Abstract:
The title of my presentation reformulates the central postulate of Piaget’s theory: that a new structure can only be built on the basis of an existing structure related to the same kind of problem. I will first explain how Piaget used this postulate as the foundation for his work on the birth of intelligence. Then I will illustrate this process by means of the development of prehension behavior. I will examine several oppositions between practical intelligence and conceptual intelligence, and reconsider the use of the terms “practical” and “conceptual” to differentiate between systems of knowledge at different levels of development. My research on the construction of simple tools by children aged 4 to 9 years old will illustrate in more detail the process of developing new skills on the basis of pre-existing skills (that are simultaneously practical and conceptual). I will conclude my presentation by discussing some problems in the history of scientific and technical knowledge that are comparable to those discussed in relation to practical and conceptual skills in child development.

Abstract:
We study totally geodesic codimension 1 smooth foliations on Lorentzian manifold. We are in particular interested by the relations between riemannian flows and geodesic foliations. We prove that, up to a 2-cover, any Seifert bundle admit such a foliation.

Abstract:
In this article we extend the Gallot-Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over such a manifold admits a parallel symmetric 2-tensor then it is incomplete and has non zero constant curvature. An application of this result to the existence of metrics with distinct Levi-Civita connections but having the same unparametrized geodesics is given.

Abstract:
In this article, we study complete pseudo-Riemannian manifolds whose cone admits a parallel symmetric 2-tensorfield. The situation splits in three cases: nilpotent, decomposable or complex Riemannian. In the complex Riemannian and decomposable cases we provide a classification. In the nilpotent case, we are able to describe completely only a dense open subset of the manifold. To conclude, we give examples with non-constant curvature in the nilpotent case.

Abstract:
We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic foliations of codimension 1. On the 2-torus, we prove that a metric with constant curvature along one of its lightlike foliation is actually flat. This allows us to show that the restriction of the action to the set of non-flat metrics is proper and that on the set of flat metrics of volume 1 the action is ergodic. Finally, we show that, contrarily to the Riemannian case, the space of metrics without isometries is not always open.

Abstract:
We prove that for any compact manifold of dimension greater than $1$, the set of pseudo-Riemannian metrics having a trivial isometry group contains an open and dense subset of the space of metrics.

Abstract:
We study the maximal extensions of Lorentzian surfaces admitting a Killing vector field. We construct a natural family of simply connected maximal (or inextendable) surfaces, that we call ``universal extensions''. They are caracterised by a condition of symmetry, the ``reflexivity'', and a by a rather weak completeness assumption (the absence of "saddles at infinity"). These surfaces play the roles of model spaces: we study their minimal quotients, divisible open sets and conjugate points. We show uniformisation results (by an open subset of one of these universal extensions, that is uniquely determined) in the following cases: compact surfaces and analytical surfaces. It allows us to give a classification of Lorentzian tori and Klein bottles with a Killing vector field.

Abstract:
We describe the compact Lorentzian $3$-manifolds admitting a parallel lightlike vector field. The classification of compact Lorentzian $3$-manifolds admitting non-isometric affine diffeomorphisms follows, together with the complete description of these morphisms. Such a Lorentzian manifold is in some sense an equivariant deformation of a flat one.

Abstract:
We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an $S^1$-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley's Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by $S^1\times \R$. It follows that every Lorentzian surface contains a non-closed geodesic.

Abstract:
Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian M\"obius strips of non constant curvature all of whose spacelike geodesics are closed. Further the conformality problem for spacelike Zoll cylinders is studied.