Knot Theory: 3 and 4-dimensional manifolds

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This project will study knot and link concordance and its implications for 3- and 4-manifolds. We will study the smooth concordance classes of topologically slice knots via the recently defined bipolar filtration, using techniques of gauge theory, Heegaard Floer homology and von Neumann signatures. We will attempt to exhibit a primary decomposition of the knot concordance group using noncommutative localization techniques. We will also study the set of concordance classes as a metric space, under several natural metrics. We will also establish new structure in the knot concordance group, a sort of Wang sequence, arising from the failure of several conjectures of Kauffman.In its broadest terms this project is about the mathematical structure of the "shape" of 3- and 4-dimensional objects. Understanding complex shapes has many applications. The geometric structure of proteins and other complex molecules is crucial to drug development. The geometric configuration of cellular DNA is important in determining the precise mechanisms of cellular processes. The geometric shape of organs (especially the brain) is vital to the field of medical imaging. The prediction of shape from incomplete data (satellite imaging, sensor networks) has important military and commercial applications. Additionally, this project will increase the participation of U.S. citizens, permanent residents and women in mathematical research and education. This project focusses on knot theory, which is known to parametrize all 3-dimensional manifolds.