Noncommutative `geometry' utilizes non-Abelian (or nonabelian) methods for quantization of spaces through `deformation' to non-commutative 'spaces' (in fact non-commutative algebraic structures, or algebras of functions). therefore, it can be considered a subfield of non-Abelian algebraic topology (NAAT).
An alternative meaning is often given to Noncommutative Geometry (viz . A Connes et al.): that is, as a non-commutative `geometric' approach- in the relativistic sense- to quantum gravity. This approach is therefore relevant to Non-Abelian Quantum Algebraic Topology (NA-QAT, or NAQAT).
A specific example due to A. Connes is the convolution
-algebra of (discrete) groups; other examples are non-commutative
-algebras of operators
defined on Hilbert spaces
of quantum operators
and states. (Please see also the other PM entries on
-algebra and noncommutative topology.)
Professor Alain Connes is also the 1983 recipient of the Fields Medal. The following is a concise quote of his work from the Crafoord Prize announcement in 2001: “Noncommutative geometry is a new field of mathematics, and Connes played a decisive role in its creation. His work has also provided powerful new methods for treating renormalization theory and the standard model of quantum and particle physics...(SUSY)... He has demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann Hypothesis.”
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