Given a Hamiltonian action of a Lie group $G$ on a symplectic manifold $(M, \omega)$, you can understand $M$ via its image under the moment map $\mu\colon M\to \mathfrak{g}^*$, which is sometimes a rational convex polytope.

A natural question is to what spaces can be associated non-rational moment polytopes. We answer this by describing Hamiltonian stacks, which are built by taking the stacky quotient of a presymplectic manifold by its null foliation. Hamiltonian stacks come with an action of a Lie group stack $\mathcal{G}$, and a moment map taking values in the dual of the lie algebra of $\mathcal{G}$.

After developing the basic theory we construct the symplectic reduction of a Hamiltonian stack, and extend the Duistermaat-Heckman theorem.

This work is joint with Reyer Sjamaar.