Nonsmooth methods in geometric function theory and geometric measure theory on the Heisenberg group

海森堡群几何函数论和几何测度论中的非光滑方法

• 批准号: 0555869
• 负责人: Jeremy Tyson
• 金额: $ 9.97万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-15 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555869&HistoricalAwards=false
• 关键词: Nonsmooth; methods; geometric; function; theory

Abstract TysonThe proposed research centers on a suite of problems at the interface between differential geometry, geometric measure theory and geometric function theory in the Heisenberg group and more general sub-Riemannian (Carnot-Caratheodory) spaces. The unifying theme is the development and application in this context of effective tools from nonsmooth metric geometry. One series of problems focuses on the geometry of submanifolds with possible application to the celebrated Heisenberg isoperimetry conjecture of Pierre Pansu. Sub-Riemannian analogs of the classical machinery of differential geometry have recently been introduced by Garofalo et al, Pauls, and Franchi et al, among others. In joint work with Capogna and Pauls, the PI will further develop this machinery in order to gain a more intrinsic understanding of the geometry of Carnot-Caratheodory submanifolds. This investigation is currently limited to surfaces given either intrinsically or extrinsically, as level sets of or parameterized by highly regular functions. Dimension jump phenomena and the size of the characteristic set for graphs of weakly regular functions will be investigated by the PI's graduate student John Maki in his thesis. A second line of research focuses on sub-Riemannian geometric function theory and fractal geometry. This includes the search for effective symmetrization procedures, existence, extension and regularity problems for quasiconformal maps, metric regularity of rough domains (John and uniform domains, domains satisfying a quasihyperbolic growth condition), and the structure of self-affine tilings. Finally (joint with Z. M. Balogh), the nonsmooth first-order calculus of Cheeger-Keith will be investigated in connection with exotic metrics on the Heisenberg group with an eye towards constructing new examples of spaces on which such calculus can be developed.Sub-Riemannian geometry is the "geometry of constrained motion"; it provides a mathematical model for any physical situation in which allowable motion is subject to a priori geometric constraints. Historically, its roots lie in Carnot's work on thermodynamics, but the subject has progressed significantly beyond these motivating questions to a central position in modern nonsmooth geometric analysis, and has recently seen remarkable applications in numerous areas, including robotic path planning, remote control of satellites, digital image reconstruction and computer vision, neurobiology, and the mathematics of finance. There are direct links between one aspect of the proposed research (sub-Riemannian differential geometry of submanifolds and the isoperimetric problem) and emerging models for the function and structure of the mammalian visual cortex. Nonsmooth techniques and methods are essential in geometric analysis in view of the incompleteness of classical spaces of smooth functions and sets; solutions to differential equations and variational problems cannot be guaranteed unless the domain of definition is widened to a suitably large family of (nonsmooth) candidates (although in hindsight, smoothness for such solutions can often be established a posteriori). The proposal includes an outreach component, joint with Capogna and Pauls, involving cross-training of graduate students and postdocs, a series of conferences, workshops and summer schools, expository articles and monographs, and an online forum for researchers in sub-Riemannian geometry aimed at developing a North American presence in this exciting field on par with the established centers of study in Europe and Australia.

摘要类型所提出的研究集中在Heisenberg群和更一般的次黎曼(Carnot-Caratheodory)空间中的微分几何、几何测度论和几何函数论之间的一系列问题。统一的主题是在这种背景下非光滑度量几何的有效工具的发展和应用。其中一系列问题集中在子流形的几何上，并可能应用于著名的皮埃尔·潘苏的海森堡等周测猜想。最近，Garofalo等人、Pauls和Franchi等人引入了微分几何经典机械的次黎曼类比。在与Capogna和Pauls的联合工作中，PI将进一步发展这一机制，以便对卡诺-Caratheodory子流形的几何有更内在的了解。这项研究目前仅限于作为高度正则函数的水平集或由高度正则函数参数化的内在或外在给定的曲面。PI的研究生John Maki将在他的论文中研究弱正则函数图的跳维现象和特征集的大小。第二条线的研究重点是次黎曼几何函数理论和分形几何。这包括寻找有效的对称化过程，拟共形映射的存在性、延拓和正则性问题，粗糙域(John域和一致域，满足拟双曲增长条件的域)的度量正则性，以及自仿射平铺的结构。最后(与Z.M.Balogh联合)，将结合Heisenberg群上的奇异度量来研究Cheeger-Keith的非光滑一阶微积分，以期构建在其上可以发展这种微积分的空间的新例子。次黎曼几何是“约束运动的几何”；它为任何允许的运动受到先验几何约束的物理情况提供了数学模型。从历史上看，它的根源在于卡诺在热力学方面的工作，但这门学科已经大大超越了这些激励性的问题，成为现代非光滑几何分析的中心位置，最近在许多领域都有显著的应用，包括机器人路径规划、卫星远程控制、数字图像重建和计算机视觉、神经生物学和金融数学。在拟议研究的一个方面(子流形的次黎曼微分几何和等周问题)与哺乳动物视觉皮质功能和结构的新兴模型之间存在直接联系。鉴于光滑函数和集合的经典空间的不完备性，非光滑技术和方法在几何分析中是必不可少的；除非将定义范围扩大到适当大的(非光滑)候选者家族，否则不能保证微分方程和变分问题的解(尽管事后看来，此类解的光滑性通常可以后验地建立)。该提案包括与Capogna和Pauls联合开展的外联部分，涉及研究生和博士后的交叉培训、一系列会议、讲习班和暑期学校、说明性文章和专著，以及一个面向次黎曼几何研究人员的在线论坛，旨在发展北美在这一令人兴奋的领域的存在，与欧洲和澳大利亚的现有研究中心相媲美。