Singular structures in the geometry of measures: decompositions, rigidity and rectifiability

The research activity falls within the framework of the FIS2 SingMeas project, devoted to the study of the singularstructures of measures arising in Geometric Analysis, Optimal Transport, and Geometric Measure Theory. Theaim is to develop new tools to describe the fine geometric structure of measures and their singularities, withparticular attention to decomposition along low-dimensional sets, concentration phenomena, and rectifiabilityproperties. The techniques employed combine methods from metric analysis, the theory of currents, fractalgeometry, and the Calculus of Variations.