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Register a new userThe Solution of the Kadison-Singer Problem
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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.
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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 and sufficient conditions for $G$ to be a $ell^2(G)$-Pompeiu group
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 $W^{(r)}_{1,2}(x)>0$ satisfying $int_{0}^{infty}x^{n}W^{(r)}_{1,2}(x)dx = {rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${rho}_{1,2}^{(r)}(n)$, such as the product ${rho}_{1}^{(r)}(n)cdot{rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.
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 associated with linear difference equations with variable coefficients of order $p$ are explicitly represented by $p$ banded Hessenbergian solutions, built up solely of the variable coefficients. This yields banded Hessenbergian representations for the elements both of the product of companion matrices and of the determinant ratio formula of the one-sided Greens function (Greens function for short). Combining the above results, the elements of the foregoing notions are endowed with compact representations formulated here by Leibnizian and nested sum representations. We show that the elements of the fundamental solution set can be expressed in terms of the first banded Hessenbergian fundamental solution, called principal determinant function. We also show that the Greens function coincides with the principal determinant function, when both functions are restricted to a fairly large domain. These results yield, an explicit and compact representation of the Greens function restriction along with an explicit and compact solution representation of the previously stated type of difference equations in terms of the variable coefficients, the initial conditions and the forcing term. The equivalence of the Greens function solution representation and the well known single determinant solution representation is derived from first principles. Algorithms and automated software are employed to illustrate the main results of this paper.
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 class of a projector splits into several path-connected components, and describe the conjugacy classes of projectors onto subspaces of $ ell_poplusell_q $ with $ p eq q $.
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