3-manifold invariants from Virasoro TQFT and asymptotics

About This Grant

This project is centered at the interface of geometric topology and quantum topology, with an influence of the recent developments in mathematical physics and probability theory. The research builds on the primary investigator's (PI) prior work, and will be partially conducted with collaborators, who bring in relevant expertise. In addition to the research itself, the project includes an educational component, integrated within the research project, which aims at enhancing the training of young mathematicians in this general area of mathematics. This aspect of the project involves the development and expansion of a strong geometric topology group at Texas A&M University, and the nurturing of a community of young mathematicians interested in quantum topology and low-dimensional geometry through conferences and other activities. The project is articulated along two inter-connected research directions related to the asymptotics of quantum invariants and their relationship with hyperbolic geometry. (1) The PI plans to study a family of Turaev-Viro type 3-manifold invariants constructed from the Virasoro TQFT and their asymptotic behavior. This builds on recent joint work of the PI where a one-parameter family of Turaev-Viro type invariants was defined for a hyperbolic 3-manifold with totally geodesic boundary and it was proved that this family of invariants decays exponentially with a rate given by the hyperbolic volume of the manifold. This research will be developed in collaboration with Xin Sun, Baojun Wu, Tianyue Liu and Shuang Ming. (2) As a crucial step towards the Volume Conjecture for the Reshetikhin-Turaev invariants, the PI aims to study various properties of the fundamental shadow link complements. This research will be conducted in collaboration with the PI's Ph.D. student Inyoung Ryu. 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.