[Update January 2015: I see that many people reach this modest blog post in search of information about the solution of the Kadison-Singer conjecture, so I figured that it would be a good service to immediately direct them away to better sources:
There are two very recent papers that I have not read yet, but I trust:
The solution to the Kadison-Singer problem: Yet another presentation, by Dan Timotin (recommended to me by friends).
Consequences of the Marcus/Spielman/Srivastava solution of the Kadison-Singer problem, by P. Casazza and J. Tremain.
and there is Terry Tao’s post on this subject that I read and recommend.
Best regards, Orr]
Boom. In the arxiv mailing list of a few days ago appeared the following paper: “Interlacing Families II: Mixed Characteristic Polynomials and The Kadison-Singer Problem” (Markus, Spielman and Srivastava). The abstract says:
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 a projection paving conjecture of Akemann and Anderson. 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.
From the abstract it might not be immediately clear that this paper claims to solve the Kadison-Singer problem, because it says that their result implies KS via another conjecture; what they mean, however, is that the conjecture they prove was proven to be equivalent to another conjecture which has already been shown in the past to be equivalent to a positive solution to the Kadison-Singer problem.