We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, showing in particular that this group is not quasi-isometric to a line. This answers a well-known question of Kapovich-Polterovich from 2006. We show that these flats in Ham(S2)Ham(S2) stabilize to certain product four-manifolds, prove constraints on Lagrangian packing, and find new instances of Lagrangian Poincare recurrence. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. This is joint work with Leonid Polterovich.