Collaborative Research: Differential Geometry and Statistics of Deformation Tensors

合作研究：变形张量的微分几何与统计

• 批准号: 1119181
• 负责人: Sarah Titus
• 金额: $ 7.45万
• 依托单位: Carleton College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1119181&HistoricalAwards=false
• 关键词: Collaborative; Research; Differential; Geometry; Statistics

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.

构造地质数据和模型依赖于各种几何概念，例如空间方向、椭球和变形张量。在数学中，这些对象通常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。该项目代表了两个结构地质学家和一个数学家之间的合作;目标是开发李群在结构地质学问题中的三个新应用。第一种方法使用基于李群的微分方程方法来计算非稳态变形。这一方法正应用于新喀里多尼亚剪切带的现有数据集。第二个适用于建立统计椭球有限应变椭球数据从南山，马里兰州，这是一个经典的领域内的位置结构地质。第三个项目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和量化数据和模型中的不确定性。这个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、数学和其他使用相关技术（如医学成像）的领域之间的进一步交流打开了大门。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学界。该项目将促进四年制学院和研究型大学教师之间的合作和研究。这项研究将有助于这两个机构的本科教育，并将为本科研究项目提供资金。该项目由NSF地质构造计划和数学地球科学合作计划的资金支持。