Soit f: A^2 --> A^2 un automorphisme polynomial du plan complexe. Le problème de Manin-Mumford dynamique consiste à décrire les courbes algébriques C contenant une infinité de points périodiques de f.Dans un travail en commun avec R. Dujardin nous avons montré que lorsque f était dissipative, une telle courbe C n'existait jamais.

J'illustrerai, dans le cas des valuations d'Abhyankar, des méthodes d'étude des valuations utilisant la noetherianité des anneaux et conduisant à une preuve de l'uniformisation locale des valuations d'Abhyankar.

Geometries can be given in a direct semantic way, say as a complex or real manifold, or more abstractly, by their co-ordinate algebras and schemes. A duality of this kind becomes highly non-trivial in cases of schemes of arithmetic type and for non-commutative co-ordinate algebras. I will discuss these issues from model-theoretic perspective. A detailed analysis will be given to the canonical commutation relation(s) underlying quantum mechanics. Some applications will be presented.

Given a pair of models Kprec L of a first-order theory T, the pair is said to be stable if the following property holds: all types over K which are realized in L are definable. Marker and Steinhorn characterized stable pairs of models of o-minimal theories as pairs K prec L where K is Dedekind complete in L. In this talk we provide a characterization of stable pairs of algebraically closed valued fields K prec L. To get a flavor of the topic, different examples will be discussed and a […]

We establish for the category of semialgebraic sets and functions on arbitrary real closed fields a full Lebesgue measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry, valuation theory and the theory of ordered abelian groups. We set up the construction in such a way that it is uniquely determined by data that can be formulated completely in terms of the given real closed field. We apply our integration theory to questions on semialgebraic geometry and […]