In this talk we will come at the question in the title from two different angles: first, from the vantage point of model theory, a branch of logic in which for over half a century the notion of “stability" has played a central role in describing tame behaviour; secondly, from the perspective of combinatorics, where so-called regularity decompositions have enjoyed a similar level of prominence in a range of finitary settings, with remarkable applications in additive number theory and theoretical computer science. In recent years, these two fundamental motifs have been shown to interact in interesting ways. I will describe some of these developments, and explain how higher-order generalisations of regularity are pointing the way towards a long-sought generalisation of stability, opening up a new dimension of the model-theoretic classification picture. This talk aims to be accessible to a broad mathematical audience. A wine reception in the central core will follow the lecture.
