Rigidity of Lipschitz and Related Mappings on Metric Spaces

微分学最基本的背景是研究在真实的直线上光滑变化的函数。在这个项目中，更一般的，非光滑的，几何对象（度量空间），可能是非常不同的真实的线上定义的功能将被调查。这些几何可能是抽象的对象，没有合理的嵌入到任何欧几里得几何中，或者它们可能是分形的，或者它们可能在所有尺度上都有复杂的行为。非光滑分析和几何学已经不再是一种技术上的好奇心，而是在纯数学、应用数学和计算机科学的许多领域中成为重要的工具，在这些领域中可能会出现非欧几里德几何。 例如，从离散群或动力系统，作为光滑对象的限制，作为大数据集，或在计算问题。更确切地说，这个项目的目标是研究之间的关系，一方面，无穷小和整体几何的非光滑空间，另一方面，分析Lipschitz和相关类别的映射定义在这些空间。一个具体的目标是进一步了解空间，允许微分的实值Lipschitz功能（在意义上的Cheeger）：什么拓扑和几何性质可以这样的空间拥有，我们可以构建新的例子？另一个目标是理解度量空间中Lipschitz映射和可求长曲线的刚性定理。例如，当度量空间之间的Lipschitz映射必须具有更刚性（例如，bi-Lipschitz）行为的大子集？我们能否通过局部平坦性条件来刻画度量空间中的可求长曲线，就像琼斯在平面上的“分析家的旅行商定理”那样？理解这些问题涉及到从经典几何测度理论与那些较新的领域相结合的技术“度量空间的分析”。 像这样的分析研究也提供了，并应继续提供，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.