Emergent hydrodynamics in integrable quantum systems out of equilibrium

 I will present a novel framework for studying transport in integrable systems: that of emergent hydrodynamics with infinitely-many conservation laws. This bridges the conceptual gap between integrable and non-integrable quantum dynamics, and gives powerful tools for accurate studies of space-time profiles in experimentally realizable quasi-one-dimensional systems such as interacting Bose gases. I will apply it to the description of transport between heat baths, occurring when two semi-infinite halves of a system initially thermalized in different states are connected to each other. I will describe the exact current-carrying non-equilibrium steady state and the transition regions surrounding it, with the exact current and density profiles, in the (relativistic) sinh-Gordon and (non-relativistic) Lieb-Liniger models. This reproduces in various limits results from conformal field theory, and fulfills stringent inequalities - both of which I will review. If time permits, I will also describe how to take into account inhomogeneous external fields within this formalism. This is based on papers in collaboration with Olalla Castro-Alvaredo and Takato Yoshimura, and on earlier works with Denis Bernard.