Applications of Moving Frames

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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