Category: Open problem

Another one bites the dust (actually many of them)

[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.

Blog posts on the solution appeared here and here, with links to excellent references. I will add here a few remarks of my own.

The remarkable Hilbert space H^2 (part III – three open problems)

This is the last in the series of three posts on the d–shift space, which accompany/replace the colloquium talk I was supposed to give. The first two parts are available here and here. In this post I will discuss three open problems that I have been thinking about, which are formulated within the setting of $H^2_d$.