Removability in Geometric Function Theory

保角映射是局部保角的空间函数或变换。对这种映射的几何性质的研究导致了几何函数论的发展，多年来已经证明对分析、几何、概率、物理和工程中的各种问题具有根本的重要性。最近，许多注意力都集中在研究映射或函数，这是共形映射的推广，称为拟共形映射，其中允许控制量的角度失真。拟共形映射具有微妙的性质，使它们在各种各样的设置中非常有用。在许多这些应用中，人们必须处理在给定平面区域中（准）共形的映射，除了区域内可能存在的一些点的“例外集”。例外集是否可以忽略和小到可以忽略的问题导致了共形可移除性的概念，这是本研究项目的核心，与复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，随着对可移除性的更好理解的进展，许多更有成效的联系应该会被发现。这个项目还将考虑几何函数论中的其他几个问题，其中一些在数值视觉领域中有应用。所提出的研究的一个重要部分涉及数值计算和建设性的方法。除了加强计算基础设施，这有可能建立跨学科的联系。第一个提议的活动涉及研究Koebe单值化猜想中的可移除性和圆域刚性之间的关系，建立在He和Schramm的开创性工作基础上。首席研究员计划继续研究这种令人惊讶的联系。第二个建议的活动是专门研究保形焊接。主要研究者建议在几个建设性的方面以及保形焊接的非注入性和可移除性之间的关系。最后，第三个活动涉及分析能力的性质的研究。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主要研究者与一名本科生共同开发的程序进行数值计算。主要研究者还计划研究全纯运动下的分析能力的行为。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.