Applications of Moving Frames

0103944Olver本项目将专注于应用提出者关于一般李群作用的运动标架的新等变理论。重点将集中在三个关键的应用方向：几何和物理中不变变分问题和不变偏微分方程的分析和应用，近似微分不变量和积分不变微分方程的保持对称的数值算法的设计，以及基于微分不变和抗噪声联合不变特征的计算机视觉中的目标识别和对称性检测。该项目将包括进一步发展基本的移动标架理论，特别是在无限维李伪群的情况下，以及为多元数值逼近和内插建立适当的几何基础。识别和利用对称性是现代数学及其应用中的一个基本工具。该项目将继续开发一种新的、强大的几何方法，称为移动框架，用于处理具有连续对称性的系统。由PI开发的现代移动标架理论已经见证了一系列显著的新应用，包括用于对象识别和对称性检测的计算机视觉、经典代数和不变量理论的几何方法，以及保持待解决问题的基本对称性的数值积分方案的设计。分析、几何和数值进步的结合，加上实际应用，已被证明是理论和实际工具的特别有力的结合。这项研究项目将继续快速发展和应用移动标架方法，专注于与计算机视觉、几何和物理以及基于对称的数值积分方法的应用相关的理论发展

0103944OlverThe project will focus on the applications of the proposer's new equivariant theory of moving frames for general Lie group actions. Particular emphasis will be in three key applied directions: analysis and applications of invariant variational problems and invariant partial differential equations in geometry and physics, the design of symmetry-preserving numerical algorithms for approximating differential invariants and integrating invariant differential equations, and object recognition and symmetry detection in computer vision based on differential invariant and noise-resistant joint invariant signatures. The project will include further development of the underlying moving frame theory, particularly in the case of infinite-dimensional Lie pseudo-groups, and the formulation of a proper geometrical foundation for multivariate numerical approximation and interpolation.The recognition and exploitation symmetry is an essential tool in modern mathematics and its applications. This project will continue to develop a new, powerful geometric approach, known as moving frames, to systems that have continuous symmetry. The modern moving frame theory developed by the PI has already witnessed a remarkable range of new applications, including computer vision, for object recognition and symmetry detection, a geometric approach to classical algebra and invariant theory, as well as the design of numerical integration schemes that preserve the underlying symmetry of the problem to be solved. The combination of analytical, geometrical, and numerical advances, coupled with practical applications has proved to be a particularly potent blend of theory and practical tools. This research project will continue the rapid development and application of the moving frame method, concentrating on theoretical developments tied to applications in computer vision, in geometry and physics, and in symmetry-based numerical integration methods