Je décrirai quelques investigations récentes autour du théorème deDrinfeld-Grinberg-Kahzdan sur le voisinage formel d'un arc nonsingulier. C'est un travail en commun avec Julien Sebag.

The first order theory of a henselian valued field of residue characteristic zero is well-understood through the celebrated Ax-Kochen-Ershov principle, which states that it is completely determined by the theory of the residue field and the theory of the value group. For henselian valued fields of positive residue characteristic, no such general principle is known. I will report on joint work with Will Anscombe in which we study (parts of) the theory of equicharacteristic henselian valued fields and prove an Ax-Kochen-Ershov principle for existential (and slightly more general) sentences. I […]

The canonical base property (CBP) is a property of finite rank theories, which was introduced by Pillay and whose formulation was motivated by results of Campana in complex geometry. The main feature of such a property is that it provides a dichotomy for types of rank one, and in consequence one can reproduce Hrushovski's proof of Mordell-Lang for function fields in characteristic zero with considerable simplifications.In this talk, I will motivate (via Mordell-Lang) the statement of the CBP and describe some results around the CBP, in particular on definable groups.