## The work of Endre Szemerédi

March 8, 2013

A few years ago, Springer published a book called, The Abel Prize: 2003-2007 The First Five Years. A brief calculation will reveal that a second volume ought to be due soon, and that is indeed the case. I was asked to write the article about Endre Szemerédi, the 2012 winner, which I have just finished. I am glad to say that Springer’s policy with regard to this book is extremely enlightened: not only am I allowed to post my article as a preprint, but the entire book will be posted on the Norwegian Academy of Sciences website and will be freely accessible.

I was told to write the article as I pleased — the articles in the first volume are very different in style from each other — so I went for a style that was not unlike what I might have written if I had wanted to present several of Szemerédi’s results in a series of blog posts. That is, I’ve tried to explain the ideas, and when the going gets tough I have skipped the details. So it seems appropriate to post the article on this blog.

If you look at it and happen to notice any typos, false statements, wrong emphases, etc., I think it isn’t too late to make changes, so I’d be grateful to hear about them. Here is the article.

## Whither Polymath?

February 28, 2013

Over at the Polymath blog, Gil Kalai recently proposed a discussion about possible future Polymath projects. This post is partly to direct you to that discussion in case you haven’t noticed it and might have ideas to contribute, and partly to start a specific Polymathematical conversation. I don’t call it a Polymath project, but rather an idea I’d like to discuss that might or might not become the basis for a nice project. One thing that Gil and others have said is that it would be a good idea to experiment with various different levels of difficulty and importance of problem. Perhaps one way of getting a Polymath project to take off is to tackle a problem that isn’t necessarily all that hard or important, but is nevertheless sufficiently interesting to appeal to a critical mass of people. That is very much the spirit of this post.

Before I go any further, I should say that the topic in question is one about which I am not an expert, so it may well be that the answer to the question I’m about to ask is already known. I could I suppose try to find out on Mathoverflow, but I’m not sure I can formulate the question precisely enough to make a suitable Mathoverflow question, so instead I’m doing it here. This has the added advantage that if the question does seem suitable, then any discussion of it that there might be will take place where I would want any continuation of the discussion to take place.
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## Ted Odell

February 10, 2013

I was shocked and saddened to hear about a week ago that Ted Odell, a mathematician to whom I owe a lot, died suddenly on January 9th of a heart attack while he was travelling to this year’s joint AMS/MAA meeting in San Diego. He was 65, but seemed a lot younger.

Ted was a world leader in Banach space theory, and in particular in the infinite-dimensional theory. The wry and slightly enigmatic smile you see in the photo was extremely characteristic: if I imagine Ted, I automatically imagine him with exactly that smile. Less clear from the photo, though perhaps it can be guessed from the camera angle, is that he was extremely tall: he belonged to a handful of mathematicians I know who make me feel short (Tom Sanders and Alex Scott being two others).

I first met Ted when I went to my first ever conference, in Strobl am Wolfgangsee in Austria in 1989. I can’t remember how it came about, but I ended up chatting to him, and he started explaining to me in a wonderfully clear way — the kind of explanation you just can’t get from a textbook — how Tsirelson’s space worked. I read in an obituary by András Zsak (which starts on page 30 of this issue of the LMS newsletter) that Ted had a reputation for being kind and encouraging to young mathematicians. He certainly was to me at this conference, taking the time to give this explanation to a graduate student about whom he knew nothing. Most of the next section describes an argument that he sketched out for me on one of those yellow pads of paper that seem to be standard in US maths departments. (I think I’ve still got the yellow sheets that he let me keep, but I’ve no idea where they are.)
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## The Elsevier boycott one year on

January 28, 2013

A few days ago was the anniversary of the beginning of the Cost of Knowledge boycott of Elsevier. It seems a good moment to take stock of what the boycott has achieved and to think about what progress has or hasn’t been made since it started. This post is a short joint statement by many of the people who signed the original Cost of Knowledge statement last year. At some point in the not too distant future I plan to write a longer post giving a more personal view.

The Elsevier boycott: where do we now stand?

In the first few months after the boycott started, the number of signatories grew very rapidly. The growth is now much slower, but this was to be expected: given that, for understandable reasons, no editorial boards of Elsevier journals were ready to take the drastic step of leaving Elsevier, it was inevitable that further progress would depend on the creation of new publication models, which takes time and work, much of it not in the public eye. We are very pleasantly surprised by how much progress of this kind there has already been, with the setting up of Forum of Mathematics, a major new open-access journal, and the recent announcement of the Episciences Project, a new platform for overlay journals. We are also pleased by the rapid progress made by the wider Open Access movement over the last year.

