Triple
T10946697
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Atiyah–Bott fixed-point theorem |
E258614
|
entity |
| Predicate | relatedTo |
P37
|
FINISHED |
| Object |
Duistermaat–Heckman formula
The Duistermaat–Heckman formula is a result in symplectic geometry that describes how the pushforward of the Liouville measure under a moment map behaves, showing it is piecewise polynomial and linking geometry with equivariant localization techniques.
|
E895659
|
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: Duistermaat–Heckman formula | Statement: [Atiyah–Bott fixed-point theorem, relatedTo, Duistermaat–Heckman formula]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Duistermaat–Heckman formula Context triple: [Atiyah–Bott fixed-point theorem, relatedTo, Duistermaat–Heckman formula]
-
A.
Strominger–Yau–Zaslow conjecture
The Strominger–Yau–Zaslow conjecture is a proposal in mirror symmetry stating that mirror pairs of Calabi–Yau manifolds can be understood as dual special Lagrangian torus fibrations, providing a geometric explanation of mirror symmetry.
-
B.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
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.
Rozansky–Witten theory
Rozansky–Witten theory is a three-dimensional topological quantum field theory associated with hyperkähler manifolds that yields invariants of 3-manifolds and links via holomorphic symplectic 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: Duistermaat–Heckman formula Triple: [Atiyah–Bott fixed-point theorem, relatedTo, Duistermaat–Heckman formula]
Generated description
The Duistermaat–Heckman formula is a result in symplectic geometry that describes how the pushforward of the Liouville measure under a moment map behaves, showing it is piecewise polynomial and linking geometry with equivariant localization techniques.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Duistermaat–Heckman formula Target entity description: The Duistermaat–Heckman formula is a result in symplectic geometry that describes how the pushforward of the Liouville measure under a moment map behaves, showing it is piecewise polynomial and linking geometry with equivariant localization techniques.
-
A.
Strominger–Yau–Zaslow conjecture
The Strominger–Yau–Zaslow conjecture is a proposal in mirror symmetry stating that mirror pairs of Calabi–Yau manifolds can be understood as dual special Lagrangian torus fibrations, providing a geometric explanation of mirror symmetry.
-
B.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
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.
Rozansky–Witten theory
Rozansky–Witten theory is a three-dimensional topological quantum field theory associated with hyperkähler manifolds that yields invariants of 3-manifolds and links via holomorphic symplectic 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_69d6aa8769b4819082bfe5e61b9017f0 |
completed | April 8, 2026, 7:20 p.m. |
| NER | Named-entity recognition | batch_69d770eaaea08190b06e508600d8a305 |
completed | April 9, 2026, 9:27 a.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69e23c3c885081908edcece772b2e759 |
completed | April 17, 2026, 1:57 p.m. |
| NEDg | Description generation | batch_69e24542b4f081909c97621f04da8ecc |
completed | April 17, 2026, 2:35 p.m. |
| NED2 | Entity disambiguation (via description) | batch_69e248f7f96481909fa6e6cd07891566 |
completed | April 17, 2026, 2:51 p.m. |
Created at: April 8, 2026, 9:23 p.m.