Nonsmooth Structures and Geometric Function Theory

非光滑结构与几何函数理论

• 批准号: 0353549
• 负责人: Mario Bonk
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0353549&HistoricalAwards=false
• 关键词: Nonsmooth; Structures; Geometric; Function; Theory

AbstractHeinonenThe PI will search for conditions that help recognizing when a given metric space can be parametrized by a homeomorphism that changes distances only in a controlled manner; that is, we ask for (local) bi-Lipschitz parametrizations by Euclidean space. This question is not suficiently understood even for two dimensional surfaces lying in Euclidean three space. There is a direct link from the parametrization problem to the problem of understanding what measurable structures in Euclidean space are locally standard in that they arise as pullbacks of the standard structure by a homeomorphism. This latter question can be asked both in bi-Lipschitz and quasiconformal categories. To that end, the PI propose new integrability conditions for overdetermined systems that may be solvable in a nontraditional sense by geometric methods. Closely related also is the nonlinear problem of recognizing Jacobian determinants of quasiconformal transformations in Euclidean space. Finally, the PI will discuss to what extend certain nonsmoothable four manifolds could be brought to bear some first order differential analysis; while this cannot be accomplished via traditional charts, it could be possible to exhibit metric structures that allow for such analysis. The main intellectual merit of this proposal lies in the synthesis and the common geometric point of view for seemingly separate problems. To that end, nontraditional and venturesome approaches and solutions are proposed.The broader impacts resulting from the proposed activity constitute of bringing together different fields of mathematics, as well as mathematicians of different training and expertise. Students and postdoctoral assistants will be trained as well as learned from, and a diverse group of visitors are brought in for consultation.

Apptractheinonenthe Pi将搜索有助于识别给定度量空间何时可以通过同态形态进行参数的条件，该标准形态仅以受控方式改变距离；也就是说，我们要求欧几里得空间（局部）Bi-Lipschitz参数化。即使对于位于欧几里得三个空间中的二维表面，这个问题也不是统一的理解。从参数化问题到理解欧几里得空间中哪些可测量结构的问题是当地标准标准的问题，因为它们是同态形态的标准结构的回调。可以在Bi-Lipschitz和准信息类别中提出后一个问题。为此，PI为过度确定的系统提出了新的可集成性条件，这些条件可以通过几何方法在非传统意义上解决。密切相关的也是识别欧几里得空间中准形式转化的雅各布决定因素的非线性问题。最后，PI将讨论扩展某些不合同的四个流形的方法，以进行一阶差分分析。尽管这不能通过传统图表来完成，但可以展示允许进行这种分析的度量结构。该提案的主要知识优点在于综合和看似独立的问题的常见几何观点。为此，提出了非传统和冒险的方法和解决方案。拟议的活动构成了不同的数学领域，以及不同培训和专业知识的数学家所产生的更广泛的影响。学生和博士后助理将接受培训和学习，并从中学到了一群访客进行咨询。