Rigidity of Lipschitz and Related Mappings on Metric Spaces

经典微积分研究欧几里德空间内函数、曲线和曲面的平滑变化。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学习微积分。这包括非光滑函数或集合、通常情况下的抽象几何、欧几里得、几何和允许在许多尺度上具有复杂行为的分形空间。该项目旨在通过将这些物体分解成更简单的片段，将它们嵌入到经典几何中，或者用线性物体近似来理解这些物体。非光滑分析和几何在纯数学和应用数学和计算机科学的许多领域中都很重要，因为在研究大数据集、计算问题和光滑问题的极限情况下会出现非光滑问题。该项目还将强调“定量”的结果：提供与正在研究的特定函数或几何无关的有保证的估计。更准确地说，该项目侧重于Lipschitz(和相关的)映射。在他最近合作工作的基础上，这位首席研究员将研究到度量空间的Lipschitz映射何时可以分解为更简单的映射或由更简单的映射很好地逼近，比如线性映射或通过树的因子映射。然后，首席研究人员计划应用这些技术，在通过Lipschitz映射定义的几何对象上提供新的结构结果，如距离球体或中轴。主要研究人员还将研究与度量空间中的曲线有关的类似问题，继续研究将曲线长度与定量平坦性条件联系起来的所谓“分析师的旅行商定理”的扩展。此外，主要研究者将研究非光滑空间上Lipschitz函数的可微性的概念，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有界失真的欧几里德空间？”)有关，该问题在几何和计算机科学中具有重要意义，也将在项目期间进行研究。所有这些相关的调查将结合在一起，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，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.