Abstract. In 1926, Hopf proved that any compact, simply connected Riemannian manifold with constant curvature 1 is isometric to the standard sphere. Motivated by this result, Hopf posed the question of whether a compact, simply connected manifold with suitably pinched curvature is topologically a sphere. This question has been studied by many authors over the past six decades, a milestone being the Topological Sphere Theorem proved by Berger and Klingenberg in 1960.

In this lecture, I will discuss the history of this problem, and describe the proof (joint with R. Schoen) of the Differentiable Sphere Theorem. This theorem classifies all manifolds with 1/4-pinched curvature up to diffeomorphism. The distinction between homeomorphism and diffeomorphism is significant in light of the exotic spheres constructed by Milnor; the proof uses the Ricci flow technique introduced by Hamilton.