form an algebraic object called the intersection form. Topological four-manifolds can realise forms that smooth ones cannot. Donaldson studied spaces of solutions to the anti-self-dual Yang–Mills equations and extracted consequences for smooth manifolds. His diagonalisation theorem says that a definite intersection form of a closed smooth simply connected four-manifold is diagonal over the integers. Comparing this with topological existence theorems shows that some spaces cannot carry any smooth structure. The striking conclusion is that asking for well-defined calculus changes the possibilities for a space.
Why it matters
The work transformed four-dimensional geometry, showing that smooth and topological classifications diverge sharply and establishing gauge theory as a central source of geometric invariants and obstructions.