The theory of graph (and structure) convergence gained recently a substantial attention. Various notions of convergence were proposed, adapted to different contexts, including Lovasz et al. theory of dense graph limits based on the notion of left convergence and Benjamini–Schramm theory of bounded degree graph limits based on the notion of local convergence. The latter approach can be extended into a notion of local convergence for graphs (stronger than left convegence) as follows: A sequence of graphs is local convergent if, for every local first-order formula, the probability that the formula is satisfied for a random (uniform independent) assignment of the free variables converge as n grows to infinity. In this talk, we show that the local convergence of a sequence of graphs allows to decompose the graphs in the sequence in a coherent way, into concentration clusters (intuitively corresponding to the limit non-zero measure connected components), a residual cluster, and a negligible set. Also, we mention that if we consider a stronger notion of local-global convergence extending Bollobas and Riordan notion of local-global convergence for graphs with bounded degree, we can further refine our decomposition by exhibiting the expander-like parts.

graphs - structural limit - graph limit - asymptotic connectivity[-]

We will present some of the original definitions, results, and proof techniques about Pfaffian functions on the reals by Khovanskii.
A simple example of a Pfaffian function is an analytic function $f$ in one variable $x$ satisfying a differential equation $f^\prime = P(x,f)$ where $P$ is a polynomial in two variables. Khovanskii gives a notion of complexity of Pfaffian functions which in the example is just the degree of $P$. Using this complexity, he proves analogues of Bézout's theorem for Pfaffian curves (say, zero loci of Pfaffian functions in two variables), with explicit upper bounds in terms of the ocurring complexities.
We explain a recent application by J. Pila and others to a low-dimensional case of Wilkie's conjecture on rational points of bounded height on restricted Pfaffian curves. The result says that the number of rational points of height bounded by $T$, on a transcendental restricted Pfaffian curve, grows at most as a power of log$(T)$ as $T$ grows. This improves the typical upper bound $T^\epsilon$ in Pila-Wilkie's results in general o-minimal structures, the improvement being due to extra geometric Bézout-like control.
In the non-archimedean setting, I will explain analogues of some of these results and techniques, most of which are (emerging) work in progress with L. Lipshitz, F. Martin and A. Smeets. Some ideas in this case come from work by Denef and Lipshitz on variants of Artin approximation in the context of power series solution.[-]

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.[-]

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.[-]

Joint work with Silvain Rideau-Kikuchi.
Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this talk, we propose a unified framework for studying them: the class of pseudo $T$ -closed fields, where $T$ is an enriched theory of fields. These fields verify a 'local-global' principle for the existence of points on varieties with respect to models of $T$ . This approach also enables a good description of some fields equipped with multiple V -topologies, particularly pseudo algebraically closed fields with a finite number of valuations. An important result that will be discussed in this talk is a (model theoretic) classification theorem for bounded pseudo T -closed fields, in particular we show that under specific hypotheses on $T$ , these fields are NTP2 of finite burden.[-]