I will discuss the influence of the seed in models of randomly growing trees; in particular, I will focus on the preferential attachment and uniform attachment models. In both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. I will discuss the differences and similarities in proving this for the two models. This is based on joint work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel.

We consider the question: ``How bad can the deformation space of an object be?'' (Alternatively: ``What singularities can appear on a moduli space?'') The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad.'' We show this for a number of important moduli spaces. More precisely, up to smooth parameters, every singularity that can be described by equations with integer coefficients appears on moduli spaces parameterizing: smooth projective surfaces (or higher-dimensional manifolds); smooth curves in projective space (the space of stable maps, or the Hilbert scheme); plane curves with nodes and cusps; stable sheaves; isolated threefold singularities; and more. The objects themselves are not pathological, and are in fact as nice as can be. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. I will begin by telling you what ``moduli spaces'' and ``deformation spaces'' are. The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. This talk is intended to be (mostly) comprehensible to a broad audience.

Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.

The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry.