Poisson brackets and symplectic invariants

We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ ³-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.