Applications of Moving Frames

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Olver DMS-0807317 The aim of this project is to further develop and extend the range of applications of the investigator's equivariant method of moving frames, for both finite-dimensional and infinite-dimensional group actions. The specific areas of research in the proposal are as follows: Development of computational algorithms for differential invariant algebras, with particular emphasis on methods for determining minimal generating sets; exploitation of the invariant variational calculus based on moving frames to analyze minimizers of invariant variational problems and the behavior of invariant submanifold flows, focusing on the evolution of differential invariants and their signatures; algorithms for analyzing the structure of symmetry pseudo-groups, their use in classifying and finding explicit solutions to nonlinear partial differential equations, with applications in fluid mechanics, gauge theories, integrable systems, and meteorological models; further development of practical tools for object recognition and symmetry detection in digital images based on differential invariant and joint invariant signature methods; and exploration and application of new invariant numerical algorithms for integrating nonlinear differential equations and approximating differential invariants such as curvatures, torsion, etc. Exploitation of symmetry, equivalence and invariance properties remains one of the most important mathematical methods for tackling the complex nonlinear systems that arise in a wide variety of applications. The investigator's reformulation of the geometrically-based method of moving frames has led to a wide range of new and, as yet, underexploited computational tools, as well as new and unexpected areas of application, including computer vision, recognition and surveillance, materials science and physical gauge theories, fluid mechanics, interface flows, DNA mechanics, relativity, and elsewhere. The project continues to develop this powerful new approach to analyzing very general group actions and their invariants, directly driven by current needs in these areas of application.