Topological Consequences of Distortion-type Estimates

该项目的重点是调查如何在全球范围内的空间变换的小尺度失真的信息影响有关的转换信息。当变换是保形变换时，没有变形的理想情况发生，保形变换是一种在物理和工程中广泛使用的变换。对变形限制不那么严格的变换也在弹性体研究等领域得到了应用。该项目将加深目前对这种扭曲限制的较弱形式的理解。所采用的方法混合了多个数学领域，包括非光滑分析，拓扑学和微分几何，并有可能加强所有这些独立领域之间的联系。该项目的一个辅助目标是制作临时文本，使非光滑分析专家更容易使用其他数学领域的工具。本计画的主要目的是探讨在高维拟共形分析领域所产生的各种拓扑阻碍现象。这些问题涉及，例如，存在问题的拟正规映射之间的流形和限制的可能拓扑形状的点原像映射下的有限失真。该项目还预计将导致进一步发展最近构思的理论准正规价值观，它提供了单值版本的经典结果在准共形分析。该项目的另一个目标是改进和寻找未充分利用的准共形分析工具的应用，如共形de Rham上同调理论和表面族的模量。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。

The project centers on the investigation of how information about small scale distortion of a spatial transformation influences information about the transformation on a global scale. The ideal case of no distortion occurs when the transformation is conformal, a type of transformation with widespread uses in physics and engineering. Transformations with less strict restrictions on their distortion have also found uses in fields such as the study of elastic bodies. The project will deepen current understanding of the weaker forms of such distortion restrictions. The methods to be employed mix multiple fields of mathematics including non-smooth analysis, topology, and differential geometry, with potential to strengthen the connections between all of these separate fields. An auxiliary goal of the project is to produce expository texts that make tools from other mathematical fields more accessible to experts in non-smooth analysis. The primary aim of this project is to investigate various topological obstruction phenomena arising in the field of higher-dimensional quasiconformal analysis. These problems involve, for instance, existence questions for quasiregular maps between manifolds and limitations on the possible topological shapes of point-preimages under mappings of finite distortion. The project is also expected to result in further development of the recently conceived theory of quasiregular values, which provides single-value versions of classical results in quasiconformal analysis. A further objective of the project is to improve upon and find applications for underutilized tools of quasiconformal analysis, such as conformal de Rham cohomology theories and the moduli of surface families.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.