The mathematical work of Simon Donaldson is on the application of mathematical analysis to problems of differential geometry and algebraic geometry, especially through the study of partial differential equations coming from mathematical physics. We mention the most revolutionary contributions:
• Application of Yang-Mills theory to smooth 4-manifolds, which served to prove that many topological 4-manifolds do not admit smooth structures. The construction of the moduli of instantons yielded the now known as "Donaldson invariants", which serve to distinguish smooth structures on 4-manifolds that are homeomorphic. These invariants have numerous applications in topology of dimensions 3 and 4, and have produced developments of many kinds.
• Characterization of the stability of holomorphic vector bundles on a smooth projective algebraic variety as those that support a Hermite-Einstein metric. The implications on the theory of moduli spaces of bundles in algebraic geometry are very extensive with many ramifications.
• Asymptotically holomorphic theory for symplectic manifolds. Striking consequences are the existence of symplectic submanifolds in very general cases or the construction of symplectic Lefschetz fibrations in any symplectic manifold. This is a true breakthrough in symplectic geometry that has deepened the understanding of the topology of symplectic manifolds.
• G2 and Spin(7) geometry. S. Donaldson and R. Thomas introduced the notion of G2-instanton and Spin(7)-instanton, and proposed a revolutionary program aimed at studying gauge theories in manifolds of dimensions 6, 7, 8 with special geometric structures. Many students of S. Donaldson have done research in this area, producing real progress in this program.
• In the last decade, Simon Donaldson's work has focused on the relationship between the algebro-geometric condition of stability of complex projective varieties and the differential-geometric condition of the existence of extremal Kähler metrics, typically of constant scalar Kähler curvature. Jointly with X. Chen and S. Sun, they have completed the case of Kähler-Einstein metrics in 2012, representing an impressive result that will change the area for the future.
According to Mathscinet, Simon Donaldson has almost 100 articles with more than 3500 citations by almost 1,500 authors, and an h-index of 35. Simon Donaldson has been supervisor of 44 doctoral students and has 119 descendants, according to the Math Project Genealogy page.