Minimal surfaces: variational theory and applications

Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more remarkably, minimal surfaces (or minimal submanifolds) have encountered striking applications in other fields, like three-dimensional topology, mathematical physics, conformal geometry, among others. Even though it has been the subject of intense activity, many basic open problems still remain. In this lecture we will survey recent advances in this area and discuss some future directions. We will give special emphasis to the variational aspects of the theory as well as to the applications to other fields.