Hyperbolic Geometry and Quantum Invariants

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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.