The brain constructs and combines modular structures for flexible computation. I will describe recent progress in characterizing the rigid and low-dimensional nature of some of these representations, using theoretical approaches including fully unsupervised topological characterization of neural population codes. I will then discuss models of how these rigid and modular circuits can emerge, and how they can generate, with high capacity and high data-efficiency without rewiring recurrent circuitry, cognitive maps across different variables (e.g. spatial and non-spatial) as well as across varied input dimensions.