Geometric measure theory provides an analytical toolkit to detect hidden structure in high-dimensional data sets and to describe non-smooth phenomena such as the formation of corners and singularities in soap bubble clusters. The term measure refers to an abstract generalization of length, area, and volume, which assigns a notion of size to each mathematical set. The pervasiveness of measures within contemporary mathematics notwithstanding, there are presently only a few tools available to analyze measures in regimes with low regularity. The proposed investigation seeks to develop novel and robust quantitative methods to study the geometry of general sets and measures in the absence of traditional simplifying hypotheses. The project will explore applications of these tools and methods to the analysis of harmonic functions and temperatures (solutions to the heat equation) in domains with rough boundary, both in ideal settings and inside non-homogenous media. The project will also provide training to graduate research assistants at the University of Connecticut and to visiting Ph.D. students working in geometric measure theory, harmonic analysis, and partial differential equations.
The project will develop four interrelated threads of research on the fine geometry of sets, the structure and regularity of measures, and the solutions of partial differential equations. The first line of inquiry pursues questions about subsets of rectifiable curves in infinite dimensional Banach spaces, with a concrete goal of solving the Analyst's Traveling Salesman Problem in a non-Hilbert setting. A second line of inquiry concerns the parameterization problem for Lipschitz images of the plane and the classification of 2-rectifiable Radon measures. Additional work will focus on the structure of measures supported on the graphs of Hölder continuous functions. A third direction of study will involve improved upper bounds on the Hausdorff dimension of harmonic and caloric measures on general domains, along with the relationship between the static and time-dependent cases. The final strand of the research program will confront contemporary challenges and initiate new inquiries in the theory of non-variational free boundary problems for harmonic and elliptic measures with two or more phases.