Belles paires of valued fields and analytification

In their work, Hrushovski and Loeser proposed the space V̂ of generically stable types concentrating on V to study the homotopy type of the Berkovich analytification of V. An important feature of V̂ is that it is canonically identified as a projective limit of definable sets in ACVF, which grants them tools from model theory. In this talk, we will give a brief introduction to this object and present an alternative approach to internalize various spaces of definable types, motivated by Poizat’s work on belles paires of stable theories. Several results of interest to model theorists will also be discussed. Particularly, we recover the space V̂ is strict pro-definable and we propose a model-theoretic counterpart Ṽ of Huber’s analytification. Time permitting, we will discuss some comparison and lifting results between V̂ and Ṽ. This is a joint project with Pablo Cubides Kovacsics and Martin Hils.