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

奇异（或非光滑）对象在数学中无处不在。仅举几个例子，它们在以下情况下出现：作为光滑对象（如黎曼流形或光滑函数的空间）的极限；作为有限生成群的渐近“无穷球”，反映群在大尺度上行为的对象；或者在集合的精细结构中，甚至在具体和实际的环境中（例如不规则的晶体或其他材料）。首席研究员研究与单一物体及其几何形状有关的问题，并对这些物体进行分析。具体地说，他试图理解古典微分（一阶）分析的概念在多大程度上可以引入这样的空间。例如，我们希望在某些奇异空间上有一个定义良好的Sobolev空间，它包含弱可微函数。同样重要的是要理解哪些潜在的非常奇异的空间可以通过“好的”空间（比如欧几里得空间）通过在固定范围内扭曲基本度量结构的变换来参数化。首席研究员和他的学生正在开发解决这类问题的新工具。最后，用有限维欧几里德空间的可嵌入性要求代替欧几里德空间的参数化问题。项目中出现的方法也应该澄清这个问题。拟议的研究涉及两个方面的应用。首先，奇点（或杂质）在自然界中无处不在，从材料的局部微观结构到宇宙的大尺度特征。理解和处理这样的奇点是现代数学和科学的中心目标之一。首席研究员对这个问题的解决做出了贡献，在这种情况下，奇点可以被分析，然后以最小的成本转换为性能更好的模型。其次，尽管与该项目没有直接关系，但该研究在理论计算机科学中有潜在的应用，其中需要将大型复杂数据集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.