CAREER: Algebraic Methods in Low-Dimensional Topology

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This project seeks to resolve specific questions in the area of 3- dimensional manifolds, knot theory, mapping class groups, and contact topology using a combination of geometric, topological and algebraic techniques. Tools from functional analysis and von Neumann algebras are also being used. The PI proposes to undertake the following projects. (1) Make significant advances toward the classification of the knot concordance group. In particular, classify the successive quotients of its (n)-solvable filtration and find new structure in the group. (2) Establish a higher-order Heegaard Floer homology theory that categorifies the higher-order Alexander polynomials defined by the PI. Use this to show that certain classical families of topologically slice knots are not smoothly slice. (3) Define new interesting canonical subgroups of the mapping class group related to the generalized Johnson subgroups and show their homology groups are infinitely generated. (3) Determine the precise relationship between certain subgroups of the mapping class group of a surface and the topology of their mapping tori (which are 3-manifolds). (4) Understand a precise relationship between transverse knots in S^3 and contact structures of arbitrary 3-manifolds that arise as cyclic and simple branched covers. Use this relationship to better understand the geometric invariants of a contact structure such as the support genus and binding number.Understanding the geometric structure of objects in 3-dimensional space is of crucial scientific importance. From cancer treatments based on the knotting of cellular DNA, to antiviral drugs based on the geometrical shapes of proteins, to non-invasive visualization of the shape of the heart, to contemplating the ``shape'' of space-time itself, we seek precise mathematical descriptions of 3-dimensional objects. When one thinks of a precise mathematical description, one often thinks in terms of numbers, but ordinary numbers are insufficient to capture the complexities of our world. Multiplication of ordinary numbers is ``commutative.'' However, the physics of the twentieth century has taught us that matter and energy cannot be described merely by numbers. Rather, vectors and matrices are required, and multiplication of matrices is not commutative, that is AB does not usually equal BA. Every physical interaction is thus based on noncommutative algebra. This project is investigating how this noncommutative algebra yields a mathematical description of the geometric structure of 3-dimensional space and of objects in 3- dimensional space. Of particular importance is the manner in which closed strings in 3-dimensional space are knotted in 3- and in 4- dimensions. The PI will use non-commutative mathematical objects to better understand the knotting of strings and 3-dimensional spaces in general.