 In a series of papers with several authors, Atiyah classified all instantons on 4-dimensional Euclidean space.It is more convenient to classify instantons on a sphere as this is compact, and this is essentially equivalent to classifying instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant.With Hitchin and Singer he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principal bundle over a compact 4-dimensional Riemannian manifold (the Atiyah–Hitchin–Singer theorem).