In recent papers by Alon et al. and Fox et al. it is demonstrated that families of graphs with a semialgebraic edge relation of bounded complexity have strong regularity properties and can be decomposed into very homogeneous semialgebraic pieces up to a small error (typical example is the incidence relation between points and lines on a real plane, or higher dimensional analogues). We show that in fact the theory can be developed for families of graphs definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of measures (this applies in particular to graphs definable in arbitrary o-minimal theories and in p-adics). (Joint work with Sergei Starchenko.)[-]

If CCM denotes the theory of compact complex spaces in the langauge of complex-analytic sets, then the theory of models of CCM equipped with an automorphism has a model companion, denoted by CCMA. The relationship to meromorphic dynamical systems is the same as that of ACFA to rational dynamical systems. I will discuss recent joint work with Martin Bays and Martin Hils that begins a systematic study of CCMA as an expansion of ACFA. Particular topics we consider include: stable embeddedness, imaginaries, and the Zilber dichotomy.[-]

In previous work with Fehm we found that the existential theory of an equicharacteristic henselian valued field is axiomatised using the existential theory of its residue field. From this we deduced a transfer of decidability: for a complete theory $T$ of residue fields, the existential consequences of $T$ are decidable if and only if the existential consequences of the theory $H(T)$ are decidable, where $H(T)$ is 'equicharacteristic, henselian, and residue field models $T^{\prime}$. In more recent work with Dittmann and Fehm we considered a similar problem in which $H(T)$ is expanded to a theory that distinguishes a uniformizer, using an additional constant symbol. In this case Denef and Schoutens gave a transfer of existential decidability conditional on Resolution of Singularities. We introduce a consequence of Resolution and prove that it implies a similar transfer of existential decidability.In this talk I'll explain these results and describe ongoing work with Fehm in which we broaden the above setting to obtain versions of these transfer results that allow incomplete theories $T$. Consequently we find several existential theories Turing equivalent to the existential theory of $\mathbb{Q}$, including the existential theory of large fields.[-]

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.[-]

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.[-]

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.[-]

Which Noetherian integral domains are NIP? This is a natural question to ask, given the prominence of Noetherian rings in commutative algebra. We cannot hope to answer this question in full generality any time soon, as it includes other hard problems such as the conjectures on stable fields and NIP fields. Nevertheless, we present some interesting partial results which begin to paint a picture of NIP Noetherian rings. Let $R$ be a Noetherian domain which is NIP. Then either $R$ is a field, or $R$ is a semilocal domain of Krull dimension 1 and characteristic 0. Assuming the henselianity conjecture on NIP valued fields, $R$ is a henselian local ring. In the dp-minimal case, one can give a complete classification. Specifically, every dp-minimal Noetherian domain is a finite index subring of a dp-minimal discrete valuation ring. The situation in dp-rank 2 seems to be significantly worse, but a classification may still be possible in terms of differential valued fields.[-]

(joint with Rideau-Kikuchi)
One of the most striking results of the model theory of henselian valued fields is the Ax-Kochen/Ershov principle, which roughly states that the first order theory of a henselian valued field that is unramified is completely determined by the first order theory of its residue field and the first order theory of its value group. Our leading question is: Can one obtain an Imaginary Ax-Kochen/Ershov principle? In previous work, I showed that the complexity of the value group requires adding the stabilizer sorts. In previous work, Hils and Rideau-Kikuchi showed that the complexity of the residue field reflects by adding the interpretable sets of the linear sorts. In this talk we present recent results on weak elimination of imaginaries that combine both strategies for a large class of henselian valued fields of equicharacteristic zero. Examples include, among others, henselian valued fields with bounded galois group and henselian valued fields whose value group has bounded regular rank (with an angular component map).[-]

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.[-]

(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.[-]