Spectral action in noncommutative geometry and global pseudodifferential calculus

In this thesis, we studied certain mathematical issues related to the computation of the Chamseddine--Connes spectral action on some fundamental noncommutative spectral triples, such as the noncommutative torus and the quantum 3-sphere SUq(2). We showed in particular that a Diophantine condition on the deformation matrix of the torus is crucial to obtain the spectral action with real structure. We also studied the question of existence of tadpoles (linear terms in the gauge potential of the fluctuation of the metric in the spectral action) for commutative Riemannian geometries, and the construction of a symbolic global pseudodifferential calculus allowing a generalization of the Weyl--Moyal product on a Schwartz space of rapidly decaying sections on a cotangent bundle of a manifold with linearization.