Applications of Symmetry & Invariants

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DMS-9500931 Olver, Peter The research project will be devoted to the theory and applications of Lie groups and the Cartan equivalence method to a variety of problems arising in physics and engineering. Basic theoretical research goals include the classification of the Lie transformation groups in three-dimensional space, their differential invariants, and the associated Lie algebra cohomology. The applications of these results fall into three broad categories. In computer vision, a new paradigm of image processing based on differential geometric diffusion equations relies on classifying the differential invariant signatures of visual symmetry groups. In quantum mechanics, the analysis of quasi-exactly solvable Schrodinger operators continues to produce significant new developments in the study of Lie algebras of differential operators, including the remarkable phenomenon known as "quantization of cohomology"; in the present project, the concentration will shift to real planar quantum operators, as well as extensions to a fully three-dimensional theory. In continuum mechanics, the application of equivalence methods and exterior differential systems to the classification of variational problems will be further analyzed with a view to providing canonical forms, geometric invariants, and conservation laws for nonlinear variational problems arising in elasticity. The goals of this research project are to further our understanding of the mathematical theory of symmetry and invariants in three-dimensional space, motivated by several key physical applications, of importance in physics, engineering and medicine. In computer vision, new methods of image processing promise to have an immediate practical impact in medical imaging, such as the ultrasound detection of breast tumors. In elasticity, the classification of simple forms and symmetries of general materials should have significant consequences in the study of crack propagation and waves. In quantum mechanics, new types of problems some of whose fundamental states are classified by algebraic tools have known applications to molecular spectroscopy and scattering theory.