Mappings and Measures in Sub-Riemannian and Metric Spaces

这个项目的目标是研究具有粗糙或分形结构的空间中的几何和分析。该项目特别关注经典欧几里德语言描述不佳的空间，如分数维和次黎曼空间。后者作为机器人学、神经生物学、天体力学和其他物理系统中问题的数学模型出现。在每个模型中，底层几何图形本质上都是非平滑的。因此，拟议的研究将有助于发展同时作为各种物理应用的基础的数学框架。在几何方面，我们将研究曲率的非光滑概念。曲率是现代几何中的一个基本概念。例如，广义相对论预测，曲率是空间产生重力的主要几何特征。曲率的非光滑推广及其与测度结构的关系也将被研究。度量论是对长度、面积和体积等基本概念的广泛数学概括。最后，分析是指作用于空间之间的变换的动态特性。由于经典的牛顿-莱布尼茨导数理论不能很好地适应非光滑空间的变换，目前人们对开发一种新的分析工具包以扩展微积分机器的通常背景很感兴趣。首席研究员将继续培训研究生、博士后和职业生涯早期的研究人员。这项研究计划的结果将通过在会议和研讨会上的演讲、发表期刊文章和向普通数学受众进行阐述来传播。PI目前正在合著一本关于非光滑空间分析的研究生教科书。正在进行的这本书的准备工作将与该领域的进展相协调，包括作为本研究计划的一部分而获得的进展。该项目涉及次黎曼流形中的度量和子流形的几何以及次黎曼流形中的映射。将审议三个协调主题。第一组问题涉及测度的密度和子流形的几何。其主要目的是通过密度来刻画可整性(光滑性的一种测度论概念)的一个著名定理的次黎曼模拟。这方面的研究将延伸到次黎曼集，欧几里得几何测度论的一些基本方面与子流形的曲率概念有关。这项研究的主要方法涉及黎曼空间对次黎曼空间的逼近，以及对黎曼逼近中几何量的极限行为的分析。第二个主题涉及共形和拟正则映射的迭代。这种迭代方案为分形对象提供了一个基本的组合模型。它们自然而然地出现在双曲几何和数论中。在他之前关于相似映射的工作的基础上，这位首席研究员将把分析扩展到共形映射。本课程的目的是详细描述次黎曼空间中的自共形极限集的几何以及此类集合上的自然测度。最后，主要研究者将研究次黎曼拟正则映象的类型问题。这里的目标是理解准正则映射的灵活性以及构造它们所固有的几何障碍。主要研究者还将研究导致最优测度和维度偏差的Sobolev映射的新构造，以及具有次Riemannian目标的Soblev映射空间中Lipschitz映射的稠密性的准则。首席调查员将利用他以前在上述所有领域的专业知识以及相关领域的最新进展，寻求对拟议问题的答案。拟议的研究将产生广泛的影响；它解决了不同数学领域交叉的广泛问题。首席调查员将使用这个广泛的框架来确定适合研究生和博士后初学者的具体问题。

The goal of this project is to investigate geometry and analysis in spaces with rough or fractal structure. The project focuses especially on spaces which are poorly described by classical Euclidean language, such as fractals and sub-Riemannian spaces. The latter occur as mathematical models for problems in robotics, neurobiology, celestial mechanics and other physical systems. In each of these models, the underlying geometry is inherently nonsmooth. The proposed research will consequently contribute to developing the mathematical framework which simultaneously underlies a variety of physical applications. On the geometric side, nonsmooth notions of curvature will be studied. Curvature is a fundamental concept in modern geometry. For instance, general relativity predicts that curvature is the primary geometric feature of space responsible for gravity. Nonsmooth generalizations of curvature and their relationship to the structure of measures will also be studied. Measure theory is a wide-ranging mathematical generalization of fundamental concepts of length, area, and volume. Finally, analysis refers to the dynamic properties of transformations acting between spaces. Since the classical Newton--Leibniz theory of the derivative is not well-adapted to transformations of nonsmooth spaces, there is currently significant interest in developing a new analytic toolkit to extend the machinery of calculus beyond its usual context. The principle investigator will continue to train graduate students, postdocs and early-career researchers. Results of this research program will be disseminated via talks at conferences and workshops, publication of journal articles, and exposition for a general mathematical audience. The PI is currently coauthoring a graduate textbook on analysis in nonsmooth spaces. The ongoing preparation of this book will be coordinated with advances in the field, including those obtained as part of this research program.The project concerns the geometry of measures and submanifolds in, and mappings between, sub-Riemannian manifolds. Three coordinating themes will be considered. The first set of problems concerns density of measures and the geometry of submanifolds. The main goal is a sub-Riemannian analog of a celebrated theorem of David Preiss characterizing rectifiability (a measure-theoretic notion of smoothness) via densities. This line of investigation will extend to the sub-Riemannian setting some fundamental aspects of Euclidean geometric measure theory tied to notions of curvature for submanifolds. The primary methodology for this investigation involves the approximation of sub-Riemannian spaces by Riemannian spaces, and an analysis of the limiting behavior of geometric quantities in the Riemannian approximations. The second theme concerns iteration of conformal and quasiregular mappings. Such iterative schemes provide a basic combinatorial model for fractal objects. They arise naturally in both hyperbolic geometry and number theory. Building on his previous work for similarity mappings, the principle investigator will extend the analysis to conformal mappings. The goal is a detailed description of the geometry of self-conformal limit sets in sub-Riemannian spaces and natural measures on such sets. Finally, the principle investigator will study the type problem for sub-Riemannian quasiregular mappings. The goal here is to understand both the flexibility of quasiregular mappings as well as inherent geometric obstructions to their construction. The principle investigator will also investigate new constructions of Sobolev mappings inducing optimal measure and dimension distortion, as well as criteria for the density of Lipschitz mappings in spaces of Sobolev mappings with sub-Riemannian targets. The principle investigator will take advantage of his prior expertise in all of the aforementioned areas, as well as the latest advances in related fields, to pursue answers to the proposed problems. The proposed research will have a broad impact; it addresses a wide range of questions at the intersection of different mathematical areas. The principle investigator will use this broad framework to identify specific problems suitable for beginning graduate students and postdocs.