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.

抽象Heinonen PI将搜索有助于识别给定度量空间何时可以通过仅以受控方式改变距离的同胚来参数化的条件;也就是说，我们要求通过欧几里得空间进行（局部）双Lipschitz参数化。这个问题是不充分理解，即使是二维曲面躺在欧几里德三空间。从参数化问题到理解欧几里得空间中哪些可测结构是局部标准的问题有一个直接的联系，因为它们是标准结构通过同胚的拉回而产生的。后一个问题可以在双Lipschitz和拟共形范畴中提出。为此，PI提出了新的可积性条件的超定系统，可以解决在非传统意义上的几何方法。密切相关的也是非线性问题的识别雅可比行列式的拟共形变换在欧氏空间。最后，PI将讨论在何种程度上某些非光滑的四流形可以承受一些一阶微分分析;虽然这不能通过传统的图表来完成，但可以展示允许这种分析的度量结构。这一建议的主要智力价值在于合成和共同的几何观点，看似独立的问题。为此，提出了非传统和冒险的方法和解决方案。拟议活动产生的更广泛的影响包括汇集不同数学领域以及不同培训和专业知识的数学家。学生和博士后助理将接受培训和学习，并邀请不同的访问者进行咨询。