Rigidity of Lipschitz and Related Mappings on Metric Spaces

微积分最基本的背景是研究函数在实线上的光滑变换。在这个项目中，将研究定义在几何对象(度量空间)上的更一般的、非光滑的函数，这些几何对象(度量空间)可能与真实的直线非常不同。这些几何可能是抽象的对象，没有合理地嵌入到任何欧几里德几何中，或者它们可能是分形的，或者它们可能在所有尺度上允许复杂的行为。非光滑分析和几何现在已经成为纯数学和应用数学以及计算机科学的许多领域中的重要工具，在这些领域可能会出现非欧几里德几何，而不是技术上的好奇。例如，从离散群或动力系统，作为光滑对象的极限，作为大型数据集，或在计算问题中。更准确地说，这个项目的目标是研究非光滑空间的无穷小和全局几何之间的关系，另一方面，分析定义在这些空间上的Lipschitz和相关的映射类。一个具体的目标是进一步理解允许实值Lipschitz函数(在Cheeger意义下)可微的空间：这样的空间可以具有什么拓扑和几何性质，我们可以构造新的例子吗？另一个目的是了解度量空间中Lipschitz映射和可直曲线的刚性定理。例如，度量空间之间的Lipschitz映射何时必须在其定义域的大子集上具有更严格的(例如，双Lipschitz)行为？我们能否通过局部平坦性条件来刻画度量空间中的可修曲线，就像琼斯在平面上的“分析师推销员定理”中那样？理解这些问题需要将经典几何测度论中的技术与较新的“度量空间分析”领域的技术相结合。像这样的分析性调查也提供了，并应该继续提供对非光滑几何形状的洞察。

The most basic setting for differential calculus is the study of smoothly changing functions on the real line. In this project, more general, non-smooth, functions that are defined on geometric objects (metric spaces) that may be very different from the real line will be investigated. These geometries may be abstract objects with no reasonable embedding into any Euclidean geometry, or they may be fractal, or they may otherwise admit complicated behavior at all scales. Far from being a technical curiosity, non-smooth analysis and geometry have now become important tools in many areas of pure and applied mathematics and computer science, where non-Euclidean geometries may arise. For example, from discrete groups or dynamical systems, as limits of smooth objects, as large data sets, or in computational problems.In more precise terms, the goal of this project is to study the relationship between, on the one hand, the infinitesimal and global geometry of non-smooth spaces and, on the other hand, the analysis of Lipschitz and related classes of mappings defined on these spaces. One specific goal is to further understand the spaces which allow for differentiation of real-valued Lipschitz functions (in the sense of Cheeger): what topological and geometric properties can such spaces possess, and can we construct new examples? Another goal is to understand rigidity theorems for Lipschitz mappings and rectifiable curves in metric spaces. For example, when must Lipschitz mappings between metric spaces have more rigid (e.g., bi-Lipschitz) behavior on large subsets of their domains? Can we characterize rectifiable curves in metric spaces via local flatness conditions, as in the "Analyst's Traveling Salesman Theorem" of Jones in the plane? Understanding these questions involves combining techniques from classical geometric measure theory with those of the newer field of "analysis on metric spaces". Analytic investigations like these have also provided, and should continue to provide, insights into the geometry of the non-smooth.