In mathematics, the Duistermaat–Heckman formula, due to , states that the
pushforward of the canonical (Liouville) measure on a symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

 under the moment map

Moment map

In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum...

 is a piecewise polynomial measure. Equivalently, the Fourier transform of the canonical measure is given exactly by the stationary phase approximation.

and, independently, showed how to deduce the Duistermaat–Heckman formula from a localization theorem for equivariant cohomology

Equivariant cohomology

In mathematics, equivariant cohomology is a theory from algebraic topology which applies to spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory....

.