theory of D-modules

E934439

branch of algebraic analysis branch of algebraic geometry mathematical theory

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.

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Observed surface forms (1)

Surface form	Occurrences
D-modules	1

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 ⓘ

Referenced by (2)

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

Joseph Bernstein → influenced → theory of D-modules ⓘ

Bernard Malgrange → fieldOfWork → theory of D-modules ⓘ

this entity surface form: D-modules