Facets of the topology and geometry of 3-manifolds

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The PI's research will focus on the following three issues. The first is whether the Heegaard genus of a hyperbolic 3-manifold is determined by the rank of its fundamental group. There, he will try to find examples where these differ via a computer search along several different avenues, including among congruence covers of arithmetic 3-manifolds where there is concrete control on the genus. The second goal is to understand the computational complexity of certain topological questions, and in particular whether there is a polynomial-time algorithm to determine the genus of a knot in the 3-sphere. The third project is to explain why certain twisted Alexander polynomials are so good at detecting topological properties by connecting them to the Culler-Shalen theory of character varieties.Topology is the study of objects up to rubbery stretching, and geometry the study of rigid bodies. The goal of this project is to understand certain fundamental problems in these areas by combining surprising relationships between them with deep connections to other areas of mathematics and computer science. Both topology and geometry are becoming more important to applications such as data mining, and this project includes collaboration with computer scientists as well as developing software for exploring aspects of these problems which will be freely available to other researchers via the web.