01 · The idea
In everyday language
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.
