Topological Consequences of Distortion-type Estimates

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.