Noncommutative geometry and theoretical physics

Abstract

The structure of a manifold can be encoded in the commutative algebra of functions on the manifold it self - this is usual -. In the case of a non commutative algebra there is no underlying manifold and the usual concepts and tools of differential geometry (differential forms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory …) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The present paper is an introduction for the non specialist and a review of the principal results on the field.