Algèbre et géométrie

Si X est un espace k-affinoïde (k étant un corps non-archimédien), un sous-ensemble S de X est dit sous-analytique surconvergent si on peut ?Roeessentiellement?R l'écrire S=f(Y) où f est un morphisme surconvergent d'espaces affinoïdes.Nous expliquerons d'abord comment décrire ces ensembles en n'utilisant que des fonctions de X, i.e. sans avoir recours à une projection. Il s'agit d'une version géométrique d'un résultat de H. Schoutens qui utilise l'élimination des quantificateurs dans ACVF.Nous montrerons ensuite que les ensembles sous-analytiques surconvergents peuvent être définis localement pour la topologie de Berkovich, mais pas pour […]

Sheaf theory is not well suited to study objects which are not defined by local properties. It is the case, for example, of functional spaces with growth conditions, as tempered distributions. Since the study of the solutions of a system of PDE in these spaces is of great importance (solutions of irregular D-modules, Laplace transform, etc.), many ways have been explored by the specialists to overcome this problem. For this purpose Kashiwara and Schapira introduced the subanalytic site and proved that some of these spaces can be realized as sheaves […]

It is now well-known what sorts have to be added to a valued field in order to achieve elimination of imaginaries. It is also known that these sorts do not suffice to eliminate imaginaries when the field is enhanced by restricted analytic functions, despite the fact that the theories still have quantifier elimination. In this talk, I will attempt to convey the intuition about the definable sets in a valued field that underlies all of these results (while explaining the model-theoretic terminology in the above).

For algebraic varieties defined over the complex numbers, one can study geometry using both algebraic and analytic methods. Over a non-Archimedean field, one can try to do the same thing using Berkovich spaces. I will discuss positivity notions for metrics on line bundles on varieties defined over discretely or trivially valued fields.

I will discuss problems around definably amenable groups in NIP theories, informed by some invariants coming from topological dynamics.