Computational topology is an exciting new area which is emerging at the intersection of theoretical computer science and mathematics. Algorithmic techniques in topology are far from new, but the increasing use of computational geometry in many disciplines has led to an explosion of new results in both computer science and mathematics. There is currently great demand for algorithms which can make topological guarantees, as well as great opportunity to apply algorithmic proofs in topology. Problems in computational topology are motivated by a wide variety of areas; applications abound in areas such as shape modeling in graphics, pathfinding in motion planning, geometric and topological modeling for protein docking prediction, encoding low dimensional spans for high dimensional data in statistical analysis, and modeling networks using cell complexes and meshes. Many topological questions are provably hard or even unsolvable in the most general settings, making it necessary to examine specific instances. My research focuses on several instances in this larger setting of computational topology. In each instance, the goal is to find an interesting topological feature, usually a cycle or path with certain desired properties. Combinatorial Surface
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