families index theorem
E391907
The families index theorem is a generalization of the Atiyah–Singer index theorem that computes the index of a continuous family of elliptic operators as a characteristic class in the K-theory of the parameter space.
All labels observed (1)
| Label | Occurrences |
|---|---|
| families index theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3821404 — 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: families index theorem Context triple: [Atiyah–Singer index theorem, hasVariant, families index theorem]
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A.
Six family
The Six family is a prominent Belgian business dynasty best known for controlling and leading the international construction and real estate group BESIX.
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B.
Section for the Family
Section for the Family is a department within the Vatican’s Dicastery for the Laity, Family and Life that focuses on promoting and supporting pastoral care, doctrine, and initiatives related to marriage and family life in the Catholic Church.
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C.
Gauss family
The Gauss family is a notable lineage associated with individuals bearing the Gauss surname, most famously linked to the legacy of mathematician Carl Friedrich Gauss.
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D.
Fricke family
The Fricke family is a benefactor family associated with the University of California, Berkeley, whose philanthropy is commemorated through the naming of Levine-Fricke Field.
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E.
The Family
The Family is a funk and R&B band formed by Prince in the mid-1980s, best known for originating the song "Nothing Compares 2 U."
- 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: families index theorem Target entity description: The families index theorem is a generalization of the Atiyah–Singer index theorem that computes the index of a continuous family of elliptic operators as a characteristic class in the K-theory of the parameter space.
-
A.
Six family
The Six family is a prominent Belgian business dynasty best known for controlling and leading the international construction and real estate group BESIX.
-
B.
Section for the Family
Section for the Family is a department within the Vatican’s Dicastery for the Laity, Family and Life that focuses on promoting and supporting pastoral care, doctrine, and initiatives related to marriage and family life in the Catholic Church.
-
C.
Gauss family
The Gauss family is a notable lineage associated with individuals bearing the Gauss surname, most famously linked to the legacy of mathematician Carl Friedrich Gauss.
-
D.
Fricke family
The Fricke family is a benefactor family associated with the University of California, Berkeley, whose philanthropy is commemorated through the naming of Levine-Fricke Field.
-
E.
The Family
The Family is a funk and R&B band formed by Prince in the mid-1980s, best known for originating the song "Nothing Compares 2 U."
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
index theorem
ⓘ
mathematical theorem ⓘ result in K-theory ⓘ result in global analysis ⓘ |
| alsoKnownAs |
family index theorem
ⓘ
index theorem for families of elliptic operators ⓘ |
| appliesTo |
families of Dirac-type operators
ⓘ
families of Dolbeault operators ⓘ families of signature operators ⓘ |
| codomain | K-theory of the parameter space ⓘ |
| defines |
analytic index as an element of K-theory of the parameter space
ⓘ
topological index via characteristic classes and pushforward ⓘ |
| describes | index of a continuous family of elliptic operators ⓘ |
| developedBy |
Isadore Singer
ⓘ
Michael Atiyah ⓘ |
| domain |
families of elliptic operators parametrized by a base space
ⓘ
smooth fiber bundles with compact fibers ⓘ |
| field |
K-theory
ⓘ
differential geometry ⓘ global analysis ⓘ operator theory ⓘ topology ⓘ |
| generalizationOf | Atiyah–Singer index theorem ⓘ |
| implies |
cohomological index formulas for families
ⓘ
local index theorems for families ⓘ |
| influenced |
Bismut’s local families index theorem
ⓘ
Quillen’s superconnection formalism ⓘ anomaly formulas in quantum field theory ⓘ theory of determinant line bundles ⓘ |
| mathematicalSubjectClassification |
19K56
ⓘ
58J20 ⓘ |
| relatedTo |
Atiyah–Singer index theorem
ⓘ
surface form:
Atiyah–Singer index theorem for a single operator
Grothendieck–Riemann–Roch theorem ⓘ Riemann–Roch theorem ⓘ
surface form:
Riemann–Roch theorem for families
|
| relates |
analytic index of a family of elliptic operators
ⓘ
topological index as a K-theory class ⓘ |
| statesThat | the analytic index of a family of elliptic operators equals its topological index in K-theory ⓘ |
| timePeriod | 1960s ⓘ |
| usesConcept |
Chern character
ⓘ
Fredholm operator ⓘ K-theory ⓘ
surface form:
K-theory of topological spaces
Todd class ⓘ characteristic classes ⓘ continuous family of operators ⓘ elliptic differential operator ⓘ family of manifolds ⓘ fiber bundle ⓘ pushforward in K-theory ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: families index theorem Description of subject: The families index theorem is a generalization of the Atiyah–Singer index theorem that computes the index of a continuous family of elliptic operators as a characteristic class in the K-theory of the parameter space.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.