Duncan Haldane talks about Quantum Geometry, Exclusion Statistics, and the Geometry of "Flux Attachment" in 2D Landau levels.
The degenerate partially-filled 2D Landau level is a remarkable environment in which kinetic energy is replaced by "quantum geometry" (or an uncertainty principle) that quantises the space occupied by the electrons quite differently from the atomic-scale quantisation by a periodic arrangement of atoms. In this arena, when the short-range part of the Coulomb interaction dominates, it can lead to "flux attachment", where a particle (or cluster of particles) exclusively occupies a quantised region of space. This principle underlies both the incompressible fractional quantum Hall fluids an the composite fermion Fermi liquid states that occur in such systems.
