Facets of the topology and geometry of 3-manifolds

PI的研究将集中在以下三个问题上。第一个问题是双曲三维流形的Heegaard亏格是否由它的基本群的秩所决定。在那里，他将试图通过计算机搜索沿着几个不同的途径找到这些不同的例子，包括在算术3-流形的同余覆盖中，那里有对亏格的具体控制。第二个目标是了解某些拓扑问题的计算复杂性，特别是是否有多项式时间算法来确定3-球面上的结亏格。第三个项目是解释为什么某些扭曲的亚历山大多项式如此擅长通过将它们与卡勒-沙伦特征变量理论联系起来来检测拓扑属性。拓扑学是研究直到橡胶拉伸的对象，几何学是研究刚体。这个项目的目标是通过将这些领域之间令人惊讶的关系与数学和计算机科学的其他领域的深层联系结合起来，来理解这些领域的某些基本问题。拓扑学和几何学对数据挖掘等应用程序都变得越来越重要，这个项目包括与计算机科学家合作，以及开发软件来探索这些问题的各个方面，其他研究人员可以通过网络免费获得这些软件。

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.