Abstract: 

We investigate beautiful pairs in various model-theoretic contexts. By classical results of Poizat, the theory of beautiful pairs of models of a stable theory T is meaningful precisely when the space of (definable) types in T is strict pro-definable. Specializing to algebraically closed fields, this yields that algebraic varieties, when viewed as schemes, naturally carry the structure of strict pro-definable sets.

In joint work with Cuibides Kovacsics and Hils, we establish Ax–Kochen–Ershov principles for several classes of beautiful pairs of valued fields, showing in particular that the corresponding theories are meaningful in the above sense. As a consequence, we obtain strict pro-definability results for a range of geometrically motivated spaces of definable types in this setting, including adic spaces and real analytifications of varieties.

More recently, in collaboration with Hils, Hrushovski, and Zou, we have initiated the study of beautiful pairs of valued difference fields. We will discuss an extension of Hrushovski’s non-standard Frobenius to this context and demonstrate how it can be used to construct a meaningful model-theoretic non-archimedean analytification of difference varieties.

Zoom: 717 463 6082 

Venue: Quan 29