Abstract: A linear subspace of a space of matrices is said to be of bounded rank r, if every matrix in the space has rank at most r and some matrix has rank equal to r. The study of such spaces dates back at least to Marcus and Flanders in 1962. In 1997 Eisenbud and Harris observed that the question could be studied from the perspective of algebraic geometry. I will describe very recent progress on this study with JM Landsberg where we combine classical and algebro-geometric techniques. If time permits, I will explain our motivation from questions in theoretical computer science and quantum information theory.
