Geometry of Sets and Measures

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Geometric measure theory is a field of mathematics that developed starting in the 1920s and 1930s, growing out of a practical need to describe nonsmooth phenomena such as the formation of corners in soap bubble clusters. The term "measure" refers to an abstract generalization of length, area, or volume, which assigns a size value to every mathematical set. Traditional outlets for geometric measure theory, such as the calculus of variations and geometric analysis, have expanded in recent decades to include partial differential equations and harmonic analysis. The widespread utility and current use of geometric measure theory in different areas of analysis justifies its continued development. The proposed investigation on the geometry of sets and measures seeks to develop new techniques that will expand the toolbox that geometric measure theory provides for researchers in adjacent areas in analysis and geometry.This project focuses on two groups of questions about the geometry of sets and measures in Euclidean space. The first group of questions concerns rectifiable measures, one of the core objects of study in geometric measure theory. Specifically, these questions are aimed at increased understanding of rectifiable measures in the absence of a standing regularity assumption that has been assumed in the past. The main approach entails adapting quantitative techniques developed in the 1990s by Jones and David-Semmes to study the qualitative rectifiability of measures. The second group of questions are designed to examine the geometry of Reifenberg-type sets, which are sets that can be approximated at all locations and scales by one or more kinds of model sets. Instances where Reifenberg-type sets occur include geometric minimization problems and free boundary problems for elliptic partial differential equations. A general goal of this inquiry is to determine in what situations and to what extent good properties of solutions to problems in ideal models (smooth settings) persist under controlled perturbation (weak regularity).