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.
This project studied the topology and geometry of 3-dimensional spaces. These are spaces, like the ordinary 3-dimensional space we live in, but which are twisted up globally, much like how the earth is 2-dimensional yet one never falls off the edge no matter how far one walks. Two of the main themes of this project were randomness versus structure and computer experimentation. For example, the one subproject concerned the distribution of quantum invariants for random 3-manifolds. Another explored the extent to which one can vary the topology and geometry of a 3-manifold independently. Several subprojects searched for examples of particular phenomena amongst 3-dimensional manifolds using large-scale computer experimentation. The project also studied how hard it is for computers to solve certain topological and geometric questions about them. For example, consider a closed loop of wire in ordinary 3-space. No matter how complicated and bent the wire loop, it is an old theorem that it always bounds a surface, just like a round loop is boundary of a disc. With Hirani, the PI showed that one always compute the least-area such surface reasonable quickly, both in theory and in practice. The PI also contributed to the education of several graduate students as well as the semester long research program at ICERM on computation and experiment in low-dimensional topology, geometry, and dynamics. The PI continued development of the open-source program SnapPy for exploring the topology and geometry of 3-dimensional manifolds.
