In this talk I will present some new result concerning the structure of measure satisfying a linear PDE constraint. In 2016, in collaboration with Filip Rindler, we prove a first structural result concerning the singular part of measure subject to PDE constraint. This turned out to have several applications in GMT and in Geometric Analysis.
In a joint work with Adolfo Arroyo Rabasa, Jonas Hirsch and Filip
Rindler we improve upon this result proving a more precise
structure on the "low" dimensional part of the measure. As a
corollary we recover several known rectifiability results. In this talk I will try to give an overview of both these results and of their applications.