Geometric evolution equations describe how geometric objects such as curves, surfaces, or metrics evolve toward more symmetric or optimal shapes. Among the most fundamental examples are the mean curvature flow and the Ricci flow, which have played central roles in modern differential geometry and topology. In this talk, I will give an introduction to these flows, explaining how curvature acts as a driving mechanism that smooths and reshapes geometry. I will also outline the key ideas behind Perelman’s proof of the Poincaré conjecture, focusing on the role of singularity formation and the classification of canonical neighborhoods. Finally, I will discuss the problem of classifying singularity models arising under geometric flows and present some recent progress on the classification of ancient oval solutions, together with possible further directions.
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