Quantum topology, via re-normalized invariants

这项研究的第一个主题是进一步探索首席调查者(PI)重新归一化的Reshetikhin-Turaev 3-流形不变量。特别是，PI将与这些重新归一化的不变量和其他拓扑对象联系起来，包括Reidemister挠率、拓扑量子场论(TQFT)和映射类群表示。本文还将继续PI对广义Kashaev量子三维流形不变量的发展和研究。这些不变量是以卡沙耶夫的基础工作为模型的，卡沙耶夫的基础工作导致了体积猜想的原始公式。这项研究的最后一个主题是通过定义3-流形不变量来推广他的重新规范化的Reshetikhin-Turaev不变量，这些不变量是通过在补集中具有平坦连接的环的外科表示来定义的。这项工作既是几何性质的，也是物理性质的，与陈-西蒙斯理论有关。自然界中数学产生的基本方式之一是通过几何，更广泛地说是通过拓扑学。特别是，被称为流形的几何对象被用来模拟广义相对论中的空间和时间。令人惊讶的是，流形的几个特征可以通过它们的基本拓扑来研究。1984年V.Jones发现了Jones多项式，1989年E.Witten解释了该多项式的三维量子场论，为使用新的代数方法研究拓扑学打开了大门。这些发展导致了一个新的数学分支，被称为“量子拓扑学”。这一领域与理论物理有关，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，量子拓扑学中的许多成熟技术都变成了零。本项目致力于开发和研究当量子维度为零时重新归一化拓扑不变量的新的系统策略。

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.