Asymptotics of Quantum Invariants

University Of Southern California, Los Angeles CA

Investigators

Abstract

The project is part of the general field of 3-dimensional topology and geometry, which analyzes 3-dimensional spaces with a high level of complexity and includes the study of the knotting phenomena that occur for curves in three dimensions. The underlying problems often arise from practical applications in the three-dimensional world we live in, but also from more theoretical considerations in mathematical physics. The project investigates a surprising connection between two very different approaches to 3-dimensional topology. The first approach is very algebraic and involves so-called quantum invariants for knotted curves in 3-dimensional spaces. The second approach is more geometric and analytic, and uses the non-euclidean geometry of the 3-dimensional hyperbolic space. The Kashaev Volume Conjecture, experimentally verified on many examples but still without a mathematical proof guaranteeing that it holds in all cases, provides an unexpected connection between these two viewpoints. The award provides support for graduate students who will be involved in related research. More precisely, the Kashaev Volume Conjecture connects the asymptotics of the colored Jones polynomial of a knot to the hyperbolic volume of its complement. This tantalizing conjecture is now 25-year old, and supported by much heuristic and experimental evidence. However, the corresponding property has been rigorously proved for only a very small number of cases. The goal of the project is to develop various steps in a roadmap towards a proof of the Kashaev Volume Conjecture, with a special emphasis on its analytic subtleties. As a first step, the project is focused on a closely related conjecture for surface bundles over the circle, where the combinatorics of quantum invariants are more clearly connected to the geometry of the underlying manifold. The PI and his collaborators will attack this conjecture in the even more special case of punctured torus bundles, in an effort to be very concrete while tackling the analytic difficulties that arise in this context. The PI will then build on the experience, insights and technical expertise gathered in this first case to proceed with the next steps in the roadmap, one after the other. 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.