Topological Consequences of Distortion-type Estimates

该项目的重点是调查关于空间转换的小尺度扭曲的信息如何影响关于全球尺度转换的信息。当变换为保角变换时，无畸变的理想情况发生，这是一种在物理和工程中广泛应用的变换。对变形限制不那么严格的变换在弹性体研究等领域也有应用。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、拓扑和微分几何，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产生说明性文本，使非光滑分析专家更容易使用来自其他数学领域的工具。本项目的主要目的是研究高维拟共形分析领域中出现的各种拓扑阻塞现象。这些问题包括流形间拟正则映射的存在性问题和有限畸变映射下点原像可能拓扑形状的限制。该项目还有望进一步发展最近构想的拟正则值理论，该理论提供了拟共形分析中经典结果的单值版本。该项目的进一步目标是改进和发现未充分利用的拟共形分析工具的应用，如共形de Rham上同调理论和曲面族的模。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。

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.