Geometric analysis in Carnot groups

该方案的智力核心结合了次黎曼空间和抽象度量空间中的几何测度论、几何函数论和微分几何。该提案由三个部分组成。第一部分主要介绍次黎曼几何测度论，特别是关于Carnot-Caratheodory和欧几里德Hausdorff测度维的比较定理。应用包括多项式类型的非线性欧几里得迭代函数系统的精确维计算。相关项目涉及卡诺群中超曲面的特征可忽略性。一个长期的目标是对喷流空间群中的常值平均曲率曲面进行分类，并着眼于识别其等周不等长的候选极值。第二部分讨论次黎曼几何函数理论，特别是Tukia-Vaisala拟共形延拓定理的Heisenberg类比和一个Heinonen-Semmes问题。在第三部分，PI研究了度量空间上的高度正则满射。这一研究起源于Peano，Lebesgue和Hahn-Mazurkiewicz的经典点集拓扑学结果，同时也受到Morse-Sard理论和可纠正性研究的影响。PI构造了从足够高维的欧几里德空间到双倍测地空间的高度正则满射。未来要考虑的问题包括边界正则性、无限维类比和其他正则性类(Holder和Sobolev映射)。研究了具有次黎曼目标的Soblev空间中Lipschitz映射的稠密性问题。提出的研究涵盖了非光滑几何分析中的一系列主题，但仍由一个公共框架统一。几何研究任意复杂和维度的空间的静态结构，而分析研究此类空间的动态性质和功能相互关系。形容词非光滑暗示非欧几里得的设置：分形图、分层(次黎曼)流形和其他抽象空间。次黎曼几何是“受约束运动的几何学”：它模拟运动受制于先验几何约束的物理情况。它的应用范围非常广泛，包括机器人运动、数字图像重建、计算机视觉、神经生物学和金融数学。次黎曼分析涉及光滑和非光滑技术的复杂混合，因为这些空间既包括光滑结构(在受限方向上)，也包括分形结构(在一般方向上)。该提案在多个层面上整合了研究、教学、服务和推广。研究生培训通过暑期研究计划、研究生核心课程和专题课程的教学以及博士指导进行。建议在本科和中学两级提供与研究有关的教育机会和推广活动。派？S的合作者遍布美国和欧洲。教师、博士后和学生访问这些机构将创造新的合作机会，并提高该领域的知名度。为此目的，国际和平研究所还将继续组织关于次黎曼几何和分析的会议和讲习班。

The intellectual core of the proposal combines geometric measure theory, geometric function theory and differential geometry in sub-Riemannian spaces and abstract metric spaces. The proposal consists of three parts. Part I focuses on sub-Riemannian geometric measure theory, specifically dimension comparison theorems for Carnot-Caratheodory and Euclidean Hausdorff measure and dimension. Applications include sharp dimension computations for nonlinear Euclidean iterated function systems of polynomial type. Related projects concern characteristic negligibility for hypersurfaces in Carnot groups. A long-term goal is to classify constant mean curvature surfaces in jet space groups with an eye to identifying candidate extremals for their isoperimetric inequalities. Part II considers sub-Riemannian geometric function theory, specifically Heisenberg analogs of the Tukia-Vaisala quasiconformal extension theorems and a problem of Heinonen-Semmes. In Part III, the PI studies highly regular surjections to metric spaces. This line of research originates in classical point-set topology results of Peano, Lebesgue and Hahn-Mazurkiewicz and is also influenced by recent work on Morse-Sard theory and rectifiability. The PI has constructed highly regular surjections from Euclidean spaces of sufficiently high dimension onto doubling geodesic spaces. Future problems to be considered include borderline regularity, infinite-dimensional analogs and other regularity classes (Holder and Sobolev maps). A problem of Gromov on density of Lipschitz maps in Sobolev spaces with sub-Riemannian target will be studied. The proposed research encompasses a range of topics within nonsmooth geometric analysis, yet remains unified by a common framework.Geometry studies the static structure of spaces of arbitrary complexity and dimension, while analysis studies dynamic properties and functional interrelations of such spaces. The adjective nonsmooth suggests non-Euclidean settings: fractals, stratified (sub-Riemannian) manifolds, and other abstract spaces. Sub-Riemannian geometry is the `geometry of constrained motion?: it models physical situations where motion is subject to a priori geometric constraints. It features in a remarkably broad spectrum of applications including robotic motion, digital image reconstruction, computer vision, neurobiology, and the mathematics of finance. Sub-Riemannian analysis involves an intricate blend of smooth and nonsmooth techniques as these spaces admit both smooth structure (in restricted directions) and fractal structure (in generic directions). The proposal integrates research, teaching, service and outreach on multiple levels. Graduate student training occurs via summer research programs, teaching of graduate core and topics courses, and Ph.D. supervision. Educational opportunities and outreach related to the research are proposed at the undergraduate and secondary school levels. The PI?s collaborators are located across the U.S. and Europe. Visits to and from these institutions by faculty, postdocs and students will generate new opportunities for collaboration and increase the visibility of the area. To this end, the PI will also continue to organize conferences and workshops in sub-Riemannian geometry and analysis.