Quantum topology, via re-normalized invariants

提出的研究的第一个主题是进一步探索主要研究者（PI）重新归一化Reshetikhin-Turaev 3流形不变量。特别是，PI将绘制与这些再归一化不变量和其他拓扑对象的连接，包括Reidemeister扭转，拓扑量子场论（TQFTs）和映射类群表示。建议的研究也将集中在继续PI的发展和研究广义Kashaev量子3流形不变量。这些不变量是根据卡沙耶夫的基础工作建立的，这些工作导致了体积猜想的原始公式。本研究的最后一个主题是通过在补中具有平坦连接的连杆的手术表示来定义3流形不变量，从而推广他的再标准化Reshetikhin-Turaev不变量。这项工作在本质上是几何和物理的，并与陈-西蒙斯理论有关。数学在自然界中产生的基本方式之一是通过几何学和更普遍的拓扑学。特别是，在广义相对论中，被称为流形的几何物体被用来模拟空间和时间。令人惊讶的是，流形的一些特征可以通过它们的底层拓扑来研究。1984年Jones多项式的发现和1989年E. Witten对其三维量子场论的解释为使用新的代数技术来研究拓扑学打开了大门。这些发展导致了一个被称为“量子拓扑学”的数学新分支。该领域与理论物理有联系，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将变为零。本项目的重点是开发和研究当量子维为零时拓扑不变量再归一化的新系统策略。

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.