Quantum groups and noncommutative geometry

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geomet ry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century se es at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical p redictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of space-time, in particular, amounts to a postulated new force or physical effect called cogravity. (C) 2000 American Institute of P hysics. [S0022-2488(00)00406-0].