Collaborative Research: Taut foliations and contact topology

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Contact and symplectic topology are branches of mathematics that are motivated by Physics, specifically by classical mechanics and thermodynamics. Three-dimensional manifolds are modeled on the space we live in, and contact structures arise naturally in the study of three-dimensional fluid flows. It is the mathematical structure of physical fluid flows that gives rise to contact topology. This National Science Foundation funded project seeks to extend the application of physical phenomena to the study of three-dimensional topology.One of the most important tools in the study of three-dimensional manifolds is an analysis of the codimension one structures they support. These include surfaces, foliations, and contact structures. These structures are most revealing of the ambient structure when they are, respectively, incompressible, taut, and tight. Some of the major advances in the field have been made when people gained insight into how these structures interact. Gabai and Thurston made major advances relating surfaces and foliations in the 1980's. Giroux's discovery of the interplay between convex surfaces and contact topology in the 1990's has been extremely useful. In 1998, Eliashberg and Thurston discovered a surprising connection between foliations and contact topology that has been very influential and is the starting point for the proposed research. The PIs propose to better understand the relationship between foliations and contact topology. This includes extending the applicability of approximation theorems, and sharpening the conclusions of such theorems. This includes also an investigation of existence and uniqueness questions, for both taut foliations and tight contact structures.