Bounds on transport from univalence

Bounds on transport represent a way of understanding allowable regimes of quantum and classical dynamics. Numerous such bounds have been proposed, either for classes of theories or (by using general arguments) universally for all theories. Few are exact and inviolable. In this talk, I will present a new set of methods and sufficient conditions for deriving exact, rigorous, and sharp bounds on all coefficients of hydrodynamic dispersion relations, including diffusivity and the speed of sound. These general techniques combine analytic properties of hydrodynamics and the theory of univalent (complex holomorphic and injective) functions. Particular attention will be devoted to bounds relating transport to quantum chaos, which can be established through pole-skipping in theories with holographic duals. I will also discuss applications of univalence methods to bounds without relation to chaos, such as the conformal bound on the speed of sound.