**Title: ** Landau Damping in a Periodic Box

**Speaker: **Jacob Bedrossian (UMD)

Landau damping is a fundamental stability mechanism in effectively collisionless plasmas caused by the mixing, and subsequent spatial homogenization, of charged particles. Despite being one of the simplest, physically relevant, examples of stability due to mixing, it largely resisted mathematically rigorous study on the nonlinear level due to subtle regularity issues intertwined with a special set of unusual nonlinear resonances known as plasma echoes. A major breakthrough was due to Mouhot and Villani in 2011, however, their work was far too unwieldy to be generalized to related mixing problems in fluid mechanics and elsewhere in kinetic theory. I will present a more recent proof due to Masmoudi, Mouhot and myself in 2013 which makes use of a few paradifferential calculus techniques (adapted from Masmoudi and I's work on mixing in fluid mechanics) to provide a more robust and significantly simplified proof of Landau damping in the natural regularity classes.