Structure of branch sets of harmonic functions and minimal submanifolds

I will discuss the fine structure of the branch set of multiple-valued solutions to the Laplace equation and minimal surface system. It was previously known that the branch set of a multiple-valued solution on a domain in $\mathbb{R}^n$ has Hausdorff dimension at most $n-2$. In joint work with Neshan Wickramasekera, we show that the branchset is countably $(n-2)$-rectifiable. This result follows from the asymptotic behavior of solutions near branch points, which we establish using a modification of the frequency function monotonicity formula due to F. J. Almgren and an adaptation to higher-multiplicity of a "blow-up" method due to L. Simon that was originally applied to "multiplicity one" classes of minimal submanifolds satisfying an integrability hypothesis. In ongoing independent work, I show that the branch set decomposes into locally real analytic submanifolds along which the frequency is constant and points where the frequency is not lower semicontinuous.