Category Archives: Expository

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

Expository, Fun stuff, Functional equations, Thoughts on mathematics

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

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

and more generally

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

(where $latex delta_1$ and $latex delta_2$ satisfy some additional conditions) and my long journey to discover that these equations have, and now I will give it away… Continue reading Where have all the functional equations gone (the end of the story and the lessons I’ve learned) →

Where have all the functional equations gone (part III)

Expository, Functional equations, Paneah, Thoughts on mathematics

The last post ended with the following problem:

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

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

In the previous post I explained why all continuously differentiable solutions of the functional equation (FE) are linear, that is, of the form $latex f(x) = cx$, 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 $latex C^1[-1,1]$ to the much larger $latex C[-1,1]$, then new solutions will appear. On the other hand, the dynamical system affiliated with this problem (the dynamical space generated by the maps $latex delta_1(t) = frac{t+1}{2}$ and $latex delta_2(t) = frac{t-1}{2}$ on the space $latex [-1,1]$) 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.

Continue reading Where have all the functional equations gone (part III) →

Where have all the functional equations gone (part II)

Dynamical systems, Expository, Fun stuff, Functional equations, Thoughts on mathematics,

I’ll start off exactly where I stopped in the previous post: I will tell you my solution to the problem my PDEs lecturer (and later master’s thesis advisor) Paneah gave us:

Problem: Find all continuously differentiable solutions to the following functional equation:

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

Before writing a solution, let me say that I think it is a fun exercise for undergraduate students, and only calculus is required for solving it, so if you want to try it now is your chance.

Continue reading Where have all the functional equations gone (part II) →

Where have all the functional equations gone (part I)

Expository, Functional equations, Paneah, Thoughts on mathematics

My first encounter with research mathematics was in the last term of my undergraduate studies (spring 2003). My professor in the course “Introduction to Partial Differential Equations”, Prof. Boris Paneah, thought that it is pointless to give standard homework problems to students of pure mathematics, and instead he gave us several problems which were either extremely challenging, related to his research or related to advanced courses that he was going to give. This was a thrilling experience for me, and is one of the reasons why I decided not long after to do my master’s thesis under his supervision, since no other faculty member came even close to engaging us like Paneah (another reason was that the lectures themselves were fantastic). For example he suggested that we explore the ultrahyperbolic equation

$latex u_{tt} + u_{ss} – u_{xx} – u_{yy} = 0 , $    in     $latex mathbb{R}^4$,

or that we try to prove the existence of solutions to the two dimensional heat equation in a non-rectangular bounded region of the plane. I remember spending hours on the heat equation, unsuccessfully of course (if I was successful I would have probably become a PDE person). Especially memorable is the one time that he ended a lecture with the following three problems, which were, as you may guess, quite unrelated to the content of the lecture: Continue reading Where have all the functional equations gone (part I) →

Arveson memorial article

Arveson, Expository, memorial, , ,

Palle Jorgensen and Daniel Markiewicz have put together a beautiful tribute to the late Bill Arveson, with contributions from about a dozen mathematicians as well as a more personal piece by Lee Ann Kaskutas. This memorial article might appear later elsewhere in shorter form, but I think it would be interesting for many people to see the full tribute, with all the various points of view and pieces of life that it contains. I wrote a post dedicated to Arveson’s memory about half a year ago, where I put links to two recent surveys (1 by Davidson and and 2 by Izumi), and in about a month there will be a big conference in Berkeley dedicated to Arveson’s legacy; still I feel that this tribute really fills a hole, and conveys in broader, fuller way what a remarkable mathematician he was, and how impacted so many so strongly. Please share the link with people who might be interested.

Forty five years later, a major open problem in operator algebras is solved

Arveson, Expository, Operator algebras

A couple of days ago, Ken Davidson and Matt Kennedy posted a preprint on the arxiv, “The Choquet boundary of an operator system“. In this paper they solve a major open problem in operator algebras, showing that every operator system has sufficiently many boundary representations. 

In 1969, William Arveson published the seminal paper, [“Subalgebras of C*-algebras”, Acta Math. 123, 1969], which is one of the cornerstones, (if not the cornerstone) of the theory of operator spaces and nonself-adjoint operator algebras. In that paper, among other things, Arveson introduced and put to good use the notion of a boundary representation. I wrote on “Subalgebras of C*-algebras” in a previous post dedicated to Arveson, and for some background material the reader is invited to look into that old post. I did not, however, write much about boundary representations (because I was emphasizing his contributions rather what he has left open). Below I wish to explain what are boundary representations, what does it mean that there are sufficiently many of these, and where Davidson and Kennedy’s new results fits in the chain of results leading to the solution of the problem. The paper itself is accessible to anyone who understands the problem, and the main ideas are clearly presented in its introduction.

