Hyperbolic Geometry and Quantum Invariants

120年来，量子物理学深刻地改变了世界。它对数学的影响也是深远而巨大的。该项目旨在将三维空间的几何与量子物理联系起来。这些问题的肯定解决将对我们对三维宇宙的理解产生影响。所研究对象的性质也处于几个不同数学领域的交叉点。研究者计划研究所有这些方面，并期望每个领域的技术将为其他相关领域带来新的启示。研究的中心是双曲几何和量子不变量的界面。PI在这两个截然不同的领域的双重专长使他为解决这些问题做好了充分的准备。该项目沿着三个相互关联的研究主题进行阐述，涉及量子不变量的渐近性及其与双曲几何的关系。具体的目标包括：(1)用流形的几何三角边的长度求出封闭双曲3流形的伴扭Reidemeister扭转的显式公式。(2)讨论了面向封闭的双曲3-流形Reshetikhin-Turaev不变量和具有完全测地边界的双曲3-流形的Turaev-Viro不变量的体积猜想和渐近展开猜想。(3)研究了Bonahon和Liu引入的曲面自同胚的量子不变量的渐近性，并将其与同胚的（三维）映射环面的双曲体积联系起来。这项研究将部分由几个合作者进行，他们的专业知识可以成为该计划的资产。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Quantum physics has profoundly changed the world since 120 years ago. Its impacts on mathematics is also deep and huge. The project aims to relate geometry of 3-dimensional space and quantum physics. The affirmative resolution of the problems will have impact on our understanding of the 3-dimensional universe. The nature of the objects under study is also at the intersection of several different areas of mathematics. The investigator plans on studying all of these aspects, and expects that techniques from each area will shed new light on the other areas involved.The research is centered at the interface of hyperbolic geometry and quantum invariants. The PI’s dual expertise in these two very different fields makes him well prepared to tackle the problems. The project is articulated along three inter-related research themes involving the asymptotics of quantum invariants and their relationship with hyperbolic geometry. Specific goals include: (1) Find an explicit formula of the adjoint twisted Reidemeister torsion of a closed hyperbolic 3-manifold in terms of the length of the edges of a geometric triangulation of the manifold. (2) Attack the Volume Conjecture and the Asymptotic Expansion Conjecture for Reshetikhin-Turaev invariants of closed oriented hyperbolic 3-manifolds and for Turaev-Viro invariants for hyperbolic 3-manifolds with totally geodesic boundary. (3) Study the asymptotics of the quantum invariants of self-homeomorphisms of a surface introduced by Bonahon and Liu, and relate them to the hyperbolic volume of the (3-dimensional) mapping torus of the homeomorphism. This research will be partially conducted with a few collaborators, whose expertise can be an asset to the program.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.