Limit Sets as Examples in Noncommutative Geometry

Limit Sets as Examples in Noncommutative Geometry

The fundamental group of a hyperbolic manifold acts on the limit set, giving rise to a cross-product C∗-algebra. We construct nontrivial K-cycles for the cross-product algebra, thereby extending some results of Connes and Sullivan to higher dimensions. We also show how the Patterson-Sullivan measure on the limit set can be interpreted as a center-valued KMS state.

Noncommutative geometry as a functor

In this note the noncommutative geometry is interpreted as a functor, whose range is a family of the operator algebras. Some examples are given and a program is sketched.

Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets

An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree [28]. This tree allows to define a family of spectral triples (CLip(C),H, D) using the l space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here CLip...

Examples of noncommutative instantons

These notes aim at a pedagogical introduction to recent work on deformation of spaces and deformation of vector bundles over them, which are relevant both in mathematics and in physics, notably monopole and instanton bundles. We first decribe toric noncommutative manifolds (also known as isospectral deformations). These come from deforming the usual Riemannian geometry of a manifold along a tor...

Noncommutative geometry