Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? This question was originally asked by Poincaré and was initially studied in the 60s. However, various facets of it remain largely open. Recently, several advances were made in the context of Hamiltonian and contact flows. I will discuss a connection between this problem and Gromov-Witten invariants, which are "counts" of holomorphic curves. This is based on a joint work with Julian Chaidez.