Ancient solutions to geometric flows are fundamental to understanding singularity formation—the main barrier to long-time existence of such flows. A central component in developing weak formulations, such as flows with surgery, is obtaining a precise geometric description of the singularity models, which are typically represented by ancient solutions. These solutions, which exist for all times in the past, have a persistent structure that makes them amenable to classification, especially under natural geometric assumptions.

 

In this talk, we will present a construction method for ancient solutions in the context of mean curvature flow and show how this approach extends to related flows, including the intrinsically defined Ricci flow. We will also address rigidity phenomena. Furthermore, in the setting of free boundary problems, we will demonstrate how these results can be generalized and will outline a comprehensive theory for mean curvature flow encompassing construction, uniqueness, and stability.