DHJ write-up and other matters

This short post is in response to Jozsef Solymosi’s request for a new DHJ thread, since the previous one has become rather long and unwieldy. We’ve stopped numbering comments now, and the main purpose of the post is so that people can continue the discussion of the write-up of the proof of DHJ(k). Thanks mostly to the efforts of Ryan O’Donnell, we now have a complete draft. See also this write-up of DHJ(3) by Jozsef.

While I’m writing, I thought I’d take the opportunity to say that I am not intending to post much over the next two or three months, either here on on the Tricki. That’s because I have three more or less completed research projects that need to be properly finished (one of which is DHJ) and I owe it to my coauthors to get them done. So the plan is to clear my backlog over the summer and then come back, refreshed and ready to go, in the autumn. At that point I plan several Tricki articles (more advanced than most of the ones I’ve written so far). I also plan to start a new polymath project. Or rather, I have a file in which I have written plans for ten polymath projects, so what I’ll probably do is explain briefly what they are and try to get some idea of what appeals to people most. I am excited about several of these possible projects, so whatever we do I will be disappointed about the ones we don’t do. I may well have an online vote about it, but first I have to decide what the results of the vote will be.

Why aren’t all functions well-defined?

I’m in the happy state of just having finished marking exams for this year. There is very little of interest to say about the week that was removed from my life: it would be fun to talk about particularly bizarre mistakes, but I can’t really do that, especially as the results are not yet known (or even fully decided). However, one general theme emerged that made no difference to anybody’s marks. There seems to be a common misconception amongst many Cambridge undergraduates that I’d like to discuss here in the hope that I can clear things up for a few people. (It is an issue that I have discussed already on my web page, but rather than turning that into a blog post I’m starting again.)

The question where the misconception made itself felt was one about functions, injections, surjections, etc. I noticed that a lot of people wrote things like, “If then so is well defined.” Now if you fully understand what a function is, then you will find this quite amusing: if then trivially by the very basic principle that you can substitute something for something else if the two things are equal to each other. (A famous type of counterexample to this from philosophy: two years ago, Michelle Obama was the wife of Barack Obama; Barack Obama is the president of the United States; two years ago, Michelle Obama was not the wife of the president of the United States. Yes yes, there are ways of explaining why this isn’t a real counterexample.)

But it seems only fair, if one is going to laugh at such sentences, to provide examples of functions that are well defined and functions that aren’t, so that the difference can be made clear. But now we have a problem: any putative example of a function that is not well defined is not a function at all. So it begins to seem as though all functions are well defined. But in that case, what are people doing when they check that a function is well defined?

Swine flu and British public health policy

One of my children has just recovered from swine flu, as a result of which I now have a clearer idea of what British policy is towards outbreaks. Much of it was perfectly sensible, but not quite all. Since there’s a small amount of mathematics involved, and since I wanted to get this off my chest, I thought I’d blog about it.

The good part was that everyone who had been in close contact with the child who had swine flu was immediately put on Tamiflu, which seems to have stopped any of the rest of us getting it. (It’s now been long enough that we can be almost certain of this.) The less good part was the piece of advice that I mainly want to discuss. The main question I had was, of course, to what extent I and my family should avoid contact with other people. The advice I was given, which, it was made clear to me, was the official policy and not just the whim of the public health official I spoke to, was that we should continue to lead our lives as normal for as long as we did not show any symptoms.