Topology of Nonisolated Singularities and the Geometry of Functions

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9803467 Damon Professor Damon's research will apply infinitesimal methods from singularity theory (originally introduced by Mather) to investigate the topological structure of a general class of highly nonisolated singularities arising as nonlinear sections of varieties. This class includes discriminants of mappings, bifurcation sets, nonlinear arrangements of hypersurfaces, etc. He will use analogues of Morse singularities for sections of varieties to derive general algebraic formulae for certain fundamental topological invariants. He will also apply these ideas involving sections of varieties to understand the geometry of functions using the notion of "relative critical sets," which extend the notion of ridges and "cores" used in computer imaging. He will combine these geometric techniques with other singularity-theoretic methods developed for determining generic properties of solutions to partial differential equations. Together they will be applied to "pixel intensity functions" representing medical images. In conjunction with Gaussian blurring or nonlinear blurring techniques, they will associate certain geometric structures that can be used for addressing various questions in medical imaging. The first part of Professor Damon's research project will concern qualitative properties of spaces defined as the set of solutions of specific types of systems of nonlinear equations. The specific form of the equations will allow the investigator to use methods from the field of mathematics called singularity theory. Specifically, this allows the analysis of the set of solutions by perturbing the equations and reducing to the analysis of certain basic cases. The total contributions from the basic cases can be determined algebraically, yielding a qualitative understanding of the structure of the set of solutions. Second, such systems of equations will be specifically applied to analyze medical images. They allow one to associate to a medical image a g eometric structure, which captures essential features of the image. The geometric structure is defined using systems of equations of the above type, and it even allows one first to remove "noise" from the image by applying appropriate filters. The basic properties of the geometric structure obtained from the first part will allow it to be used to address several questions in medical imaging. ***