Three-Manifolds, Singularities, and Invariants

对于双曲流形，流形的几何不变量、表示论不变量和基于量子的不变量之间存在假定的联系。PI将致力于与这些不变量相关的特定猜想；特别是，他将研究体积猜想的剪刀同余方法，并将继续他的猜想，将A多项式与Bloch不变量的变量联系起来。对这些联系的研究涉及到与数论和代数K理论有关的不变量，而机器计算一直是研究的重要组成部分。主要工具是程序快照，而本项目的另一个重要方面是该程序的进一步开发。PI还将继续他对3-流形的可公度性的研究。这一建议解决了关于3维空间形式的一些长期未解决的问题。这些问题涉及跨越不同数学领域的不变量，创造了低维拓扑与代数、数论和理论物理的相互作用。不同领域之间的联系在很大程度上提供了数学的力量，而加强这些联系会增加这种力量。该项目还将导致强大的3-流形软件的重大增强，该软件最初是由Pi和C.Hodgson领导的一个项目开发的，是研究这一建议的重要工具，也在数学界广泛使用。

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.