Quantum Invariants and Geometric Structures
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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.