We will discuss the proof and applications of a recent result of Friedland and Youssef, who utilize the solution of the Kadison-Singer problem by Marcus, Spielman and Srivastava to deal with the following question: given an n by m matrix A, can we ignore but only a few of the columns of A and still approximate well how A acts on R^m?
Friedland and Youssef show that, through Kadison-Singer, one can provide an optimal bound for the number of retained columns in terms of the stable rank of A.

One application of this result is to John decompositions of the identity: Friedland and Youssef, improving/complementing previous results of Rudelson and Srivastava, show that, arbitrarily close to any convex body K in R^n (even not necessarily symmetric), we can find another convex body Q that has only linear in n contact points with its John ellipsoid. Speaker(s): Beatrice Vritsiou (University of Michigan)