Asymptotic and quantitative geometry of groups and spaces

摘要奖：DMS 1612061，首席研究员：Robert Young这个项目提出研究某些空间中的曲面几何，源于数学和计算机科学。曲面是几何体中的基本对象，空间中曲面的形状提供了对空间的洞察。极小或接近极小的曲面的行为尤其重要。例如，像肥皂膜这样的极小表面在许多情况下都是光滑的，但只有接近最小表面的表面可能是粗糙的。PI打算开发新的工具来研究各种空间中的极小和接近极小的曲面，并使用它们来研究几何和分析问题。这些工具的一个拟议应用是研究Goemans-Linial算法的精度，该算法近似以最佳方式将对象切割成两个大致相等的部分。这个算法是许多“分而治之”算法的关键部分，这些算法通过将复杂问题分解成更简单的问题来解决。PI计划开发新的定量工具来研究群和空间中的曲面几何，并将它们应用于几何群论、几何测度论和理论计算机科学中的问题。首先，该项目旨在证明Gromov和瑟斯顿关于对称空间中格的填充函数的一个猜想，从而提供对这些空间的大规模几何的新的理解。其次，该项目将探索欧几里德空间中嵌入的曲面的分解，并尝试使用几何测度论中的新工具来约束这些曲面的几何和拓扑。第三，该项目将分析Heisenberg群和其他幂零群中有限周长的曲面。如果最后一个目标成功，它将导致Heisenberg群嵌入Banach空间的锐界，以及最稀疏割问题最知名近似解的精度的锐界。

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.