Algèbre et géométrie

Fix an abstract field K. For each K-variety V, we will define an “étale-open” topology on the set V(K) of rational points of V. This notion uniformly recovers (1) the Zariski topology on V(K) when K is algebraically closed, (2) the analytic topology on V(K) when K is the real numbers, (3) the valuation topology on V(K) when K is almost any henselian field. On pseudo-finite fields, the étale-open topology seems to be new, and has some interesting properties. The étale-open topology is mostly of interest when Kis large (also […]

Groups definable in o-minimal structures have been studied by many authors in the last 30 years and include algebraic groups over algebraically closed fields of characteristic 0, semi-algebraic groups over real closed fields, important classes of real Lie groups such as abelian groups, compact groups and linear semisimple groups. In this talk I will present results on groups definable in o-minimal structures, demonstrating a strong analogy with topological decompositions of linear algebraic groups. Limitations of this analogy will be shown through several examples.

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f+g, fg and f^g are also in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2^(2^x). They did so by studying the possible limits at infinity of […]

Roth's theorem on arithmetic progression states that a subset A of the natural numbers of positive upper density contains an arithmetic progression of length 3, that is, the equation x+z=2y has a solution in A.Finitary versions of Roth's theorem study subsets A of {0, ... , N}, and ask whether the same holds for sufficiently large N, for a fixed lower bound on the density. In a similar way, concerning finite groups, one may study whether or not sufficiently large sets of a finite group contain solutions of an equation, […]

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 […]