Archive for June, 2010

A mathematician watches tennis II

June 24, 2010

This has been a year to remember for anybody whose interest in tennis is more that of a nerd than that of a tennis player (which, given the uselessness of my serve, very much applies to me), in that it has given us two records that may well never be beaten. First we have Roger Federer’s record of 23 consecutive Grand Slam semi-finals (set at the Australian Open, and finally fixed at 23 when he lost in the quarter-finals at Roland Garros), and now, something I’ve been hoping for all my life: a seemingly endless match. At the time of writing, John Isner and Nicolas Mahut are waiting to resume a match that has gone into a third day. They will do so later today, with the score standing at 59-59 in the final set. This doesn’t just beat previous records — it utterly smashes them. This set is way more than twice as long as the previous longest set in a Grand Slam, it alone is far longer than the previous longest ever full match in professional tennis, both players have served far more aces in a single match (95 for Mahut, 98 for Isner) than anybody before, and so on. And if you also take account of the fact that the previous two sets had to be settled by tie-breaks, with no breaks of serve in either, then we have had 142 games in a row with no breaks of serve. (I can’t remember when the break occurred in the second set, but even this number 142 can probably be improved slightly.) [Update. The match is now over, with Isner winning 70-68, so the eventual number of consecutive unbroken service games was 137 in the final set, 161 if you include the previous two sets, and a few more still, I think, if you include the last few games of the second set. The number of aces for both players ended up well into triple figures.]

Isner said, with some justification, that nothing like this will ever happen again. But with how much justification? As ever, to answer this question involves choosing some kind of probabilistic model, and it is far from obvious how to choose an appropriate one. But it is possible to get some feel for the probabilities by looking at a crude model, while being fully aware that it is not realistic. (more…)

EDP15 — finding a diagonal matrix

June 21, 2010

It is not at all clear to me what should now happen with the project to solve the Erdős discrepancy problem. A little while ago, it seems that either the project ran out of steam, or else several people, at roughly the same time, decided that other things they were doing should take higher priority for a while. Perhaps those are two ways of saying the same thing.

But they are not quite the same: as an individual researcher I often give up on problems with the strong intention of returning to them, and I often find that if I do return to them then the break has had an invigorating effect. For instance, it can cause me to forget various ideas that weren’t really working, while remembering the important progress. I imagine these are common experiences. But do they apply to Polymath? Is it possible that the EDP project could regain some of its old vigour and that we could push it forward and make further progress? Is it conceivable that it could go into a different mode, where people contributed to it only occasionally? (A problem with that is that one of the appeals of a polymath project is checking in to see whether there are new ideas, or whether people have reacted to your comments. It is not clear to me that this appeal works if it is substantially slowed down.)

Anyhow, as Terry Tao might put it, this situation can be regarded as an opportunity to add a new datapoint to the general polymath experiment. Recently the following conjunction of circumstances occurred: I found myself on a plane, my laptop battery lasts a fraction of the time it should, and all the films on offer were either unappealing or too appealing (meaning that I’d been looking forward to seeing them but didn’t want to waste them by watching them in aeroplane conditions). I therefore found myself with several hours to think about mathematics. It was just like the old days when I didn’t have a laptop and there would only be a couple of films, both terrible. So I thought about EDP. Now I would like to avail myself of the opportunity to obtain a new datapoint by writing my thoughts, such as they were, down in a post and seeing whether anyone feels like joining or rejoining the discussion. I have plenty of questions, some of which may be fairly easy: I hope that will be a good way of tempting people. (more…)