Recent progress in non-commutative geometry

Abstract

We expose here some recent results of the common work performed with M. Dubois-Violette and John Madore. The leading idea of a non-commutative geometry is to replace the commutative algebra of smooth functions on a differential manifold by a more general, non-commutative associative algebra with unity. The maximal ideals can be identified as points, and the corresponding “manifolds” reduce usually to a discrete set of points in the non-commutative case. Nevertheless, the notions of vector fields, exterior forms, metric and connection can be easily generalized. A new version of the Kaluza-Klein type of theory can be set up, introducing such a non-commutative structure instead of internal space.

Talk at the XVIIIth ICGTMP Conference in Moscow, June 1990.