Triple

T2314514
Position Surface form Disambiguated ID Type / Status
Subject Michael Atiyah E51031 entity
Predicate notableWork P4 FINISHED
Object “K-Theory” (book with Friedrich Hirzebruch and others)
“K-Theory” is a foundational mathematical monograph co-authored by Michael Atiyah, Friedrich Hirzebruch, and others that systematically develops topological K-theory and its applications in geometry and topology.
E255577 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: “K-Theory” (book with Friedrich Hirzebruch and others) | Statement: [Michael Atiyah, notableWork, “K-Theory” (book with Friedrich Hirzebruch and others)]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: “K-Theory” (book with Friedrich Hirzebruch and others)
Context triple: [Michael Atiyah, notableWork, “K-Theory” (book with Friedrich Hirzebruch and others)]
  • 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. Grothendieck–Riemann–Roch theorem
    The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
  • C. Categories for the Working Mathematician
    Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
  • D. 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.
  • E. Grothendieck group
    The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic 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: “K-Theory” (book with Friedrich Hirzebruch and others)
Triple: [Michael Atiyah, notableWork, “K-Theory” (book with Friedrich Hirzebruch and others)]
Generated description
“K-Theory” is a foundational mathematical monograph co-authored by Michael Atiyah, Friedrich Hirzebruch, and others that systematically develops topological K-theory and its applications in geometry and topology.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: “K-Theory” (book with Friedrich Hirzebruch and others)
Target entity description: “K-Theory” is a foundational mathematical monograph co-authored by Michael Atiyah, Friedrich Hirzebruch, and others that systematically develops topological K-theory and its applications in geometry and topology.
  • 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. Grothendieck–Riemann–Roch theorem
    The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
  • C. Categories for the Working Mathematician
    Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
  • D. 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.
  • E. Grothendieck group
    The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic 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_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.