Teoremi di indice e calcolo pseudodifferenziale su varietà

In the study of connections between analysis and geometry, one discovers that the topology of a manifold is deeply intertwined with the theory of elliptic pseudodifferential operators. Namely, for a given compact boundaryless manifold M, we can study its cohomology (that is, the topology) via the study of elliptic complexes on M, and this generalizes to the case of fibre bundles having M as base manifold as well. The culmination of such ideas was in the works of M.Atiyah and I.Singer about the interpretation of elliptic operators as K-theory maps and the proof of the Index theorem which bears their name. First, this thesis presents the construction of the class of pseudodifferential operators Psi^m(M) and their symbols S^m(M), with attention towards the problem of invariance of the principal symbol. Secondly, we study elliptic complexes and their relationship with the cohomology of a vector bundle over M. This leads us to characteristic classes, which are cohomology classes obtained from the curvature form via some generating function. These are the third object of study of this thesis. The final part of this work is devoted to the statement of the Atiyah-Singer index theorem and to work through the computations for a particular case, namely the de Rham complex for a simple manifold.