Collaborative Research: Differential Geometry and Statistics of Deformation Tensors

构造地质数据和模型依赖于各种几何概念，如空间方向、椭球体和变形张量。在数学中，这些对象通常被表示为李群及其相关对称空间的元素。这一庞大的数学理论仅在少数情况下应用于地质学。这个项目代表了两位构造地质学家和一位数学家之间的合作；目标是开发李群在构造地质学问题上的三个新应用。第一种是用基于李群的微分方程组方法计算非定常变形。这一方法正在应用于新喀里多尼亚剪切带的现有数据集。第二种方法将已建立的椭球体统计量应用于来自马里兰州南山的有限应变椭球体数据，这是构造地质学领域中的一个经典野外位置。第三个项目发展了形变张量的统计理论，并将其用于计算最佳拟合形变并量化数据和模型中的不确定性。该项目丰富了地质学家用于描述数据和根据数据进行推断的基本工具集。它打开了利用相关技术，如医学成像，进一步在地质学、数学和其他领域之间进行交叉培养的大门。该项目的成果将通过学术出版物、短期课程和分发给地质界的免费软件进行传播。该项目将促进一所四年制学院和一所研究密集型大学的教师之间的合作和研究。这项研究将为这两所大学的本科生教育做出贡献，并为本科生研究项目提供资金。该项目得到了NSF EAR构造计划和数学地球科学合作计划的资金支持。

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.