Isometric Embedding and Other Problems in Geometry and Differential Equations

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AbstractAward: DMS 1206272, Principal Investigator: Jeanne Nielsen ClellandThe principal investigator plans to apply the theory of exterior differential systems and Cartan's method of equivalence to study several problems in differential geometry and differential equations, both individually and with various collaborators. In joint work with Gui-Qiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang, she will investigate the local existence problem for isometric embedding of higher-dimensional Riemannian manifolds into Euclidean space, incorporating ideas from elastostatics and hyperbolic conservation laws to find conditions which guarantee local existence of isometric embeddings. She will continue her joint work with Thomas Ivey on classification results for Backlund transformations for certain categories of PDEs, paying particular attention to auto-Backlund transformations, which may be used to generate new solutions of a PDE from a known solution of the same PDE. She will expand upon her joint work with Christopher Moseley and George Wilkens on sub-Finsler geometry, and investigate geometric properties of metric structures associated to affine distributions, as well as potential applications to optimal control theory. In joint work with her former Ph.D. student Matthew Stackpole, she will investigate the phenomenon of dynamic equivalence between control systems, with a focus on finding ways to compute invariants to distinguish between dynamically inequivalent systems. In joint work with her former M.A. student Nathaniel Bushek, she will investigate geometric properties of surfaces in 5-dimensional centroaffine space, including the classification of homogeneous examples and other special families.Differential equations have an enormous range of applications, from engineering and physics to biology and finance, just to name a few. A common theme in all the proposed projects is the use of geometric techniques to study structural features of differential equations which may be obscured by the expression of an equation in a particular choice of coordinates. This approach, pioneered in the early 20th century by Elie Cartan, has enjoyed great success in furthering the understanding of many types of differential equations and their solutions. The proposed research on isometric embedding addresses one of the 23 mathematical challenges recently posed by DARPA, due to potential applications to the science of protein folding. The proposed research on Backlund transformations is related to integrable systems, whose applications include efficient signal propagation over long distances with minimal distortion in signal quality. The proposed projects on affine distributions and dynamic equivalence are related to control theory, whose applications include areas such as robotics and quantum computing.