Triple

T3821387
Position Surface form Disambiguated ID Type / Status
Subject Atiyah–Singer index theorem E84379 entity
Predicate usesConcept P531 FINISHED
Object Chern character
The Chern character is a fundamental homomorphism from K-theory to cohomology that translates vector bundles into characteristic classes, playing a central role in index theory and algebraic topology.
E391904 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: Chern character | Statement: [Atiyah–Singer index theorem, usesConcept, Chern character]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Chern character
Context triple: [Atiyah–Singer index theorem, usesConcept, Chern character]
  • A. Chern classes
    Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
  • 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. Characteristic Classes
    Characteristic Classes is a foundational mathematical text in differential topology and geometry that systematically develops the theory of characteristic classes for vector bundles and fiber bundles.
  • D. Hirzebruch–Riemann–Roch theorem
    The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
  • E. 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.
  • 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: Chern character
Triple: [Atiyah–Singer index theorem, usesConcept, Chern character]
Generated description
The Chern character is a fundamental homomorphism from K-theory to cohomology that translates vector bundles into characteristic classes, playing a central role in index theory and algebraic topology.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Chern character
Target entity description: The Chern character is a fundamental homomorphism from K-theory to cohomology that translates vector bundles into characteristic classes, playing a central role in index theory and algebraic topology.
  • A. Chern classes
    Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
  • 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. Characteristic Classes
    Characteristic Classes is a foundational mathematical text in differential topology and geometry that systematically develops the theory of characteristic classes for vector bundles and fiber bundles.
  • D. Hirzebruch–Riemann–Roch theorem
    The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
  • E. 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.
  • 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_69aed931f5908190be2c07af66d4df25 completed March 9, 2026, 2:29 p.m.
NER Named-entity recognition batch_69aeea62cdfc81909a3bf458b73d60e7 completed March 9, 2026, 3:42 p.m.
NED1 Entity disambiguation (via context triple) batch_69b4fb4998248190b4174dd80a8e790c completed March 14, 2026, 6:08 a.m.
NEDg Description generation batch_69b4ffcf7e24819098cf2e46b92bed4a completed March 14, 2026, 6:27 a.m.
NED2 Entity disambiguation (via description) batch_69b500596e308190a31e44c24de3f31d completed March 14, 2026, 6:29 a.m.
Created at: March 9, 2026, 3:17 p.m.