Volume Conjecture for Surface Diffeomorphisms

Explore a 54-minute lecture on the Volume Conjecture for Surface Diffeomorphisms, presented by Helen Wong at the Centre de recherches mathématiques (CRM) workshop on Quantum symmetries. Delve into the representation theory of the Kauffman bracket skein algebra of a punctured surface to understand the derivation of a quantum invariant for surface diffeomorphisms. Examine the conjecture that, at a specific root of unity, the asymptotic growth of this quantum invariant detects the volume of the mapping torus of the surface diffeomorphism. Analyze supporting evidence, including numerical data and a proof for the simplest case where the mapping torus is the figure-eight knot complement. Gain insights into this collaborative research conducted with Francis Bonahon and Tian Yang, bridging concepts from quantum topology, representation theory, and geometric topology.