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Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics that studies geometric ideas through noncommutative algebras. In ordinary geometry, a space can often be studied by means of a commutative algebra of functions on it; noncommutative geometry extends this viewpoint to algebras in which the product of two elements need not commute. Such algebras are…

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Motivation

Operator-algebraic noncommutative spaces

Spectral triples and noncommutative differentiable manifolds

Differential calculi and connections

Noncommutative affine and projective schemes

Invariants and index theory

Examples of noncommutative spaces

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Noncommutative geometry

Nodes46
Edges45
Triples73
Avg. degree1.96
Density0.043478
Components1

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Noncommutative geometry

Top relations

related to Further reading · 21
Noncommutative geometry → Alain, An, Arithmetic, Arithmetic Noncommutative Geometry, Baltimore, Caterina, Connes, Consani, Ginzburg, ISBN, Johns Hopkins University Press, Khalkhali, Lectures, Marcolli, Masoud, Masson, Matilde, Related Topics, Thierry, Very Basic Noncommutative Geometry
related to Invariants and index theory · 14
Noncommutative geometry → Chern, Connes, Cyclic, Fredholm, In, JLO, K-homology, K-theory, Lesniewski, Operator K-theory, Osterwalder, Rham, The Chern, The Jaffe
related to Motivation · 9
Noncommutative geometry → Addition, Gelfand, Hausdorff, If, In, Noncommutative, Rather, Similarly, The
has application · 7
Noncommutative geometry → Hall, In, Lagrangians, Noncommutative, Standard Model, The, These
related to history · 5
Noncommutative geometry → Alain Connes, Deformation, Ideas, In, The
related to Motivation from ergodic theory · 4
Noncommutative geometry → Earlier, George Mackey's, In, Some
related to External links · 2
Noncommutative geometry → Connection, MathOverflowNoncommutative
is a · 1
Noncommutative geometry → only mathematical framework for the corresponding physical problems.The fuzzy sphere has also been used as a finite-dimensional regularization in numerical and theoretical studies

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noncommutative geometry algebras algebra functions theory spaces spectral space triples commutative displaystyle differential index ordinary analogues cyclic -algebras one geometric

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Noncommutative geometryis aonly mathematical framework for the corresponding physical problems.The fuzzy sphere has also been used as a finite-dimensional regularization in numerical and theoretical studies0.90text
George Mackey's virtual subgroup theory anticipated the use of operator algebras to treat ergodic actions as generalized homogeneous spacesinstance ofEarlier ideas0.80text
noncommutative integrationinstance ofThis viewpoint is useful in areas0.80text
Tomitainstance ofThis viewpoint is useful in areas0.80text
noetherianityinstance ofextends features of projective geometry to noncommutative graded algebras under hypotheses0.80text
regularity.Many familiar theorems have analogues in this settinginstance ofextends features of projective geometry to noncommutative graded algebras under hypotheses0.80text
the Atiyahinstance ofgives a Chern character for certain Fredholm modules and spectral triples.Noncommutative index theory extends classical results0.80text
the Gauss map on continued fractionsinstance ofincluding actions connected with number theory0.80text
provide examples where orbit spaces are studied by noncommutative algebras.The fuzzy sphere replaces the algebra of functions on the two-sphere by finite-dimensional matrix algebrasinstance ofincluding actions connected with number theory0.80text
is used as a finite-mode approximation to geometricinstance ofincluding actions connected with number theory0.80text
field-theoretic models.Quantum spacetime models replace commuting coordinate functions by noncommuting operatorsinstance ofincluding actions connected with number theory0.80text
Noncommutative geometryhas applicationNoncommutative0.60section

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