We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. [BDWY11] in which they were used to answer questions regarding point configurations. In this work we derive near-optimal rank bounds for these matrices and use them to obtain asymptot...

In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation o...