Persistence, Combinatorial Morse Functions and Shape Spaces and their Applications

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This proposal concerns the exploration of new approaches to computational problems that arise in robotics, geographic information systems, and in biological applications. These approaches involve techniques that lie at the intersection of mathematics and computer science and the application of mathematical perspectives in the development on new algorithms to solve a variety of problems. The primary goal is to use techniques from Morse theory, homology theory, geometric group theory and combinatorics to study problems of shape, configuration, motion planning, and structure in computer science. Applications include computer graphics, visualization of scientific data, computational analysis of molecular docking problems and robotics. A key element is that this work will be done within an interdisciplinary team that includes researchers in computer science, biochemistry and chemistry at Duke, UNC and Stanford.The research in this project will be important to advance our understanding in a number of important areas. Techniques developed will give new approaches to analyzing noise in data sets such as those from x-ray Crystallography, brain scans, satellite images, and ocean temperature. The robotics work will enhance our ability to create automated devices that perform tasks in dangerous or remote environments. And the biology work has as its long term goal the development of better computational tools to help biomedical researchers find appropriate molecules that can interact with