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 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.[-]

We will describe a new approach to the study of discrete dynamical systems on the real line, which consists in considering their orbits as "fractal objetcs". In particular, the formal classification of analytic systems can be reproven with this method. We will also explain the main lines of a program devoted to the study of some non analytic systems. These are generated by maps which admit a specific type of transseries (Dulac's transseries) as asymptotic expansions.[-]

O-minimalism is the first-order theory of o-minimal structures, an important class of models of which are the ultraproducts of o-minimal structures. A complete axiomatization of o-minimalism is not known, but many results are already provable in the weaker theory DCTC given by definable completeness and type completeness (a small extension of local o-minimality). In DCTC, we can already prove how many results from o-minimality (dimension theory, monotonicity, Hardy structures) carry over to this larger setting upon replacing ‘finite' by ‘discrete, closed and bounded'. However, even then cell decomposition might fail, giving rise to a related notion of tame structures. Some new invariants also come into play: the Grothendieck ring is no longer trivial and the definable, discrete subsets form a totally ordered structure induced by an ultraproduct version of the Euler characteristic. To develop this theory, we also need another first-order property, the Discrete Pigeonhole Principle, which I cannot yet prove from DCTC. Using this, we can formulate a criterion for when an ultraproduct of o-minimal structures is again o-minimal.[-]