Mathematical Sciences: Topological Properties of Singularities and Solutions of Nonlinear Equations

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9400930 Damon Professor Damon has been developing analogues of infinitesimal methods introduced by Mather for answering questions about the structure of nonlinear mappings and the topological structure of special but important classes of highly nonisolated singularities, such as occur in discriminants of mappings. In quite different circumstances, he has also discovered how to modify these infinitesimal methods for understanding the generic behavior of "pixel intensity functions" under Gaussian blurring used in computer vision. He intends to build on his earlier results and derive general algebraic formulae for certain fundamental topological invariants for highly singular spaces. Second, he will further develop methods for determining the generic properties of solutions to partial differential equations and, in particular, apply them to recent nonlinear versions of Gaussian blurring. Third, he will apply the results on the topology of singular spaces to understand how the qualitative behavior of the solutions can be deduced from the singularities which occur. This research project will investigate certain qualitative properties of the set of solutions of systems of nonlinear equations. Such properties will be applied to generally appearing solutions of partial differential equations. Professor Damon has been developing methods for determining such qualitative properties. Moreover, he has begun applying them to understand the effects of Gaussian blurring applied to "pixel intensity functions" for analyzing images in computer vision. He intends to develop these methods further so that they apply to general partial differential equations. This will include the investigation of relations between certain invariants of the solutions and their qualitative properties. One particularapplication will be to the increasingly sophisticated nonlinear blurring schemes which are being developed for identifying essential features and properties of images suc h as edges, ridges, etc. ***