Rigidity of Lipschitz and Related Mappings on Metric Spaces

微分学最基本的设置是研究在实线上平滑变化的函数。在这个项目中，将研究在几何对象（度量空间）上定义的更一般，非光滑的函数，这些函数可能与实际线非常不同。这些几何图形可能是抽象的物体，没有合理地嵌入任何欧几里得几何，或者它们可能是分形的，或者它们可能在所有尺度上都承认复杂的行为。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许多领域的重要工具，而非欧几里得几何可能在这些领域出现。例如，从离散群或动力系统，作为光滑物体的极限，作为大数据集，或在计算问题。更精确地说，这个项目的目标是一方面研究非光滑空间的无穷小几何和整体几何之间的关系，另一方面分析在这些空间上定义的Lipschitz和相关类映射。一个具体的目标是进一步理解允许实值Lipschitz函数微分的空间（在Cheeger的意义上）：这样的空间可以拥有什么样的拓扑和几何性质，我们可以构建新的例子吗？另一个目标是理解度量空间中Lipschitz映射和可整流曲线的刚性定理。例如，度量空间之间的Lipschitz映射何时必须在其域的大子集上具有更严格的（例如，双Lipschitz）行为？我们能否像琼斯在平面上的“分析家旅行推销员定理”那样，通过局部平坦性条件来表征度量空间中的可矫正曲线？理解这些问题需要将经典几何测量理论的技术与新领域“度量空间分析”的技术相结合。像这样的分析研究也提供了，并将继续提供，对非光滑几何的见解。

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.