Multiscale Methods in Quantitative Geometry

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This NSF award provides funding for a project to develop new methods for working with objects with complicated geometry at many different scales. Coastlines, clouds, and leaves are examples of such objects that occur in nature. The famous coastline paradox states that since a coastline is so rough at so many different scales, it has no defined length. The PI plans to develop and study new methods to build such multiscale objects and to break them down into easily handled pieces. New ways to decompose objects often lead to great advances in mathematics, where simple problems can have complicated, multiscale solutions. The project’s three-prong approach includes studying ways to break down complex objects, build complex objects out of simple pieces, and to measure multiscale complexity. In addition to the research, the PI will train graduate students and postdocs in the techniques developed by this project through advising, seminars, minicourses, and reading groups and disseminate material on his webpage for public access.The project aims to study problems in metric geometry, geometric measure theory, and harmonic analysis related to maps and surfaces with multiscale structure, that is, objects like Lipschitz and Holder maps or fractals, that are hard to approximate by affine maps or by planes. One focus is the study of geometric problems that have non-smooth solutions but no smooth solutions, such as the Nash Embedding Theorem. Solutions to these problems often use multiscale or self-similar structure to break rules that smooth maps have to satisfy, and one aim of this project is to understand when and why this phenomenon occurs. Another focus is quantitative and uniform rectifiability. Uniform rectifiability has been a powerful tool for studying singular integrals and geometric measure theory in Euclidean space. Recent work has made it possible to find sharp bounds on the quantitative rectifiability of surfaces in the Heisenberg group, and this project will explore the possibility of extending notions of uniform rectifiability to the Heisenberg group and using them to solve problems in the geometry of Euclidean space and the Heisenberg group.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.