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In this article, recent results on the theory of intersection spaces and their cohomology groups are reviewed. The focus is on the construction of intersection spaces for non-isolated singularities and stratification depth greater than one as well as on the de Rham, sheaf theoretic and algebraic approaches towards intersection space cohomology. At the end, a list of open problems is provided.

In this note we collect some results in R^n about (p,q) Poincaré and Sobolev inequalities for differential forms obtained in a joint research with Franchi and Pansu. In particular, we focus to the case p=1. From the geometric point of view, Poincaré and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. As an application of the results obtained in the case p=1 we obtain Poincaré and Sobolev inequalities for Euclidean currents.

Lanzani & Stein consider a class of div-curl inequalities in de Rham's complex. In this note we examine the natural counterpart of that kind of inequalities for dierential forms in Heisenberg groups H1 and H2.

In the context of generalized geometry we first show how the Courant bracket helps to define connections with skew torsion and then investigate a five-dimensional invariant functional and its associated geometry. A Hamiltonian flow arising from this corresponds to a version of the Nahm equations using the Courant bracket, and we investigate the six-dimensional geometrical structure this describes., Comment: 26 pages; dedicated to the memory of S.S.Chern