Topology of Nonisolated Singularities and Scale-based Geometry

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DMS-0103862James N. DamonProfessor Damon's research will apply infinitesimal and stratification methods from singularity theory to investigatethe topology and deformation properties of a general class of highly nonisolated singular spaces. These spaces arise as nonlinear sections of certain natural universal varieties. For instance, geometric and deformation properties of mappings under various equivalences can be captured by such universal varieties. The structure of the tangent vector fields will be used to deduce algebraic formulae for certain fundamental topological invariants. These will be given in terms of certain natural algebraic and geometric multiplicities, measuring the singular behavior of mappings relative to associated geometric structures such as foliations Second, he will refine these ideas for questions in computer imaging by developing geometric structures associated to objects and features in images in terms of such highly singular spaces. The presence of discreteness, noise, and distortions in images require a "scale-based geometry" which is applicable to nondifferentiable functions, measures, and even distributions. Such a geometry will apply to "almost all" objects in a given type, and will yield stable geometric structures in scale space. This will allow the geometric analysis of images using functions and measures discriminating various features in images.The first part of Professor Damon's research will determine for specific types of systems of nonlinear equations, the qualitative properties of the set of solutions. These can be obtained from certain universal systems of equations. He proposes to use certain infinitesimal symmetries of the equations to deduce properties of the set of solutions in terms of algebraic invariants which reflect both properties of the universal equations and how the specific equations relate to the universal ones. Second, the research will be applied to problems in computer imaging. To objects and features in images, one may associate geometric structures capturing their properties for various imaging purposes. Such structures are defined using systems of equations as above. The presence of discreteness, noise and distortions in images interferes with identifying geometric features. The research will refine the methods described above via "scale-based" versions which introduce robust geometric structures overcoming these difficulties, which can then beused for a variety of computer imaging problems.