The National Science Foundation awarded $376,018 under the Mathematical and Physical Sciences program (CFDA 47.049) to Carnegie Mellon University for the project "Stability and Regularity: From Analysis to Geometry" from September 1, 2022 to August 31, 2025.
The project focuses on using functional and geometric inequalities to explore the geometry of manifolds and domains. Such inequalities describe ground states for physical systems in acoustics and materials science. Researchers will examine stability and regularity estimates for Riemannian manifolds with scalar curvature bounds and Laplace spectra, as well as rectifiability results using the Alt-Caffarelli-Friedman monotonicity formula. The award will also support quantitative estimates for the Yamabe problem in conformal geometry. By bringing together ideas from geometric analysis, partial differential equations, and calculus of variations, the project aims to advance understanding of cross-cutting problems at the intersection of analysis and geometry. Graduate student research training is also supported.