Rigidity of Lipschitz and Related Mappings on Metric Spaces

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Classical calculus studies smoothly changing functions, curves, and surfaces inside the Euclidean space. In this project, the principal investigator will study calculus in arenas where the classical tools do not always apply. This includes non-smooth functions or sets, abstract geometries outside of the usual, Euclidean, geometry, and fractal spaces admitting complex behavior at many scales. The project aims to understand these objects by decomposing them into simpler pieces, embedding them into classical geometries, or approximating them by linear objects. Non-smooth analysis and geometry are important in many areas of pure and applied mathematics and computer science, since non-smooth problems arise in studying large data sets, in computational questions, and as limiting cases of smooth problems. The project will also emphasize results that are "quantitative": providing guaranteed estimates independent of the particular function or geometry being studied.More precisely, the project focuses on Lipschitz (and related) mappings. Building on his recent collaborative work, the principal investigator will study when Lipschitz mappings into metric spaces can be decomposed into or well-approximated by simpler mappings, like linear maps or maps that factor through trees. The principal investigator then plans to apply these techniques to provide new structural results on geometric objects defined through Lipschitz maps, like distance spheres or medial axes. The principal investigator will also investigate similar questions related to curves in metric spaces, continuing work on extensions of the so-called “Analyst’s Traveling Salesman Theorem” that links lengths of curves to quantitative flatness conditions. In addition, the principal investigator will investigate notions of differentiability of Lipschitz functions on non-smooth spaces, and how they constrain geometry. This is connected to the bi-Lipschitz embedding problem (“which spaces can be embedded in Euclidean space with bounded distortion?”), which is of major importance in geometry and computer science, and which will also be studied during the project. All these linked investigations will combine to improve our understanding of the analysis and geometry of non-smooth objects.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.