0.1 Non-commutative Geometry and Non-Abelian Algebraic Topology

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 C^∗-algebra of (discrete) groups; other examples are non-commutative C^∗-algebras of operators defined on Hilbert spaces of quantum operators and states. (Please see also the other PM entries on C^∗-algebra and noncommutative topology.)

0.2 Recent Developments in NCG

• 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”. (Crafoord Prize in 2001 in Noncommutative Geometry and Quantum Operator Algebras).

  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.”

• “The Crafoord Prize prize consisted of a gold medal and US dollars 500,000. The Anna-Greta 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 Shing-Tung Yau (1994).