String theory predicts the existence of several spatial dimensions in addition to the three of our everyday experience. The space spanned by these dimensions might be small enough to have escaped detection so far.The aim of this project is to characterize which spaces are allowed by the dynamics of the theory, and what physics they give rise to. A few years ago, I discovered a reformulation of supergravity in terms of differential forms, based on the so-called generalized complex geometry. This method was originally limited to string theory vacuum solutions, and over the years it has permitted to find many of them, often with applications to AdS/CFT. Recently, I was able to extend it to deal with any kind of spacetime dependence; this will allow to probe the choice of extra dimensions more extensively, for example by studying black hole solutions. It will help single out interesting geometries for the extra dimensions, even before one sets out to understand the effective four-dimensional Lagrangian that would result from compactifying string theory on it.Moreover, I plan to extend the method even further, to deal with controlled supersymmetry breaking. That would open the possibility of producing systematically vacuum solutions which have a positive cosmological constant. The vacua obtained in this way would be fully classical, and under better control than current models.