Explore the cutting-edge developments in mathematical fluid dynamics through this Stony Brook Mathematics Colloquium talk. Delve into the unexpected progress made in understanding non-uniqueness of solutions to fundamental partial differential equations, specifically the Euler and Navier-Stokes equations. Examine the state-of-the-art research in this field, with a particular emphasis on the relationship between instability and non-uniqueness. Survey parallel programs by prominent researchers, including Jia-Svěrák-Guillod, Vishik, Bressan-Murray-Shen, and Albritton-Brué-Colombo. Learn about the groundbreaking proof that Leray-Hopf solutions of the forced Navier-Stokes equations are not unique. Gain insights into the connections between physics and "spontaneous stochasticity" that are yet to be fully understood in this complex area of study.