Tannaka duality for proper Lie groupoids

The main contribution of this thesis is a Tannaka duality theorem for proper Lie groupoids. This result is obtained by replacing the category of smooth vector bundles over the base manifold of a Lie groupoid with a larger category, the category of smooth Euclidean fields, and by considering smooth actions of Lie groupoids on smooth Euclidean fields. The notion of smooth Euclidean field that is introduced here is the smooth, finite dimensional analogue of the familiar notion of continuous Hilbert field. In the second part of the thesis, ordinary smooth representations of Lie groupoids on smooth vector bundles are systematically studied from the point of view of Tannaka duality, and various results are obtained in this direction.