Much of classical Riemannian geometry and its applications relied on the study
of geodesics and their variational properties. It was therefore natural to look at what one
can do using minimal surfaces which are higher dimensional analogues of geodesics,
submanifolds which minimize volume in a small neighborhood of each point. There has
been dramatic progress in recent decades both in the understanding of existence of minimal
surfaces and also of their place in the study of curved manifolds. The minimal hypersurface
case has had profound applications to the study of positive scalar curvature and related questions
in general relativity. The general case of higher codimension is much less understood.
This talk will highlight recent work on the case of two dimensional surfaces in higher
codimensions.