Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.
In noncommutative geometry the basic idea is that the commutative algebra of functions on an ordinary space (e.g. a topological space, a differential manifold, an algebraic variety) is to be replaced by a noncommutative algebra which then describes a noncommutative analog of the previous commutative space. Here various scenarios are possible, depending on what class of algebras and spaces one is interested in. In noncommutative geometry one then tries to study noncommutative algebras from the point of view of geometry by establishing a dictionary between the geometry and algebra side. Important applications of noncommutative geometry can be found in particular in contemporary mathematical physics.
