EDP28 — problem solved by Terence Tao!

September 20, 2015

I imagine most people reading this will already have heard that Terence Tao has solved the Erdős discrepancy problem. He has blogged about the solution in two posts, a first that shows how to reduce the problem to the Elliott conjecture in number theory, and a second that shows (i) that an averaged form of the conjecture is sufficient and (ii) that he can prove the averaged form. Two preprints covering (i) and (ii) are here and here: the one covering (i) has been submitted to Discrete Analysis.

This post is therefore the final post of the polymath5 project. I refer you to Terry’s posts for the mathematics. I will just make a few comments about what all this says about polymath projects in general.
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Discrete Analysis — an arXiv overlay journal

September 10, 2015

This post is to announce the start of a new mathematics journal, to be called Discrete Analysis. While in most respects it will be just like any other journal, it will be unusual in one important way: it will be purely an arXiv overlay journal. That is, rather than publishing, or even electronically hosting, papers, it will consist of a list of links to arXiv preprints. Other than that, the journal will be entirely conventional: authors will submit links to arXiv preprints, and then the editors of the journal will find referees, using their quick opinions and more detailed reports in the usual way in order to decide which papers will be accepted.

Part of the motivation for starting the journal is, of course, to challenge existing models of academic publishing and to contribute in a small way to creating an alternative and much cheaper system. However, I hope that in due course people will get used to this publication model, at which point the fact that Discrete Analysis is an arXiv overlay journal will no longer seem interesting or novel, and the main interest in the journal will be the mathematics it contains.

The members of the editorial board so far — but we may well add further people in the near future — are Ernie Croot, me, Ben Green, Gil Kalai, Nets Katz, Bryna Kra, Izabella Laba, Tom Sanders, Jozsef Solymosi, Terence Tao, Julia Wolf, and Tamar Ziegler. For the time being, I will be the managing editor. I interpret this as meaning that I will have the ultimate responsibility for the smooth running of the journal, and will have to do a bit more work than the other editors, but that decisions about journal policy and about accepting or rejecting papers will be made democratically by the whole editorial board. (For example, we had quite a lot of discussion, including a vote, about the title, and the other editors have approved this blog post after suggesting a couple of minor changes.)

I will write the rest of this post as a series of questions and answers.
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Is Nick Clegg a Liberal Democrat?

April 28, 2015

All my life I have found that Liberal Democrat policies (and before that, Liberal-SDP Alliance policies, and before that, Liberal policies) have been, if not a perfect match to my own views, then at least the closest to them. In particular, I am broadly centrist, tilting somewhat to the left, strongly in favour of voting reform, strongly in favour of remaining part of the European Union, and very keen to take much more radical action to combat climate change. However, Nick Clegg is doing his best to persuade me that to vote Liberal Democrat is no longer a good way of promoting these values. Here is what he has said about building coalitions after the election:

As we have always said, the party with the most votes and the most seats in this election has the first right to seek to form a Government. The British people would rightly question the legitimacy of a coalition that didn’t allow the party with the largest number of seats and votes the opportunity to attempt to form a Government first.

I’m proud that the Liberal Democrats have proved we can form a strong and stable coalition government, able to bring prosperity to Britain.

Just like we would not put UKIP in charge of Europe, we are not going to put the SNP in charge of Britain – a country they want to rip apart.

The current projections at Five Thirty-Eight put the Conservatives on 281 seats, Labour on 268, the Scottish Nationalists on 49 and the Liberal Democrats on 26. If these are correct, then Clegg is saying that he will try first to form a Government with the Conservatives. I claim that this is inconsistent with all four of the fundamental Liberal values I mentioned.
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USS changes — don’t be fooled

March 12, 2015

This post is meant for anybody who will be affected by proposed changes to the Universities Superannuation Scheme, the body to which I and many other UK academics have paid our pension contributions and that now proposes to change the rules to deal with the fact that it has a large deficit as a result of the financial crisis. (Or rather, it says it has a large deficit, but there are arguments that the amount by which it is in deficit or surplus is highly volatile, so major changes are not necessarily justified.)

