The Mordell-Lang conjecture (now a theorem, proved by Faltings, Vojta, McQuillan,...) asserts that if G is a semiabelian variety G defined over an algebraically closed field of characteristic zero, X is a subvariety of G, and Γ is a finite rank subgroup of G, then X ∩ Γ is a finite union of cosets of Γ. In positive characteristic, the naive translation of this theorem does not hold, however Hrushovski, using model theoretic techniques, showed that in some sense all counterexamples arise from semiabelian varieties defined over finite fields (the […]

It is well-known that certain algebraic differential equations restrain in an essential way the algebraic relations that their solutions share. For example, the solutions of the first equation of Painlevé y'' = 6y^2 + t are “new” transcendental functions of order two which whenever distinct are algebraically independent (together with their derivatives).I will first describe an account of such phenomena using the language of geometric stability theory in a differentially closed field. I will then explain how linearization procedures and geometric stability theory fit together to study such transcendence results […]

In statistical learning theory, the notion of VC-dimension was developed by Vapnik and Chervonenkis in the context of approximating probabilities of events by the relative frequency of random test points. This notion has been widely used in combinatorics and computer science, and is also directly connected to model theory through the study of NIP theories. This talk will start with an overview of VC-dimension, with examples motivated by discrete geometry and additive combinatorics. I will then present several model theoretic applications of VC-dimension. The selection of topics will focus on […]

Assume that Q is a relation on R^s of arity s definable in an o-minimal expansion of R. I will discuss how certain extremal asymptotic behaviors of the sizes of the intersections of Q with finite n × ... × n grids, for growing n, can only occur if Q is closely connected to a certain algebraic structure.On the one hand, if the projection of Q onto any s-1 coordinates is finite-to-one but Q has maximal size intersections with some grids (of size >n^(s-1 - ε)), then Q restricted to […]

Let T be the theory of fields K of characteristic 0 such that the equation x^4 + y^4 = 1 has only four solutions in K. We show that T has a model companion. More generally, if K_0 is a field of characteristic 0 and C is a curve (affine or projective) of genus ≥ 2 with C(K_0) = ∅, then there is a model companion CXF of the theory of fields K extending K_0 with C(K) = ∅. We can use this theory to construct a field K with […]

Joint work with Samaria MontenegroThe striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a “local-global” principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in […]

An open problem in theoretical computer science asks to characterize tameness for hereditary classes of finite structures. The notion of bounded twin-width was proposed and studied recently by Bonnet, Geniet, Kim, Thommasé and Watrignant. Classes of graphs of bounded twin-width have many desirable properties. In particular, they are monadically NIP (remain NIP after naming arbitrary unary predicates). In joint work with Szymon Torunczyk we show the converse for classes of ordered graphs. We then obtain a very clear dichotomy between tame (slow growth, monadically NIP, algorithmically simple ...) and wild […]

I will discuss recent work in collaboration with Edward Bierstone on transformation of a mapping to monomial form (with respect to local coordinates) by simple modifications of the source and target. Our techniques apply in a uniform way to the algebraic and analytic categories, as well as to classes of infinitely differentiable real functions that are quasianalytic or definable in an o-minimal structure. Our results in the real cases are best possible. The talk will focus on real phenomena and on an application to quantifier elimination of certain o-minimal polynomially […]