In this talk I want to present a new version of Brakke's local regularity theorem. We consider a weak mean curvature flow given for times in [0,T] in some ball. Suppose the flow lies in a narrow slab, has density ratios less than two planes and does not vanish at time T. Then Brakke's local regularity theorem says that in some small neighbourhood the flow is smooth and graphical for times in (C,T-C) for some constant C. Here we will discuss that this actually holds for times in (C,T). One key observation is that a non-vanishing weak mean curvature flow that is initially locally graphical with small Lipschitz constant will stay graphical for some time.

In this talk we present the new regularity results proved for the singular sets of minimizing and stationary harmonic maps in collaboration with Aaron Naber (see arXiv:1504.02043). We prove that the singular set of a minimizing harmonic map is rectifiable with effective n-2 volume estimates. The results are based on an improved quantitative stratification technique, which consists in a detailed analysis of the symmetries and almost symmetries of the map u and its blow-ups at different scales, and rely on a new W^{1,p} version of Reifenberg's topological disk theorem. The application of this theorem in the situation of harmonic maps hinges on the monotonicity formula for the normalized energy. Similar results are available for minimizing and stationary currents (see arXiv:1505.03428)