Asymptotic and quantitative geometry of groups and spaces

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AbstractAward: DMS 1612061, Principal Investigator: Robert YoungThis project proposes to study the geometry of surfaces in some spaces arising from mathematics and computer science. Surfaces are fundamental objects in geometry, and the shape of surfaces in a space provides insight into the space. The behavior of surfaces that are minimal or near-minimal is especially important. For example, minimal surfaces, like soap films, are known to be smooth under many conditions, but surfaces that are only close to minimal can be rough. The PI intends to develop new tools to study minimal and close-to-minimal surfaces in a variety of spaces and use them to study geometric and analytical problems. One proposed application of these tools is the study of the accuracy of the Goemans-Linial algorithm, an algorithm to approximate the best way of cutting an object into two roughly equal pieces. This algorithm is a key ingredient in many "divide-and-conquer" algorithms, which solve complex problems by decomposing them into simpler ones.The PI plans to develop new quantitative tools for studying the geometry of surfaces in groups and spaces and apply them to questions in geometric group theory, geometric measure theory, and theoretical computer science. First, the project aims to prove a conjecture of Gromov and Thurston on the filling functions of lattices in symmetric spaces, providing new understanding of the large-scale geometry of these spaces. Second, the project will explore decompositions of surfaces embedded in Euclidean space and try to use new tools from geometric measure theory to bound the geometry and topology of these surfaces. Third, the project will analyze surfaces of finite perimeter in the Heisenberg group and other nilpotent groups. If this last goal is successful, it would lead to sharp bounds on embeddings of the Heisenberg group into Banach spaces and sharp bounds on the accuracy of the best known approximate solution to the Sparsest Cut problem.