Applications of Geometric Flows

Geometric flows describe the evolution of shapes driven by geometric quantities such as curvature.While originating in pure mathematics, where they played a central role in milestones like the proof of the Poincar conjecture, these flows also have a rich history of applications across science and engineering.In this paper, we provide a brief overview of the key ideas behind geometric flows, focusing on curve and surface evolution in Euclidean space, and highlight their relevance to modeling dislocation lines in crystals, vortex filaments in fluids, and optimal structural forms in engineering.Our aim is to provide a concise but selfcontained exposition, emphasizing the conceptual beauty and broad utility of this mathematical framework.