Teoría de Modelos
We study expansions of a vector space V over a field F, possibly with extra structure, with a generic submodule over a subring of F. We show that this expansions preserve tame model theoretic properties such as stability, NIP, NTP1, NTP2 and NSOP1.
One of the most striking results in the model theory of henselian valued fields is the Ax-Kochen theorem, which roughly states that the first order theory of a henselian valued field of equicharacteristic zero, or of mixed characteristic, unramified and with perfect residue field is determined by the first order theory of the residue field and its value group.
A model theoretic principle follows from this theorem: any model theoretic question about the valued field can be reduced into a question to its residue field, its value groups and their interaction in the field.
A fruitful application of this theorem has been applied to describe the class of definable sets in a valued field, for example Pas proved elimination of field quantifiers relative to the residue field and the value group once we add an angular component in the equicharacteristic zero case.
One can therefore ask the following question: Can one obtain an Ax-Kochen style theorem to eliminate imaginaries in a henselian valued field?
Following the Ax-Kochen principle, it seems natural to look at the problem in two orthogonal directions: one can either make the residue field extremely tame and understand the problems that the value group brings naturally to the picture, or one can assume the value group to be very tame and study the issues that the residue field would contribute to the problem.
In this talk we will address the first approach. I will present how to eliminate imaginaries in henselian valued fields of equicharacteristic zero with residue field algebraically closed. The results obtained are sensitive to the complexity of the value group. I will start by introducing the problem of imaginaries in order abelian groups according to their combinatorial complexity. Once the picture has been clarified for this setting, we will present how to solve the question for valued fields.
Joint with G. Conant, C. D'Élbée, Y. Halevi and L. Jimenez
This all started with a question on the possible unary expansions of the group of integers: all of the known stable examples are superstable of rank omega and strictly simple examples use highly non trivial results in number theory. As it turns out, that was all just a coincidence and any graph can be interpreted in some enrichment of (Z,+) by a single unary predicate.
The main tool to build these examples is that adding structure to a stably embedded set does make it much more complicated (classification wise) than the initial structure or the added structure were. These « resplendence » phenomenons were implicit in work of Chernikov and Hils on NTP_2 and were later made explicit in work on Jahnke and Simon on NIP.
In this talk I will explain how these results extend to other classes, among which stability, superstability, simplicity and NSOP_1. And I will explain how this relates to the (apparently unrelated) initial question!