  This note discusses recent new approaches to studying flopping curves on
3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold
invariant, the contraction algebra, which may be associated to such curves. It
characterises their geometric and homological properties in a unified manner,
using the theory of noncommutative deformations. Toda has now clarified the
enumerative significance of the contraction algebra for flopping curves,
calculating its dimension in terms of Gopakumar-Vafa invariants. Before
reviewing these results, and others, I give a brief introduction to the rich
geometry of flopping curves on 3-folds, starting from the resolutions of
Kleinian surface singularities. This is based on a talk given at VBAC 2014 in
Berlin.
