Operator scaling has wide applications in theoretical computer science including permanent approximation, signed rank lower bounds, linear matroid intersection, etc, and is recently receiving a lot of attentions due to its connection with non-commutative ranks and Brascamp Lieb inequality.  Recently, we use techniques in operator scaling to solve a basic open problem in frame theory named the Paulsen problem: given vectors u_1, …, u_n in R^d that are nearly Parseval and nearly equal norm, can we change the vectors slightly to v_1, …, v_n so that they become exactly Parseval and equal norm?  The fundamental question is whether the total change sum_i ||u_i – v_i||^2 need to depend on the number of vectors n.  We answer this question affirmatively.

In this talk, I will present our result on the Paulsen problem, and discuss our recent progress on the spectral analysis of operator scaling.