A pseudoline arrangement is a collection of unbounded curves in the plane that resemble non-parallel straight lines: each pair of curves intersects exactly once. Despite their simple definition, pseudoline arrangements are deeply connected to a wide range of combinatorial structures, including permutations, sorting networks, standard Young tableaux, rhombic tilings, oriented matroids, and the orientations of certain polytope skeletons.

In this talk, we will explore some of these surprising and elegant correspondences. Time permitting, we will also discuss how pseudoline arrangements can be randomly sampled using Markov chain techniques.