A guide to growing random geometries

In this talk we will discuss the problem of providing "uniform growth schemes" for various types of planar maps — namely, of coupling a uniform map with n faces with a uniform map with n+1 faces in such a way that the smaller map is always obtained from the larger by collapsing a single face. We will explore the connection of this question to the idea of growing trees by the leaves and briefly touch on some applications: on one side, some implications for mixing time questions for edge flip chains, and on the other, some results about where the Brownian tree naturally grows leaves. Based on joint works with Alexandre Stauffer, and with Nicolas Curien and Robin Stephenson.