Continue reading Forty five years later, a major open problem in operator algebras is solved →

Mathematics on mathematics

Expository, Thoughts on mathematics, Uncategorized

 

This post is the outline of a talk (or perhaps the talk is an outline of this post) that I will give on February 28 on our “open day” to prospective students in our department. This is supposed to be a story, it is intended to give a flavor, and neither the history nor the math are 100% precise, because it is a 15 minute talk! The big challenge is to take some rough ideas from this post, throw away the rest, and make that into an interesting quarter of an hour lecture. Comments are very welcome.

1. Introduction

What is mathematics? I am not going to answer that. You have all met mathematics in your life, in school, but also in other places as well, because math is everywhere. So you have some kind of idea what mathematics is about. However, I suspect that the most profound aspects of math have been hidden from you. I am here today to try to give you a taste of this mathematics which you have not yet seen. It is only fair to let you know that — for better and for worse — the mathematician’s mathematics, the mathematics that you will study if you do an undergraduate degree in math, is of a dramatically different nature from the math you learn in high-school or the math-is-everywhere kind of math which you meet in various popular accounts.

You have met various different kinds of mathematics: combinatorics, geometry, algebra, integral and differential calculus (aka HEDVA — which literally means “joy” in Hebrew). It seems as if mathematics splits into various branches, where in each branch there are different tasks that one should do. The objects of study of geometry are triangles, circles, trapezoids, etc; one has to prove that a certain triangle has this or that property, or one has to compute some angle or length or area. The objects of study in algebra are certain symbolic expressions or equations; one has to find the root of an equation, or to simplify an expression. In HEDVA the objects of study are functions; one has to compute the minimum of a function, or its anti-derivative, and so on.

The theme of this talk is that the objects of study in mathematics do not have to be only triangles or functions or equations, but they can also be geometry or analysis or algebra. Mathematics can also be used to study mathematics itself. This is profound. But perhaps more surprisingly, this has practical consequences.

Of course, I have no time to tell you precisely how this works. For this, I recommend that you come here and study mathematics. Continue reading Mathematics on mathematics →

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

d shift space, Expository, Open problem, Operator theory, Research, Talks, ,

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 $latex H^2_d$.

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

Matthew Kennedy awarded CMS 2012 Doctoral Prize

Awards, Expository, Operator theory

One of the nicest things that happen on math blogs is that people write expository posts on other people’s work. Tim Gowers and Terry Tao have set a fine example in their expositions of the works of Fields Medalists or Abel Prize laureates. These are among the most interesting and important posts out there, I think.

Recently, at the Canadian Mathematical Society’s Winter Meeting, Matthew Kennedy was awarded the CMS 2012 Doctoral Prize (at the meeting several other prizes were awarded by the CMS as you can see on the meeting’s homepage). See here for the media release, and here for a description of the prize and a list of past recipients.

Matt gave a plenary lecture at the CMS meeting surveying (some of) his work. Here are the slides, which are certainly worth looking at (thanks to Matt for allowing me to post them). Note that on the last slide there are photos of two gentlemen with no captions; these are Ken Davidson (on the left, Matt’s PhD supervisor) and Heydar Radjavi (Matt’s undergraduate research supervisor).

In this post I will describe a couple of Matt’s first really big results, one of which didn’t make it into his talk. These are the existence of wandering vectors for (certain) free semigroup algebras, and the reflexitivity of free semigroup algebras. These result appeared in the tour-de-force paper “Wandering vectors and the reflexivity of free semigroup algebras”; here are links to arxiv, mathscinet, and the official version in Crelle.  Continue reading Matthew Kennedy awarded CMS 2012 Doctoral Prize →

The remarkable Hilbert space H^2 (Part II – multivariable operator theory and model theory)

d shift space, Expository, Operator theory, Talks,

This post is the second post in the series of posts on the d–shift space, a.k.a. the Drury–Arveson space, a.k.a. $latex H^2_d$ (see this previous post about the space $latex H^2$).

Continue reading The remarkable Hilbert space H^2 (Part II – multivariable operator theory and model theory) →