My goal, in this talk, is to explain a new notion of minimality for (characteristic zero) Henselian fields, which generalizes C-minimality, P-minimality and V-minimality and puts no restriction on the residue field or valued group contrary to these previous notions. This new notion, h-minimality, can be defined, analogously to other minimality notions, by asking that 1-types, over algebraically closed sets, are entirely determined by their reduct to some sublanguage - in that case the pure language of valued fields. However, contrary to what happens with other minimality notions, particular care […]

In this talk, we characterize NIP henselian valued fields modulo the theory of their residue field. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.

The famous Mumford-Tate conjecture asserts that, for every prime number l, Hodge cycles are Q_l linear combinations of Tate cycles, through Artin's comparisons theorems between Betti and étale cohomology. The algebraic Sato-Tate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized Sato-Tate conjecture. This previous conjecture is related with the equidistribution of Frobenius traces.Our main goal is to prove that the Mumford-Tate conjecture for an abelian variety A implies the algebraic Sato-Tate conjecture for A. The […]

Étant donné un anneau de valuation V de corps résiduel F et contenant un corps k, et une extension k' de k, on cherche à construire une extension V' de V contenant k', d'idéal maximal engendré par celui de V, et de corps résiduel composé de F et k'. On y parvient notamment si F ou k' est séparable sur k.

Lipschitz geometry is a branch of singularity theory that studies the metric data of a germ of a complex analytic space.I will discuss a new approach to the study of such metric germs, and in particular of an invariant called Lipschitz inner rate, based on the combinatorics of a space of valuations, the so-called non-archimedean link of the singularity. I will describe completely the inner metric structure of a complex surface germ showing that its inner rates both determine and are determined by global geometric data: the topology of the […]