LOCAL-GLOBAL INTERACTION IN NONCOMMUTATIVE GEOMETRY

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ABSTRACTThe foremost objective of this project is to extend the local representation of thecharacteristic classes of noncommutative spaces to the more intricate spaces described by the spectral triples of type III introduced by Connes and the PI. The resulting local-global theory will be applied to the transverse geometry offoliations, to modular forms, Hecke operators and Rankin-Cohen brackets, and to the refinement of the mathematical formulation of the standard model of particle physics. In the process, the Hopf cyclic theory of characteristic classes for foliations will be extended to all classical types of transverse geometries. In a different direction, the concept of boundary will be incorporated in the spectral triple approach, and the characteristic classes of the noncommutative spaces with boundary will be developed by means of relative cyclic cohomology. In various branches of science, the concepts of local and global form two markedly different but coexisting facets of a theory, which are often correlated in an interesting way by a local-global principle. The present project will develop newtools for the implementation of this principle in the setting of noncommutative geometry, a modern variant of geometry inspired by quantum mechanics whichallows for non-commuting operator-valued coordinates. Although this feature precludes ab initio any naive spatial conceptualization based on points, there is a more subtle interpretation of the notion of locality, inspired by the Bohr correspondence principle in quantum mechanics, which will be fully exploited. The proposed work will engender new connections between several fields of mathematics and physics, thus stimulating their mutually enriching interaction.