The complexity of group presentations, manifolds, and the Andrews-Curtis conjecture

Many natural problems concerning the geometry and topology of manifolds are intimately connected with the nature of presentations for the fundamental groups of the manifolds. I shall illustrate this theme with various specific results, then focus on balanced presentations. I'll explain the (open) Andrews-Curtis conjecture and it's relation to the smooth 4-dimensional Poincare conjecture, and I'll present a construction that gives (huge) lower bounds on how hard it is to distinguish a homology 4-sphere from a genuine sphere.