In this post, I am going to give you a simple, self-contained, and fruitful demonstration of a recently introduced proof technique called the method of interlacing families of polynomials, which was also mentioned in an earlier post. This method, which may be seen as an incarnation of the probabilistic method, is relevant in situations when … Continue reading Restricted Invertiblity by Interlacing Polynomials

Discrepancy theory seeks to understand how well a continuous object can be approximated by a discrete one, with respect to some measure of uniformity. For instance, a celebrated result due to Joel Spencer says that given any set family $latex {S_1,\ldots,S_n\subset [n]}&fg=000000$, it is possible to color the elements of $latex {[n]}&fg=000000$ Red and Blue … Continue reading Discrepancy, Graphs, and the Kadison-Singer Problem