Classical and quantum homomorphisms from discrete groups to Lie groups

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The modern formulation of various physical problems involves families of matrices (that is tables of numbers) called Lie groups. This is true of the foundations of classical mechanics, and even more so for general relativity and quantum physics. For this reason much mathematics has been developed in this framework for the past 150 years. In the 1980s, several breakthroughs have emphasized the role of specific Lie groups in various geometric problems. This includes the surprisingly effective use of non-euclidean hyperbolic geometry to analyze the knotting of curves in space. Other developments, drawing their inspiration from quantum physics, have provided tools to attack the same knotting problems based on certain deformations of Lie groups called quantum groups. The Project investigates several problems involving classical Lie groups and their quantum group deformations, and their applications to the study of knots and 3-dimensional spaces. It draws its motivation and technical tools from several different branches of mathematics, including geometry, topology, algebra and dynamical systems. The Project is articulated along two themes that are very different in nature, but united by the fact that hyperbolic geometry can be used as an intellectual guide in each of them. The Project also has a strong educational component, as its research themes are designed so that they can nurture the doctoral work of several graduate students, and provide a broad postdoctoral training to junior faculty in the research group of the Principal Investigator (PI). The first theme of the Project is focused on the classical geometry of homomorphisms from the fundamental group of a surface to a Lie group. When the Lie group is split real, for instance for the special linear group SL(n,R), the so-called Hitchin homomorphisms satisfy many important geometric and dynamical properties. A first goal of the proposal is to develop a differential calculus for the spectrum of the images under Hitchin homomorphisms of simple closed curves on the surface. This includes the development of a parametrization of the space of Hitchin homomorphisms that is well-adapted to such a calculus. The Project will also use these methods to investigate the boundary at infinity of the space of Hitchin homomorphisms. Moving to the complex set-up, the PI will investigate the geometry of homomorphisms valued in complex Lie groups but close to real Hitchin homomorphisms. The second theme of the Project involves quantum group invariants of knots in 3-dimensional manifolds. The PI will continue his investigation of representations of the Kauffman skein algebra on a surface, considered as points of a quantization of the space of homomorphism from the fundamental group of the surface to the Lie group SL(2,C). He will then build on the results and tools developed in this investigation, and on earlier work of Kashaev-Baseilhac-Benedetti, to build a (2+1)-dimensional topological quantum field theory that mixes quantum topology and hyperbolic geometry. The long term goals of this work is to provide conceptual and technical tools to attack the Volume Conjecture, which predicts a precise relationship between the asymptotic behavior of certain quantum invariants of a knot in 3-space and the hyperbolic volume of its complement. The technology provided by the Kauffman skein algebra is more intrinsic than earlier approaches, and should be particularly useful.