This thesis explores the geometric principles underlying many of the known Trichotomy Theorems. The main aims are to unify the field construction in non-linear o-minimal structures and generalizations of Zariski Geometries as well as to pave the road for completely new results in this direction. In the first part of this thesis we introduce a new axiomatic framework in which all the relevant structures can be studied uniformly and show that these axioms are preserved under elementary extensions. A particular focus is placed on the study of a smoothness condition which generalizes the presmoothness condition for Zariski Geometries. We also modify Zilber's notion of universal specializations to obtain a suitable notion of infinitesimals. In addition, families of curves and the combinatorial geometry of one-dimensional structures are studied to prove a weak trichotomy theorem based on very weak one-basedness. It is then shown that under suitable additional conditions groups and group actions can be constructed in canonical ways. This construction is based on a notion of ``geometric calculus'' and can be seen in close analogy with ordinary differentiation. If all conditions are met, a definable distributive action of one one-dimensional type-definable group on another are obtained. The main result of this thesis is that both o-minimal structures and generalizations of Zariski Geometries fit into this geometric framework and that the latter always satisfy the conditions required in the group constructions. We also exhibit known methods that allow us to extract fields from this. In addition to unifying the treatment of o-minimal structures and Zariski Geometries, this also gives a direct proof of the Trichotomy Theorem for "type-definable" Zariski Geometries as used, for example, in Hrushovski's proof of the relative Mordell-Lang conjecture.

Le travail de cette thèse a pour objet les interactions entre deux approches d'étude des équations différentielles: la théorie des modèles des corps différentiellement clos d'une part et l'étude dynamique des équations différentielles réelles d'autre part. Dans le premier chapitre, on présente un formalisme d'algèbre différentielle, en termes de D-schémas à la Buium au-dessus du corps des nombres réels (muni de la dérivation triviale), qui permet de rendre compte de ces deux approches d'étude en même temps. Le résultat principal est un critère d'orthogonalité aux constantes pour le type générique d'une D-variétés réelle absolument irréductible, basé sur la dynamique topologique de son flot réel analytique associé. Le deuxième chapitre est consacré aux équations différentielles algébriques décrivant le flot géodésique de variétés algébriques réelles munies de 2-formes symétriques non-dégénérées. A l'aide du critère précédent, on démontre un théorème d'orthogonalité aux constantes "en courbure strictement négative'', s'appuyant sur les résultats d'Anosov et de ses successeurs concernant la dynamique topologique - la propriété de mélange topologique faible - du flot géodésique d'une variété riemannienne compacte à courbure strictement négative. En dimension 2, on conjecture en fait une description plus précise - son type générique est minimal de prégéométrie triviale - de la structure associée aux équations différentielles géodésiques unitaires. On présente, dans le troisième chapitre, des motivations et des résultats partiels concernant cette conjecture. / This thesis is dedicated to studying the interactions between two different approaches regarding differential equations: the model-theory of differentially closed fields on the one side and the dynamical analysis of real differential equations, on the other side. In the first chapter, we present a formalism from differential algebra, in terms of D-varieties à la Buium over the field of real numbers (endowed with the trivial derivation), that allows one to realise both approaches at the same time. The main result is a criterion of orthogonality to the constants, based on the topological dynamic of its associated real analytic flow. The second chapter is dedicated to the algebraic differential equations describing the (unitary) geodesic flow of a real algebraic variety endowed with an algebraic, non-degenerated symmetric 2-form. Using the previous criterion, we prove a theorem of orthogonality to the constants "in negative curvature'', that relies on the results of Anosov and of his followers, regarding the topological dynamic - the weakly mixing topological property - for the geodesic flow of a compact Riemannian manifold with negative curvature. In dimension 2, we conjecture a more precise description - its generic type is minimal and has a trivial pregeometry- for the structure associated to the unitary geodesic equation. In the third chapter, we present some motivations and partial results on this conjecture.