Quantum topology, via re-normalized invariants

所提出的研究的第一个主题是进一步探索主要研究者（PI）重新规范化的Reshetikhin-Turaev 3-流形不变量。 特别是，PI将与这些重新规范化的不变量和其他拓扑对象（包括Reidemeister扭转，拓扑量子场论（TQFT）和映射类群表示）建立联系。 拟议的研究还将集中在继续PI的发展和广义Kashaev量子3流形不变量的研究。 这些不变量是仿照卡沙耶夫的基础工作，导致原来的制定体积猜想。 建议的研究的最后一个主题是推广他的重新规范化Reshetikhin-Turaev不变量通过定义3流形不变量通过手术介绍的链接与平坦的连接，在他们的补充。 这项工作是几何和物理的性质，并与陈-西蒙斯理论。 数学在自然界中产生的基本方式之一是通过几何和更一般的拓扑学。 特别是，称为流形的几何对象被用来在广义相对论中模拟空间和时间。 令人惊讶的是，流形的几个特征可以通过它们的底层拓扑来研究。 1984年V. Jones发现了Jones多项式，并由E.维滕在1989年已经打开了大门，使用新的代数技术来研究拓扑结构。 这些发展导致了一个新的数学分支，称为“量子拓扑学”。“这个领域与理论物理学有关，包括量子引力、拓扑量子场论和量子计算。 当量子维度消失时，量子拓扑学中许多已建立的技术变为零。 该项目的重点是开发和研究新的系统策略，以重新规范化量子维数为零时的拓扑不变量。

The first main theme of the proposed research is further exploration of the Principal Investigators (PI) re-normalized Reshetikhin-Turaev 3-manifold invariants. In particular, the PI will draw connections with these re-normalized invariants and other topological objects including Reidemeister torsion, Topological Quantum Field Theories (TQFTs) and mapping class group representations. The proposed research will also focus on continuing the PI's development and research of generalized Kashaev quantum 3-manifold invariants. These invariants are modeled on Kashaev's foundational work that led to the original formulation of the Volume Conjecture. The final topic of the proposed research is to generalize his re-normalized Reshetikhin-Turaev invariants by defining 3-manifold invariants via surgery presentations of links with flat connections in their complements. This work is both geometric and physical in nature and related to Chern-Simons theory. One of the fundamental ways mathematics arises in nature is through geometry and more generally topology. In particular, geometric objects called manifolds are used to model space and time in general relativity. Surprisingly, several features of manifolds can be studied via their underlying topology. The discovery of the Jones polynomial by V. Jones in 1984 and its 3-dimensional quantum field theory interpretation by E. Witten in 1989 have opened the door for the use of new algebraic techniques to study topology. These developments led to a new branch of mathematics known as "quantum topology." This area has connections to theoretical physics, including quantum gravity, topological quantum field theory and quantum computing. Many established techniques in quantum topology become zero when quantum dimensions vanish. This project focuses on developing and studying new systematic strategies to re-normalize topological invariants when quantum dimensions are zero.