Archive for July, 2009

PCM errata III

July 30, 2009

A quick note to say that I’ve just been asked by PUP to send all the corrections I have, so I have sent the corrections that are listed in comments on the post PCM errata II. Further errata should therefore ideally be pointed out here. I think this means that a reprint with corrections should appear in the next few months, but I’m not sure of the exact time scale.

Once again, many thanks to those who have notified me of errors. Particular thanks this time round go to Axel Boldt, who sent a very long list. He was to PCM errata II as Joseph Myers was to PCM errata I. But others too made observations that will lead to significant improvements when the reprint comes out. It has been disconcerting to see just how many mistakes remained after the effort we spent on eliminating them, but I suppose in a book this size it is inevitable, and I imagine that there are several more lurking there. So do please continue to let me know of any errors that you find. At some point in the not too distant future I’ll try to merge the second list with the first to make it easier for people to see which mistakes have already been spotted. For now, after the break, here is a copy of the instructions I have sent to PUP. (more…)

The next phase of Polymath

July 28, 2009

After the surprising success of the collaborative attack on the density Hales-Jewett theorem, many people have asked me when there would next be a Polymath project. The answer, I hope, is that it will be some time in October, though it may even be sooner if Gil Kalai goes ahead with his proposal to tackle the polynomial Hirsch conjecture polymathematically. There are many issues that seem worth discussing before further Polymath projects get going, and Terence Tao has set up a new Polymath blog in order for such discussions to take place, and also, at some point, to host the projects themselves.

So if you have any views about this, then please go over there and join the discussion. The kinds of questions we are discussing are things like how we should choose the next project, whether there are enough potential participants to support more than one project at once, whether anything can (or should) be done to broaden participation and make it easier for more people to keep up with what is going on, and so on. One obvious question is whether the blog format is the right one. However, that is not one of the main questions under discussion at the moment. Almost certainly it will become one after a few more projects have been attempted and we’ve gained a bit more experience in this way of doing things, but for now the blog, for all its limitations, seems a reasonable way of continuing.

Help — I’m stuck in my ivory tower!

July 11, 2009

The UK Qualifications and Curriculum Authority is considering introducing a new A’level course (in Britain, A’level is the exam that is taken at the end of high school) called “Use of Mathematics”. As one might expect, this idea has not met with universal approval, and there is now a campaign to stop the idea in its tracks. (I should warn you that the preceding link is to a Word file rather than to a web page.)

The General Secretary of the National Association of Headteachers has this to say to the campaigners:

They should get down from their ivory towers. They should be out in the world where young people live and exist and they should be appreciative that young people have great skills in the use of technology and we have to latch on to that.

We cannot continue teaching an out dated 19th century curriculum. This is simply turning many children off education because it is completely not relevant to them at all.

Some sample papers for the new course have been made available, so let’s have a look at the up-to-date 21st-century curriculum that will enthuse a new generation of British schoolchildren. I’ll concentrate on one or two questions but if you want to see more, then the sample papers can be found at the bottom of this page. (Update: unfortunately, these sample papers have been taken down. I can’t help wondering why. Further update: at least some sample papers are now available at the bottom of this page.) (more…)

A mathematician watches tennis

July 4, 2009

Because the French Open and Wimbledon have been available on the BBC website I’ve been watching a lot of tennis recently. And as I do so I can’t help thinking about whether mathematics has anything to say about the tactics that the players should adopt in various situations. And the more I think (or rather, idly muse) about this question, the more it becomes clear that the modelling problem it presents is a pretty hard one. Most of this post will be a discussion of questions rather than a serious attempt to supply answers.

Just to make the discussion more concrete, here are a couple of more specific questions, which I’ll come back to later. The first one is fairly simple.

1. It is generally held to be a slight advantage to serve first in a set. The reasoning goes like this. Let’s suppose (for simplicity) that the game goes with serve till 4-4. If you are serving first, then you will be in a very dangerous position if your serve is broken, since you will then have to break back immediately or lose the set. However, at least you won’t have lost. By contrast, if you are serving second and the score is 4-5, then you can’t afford to be broken — if you are broken then you lose the set and do not get even a small chance to redeem yourself. And if you have just broken your opponent so that it’s 5-4, then you still have the task of serving for the set.

However, a simple model would suggest that this reasoning is flawed. If you have a probability p of winning a game on your serve and a probability q of winning it on your opponent’s serve, then over the next two games you have a probability pq of winning both, p(1-q)+q(1-p) of winning one, and (1-p)(1-q) of losing both, and the order the games are played in makes no difference. (more…)