In one respect the boycott has been an unqualified success: it has helped to raise awareness of the concerns we have about academic publishing. This, we believe, will make it easier for new publishing initiatives to succeed, and we strongly encourage further experimentation. We believe that commercial publishers could in principle play a valuable role in the future of mathematical publishing, but we would prefer to see publishers as “service providers”: that is, mathematicians would control journals, publishers would provide services that mathematicians deemed necessary, and prices would be kept competitive since mathematicians would have the option of obtaining these services elsewhere.

We welcome the moves that Elsevier made last year in the months that followed the start of the boycott: the dropping of support for the Research Works Act, the fact that back issues for many journals have now been made available, a clear statement that authors can post preprints on the arXiv that take into account comments by referees, and some small price reductions. However, the fundamental problems remain. Elsevier still has a stranglehold over many of our libraries as a result of Big Deals (a.k.a. bundling) and this continues to do real damage, such as forcing them to cancel subscriptions to more independent journals and to reduce their spending on books. There has also been no improvement in transparency: it as hard as ever to know what libraries are paying for Big Deals. We therefore plan to continue boycotting Elsevier and encourage others to do the same.

The problem of expensive subscriptions will not be solved until more libraries are prepared to cancel subscriptions and Big Deals. To be an effective negotiating tactic this requires support from the community: we must indicate that we would be willing to put up with cancelling overly expensive subscriptions. The more papers are made freely available online (e.g., through the arXiv), the easier that will be. Many already are, and we regard it as a moral duty for mathematicians to make their papers available when publishers allow it. Unfortunately, since mathematics papers are bundled together with papers in other subjects, real progress on costs will depend on coordinated action by mathematicians and scientists, many of whom have very different publication practices. However, a statement by mathematicians that they would not be unduly inconvenienced by the cancelling of expensive subscriptions would be a powerful one.

We are well aware that the problems mentioned above are not confined to Elsevier. We believe that the boycott has been more successful as a result of focusing attention on Elsevier, but the problem is a wider one, and many of us privately try to avoid the other big commercial publishers. We realize that this is not easy for all researchers. When there are more alternatives available, it will become easier: we encourage people to support new ventures if they are in a position do so without undue risk to their careers.

We acknowledge that there are differing opinions about what an ideal publishing system would be like. In particular, the issue of article processing charges is a divisive one: some mathematicians are strongly opposed to them, while others think that there is no realistic alternative. We do not take a collective position on this, but we would point out that the debate is by no means confined to mathematicians: it has been going on in the Open Access community for many years. We note also that the advantages and disadvantages of article processing charges depend very much on the policies that journals have towards fee waivers: we strongly believe that editorial decisions should be independent of an author’s access to appropriate funds, and that fee-waiver policies should be designed to ensure this.

To summarize, we believe that the boycott has been a success and should be continued. Further success will take time and effort, but there are simple steps that we can all take: making our papers freely available, and supporting new and better publication models when they are set up.

Doug Arnold, John Baez, Folkmar Bornemann, Danny Calegari, Henry Cohn, Ingrid Daubechies, Jordan Ellenberg, Marie Farge, David Gabai, Timothy Gowers, Michael Harris, Frédéric Hé lein, Rolf Jeltsch, Rob Kirby, Vincent Lafforgue, Randall J. LeVeque, Peter Olver, Olof Sisask, Terence Tao, Richard Taylor, Nick Trefethen, Marie-France Vigneras, Wendelin Werner, Günter M. Ziegler

## Why I’ve also joined the good guys

January 16, 2013

For some months now I have known of a very promising initiative that until recently I have been asked not to publicize too widely, because the people in charge of it did not have a good estimate for when it would actually come to fruition. But now those who know about it have been given the green light. The short version of what I want to say in this post is that a platform is to be created that will make it very easy to set up arXiv overlay journals.

What is an arXiv overlay journal? It is just like an electronic journal, except that instead of a website with lots of carefully formatted articles, all you get is a list of links to preprints on the arXiv. The idea is that the parts of the publication process that academics do voluntarily — editing and refereeing — are just as they are for traditional journals, and we do without the parts that cost money, such as copy-editing and typesetting.
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## Why I’ve joined the bad guys

January 14, 2013

A few months ago I was alerted by a pingback to the existence of a blog post by Orr Shalit entitled Worse than Elsevier which included the assertion that Terence Tao and I had “joined the bad guys”. That is an allusion to the fact that we are editors for Forum of Mathematics, CUP’s new open-access journal. This post serves a dual purpose: to draw attention to the fact that Forum of Mathematics is now accepting submissions, and to counter some of the many objections that have been raised to it. In particular, I want to separate out the objections that are based on misconceptions from the objections that have real substance. Both kinds exist, and unfortunately they tend to get mixed up.

