This project is centered on the further development of the theoretical tools and extension of the range of applications of the equivariant method of moving frames, that was first introduced in the late 1990's by the principal investigator in collaboration with a visiting postdoc, and since developed in many directions by a number of research groups worldwide. Of particular note are: First, new directions in image processing, object recognition and shape matching, involving combinations and comparisons of moving frame-based signatures and joint invariant histograms. Second, analysis and classification of invariant variational problems and invariant curve and surface flows using moving frame techniques applied to the invariant variational bicomplex, motivated by applications in geometric (bio-)mechanics, image denoising and smoothing, interface dynamics, and integrable soliton differential equations. Third, applications of the newly completed moving frame-based structure theory for infinite-dimensional Lie pseudo-groups that arise as symmetry groups of a wide variety of physical systems, including fluid mechanics, gauge theories, and solitons. Fourth, a detailed analysis of the ramifications of a new and surprising phenomenon of "dispersive quantization" that has been recently shown to arise in very basic linear models of dispersive wave equations on periodic domains, as well as, previously, in quantum mechanical systems and optics, where it is known as the Talbot effect.
The project is based on the exploitation of symmetry in a wide range of mathematics and its applications, through new, powerful methods that were inspired by classical tools coming from pure geometry. Besides further development of the underlying mathematical theory and tools, the primary focus is on three interconnected areas of application: First, object recognition in digital images from various sources, under various notions of when two objects can be considered the same: e.g., under rigid motions; under change in camera viewing angle; under deformations of a prescribed type, etc. Second, analysis of dynamical equations incorporating intrinsic physical and mathematical symmetries that arise in a wide range of applications in biomechanics and materials, fluid mechanics, image processing, and interface motions. Third, understanding and exploring a new, surprising and potentially important phenomenon, in which, under the assumption of periodicity, solutions are "fractalized" at irrational times and "quantized/localized" at rational times, that was recently shown to arise in many basic linearly dispersive wave models governing a very broad range of wave motions, including quantum mechanical systems.
