The mathematics of lecture hall partitions

Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to lecture hall partitions have used polyhedral geometry to discover further properties of these rich combinatorial objects. In this paper we give an overview of some of the surprising connections that have surfaced in the process of trying to understand the lecture hall partitions.