Mathematical Conversations

The end of the previous century saw radical changes to three-dimensional topology, which arose from two completely different approaches. One breakthrough came from Thurston's introduction of hyperbolic geometry into the field. The second one came from the discovery of new quantum invariants for knots and three-dimensional manifolds, and brought in insight and techniques from mathematical physics and quantum groups. The Volume Conjecture proposes a surprising connection between these two approaches, but has proved quite elusive. We will explain its relevance and current progress (or lack thereof).