MSI Colloquium 2018: Ancient solutions to Geometric Flows

Some of the most important problems in geometric flows are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time  $- \infty < t \leq T$ for some $T \leq +\infty$.

We will refer to them as Ancient solutions. The classification of such solutions often sheds new insight to the singularity analysis. In this lecture we will give an overview of Uniqueness Theorems for ancient solutions to geometric partial differential equations such as the Mean curvature flow. and the Ricci flow and the Yamabe flow. This often involves the understanding of the geometric properties of such solutions.

We will also discuss the construction of new ancient solutions from the parabolic gluing of one or more solitons.