Surprisingly many different problems of Analysis naturally lead to questions about singularities in (vector) measures. These problems come from both "pure" Analysis, such as the question for which measures Rademacher's theorem on the differentiability of Lipschitz functions holds, and its non-Euclidean analogues, as well as from "applied" Analysis, for example the problem to determine the fine structure of slip lines in elasto-plasticity. It is a remarkable fact that many of the (vector) measures that naturally occur in these questions satisfy an (under-determined) PDE constraint, e.g., divergence- or curl-freeness. The crucial task is then to analyse the fine properties of these PDE-constrained measures, in particular to determine the possible singularities that may occur. It turns out that the PDE constraint imposes strong restrictions on the shape of these singularities, for instance that they can only occur on a set of bounded Hausdorff-dimension, or even that the measure is k-rectifiable where its upper k-density is positive. The essential difficulty in the  analysis of PDE-constrained measures is that many standard methods from harmonic analysis are much weaker in an L¹-context and thus new strategies are needed. In this talk, I will survey recent and ongoing work on this area of research.

Seminar website https://ncts.ntu.edu.tw/gmt-seminar.html