Contemporary Mathematics Singularities at t = in Equivariant Harmonic Map Flow Summary: Contemporary Mathematics Singularities at t = in Equivariant Harmonic Map Flow Sigurd Angenent and Joost Hulshof 1. Introduction Many nonlinear parabolic equations in Geometry and Applied Mathematics (mean curvature flow, Ricci flow, harmonic map flow, the Yang-Mills flow, reaction diffusion equations such as ut = u + up ) have solutions which become singular either in finite or infinite time, meaning either that the evolving object (map, metric, surface, or function) becomes unbounded, or that one of its derivatives becomes unbounded. The analysis of the asymptotic behaviour of a solution of a nonlinear parabolic equation just before it becomes singular is known to be a difficult problem. The main general point of this note is that this analysis is considerably easier in the case where the singularity occurs in infinite time. The reason for this is that infinite time singularities are a "stable phenomenon" in the following sense. Given an initial data whose solution becomes singular at t = , a slight modification of this initial data will generally still produce a solution which becomes singular at the same time (namely, t = ). In contrast, if a solution becomes singular at time t = T < , then a small perturbation of the initial data will generally still produce a solution which becomes singular in finite time, but, usually, at a Collections: Mathematics