Asymptotics of Quantum Invariants of Surface Homeomorphisms

The Kashaev Volume Conjecture is a tantalizing problem relating two very different areas of 3-dimensional topology: the combinatorial/algebraic theory of quantum invariants of knots and links in 3-dimensional space; and the geometric/analytic construction of unique hyperbolic metrics on the complement of these knots and links. I will discuss a closely related conjecture in one dimension lower, involving homeomorphisms of surfaces but also leading to 3-dimensional hyperbolic geometry. In particular, I will present numerical evidence and rigorous theorems obtained in collaboration with Helen Wong and Tian Yang.