Seminar

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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 Pablo Cubides Kovacsics and Jinhe Ye.

Spaces Of Definable Types And Beautiful Pairs In Unstable Theories