Geometric analysis in Carnot groups

卡诺群中的几何分析

• 批准号: 0901620
• 负责人: Jeremy Tyson
• 金额: $ 23.14万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901620&HistoricalAwards=false
• 关键词: Geometric; analysis; Carnot; groups

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.

该建议的智能核心结合几何测度理论，几何函数理论和微分几何在次黎曼空间和抽象度量空间。该建议包括三个部分。第一部分着重于次黎曼几何测度理论，特别是卡诺-Caratheodory和欧几里德Hausdorff测度和维数的维数比较定理。应用程序包括尖锐的尺寸计算非线性欧几里德迭代函数系统的多项式类型。相关项目关注卡诺群中超曲面的特征可忽略性。一个长期的目标是分类恒定的平均曲率表面喷射空间组的眼睛，以确定候选极值等周不等式。第二部分考虑次黎曼几何函数理论，特别是海森堡类似的Tukia-Vaisala拟共形扩张定理和问题的海诺南-塞姆斯。在第三部分中，PI研究度量空间的高度正则满射。这条研究线起源于经典的点集拓扑结果的Peano，Lebesgue和Hahn-Mazurkiewicz，也受到最近的工作Morse-Sard理论和rectifiability。PI已经从足够高维的欧氏空间到加倍测地空间构造了高度正则满射。未来要考虑的问题包括边界规则性、无限维类似物和其他规则性类（保持器和索博列夫映射）。研究了Sobolev空间中具有次黎曼目标的Lipschitz映射的密度的Gromov问题。拟议的研究包括一系列的主题在非光滑几何分析，但仍然统一的一个共同的framework.Geometry研究静态结构的空间的任意复杂性和尺寸，而分析研究动态特性和功能的相互关系，这样的空间。形容词nonsmooth表示非欧几里德环境：分形，分层（亚黎曼）流形和其他抽象空间。亚黎曼几何是“受约束运动的几何？”它对运动受到先验几何约束的物理情况进行建模。它具有非常广泛的应用，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，因为这些空间既允许光滑结构（在限制方向上），也允许分形结构（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。研究生培训通过夏季研究计划，研究生核心和主题课程的教学，以及博士学位。监管建议在大学和中学两级提供与研究有关的教育机会和外联活动。私家侦探？的合作者遍布美国和欧洲。教师，博士后和学生对这些机构的访问将产生新的合作机会，并提高该地区的知名度。为此，PI还将继续组织亚黎曼几何和分析方面的会议和研讨会。