Geometric Mapping Theory in Sub-Riemannian and Metric Spaces

PI将研究子黎曼空间和更一般的度量空间中的映射。一个反复出现的主题是研究一般的维数失真性质的映射。几个问题提出了关于精确的，定量的陈述的频率与给定的豪斯多夫维数界举行相对于一个指定的参数空间：这样的频率是衡量措施的参数空间（在许多情况下，这些参数化措施本身豪斯多夫措施）。前面的主题将在两个背景下展开：（i）度量空间值Sobolev映射，和（ii）定义在Heisenberg群和其他Carnot群上的线性和非线性投影型映射。另一个主题所涵盖的拟议研究计划关注新的方面的分析理论的拟共形和拟正则映射的海森堡群。这既包括新的建设等映射以及新兴的连接到偏微分方程和次黎曼外微分公式化的鲁明。其他议题包括在建议包括密度的Lipschitz映射在空间的海森堡群值的Sobolev映射，和豪斯多夫措施可忽略的高度特征点的子流形在sub-Riemannian卡诺集团。这个研究项目处于几何学和分析学的交叉点。几何学是研究空间结构的学科，既有古典时代熟悉的欧几里德空间，也有近代的奇异空间。分析学是对变化和运动的研究：它起源于牛顿和莱布尼茨在17世纪发展的微积分理论，在19世纪后半叶由柯西，维尔斯特拉斯，黎曼和其他人阐明和严格制定，并最终在20世纪在勒贝格，希尔伯特，哈代和（后来的）索博列夫的工作中扩展到现代形式。这两个学科广泛地相互作用：几何结构既影响空间之间映射的动态分析行为，也受其影响。Sobolev空间提供了一个框架来量化“非光滑”函数和映射的分析性质。它们是现代偏微分方程方法的基础工具。理解对这种空间的分析是重要的，原因有几个。首先，这样的理解提供了新的视角的经典理论，突出相关的和必要的功能的基础几何。同时，这种抽象的方法具有更大的适用范围。度量空间的分析技术和结果已经在度量图和网络、分形和其他几何环境（如次黎曼空间）的研究中得到应用。亚黎曼几何出现在“外部”应用的数学模型的控制理论，机器人路径规划和神经生物学，以及“内部”应用的其他分支的纯数学，如几个复杂的变量，辛和接触几何和几何群论。所提出的研究涉及次黎曼映射理论的一个方面涉及阻抗断层成像和图像重建中的数学模型。

The PI will study mappings in and between sub-Riemannian spaces and more general metric spaces. A recurring theme is the study of generic dimension distortion properties of mappings. Several problems are posed concerning precise, quantitative statements of the frequency with which given Hausdorff dimension bounds hold relative to a specified parameter space: such frequency is measured with respect to measures on the parameter space (in many cases, these parameterizing measures are themselves Hausdorff measures). The preceding themes will be developed in two contexts: (i) metric space-valued Sobolev mappings, and (ii) linear and nonlinear projection-type mappings defined on the Heisenberg group and other Carnot groups. Another theme covered by the proposed research program concerns new aspects of the analytic theory of quasiconformal and quasiregular mappings on the Heisenberg group. This includes both novel constructions of such mappings as well as emerging connections to partial differential equations and sub-Riemannian exterior differential calculus as formulated by Rumin. Additional topics included in the proposal include density of Lipschitz mappings in spaces of Heisenberg group-valued Sobolev mappings, and Hausdorff measure negligibility of highly characteristic points on submanifolds in sub-Riemannian Carnot groups. This research program lies at the intersection of geometry and analysis. Geometry is the study of the structure of space, both the familiar Euclidean spaces of classical antiquity as well as more recent, exotic spaces. Analysis is the study of change and motion: its origins lie in the theory of differential and integral calculus developed by Newton and Leibniz in the 17th century, clarified and rigorously formulated by Cauchy, Weierstrass, Riemann and others in the second half of the 19th century, and finally extended to its modern form in the 20th century in work of Lebesgue, Hilbert, Hardy and (later) Sobolev. The two subjects interact extensively: geometric structure both influences and is influenced by the dynamic, analytic behavior of mappings between spaces. Sobolev spaces provide a framework for quantifying the analytic properties of `non-smooth' functions and mappings. They are a foundational tool in modern approaches to partial differential equations. Understanding analysis on such spaces is important for several reasons. First, such understanding provides new perspectives on the classical theory, highlighting relevant and necessary features of the underlying geometry. At the same time, this abstract approach has a greater range of applicability. Techniques and results from analysis on metric spaces have found application in the study of metric graphs and networks, fractals and other geometric environments such as sub-Riemannian spaces. Sub-Riemannian geometries arise in `external' applications to mathematical models of control theory, robotic path planning and neurobiology, as well as `internal' applications to other branches of pure mathematics such as several complex variables, symplectic and contact geometry and geometric group theory. One aspect of the proposed research involving sub-Riemannian mapping theory relates to mathematical models in impedance tomography and image reconstruction.