Abstract: An old question of Poincaré concerns creating periodic points via perturbations of Hamiltonian diffeomorphism. Hamiltonian diffeomorphisms are solutions of a certain ODE and arise from classical mechanics. In the first part of the talk, I will provide an overview of results related to this problem. In the second part, I will discuss how holomorphic curves—complex 1-dimensional submanifolds—can be used to study the existence and properties of periodic points of Hamiltonian diffeomorphisms. The talk is based on a joint work with Julian Chaidez.