"Geometers have studied the topology of closed surfaces and their higher- dimensional analogues (manifolds) for a long time. But a remarkable breakthrough came in the early 1980s when Simon Donaldson found some totally new and unexpected invariants of four-dimensional manifolds. These were based on the Yang–Mills equations of physics but it was not until later that Edward Witten again showed how to interpret Donaldson's invariants in terms of quantum field theory. Later still, using duality ideas from string theory, Witten and Seiberg made a significant improvement of Donaldson theory which led to solutions of old problems."