Coverings, correspondences, and noncommutative geometry

Unbounded Bivariant K-theory and Correspondences in Noncommutative Geometry

By adapting the algebraic notion of universal connection to the setting of unbounded KK-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable C∗-module. The theory of operator spaces provides the required tools. Finally, the above...

Noncommutative Coverings of Quantum Tori

We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

Noncommutative geometry

Noncommutative Geometry and Quantization

The purpose of these lectures is to survey several aspects of noncommutative geometry, with emphasis on its applicability to particle physics and quantum field theory. By now, it is a commonplace statement that spacetime at short length —or high energy— scales is not the Cartesian continuum of macroscopic experience, and that older methods of working with functions on manifolds must be rethough...

Noncommutative geometry and reality

We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR-theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the “Connes-Lott” description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincare duality and in the unimodularity condition.