The comprehensive goal of this project is to achieve a conceptual understanding of the local invariants for noncommutative spaces and of the way the local and global properties interact and condition each other. Of primary interest is the concept of intrinsic curvature, which lies at the very core of the classical geometry but remained for a long time intangible in noncommutative geometry. One of the main thrusts of the present project is to build on the latest advances in this direction, which led to the uncovering of the nonunimodular character of the conformal geometry of the noncommutative torus. Other themes concern the continuation of the development of new techniques for computing Hopf algebra cyclic cohomology, and applying them to the determination of explicit expressions for the transverse characteristic classes of a large class of noncommutative spaces occurring in geometry and number theory; the investigation of the invariants called higher indices for manifolds with boundary in order to elucidate their role in detecting geometric and topological properties of these spaces; the exploration of the potential arithmetic implications of the obtained results in connection with several open problems at the interface between noncommutative geometry and number theory.
In noncommutative geometry, the paradigm of space as a manifold formed of points labeled by numerical coordinates is replaced by one of a much more general nature, in which the coordinates are operator-valued and may no longer commute, as in quantum physics. This new foundational principle allows for a substantial enrichment of the class of objects that can be treated as geometric spaces and has already found spectacular applications in both mathematics and theoretical physics. While the most basic algebraic, topological, and analytical tools for the global treatment of such spaces have been already successfully developed, the local treatment, and even the meaning of locality in the absence of the deeply ingrained spatial conceptualization, is still very much under construction. Its truly conceptual understanding, which makes the primary object of this project, engenders new applications to several central fields of contemporary mathematics and is quite likely of considerable relevance for the latest developments in particle and high-energy physics.
