Many applications in science and engineering encounter the problem of identifying and processing topologically interesting features in the digital representation of a geometry or data. Such features often need to be optimal with respect to some metric (measurement). It is recognized that homology groups from algebraic topology play an essential role in these computations. Although the study of structural properties of the homology groups has a rich history in mathematics, their computations in combination with geometry are not that well studied. The principal investigators (PIs) propose to study these fundamental questions thoroughly, along with their connections to practical problems from science and engineering. Intellectual merit: Efficient solutions of the optimality questions in homology computations require both mathematical and algorithmic developments. The PIs bring aboard these required expertise. Apart from the synergistic effect of the proposed study on mathematics and theoretical computer science, the close ties with various applications in science and engineering will play a synergistic role between computational fields such as computer graphics, computer vision, sensor networks, computer aided design, and scientific fields such as biology, physics, chemistry, and others. Broader impacts: Optimization of aspects of homology groups provides important insights in many scientific and engineering applications ranging from tunnels in protein molecules to voids in large machines. Solutions of such problems can aid in the manufacturing of better machines, designing of new drugs, and rapid modeling of customized objects. The educational impact of this project is in a large synergy between mathematics and computer science motivated by real applications. Course notes, internet distributions, and software systems developed through the project will enable the scientific community to study challenging problems in geometry, topology, and algorithms. Graduate students supported by the project will develop skills in mathematics and theoretical computer science and also in writing robust, efficient, and user-friendly software.
