An old question of Poincaré concerns creating periodic orbits via perturbations of a flow/diffeomorphism. While pseudoholomorphic methods have successfully addressed this question in dimensions 2-3, the higher-dimensional case remains less understood. I will describe a connection between this question and Gromov–Witten invariants, which goes through a new class of invariants of symplectic cobordisms. This is a joint work with Julian Chaidez.