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.[-]

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 concept of a "transseries" is a natural extension of that of a Laurent series, allowing for exponential and logarithmic terms. Transseries were introduced in the 1980s by the analyst Écalle and also, independently, by the logicians Dahn and Göring. The germs of many naturally occurring real-valued functions of one variable have asymptotic expansions which are transseries. Since the late 1990s, van den Dries, van der Hoeven, and myself, have pursued a program to understand the algebraic and model-theoretic aspects of this intricate but fascinating mathematical object. A differential analogue of “henselianity" is central to this program. Last year we were able to make a significant step forward, and established a quantifier elimination theorem for the differential field of transseries in a natural language. My goal for this talk is to introduce transseries without prior knowledge of the subject, and to explain our recent work.[-]

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.)[-]

It is by now well known that collections of compact (real-)analytic vector fields and locally connected trajectories thereof are mutually well behaved in a way that can be made precise via notions from mathematical logic, namely, by saying that the structure on the real field generated by the collection is o-minimal (that is, every subset of the real numbers definable in the structure is a finite union of points and open intervals). There are also many examples known where the assumption of analyticity or compactness can be removed, yet o-minimality still holds. Less well known is that there are examples where o-minimality visibly fails, but there is nevertheless a well-defined notion of tameness in place. In this talk, I will: (a) make this weaker notion of tameness precise; (b) describe a class of examples where the weaker notion holds; and (c) present evidence for conjecturing that there might be no other classes of examples of “non-o-minimal tameness”. (Joint work with Patrick Speissegger.)
A few corrections and comments about this talk are available in the PDF file at the bottom of the page.[-]

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.[-]

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.[-]

I will talk about a joint work with Novikov on 'complex cells', which are a complexification of the cells/cylinders used in o-minimality theory. It turns out that complex cells admit a canonical hyperbolic metric which is not directly accessible in the real setting, leading to a much richer structure theory. In particular, complex cells are closer than real cells to resolution of singularities - and many of their basic properties are inspired by this connection. Our main motivation for introducing complex cells was to prove a sharper form of the Yomdin-Gromov lemma, leading to some applications in dynamics and number theory. I will outline how complex cells can be used to achieve this, and in particular how their hyperbolic structure leads to much sharper constructions compared to the previously existing methods.[-]

I will talk about a joint work with Novikov on 'complex cells', which are a complexification of the cells/cylinders used in o-minimality theory. It turns out that complex cells admit a canonical hyperbolic metric which is not directly accessible in the real setting, leading to a much richer structure theory. In particular, complex cells are closer than real cells to resolution of singularities - and many of their basic properties are inspired by this connection. Our main motivation for introducing complex cells was to prove a sharper form of the Yomdin-Gromov lemma, leading to some applications in dynamics and number theory. I will outline how complex cells can be used to achieve this, and in particular how their hyperbolic structure leads to much sharper constructions compared to the previously existing methods.[-]