Logic Seminar

Hrushovski constructions are a variant of amalgamation methods. They were invented to construct new examples of strongly minimal theories. The method was later adapted to expansions of fields, including colored fields and powered fields. In this talk, I will present my attempt to apply Hrushovski constructions to ordered fields. I will construct an expansion of RCF by a dense multiplicative subgroup (green points). Hrushovski constructions induce a back-and-forth system, enabling us to study the dp-rank and the open core of this structure. I will also introduce my recent progress on powered fields, an expansion of RCF by "power functions" on the unit circle, and my plan to axiomatize expansions of the real field using Hrushovski constructions.

Work of Kuhlmann and coauthors has established AKE principles for tame and separably tame valued fields, extending for example the work of Delon on the narrower class of algebraically (or separable-algebraically) maximal Kaplansky valued fields. These principles, and their underlying methods, have had striking applications, for example to existential theories of henselian valued fields, the transfer of NIP from residue field to valued field, and the recent work of Jahnke and Kartas on theories of perfectoid fields. The "Generalized Stability Theorem" is even an ingredient in Temkin's inseparable local uniformization. In this talk I want to explain some extensions of the known AKE principles, and related partial results on relative quantifier elimination, all in various special cases. This includes work joint with Boissonneau, and work of Soto Moreno.