Milton Jara: NESS and Markov chains Non-equilibrium stationary states (NESS) are ubiquitous in nature, and their description presents striking challenges to mathematicians. Roughly speaking, NESS are stationary states on which the presence of currents prevents the system to be in (statistical) equilibrium. A possible way to describe NESS is through Markov chains. We say that the invariant measure of a Markov chain is a NESS if it is not reversible. Therefore, we can restate the study of NESS as the study of non-reversible Markov chains and their invariant measures. Donsker-Varadhan theory of large deviations of Markov chains is an example of a general theory that can be used to achieve this goal. One successful example of application of this strategy is the Macroscopic Fluctuation Theory (MFT) of Bertini, De Solé, Gabrielli, Jona-Lasinio and Landim, which describes the large fluctuations of NESS for driven-diffusive systems in terms of certain thermodynamic variables. In recent works, we have developed a theory of quantitative hydrodynamics that allows us to describe the CLT fluctuations of NESS for driven-diffusive systems, that confirms the prediction of MFT also at the level of the CLT.