This project will continue the program initiated several years ago by the principal investigator in collaboration with Alain Connes, providing a general setting for the understanding of transverse geometry. A main new objective consists in the application of this program to the context of modular forms and modular correspondences, in order to uncover the hidden compatibility between the pointwise product of modular forms and the action of the Hecke operators. Another major goal of the proposed research is the elaboration of a K-theoretic interpretation for the Godbillon-Vey class and for other secondary invariants, by means of a refinement of the operator-theoretic local index formula that lies at the core of the program. The proposed work will contribute to the development of new mathematical tools for the treatment of a multitude of non-classical spaces, which have a discernible geometric meaning but cannot be adequately described by means of commuting coordinates. Such spaces, arising in many areas of mathematics and physics, have in common the feature that their observable local parameters behave like infinite-dimensional matrices rather than numerical variables. To deal with them efficiently much stretching and rethinking of the known geometric techniques is required, process that has led to the development of noncomutative geometry. A distinctive feature of the present project is the systematic utilization of non-classical symmetry principles.