A toric origami manifold is a generalization of a symplectic toric manifold, where the origami symplectic form is allowed to degenerate in a good controllable way.
As symplectic toric manifolds are encoded by Delzant polytopes, toric origami manifolds also bijectively correspond to special combinatorial structures, called origami templates, via moment maps. To describe the cohomology ring and T-equivariant cohomology ring of a toric origami manifold in terms of the corresponding origami template is a problem of certain interest. In this talk, I will present some partial results concerning this problem. This talk is based on a joint work with A. Ayzenberg, M. Masuda, and H. Zeng.