Mathematical Sciences: Quantum Invariants of 3-Manifolds

研究者将继续他对3流形的量子不变量的研究，首先由Witten（在量子场论的背景下）和Reshetikhin和Turaev（在量子群的背景下）定义。该项目的总体目标是：1)研究已知量子不变量的结构及其改进，并探索可能的推广；2)研究量子不变量与其他三流形不变量之间的联系；3)探索量子不变量与数论的关系；4)将量子不变量应用于低维拓扑问题。数学家研究物理问题有悠久的传统，常常使这两门学科相互丰富。工具是用来解决物理问题的，后来被证明具有更大的通用性，并被广泛应用于数学。3流形的新量子不变量的故事是类似的。对量子场论问题感兴趣的数学家们得到了某种几何结构。这种构造反过来又导致了一个不变量，它只依赖于底层流形的拓扑特征，而不是完全的几何特征。现在有了原始结构的变体和许多由此产生的量子不变量。它们的起源与其他已知的不变量有很大的不同，因此它们可以探测到传统不变量无法探测到的东西。现在拓扑学家应该证明这一点，并整理出不同的量子不变量以及它们与更传统的不变量的关系。

The investigator will continue his research on the quantum invariants of 3-manifolds, first defined by Witten (in the context of quantum field theory) and by Reshetikhin and Turaev (in the context of quantum groups). The overall objectives of the project are: 1) to study the structure of the known quantum invariants and their refinements, and to explore possible generalizations; 2) to study connections between quantum invariants and other 3- manifold invariants; 3) to explore the relationship of quantum invariants with number theory; 4) to apply quantum invariants to problems in low dimensional topology. There is a long tradition of mathematicians involving themselves in the problems of physics, often to the mutual enrichment of both subjects. Tools are developed to solve physical problems and then turn out to have much greater generality and become widely used in mathematics. The story of the new quantum invariants of 3-manifolds is similar. Mathematicians interesting themselves in problems of quantum field theory were led to a certain geometric construction. The construction led in turn to an invariant that depended only on the topological character and not the full geometric character of the underlying manifold. There are now variations on the original construction and numerous resulting quantum invariants. Their origin is sufficiently different from that of other previously known invariants that they can be expected to detect things the traditional invariants cannot. It now behooves topologists to demonstrate this as well as to sort out the different quantum invariants and their relationships to more traditional invariants.