A mathematician watches tennis

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 of winning a game on your serve and a probability of winning it on your opponent’s serve, then over the next two games you have a probability of winning both, of winning one, and of losing both, and the order the games are played in makes no difference.

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.

What is Wolfram Alpha good for?

It’s early days and this isn’t meant to be a carefully considered review of Wolfram’s “computational knowledge engine”. Rather, I just want to point out, for the benefit of anyone who might not yet know, that one small part of what it does is genuinely useful in a certain circumstance that comes up from time to time. Suppose that for some reason you want a list of primes, or to know to 100 decimal places, or the 100th power of 2. Previously I would have used Google for the first two, banking on someone somewhere having put the information online, and I might have struggled to understand just enough Mathematica to do the third. (However, I have just discovered that powers of 2 can also be found quite easily with the help of Google, so a more complicated example might be needed.)

Anyhow, with Wolfram Alpha one can type in some reasonable text such as “The first hundred powers of 2″ or “pi to 100 places” and it works out what you mean and gives you the answer. That alone won’t change my life, but it is convenient and it will occasionally help me with things like preparing lectures for a general audience, which I think is just about enough to make it worth it to me to bookmark the site, though I haven’t yet done so. It will also sketch graphs and simplify mathematical expressions without one having to learn any special language to put them in — you just guess what to write and if your guess isn’t too perverse it can work out what you mean.

What else does it do? Typing in “father of Barack Obama” gives “Wolfram|Alpha isn’t sure what to do with your input”. Just typing “Barack Obama” gives you his full name and his date and place of birth. Typing “England” gives you various basic facts about England. Typing “capital of Uruguay” gives you Montevideo and various facts such as its population, current weather, etc. After noodling about like this for a short time, I did what any non-saint would do and typed in my own name. To be precise, I typed in “Gowers”. The result was “Wolfram|Alpha isn’t sure what to do with your input”, together with the helpful suggestion that perhaps I had meant “powers”.

I think that gives a fairly good idea of what it does and what it doesn’t do. Perhaps one should regard the latter as a truly positive and innovative aspect of Wolfram Alpha: a New Kind of Search Engine (or whatever it should be called) that doesn’t waste hours of your time by tempting you to look yourself up.

A solution to an exposition problem

Let me explain the title of this post by quoting from Timothy Chow’s highly recommended expository article A beginner’s guide to forcing: “All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.” He goes on to claim that forcing was an open exposition problem, since there was no explanation in the literature that had these qualities.

I have just finished giving a graduate course in Cambridge on two highlights of theoretical computer science: Razborov’s lower bound for the monotone circuit complexity of the clique function, and Shor’s quantum algorithm for factorizing integers. Because nobody was taking an exam on the course, I was free to be somewhat informal in the lectures, but at one or two points it got slightly too informal so I rashly promised that I would produce notes on Razborov’s theorem. However, the document that resulted ended up being less a set of lecture notes in the usual sense and more an attempt to solve an exposition problem in Chow’s sense. Very briefly, Razborov produces a lattice of a certain kind, with a rather strange definition, and it goes on to do the job it is supposed to do in a seemingly miraculous way. What I have written is an attempt to solve the problem, “Where did Razborov’s definition come from?”

When normality is abnormal

Suppose you were reading a novel, or watching a play or film, that included a fictional mathematician …

My guess is that the moment you read the two words “fictional mathematician” a second or two ago, your mind leapt ahead and you had a pretty good idea of what he—yes he, since even if there are female fictional mathematicians out there, femaleness is unlikely to be part of your instant and not fully conscious reaction to the phrase—was like: a social misfit who is prone to flashes of extraordinary insight that completely baffle everybody else, or perhaps a social misfit who would like to have those flashes but doesn’t and goes mad instead, or perhaps a social misfit who does have the insights but with madness the huge price he has to pay.

So here is a question: is there any example of a mathematician in literature, theatre or cinema who is a fairly normal person socially, and pretty good at maths but not astoundingly so? Some examples that do not work are Uncle Petros, from Uncle Petros and Goldbach’s Conjecture, both the father and the daughter in Proof, and Will from Good Will Hunting: they’re all either ridiculously good at maths (usually without having to do all that routine stuff like learning the proof of Schur’s lemma, or the open mapping theorem, or the Gram-Schmidt orthogonalization process etc.) or mad, or both. I also don’t count characters if they are colleagues of a crazy genius and their main role in the book/play/film is to marvel at how clever the crazy genius is. Let’s say that the character has to be the main one, or at least the main mathematical one.

Cohomology for amateurs—by an amateur

I have just finished a Tricki article about how to recognise situations where homology and cohomology can help you. (It is aimed at people who might have seen the definitions but felt uncomfortable about how to apply them.) Ages ago, I sort of promised to write something on this topic on this blog, so I am posting the article. This will have the added, though no doubt painful, benefit that people who know far more about the topic will be able to point out mistakes, misleading statements, unnecessarily complicated ways of thinking about things, and so on. But the one thing I do like is the (admittedly old-fashioned) way of looking at homology and cohomology as “soap-bubble homotopy”. See below for an explanation.

Tricki now fully live

Update (25/4/09). Since the launch, the number of pages on the Tricki has doubled (from 104 to 208), and is increasing fast.

Main post. If you have visited the Tricki recently, then you will already know that it has gone live. I’ve delayed posting about it until we were sure that everything was fully transferred: if you visit the prelive site you are now automatically redirected to the proper site, which you can also get to by clicking here. The URL is http://www.tricki.org.

A few small points to note here. In response to comments, we have introduced some new features. One is a feature for marking an article as a stub. Our working definition of a stub is that it should have no substantial mathematical content, and should not link forwards to any articles with substantial mathematical content. (That is, a parent of a non-stub is always a non-stub.) The thought behind this is that there are two directed graphs of interest: one with all articles, whether written or unwritten, and the other the set of all ancestors of articles with interesting content. The stub feature allows one to explore either of these trees with ease, because if an article is marked as a stub, then all links to that article are clearly marked as well, with a little leaf symbol.

Tricki available for viewing

It’s been a long time coming, but the Tricki is now on the point of going fully live. If you need convincing that this is a stronger statement than earlier and almost identical statements I have made on this blog, then click here to be taken to the site.

At the moment the site is read-only. This is for two reasons. First, we would like to give people a chance to spot flaws with the site as it now is, while it is still relatively easy to correct them. These can be anything from technical bugs to the content and organization of the articles. Any suggestions for improvement will be greatly welcomed: the best way of making them is to click on “Forums” at the top of any page on the site and to start or continue a forum topic. Of course, you are also welcome to make comments on this blog post.

The second reason is that I will be on holiday for the next week or so, and I want to be on hand when articles start coming in, in case work needs to be done in fitting them into the organizational hierarchy of the Tricki, or making sure that they are consistent with the Tricki house style.

An advantage of this final delay is that if you will have a chance to browse the site and get an idea of what it is like before contributing an article, if you have a topic that might be appropriate. If you click on “Help” and then on “Formatting on the Tricki”, you will discover that writing an article is extremely easy (at least if you know what you want to say). In particular, if you want to type in mathematical symbols, you just have to write them in TeX or LaTeX and enclose them in dollars. I hope you will agree with me that Alex Frolkin and Olof Sisask have done an amazing job and will enjoy using and contributing to the site as much as I have.