Geometric Mapping Theory in Sub-Riemannian and Metric Spaces

PI将研究亚黎曼空间和更一般度量空间之间的映射。一个反复出现的主题是对映射的一般维度畸变特性的研究。关于给定的豪斯多夫维边界相对于特定参数空间保持的频率的精确、定量表述提出了几个问题：这种频率是相对于参数空间上的度量来测量的（在许多情况下，这些参数化度量本身就是豪斯多夫度量）。前面的主题将在两个背景下发展：(i)度量空间值Sobolev映射，（ii）在Heisenberg群和其他卡诺群上定义的线性和非线性投影型映射。另一个被提议的研究计划所涵盖的主题是关于海森堡群上的拟共形和拟正则映射的解析理论的新方面。这既包括这种映射的新构造，也包括与偏微分方程和Rumin制定的次黎曼外微分的新联系。本文还讨论了Heisenberg群值Sobolev映射空间中Lipschitz映射的密度，以及子黎曼卡诺群中子流形上高特征点的Hausdorff测度可忽略性。这个研究项目是几何学和分析学的交叉。几何是对空间结构的研究，既包括古典时代熟悉的欧几里得空间，也包括近代的奇异空间。分析是对变化和运动的研究：它起源于17世纪由牛顿和莱布尼茨发展的微分和积分理论，19世纪下半叶由柯西、魏尔斯特拉斯、黎曼等人澄清和严格表述，最终在20世纪勒贝格、希尔伯特、哈代和（后来的）索博列夫的工作中扩展到其现代形式。这两个主题广泛地相互作用：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空间为量化“非光滑”函数和映射的解析性质提供了一个框架。它们是现代偏微分方程方法的基本工具。理解对这些空间的分析很重要，原因如下。首先，这种理解为经典理论提供了新的视角，突出了基础几何的相关和必要特征。同时，这种抽象的方法具有更大的适用范围。度量空间分析的技术和结果在度量图和网络、分形和其他几何环境（如亚黎曼空间）的研究中得到了应用。亚黎曼几何出现在控制理论、机器人路径规划和神经生物学的数学模型的“外部”应用中，也出现在纯数学的其他分支的“内部”应用中，如几个复杂变量、辛几何和接触几何以及几何群论。提出的研究涉及亚黎曼映射理论的一个方面涉及阻抗层析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.