Geometric measure theory has roots going back to ancient Greek mathe- matics. For considerations of the isoperimetric problem (to find the planar domain of given perimeter having greatest area) led naturally to questions about spatial regions and boundaries.
In more modern times, the Plateau problem is considered to be the well- spring of questions in geometric measure theory. Named in honor of the nineteenth century Belgian physicist Joseph Plateau who studied surface tension phenomena in general, and soap films and soap bubbles in particu- lar, the question (in its original formulation) was to show that a fixed, simple closed curve in three-space will bound a surface of the type of a disc and hav- ing minimal area. Further, one wishes to study uniqueness for this minimal surface, and also to determine its other properties.
Jesse Douglas solved the original Plateau problem by considering the minimal surface to be a harmonic mapping (which one sees by studying the Dirichlet integral). For this effort he was awarded the Fields Medal in 1936.
Unfortuately, Douglas’s methods do not adapt well to higher dimensions, so it is desirable to find other techniques with broader applicability. Enter the theory of currents. Currents are continuous linear functionals on spaces of differential forms. Brought to fruition by Federer and Fleming in the 1950s, currents turn out to be a natural language in which to formulate the sorts of extremal problems that arise in geometry. One can show that the natural differential operators in the subject are closed when acting on spaces of currents, and one can prove compactness and structure theorems for spaces of currents that satisfy certain natural bounds. These two facts are key to the study of generalized versions of the Plateau problem and related questions of geometric analysis. As a result, Federer and Fleming were able to prove the existence of a solution to the general Plateau problem in all dimensions and codimensions in 1960.