Of course, any change will have to be in the direction of making the deal less generous for those with pensions. Indeed, changes have already been made. Until a few years ago, the amount you got at the end was based on your final salary. More precisely, you got one 80th of your final salary per year after retirement for each year that you contributed to the scheme, up to a maximum of 40 years of contributions (and thus a maximum of half your final salary when you retire). But a few years ago they closed this final-salary scheme to new entrants, because (they said) it had become too expensive. This was partly because now a much larger proportion of academics end up as professors, so their final salaries are higher, and also of course because people live for longer.
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Sums of kth powers

November 4, 2014

Recently I was giving a talk at a school in London and had cause to think about sums of kth powers, because I wanted to show people the difference between unenlightening proofs and enlightening ones. (My brief was to try to give them some idea of what proofs are and why they are worth bothering about.) On the train down, I realized that there is a very simple general argument that can be used to work out formulae for sums of kth powers. After a brief look online, I haven’t found precisely this method, though I’ve found plenty of methods in the vicinity of it. It’s bound to be in the literature somewhere, so perhaps somebody can point me towards a reference. But even so, it’s worth a short post.

I’ll start with the case k=1. I want to phrase a familiar argument in a way that will make it easy to generalize in the way I want to generalize it. Let R_2(n) be the rectangle consisting of all integer points (x,y) such that 1\leq x\leq n and 1\leq y\leq n+1. We can partition R_2(n) into those points for which x\geq y and those points for which x<y. The number of points of the first kind is 1 + 2 + ... + n, since for each x we get x possibilities for y. The number of points of the second kind is 0 + 1 + ... + n, since for each y we get y-1 possibilities for x. Therefore, 2(1 + 2 + ... + n) = n(n+1) and we get the familiar formula. Read the rest of this entry »

ICM2014 — Bhargava, Gentry, Sanders

September 7, 2014

On my last day at the ICM I ended up going to fewer talks. As on the previous two days the first plenary lecture was not to be missed — it was Maryam Mirzakhani — so despite my mounting tiredness I set my alarm appropriately. I was a little surprised when I got there by just how empty it was, until eventually I saw that on the screens at the front it said that the lecture was cancelled because of her Fields medallist’s lecture the following Tuesday. I belonged to the small minority that had not noticed this, partly because I have had a lot of trouble with my supposedly-smart phone so was there with a temporary and very primitive replacement which was not the kind of phone on to which one could download a special ICM app that kept one up to date with things like this. I had planned to skip the second lecture of the morning, so I slightly rued my lost couple of hours of potential sleep, while also looking forward to being able to use those hours to work, or perhaps make progress with writing these posts — I can’t remember which of the two I ended up doing.

As a result, the first talk I went to was Manjul Bhargava’s plenary lecture, which was another superb example of what a plenary lecture should be like. Like Jim Arthur, he began by telling us an absolutely central general problem in number theory, but interestingly it wasn’t the same problem — though it is related.
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ICM2014 — Kollár, Conlon, Katz, Krivelevich, Milnor

September 3, 2014

As the ICM recedes further into the past, these posts start to feel less and less fresh. I’ve had an enforced break from them as over the course of three days I drove my family from the south of France back to Cambridge. So I think I’ll try to do what I originally intended to do with all these posts, and be quite a lot briefer about each talk.

As I’ve already mentioned, Day 3 started with Jim Arthur’s excellent lecture on the Langlands programme. (In a comment on that post, somebody questioned my use of “Jim” rather than “James”. I’m pretty sure that’s how he likes to be known, but I can’t find any evidence of that on the web.) The next talk was by Demetrios Christodoulou, famous for some extraordinarily difficult results he has proved in general relativity. I’m not going to say anything about the talk, other than that I didn’t follow much of it, because he had a series of dense slides that he read word for word. The slides may even have been a suitably chopped up version of his article for the ICM proceedings, but I have not been able to check that. Anyhow, after a gentle introduction of about three or four minutes, I switched off.
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ICM2014 — Jim Arthur plenary lecture

August 27, 2014

The main other thing I did on day two of the congress was go to a reception in the evening hosted by the French Embassy. It was less formal than that makes it sound, and as I circulated I met a number of people I hadn’t seen for quite a while, as well as others I had got to know at the congress itself. The French ambassador, who was disconcertingly young looking, gave a speech, as did Artur Avila (as you know, Avila, like Ngo four years ago, is French), and one other person, whose name I’ve annoyingly forgotten. One interesting nugget of information to come out of those speeches was that Paris is planning to bid for the 2022 ICM. If that bid is successful, then Avila will have two successive ICMs in his home country. There was plenty of food at the reception, so, as I had hoped, I didn’t need to think about finding supper. When we arrived, we were asked for our business cards. In common with approximately 99.9% of mathematicians, I don’t have a business card, but for cardless people it was sufficient to write our names on little bits of paper. This, it turned out, was to be entered in a draw for a bottle of wine. When the time came, it was Avila who drew out the pieces of paper. Apparently, this is a Korean custom. There were in fact two bottles going, so two chances to win, which sort of became three chances when the person who should have won the first bottle wasn’t there to claim it. And so, for the first time in my life … not really. I have never won anything in a raffle and that puzzling sequence continued.

