Classical and quantum hyperbolic geometry

University Of Southern California, Los Angeles CA

Investigators

Abstract

In the last thirty years, much of the progress in our understanding of 3-dimensional topology has been grounded in hyperbolic (non- euclidean) geometry and in topological quantum field theory. However, these two branches of mathematics have long evolved in parallel, without much interaction with each other. Hypothetical bridges between these two fields are now beginning to emerge. One of them is the Volume Conjecture, experimentally verified by a few computations, which connects the asymptotic growth of the Jones polynomials of a knot to the volume of the canonical hyperbolic metric of its complement. The goal of the Project is to develop a conceptual framework combining the two points of view. The Project has a 2- dimensional component, focussed on the representation theory of the quantum Teichmuller space of a surface, as introduced by physicists to model quantum gravity in dimension 2+1. A second part of the Project builds on the insight gained in dimension 2 to develop invariants of hyperbolic 3-dimensional manifolds, closely related to the objects appearing in the Volume Conjecture. The Project aims at gaining a better understanding of 3-dimensional geometry, such as the problem of deciding when two knotted curves in space can be deformed to each other. One traditional approach to this problem involves the algebraic manipulation of certain polynomials associated to pictorial descriptions of these curves. Another powerful technique, with widely used software implementation, uses hyperbolic non-euclidean geometry. Experimental evidence suggests an unexpected connection between these two points of view. The goal of the Project is to develop technical and conceptual tools to confirm (or disprove) this connection, and to better understand the phenomena involved.