Séminaires : Géométrie et Théorie des Modèles

Equipe(s) : lm,
Responsables :Zoé Chatzidakis, Raf Cluckers, Silvain Rideau.
Email des responsables : zoe.chatzidakis@imj-prg.fr
Salle :
Adresse :ENS



Pour recevoir le programme par e-mail, écrivez à : zchatzid_at_dma.ens.fr.Retour ligne automatique
Pour les personnes ne connaissant pas du tout de théorie des modèles, des notes introduisant les notions de base (formules, ensembles définissables, théorème de compacité, etc.) sont disponibles ici : http://www.logique.jussieu.fr/~zoe/papiers/MTluminy.dvi. Ces personnes peuvent aussi consulter les premiers chapitres du livre Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Notes in Mathematics 1696, Berlin 1998.Retour ligne automatique
Les notes de quelques-uns des exposés sont disponibles.

Orateur(s) Artem Chernikov - UCLA,
Titre Recognizing groups and fields in Erdős geometry and model theory
Horaire16:30 à 18:00
RésumeAssume 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 some open set is, up to coordinatewise homeomorphisms, of the form x_1+...+x_s=0. This is a special case of the recent generalization of the Elekes-Szabó theorem to any arity and dimension in which general abelian Lie groups arise (joint work with Kobi Peterzil and Sergei Starchenko). On the other hand, if Q omits a finite complete s-partite hypergraph but can intersect finite grids in more that than n^(s-1 + ε) points, then the real field can be definably recovered from Q (joint work with Abdul Basit, Sergei Starchenko, Terence Tao and Chieu-Minh Tran).
I will explain how these results are connected to the model-theoretic trichotomy principle and discuss variants for higher dimensions, and for stable structures with distal expansions.