In this talk, we consider the hyperbolic nonlinear Schr¨odinger equations (HNLS) on R×T. Our primary goal is to present local and global Strichartz estimates for typical hyperbolic propagators. As an application to such estimates, we obtain the local/global existence of solutions to HNLS in critical Sobolev spaces. Additionally, we also mention the analytical connection between solutions of the hyperbolic Schr¨odinger equation and those of the Boltzmann equation. This is joint work (https://arxiv.org/abs/2504.15836) with Chenmin Sun (Universite Paris- Est Creteil), Nikolay Tzvetkov (ENS Lyon), and Yuzhao Wang (University of Birmingham).