The free boundary problem for minimal disks and applications

In modern Riemannian geometry, minimal surfaces have proven to have powerful and far-reaching applications related to the interconnection between the geometry and topology of manifolds. This will be a survey talk in which we will describe the variational theory for the free boundary problem for minimal disks. This is related to celebrated work of Sacks-Uhlenbeck and Micallef-Moore on existence of minimal two-spheres. We will also describe applications of this theory to convex domains, manifolds of positive isotropic curvature and holomorphic curves.