Mathematics

Neckpinch singularities are a prevalent feature of Ricci flow, and recent work has given us a good picture of their asymptotic behavior, so long as the geometries are rotationally symmetric. We discuss this asymptotic behavior, both for degenerate and non-degenerate neckpinches. It has been conjectured that neckpinch singularities which develop in non-rotationally symmetric Ricci flows do asymptotically approach roundness, and consequently have very similar asymptotic behavior to those which are rotationally symmetric. We discuss very recent work which supports this conjecture.