An Introduction to Noncommutative Differential Geometry and its Physical Applications (London Mathematical Society Lecture Note Series, Series Number 206)

A significant amount of the differential structure of a smooth manifold can be encoded in the algebra of smooth functions defined on it. A noncommutative geometry is what one obtains when one replaces this algebra by a noncommutative associative algebra. Of particular interest is the case when the algebra is of finite dimension, for example an algebra of matrices. The resulting geometry is rather trivial from the point of view of analysis and can serve as a simple introduction to some of the more elementary aspects of the noncommutative geometries which one can obtain by considering more general infinite-dimensional algebras. It also has certain specific additional properties which makes it well suited to the construction of finite models of space-time. A more or less complete survey of this geometry is given as well as some possible applications to elementary particle physics and field theory. This book arose from the 1994 LMS invited lectures and will be essential to mathematicians and theoretical physicists with an interest in noncommutative geometry.