Isoperimetric Problem and Group Structures in Geometric Analysis

Among all shapes that enclose a given volume, which ones have the smallest possible surface area? This classical question, known as the isoperimetric problem, dates back to ancient Greece and appears naturally in phenomena such as soap bubbles, crystals, and models of the universe in physics. This project aims to deepen our understanding of how geometric properties of a space -- especially curvature -- influence solutions to geometric optimization problems. The broader impact of the project includes mentoring undergraduate and graduate students and organizing seminars to engage the local mathematical community and foster the development of young researchers. The project uses tools from Geometric Measure Theory to study the isoperimetric structure of spaces with geometric constraints. A central theme is the development of a theory for spaces with curvature bounded below in a spectral sense, with applications to classical problems like the classification of stable minimal surfaces in Euclidean space. Another major goal is to analyze the existence and uniqueness of isoperimetric sets in nonnegatively curved spaces, including specific cases like the Euclidean unit cube and manifolds with nonnegative scalar curvature -- a setting that is also relevant in Mathematical Relativity. The project also addresses problems at the intersection of Algebra, Analysis, and Geometry, including the rectifiability of metric spaces with the same tangents almost everywhere, and the quasi-isometric classification of nilpotent groups. 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.