  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. I 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 is devoted to bounds relating transport to
quantum chaos, which can be established through pole-skipping in theories with
holographic duals. Examples of such bounds are shown along with holographic
theories that can demonstrate the validity of the conditions involved. I also
discuss potential applications of univalence methods to bounds without relation
to chaos, such as for example the conformal bound on the speed of sound.
