noncommutative geometry

noncommutative geometry

1

Topic on Non-commutative Geometry (NCG)

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).

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

.

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.

1.1

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

”.
(

http://www.ams.org/notices/200105/comm-crafoord.pdf

Crafoord 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.

”

•

“

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).

Title

noncommutative geometry

Canonical name

NoncommutativeGeometry

Date of creation

2013-03-22 18:13:53

Last modified on

2013-03-22 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

non-commutative geometry

Synonym

non-Abelian geometry

Synonym

anabelian geometry

Synonym

non-Abelian topology (NAAT)

Synonym

noncommutative topology

Synonym

non-commutative 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 non-commutative algebras of functions and quantum operators

Defines

or ‘spectral (quantum) triples’

Generated on Sat Feb 10 13:07:15 2018 by

LaTeXML