Connes–Moscovici index theorem

E286301

index theorem mathematical theorem

The Connes–Moscovici index theorem is a fundamental result in noncommutative geometry that generalizes the classical Atiyah–Singer index theorem to the setting of foliations and noncommutative spaces.

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All labels observed (3)

Label	Occurrences
Connes–Moscovici index theorem canonical	1
index formula for elliptic operators on foliated manifolds	1
noncommutative local index formula	1

How this entity was disambiguated

This entity first appeared as the object of triple T2648172 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Connes–Moscovici index theorem
Context triple: [Alain Connes, knownFor, Connes–Moscovici index theorem]

A. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
B. Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
C. Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
D. Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
E. Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Connes–Moscovici index theorem
Target entity description: The Connes–Moscovici index theorem is a fundamental result in noncommutative geometry that generalizes the classical Atiyah–Singer index theorem to the setting of foliations and noncommutative spaces.

A. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
B. Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
C. Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
D. Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
E. Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	index theorem ⓘ mathematical theorem ⓘ
appliesTo	foliations ⓘ longitudinal elliptic operators on foliations ⓘ noncommutative spaces ⓘ transversely elliptic operators ⓘ
characteristic	expresses index as pairing between K-theory and cyclic cohomology ⓘ local formula for the index in terms of cyclic cocycles ⓘ
context	leaf space of a foliation ⓘ noncommutative leaf space ⓘ
developedBy	Alain Connes ⓘ Henri Moscovici ⓘ
equates	analytic index and topological index in the noncommutative setting ⓘ
field	foliation theory ⓘ global analysis ⓘ noncommutative geometry ⓘ operator algebras ⓘ
generalizes	Atiyah–Singer index theorem ⓘ
hasVersion	higher index theorem for foliations ⓘ index theorem for transversely hypoelliptic operators on foliations ⓘ
influenced	applications of noncommutative geometry to foliation theory ⓘ development of cyclic cohomology ⓘ index theory on groupoids ⓘ
involves	Chern character in cyclic cohomology ⓘ Godbillon–Vey class ⓘ linked to: Godbillon–Vey invariant characteristic classes of foliations ⓘ transverse fundamental class ⓘ
language	C*-algebraic formulation of index theory ⓘ noncommutative differential geometry ⓘ
motivation	extension of Atiyah–Singer index theorem to foliations ⓘ formulation of index theory on noncommutative spaces ⓘ
namedAfter	Alain Connes ⓘ Henri Moscovici ⓘ
provides	index formula for elliptic operators on foliated manifolds ⓘ linked to: Connes–Moscovici index theorem local index formula in noncommutative geometry ⓘ
relatedTo	Baum–Connes conjecture ⓘ noncommutative local index formula ⓘ linked to: Connes–Moscovici index theorem
relates	analytic index ⓘ topological index ⓘ
usesConcept	C-algebras ⓘ K-homology ⓘ K-theory ⓘ cyclic cohomology ⓘ groupoid C-algebras ⓘ pseudodifferential operators ⓘ von Neumann algebras ⓘ
yearProposed	1980s ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alain Connes → knownFor → Connes–Moscovici index theorem ⓘ

Connes–Moscovici index theorem → provides → index formula for elliptic operators on foliated manifolds ⓘ

linked to: Connes–Moscovici index theorem

Connes–Moscovici index theorem → relatedTo → noncommutative local index formula ⓘ