Triple

T10931404
Position Surface form Disambiguated ID Type / Status
Subject Nolan Wallach E258214 entity
Predicate hasWrittenWork P4 FINISHED
Object Harmonic Analysis on Homogeneous Spaces
Harmonic Analysis on Homogeneous Spaces is a mathematical monograph by Nolan Wallach that develops the theory of harmonic analysis and representation theory on Lie groups and their homogeneous spaces.
E893438 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: Harmonic Analysis on Homogeneous Spaces | Statement: [Nolan Wallach, hasWrittenWork, Harmonic Analysis on Homogeneous Spaces]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Harmonic Analysis on Homogeneous Spaces
Context triple: [Nolan Wallach, hasWrittenWork, Harmonic Analysis on Homogeneous Spaces]
  • A. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
    "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" is a foundational graduate-level textbook by Elias Stein that systematically develops modern harmonic analysis using real-variable techniques, emphasizing singular integrals, Littlewood–Paley theory, and oscillatory integral methods.
  • B. Harmonic Analysis and the Theory of Probability
    Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
  • C. Introduction to Abstract Harmonic Analysis
    Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
  • D. Three regularity results in harmonic analysis
    "Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
  • E. Paley–Wiener theorem for real reductive groups
    The Paley–Wiener theorem for real reductive groups is a fundamental result in harmonic analysis that characterizes the image of compactly supported smooth functions under the group Fourier transform in terms of holomorphic functions with specific growth and support conditions.
  • 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: Harmonic Analysis on Homogeneous Spaces
Triple: [Nolan Wallach, hasWrittenWork, Harmonic Analysis on Homogeneous Spaces]
Generated description
Harmonic Analysis on Homogeneous Spaces is a mathematical monograph by Nolan Wallach that develops the theory of harmonic analysis and representation theory on Lie groups and their homogeneous spaces.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Harmonic Analysis on Homogeneous Spaces
Target entity description: Harmonic Analysis on Homogeneous Spaces is a mathematical monograph by Nolan Wallach that develops the theory of harmonic analysis and representation theory on Lie groups and their homogeneous spaces.
  • A. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
    "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" is a foundational graduate-level textbook by Elias Stein that systematically develops modern harmonic analysis using real-variable techniques, emphasizing singular integrals, Littlewood–Paley theory, and oscillatory integral methods.
  • B. Harmonic Analysis and the Theory of Probability
    Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
  • C. Introduction to Abstract Harmonic Analysis
    Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
  • D. Three regularity results in harmonic analysis
    "Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
  • E. Paley–Wiener theorem for real reductive groups
    The Paley–Wiener theorem for real reductive groups is a fundamental result in harmonic analysis that characterizes the image of compactly supported smooth functions under the group Fourier transform in terms of holomorphic functions with specific growth and support conditions.
  • 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_69d6aa8769b4819082bfe5e61b9017f0 completed April 8, 2026, 7:20 p.m.
NER Named-entity recognition batch_69d770a062f481908beb76c6dbaeb6a6 completed April 9, 2026, 9:25 a.m.
NED1 Entity disambiguation (via context triple) batch_69e2175711d4819088f93bdf64ba4d3f completed April 17, 2026, 11:19 a.m.
NEDg Description generation batch_69e21e54e22881908726f088643edfd2 completed April 17, 2026, 11:49 a.m.
NED2 Entity disambiguation (via description) batch_69e21f1134d0819082fcc735da61d347 completed April 17, 2026, 11:52 a.m.
Created at: April 8, 2026, 9:23 p.m.