In his PhD thesis, A. Woerheide constructed well-behaved homology groups for definable sets in o-minimal expansions of real closed fields. The question arises whether such groups exist in o-minimal reducts, such as ordered vector spaces over ordered division rings. Why is this question interesting? A positive answer, combined with the work of Hrushovski-Loeser on stable completions, forms the basis for defining homology groups of definable sets in algebraically closed valued fields (ACVF). As an application, one can recover and extend results of S. Basu and D. Patel concerning uniform bounds of Betti numbers in ACVF. In this talk, I will present results and advancements on this topic. This is an ongoing joint work with Mario Edmundo, Pantelis Eleftheriou and Jinhe Ye.[-]

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.[-]

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.[-]

My main goal is to present a somewhat technical result, joint with Karim Adiprasito and Ehud Hrushovski, about the existence of a family of polynomials whose zeros (at a certain desired family of sets) increase faster than their degree.I will explain the 'local' and 'global' consequences of this result. In the unlikely case I have enough time, I shall attempt to elaborate on the relation to quantifier-free stability in Globally Valued Fields.This will be in some sense complementary to my recent talk in Banff.[-]

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.[-]

Skeletons are subsets of non-archimedean spaces (in the sense of Berkovich) that inherit from the ambiant space a natural PL (piecewise-linear) structure, and if $S$ is such a skeleton, for every invertible holomorphic function $f$ defined in a neighborhood of $S$, the restriction of $\log |f|$ to $S$ is $\mathrm{PL}$.In this talk, I will present a joint work with E. Hrushovski, F. Loeser and J. Ye in which we consider an irreducible algebraic variety $X$ over an algebraically closed, non-trivially valued and complete non-archimedean field $k$, and a skeleton $S$ of the analytification of $X$ defined using only algebraic functions, and consisting of Zariski-generic points. If $f$ is a non-zero rational function on $X$ then $\log |f|$ indices a $\mathrm{PL}$ function on $S$, and if we denote by $E$ the group of all $\mathrm{PL}$ functions on $S$ that are of this form, we prove the following finiteness result on the group $E$ : it is stable under min and max, and there exist finitely many non-zero rational functions $f_1, \ldots, f_m$ on $X$ such that $E$ is generated, as a group equipped with min and max operators, by the $\log \left|f_i\right|$ and the constants $|a|$ for a in $k^*$. Our proof makes a crucial use of Hrushovski-Loesers theory of stable completions, which are model-theoretic avatars of Berkovich spaces.[-]

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.[-]