RUI: Algebraic topology of knot and link spaces

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AbstractAward: DMS 1205786, Principal Investigator: Ismar VolicInsert abstract here for an award recommendation.The main goal of this project is a better understanding of the topology of knot and link spaces as well as more general embedding spaces. The principal investigator proposes to use techniques such as calculus of functors, cosimplicial spaces, operads, and configuration space integrals to prove results about homology and homotopy of knots, links, homotopy links, and braids in Euclidean spaces of various dimensions. In particular, these proposed projects include a program to describe the rational homotopy type of these spaces; to combine configuration space integrals with the theory of Milnor invariants; and to better understand, if not complete resolve, of the issue of the separation of knots and links by finite type invariants. Moreover, the principal investigator plans to unify the various ways in which operads appear in the applications of calculus of functors in knot theory and further the understanding and uses of configuration space integrals. Some of his long-term projects will attempt to connect calculus of functors point of view in embedding theory to Khovanov homology as well as to the study of embeddings of surfaces and the mapping class group.Knot and link spaces represent some of the most interesting objects of study in topology because they are easy to define and visualize and because they are of interest to physicists and chemists. Some fundamental questions about knots, such as their classification, or construction of efficient ways of telling them apart, still generate a wealth of exciting research. The principal investigator's proposed work intends to bring us closer to answering these questions. Furthermore, the techniques he plans to use are quite general and point to new connections between topology, geometry, combinatorics, and physics. These connections will potentially help answer several important conjectures about the structure of knot and link spaces and introduce new points of view in algebraic topology in general.