"Noncommutative geometry, as developed by Connes starting in the early ’80s ..., extends the tools of ordinary geometry to treat spaces that are quotients, for which the usual “ring of functions”, defined as functions invariant with respect to the equivalence relation, is too small to capture the information on the “inner structure” of points in the quotient space. Typically, for such spaces functions on the quotients are just constants, while a nontrivial ring of functions, which remembers the structure of the equivalence relation, can be defined using a noncommutative algebra of coordinates, analogous to the non- commuting variables of quantum mechanics."