Exploring Large-Scale Geometry via Local and Nonlocal Potential Theory

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This project will develop new mathematical tools for the analysis of metric measure spaces – that is, spaces equipped (like Euclidean space) with notions of distance and volume – with a focus on metric measure spaces that (unlike Euclidean space) lack smooth structure. The analysis of non-smooth spaces is a vital area of research with diverse applications across the mathematical and physical sciences, including fluid mechanics, neurophysiology, and fractal geometry. The PI will investigate the large-scale geometric behavior of objects in these spaces using the mathematical tools of local and nonlocal energies. Given a function measuring a physical phenomenon, such as temperature or momentum, local energies measure the function’s nearby or small-scale oscillations, while nonlocal energies measure its variations over long distances. A primary goal of this work is to develop much-needed mathematical tools for analyzing nonlocal energies. The project will also enhance the professional training of graduate students and postdoctoral scholars, through collaborative research projects, instruction in effective mathematical communication, and opportunities for research interactions with undergraduate students. The primary objects of study in this project are represented as metric measure spaces that lack smooth structure. The finite dimensionality of the ambient space is represented by the property of supporting a doubling Radon measure. In such a setting, nearby or asymptotic oscillation of a function is measured using upper gradients, which are viable substitutes for the derivative of a function, and the local energy is associated with the collection of functions on the object, called Sobolev functions. The large-scale variation energy is associated with the collection called Besov space of functions. Recent research has uncovered a connection between local energies on a region in a metric measure space and nonlocal energies on the boundary of the region. The project will leverage this connection to explore the large-scale geometry of nonlocal energies on the boundary of the region by linking them with small-scale behavior of local energies on the region itself. In particular, connections between Dirichlet-type boundary value problems and Neumann-type boundary value problems will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.