Ursula HAMENSTÄDT - The geometry of 3 - manifolds before and after Perelman

The rank of a hyperbolic manifold is the smallest number of generators of its fundamental group. McMullen conjectured that for all $k\geq 2$, the pointwise injectivity radius of a closed hyperbolic 3-manifold of rank at most k is uniformly bounded from above. We explain some methods which were introduced before and after the foundational work of Perelman to study these manifolds, and we show how these methods can be used to prove McMullen's conjecture in many cases inc luding random 3-manifolds.