AF: Medium: Collaborative Research: Optimality in Homology - Algorithms and Applications

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Many applications in science and engineering encounter the problem ofidentifying and processing topologically interesting features in thedigital representation of a geometry or data. Such features often needto be optimal with respect to some metric (measurement). It isrecognized that homology groups from algebraic topology play anessential role in these computations. Although the study of structuralproperties of the homology groups has a rich history in mathematics,their computations in combination with geometry are not that wellstudied. The principal investigators (PIs) propose to study thesefundamental questions thoroughly, along with their connections topractical problems from science and engineering.Intellectual merit: Efficient solutions of the optimality questions inhomology computations require both mathematical and algorithmicdevelopments. The PIs bring aboard these required expertise. Apartfrom the synergistic effect of the proposed study on mathematics andtheoretical computer science, the close ties with various applicationsin science and engineering will play a synergistic role betweencomputational fields such as computer graphics, computer vision,sensor networks, computer aided design, and scientific fields such asbiology, physics, chemistry, and others.Optimization of aspects of homology groups providesimportant insights in many scientific and engineering applicationsranging from tunnels in protein molecules to voids in largemachines. Solutions of such problems can aid in the manufacturing ofbetter machines, designing of new drugs, and rapid modeling ofcustomized objects. The educational impact of this project is in alarge synergy between mathematics and computer science motivated byreal applications. Course notes, internet distributions, and softwaresystems developed through the project will enable the scientificcommunity to study challenging problems in geometry, topology, andalgorithms. Graduate students supported by the project will developskills in mathematics and theoretical computer science and also inwriting robust, efficient, and user-friendly software.