A brief introduction to the mathematics of Landau damping

In these short, rather informal, expository notes I review the current state of the field regarding the mathematics of Landau damping, based on lectures given at the CIRM Research School on Kinetic Theory, November 14--18, 2022. These notes are mainly on Vlasov-Poisson in $(x,v) \in \mathbb T^d \times \mathbb R^d$ however a brief discussion of the important case of $(x,v) \in \mathbb R^d \times \mathbb R^d$ is included at the end. The focus will be nonlinear and these notes include a proof of Landau damping on $(x,v) \in \mathbb T^d \times \mathbb R^d$ in the Vlasov--Poisson equations meant for graduate students, post-docs, and others to learn the basic ideas of the methods involved. The focus is also on the mathematical side, and so most references are from the mathematical literature with only a small number of the many important physics references included. A few open problems are included at the end. These notes are not currently meant for publication so they may not be perfectly proof-read and the reference list might not be complete. If there is an error or you have some references which you think should be included, feel free to send me an email and I will correct it when I get a chance.