  The best known finite-time local Ricci flow singularity is the neckpinch, in
which a proper subset of the manifold becomes geometrically close to a portion
of a shrinking cylinder. In this paper, we prove precise asymptotics for
rotationally symmetric Ricci flow neckpinches. We then compare these rigorous
results with formal matched asymptotics for fully general neckpinch
singularities.
