The Plateau problem in \mbbR3 begins with a given simple, closed curve γ, and asks to find a surface M with ∂M=γ that minimizes area among all surfaces with γ as their boundary. In 1960 Federer and Fleming generalized this idea and analyzed the currents developed by De Rham. They proved certain subclasses of currents (in particular, the integral currents) can be used as a powerful tool in area and volume minimization problems. In this thesis, we approach two generalizations--first, we prove a Homological Plateau problem in the singular setting of semi-algebraic geometry using the tools of geometric measure theory. We obtain similar results to those of Federer and Fleming even in this more singular case.

Second, an earlier solution to the Plateau problem was achieved independently by Douglas and Rado in 1931 and 1933, respectively, using mappings from the two-dimensional disk. In the second chapter we generalize this mapping to a so-called ``multiple-valued'' mapping of the disk. Multiple-valued maps are a cornerstone of the regularity theorems of F. Almgren and are interesting in their own right for many problems in the geometric calculus of variations. We prove existence and regularity for these Plateau solutions under fairly general conditions. We also produce a class of examples and analyze a degenerate case.