Mathematical Sciences: Quantum Invariants of 3-Manifolds

研究者将继续他对三维流形的量子不变量的研究，首先是由Witten(在量子场论的背景下)和Reshetikhin和Turaev(在量子群的背景下)定义的。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3-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。数学家投身于物理学问题的传统由来已久，这往往有助于两门学科的相互充实。工具是为了解决物理问题而开发的，后来被证明具有更大的普遍性，并在数学中得到广泛应用。3-流形的新量子不变量的故事也是类似的。对量子场论问题感兴趣的数学家们得到了某种几何结构。这种构造又导致了一个不变量，该不变量只依赖于基本流形的拓扑特征，而不依赖于基本流形的全部几何特征。现在，在原始结构的基础上有了一些变化，并由此产生了许多量子不变量。它们的起源与以前已知的其他不变量的起源有很大的不同，因此可以预期它们可以检测到传统不变量无法检测的东西。现在，拓扑学家应该证明这一点，并理清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.