Suppose $M$ is a minimal surface (a surface which is a critical point for the area functional) in the unit ball and that it is very close to the flat unit disc in the $x$-$y$ plane. If the area of $M$ is close to the area of the disc, then $M$ must be a smooth graph over the disc and many useful estimates hold. This is a kind of regularity result for minimal surfaces and results like this are prototypical in the wider world of geometric variational problems.

An open and fundamental problem in regularity theory relates to the situation in which $M$ is close to the disc but where the area of $M$ is close to $2\ \times$ the area of the disc. In this situation, it is not known exactly how complicated $M$ can be. In a forthcoming series of papers, written jointly with Neshan Wickramasekera, a special case is analyzed using techniques from geometric measure theory: the case in which $M$ corresponds to a two-sheeted Lipschitz graph (over some other plane). Central to the method is a graphical linearization procedure (in the spirit of that performed by \emph{e.g.} W. Allard, F. Almgren, or L. Simon) which produces a class of `two-valued harmonic functions' that must be studied in detail. Significant new results are obtained in this case and in the course of so doing, more is revealed - at a technical level at least - about the general case (when $M$ is not a two-sheeted graph). I will give an overview of the problem and this work.