Mappings and Measures in Sub-Riemannian and Metric Spaces

这个项目的目标是在粗糙或分形结构的空间中研究几何和分析。该项目特别关注古典欧几里得语言难以描述的空间，如分形和亚黎曼空间。后者作为机器人、神经生物学、天体力学和其他物理系统问题的数学模型出现。在这些模型中，底层的几何结构本质上是不光滑的。因此，拟议的研究将有助于发展数学框架，同时成为各种物理应用的基础。在几何方面，曲率的非光滑概念将被研究。曲率是现代几何中的一个基本概念。例如，广义相对论预言曲率是空间的主要几何特征，负责引力。曲率的非光滑推广及其与测度结构的关系也将被研究。测量理论是对长度、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来扩展微积分的机制非常感兴趣。首席研究员将继续培养研究生、博士后和早期职业研究人员。这项研究计划的结果将通过会议和研讨会的演讲、期刊文章的出版和一般数学观众的展览来传播。PI目前正在与人合作编写一本关于非光滑空间分析的研究生教科书。本书正在进行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一部分获得的进展。该项目关注子黎曼流形中的度量和子流形的几何，以及子黎曼流形之间的映射。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。主要目标是用亚黎曼类比David Preiss的一个著名定理，该定理通过密度来描述可整流性（平滑的测量理论概念）。这条研究路线将扩展到亚黎曼设置欧几里德几何测量理论的一些基本方面，这些方面与子流形的曲率概念有关。本研究的主要方法包括用黎曼空间逼近次黎曼空间，并分析黎曼近似中几何量的极限行为。第二个主题是关于共形映射和拟正则映射的迭代。这种迭代方案为分形对象提供了一种基本的组合模型。它们在双曲几何和数论中都很自然地出现。在他之前的相似性映射工作的基础上，主要研究者将把分析扩展到保角映射。目的是详细描述亚黎曼空间中自共形极限集的几何形状及其上的自然测度。最后，主要研究者将研究亚黎曼拟正则映射的类型问题。这里的目标是理解准正则映射的灵活性以及它们构造的固有几何障碍。主要研究者还将研究Sobolev映射的新构造，包括最优测量和维度畸变，以及在Sobolev映射空间中具有亚黎曼目标的Lipschitz映射的密度标准。首席研究员将利用他在上述所有领域的专业知识以及相关领域的最新进展来寻求所提出问题的答案。拟议的研究将产生广泛的影响；它在不同数学领域的交叉点上解决了广泛的问题。主要研究者将使用这个广泛的框架来确定适合初级研究生和博士后的具体问题。

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.