Stability and Regularity: From Analysis to Geometry

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

This project focuses on several problems at the crossroads of analysis and geometry that are united by a central theme: the use of functional and geometric inequalities as a tool to explore the geometry of a manifold or domain. These kinds of inequalities are used to describe ground states for physical systems, for instance, in acoustics and materials science; a classical example is the isoperimetric inequality, which mathematically describes why soap bubbles take a spherical shape. The project provides research training opportunities for graduate students. The results obtained as part of the project will be broadly disseminated to the scientific community. The project centers on stability and regularity estimates in different settings, including regularity of Riemannian manifolds with scalar curvature bounds and stability properties of their Laplace spectra; rectifiability results using the Alt-Caffarelli-Friedman monotonicity formula; and quantitative estimates for the Yamabe problem in conformal geometry. The research will bring together ideas from geometric analysis, partial differential equations and the calculus of variations, in terms of both techniques and applications. 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.