Topological Consequences of Distortion-type Estimates

该项目的中心是调查关于空间变换的小尺度失真的信息如何影响全球尺度上的关于变换的信息。理想情况下，当变换是保形变换时不会出现失真，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研究弹性体等领域中得到了应用。该项目将加深目前对此类扭曲限制的较弱形式的理解。所采用的方法混合了包括非光滑分析、拓扑学和微分几何在内的多个数学领域，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说明性文本，使非流畅分析的专家更容易获得来自其他数学领域的工具。这个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碍现象。例如，这些问题涉及流形之间的拟正则映射的存在性问题，以及在有限变形映射下对点前像可能的拓扑形状的限制。该项目还有望导致最近构思的拟正则值理论的进一步发展，该理论提供了拟共形分析中经典结果的单值版本。该项目的另一个目标是改进和寻找未得到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数。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.