A classical problem going back to ancient Greece is to find the shortest curve in the plane enclosing a given area: the isoperimetric problem. A similar question is whether given a curve on a surface it can be deformed to a shortest one. Whilst the solutions to these classical problems are well-known, natural generalisations in higher dimensions are mostly unsolved. I will explain how this leads us to the study of minimal Lagrangians and the question of how to find them, which will take us to the interface between symplectic topology, Riemannian geometry and analysis of nonlinear PDEs, with links to theoretical physics.