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。这个庞大的数学理论只在少数几个例子中应用于地质学。这个项目代表了两位构造地质学家和一位数学家的合作；目标是开发李群在构造地质学问题中的三种新应用。第一种方法采用基于李群的微分方程方法计算非稳态变形。该方法正在应用于新喀里多尼亚剪切带的现有数据集。二是将已建立的椭球统计量应用于构造地质领域内典型的场位——马里兰州南山有限应变椭球数据。第三章发展了变形张量的统计理论，并使用它们来计算最佳拟合变形，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来描述数据和从数据中进行推断的基本工具集。它为地质学、数学和其他相关技术（如医学成像）之间的进一步交叉施肥打开了大门。该项目的成果将以学术出版物、短期课程和免费软件的形式分发给地质社区。该项目将促进来自四年制学院和研究密集型大学的教师之间的合作和研究。这项研究将有助于两所院校的本科教育，并将为本科生研究项目提供资金。本项目由NSF EAR构造学项目和数学地球科学合作项目资助。

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.