One way of looking at Cauchy’s theorem

Cauchy’s theorem is the assertion that the path integral of a complex-differentiable function around a closed curve is zero (as long as there aren’t any holes inside the curve where the function has singularities or isn’t defined). This theorem, which is fundamental to complex analysis, can be vastly generalized and seen from many different points of view. This post is about a little idea that occurred to me once when I was teaching complex analysis. I meant to include it as a Mathematical Discussion on my web page but didn’t get round to it. Now, while the novelty of having a blog still hasn’t worn off, I find I have the energy to put it here.

Let’s start with a simpler fact: that if is a function from to and the derivative of is everywhere zero, then is constant. What is the natural generalization of this fact to functions defined on the plane?

About this blog

As I mentioned in my first post, I am very busy at the moment, so my rate of posting will soon go right down, probably until about February. However, I wanted to have at least one post that had mathematical content (as opposed to remarks about the presentation of mathematics) since I hope that in the end that will be the main focus of the blog. So the post about cubics is supposed to be more representative of what this blog is about. Thanks to all who have contributed so far. Although I had read about the power of blogs it nevertheless came as a surprise to see from the stats how many more visits a blog gets than a homepage, and also to discover how easy it is to get useful advice. I’ve even had a serious offer of technical help with setting up new wiki-style websites, so there’s a good chance that some of those ideas will actually be realized.

Discovering a formula for the cubic

In this post I want to revisit a topic that I first discussed on my web page here. My aim was to present a way in which one might discover a solution to the cubic without just being told it. However, the solution that arose was not very nice, and at the end I made the comment that I did not know a way of removing the rabbit-out-of-a-hat feeling that the usual much neater formula for the cubic (together with its derivation) left me with.

A couple of years ago, I put that situation right by stumbling on a very simple idea about quadratic equations that generalizes easily to cubics. More to the point, the stumble wasn’t completely random, so the entire approach can be justified as the result of standard and easy research strategies. I am no historian, but I would imagine that this idea is pretty similar to the idea (in some equivalent form) that first led to this solution.

Alternative maths reviews

Here’s another idea for a wiki-style website, one that might bring closer the day when mathematicians ceased to bother about print journals. It’s a site where people can post reviews of mathematical papers. Such a site, if it did what I have in mind, would have one disadvantage and two advantages over Math Reviews. The disadvantage, which is also one of the advantages actually, is that by no means every paper would be included. If you want a list of all published papers in mathematics, then Math Reviews (or Zentralblatt) does the job very well. However, it’s not really a site where one would browse for fun, and part of the reason is that all papers are given equal status, so if one is looking for an interesting paper one has to look amongst a whole lot of uninteresting ones. With a bit of skill and prior knowledge one can find interesting things of course, but that’s not really what I’d call browsing, in the sense of just having a look at what’s there and finding all sorts of gems.

How should vector spaces be introduced?

This question arose in the discussion of my previous post, but deserves a place to itself because (in my opinion, which I shall try to justify) it involves different issues. For example, how does one explain the point of the abstract notion of finite-dimensional vector spaces when, unlike with groups, you don’t seem to have an interesting collection of different spaces? Why not just use ? I addressed this point on my home page here, so won’t discuss it further on this post. But another point, which was raised in the previous discussion, concerns the relationship between theory and computation. I think it’s pretty uncontroversial to say that if you don’t know how to invert a matrix, or extend a linearly independent subset of to a basis, then you don’t truly understand linear algebra, even if you can state and prove conditions for a linear map to be invertible and can prove that every linearly independent set extends to a basis. Equally, as was pointed out, if you can multiply matrices but don’t understand their connection with linear maps, then you don’t truly understand matrix multiplication. (For example, it won’t be obvious to you that it is associative.) But how does one get people to understand the theory, be able to carry out computations, and see the links between the two? This is another situation where my own experience was not completely satisfactory: I’d be taught the theory in lectures and given computational questions to do, as though once I knew the theory I’d immediately see its relevance. But in fact I found the computational questions pretty hard, and some of the links to the theory were things I didn’t appreciate until years later and I found myself having to explain the subject to others.

How should logarithms be taught?

Having a blog gives me a chance to defend myself against a number of people who took issue with a passage in Mathematics, A Very Short Introduction, where I made the tentative suggestion that an abstract approach to mathematics could sometimes be better, pedagogically speaking, than a concrete one — even at school level. This was part of a general discussion about why many people come to hate mathematics.

The example I chose was logarithms and exponentials. The traditional method of teaching them, I would suggest, is to explain what they mean and then derive their properties from this basic meaning. So, for example, to justify the rule that x^a+b=x^ax^b one would say something like that if you have a xs followed by b xs and you multiply them all together then you are multiplying a+b xs all together.

What might an expository mathematical wiki be like?

This post has its origins in a discussion that arose as a result of a very interesting post of Terence Tao. Both the post and the discussion can be found here . The post outlines a rather general idea, or trick, that can be used in many mathematical situations. With such tricks, it is usually difficult, and in any case not desirable, to formalize them as lemmas: if you try to do so then almost certainly your formal lemma will not apply in all the situations where the trick does. This has the unfortunate consequence that they are relegated to something like “folklore,” transmitted orally (to a lucky few) or rediscovered over and over again (the more usual experience).

The Princeton Companion to Mathematics

I have decided to follow the excellent example of Terence Tao and start up a blog. For the moment I am too busy to do this properly, because, with the help of June Barrow-Green and Imre Leader, I am editing a book called The Princeton Companion to Mathematics. However, it is partly for that very reason that I want to set up the blog. It is somewhat hard to explain what the book is, but if you want to get quite a good idea, there is a substantial (though out of date) description of it, with several sample articles available here. You can get into this site with userid Guest and password PCM. Comments welcome. A sufficiently sensible comment could even influence what goes into the book, but I should warn that, because we are at a rather late stage of the editing process, I no longer have much room for manoeuvre. So I may end up having to say, “Yes, great point, but unfortunately it’s too late to do anything about it.”

Actually, I hope that the PCM blog will really come into its own after the book comes out. In particular, if you feel that there are unfortunate gaps (as there undoubtedly will be) then maybe it will be possible to do something about it online — I might even start up a wiki consisting of PCM supplements. (The distinction between that and the regular mathematics articles in Wikipedia would be some kind of certification that an article had reached PCM levels of comprehensibility. I’d probably be unwilling to put in the sort of editorial efforts I’ve been putting in over the last few years, but would try to distribute that task by using the blog medium. If you are reading this, maybe you will have a suggestion about how to go about it — in particular, I don’t yet know anything about the technicalities of this kind of thing.)