Removability in Geometric Function Theory

几何函数理论中的可移性

• 批准号: 2050113
• 负责人: Malik Younsi
• 金额: $ 12.14万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2050113&HistoricalAwards=false
• 关键词: Removability; Geometric; Function; Theory

Conformal maps are functions or transformations of space that, locally, preserve angles. The study of the geometric properties of such maps has led to the development of Geometric Function Theory and has proven over the years to be of fundamental importance to a wide variety of problems in analysis, geometry, probability, physics, and engineering. More recently, much attention has been devoted to the study of maps, or functions, that are a generalization of conformal maps, called of quasiconformal maps, where a controlled amount of angle distortion is allowed. Quasiconformal mappings possess subtle properties, making them very useful in a wide variety of settings. In many of these applications, one must deal with maps that are (quasi)conformal in a given planar region except possibly for some “exceptional set” of points inside the region. The question whether the exceptional set is negligible and small enough to be ignored leads to the notion of conformal removability, central to this research project and closely related to fundamental questions in complex analysis and other areas. The principal investigator has no doubt that many more fruitful connections should come to light as progress is made toward a better understanding of removability. This project will also consider several other problems in Geometric Function Theory, some with applications in the field of numerical vision. An important portion of the proposed research involves numerical computations and constructive methods. In addition to enhancing computational infrastructure, this has the potential to build interdisciplinary connections. The first proposed activity deals with the study of the relationship between removability and the rigidity of circle domains in Koebe’s uniformization conjecture, building upon the pioneering work of He and Schramm. The principal investigator plans to pursue the study of this surprising connection. The second proposed activity is devoted to the study of conformal welding. The principal investigator proposes to work on several constructive aspects as well as on the relationship between the non-injectivity of conformal welding and removability. Finally, the third proposed activity involves the study of the properties of analytic capacity. The principal investigator first suggests to further investigate the subadditivity of analytic capacity. This part of the proposal involves numerical computations using a program developed by the principal investigator together with an undergraduate student. The principal investigator also plans to study the behavior of analytic capacity under holomorphic motions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

共形图是局部保留角度的空间的函数或变换。对此类地图的几何特性的研究导致了几何函数理论的发展，并且事实证明，对于分析，几何，几何，概率，物理和工程方面的各种问题而言，这对于多种问题至关重要。最近，已经大量关注的是研究图或函数的研究，这些图或函数是共形图的概括，称为准文化图，其中允许受控的角度变形。准文明映射潜在的微妙特性，使其在各种环境中非常有用。在许多这些应用中，必须处理给定平面区域中（准）共形的地图，除非该区域内一些“特殊集”点。这个问题是否可以忽略不计，足以忽略足以忽略保形可移动性的概念，这是该研究项目的核心，并且与复杂分析和其他领域的基本问题密切相关。首席调查员毫无疑问，随着进步是为了更好地理解可移动性，因此应该揭示许多更富有成果的联系。该项目还将考虑几何函数理论中的其他几个问题，其中一些在数值视觉领域中具有应用。提出的研究的重要部分涉及数值计算和建设性方法。除了增强计算基础架构外，这还具有建立跨学科连接的潜力。第一个提议的活动涉及研究在Koebe的统一协议中取消与圆形域之间的关系的研究，这是基于HE和Schramm的开拓性工作。主要研究人员计划研究这种令人惊讶的联系。第二个提出的活性致力于共形焊接研究。主要研究者的提议提出了一些建设性方面以及共形焊接和去除的非注射率之间的关系。最后，第三个提出的活动涉及研究分析能力的性质。首先研究人员首先建议进一步研究分析能力的亚功能。该提案的这一部分涉及使用主要研究人员与本科生一起开发的计划进行数值计算。首席研究人员还计划研究霍明型动议下的分析能力的行为。该奖项反映了NSF的法定使命，并使用基金会的知识分子优点和更广泛的影响评估标准，被视为通过评估而被视为珍贵的支持。