Floer theory
E911359
Floer theory is a branch of symplectic geometry and low-dimensional topology that extends Morse-theoretic ideas to infinite-dimensional spaces, providing powerful tools for studying periodic orbits, Lagrangian intersections, and invariants such as Floer homology.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Floer homology | 1 |
| Floer theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219562 — 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: Floer theory Context triple: [Morse theory, hasVariant, Floer theory]
-
A.
Arnold conjecture
The Arnold conjecture is a central statement in symplectic geometry predicting a lower bound on the number of fixed points of Hamiltonian diffeomorphisms in terms of the topology of the underlying manifold.
-
B.
Morse Theory
Morse Theory is a branch of differential topology that studies the relationship between the topology of manifolds and the critical points of smooth real-valued functions defined on them.
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C.
Introduction to Symplectic Topology
Introduction to Symplectic Topology is a foundational graduate-level textbook that systematically develops the theory and applications of symplectic manifolds and symplectic geometry.
-
D.
McDuff–Salamon theory of J-holomorphic curves
The McDuff–Salamon theory of J-holomorphic curves is a foundational framework in symplectic geometry that systematically develops the analysis, topology, and applications of pseudoholomorphic curves in symplectic manifolds.
-
E.
Lefschetz fibration
A Lefschetz fibration is a smooth map from a higher-dimensional manifold to a lower-dimensional one whose singularities are modeled on complex Morse-type critical points, playing a central role in symplectic and complex geometry.
- 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: Floer theory Target entity description: Floer theory is a branch of symplectic geometry and low-dimensional topology that extends Morse-theoretic ideas to infinite-dimensional spaces, providing powerful tools for studying periodic orbits, Lagrangian intersections, and invariants such as Floer homology.
-
A.
Arnold conjecture
The Arnold conjecture is a central statement in symplectic geometry predicting a lower bound on the number of fixed points of Hamiltonian diffeomorphisms in terms of the topology of the underlying manifold.
-
B.
Morse Theory
Morse Theory is a branch of differential topology that studies the relationship between the topology of manifolds and the critical points of smooth real-valued functions defined on them.
-
C.
Introduction to Symplectic Topology
Introduction to Symplectic Topology is a foundational graduate-level textbook that systematically develops the theory and applications of symplectic manifolds and symplectic geometry.
-
D.
McDuff–Salamon theory of J-holomorphic curves
The McDuff–Salamon theory of J-holomorphic curves is a foundational framework in symplectic geometry that systematically develops the analysis, topology, and applications of pseudoholomorphic curves in symplectic manifolds.
-
E.
Lefschetz fibration
A Lefschetz fibration is a smooth map from a higher-dimensional manifold to a lower-dimensional one whose singularities are modeled on complex Morse-type critical points, playing a central role in symplectic and complex geometry.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
branch of low-dimensional topology
ⓘ
branch of symplectic geometry ⓘ mathematical theory ⓘ |
| appliesTo |
Hamiltonian diffeomorphisms
ⓘ
pairs of Lagrangian submanifolds ⓘ symplectic manifolds ⓘ |
| basedOn | Morse theory NERFINISHED ⓘ |
| coreConcept |
action functional on loop space
ⓘ
continuation maps ⓘ moduli spaces of solutions to Floer equations ⓘ spectral invariants ⓘ transversality and perturbations ⓘ |
| defines |
Floer chain complex
NERFINISHED
ⓘ
Floer differential NERFINISHED ⓘ Floer homology groups NERFINISHED ⓘ |
| developedBy | Andreas Floer NERFINISHED ⓘ |
| extends | Morse theory to infinite-dimensional spaces ⓘ |
| fieldOfStudy |
Hamiltonian dynamics
ⓘ
Morse theory ⓘ low-dimensional topology ⓘ symplectic geometry ⓘ |
| hasVariant |
Hamiltonian Floer homology
ⓘ
Lagrangian Floer homology NERFINISHED ⓘ instanton Floer homology ⓘ monopole Floer homology ⓘ symplectic Floer homology ⓘ |
| historicalPeriod | late 20th century ⓘ |
| inspired |
Heegaard Floer homology
NERFINISHED
ⓘ
embedded contact homology ⓘ symplectic field theory NERFINISHED ⓘ |
| mainTool | Floer homology NERFINISHED ⓘ |
| namedAfter | Andreas Floer NERFINISHED ⓘ |
| notableResult | proofs of cases of the Arnold conjecture for symplectic fixed points ⓘ |
| relatedTo |
Gromov–Witten theory
NERFINISHED
ⓘ
Heegaard Floer homology NERFINISHED ⓘ Seiberg–Witten theory NERFINISHED ⓘ instanton Floer homology ⓘ monopole Floer homology ⓘ |
| studies |
Lagrangian intersections
ⓘ
periodic orbits of Hamiltonian systems ⓘ pseudo-holomorphic curves ⓘ symplectic invariants ⓘ |
| usedFor |
Arnold conjecture
NERFINISHED
ⓘ
Weinstein conjecture NERFINISHED ⓘ classification of symplectic manifolds ⓘ construction of invariants of 3-manifolds ⓘ construction of invariants of 4-manifolds ⓘ |
| uses |
Fredholm theory
NERFINISHED
ⓘ
compactness results for moduli spaces ⓘ elliptic partial differential equations ⓘ gradient flow lines of an action functional ⓘ |
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: Floer theory Description of subject: Floer theory is a branch of symplectic geometry and low-dimensional topology that extends Morse-theoretic ideas to infinite-dimensional spaces, providing powerful tools for studying periodic orbits, Lagrangian intersections, and invariants such as Floer homology.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Morse theory
subject surface form:
Morse theory
this entity surface form:
Floer homology