Quantum Invariants and Geometric Structures

量子物理学在过去的120年里深刻地改变了世界。它对数学的影响也很深。这个国家科学基金会资助的项目旨在将三维空间的几何与量子物理学联系起来。该项目中问题的肯定性解决将有助于我们对三维宇宙的理解。这个项目是在两个数学领域的纽带，被称为量子拓扑和几何拓扑。作为这两个领域为数不多的专家之一，调查人员计划使用其中一个领域的技术来揭示另一个领域的新情况。Kashaev的体积猜想、维滕的渐近展开猜想及其推广证明了三维流形的某些量子不变量的渐近行为提供了流形的拓扑和几何信息。在Kashaev和维滕的理论的各种推广中，最近由陈庆涛和PI关于Turaev-Viro不变量及其与三维流形的双曲体积的关系的推广引起了专家们的极大关注。在这个项目中，PI计划通过研究Turaev-Viro不变量的构建块、量子6 j符号和引导构建块组装的几何结构的渐近行为来研究Turaev-Viro不变量的渐近行为。其中一个关键因素是PI和合作者开发的双曲锥度量的刚性。在一个相关的项目中，PI将试图解决Andersen，Masbaum和Ueno（AMU）的猜想，该猜想声称量子表示检测表面映射类的Nielson-Thurston分类。最近，PI和一个合作者意识到AMU猜想是Chen和PI猜想的一个较弱版本的直接结果。他们的方法可以解决较弱版本的猜想，从而解决AMU猜想。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Quantum physics has profoundly changed the world in the last 120 years. Its impact on mathematics is also deep. This National Science Foundation funded project aims to relate the geometry of the three-dimensional space to quantum physics. The affirmative resolution of the problems in the project will benefit our understanding of our three-dimensional universe. This project is at the nexus of two areas of mathematics, known as quantum topology and geometric topology. Being one of the few experts in both these areas, the investigator plans to use techniques from one to shed new light on the other. Kashaev's Volume Conjecture, Witten's Asymptotic Expansion Conjecture and their generalizations assert that the asymptotic behavior of certain quantum invariants of a 3-manifold provides topological and geometric information of the manifold. Among various generalizations of Kashaev's and Witten's conjectures, the recent one by Qingtao Chen and the PI on the Turaev-Viro invariants and their relationship with the hyperbolic volume of the 3- manifold attracts massive attention from the experts. In this project, the PI plans to study the asymptotic behavior of the Turaev-Viro invariants by studying the asymptotic behavior of their building blocks, the quantum 6j-symbols, and the geometry that guides the assembling of the building blocks. A key ingredient is the rigidity of hyperbolic cone metrics developed by the PI and a collaborator. In a related project the PI will attempt to solve a conjecture of Andersen, Masbaum and Ueno (AMU) which asserts that the quantum representations detect the Nielson-Thurston classifications of the mapping classes of a surface. Recently, the PI and a collaborator realized that the AMU conjecture is an immediate consequence of a weaker version of the conjecture by Chen and the PI. Their approach could settle the weaker version of the conjecture, and thereby solve the AMU conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.