These lectures are devoted to recent work on deformation of spaces, in particular spheres, and deformation of vector bundles which are relevant in physics, notably monopole and instanton bundles.
A first class of examples consists of toric noncommutative manifolds which were introduced out of cycles in homology. These examples have non-trivial global features and can be endowed with a structure of noncommutative manifolds, in terms of a spectral triple. In particular, noncommutative spheres are isospectral deformations of usual spherical geometries. Instantonic bundles and Yang-Mills duality equations can be introduced and a geometric analysis paralleling the one on commutative space can be performed.
A second class of examples is made of spaces coming from quantum groups, some of which can again be endowed with a spectral triple. Also one can construct generalizations of instanton bundles.
Finally, we also review some work on harmonic maps and non-linear sigma-models in noncommutative geometry, with particular attention on models on noncommutative tori.
Fredholm modules and spectral triples
Isospectral deformations
Gauge instantons on isospectral deformations
Noncommutative geometry of quantum groups
Noncommutative sigma-model instantons
The noncommutative geometry of quantum groups
Updated 14/09/2005