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Non-commutative geometry deals with spaces where the uncertainty principle of quantum mechanics thwarts even one's ability to simultaneously measure two position co-ordinates. It finds applications in models where geometry is emergent including the matrix model approach to string theory.

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The physics community's take on non-commutative geometry

can the NCG picture be successfully modified to fit the actual Higgs mass? arXiv:1208.1030 "Resilience of the Spectral Standard Model" arXiv:1208.5023 "Asymptotic safety, hypergeometric functio …
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