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.[-]

The field of Laurent series (with real coefficients, say) has a natural derivation but is too small to be closed under integration and other natural operations such as taking logarithms of positive elements. The field has a natural extension to a field of generalized series, the ordered differential field of transseries, where these defects are remedied in a radical way. I will sketch this field of transseries. Recently it was established (Aschenbrenner, Van der Hoeven, vdD) that the differential field of transseries also has very good model theoretic properties. I hope to discuss this in the later part of my talk.[-]

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.[-]

In the mid-90's, generalising a theorem of Jouanolou, Hrushovski proved that if a D-variety over the constant field C has no non-constant D-rational functions to C, then it has only finitely many D-subvarieties of codimension one. This theorem has analogues in other geometric contexts where model theory plays a role: in complex analytic geometry where it is an old theorem of Krasnov, in algebraic dynamics where it is a theorem of Bell-Rogalski-Sierra, and in meromorphic dynamics where it is a theorem of Cantat. I will report on work-in-progress with Jason Bell and Adam Topaz toward generalising and unifying these statements.[-]

Although the class of henselian valued fields of fixed mixed characteristic and fixed finite initial ramification is algebraically well-behaved, it harbours some model-theoretic surprises - for instance, some members of the class fail to be existentially decidable even though their residue field and algebraic part are. I will discuss how to rectify the situation by endowing residue fields with a canonical enrichment of the pure field structure, and how this gives rise to Ax-Kochen-Ershov principles (among other things, describing existential theories and full theories of valued fields in terms of value groups and residue fields). This is joint work with Sylvy Anscombe and Franziska Jahnke.[-]

(joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou)
We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal tran-scendence degree.
The proof uses a strong form of the Lang-Weil estimates and, as key ingredi-ent to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.[-]

Given a perfectoid field, we find an elementary extension and an especially nice valuation on it whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of $\mathbb{F}_p(t)^h$ is an elementary substructure of the perfect hull of $\mathbb{F}_p((t))$. Joint work with Franziska Jahnke.[-]