Noncommutative geometry governs the physics of quantum Hall (QH) effects. The Hamiltonian is not local: It is constructed on a von Neumann lattice. Noncommutative geometry leads to a spontaneous development of quantum coherence. As coherent excitations there arise isospin waves and topological solitons (skyrmions). It leads to a Josephson-like effect in a bilayer system. We expect SU(2N) quantum coherence and CP~(2N-1) skyrmions in N-layer QH systems in general. QH effects present experimental tests of various ideas inherent to noncommutative geometry.
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