This talk is based on a joint work with Vincent Jinhe Ye. I will define various classes of hyperbolic varieties (Broody hyperbolic, algebraically hyperbolic, bounded, groupless) and discuss existence of model companions of classes of fields that exclude them. This is related to moduli spaces of maps to hyperbolic varieties and to the (open) question whether the above mentioned hyperbolicity notions are in fact equivalent.

The space of measured laminations on a hyperbolic surface is a generalisation of the set of weighted multi curves. The action of the mapping class group on this space is an important tool in the study of the geometry of the surface. 
For orientable surfaces, orbit closures are now well-understood and were classified by Lindenstrauss and Mirzakhani. In particular, it is one of the pillars of Mirzakhani’s curve counting theorems. 
For non-orientable surfaces, the behaviour of this action is very different and the classification fails. In this talk I will review some of these differences and describe mapping class group orbit closures of (projective) measured laminations for non-orientable surfaces. This is joint work with Erlandsson, Gendulphe and Souto.

If a group quasiisometrically embeds into a finite product of infinite valence trees then a number of things are implied; for example, the group will have finite Assouad-Nagata dimension and finite asymptotic dimension. An even stronger statement is that the group quasiisometrically embeds into a finite product of uniformly bounded valence trees. The research on which groups quasiisometrically embed into finite products of uniformly bounded valence trees is limited, however a notable result of Buyalo, Dranishnikov and Schroeder from 2007 proves that all hyperbolic groups do admit these quasiisometric embeddings. In a recently released preprint, I extend their result to cover groups which are relatively hyperbolic with respect to virtually abelian peripheral subgroups. 

This talk will focus on the ideas at the core of Buyalo, Dranishnikov and Schroeder’s result and the extension that I proved, and in particular I will attempt to provide a general framework for upgrading quasiisometric embeddings into infinite valence trees so that they are now quasiisometric embeddings into uniformly bounded valence trees. The central concept is called a diary which I will define.