Elia Bruè, of the Institute for Advanced Study, presents a seminar on Bressan's mixing conjecture, an open problem from 2003 about how efficiently incompressible flows can mix a substance. Bruè surveys what progress has been made toward resolving it and then presents a new result establishing the conjecture's asymptotic validity for time-periodic velocity fields. The core technical move is extending tools from smooth dynamical systems, notably Lyapunov exponents, into the rougher setting of DiPerna-Lions flows, where velocity fields need not be smooth enough for classical trajectory arguments. Bruè connects this construction to bounds on metric entropy and to generalizations of the Ruelle inequality, linking the mixing question to broader themes in ergodic theory. Delivered at the Institute for Advanced Study's Analysis and Mathematical Physics seminar, the talk assumes graduate-level familiarity with PDE and dynamical systems and moves at a research pace, aimed at specialists tracking the current state of a two-decade-old open problem in fluid dynamics.