Title: Types spaces in local positive logic



Abstract: In 2020, in a paper entitled "Beyond the Lascar group", Hrushovski solved a conjecture proposed by Massicot and Wagner, providing an abstract classification of all approximate subgroups and giving some meaningful applications in the case of approximate lattices. This paper is, in fact, a continuation of an earlier paper from 2019 entitled "Definability patterns and their symmetries", in which Hrushovski develops a relational structure for type spaces carrying significant stability-like information about the theory and with important connections to topological dynamics.

In order to generalise these results to hyperdefinable sets and metric approximate subgroups, I will present local positive logic, a new (first-order) logic that mixes positive logic (in which negation is not allowed) and local logic (in which quantifiers are bounded). We will start by recalling some basic notions, and continue with the study of type spaces and definability patterns for this new logic. Joint work with Ori Segel.