In this talk there is no valued field but we try to find one. Or, to be more modest, we try first to find a group. Our problematic is the trichotomy of Zilber. Given an abstract structure which shares certain model theoretical properties with an infinite group (or with an infinite field) can we define an infinite group (or an infinite field) in this structure?
The initial conjecture was about strongly minimal structures and it turned out to be wrong. It becomes correct in the framework of Zariski structures. These are minimal structures in which some definable sets are identified as closed, the connection between closed and definable sets being similar to what happens in algebraically closed fields with the topologies of Zariski. This is the content of a large volume of work by Ehud Hrushovski and Boris Zilber. O-minimal structures and their Cartesian powers arrive equipped with a topology. Although these topologies are definitely not Noetherian, the situation presents great analogies with Zariski structures. Now, Kobi Peterzil and Sergei Starchenko have shown Zilber's Conjecture in this setting (up to a nuance).
The question then arises naturally in $C$-minimal structures. Let us recall what they are. $C$-sets can be understood as reducts of ultrametric spaces: if the distance is $d$, we keep only the information given by the ternary relation $C(x, y, z)$ iff $d(x, y)=d(x, z)>d(y, z)$. So, there is no longer a space of distances, we can only compare distances to a same point. A $C$-minimal structure $M$ is a $C$-set possibly with additional structure in which every definable subset is a Boolean combination of open or closed balls, more exactly of their generalizations in the framework of $C$-relations, cones and 0-level sets. Moreover, this must remain true in any structure $N$ elementary equivalent to $M$. Zilber's conjecture only makes sense if the structure is assumed to be geometric. Which does not follow from $C$-minimality.
Nearly 15 years ago Fares Maalouf has shown that an inifinite group is definable in any nontrivial locally modular geometric $C$-minimal structure. Fares, Patrick Simonetta and myself do the same today in a non-modular case. Our proof draws heavily on that of Peterzil and Starchenko.[-]

The idea of stable domination for types in a theory was proposed and developed for algebraically closed valued fields in the eponymous book by Haskell, Hrushovski and Macpherson (2008). With the observation both that valued fields that are not algebraically closed generally have no stable part and that the stable part of an algebraically closed valued field is closely linked to the residue field, it seemed appropriate to consider a notion of residue field domination. In this talk, I will illustrate the idea of residue field domination with various examples, and then present some theorems which apply to some henselian valued fields of characteristic zero. These results are presented in a recent preprint of Ealy, Haskell and Simon, with similar results in a preprint of Vicaria.[-]

Joint work with Silvain Rideau-Kikuchi.
Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this talk, we propose a unified framework for studying them: the class of pseudo $T$ -closed fields, where $T$ is an enriched theory of fields. These fields verify a 'local-global' principle for the existence of points on varieties with respect to models of $T$ . This approach also enables a good description of some fields equipped with multiple V -topologies, particularly pseudo algebraically closed fields with a finite number of valuations. An important result that will be discussed in this talk is a (model theoretic) classification theorem for bounded pseudo T -closed fields, in particular we show that under specific hypotheses on $T$ , these fields are NTP2 of finite burden.[-]

We extend the notion of beautiful pairs by Poizat to unstable theories via definable types, with a specific interest in such pairs of valued fields. In particular, we establish an analogue of Ax-Kochen-Ershov principles in for certain pairs of valued fields. In the specific case of ACVF, we classify all such pairs and deduce the strict pro-definability of various spaces of definable types, such as the stable completion introduced by Hrushovski-Loeser and a model theoretic analogue of the Huber analytification of an algebraic variety. This is joint with Pablo Cubides Kovacsics and Martin Hils.[-]

A valuation $v$ on a field $K$ is said to be definable (in a specified language) if its corresponding valuation ring is a definable subset of $K$. Historically, the study of definable valuations on certain fields was motivated by the general analysis of definable subsets of fields and related decidability questions, but has also re-emerged lately in the context of classifying NIP fields. In my talk, I will present some recent progress in the study of definable valuations on ordered fields ([1] to [4]), where definability is considered in the language of rings as well as the richer language of ordered rings. Within this framework, the focus lies on convex valuations, that is, valuations whose valuation ring is convex with respect to the linear ordering on the field. The most important examples of such valuations are the henselian ones, which are convex with respect to any linear ordering on the field. I will present topological conditions on the value group and the residue field ensuring the definability of the corresponding valuation. Moreover, I will outline some definability and non-definability results in the context of specific classes of ordered fields such as t-henselian, almost real closed, and strongly dependent ones.[-]

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.[-]