Explore the fascinating world of geometry in this 55-minute lecture by Simon Brendle, part of the Aisenstadt Chair Lecture Series at the Centre de recherches mathématiques (CRM). Delve into the historical significance of the isoperimetric inequality and discover its generalization to submanifolds in Euclidean space. Examine a sharp isoperimetric inequality for minimal submanifolds of codimension at most 2, addressing a long-standing question rooted in Carleman's work. Gain insights into the proof methodology, which draws inspiration from optimal transport without directly employing it. Enhance your understanding of minimal surfaces and their relationship to isoperimetric inequalities in this advanced mathematical exploration.