Analysis and Geometry in Metric Spaces

这个项目寻求认识度量空间的几何和拓扑性质，这些性质允许发展类似于欧几里德空间的分析理论。虽然拓扑学、几何学和分析在二维平面上是统一的，但高维欧几里得空间或抽象度量空间缺乏这样强大的工具。这些考虑促使了“度量空间分析”领域的发展，其中一阶微积分和几何测度论从经典的欧几里得或黎曼环境扩展到没有先验光滑结构的空间领域(如分形)。这一领域的结果和技巧在几何群论、流形结构和分形学分析中都有重要的应用。此外，除了它们在数学上的重要性，这些理论的物理应用范围从大数据集中缺失数据的重建，到数据存储和访问的方法，以及薄膜的研究。这个项目试图开发技术来解决度量空间分析和几何测量理论领域中的几个长期存在的问题。第一个目标是将非光滑流形上离散形式的曲率的积分界与局部欧氏双Lipschitz参数化联系起来。这种参数化在两个维度上是很好理解的，但到目前为止，在大于或等于三维的维度上是难以捉摸的。另一个目标是找出2-可纠性的充分条件，即了解正方形的Lipschitz映象中包含哪些集合。这一方向的结果将反过来导致对2--可纠正措施的更好理解。最后，该项目解决了欧几里德嵌入问题，即刻画那些允许嵌入到有限维欧几里德空间中的度量空间，并且该空间不会太大地扭曲空间的几何。除了提供对度量空间的几何和分析的更好的理解外，这种嵌入的存在对理论计算机科学和图形成像的最新进展起到了重要作用。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，被认为是值得支持的。

This project seeks to recognize the geometric and topological qualities of metric spaces that allow for the development of a theory of analysis similar to that of Euclidean spaces. While topology, geometry, and analysis are united in the two-dimensional plane, higher dimensional Euclidean spaces or abstract metric spaces lack such powerful tools. Such considerations prompted the development of the field of "analysis on metric spaces," in which first-order differential calculus and geometric measure theory are extended from the classical Euclidean or Riemannian setting to the realm of spaces without a priori smooth structure (such as fractals). Results and techniques in this field have found important applications in geometric group theory, in the structure of manifolds, and in analysis on fractals. Furthermore, besides their mathematical importance, physical applications of these theories range from the reconstruction of missing data in large data sets, to methodologies for data storage and access, and to the study of thin films.This project seeks to develop techniques to address several long-standing questions in the field of analysis on metric spaces and geometric measure theory. The first goal is to relate integral bounds for discrete forms of curvature on non-smooth manifolds with locally Euclidean bi-Lipschitz parameterizations. Such parameterizations are well understood in two dimensions but have so far been elusive in dimensions greater or equal to three. Another goal is to identify sufficient conditions for 2-rectifiability, that is, to understand which sets are contained within the Lipschitz image of a square. Results in this direction will in turn lead to an improved understanding of 2-rectifiable measures. Finally, the project addresses the Euclidean embedding question, namely, to characterize those metric spaces that admit an embedding into a finite-dimensional Euclidean space that does not distort the geometry of the space too much. Apart from providing a better understanding of the geometry and analysis of metric spaces, the existence of such embeddings has been instrumental in recent advances in theoretical computer science and graphic imaging.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.