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Explore the evolution of planar polygonal curves under crystalline elastic flow in this 40-minute lecture from the Erwin Schrödinger International Institute for Mathematics and Physics. Discover how this geometric gradient flow, associated with crystalline perimeter, serves as a natural perturbation of crystalline curvature flow where polygonal sides evolve through parallel translation. Learn about the long-time existence and uniqueness results for immersed polygonal curves, including potentially unbounded configurations. Examine how the flow can be restarted beyond singularities for closed polygons, creating a global evolution that preserves the topological index of the curve. Investigate the long-time behavior through a Lojasiewicz–Simon type inequality that proves convergence to stationary configurations. Understand the classification of stationary and translating solutions specifically in cases of square anisotropy. This research represents collaborative work with Giovanni Bellettini from the University of Siena and ICTP Trieste, and Matteo Novaga from the University of Pisa, presented as part of the Thematic Programme on Free Boundary Problems.
