noncommutative geometry
1 Topic on Noncommutative Geometry (NCG)
Noncommutative geometry^{} utilizes nonAbelian^{} (or nonabelian^{}) methods for quantization of spaces through deformation^{} to noncommutative ’spaces’ (in fact noncommutative algebraic structures^{}, or algebras of functions).
An alternative meaning is often given to noncommutative geometry (viz . A Connes et al.): that is, as a noncommutative ‘geometric’ approach– in the relativistic sense– to quantum gravity.
A specific example due to A. Connes is the convolution ${C}^{*}$algebra^{} of (discrete) groups; other examples are noncommutative ${C}^{*}$algebras of operators defined on Hilbert spaces^{} of quantum operators and states.
1.1 Recent Developments in NCG

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The Royal Swedish Academy of Sciences has awarded the 2001 Crafoord Prize in mathematics to Professor Alain Connes of the Institut des Hautes Études Scientifiques (IHES) and the Collége de France, Paris, “for his penetrating work on the theory of… (quantum)… operator algebras and for having been a founder of noncommutative geometry”. (http://www.ams.org/notices/200105/commcrafoord.pdfCrafoord Prize in 2001 in Noncommutative Geometry and Quantum Operator Algebras^{}).
Professor Alain Connes is also the 1983 recipient of the Field 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) (http://planetmath.org/SpinGroup)… He has demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann Hypothesis.”

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“The Crafoord Prize prize consisted of a gold medal and US dollars 500,000. The AnnaGreta and Holger Crafoord Foundation was established in 1980 for promoting basic research in mathematics, astronomy, the biosciences (particularly ecology), the geosciences, and polyarthritis (joint rheumatism)”. Previous (‘Nobel style’), Crafoord Laureates in Mathematics were: Vladimir I. Arnold and Louis Nirenberg in 1982, Alexandre Grothendieck (who publicly declined the prize) and Pierre Deligne–who accepted the prize in 1988, and Simon Donaldson and ShingTung Yau (1994).
Title  noncommutative geometry 
Canonical name  NoncommutativeGeometry 
Date of creation  20130322 18:13:53 
Last modified on  20130322 18:13:53 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  31 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 81T75 
Synonym  nonabelian algebraic topology 
Synonym  noncommutative geometry^{} 
Synonym  nonAbelian geometry 
Synonym  anabelian geometry 
Synonym  nonAbelian topology (NAAT) 
Synonym  noncommutative topology^{} 
Synonym  noncommutative topology 
Related topic  CAlgebra 
Related topic  SpinGroup 
Related topic  FieldsMedal 
Related topic  CrafoordPrize 
Related topic  CAlgebra3 
Related topic  NuclearCAlgebra 
Related topic  QuantumGravityTheories 
Related topic  MathematicalProgrammesForDevelopingQuantumGravityTheories 
Related topic  QuantumGeometry 
Related topic  QuantumGeometry2 
Related topic  AlgebraicTopology 
Related topic  NoncommutativeTopology 
Defines  ‘Geometry’ of quantum spaces in terms of noncommutative algebras of functions and quantum operators 
Defines  or ‘spectral (quantum) triples’ 