Embedded and Immersed Surfaces in Three-Dimensional Topology

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The guiding question in three-dimensional topology is the classification problem for three-dimensional manifolds - to construct a list in which every homeomorphism type appears exactly once. The PI will both address this question and explore applications of the resulting methods to the topology of large data sets in three components: First, the PI will expand and refine techniques developed recently by Minsky, Namazi, Souto and others to related hyperbolic geometry to topology via Teichmuller theory. Second, the PI will generalize these techniques to answer questions about finite covers of three-dimensional manifolds suggested by the recent proof of the Virtual Haken Conjecture. Third, the PI will explore applications of these topological techniques to data analysis in order to develop and refine new algorithms that will be useful to scientists in a broad range of fields.A 3-dimensional manifold is a topological space that models the 3-dimensional universe in which we live. Heegaard splittings are topological structures that allow one to see such a space as a pair of simple pieces that have been combined in a possibly complicated manner. While Heegaard splittings have been studied for over a century, our knowledge of Heegaard splittings has only in the last two decades matured to the point where they can be thoroughly understood. They are now an integral part of understanding geometric structures on 3-manifolds and knots. Moreover, recent developments have demonstrated that techniques (particularly one called thin position) used in the abstract study of Heegaard splittings can be adapted to work on certain problems in applied mathematics, particularly the analysis of large data sets. The research funded by this grant will develop both the abstract and the applied aspects of this field.