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.

保角映射是局部保持角度的空间函数或变换。对这种映射的几何性质的研究导致了几何函数理论的发展，并且多年来被证明对分析、几何、概率、物理和工程中的各种问题具有根本性的重要性。最近，很多注意力都集中在映射或函数的研究上，这些映射是共形映射的推广，称为准共形映射，其中允许控制角度畸变的量。拟共形映射具有微妙的属性，使它们在各种设置中非常有用。在许多这样的应用中，我们必须处理在给定平面区域内（拟）保角的映射，除非该区域内可能有一些点的“例外集”。异常集是否可以忽略，是否小到可以忽略的问题导致了保形可移除性的概念，这是本研究项目的核心，与复杂分析和其他领域的基本问题密切相关。首席研究员毫不怀疑，随着对可移除性的进一步了解，将会有更多卓有成效的联系出现。本项目还将考虑几何函数理论中的其他几个问题，其中一些问题在数值视觉领域有应用。所提出的研究的一个重要部分涉及数值计算和构造方法。除了加强计算基础设施，这有可能建立跨学科的联系。第一个提议的活动是在He和Schramm的开创性工作的基础上，研究Koebe均匀化猜想中可移除性和圆域刚性之间的关系。首席研究员计划继续研究这种令人惊讶的联系。第二个提议的活动是致力于研究保形焊接。首席研究员建议在几个建设性的方面以及在保形焊接的非注入性和可移除性之间的关系。最后，第三个提议的活动涉及分析能力性质的研究。主要研究者首先建议进一步研究分析能力的次可加性。提案的这一部分涉及使用由首席研究员和本科生共同开发的程序进行数值计算。首席研究员还计划研究全纯运动下分析能力的行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。