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Non-commutative geometry is a recent and rapidly developing branch of mathematics. Its subject is the study of the geometry of "quantum spaces". Put a little more prosaically, this means the "geometric properties" of non-commutative algebras. While non-commutative algebras are the basic tool of quantum mechanics, their serious applications to geometry are relatively new.

The underlying philosophy is based on the observation that various categories of spaces can be completely described by the algebras of functions on them. For example, a locally compact space is described by the algebra of continuous functions, a smooth manifold by the algebra of smooth functions, an affine algebraic variety by its coordinate ring.

The idea then is that a non-commutative algebra can be viewed as an algebra of functions on a virtual "non-commutative space".

This approach is very flexible: For instance, it puts on the same footing the algebra of functions on a manifold, the algebra of pseudo-differential operators and the convolution algebra of a groupoid. As the result, the techniques coming from different fields of mathematics come together in a novel and fascinating way.

For a more detailed description of the subject we refere to an article of Alain Connes (pdf)

For a brief non-scientific description of the subject click here(pdf)

(Discription in Danish) For en kort populær beskrivelse af emnet se her(pdf)