Precise Asymptotics of the Ricci Flow Neckpinch

1.1. Antecedents. In virtually all known applications of the Ricci flow, it is valuable to have a good understanding of singularity formation. Heuristically, there are at least three reasons for this. The first is that one expects finite-time singularities to form for a broad spectrum of initial data. Indeed, such singularities are inevitable if the scalar curvature is strictly positive. The second reason is that one expects the geometry of a solution to resemble a standard model (for example, a self-similar solution) in a space-time neighborhood of a developing singularity. The third reason is that having a sufficiently detailed picture of a developing singularity facilitates the geometric-topological surgeries by which Ricci flow decomposes a given manifold.