Hyperbolic Geometry and Quantum Invariants

自120年前以来，量子物理已经深刻地改变了世界。它对数学的影响也是深刻而巨大的。该项目旨在将三维空间的几何学与量子物理联系起来。这些问题的肯定解决将影响我们对三维宇宙的理解。研究对象的性质也是几个不同数学领域的交集。研究人员计划研究所有这些方面，并期待每个领域的技术将为其他相关领域带来新的曙光。研究集中在双曲几何和量子不变量的界面上。私家侦探在这两个截然不同的领域拥有双重专业知识，这使他做好了应对这些问题的准备。该项目围绕三个相互关联的研究主题展开，涉及量子不变量的渐近性及其与双曲几何的关系。具体目标包括：(1)根据流形的几何三角剖分的边的长度，求出闭双曲3-流形的伴随扭转Reidomeister挠率的显式公式。(2)对闭定向双曲三维流形的Reshetikhin-Turaev不变量和具有全测地边界的双曲三维流形的Turaev-Viro不变量的体积猜想和渐近展开猜想进行了反驳。(3)研究了由Bonahon和Liu引入的曲面自同胚的量子不变量的渐近性，并将它们与自同胚的(3维)映射环面的双曲体积联系起来。这项研究将部分与一些合作者一起进行，他们的专业知识可能是该计划的一项资产。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.