LocalStructure · Local Structure of Sets, Measures and Currents

The objective of this research proposal is to develop new methods to answer a number of fundamental questions generated by the recent development of modern analysis. The questions we are interested in are specifically related to the study of local structure of sets and functions in the classical Euclidean setting, in infinite dimensional Banach spaces and in the modern setting of analysis on metric spaces. The main areas of study will be:(a) Structure of null sets and representation of (singular) measures, one of the key motivations being the differentiability of Lipschitz functions in finite dimensional spaces.(b) Nonlinear geometric functional analysis, with particular attention to the differentiability of Lipschitz functions in infinite dimensional Hilbert spaces and Banach spaces with separable dual.(c) Foundations of analysis on metric spaces, the key problems here being representation results for Lipschitz differentiability spaces and spaces satisfying the Poincar\'e inequality.(d) Uniqueness of tangent structure in various settings, where the ultimate goal is to contribute to the fundamental problem whether minimal surfaces (in their geometric measure theoretic model as area minimizing integral currents) have a unique behaviour close to any point.