Differential schemes

This thesis focuses on the theory --- still under construction --- of differential schemes. The aim of our work is to provide two new perspectives to this theory. The first perspective is geometric and consists in considering schemes endowed with vector fields instead of differential rings. In this context, we define what is a leaf and what is the trajectory of a point. With the help of these tools, we reinvest and generalize some results of differential Galois theory. Similarly, we show that the Carrà Ferro sheaf is the natural sheaf of the space of leaves of a scheme with vector field. It is also this approach that lead us to prove that, in the reduced case, the Kovacic and Keigher sheaves are isomorphic and that they have the same constant as the Carrà Ferro sheaf. The second perspective is functorial, and is based on the notion of scheme due to Toën and Vaquié [6]. We prove that the category of differential schemes in the sense similar to the one of these authors is equivalent to the category of schemes endowed with a vector field.