The Geometry of Positive Semidefinite Rank

Positive semidefinite rank is a generalization of regular and nonnegative matrix ranks. We are interested in the semialgebraic set and boundaries of the set of matrices of rank at most r and of psd rank at most k, where r and k are small. In particular, we describe the algebraic boundary of this semialgebraic set for r=3 and k=2 and conjecture a characterization of the boundary for r=k+1. I will also explain a geometric version of our conjecture in terms of polytopes and spectrahedral shadows. This talk is based on joint work with Elina Robeva and Richard Z. Robinson.