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This book starts with an introduction by V. Pasquier on the usefulness of non-commutative geometry, especially in the condensed matter context of the Hall effect. This theme is further developed by A. Polychronakos, who together with L. Susskind introduced the concept of non-commutative fluids. Jean-Michel Maillet compares the experimental results on one dimensional magnetic chains to the theoretical predictions based on the algebraic Bethe Ansatz and related to quantum group symmetries that he actively developed with many collaborators. On the high energy side of non-commutative geometry the

Noncommutative differential geometry --- Quantum Hall effect --- Space and time --- Quantum theory --- Differential geometry, Noncommutative --- Geometry, Noncommutative differential --- Non-commutative differential geometry --- Infinite-dimensional manifolds --- Operator algebras --- Quantum theory. --- Quantum Physics. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Quantum physics.