Mohan Ravichandran - Mimar Sinan University I'll talk about how the method of interlacing polynomials can be used to show the existence of common pavings of asymptotically optimal size for finite collections of matrices. This complements Michael Cohen's extension of the main theorem of Marcus, Spielman and Srivastava to higher rank random matrices. As an application, I will present a simplified proof of a strengthening of Johnson, Ozawa and Schechtman's quantitative commutator theorem for zero trace matrices. This is joint work with Nikhil Srivastava.