Flat Forms, Bi-Lipschitz Parametrizations, and Calculus on Singular Spaces

奇异(或非光滑)物体在数学中随处可见。仅举几个例子，它们出现在以下情况中：作为光滑对象(如黎曼流形或光滑函数等空间)的极限；作为有限生成群的渐近“无限球体”，即在大尺度上反映群的行为的对象；或在集合的精细尺度结构中，甚至在具体和实际情况下(例如，不规则晶体或其他材料)。首席研究员研究与奇异物体及其几何有关的问题，并对这些物体进行分析。具体地说，他试图了解经典微分(一阶)分析的概念可以在多大程度上被引入这样的空间。例如，人们希望在某些奇异空间上有一个由弱可微函数组成的定义良好的Soblev空间。同样重要的是，了解哪些潜在的非常奇异的空间可以通过仅在固定界限内扭曲基本度量结构的变换，被“好的”空间(例如，通过欧几里德空间)参数化。首席研究员和他的学生正在开发处理这类问题的新工具。最后，将欧氏空间的参数化问题转化为对有限维欧氏空间的可嵌入性要求。从项目中出现的方法也应该澄清这个问题。拟议的研究在两个方面涉及到应用。首先，奇点(或杂质)在自然界中无处不在，从物质的局部微结构到宇宙的大尺度特征。理解和处理这种奇点是现代数学和科学的中心目标之一。在奇点可以被分析，然后以最小的成本转换成行为更好的模型的情况下，首席研究员为解决这个问题做出了贡献。其次，尽管与该项目没有直接关系，但这项研究在理论计算机科学中有潜在的应用，在理论计算机科学中，大型和复杂的数据集需要转换并以更简单的形式存储。

Singular (or nonsmooth) objects arise everywhere in mathematics. To mention just a few instances, they turn up in the following situations: as limits of smooth objects (spaces such as Riemannian manifolds, or smooth functions); as asymptotic "spheres at infinity" of finitely generated groups, objects that reflect the behavior of the groups at large scales; or in the fine-scale structure of sets, even in concrete and practical circumstances (irregular crystals, for instance, or other materials). The principal investigator studies questions that relate to singular objects and their geometry, and he does analysis on such objects. Specifically, he seeks to understand the extent to which the concepts of classical differential (first-order) analysis can be introduced into such spaces. For example, one would like to have a well-defined Sobolev space of weakly differentiable functions on certain singular spaces. It is also important to understand which potentially very singular spaces can be parametrized by "nice" spaces (say, by Euclidean spaces) via transformations that distort the basic metric structure only within fixed bounds. The principal investigator and his students are developing new tools for approaching this type of problem. Finally, the question of parametrization by a Euclidean space can be replaced with the requirement of embedability in some finite-dimensional Euclidean space. The methods that emerge from the project should clarify this problem as well.The proposed research relates to applications in two ways. First, singularities (or impurities) occur everywhere in nature, from the local microstructure of materials to the large-scale features of the universe. Understanding and dealing with such singularities is one of the central objectives of modern mathematics and science. The principal investigator has made contributions to the solution of this problem in cases where the singularities can be analyzed and then transformed, with minimal cost, to better behaved models. Second, although not directly related to the project, there are potential applications of the research to theoretical computer science, where large and complex data sets need to be transformed and stored in simpler form.