Floer theory

E911359

branch of low-dimensional topology branch of symplectic geometry mathematical theory

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.

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Observed surface forms (1)

Surface form	Occurrences
Floer homology	1

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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Morse Theory → hasVariant → Floer theory ⓘ

subject surface form: Morse theory

Morse Theory → inspired → Floer theory ⓘ

subject surface form: Morse theory

this entity surface form: Floer homology