theory of D-modules
E934439
The theory of D-modules is a branch of algebraic analysis and algebraic geometry that studies modules over rings of differential operators, providing a powerful framework for understanding systems of linear differential equations and their geometric and representation-theoretic properties.
All labels observed (2)
| Label | Occurrences |
|---|---|
| D-modules | 1 |
| theory of D-modules canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11576271 — 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: theory of D-modules Context triple: [Joseph Bernstein, influenced, theory of D-modules]
-
A.
Beilinson–Bernstein localization theorem
The Beilinson–Bernstein localization theorem is a fundamental result in geometric representation theory that realizes representations of semisimple Lie algebras as sheaves of differential operators on flag varieties, establishing an equivalence between algebraic and geometric categories.
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B.
Picard–Vessiot theory
Picard–Vessiot theory is a branch of differential Galois theory that studies linear differential equations via the symmetries of their solution fields, analogous to classical Galois theory for polynomial equations.
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C.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
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D.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
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E.
Hilbert scheme theory
Hilbert scheme theory is a branch of algebraic geometry that studies parameter spaces representing families of subschemes of projective space, capturing how such geometric objects vary in moduli.
- 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: theory of D-modules Target entity description: The theory of D-modules is a branch of algebraic analysis and algebraic geometry that studies modules over rings of differential operators, providing a powerful framework for understanding systems of linear differential equations and their geometric and representation-theoretic properties.
-
A.
Beilinson–Bernstein localization theorem
The Beilinson–Bernstein localization theorem is a fundamental result in geometric representation theory that realizes representations of semisimple Lie algebras as sheaves of differential operators on flag varieties, establishing an equivalence between algebraic and geometric categories.
-
B.
Picard–Vessiot theory
Picard–Vessiot theory is a branch of differential Galois theory that studies linear differential equations via the symmetries of their solution fields, analogous to classical Galois theory for polynomial equations.
-
C.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
-
D.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
E.
Hilbert scheme theory
Hilbert scheme theory is a branch of algebraic geometry that studies parameter spaces representing families of subschemes of projective space, capturing how such geometric objects vary in moduli.
- F. None of above. chosen
Statements (58)
| Predicate | Object |
|---|---|
| instanceOf |
branch of algebraic analysis
ⓘ
branch of algebraic geometry ⓘ mathematical theory ⓘ |
| appliesTo |
algebraic varieties
ⓘ
analytic spaces ⓘ complex manifolds ⓘ |
| basedOn | rings of differential operators ⓘ |
| developedBy |
Alexander Beilinson
NERFINISHED
ⓘ
Bernard Malgrange NERFINISHED ⓘ Joseph Bernstein NERFINISHED ⓘ Masaki Kashiwara NERFINISHED ⓘ Pierre Deligne NERFINISHED ⓘ Takuro Shintani NERFINISHED ⓘ Zoghman Mebkhout NERFINISHED ⓘ |
| emergedIn | 1970s ⓘ |
| fieldOfStudy | D-modules ⓘ |
| formalizes | systems of linear partial differential equations ⓘ |
| frameworkFor | algebraic analysis of differential equations ⓘ |
| hasApplication |
classification of linear differential equations with regular singularities
ⓘ
geometric Langlands program NERFINISHED ⓘ index theorems in analysis ⓘ representation theory of Lie algebras ⓘ representation theory of algebraic groups ⓘ study of singularities of differential equations ⓘ |
| hasImportantResult |
Riemann–Hilbert correspondence for regular holonomic D-modules
ⓘ
equivalence between regular holonomic D-modules and perverse sheaves on complex algebraic varieties ⓘ |
| hasKeyObject |
Riemann–Hilbert correspondence
NERFINISHED
ⓘ
characteristic varieties ⓘ coherent D-modules ⓘ de Rham functor NERFINISHED ⓘ flat connections ⓘ holonomic D-modules ⓘ integrable connections ⓘ left D-modules ⓘ local systems ⓘ regular holonomic D-modules ⓘ right D-modules ⓘ singular support ⓘ solution functor ⓘ |
| relatedTo |
Hodge theory
NERFINISHED
ⓘ
algebraic topology ⓘ geometric representation theory ⓘ microlocal geometry ⓘ representation theory ⓘ symplectic geometry ⓘ |
| studies |
algebraic aspects of differential equations
ⓘ
analytic aspects of differential equations ⓘ geometric properties of differential equations ⓘ modules over rings of differential operators ⓘ representation-theoretic properties of differential equations ⓘ systems of linear differential equations ⓘ |
| usesConcept |
algebraic geometry
ⓘ
coherent sheaves ⓘ derived categories ⓘ homological algebra ⓘ microlocal analysis ⓘ perverse sheaves ⓘ sheaves ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: theory of D-modules Description of subject: The theory of D-modules is a branch of algebraic analysis and algebraic geometry that studies modules over rings of differential operators, providing a powerful framework for understanding systems of linear differential equations and their geometric and representation-theoretic properties.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
D-modules