Poincaré conjecture
E156188
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Poincaré Conjecture | 5 |
| Poincaré conjecture canonical | 5 |
| Clay Millennium Prize Problem on the Poincaré conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358646 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Poincaré conjecture Context triple: [Henri Poincaré, notableWork, Poincaré conjecture]
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A.
Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
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B.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
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C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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D.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
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E.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Poincaré conjecture Target entity description: The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
A.
Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
-
B.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
-
C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
D.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
-
E.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
topology conjecture ⓘ |
| concerns |
3-manifolds
ⓘ
three-dimensional sphere ⓘ topological characterization of the 3-sphere ⓘ |
| difficulty | famously difficult problem ⓘ |
| dimension | 3 ⓘ |
| field |
differential topology
ⓘ
geometric topology ⓘ topology ⓘ |
| fullyResolvedIn | early 21st century ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| impliedBy | geometrization conjecture ⓘ |
| importance |
central result in 3-manifold theory
ⓘ
first solved Millennium Prize Problem ⓘ landmark problem in topology ⓘ |
| influenced |
development of geometric analysis
ⓘ
study of Ricci flow ⓘ |
| involvesConcept |
3-sphere S^3
ⓘ
closed manifold ⓘ fundamental group ⓘ homology ⓘ homotopy ⓘ simply connected space ⓘ |
| millenniumPrizeDecision | prize declined by Grigori Perelman ⓘ |
| millenniumPrizeStatus | solution verified ⓘ |
| namedAfter | Henri Poincaré ⓘ |
| originallyFormulatedBy | Henri Poincaré ⓘ |
| originalPublicationYear | 1904 ⓘ |
| partOf |
Millennium Prize Problem
ⓘ
surface form:
Millennium Prize Problems
|
| proofPublishedAs | series of preprints on arXiv ⓘ |
| recognizedBy | Clay Mathematics Institute ⓘ |
| relatedConjecture | geometrization conjecture ⓘ |
| solutionBasedOn |
Ricci flow
ⓘ
surface form:
Richard S. Hamilton's Ricci flow program
|
| solutionMethod | Ricci flow with surgery ⓘ |
| solutionYears |
2002
ⓘ
2003 ⓘ 2004 ⓘ |
| solvedBy | Grigori Perelman ⓘ |
| statementForm | Every closed simply connected 3-manifold is homeomorphic to the 3-sphere ⓘ |
| status | proved ⓘ |
| subfield | 3-manifold theory ⓘ |
| topic |
homeomorphism
ⓘ
homotopy 3-sphere ⓘ simply connected manifolds ⓘ topological equivalence ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Poincaré conjecture Description of subject: The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré Conjecture
this entity surface form:
Poincaré Conjecture
this entity surface form:
Poincaré Conjecture
this entity surface form:
Poincaré Conjecture
this entity surface form:
Poincaré Conjecture
this entity surface form:
Clay Millennium Prize Problem on the Poincaré conjecture