Duistermaat–Heckman formula
E895659
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Duistermaat–Heckman formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10946697 — 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: Duistermaat–Heckman formula Context triple: [Atiyah–Bott fixed-point theorem, relatedTo, Duistermaat–Heckman formula]
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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.
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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.
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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.
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
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in symplectic geometry ⓘ |
| appliesTo |
Hamiltonian actions of compact Lie groups on symplectic manifolds
ⓘ
Hamiltonian torus actions ⓘ |
| assumes | compactness conditions on the symplectic manifold or properness of the moment map ⓘ |
| characterizes | pushforward of symplectic volume under the moment map ⓘ |
| connectedTo |
Atiyah–Bott localization theorem
NERFINISHED
ⓘ
Berline–Vergne localization formula NERFINISHED ⓘ Kirwan convexity theorem NERFINISHED ⓘ moment polytope ⓘ |
| describes | behavior of the pushforward of the Liouville measure under a moment map ⓘ |
| field |
Hamiltonian group actions
ⓘ
equivariant cohomology ⓘ symplectic geometry ⓘ |
| hasApplicationIn |
geometric quantization
ⓘ
integrable systems ⓘ mathematical physics ⓘ representation theory of compact Lie groups ⓘ |
| implies |
density of the pushforward measure is polynomial on each chamber of regular values of the moment map
ⓘ
jumps in the density occur when crossing walls of singular values of the moment map ⓘ |
| inspired | developments in equivariant localization techniques ⓘ |
| involvesConcept |
Fourier transform of symplectic volume
ⓘ
Liouville measure NERFINISHED ⓘ coadjoint orbits ⓘ equivariant localization ⓘ fixed points of group actions ⓘ moment map ⓘ piecewise polynomial density ⓘ symplectic volume ⓘ |
| language | mathematical English ⓘ |
| namedAfter |
Gert Heckman
NERFINISHED
ⓘ
Johannes J. Duistermaat NERFINISHED ⓘ |
| originalAuthors |
Gert Heckman
NERFINISHED
ⓘ
Johannes J. Duistermaat NERFINISHED ⓘ |
| originalPublication | Acta Mathematica NERFINISHED ⓘ |
| relates |
geometry of moment map to combinatorics of polytopes
ⓘ
symplectic geometry to equivariant cohomology ⓘ symplectic geometry to representation theory ⓘ |
| states | the pushforward of the Liouville measure by the moment map has a piecewise polynomial density ⓘ |
| typeOf | localization formula ⓘ |
| usedFor |
computing distributions of values of the moment map
ⓘ
computing symplectic volumes of reduced spaces ⓘ studying Hamiltonian torus actions on compact symplectic manifolds ⓘ studying symplectic reduction ⓘ |
| yearProved | 1982 ⓘ |
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Subject: Duistermaat–Heckman formula Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.