Lagrangian configurations and Hamiltonian maps

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, prove constraints on Lagrangian packing, and find new instances of Lagrangian Poincare recurrence. In particular, this answers a question of Kapovich-Polterovich from 2006 that appears as Problem 21 in McDuff-Salamon's list of open problems. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. This is joint work with Leonid Polterovich.