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.[-]

A real analytic function can always be continued holomorphically to some domain. However, the holomorphic continuations of definable functions in an o-minimal structure may not be definable. I will present joint work with P. Speissegger in which we study holomorphic continuations of functions definable in two o-minimal expansions of the real field. I will also discuss how to apply these results to the complex Gamma function and Riemann zeta function.[-]

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.[-]

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.[-]

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.[-]