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These lecture notes are meant to accompany two lectures given at the CDM 2016 conference, about the Kadison-Singer Problem. They are meant to complement the survey by the same authors (along with Spielman) which appeared at the 2014 ICM. In the first part of this survey we will introduce the Kadison-Singer problem from two perspectives ($C^*$ algebras and spectral graph theory) and present some examples showing where the difficulties in solving it lie. In the second part we will develop the framework of interlacing families of polynomials, and show how it is used to solve the problem. None of the results are new, but we have added annotations and examples which we hope are of pedagogical value.
We consider a two-sided Pompeiu type problem for a discrete group $G$. We give necessary and sufficient conditions for a finite set $K$ of $G$ to have the $mathcal{F}(G)$-Pompeiu property. Using group von Neumann algebra techniques, we give necessary
We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${rho}_{1}^{(r)}(n)=(2rn)!$ and ${rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ sa
Leibniz combinatorial formula for determinants is modified to establish a condensed and easily handled compact representation for Hessenbergians, referred to here as Leibnizian representation. Alongside, the elements of a fundamental solution set ass
In this note, we give a short solution of the kissing number problem in dimension three.
We characterize the projectors $ P $ on a Banach space $ E $ having the property of being connected to all the others projectors obtained as a conjugation of $ P $. Using this characterization we show an example of Banach space where the conjugacy cl