I will discuss some recent results on the structure of the branch set of multiple valued solutions to the Laplace's equation and the minimal surface system. It is known that the branch set has Hausdorff dimension $n-2$. I will show the branch set is countably $(n-2)$-rectifiable using a method for establishing asymptotics near branch points, which is based on 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. This is joint work with Neshan Wickramasekera.

The aim of this talk is twofold. In the first part we establish a quantitative isoperimetric inequality in higher codimension. In a certain sense such an inequality can be regarded as a stability result (second order Taylor approximation) of Almgren's optimal isoperimetric. In particular, we prove that for any closed $(n-1)$-dimensional manifold $\Gamma$ in $\mathbb R^{n+k}$ the following inequality $$ \mathbf D(\Gamma)\ge C \mathbf d^2(\Gamma) $$ holds true. Here, $\mathbf D(\Gamma)$ stands for the isoperimetric gap of $\Gamma$, i.e. the deviation in measure of $\Gamma$ from being a round $(n-1)$-dimensional sphere and $\mathbf d(\Gamma )$ denotes a natural generalization of the Fraenkel asymmetry index of $\Gamma$ to higher codimensions. It measures in a generalized sense the distance of $\Gamma$ from spheres enclosing the same volume as $\Gamma$.   In the second part of the talk we are interested in the isoperimetric inequality on manifolds. As the simplest manifold we will consider the sphere. It was shown by E.~Schmidt that geodesic balls are the unique isoperimetric sets on the sphere. We will present a strong quantitative version of this result. More precisely, we prove that for any set $E\subset S^n$ of finite perimeter there holds $$ \mathbf D(E) := \frac{\mathbf P(E)-\mathbf P(B_\vartheta)}{\mathbf P(B_\vartheta)} \ge C \boldsymbol\beta^2(E), $$ where $B_\vartheta\subset S^n$ is a geodesic ball with the same volume as $E$ and $\boldsymbol\beta$ denotes the $L^2$-oscillation index of $\partial E$. In principle the oscillation index is given by $$ \boldsymbol\beta (E) := \boldsymbol\alpha (E) + \mbox{term counting the $L^2$-oscillation of $\partial E$,} $$ where $\boldsymbol\alpha$ denotes the Fraenkel asymmetry index. The advantage of such a quantity stems from the fact that the above quantitative isoperimetric inequality is sharp in the sense that also the reverse inequality holds true. Both results are joint work with Frank Duzaar and Nicola Fusco.