If you are not already familiar with this debate, the aspect of Forum of Mathematics that many people dislike is that it will be funded by means of article processing charges (which I shall abbreviate to APCs) rather than subscriptions. For the next three years, these charges will be waived, but after that there will be a charge of £500 per article. Let me now consider a number of objections that people have to APCs.
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## What maths A-level doesn’t necessarily give you

November 20, 2012

I had a mathematical conversation yesterday with a 17-year-old boy who is in his second year of doing maths A-level. Although a sample of size 1 should be treated with caution, I’m pretty sure that the boy in question, who is very intelligent and is expected to get at least an A grade, has been taught as well as the vast majority of A-level mathematicians. If this is right, then what I discovered from talking to him was quite worrying.

The purpose of the conversation was to help him catch up with some work that he had missed through illness. The particular topics he wanted me to cover were integrating $\log x$, or $\ln x$ as he called it, and integration by parts. (Actually, after I had explained integration by parts to him, he told me that that hadn’t been what he had meant, but I don’t think any harm was done.) But as we were starting, he asked me why the derivative of $e^x$ was $e^x$, and what was special about $e$.
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## What actually happened

November 9, 2012

The short version is that I’ve had the ablation (see previous post) and the surgeon who did it says that he has a good feeling about it. It’s taken till now to write this because, unlike most people who have ablations, I felt terrible for two days after it — with a headache (normal) and a fever (less normal but not unheard of). The fever was not very high, but high enough to be unpleasant, and meant that the only thing I could bear to do was go to bed, except that on the second night after the operation I had to spend part of the night sitting up on a sofa because my chest hurt too much when I was horizontal. (That was normal, and nothing to worry about.) So today is the first day that I am well enough to do anything as strenuous as writing a blog post.
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## Mathematics meets real life

November 5, 2012

I’ve been in two minds about whether to post this. On the one hand, I try to keep personal matters out of this blog — though there has been the occasional exception — but on the other hand I have a topic that fits quite nicely with some of what I’ve been writing about recently, since it concerns a fairly important medical decision that I have had to make based on what felt like inadequate information. Since that is quite an interesting situation from a mathematical point of view, and even a philosophical point of view, and since most people have to make similar decisions at some point in their lives, I have opted to write the post.

The background is that over the last fifteen years or so I have had occasional bouts of atrial fibrillation, a condition that causes the heart to beat irregularly and not as strongly as it should. It is quite a common condition: I’ve just read that 2.3% of people over the age of 40 have it, and 5.9% of people over 65. Some people have no symptoms. I myself have mild symptoms — I can feel a slightly strange, and instantly recognisable, feeling in my chest, and I experience a few seconds of dizziness almost every time I stand up from a relaxed seated position — otherwise known as orthostatic hypotension, which I often used to get anyway (as do many people).
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## EDP27 — the modular version of Roth’s AP-discrepancy theorem

September 19, 2012

Recall from earlier posts Gil’s modular conjecture for HAPs. It states that if $n$ is large enough and $f$ is a function from $\{1,2,\dots,n\}$ to $\mathbb{Z}_p$ that never takes the value 0, then for every $a$ there exists a HAP $P$ such that $\sum_{x\in P}f(x)\equiv a$ mod $p$. It is easy to see that this implies EDP, so it may well be very hard, or even false. However, one can hold out a little hope that, as with some strengthenings of statements, it is in fact easier, because it is in some way more symmetrical and fundamental. Given that, it makes good sense, as Gil has suggested, to try to prove modular versions of known discrepancy theorems, in the hope of developing general techniques that can then be tried out on the modular EDP conjecture.

A very obvious candidate for a discrepancy theorem that we could try to modularize is Roth’s theorem, which asserts that for any $\pm 1$-valued function $f$ on $\{1,2,\dots,n\}$ there exists an arithmetic progression $P$ such that $|\sum_{x\in P}f(x)|\geq cn^{1/4}$. That gives rise to the following problem.

Problem. Let $p$ be a prime. What is the smallest $n$ such that for every function $f:\{1,2,\dots,n\}\to\mathbb{Z}_p$ that never takes the value 0, every $a\in\mathbb{Z}_p$ can be expressed as $\sum_{x\in P}f(x)$ for some arithmetic progression $P$?