Fine structure and higher regularity of the branch sets

I will discuss work on the structure of the branch set of two-valued solutions to the Laplace’s equation and the minimal surface system. Previously, the dimension of the branch set was known; we consider the fine structure of the branch set. In joint work with Neshan Wickramasekera, we show that the branch set is countably $(n-2)$-rectifiable. Moreover, I have independently shown that the branch set is locally real analytic on a relatively open dense subset of the branch set. Essential ingredients for both results include the monotonicity formula for frequency functions due to F. J. Almgren and a blow-up method, which was originally applied by Leon Simon to multiplicity one classes of minimal submanifolds. The analyticity result requires applying the blow-up method in a new setting where tangent maps are not necessarily translation invariant along $(n-2)$-dimensional subspaces.