Noncommutative Analysis

Category: Thoughts on mathematics

Journal of Xenomathematics

I am happy to advertise the existence of a new electronic journal/forum/website: Journal of Xenomathematics. Don’t worry, it’s not another new research journal. The editor is John E. McCarthy. The purpose is to discuss mathematics that is out of this world. Aren’t you curious?

Thirty one years later: a counterattack on Halmos’s critique of non-standard analysis

As if to celebrate in an original way the fifty year anniversary of Bernstein and Robinson’s solution to (a generalization of) the Smith-Halmos conjecture (briefly, that if T is an operator such that p(T) is compact for some polynomial p, then T has an invariant subspace), several notable mathematicians posted a interesting and very nonstandard (as they say) paper on the arxiv.

This paper briefly tells the story regarding the publication of this paper, in which Bernstein and Robinson use Robinson’s new theory of non-standard analysis (NSA) to prove the above mentioned conjecture in operator theory. This was one of the first major successes of NSA, and perhaps one would think that all of the operator theory community should have accepted the achievement with nothing but high praise. Instead, it was received somewhat coldly: Halmos went to work immediately to translate the NSA proof and published a paper proving the same result, with a proof in “standard” operator theoretic terms. (See the paper, I am leaving out the juicy parts). And then, since 1966 until 2000 (more or less), Halmos has been apparently at “war” with NSA (in the paper the word “battle” is used), and has also had criticism of logic; for example, it is implied in his book that he did not always consider logic to be a part of mathematics, worse, it seems that he has not always considered logicians to be mathematicians. (When I wrote about Halmos’s book a few months ago, I wrote that I do not agree with all the opinions expressed in the book, and I remember having the issue with logic and logicians in my mind when writing that).

In the paper that appeared on the arxiv today, the authors take revenge on Halmos. Besides a (convincing) rebuttal of Halmos’s criticisms, the seven authors hand Halmos at least seven blows, not all of them below the belt. The excellent and somewhat cruel title says it all: A non-standard analysis of a cultural icon: the case of Paul Halmos.

Besides some feeling of uneasiness in seeing a corpse being metaphorically stabbed (where have you been in the last thirty years?), the paper raises interesting issues (without wallowing too much on either one), and may serve as a lesson to all of us. There is nothing in this story special to operator theory versus model theory, or NSA, or logic. The real story here is the suspicion and snubbish-ness of mathematicians towards fields in which they do not work, and towards people working in these fields.

I see it all the time. Don’t kid me: you have also seen quite a lot of it. It is possible, I confess, that I have exercised myself a small measure of suspicion and contempt to things that I don’t understand. As the authors of the paper hint, these things are worse than wrong – they might actually hurt people.

Anyway, many times people who are ignorantly snobbish to other fields end up looking like idiots. Stop doing that, or thirty years from now a mob of experts will come and tear you to shreds.

P.S. – It seems that the question of who was the referee of the Bernstein-Robinson paper is not settled, though some suspect it was Halmos. Well, if someone could get their hands on the (anonymous!) referee report (maybe Bernstein or Robinson kept the letter?), I am quite sure that if it was Halmos, it would be clear. In other words, if Bernstein or Robinson suspected that it was him on account of the style, then I bet it was.

 

P.P.S. – regarding the theorem starting this discussion the quickest way to understand it is via Lomonosov’s theorem. The invariant subspace theorem proved by Bernstein and Robinson (polynomially compact operator has an invariant subspace) is now superseded by Lomonosov’s theorem (google it for a simple proof), which says that every bounded operator on a Banach space that commutes with a nonzero compact operator has a non-trivial invariant subspace.

Daniel Spielman talks at HUJI – thoughts

I got an announcement in the email about the “Erdos Lectures”, that will be given by Daniel Spielman in the Hebrew University of Jerusalem next week (here is the poster on Gil Kalai’s blog). The title of the first lecture is “The solution of the Kadison-Singer problem”. Recall that not long ago Markus, Spielman and Srivastava proved Weaver’s KS2 conjecture, which implies a positive solution to Kadison-Singer (the full story been worked out to expository perfection on Tao’s blog).

My immediate response to this invitation was to start planning a trip to Jerusalem on Monday – after all it is not that far, it’s about a solution of a decades old problem, and Daniel Spielman is sort of a Fields medalist. I highly recommend to everyone to go hear great scientists live whenever they have the opportunity. At worst, their lectures are “just” inspiring. It is not for the mathematics that one goes for in these talks, but for all the stuff that goes around mathematics (George Mostow’s unusual colloquium given at BGU on May 2013 comes to mind).

But then I remembered that I have some obligations on Monday, so I searched and found a lecture by Daniel Spielman with the same title online: here. Watching the slides with Spielman’s voice is not as inspiring as hearing and seeing a great mathematician live, but quite good. He makes it look so easy!

In fact, Spielman does not discuss KS at all. He says (about a minute into the talk) “Actually, I don’t understand, really, the Kadison-Singer problem”. A minute later he has a slide where the problem is written down, but he says “let me not explain what it is”, and sends the audience to read Nick Harvey’s survey paper (which is indeed very nice). These were off-hand remarks, and I should not catch someone at his spoken word, (and I am sure that even things that Spielman would humbly claim to “not understand, really”, he probably understands as well as I do, at least), but the naturality in which the KS problem was pushed aside in a talk about KS made we wonder.

In the post I put up soon after appearance of the paper I wrote (referring to the new proof of KS2) that “… this looks like a very nice celebration of the Unity of Mathematics”. I think that in a sense the opposite is also true. I will try to reformulate what I wrote.

“The solution of KS is a beautiful and intriguing manifestation of the chaotic, sticky, psychedelic, thickly interwoven, tangled, scattered, shattered and diffuse structure of today’s mathematics.”

I don’t mean that in a bad way. I mean that a bunch of deep conjectures, from different fields, most of which, I am guessing, MSS were not worried about, were shown over several decades to be equivalent to each other, and were ultimately reduced (by Weaver) to a problem on the arrangement of vectors in finite dimensional spaces (Discrepancy Theory), and eventually solved, following years of hard work, by three brilliant mathematicians using ingenious yet mostly elementary tools. The problem solved is indeed interesting in itself, and the proof is also very interesting, but it seems that the connection with “Kadison-Singer” is more a trophy than a true reward.

It would be very interesting now to think of all the equivalent formulations with hindsight, and seek the unifying structure, and to try to glean a reward.

 

 

Measure theory is a must

[This post started out as an introduction to a post I was planning to write on convergence theorems for the Riemann integral. The introduction kind of got out hand, so I decided to post it separately. Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break (but in any case before we need them for the course I am teaching this term, Calculus 2)].

 

Mathematicians love to argue about subjective opinions. One of the most tiresome and depressing subjects of debate is “What should an undergraduate math major curriculum contain?”

 

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Thoughts following the Notices opinion article

The December issue of the Notices of the AMS has quite a thought provoking opinion article by Doron Zeilberger. In fact I already read this piece earlier in Opinions of Dr. Z, but re-finding it in the notices re-kindled a feeling that I get so often: we (mathematicians) are lost. More precisely: we have lost contact with the ground. Read the rest of this entry »

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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Where have all the functional equations gone (part II)

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) f(t) = f\left(\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.

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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

u_{tt} + u_{ss} - u_{xx} - u_{yy} = 0 ,    in     \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: Read the rest of this entry »

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. Read the rest of this entry »