Rigidity of Lipschitz and Related Mappings on Metric Spaces

经典微积分研究欧氏空间内平滑变化的函数、曲线和曲面。在这个项目中，首席研究员将在经典工具并不总是适用的领域研究微积分。这包括非光滑函数或集合，通常之外的抽象几何，欧几里得几何和分形空间在许多尺度上承认复杂的行为。该项目旨在通过将这些对象分解为更简单的片段，将其嵌入经典几何形状或通过线性对象近似来理解它们。非光滑分析和几何在纯数学、应用数学和计算机科学的许多领域都很重要，因为非光滑问题出现在研究大数据集、计算问题和光滑问题的限制情况中。该项目还将强调“定量”的结果：提供与所研究的特定函数或几何无关的有保证的估计。更确切地说，该项目侧重于Lipschitz（及相关）映射。在他最近的合作工作的基础上，首席研究员将研究Lipschitz映射到度量空间时可以分解为或近似为更简单的映射，如线性映射或通过树因子的映射。然后，首席研究员计划应用这些技术来提供通过Lipschitz映射定义的几何对象的新结构结果，如距离球或中轴。首席研究员还将研究与度量空间中的曲线相关的类似问题，继续研究所谓的“分析师旅行推销员定理”的扩展，该定理将曲线的长度与定量平坦条件联系起来。此外，首席研究员将研究非光滑空间上Lipschitz函数的可微性概念，以及它们如何约束几何。这与双Lipschitz嵌入问题（“哪些空间可以嵌入到具有有界失真的欧几里得空间中？”）有关，这在几何学和计算机科学中非常重要，也将在项目期间进行研究。所有这些相关的研究将联合收割机结合起来，以提高我们对非光滑物体的分析和几何形状的理解。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.