Non-specialists may, I think, be forgiven for feeling confused by the title of J. Madore's An Introduction to Noncommutative Differential Geometry and its Physical Applications. It's not too easy to see in what sense the differential geometry we know and love is "commutative" and even harder to imagine what a "noncommutative" geometry might look like. The first words of the introduction help us out with the first question. They point out that if V is a set of points then the set of complex-valued functions on V is a (finite-dimensional) commutative (and associative) algebra. If V is a compact space, then we can restrict to continuous complex-valued functions on V and we get an algebra C0(V) which is in fact a "C*-algebra," and if V is a smooth manifold we can look at smooth functions, and so on. It turns out that much of classical differential geometry can be expressed in terms of such algebras, and the idea of "noncommutative geometry" is to generalize this version of differential geometry to the case of noncommutative algebras. Amazingly, this turns out to yield a theory that is not only interesting mathematically but also useful in understanding the mathematics of quantum field theory. This book, volume 257 in the traditional "London Mathematical Society Lecture Note Series", is intended as an accessible introduction to the subject for non-specialists. It looks to me that the author has done a good job of opening the way to understanding a difficult theory. This is the second edition of a book first published in 1995, and the very fact that a new edition has appeared so soon is an indication that the book has been successful. Not for the faint of heart, but worth a look.
1. Introduction; 2. Differential geometry; 3. Matrix geometry; 4. Non-commutative geometry; 5. Vector bundles; 6. Cyclic homology; 7. Modifications of space-time; 8. Extensions of space-time.