Workshop on Minimal Surfaces, Sub-Elliptic PDE's and Geometric Analysis

最小曲面、次椭圆偏微分方程和几何分析研讨会

• 批准号: 0503695
• 负责人: Scott Pauls
• 金额: $ 2万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-04-15 至 2006-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503695&HistoricalAwards=false
• 关键词: Workshop; Minimal; Surfaces; Sub; Elliptic

Workshop on Minimal Surfaces, Sub-Elliptic PDE's and Geometric Analysis. The study of PDE's, harmonic analysis, and geometric analysis in the sub-Riemannian setting has reached a critical juncture: recently, researchers from disparate fields have made significant progress in this area and have opened up many new avenues of research. The conference will focus on contemporary developments in the study of several problems from analysis and geometry in the setting of Carnot-Carath\'eodory metrics. Most of the invited lecturers will address a variety of interrelated topics, such as: ``best-constant'' type problems concerning Sobolev and isoperimetric inequalities; the study of minimal and constant-curvature submanifolds; rectifiability and geometric measure theory; quasiconformal maps and potential theory; geometric flows and applications. Analysis in Carnot-Carath\'eodory spaces is an important component in the general theory of abstract, non-smooth analysis which has seen extensive development in recent years. The conference, as envisioned by the PI's, will foster the collaboration of different research groups and provide a ground for discussion. The study of systems whose dynamics is subject to physical constraints has been a focus of attention for a long time, both from the point of view of pure mathematics and from the point of view of engineering and physics. Motivation for these inquiries stems from the wide variety of applications to problems in control theory, robotic planning, the structure of crystalline materials, image reconstruction, nonholonomic mechanics, and others. In mathematical terms such systems are represented by Carnot-Carath\'eodory (sub-Riemannian) spaces. These are manifolds with a preferred set of directions at every point. These preferred directions represent the constraints; motion is only allowed in these directions. The study of geometry and analysis on CC spaces is based on techniques from several mathematical disciplines: several complex variables, contact geometry, partial differential equations, harmonic analysis and geometric function theory. In turn, new results in the sub-Riemannian context often yield important progress in these areas.

最小曲面、次椭圆偏微分方程和几何分析研讨会。亚黎曼背景下的偏微分方程、调和分析和几何分析的研究已经到了一个关键时刻：最近，来自不同领域的研究人员在这一领域取得了重大进展，并开辟了许多新的研究途径。会议将集中在当代发展的几个问题的研究，从分析和几何设置的卡诺-卡拉思\'eodory度量。大多数受邀讲师将解决各种相互关联的主题，如：“最佳常数”型问题有关Sobolev和等周不等式;研究最小和常数曲率子流形;可求正性和几何测度理论;拟共形映射和潜在的理论;几何流和应用。Carnot-Carath\'eodory空间中的分析是抽象非光滑分析一般理论的重要组成部分，近年来得到了广泛的发展。正如PI所设想的那样，会议将促进不同研究小组的合作，并为讨论提供基础。 长期以来，无论是从纯数学的角度，还是从工程和物理的角度，对动力学受物理约束的系统的研究一直是人们关注的焦点。这些调查的动机源于各种各样的应用问题的控制理论，机器人规划，晶体材料的结构，图像重建，非完整力学等。在数学术语中，这样的系统由Carnot-Carath\'eodory（次黎曼）空间表示。这些流形在每一点上都有一组优选的方向。这些首选方向表示约束;仅允许在这些方向上运动。几何和分析CC空间的研究是基于几个数学学科的技术：几个复变量，接触几何，偏微分方程，调和分析和几何函数理论。反过来，在亚黎曼背景下的新结果往往在这些领域产生重要的进展。