|Abstract:|| Isogeometric analysis is an emerging technology that intends to advance towards the integration of CAD and CAE technologies. The idea is, invoking the isoparametric concept, to use the same kind of basis functions for the geometry description given by CAD and the test and trial spaces in the discrete problem to be solved by CAE. The functions most commonly used up to now are (rational) B-splines and their non-tensor-product generalizations, such as hierarchical splines. These are a set of splines with multi-level structure, that provide a very promising framework for adaptive simulation, due to their simple construction and their capability to deal with complicated geometries. By abandoning the isoparametric concept, it is possible to extend isogeometric analysis to the definition of discrete differential forms based on spline spaces. This kind of discrete spaces are useful for the construction of structure-preserving discretizations, with applications in computational electromagnetics and fluid mechanics, for instance. In some sense, they can be seen as a generalization of edge and face elements with higher continuity. In the first part of this talk I will review the definition and properties of isogeometric differential forms based on tensor-product B-splines. In the second part, I will extend their construction to multi-level hierarchical splines. In this case, simple numerical tests show that not all the hierarchical meshes guarantee the structure-preserving property. Using cohomology theory, I will provide sufficient and necessary conditions on the hierarchical mesh that ensure that the structure-preserving property is satisfied. This presentation is from joint works with Annalisa Buffa, John Evans, Giancarlo Sangalli, Michael Scott and Derek Thomas.