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.

共形映射是局部保留角度的空间函数或变换，对此类映射的几何特性的研究导致了几何函数理论的发展，并且多年来已被证明对解决各种问题具有根本重要性。最近，在分析、几何、概率、物理和工程领域，人们越来越关注映射或函数的研究，它们是共形映射的推广，称为拟共形映射，其中受控的角度畸变量为允许。拟共形映射具有微妙的属性，使其在各种设置中非常有用，在许多此类应用中，人们必须处理给定平面区域中的（准）共形映射，但内部可能存在一些“特殊的”点集。异常集是否微不足道且小到可以忽略的问题引出了共形可去除性的概念，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。许多随着对可移除性的更好理解的进展，应该会发现更多富有成效的联系，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的应用是拟议研究的一个重要部分涉及数值。除了增强计算基础设施之外，这还有可能建立跨学科联系，以 Koebe 均匀化猜想为基础，研究圆域的可移除性和刚性之间的关系。他和施拉姆的开创性工作计划继续研究这种令人惊讶的联系。最后，第三项提议的活动涉及分析能力的性质，主要研究人员首先建议进一步研究分析能力的次可加性。使用由首席研究员与本科生一起开发的程序进行计算。首席研究员还计划研究全纯运动下的分析能力行为。该奖项是法定使命，并通过使用基金会的智力价值和评估进行评估，被认为值得支持。更广泛的影响审查标准。