Triple

T12027063
Position Surface form Disambiguated ID Type / Status
Subject Noncommutative Geometry, Quantum Fields and Motives E286304 entity
Predicate topic P261 FINISHED
Object Riemann–Hilbert correspondence
The Riemann–Hilbert correspondence is a fundamental result in mathematics that establishes an equivalence between certain differential equations (or flat connections) on complex manifolds and representations of their fundamental groups, linking analytic and topological data.
E959837 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Riemann–Hilbert correspondence | Statement: [Noncommutative Geometry, Quantum Fields and Motives, topic, Riemann–Hilbert correspondence]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Riemann–Hilbert correspondence
Context triple: [Noncommutative Geometry, Quantum Fields and Motives, topic, Riemann–Hilbert correspondence]
  • A. Beilinson–Drinfeld Grassmannian
    The Beilinson–Drinfeld Grassmannian is a geometric object in algebraic geometry and representation theory that generalizes the affine Grassmannian to configurations of multiple points, playing a central role in the geometric Langlands program.
  • B. Hodge–Riemann bilinear relations
    The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
  • C. 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.
  • D. Hitchin fibration
    The Hitchin fibration is a fundamental geometric structure in the theory of Higgs bundles that organizes their moduli space into an algebraically completely integrable system with deep connections to representation theory and the geometric Langlands program.
  • E. Picard–Lefschetz theory
    Picard–Lefschetz theory is a branch of algebraic and symplectic geometry that studies how the topology of complex algebraic varieties changes under deformation, particularly via vanishing cycles and monodromy around singularities.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Riemann–Hilbert correspondence
Triple: [Noncommutative Geometry, Quantum Fields and Motives, topic, Riemann–Hilbert correspondence]
Generated description
The Riemann–Hilbert correspondence is a fundamental result in mathematics that establishes an equivalence between certain differential equations (or flat connections) on complex manifolds and representations of their fundamental groups, linking analytic and topological data.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Riemann–Hilbert correspondence
Target entity description: The Riemann–Hilbert correspondence is a fundamental result in mathematics that establishes an equivalence between certain differential equations (or flat connections) on complex manifolds and representations of their fundamental groups, linking analytic and topological data.
  • A. Beilinson–Drinfeld Grassmannian
    The Beilinson–Drinfeld Grassmannian is a geometric object in algebraic geometry and representation theory that generalizes the affine Grassmannian to configurations of multiple points, playing a central role in the geometric Langlands program.
  • B. Hodge–Riemann bilinear relations
    The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
  • C. 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.
  • D. Hitchin fibration
    The Hitchin fibration is a fundamental geometric structure in the theory of Higgs bundles that organizes their moduli space into an algebraically completely integrable system with deep connections to representation theory and the geometric Langlands program.
  • E. Picard–Lefschetz theory
    Picard–Lefschetz theory is a branch of algebraic and symplectic geometry that studies how the topology of complex algebraic varieties changes under deformation, particularly via vanishing cycles and monodromy around singularities.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69d6ab4669e48190b59246358b0383ab completed April 8, 2026, 7:23 p.m.
NER Named-entity recognition batch_69d903f02638819091e0cc0e93fa5ea7 completed April 10, 2026, 2:06 p.m.
NED1 Entity disambiguation (via context triple) batch_69f48b8111b88190a42a8904a2d26862 completed May 1, 2026, 11:16 a.m.
NEDg Description generation batch_69f48fc7a8848190a06b34cc45db4789 completed May 1, 2026, 11:34 a.m.
NED2 Entity disambiguation (via description) batch_69f495f069c48190a6e5856c272420c0 completed May 1, 2026, noon
Created at: April 8, 2026, 9:47 p.m.