The next morning kicked off (after breakfast at the place on the corner opposite my hotel, which served decent espressos) with Jim Arthur, who gave a talk about the Langlands programme and his role in it. He told us at the beginning that he was under strict instructions to make his talk comprehensible — which is what you are supposed to do as a plenary lecturer, but this time it was taken more seriously, which resulted in a higher than average standard. Ingrid Daubechies deserves a lot of credit for that. He explained that in response to that instruction, he was going to spend about two thirds of his lecture giving a gentle introduction to the Langlands programme and about one third talking about his own work. In the event he messed up the timing and left only about five minutes for his contribution, but for everybody except him that was just fine: we all knew he was there because he had done wonderful work, and most of us stood to learn a lot more from hearing about the background than from hearing about the work itself.
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ICM2014 — Barak, Guralnick, Brown

August 26, 2014

Here’s a little puzzle to get this post started. Of the fourteen 21st-century Fields medallists (if you count Perelman), seven — Lafforgue, Voevodsky, Tao, Werner, Smirnov, Avila and Mirzakhani — have something interesting in common that the others lack. What is this property?

That’s a fairly easy question, so let’s follow it up with another one: how surprised should we be about this? Is there unconscious bias towards mathematicians with this property? Of this year’s 21 plenary lecturers, the only one with the property was Mirzakhani, and out of the 20 plenary lecturers in 2010, the only one with the property was Avila. What is going on?

On to more serious matters. After Candès’s lecture I had a solitary lunch in the subterranean mall (Korean food of some description, but I’ve forgotten exactly what) and went to hear Martin Hairer deliver his Fields medal lecture, which I’m not going to report on because I don’t have much more to say about his work than I’ve already said.

By and large, the organization of the congress was notably good — for example, I almost never had to queue for anything, and never for any length of time — but there was a little lapse this afternoon, in that Hairer’s lecture was scheduled to finish at 3pm, exactly the time that the afternoon’s parallel sessions started. In some places that might have been OK, but not in the vast COEX Seoul conference centre. I had to get from the main hall to a room at the other end of the centre where theoretical computer science talks were taking place, which was probably about as far as walking from my house in Cambridge to the railway station. (OK, I live close to the station, but even so.)
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ICM2014 — Emmanuel Candès plenary lecture

August 25, 2014

If you are a mathematical researcher, do you ever stop to ask yourself what the point is of all your research? Do you worry that the world could get along just fine without it?

One person who doesn’t lose any sleep over doubts like this is Emmanuel Candès, who gave the second plenary lecture I went to. He began by talking a little about the motivation for the kinds of problems he was going to discuss, which one could summarize as follows: his research is worthwhile because it helps save the lives of children. More precisely, it used to be the case that if a child had an illness that was sufficiently serious to warrant an MRI scan, then doctors faced the following dilemma. In order for the image to be useful, the child would have to keep completely still for two minutes. The only way to achieve that was to stop the child’s breathing for those two minutes. But depriving a child’s brain (or indeed any brain, I’d imagine) of oxygen for two minutes is not without risk, to put it mildly.

Now, thanks to the famous work of Candès and others on compressed sensing, one can reconstruct the image using many fewer samples, which reduces the time the child must keep still to 15 seconds. Depriving the brain of oxygen for 15 seconds is not risky at all. Candès told us about a specific boy who had something seriously wrong with his liver (I’ve forgotten the details) who benefited from this. If you want a ready answer for when people ask you about the point of doing maths, and if you’re sick of the Hardy-said-number-theory-useless-ha-ha-but-what-about-public-key-cryptography-internet-security-blah-blah example, then I recommend watching at least some of Candès’s lecture, which is available here, and using that instead. Then you’ll really have seized the moral high ground.

Actually, I recommend watching it anyway, because it was a fascinating lecture from start to finish. In that case, you may like to regard this post as something like a film review with spoilers: if you mind spoilers, then you’d better stop reading here.
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