Where have all the functional equations gone (part I)

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

   in     ,

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:

Arveson memorial article

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.

A sneaky proof of the maximum modulus principle

The April 2013 issue of the American Mathematical Monthly has just appeared, and with it my small note “A Sneaky Proof of the Maximum Modulus Principle”. Here is a link to the current issue on the journal’s website, and here is a link to a version of the paper on my homepage. As the title suggest, the note contains a new proof — which I find extremely cool — for the maximum modulus principle from the theory of complex variables. The cool part is that the proof is based on some basic linear algebra. The note is short and very easy, and I am not going to say anything more about the proof, except that it relates to some of my “real” research (the way in which it relates can be understood by reading the Note and its references).

I am writing this post not only to publicize this note, but also to record somewhere my explanation why I have been behaving in a sneaky fashion. Indeed, this is the first paper that I wrote which I did not post on the Arxiv. Why?

Unlike research journals, the American Mathematical Monthly is a journal which has, if I am not mistaken, actual subscribers. I mean real people, some of them perhaps old school (like myself), and I could see them waiting to receive their copy in the mailbox, and then when the new issue finally arrives they gently open the envelope — or perhaps they tear it open, depending on their custom — after which they sit down and browse through the fresh issue. I could believe that there are such persons (for I myself am such a person) that do not look at the online version of the journal even though they have access, because that would spoil their fun with the paper copy which is to arrive a few days later.

Now I wouldn’t like to spoil a small pleasure of a subscriber, somewhere out there. So I did not post the Note on the Arxiv, lest it pop up on somebody’s mailing list. “Oh, this I have already seen…”. I shall not be resposible for such spoilers! So I decided to keep my note relatively secret, putting it on my homepage, but putting off the Arxiv until the journal really gets published and all the physical copies are safely in the mailboxes of all subscribers. I made this decision about a year ago from now, and to tell the truth I felt that a year is a terribly long time to wait. In the end, this year appears much much shorter from this end than from the other one.

(I guess that it does not matter much if I put it on the Arxiv now: in the meanwhile I discovered that google scholar has managed to figure out that such a note exists on somebody’s webpage. Probably I will post it on the Arxiv, for the sake of all things being in good order).

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

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.

Mathematics on mathematics

 

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.

Partial results

What is more wonderful: something wonderful, or the moment before the wonderful something happens?

Recently I proved a result that can best be described as a partial result. This is certainly not what I planned to obtain. My goal was to prove C. The plan was as follows:

1. Prove A
2. Prove B
3. Prove that A and B together imply C

Step 3 is easy and I had it from the start. I’ve been trying to prove A and B for a several months now. I was sure that step 1 is the easier part and that step 2 is harder. I was quite hoping that step 1 follows from a general theorem which I also wanted to prove, but would have been just as happy to find it in the books.

Last week somebody showed me a counter example to the general theorem that would have implied A. Two days later I proved B. Now it remains to prove A, but A is not going to be true for the reasons I thought it would. Have to try a different approach…

But wait! I don’t want to prove A yet (not that I believe that there is a real “danger” in that happening). I want to enjoy B. B is beautiful. If I prove A, then C follows, and B is a triviality compared to C.

Worse, if A proves to be impenetrable, then as far as C goes, B is useless.

The role of B as a step in the proof of C can be appreciated a little more if we wait. Even if A is never proved, I wish to take some time enjoy B.

…

Alright, time out over! Back to A.

Some links and announcements

• The course “Advanced Analysis” is over. The lecture notes (the part that I prepared) are available here. Comments are very welcome. I hope to teach this course again in the not too far future and complete the lecture notes (add notes on Banach and C*-algebras, spectral theory and Fredholm theory). The homework exercises are available here, at the bottom of the page (the webpage is in Hebrew but the exercises are in English). 
• In April Ken Davidson will be visiting our department at BGU. On this occasion we will hold a short conference, dates: April 9-10. Here is the conference page. Contact me for more details.
• There are some interesting discussions going on in Gowers’s Weblog (see “Why I’ve joined the bad guys” and “Why I’ve joined the good guys” and some of the comments), regarding journals, publishing, new ideas, APCs, and so forth. The big news is that Gowers (after he kind of admits that being an editor of Forum of Mathematics makes him one of the bad guys) is now connected to another publishing adventure, that of epijournals, or arxiv overlay journals, which makes him one of the good guys (Just to set things straight: I think Gowers is a good guy). BTW: Gowers makes it clear that the credit for this initiative does not belong to him but to others, see his post.
• I promised myself to stop writing about this topic, but I guess I am still allowed to put a link to something that I wrote about this in the past. So here is a link to a letter (also other letters) I sent to Letters to the Editor of the Notices. It is a response to this article by Rob Kirby.

Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation theorem)

In this final lecture we will give a proof of Pick’s interpolation theorem that is based on operator theory.

Theorem 1 (Pick’s interpolation theorem): Let , and be given. There exists a function satisfying and 

if and only if the following matrix inequality holds:

Note that the matrix element appearing in the theorem is equal to , where is the reproducing kernel for the Hardy space (this kernel is called the Szego kernel). Given , the matrix

is called the Pick matrix, and it plays a central role in various interpolation problems on various spaces.

I learned this material from Agler and McCarthy’s monograph [AM], so the following is my adaptation of that source.

(A very interesting article by John McCarthy on Pick’s theorem can be found here).

Advanced Analysis, Notes 16: Hilbert function spaces (basics)

In the final week of the semester we will study Hilbert function spaces (also known as reproducing kernel Hilbert spaces) with the goal of presenting an operator theoretic proof of the classical Pick interpolation theorem. Since time is limited I will present a somewhat unorthodox route, and ignore much of the beautiful function theory involved. BGU students who wish to learn more about this should consider taking Daniel Alpay’s course next semester. Let me also note the helpful lecture notes available from Vern Paulsen’s webpage and also this monograph by Jim Agler and John McCarthy (in this post and the next one I will refer to these as [P] and [AM] below).

(Not directly related to this post, but might be of some interest to students: there is an amusing discussion connected to earlier material in the course (convergence of Fourier series) here).

Souvenirs from Bangalore

I recently returned from the two week long workshop and conference Recent Advances in Operator Theory and Operator Algebras which took place in ISI Bangalore. As I promised myself before going, I was on the look-out for something new to be excited about and to learn. The event (beautifully organized and run) was made of two parts: a workshop, which was a one week mini-school on several topics (see here for topics) and a one week conference. It was very very broad, and there were several talks (or informal discussions) which I plan to pursue further.

In this post and also perhaps in a future one I will try to work out (for my own benefit, mostly) some details of a small part of the research presented in two of the talks. The first part is the Superproduct Systems which arise in the theory of E_0-semigroups on type II_1 factors (following the talk of R. Srinivasan). The second (which I will not discuss here, but perhpas in the future) is the equivalence between the Baby Corona Theorem and the Full Corona Theorem (following the mini-course given by B. Wick). In neither case will I describe the most important aspect of the work, but something that I felt was urgent for me to learn.