This is the first of two Stelson lectures on Quantum Hyperbolic Geometry. In the past 30 years, a lot of the activity in low-dimensional topology has occurred in hyperbolic geometry and in topological quantum field theory. However, these two branches of mathematics have largely evolved in parallel, without much interaction. For instance, proofs in hyperbolic geometry tend to be analytic, whereas topological quantum field theory has a more combinatorial/algebraic flavor. The so-called Volume Conjecture now provides an exciting conjectural bridge between these two domains. Technically and conceptually, the challenge is to figure out how these two fields can fit together in a common context. We will discuss some of these issues, and briefly sketch a framework which combines hyperbolic geometry and topological quantum field theory.
We will provide a more detailed illustration of the principles discussed in the first talk. This second talk will be focused on a punctured surface S. The quantum Teichmüller space of S is a certain non-commutative deformation of the algebra of rational functions on the space of (2-dimensional) hyperbolic metrics on S. This is a purely algebraic object, closely related to the combinatorics of the Harer-Penner complex of ideal cell decompositions of the surface. It turns out that the finite-dimensional representation theory of this algebraic object is controlled by the same type of data as hyperbolic metrics on the 3-dimensional manifold product of S with the real line. We will use this correspondence to exhibit strange invariants of surface diffeomorphisms, and speculate on the relevance of this construction to the Volume Conjecture.
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