Explore fundamental mathematical concepts in plasma physics through this 58-minute colloquium that delves into the stability and instability of Vlasov-type equations, essential for understanding charged particle behavior in plasmas. Learn about kinetic theory fundamentals, making complex plasma mathematics accessible to mathematicians of various backgrounds. Examine critical mathematical principles including well-posedness, stability analysis, and solution behaviors in singular limits, while discovering a novel class of Wasserstein-type distances that provides fresh insights into kinetic equation stability. Gain valuable knowledge about the mathematical challenges and recent breakthroughs in plasma physics research through this presentation by Mikaela Lacobelli at the Institute for Advanced Study.