The theory of minimal surfaces arose historically from work of J. L. Lagrange and physical observations of J. Plateau almost 200 years ago. Rigorous mathematical theory was developed in the 20th century. In more recent times the theory has found important applications to diverse areas of geometry and relativity. In this talk, which is aimed at a general mathematical audience, we will introduce the subject and describe a few recent applications of the theory.

Consider a permutation $\tau$ of the set $\{1,2,\dots,n,\}$.  If we divide the unit interval $[0,1)$ into $n$ half-open subintervals, we can consider the map $f$ which rearranges the subinterval according to the permutation $\tau$.  Such maps are called interval exchange transformations (IETs) and are the order preserving piecewise isometries of intervals, and preserve the Lebesgue measure.  IETs were first studied by Sinai in 1973, and then Keane in 1977, who showed that each minimal IET had a finite number of ergodic measures and conjectured that the Lebesgue measure was in fact the only ergodic invariant measure for such maps.  Much of the following research on IETs was based around proofs of this conjecture and will be discussed in the talk.

The uniformization conjecture states that any complete noncompact Kahler manifold with positive bisectional curvature is biholomorphic to C^n.  Perhaps one of reasons that the problem is difficult is lack of examples. Recently assuming U(n) symmetry Wu and Zheng gave a systematic construction on examples of such metrics,  we will talk about some related results.