Explore a seminar on spectral geometry focusing on the Riesz inequality for polygons, presented by Ilaria Fragalà from Politecnico di Milano. Delve into counterparts of the classical Hardy-Littlewood and Riesz inequalities when applied to polygons with a fixed number of sides, which corresponds to studying the polygonal isoperimetric problem in a nonlocal version. Learn about the concepts of symmetry and symmetry breaking in this context. The talk, part of the "Spectral Geometry in the clouds" series, is based on joint work with Beniamin Bogosel and Dorin Bucur. Gain insights into this advanced mathematical topic over the course of 1 hour and 12 minutes, presented at the Centre de recherches mathématiques (CRM).