The foremost objective of this project is to extend the local representation of the characteristic 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 of foliations, 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 new tools for the implementation of this principle in the setting of noncommutative geometry, a modern variant of geometry inspired by quantum mechanics which allows 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.
In noncommutative geometry the paradigm of space as a manifold formed of points labeled by numerical coordinates is replaced by one of a more general nature, as in quantum physics, in which the coordinates are operator-valued and may no longer commute with each other. This novel foundational principle allows for a vast enrichment of the class of objects which qualify for being treated as spaces, even though they may lack any meaningful notion of points. Basic algebraic-topological and analytical tools for the global treatment of the usual spaces have been successfully upgraded to the noncommutative context. However the meaning of locality in the absence of the deep-rooted ``pointillistic" intuition, and the relationship between local and global spacial properties pose quite a challenge. The main outcomes of the project address precisely these aspects. The first outcome has to do with the fundamental concept of intrinsic curvature, which lies at the very core of classical geometry but remained until recently mysterious for noncommutative spaces. In collaboration with A. Connes we found the precise expression, as well as its meaning, for the curvature of the noncommutative torus, the prototypical noncommutative surface. The second outcome has been a significant refinement (developed in collaboration with B. Rangipour) of Hopf cyclic cohomology, a technology initiated earlier by A. Connes and the principal investigator, which allows to relate the local and global invariants of a generic class of noncommutative spaces, those which describe the transverse geometry of foliations. The third outcome (obtained in joint work with M. Lesch and M. Pflaum) exploits tools specific to noncommutative geometry to obtain new relations between the geometry of the interior and that of the boundary for a standard class of usual geometric spaces, namely the compact manifolds with boundary. The shared intellectual merit of these works is that they contribute, each in its own distinct way, to a deeper understanding of the relationship between local and global properties in noncommutative geometry, and at the same time engender new interactions with other areas of mathematics and physics. Being inherently connected with several different areas, the above research was well-suited to be integrated in the process of instruction. Under the supervision of the principal investigator, three graduate students are currently completing their doctoral dissertations stemming from the above topics, and one postdoctoral? fellow has finished two papers inspired in part by the work reported above.
