Collaborative Research: Differential Geometry and Statistics of Deformation Tensors

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Structural geology data and models rely on a variety of geometric concepts, such as directions in space, ellipsoids, and deformation tensors. In mathematics, these objects are often represented as elements of Lie groups and their associated symmetric spaces. This vast mathematical theory has been applied to geology in only a few instances. This project represents collaboration between two structural geologists and a mathematician; the goal is to develop three new applications of Lie groups to problems within structural geology. The first uses Lie group-based differential equations methods to compute non-steady deformations. This method is being applied to an existing dataset from a shear zone in New Caledonia. The second applies established statistics of ellipsoids to finite strain ellipsoid data from South Mountain, Maryland, which is a classic field location within the field of structural geology. The third develops a statistical theory of deformation tensors, and uses them to compute best-fit deformations and to quantify the uncertainty in data and models.This project enriches the fundamental tool set that geologists use to describe data and make inferences from data. It opens the door to further cross-fertilization among geology, mathematics, and other fields using related techniques, such as medical imaging. The products of the project will be disseminated in scholarly publications, short courses, and free software distributed to the geology community. The project will facilitate collaboration and research between faculty from a four-year college and a research-intensive university. The study will contribute to undergraduate education at both institutions, and will provide funding for undergraduate research projects. This project is supported by funds from the NSF EAR Tectonics Program and the Collaborations in Mathematical Geosciences Program.