Noncommutative Geometry

Noncommutative Geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces which are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm to be possibly carried by the noncommutative algebra of functions. The leading direction in noncommutative geometry has been laid by French mathematician Alain Connes since his involvement from about 1979.