Mathematical Conversations

I will discuss a 1965 conjecture of J. Andrews and M. Curtis---a beguilingly straightforward statement about presentations of the trivial group, which has striking significance in low-dimensional topology: e.g. in relation to the 3-dimensional Poincaré Conjecture, the smooth 4-dimensional Poincaré Conjecture (which remains open), and to the Zeeman Conjecture (also still open).