Three-Manifolds, Singularities, and Invariants

对于双曲流形，几何不变量、表示论不变量和基于量子的流形不变量之间存在假定的联系。 PI将致力于具体的approachetures relatingthese不变量;特别是，他将调查一个scissorscongruence方法的体积猜想，他将继续工作对他的猜想有关的A-多项式与变化的Blochinvariables。对这些联系的研究涉及到与数论和代数K-理论相关的不变量，机器计算是其中的一个重要组成部分。主要工具是Snap程序，该项目的另一个重要方面是该程序的进一步开发。PI也将继续他对三维流形的可拓性的研究。这个提议解决了一些关于三维空间形式的长期悬而未决的问题。这些问题涉及跨越不同数学领域的不变量，创造低维拓扑与代数、数论和理论物理的相互作用。 不同领域之间的联系提供了数学的大部分力量，加强这些联系会增加数学的力量。该项目还将导致一个强大的3歧管软件，最初在PI和C领导的项目开发的重大增强。Hodgson，这是研究这一建议的重要工具，也是数学界广泛使用的.

For hyperbolic manifolds there are postulated connections betweengeometric, representation-theoretic, and quantum based invariants ofmanifolds. The PI will work towards specific conjectures relatingthese invariants; in particular, he will investigate a scissorscongruence approach to the volume conjecture and he will continue workon his conjecture relating the A-polynomial with variation of Blochinvariants. The research on these connections involves invariants thatare related to number theory and algebraic K-theory, and machinecomputation has been an important component of the research. The maintool is the program Snap, and another important aspect of the projectis further development of this program. The PI will also continue hisinvestigation of commensurability properties of 3-manifolds.This proposal addresses some long-term open questions concerning3-dimensional space forms. These questions involve invariants thatspan different areas of mathematics, creating interactions of lowdimensional topology with algebra, number theory, and theoreticalphysics. Links between disparate areas provide much of the power ofmathematics, and strengthening these links increases the power. Theproject will also lead to a major enhancement of powerful 3-manifoldsoftware, originally developed in a project led by PI and C. Hodgson,that is an important tool for the research of this proposal and isalso widely used in the mathematical community.