Noncommutative Analysis

Month: June, 2013

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.

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Where have all the functional equations gone (the end of the story and the lessons I’ve learned)

This will be the last of this series of posts on my love affair with functional equations (here are links to parts one, two and three).

1. A simple solution of the functional equation

In the previous posts, I told of how I came to know of the functional equations

(*)  f(t) = f\left(\frac{t+1}{2}\right) + f \left( \frac{t-1}{2}\right) \,\, , \,\, t \in [-1,1]

and more generally

(**) f(t) = f(\delta_1(t)) + f(\delta_2(t)) \,\, , \,\, t \in [-1,1]

(where \delta_1 and \delta_2 satisfy some additional conditions) and my long journey to discover that these equations have, and now I will give it away… Read the rest of this entry »

Where have all the functional equations gone (part III)

The last post ended with the following problem:

Problem: Find all continuous solutions to the following functional equation:

(FE) f(t) = f\left(\frac{t+1}{2} \right) + f \left(\frac{t-1}{2} \right) \,\, , \,\, t \in [-1,1] .

In the previous post I explained why all continuously differentiable solutions of the functional equation (FE) are linear, that is, of the form f(x) = cx, but now we remove the assumption that the solution be continuously differentiable and ask whether the same conclusion holds. I found this problem to be extremely interesting, and at this point I will only give away that I eventually solved it, but after five (!) years.

In principle, it is plausible that, when one enlarges the space of functions in which one is searching for a solution from C^1[-1,1] to the much larger C[-1,1], then new solutions will appear. On the other hand, the dynamical system affiliated with this problem (the dynamical space generated by the maps \delta_1(t) = \frac{t+1}{2} and \delta_2(t) = \frac{t-1}{2} on the space [-1,1]) is minimal, and therefore one expects the functional equation to be rigid enough to allow only for the trivial solutions (at least under some mild regularity assumptions). In short, a good case can be made in favor of either a conjecture that all the continuous solutions are linear or a conjecture that there might be new, nonlinear solutions.

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