The celebrated proof of the Kadison-Singer problem in 2013 by Adam Marcus, Daniel Spielman and Nikhil Srivastava involved a variety of spectacular insights. In this talk, I will show how three of their innovations: An algebraic technique called the Method of Interlacing polynomials, a combinatorial technique to calculate expected characteristic polynomials of random matrices and an analytic technique called the Multivariate Barrier method can be generalized, leading notonly to better quantitative estimates but also some intriguing open questions. I will in particular, mention applications to probability and convex geometry. The talk will be self-contained.