Triple

T2314516
Position Surface form Disambiguated ID Type / Status
Subject Michael Atiyah E51031 entity
Predicate notableWork P4 FINISHED
Object “Elliptic Operators and Compact Groups” (with Raoul Bott)
“Elliptic Operators and Compact Groups” (with Raoul Bott) is a seminal mathematical paper that develops the theory of equivariant index for elliptic operators under compact Lie group actions, laying key foundations for modern index theory and representation theory.
E255579 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: “Elliptic Operators and Compact Groups” (with Raoul Bott) | Statement: [Michael Atiyah, notableWork, “Elliptic Operators and Compact Groups” (with Raoul Bott)]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: “Elliptic Operators and Compact Groups” (with Raoul Bott)
Context triple: [Michael Atiyah, notableWork, “Elliptic Operators and Compact Groups” (with Raoul Bott)]
  • 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. 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.
  • C. L’intégration dans les groupes topologiques et ses applications
    L’intégration dans les groupes topologiques et ses applications is a foundational mathematical monograph by André Weil that develops the theory of integration on topological groups and explores its far-reaching applications in analysis and number theory.
  • D. The Poincaré-Birkhoff-Witt theorem in ring theory
    "The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
  • E. Produits tensoriels topologiques et espaces nucléaires
    "Produits tensoriels topologiques et espaces nucléaires" is a foundational 1953 doctoral thesis in functional analysis that introduced and developed the theory of nuclear spaces and topological tensor products.
  • 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: “Elliptic Operators and Compact Groups” (with Raoul Bott)
Triple: [Michael Atiyah, notableWork, “Elliptic Operators and Compact Groups” (with Raoul Bott)]
Generated description
“Elliptic Operators and Compact Groups” (with Raoul Bott) is a seminal mathematical paper that develops the theory of equivariant index for elliptic operators under compact Lie group actions, laying key foundations for modern index theory and representation theory.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: “Elliptic Operators and Compact Groups” (with Raoul Bott)
Target entity description: “Elliptic Operators and Compact Groups” (with Raoul Bott) is a seminal mathematical paper that develops the theory of equivariant index for elliptic operators under compact Lie group actions, laying key foundations for modern index theory and representation theory.
  • 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. 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.
  • C. L’intégration dans les groupes topologiques et ses applications
    L’intégration dans les groupes topologiques et ses applications is a foundational mathematical monograph by André Weil that develops the theory of integration on topological groups and explores its far-reaching applications in analysis and number theory.
  • D. The Poincaré-Birkhoff-Witt theorem in ring theory
    "The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
  • E. Produits tensoriels topologiques et espaces nucléaires
    "Produits tensoriels topologiques et espaces nucléaires" is a foundational 1953 doctoral thesis in functional analysis that introduced and developed the theory of nuclear spaces and topological tensor products.
  • 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_69a88b074b908190ae983dbca7757d88 completed March 4, 2026, 7:41 p.m.
NER Named-entity recognition batch_69abc61d41f88190983f8947667b4c7a completed March 7, 2026, 6:30 a.m.
NED1 Entity disambiguation (via context triple) batch_69ae895f5420819087b403e9772dce9a completed March 9, 2026, 8:48 a.m.
NEDg Description generation batch_69ae8af65eb88190b17d74e7411967cc completed March 9, 2026, 8:55 a.m.
NED2 Entity disambiguation (via description) batch_69ae8ba02cec8190917c0e17d3fedb0e completed March 9, 2026, 8:58 a.m.
Created at: March 4, 2026, 7:49 p.m.