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.

The Christian church was begun in 19Ih century in the traditionally multi-religious society in Korea. Christianity holds a major position in Korea today and has for the past 20 years been growing rapidly in numbers. Despite its phenomenal growth, the churches. and Christianity in general, have been suffering from several ailments, of which dualistic thinking is not the least. Anthropological dualism amounts to not only distinguishing between soul and body, but also ascribing a separate and independent existence to each of these "components" of the human being. This dualism (as well as others) developed in the church under the influence of traditional Korean religions such as Buddhism, Confucianism and Shamanism that have been teaching such dualisms. The Korean Christian mindset has to this day been dualistic both as a result of such cultural and philosophical influences and o€ ideas brought by the early missionaries to Korea. The influence of Platonic dualism is still widespread in the conservative and gospel church. This study focuses on: + examining the nature of the problem of anthropological (and other forms of) dualism + how the problem has been manifesting itself in Korean churches and in church education + the most momentous influences on Korean Christianity and churches resulting in a dualistic mindset regarding life in general and the human being in particular + the impact of anthropological dualism on church life and especially on education in the context of the church + the Biblical view of the human being, and on + how the pervasive problem of anthropological dualism can be eradicated. It was found that, although the Bible uses a whole variety of words that somehow relate to or describe the human being, these words or t e n s do not refer to "parts" or "components" but rather to different facets of the human being, much like one can refer to the different facets of a polished diamond. Whenever a word is used, it refers to a particular perspective from which the human being is approached or viewed but in the final analysis, it refers to the whole being. Discovery of this perspective was important in view of the dualistic tendencies in Korean churches and in church education. Application of a holistic view of the human being enables one to approach education as the guiding, leading, enabling, equipping and discipling of educands (those who are being guided etc.) as whole, total and integrated persons. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2007.

The Christian church was begun in 19Ih century in the traditionally multi-religious society in Korea. Christianity holds a major position in Korea today and has for the past 20 years been growing rapidly in numbers. Despite its phenomenal growth, the churches. and Christianity in general, have been suffering from several ailments, of which dualistic thinking is not the least. Anthropological dualism amounts to not only distinguishing between soul and body, but also ascribing a separate and independent existence to each of these "components" of the human being. This dualism (as well as others) developed in the church under the influence of traditional Korean religions such as Buddhism, Confucianism and Shamanism that have been teaching such dualisms. The Korean Christian mindset has to this day been dualistic both as a result of such cultural and philosophical influences and o€ ideas brought by the early missionaries to Korea. The influence of Platonic dualism is still widespread in the conservative and gospel church. This study focuses on: + examining the nature of the problem of anthropological (and other forms of) dualism + how the problem has been manifesting itself in Korean churches and in church education + the most momentous influences on Korean Christianity and churches resulting in a dualistic mindset regarding life in general and the human being in particular + the impact of anthropological dualism on church life and especially on education in the context of the church + the Biblical view of the human being, and on + how the pervasive problem of anthropological dualism can be eradicated. It was found that, although the Bible uses a whole variety of words that somehow relate to or describe the human being, these words or t e n s do not refer to "parts" or "components" but rather to different facets of the human being, much like one can refer to the different facets of a polished diamond. Whenever a word is used, it refers to a particular perspective from which the human being is approached or viewed but in the final analysis, it refers to the whole being. Discovery of this perspective was important in view of the dualistic tendencies in Korean churches and in church education. Application of a holistic view of the human being enables one to approach education as the guiding, leading, enabling, equipping and discipling of educands (those who are being guided etc.) as whole, total and integrated persons. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2007.

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L’objectif de cette thèse est d’étudier et de donner des outils pour la compréhension des problèmes de perturbations singulières pour des modèles épidémiques et des problèmes de dynamiques de populations. Les modèles considérés sont des équations structurées en âge qui peuvent dans certains cas se réécrire comme des équations à retard. L’étude de ces classes d’exemples s’est faite avec succès et a permis de comprendre et de mettre en évidence toute la complexité et l’étendue de ces problèmes. Comme on peut le remarquer dans la littérature, l’une des clés fondamentales à la compréhension de ces problèmes est l’étude des variétés normalement hyperboliques en dimension infinie que nous avons largement étudiées dans cette thèse. L’approche utilisée est la méthode de Lyapunov-Perron. Ce qui nous a amené à étudier les problèmes de persistance et d’existence de trichotomie (dichotomie) exponentielle qui sont des éléments fondamentaux dans l’utilisation de cette méthode. / In this thesis we aim to give tools to understand singular perturbations in epidemic model sand population dynamic models. We study some singularly perturbed delay differential equation which does not enter into the class frame work of geometric singular perturbation for delay differential equations. An example of singularly perturbed age structured model is also studied. The study of these examples allowed us to understand and highlight some complexities of these problems. One of the main tools in understanding such questions is the normally hyperbolic manifolds theory which is our central focus in this thesis. The approach used here is the Lyapunov-Perron method. Therefore the problems of persistence and existence of exponential trichotomy (dichotomy) are also stressed since there are one of the mainingredients of this method.

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.