Fundamental to the study of hyperbolic groups is their Gromov boundaries. The classical Cannon--Thurston map for a closed fibered hyperbolic 3-manifolds relates two such boundaries: it gives a continuous surjection from the boundary of the surface group (a circle) to the boundary of the 3-manifold group (a 2-sphere). Mj (Mitra) generalized this to all hyperbolic groups with hyperbolic normal subgroups. A generalization of the Gromov boundary to all finitely generated groups is called the Morse boundary. It collects all the "hyperbolic-like" rays in a group. In this talk we will discuss Cannon--Thurston maps for Morse boundaries. This is joint work with Ruth Charney, Antoine Goldsborough, Alessandro Sisto and Stefanie Zbinden.

(EFC-DGAs) lead to an algebraic approach to derived analytic geometry, pioneered for more general Fermat theories by Carchedi and Roytenberg.
 
They are well-suited to modelling finite-dimensional analytic spaces, and classical theorems in analysis ensure they give a largely equivalent theory to Lurie's more involved approach via pregeometries. DG dagger affinoid spaces provide a well-behaved class of geometric building blocks whose homotopy theory is governed by the underlying EFC-DGAs. 

Time permitting, I might also say a little about non-commutative generalisations.
 

I will describe the contents of a joint project with Pablo Destic and Nuno Hultberg. In the paper we confirm a conjecture of Roberto Gualdi regarding a formula for the average height of the intersection of twisted (by roots of unity) hyperplanes in a toric variety. I will introduce the 'GVF analytification' of a variety, which is defined similarly as the Berkovich analytification, but with norms replaced by heights. Moreover, I will discuss some motivations coming from (continuous) model theory and Arakelov geometry.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).