I am a bit suprised and disappointed to see that the online maths communities I lurk around seem largely oblivious to this recent preprint 1306.3969. Here is the abstract: the added emphasis is mine.
We use the method of interlacing families of polynomials to prove Weaver’s conjecture KS2, which is known to imply a positive solution to the Kadison-Singer problem via Anderson’s Paving Conjecture. Our proof goes through an analysis of the largest roots of a family of polynomials that we call the “mixed characteristic polynomials” of a collection of matrices.
(A few years ago, the 2nd and 3rd authors of that preprint recently made a dramatic improvement in our understanding of a theorem of Bourgain and Tzafriri, see arXiv 0911.1114. So this paper is certainly worth taking seriously at the very least.)
Over on G+, Willie Wong quite sensibly asked for some brief explanation of what the problem said, and why people care(d). I must confess that the full background to the Kadison-Singer conjecture/problem is well outside my area of technical expertise, possibly outside my area of competence. Nevertheless, I can at least link to this article by Casazza and Tremain, which mentions some other conjectures in functional analysis that are known to be equivalent to the Paving Conjecture, and hence (by work of Anderson) to the Kadison-Singer conjecture.
P. G. Casazza, J. C. Tremain. The Kadison–Singer Problem in mathematics and engineering. PNAS vol. 103 (2006) no. 7, 2032–2039
Here is a link to some web material for an AIM workshop on the Kadison-Singer problem, which may give the general audience some idea of work in recent years.
The paper of Weaver which the preprint refers to is:
The MathReview of Weaver’s paper, by P. J. Stacey, is short enough that it can be reproduced here:
In [Amer. J. Math. 81 (1959), 383–400; MR0123922 (23 #A1243)], R. V. Kadison and I. M. Singer asked if every pure state on an atomic maximal abelian subalgebra of B(H), the algebra of bounded operators on a separable Hilbert space H, extends uniquely to a pure state on B(H). Developing the approach in [C. A. Akemann and J. Anderson, Mem. Amer. Math. Soc. 94 (1991), no. 458, iv+88 pp.; MR1086563 (92e:46113)], the author formulates a combinatorial version of the Kadison-Singer problem, in terms of unit vectors in Ck. Some positive partial results are then obtained using discrepancy theory.
Perhaps I will keep this blog post updated with some more links, if anyone has suggestions. Though really it should be left to the operator theorists, operator algebraists, and combinatorists to write some expositions in the weeks to come.
I see there is some attention now that Terence Tao has mentioned this on G+ and thence on the Selected Papers Network. (I admit that when I mentioned the paper on G+, I didn’t tag it with #spnetwork, mainly because I didn’t feel I had anything intelligent to say at the time; and if this #spnetwork is to become useful to the community of research mathematicians, it needs less noise from spectators, and more commentary from people who understand some ideas in the papers under discussion!)
Gil Kalai has a blogpost which says a little more about how the paper of Marcus, Spielman and Srivastava relates to the previous results of Bourgain and Tzafriri, and mentions that Spielman and Srivastava had previously given a new proof – an improved proof? – of Bourgain-Tzafriri’s restricted invertibility theorem.
Orr Shallit has also picked up on this, and offers some thoughts from the perspective of an operator algebraist/operator theorist.