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

平面形式、Bi-Lipschitz 参数化和奇异空间上的微积分

• 批准号: 1058283
• 负责人: Mario Bonk
• 金额: $ 11.61万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1058283&HistoricalAwards=false
• 关键词: Flat; Forms; Bi; Lipschitz; Parametrizations

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.

数学中到处都会出现奇异（或非滑动）对象。仅提及几个实例，它们在以下情况下出现：作为光滑对象的限制（诸如Riemannian歧管或光滑功能之类的空间）；作为有限生成的组的渐近“在无穷大的”中，反映了群体在大尺度上的行为的对象；或在混凝土和实际情况（例如不规则的晶体或其他材料）的精细尺度结构中。主要研究者研究与单数对象及其几何形状有关的问题，他对此类对象进行了分析。具体而言，他试图了解可以将经典差异（一阶）分析的概念引入此类空间的程度。例如，一个人希望在某些单数空间上拥有一个弱不同函数的Sobolev空间。同样重要的是要了解哪些潜在的非常奇异的空间可以通过“尼斯”空间（例如，通过欧几里得空间）进行参数化，从而使仅在固定边界内扭曲基本度量结构的转换。首席研究员和他的学生正在开发解决此类问题的新工具。最后，可以在某些有限维欧几里得空间中嵌入性的要求代替欧几里得空间的参数化问题。从项目中出现的方法也应阐明此问题。拟议的研究以两种方式与应用有关。首先，从材料的局部微观结构到宇宙的大规模特征。理解和处理这种奇异性是现代数学和科学的核心目标之一。在可以分析奇异性然后以最低成本转换以更好地表现模型的情况下，主要研究者为解决该问题的解决方案做出了贡献。其次，尽管与项目没有直接相关，但研究对理论计算机科学有潜在的应用，其中大而复杂的数据集需要以更简单的形式进行转换和存储。