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| \section{Description} |
Noncommutative `geometry' utilizes non-Abelian (or nonabelian) methods for quantization of spaces through `deformation' to non-commutative 'spaces' (in fact {\em non-commutative} algebraic structures, or algebras of functions). |
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\emph{An alternative meaning is often given to Noncommutative Geometry (viz . A Connes et al.)}: |
| Noncommutative `geometry' utilizes non-Abelian (or nonabelian) methods for quantization of spaces through `deformation' to non-commutative 'spaces' (in fact {\em non-commutative} algebraic structures, or algebras of functions). |
i.e., as a non-commutative `geometric' approach-- \emph{in the relativistic sense}-- to Quantum Gravity.\\ |
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A specific example due to A. Connes is the convolution $C^*$-algebra of (discrete) groups; |
| \emph{An alternative meaning is often given to Noncommutative Geometry (viz . A Connes et al.)}: |
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| i.e., as a non-commutative `geometric' approach-- \emph{in the relativistic sense}-- to Quantum Gravity. |
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| A specific example due to A. Connes is the convolution $C^*$-algebra of (discrete) groups; |
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| other examples are non-commutative $C^*$-algebras of operators defined on Hilbert spaces of |
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 |
quantum operators and states. (Please see also the other PM entries on $C^*$-algebra and |
| Noncommutative Topology.) |
Noncommutative Topology.) |
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\subsection{Remarks}
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\textbf{Notes:}
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| \item The Royal Swedish Academy of Sciences has awarded the 2001 Crafoord Prize in mathematics |
\item The Royal Swedish Academy of Sciences has awarded the 2001 Crafoord Prize in mathematics |
| to Professor Alain Connes of the Institut des Hautes \'Etudes Scientifiques (IHES) and the |
to Professor Alain Connes of the Institut des Hautes \'Etudes Scientifiques (IHES) and the |
| Coll\'ege de France, Paris, ``for his penetrating work on the theory of... (quantum)... operator algebras |
Coll\'ege de France, Paris, ``for his penetrating work on the theory of... (quantum)... operator algebras |
| and for having been a founder of \emph{noncommutative geometry}''. |
and for having been a founder of \emph{noncommutative geometry}''. |
| (\PMlinkexternal{Crafoord Prize in 2001 in Noncommutative Geometry and Quantum Operator Algebras}{http://www.ams.org/notices/200105/comm-crafoord.pdf}). |
(\PMlinkexternal{Crafoord Prize in 2001 in Noncommutative Geometry and Quantum Operator Algebras}{http://www.ams.org/notices/200105/comm-crafoord.pdf}). |
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| Professor Alain Connes is also the 1983 recipient of the Field Medal. The following is a concise quote of his work |
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: ``{\em Noncommutative geometry is a new field of mathematics, and |
from the Crafoord Prize announcement in 2001: ``{\em 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...\PMlinkname{(SUSY)}{SpinGroup}... |
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...\PMlinkname{(SUSY)}{SpinGroup}... |
| He has demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann Hypothesis.}'' |
He has demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann Hypothesis.}'' |
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| \item ``\emph{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 |
\item ``\emph{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 |
(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). |
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). |
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