Hideki Omori 
Introduction to noncommutative differential geometry 
(Lobachevskii Journal of Mathematics, Vol.4, pp.13-46)

It is getting a major understanding that the {\it function ring} of the underlying space of the quantum world is {\it not} a commutative ring, and the underlying space itself is not even a point set. \par In spite of this, the underlying space of the quantum world should be a {\it continuum}, on which one can make a calculus. We hope we can see such mathematics in the coming century. This article is a survey of such effort or trial to make such calculus. However, this article does not give a bird-eye-view of the noncommutative world but this gives several toys which come from the noncommutative world.

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