Asymptotic Behavior of Ricci Flow near Cylinders. 

The cylindrical soliton solution arises naturally in Ricci flow. It models the behavior of the flow near many singularities (for instance when a "neck region" is pinching off during the evolution of a dumbbell-shaped surface in its middle part) and it describes the asymptotic shape of ancient solution after rescaling such as Bryant soliton and Perelman's "sausage" solution. It is therefore important to understand Ricci flow on manifolds close to the cylinder if one wants to classify singularity models or ancient solutions. We describe some classical techniques coming from blow up analysis of semi-linear heat equation. We then apply these ideas to Ricci flow and its linearization around the cylinder. This leads to rigidity results regarding the precise geometry of Ricci flow solutions.