Appears in collection : 2015 - T1 - Disordered systems, random spatial processes and some applications

In 1986, Kardar, Parisi and Zhang introduced a non-linear stochastic PDE to describe dynamics of interface motion. This KPZ equation has been studied intensively and extensively since then, but recently its one-dimensional version has been attracting particular attention because of its tractability and connections to various areas of mathematics and physics. In this lecture I will explain part of these developments . I will mainly focus on the height distribution for the KPZ equation. I will also explain the underlying algebraic structure such as random matrix theory, Schur process and Macdonald process in connection to related discrete interacting particle systems like ASEP and q-boson zero range process.