It is a relatively recent discovery in geometric topology that optimization problems of certain topological complexity are connected to important geometric and topological information. One example is the Gromov-Thurstonnorm, which is the minimal complexity of surfaces representing a given second homology class, and it is closely related to fibrations of 3-manifolds as surface bundles. This is an introductory talk on a relative version of the Gromov-Thurston norm called the stable commutator length (scl), which is the minimal complexity of surfaces bounding a given loop. I will explain its computation and sharp lower bounds in relation tolinear programming, as well as its close relation to various problems in topology and group theory.
