Geometric mapping theory in Euclidean space and metric spaces

This project concerns the structure of metric spaces and mappings between them. A metric space is an abstract representation of a data set based upon the distance between the elements of the set. In applications, such a data set might have little a priori structure, and so it is desirable to embed it in a nicer, well-understood model space, such as Euclidean space, that is more amenable to analysis. The basic objective of geometric mapping theory is to decide when a given metric space can be mapped to another space with controlled distortion. This type of problem arises in a variety of fields, including geometric group theory, theoretical computer science and data science. In addition to the advancement of knowledge, this project will support student research and career preparation. In this project, the investigator will study the existence and properties of geometry-preserving classes of mappings, such as bi-Lipschitz and quasisymmetric mappings, in both the Euclidean and metric space settings. The research is divided into three broad parts. The first concerns conformal dimension, one of the main analytic tools in the field. One main goal is to understand when a set is minimal for conformal dimension. Such sets have good analytic and geometric properties, yet currently there is a limited understanding of which sets are minimal. A further goal is to develop the method of holomorphic motions in connection with conformal dimension. The second part concerns the uniformization problem, focusing on the uniformization of surfaces and of planar domains. While the uniformization problem has been intensively studied in recent years, the project focuses on several areas that are still open, including the existence of Lipschitz parametrizations and how to recognize quasispheres. The final part considers computational approaches to several problems in metric geometry, such as the metric complexity of graphs and how to efficiently embed sets into lower dimensional spaces. These problems typically involve computer exploration and are especially appropriate for students. 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.