Low-dimensional manifolds and computation

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Abstract for NSF proposalLow Dimensional Manifolds and ComputationThis project concerns problems in low-dimensional topology and geometry, with an emphasis on computational aspects. Manifolds in two and three dimensions provide the geometric models for many physical phenomena. Computational issues are playing an increasing role in mathematical investigations in these areas. Research in low-dimensional manifolds in turn is making contributions to computational geometry and topology. It alsoimpacts computational complexity theory, which studies questions such as how long it takes a computer to solve a problem. Techniques of computational topology have already led to improvements in algorithmsin computer graphics, visualization, medical and molecular modeling and image recognition. Several problems on the theme of computational topology will be investigated in this project. The computational complexity of many important problems in topology and geometry is unknown, even when explicit algorithms exist. Recent results have revealed intriguing connections between complexity theory, minimal surface theory, normal surface theory and isoperimetric problems. The project will pursue research in this direction, examining the complexity of Knot Recognition, Knot Genus, Homology Genusand related topological problems. The project also includes plans to investigate the generalized theory of normal surfaces. Normal surfacesare particularly simple surfaces relative to a fixed triangulation ofa 3-manifold. They are the analogs of minimal surfaces in the PL Category. The collection of such surfaces is discrete and well suited to computation, leading to applications in classification, complexity, enumeration and algorithmic recognition. Index-one, or almost normal surfaces, were applied with spectacular success in recent work on manifold recognition. A third focus of the project concerns multi-region isoperimetric problems, which ask what are the shortest boundaries enclosing multiple regions of a given size. The goal is to give a mathematical proof that the configurations assumed by soap bubbles are optimal. Questions such as finding the shortest curve enclosing three given areas in the plane remain open. Also missing is an understanding of general properties of stable soap-bubble-like configurations, such as whether the minimizing configurations always have connected regions. Finally, the project examines questions of curve and surface evolution, using ideas based on normal surfaces. Such evolution methods have the potential for practical applications in many areas, similar to the many applications of mean curvature evolution.