CAREER: Geometric Aspects of Isoperimetric and Sobolev-type Inequalities

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

Isoperimetric and Sobolev-type inequalities play a central role in the mathematical fields of analysis and geometry and provide a mathematical framework to describe optimal configurations for various engineering problems and physical systems. For example, one isoperimetric-type inequality gives the mathematical justification that a metal rod will have the strongest resistance to twisting forces if its cross-sections are circular. This project investigates several geometric questions related to isoperimetric and Sobolev-type inequalities, including the following: if one only has measurements of a given rod's resistance to twisting forces, how much geometric information can be recovered about the shape of the rod's cross-sections? Questions of this type have powerful and sometimes unexpected applications in other branches of mathematics. A fundamental part of the project is a two-pillared educational component. First, the Principal Investigator (PI) will organize a workshop in analysis at Carnegie Mellon University, integrating research and education through mini-courses, research talks, and opportunities for junior researchers. Second, the PI will initiate a joint Directed Reading Program between Carnegie Mellon University and the neighboring University of Pittsburgh, delivering vertically integrated mathematical and professional development and a timely opportunity to rebuild bridges between the two departments post-pandemic. This project, rooted in the calculus of variations and partial differential equations, develops novel applications of isoperimetric and Sobolev-type inequalities to attack central questions in analysis and geometry, and explores the interplay between geometry and optimal constants and equality cases for the inequalities. The PI will develop a framework for proving a new type of broadly applicable quantitative stability estimate in the context of isoperimetric problems for shape functionals driven by partial differential equations; prove quantitative descriptions of local minimizers of isoperimetric problems in Riemannian manifolds and Euclidean domains, expanding the toolbox in this area; via doubly-constrained Sobolev-type minimization problems, build constant scalar curvature conformal metrics with constant mean curvature boundary of prescribed area; and prove localized versions of epsilon-regularity theorems for Riemannian manifolds with lower bounds on scalar curvature, paving the way for the analysis of singularities. 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.