Spaces of Definable Types and Beautiful Pairs in Unstable Theories

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Abstract

We study beautiful pairs in the context of unstable theories. By classical results of Poizat, the theory of beautiful pairs of models of a stable theory T is "meaningful" precisely when the set of all definable types in T is strict pro-definable, which is the case if and only if T does not have the finite cover property. We establish Ax-Kochen-Ershov principles for various questions concerning beautiful pairs of henselian valued fields of equicharacteristic 0. Using this, we show that the theory of beautiful pairs of models of ACVF is "meaningful" and infer the strict pro-definability of various spaces of definable types in ACVF of a geometric origin, e.g., the stable completion introduced by Hrushovski-Loeser, and a model theoretic analogue of the Huber analytification of an algebraic variety. This is work in progress, joint with Pa We study beautiful pairs in the context of unstable theories. By classical results of Poizat, the theory of beautiful pairs of models of a stable theory T is "meaningful" precisely when the set of all definable types in T is strict pro-definable, which is the case if and only if T does not have the finite cover property. We establish Ax-Kochen-Ershov principles for various questions concerning beautiful pairs of henselian valued fields of equicharacteristic 0. Using this, we show that the theory of beautiful pairs of models of ACVF is "meaningful" and infer the strict pro-definability of various spaces of definable types in ACVF of a geometric origin, e.g., the stable completion introduced by Hrushovski-Loeser, and a model theoretic analogue of the Huber analytification of an algebraic variety. This is work in progress, joint with Pablo Cubides Kovacsics and Jinhe Ye.

Speakers from the University of Münster

Hils, Martin

Professorship for Mathematical Logic (Prof. Hils)

Projects the talk is about

Geometric and Combinatorial Configurations in Model Theory (GeoMod) Duration: 01/01/2020 - 31/12/2024 Funded by: DFG - Individual Grants Programme Type of project: Individual project

EXC 2044 - A2: Groups, model theory and sets Duration: 01/01/2019 - 31/12/2025 | 1st Funding period Funded by: DFG - Cluster of Excellence Type of project: Subproject in DFG-joint project hosted at University of Münster

Publications referred to in the talk

Beautiful pairs Cubides Kovacsics Pablo, Hils Martin, Ye Jinhe (2021) In: ((Bitte Journal prüfen)), 2021. Research article (journal) | submitted / under review