Introduction to noncommutative differential geometry
(Lobachevskii Journal of Mathematics, Volume IV)
It is getting a major understanding that the function ring of the underlying space of the quantum world is not a commutative ring, and the underlying space itself is not even a point set.
In spite of this, the underlying space of the quantum world should be a 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.