Abstract
We present some results on root bounds for convolutions of real stable polynomials and use these to get improved estimates in Kadison-Singer type problems. Combined with a simple trick that can yield control over both top and bottom eigenvalues, these can be used to prove some generalizations of results of Marcus, Spielman and Srivastava. Several problems will also be posed. This includes work done jointly with Jonathan Leake and Nikhil Srivastava.